Research Article A Generalized Fuzzy Integer Programming...

Research ArticleA Generalized Fuzzy Integer Programming Approach forEnvironmental Management under Uncertainty

Y R Fan1 G H Huang12 K Huang1 L Jin3 and M Q Suo4

1 Faculty of Engineering and Applied Science University of Regina Regina SK Canada S4S 0A22MOE Key Laboratory of Regional Energy Systems Optimization Resources and Environmental Research AcademyNorth China Electric Power University Beijing 102206 China

3 College of Environmental Science and Engineering Xiamen University of Technology Xiamen Fujian 361024 China4College of Urban Construction Hebei University of Engineering Handan Hebei 056038 China

Correspondence should be addressed to G H Huang huanggiseisorg

Received 10 March 2014 Revised 6 June 2014 Accepted 17 June 2014 Published 14 September 2014

Academic Editor Hang Xu

Copyright copy 2014 Y R Fan et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study a generalized fuzzy integer programming (GFIP) method is developed for planning waste allocation and facilityexpansion under uncertaintyThedevelopedmethod can (i) deal with uncertainties expressed as fuzzy sets with knownmembershipfunctions regardless of the shapes (linear or nonlinear) of these membership functions (ii) allow uncertainties to be directlycommunicated into the optimization process and the resulting solutions and (iii) reflect dynamics in terms of waste-flow allocationand facility-capacity expansion A stepwise interactive algorithm (SIA) is proposed to solve the GFIP problem and generatesolutions expressed as fuzzy sets The procedures of the SIA method include (i) discretizing the membership function grade offuzzy parameters into a set of 120572-cut levels (ii) converting the GFIP problem into an inexact mixed-integer linear programming(IMILP) problem under each 120572-cut level (iii) solving the IMILP problem through an interactive algorithm and (iv) approximatingthe membership function for decision variables through statistical regression methods The developed GFIP method is applied toa municipal solid waste (MSW) management problem to facilitate decision making on waste flow allocation and waste-treatmentfacilities expansion The results which are expressed as discrete or continuous fuzzy sets can help identify desired alternatives formanaging MSW under uncertainty

1 Introduction

Municipal solid waste (MSW) management is a priority formany developed and developing countries throughout theworld Effective planning of MSW is critical for supportingsustainable socioeconomic development in urban commu-nities However extensive uncertainties may exist in manysystem components and impact factors For example wastegeneration ratewithin a city is related tomany socioeconomicand environmental factors and exhibits various uncertainfeatures Such uncertainties and their interactions can leadto increased complexities in the related planning effortsand will affect consequent decision processes Besides theseuncertaintiesmay be furthermultiplied becausemany systemcomponents are of multiperiod multilayer and multiob-jective features Li and Huang [1] Moreover waste-disposal

facilities in a MSW management system usually have overallcumulative or daily operating-capacity limits Increasingwaste generation rates as a result of population explosionand economic development lead to intensified conflicts withdecreasing waste-treatmentdisposal capacities Therefore itis desired that the above uncertain and dynamic complexitiesbe reflected in efforts for identifying effective environmentalmanagement alternatives

In the past decades a number of inexact optimizationtechniques were developed to deal with uncertainties anddynamics in MSWmanagement They were mainly classifiedas fuzzy stochastic and interval mathematical programming(FMP SMP and IMP resp) [2ndash6] For example Li andHuang [7] proposed an inexact two-stage mixed-integerlinear programming (ITSMILP) method for solid wastemanagement in the city of Regina through incorporating

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 486576 16 pageshttpdxdoiorg1011552014486576

2 Mathematical Problems in Engineering

interval linear programming (ILP) two-stage stochastic pro-gramming (TSP) and mixed-integer programming (MIP)within a general mathematical programming frameworkHowever IMP could merely deal with interval uncertaintieswithout distributional information SMP was inapplicable tolarge-scale problems due to its stringent requirements forinformation of probabilistic distributions

Fuzzy mathematical programming (FMP) as a branchof fuzzy set theory could generally deal with uncertaintiesexpressed as fuzzy sets or fuzzy goalsconstraints [8ndash11]Recently various FMP methods were employed to deal withuncertainties in MSW systems [12ndash15] For example Fan etal [12] explored a fuzzy linear programming (FLP) methodfor dealing with uncertainties expressed as fuzzy sets thatexist in the constraintsrsquo left-hand and right-hand sides andthe objective function however this method was unable toreflect dynamic complexities related to capacity expansionschemes for waste-treatment facilities Srivastava and Nema[13] proposed a fuzzy parametric programming model foridentifying desired treatmentdisposal facilities planningwaste management capacities and allocating waste flowsunder uncertainty however the proposed method generateddeterministic waste allocation schemes without provision ofbases for supporting generation of multiple decision optionscorresponding to the uncertain system conditions

Generalized fuzzy linear programming (GFLP) (or fullyfuzzy linear programming (FFLP)) methods extended tradi-tional FMP approaches through permitting uncertain infor-mation in the optimization process and resulting solutionsRecently several GFLP (or FFLP) methods were proposedto deal with uncertain information in both parameters anddecision variables [16ndash21] For example Hosseinzadeh Lotfiet al [18] developed a lexicography method to solve the FFLPproblem and generate approximate solutions presented asfuzzy sets However previous studies on fuzzy variables inFMP problemsmainly focused on some special types of fuzzysets (such as symmetric triangular or trapezoidal fuzzy sets)Furthermore some of them might lead to complicated inter-mediate models and thus were not applicable for large-scaleproblems Fan et al [19 20] proposed another kind of fuzzyprogramming (named generalized fuzzy linear programming(GFLP) method) to deal with fuzzy uncertainty in bothparameters and variables in which all fuzzy sets with knownmembership functions can be treated through defuzzificationmethod (ie 120572-cut method) However the GFLP approachcannot reflect dynamic features in environmental manage-ment problemsMoreover no previous studywas reported oncapacity expansion issues under fuzzy conditions through thegeneralized fuzzy optimization approach where expansionschemes were desired under multiple scenarios and 120572-cutlevels

As an extension of developed GFLP approach a gen-eralized fuzzy integer programming (GFIP) method wouldbe proposed for MSW management under uncertainty Theproposed GFIP approach integrates the techniques of gener-alized fuzzy linear programming (GFLP) and mixed-integerprogramming (MIP) within an optimization framework Indetail (i) the GFIP method can deal with uncertaintiesexpressed as fuzzy sets with known membership functions

regardless of whether these functions are linear or nonlinear(ii) the proposed GFIP method can allow uncertainties tobe directly communicated into the optimization process andthe resulting solutions (iii) the GFIP method can reflectdynamics in terms of waste-flow allocation and facility-capacity expansion (iv) compared with other inexact mixed-integer programming approaches (eg ITSMILP by Li andHuang [7]) the GFIP can analyze the inherent interrelation-ship between the uncertainty of fuzzy parameters (ie 120572-cutlevels) and capacity expansion options of waste managementfacilities and such analysis can help decision makers maketradeoffs between system reliability and system cost Thena case study will be provided to demonstrate applicabilityof the GFIP method to support dynamic analysis for MSWmanagement under uncertainty The results will be usedfor generating different decision alternatives under varioussystem conditions and thus for helping identify desired wastemanagement policies

2 Methodology

21 Formulation of the Generalized Fuzzy Integer Program-ming A GFIP model with ambiguous coefficients and deci-sion variables expressed as fuzzy sets can be formulated asfollows

Max 119891 = 119888 times 119883 (1a)

subject to

119860 times 119883 le119887 (1b)

119883 ge 0 (1c)

119909119895 = fuzzy continuous variables 119909119895 isin 119883

119895 = 1 2 119901 (119901 lt 119899)

(1d)

119909119895 = fuzzy integer variables 119909119895 isin 119883

119895 = 119901 + 1 119901 + 2 119899

(1e)

where 119888 isin 1times119899119883 isin 119899times1 119887 isin 119898times1 and 119860 isin

119898times119899 denotes a set of fuzzy sets 119888 = (1198881 1198882 119888119899) 119883

119879=

(1199091 1199092 119909119899) 119887119879= (

1198871

1198872

119887119899) and119860 = (119886119894119895)119898times119899 for

all 119894 isin 119898 119895 isin 119899 A fuzzy set (119860) in 119883 can be defined as119909 120583119860(119909) | 119909 isin 119883 120583119860(119909) 119883 rarr [0 1] where 120583119860(119909) isthe membership function or grade of membership [22] If allelements in119860 are integers and120583119860(119909) is a discretemembershipfunction then119860 is a fuzzy integer set [23]The value of 120583119860(119909)varies between 0 and 1 indicating the possibility of an element119909 belonging to 119860 120583119860(119909) = 1 means that 119909 definitely belongsto the fuzzy set (119860) while 120583119860(119909) = 0 denotes that 119909 does notbelong to 119860 The closer 120583119860(119909) is to 1 the more likely that 119909belongs to119860 conversely the closer 120583119860(119909) is to 0 the less likelythat 119909 belongs to 119860 [22 24] An 120572-cut of 119860 can be defined asan ordinary set (denoted as [119860]120572) in which the membershipdegrees of elements exceed 120572 [119860]120572 is usually a continuous or

Mathematical Problems in Engineering 3

discrete fuzzy interval Consequently through the concept of120572-cut each fuzzy parameter can be characterized as a series ofintervals under different 120572-cut levels Then interval analysismethods can be applied to process these fuzzy intervals

22 Solution Method of GFIP Model through Stepwise Interac-tive Algorithm If the parameters and variables inmodel (1a)ndash(1e) are triangular fuzzy numbers several methods can beapplied to solve the model such as the lexicography methodproposed by Hosseinzadeh Lotfi et al [18] and the methodsof Fan et al [12] and Kumar et al [21] However whenthe parameters of model (1a)ndash(1e) are expressed throughother kinds of fuzzy numbers the above methods are notapplicable Consequently in this study a newmethod namedstepwise interactive algorithm (SIA) will be proposed to solvemodel (1a)ndash(1e) This algorithm is based on computationalprinciples related to fuzzy intervals [25ndash28] (see Appendixsection) The detailed proof of the solution algorithm can befound in Fan et al [20] The inherent idea of the stepwiseinteractive algorithm is based on the design of experimentin which the optimization model would be considered as anexperiment with the 120572-cut levels being the inputs and theoptimal solutions being the outputs The detailed proceduresof the SIA method include (i) discretizing the membershipfunction grade of fuzzy parameters into a set of 120572-cut levels(ii) converting the GFIP problem into an inexact mixed-integer linear programming (IMILP) problem under each 120572-cut level (iii) solving the IMILP problem through an inter-active algorithm and (iv) approximating the membershipfunction for decision variables through statistical regressionmethods Compared with the previous methods SIA canallow uncertainties to be directly communicated into theoptimization process Moreover it will not lead to complexintermediate submodels and thus lead to a relatively lowcomputational requirementThis is meaningful when the SIAmethod is applied to solve large-scale management modelsFinally the proposed SIA method can generate solutionsexpressed as fuzzy sets

Since the parameters in model (1a)ndash(1e) are expressed asfuzzy sets these parameters will be defuzzified before themodel is solved Various defuzzification methods have beenproposed to convert fuzzy sets into crisp sets including 120572-cut max-membership principle centroid weighted averagemean-max membership center of sums center of largestand first of maxima or last of maxima methods In thisstudy the 120572-cut would be applied to defuzzify the fuzzyparameters in model (1a)ndash(1e) due to its popularity and easeof implementation The concept of 120572-cut is important inreflecting the relationship between fuzzy sets and crisp setsEach fuzzy set can be uniquely represented by all of its 120572-cuts As stated by Kreinovich [29] fuzzy data processing iscomputable for 120572-cuts but in general not computable formembership functions Consequently the fuzzy parametersand decision variables in model (1a)ndash(1e) are defuzzifiedthrough the 120572-cut method instead of their membershipfunctions Through the 120572-cut method the fuzzy parametersand decision variables in model (1a)ndash(1e) will be convertedinto the related fuzzy intervals The optimization model with

interval parameters can then be transformed into determinis-tic submodels which can be solved through ordinary solutionmethods (eg simplex method) Therefore before solvingmodel (1a)ndash(1e) a set of 120572-cut levels (ie 1205721 1205722 120572119902) areselected from the unit interval [0 1]Then for any 120572119894 isin [0 1]the associated 120572-cuts for 119888119895 119909119895 119886119894119895 and

119887119894 can be expressedas (119888119895)120572119894 = [(119888119895)

minus120572119894 (119888119895)+120572119894] (119909119895)120572119894 = [(119909119895)

minus120572119894 (119909119895)+120572119894] (119886119894119895)120572119894 =

[(119886119894119895)minus120572119894 (119886119894119895)+120572119894] and (119887119894)120572119894 = [(119887119894)

minus120572119894 (119887119894)+120572119894]

Rank these 120572-cut levels into an increasing sequence120572(1) 120572(2) 120572(119902) where 120572(1) le 120572(2) le sdot sdot sdot le 120572(119902) Theminimum 120572-cut level [ie 120572(1)] will be appointed firstly tocut model (1a)ndash(1e) Then an inexact mixed-integer linearprogramming (IMILP) model can be formulated as follows

Max (119891)plusmn

120572(1)=

119899

sum

119895=1

(119888119895)plusmn

120572(1)times (119909119895)

plusmn

120572(1)(2a)

subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(1)times (119909119895)

plusmn

120572(1)le (119887119894)

plusmn

120572(1)119894 = 1 2 119898 (2b)

(119909119895)plusmn

120572(1)ge 0 119895 = 1 2 119898 (2c)

(119909119895)plusmn

120572(1)= interval continuous variables (119909119895)

plusmn

120572(1)isin (119883)

plusmn120572(1)

= 1 2 119901 (119901 lt 119899)

(2d)

(119909119895)plusmn

120572(1)= interval integer variables (119909119895)

plusmn

120572(1)isin (119883)

plusmn120572(1)

= 119901 + 1 119901 + 1 119899

(2e)

where (119891)plusmn120572(1)

(119888119895)plusmn120572(1)

(119909119895)plusmn120572(1)

(119886119894119895)plusmn120572(1)

and (119887119894)plusmn120572(1)

are fuzzyintervals under 120572(1) (119888119895)

plusmn120572(1)

= [(119888119895)minus120572(1) (119888119895)+120572(1)] (119909119895)

plusmn120572(1)

=

[(119909119895)minus120572(1) (119909119895)+120572(1)] (119886119894119895)

plusmn120572(1)

= [(119886119894119895)minus120572(1) (119886119894119895)+120572(1)] and (119887119894)

plusmn120572(1)

=

[(119887119894)minus120572(1) (119887119894)+120572(1)] Fuzzy intervals under other 120572-cut levels also

have similar expressions Furthermore an interval number(119886plusmn) can be defined as 119886plusmn = [119886

minus 119886+] = 119905 | 119886

minusle 119905 le 119886

+

Model (2a)ndash(2e) shows the formulation of intervalmixed-integer linear programming (IMILP)method with all param-eters expressed as interval numbers The IMILP modelwas developed through introducing the concept of intervalanalysis into amixed-integer linear programming framework[3] It allowed uncertainties to be directly communicated intothe optimization processes and resulting solutions and didnot lead to complicated intermediate models [3]

Since model (2a)ndash(2e) is an inexact optimization modelwith all parameters expressed as intervals it can be solvedthrough the interactive algorithm proposed by Huang et al[3] Assume that the former 1198961 coefficients of model (2a)ndash(2e) are positive and the latter 1198962 coefficients are negative(1198961 + 1198962 = 119899) Then model (2a)ndash(2e) can be converted into


two submodels In detail the first submodel correspondingto (119891)+120572(1)can be formulated as

Max (119891)+

120572(1)=

1198961

sum

119895=1

(119888119895)+

120572(1)(119909119895)+

120572(1)+

119899

sum

119895=1198961+1

(119888119895)+

120572(1)(119909119895)minus

120572(1)

(3a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

le (119887119894)+

120572(1) forall119894

(3b)

(119909119895)plusmn

120572(1)= interval continuous variables

119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(3c)

(119909119895)plusmn

120572(1)= interval discrete variables

119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(3d)

(119909119895)plusmn

120572(1)ge 0 forall119895 (3e)

Solutions of (119909119895opt)+120572(1)(119895 = 1 2 1198961) and (119909119895opt)

minus120572(1)(119895 =

1198961+1 1198961+2 119899) can be obtained from submodel (3a)ndash(3e)Then the second submodel corresponding to (119891)

minus120572(1)

can beformulated based on solutions from the first submodel whichcan be expressed as follows

Max (119891)minus

120572(1)=

119896

sum

119895=1

(119888119895)minus

120572(1)(119909119895)minus

120572(1)+

119899

sum

119895=119896+1

(119888119895)minus

120572(1)(119909119895)+

120572(1)

(4a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

le (119887119894)minus

120572(1) for all 119894

(4b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(4c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(4d)

(119909119895)minus

120572(1)le (119909119895opt)

+

120572(1) 119895 = 1 2 1198961 (4e)

(119909119895)+

120572(1)ge (119909119895opt)

minus

120572(1) 119895 = 1198961 + 1 1198961 + 2 119899 (4f)

(119909119895)plusmn

120572(1)ge 0 forall119895 (4g)

Hence solutions of (119909119895)minus120572(1)(119895 = 1 2 1198961) and

(119909119895)+120572(1)(119895 = 1198961 + 1 1198961 + 2 119899) can be obtained from

submodel (4a)ndash(4g) Therefore the final solutions for model(2a)ndash(2e) can be generated which are presented as follows

(119909119895opt)plusmn

120572(1)= [(119909119895opt)

minus

120572(1)(119909119895opt)

+

120572(1)] (5a)

(119891opt)plusmn

120572(1)= [(119891opt)

minus

120572(1)(119891opt)

+

120572(1)] (5b)

Formulas ((3a)ndash(3e)) to ((5a)-(5b)) show the detailedsolution process of an IMILP model through the interactivealgorithm (also named two-stepmethod) Based on the inter-active algorithm the original IMILP model is firstly refor-mulated into two submodels corresponding respectively toits upper and lower bounds of objective function the twosubmodels are then solved separately one after another [30]The sequence to solve two submodels is subject to the natureof objective function (max ormin) For amaximized problem[ie model (2a)ndash(2e)] the submodel corresponding to theupper bound of the objective function is solved first followedby solving the submodel corresponding to the lower bound ofthe objective function besides the optimal solutions from thefirst submodel should be used as constraints for the secondsubmodel [30]

Based on solutions of model (2a)ndash(2e) we will select 120572(2)to 120572(119902) in sequence and then formulate corresponding IMILPmodels as follows

Max (119891)plusmn

120572(119897)=

119899

sum

119895=1

(119888119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)(6a)


subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)le (119887119894)

plusmn

120572(119897) for 119894 = 1 2 119898 (6b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(6c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(6d)

(119909119895)plusmn

120572(119897)sube (119909119895opt)

plusmn

120572(119897minus1)(6e)

(119909119895)plusmn

120572(1)ge 0 forall119895 (6f)

where 120572(119897) isin 120572(2) 120572(119902) and (119909119895opt)plusmn

120572(119897minus1)are the optimal

solutions obtained from the IMILP model under 120572(119897minus1)Formula (6e) is proposed to reflect the property of the fuzzynumber that (119909119895)

plusmn1205721

supe (119909119895)plusmn1205722

holds when 1205721 1205722 isin [0 1] and1205721 le 1205722

Based on the interactive algorithm model (6a)ndash(6f) willbe converted into two submodels as follows

Submodel 1

Max (119891)+

120572(119897)=

1198961

sum

119895=1

(119888119895)+

120572(119897)(119909119895)+

120572(119897)+

119899

sum

119895=1198961+1

(119888119895)+

120572(119897)(119909119895)minus

120572(119897)

(7a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

le (119887119894)+

120572(119897) forall119894

(7b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(7c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(7d)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1 2 1198961 (7e)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (7f)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (7g)

Submodel 2

Max (119891)minus

120572(119897)=

119896

sum

119895=1

(119888119895)minus

120572(119897)(119909119895)minus

120572(119897)+

119899

sum

119895=119896+1

(119888119895)minus

120572(119897)(119909119895)+

120572(119897)(8a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

le (119887119894)minus

120572(119897) forall119894

(8b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(8c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(8d)

(119909119895)minus

120572(119897)le (119909119895opt)

+

120572(119897) 119895 = 1 2 1198961 (8e)

(119909119895)+

120572(119897)ge (119909119895opt)

minus

120572(119897) 119895 = 1198961 + 1 1198961 + 2 119899 (8f)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1 2 1198961 (8g)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (8h)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (8i)


From submodels ((7a)ndash(7g)) and ((8a)ndash(8i)) we canobtain the final solutions for model (6a)ndash(6f) under 120572(119897) (119897 =2 3 119902) as follows

(119909119895opt)plusmn

120572(119897)= [(119909119895opt)

minus

120572(119897)(119909119895opt)

+

120572(119897)] (9a)

(119891opt)plusmn

120572(119897)= [(119891opt)

minus

120572(119897)(119891opt)

+

120572(119897)] (9b)

Based on formulas ((2a)ndash(2e))ndash((9a)-(9b)) we can obtaina series of fuzzy interval solutions for model (1a)ndash(1e)under different 120572-cut levels Then we can approximate themembership function for continuous decision variables bystatistical regression methods In this procedure the GFIPmodel is supposed to be an experiment with 120572-cut levelsbeing its inputs (ie independent variables) and the lowerand upper bounds of decision variables being its outputs (iedependent variables) Take (119909119895opt)

minus120572 as an example we can

obtain a regression function between (119909119895opt)minus120572 and 120572 based on

the fuzzy interval solutions Such a regression functionwill beconsidered as the inverse function of the left shape functionfor 119909119895 denoted as 119871minus1(119909) then we can acquire the left shapefunction for 119909119895 expressed as 119871(119909) In the same way we canobtain the right shape function for 119909119895 expressed as 119877(119909)

3 Case Study

A hypothetical municipal solid waste (MSW) managementproblem is used to illustrate the applicability of GFIPapproachThe studied system includes threemunicipal citiesA planning horizon of 15 years is divided into three periodswith each one having a time interval of 5 years Two typesof facilities can be available for waste treatmentdisposal Alandfill is considered in the proposed case due to its crucialrole for MSW disposal in both developed and developingcountries For example more than 54 percent of MSW waslandfilled in the United States during 2009 [31] while 893percent of the generated MSW (7404 million tonnes) waslandfilled in China in 2002 [32] The landfill is typicallycharacterized as an overall capacity limit Also a waste-to-energy (WTE) facility which can effectively minimize landdepletion caused by landfilling is employed to serve waste-disposal needs It is characterized as a daily capacity limit

In fact a MSW management system involves severalprocesses with socioeconomic and environmental implica-tions such aswaste generation transportation treatment anddisposal [33] Extensive uncertainties usually exist in theseprocesses due to impacts of the economic development pop-ulation growth and human activitiesMoreover probabilisticmethods are not applicable to quantify these uncertaintieswhen data are insufficient Consequently adoption of fuzzyset theory would be a potential alternative especially whenuncertainties can be consciously assumed by decisionmakersor experts Furthermore uncertain inputs in the MSWmanagement system would lead to variations in the resultingsolutions Therefore the GFIP method will be desired toreflect uncertain and dynamic complexities in the MSWmanagement system and generate solutions expressed asfuzzy sets

Table 1 shows related waste generation levels and costcoefficients including waste generation rates in three citiesoperation costs of two facilities and transportation costs forshippingwaste flowsThese parameters are estimated as trian-gular fuzzy numbers with knownmost possible values as wellas left and right spreads Table 2 presents capacity expansionoptions and related costs forwaste disposal facilitiesThe totalcapacity of landfill is (18 times 10

6 02 times 10

5 03 times 10

5) tonnewhichmeans themost possible capacity of landfill is 18times106tonne and the lower and upper bound is 178 times 10

6 and183 times 10

6 tonne respectively The daily capacity of WTEfacility is (390 20 and 20) tonneday which means the mostpossible capacity and lower and upper bound is 390 370 and410 tonneday respectively The WTE facility will generateresidues of about 30 (10 as its left and right spread) of theincoming waste stream The revenue from the WTE facilityis approximately $20tonne with its left and right spreadsbeing $2tonne In this study all parameters are assumed tobe triangular fuzzy numbers The triangular fuzzy numbersare considered in this study because (i) the triangular formis the simplest type of fuzzy numbers (ii) many other typesof fuzzy numbers can be estimated through the triangularfuzzy numbers and (iii) triangular membership functioncan provide the most important information for a fuzzyset lower-bound value upper-bound value and the mostpossible value [34] Also other kinds of fuzzy numbers canbe treated through the proposed GFIP approach if theirmembership functions are known

The problem under consideration is how to effectivelyallocate waste flows and choose appropriate capacity expan-sion options of waste-disposal facilities under a numberof environmental economic and treatmentdisposal con-straints in order to minimize the overall system cost A GFIPmodel can thus be formulated to solve this problem

In this study decision variable 119909119894119895119896 represents the amountof waste flow from city 119895 to waste-treatment facility 119894 inperiod 119896Theobjective is tominimize the systemcost througheffectively allocating waste flows from three cities to twodisposal facilities and choosing appropriate waste-disposal-facility options for excessive waste-disposal requirementsThe constraints involve relationships between decision vari-ables and waste generationmanagement conditions Thus aGFIP model can be formulated as follows

Min 119891 = 1825

3

sum

119895=1

3

sum

119896=1

2

sum

119894=1

119909119894119895119896 (TR119894119895119896 +OP119894119896) + 1199092119895119896

times [FE (FT119896 +OP1119896)

minusRE119896]

+

3

sum

119896=1

FLC119896119896 +3

sum

119897=1

3

sum

119896=1

FTC119897119896119885119897119896

(10a)

subject to


Table 1 Waste generation levels and cost coefficients

Time period119896 = 1 119896 = 2 119896 = 3

Waste generationWG119895119896 (tday)City 1 (225 25 25) (250 25 25) (275 25 25)City 2 (375 25 25) (400 25 25) (425 25 25)City 3 (300 25 25) (325 25 25) (350 25 25)

Cost of transportation to landfillTR1119895119896($t)

City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)

Cost of transportation to landfill FT119896($t)

Waste-to-energy facility (10 1 1) (12 1 1) (14 1 1)Cost of transportation towaste-to-energy facilityTR2119895119896 ($t)

City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)

Operation costs OP119894119896 ($t)Landfill (50 8 7) (58 10 10) (68 15 15)Waste-to-energy facility (60 10 10) (68 10 10) (75 10 10)

Table 2 Capacity expansion options and costs for landfill and WTE facilities

Data Time period119896 = 1 119896 = 2 119896 = 3

Capacity expansion options for WTEfacility (tonneday)

ΔTC1 (option 1) 150 150 150ΔTC2 (option 1) 200 200 200ΔTC3 (option 1) 250 250 250

Capacity expansion options for thelandfill facility (106 tonne)

ΔTC (031 001 0015) (031 001 0015) (031 001 0015)

Capital cost for WTE expansion ($106)FTC1119896 (option 1) 105 83 65FTC2119896 (option 1) 152 119 93FTC3119896 (option 1) 198 155 122

Capital cost for landfill expansion($106)

FLC119896 (14 1 1) (14 1 1) (14 1 1)

(1) Landfill capacity constraint

1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)

(2) WTE facility-capacity constraints

3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)

(3) Waste disposal demand constraints2

sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)


(4) Nonnegativity constraints

119909119894119895119896 ge 0 forall119894 119895 119896 (10e)

(5) Nonnegativity and binary constraints

119896 =

le 1

ge 0

= integer forall119896

(10f)

119885119897119896 =

le 1

ge 0

= integer forall119897 119896

(10g)

(6) Landfill expansion constraint

3

sum

119896=1

119896 le 1 (10h)

(7) WTE facility expansion constraints

3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)

where FE is the residue flow from WTE to landfill ( ofincoming mass to WTE facility) FLC119896 is the capital cost oflandfill expansion in period 119896 ($) FT119896 is the transportationcost of waste flow fromWTE to landfill in period 119896 ($tonne)FTC119897119896 is the capital cost of expanding WTE by option 119897

in period 119896 ($) OP119894119896 is the operating cost of facility 119894 inperiod 119896 ($tonne) RE119896 is the revenue from WTE in period119896 ($tonne) TE is the maximum capacity of WTE facility(tonneday) TL is the capacity of landfill (tonne) TR119894119895119896 isthe transportation cost for waste flow from city 119895 to facility119894 during period 119896 ($tonne) WG119895119896 is the waste generationrate in city 119895 during period 119896 (tonneday) ΔTC is the totalamount of expansion capacity for landfill (tonne) ΔTC119897 isthe amount of the 119897th type of expansion capacity for WTE(toneday) 119909119894119895119896 is the waste-flow rate from city 119895 to facility 119894in period 119896 (tonneday) 119894 = 1 2 j = 1 2 3 119896 = 1 2 3 119896 is thebinary decision variable for landfill expansion at the start ofperiod 119896 119885119897119896 is the binary decision variable for WTE facilitywith expansion option 119897 at the start period of 119896 119894 is the indexfor facility (119894 = 1 for landfill and 119894 = 2 for WTE facility) 119895 isthe index for three cities (119895 = 1 2 3) 119896 is the index for timeperiods (119896 = 1 2 3)

In model (10a)ndash(10i) the objective (ie formula (10a)) isto minimize the total cost of facility expansion and waste-flow disposal over the entire planning horizon which willcover expenses of handling waste flows charges of expandingfacilities and revenues from the WTE facility Constraint(10b) specifies that the total amount of waste allocated to thelandfill must not exceed its existing and expanded capacitiesIn this study one year is assumed to have 365 days andthere are 5 years in each period Consequently the coefficient

in constraint (10b) would be 1825 (ie 365 times 5) Constraint(10c) means that the actual daily waste flows shipped to theWTE facility should not exceed its existing and expandedcapacities Constraint (10d) indicates that for each city ineach period the waste flows transported to the landfill andWTEmust be not less than its waste-disposal demand in thisperiodThis constraint also assumes that all solid wastes haveto be shipped to a disposal site within a certain period after itsgeneration and nomass loss is incurred in the transportationprocess Constraints (10f)sim(10i) define the binary variablesrelated to capacity expansion decisions constraint (10h)denotes that the landfill can only be expanded once withinthe entire planning horizon and constraint (10i) means thatthe WTE can be expanded once in each period

Figure 1 shows the schematic of theGFIPmodel forMSWmanagement Obviously the GFIP model is an integration ofgeneralized fuzzy linear programming (GFLP) and mixed-integer linear programming (MILP) methods Each methodhas a unique contribution in enhancing the capability ofGFIPin dealing with uncertainties and dynamic features in solidwaste management For example fuzzy parameters can behandled by GFLP and waste management facility expansioncan be addressed by MILP Besides a stepwise interactivealgorithm (SIA) is proposed for solving the proposed GFIPmodel which can permit uncertainty to be directly commu-nicated into the optimization process and resulting solutionsThrough SIA the developed GFIP model will firstly beconverted into several IMILP submodels and then be furthertransformed into mixed-integer linear programming (MILP)submodels Consequently the computational complexity ofthe GFIP would be reasonable For example if 119899 120572-cut levelsare identified in solving the GFIPmodel 119899 IMILP submodelswill be firstly generated According to interactive algorithmeach IMILP submodel can be further converted into twoMILP submodels thus the GFIP model will finally result in2119899MILP submodels with deterministic parameters

4 Result Analysis

In this study a GFIP model is developed for supportingdecision making in MSW management A stepwise interac-tive algorithm (SIA) is proposed to solve the GFIP modelBased on SIA six 120572-cut levels (ie 0 03 05 07 085and 1) would be considered Under each 120572-cut level thefuzzy parameters presented in Tables 1 and 2 would beconverted into corresponding fuzzy intervals and model(10a)ndash(10i) would also be transformed into an inexact mixed-integer linear programming (IMILP) model Tables 3 to 5present waste-flow diversion schemes capacity expansionoptions and related system costs obtained through IMILPmodels under the selected 120572-cut levels The results indicatethat waste-flow patterns and capacity expansion optionswould vary due to temporal and spatial variations in wastegeneration rates and waste management conditions

In period 1 waste from city 1 would be initially shippedto WTE with a flow amount of 195 tonneday For city 1the WTE facility would be the first choice to serve its wastedisposal requirement Then the remaining waste would be


MSW management under uncertainty

Fuzzy parameters (eg wastegeneration facility capacity etc)

Permitting uncertaininformation in solutions

Generalized fuzzy linearprogramming (GFLP)

Generalized fuzzy integer programming(GFIP) model for MSW management

Discretize membershipgrade into (1205721 1205722

120572q)

Reorder the 120572-cut levels into

Use 120572(i) to cut fuzzyparameters in GFIP model

Convert the GFIP model intoan inexact mixed integer linearprogramming (IMILP) model

Interactive algorithm

Lower-bound submodel(fminus) under 120572(i)-cut level

Upper-bound submodel(f+) under 120572(i)-cut level

Obtain interval solutions for objectiveand decision variables under 120572(i)-cut level

Generate membership function forcontinuous decision variables

Generate optimal MSW management policies

Step

wise

inte

ract

ive a

lgor

ithm

MSW disposalfacility expansion

Mixed integer linearprogramming (MILP)

a sequence 120572(1) 120572(2) 120572(q) where 120572(1) le 120572(2) le middot middot middot le 120572(q)

i = i + 1 i = 1

Figure 1 The schematic of the GFIP model for MSWmanagement

allocated to the landfill with a waste flow of (30 25 and 25)tonneday The (30 25 and 25) indicates a triangular fuzzynumber with 30 5 and 55 as its most possible value andlower and upper bound respectivelyThe fuzzy characteristicof the waste flow to the landfill indicates that the variation inthe waste generation rate of city 1 would be handled throughlandfilling The waste disposal scheme for city 2 is muchdifferent from that of city 1 The waste-treatment demand ofcity 2 can be satisfied through landfilling Consequently nowaste would flow to WTE in this period Conversely all ofthe generated waste in city 3 would be shipped to WTE inspite of its variation in waste generation rate

The waste allocation schemes for three cities in period 2would be similar to those in period 1 The majority of wastefrom city 1 would be allocated to WTE with the residues

being shipped to landfill However compared with the wasteflows in period 1morewastewould be transported toWTE inperiod 2 as a result of temporally increasing waste generationrate Moreover the amount of waste allocated to WTE fromcity 1 would fluctuate within small intervals under lowplausibilities ((225 2366) and (225 2299) tonneday under120572 = 0 and 03 resp)This is because the waste generation ratewould vary within significant ranges under low plausibilitiesMeanwhile landfill would be the only choice to satisfy thewaste-treatment demand of city 2 while all waste from city 3would be shipped to WTE

The waste-flow patterns would be changed significantlyin period 3 All waste from the three cities would be deliveredto the landfill due to its lower operation cost In detail wasteflows shipped to the landfill from cities 1 2 and 3 would be


X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0

The upper bound of waste flow under different 120572-cut values

Figure 2 The upper bounds of waste flows under different 120572-cutlevels

(250 275 and 300) (400 425 and 450) and (325 350 and375) tonneday respectively Also the (250 275 and 300)(400 425 and 450) and (325 350 and 375) indicates threetriangular fuzzy numbers reflecting uncertainty in resultingsolutions derived from uncertain inputs

Since parameters in model (10a)ndash(10i) are expressed asfuzzy sets the fluctuating ranges of these inputs would bevaried under different plausibilities (120572-cut levels) and thusresult in variations in the generated solutions For exampleunder 120572 = 0 (the lowest plausibility degree) the amount ofwaste allocated to the landfill from city 1 [denoted as (119883111)

plusmn120572]

would be [5 55] tonneday in comparison under 120572 = 1

(the highest plausibility degree) this waste flow would be 30tonneday As the value of 120572-cut level increases from 0 to 1the lower bound of (119883111)

plusmn120572 would also increase (ie 125 175

225 and 2625 tonneday under 120572 = 03 05 07 and 085resp) while the upper bound of (119883111)

plusmn120572 would decrease (ie

475 425 375 and 3375 tonneday under 120572 = 03 05 07and 085 resp) Figures 2 and 3 show the lower and upperbounds of waste-flow patterns under different 120572-cut levelsThey indicate that solutions of waste diversion schemes fromthree cities would vary as the variation in 120572-cut levels Thelower bound would increase and the upper bound woulddecrease when the 120572-cut level increases from 0 to 1 Suchvariations in waste-flow patterns would stem from the inputfuzziness of model (10a)ndash(10i)

Multiple capacity expanding options are considered inresponse to fuzzy characteristics of the input parametersTable 4 shows capacity expanding options for the landfillDifferent capacity options would be applied under differentplausibilities (120572-cut levels) In detail the landfill would beexpanded in period 1 under 120572 = 0 When 120572 = 0 model(10a)ndash(10i) would consider all possible values of the wastegeneration rates as a result the landfill would be expandedin period 1 to tackle the variations in waste generationrates As the 120572-cut level increases uncertainties of the inputswould decrease leading to adaptation of expanding optionfor the landfill When 120572 = 03 landfill expansion would beapplied in period 1 under demanding conditions (ie the

X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0

The lower bounds of waste flow under different 120572-cut values

Figure 3 The lower bounds of waste flows under different 120572-cutlevels

capacity of landfill achieves its lower bound and the wastegeneration rates reach their upper bounds) This means thatthe existing capacity of landfill would be sufficient to disposeof the solid waste under advantageous conditions (ie thecapacity of landfill achieves its upper bound and the wastegeneration rates reach their lower bounds) However when120572-cut level increases to 05 the landfill would require anexpansion in period 2 under advantageous conditions due tothe increase in the lower bounds of waste generation rates andthe decrease in the upper bound of landfill capacity Under120572 = 07 expansion option of the landfill would be similarto that under 120572 = 03 except the option being applied inperiod 2 As shown in Table 4 when 120572 ge 085 the landfillwould be expanded in period 2 under both demandingand advantageous conditions In brief the results in Table 4suggest that (i) expansion of the landfill in period 1 leads tolow risk but high system cost and (ii) the landfill at least isexpanded in period 2

As shown in Table 5 the WTE facility would generallybe expanded in periods 1 and 2 In period 1 option 1 (ie150 tonneday) would be applied for WTE regardless ofthe impacts of uncertain inputs However in period 2 thecapacity expanding options would be influenced significantlyby system uncertainty The WTE facility would be expandedwith different options under different plausibilities in thisperiod When 120572 = 0 option 1 is considered to be suitablefor WTE under advantageous conditions while option 3is chosen under demanding conditions As the 120572-cut levelvaries between 03 and 07 options 1 and 2 are applicablefor WTE under advantageous and demanding conditionsFurthermore as shown in Table 5 the WTE facility would atleast be expanded with option 2 in period 2

Based on the waste-flow diversion schemes underselected 120572-cut levels (as shown in Table 3) we can approxi-mate their membership functions through statistical regres-sion Figure 4 shows themembership functions for the waste-flow schemes It indicates that these membership functionscan be well fitted based on the results in Table 3 However


for the variables indicating facility expansion options itcan hardly obtain their membership function since they arebinary variables But decisionmakers can still generate appro-priate waste-flow diversion schemes and facility expansionoptions based on Figure 4 and Tables 4 and 5 For exampleif a decision maker wants to identify the waste managementpolicy under an 120572-cut level of 06 the waste-flow patterns canbe obtained through the membership functions in Figure 4(eg (119883111)

minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus

06)04 = 40) the facility expansion options can be adoptedbased on results under 120572 = 05 and 07 in Tables 4 and 5

Table 2 also provides the total system costs (denoted asthe objective function) obtained from the GFIP model undersix 120572-cut levelsThe results suggest that different plausibilitiesof uncertain inputs lead to varied system costs The lowerbounds of the objective function correspond to advantageousconditions while the upper bounds are associated withdemanding conditions When 120572 = 0 the system costwould be $[609 802] times 10

9 Under 120572 = 03 the lowerbound of system cost would be $618 times 10

9 and the upperbound would be $727 times 109 leading to a fluctuating intervalof $[618 727] times 10

9 for system cost However the valueof the objective function does not necessarily hold such acharacteristic that as the 120572-cut level increases the lowerbound increases and the upper bound decreases (as shownin Figure 5) For example the lower bound of the objectivefunction under 120572 = 05 and 07 is $624 times 10

9 and $38 times109 respectively This is due to the variation in capacity-

expanding options under these two 120572-cut levels When120572 = 05 capacity-expanding for landfill is considered underboth demanding and advantageous conditions in contrastthe landfill would not be expanded under advantageousconditionswhen120572 = 07 which leads to a significant decreaseof the objective function value

Generally speaking the conventional inexact mixed-integer linear programming (IMILP)methodwould consideruncertain parameters with known lower and upper boundsFor the IMILP method it cannot consider any distributionalinformation between the lower and upper bounds In thisstudy the solutions of GFIP model under 120572 = 0 areidentical to the solutions obtained from the IMILP modelAs shown in Tables 3 to 5 the solutions of IMILP modelcan only provide interval values for waste allocation schemesand neglect distributional information within these intervalsParticularly when distributional information is available foruncertain inputs the IMILP method can hardly reflect therelationships between uncertainty of the inputs and theresulting solutions Conversely the GFIP method can notonly provide fluctuating intervals for waste-flow allocationschemes (ie lower and upper bounds) as well as correspond-ing capacity expanding options for waste-treatment facilitiesbut also afford plausibilities for such waste managementalternatives

5 Conclusions

In this study a generalized fuzzy integer programming(GFIP) method was developed for solid waste management

under uncertainty The developed GFIP could deal withuncertainties expressed as fuzzy sets that exist in the con-straintsrsquo left- and right-sides and the objective function Astepwise interactive algorithm (SIA) was proposed to solvethe GFIP model and generate solutions expressed as fuzzysets The SIA firstly discretized the membership functiongrade into a set of 120572-cut levels Then the GFIP model wasconverted into a series of IMILP submodels The interactivealgorithm proposed by Huang et al [3] was applied tosolve the IMILP submodels and generate interval solutionsunder each 120572-cut level The membership functions for fuzzycontinuous variables were finally obtained through statisticalregression method based on those interval solutions

The developedmethod was applied to a case of municipalsolid waste management to illustrate the applicability ofGFIP methodThe solutions for binary variables which wereobtained as discrete fuzzy sets provided different capacityexpansion alternatives for waste-treatment facilities underdifferent plausibilities of uncertain inputs The solutions forcontinuous variables which were expressed as fuzzy sets withknown membership functions provided optimal schemesfor waste-flow allocations These solutions were used forgenerating decision alternatives and thus helping decisionmakers to identify desired policies for MSW managementunder uncertainty Although the case study in this study isjust a hypothetical one this case involved the main factors(eg transportation operation and capacity expansion costs)in MSW management Consequently the developed GFIPmethod can also be applied to real-world MSWmanagementproblems Through the hypothetical (MSW) case study thefollowing advantages of the GFIP approach were presented(i) GFIP could deal with fuzzy parameters with any kind ofmembership function shape (ii) the solution process of GFIPwould not generate complicated intermediate submodels (iii)the computational requirement was reasonable and (iv) thesolutions of GFIP exhibited some distributional informationwhich was helpful for decisionmakingThe above advantagesindicated the usefulness and applicability of the developedGFIP approach in real-world MSWmanagement problem

Compared with the inexact mixed-integer linear pro-gramming (IMILP) method uncertainties presented as fuzzysets were incorporated within the GFIPrsquos optimization pro-cesses Solutions obtained fromGFIPmethod specified somedistributional information which contained not only thelower and upper bounds with associated plausibility degreesbut also the most possible values

The developed GFIP could deal with various fuzzy setswith known membership functions regardless of the shapesof these functions However it had difficulties in dealingwith other kinds of uncertainties expressed as probabilisticdistributions Therefore improvement for GFIP was furtherdesired to enhance its capability of dealing with multi-ple uncertainties through incorporating interval program-ming or stochastic programming into the GFIP frameworkBesides the developed GFIP method cannot be used forevaluating the detailed effects of interactions amongmultipleuncertain inputs Consequently the factorial analysismethodcould then be integrated into the GFIP framework to addressthe above issue


1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583

L(x) = 004x minus 4E minus 16

R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50

Lower boundLower bound

Upper boundUpper bound



250 260 270 280 290 300 310

L(x) = 004x minus 10

R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)

Figure 4 The membership functions for fuzzy variables

Table 3 Solutions for the GFIP under each 120572-cut level

Waste allocation 120572-cut level0 03 05 07 085 1

(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix

Definition A1 (fuzzy set) Let 119883 denote a universal set Thena fuzzy set 119860 in119883 can be defined by a membership functionas follows

120583119860 119883 997888rarr [0 1] (A1)

For each 119909 isin 119883 the value of 120583119860(119909) shows the grade (ordegree) of membership of the element 119909 of119883 in fuzzy set 119860

DefinitionA2 (120572-cut) Given a fuzzy set119860defined on119883 and aparticular number 120572 in the unit interval [0 1] the 120572-cut of119860

denoted as [119860]120572 is a crisp set that consists of all elements of119883 whose membership degrees in 119860 are greater than or equalto 120572

[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)

Definition A3 (fuzzy number) A fuzzy number is a convexnormalized fuzzy set in the real number whose membershipfunction is piecewise continuous


Table 4 Capacity expanding options for the landfill under each 120572-cut level

Expanding options 120572-cut level0 03 05 07 085 1

(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0

Table 5 Capacity expanding options for the WTE facility under each 120572-cut level


(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583

Lower boundUpper bound

f(times109)

Figure 5The objective function values under different 120572-cut levels

Definition A4 (L-R fuzzy numbers) A fuzzy number119872 is aso-called 119871-119877 fuzzy 119872 = (119898 120572 120573)119871119877 if the correspondingmembership function satisfies for all 119909 isin 119877

120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)

where 119898 is the mean value of 119872 120572 gt 0 and 120573 gt 0 areleft and right spread respectively 119871 and 119877 are called theleft and right shape function respectively which are strictlydecreasing continuous functions from [0 1] to [0 1] such that119871(0) = 119877(0) = 1 and 119871(1) = 119877(1) = 0 If 119871(119909) and 119877(119909) arelinear functions then the corresponding 119871-119877 fuzzy numberis considered as a triangular fuzzy number

Definition A5 Let 1198650(119877) denote the set of all fuzzy numbersin 119877 For any 119886 isin 1198650(119877) an 120572-cut of 119886 can be expressed as aclosed interval

119886plusmn120572 = [119886

minus120572 119886+120572 ] for any 120572 isin [0 1] 119886

minus120572 le 119886+120572 (A4)

Remark A6 For any 119886 isin 1198650(119877) suppose two 120572-cut levels 12057211205722 are selected to cut 119886 then these two 120572-cuts can beformulated as

119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have

119886plusmn1205721sube 119886plusmn1205722 namely 119886minus1205721 ge 119886

minus1205722 119886+1205721le 119886+1205722 (A6)

Definition A7 For 119886plusmn120572 = [119886minus120572 119886+120572 ] and 119887

plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]

(2) 119886plusmn120572 minus 119887plusmn120572 = [119886

minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886

minus120572 minus 119887+120572 119886+120572 minus 119887minus120572 ]

(3) 119886plusmn120572 sdot 119887plusmn120572 = [119886

minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886

minus120572119887minus120572 and 119886minus120572119887+120572 and 119886+120572119887minus120572 and

119886+120572119887+120572 119886minus120572119887minus120572 or 119886minus120572119887+120572 or 119886+120572119887minus120572 or 119886+120572119887+120572 ]

(4) The order relation ldquolerdquo is defined by

[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886

minus120572 le 119887minus120572 119886+120572 le 119887+120572 (A7)

(5) Let [119886minus120572119894 119886+120572119894] sub 119877 119894 isin 119868 119868 is the index set then

and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and

119894isin119868119886minus120572119894gt minusinfin

or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)

Definition A8 Let 119886119894 | 119894 isin 119868 sub 1198650(119877) 120572 isin [0 1] then


(1) 119891 = and119894isin119868119886119894 is defined by a fuzzy number 119886119894 isin 1198650(119877)

such that 119891120572 = and119894isin119868(119886119894)120572(2) 119892 = or119894isin119868119886119894 is defined by a fuzzy number 119886119894 isin 1198650(119877)

such that 119892120572 = or119894isin119868(119886119894)120572

Definition A9 Let 119886 119887 isin 1198650(119877) Then for any 120572 isin (0 1] wehave

(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)

where lowastmay be any continuous algebraic operation

Conflict of Interests

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

Acknowledgments

This research was supported by the Major Project Programof the Natural Sciences Foundation (51190095) the Programfor Innovative Research Team in University (IRT1127) andthe MOE Key Project Program (311013) The authors areextremely grateful to the editor and the anonymous reviewersfor their insightful comments and suggestions

References

[1] Y P Li and G H Huang ldquoDynamic analysis for solid wastemanagement systems An inexact multistage integer program-ming approachrdquo Journal of the Air amp Waste ManagementAssociation vol 59 no 3 pp 279ndash292 2009

[2] B W Baetz ldquoCapacity planning for waste management sys-temsrdquo Civil Engineering Systems vol 7 pp 229ndash235 1990

[3] G H Huang B W Baetz and G G Patry ldquoGrey integerprogramming an application to waste management planningunder uncertaintyrdquo European Journal of Operational Researchvol 83 no 3 pp 594ndash620 1995

[4] E Davila N Chang and S Diwakaruni ldquoLandfill space con-sumption dynamics in the Lower Rio Grande Valley by greyinteger programming-based gamesrdquo Journal of EnvironmentalManagement vol 75 no 4 pp 353ndash365 2005

[5] Y R Fan G H Huang P Guo and A L Yang ldquoInexacttwo-stage stochastic partial programming application to waterresources management under uncertaintyrdquo Stochastic Environ-mental Research and Risk Assessment vol 26 no 2 pp 281ndash2932012

[6] Q Hu G H Huang Y P Cai andW Sun ldquoPlanning of electricpower generation systems under multiple uncertainties andconstraint-violation levelsrdquo Journal of Environmental Informat-ics vol 23 no 1 pp 55ndash64 2014

[7] Y P Li and G H Huang ldquoAn inexact two-stage mixed integerlinear programmingmethod for solid wastemanagement in theCity of Reginardquo Journal of Environmental Management vol 81no 3 pp 188ndash209 2006

[8] M Delgado J L Verdegay andM A Vila ldquoA general model forfuzzy linear programmingrdquo Fuzzy Sets and Systems vol 29 no1 pp 21ndash29 1989

[9] M Delgado F Herrera J L Verdegay and M A Vila ldquoPost-optimality analysis on the membership functions of a fuzzy

linear programming problemrdquo Fuzzy Sets and Systems vol 53no 3 pp 289ndash297 1993

[10] H Rommelfanger ldquoFuzzy linear programming and applica-tionsrdquo European Journal of Operational Research vol 92 no 3pp 512ndash527 1996

[11] H RMalekiM Tata andMMashinchi ldquoLinear programmingwith fuzzy variablesrdquo Fuzzy Sets and Systems vol 109 no 1 pp21ndash33 2000

[12] Y R Fan G H Huang Y P Li M F Cao and G H Cheng ldquoAfuzzy linear programming approach for municipal solid-wastemanagement under uncertaintyrdquo Engineering Optimization vol41 no 12 pp 1081ndash1101 2009

[13] A K Srivastava and A K Nema ldquoFuzzy parametric pro-grammingmodel for integrated solid waste management underuncertaintyrdquo Journal of Environmental Engineering vol 137 no1 pp 69ndash83 2011

[14] A K Srivastava and A K Nema ldquoFuzzy parametric program-ming model for multi-objective integrated solid waste manage-ment under uncertaintyrdquo Expert Systems with Applications vol39 no 5 pp 4657ndash4678 2012

[15] T Y Xu andX SQin ldquoSolvingwater qualitymanagement prob-lem through combined genetic algorithmand fuzzy simulationrdquoJournal of Environmental Informatics vol 22 no 1 pp 39ndash482013

[16] J J Buckley and T Feuring ldquoEvolutionary algorithm solutionto fuzzy problems fuzzy linear programmingrdquo Fuzzy Sets andSystems vol 109 no 1 pp 35ndash53 2000

[17] S M Hashemi M Modarres E Nasrabadi and M MNasrabadi ldquoFully fuzzified linear programming solution anddualityrdquo Journal of Intelligent and Fuzzy Systems vol 17 no 3pp 253ndash261 2006

[18] F Hosseinzadeh Lotfi T Allahviranloo M Alimardani Jond-abeh and L Alizadeh ldquoSolving a full fuzzy linear programmingusing lexicography method and fuzzy approximate solutionrdquoApplied Mathematical Modelling vol 33 no 7 pp 3151ndash31562009

[19] Y Fan G Huang and A Veawab ldquoA generalized fuzzy linearprogramming approach for environmental management prob-lem under uncertaintyrdquo Journal of the Air ampWaste ManagementAssociation vol 62 no 1 pp 72ndash86 2012

[20] Y R Fan G H Huang and A L Yang ldquoGeneralized fuzzylinear programming for decision making under uncertaintyfeasibility of fuzzy solutions and solving approachrdquo InformationSciences vol 241 pp 12ndash27 2013

[21] A Kumar J Kaur and P Singh ldquoA newmethod for solving fullyfuzzy linear programming problemsrdquo Applied MathematicalModelling vol 35 no 2 pp 817ndash823 2011

[22] H Zimmermann Fuzzy Set Theorymdashand Its ApplicationsKluwerndashNijhoff Dordrecht The Netherlands 1985

[23] J J Buckley and L J Jowers Monte Carlo Methods in FuzzyOptimization Studies in Fuzziness and SoftComputing SpringerBerlin Germany 2008

[24] Y J Lai and C L Hwang Fuzzy Mathematical ProgrammingSpringer Berlin Germany 1992

[25] A Kaufmann and M Cupta Fuzzy Mathematical Models inEngineering and Many Science North Holland PublishingAmsterdam The Netherlands 1988

[26] E E Ammar ldquoOn solutions of fuzzy random multiobjectivequadratic programming with applications in portfolio prob-lemrdquo Information Sciences vol 178 no 2 pp 468ndash484 2008


[27] M Dehghan B Hashemi and M Ghatee ldquoComputationalmethods for solving fully fuzzy linear systemsrdquo Applied Mathe-matics and Computation vol 179 no 1 pp 328ndash343 2006

[28] M Dehghan M Ghatee and B Hashemi ldquoInverse of a fuzzymatrix of fuzzy numbersrdquo International Journal of ComputerMathematics vol 86 no 8 pp 1433ndash1452 2009

[29] V Kreinovich ldquoMembership functions or 120572-Cuts Algorithmic(constructivist) analysis justifies an interval approachrdquo Mathe-matical Problems of Computer Science vol 38 pp 70ndash71 2012

[30] R Zou Y Liu L Liu and H Guo ldquoREILP approach foruncertainty-based decision making in civil engineeringrdquo Jour-nal of Computing in Civil Engineering vol 24 no 4 pp 357ndash3642010

[31] US Environmental Protection Agency (USEPA) ldquoMunicipalSolid Waste in the United States 2009 Facts and Figuresrdquo 2009

[32] Q Huang Q Wang L Dong B Xi and B Zhou ldquoThe currentsituation of solid waste management in Chinardquo Journal ofMaterial Cycles and Waste Management vol 8 no 1 pp 63ndash692006

[33] D C Wilson ldquoLong-term planning for solid waste manage-mentrdquoWasteManagement amp Research vol 3 no 1 pp 203ndash2161985

[34] N van Hop ldquoSolving fuzzy (stochastic) linear programmingproblems using superiority and inferiority measuresrdquo Informa-tion Sciences vol 177 no 9 pp 1977ndash1991 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


interval linear programming (ILP) two-stage stochastic pro-gramming (TSP) and mixed-integer programming (MIP)within a general mathematical programming frameworkHowever IMP could merely deal with interval uncertaintieswithout distributional information SMP was inapplicable tolarge-scale problems due to its stringent requirements forinformation of probabilistic distributions

Fuzzy mathematical programming (FMP) as a branchof fuzzy set theory could generally deal with uncertaintiesexpressed as fuzzy sets or fuzzy goalsconstraints [8ndash11]Recently various FMP methods were employed to deal withuncertainties in MSW systems [12ndash15] For example Fan etal [12] explored a fuzzy linear programming (FLP) methodfor dealing with uncertainties expressed as fuzzy sets thatexist in the constraintsrsquo left-hand and right-hand sides andthe objective function however this method was unable toreflect dynamic complexities related to capacity expansionschemes for waste-treatment facilities Srivastava and Nema[13] proposed a fuzzy parametric programming model foridentifying desired treatmentdisposal facilities planningwaste management capacities and allocating waste flowsunder uncertainty however the proposed method generateddeterministic waste allocation schemes without provision ofbases for supporting generation of multiple decision optionscorresponding to the uncertain system conditions

Generalized fuzzy linear programming (GFLP) (or fullyfuzzy linear programming (FFLP)) methods extended tradi-tional FMP approaches through permitting uncertain infor-mation in the optimization process and resulting solutionsRecently several GFLP (or FFLP) methods were proposedto deal with uncertain information in both parameters anddecision variables [16ndash21] For example Hosseinzadeh Lotfiet al [18] developed a lexicography method to solve the FFLPproblem and generate approximate solutions presented asfuzzy sets However previous studies on fuzzy variables inFMP problemsmainly focused on some special types of fuzzysets (such as symmetric triangular or trapezoidal fuzzy sets)Furthermore some of them might lead to complicated inter-mediate models and thus were not applicable for large-scaleproblems Fan et al [19 20] proposed another kind of fuzzyprogramming (named generalized fuzzy linear programming(GFLP) method) to deal with fuzzy uncertainty in bothparameters and variables in which all fuzzy sets with knownmembership functions can be treated through defuzzificationmethod (ie 120572-cut method) However the GFLP approachcannot reflect dynamic features in environmental manage-ment problemsMoreover no previous studywas reported oncapacity expansion issues under fuzzy conditions through thegeneralized fuzzy optimization approach where expansionschemes were desired under multiple scenarios and 120572-cutlevels

As an extension of developed GFLP approach a gen-eralized fuzzy integer programming (GFIP) method wouldbe proposed for MSW management under uncertainty Theproposed GFIP approach integrates the techniques of gener-alized fuzzy linear programming (GFLP) and mixed-integerprogramming (MIP) within an optimization framework Indetail (i) the GFIP method can deal with uncertaintiesexpressed as fuzzy sets with known membership functions

regardless of whether these functions are linear or nonlinear(ii) the proposed GFIP method can allow uncertainties tobe directly communicated into the optimization process andthe resulting solutions (iii) the GFIP method can reflectdynamics in terms of waste-flow allocation and facility-capacity expansion (iv) compared with other inexact mixed-integer programming approaches (eg ITSMILP by Li andHuang [7]) the GFIP can analyze the inherent interrelation-ship between the uncertainty of fuzzy parameters (ie 120572-cutlevels) and capacity expansion options of waste managementfacilities and such analysis can help decision makers maketradeoffs between system reliability and system cost Thena case study will be provided to demonstrate applicabilityof the GFIP method to support dynamic analysis for MSWmanagement under uncertainty The results will be usedfor generating different decision alternatives under varioussystem conditions and thus for helping identify desired wastemanagement policies

2 Methodology

21 Formulation of the Generalized Fuzzy Integer Program-ming A GFIP model with ambiguous coefficients and deci-sion variables expressed as fuzzy sets can be formulated asfollows

Max 119891 = 119888 times 119883 (1a)

subject to

119860 times 119883 le119887 (1b)

119883 ge 0 (1c)

119909119895 = fuzzy continuous variables 119909119895 isin 119883

119895 = 1 2 119901 (119901 lt 119899)

(1d)

119909119895 = fuzzy integer variables 119909119895 isin 119883

119895 = 119901 + 1 119901 + 2 119899

(1e)

where 119888 isin 1times119899119883 isin 119899times1 119887 isin 119898times1 and 119860 isin

119898times119899 denotes a set of fuzzy sets 119888 = (1198881 1198882 119888119899) 119883

119879=

(1199091 1199092 119909119899) 119887119879= (

1198871

1198872

119887119899) and119860 = (119886119894119895)119898times119899 for

all 119894 isin 119898 119895 isin 119899 A fuzzy set (119860) in 119883 can be defined as119909 120583119860(119909) | 119909 isin 119883 120583119860(119909) 119883 rarr [0 1] where 120583119860(119909) isthe membership function or grade of membership [22] If allelements in119860 are integers and120583119860(119909) is a discretemembershipfunction then119860 is a fuzzy integer set [23]The value of 120583119860(119909)varies between 0 and 1 indicating the possibility of an element119909 belonging to 119860 120583119860(119909) = 1 means that 119909 definitely belongsto the fuzzy set (119860) while 120583119860(119909) = 0 denotes that 119909 does notbelong to 119860 The closer 120583119860(119909) is to 1 the more likely that 119909belongs to119860 conversely the closer 120583119860(119909) is to 0 the less likelythat 119909 belongs to 119860 [22 24] An 120572-cut of 119860 can be defined asan ordinary set (denoted as [119860]120572) in which the membershipdegrees of elements exceed 120572 [119860]120572 is usually a continuous or







minus120572119894 (119888119895)+120572119894] (119909119895)120572119894 = [(119909119895)

minus120572119894 (119909119895)+120572119894] (119886119894119895)120572119894 =

[(119886119894119895)minus120572119894 (119886119894119895)+120572119894] and (119887119894)120572119894 = [(119887119894)

minus120572119894 (119887119894)+120572119894]


Max (119891)plusmn

120572(1)=

119899

sum

119895=1

(119888119895)plusmn

120572(1)times (119909119895)

plusmn

120572(1)(2a)

subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(1)times (119909119895)

plusmn

120572(1)le (119887119894)

plusmn

120572(1)119894 = 1 2 119898 (2b)

(119909119895)plusmn

120572(1)ge 0 119895 = 1 2 119898 (2c)

(119909119895)plusmn


plusmn

120572(1)isin (119883)

plusmn120572(1)

= 1 2 119901 (119901 lt 119899)

(2d)

(119909119895)plusmn


plusmn

120572(1)isin (119883)

plusmn120572(1)

= 119901 + 1 119901 + 1 119899

(2e)

where (119891)plusmn120572(1)

(119888119895)plusmn120572(1)

(119909119895)plusmn120572(1)

(119886119894119895)plusmn120572(1)

and (119887119894)plusmn120572(1)


plusmn120572(1)

= [(119888119895)minus120572(1) (119888119895)+120572(1)] (119909119895)

plusmn120572(1)

=

[(119909119895)minus120572(1) (119909119895)+120572(1)] (119886119894119895)

plusmn120572(1)

= [(119886119894119895)minus120572(1) (119886119894119895)+120572(1)] and (119887119894)

plusmn120572(1)

=



minus 119886+] = 119905 | 119886

minusle 119905 le 119886

+





Max (119891)+

120572(1)=

1198961

sum

119895=1

(119888119895)+

120572(1)(119909119895)+

120572(1)+

119899

sum

119895=1198961+1

(119888119895)+

120572(1)(119909119895)minus

120572(1)

(3a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

le (119887119894)+

120572(1) forall119894

(3b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(3c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(3d)

(119909119895)plusmn

120572(1)ge 0 forall119895 (3e)


minus120572(1)(119895 =


minus120572(1)


Max (119891)minus

120572(1)=

119896

sum

119895=1

(119888119895)minus

120572(1)(119909119895)minus

120572(1)+

119899

sum

119895=119896+1

(119888119895)minus

120572(1)(119909119895)+

120572(1)

(4a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

le (119887119894)minus

120572(1) for all 119894

(4b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(4c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(4d)

(119909119895)minus

120572(1)le (119909119895opt)

+

120572(1) 119895 = 1 2 1198961 (4e)

(119909119895)+

120572(1)ge (119909119895opt)

minus

120572(1) 119895 = 1198961 + 1 1198961 + 2 119899 (4f)

(119909119895)plusmn

120572(1)ge 0 forall119895 (4g)




(119909119895opt)plusmn

120572(1)= [(119909119895opt)

minus

120572(1)(119909119895opt)

+

120572(1)] (5a)

(119891opt)plusmn

120572(1)= [(119891opt)

minus

120572(1)(119891opt)

+

120572(1)] (5b)



Max (119891)plusmn

120572(119897)=

119899

sum

119895=1

(119888119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)(6a)


subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)le (119887119894)

plusmn

120572(119897) for 119894 = 1 2 119898 (6b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(6c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(6d)

(119909119895)plusmn

120572(119897)sube (119909119895opt)

plusmn

120572(119897minus1)(6e)

(119909119895)plusmn

120572(1)ge 0 forall119895 (6f)




plusmn1205721

supe (119909119895)plusmn1205722



Submodel 1

Max (119891)+

120572(119897)=

1198961

sum

119895=1

(119888119895)+

120572(119897)(119909119895)+

120572(119897)+

119899

sum

119895=1198961+1

(119888119895)+

120572(119897)(119909119895)minus

120572(119897)

(7a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

le (119887119894)+

120572(119897) forall119894

(7b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(7c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(7d)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1 2 1198961 (7e)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (7f)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (7g)

Submodel 2

Max (119891)minus

120572(119897)=

119896

sum

119895=1

(119888119895)minus

120572(119897)(119909119895)minus

120572(119897)+

119899

sum

119895=119896+1

(119888119895)minus

120572(119897)(119909119895)+

120572(119897)(8a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

le (119887119894)minus

120572(119897) forall119894

(8b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(8c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(8d)

(119909119895)minus

120572(119897)le (119909119895opt)

+

120572(119897) 119895 = 1 2 1198961 (8e)

(119909119895)+

120572(119897)ge (119909119895opt)

minus

120572(119897) 119895 = 1198961 + 1 1198961 + 2 119899 (8f)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1 2 1198961 (8g)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (8h)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (8i)



(119909119895opt)plusmn

120572(119897)= [(119909119895opt)

minus

120572(119897)(119909119895opt)

+

120572(119897)] (9a)

(119891opt)plusmn

120572(119897)= [(119891opt)

minus

120572(119897)(119891opt)

+

120572(119897)] (9b)





3 Case Study




6 02 times 10

5 03 times 10


6 and183 times 10




Min 119891 = 1825

3

sum

119895=1

3

sum

119896=1

2

sum

119894=1

119909119894119895119896 (TR119894119895119896 +OP119894119896) + 1199092119895119896

times [FE (FT119896 +OP1119896)

minusRE119896]

+

3

sum

119896=1

FLC119896119896 +3

sum

119897=1

3

sum

119896=1

FTC119897119896119885119897119896

(10a)

subject to



Time period119896 = 1 119896 = 2 119896 = 3



City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)



City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)



Data Time period119896 = 1 119896 = 2 119896 = 3




ΔTC (031 001 0015) (031 001 0015) (031 001 0015)



FLC119896 (14 1 1) (14 1 1) (14 1 1)


1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)


3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)


sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)



119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















minus120572119894 (119888119895)+120572119894] (119909119895)120572119894 = [(119909119895)

minus120572119894 (119909119895)+120572119894] (119886119894119895)120572119894 =

[(119886119894119895)minus120572119894 (119886119894119895)+120572119894] and (119887119894)120572119894 = [(119887119894)

minus120572119894 (119887119894)+120572119894]


Max (119891)plusmn

120572(1)=

119899

sum

119895=1

(119888119895)plusmn

120572(1)times (119909119895)

plusmn

120572(1)(2a)

subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(1)times (119909119895)

plusmn

120572(1)le (119887119894)

plusmn

120572(1)119894 = 1 2 119898 (2b)

(119909119895)plusmn

120572(1)ge 0 119895 = 1 2 119898 (2c)

(119909119895)plusmn


plusmn

120572(1)isin (119883)

plusmn120572(1)

= 1 2 119901 (119901 lt 119899)

(2d)

(119909119895)plusmn


plusmn

120572(1)isin (119883)

plusmn120572(1)

= 119901 + 1 119901 + 1 119899

(2e)

where (119891)plusmn120572(1)

(119888119895)plusmn120572(1)

(119909119895)plusmn120572(1)

(119886119894119895)plusmn120572(1)

and (119887119894)plusmn120572(1)


plusmn120572(1)

= [(119888119895)minus120572(1) (119888119895)+120572(1)] (119909119895)

plusmn120572(1)

=

[(119909119895)minus120572(1) (119909119895)+120572(1)] (119886119894119895)

plusmn120572(1)

= [(119886119894119895)minus120572(1) (119886119894119895)+120572(1)] and (119887119894)

plusmn120572(1)

=



minus 119886+] = 119905 | 119886

minusle 119905 le 119886

+





Max (119891)+

120572(1)=

1198961

sum

119895=1

(119888119895)+

120572(1)(119909119895)+

120572(1)+

119899

sum

119895=1198961+1

(119888119895)+

120572(1)(119909119895)minus

120572(1)

(3a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

le (119887119894)+

120572(1) forall119894

(3b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(3c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(3d)

(119909119895)plusmn

120572(1)ge 0 forall119895 (3e)


minus120572(1)(119895 =


minus120572(1)


Max (119891)minus

120572(1)=

119896

sum

119895=1

(119888119895)minus

120572(1)(119909119895)minus

120572(1)+

119899

sum

119895=119896+1

(119888119895)minus

120572(1)(119909119895)+

120572(1)

(4a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

le (119887119894)minus

120572(1) for all 119894

(4b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(4c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(4d)

(119909119895)minus

120572(1)le (119909119895opt)

+

120572(1) 119895 = 1 2 1198961 (4e)

(119909119895)+

120572(1)ge (119909119895opt)

minus

120572(1) 119895 = 1198961 + 1 1198961 + 2 119899 (4f)

(119909119895)plusmn

120572(1)ge 0 forall119895 (4g)




(119909119895opt)plusmn

120572(1)= [(119909119895opt)

minus

120572(1)(119909119895opt)

+

120572(1)] (5a)

(119891opt)plusmn

120572(1)= [(119891opt)

minus

120572(1)(119891opt)

+

120572(1)] (5b)



Max (119891)plusmn

120572(119897)=

119899

sum

119895=1

(119888119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)(6a)


subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)le (119887119894)

plusmn

120572(119897) for 119894 = 1 2 119898 (6b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(6c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(6d)

(119909119895)plusmn

120572(119897)sube (119909119895opt)

plusmn

120572(119897minus1)(6e)

(119909119895)plusmn

120572(1)ge 0 forall119895 (6f)




plusmn1205721

supe (119909119895)plusmn1205722



Submodel 1

Max (119891)+

120572(119897)=

1198961

sum

119895=1

(119888119895)+

120572(119897)(119909119895)+

120572(119897)+

119899

sum

119895=1198961+1

(119888119895)+

120572(119897)(119909119895)minus

120572(119897)

(7a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

le (119887119894)+

120572(119897) forall119894

(7b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(7c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(7d)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1 2 1198961 (7e)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (7f)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (7g)

Submodel 2

Max (119891)minus

120572(119897)=

119896

sum

119895=1

(119888119895)minus

120572(119897)(119909119895)minus

120572(119897)+

119899

sum

119895=119896+1

(119888119895)minus

120572(119897)(119909119895)+

120572(119897)(8a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

le (119887119894)minus

120572(119897) forall119894

(8b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(8c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(8d)

(119909119895)minus

120572(119897)le (119909119895opt)

+

120572(119897) 119895 = 1 2 1198961 (8e)

(119909119895)+

120572(119897)ge (119909119895opt)

minus

120572(119897) 119895 = 1198961 + 1 1198961 + 2 119899 (8f)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1 2 1198961 (8g)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (8h)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (8i)



(119909119895opt)plusmn

120572(119897)= [(119909119895opt)

minus

120572(119897)(119909119895opt)

+

120572(119897)] (9a)

(119891opt)plusmn

120572(119897)= [(119891opt)

minus

120572(119897)(119891opt)

+

120572(119897)] (9b)





3 Case Study




6 02 times 10

5 03 times 10


6 and183 times 10




Min 119891 = 1825

3

sum

119895=1

3

sum

119896=1

2

sum

119894=1

119909119894119895119896 (TR119894119895119896 +OP119894119896) + 1199092119895119896

times [FE (FT119896 +OP1119896)

minusRE119896]

+

3

sum

119896=1

FLC119896119896 +3

sum

119897=1

3

sum

119896=1

FTC119897119896119885119897119896

(10a)

subject to



Time period119896 = 1 119896 = 2 119896 = 3



City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)



City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)



Data Time period119896 = 1 119896 = 2 119896 = 3




ΔTC (031 001 0015) (031 001 0015) (031 001 0015)



FLC119896 (14 1 1) (14 1 1) (14 1 1)


1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)


3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)


sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)



119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Max (119891)+

120572(1)=

1198961

sum

119895=1

(119888119895)+

120572(1)(119909119895)+

120572(1)+

119899

sum

119895=1198961+1

(119888119895)+

120572(1)(119909119895)minus

120572(1)

(3a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

le (119887119894)+

120572(1) forall119894

(3b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(3c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(3d)

(119909119895)plusmn

120572(1)ge 0 forall119895 (3e)


minus120572(1)(119895 =


minus120572(1)


Max (119891)minus

120572(1)=

119896

sum

119895=1

(119888119895)minus

120572(1)(119909119895)minus

120572(1)+

119899

sum

119895=119896+1

(119888119895)minus

120572(1)(119909119895)+

120572(1)

(4a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(1)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(1))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(1)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(1)

le (119887119894)minus

120572(1) for all 119894

(4b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(4c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(4d)

(119909119895)minus

120572(1)le (119909119895opt)

+

120572(1) 119895 = 1 2 1198961 (4e)

(119909119895)+

120572(1)ge (119909119895opt)

minus

120572(1) 119895 = 1198961 + 1 1198961 + 2 119899 (4f)

(119909119895)plusmn

120572(1)ge 0 forall119895 (4g)




(119909119895opt)plusmn

120572(1)= [(119909119895opt)

minus

120572(1)(119909119895opt)

+

120572(1)] (5a)

(119891opt)plusmn

120572(1)= [(119891opt)

minus

120572(1)(119891opt)

+

120572(1)] (5b)



Max (119891)plusmn

120572(119897)=

119899

sum

119895=1

(119888119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)(6a)


subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)le (119887119894)

plusmn

120572(119897) for 119894 = 1 2 119898 (6b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(6c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(6d)

(119909119895)plusmn

120572(119897)sube (119909119895opt)

plusmn

120572(119897minus1)(6e)

(119909119895)plusmn

120572(1)ge 0 forall119895 (6f)




plusmn1205721

supe (119909119895)plusmn1205722



Submodel 1

Max (119891)+

120572(119897)=

1198961

sum

119895=1

(119888119895)+

120572(119897)(119909119895)+

120572(119897)+

119899

sum

119895=1198961+1

(119888119895)+

120572(119897)(119909119895)minus

120572(119897)

(7a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

le (119887119894)+

120572(119897) forall119894

(7b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(7c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(7d)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1 2 1198961 (7e)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (7f)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (7g)

Submodel 2

Max (119891)minus

120572(119897)=

119896

sum

119895=1

(119888119895)minus

120572(119897)(119909119895)minus

120572(119897)+

119899

sum

119895=119896+1

(119888119895)minus

120572(119897)(119909119895)+

120572(119897)(8a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

le (119887119894)minus

120572(119897) forall119894

(8b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(8c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(8d)

(119909119895)minus

120572(119897)le (119909119895opt)

+

120572(119897) 119895 = 1 2 1198961 (8e)

(119909119895)+

120572(119897)ge (119909119895opt)

minus

120572(119897) 119895 = 1198961 + 1 1198961 + 2 119899 (8f)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1 2 1198961 (8g)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (8h)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (8i)



(119909119895opt)plusmn

120572(119897)= [(119909119895opt)

minus

120572(119897)(119909119895opt)

+

120572(119897)] (9a)

(119891opt)plusmn

120572(119897)= [(119891opt)

minus

120572(119897)(119891opt)

+

120572(119897)] (9b)





3 Case Study




6 02 times 10

5 03 times 10


6 and183 times 10




Min 119891 = 1825

3

sum

119895=1

3

sum

119896=1

2

sum

119894=1

119909119894119895119896 (TR119894119895119896 +OP119894119896) + 1199092119895119896

times [FE (FT119896 +OP1119896)

minusRE119896]

+

3

sum

119896=1

FLC119896119896 +3

sum

119897=1

3

sum

119896=1

FTC119897119896119885119897119896

(10a)

subject to



Time period119896 = 1 119896 = 2 119896 = 3



City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)



City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)



Data Time period119896 = 1 119896 = 2 119896 = 3




ΔTC (031 001 0015) (031 001 0015) (031 001 0015)



FLC119896 (14 1 1) (14 1 1) (14 1 1)


1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)


3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)


sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)



119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










subject to

119899

sum

119895=1

(119886119894119895)plusmn

120572(119897)times (119909119895)

plusmn

120572(119897)le (119887119894)

plusmn

120572(119897) for 119894 = 1 2 119898 (6b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(6c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(6d)

(119909119895)plusmn

120572(119897)sube (119909119895opt)

plusmn

120572(119897minus1)(6e)

(119909119895)plusmn

120572(1)ge 0 forall119895 (6f)




plusmn1205721

supe (119909119895)plusmn1205722



Submodel 1

Max (119891)+

120572(119897)=

1198961

sum

119895=1

(119888119895)+

120572(119897)(119909119895)+

120572(119897)+

119899

sum

119895=1198961+1

(119888119895)+

120572(119897)(119909119895)minus

120572(119897)

(7a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

le (119887119894)+

120572(119897) forall119894

(7b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(7c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(7d)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1 2 1198961 (7e)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (7f)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (7g)

Submodel 2

Max (119891)minus

120572(119897)=

119896

sum

119895=1

(119888119895)minus

120572(119897)(119909119895)minus

120572(119897)+

119899

sum

119895=119896+1

(119888119895)minus

120572(119897)(119909119895)+

120572(119897)(8a)

subject to

119896

sum

119895=1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

+

(119909119895)minus

120572(119897)

+

119899

sum

119895=119896+1

Sign((119886119894119895)plusmn

120572(119897))

1003816100381610038161003816100381610038161003816

(119886119894119895)120572(119897)

1003816100381610038161003816100381610038161003816

minus

(119909119895)+

120572(119897)

le (119887119894)minus

120572(119897) forall119894

(8b)

(119909119895)plusmn


119895 = 1 2 1199011 1198961 + 1 1198961 + 2 1198961 + 1199012

(1199011 le 1198961 1199012 le 1198962 1198961 + 1198962 = 119899)

(8c)

(119909119895)plusmn


119895 = 1199011 + 1 1199011 + 2 1198961

1198961 + 1199012 + 1 1198961 + 1199012 + 2 119899

(8d)

(119909119895)minus

120572(119897)le (119909119895opt)

+

120572(119897) 119895 = 1 2 1198961 (8e)

(119909119895)+

120572(119897)ge (119909119895opt)

minus

120572(119897) 119895 = 1198961 + 1 1198961 + 2 119899 (8f)

(119909119895)minus

120572(119897)ge (119909119895opt)

minus

120572(119897minus1) 119895 = 1 2 1198961 (8g)

(119909119895)+

120572(119897)le (119909119895opt)

+

120572(119897minus1) 119895 = 1198961 + 1 1198961 + 2 119899 (8h)

(119909119895)plusmn

120572(119897)ge 0 forall119895 (8i)



(119909119895opt)plusmn

120572(119897)= [(119909119895opt)

minus

120572(119897)(119909119895opt)

+

120572(119897)] (9a)

(119891opt)plusmn

120572(119897)= [(119891opt)

minus

120572(119897)(119891opt)

+

120572(119897)] (9b)





3 Case Study




6 02 times 10

5 03 times 10


6 and183 times 10




Min 119891 = 1825

3

sum

119895=1

3

sum

119896=1

2

sum

119894=1

119909119894119895119896 (TR119894119895119896 +OP119894119896) + 1199092119895119896

times [FE (FT119896 +OP1119896)

minusRE119896]

+

3

sum

119896=1

FLC119896119896 +3

sum

119897=1

3

sum

119896=1

FTC119897119896119885119897119896

(10a)

subject to



Time period119896 = 1 119896 = 2 119896 = 3



City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)



City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)



Data Time period119896 = 1 119896 = 2 119896 = 3




ΔTC (031 001 0015) (031 001 0015) (031 001 0015)



FLC119896 (14 1 1) (14 1 1) (14 1 1)


1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)


3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)


sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)



119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119909119895opt)plusmn

120572(119897)= [(119909119895opt)

minus

120572(119897)(119909119895opt)

+

120572(119897)] (9a)

(119891opt)plusmn

120572(119897)= [(119891opt)

minus

120572(119897)(119891opt)

+

120572(119897)] (9b)





3 Case Study




6 02 times 10

5 03 times 10


6 and183 times 10




Min 119891 = 1825

3

sum

119895=1

3

sum

119896=1

2

sum

119894=1

119909119894119895119896 (TR119894119895119896 +OP119894119896) + 1199092119895119896

times [FE (FT119896 +OP1119896)

minusRE119896]

+

3

sum

119896=1

FLC119896119896 +3

sum

119897=1

3

sum

119896=1

FTC119897119896119885119897119896

(10a)

subject to



Time period119896 = 1 119896 = 2 119896 = 3



City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)



City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)



Data Time period119896 = 1 119896 = 2 119896 = 3




ΔTC (031 001 0015) (031 001 0015) (031 001 0015)



FLC119896 (14 1 1) (14 1 1) (14 1 1)


1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)


3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)


sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)



119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Time period119896 = 1 119896 = 2 119896 = 3



City 1 (141 2 2) (155 22 22) (17 24 25)City 2 (12 15 2) (13 19 19) (15 22 19)City 3 (151 24 19) (165 25 22) (18 26 26)



City 1 (112 16 16) (123 15 15) (135 18 2)City 2 (118 17 16) (129 18 18) (142 2 2)City 3 (102 14 15) (113 14 15) (126 17 17)



Data Time period119896 = 1 119896 = 2 119896 = 3




ΔTC (031 001 0015) (031 001 0015) (031 001 0015)



FLC119896 (14 1 1) (14 1 1) (14 1 1)


1825

3

sum

119895=1

1198961015840

sum

119896=1

(1199091119895119896 + 1199092119895119896FE) le TL

+ΔTC1198961015840

sum

119896=1

119896 1198961015840= 1 2 3

(10b)


3

sum

119895=1

11990921198951198961015840 leTE +

3

sum

119897=1

1198961015840

sum

119896=1

ΔTCl

119885119897119896 119896

1015840= 1 2 3 (10c)


sum

119894=1

119909119894119895119896 geWG119895119896 forall119895 119896 (10d)



119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119894119895119896 ge 0 forall119894 119895 119896 (10e)


119896 =

le 1

ge 0


(10f)

119885119897119896 =

le 1

ge 0


(10g)


3

sum

119896=1

119896 le 1 (10h)


3

sum

119897=1

119885119897119896 le 1 forall119896 (10i)






4 Result Analysis










120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















120572q)










Step

wise

inte

ract

ive a

lgor

ithm




i = i + 1 i = 1







X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07

120572 = 05120572 = 03

120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0





plusmn120572]








X111

X112

X113

X121

X122

X123

X131

X132

X133

X211

X212

X213

X221

X222

X223

X231

X232

X233

120572 = 1

120572 = 07120572 = 05

120572 = 03120572 = 0

0100200300400500

120572 = 1

120572 = 085

120572 = 085

120572 = 07

120572 = 05

120572 = 03

120572 = 0








minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus06 = (06 + 02)004 = 20 (119883111)

+06 = (22 minus









5 Conclusions







1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

00 10 20 30 40 50 60

X111

L(x) = 004x minus 02R2 = 1

R2 = 1R(x) = minus004x + 22

120583

1

08

06

04

02

0

120583


R2 = 1

R2 = 1

R2 = 1R(x) = minus004x + 2

X112

R(x) = 11607x2 minus 88653x + 16927

0 10 20 30 40 50





250 260 270 280 290 300 310


R2 = 1R2 = 1

R(x) = minus004x + 12006

X113

240

1

08

06

04

02

0

120583


R2 = 1R2 = 1

R(x) = minus004x + 16

X121

340 350 360 370 380 390 400 410

1

08

06

04

02

0

120583













1

08

06

04

02

0

120583


R2 = 1 R2 = 1

R(x) = minus004x + 17

370 380 390 400 410 420 430

X122

1

08

06

04

02

0

120583

390 400 410 420 430 440 450 460

X123


R2 = 1

R2 = 1

R(x) = minus004x + 18

1

08

06

04

02

0

120583

320 330 340 350 360 370 380


R2 = 1 R2 = 1

R(x) = minus004x + 15

X133

R2 = 1


R2 = 1R(x) = minus004x + 13

1

08

06

04

02

0

120583

270 280 290 300 310 320 330

X231

(a)

Figure 4 Continued



R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











R2 = 1 R2 = 1R(x) = minus004x + 14

290 300 310 320 330 340 350 360

X232

1

08

06

04

02

0

120583



(b)




(119883111)plusmn

120572 [5 55] [125 475] [175 425] [225 375] [2625 3375] 30(119883121)

plusmn

120572 [350 400] [3575 3925] [3625 3875] [3675 3825] [37125 37875] 375(119883131)

plusmn

120572 0 0 0 0 0 0(119883112)

plusmn

120572 [0 384] [75 376] [125 375] [175 325] [2125 2875] 25(119883122)

plusmn

120572 [375 425] [3825 4175] [3875 4125] [3925 4075] [39625 40375] 400(119883132)

plusmn

120572 0 0 0 0 0 0(119883113)

plusmn

120572 [250 300] [2575 2925] [2625 2875] [2675 2825] [27125 27875] 275(119883123)

plusmn

120572 [400 450] [4075 4425] [4125 4375] [4175 4325] [42125 42875] 425(119883133)

plusmn

120572 [325 375] [3325 3675] [3375 3625] [3425 3575] [34625 35375] 350(119883211)

plusmn

120572 195 195 195 195 195 195(119883221)

plusmn

120572 0 0 0 0 0 0(119883231)

plusmn

120572 [275 325] [2825 3175] [2875 3125] [2925 3075] [29625 30375] 300(119883212)

plusmn

120572 [225 2366] [225 2299] 225 225 225 225(119883222)

plusmn

120572 0 0 0 0 0 0(119883232)

plusmn

120572 [300 350] [3075 3425] [3125 3375] [3175 3325] [32125 32875] 325(119883213)

plusmn

120572 0 0 0 0 0 0(119883223)

plusmn

120572 0 0 0 0 0 0(119883233)

plusmn

120572 0 0 0 0 0 0(119891)plusmn

120572 (times109) [609 802] [618 727] [624 72] [38 714] [7 71] 705

Appendix


120583119860 119883 997888rarr [0 1] (A1)




[119860]120572= 119909 | 120583119860 (119909) ge 120572 120572 isin [0 1] (A2)





(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(1198841)plusmn

120572 [1 1] [0 1] [0 1] [0 0] 0 0(1198842)plusmn

120572 0 0 [1 0] [0 1] [1 1] 1(1198843)plusmn

120572 0 0 0 0 0 0



(11988511)plusmn

120572 1 1 1 1 1 1(11988521)plusmn

120572 0 0 0 0 0 0(11988531)plusmn

120572 0 0 0 0 0 0(11988512)plusmn

120572 [1 0] [1 0] [1 0] [1 0] 0 0(11988522)plusmn

120572 0 [0 1] [0 1] [0 1] 1 1(11988532)plusmn

120572 [0 1] 0 0 0 0 0(11988513)plusmn

120572 0 0 0 0 0 0(11988523)plusmn

120572 0 0 0 0 0 0(11988533)plusmn

120572 0 0 0 0 0 0

9

8

7

6

5

4

30 02 04 06 08 1 12

120583


f(times109)



120583119872 (119909) =

119871(

119898 minus 119909

120572

) for 119898 minus 120572 le 119909 le 119898

119877(

119909 minus 119898

120573

) for 119898 le 119909 le 119898 + 120573

0 else

(A3)



119886plusmn120572 = [119886


minus120572 le 119886+120572 (A4)


119886plusmn1205721= [119886minus1205721 119886+1205721] 119886

plusmn1205722= [119886minus1205722 119886+1205722] (A5)

If 1205721 ge 1205722 we have


minus1205722 119886+1205721le 119886+1205722 (A6)


plusmn120572 = [119887

minus120572 119887+120572 ] we can

define

(1) 119886plusmn120572 + 119887plusmn120572 = [119886

minus120572 119886+120572 ] + [119887

minus120572 119887+120572 ] = [119886

minus120572 + 119887minus120572 119886+120572 + 119887+120572 ]


minus120572 119886+120572 ] minus [119887

minus120572 119887+120572 ] = [119886



minus120572 119886+120572 ] sdot [119887

minus120572 119887+120572 ] = [119886




[119886minus120572 119886+120572 ] le [119887

minus120572 119887+120572 ] iff 119886



and

119894isin119868[119886minus120572119894 119886+120572119894] = [and

119894isin119868119886minus120572119894 and

119894isin119868119886+120572119894] if and


or

119894isin119868[119886minus120572119894 119886+120572119894] = [or

119894isin119868119886minus120572119894 or

119894isin119868119886+120572119894] if or

119894isin119868119886+120572119894lt infin

(A8)





such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












such that 119892120572 = or119894isin119868(119886119894)120572


(119886 lowast 119887)120572 = 119886120572 lowast 119887120572 (A9)




Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Generalized Fuzzy Integer Programming...

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