Int. J. Production Economics 135 (2012) 299–307

Contents lists available at ScienceDirect

Int. J. Production Economics

0925-52

doi:10.1

n Corr

E-m

demirli@1 Te

journal homepage: www.elsevier.com/locate/ijpe

Integrated safety stock optimization for multiple sourced stockpointsfacing variable demand and lead time

Hany Osman a,n, Kudret Demirli b,1

a Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd.W., EV13.119, Montreal, Quebec, Canada H3G 1M8b Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd.W., EV4.187, Montreal, Quebec, Canada H3G 1M8

a r t i c l e i n f o

Article history:

Received 16 March 2011

Accepted 2 August 2011Available online 7 August 2011

Keywords:

Supply chain

Safety stock

Order statistics

Variable demand and lead time

Benders decomposition technique

73/$ - see front matter & 2011 Elsevier B.V. A

016/j.ijpe.2011.08.004

esponding author. Tel.: þ1 514 848 2424x72

ail addresses: h_most@encs.concordia.ca (H. O

encs.concordia.ca (K. Demirli).

l.: þ1 514 848 2424x3160; fax: þ1 514 848

a b s t r a c t

The safety stock placement problem of a multi-stage supply chain comprising multiple sourced

stockpoints is addressed in this paper. Each stockpoint faces variability in its downstream demand and

suppliers’ lead time. The maximum among these suppliers’ lead time is determined by employing

concepts of order statistics. It is required to find the fill rate and safety stocks at each stockpoint that

leads to satisfying the end customer service level at minimum safety stock placement cost. Hence, the

fill rates and the safety amounts are decided from a global supply chain perspective. Two models are

proposed; a decentralized safety stock placement model and a centralized consolidation model. The

decentralized model finds the safety amounts at each stockpoint required to face its underlying lead

time demand variability. The consolidation model finds the consolidated safety amounts that will be

kept in the relevant consolidation center at each stage. A Benders decomposition technique is

developed to handle the nonlinearity and binary restrictions involved in the safety stock consolidation

model. Strategies proposed by the consolidation model achieve 45.2–62% reduction in safety amounts

that results in a cost savings ranging between 22.2–44.2% as compared to the strategies proposed by

the decentralized model.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

In this paper, the safety stock placement (SSP) problem of asupply chain including multiple sourced stockpoints is tackled.Each existing member in this supply chain faces variability in theupstream lead time and in the downstream demand. A graphicalrepresentation of the supply chain under consideration isdepicted in Fig. 1. The chain is composed of an assembly facilityat the most downstream stage, Tier1-suppliers at the intermedi-ate stage, and Tier2-suppliers at the initial stage.

The supplying and inventory strategies, currently employedthroughout the supply chain, failed to satisfy the promiseddelivery dates of the end items. The primary reason behind thisshortfall is the existence of unreliable suppliers that are unable todeliver materials on time. The secondary reason concerns theinventory systems employed throughout the chain. These systemsare established based on random ordering decisions, which oftenlead to a stockout occurrence. Moreover, each member of thechain does not employ a proper safety stock policy to face thefluctuation of customer demand and suppliers’ lead time.

ll rights reserved.

24; fax: þ1 514 848 3175.

sman),

3175.

A three-stage research is conducted to resolve this problem. Inthe first stage, the supply chain is reconfigured and materials areredistributed to the highly reliable and coordinated suppliers(Osman and Demirli, 2010). In the second stage, a joint supplychain inventory–production system is proposed based on thedeterministic assumptions of customer demand and supplier’s leadtime (Osman, 2011). At the third stage, presented in this paper, thevariability of downstream demand and upstream lead time isconsidered while developing a safety stock placement policythrough the supply chain. Such an integrated policy, establishedfrom a supply chain perspective, should specify sufficient safetyamounts and fill rate at each stockpoint to achieve a prespecifiedend customer service level at minimum cost.

Two SSP models are developed to establish two different SSPpolicies. The decentralized policy, characterizing the first model,allows each stockpoint to independently handle the variability ofits lead time demand. The centralized policy, characterizing thesecond model, aims at reducing the variability of lead time demandat each stage by pooling this variability at one stockpoint. Orderstatistics concepts are applied to determine the parameters of thelead time probability distribution at each multiple sourced stock-point. The two policies can be implemented after deciding on cycletime and order amount of deterministic inventory systems. Thedecentralized model is solvable to optimality using the nonlinearcommercial solver Minos, whereas a decomposition algorithm

T1-Suppliers(Machining)

T2-Suppliers(Components)

AssemblyCompany

Raw Components Machined Components

Demandforecast

Raw componentsforecast

Fig. 1. Configuration of the supply chain under consideration.

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307300

based on Benders decomposition technique is developed to solvethe centralized safety stock consolidation (SSC) model.

A review of the literature of the SSP problem is given in thefollowing section. The specific problem under study is defined inSection 3. Section 4 discusses the application of order statisticsconcepts to obtain the parameters of the lead time probabilitydistribution at a multiple sourced stockpoint. The decentralizedSSP model and the centralized SSC model are given in Sections5 and 6, respectively. The Benders decomposition algorithmdeveloped to handle the difficulty of the mixed integer nonlinearSSC model is explained in Section 7. A comparison between theproposed centralized and decentralized strategies is demon-strated in Section 8. This section also shows the computationalefficiency of the developed decomposition method in solving theSSC model. The paper ends with summary and future extensions.

2. Literature review

Simpson’s (1958) model can be considered as one of the initialworks that dealt with demand uncertainty in multi-stage produc-tion and inventory systems. The model determines the combina-tion of service times offered by each stage to satisfy customerorders at a predetermined service level. Inderfurth (1991) extendsthis work and establishes the optimal policy for divergent supplychains taking into account the impact of demand correlation onSSP using risk pooling effects. Inderfurth and Minner (1998) dealwith different service measures that restrict the amount of safetystock. These measures are the probability of stockout occurrenceand the probability of stockout amount. Their proposed modelfinds the safety amount that covers the demand fluctuationsduring a time period called the coverage time. Minner (1997)derives the forward and backward recursive formulas to find theoptimal policy of these coverage times.

Graves and Willems (2000) simplify the SSP problem from itsstochastic nature to be a deterministic optimization problem. Thecustomer demand is considered as a random variable while thereplenishment lead time is assumed to be deterministic. Thestochastic lead time demand is assumed to be bounded by amaximum value obtained from its mean and standard deviation.They consider only supply chains that can be represented as aspanning tree while general supply chains under the sameassumptions are handled by Graves and Lesnaia (2004). Sitompuland Aghezzaf (2006) extend the problem addressed in Graves andWillems (2000) to consider capacity limitations. They state that thesafety stock amounts have to be updated by a tabulated correctionfactor that relates safety stock with a measure representing thedegree of capacity that covers demand variations. Further relation-ships between demand variability, capacity, delivery lead time, andsafety stocks are investigated in Sitompul et al. (2008).

Jung et al. (2008) propose a linear programming model thatdetermines the base stock level under the dependency of service

measures at different stages of a supply chain. The inventory level atproduction facilities and the base stock level at warehouses are alsoconstrained by the safety production capacity limit. Kim et al. (2005)deal with two echelon supply chains comprising single supplier andmultiple retailers under non-stationary demand pattern. Two mod-els are proposed based on the centralized and decentralizedapproaches of inventory control. Boulaksil et al. (2009) develop amathematical model that allows backordering at each stage. Themodel does not state any assumptions about the demand distribu-tion. Louly and Dolgui (2009) consider an assembly system facingconstant demand and discrete distributed random lead time ofdelivering the assembled components. The model is valid for anydiscrete probability distribution. Persona et al. (2007) tackle thesafety stock determination problem for assemble-to-order andmake-to-order systems. Their cost-based analytical models considerdemand as a normally distributed random variable while theyconsider a constant lead time of subassemblies delivery.

The case of stochastic lead time and customer demand for asingle stock is handled by Eppen and Martin (1988). An exponen-tial smoothing model is proposed to estimate the unknowndistribution parameters of lead time and demand. Hayya et al.(2009) develop a regression equation that represents optimal cost,order quantity, and safety stock factor in terms of cost parameters,standard deviation of demand, and standard deviation of lead time.Ettl et al. (2000) propose an inventory-queue model that estab-lishes the base stock level at each store of a supply network. Thenominal lead time at each store is assumed to be independent andidentically distributed while demand is considered to be non-stationary. Saharidis et al. (2009) analyze two control polices fortwo echelon supply chains: base stock control and echelon basestock control. The demand follows Poisson distribution while theproduction time is exponentially distributed. Simchi-Levi and Yao(2005) propose a unified framework that integrates stages employ-ing continuous review base stock inventory control. The lead timeis assumed to be stochastic, sequential and exogenously deter-mined with known probability distribution while the customerdemand follows independent Poisson process.

This paper addresses the problem of placing safety stocks inmultiple sourced stockpoints located in a multi-stage supplychain. Each stockpoint in the supply chain undergo uncertaintyin upstream lead time and downstream demand. The lead timeand demand are considered as two independent normally dis-tributed random variables. The overall objective is to find theoptimal SSP policy that achieves the end customer service level atminimum cost. Two SSP models are proposed to solve thisproblem. The two models establish the decentralized and cen-tralized SSP strategies from a global supply chain viewpoint. Ineach strategy, the fill rates required at connected stockpoints aredecided simultaneously in a sense that stockpoints possessinghigh holding cost are assigned low fill rates and stockpointspossessing low holding costs are assigned high fill rates. Prior toapply these models, parameters of the maximum lead time

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307 301

probability distribution at each stockpoint are determined byapplying order statistics concepts. The problem is described indetails in the following section.

3. Problem description and assumptions

At each supplier and the assembly facility depicted in Fig. 1,there is a stockpoint facing a stochastic environment. Such anenvironment is characterized by volatile downstream demandand upstream lead time. The lead time is defined here as the timeelapsed in shipping materials from one stockpoint to its succes-sor. An integrated inventory–production system is designedthroughout the chain to determine the optimal delivery intervals(T), replenishment sizes (Q), and production sequence, Osman(2011). This integrated system ignores variability of downstreamdemand and upstream lead time.

A new strategy has to be established so as to allocate sufficientsafety amounts at appropriate places throughout the supply chain.The overall objective of the new safety stock strategy is to fulfill theend customer demand at a predefined service level representing apercentage of demand satisfied from stock. The new strategy,which minimizes the SSP costs, should specify the optimal fill rateand the safety stock required at each stockpoint from a supplychain point of view. Having established the order amounts (Q)based on deterministic assumption of customer demand, the newstrategy proposed by this paper specifies the reorder point (r).

The supply chain possesses three settings related to the replen-ishment and production polices, these settings are stated below:

1.

A single item can be replenished from multiple suppliers. 2. Replenishment cycle time (T) is common among members of a

given stage.

3. Production can start upon receiving shipments ordered from

the multiple sources.

The first and third assumptions call for applying order statis-tics concepts to find the delivery time used in safety stockscalculations. Since each stockpoint receives material from multi-ple sources, the lead time for receiving an item is considered asthe maximum among these multiple sources lead times. Conse-quently, this maximum is considered as a random variable havinga probability distribution with certain parameters. Section 4illustrates the application of order statistics to find parametersof the probability distribution of the maximum lead time.

The problem is solved in two different approaches; in eachapproach a safety stock placement model is developed. The firstmodel, called the SSP model, is established based on the decen-tralized approach of allocating safety stocks. Following thisapproach, each stockpoint is required to keep sufficient safetyamounts from each item at its site to face the variability of leadtime demand. The second model, called the SSC model, isformulated based on the centralization principles of safety stockdistribution. In this approach, the safety amounts required at allstockpoints of a given stage are consolidated at the most relevantstockpoint at this stage. This stockpoint is chosen based on itscapacity limitation, the holding cost, and the paid credits to thestockpoints. Stockpoints preferred to be consolidation centers arerequired to cope with the variations of lead time demand of otherstockpoints by shipping a sufficient amount of stock to theirdownstream stage. In return, a consolidation center will be givenan amount of credits to cover their responsibilities for holding theconsolidated safety amounts. Consolidating safety stocks willresult in smaller amounts of safety stocks compared to thedecentralized approach. Reduction in safety stocks is a conse-quence of pooling the lead time demand variability at each stage.

4. Stochastic lead time of multiple-sourced stockpoints

The problem of multiple sourced inventory systems has beeninvestigated by researchers to figure out the effect of ordersplitting on the lead time distribution, in which the entire orderis distributed among multiple sources instead of being replen-ished from a sole source (Sculli and Wu, 1981; Pan, 1987; Panet al., 1991; Ramasesh et al., 1991). The effect of order splitting onminimizing the stockout risk is investigated in Kelle and Miller(2001). Another problem relating to a multiple-sourced stochasticinventory system arises when the lead time is considered as themaximum among the multiple sources lead times. An assemblysystem is a common example that exhibits this approach ofcalculating lead time, where assembly cannot start until all therequired components have been received. Also, a productionbatch of a single item may require all the raw material suppliedfrom multiple vendors to be processed in one production run.

In the problem studied in this paper, a multiple-sourced stock-point faces the problem of determining delivery lead time of itsinput material. If a stockpoint receives material from n sourcesconsidering their lead time as independent and identically nor-mally distributed random variables, the maximum among these n

variables equals the maximum of a random sample of size n takenfrom a normally distributed population (Clark, 1961). The deter-mination of this maximum can be found by consulting orderstatistics distributions and moments (David and Nagaraja, 2003).The mean of this random variable is the expected lead time thatwill be used along with its variance in calculating the reorder pointand the safety stock. The expected value of the ith order statisticsfor a set of independent standard normal random variables X1,X2, y, Xn is given by Eq. (1) where i represents the order. If i equalsn, it represents the maximum of this order statistics.

EðXiÞ ¼n!

ði�1Þ!ðn�iÞ!

Z 1�1

xffðxÞgi�1f1�fðxÞgn�if ðxÞdx ð1Þ

Godwin (1949) establishes tables of mean, variance and covar-iance of normally distributed order statistics of size 10 or less. Forsamples of 20 or less, tables of the expected value of the ith orderstatistics are established by Teichroew (1956). For larger samplesizes of 2(1) 100(25) 250(50) 400, Harter (1961) presents theexpected values of normally distributed order statistics. Federer(1951), Blom (1958), Westcott (1977), and Royston (1982) intro-duce algorithms to approximate the expected values of orderstatistics. These algorithms apply numerical methods and do notprovide closed form solutions to find moments of order statistics.

Simchi-Levi and Yao (2005) consider such a case of lead timerepresentation in their model and apply the approximationmethod introduced by Clark (1961) to determine the lead timeat assembly facilities. Clark (1961) finds the maximum among afinite set of random variables through successive iterations thatrequire searching in the standard normal table each time. How-ever, searching in the normal table is time-consuming and isfound to be difficult to put into a computer code. Furtherinventory models that incorporate explicit forms for determiningthe maximum lead time at assembly facilities have to be intro-duced to facilitate handling the difficulty of variable lead time.

The algorithm introduced by Ozturk and Aly (1991) is appliedhere to approximate parameters of the normally distributed leadtime at each stockpoint. The algorithm approximates the expectedvalue and variance of normally distributed order statistics using thegeneralized lambda distribution. In particular, the moments ofgeneralized lambda distributed order statistics are used as anapproximation to the moments of standard normally distributedorder statistics. In addition to providing results with small errors,the algorithm proposed by Ozturk and Aly (1991) is straightforwardand needs less computational efforts compared to Clark’s (1961).

Table 1Comparison between the absolute error in estimating the

mean of the maximum among n standard normal random

variables using Ozturk and Aly (1991) and Clark (1961).

n Ozturk and Aly (1991) Clark (1961)

2 0.00019 0.00000

3 0.00029 0.00130

4 0.00074 0.00260

5 0.00122 0.00130

6 0.00146 0.00070

7 0.00151 0.00000

8 0.00171 0.00060

9 0.00195 0.00130

10 0.00208 0.00210

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307302

Table 1 shows a comparison between these two approximationmethods. The second column of the table represents the upperbound of the error in estimating the maximum among a set ofrandom variables using the algorithm proposed by Ozturk and Aly(1991). Clark’s (1961) error values are depicted in the third column.Clark’s (1961) error results are subjected to increase if an approx-imation method is used instead of searching in the normal table.

The inverse distribution function of the generalized lambdadistribution proposed by Ramberg and Schmeiser (1972) is shownin Eq. (2) where l1, l2, l3 and l4 are the parameters of thedistribution. For 0, 0.1975, 0.1349 and 0.1349 given values ofthese parameters, Schmeiser (1977) shows that the maximumabsolute error through approximating the standard normal dis-tribution by the generalized lambda distribution is 0.001. Eqs.(3) and (5) represent the closed form, given by Ozturk and Aly(1991), to approximate the mean and the variance of standardnormally distributed order statistics. The b function used tocalculate the variance is shown in Eq. (7).

F�1ðpÞ ¼ l1þpl3�ð1�pÞl4

l2ð2Þ

mi ¼Ci�Cn�iþ1

l2Cnþ1ð3Þ

where Cr ¼ rYr

k ¼ 1

1þl3�1

k

� �ð4Þ

vi ¼bð2l3þ i,tÞ�2bðl3þ i,tþl4Þþbði,tþ2l4Þ

l22bði,tÞ

�ðmi�l1Þ2

ð5Þ

Where t¼ n�iþ1 ð6Þ

and bðx,yÞ ¼ðx�1Þ!ðy�1Þ!

ðxþy�1Þ!ð7Þ

The parameters mi and vi of the standard normally distributedorder statistics are used to drive the mean E(Xi) and the varianceVar(Xi) of the original order statistics. If the n lead times at a givenstockpoint are represented by identical normal distributions hav-ing mean m and variance s2, parameters of the maximum lead timedistribution are given by Eqs. (8) and (9) where i equals n.

EðXiÞ ¼ mþsmi ð8Þ

VarðXiÞ ¼ s2vi ð9Þ

5. Decentralized safety stock placement model

The proposed SSP model is discussed in this section. Thecontribution of this model is the incorporation of the serviceper units demanded (i.e., fill rate) as a measure of service in

a multi-stage supply chain. Another original aspect of this modelconcerns the relation between the fill rates that are establishedfrom a supply chain perspective. Each item moves throughout thenetwork on a path that starts from the T2-suppliers stage until itreaches the company. The expected fill rates at the stockpointsplaced on a given path should result in satisfying the endcustomer service level. This is ensured through satisfying theconstraint setting that service level as a lower bound on themultiplication of these fill rates. Parameters and decision vari-ables of the proposed nonlinear model are defined below.

5.1. Nomenclature

Indices and sets

i index set of supply chain stages, i¼1,2,y,Ij index set of stockpoints at a given stage j¼1,2,y,Ji

k index set of items, k¼1,2,y,K

Parameters

hijk holding cost of item k at stockpoint j in stage i perunit time

slk service level required to be met of item k

qijk order quantity of item k at stockpoint j in stage i

dijk mean demand of item k at stockpoint j in stage i

lijk mean lead time of item k at stockpoint j in stage i

sdijk standard deviation of demand for item k at stockpoint j

in stage i

slijk standard deviation of lead time for item k at stockpoint j

in stage i

sijk standard deviation of lead time demand for item k atstockpoint j in stage i

Ukrst 0–1 matrix that specifies whether or not item k passesthrough the stockpoint r at the most downstream stageand the stockpoint s at the intermediate stage, and thestockpoint t at the most upstream stage

Decision variables

Fijk fill rate of item k at stockpoint j in stage i

EðZÞijk standardized stockout quantity for a standard normaldistribution for item k at stockpoint j in stage i

Zijk standard normal deviate for item k at stockpoint j in stage i

The mean lijk and variance slijk of the delivery time of item k

at stockpoint j placed in stage i are calculated using Eqs. (8) and(9), respectively. Eq. (10) represents the standard deviation oflead time demand in the case of variable demand and variablelead time. Eq. (11) shows the relationship between the standar-dized stockout quantity per order cycle and the fill rate of a singleitem at a given stockpoint, Tersine (1988). This equation isextended in the model to determine the fill rates at the stock-points existing in the supply chain.

sijk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilijksd2

ijkþd2ijksl2ijk

qð10Þ

sl¼ 1�sEðZÞ

qð11Þ

Eqs. (12)–(16) show the decentralized SSP model. The model iscoded using AMPL, Fourer et al. (2003), and is solvable tooptimality using Minos.

Min SSHC ¼XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

hijksijkZijk ð12Þ

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307 303

Subject to

F1rk � F2sk � F3tkZslk k¼ 1,2,. . .K , r¼ 1, s¼ 1,2,. . .J2,

t¼ 1,2,. . .,J3 : Ukrst ¼ 1 ð13Þ

EðZijkÞ ¼ð1�FijkÞqijk

sijki¼ 1,2,. . .,I j¼ 1,2,. . .,Ji k¼ 1,2,. . .,K ð14Þ

Zijk ¼ EðZÞijk�0:38984228h i

�

�1:75294 þ 0:4442135EðZÞijk�0:07061455EðZÞ2ijk

� 0:17592241EðZÞijkþ 0:044212641�

0:0012267386EðZÞijkþ 0:00030570313

24

35

i¼ 1,2,. . .I j¼ 1,2,. . .,Ji k¼ 1,2,. . .,K ð15Þ

EðZÞijk, Fijk, ZijkZ0 i¼ 1,2,. . .,I j¼ 1,2,. . .,Ji k¼ 1,2,. . .,K ð16Þ

Objective function (12) minimizes the safety stock holdingcost throughout the supply chain. The amount sijk Zijk is the safetystock of item k that should be kept at stockpoint j in stage i percycle. The desired service level of item k is satisfied through Eq.(13). This constraint is considered only when the parameter Ukrst

equals 1. For the case handled in this paper, the entries Ukrst

specify whether or not item k passes through stockpoint r locatedat the most downstream stage where i ¼1, and stockpoint s

located at the intermediate stage where i¼2, and stockpoint t

located at the most upstream stage where i¼3. This equationensures that each path of a given item k on the network will yielda service level greater than or equal to the desired one. Themultiplication of the fill rates on a given path gives the model theflexibility to assign high fill rate to the stockpoints having lowholding cost and assign lower fill rate at the stockpoints havinghigh holding cost. Eq. (14), which is driven from Eq. (11),calculates the standard stockout quantity E(Z)ijk of item k atstockpoint j in stage i. Brown’s (1967) nonlinear approximationis shown in Eq. (15). This convex nonlinear approximation isapplied instead of searching in statistical tables to find value ofZijk for a given value of E(Z)ijk. The drawback of this function isthat it does not provide a reasonable approximation for the largeabsolute values of E(Z) close to 4.5. The non-negative restrictionon the decision variables is insured by the last constraint. Thereorder point of each item k at stockpoint j in stage i can bedirectly determined by adding the safety stock sijkZijk to theaverage lead time demand lijkdijk.

When a stockpoint undergoes order crossover effects, resultingfrom receiving a recently placed order before an order placedearlier, the lead time demand distribution should be replaced byeither the effective lead time demand distribution or the shortfalldistribution. If the effective lead time demand distribution isemployed, m and s2 appearing in Eqs. (8) and (9) should refer tothe effective lead time distribution. The effective lead time can beobtained from the original lead time by considering the timeelapsed between placing the first order and receiving the firstdelivery, Hayya et al. (2009). Mean and variance of the shortfalldistribution are calculated in Robinson et al. (2001). Shortfall andlead time demand distributions have the same mean, while thevariance of the shortfall distribution is less than or equal to thevariance of the lead time demand distribution. So, if the shortfalldistribution is employed, the standard deviation of lead timedemand sijk in Eq. (14) is replaced by the standard deviation ofthe shortfall distribution. So, either to work on the effective leadtime demand or the shortfall distributions, our proposed model isstill valid to handle stochastic lead times with order crossover.

6. Centralized safety stock consolidation model

The inventory consolidation problem has been studied in theliterature to examine the effect of consolidation on inventory cyclestock and safety stock savings (Maister, 1976; Zinn et al., 1989;Mahmoud, 1992; Evers and Beier, 1993; Tallon, 1993; Evers, 1995;Caron and Marchet, 1996; Evers and Beier, 1998; Tyagi and Das,1998; Das and Tyagi, 1999; Ballou and Burnetas, 2003; Ballou,2005; Wanke, 2009). The consolidation models discussed in thesepapers are developed given that both cycle stock and safety stockof decentralized locations are consolidated in one or more cen-tralized locations. In this case, the mean and variance of demand ata given consolidation center are calculated based on the portions ofaverage demand moved from the decentralized locations to thatcenter. Thus, the standard deviation of lead time demand at aconsolidation center is affected by these portions.

Our consolidation model conceptually handles another case, inwhich the cycle stock is kept at the decentralized locations andconsolidation is applied only on the safety stocks. This is because ourproposed model is designed to deal with the variability of suppliers’lead time and customer demand of an integrated productioninventory system. In such a case, the cycle stock is preferred to beclose enough to the production line in order to facilitate shipping onthe promised delivery dates. So, the variability of lead time demandat each stockpoint will be satisfied by the safety amounts consoli-dated at the centralized locations. The standard deviation of leadtime demand at a consolidation center is calculated through poolingthe variability of lead time demand of the decentralized locations. Acommon instance of an integrated production inventory system is amulti-stage supply chain facing the economic lot and deliveryscheduling problem. In this problem, it is required to decide oncycle times, cycle stocks, and production sequence at each produc-tion facility. The deterministic case of the economic lot and deliveryscheduling problem (ELDSP) has been studied in Hahm and Yano(1992, 1995a, 1995b, 1995c), Khouja et al. (1998), Khouja (2000,2003), Jensen and Khouja (2004), Clausen and Ju (2006), Ghomi et al.(2006), Kim et al. (2006), Torabi and Jenabi (2009), and Osman(2011). The stochastic case of the ELDSP has not been tackled yet.

In addition to this unique way of consolidating safety stock, theproposed SSC model differs significantly in four aspects from thoseconsolidation models studied in Maister (1976), Zinn et al. (1989),Mahmoud (1992), Evers and Beier (1993, 1998), Tallon (1993),Evers (1995), Caron and Marchet (1996), Tyagi and Das (1998), Dasand Tyagi (1999), Ballou and Burnetas (2003), Ballou (2005), andWanke (2009). First, the service level considered in the proposedSSC model represents the probability of stockout amount while theservice level employed in these papers is the probability of stock-out occurrence. Probability of stockout amount is more informativethan probability of stockout occurrence since it shows how manyof the demanded units are not satisfied. Second, the safety factorassociated with the service level in the proposed SSC model is adecision variable while that used in other models appearing in theabove cited papers is a known parameter. Moreover, in theproposed SSC model, the value of that factor is decided upon fromthe viewpoint of the whole supply chain while its value is set byeach consolidation center prior to solving the models proposed bythese papers. Third, none of the above cited models applies thecriteria used in our model to select a consolidation center. Theyfocus only in the reduction in cycle and safety stocks. In our model,a consolidation center is chosen according to its capacity limit, theholding costs and the amount of credits that will be given to thecenter. Fourth, our model handles a multistage supply chain inwhich at each stage there will be a consolidation center. The fillrates established at these consolidation centers are determinedfrom a supply chain perspective, not from an individual perspec-tive, to achieve the end customer service level at minimum cost.

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307304

In the SSP model proposed in Section 5, each stockpoint isresponsible for dealing with its ongoing fluctuations in demandand lead time by keeping adequate safety amounts. Consolidationis applied here to centralize the safety amount of each item to belocated in one place at each stage. As such, if any stockpoint facesdemand or lead time positive variations, the consolidation centeris required to ship an amount sufficient to meet such variations tothe downstream stage. The impact of consolidation is a reductionin the total safety amounts of each item stored at each stage. Thisreduction is a result of applying variability pooling concept to thevariability of lead time demand. Eq. (17) shows the pooledvariability at stage i for each item k, where each entry s2

ijk isobtained from Eq. (10).

sik ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXJ

j ¼ 1

s2ijk

vuut ð17Þ

Throughout the SSC model given by Eqs. (18)–(26), the pre-viously defined decision variables are used to represent eachstage instead of each stockpoint. For example, the decisionvariable Fijk appears here as Fik to symbolize the fill rate of itemk required at stage i to meet the service level slk of this item k. Abinary decision variable Xijk is defined to decide on which stock-point j is used to hold the consolidated safety amount of eachitem k at stage i. Two more parameters are introduced, cij thatrepresents the total capacity of stockpoint j in stage i, and themotivation cost wijk that indicates the amount of money paid bythe supply chain partners to stockpoint j in stage i as an incentiveto take the burden of handling the consolidated safety stock ofitem k. Through objective function (18), these motivating dollarsalong with the holding cost are used to select the most relevantstockpoint among the feasible candidates to be the safety con-solidation center of item k at stage i. Constraint, given by Eq. (19),ensures that the consolidated safety amount of item k is assignedto only one stockpoint among the available Ji stockpoints at stagei. Capacity restriction of stockpoint j at stage i to hold one or moreitems is satisfied through Eq. (20).

Min SSCC ¼XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

wijkXijkþhijksikZikXijk ð18Þ

Subject to

XJj

j ¼ 1

Xijk ¼ 1 i¼ 1,2,. . .,I k¼ 1,2,. . .,K ð19Þ

XK

k ¼ 1

sikZikXijkrcij i¼ 1,2,. . .,I j¼ 1,2,. . .,Ji ð20Þ

YI

i ¼ 1

FikZslk k¼ 1,2,. . .,K ð21Þ

EðZikÞ ¼ð1�FikÞ

PJi

j ¼ 1 qijkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPJi

j ¼ 1 s2jik

q i¼ 1,2,. . .,I k¼ 1,2,. . .,K ð22Þ

Zik ¼ EðZÞik�0:38984228� �

��1:75294þ0:4442135EðZÞik�0:07061455EðZÞ2ik

� 0:17592241EðZÞikþ0:044212641�

0:0012267386EðZÞikþ :00030570313

24

35

i¼ 1,2,. . .,I k¼ 1,2,. . .,K ð23Þ

Xijk is binary i¼ 1,2,. . .,I j¼ 1,2,. . .,Ji k¼ 1,2,. . .,K ð24Þ

EðZÞik, Fik, ZikZ0 i¼ 1,2,. . .I k¼ 1,2,. . .K ð25Þ

7. Decomposition of the centralized safety stockconsolidation model

The difficulty of the SSC model lies in the binary variable Xijk,and Brown’s nonlinear approximation given by (23). The binaryvariable hinders Minos from solving this model since it is anonlinear solver, while Eq. (23), prevents Cplex from solving themodel directly. Consequently, the model is decomposed into abinary master problem solved using Cplex and a continuousnonlinear sub-problem solved using Minos. This kind of decom-position follows the generalized Benders decomposition techni-que (Geoffrion, 1972).

7.1. Master problem

The master problem is solved to find the optimal values of thecomplicating binary variables Xijk, where these values are thensent to the sub-problem. If the sub-problem is infeasible to thosegiven values of the complicating variables Xijk, a feasibility cutshown in Eq. (27) is added to the master problem. The feasibilitycut used here is the combinatorial cut proposed by Codato andFischetti (2006). According to the model constraints, it is better toapply this cut only to those stockpoints that show insufficientcapacity in a previous infeasible iteration s. This accelerates themaster problem toward reaching a feasible 0–1 combination byminimizing the number of Xijk candidates included in the cut. Theparameters Xs

ijk and Zsik are the recorded values of the binary

variable Xijk and the standard normal Zik at the infeasible iterations. The cut searches for new combinations of 0–1 values of thebinary variables that were not considered infeasible before.

To guide the master problem to the best Xijk combination forthe sub-problem, the optimality cut given by Eq. (28) is addedafter each feasible iteration of the sub-problem. The multiplier lijk

appearing in this cut reflects the change in the sub-problemobjective function when the associated Xijk changes from 0 to 1.This multiplier lijk is calculated as follows: since each stage i

accepts only one Xijk to be 1 over the subscript j, the sub-problemis solved ik(j�1) times to evaluate the change lijk due to replacingthe Xijk leveled at 1 by the other binary variables Xijk leveled at 0.Based on the passed values lijk from the sub-problem, theoptimality cut (28) gives more opportunity to assign 1 to the Xijk

variable that has minimum value of the multiplier lijk. In theclassical BD approach, this multiplier is the dual variable asso-ciated with constraint (28). The dual variable cannot be used inthe optimality cut (28) because its returned values by the sub-problem are non-negative. This non-negativity is a result of thepositive increase hijk Xijk in objective function (29) associated withincreasing Xijk from 0 to 1. The master problem is thus stated asfollows:

Min MP¼XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

wijkXijkþa ð26Þ

Subject toConstraints (19), (24)

XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1:Xsijk¼ 0

XijkþXI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1:Xsijk¼ 1

1�XijkZ1

s¼ 1,2,. . .,S i¼ 1,2,. . .,I j¼ 1,2,. . .,J :XK

k ¼ 1

sikZsikXs

ijk4cij ð27Þ

aZSPtþXI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

ltijkðXijk�Xt

ijkÞ t¼ 1,2,. . .,T ð28Þ

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307 305

7.2. Sub-problem

The sub-problem finds the optimal values of E(Z)ik, Zik, and Fik

for the given values Xtijk of the complicating variables Xijk at any

feasible iteration t. The sub-problem is given by the followingequations:

Min SP¼XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

hijksikZikXijk ð29Þ

Subject toConstraints (20)–(23) and (25)

Xijk ¼ Xtijk : lijk i¼ 1,2,. . .,I j¼ 1,2,. . .,Ji k¼ 1,2,. . .,K ð30Þ

The algorithm iterates between both problems until an opti-mal solution is obtained. This can be attained when the lowerbound (31) obtained from the relaxed master problem equals theupper bound (32) resulting from the restricted sub-problem.Equality of both bounds implies that no more improvement inthe values of the complicating variables can be achieved.

LB¼XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

wijkXtijkþa ð31Þ

UB¼XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

wijkXtijkþ

XI

i ¼ 1

XJi

j ¼ 1

XK

k ¼ 1

hijksikZikXtijk ð32Þ

8. Computational experiments

The computational efficiency of solving the SSC model throughdecomposition as well as the savings that can be obtained fromdistributing safety stock based on the obtained results of thismodel are evaluated in this section. Experiments were conductedon AMD SempronTM processor 3400þ 1.8 GHz and 1 GB of RAM.

A comparison between the SSP model and the SSC model isshown in Table 2. Results are tabulated for seven different supplychain configurations. The first stage of the chain is a singlestockpoint while the number of stockpoints at Tier-1 and Tier-2stages are shown in the second and third columns. The SSC modelgives less safety amount than the SSP model as a result ofintroducing the safety pooling concept into the consolidationmodel. The motivation cost, shown in the ninth column, is thefirst term of the objective function (18) which represents theamount of money paid to the selected centers. The assumed rangein which the annual holding cost per unit takes its input values is$50–$150, while the assumed range of the motivation cost perclass of item per year is $30,000–$50,000.

The third-to-last column shows the cost savings that can beattained by applying the SSC model. Based on the assigned ranges

Table 2Comparison between the decentralized safety stock placement model and the centrali

Problem

number

Number of

Tier-1

Number of

Tier-2

Number of

Items

Decentralized safety

stock placement

model

Cent

Safety

amount

Holding

cost

Safet

amo

1 3 4 7 17213 1,612,850 853

2 5 4 6 27381 2,553,660 1190

3 4 5 10 32252 2,885,072 1760

4 5 5 8 38302 3,414,066 1742

5 7 3 9 32632 2,707,316 1560

6 8 6 5 28811 2,512,972 1104

7 9 7 4 25702 1,446,542 975

of cost parameters, 22.2–44.2% cost savings can be achievedannually as shown in the second-to-last column. The last columnshows the percentage reduction in safety stock size attainablethrough applying the centralization model. This percentagereflects the portfolio effect of SSC, introduced by Zinn et al.(1989), shown in equation (33). Up to 62% of the safety amountsresulting from the decentralized model can be saved if theconsolidation model is employed.

Portfolio Effect¼ 1�Sum of safetystock at consolidation centers

Sum of safety stock at decenterlized locations

ð33Þ

Each of the seven problem instances of the multi-item model issolved directly using Minos in less than a second. Table 3 illustratesthe computational experiments regarding the SSC model. Tendifferent problems with different values of input parameters aretested to check the computational efficiency of the proposedconsolidation model and solution methodology. The number offeasibility and optimality cuts added to the master problem differsfrom problem to another. For example, in the sixth problem all thestockpoints have enough capacity to handle any number ofproducts. Thus, no feasibility cuts are added to the master problemas depicted in the fifth column. In contrast, in the seventh problem,219 feasibility cuts are added to the master problem in order toprovide feasible Xijk solutions to the sub-problem.

Because of the efficiency of the lijk multiplier used in for-mulating the optimality cuts, the number of these cuts through-out the experiments is considered to be low, ranging from 4–30cuts as shown in the sixth column. The drawback of the multiplierlijk is the time it takes to be calculated. As shown in the last threecolumns of the table, the solution time of the master and sub-problems is very short compared to the total time of solving aproblem. This indicates that most of the solution time of a givenproblem is consumed in calculating the multipliers at eachiteration. The third-to-last column demonstrates the efficiencyof the proposed BD method to reach the optimal solution of theSSC model. The solution time elapsed to solve any of the 10different problems is very short, between 10 and 83 s.

9. Summary and future extensions

In this paper, two safety stock placement models are proposedto place safety amounts in a multi-stage supply chain. The supplychain comprises multiple sourced stockpoints facing variabledemand and lead time. An explicit form is applied to determinethe characteristics of the lead time at the multiple-sourcedstockpoints by applying order statistics concepts. The fill rate ateach stockpoint is decided from a supply chain perspective tominimize the safety stock placement cost throughout the entire

zed safety stock consolidation model.

ralized safety consolidation model Cost

savings ($)

% Saving

y

unt

Holding

cost ($)

Motivation

cost ($)

Total cost

($)

Cost Safety

stock size

6 820,520 423,100 1,243,620 369,230 22.89 50.40957

5 1,180,046 406,000 1,586,046 967,614 37.89 56.52095

3 1,667,517 578,020 2,245,537 639,535 22.17 45.42044

3 1,623,960 524,200 2,148,160 1,265,906 37.08 54.51151

0 1,261,791 580,860 1,842,651 864,665 31.94 52.19417

0 1,066,621 373,000 1,439,621 1,073,351 42.71 61.6813

3 511,836 296,000 807,836 638,706 44.15 62.05354

Table 3Computational efficiency of the decomposition method used to solve the centralized safety stock consolidation model.

Number Tier-1 Tier-2 Items Added cuts Solution time (s)

Feasibility Optimality Total Master-problem Sub-problem

1 3 4 7 52 17 21 3 2.5

2 6 8 11 4 6 29 0.46 0.35

3 4 5 10 25 18 40 2 2.5

4 8 6 5 31 30 46 1.4 2.5

5 9 7 4 5 7 10 0.4 0.3

6 12 9 12 0 4 26 0.15 0.12

7 5 5 8 219 18 59 20 9

8 7 13 8 10 4 16 0.6 0.5

9 10 11 7 28 23 83 2.5 2

10 10 12 9 31 5 17 1.5 0.8

H. Osman, K. Demirli / Int. J. Production Economics 135 (2012) 299–307306

supply chain. The minimum cost can be achieved by assigninghigh fill rates to stockpoints possessing low holding costs and lowfill rates to stockpoints having high holding costs. The first modelis developed based on the decentralized approach of safety stockallocation, in which each stockpoint keeps sufficient safetyamounts to face its underlying variability in lead time demand.The second model benefits from variability reduction, resultingfrom pooling the lead time demand variability, to consolidatesafety stocks in consolidation centers. The consolidation center ata given stage is responsible for dealing with the variability of leadtime demand encountered at that stage. In return to this respon-sibility, each consolidation center will be paid an amount ofcredits. These credits along with the capacity restrictions andholding costs at each center are the criteria deciding on selectingthe consolidation center at each stage.

The safety consolidation model includes nonlinear and inte-grality constraints that inhibit commercial solvers from handlingit directly. A Benders decomposition method is developed todecompose this model into two problems. The resulting masterand sub-problem problems are solvable directly by Cplex andMinos respectively. Computational experiments recorded inTable 3 show that the proposed decomposition method solvesthe nonlinear mixed integer safety consolidation model in a veryshort time, between 10 and 83 s. The comparison made betweenthe decentralized and the safety consolidation models illustratedin Table 2 shows that cost savings between 22.17% and 44.15%can be accomplished by employing the consolidation strategies. Areduction, ranging between 45.4% and 62%, in safety stock can beachieved by applying the consolidation policy.

The proposed safety stock placement models could beextended by considering other probability distributions ratherthan the normal distribution to represent lead time and demandvariability. The SSC model assumes that stockpoints placed at agiven stage employ the same cycle time. By relaxing this assump-tion the problem can be investigated with different cycle times atthe stockpoints in the same stage. The decentralized and centra-lized approaches are compared based on the amount of safetystocks and the safety stock placement costs. Moving the decen-tralized safety amounts to the consolidation centers may affect onthe delivery date at a downstream stage. The problem could beinvestigated further to consider the delivery performance inselecting a consolidation center and to study the effect of safetystock consolidation on the delivery time.

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