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Research Article An Approximation Solution to Refinery Crude Oil Scheduling Problem with Demand Uncertainty Using Joint Constrained Programming Qianqian Duan, 1 Genke Yang, 1 Guanglin Xu, 2 and Changchun Pan 1 1 Department of Automation, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, China 2 College of Mathematics and Information, Shanghai Lixin University of Commerce, China Correspondence should be addressed to Genke Yang; [email protected] and Changchun Pan; [email protected] Received 1 December 2013; Accepted 20 January 2014; Published 16 March 2014 Academic Editors: Z. Chen, W.-C. Hong, and K.-C. Ying Copyright © 2014 Qianqian Duan et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is devoted to develop an approximation method for scheduling refinery crude oil operations by taking into consideration the demand uncertainty. In the stochastic model the demand uncertainty is modeled as random variables which follow a joint multivariate distribution with a specific correlation structure. Compared to deterministic models in existing works, the stochastic model can be more practical for optimizing crude oil operations. Using joint chance constraints, the demand uncertainty is treated by specifying proximity level on the satisfaction of product demands. However, the joint chance constraints usually hold strong nonlinearity and consequently, it is still hard to handle it directly. In this paper, an approximation method combines a relax- and-tight technique to approximately transform the joint chance constraints to a serial of parameterized linear constraints so that the complicated problem can be attacked iteratively. e basic idea behind this approach is to approximate, as much as possible, nonlinear constraints by a lot of easily handled linear constraints which will lead to a well balance between the problem complexity and tractability. Case studies are conducted to demonstrate the proposed methods. Results show that the operation cost can be reduced effectively compared with the case without considering the demand correlation. 1. Introduction In recent years refineries have to explore all potential cost- saving strategies due to intense competition arising from fluctuating product demands and ever-changing crude prices. Scheduling of crude oil operations is a critical task in the overall refinery operations [1–3]. Basically, the optimization of crude oil scheduling oper- ations consists of three parts [4]. e first part involves the crude oil unloading, mixing, transferring, and multilevel crude oil inventory control process. e second part deals with fractionation, reaction scheduling, and a variety of intermediate product tanks control. e third part involves the finished product blending and distributing process. In this paper, we focus on the first part, as it is a critical component for refinery scheduling operations. e crude oil scheduling problem has received consid- erable attention from researchers and different models have been developed on the basis of deterministic mathematical programming techniques. Lee et al. [5] developed a mixed- integer linear programming (MILP) model to solve a short- term crude oil scheduling problem, in which the linearity of the bilinear constraints is maintained by replacing bilinear terms with individual component flows, but it can lead to composition discrepancy. To overcome the composition discrepancy problem, Wenkai et al. [6] and Reddy et al. [7] proposed an iterative MIP-NLP model and an iterative discrete-time MIP model, respectively. However the models rely on time discretization representations. Recently, most mathematical models of crude oil scheduling operations put emphasis on continuous-time formulation so as to shorten the gap between theoretical research and real-world oper- ation. Chryssolouris et al. [8] studied the similar problem as Lee et al. [5] and took the temperature cut-points into consideration for each distillation. Jia and Kelly [4] for- mulated the same problem by a state-task-network (STN) Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 730314, 13 pages http://dx.doi.org/10.1155/2014/730314

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Research ArticleAn Approximation Solution to Refinery Crude OilScheduling Problem with Demand Uncertainty Using JointConstrained Programming

Qianqian Duan1 Genke Yang1 Guanglin Xu2 and Changchun Pan1

1 Department of Automation Shanghai Jiao Tong University Dongchuan Road 800 Shanghai China2 College of Mathematics and Information Shanghai Lixin University of Commerce China

Correspondence should be addressed to Genke Yang sjtu1019163com and Changchun Pan 390635304qqcom

Received 1 December 2013 Accepted 20 January 2014 Published 16 March 2014

Academic Editors Z Chen W-C Hong and K-C Ying

Copyright copy 2014 Qianqian Duan et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to develop an approximationmethod for scheduling refinery crude oil operations by taking into considerationthe demand uncertainty In the stochastic model the demand uncertainty is modeled as random variables which follow a jointmultivariate distribution with a specific correlation structure Compared to deterministic models in existing works the stochasticmodel can be more practical for optimizing crude oil operations Using joint chance constraints the demand uncertainty is treatedby specifying proximity level on the satisfaction of product demands However the joint chance constraints usually hold strongnonlinearity and consequently it is still hard to handle it directly In this paper an approximation method combines a relax-and-tight technique to approximately transform the joint chance constraints to a serial of parameterized linear constraints so thatthe complicated problem can be attacked iteratively The basic idea behind this approach is to approximate as much as possiblenonlinear constraints by a lot of easily handled linear constraints which will lead to a well balance between the problem complexityand tractability Case studies are conducted to demonstrate the proposed methods Results show that the operation cost can bereduced effectively compared with the case without considering the demand correlation

1 Introduction

In recent years refineries have to explore all potential cost-saving strategies due to intense competition arising fromfluctuating product demands and ever-changing crude pricesScheduling of crude oil operations is a critical task in theoverall refinery operations [1ndash3]

Basically the optimization of crude oil scheduling oper-ations consists of three parts [4] The first part involves thecrude oil unloading mixing transferring and multilevelcrude oil inventory control process The second part dealswith fractionation reaction scheduling and a variety ofintermediate product tanks control The third part involvesthe finished product blending and distributing process Inthis paper we focus on the first part as it is a criticalcomponent for refinery scheduling operations

The crude oil scheduling problem has received consid-erable attention from researchers and different models have

been developed on the basis of deterministic mathematicalprogramming techniques Lee et al [5] developed a mixed-integer linear programming (MILP) model to solve a short-term crude oil scheduling problem in which the linearity ofthe bilinear constraints is maintained by replacing bilinearterms with individual component flows but it can leadto composition discrepancy To overcome the compositiondiscrepancy problem Wenkai et al [6] and Reddy et al[7] proposed an iterative MIP-NLP model and an iterativediscrete-time MIP model respectively However the modelsrely on time discretization representations Recently mostmathematical models of crude oil scheduling operations putemphasis on continuous-time formulation so as to shortenthe gap between theoretical research and real-world oper-ation Chryssolouris et al [8] studied the similar problemas Lee et al [5] and took the temperature cut-points intoconsideration for each distillation Jia and Kelly [4] for-mulated the same problem by a state-task-network (STN)

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 730314 13 pageshttpdxdoiorg1011552014730314

2 The Scientific World Journal

continuous-time representation Hu and Zhu [9] extendedthe event-basedmodel of Jia et al [10 11] to the slotmodel byeliminating the redundant event points on others reducingthe size of the model and hence the solution time of theproblem

Most of the current plant planning and schedulingmodels are based on deterministic programming Howeverdue to the volatile raw material prices fluctuating productsdemands and other changing market conditions manyparameters in a planning and scheduling model are usuallyuncertain Neiro and Pinto [12] constructed a corporateplanning model for multiple refineries using scenario basedapproach Neiro and Pinto [12] developed a multiperiodMINLP model to deal with uncertainty in product priceand crude price However the scenario based approachprovides no obvious information on the relation betweenreliability and profitability which is crucial for decisionmakers Several recent papers applied chance constrainedprogramming models to the refinery short-term crude oilscheduling problem [13ndash16]

All of the aforementionedmodels in the domestic refineryoptimization are either deterministic [5] or stochastic withindependent demand distribution [14 15] Cao et al [15]considered the demand correlation for different crude mixin the same time period However they did not consider thecorrelation for the same crude mix in different time periodsIn this paper we will consider the crude mix demand correla-tion in demands not only for different crude mix in the sametime period but also for the same crude mix indifferent timeperiodsThese considerations have practical significance Forexample if two crude mixes are predominantly used as rawmaterials in another process their demands will be positivelycorrelated in each time period Alternatively unusually highdemand for a crude mix in one time period more often thannot is followed by lower than normal demand in the nextperiod implying negative correlation Taking into accountsuch information whenever it is available enables a moreefficient allocation of crude mix capacity to minimize costand meet certain marketing objectives

In this paper we will propose a stochastic multiperiodmodel with considering the uncertain crude mix demandcorrelations The model employs two-level time structureformulation inwhich the entire scheduling horizon is dividedinto several interrelated macroperiods Each macroperiodwith fluctuation demand consists of time intervals with fixedlength A chance constrained programming formulation isdeveloped for solving the problem The deterministic formof the stochastic constraint is used in solving the problemiteratively In real-world situations the future demand alwayschanges as time rolls forward To deal with the uncertaintyof the model it is important to adjust the planning policyand update the corresponding schedule and the correlationstructure of the demand at the end of each time period basedon the real sales [17]

The rest of the paper is organized as followsThe schedul-ing problem is specified in Section 2 In Section 3 a stochasticmultiperiodmodel with two-level time structure formulationis formulated In Section 4 the deterministic representationof the model is presented In Section 5 an approximation

Vessels

Storage tanks Charging

tanksCDUS

Dock

middot middot middot

Figure 1 Typical flow process for the refinery crude oil operations

method combining relax-and-tight technique is developed tosolve the joint chance constrained problem above By relaxwe mean that the ΣA approach [18] is used to approximatethe original-covariance matrix with a new one of simplifiedcovariance structure By tight we mean that the joint chanceconstrains are transformed into several linear constraintswith parameterized dependent which aremore stringent thanthe original constraints Moreover an update policy uponthe realization of the random demands of crude mix isdescribed A test problem involving correlated random crudemix demands is solved in Section 6 highlighting variousmodeling and algorithmic issues Section 7 summarizes thework and provides some concluding remarks

2 Problem Statements and Operation Rules

21 Problem Statements The problem studied involves crudeoil unloading process from vessels to storage tanks transfer-ring process from storage tanks to charging tanks (where sev-eral crude oils aremixed) and charging process from chargingtanks to crude oil distillations (CDUs) Figure 1 shows thetypical processes During a given scheduling horizon crudevessels arrive in the vicinity of the refinery docking stationand according to FCFS wait for unloading of the precedingvessel in the docking station At the docking station crude oilis unloaded into storage tanks Crude oil is then transferredfrom storage tanks to charging tanks which are buffers toproduce a crude mix of which component compositionswere determined at the planning level The crude oil mix ineach charging tank is then charged into a CDU Given theconfiguration of themultistage system aswell as the uncertainarrival times of vessels equipment capacity limitations andkey component concentration ranges the problem is thento determine the following operating variables to minimizeoperating costs (a) waiting time of each vessel at sea (b)unloading time of each vessel (c) crude unloading rate fromvessels to storage tanks (d) crude oil transfer andmixing ratefrom storage tank to charging tanks (e) inventory levels ofstorage and charging tanks (f) CDU charging rates and (g)sequence for charging mixed crude into each CDU

A basic deterministic model assumption was presentedin [5] However in this paper we will extend it to a

The Scientific World Journal 3

newmixed-integer nonlinear stochastic programmingmodelwith chance constraints The demands for different crudemix are uncertain andpossibly correlated reflecting changingmarket conditions and periodic variation in customer ordersAn optimal planning policy for minimizing expected cost isdeveloped This is achieved by letting crude mix demands besatisfied with at least a prespecified probability level

The proposed stochastic model involves the followingfeatures and assumptions

(1) A two-level time structure introduced by Fleis-chmann andMeyr [19] is adopted formodeling a gen-eral stochastic system The entire planning intervalis divided into macroperiods each with fluctuationdemand and eachmacroperiod consists of time inter-vals with fixed length

(2) Thedemands for different crudemixes aremodeled asmultivariate normally distributed random variablesThenormality assumption has beenwidely invoked inthe literature because it captures the essential featuresof demand uncertainty and it is convenient to useThe use of more ldquocomplexrdquo probability distributionsis hindered by the fact that statistical informationapart frommean and covariance estimates of productdemands is rarely available

(3) The demand correlation in the problem formulationstands for correlation of demands for different crudemixes in the same time period and demands for thesame crude mix in different time periods

(4) The penalty for some crude mix shortfalls is propor-tional to the amount of underproduction

22 Operation Rules The operation rules are shown asfollows (1) the refinery uses only one docking station anda new arriving vessel has to wait at sea until the anteriorvessel leaves the docking station (2) while a charging tankis charging CDU crude oil from the storage tanks cannot befed into the charging tank and vice versa (3) each chargingtank can only charge one CDU at each time interval (4)each CDU can only be charged by one charging tank ateach time interval (5) CDUs must be operated continuouslythroughout the scheduling time horizon

3 Mathematical Model

Indices and Sets

119894 isin 1 119873119878 = crude oil storage tank and the crude

oil in it119895 isin 1 119873

119861 = crude oil charging tank and the

crude oil mix in the charging tank119896 isin 1 119873

119862 = key component of crude oil

119897 isin 1 119873CDU

= crude distillation unit119898 isin 1 119873

119872 = macroperiod

119905 isin 1 119873SCH

= time interval 119873SCH denotesscheduling horizon

119879119898= the starting time of the macroperiodm

V isin 1 119873119881 = crude vessels

Variables

119863119895119897119905

= 0-1 variable to denote if the crude oil mix incharging tank 119895 charges CDU 119897 at time t119883119865

V119905 = 0-1 variable to denote if vessel V starts unload-ing at time t119883119871

V119905 = 0-1 variable to denote if vessel V just completesunloading at time t119883119882

V119905 = 0-1 continuous variable to denote if vessel v isunloading its crude oil at time t1198851198951198951015840 119897119905 = 0-1 continuous variable to denote if transi-tion from crude mix (or charging tank) j to 1198951015840 at timet in CDU l119891SB119894119895119896119905

= volumetric flow rate of component 119896 fromstorage tank 119894 to charging tank 119895 at time t119891BC119895119897119896119905

= volumetric flow rate of component 119896 fromcharging tank 119895 to CDU 119897 at time t119865VSV119894119905 = volumetric flow rate of crude oil from vessel V

to storage tank 119894 at time t119865SB119894119895119905

= volumetric flow rate of crude oil from storagetank 119894 to charging tank 119895 at time t119865BC119895119897119905

= volumetric flow rate of crude oil mix fromcharging tank 119895 to CDU 119897 at time t119879119865

V = vessel v unloading initiation time119879119871

V = vessel v unloading completion and departuretimeV119861119895119896119905

= volume of component k in charging tank j attime t119881119881

V119905 = volume of crude oil in crude vessel v at time t

119881119878

119894119905= volume of crude oil in storage tank i at time t

119881119861

119895119905= volume of mixed oil in charging tank j at time

t

Parameters

119862UNLOAD119881

= unloading cost of vessel V per time inter-val119862SEA119881

= sea waiting cost of vessel V per time interval119862INVST119894

= inventory cost of storage tank 119894 per time pervolume119862INVBL119895

= inventory cost of charging tank 119895 per timeper volume119862SETUP1198951198951015840 = changeover cost for transition from crude

mix j to 1198951015840 in CDU119862PEN119895

= penalty cost for crude mix 119895 shortfalls

119863119872

119895119898= stochastic demand of crude mix 119895 by CDUs

during the macroperiodm

4 The Scientific World Journal

119865VSminV119894 =minimumcrude oil transfer rate from vessel

V to storage tank i

119865VSmaxV119894 = maximum crude oil transfer rate from

vessel V to storage tank i

119865SBmin119894119895

= minimum crude oil transfer rate from stor-age tank 119894 to charging tank j

119865SBmax119894119895

= maximum crude oil transfer rate fromstorage tank 119894 to charging tank j

119865BCmin119895119897

= minimum crude oil charging rate fromcharging tank 119895 to CDU l

119865BCmax119895119897

= maximum crude oil charging rate fromcharging tank 119895 to CDU l

119879ARRV = crude vessel V arrival time around the docking

station

119881119881

V0 = initial volume of crude oil in crude vessel v

119881119878min119894

=minimum crude oil volume of storage tank i

119881119878max119894

= maximum crude oil volume of storage tanki

119881119878

1198940= initial crude oil volume of storage tank i

119881119861min119895

= minimummixed crude oil volume of charg-ing tank j

119881119861max119895

=maximummixed crude oil volume of charg-ing tank j

120576119878

119894119896= concentration of component 119896 in the crude oil

of storage tank i

120576119861

1198951198960= initial concentration of component 119896 in the

crude mix of charging tank j

120576119861min119895

= minimum concentration of component 119896 inthe crude mix of charging tank j

120576119861max119895

= maximum concentration of component 119896 inthe crude mix of charging tank j

In this paper a two-level time structure is used tomodel a general dynamic production system We considerthe crude mix 119895 to be scheduled over a finite planninghorizon consisting of macroperiods 119898 = 1 sdot sdot sdot 119873

119872 Eachmacroperiod 119898 is divided into time intervals with fixedlength where 119879

119898represents the starting time of macrope-

riod 119898 Let 119863119872119895119898

a random variable denote the demand ofcrude mix 119895 in macroperiod 119898 We approximate the crudemix demand as a multivariate normal distribution with aspecific correlation structure 119863119872 sim 119873(120583 Σ) with covariancematrix Σ where 119863119872 def

= [119863119872

119895119898] 120583

def= [120583119895119898] It means that the

demands for different mix crudes not only are correlated butalso are the demands for the same mixed crude in differentmacroperiods

The mathematical model is formulated as follows(1) Operating Cost

MinimizeOperating cost = unloading cost for the crude vessel+ cost for vessel waiting in the sea + inventory costfor storage and charging tanks + changeover cost +underproduction penalty cost

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873BT

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873119861119879

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840119897119905

+

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

(1)

Subject to

(2) Vessel Arrival and Departure Operation Rules Each vesselarrives at the docking station for unloading only oncethroughout the scheduling horizon

119873SCH

sum

119905=1

119883119865

V119905 = 1 forallV (2a)

Each vessel leaves the docking station only once throughoutthe scheduling horizon Consider

119873SCH

sum

119905=1

119883119871

V119905 = 1 forallV (2b)

Equation for unloading initiation time

119879119865

V =

119873SCH

sum

119905=1

119905119883119865

V119905 forallV (2c)

Equation for unloading completion time

119879119871

V =

119873SCH

sum

119905=1

119905119883119871

V119905 forallV (2d)

Each crude vessel should start unloading after arrival time setin the planning level

119879119865

V ge 119879ARRV forallV (2e)

The Scientific World Journal 5

Duration of the vessel unloading is bounded by the initial vol-ume of oil in the vessel divided by the maximum unloadingrate

119879119871

V minus 119879119865

V ge lceil119881119881

V0

max lceil119865119881119878maxV119894 rceil

rceil forallV (2f)

lceilrceil corresponds to round-up of the next highest integer valueVessel in the sea cannot arrive at the docking station forunloading unless the preceding vessel leaves

119879119865

V+1 ge 119879119871

V forallV (2g)

Unloading is possible between time 119879119865V and 119879119871V

119883119882

V119905 le

119905

sum

120591=1

119883119865

V120591 119883119882

V119905 le

SCHsum

120591=119905

119883119871

V120591 forallV 119905 (2h)

(3) Material Balance Equations for the Vessel For vessel vthe oil volume at time 119905 equals the initial volume minustransferred volume from vessel V to storage tanks up to timet

119881119881

V119905 = 119881119881

V0 minus

119873ST

sum

119894=1

119905

sum

120591=1

119865119881119878

V119894120591 forallV 119905 (3a)

Operating constraints on crude oil transfer rate from vessel Vto storage tank 119894 at time t

119865VSminV119894 119883

119882

V119905 le 119865VSV119894119905 le 119865

VSmaxV119894 119883

119882

V119905 forallV 119894 119905 (3b)

The volume of crude oil transferred from vessel V to storagetanks during the scheduling horizon equals the initial crudeoil volume of vessel v

119873ST

sum

119894=1

119873SCH

sum

119905=1

119865VSV119894119905 = 119881

119881

V0 forallV 119894 119905 (3c)

(4) Material Balance Equations for Storage Tanks The oilvolume in storage tank 119894 at time 119905 equals the initial oil volumein tank 119894 plus the oil volume transferred from vessels to tank119894 up to the time 119905 and minus oil volume removed from tank 119894to charging tanks up to the time t

119881119878

119894119905= 119881119878

1198940+

119873119881

sum

V=1

119905

sum

120591=1

119865VSV119894120591 minus

119873BT

sum

119895=1

119905

sum

120591=1

119865SB119894119895120591

forall119894 119905 (4a)

Operating constraints on crude oil transfer rate from storagetank 119894 to charging tank 119895 at time t the term 1 minus sum

NCDU119897=1

119863119895119897119905

denotes that if charging tank 119895 is charging any CDU there isno oil transfer from storage tank 119894 to charging tank j Consider

119865SBmin119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905)

le 119865SB119894119895119905

le 119865SBmax119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905) forall119894 119895 119905

(4b)

Volume capacity limitations for storage tank 119894 at time t

119881119878min119894

le 119881119878

119894119905le 119881119878max119894

forall119894 119905 (4c)

(5) Material Balance Equations for Charging TanksThe crudeoilmix in charging tank 119895 at time 119905 equals the initial oil volumein charging tank 119895 plus the crude oil transferred from storagetanks to charging tank j up to time t and minus the crude oilmix 119895 charged into CDUs up to time t

119881119861

119895119905= 119881119861

1198950+

119873ST

sum

119894=1

119905

sum

120591=1

119865SB119894119895120591

minus

119873CDU

sum

119897=1

119905

sum

120591=1

119865BC119895119897120591

forall119895 119905

(5a)

Operating constraints on mixed oil transfer rate from charg-ing tank 119895 to CDU 119897 at time t

119865BCmin119895119897

119863119895119897119905

le 119865BC119895119897119905

le 119865BCmax119895119897

119863119895119897119905

forall119895 119897 119905 (5b)

Volume capacity limitations for charging tank 119895 at time t

119881119861min119895

le 119881119861

119895119905le 119881119861max119895

forall119895 119905 (5c)

(6) Material Balance Equations for Component 119896 in ChargingTanks The volume of component 119896 in charging tank 119895 attime 119905 equals the initial component 119896 in charging tank 119895 pluscomponent 119896 in crude oil transferred from storage tanks tocharging tank 119895 up to the time 119905 and minus component 119896 incrude oil mix 119895 transferred to CDUs up to the time t

V119861119895119905= V1198611198950+

119905

sum

120591=1

119873ST

sum

119894=1

119891SB119894119895120591

minus

119905

sum

120591=1

119873CDU

sum

119897=1

119891BC119895119897120591

forall119895 119896 119905 (6a)

Operating constraints on volumetric flow rate of component119896 from storage tank 119894 to charging tank j

119891SB119894119895119896119905

= 119865SB119894119895119905120576119878

119894119896forall119894 119895 119896 119905 (6b)

Operating constraints on volumetric flow rate of component119896 from charging tank 119895 to CDU l

119865BC119895119897119905120576119861min119895119896

le 119891BC119895119897119896119905

le 119865BC119895119897119905120576119861max119895119896

forall119895 119897 119896 119905 (6c)

Volume capacity limitations for component 119896 in chargingtank 119895 at time t

119881119861

119895119905120576119861min119895119896

le V119861119895119896119905

le 119881119861

119895119905120576119861max119895119896

forall119895 119897 119896 119905 (6d)

(7) Operating Rules for Crude Oil Charging Charging tank 119895can charge at most one CDU at any time t

119873CDU

sum

119897=1

119863119895119897119905

le 1 forall119895 119905 (7a)

6 The Scientific World Journal

CDU 119897 can be charged only by one charging tank at any timet

119873BT

sum

119895=1

119863119895119897119905

= 1 forall119897 119905 (7b)

If CDU 119897 is charged by crude oilmix 119895 at time t minus 1 and chargedby 1198951015840 at time t changeover cost is involved

1198851198951198951015840119897119905ge 1198631198951015840119897119905+ 119863119895119897119905minus1

minus 1

119895 1198951015840(119895 = 1198951015840) = 1 119873

119861forall119897 119905

(7c)

The proposed formulation involves stochastic expres-sions which requires a different course of action for trans-forming it into an equivalent deterministic form The deter-ministic equivalent representation of the expectation of theobjective function is examined in the next section

4 Chance Constraint BasedDeterministic Transformation

41 Expectation of Penalty Cost The expectation of theunderproduction penalty cost term for crude mix 119895 inmacroperiod119898 is equal to

119864[

[

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)]

]

(8)

where

119862119895119898

≜ max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

= minusmin(0119873

CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

minus 119863119872

119895119898)

= minus[

[

min(119863119872119895119898

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905) minus 119863

119872

119895119898]

]

(9)

To facilitate the calculation of the expectation the stan-dardization of the normally distributed variables 119863119872

119895119898and

variables sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

is performed first Normal ran-dom variables can be recast into the standardized normalform with a mean of zero and a variance of 1 by subtractingtheir mean and dividing by their standard deviation (squareroot of variance) This defines the standardized normalvariables

119909119895119898 =

119863119872

119895119898minus119863119872

119895119898

120590119895119898

(10)

where 119863119872119895119898

denotes the mean of 119863119872119895119898

and 120590119895119898

the squareroot of its variance In the same way the ldquostandardizationrdquo ofthe variables sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

defines

119870119895119898

=

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

(11)

Using the method in paper [17] the underproductionpenalty term 119862

119895119898is charging into the following form

119862119895119898 = minus [minus119891 (119870119895119898) + (1 minus Φ (119870119895119898))119870119895119898] (12)

where 119891 is the standardized normal distribution functionand Φ denotes the cumulative probability function of astandard normal random variable

42 Satisfaction Level for Single Crude Mix Demand Theminimization of the objective function (1) as defined aboveestablishes the production and planned sales policy whichmost appropriately balances profits with inventory costs andunderproduction shortfalls A crudemix demand satisfactionlevel is not explicitly specified but rather it is the outcomeof the minimization of the profit function While highervalues of the parameter 119862

PEN119895

conceptually increase theprobability of demand satisfaction this strategymay still leadto unacceptably low probabilities of satisfying certain crudemix demands (see examples)Therefore the setting of explicitprobability targets on crudemix demand satisfaction is muchmore desirable A systematic way to accomplish this is toimpose explicit lower bounds on the probabilities of satisfyinga single crude mix demand This requirement for crude mix119895 in macroperiod119898 assumes the following form

Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898]

]

ge 120573119895119898 (13)

This constraint known as a chance constraint imposes a lowerbound 120573

119895119898on the probability that the crude mix demand

realization will be greater than the planned sales 119863119872119895119898

forcrude mix 119895 in macroperiod119898 The deterministic equivalentrepresentation can be obtained based on the concepts intro-duced by Charnes and Cooper [20]

Specifically by subtracting the mean and dividing bythe standard deviation of 119863119872

119895119898 the chance constraint can be

written equivalently as

Pr[

[

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

ge

119863119872

119895119898minus119863119872

119895119898

120590119895119898

]

]

ge 120573119895119898

Pr lceil119870119895119898

ge 119909119895119898rceil ge 120573119895119898

(14)

The right-hand side of the inequality within the proba-bility sign is a normally distributed random variable with amean of zero and a variance of 1 This implies that the chance

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

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Page 2: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

2 The Scientific World Journal

continuous-time representation Hu and Zhu [9] extendedthe event-basedmodel of Jia et al [10 11] to the slotmodel byeliminating the redundant event points on others reducingthe size of the model and hence the solution time of theproblem

Most of the current plant planning and schedulingmodels are based on deterministic programming Howeverdue to the volatile raw material prices fluctuating productsdemands and other changing market conditions manyparameters in a planning and scheduling model are usuallyuncertain Neiro and Pinto [12] constructed a corporateplanning model for multiple refineries using scenario basedapproach Neiro and Pinto [12] developed a multiperiodMINLP model to deal with uncertainty in product priceand crude price However the scenario based approachprovides no obvious information on the relation betweenreliability and profitability which is crucial for decisionmakers Several recent papers applied chance constrainedprogramming models to the refinery short-term crude oilscheduling problem [13ndash16]

All of the aforementionedmodels in the domestic refineryoptimization are either deterministic [5] or stochastic withindependent demand distribution [14 15] Cao et al [15]considered the demand correlation for different crude mixin the same time period However they did not consider thecorrelation for the same crude mix in different time periodsIn this paper we will consider the crude mix demand correla-tion in demands not only for different crude mix in the sametime period but also for the same crude mix indifferent timeperiodsThese considerations have practical significance Forexample if two crude mixes are predominantly used as rawmaterials in another process their demands will be positivelycorrelated in each time period Alternatively unusually highdemand for a crude mix in one time period more often thannot is followed by lower than normal demand in the nextperiod implying negative correlation Taking into accountsuch information whenever it is available enables a moreefficient allocation of crude mix capacity to minimize costand meet certain marketing objectives

In this paper we will propose a stochastic multiperiodmodel with considering the uncertain crude mix demandcorrelations The model employs two-level time structureformulation inwhich the entire scheduling horizon is dividedinto several interrelated macroperiods Each macroperiodwith fluctuation demand consists of time intervals with fixedlength A chance constrained programming formulation isdeveloped for solving the problem The deterministic formof the stochastic constraint is used in solving the problemiteratively In real-world situations the future demand alwayschanges as time rolls forward To deal with the uncertaintyof the model it is important to adjust the planning policyand update the corresponding schedule and the correlationstructure of the demand at the end of each time period basedon the real sales [17]

The rest of the paper is organized as followsThe schedul-ing problem is specified in Section 2 In Section 3 a stochasticmultiperiodmodel with two-level time structure formulationis formulated In Section 4 the deterministic representationof the model is presented In Section 5 an approximation

Vessels

Storage tanks Charging

tanksCDUS

Dock

middot middot middot

Figure 1 Typical flow process for the refinery crude oil operations

method combining relax-and-tight technique is developed tosolve the joint chance constrained problem above By relaxwe mean that the ΣA approach [18] is used to approximatethe original-covariance matrix with a new one of simplifiedcovariance structure By tight we mean that the joint chanceconstrains are transformed into several linear constraintswith parameterized dependent which aremore stringent thanthe original constraints Moreover an update policy uponthe realization of the random demands of crude mix isdescribed A test problem involving correlated random crudemix demands is solved in Section 6 highlighting variousmodeling and algorithmic issues Section 7 summarizes thework and provides some concluding remarks

2 Problem Statements and Operation Rules

21 Problem Statements The problem studied involves crudeoil unloading process from vessels to storage tanks transfer-ring process from storage tanks to charging tanks (where sev-eral crude oils aremixed) and charging process from chargingtanks to crude oil distillations (CDUs) Figure 1 shows thetypical processes During a given scheduling horizon crudevessels arrive in the vicinity of the refinery docking stationand according to FCFS wait for unloading of the precedingvessel in the docking station At the docking station crude oilis unloaded into storage tanks Crude oil is then transferredfrom storage tanks to charging tanks which are buffers toproduce a crude mix of which component compositionswere determined at the planning level The crude oil mix ineach charging tank is then charged into a CDU Given theconfiguration of themultistage system aswell as the uncertainarrival times of vessels equipment capacity limitations andkey component concentration ranges the problem is thento determine the following operating variables to minimizeoperating costs (a) waiting time of each vessel at sea (b)unloading time of each vessel (c) crude unloading rate fromvessels to storage tanks (d) crude oil transfer andmixing ratefrom storage tank to charging tanks (e) inventory levels ofstorage and charging tanks (f) CDU charging rates and (g)sequence for charging mixed crude into each CDU

A basic deterministic model assumption was presentedin [5] However in this paper we will extend it to a

The Scientific World Journal 3

newmixed-integer nonlinear stochastic programmingmodelwith chance constraints The demands for different crudemix are uncertain andpossibly correlated reflecting changingmarket conditions and periodic variation in customer ordersAn optimal planning policy for minimizing expected cost isdeveloped This is achieved by letting crude mix demands besatisfied with at least a prespecified probability level

The proposed stochastic model involves the followingfeatures and assumptions

(1) A two-level time structure introduced by Fleis-chmann andMeyr [19] is adopted formodeling a gen-eral stochastic system The entire planning intervalis divided into macroperiods each with fluctuationdemand and eachmacroperiod consists of time inter-vals with fixed length

(2) Thedemands for different crudemixes aremodeled asmultivariate normally distributed random variablesThenormality assumption has beenwidely invoked inthe literature because it captures the essential featuresof demand uncertainty and it is convenient to useThe use of more ldquocomplexrdquo probability distributionsis hindered by the fact that statistical informationapart frommean and covariance estimates of productdemands is rarely available

(3) The demand correlation in the problem formulationstands for correlation of demands for different crudemixes in the same time period and demands for thesame crude mix in different time periods

(4) The penalty for some crude mix shortfalls is propor-tional to the amount of underproduction

22 Operation Rules The operation rules are shown asfollows (1) the refinery uses only one docking station anda new arriving vessel has to wait at sea until the anteriorvessel leaves the docking station (2) while a charging tankis charging CDU crude oil from the storage tanks cannot befed into the charging tank and vice versa (3) each chargingtank can only charge one CDU at each time interval (4)each CDU can only be charged by one charging tank ateach time interval (5) CDUs must be operated continuouslythroughout the scheduling time horizon

3 Mathematical Model

Indices and Sets

119894 isin 1 119873119878 = crude oil storage tank and the crude

oil in it119895 isin 1 119873

119861 = crude oil charging tank and the

crude oil mix in the charging tank119896 isin 1 119873

119862 = key component of crude oil

119897 isin 1 119873CDU

= crude distillation unit119898 isin 1 119873

119872 = macroperiod

119905 isin 1 119873SCH

= time interval 119873SCH denotesscheduling horizon

119879119898= the starting time of the macroperiodm

V isin 1 119873119881 = crude vessels

Variables

119863119895119897119905

= 0-1 variable to denote if the crude oil mix incharging tank 119895 charges CDU 119897 at time t119883119865

V119905 = 0-1 variable to denote if vessel V starts unload-ing at time t119883119871

V119905 = 0-1 variable to denote if vessel V just completesunloading at time t119883119882

V119905 = 0-1 continuous variable to denote if vessel v isunloading its crude oil at time t1198851198951198951015840 119897119905 = 0-1 continuous variable to denote if transi-tion from crude mix (or charging tank) j to 1198951015840 at timet in CDU l119891SB119894119895119896119905

= volumetric flow rate of component 119896 fromstorage tank 119894 to charging tank 119895 at time t119891BC119895119897119896119905

= volumetric flow rate of component 119896 fromcharging tank 119895 to CDU 119897 at time t119865VSV119894119905 = volumetric flow rate of crude oil from vessel V

to storage tank 119894 at time t119865SB119894119895119905

= volumetric flow rate of crude oil from storagetank 119894 to charging tank 119895 at time t119865BC119895119897119905

= volumetric flow rate of crude oil mix fromcharging tank 119895 to CDU 119897 at time t119879119865

V = vessel v unloading initiation time119879119871

V = vessel v unloading completion and departuretimeV119861119895119896119905

= volume of component k in charging tank j attime t119881119881

V119905 = volume of crude oil in crude vessel v at time t

119881119878

119894119905= volume of crude oil in storage tank i at time t

119881119861

119895119905= volume of mixed oil in charging tank j at time

t

Parameters

119862UNLOAD119881

= unloading cost of vessel V per time inter-val119862SEA119881

= sea waiting cost of vessel V per time interval119862INVST119894

= inventory cost of storage tank 119894 per time pervolume119862INVBL119895

= inventory cost of charging tank 119895 per timeper volume119862SETUP1198951198951015840 = changeover cost for transition from crude

mix j to 1198951015840 in CDU119862PEN119895

= penalty cost for crude mix 119895 shortfalls

119863119872

119895119898= stochastic demand of crude mix 119895 by CDUs

during the macroperiodm

4 The Scientific World Journal

119865VSminV119894 =minimumcrude oil transfer rate from vessel

V to storage tank i

119865VSmaxV119894 = maximum crude oil transfer rate from

vessel V to storage tank i

119865SBmin119894119895

= minimum crude oil transfer rate from stor-age tank 119894 to charging tank j

119865SBmax119894119895

= maximum crude oil transfer rate fromstorage tank 119894 to charging tank j

119865BCmin119895119897

= minimum crude oil charging rate fromcharging tank 119895 to CDU l

119865BCmax119895119897

= maximum crude oil charging rate fromcharging tank 119895 to CDU l

119879ARRV = crude vessel V arrival time around the docking

station

119881119881

V0 = initial volume of crude oil in crude vessel v

119881119878min119894

=minimum crude oil volume of storage tank i

119881119878max119894

= maximum crude oil volume of storage tanki

119881119878

1198940= initial crude oil volume of storage tank i

119881119861min119895

= minimummixed crude oil volume of charg-ing tank j

119881119861max119895

=maximummixed crude oil volume of charg-ing tank j

120576119878

119894119896= concentration of component 119896 in the crude oil

of storage tank i

120576119861

1198951198960= initial concentration of component 119896 in the

crude mix of charging tank j

120576119861min119895

= minimum concentration of component 119896 inthe crude mix of charging tank j

120576119861max119895

= maximum concentration of component 119896 inthe crude mix of charging tank j

In this paper a two-level time structure is used tomodel a general dynamic production system We considerthe crude mix 119895 to be scheduled over a finite planninghorizon consisting of macroperiods 119898 = 1 sdot sdot sdot 119873

119872 Eachmacroperiod 119898 is divided into time intervals with fixedlength where 119879

119898represents the starting time of macrope-

riod 119898 Let 119863119872119895119898

a random variable denote the demand ofcrude mix 119895 in macroperiod 119898 We approximate the crudemix demand as a multivariate normal distribution with aspecific correlation structure 119863119872 sim 119873(120583 Σ) with covariancematrix Σ where 119863119872 def

= [119863119872

119895119898] 120583

def= [120583119895119898] It means that the

demands for different mix crudes not only are correlated butalso are the demands for the same mixed crude in differentmacroperiods

The mathematical model is formulated as follows(1) Operating Cost

MinimizeOperating cost = unloading cost for the crude vessel+ cost for vessel waiting in the sea + inventory costfor storage and charging tanks + changeover cost +underproduction penalty cost

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873BT

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873119861119879

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840119897119905

+

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

(1)

Subject to

(2) Vessel Arrival and Departure Operation Rules Each vesselarrives at the docking station for unloading only oncethroughout the scheduling horizon

119873SCH

sum

119905=1

119883119865

V119905 = 1 forallV (2a)

Each vessel leaves the docking station only once throughoutthe scheduling horizon Consider

119873SCH

sum

119905=1

119883119871

V119905 = 1 forallV (2b)

Equation for unloading initiation time

119879119865

V =

119873SCH

sum

119905=1

119905119883119865

V119905 forallV (2c)

Equation for unloading completion time

119879119871

V =

119873SCH

sum

119905=1

119905119883119871

V119905 forallV (2d)

Each crude vessel should start unloading after arrival time setin the planning level

119879119865

V ge 119879ARRV forallV (2e)

The Scientific World Journal 5

Duration of the vessel unloading is bounded by the initial vol-ume of oil in the vessel divided by the maximum unloadingrate

119879119871

V minus 119879119865

V ge lceil119881119881

V0

max lceil119865119881119878maxV119894 rceil

rceil forallV (2f)

lceilrceil corresponds to round-up of the next highest integer valueVessel in the sea cannot arrive at the docking station forunloading unless the preceding vessel leaves

119879119865

V+1 ge 119879119871

V forallV (2g)

Unloading is possible between time 119879119865V and 119879119871V

119883119882

V119905 le

119905

sum

120591=1

119883119865

V120591 119883119882

V119905 le

SCHsum

120591=119905

119883119871

V120591 forallV 119905 (2h)

(3) Material Balance Equations for the Vessel For vessel vthe oil volume at time 119905 equals the initial volume minustransferred volume from vessel V to storage tanks up to timet

119881119881

V119905 = 119881119881

V0 minus

119873ST

sum

119894=1

119905

sum

120591=1

119865119881119878

V119894120591 forallV 119905 (3a)

Operating constraints on crude oil transfer rate from vessel Vto storage tank 119894 at time t

119865VSminV119894 119883

119882

V119905 le 119865VSV119894119905 le 119865

VSmaxV119894 119883

119882

V119905 forallV 119894 119905 (3b)

The volume of crude oil transferred from vessel V to storagetanks during the scheduling horizon equals the initial crudeoil volume of vessel v

119873ST

sum

119894=1

119873SCH

sum

119905=1

119865VSV119894119905 = 119881

119881

V0 forallV 119894 119905 (3c)

(4) Material Balance Equations for Storage Tanks The oilvolume in storage tank 119894 at time 119905 equals the initial oil volumein tank 119894 plus the oil volume transferred from vessels to tank119894 up to the time 119905 and minus oil volume removed from tank 119894to charging tanks up to the time t

119881119878

119894119905= 119881119878

1198940+

119873119881

sum

V=1

119905

sum

120591=1

119865VSV119894120591 minus

119873BT

sum

119895=1

119905

sum

120591=1

119865SB119894119895120591

forall119894 119905 (4a)

Operating constraints on crude oil transfer rate from storagetank 119894 to charging tank 119895 at time t the term 1 minus sum

NCDU119897=1

119863119895119897119905

denotes that if charging tank 119895 is charging any CDU there isno oil transfer from storage tank 119894 to charging tank j Consider

119865SBmin119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905)

le 119865SB119894119895119905

le 119865SBmax119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905) forall119894 119895 119905

(4b)

Volume capacity limitations for storage tank 119894 at time t

119881119878min119894

le 119881119878

119894119905le 119881119878max119894

forall119894 119905 (4c)

(5) Material Balance Equations for Charging TanksThe crudeoilmix in charging tank 119895 at time 119905 equals the initial oil volumein charging tank 119895 plus the crude oil transferred from storagetanks to charging tank j up to time t and minus the crude oilmix 119895 charged into CDUs up to time t

119881119861

119895119905= 119881119861

1198950+

119873ST

sum

119894=1

119905

sum

120591=1

119865SB119894119895120591

minus

119873CDU

sum

119897=1

119905

sum

120591=1

119865BC119895119897120591

forall119895 119905

(5a)

Operating constraints on mixed oil transfer rate from charg-ing tank 119895 to CDU 119897 at time t

119865BCmin119895119897

119863119895119897119905

le 119865BC119895119897119905

le 119865BCmax119895119897

119863119895119897119905

forall119895 119897 119905 (5b)

Volume capacity limitations for charging tank 119895 at time t

119881119861min119895

le 119881119861

119895119905le 119881119861max119895

forall119895 119905 (5c)

(6) Material Balance Equations for Component 119896 in ChargingTanks The volume of component 119896 in charging tank 119895 attime 119905 equals the initial component 119896 in charging tank 119895 pluscomponent 119896 in crude oil transferred from storage tanks tocharging tank 119895 up to the time 119905 and minus component 119896 incrude oil mix 119895 transferred to CDUs up to the time t

V119861119895119905= V1198611198950+

119905

sum

120591=1

119873ST

sum

119894=1

119891SB119894119895120591

minus

119905

sum

120591=1

119873CDU

sum

119897=1

119891BC119895119897120591

forall119895 119896 119905 (6a)

Operating constraints on volumetric flow rate of component119896 from storage tank 119894 to charging tank j

119891SB119894119895119896119905

= 119865SB119894119895119905120576119878

119894119896forall119894 119895 119896 119905 (6b)

Operating constraints on volumetric flow rate of component119896 from charging tank 119895 to CDU l

119865BC119895119897119905120576119861min119895119896

le 119891BC119895119897119896119905

le 119865BC119895119897119905120576119861max119895119896

forall119895 119897 119896 119905 (6c)

Volume capacity limitations for component 119896 in chargingtank 119895 at time t

119881119861

119895119905120576119861min119895119896

le V119861119895119896119905

le 119881119861

119895119905120576119861max119895119896

forall119895 119897 119896 119905 (6d)

(7) Operating Rules for Crude Oil Charging Charging tank 119895can charge at most one CDU at any time t

119873CDU

sum

119897=1

119863119895119897119905

le 1 forall119895 119905 (7a)

6 The Scientific World Journal

CDU 119897 can be charged only by one charging tank at any timet

119873BT

sum

119895=1

119863119895119897119905

= 1 forall119897 119905 (7b)

If CDU 119897 is charged by crude oilmix 119895 at time t minus 1 and chargedby 1198951015840 at time t changeover cost is involved

1198851198951198951015840119897119905ge 1198631198951015840119897119905+ 119863119895119897119905minus1

minus 1

119895 1198951015840(119895 = 1198951015840) = 1 119873

119861forall119897 119905

(7c)

The proposed formulation involves stochastic expres-sions which requires a different course of action for trans-forming it into an equivalent deterministic form The deter-ministic equivalent representation of the expectation of theobjective function is examined in the next section

4 Chance Constraint BasedDeterministic Transformation

41 Expectation of Penalty Cost The expectation of theunderproduction penalty cost term for crude mix 119895 inmacroperiod119898 is equal to

119864[

[

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)]

]

(8)

where

119862119895119898

≜ max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

= minusmin(0119873

CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

minus 119863119872

119895119898)

= minus[

[

min(119863119872119895119898

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905) minus 119863

119872

119895119898]

]

(9)

To facilitate the calculation of the expectation the stan-dardization of the normally distributed variables 119863119872

119895119898and

variables sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

is performed first Normal ran-dom variables can be recast into the standardized normalform with a mean of zero and a variance of 1 by subtractingtheir mean and dividing by their standard deviation (squareroot of variance) This defines the standardized normalvariables

119909119895119898 =

119863119872

119895119898minus119863119872

119895119898

120590119895119898

(10)

where 119863119872119895119898

denotes the mean of 119863119872119895119898

and 120590119895119898

the squareroot of its variance In the same way the ldquostandardizationrdquo ofthe variables sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

defines

119870119895119898

=

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

(11)

Using the method in paper [17] the underproductionpenalty term 119862

119895119898is charging into the following form

119862119895119898 = minus [minus119891 (119870119895119898) + (1 minus Φ (119870119895119898))119870119895119898] (12)

where 119891 is the standardized normal distribution functionand Φ denotes the cumulative probability function of astandard normal random variable

42 Satisfaction Level for Single Crude Mix Demand Theminimization of the objective function (1) as defined aboveestablishes the production and planned sales policy whichmost appropriately balances profits with inventory costs andunderproduction shortfalls A crudemix demand satisfactionlevel is not explicitly specified but rather it is the outcomeof the minimization of the profit function While highervalues of the parameter 119862

PEN119895

conceptually increase theprobability of demand satisfaction this strategymay still leadto unacceptably low probabilities of satisfying certain crudemix demands (see examples)Therefore the setting of explicitprobability targets on crudemix demand satisfaction is muchmore desirable A systematic way to accomplish this is toimpose explicit lower bounds on the probabilities of satisfyinga single crude mix demand This requirement for crude mix119895 in macroperiod119898 assumes the following form

Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898]

]

ge 120573119895119898 (13)

This constraint known as a chance constraint imposes a lowerbound 120573

119895119898on the probability that the crude mix demand

realization will be greater than the planned sales 119863119872119895119898

forcrude mix 119895 in macroperiod119898 The deterministic equivalentrepresentation can be obtained based on the concepts intro-duced by Charnes and Cooper [20]

Specifically by subtracting the mean and dividing bythe standard deviation of 119863119872

119895119898 the chance constraint can be

written equivalently as

Pr[

[

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

ge

119863119872

119895119898minus119863119872

119895119898

120590119895119898

]

]

ge 120573119895119898

Pr lceil119870119895119898

ge 119909119895119898rceil ge 120573119895119898

(14)

The right-hand side of the inequality within the proba-bility sign is a normally distributed random variable with amean of zero and a variance of 1 This implies that the chance

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

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Page 3: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

The Scientific World Journal 3

newmixed-integer nonlinear stochastic programmingmodelwith chance constraints The demands for different crudemix are uncertain andpossibly correlated reflecting changingmarket conditions and periodic variation in customer ordersAn optimal planning policy for minimizing expected cost isdeveloped This is achieved by letting crude mix demands besatisfied with at least a prespecified probability level

The proposed stochastic model involves the followingfeatures and assumptions

(1) A two-level time structure introduced by Fleis-chmann andMeyr [19] is adopted formodeling a gen-eral stochastic system The entire planning intervalis divided into macroperiods each with fluctuationdemand and eachmacroperiod consists of time inter-vals with fixed length

(2) Thedemands for different crudemixes aremodeled asmultivariate normally distributed random variablesThenormality assumption has beenwidely invoked inthe literature because it captures the essential featuresof demand uncertainty and it is convenient to useThe use of more ldquocomplexrdquo probability distributionsis hindered by the fact that statistical informationapart frommean and covariance estimates of productdemands is rarely available

(3) The demand correlation in the problem formulationstands for correlation of demands for different crudemixes in the same time period and demands for thesame crude mix in different time periods

(4) The penalty for some crude mix shortfalls is propor-tional to the amount of underproduction

22 Operation Rules The operation rules are shown asfollows (1) the refinery uses only one docking station anda new arriving vessel has to wait at sea until the anteriorvessel leaves the docking station (2) while a charging tankis charging CDU crude oil from the storage tanks cannot befed into the charging tank and vice versa (3) each chargingtank can only charge one CDU at each time interval (4)each CDU can only be charged by one charging tank ateach time interval (5) CDUs must be operated continuouslythroughout the scheduling time horizon

3 Mathematical Model

Indices and Sets

119894 isin 1 119873119878 = crude oil storage tank and the crude

oil in it119895 isin 1 119873

119861 = crude oil charging tank and the

crude oil mix in the charging tank119896 isin 1 119873

119862 = key component of crude oil

119897 isin 1 119873CDU

= crude distillation unit119898 isin 1 119873

119872 = macroperiod

119905 isin 1 119873SCH

= time interval 119873SCH denotesscheduling horizon

119879119898= the starting time of the macroperiodm

V isin 1 119873119881 = crude vessels

Variables

119863119895119897119905

= 0-1 variable to denote if the crude oil mix incharging tank 119895 charges CDU 119897 at time t119883119865

V119905 = 0-1 variable to denote if vessel V starts unload-ing at time t119883119871

V119905 = 0-1 variable to denote if vessel V just completesunloading at time t119883119882

V119905 = 0-1 continuous variable to denote if vessel v isunloading its crude oil at time t1198851198951198951015840 119897119905 = 0-1 continuous variable to denote if transi-tion from crude mix (or charging tank) j to 1198951015840 at timet in CDU l119891SB119894119895119896119905

= volumetric flow rate of component 119896 fromstorage tank 119894 to charging tank 119895 at time t119891BC119895119897119896119905

= volumetric flow rate of component 119896 fromcharging tank 119895 to CDU 119897 at time t119865VSV119894119905 = volumetric flow rate of crude oil from vessel V

to storage tank 119894 at time t119865SB119894119895119905

= volumetric flow rate of crude oil from storagetank 119894 to charging tank 119895 at time t119865BC119895119897119905

= volumetric flow rate of crude oil mix fromcharging tank 119895 to CDU 119897 at time t119879119865

V = vessel v unloading initiation time119879119871

V = vessel v unloading completion and departuretimeV119861119895119896119905

= volume of component k in charging tank j attime t119881119881

V119905 = volume of crude oil in crude vessel v at time t

119881119878

119894119905= volume of crude oil in storage tank i at time t

119881119861

119895119905= volume of mixed oil in charging tank j at time

t

Parameters

119862UNLOAD119881

= unloading cost of vessel V per time inter-val119862SEA119881

= sea waiting cost of vessel V per time interval119862INVST119894

= inventory cost of storage tank 119894 per time pervolume119862INVBL119895

= inventory cost of charging tank 119895 per timeper volume119862SETUP1198951198951015840 = changeover cost for transition from crude

mix j to 1198951015840 in CDU119862PEN119895

= penalty cost for crude mix 119895 shortfalls

119863119872

119895119898= stochastic demand of crude mix 119895 by CDUs

during the macroperiodm

4 The Scientific World Journal

119865VSminV119894 =minimumcrude oil transfer rate from vessel

V to storage tank i

119865VSmaxV119894 = maximum crude oil transfer rate from

vessel V to storage tank i

119865SBmin119894119895

= minimum crude oil transfer rate from stor-age tank 119894 to charging tank j

119865SBmax119894119895

= maximum crude oil transfer rate fromstorage tank 119894 to charging tank j

119865BCmin119895119897

= minimum crude oil charging rate fromcharging tank 119895 to CDU l

119865BCmax119895119897

= maximum crude oil charging rate fromcharging tank 119895 to CDU l

119879ARRV = crude vessel V arrival time around the docking

station

119881119881

V0 = initial volume of crude oil in crude vessel v

119881119878min119894

=minimum crude oil volume of storage tank i

119881119878max119894

= maximum crude oil volume of storage tanki

119881119878

1198940= initial crude oil volume of storage tank i

119881119861min119895

= minimummixed crude oil volume of charg-ing tank j

119881119861max119895

=maximummixed crude oil volume of charg-ing tank j

120576119878

119894119896= concentration of component 119896 in the crude oil

of storage tank i

120576119861

1198951198960= initial concentration of component 119896 in the

crude mix of charging tank j

120576119861min119895

= minimum concentration of component 119896 inthe crude mix of charging tank j

120576119861max119895

= maximum concentration of component 119896 inthe crude mix of charging tank j

In this paper a two-level time structure is used tomodel a general dynamic production system We considerthe crude mix 119895 to be scheduled over a finite planninghorizon consisting of macroperiods 119898 = 1 sdot sdot sdot 119873

119872 Eachmacroperiod 119898 is divided into time intervals with fixedlength where 119879

119898represents the starting time of macrope-

riod 119898 Let 119863119872119895119898

a random variable denote the demand ofcrude mix 119895 in macroperiod 119898 We approximate the crudemix demand as a multivariate normal distribution with aspecific correlation structure 119863119872 sim 119873(120583 Σ) with covariancematrix Σ where 119863119872 def

= [119863119872

119895119898] 120583

def= [120583119895119898] It means that the

demands for different mix crudes not only are correlated butalso are the demands for the same mixed crude in differentmacroperiods

The mathematical model is formulated as follows(1) Operating Cost

MinimizeOperating cost = unloading cost for the crude vessel+ cost for vessel waiting in the sea + inventory costfor storage and charging tanks + changeover cost +underproduction penalty cost

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873BT

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873119861119879

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840119897119905

+

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

(1)

Subject to

(2) Vessel Arrival and Departure Operation Rules Each vesselarrives at the docking station for unloading only oncethroughout the scheduling horizon

119873SCH

sum

119905=1

119883119865

V119905 = 1 forallV (2a)

Each vessel leaves the docking station only once throughoutthe scheduling horizon Consider

119873SCH

sum

119905=1

119883119871

V119905 = 1 forallV (2b)

Equation for unloading initiation time

119879119865

V =

119873SCH

sum

119905=1

119905119883119865

V119905 forallV (2c)

Equation for unloading completion time

119879119871

V =

119873SCH

sum

119905=1

119905119883119871

V119905 forallV (2d)

Each crude vessel should start unloading after arrival time setin the planning level

119879119865

V ge 119879ARRV forallV (2e)

The Scientific World Journal 5

Duration of the vessel unloading is bounded by the initial vol-ume of oil in the vessel divided by the maximum unloadingrate

119879119871

V minus 119879119865

V ge lceil119881119881

V0

max lceil119865119881119878maxV119894 rceil

rceil forallV (2f)

lceilrceil corresponds to round-up of the next highest integer valueVessel in the sea cannot arrive at the docking station forunloading unless the preceding vessel leaves

119879119865

V+1 ge 119879119871

V forallV (2g)

Unloading is possible between time 119879119865V and 119879119871V

119883119882

V119905 le

119905

sum

120591=1

119883119865

V120591 119883119882

V119905 le

SCHsum

120591=119905

119883119871

V120591 forallV 119905 (2h)

(3) Material Balance Equations for the Vessel For vessel vthe oil volume at time 119905 equals the initial volume minustransferred volume from vessel V to storage tanks up to timet

119881119881

V119905 = 119881119881

V0 minus

119873ST

sum

119894=1

119905

sum

120591=1

119865119881119878

V119894120591 forallV 119905 (3a)

Operating constraints on crude oil transfer rate from vessel Vto storage tank 119894 at time t

119865VSminV119894 119883

119882

V119905 le 119865VSV119894119905 le 119865

VSmaxV119894 119883

119882

V119905 forallV 119894 119905 (3b)

The volume of crude oil transferred from vessel V to storagetanks during the scheduling horizon equals the initial crudeoil volume of vessel v

119873ST

sum

119894=1

119873SCH

sum

119905=1

119865VSV119894119905 = 119881

119881

V0 forallV 119894 119905 (3c)

(4) Material Balance Equations for Storage Tanks The oilvolume in storage tank 119894 at time 119905 equals the initial oil volumein tank 119894 plus the oil volume transferred from vessels to tank119894 up to the time 119905 and minus oil volume removed from tank 119894to charging tanks up to the time t

119881119878

119894119905= 119881119878

1198940+

119873119881

sum

V=1

119905

sum

120591=1

119865VSV119894120591 minus

119873BT

sum

119895=1

119905

sum

120591=1

119865SB119894119895120591

forall119894 119905 (4a)

Operating constraints on crude oil transfer rate from storagetank 119894 to charging tank 119895 at time t the term 1 minus sum

NCDU119897=1

119863119895119897119905

denotes that if charging tank 119895 is charging any CDU there isno oil transfer from storage tank 119894 to charging tank j Consider

119865SBmin119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905)

le 119865SB119894119895119905

le 119865SBmax119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905) forall119894 119895 119905

(4b)

Volume capacity limitations for storage tank 119894 at time t

119881119878min119894

le 119881119878

119894119905le 119881119878max119894

forall119894 119905 (4c)

(5) Material Balance Equations for Charging TanksThe crudeoilmix in charging tank 119895 at time 119905 equals the initial oil volumein charging tank 119895 plus the crude oil transferred from storagetanks to charging tank j up to time t and minus the crude oilmix 119895 charged into CDUs up to time t

119881119861

119895119905= 119881119861

1198950+

119873ST

sum

119894=1

119905

sum

120591=1

119865SB119894119895120591

minus

119873CDU

sum

119897=1

119905

sum

120591=1

119865BC119895119897120591

forall119895 119905

(5a)

Operating constraints on mixed oil transfer rate from charg-ing tank 119895 to CDU 119897 at time t

119865BCmin119895119897

119863119895119897119905

le 119865BC119895119897119905

le 119865BCmax119895119897

119863119895119897119905

forall119895 119897 119905 (5b)

Volume capacity limitations for charging tank 119895 at time t

119881119861min119895

le 119881119861

119895119905le 119881119861max119895

forall119895 119905 (5c)

(6) Material Balance Equations for Component 119896 in ChargingTanks The volume of component 119896 in charging tank 119895 attime 119905 equals the initial component 119896 in charging tank 119895 pluscomponent 119896 in crude oil transferred from storage tanks tocharging tank 119895 up to the time 119905 and minus component 119896 incrude oil mix 119895 transferred to CDUs up to the time t

V119861119895119905= V1198611198950+

119905

sum

120591=1

119873ST

sum

119894=1

119891SB119894119895120591

minus

119905

sum

120591=1

119873CDU

sum

119897=1

119891BC119895119897120591

forall119895 119896 119905 (6a)

Operating constraints on volumetric flow rate of component119896 from storage tank 119894 to charging tank j

119891SB119894119895119896119905

= 119865SB119894119895119905120576119878

119894119896forall119894 119895 119896 119905 (6b)

Operating constraints on volumetric flow rate of component119896 from charging tank 119895 to CDU l

119865BC119895119897119905120576119861min119895119896

le 119891BC119895119897119896119905

le 119865BC119895119897119905120576119861max119895119896

forall119895 119897 119896 119905 (6c)

Volume capacity limitations for component 119896 in chargingtank 119895 at time t

119881119861

119895119905120576119861min119895119896

le V119861119895119896119905

le 119881119861

119895119905120576119861max119895119896

forall119895 119897 119896 119905 (6d)

(7) Operating Rules for Crude Oil Charging Charging tank 119895can charge at most one CDU at any time t

119873CDU

sum

119897=1

119863119895119897119905

le 1 forall119895 119905 (7a)

6 The Scientific World Journal

CDU 119897 can be charged only by one charging tank at any timet

119873BT

sum

119895=1

119863119895119897119905

= 1 forall119897 119905 (7b)

If CDU 119897 is charged by crude oilmix 119895 at time t minus 1 and chargedby 1198951015840 at time t changeover cost is involved

1198851198951198951015840119897119905ge 1198631198951015840119897119905+ 119863119895119897119905minus1

minus 1

119895 1198951015840(119895 = 1198951015840) = 1 119873

119861forall119897 119905

(7c)

The proposed formulation involves stochastic expres-sions which requires a different course of action for trans-forming it into an equivalent deterministic form The deter-ministic equivalent representation of the expectation of theobjective function is examined in the next section

4 Chance Constraint BasedDeterministic Transformation

41 Expectation of Penalty Cost The expectation of theunderproduction penalty cost term for crude mix 119895 inmacroperiod119898 is equal to

119864[

[

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)]

]

(8)

where

119862119895119898

≜ max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

= minusmin(0119873

CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

minus 119863119872

119895119898)

= minus[

[

min(119863119872119895119898

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905) minus 119863

119872

119895119898]

]

(9)

To facilitate the calculation of the expectation the stan-dardization of the normally distributed variables 119863119872

119895119898and

variables sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

is performed first Normal ran-dom variables can be recast into the standardized normalform with a mean of zero and a variance of 1 by subtractingtheir mean and dividing by their standard deviation (squareroot of variance) This defines the standardized normalvariables

119909119895119898 =

119863119872

119895119898minus119863119872

119895119898

120590119895119898

(10)

where 119863119872119895119898

denotes the mean of 119863119872119895119898

and 120590119895119898

the squareroot of its variance In the same way the ldquostandardizationrdquo ofthe variables sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

defines

119870119895119898

=

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

(11)

Using the method in paper [17] the underproductionpenalty term 119862

119895119898is charging into the following form

119862119895119898 = minus [minus119891 (119870119895119898) + (1 minus Φ (119870119895119898))119870119895119898] (12)

where 119891 is the standardized normal distribution functionand Φ denotes the cumulative probability function of astandard normal random variable

42 Satisfaction Level for Single Crude Mix Demand Theminimization of the objective function (1) as defined aboveestablishes the production and planned sales policy whichmost appropriately balances profits with inventory costs andunderproduction shortfalls A crudemix demand satisfactionlevel is not explicitly specified but rather it is the outcomeof the minimization of the profit function While highervalues of the parameter 119862

PEN119895

conceptually increase theprobability of demand satisfaction this strategymay still leadto unacceptably low probabilities of satisfying certain crudemix demands (see examples)Therefore the setting of explicitprobability targets on crudemix demand satisfaction is muchmore desirable A systematic way to accomplish this is toimpose explicit lower bounds on the probabilities of satisfyinga single crude mix demand This requirement for crude mix119895 in macroperiod119898 assumes the following form

Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898]

]

ge 120573119895119898 (13)

This constraint known as a chance constraint imposes a lowerbound 120573

119895119898on the probability that the crude mix demand

realization will be greater than the planned sales 119863119872119895119898

forcrude mix 119895 in macroperiod119898 The deterministic equivalentrepresentation can be obtained based on the concepts intro-duced by Charnes and Cooper [20]

Specifically by subtracting the mean and dividing bythe standard deviation of 119863119872

119895119898 the chance constraint can be

written equivalently as

Pr[

[

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

ge

119863119872

119895119898minus119863119872

119895119898

120590119895119898

]

]

ge 120573119895119898

Pr lceil119870119895119898

ge 119909119895119898rceil ge 120573119895119898

(14)

The right-hand side of the inequality within the proba-bility sign is a normally distributed random variable with amean of zero and a variance of 1 This implies that the chance

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

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Stochastic AnalysisInternational Journal of

Page 4: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

4 The Scientific World Journal

119865VSminV119894 =minimumcrude oil transfer rate from vessel

V to storage tank i

119865VSmaxV119894 = maximum crude oil transfer rate from

vessel V to storage tank i

119865SBmin119894119895

= minimum crude oil transfer rate from stor-age tank 119894 to charging tank j

119865SBmax119894119895

= maximum crude oil transfer rate fromstorage tank 119894 to charging tank j

119865BCmin119895119897

= minimum crude oil charging rate fromcharging tank 119895 to CDU l

119865BCmax119895119897

= maximum crude oil charging rate fromcharging tank 119895 to CDU l

119879ARRV = crude vessel V arrival time around the docking

station

119881119881

V0 = initial volume of crude oil in crude vessel v

119881119878min119894

=minimum crude oil volume of storage tank i

119881119878max119894

= maximum crude oil volume of storage tanki

119881119878

1198940= initial crude oil volume of storage tank i

119881119861min119895

= minimummixed crude oil volume of charg-ing tank j

119881119861max119895

=maximummixed crude oil volume of charg-ing tank j

120576119878

119894119896= concentration of component 119896 in the crude oil

of storage tank i

120576119861

1198951198960= initial concentration of component 119896 in the

crude mix of charging tank j

120576119861min119895

= minimum concentration of component 119896 inthe crude mix of charging tank j

120576119861max119895

= maximum concentration of component 119896 inthe crude mix of charging tank j

In this paper a two-level time structure is used tomodel a general dynamic production system We considerthe crude mix 119895 to be scheduled over a finite planninghorizon consisting of macroperiods 119898 = 1 sdot sdot sdot 119873

119872 Eachmacroperiod 119898 is divided into time intervals with fixedlength where 119879

119898represents the starting time of macrope-

riod 119898 Let 119863119872119895119898

a random variable denote the demand ofcrude mix 119895 in macroperiod 119898 We approximate the crudemix demand as a multivariate normal distribution with aspecific correlation structure 119863119872 sim 119873(120583 Σ) with covariancematrix Σ where 119863119872 def

= [119863119872

119895119898] 120583

def= [120583119895119898] It means that the

demands for different mix crudes not only are correlated butalso are the demands for the same mixed crude in differentmacroperiods

The mathematical model is formulated as follows(1) Operating Cost

MinimizeOperating cost = unloading cost for the crude vessel+ cost for vessel waiting in the sea + inventory costfor storage and charging tanks + changeover cost +underproduction penalty cost

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873BT

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873119861119879

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840119897119905

+

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

(1)

Subject to

(2) Vessel Arrival and Departure Operation Rules Each vesselarrives at the docking station for unloading only oncethroughout the scheduling horizon

119873SCH

sum

119905=1

119883119865

V119905 = 1 forallV (2a)

Each vessel leaves the docking station only once throughoutthe scheduling horizon Consider

119873SCH

sum

119905=1

119883119871

V119905 = 1 forallV (2b)

Equation for unloading initiation time

119879119865

V =

119873SCH

sum

119905=1

119905119883119865

V119905 forallV (2c)

Equation for unloading completion time

119879119871

V =

119873SCH

sum

119905=1

119905119883119871

V119905 forallV (2d)

Each crude vessel should start unloading after arrival time setin the planning level

119879119865

V ge 119879ARRV forallV (2e)

The Scientific World Journal 5

Duration of the vessel unloading is bounded by the initial vol-ume of oil in the vessel divided by the maximum unloadingrate

119879119871

V minus 119879119865

V ge lceil119881119881

V0

max lceil119865119881119878maxV119894 rceil

rceil forallV (2f)

lceilrceil corresponds to round-up of the next highest integer valueVessel in the sea cannot arrive at the docking station forunloading unless the preceding vessel leaves

119879119865

V+1 ge 119879119871

V forallV (2g)

Unloading is possible between time 119879119865V and 119879119871V

119883119882

V119905 le

119905

sum

120591=1

119883119865

V120591 119883119882

V119905 le

SCHsum

120591=119905

119883119871

V120591 forallV 119905 (2h)

(3) Material Balance Equations for the Vessel For vessel vthe oil volume at time 119905 equals the initial volume minustransferred volume from vessel V to storage tanks up to timet

119881119881

V119905 = 119881119881

V0 minus

119873ST

sum

119894=1

119905

sum

120591=1

119865119881119878

V119894120591 forallV 119905 (3a)

Operating constraints on crude oil transfer rate from vessel Vto storage tank 119894 at time t

119865VSminV119894 119883

119882

V119905 le 119865VSV119894119905 le 119865

VSmaxV119894 119883

119882

V119905 forallV 119894 119905 (3b)

The volume of crude oil transferred from vessel V to storagetanks during the scheduling horizon equals the initial crudeoil volume of vessel v

119873ST

sum

119894=1

119873SCH

sum

119905=1

119865VSV119894119905 = 119881

119881

V0 forallV 119894 119905 (3c)

(4) Material Balance Equations for Storage Tanks The oilvolume in storage tank 119894 at time 119905 equals the initial oil volumein tank 119894 plus the oil volume transferred from vessels to tank119894 up to the time 119905 and minus oil volume removed from tank 119894to charging tanks up to the time t

119881119878

119894119905= 119881119878

1198940+

119873119881

sum

V=1

119905

sum

120591=1

119865VSV119894120591 minus

119873BT

sum

119895=1

119905

sum

120591=1

119865SB119894119895120591

forall119894 119905 (4a)

Operating constraints on crude oil transfer rate from storagetank 119894 to charging tank 119895 at time t the term 1 minus sum

NCDU119897=1

119863119895119897119905

denotes that if charging tank 119895 is charging any CDU there isno oil transfer from storage tank 119894 to charging tank j Consider

119865SBmin119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905)

le 119865SB119894119895119905

le 119865SBmax119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905) forall119894 119895 119905

(4b)

Volume capacity limitations for storage tank 119894 at time t

119881119878min119894

le 119881119878

119894119905le 119881119878max119894

forall119894 119905 (4c)

(5) Material Balance Equations for Charging TanksThe crudeoilmix in charging tank 119895 at time 119905 equals the initial oil volumein charging tank 119895 plus the crude oil transferred from storagetanks to charging tank j up to time t and minus the crude oilmix 119895 charged into CDUs up to time t

119881119861

119895119905= 119881119861

1198950+

119873ST

sum

119894=1

119905

sum

120591=1

119865SB119894119895120591

minus

119873CDU

sum

119897=1

119905

sum

120591=1

119865BC119895119897120591

forall119895 119905

(5a)

Operating constraints on mixed oil transfer rate from charg-ing tank 119895 to CDU 119897 at time t

119865BCmin119895119897

119863119895119897119905

le 119865BC119895119897119905

le 119865BCmax119895119897

119863119895119897119905

forall119895 119897 119905 (5b)

Volume capacity limitations for charging tank 119895 at time t

119881119861min119895

le 119881119861

119895119905le 119881119861max119895

forall119895 119905 (5c)

(6) Material Balance Equations for Component 119896 in ChargingTanks The volume of component 119896 in charging tank 119895 attime 119905 equals the initial component 119896 in charging tank 119895 pluscomponent 119896 in crude oil transferred from storage tanks tocharging tank 119895 up to the time 119905 and minus component 119896 incrude oil mix 119895 transferred to CDUs up to the time t

V119861119895119905= V1198611198950+

119905

sum

120591=1

119873ST

sum

119894=1

119891SB119894119895120591

minus

119905

sum

120591=1

119873CDU

sum

119897=1

119891BC119895119897120591

forall119895 119896 119905 (6a)

Operating constraints on volumetric flow rate of component119896 from storage tank 119894 to charging tank j

119891SB119894119895119896119905

= 119865SB119894119895119905120576119878

119894119896forall119894 119895 119896 119905 (6b)

Operating constraints on volumetric flow rate of component119896 from charging tank 119895 to CDU l

119865BC119895119897119905120576119861min119895119896

le 119891BC119895119897119896119905

le 119865BC119895119897119905120576119861max119895119896

forall119895 119897 119896 119905 (6c)

Volume capacity limitations for component 119896 in chargingtank 119895 at time t

119881119861

119895119905120576119861min119895119896

le V119861119895119896119905

le 119881119861

119895119905120576119861max119895119896

forall119895 119897 119896 119905 (6d)

(7) Operating Rules for Crude Oil Charging Charging tank 119895can charge at most one CDU at any time t

119873CDU

sum

119897=1

119863119895119897119905

le 1 forall119895 119905 (7a)

6 The Scientific World Journal

CDU 119897 can be charged only by one charging tank at any timet

119873BT

sum

119895=1

119863119895119897119905

= 1 forall119897 119905 (7b)

If CDU 119897 is charged by crude oilmix 119895 at time t minus 1 and chargedby 1198951015840 at time t changeover cost is involved

1198851198951198951015840119897119905ge 1198631198951015840119897119905+ 119863119895119897119905minus1

minus 1

119895 1198951015840(119895 = 1198951015840) = 1 119873

119861forall119897 119905

(7c)

The proposed formulation involves stochastic expres-sions which requires a different course of action for trans-forming it into an equivalent deterministic form The deter-ministic equivalent representation of the expectation of theobjective function is examined in the next section

4 Chance Constraint BasedDeterministic Transformation

41 Expectation of Penalty Cost The expectation of theunderproduction penalty cost term for crude mix 119895 inmacroperiod119898 is equal to

119864[

[

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)]

]

(8)

where

119862119895119898

≜ max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

= minusmin(0119873

CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

minus 119863119872

119895119898)

= minus[

[

min(119863119872119895119898

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905) minus 119863

119872

119895119898]

]

(9)

To facilitate the calculation of the expectation the stan-dardization of the normally distributed variables 119863119872

119895119898and

variables sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

is performed first Normal ran-dom variables can be recast into the standardized normalform with a mean of zero and a variance of 1 by subtractingtheir mean and dividing by their standard deviation (squareroot of variance) This defines the standardized normalvariables

119909119895119898 =

119863119872

119895119898minus119863119872

119895119898

120590119895119898

(10)

where 119863119872119895119898

denotes the mean of 119863119872119895119898

and 120590119895119898

the squareroot of its variance In the same way the ldquostandardizationrdquo ofthe variables sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

defines

119870119895119898

=

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

(11)

Using the method in paper [17] the underproductionpenalty term 119862

119895119898is charging into the following form

119862119895119898 = minus [minus119891 (119870119895119898) + (1 minus Φ (119870119895119898))119870119895119898] (12)

where 119891 is the standardized normal distribution functionand Φ denotes the cumulative probability function of astandard normal random variable

42 Satisfaction Level for Single Crude Mix Demand Theminimization of the objective function (1) as defined aboveestablishes the production and planned sales policy whichmost appropriately balances profits with inventory costs andunderproduction shortfalls A crudemix demand satisfactionlevel is not explicitly specified but rather it is the outcomeof the minimization of the profit function While highervalues of the parameter 119862

PEN119895

conceptually increase theprobability of demand satisfaction this strategymay still leadto unacceptably low probabilities of satisfying certain crudemix demands (see examples)Therefore the setting of explicitprobability targets on crudemix demand satisfaction is muchmore desirable A systematic way to accomplish this is toimpose explicit lower bounds on the probabilities of satisfyinga single crude mix demand This requirement for crude mix119895 in macroperiod119898 assumes the following form

Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898]

]

ge 120573119895119898 (13)

This constraint known as a chance constraint imposes a lowerbound 120573

119895119898on the probability that the crude mix demand

realization will be greater than the planned sales 119863119872119895119898

forcrude mix 119895 in macroperiod119898 The deterministic equivalentrepresentation can be obtained based on the concepts intro-duced by Charnes and Cooper [20]

Specifically by subtracting the mean and dividing bythe standard deviation of 119863119872

119895119898 the chance constraint can be

written equivalently as

Pr[

[

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

ge

119863119872

119895119898minus119863119872

119895119898

120590119895119898

]

]

ge 120573119895119898

Pr lceil119870119895119898

ge 119909119895119898rceil ge 120573119895119898

(14)

The right-hand side of the inequality within the proba-bility sign is a normally distributed random variable with amean of zero and a variance of 1 This implies that the chance

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

The Scientific World Journal 5

Duration of the vessel unloading is bounded by the initial vol-ume of oil in the vessel divided by the maximum unloadingrate

119879119871

V minus 119879119865

V ge lceil119881119881

V0

max lceil119865119881119878maxV119894 rceil

rceil forallV (2f)

lceilrceil corresponds to round-up of the next highest integer valueVessel in the sea cannot arrive at the docking station forunloading unless the preceding vessel leaves

119879119865

V+1 ge 119879119871

V forallV (2g)

Unloading is possible between time 119879119865V and 119879119871V

119883119882

V119905 le

119905

sum

120591=1

119883119865

V120591 119883119882

V119905 le

SCHsum

120591=119905

119883119871

V120591 forallV 119905 (2h)

(3) Material Balance Equations for the Vessel For vessel vthe oil volume at time 119905 equals the initial volume minustransferred volume from vessel V to storage tanks up to timet

119881119881

V119905 = 119881119881

V0 minus

119873ST

sum

119894=1

119905

sum

120591=1

119865119881119878

V119894120591 forallV 119905 (3a)

Operating constraints on crude oil transfer rate from vessel Vto storage tank 119894 at time t

119865VSminV119894 119883

119882

V119905 le 119865VSV119894119905 le 119865

VSmaxV119894 119883

119882

V119905 forallV 119894 119905 (3b)

The volume of crude oil transferred from vessel V to storagetanks during the scheduling horizon equals the initial crudeoil volume of vessel v

119873ST

sum

119894=1

119873SCH

sum

119905=1

119865VSV119894119905 = 119881

119881

V0 forallV 119894 119905 (3c)

(4) Material Balance Equations for Storage Tanks The oilvolume in storage tank 119894 at time 119905 equals the initial oil volumein tank 119894 plus the oil volume transferred from vessels to tank119894 up to the time 119905 and minus oil volume removed from tank 119894to charging tanks up to the time t

119881119878

119894119905= 119881119878

1198940+

119873119881

sum

V=1

119905

sum

120591=1

119865VSV119894120591 minus

119873BT

sum

119895=1

119905

sum

120591=1

119865SB119894119895120591

forall119894 119905 (4a)

Operating constraints on crude oil transfer rate from storagetank 119894 to charging tank 119895 at time t the term 1 minus sum

NCDU119897=1

119863119895119897119905

denotes that if charging tank 119895 is charging any CDU there isno oil transfer from storage tank 119894 to charging tank j Consider

119865SBmin119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905)

le 119865SB119894119895119905

le 119865SBmax119894119895

(1 minus

119873CDU

sum

119897=1

119863119895119897119905) forall119894 119895 119905

(4b)

Volume capacity limitations for storage tank 119894 at time t

119881119878min119894

le 119881119878

119894119905le 119881119878max119894

forall119894 119905 (4c)

(5) Material Balance Equations for Charging TanksThe crudeoilmix in charging tank 119895 at time 119905 equals the initial oil volumein charging tank 119895 plus the crude oil transferred from storagetanks to charging tank j up to time t and minus the crude oilmix 119895 charged into CDUs up to time t

119881119861

119895119905= 119881119861

1198950+

119873ST

sum

119894=1

119905

sum

120591=1

119865SB119894119895120591

minus

119873CDU

sum

119897=1

119905

sum

120591=1

119865BC119895119897120591

forall119895 119905

(5a)

Operating constraints on mixed oil transfer rate from charg-ing tank 119895 to CDU 119897 at time t

119865BCmin119895119897

119863119895119897119905

le 119865BC119895119897119905

le 119865BCmax119895119897

119863119895119897119905

forall119895 119897 119905 (5b)

Volume capacity limitations for charging tank 119895 at time t

119881119861min119895

le 119881119861

119895119905le 119881119861max119895

forall119895 119905 (5c)

(6) Material Balance Equations for Component 119896 in ChargingTanks The volume of component 119896 in charging tank 119895 attime 119905 equals the initial component 119896 in charging tank 119895 pluscomponent 119896 in crude oil transferred from storage tanks tocharging tank 119895 up to the time 119905 and minus component 119896 incrude oil mix 119895 transferred to CDUs up to the time t

V119861119895119905= V1198611198950+

119905

sum

120591=1

119873ST

sum

119894=1

119891SB119894119895120591

minus

119905

sum

120591=1

119873CDU

sum

119897=1

119891BC119895119897120591

forall119895 119896 119905 (6a)

Operating constraints on volumetric flow rate of component119896 from storage tank 119894 to charging tank j

119891SB119894119895119896119905

= 119865SB119894119895119905120576119878

119894119896forall119894 119895 119896 119905 (6b)

Operating constraints on volumetric flow rate of component119896 from charging tank 119895 to CDU l

119865BC119895119897119905120576119861min119895119896

le 119891BC119895119897119896119905

le 119865BC119895119897119905120576119861max119895119896

forall119895 119897 119896 119905 (6c)

Volume capacity limitations for component 119896 in chargingtank 119895 at time t

119881119861

119895119905120576119861min119895119896

le V119861119895119896119905

le 119881119861

119895119905120576119861max119895119896

forall119895 119897 119896 119905 (6d)

(7) Operating Rules for Crude Oil Charging Charging tank 119895can charge at most one CDU at any time t

119873CDU

sum

119897=1

119863119895119897119905

le 1 forall119895 119905 (7a)

6 The Scientific World Journal

CDU 119897 can be charged only by one charging tank at any timet

119873BT

sum

119895=1

119863119895119897119905

= 1 forall119897 119905 (7b)

If CDU 119897 is charged by crude oilmix 119895 at time t minus 1 and chargedby 1198951015840 at time t changeover cost is involved

1198851198951198951015840119897119905ge 1198631198951015840119897119905+ 119863119895119897119905minus1

minus 1

119895 1198951015840(119895 = 1198951015840) = 1 119873

119861forall119897 119905

(7c)

The proposed formulation involves stochastic expres-sions which requires a different course of action for trans-forming it into an equivalent deterministic form The deter-ministic equivalent representation of the expectation of theobjective function is examined in the next section

4 Chance Constraint BasedDeterministic Transformation

41 Expectation of Penalty Cost The expectation of theunderproduction penalty cost term for crude mix 119895 inmacroperiod119898 is equal to

119864[

[

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)]

]

(8)

where

119862119895119898

≜ max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

= minusmin(0119873

CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

minus 119863119872

119895119898)

= minus[

[

min(119863119872119895119898

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905) minus 119863

119872

119895119898]

]

(9)

To facilitate the calculation of the expectation the stan-dardization of the normally distributed variables 119863119872

119895119898and

variables sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

is performed first Normal ran-dom variables can be recast into the standardized normalform with a mean of zero and a variance of 1 by subtractingtheir mean and dividing by their standard deviation (squareroot of variance) This defines the standardized normalvariables

119909119895119898 =

119863119872

119895119898minus119863119872

119895119898

120590119895119898

(10)

where 119863119872119895119898

denotes the mean of 119863119872119895119898

and 120590119895119898

the squareroot of its variance In the same way the ldquostandardizationrdquo ofthe variables sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

defines

119870119895119898

=

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

(11)

Using the method in paper [17] the underproductionpenalty term 119862

119895119898is charging into the following form

119862119895119898 = minus [minus119891 (119870119895119898) + (1 minus Φ (119870119895119898))119870119895119898] (12)

where 119891 is the standardized normal distribution functionand Φ denotes the cumulative probability function of astandard normal random variable

42 Satisfaction Level for Single Crude Mix Demand Theminimization of the objective function (1) as defined aboveestablishes the production and planned sales policy whichmost appropriately balances profits with inventory costs andunderproduction shortfalls A crudemix demand satisfactionlevel is not explicitly specified but rather it is the outcomeof the minimization of the profit function While highervalues of the parameter 119862

PEN119895

conceptually increase theprobability of demand satisfaction this strategymay still leadto unacceptably low probabilities of satisfying certain crudemix demands (see examples)Therefore the setting of explicitprobability targets on crudemix demand satisfaction is muchmore desirable A systematic way to accomplish this is toimpose explicit lower bounds on the probabilities of satisfyinga single crude mix demand This requirement for crude mix119895 in macroperiod119898 assumes the following form

Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898]

]

ge 120573119895119898 (13)

This constraint known as a chance constraint imposes a lowerbound 120573

119895119898on the probability that the crude mix demand

realization will be greater than the planned sales 119863119872119895119898

forcrude mix 119895 in macroperiod119898 The deterministic equivalentrepresentation can be obtained based on the concepts intro-duced by Charnes and Cooper [20]

Specifically by subtracting the mean and dividing bythe standard deviation of 119863119872

119895119898 the chance constraint can be

written equivalently as

Pr[

[

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

ge

119863119872

119895119898minus119863119872

119895119898

120590119895119898

]

]

ge 120573119895119898

Pr lceil119870119895119898

ge 119909119895119898rceil ge 120573119895119898

(14)

The right-hand side of the inequality within the proba-bility sign is a normally distributed random variable with amean of zero and a variance of 1 This implies that the chance

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

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Stochastic AnalysisInternational Journal of

Page 6: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

6 The Scientific World Journal

CDU 119897 can be charged only by one charging tank at any timet

119873BT

sum

119895=1

119863119895119897119905

= 1 forall119897 119905 (7b)

If CDU 119897 is charged by crude oilmix 119895 at time t minus 1 and chargedby 1198951015840 at time t changeover cost is involved

1198851198951198951015840119897119905ge 1198631198951015840119897119905+ 119863119895119897119905minus1

minus 1

119895 1198951015840(119895 = 1198951015840) = 1 119873

119861forall119897 119905

(7c)

The proposed formulation involves stochastic expres-sions which requires a different course of action for trans-forming it into an equivalent deterministic form The deter-ministic equivalent representation of the expectation of theobjective function is examined in the next section

4 Chance Constraint BasedDeterministic Transformation

41 Expectation of Penalty Cost The expectation of theunderproduction penalty cost term for crude mix 119895 inmacroperiod119898 is equal to

119864[

[

119873BT

sum

119895=1

119862PEN119895

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)]

]

(8)

where

119862119895119898

≜ max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905)

= minusmin(0119873

CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

minus 119863119872

119895119898)

= minus[

[

min(119863119872119895119898

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905) minus 119863

119872

119895119898]

]

(9)

To facilitate the calculation of the expectation the stan-dardization of the normally distributed variables 119863119872

119895119898and

variables sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

is performed first Normal ran-dom variables can be recast into the standardized normalform with a mean of zero and a variance of 1 by subtractingtheir mean and dividing by their standard deviation (squareroot of variance) This defines the standardized normalvariables

119909119895119898 =

119863119872

119895119898minus119863119872

119895119898

120590119895119898

(10)

where 119863119872119895119898

denotes the mean of 119863119872119895119898

and 120590119895119898

the squareroot of its variance In the same way the ldquostandardizationrdquo ofthe variables sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

defines

119870119895119898

=

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

(11)

Using the method in paper [17] the underproductionpenalty term 119862

119895119898is charging into the following form

119862119895119898 = minus [minus119891 (119870119895119898) + (1 minus Φ (119870119895119898))119870119895119898] (12)

where 119891 is the standardized normal distribution functionand Φ denotes the cumulative probability function of astandard normal random variable

42 Satisfaction Level for Single Crude Mix Demand Theminimization of the objective function (1) as defined aboveestablishes the production and planned sales policy whichmost appropriately balances profits with inventory costs andunderproduction shortfalls A crudemix demand satisfactionlevel is not explicitly specified but rather it is the outcomeof the minimization of the profit function While highervalues of the parameter 119862

PEN119895

conceptually increase theprobability of demand satisfaction this strategymay still leadto unacceptably low probabilities of satisfying certain crudemix demands (see examples)Therefore the setting of explicitprobability targets on crudemix demand satisfaction is muchmore desirable A systematic way to accomplish this is toimpose explicit lower bounds on the probabilities of satisfyinga single crude mix demand This requirement for crude mix119895 in macroperiod119898 assumes the following form

Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898]

]

ge 120573119895119898 (13)

This constraint known as a chance constraint imposes a lowerbound 120573

119895119898on the probability that the crude mix demand

realization will be greater than the planned sales 119863119872119895119898

forcrude mix 119895 in macroperiod119898 The deterministic equivalentrepresentation can be obtained based on the concepts intro-duced by Charnes and Cooper [20]

Specifically by subtracting the mean and dividing bythe standard deviation of 119863119872

119895119898 the chance constraint can be

written equivalently as

Pr[

[

sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

minus119863119872

119895119898

120590119895119898

ge

119863119872

119895119898minus119863119872

119895119898

120590119895119898

]

]

ge 120573119895119898

Pr lceil119870119895119898

ge 119909119895119898rceil ge 120573119895119898

(14)

The right-hand side of the inequality within the proba-bility sign is a normally distributed random variable with amean of zero and a variance of 1 This implies that the chance

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

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Page 7: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

The Scientific World Journal 7

constraint can be replaced with the following deterministicequivalent expression

120601 (119870119895119898) ge 120573119895119898 (15)

The application of the inverse of the normal cumulativedistribution function 120601minus1 which is a monotonically increas-ing function yields

119870119895119898 ge 120601minus1(120573119895119898) (16)

Inspection of the deterministic equivalent constraintreveals that it is linear in the deterministic variables and iscomposed of the mean of the original constraint augmentedby the squared root of its variance times 120601minus1(120573

119895119898)

43 Satisfaction Level for All Crude Mix Demand In somecases a probability target is desired for the demand satisfac-tion of a group of crude mix in different macroperiod Forexample when only up to 10 unsatisfied crudemix demandcan be tolerated throughout the entire horizon withoutdistinguishing between different crude mixes While single-product chance constraints are unaffected by correlationsbetween crude mix demands this is not always the case forjoint chance constraints

This gives rise to joint chance constraints Joint chanceconstraints impose a probability target of simultaneouslysatisfying the demands for a group of crudemixes in differentmacroperiods as follows

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 120573119898

119868119898= (119895119898

1015840) | 1 le 119895 le 119873

BT 119898 le 119898

1015840le 119872

(17)

where 119868119898is the set of product-period (119895 119898) combinations

whose simultaneous demand satisfaction with probability ofat least 120573119898 is sought and 119868119898 = (119895 119898

1015840) | 1 le 119895 le 119873

BT 119898 le

1198981015840le 119872 is the set of all joint chance constraint The joint

probability 120573119898 is the probability of intersection of individualconstraints to be satisfied

We use the approach in [21] to solve the joint chanceconstrained problem The joint chance constraint equation(17) can be written as separated equivalent deterministicequations Initially we choose the 119879 constraints in a mannersuch that together they are more stringent than (17) Theinitial set of individual linear constraints equation (18) replac-ing (17) was obtained using the following argument First wedenote the event sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865119861119862

119895119897119905ge 119863119872

119895119898by 119860119895119898

and its

complement event sum119873CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905

lt 119863119872

119895119898by 119860119862119895119898

FromBoolersquos inequality of probability theory [22] it is well knownthat

119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

le

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119895119898] (18)

Thus it follows that if 119875[119860119862119905] le (120573

119898119879)119898 = 1 2 119872 then

119875[

[

⋂

(119895119898)isin119868119898

119860119895119898]

]

= 1 minus 119875[

[

⋃

(119895119898)isin119868119898

119860119895119898]

]

ge 1 minus

119869

sum

119895=1

119872

sum

119898=1

119875 [119860119862

119895119898] ge 1 minus (1 minus 120573

119898)

(19)

Now because 119863119872119895119898

is normally distributed with mean 120583119895119898

and standard deviation 120590119895119898 119875[119860119888

119895119898] le (1 minus 120573

119898)119879 is

equivalent to 119875[119863119872119895119898

gt sum119873

CDU

119897=1sum119879119898+1minus1

119879119898

119865BC119895119897119905] le (1 minus 120573

119898)119879

which in its turn is equivalent to

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120572119879120590119895119898 (20)

where 119911120572119879

is the 100(1 minus ((1 minus 120573119898)119879))th percentile of the

standard normal distribution Subsequently we will suppressthe subscript for z and replace (20) by the following equation

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905ge 120583119895119898

+ 119911120590119895119898 (21)

The initial value of 119911 will be chosen to be 119911 = 119911120572119879

Using the individual constraints equation (21) results in a

more conservative solution than the joint chance constraintequation (17) Therefore it is required to update the valueof z in order to find a solution which satisfies the requiredconfidence levelThemethod in updating z proposed in [23]is used which is based on the interpolation of z values

The deterministic forms of the objective function andstochastic constraint are used in solving the joint chanceconstrained problem iteratively by using a different z valueThe algorithm is examined in the next section

5 The Algorithm and Update Policy for theJoint Chance Constrained Problem

In this section a method is developed to solve the jointchance constrained problemabove and then anupdate policyupon the realization of the random demands of crude mix isdescribed

51 Algorithm for the Joint Chance Constrained Problem Thefollowing three steps are performed iteratively

Step 1 (initialization) Problem is solved with the equivalentdeterministic objective function and the constraint equation(21) is replaced with (17) for a fixed scalar z to determinethe set of new controlled variables The algorithm starts bychoosing a high value for the initial 119911-value as in (21) whichmakes the corresponding solution satisfy the demand with aprobability level higher thanPtarget = 120573119898

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

8 The Scientific World Journal

Step 2 (MINLP solution) The problem is implemented withGAMS (General Algebraic Modeling System) The CPLEX isused to solve this deterministic problem

Step 3 (evaluate probability) Evaluate the multivariate nor-mal probability (17) A subregion adaptive algorithm pro-posed by Genz and Bretz [24] is employed to carry outmultivariate integrationwhichmakes this evaluation feasible

Step 4 (updatez-value) If the evaluated probability level isin the 120574 neighborhood of Ptarget the algorithm terminatessince the goal of finding a schedule that satisfies the demandwith a probability ofPtarget is accomplished otherwise the 119911value is updated and the previous steps are repeated to obtainanother schedule

To update the z-value the following algorithm is usedThe goal is to find a z-value in (21) that provides a schedulesuch that the demand can be satisfied with a probability ofPtarget = 120573119898 over the entire time horizon The z-value needsto be obtained iteratively

Step 1 (obtain the upper and lower bounds of z) First wechoose z = z

120572in (21) Obviously it yields a lower bound for

the correctz-valueWe call itzlowerWe now run Step 2 of thealgorithm for this lower bound and obtain an estimate of theprobability with which the demand is being met We call thisprobabilityPlower Next we choose an arbitrary large value forz We denote it by zupper and obtain a similar estimate of theprobability of demand satisfactionWedenote this probabilitybyPupper

Step 2 (obtain the upper and lower bounds of Ptarget) Inthis step we obtain the upper percentiles of the univariatestandard normal distribution for these probabilities anddenote Pupper and Plower by z

2and z

1 respectively We

also denote the corresponding percentile for the Ptarget valueby ztarget

Step 3 (update z-value) Based on these values the updatedz-value is obtained using the following linear interpolationformula

znew = zlower + (ztarget minus z1

z2minus z1

) (Zupper minusZlower) (22)

If the Znew value is lower than Z2and higher than Ztarget

then we replace Z2byZnew If it is lower thanZtarget and

higher than Z1 we replace Z

1by Znew We repeat this

process using (17) until Ptarget is reached

52 Update of the Planning and Scheduling Policy As dis-cussed in Section 3 the demand 119863119872 sim 119873(120583 Σ) follows amultivariate normal distribution with a specific correlationstructure In order to decrease the uncertainty associatedwiththe random variables 119863119872

119895119898and thus increase the predictive

power of the stochastic model it is proposed to update thecorresponding schedule at the end of eachmacroperiod basedon the realized production [17] The update of the scheduleis accomplished by recalculating the conditional multivariate

probability function given the demand realizations in previ-ous periods

After solving the scheduling problem only the decisionvariables associated with the immediately following periodmay be taken as final The decision variables referring tosuccessive periods can be used for planning and activitiesrelated to the operation

The update of the planning policy and schedule at the endof macroperiod 119898 is described in the following steps

Step 1 (update statistics) Update future demand statisticsbased on present and previous product demand realizations

Step 2 (resolve the problem) Resolve the scheduling problemfor1198981015840 = 119898 + 1 119872

Step 3 (updatem) Set119898 larr 119898+1 If119898 le 119872 return to Step 1

The first step allows the use of updated demand forecast-ing information If there are no new demand forecasts avail-able the conditionalmultivariate probability density functionbased on the realizations of the demands in previous periodscan be utilized The procedure for finding the conditionaldistribution function is outlined below Step 2 involves thesolution of the problem which in turn determines the newplanning and scheduling policies

The derivation of the conditional probability distributionbased on the realization of previous random variables can beaccomplished as follows First the random variable 119863119872

119895119898is

partitioned into two sets The first set

119875 = 119863119872

1198951198981015840 1 le 119898

1015840le 119898119898 = 1 119872 (23)

contains the crude mix demands realized in the past periods1198981015840= 1 119898 The second set

119865 = 119863119872

1198951198981015840 119898 + 1 le 119898

1015840le 119872 119894 = 1 119872 (24)

contains the uncertain crude mix demands for the futureperiods Based on this partitioning the variance-covariancematrix can be expressed as follows

Σ = (Σ119875119875 Σ119875119865

Σ119865119875

Σ119865119865

) (25)

where ΣPP and ΣFF are the variance-covariance submatricesof the crude mix demands belonging to the sets 119875 and Frespectively The elements of submatrices ΣPF = (ΣFP)

119879 arethe covariances between the elements of 119875 and 119865 Let

Ξ = (1205851198941 1205851198942 120585

119894119905) (26)

denote the realizations (outcomes) of the crudemix demandsof set 119875 associated with past periods The conditional meansof the uncertain future crude mix demands include

120583119865119875

= 120583119865+ Σ119865119875Σ119875119875

minus1(Ξ minus 120583

119875) (27)

where 120583119865119875

denotes the conditional demand expectationsand 120583

119875and 120583

119865are the past and future mean values The new

(conditional) variance-covariance matrix is equal to

Σ119865119860

= Σ119865119865minus Σ119865119875Σ119875119875

minus1Σ119865119875 (28)

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

The Scientific World Journal 9

Table 1 System information

Scheduling horizon 30Number of vessel arrivals 9

Vessel Arrival time Amount of crude Concentration of keycomponent 1

Concentration of keycomponent 2

Vessel 1 1 100 001 004Vessel 2 4 100 003 002Vessel 3 7 100 005 001Vessel 4 11 100 001 004Vessel 5 14 100 003 002Vessel 6 17 100 005 001Vessel 7 21 100 001 004Vessel 8 24 100 003 002Vessel 9 27 100 005 001

Number of storage tanks 3

Storage tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Storage tank 1 100 20 00167 00333Storage tank 2 100 50 003 00225Storage tank 3 100 70 00433 00133

Number of charging tanks 3

Charging tank Capacity Initial oilamount

Initial concentrationof key component 1

Initial concentrationof key component 2

Charging tank 1 100 30 00167 00333Charging tank 2 100 50 003 00225Charging tank 3 100 30 00433 00133

Number of CDUs 2Costs involved in vessel operation Unloading cost 8 sea waiting cost 5

Tank inventory costs Storage tank 005 charging tank 008Changeover cost for charged oil switch 50The penalty cost for crude mix shortfalls 05

A detailed treatise of conditional multivariate normal prob-ability functions can be found in Tong 1990 [18] Notethat while the conditional mean value 120583119865119860 depends onthe demand realizations in past periods the conditionalcovariance matrix Σ119865119860 is independent of the demandrealizations Ξ Updating the probability distribution at theend of each period decreases the uncertainty (variances) asso-ciated with the remaining random variables thus increasingthe predictive power of the stochastic modelThis is observedin the case study in Section 6

6 Case Study

In the case study three different crude mixes are to beproduced The detailed data of the case study is given inTable 1 The time horizon of 30 days is divided into 30 equaltimesThe schedule involves threemacroperiodswith 10 daysThe product demands in each macroperiod are described bynormal multivariate probability distributions The expected(mean) values of the demands are given in Table 2 and theirstandard deviation is assumed to be 25 of theirmean values

Table 2 Mean values of the product demand

Macroperiod Product 1 Product 2 Product 31 90 98 952 100 80 983 100 90 98

Table 3 Employed realizations of the random product demands

Macroperiod Product 1 Product 2 Product 31 925504 1070773 8369012 1042967 827651 9215863 985225 935261 1140730

The alternative model formulation is defined and solvedusing the proposed solution procedure

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

10 The Scientific World Journal

Table 4 Updates of the product demand expectations during theprocess of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 90 98 952 100 80 983 100 90 98

2 2 1000438 810850 9806503 100 923 996155

3 3 1007265 921073 986984

Table 5 Updates of the product demand standard deviations duringthe process of policy revision

Policyrevision MacroperiodM Product 1 Product 2 Product 3

11 47434 49497 487342 5 44721 494973 5 47434 49497

2 2 48947 42417 471203 50 36691 48497

3 3 48821 33937 48020

Mode

119862COST =119873119881

sum

119881=1

119862UNLOAD119881

(119879119871

V minus 119879119865

V ) +

119873119881

sum

V=1119862SEA119881

(119879119865

V minus 119879ARRV )

+

119873ST

sum

119894=1

119862INVST119894

119873SCH

sum

119905=1

(119881119878

119894119905+ 119881119878

119894119905minus1

2)

+

119873119861119879

sum

119895=1

119862INVBL119895

119873SCH

sum

119905=1

(

119881119861

119895119905+ 119881119861

119895119905minus1

2)

+

119873SCH

sum

119905=1

119873BT

sum

119895=1

119873BT

sum

1198951015840=1

119862SETUP1198951198951015840

119873CDU

sum

119897=1

1198851198951198951015840 119897119905 +

119873BT

sum

119895=1

119862PEN119895

times

119873119872

sum

119898=1

max(0119863119872119895119898

minus

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865119861119862

119895119897119905)

Subject to Pr[

[

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898]

]

ge 120573119895119898

[

[

Pr ⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872119895119898)]

]

ge 120573119898

others (1ndash7) (29)

Model incorporates not only single but also joint chanceconstraints for setting probability targets for the satisfactionof a group of products First the effect of correlation onthe economic parameters and the optimal product mix isevaluated and discussed and next the proposed update policyis applied on the example problem

The problem is implemented with GAMSThe CONOPTand CPLEX 40 solvers are used to solve the formulatedmodels above

61 Use of Joint Probability Constraints Two separate casesare considered for model involving (i) uncorrelated and (ii)arbitrarily correlated crude mix demands For the correlatedcrude mix demands a variance-covariance matrix is con-structed and shown in (30) The matrix has sparsity of 60and its off-diagonal elements vary between minus03 and 06 Thebias toward positive covariance is introduced to maintainsemipositive definiteness which is a property of all variance-covariance matrices Consider

Σ =

((((((

(

10 00 minus02 02 00 03 00 00 00

00 10 00 00 03 00 00 06 02

minus02 00 10 00 01 00 00 02 00

02 00 00 10 00 00 00 02 00

00 03 01 00 10 02 02 03 00

03 00 00 00 02 10 00 02 01

00 00 00 00 02 00 10 02 02

00 06 02 02 03 02 02 10 00

00 02 00 00 00 01 02 00 10

))))))

)

(30)

The resulting MILP problem involves the objective func-tion and a set of linear constraints representing the deter-ministic equivalent representation of the individual and jointprobability constraints The joint probability index 120573119898 forthe set of all 9 crude oil demands is set to 50 For the valueof 120573119898is identified the expectation of the cost is compared

with those without considering demand correlation Figure 2shows the minimization expected cost for the two casesThese results indicate that the expectation of the total cost inthe model considering the correlation decreases

The lower expectation of the total cost results from thefact that the joint probability target can be met with smallerproduction levels than the ones ignoring correlations

62 Revision of Planning Policy Using the correlation matrixin (30) a demand realization is randomly generated usingthe method presented by Tong (1980) [18] The crude mixdemands realizations are given in Table 3 a joint probabilitytarget of 50 is set to be the demand satisfaction of each crudemix in all time periods This introduces joint probabilityconstraints in the following form

Pr[

[

⋂

(119895119898)isin119868119898

(

119873CDU

sum

119897=1

119879119898+1minus1

sum

119879119898

119865BC119895119897119905

ge 119863119872

119895119898)]

]

ge 05 (31)

The resulting optimization problem is solved in the entiretime horizon Next based on (i) the demand realizations in

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

The Scientific World Journal 11

Table 6 Updates of the product demand correlation matrix

Policy revision

1

10 00 minus02 02 00 03 00 00 0000 10 00 00 03 00 00 06 02minus02 00 10 00 01 00 00 02 0002 00 00 10 00 00 00 02 0000 03 01 00 10 02 02 03 0003 00 00 00 02 10 00 02 0100 00 00 00 02 00 10 02 0200 06 02 02 03 02 02 10 0000 02 00 00 00 01 02 00 10

2

1 minus00045 minus00671 0 02531 0minus00045 1 02146 02109 01352 minus00646minus00671 02146 1 0 02546 01072

0 02109 0 1 02586 0204102531 01352 02546 02586 1 minus015830 minus00646 01072 02041 minus01583 1

310000 02672 0231202672 10000 minus0197002312 minus01970 10000

Macroperiod 1 Macroperiod 2 Macroperiod 30

200

400

600

800

1000

1200

x-axis

y-a

xis

344035344035

667988 668103

9868361010906

Model considering correlationModel ignoring correlation

Figure 2 Comparison for the total operating cost with and withoutconsideration of demand uncertainty

the first period (ii) the correlation matrix (iii) the mean val-ues of the remaining random demands and (iv) the remain-ing inventory at the beginning of the second period a newplanning revision problem is solved for macroperiod 119898 isin

(2-3) This is repeated for all remaining macroperiods Theupdated values of the mean vectors standard deviations andcovariance matrix after each time period are given in Tables4 5 and 6

Two separate cases are considered for the models involv-ing (i) correlated crude mix demands using the proposedpolicy and (ii) correlated crude mix demands without thepolicy The actual production amount for crude mix 1 is

Macroperiod 1 Macroperiod 2 Macroperiod 380

85

90

95

100

105

110

115

120

x-axis

y-a

xis

90

115 115

100103204

100 101

105104204

The realized demandPolicy with updatingPolicy without updating

Figure 3 Comparison for the realized demand and the actualproduction for crude mix under three circumstances

shown in Figure 3 Examination of the data reveals that theactual production amount using the updating policy is closeto the actual realized demand

Clearly when the solution is updated the planning policyis much more effective because overproduction is decreasedThe proposed scheduling updating policy by updating theprobability distribution at the end of each period decreaseduncertainty associated with the remaining random variablesand thus increased the predictive power of the stochasticmodel for the subsequent periods

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

12 The Scientific World Journal

7 Conclusion

In this paper the multiperiod crude oil scheduling prob-lem under demand uncertainty was addressed A stochastictwo-level time structure model considering the uncertaincrude mix demand correlations is formulated The modelinvolves the minimization of the expected cost subject toconstraints for the satisfaction of single- andor multiple-crude mix demands with a prespecified level of probability Achance constrained programming formulation is developedfor solving the problem and the deterministic form of thestochastic constraint is used to solve the problem in aniterative way Moreover a revision strategy for the update ofthe corresponding schedule at the end of each time periodbased on the realized productionwas also presentedThis wasaccomplished by recalculating the conditional multivariateprobability function given the demand realizations in previ-ous periods

Results on the correlated case study demonstrate that theexpected profit and in particular the corresponding planningpolicy and schedule are strongly affected by the presence ofcorrelations The proposed planning revision and schedulingupdating policy by updating the probability distribution atthe end of each period decreased uncertainty associatedwith the remaining random variables and thus increased thepredictive power of the stochastic model for the subsequentperiods

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank three anonymous referees whose manyconstructive suggestions helped improve this paper signif-icantly This research is partially supported by the ChinaNational Natural Science Foundation under Grant 61203178Grant 61304214 Grant 61290322 and Grant 61290323 Theauthors also thank the financial funds from Shanghai Scienceand Technology Committee under Grant 12511501002 and theBiomedical Engineering Crossover Funds of Shanghai JiaoTong University with Grant YG2012MS55

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation Crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[6] L Wenkai C-W Hui B Hua and Z Tong ldquoSchedulingcrude oil unloading storage and processingrdquo Industrial andEngineering Chemistry Research vol 41 no 26 pp 6723ndash67342002

[7] P C P Reddy I A Karimi and R Srinivasan ldquoNovel solutionapproach for optimizing crude oil operationsrdquo AIChE Journalvol 50 no 6 pp 1177ndash1197 2004

[8] G Chryssolouris N Papakostas and D Mourtzis ldquoRefineryshort-term scheduling with tank farm inventory and distilla-tion management an integrated simulation-based approachrdquoEuropean Journal of Operational Research vol 166 no 3 pp812ndash827 2005

[9] YHu andY Zhu ldquoAn asynchronous time slot-based continuoustime for mulation approach for crude oil schedulingrdquo Comput-ers and Applied Chemistry vol 24 pp 713ndash719 2007

[10] Z Jia and M Ierapetritou ldquoEfficient short-term scheduling ofrefinery operations based on a continuous time formulationrdquoComputers and Chemical Engineering vol 28 no 6-7 pp 1001ndash1019 2004

[11] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[12] S M Neiro and J M Pinto ldquoSupply chain optimizationof petroleum refinery complexesrdquo in Proceedings of the 4thInternational Conference on Foundations of Computer-AidedProcess Operations pp 59ndash72 2003

[13] J Wang and G Rong ldquoRobust optimization model for crudeoil scheduling under uncertaintyrdquo Industrial and EngineeringChemistry Research vol 49 no 4 pp 1737ndash1748 2010

[14] C Cao and X Gu ldquoChance constrained programming withfuzzy parameters for re nery crude oil scheduling problemrdquo inFuzzy Systems and Knowledge Discovery pp 1010ndash1019 2006

[15] C Cao X Gu and Z Xin ldquoChance constrained programmingmodels for refinery short-term crude oil scheduling problemrdquoApplied Mathematical Modelling vol 33 no 3 pp 1696ndash17072009

[16] C Cao X Gu and Z Xin ldquoStochastic chance constrainedmixed-integer nonlinear programmingmodels and the solutionapproaches for refinery short-term crude oil scheduling prob-lemrdquo Applied Mathematical Modelling vol 34 no 11 pp 3231ndash3243 2010

[17] S B Petkov and C D Maranas ldquoMultiperiod Planningand Scheduling of Multiproduct Batch Plants under DemandUncertaintyrdquo Industrial and Engineering Chemistry Researchvol 36 no 11 pp 4864ndash4881 1997

[18] Y L TongProbability Inequalities forMultivariateDistributionsAcademic Press New York NY USA 1980

[19] B Fleischmann and H Meyr ldquoThe general lotsizing andscheduling problemrdquoOR Spectrum vol 19 no 1 pp 11ndash21 1997

[20] A Charnes and W W Cooper ldquoNormal deviates and chanceconstraintsrdquo Journal of the American Statistical Association vol52 p 134 1962

[21] U A Ozturk M Mazumdar and B A Norman ldquoA solutionto the stochastic unit commitment problem using chance con-strained programmingrdquo IEEE Transactions on Power Systemsvol 19 no 3 pp 1589ndash1598 2004

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

The Scientific World Journal 13

[22] P Billingsley Probability andMeasure JohnWiley amp Sons NewYork NY USA 2nd edition 1986

[23] F N Lee M-Y Lin and A M Breipohl ldquoEvaluation of thevariance of production cost using a stochastic outage capacitystate modelrdquo IEEE Transactions on Power Systems vol 5 no 4pp 1061ndash1067 1990

[24] A Genz and F Bretz ldquoComparison of methods for the compu-tation of multivariate t probabilitiesrdquo Journal of Computationaland Graphical Statistics vol 11 no 4 pp 950ndash971 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article An Approximation Solution to …downloads.hindawi.com › journals › tswj › 2014 › 730314.pdfScheduling of crude oil operations is a critical task in the overall

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


PENYELESAIAN RESOURCE-CONSTRAINED PROJECT …lib.unnes.ac.id/32208/1/4111413020.pdfScheduling Problem (RCPSP) adalah masalah penjadwalan proyek yang harus memenuhi urutan pengerjaan

PENYELESAIAN RESOURCE-CONSTRAINED PROJECT …lib.unnes.ac.id/32208/1/4111413020.pdfScheduling Problem (RCPSP) adalah masalah penjadwalan proyek yang harus memenuhi urutan pengerjaan

Approximation Algorithms - gdeepak.com · Objective of Approximation Goal of the approximation algorithms is to design an algorithm such that approximation factor is as small as possible

Approximation Algorithms - gdeepak.com · Objective of Approximation Goal of the approximation algorithms is to design an algorithm such that approximation factor is as small as possible

Linear Programming and Approximation Seminar in Approximation Algorithms.

Linear Programming and Approximation Seminar in Approximation Algorithms.

Approximation Problems

Approximation Problems

Approximation Algorithms

Approximation Algorithms

Approximation Algorithm

Approximation Algorithm

Approximation via Cost Sharing: Simpler and Better Approximation ...

Approximation via Cost Sharing: Simpler and Better Approximation ...

Approximation Algorithms - Lehigh Universitycoral.ie.lehigh.edu/wp-content/uploads/coral... · Kumar Abhishek Approximation Algorithms. university-logo Motivation Introduction Approximation

Approximation Algorithms - Lehigh Universitycoral.ie.lehigh.edu/wp-content/uploads/coral... · Kumar Abhishek Approximation Algorithms. university-logo Motivation Introduction Approximation

9-1 Chapter 9 Approximation Algorithms. NP-Complete Problem Enumeration Branch an Bound Greedy Approximation PTAS K-Approximation No Approximation 9-2.

9-1 Chapter 9 Approximation Algorithms. NP-Complete Problem Enumeration Branch an Bound Greedy Approximation PTAS K-Approximation No Approximation 9-2.

CHARACTERISTIC SPECIFIC PRIORITIZED …aircconline.com/ijcsea/V5N5/5515ijcsea01.pdfscheduling algorithm has been proposed, named as CSPDABRR (Characteristic specific Prioritized Dynamic

CHARACTERISTIC SPECIFIC PRIORITIZED …aircconline.com/ijcsea/V5N5/5515ijcsea01.pdfscheduling algorithm has been proposed, named as CSPDABRR (Characteristic specific Prioritized Dynamic

Explicit Approximation

Explicit Approximation

Numerical approximation

Numerical approximation

Approximation - courses.cs.washington.edu · This is a 2-approximation for vertex cover! Another Approximation Algorithm Let’s look at another approximation algorithm for vertex

Approximation - courses.cs.washington.edu · This is a 2-approximation for vertex cover! Another Approximation Algorithm Let’s look at another approximation algorithm for vertex

Approximation Algorithms. 2301681Approximation Algorithms2 Outlines Why approximation algorithm? Approximation ratio Approximation vertex cover.

Approximation Algorithms. 2301681Approximation Algorithms2 Outlines Why approximation algorithm? Approximation ratio Approximation vertex cover.

Research Article On Some Approximation Theorems for Power ...downloads.hindawi.com/journals/tswj/2014/513162.pdf · On Some Approximation Theorems for Power -Bounded Operators on

Research Article On Some Approximation Theorems for Power ...downloads.hindawi.com/journals/tswj/2014/513162.pdf · On Some Approximation Theorems for Power -Bounded Operators on

Trapezoidal Approximation

Trapezoidal Approximation

Approximation Algorithms:

Approximation Algorithms:

Approximation Theory and Approximation Practice · PDF file2 Approximation Theory and Approximation Practice ... (1880–1968), and Jackson ... At a more detailed level,

Approximation Theory and Approximation Practice · PDF file2 Approximation Theory and Approximation Practice ... (1880–1968), and Jackson ... At a more detailed level,

Area Approximation

Area Approximation

Approximation Theory

Approximation Theory