ReviewArticle - Hindawi Publishing Corporationdownloads.hindawi.com/journals/aor/2019/9714137.pdf(RMCGP) and a utility function as an approach to the MOTP.Inanotherpaper,theyintroducedaprocedurefor

Review ArticleA Goal Programming Approach to Multichoice MultiobjectiveStochastic Transportation Problems with ExtremeValue Distribution

Hadeel Al Qahtani 1 Ali ElndashHefnawy12 Maha M ElndashAshram2 and Aisha Fayomi1

1Department of Statistics Faculty of Science King Abdulaziz University Jeddah Saudi Arabia2Department of Statistics Faculty of Economics amp Political Science Cairo University Giza Egypt

Correspondence should be addressed to Hadeel Al Qahtani hadeel1089hotmailcom

Received 1 April 2019 Revised 10 July 2019 Accepted 17 August 2019 Published 11 September 2019

Academic Editor Yi-Kuei Lin

Copyright copy 2019 Hadeel Al Qahtani et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

is paper presents the study of a multichoice multiobjective transportation problem (MCMOTP) when at least one of theobjectives has multiple aspiration levels to achieve and the parameters of supply and demand are random variables which are notpredetermined e random variables shall be assumed to follow extreme value distribution and the demand and supplyconstraints will be converted from a probabilistic case to a deterministic one using a stochastic approach A transformationmethod using binary variables reduces the MCMOTP into a multiobjective transportation problem (MOTP) selecting oneaspiration level for each objective frommultiple levelse reduced problem can then be solved with goal programminge noveladapted approach is significant because it enables the decisionmaker to handle the many objectives and complexities of real-worldtransportation problem in one model and find an optimal solution Ultimately a mixed-integer mathematical model has beenformulated by utilizing GAMS software and the optimal solution of the proposed model is obtained A numerical example ispresented to demonstrate the solution in detail

1 Introduction

e transportation problem is a well-known specific ap-plication of linear programming in which an item is to betransported from $m$ sources to $n$ destinations [1] eavailability of the product at the $ith$ source is denoted by$ai$ where i 1 2 m and the demand required at the$jth$ destination is $bj$ where j 1 2 n e penalty$cij$ is the cost coefficient of the objective function that canrepresent the expense of transporting wares from sources todestinations which is desired to be minimized [1]

ere may be more than one objective to the problemand they could be conflicting for example minimizing thecost of transportation as well as minimizing the shippingtime Here the two goals have the same direction ieminimization but there is a trade-off For example using acar as the transport means may be lower in cost than by airfreight but will take much longer Hence goal programming

is introduced so that the decision maker (DM) may setmultiple choices for the aspiration levels in at least one goalin a transportation problem defining a multiaspiration levelgoal programming transportation problem In addition thesupply and demand parameters can be random variables soit becomes a stochastic multiaspiration level goal pro-gramming transportation problem

Mahapatra [2] considers a model of a multichoice sto-chastic transportation problem (MCSTP) where the supplyand demand parameters of the constraints follow extremevalue distribution Some of the cost coefficients of the ob-jective function are a multichoice type In an optimal so-lution the number of units to be transported should bedetermined while satisfying source and destination demandsto ensure minimum transportation costs

In this paper we will look at the problem from anotherangle by including the concept of goal programming toallow the model to deal with more than one conflicting

HindawiAdvances in Operations ResearchVolume 2019 Article ID 9714137 6 pageshttpsdoiorg10115520199714137

objective and set multiaspiration levels to certain goals enew model becomes a stochastic multiaspiration level goalprogramming transportation problemwith an extreme valuedistribution

Often one cannot determine an exact value of anyparameters in the problem because of uncertainty in supplyor demand parameters for a number of reasons For ex-ample fluctuating markets or service output levels fromsuppliers raw material defects machine performancesdelivery delays and transportation issues are among thefactors which cause uncertainty in supply assumptionsSimilarly unknown customer demand for products orservices offered by the buyer customer preferences com-petition and an unpredictable economy are among thefactors that contribute to demand uncertainty A stochasticproblem can be formulated to overcome these uncertaintiesby considering that random variables follow a specificdistribution instead of assuming fixed values Here an ex-treme value distribution will be assumed to convert theconstraints from probabilistic to deterministic with thedisjoint chance-constrained method Extreme value distri-bution is used when there is a requirement for a limitingdistribution to the maximum or minimum of a sample ofindependent and identically distributed random variablese probability density function of extreme value distri-bution type I [3] is as follows

f(x α β)

1β

eminus ((xminus α)β) exp minus e

minus ((xminus α)β)1113960 1113961 minus infinle xleinfin βgt 0

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(1)

Goal programming is an extension of linear pro-gramming which handles multiobjective optimization wherethe individual objectives are often conflicting Every one ofthese measures is assigned a goal or target value to be ac-complished Undesirable deviations from this arrangementof target values are then minimized through an achievementfunction is can be a vector or a weighted sum dependingon the goal programming variant adopted or the DMrsquosrequirements

e type of goal programming model employed is de-termined by the nature of the DMrsquos goals e initial goalprogramming formulations order the undesirable deviationsinto a hierarchy by criticality which enables more priority tobe given to minimizing deviation of the more importantfactors is is known as lexicographic (preemptive) or non-Archimedean goal programming

Lexicographic goal programming can be used whenprioritization is relevant to the goals In preemptive goalprogramming the objectives can be separated into variouspriority classes Here it is assumed that no two goals haveequal priorities Each will then be satisfied sequentially frommost important to least important e DMs can set mul-tichoice aspiration levels (MCALs) for each goal to avoidunderestimating accounting for the ldquomorehigher is betterrdquoand ldquolesslower is betterrdquo aspirations To handle thesemultiple aspiration levels multiplicative terms of binary

variables are utilized where all binary variables constitutemutually exclusive choices and only one variable is selectede number of binary variables required for a constraint isequivalent to the total number of options of that constraint

e article is subsequently organized as follows InSection 2 a problem overview will be considered themathematical model will be presented in Section 3 andSection 4 will discuss the transformation of the goal con-straint involvingmultiple aspiration levels into an equivalentform Finally a case study to demonstrate the model will bepresented in Section 5

2 Problem Overview

Contini [4] considered the first formulation of the stochasticgoal programming model He set the goals as uncertainnormally distributed variables e technique for solving theprobabilistic programming model was to convert it into anequivalent deterministic model Many approaches have beenproposed to solve the probabilistic programming model ofwhich the most common approach is chance-constrainedprogramming (CCP) developed by Charnes and Cooper[5ndash7]

Chang [8] proposed a new idea for modelling themultichoice goal programming problem usingmultiplicativeterms of binary variables to handle multiple aspiration levelsBiswal and Acharya [9] proposed transformation techniquesto transform a multichoice linear programming probleminto an equivalent mathematical model in which constraintsare associated with multichoice parameters

Many researchers have extensively studied the MCSTPBarik et al [10] presented a stochastic transportation modelinvolving Pareto distribution Roy et al [11] presented anequivalent deterministic model of MCSTP by assuming thatboth availabilities $ai$ and demands $bj$ are random var-iables following an exponential distribution Biswal andSamal [12] obtained an equivalent deterministic model ofMCSTP in which they considered that both $ai$ and $bj$follow Cauchy distribution Mahapatra [2] considered anMCSTP with extreme value distribution serving as a basis ofinspiration for this research e novel contribution of thispaper is to include the multiobjective problem within themodel and to represent the multichoice problem in terms ofaspiration levels instead of a cost coefficient parameterMahapatra [2] also studied an MCSTP model involvingWeibull distribution whereas Quddoos et al [13] presentedanMCSTP involving a general form of distribution Roy [14]introduced Lagrangersquos interpolating polynomial to deal withthe multichoice transportation problem He subsequentlypublished a paper of the transportation problem withmultichoice cost and demand parameters and stochasticsupply [15] in which he used Lagrangersquos interpolatingpolynomial to select an appropriate value for the cost co-efficients of the objective function and the demand of theconstraints in the transportation problem By adoptingstochastic programming the stochastic supply constraintswere transformed into deterministic constraints One of thekey publications of Maity and Roy [16] proposed thetechniques of revised multichoice goal programming

2 Advances in Operations Research

(RMCGP) and a utility function as an approach to theMOTP In another paper they introduced a procedure forconverting a multichoice interval transportation problem(MCITP) into a deterministic transportation problem tosolve [17] In an additional publication the same authorsdemonstrated solving a fuzzy transportation problem (FTP)using a multichoice goal programming approach [18] Royet al [19] also proposed the technique of introducing a conicscalarizing function into the MOTP in combination withRMCGP

In this study we will propose a new approach to thetransportation problem whereby the supply and demandparameters are random variables following extreme valuedistribution Rather than minimizing the cost coefficient forthe transportation problem we can minimize the time forshipping minimize the risk in shipping the items and so onAs an additional feature each objective can have multipleaspiration levels instead of only one Now the problembecomes a multichoice multiobjective stochastic trans-portation problem To overcome this difficulty first we willuse a stochastic approach to turn the probabilistic constraintinto a deterministic one Second a general transformationconsisting of binary variables is applied to select one aspi-ration level for each objective from multiple levels ereduced problem then becomes an MOTP and it will besolved with goal programming

3 Mathematical Model

Initially the classical transportation problem is consideredIf xij represents the amount transported from the source tothe destination then the transportation model can be de-fined as follows

Model 1

Findxij i 1 2 m j 1 2 n

min z 1113944ij

cijxij

Subject to (st)

(2)

1113944

n

j1xij le ai foralli (3)

1113944

m

i1xij ge bj forallj (4)

1113944i

ai 1113944j

bj (5)

xij ge 0 foralli j (6)

where cij is the transportation cost per unit xij is the amountshipped ai is the amount of supply at source i and bj is theamount of demand at destination j [20]

Now we consider a mathematical model for a stochasticmultiaspiration level goal programming transportationproblem with an extreme value distribution as follows

Model 2Lexmin ni pi1113864 1113865

st fi(x) + ni minus pi g1 g2 gq q 1 2 k

P 1113944n

j1xij le ai

⎛⎝ ⎞⎠ge 1 minus ci i 1 2 m 0le ci le 1

P 1113944m

i1xij ge bj

⎛⎝ ⎞⎠ge 1 minus δj j 1 2 n 0le δj le 1

xij ge 0 ni pi ge 0

1113944

m

i1ai 1113944

n

j1bj

(7)

where fi(x) is the linear function of the ith goal gi is theaspiration levels of the ith goal xij is the amount shipped ai

is the amount of supply at source i bj is the amount ofdemand at destination j ni is the negative deviational var-iable and pi is the positive deviational variable

31 Converting the Probabilistic Constraint into a De-terministic Constraint Using the Disjoint Chance-ConstrainedMethod FromMahapatra [2] three cases of randomness onthe right-hand side of the supply and demand constraintswere considered

(1) Only ai i 1 2 m follows extreme valuedistribution

(2) Only bj j 1 2 n follows extreme valuedistribution

(3) Both ai i 1 2 m and bj j 1 2 n followextreme value distribution

is led to three different models (for more details seeMahapatra [2] e final transformed constraint is thenconsidered here as the probabilistic constraint (4) trans-formed into a deterministic linear constraint

1113944

n

j1xij le αi minus βi ln minus ln ci( 11138571113864 11138651113858 1113859 (8)

e probabilistic constraint (5) transformed into a de-terministic linear constraint

1113944

m

i1xij ge α j minus β j ln minus ln 1 minus δj1113872 11138731113966 11139671113960 1113961 (9)

Advances in Operations Research 3

Now a deterministic multiaspiration level goal pro-gramming transportation problem with an extreme valuedistributions model will be obtained as follows



1113944

n

j1xij le αi minus βi ln minus ln ci( 11138571113864 11138651113858 1113859 i 1 2 m

1113944

m

i1xij ge α1 minus β j ln minus ln 1 minus δj1113872 11138731113966 11139671113960 1113961 j 1 2 n

xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(10)

where 1113936mi1αi minus βi[ln minus ln(ci)1113864 1113865]ge1113936

nj1αj minus βj[ln minus ln(1minus

δj)] is the feasibility condition

4 Transformation of the Goal ConstraintInvolving Multiaspiration Levels to anEquivalent Form

Considering the goal constraint with multiple aspirationlevels

fi(x) + ni minus pi g1 g2 gq (11)

A binary variable will be utilized to select a single as-piration level and we can utilize the relation ln nln 2 todeterminate the number of binary variables needed with n

aspiration levels under the given linearized constraint [21]Let x zizj where x satisfies the following inequalities

zi + zj minus 21113872 1113873 + 1le xle 2 minus zi minus zj1113872 1113873 + 1 (12)

xle zi (13)

xle zj (14)

xge 0 (15)

e inequalities are identified

(i) If zi zj 1 then x 1 (from (12))(ii) If zizj 0 then x 0 (from (13)ndash(15))

And so the new goal constraint will be

fi(x) + ni minus pi 1113944n

j1gijSij(B) i 1 2 m (16)

where Sij(B) represents the function of the binary serialnumber

A stochastic multiaspiration level goal programmingtransportation problem with extreme value distributionmodel will be as follows

Model 4

Lexmin ni pi1113864 1113865

st fi(x) + ni minus pi 1113944

n

j1gijSij(B)

1113944

n


1113944

m

i1xij ge α j minus β j ln minus ln 1 minus δj1113872 11138731113966 11139671113960 1113961 j 1 2 n

xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(17)




5 Case Study

In this section a case study from Mahapatra [2] is con-sidered with modifications and the assumption of extremevalue distribution instead of Weibull distribution In thiscase study a cold drink supply company transports colddrinks from three product centres at Dankuni Howrahand Asansol to four destination centres at JhargramKharagpur Tarkeshwar and Contai In the summerseason the cold drinks are in high demand at each ofthe four destination centres e transportation timecost is an essential factor in a transportation planningprogramme as well as the transportation cost emanufacturing time at production centres depends on theavailability of current supply machine condition skilledlabour etc Delivery time is related to the transportingmeans and seamless distribution of a product in due timeto destination centres e transportation time cost tij andcost coefficient cij from each source to each destination areconsidered in Table 1

e cold drinks supply company is seeking to reach thefollowing goals goal 1 is to minimize the transportation timecost and goal 2 is to minimize the cost of transportation etarget values are 112000 or 113000 hours and $150000 or$160000 respectively

Due to the fluctuation of the above factor a stochasticmultiaspiration level goal programming transportationproblem approach has been considered in which the supplyand demand parameters follow extreme value distributione specified probability levels with shape and scale pa-rameters for supply are listed in Table 2 and the specified


probability levels with shape and scale parameters of de-mand parameters are provided in Table 3 Utilizing the data in Tables 1ndash3 the deterministic

multiaspiration level goal programming transportationproblem is formulated as follows

Lexmin p1 p21113864 1113865

st 11139443

i11113944

4

j1xij + n1 minus p1 112000 z1 + 113000 1 minus z1( 1113857

1113944

3

i11113944

4


1113944

4

j1x1j le 2994502

1113944

4

j1x2j le 2495908

1113944

4

j1x3j le 19969989

1113944

3

i1xi1 ge 1707038

1113944

3

i1xi2 ge 150594

1113944

3

i1xi3 ge 1254452

1113944

3

i1xi4 ge 1003147

xij nq dq ge 0 i 1 2 3 j 1 2 3 4 q 1 2 zk 0 or 1 k 1 2

(18)

Table 1 Transportation time cost tij and cost coefficient cij from each source to each destination

No Route xij Transportation time cost tij (in hours) Cost coefficient cij (in dollars)

1 (11) x11 12 212 (12) x12 15 253 (13) x13 19 304 (14) x14 24 345 (21) x21 16 276 (22) x22 18 287 (23) x23 9 158 (24) x24 17 269 (31) x31 24 3410 (32) x32 12 2411 (33) x33 25 3712 (34) x34 28 40

Table 2 Values of location and scale parameter with SPL of ai

Shape parameter Scale parameter SPLα1 3000 β1 36 c1 001α2 2500 β2 30 c2 002α3 2000 β3 24 c3 003

Table 3 Values of location and scale parameter with SPL of bj

Shape parameter Scale parameter SPLα1 1700 β1 22 δ1 004α2 1500 β2 20 δ2 005α3 1250 β3 16 δ3 006α4 1000 β4 12 δ4 007


Checking that the feasibility condition is satisfied

1113944

m

i1αi minus βi ln minus ln ci( 11138571113864 11138651113858 1113859 7487399

ge 1113944n

j1α j minus β j ln minus ln 1 minus δj1113872 11138731113966 11139671113960 1113961 5470577

(19)

e deterministic linear mixed-integer problem is thensolved using GAMS (software) where the optimal solution isobtained

x12 736904

x13 1254452

x14 1003147

x22 1707038

x22 769037

p1 0

p2 0

n1 12112028

n2 2213805

where z1 0 and z2 0

(20)

e remaining decision variables are zero e resultsshow that goal 1 has an aspiration level of 113000 hours andzero positive deviation which means that the transportationtime cost achieved the aspiration level exactly and goal 2 hasan aspiration level of $160000 and zero positive deviationwhich means that transportation cost also reached the de-sired aspiration level exactly

6 Conclusion

In this paper we have explored a study of problem when thesupply and demand parameters are the stochastic type andfollow extreme value distribution ree different ap-proaches (the stochastic approach binary variable approachand goal programming approach) can be combined to reachan optimal solution to the transportation problem isprovides a new capability to handle real-life DM problemssuch as agricultural managerial economical and industrialOne numerical example has been presented to illustrate theapproach which was solved using GAMS software One canapply the proposed model to real-life problems in featurework or adapt other multiobjective techniques such as theε-constraint method weighting method or fuzzy pro-gramming methods and compare their performance

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] D R Mahapatra S K Roy and M P Biswal ldquoMulti-choicestochastic transportation problem involving extreme value

distributionrdquo Applied Mathematical Modelling vol 37 no 4pp 2230ndash2240 2013

[2] D R Mahapatra ldquoMulti-choice stochastic transportationproblem involving Weibull distributionrdquo An InternationalJournal of Optimization and Control 4eories amp Applications(IJOCTA) vol 4 no 1 pp 45ndash55 2013

[3] B S Everitt 4e Cambridge Dictionary of Statistics Cam-bridge University Press Cambridge UK 2002

[4] B Contini ldquoA stochastic approach to goal programmingrdquoOperations Research vol 16 no 3 pp 576ndash586 1968

[5] A Charnes and W W Cooper ldquoChance-constrained pro-grammingrdquoManagement Science vol 6 no 1 pp 73ndash79 1959

[6] A Charnes and W W Cooper ldquoChance constraints andnormal deviatesrdquo Journal of the American Statistical Associ-ation vol 57 no 297 pp 134ndash148 1962

[7] A Charnes andWW Cooper ldquoDeterministic equivalents foroptimizing and satisficing under chance constraintsrdquo Oper-ations Research vol 11 no 1 pp 18ndash39 1963

[8] C-T Chang ldquoMulti-choice goal programmingrdquo Omegavol 35 no 4 pp 389ndash396 2007

[9] M P Biswal and S Acharya ldquoTransformation of a multi-choice linear programming problemrdquo Applied Mathematicsand Computation vol 210 no 1 pp 182ndash188 2009

[10] S K Barik M P Biswal and D Chakravarty ldquoStochasticprogramming problems involving pareto distributionrdquoJournal of Interdisciplinary Mathematics vol 14 no 1pp 40ndash56 2011

[11] S K Roy D R Mahapatra and M P Biswal ldquoMulti-choicestochastic transportation problem with exponential distribu-tionrdquo Journal of Uncertain Systems vol 6 pp 200ndash213 2012

[12] M P Biswal and H K Samal ldquoStochastic transportationproblem with Cauchy random variables and multi choiceparametersrdquo Journal of Physical Sciences vol 17 pp 117ndash1302013

[13] A Quddoos M G Ull Hasan andMM Khalid ldquoMulti-choicestochastic transportation problem involving general form ofdistributionsrdquo Springer Plus vol 3 no 1 pp 1ndash9 2014

[14] S K Roy ldquoLagrangersquos interpolating polynomial approach tosolve multi-choice transportation problemrdquo InternationalJournal of Applied and Computational Mathematics vol 1no 4 pp 639ndash649 2015

[15] S K Roy ldquoTransportation problem with multi-choice costand demand and stochastic supplyrdquo Journal of the OperationsResearch Society of China vol 4 no 2 pp 193ndash204 2016

[16] G Maity and S K Roy ldquoSolving multi-objective trans-portation problem with interval goal using utility functionapproachrdquo International Journal of Operational Researchvol 27 no 4 pp 513ndash529 2016

[17] S K Roy and G Maity ldquoMinimizing cost and time throughsingle objective function in multi-choice interval valuedtransportation problemrdquo Journal of Intelligent amp Fuzzy Sys-tems vol 32 no 3 pp 1697ndash1709 2017

[18] G Maity and S K Roy ldquoSolving fuzzy transportation problemusing multi-choice goal programmingrdquo Discrete MathematicsAlgorithms and Applications vol 9 no 6 article 1750076 2017

[19] S K Roy G Maity G W Weber and S Z A Gok ldquoConicscalarization approach to solve multi-choice multi-objectivetransportation problem with interval goalrdquo Annals of Oper-ations Research vol 253 no 1 pp 599ndash620 2017

[20] H A Taha Operations Research An Introduction PearsonPrentice Hall Upper Saddle River NJ USA 2011

[21] C-T Chang ldquoAn efficient linearization approach for mixed-integer problemsrdquo European Journal of Operational Researchvol 123 no 3 pp 652ndash659 2000


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objective and set multiaspiration levels to certain goals enew model becomes a stochastic multiaspiration level goalprogramming transportation problemwith an extreme valuedistribution

Often one cannot determine an exact value of anyparameters in the problem because of uncertainty in supplyor demand parameters for a number of reasons For ex-ample fluctuating markets or service output levels fromsuppliers raw material defects machine performancesdelivery delays and transportation issues are among thefactors which cause uncertainty in supply assumptionsSimilarly unknown customer demand for products orservices offered by the buyer customer preferences com-petition and an unpredictable economy are among thefactors that contribute to demand uncertainty A stochasticproblem can be formulated to overcome these uncertaintiesby considering that random variables follow a specificdistribution instead of assuming fixed values Here an ex-treme value distribution will be assumed to convert theconstraints from probabilistic to deterministic with thedisjoint chance-constrained method Extreme value distri-bution is used when there is a requirement for a limitingdistribution to the maximum or minimum of a sample ofindependent and identically distributed random variablese probability density function of extreme value distri-bution type I [3] is as follows

f(x α β)

1β

eminus ((xminus α)β) exp minus e

minus ((xminus α)β)1113960 1113961 minus infinle xleinfin βgt 0

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(1)

Goal programming is an extension of linear pro-gramming which handles multiobjective optimization wherethe individual objectives are often conflicting Every one ofthese measures is assigned a goal or target value to be ac-complished Undesirable deviations from this arrangementof target values are then minimized through an achievementfunction is can be a vector or a weighted sum dependingon the goal programming variant adopted or the DMrsquosrequirements

e type of goal programming model employed is de-termined by the nature of the DMrsquos goals e initial goalprogramming formulations order the undesirable deviationsinto a hierarchy by criticality which enables more priority tobe given to minimizing deviation of the more importantfactors is is known as lexicographic (preemptive) or non-Archimedean goal programming

Lexicographic goal programming can be used whenprioritization is relevant to the goals In preemptive goalprogramming the objectives can be separated into variouspriority classes Here it is assumed that no two goals haveequal priorities Each will then be satisfied sequentially frommost important to least important e DMs can set mul-tichoice aspiration levels (MCALs) for each goal to avoidunderestimating accounting for the ldquomorehigher is betterrdquoand ldquolesslower is betterrdquo aspirations To handle thesemultiple aspiration levels multiplicative terms of binary

variables are utilized where all binary variables constitutemutually exclusive choices and only one variable is selectede number of binary variables required for a constraint isequivalent to the total number of options of that constraint

e article is subsequently organized as follows InSection 2 a problem overview will be considered themathematical model will be presented in Section 3 andSection 4 will discuss the transformation of the goal con-straint involvingmultiple aspiration levels into an equivalentform Finally a case study to demonstrate the model will bepresented in Section 5

2 Problem Overview

Contini [4] considered the first formulation of the stochasticgoal programming model He set the goals as uncertainnormally distributed variables e technique for solving theprobabilistic programming model was to convert it into anequivalent deterministic model Many approaches have beenproposed to solve the probabilistic programming model ofwhich the most common approach is chance-constrainedprogramming (CCP) developed by Charnes and Cooper[5ndash7]

Chang [8] proposed a new idea for modelling themultichoice goal programming problem usingmultiplicativeterms of binary variables to handle multiple aspiration levelsBiswal and Acharya [9] proposed transformation techniquesto transform a multichoice linear programming probleminto an equivalent mathematical model in which constraintsare associated with multichoice parameters

Many researchers have extensively studied the MCSTPBarik et al [10] presented a stochastic transportation modelinvolving Pareto distribution Roy et al [11] presented anequivalent deterministic model of MCSTP by assuming thatboth availabilities $ai$ and demands $bj$ are random var-iables following an exponential distribution Biswal andSamal [12] obtained an equivalent deterministic model ofMCSTP in which they considered that both $ai$ and $bj$follow Cauchy distribution Mahapatra [2] considered anMCSTP with extreme value distribution serving as a basis ofinspiration for this research e novel contribution of thispaper is to include the multiobjective problem within themodel and to represent the multichoice problem in terms ofaspiration levels instead of a cost coefficient parameterMahapatra [2] also studied an MCSTP model involvingWeibull distribution whereas Quddoos et al [13] presentedanMCSTP involving a general form of distribution Roy [14]introduced Lagrangersquos interpolating polynomial to deal withthe multichoice transportation problem He subsequentlypublished a paper of the transportation problem withmultichoice cost and demand parameters and stochasticsupply [15] in which he used Lagrangersquos interpolatingpolynomial to select an appropriate value for the cost co-efficients of the objective function and the demand of theconstraints in the transportation problem By adoptingstochastic programming the stochastic supply constraintswere transformed into deterministic constraints One of thekey publications of Maity and Roy [16] proposed thetechniques of revised multichoice goal programming






Model 1


min z 1113944ij

cijxij

Subject to (st)

(2)

1113944

n


1113944

m


1113944i

ai 1113944j

bj (5)






P 1113944n

j1xij le ai


P 1113944m

i1xij ge bj


xij ge 0 ni pi ge 0

1113944

m

i1ai 1113944

n

j1bj

(7)








1113944

n



1113944

m






1113944

n


1113944

m


xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(10)










xle zi (13)

xle zj (14)

xge 0 (15)








Model 4

Lexmin ni pi1113864 1113865


n

j1gijSij(B)

1113944

n


1113944

m


xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(17)




5 Case Study







Lexmin p1 p21113864 1113865

st 11139443

i11113944

4


1113944

3

i11113944

4


1113944

4

j1x1j le 2994502

1113944

4

j1x2j le 2495908

1113944

4

j1x3j le 19969989

1113944

3

i1xi1 ge 1707038

1113944

3

i1xi2 ge 150594

1113944

3

i1xi3 ge 1254452

1113944

3

i1xi4 ge 1003147


(18)



1 (11) x11 12 212 (12) x12 15 253 (13) x13 19 304 (14) x14 24 345 (21) x21 16 276 (22) x22 18 287 (23) x23 9 158 (24) x24 17 269 (31) x31 24 3410 (32) x32 12 2411 (33) x33 25 3712 (34) x34 28 40







1113944

m


ge 1113944n


(19)


x12 736904

x13 1254452

x14 1003147

x22 1707038

x22 769037

p1 0

p2 0

n1 12112028

n2 2213805

where z1 0 and z2 0

(20)


6 Conclusion




References































Journal of












Journal of







Volume 2018






Volume 2018












Model 1


min z 1113944ij

cijxij

Subject to (st)

(2)

1113944

n


1113944

m


1113944i

ai 1113944j

bj (5)






P 1113944n

j1xij le ai


P 1113944m

i1xij ge bj


xij ge 0 ni pi ge 0

1113944

m

i1ai 1113944

n

j1bj

(7)








1113944

n



1113944

m






1113944

n


1113944

m


xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(10)










xle zi (13)

xle zj (14)

xge 0 (15)








Model 4

Lexmin ni pi1113864 1113865


n

j1gijSij(B)

1113944

n


1113944

m


xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(17)




5 Case Study







Lexmin p1 p21113864 1113865

st 11139443

i11113944

4


1113944

3

i11113944

4


1113944

4

j1x1j le 2994502

1113944

4

j1x2j le 2495908

1113944

4

j1x3j le 19969989

1113944

3

i1xi1 ge 1707038

1113944

3

i1xi2 ge 150594

1113944

3

i1xi3 ge 1254452

1113944

3

i1xi4 ge 1003147


(18)



1 (11) x11 12 212 (12) x12 15 253 (13) x13 19 304 (14) x14 24 345 (21) x21 16 276 (22) x22 18 287 (23) x23 9 158 (24) x24 17 269 (31) x31 24 3410 (32) x32 12 2411 (33) x33 25 3712 (34) x34 28 40







1113944

m


ge 1113944n


(19)


x12 736904

x13 1254452

x14 1003147

x22 1707038

x22 769037

p1 0

p2 0

n1 12112028

n2 2213805

where z1 0 and z2 0

(20)


6 Conclusion




References































Journal of












Journal of







Volume 2018






Volume 2018











1113944

n


1113944

m


xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(10)










xle zi (13)

xle zj (14)

xge 0 (15)








Model 4

Lexmin ni pi1113864 1113865


n

j1gijSij(B)

1113944

n


1113944

m


xij ge 0 ni pi ge 0

0le δj le 1

0le ci le 1

(17)




5 Case Study







Lexmin p1 p21113864 1113865

st 11139443

i11113944

4


1113944

3

i11113944

4


1113944

4

j1x1j le 2994502

1113944

4

j1x2j le 2495908

1113944

4

j1x3j le 19969989

1113944

3

i1xi1 ge 1707038

1113944

3

i1xi2 ge 150594

1113944

3

i1xi3 ge 1254452

1113944

3

i1xi4 ge 1003147


(18)



1 (11) x11 12 212 (12) x12 15 253 (13) x13 19 304 (14) x14 24 345 (21) x21 16 276 (22) x22 18 287 (23) x23 9 158 (24) x24 17 269 (31) x31 24 3410 (32) x32 12 2411 (33) x33 25 3712 (34) x34 28 40







1113944

m


ge 1113944n


(19)


x12 736904

x13 1254452

x14 1003147

x22 1707038

x22 769037

p1 0

p2 0

n1 12112028

n2 2213805

where z1 0 and z2 0

(20)


6 Conclusion




References































Journal of












Journal of







Volume 2018






Volume 2018










Lexmin p1 p21113864 1113865

st 11139443

i11113944

4


1113944

3

i11113944

4


1113944

4

j1x1j le 2994502

1113944

4

j1x2j le 2495908

1113944

4

j1x3j le 19969989

1113944

3

i1xi1 ge 1707038

1113944

3

i1xi2 ge 150594

1113944

3

i1xi3 ge 1254452

1113944

3

i1xi4 ge 1003147


(18)



1 (11) x11 12 212 (12) x12 15 253 (13) x13 19 304 (14) x14 24 345 (21) x21 16 276 (22) x22 18 287 (23) x23 9 158 (24) x24 17 269 (31) x31 24 3410 (32) x32 12 2411 (33) x33 25 3712 (34) x34 28 40







1113944

m


ge 1113944n


(19)


x12 736904

x13 1254452

x14 1003147

x22 1707038

x22 769037

p1 0

p2 0

n1 12112028

n2 2213805

where z1 0 and z2 0

(20)


6 Conclusion




References































Journal of












Journal of







Volume 2018






Volume 2018









1113944

m


ge 1113944n


(19)


x12 736904

x13 1254452

x14 1003147

x22 1707038

x22 769037

p1 0

p2 0

n1 12112028

n2 2213805

where z1 0 and z2 0

(20)


6 Conclusion




References































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

ReviewArticle - Hindawi Publishing Corporationdownloads.hindawi.com/journals/aor/2019/9714137.pdf(RMCGP) and a utility function as an approach to the MOTP.Inanotherpaper,theyintroducedaprocedurefor