Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 823876 10 pageshttpdxdoiorg1011552013823876

Research ArticleA Multiobjective Optimization Model in Automotive SupplyChain Networks

Abdolhossein Sadrnia1 Napsiah Ismail1 Norzima Zulkifli1 M K A Ariffin1

Hossein Nezamabadi-pour2 and Hamed Mirabi1

1 Department of Mechanical and Manufacturing Engineering Faculty of Engineering University Putra Malaysia43400 Serdang Selangor Malaysia

2 Department of Electrical Engineering Shahid Bahonar University of Kerman Kerman Iran

Correspondence should be addressed to Abdolhossein Sadrnia hsadrniayahoocom

Received 5 February 2013 Revised 10 June 2013 Accepted 5 August 2013

Academic Editor Bijaya Panigrahi

Copyright copy 2013 Abdolhossein Sadrnia et al This 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 is properlycited

In the new decade green investment decisions are attracting more interest in design supply chains due to the hidden economicbenefits and environmental legislative barriers In this paper a supply chain network design problem with both economic andenvironmental concerns is presented Therefore a multiobjective optimization model that captures the trade-off between the totallogistics cost and CO

2emissions is proposed With regard to the complexity of logistic networks a new multiobjective swarm

intelligence algorithm known as a multiobjective Gravitational search algorithm (MOGSA) has been implemented for solvingthe proposed mathematical model To evaluate the effectiveness of the model a comprehensive set of numerical experiments isexplainedThe results obtained show that the proposedmodel can be applied as an effective tool in strategic planning for optimizingcost and CO

2emissions in an environmentally friendly automotive supply chain

1 Introduction

In conventional supply chain management the emphasis ofthe supply chain network is usually a single objective such asprofit maximization or cost minimization Despite waterwaypollution and themassive loss of biodiversity global warmingand carbon dioxide (CO

2) emissions are considered to be the

types of environmental impacts that have to be consideredwith regard to the coordinated activity of organizations ina supply chain Additionally attention must be given to thegreenhouse gases that are increasingly facilitated by globalwarming and which are the result of human activities sincethe industrial revolutionThemost significant anthropogenicgreenhouse gas is CO

2 The situation is that global CO

2

emissions have increased by about 80 during the period of1970ndash2004 mainly as a result of using fossil fuels [1]

The distances between facilities in a distribution networkhave grown dramatically due to the globalisation of supplychains which leads to increased vehicle emissions as partof transportation In the US approximately 33 of CO

2

emissions and 73 of carbon monoxide (CO) are based on

the distribution of products and transportation [2] Conse-quently there is the need to effectively and efficiently designenvironmentally friendly supply chain networks to bothimprove environmental conditions as well as the economicdimension of an organisation Elhedhli and Merrick [3]expressed network design as a logical place to start whenlooking towards greening a supply chain design Usually theplanning of a logistics network requires making decisionsregarding the followings [4]

(i) the number location capacity and technology ofmanufacturing plants warehouses and retailers

(ii) selection of suppliers(iii) assignment of product ranges to manufacturing

plants and warehouses(iv) selection of distribution channels and transportation

modes(v) flow of raw materials through the networkOver the last decade besides the economic concerns

environmental issues have interested many scholars when

2 Mathematical Problems in Engineering

they design supply chain networks The network designproblem is one of the most comprehensive strategic deci-sion problems that should be optimised for long-term effi-cient operations throughout the supply chain The problemincludes a wide range of mathematical modelling rangingfrom linear deterministic models [5ndash7] to complex nonlinearstochastic models [8] In the literature several attempts havebeen made to solve the network design problems of supplynetworks and these studies have been surveyed by BeamonandFernandes [9] Chopra [10] andPeidro et al [11] Scholarshave attempted to develop the classical quantitative model byincorporating various factors such as transportation modes[12 13] vehicle speed [14] and greenhouse gasses [15 16]

Despite the volume of work undertaken in respect to thesupply chain network design problem themodelling is usual-ly done as a single objective problem [17 18]while the ldquodesignrdquoin real cases typically involves compromises and trade-offsbetween different incompatible objectives For example theoptimum design for a supply chain network should notonly concentrate on total cost minimisation but it also hasto satisfy other objectives such as delivery flexibility [19]minimising waste [20] and minimising CO

2emissions [21]

Satisfying environmental issues and minimizing the totalvariable and fixed costs causes the logistics network designproblem to be an NP-hard problem [22 23] Solving thisproblem on a large scale by exact algorithms is very timeconsuming and sometimes in the practical case is impossibleTherefore many heuristics and metaheuristics have beendeveloped to obtain optimal solutions for these kinds of prob-lems [24] In this regard during the last decade the imple-mentation of meta-heuristics instead of exact and exhaustivemethods has been growing Approaches such as bioinspiredalgorithms such as ant colony optimization (ACO) [25ndash27] particle swarm optimization (PSO) [28] artificial weedcolonies [29] and also evolutionary algorithms such asgenetic algorithms (GA) have been used to solve a varietyof multi-objective problems in production and supply chainmanagement specially in complex environment [29 30]

Recently the consideration of supply network designwith multi-objective optimisation (generally incompatibleobjectives) is a new trend worthy of study The solutionto the above problem is not unique but a set of Paretooptimal points A Pareto optimal solution is the best solutionvector out of several numbers of solution vectors that couldbe achieved without disadvantaging other objectives Eachsolution set is known as nondominated solution set Anarray containing all the Pareto optimal solutions of a multi-objective problem is called the Pareto optimal set Thusinstead of being a unique solution to the problem the solutionof a multi-objective problem is a possibly infinite set ofPareto points [31] Compared to studies working with a singleobjective a multi-objective approach is more reasonable andmore practical in terms of actual applications such that ithas been considered by different researchers in the literatureFor example a multi-objective programmingmodel has beenproposed [32] to minimise the total cost of plant and itsdistribution centres and also to maximise customer deliverytime In addition Pishvaee et al [33] proposed a biobjectivemodel that involved a trade-off between the total costs and the

responsiveness of the supply chain network Alcada-Almeidaet al [34] proposed a multi-objective programming methodto determine the most appropriate location allocation of haz-ardous material incineration facilities and to make a mutualbalance between the societal economic and environmentalimpacts Paksoya et al [35] conducted another related studyin which the green impact on a close-looped logistics supplychain network was investigated The authors attempted todecrease and prevent greater CO

2gas emissions and also

to encourage customers to use recyclable goods They havepresented different transportation choices between echelonsaccording to CO

2emissions Another related study was

conducted byWang et al [36] who presented amathematicalmodel to minimize CO

2emissions and cost to design a green

supply chain They defined environmental protection levelvariable and CO

2emissions between supply chain facilities

Among recently published studies gaps can be found inadaptation research and there is a deficit of studies in mostdeveloping regions This is especially true for determiningalternative solutions for decision makers that can makethe trade-off between environmental impacts and economicactivities based on Pareto optimality that is known as quite anew problem in supply chain network design [37] Amongthe related researches that have been presented on themathematical model in green supply chain Wang et al [36]used the Pareto set concept to make a solution set butthey neglected the truck type for transporting material andproducts in modeling Another multi-objective green supplychain modeling has been done by Paksoya et al [35] whichfocuses on optimizing CO

2and cost but it neglects to present

the Pareto optimality set In this research a multi-objectivemixed-integer formulation is provided for the supply chainnetwork design problem The multi-objective model explic-itly considers environmental concerns by introducing thethree groups of tier-1 suppliers as the new concept of supplychain thinking which is compatible with the automotive sup-ply chain logistics Furthermore in this research choosingtruck type and facility location among the potential locationshave been considered to find Pareto optimal set solutionsand trade-off cost and CO

2emission in the logistics network

Another contribution of the research is implementation ofthe multi-objective gravitational search algorithm (MOGSA)as a new swarm algorithm for solving complex logisticalmathematicalmodels to find a set of Pareto optimal solutionsThe result obtained can be easily applied to the decisionsupport systems which the industry needs Furthermore thelast contribution of this research is the evaluation of theproposed model by using a real case study for one of thelargest car manufacturers in the Middle East

The rest of this research is organised as follows In thenext section the problem is presented in detail includingmathematical modelling with two objectives cost and CO

2

minimisation In Section 3 GSA and MOGSA theories areintroduced For a real case the objective functions are solvedby using MOGSA in MATLAB software in Section 4 Finallythe paper is concluded in Section 5 and some interestingpossible future research is also introduced

Mathematical Problems in Engineering 3

2 Problem Modeling

In this section the mathematical formulation of the designmodel for a supply chain network will be presented indetail In automotive industries the general structure ofthe proposed supply chain logistic network is illustrated inFigure 1 In the forward direction the suppliers are respon-sible for providing all the parts needed by the car manu-facturing facilities Suppliers have been grouped into threecategories as tier-1 suppliersThe first group of tier-1 suppliersis responsible for supplyingmetal-based parts such as wheelsthe main body and springs The second group of suppliersis responsible for supplying electric parts such as wiringharnesses instrument panels and engine control units Thelast tier-1 group supplies plastics- and latex-based partssuch as tires interior parts and dashboard panels The newproducts are conveyed fromplants to retailers via distributioncentres to meet customer demand All assumptions andlimitations sets parameters and variables are defined andfollow a model-detailed description

21 Assumptions and Limitations

(i) The model is deterministic so that demand is fixedand known and must be satisfied

(ii) The model selects plants warehouses and distribu-tions from a given set of potential locations More-over the model also selects the supplier and retailerfrom a set of fixed locations

(iii) Suppliers plants warehouse and distributors have acapacity limit for the product

(iv) Cost is independent of product load but is varied forany kind of transportation vehicles such as the truckroad classes

(v) Carbon emission is dependent on product load(capacity of truck type) and also is varied for any kindof transportation vehicles

(vi) All truck types in transportation are turned off whenunloading product

(vii) Theflow is only allowed to be transferred between twosequential echelons

(1) Indices and Sets

119868 = 1 119873V set of fixed location of metal-partssuppliersrsquo tier-1119869 = 1 119873

119892 set of fixed location of electric-parts

suppliersrsquo tier-1119873 = 1 119873

119889 set of fixed location of plastic-parts

suppliersrsquo tier-1119870 = 1 119873

119891 set of potential available for plants

119871 = 1 119873119908

set of potential available for ware-houses and distributors119872 = 1 119873

119903 set of fixed location of retailers

(2) Parameters

120576119905 maximum capacity in truck type 119905 (tone) 119905 isin 119879

120591119905 transportation cost mode 119905 per km ($km) 119905 isin 119879

120593119905 carbon dioxide emission by truck type 119905 (gram

km) 119905 isin 119879120597V119891119894119896 distance between metal-parts suppliersrsquo tier-1 119894 to

plant 119896 (km) that 119894 isin 119868 119896 isin 119870

120597119892119891

119895119896 distance between electric-parts suppliersrsquo tier-1 119895

to plant 119896 (km) that 119894 isin 119868 119896 isin 119870

120597119889119891

119899119896 distance between plastic-parts suppliersrsquo tier-1 119899

to plant 119896 (km) that 119899 isin 119873 119896 isin 119870

120597119891119908

119896119897 distance between plant 119896 to distributor 119897 (km)

that 119896 isin 119870 119897 isin 119871120597119908119903

119897119898 distance between warehouse and distributor 119897 to

retailer 119898 (km) that 119897 isin 119871 119896 isin 119870120596119901 weight of product (tone)

120588119898 rate of metal parts contribution in the product (0-

1)120588119890 rate of electrical parts contribution in the product

(0-1)120588119901 rate of plastic parts contribution in the product (0-

1)120578119898 product demand retailer 119898

120583V119894 maximum capacity metal-parts suppliersrsquo tier-1 119894

(tone) 119894 isin 119868120583119892

119895 maximum capacity electric-parts suppliersrsquo tier-1

119895 isin 119869120583119889

119899 maximum capacity plastic-parts suppliersrsquo tier-1 119899

(tone) 119899 isin 119873120583119891

119896 maximumproduct capacity for plant 119896 that 119896 isin 119870

120583119908

119897 maximum product capacity for warehouse and

distributor 119897 that 119897 isin 119871120574119891

119896 unit Production and material handling cost of

product in plant 119896 ($) that 119896 isin 119870120574119908

119897 unit operation cost of product in distributor 119897 ($)

that 119897 isin 119871120574119903

119898 unit operation cost of product in retailer119898 ($) that

119898 isin 119872120572119891

119896 fixed cost of opening plant 119896 ($) that 119896 isin 119870

120572119908

119897 fixed cost of opening distribution center 119897 ($) that

119897 isin 119871

(3) Decision Variables

1198831

119894119896119905 quantity of product produced in plant 119896 using

metal parts of metal-parts suppliersrsquo tier-1 119894 by trucktype 119905 119896 isin 119870 119894 isin 119868 119905 isin 1198791198832

119895119896119905 quantity of product produced in plant 119896 using

electrical parts of electric-parts suppliersrsquo tier-1 119895 bytruck type 119905 119896 isin 119870 119895 isin 119869 119905 isin 119879

4 Mathematical Problems in Engineering

Supplier group-1

Plant Distribution center Retailers

Potential available facilitiesSet of available facilities

Fossil fuel

Supplier group-3

Supplier group-2

CO2 emission

Figure 1 General structure of automotive supply chain logistic network

1198833

119899119896119905 quantity of product produced in plant 119896 using

plastic parts of plastic-parts suppliersrsquo tier-1 119899 by trucktype 119905 119899 isin 119873 119896 isin 119870 119905 isin 1198791198834

119896119897119905 quantity of product shipped from plant 119896 todis-

tribution center 119897 by truck type 119905 119896 isin 119870 119897 isin 119871 119905 isin 1198791198835

119897119898119905 quantity of product shipped fromdistribution

center 119897 to retailer 119898 by truck type 119905 119897 isin 119871 119898 isin 119872119905 isin 119879

1198841

119896=

1 if factory 119896 is opened 119896 isin 119870

0 otherwise

1198842

119897=

1 if warehouse 119897 is opened 119897 isin 119871

0 otherwise

(1)

Objective Functions With regards to the above descriptionthe first objective function that minimizes the total cost ofsupply chain network is as follows

Total Cost = Transportation Cost + Operation Cost

+ Initial Facility Cost(2)

In objective function (3) the first five terms define thetotal transportation cost of the products between each

pair suppliers plants and distributors Next two terms arereferred to operation cost in plants and distributors Fixedsetup cost for opening the facilities (factories distributor cen-ters resp) can be defined by the last two terms

Total Cost = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120591119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119890

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120591119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120591119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120591119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120591119905

+ sum

119905

sum

119897

sum

119896

1199094

119896119897119905sdot 120574119891

119896+ sum

119905

sum

119898

sum

119897

1199095

119897119898119905sdot 120574119908

119897

+ sum

119896isin119870

120572119891

119896sdot 1198841

119896+ sum

119897isin119871

120572119908

119897sdot 1198842

119897

(3)

Mathematical Problems in Engineering 5

The second objective function that should be minimizedis total CO

2emission in supply chain network due to trans-

portation between all facilities

Carbon Emission = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120593119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119899

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120593119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120593119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120593119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120593119905

(4)

ConstrainsThis section is a representation of the constraintsof the proposed model

sum

119905

sum

119897

1199095

119897119898119905= 120578119898

forall119898 isin 119872 (5)

120596119901

sdot 120588119898

sdot sum

119906

sum

119905

sum

119894

1199091

119894119896119905+ 120596119901

sdot 120588119890

sdot sum

119905

sum

119895

1199092

119895119896119905

+ 120596119901

sdot 120588119901

sum

119905

sum

119899

1199093

119899119896119905= sum

119905

sum

119897

120596119901

sdot 1199094

119896119897119905forall119896 isin 119870

(6)

sum

119905

sum

119896

1199094

119897119896119905119906= sum

119905

sum

119898

1199095

119897119898119905forall119897 isin 119871 (7)

120588119898

sdot sum

119905

sum

119896

1199091

119894119896119905le 120583

V119894

forall119894 isin 119868 (8)

120588119890

sdot sum

119905

sum

119896

1199092

119895119896119905le 120583119889

119895forall119895 isin 119869 (9)

120588119901

sdot sum

119905

sum

119896

1199093

119899119896119905le 120583119889

119895forall119899 isin 119873 (10)

sum

119905

sum

119897

1199094

119896119897119905119906le 120583119891

119896sdot 1198841

119896forall119896 isin 119870 (11)

sum

119905

sum

119898

1199095

119897119898119905119906le 120583119908

119897sdot 1198842

119897forall119897 isin 119871 (12)

1198831

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905ge 0 forall119894 119895 119896 119897 119898 119905 (13)

1198841

119896 1198842

119897isin 0 1 forall119896 119897 (14)

Equation (5) states that for each product the flow exitingthe distribution centers must satisfy the demand of all cus-tomers Equation (6) shows that for each product the sum ofthe flow entering each plant from all suppliers is equal to theflow leaving the facility to the distribution centers Equation(7) insures that the sum of the product flow entering each

distribution centre from all plants is equal to the flow leavingthis distribution centre to retailers Equation (8) ensures thatfor each product the sum of the flow exiting eachmetal-partssupplier to all plants does not exceed the supplierrsquos capacityEquation (9) ensures that for each product the sum of theflow exiting each electric-parts supplier to all plants doesnot exceed the supplierrsquos capacity Equation (10) ensures thatfor each product the sum of the flow exiting each electric-parts supplier to all plants does not exceed the supplierrsquoscapacity Equation (11) states that the sum of the flow exitingeach plant to all distribution counters does not exceed theplantrsquos capacity And finally (12) shows that the sum of theflow exiting from each distribution centre to all customersdoes not exceed the distributorrsquos capacity Constraints (13)and (14) impose binary and nonnegativity restrictions on thecorresponding decision variables respectively

3 Solving Method

In this section we first introduce the Gravitational SearchAlgorithm (GSA) and then explain the extended GSA whichis used to solve multi-objective optimisation known as themulti-objective gravitational search algorithm (MOGSA)

31 GSA One of the novel meta-heuristic stochastic opti-misation algorithms inspired by the law of gravity and massinteractions is called the Gravitational Search Algorithm(GSA) [38] In GSA the individuals are called agents and area collection of objects that interact with each other based onNewtonian gravity and the laws of motion The agents shareinformation using gravitational force to guide the searchtowards the best location in the search spaceThe high perfor-mance and the global search ability of GSA in solving variousnonlinear functions infers from the results of experimentsundertaken previously [38ndash42] At GSA searcher agents are acollection of objects All these objects attract one another bygravitational force and this force causes a global movementof all objects toward the objects with heavier masses asFigure 2 shows Hence objects cooperate using a direct formof communication through gravitational force The heavyobjects that correspond to good solutions move more slowlythan the lighter objects which guarantee the exploitationability of the algorithm

The position of the objects corresponds to a solution ofthe problem and itsmass is determined using amap of fitnessfunction By changing the velocities over time the agents arelikely to move towards the global optima In this algorithmthere are objects The position of the 119894th agent is defined by

119883119894

= (1199091

119894 119909

119889

119894 119909

119899

119894) 119894 = 1 2 119873 (15)

where 119909119889

119894presents the position of the 119894th agent in the 119889th

dimension Based on theNewton gravitation theory the forceacting on the 119894th object from the 119895th object is

119865119889

119894119895(119905) = 119866 (119905)

119872119894(119905) times 119872

119895(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905)) (16)

119866 (119905) = 1198660

lowast exp(minus119886119905

119879) (17)

6 Mathematical Problems in Engineering

Figure 2 Each object accelerates toward the result force that attractsit from the other objects

where 119872119894is the mass of object 119894 119872

119895is the mass of object 119895

and 119866(119905) is the gravitational constant at time 119905 Notably thegravitational constant 119866(119905) is important in determining theperformance of GSA 119866(119905) is defined as a decreasing function(17) of time which is set to 119866

0at the beginning and decreased

exponentially to zero as time progressesTo compute the acceleration of an agent total forces from

a set of heavier masses that apply on it should be consideredbased on law of gravity (18) which is followed by calculationof agent acceleration using law of motion (19) Afterwardthe next velocity of an agent is calculated as a fraction ofits current velocity added to its acceleration (20) Then itsposition could be calculated using

119865119889

119894(119905) = sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905) 119872119894(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

(18)

119886119889

119894(119905) =

119865119889

119894(119905)

119872119894(119905)

= sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905)

119877119894119895

(119905) + 120576

times (119909119889

119895(119905) minus 119909

119889

119894(119905))

(19)

V119889119894

(119905 + 1) = rand119894

times V119889119894

(119905) + 119886119889

119894(119905) (20)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1) (21)

where rand119894and rand

119895are two uniform randoms in the

interval [0 1] 120576 is a small value and 119877119894119895

(119905) is the Euclidiandistance between two agents 119894 and 119895 defined as 119877

119894119895(119905) =

119883119894(119905) 119883119895(119905)2 119896best is the set of first 119870 agents with the best

fitness value and biggest mass 119896best is a function of timeinitialized to 119870

0at the beginning and decreasing with time

Here 1198700is set to 119873 (total number of agents) and is decreased

linearly to 1Based on [39] the mass of each agent is calculated after

computing current populationrsquos fitness as follows

119902119894(119905) =

fit119894(119905) minus worst (119905)

best (119905) minus worst (119905)

119872119894(119905) =

119902119894(119905)

sum119904

119895=1119902119895

(119905)

(22)

where 119872119894(119905) and fit

119894(119905) represent the mass and the fitness

value of the agent 119894 at 119905 and worst(119905) and best(119905) are definedas follows (for a minimization problem)

best (119905) = min119895isin1119904

fit119895

(119905)

worst (119905) = max119895isin1119904

fit119895

(119905)

(23)

Figure 3 illustrates the flow chart of the GSA

32 MOGSA Suppose that 119883 = (1199091 1199092 119909

119899) is the vector

of decision variable 119891119894

119877119899

rarr 119877 119894 = 1 119896 are theobjective functions and 119892

119894 ℎ119894

119877119899

rarr 119877 119894 = 1 119898 119895 =

1 119901 are the constraints of the problem Pareto domain-based algorithmMOGSA that is used in this paper presentsthis problem and definition as follows

(i) Minimize 119891(119883) = [1198911(119883) 119891

2(119883) 119891

119896(119883)]

(ii) Subject to 119892119894(119883) le 0 119894 = 1 2 119898

(iii) ℎ119895(119883) = 0 119895 = 1 2 119901

Definition 1 Given two vectors 119883 119884 isin 119877119896 we say that 119883 le 119884

if 119909119894

le 119910119894 119894 = 1 119896 and that 119883 dominates 119884 (or presented

by 119883 ≺ 119884) if 119883 le 119884 and 119883 = 119884

Definition 2 We say 119883 isin 120594 sub 119877119896 is nondominated with

respect to 120594 if another 1198831015840

isin 120594 does not exist such that119891(1198831015840) ≺ 119891(119883)

Selected non-dominated solutions are stored in an exter-nal archive Archive updates in each iteration of the algo-rithm When the running algorithm is terminated externalarchive presents final results

Recently the multi-objective gravitational search algo-rithm (MOGSA) [43ndash45] has been developed to solve multi-objective problems by using the UniformMutation Operator[46] and an Elitist Policy [47] as well-known operators ingenetic algorithm (GA) into GSA MOGSA also uses anexternal archive to reserve the non-dominated solutionsand update it the same as Simple Multi-Objective PSO(SMOPSO) [47] The first operator selects one of the objectrsquosdimensions randomly and adds a signed random term to itThe total value should be in the permitted range otherwiseit will be truncated to its nearest feasible value The non-dominated solutions found are stored by the Elitist Policyin an archive that has a grid structure The grid structure iscreated as follows each dimension in the objective space isdivided into 2

119899119894 equal divisions where 119894 denotes dimensionindex and hence for a 119896th objective optimisation problemtherewill be 2

(sum119896

119894=1119899119894) different segments As long as the archive

is not full new non-dominated solutions are added to itAs soon as the number of elements in the archive meets itslimit one of the elements should be omitted from it Thestrategy to remove an element from the archive is to findthe most crowded hypercube and randomly select one of theelements in it to extract For this purpose objective functionsand solution space are divided into hyper-rectangles When

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

they design supply chain networks The network designproblem is one of the most comprehensive strategic deci-sion problems that should be optimised for long-term effi-cient operations throughout the supply chain The problemincludes a wide range of mathematical modelling rangingfrom linear deterministic models [5ndash7] to complex nonlinearstochastic models [8] In the literature several attempts havebeen made to solve the network design problems of supplynetworks and these studies have been surveyed by BeamonandFernandes [9] Chopra [10] andPeidro et al [11] Scholarshave attempted to develop the classical quantitative model byincorporating various factors such as transportation modes[12 13] vehicle speed [14] and greenhouse gasses [15 16]

Despite the volume of work undertaken in respect to thesupply chain network design problem themodelling is usual-ly done as a single objective problem [17 18]while the ldquodesignrdquoin real cases typically involves compromises and trade-offsbetween different incompatible objectives For example theoptimum design for a supply chain network should notonly concentrate on total cost minimisation but it also hasto satisfy other objectives such as delivery flexibility [19]minimising waste [20] and minimising CO

2emissions [21]

Satisfying environmental issues and minimizing the totalvariable and fixed costs causes the logistics network designproblem to be an NP-hard problem [22 23] Solving thisproblem on a large scale by exact algorithms is very timeconsuming and sometimes in the practical case is impossibleTherefore many heuristics and metaheuristics have beendeveloped to obtain optimal solutions for these kinds of prob-lems [24] In this regard during the last decade the imple-mentation of meta-heuristics instead of exact and exhaustivemethods has been growing Approaches such as bioinspiredalgorithms such as ant colony optimization (ACO) [25ndash27] particle swarm optimization (PSO) [28] artificial weedcolonies [29] and also evolutionary algorithms such asgenetic algorithms (GA) have been used to solve a varietyof multi-objective problems in production and supply chainmanagement specially in complex environment [29 30]

Recently the consideration of supply network designwith multi-objective optimisation (generally incompatibleobjectives) is a new trend worthy of study The solutionto the above problem is not unique but a set of Paretooptimal points A Pareto optimal solution is the best solutionvector out of several numbers of solution vectors that couldbe achieved without disadvantaging other objectives Eachsolution set is known as nondominated solution set Anarray containing all the Pareto optimal solutions of a multi-objective problem is called the Pareto optimal set Thusinstead of being a unique solution to the problem the solutionof a multi-objective problem is a possibly infinite set ofPareto points [31] Compared to studies working with a singleobjective a multi-objective approach is more reasonable andmore practical in terms of actual applications such that ithas been considered by different researchers in the literatureFor example a multi-objective programmingmodel has beenproposed [32] to minimise the total cost of plant and itsdistribution centres and also to maximise customer deliverytime In addition Pishvaee et al [33] proposed a biobjectivemodel that involved a trade-off between the total costs and the

responsiveness of the supply chain network Alcada-Almeidaet al [34] proposed a multi-objective programming methodto determine the most appropriate location allocation of haz-ardous material incineration facilities and to make a mutualbalance between the societal economic and environmentalimpacts Paksoya et al [35] conducted another related studyin which the green impact on a close-looped logistics supplychain network was investigated The authors attempted todecrease and prevent greater CO

2gas emissions and also

to encourage customers to use recyclable goods They havepresented different transportation choices between echelonsaccording to CO

2emissions Another related study was

conducted byWang et al [36] who presented amathematicalmodel to minimize CO

2emissions and cost to design a green

supply chain They defined environmental protection levelvariable and CO

2emissions between supply chain facilities

Among recently published studies gaps can be found inadaptation research and there is a deficit of studies in mostdeveloping regions This is especially true for determiningalternative solutions for decision makers that can makethe trade-off between environmental impacts and economicactivities based on Pareto optimality that is known as quite anew problem in supply chain network design [37] Amongthe related researches that have been presented on themathematical model in green supply chain Wang et al [36]used the Pareto set concept to make a solution set butthey neglected the truck type for transporting material andproducts in modeling Another multi-objective green supplychain modeling has been done by Paksoya et al [35] whichfocuses on optimizing CO

2and cost but it neglects to present

the Pareto optimality set In this research a multi-objectivemixed-integer formulation is provided for the supply chainnetwork design problem The multi-objective model explic-itly considers environmental concerns by introducing thethree groups of tier-1 suppliers as the new concept of supplychain thinking which is compatible with the automotive sup-ply chain logistics Furthermore in this research choosingtruck type and facility location among the potential locationshave been considered to find Pareto optimal set solutionsand trade-off cost and CO

2emission in the logistics network

Another contribution of the research is implementation ofthe multi-objective gravitational search algorithm (MOGSA)as a new swarm algorithm for solving complex logisticalmathematicalmodels to find a set of Pareto optimal solutionsThe result obtained can be easily applied to the decisionsupport systems which the industry needs Furthermore thelast contribution of this research is the evaluation of theproposed model by using a real case study for one of thelargest car manufacturers in the Middle East

The rest of this research is organised as follows In thenext section the problem is presented in detail includingmathematical modelling with two objectives cost and CO

2

minimisation In Section 3 GSA and MOGSA theories areintroduced For a real case the objective functions are solvedby using MOGSA in MATLAB software in Section 4 Finallythe paper is concluded in Section 5 and some interestingpossible future research is also introduced

Mathematical Problems in Engineering 3

2 Problem Modeling

In this section the mathematical formulation of the designmodel for a supply chain network will be presented indetail In automotive industries the general structure ofthe proposed supply chain logistic network is illustrated inFigure 1 In the forward direction the suppliers are respon-sible for providing all the parts needed by the car manu-facturing facilities Suppliers have been grouped into threecategories as tier-1 suppliersThe first group of tier-1 suppliersis responsible for supplyingmetal-based parts such as wheelsthe main body and springs The second group of suppliersis responsible for supplying electric parts such as wiringharnesses instrument panels and engine control units Thelast tier-1 group supplies plastics- and latex-based partssuch as tires interior parts and dashboard panels The newproducts are conveyed fromplants to retailers via distributioncentres to meet customer demand All assumptions andlimitations sets parameters and variables are defined andfollow a model-detailed description

21 Assumptions and Limitations

(i) The model is deterministic so that demand is fixedand known and must be satisfied

(ii) The model selects plants warehouses and distribu-tions from a given set of potential locations More-over the model also selects the supplier and retailerfrom a set of fixed locations

(iii) Suppliers plants warehouse and distributors have acapacity limit for the product

(iv) Cost is independent of product load but is varied forany kind of transportation vehicles such as the truckroad classes

(v) Carbon emission is dependent on product load(capacity of truck type) and also is varied for any kindof transportation vehicles

(vi) All truck types in transportation are turned off whenunloading product

(vii) Theflow is only allowed to be transferred between twosequential echelons

(1) Indices and Sets

119868 = 1 119873V set of fixed location of metal-partssuppliersrsquo tier-1119869 = 1 119873

119892 set of fixed location of electric-parts

suppliersrsquo tier-1119873 = 1 119873

119889 set of fixed location of plastic-parts

suppliersrsquo tier-1119870 = 1 119873

119891 set of potential available for plants

119871 = 1 119873119908

set of potential available for ware-houses and distributors119872 = 1 119873

119903 set of fixed location of retailers

(2) Parameters

120576119905 maximum capacity in truck type 119905 (tone) 119905 isin 119879

120591119905 transportation cost mode 119905 per km ($km) 119905 isin 119879

120593119905 carbon dioxide emission by truck type 119905 (gram

km) 119905 isin 119879120597V119891119894119896 distance between metal-parts suppliersrsquo tier-1 119894 to

plant 119896 (km) that 119894 isin 119868 119896 isin 119870

120597119892119891

119895119896 distance between electric-parts suppliersrsquo tier-1 119895

to plant 119896 (km) that 119894 isin 119868 119896 isin 119870

120597119889119891

119899119896 distance between plastic-parts suppliersrsquo tier-1 119899

to plant 119896 (km) that 119899 isin 119873 119896 isin 119870

120597119891119908

119896119897 distance between plant 119896 to distributor 119897 (km)

that 119896 isin 119870 119897 isin 119871120597119908119903

119897119898 distance between warehouse and distributor 119897 to

retailer 119898 (km) that 119897 isin 119871 119896 isin 119870120596119901 weight of product (tone)

120588119898 rate of metal parts contribution in the product (0-

1)120588119890 rate of electrical parts contribution in the product

(0-1)120588119901 rate of plastic parts contribution in the product (0-

1)120578119898 product demand retailer 119898

120583V119894 maximum capacity metal-parts suppliersrsquo tier-1 119894

(tone) 119894 isin 119868120583119892

119895 maximum capacity electric-parts suppliersrsquo tier-1

119895 isin 119869120583119889

119899 maximum capacity plastic-parts suppliersrsquo tier-1 119899

(tone) 119899 isin 119873120583119891

119896 maximumproduct capacity for plant 119896 that 119896 isin 119870

120583119908

119897 maximum product capacity for warehouse and

distributor 119897 that 119897 isin 119871120574119891

119896 unit Production and material handling cost of

product in plant 119896 ($) that 119896 isin 119870120574119908

119897 unit operation cost of product in distributor 119897 ($)

that 119897 isin 119871120574119903

119898 unit operation cost of product in retailer119898 ($) that

119898 isin 119872120572119891

119896 fixed cost of opening plant 119896 ($) that 119896 isin 119870

120572119908

119897 fixed cost of opening distribution center 119897 ($) that

119897 isin 119871

(3) Decision Variables

1198831

119894119896119905 quantity of product produced in plant 119896 using

metal parts of metal-parts suppliersrsquo tier-1 119894 by trucktype 119905 119896 isin 119870 119894 isin 119868 119905 isin 1198791198832

119895119896119905 quantity of product produced in plant 119896 using

electrical parts of electric-parts suppliersrsquo tier-1 119895 bytruck type 119905 119896 isin 119870 119895 isin 119869 119905 isin 119879

4 Mathematical Problems in Engineering

Supplier group-1

Plant Distribution center Retailers

Potential available facilitiesSet of available facilities

Fossil fuel

Supplier group-3

Supplier group-2

CO2 emission

Figure 1 General structure of automotive supply chain logistic network

1198833

119899119896119905 quantity of product produced in plant 119896 using

plastic parts of plastic-parts suppliersrsquo tier-1 119899 by trucktype 119905 119899 isin 119873 119896 isin 119870 119905 isin 1198791198834

119896119897119905 quantity of product shipped from plant 119896 todis-

tribution center 119897 by truck type 119905 119896 isin 119870 119897 isin 119871 119905 isin 1198791198835

119897119898119905 quantity of product shipped fromdistribution

center 119897 to retailer 119898 by truck type 119905 119897 isin 119871 119898 isin 119872119905 isin 119879

1198841

119896=

1 if factory 119896 is opened 119896 isin 119870

0 otherwise

1198842

119897=

1 if warehouse 119897 is opened 119897 isin 119871

0 otherwise

(1)

Objective Functions With regards to the above descriptionthe first objective function that minimizes the total cost ofsupply chain network is as follows

Total Cost = Transportation Cost + Operation Cost

+ Initial Facility Cost(2)

In objective function (3) the first five terms define thetotal transportation cost of the products between each

pair suppliers plants and distributors Next two terms arereferred to operation cost in plants and distributors Fixedsetup cost for opening the facilities (factories distributor cen-ters resp) can be defined by the last two terms

Total Cost = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120591119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119890

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120591119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120591119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120591119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120591119905

+ sum

119905

sum

119897

sum

119896

1199094

119896119897119905sdot 120574119891

119896+ sum

119905

sum

119898

sum

119897

1199095

119897119898119905sdot 120574119908

119897

+ sum

119896isin119870

120572119891

119896sdot 1198841

119896+ sum

119897isin119871

120572119908

119897sdot 1198842

119897

(3)

Mathematical Problems in Engineering 5

The second objective function that should be minimizedis total CO

2emission in supply chain network due to trans-

portation between all facilities

Carbon Emission = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120593119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119899

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120593119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120593119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120593119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120593119905

(4)

ConstrainsThis section is a representation of the constraintsof the proposed model

sum

119905

sum

119897

1199095

119897119898119905= 120578119898

forall119898 isin 119872 (5)

120596119901

sdot 120588119898

sdot sum

119906

sum

119905

sum

119894

1199091

119894119896119905+ 120596119901

sdot 120588119890

sdot sum

119905

sum

119895

1199092

119895119896119905

+ 120596119901

sdot 120588119901

sum

119905

sum

119899

1199093

119899119896119905= sum

119905

sum

119897

120596119901

sdot 1199094

119896119897119905forall119896 isin 119870

(6)

sum

119905

sum

119896

1199094

119897119896119905119906= sum

119905

sum

119898

1199095

119897119898119905forall119897 isin 119871 (7)

120588119898

sdot sum

119905

sum

119896

1199091

119894119896119905le 120583

V119894

forall119894 isin 119868 (8)

120588119890

sdot sum

119905

sum

119896

1199092

119895119896119905le 120583119889

119895forall119895 isin 119869 (9)

120588119901

sdot sum

119905

sum

119896

1199093

119899119896119905le 120583119889

119895forall119899 isin 119873 (10)

sum

119905

sum

119897

1199094

119896119897119905119906le 120583119891

119896sdot 1198841

119896forall119896 isin 119870 (11)

sum

119905

sum

119898

1199095

119897119898119905119906le 120583119908

119897sdot 1198842

119897forall119897 isin 119871 (12)

1198831

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905ge 0 forall119894 119895 119896 119897 119898 119905 (13)

1198841

119896 1198842

119897isin 0 1 forall119896 119897 (14)

Equation (5) states that for each product the flow exitingthe distribution centers must satisfy the demand of all cus-tomers Equation (6) shows that for each product the sum ofthe flow entering each plant from all suppliers is equal to theflow leaving the facility to the distribution centers Equation(7) insures that the sum of the product flow entering each

distribution centre from all plants is equal to the flow leavingthis distribution centre to retailers Equation (8) ensures thatfor each product the sum of the flow exiting eachmetal-partssupplier to all plants does not exceed the supplierrsquos capacityEquation (9) ensures that for each product the sum of theflow exiting each electric-parts supplier to all plants doesnot exceed the supplierrsquos capacity Equation (10) ensures thatfor each product the sum of the flow exiting each electric-parts supplier to all plants does not exceed the supplierrsquoscapacity Equation (11) states that the sum of the flow exitingeach plant to all distribution counters does not exceed theplantrsquos capacity And finally (12) shows that the sum of theflow exiting from each distribution centre to all customersdoes not exceed the distributorrsquos capacity Constraints (13)and (14) impose binary and nonnegativity restrictions on thecorresponding decision variables respectively

3 Solving Method

In this section we first introduce the Gravitational SearchAlgorithm (GSA) and then explain the extended GSA whichis used to solve multi-objective optimisation known as themulti-objective gravitational search algorithm (MOGSA)

31 GSA One of the novel meta-heuristic stochastic opti-misation algorithms inspired by the law of gravity and massinteractions is called the Gravitational Search Algorithm(GSA) [38] In GSA the individuals are called agents and area collection of objects that interact with each other based onNewtonian gravity and the laws of motion The agents shareinformation using gravitational force to guide the searchtowards the best location in the search spaceThe high perfor-mance and the global search ability of GSA in solving variousnonlinear functions infers from the results of experimentsundertaken previously [38ndash42] At GSA searcher agents are acollection of objects All these objects attract one another bygravitational force and this force causes a global movementof all objects toward the objects with heavier masses asFigure 2 shows Hence objects cooperate using a direct formof communication through gravitational force The heavyobjects that correspond to good solutions move more slowlythan the lighter objects which guarantee the exploitationability of the algorithm

The position of the objects corresponds to a solution ofthe problem and itsmass is determined using amap of fitnessfunction By changing the velocities over time the agents arelikely to move towards the global optima In this algorithmthere are objects The position of the 119894th agent is defined by

119883119894

= (1199091

119894 119909

119889

119894 119909

119899

119894) 119894 = 1 2 119873 (15)

where 119909119889

119894presents the position of the 119894th agent in the 119889th

dimension Based on theNewton gravitation theory the forceacting on the 119894th object from the 119895th object is

119865119889

119894119895(119905) = 119866 (119905)

119872119894(119905) times 119872

119895(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905)) (16)

119866 (119905) = 1198660

lowast exp(minus119886119905

119879) (17)

6 Mathematical Problems in Engineering

Figure 2 Each object accelerates toward the result force that attractsit from the other objects

where 119872119894is the mass of object 119894 119872

119895is the mass of object 119895

and 119866(119905) is the gravitational constant at time 119905 Notably thegravitational constant 119866(119905) is important in determining theperformance of GSA 119866(119905) is defined as a decreasing function(17) of time which is set to 119866

0at the beginning and decreased

exponentially to zero as time progressesTo compute the acceleration of an agent total forces from

a set of heavier masses that apply on it should be consideredbased on law of gravity (18) which is followed by calculationof agent acceleration using law of motion (19) Afterwardthe next velocity of an agent is calculated as a fraction ofits current velocity added to its acceleration (20) Then itsposition could be calculated using

119865119889

119894(119905) = sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905) 119872119894(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

(18)

119886119889

119894(119905) =

119865119889

119894(119905)

119872119894(119905)

= sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905)

119877119894119895

(119905) + 120576

times (119909119889

119895(119905) minus 119909

119889

119894(119905))

(19)

V119889119894

(119905 + 1) = rand119894

times V119889119894

(119905) + 119886119889

119894(119905) (20)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1) (21)

where rand119894and rand

119895are two uniform randoms in the

interval [0 1] 120576 is a small value and 119877119894119895

(119905) is the Euclidiandistance between two agents 119894 and 119895 defined as 119877

119894119895(119905) =

119883119894(119905) 119883119895(119905)2 119896best is the set of first 119870 agents with the best

fitness value and biggest mass 119896best is a function of timeinitialized to 119870

0at the beginning and decreasing with time

Here 1198700is set to 119873 (total number of agents) and is decreased

linearly to 1Based on [39] the mass of each agent is calculated after

computing current populationrsquos fitness as follows

119902119894(119905) =

fit119894(119905) minus worst (119905)

best (119905) minus worst (119905)

119872119894(119905) =

119902119894(119905)

sum119904

119895=1119902119895

(119905)

(22)

where 119872119894(119905) and fit

119894(119905) represent the mass and the fitness

value of the agent 119894 at 119905 and worst(119905) and best(119905) are definedas follows (for a minimization problem)

best (119905) = min119895isin1119904

fit119895

(119905)

worst (119905) = max119895isin1119904

fit119895

(119905)

(23)

Figure 3 illustrates the flow chart of the GSA

32 MOGSA Suppose that 119883 = (1199091 1199092 119909

119899) is the vector

of decision variable 119891119894

119877119899

rarr 119877 119894 = 1 119896 are theobjective functions and 119892

119894 ℎ119894

119877119899

rarr 119877 119894 = 1 119898 119895 =

1 119901 are the constraints of the problem Pareto domain-based algorithmMOGSA that is used in this paper presentsthis problem and definition as follows

(i) Minimize 119891(119883) = [1198911(119883) 119891

2(119883) 119891

119896(119883)]

(ii) Subject to 119892119894(119883) le 0 119894 = 1 2 119898

(iii) ℎ119895(119883) = 0 119895 = 1 2 119901

Definition 1 Given two vectors 119883 119884 isin 119877119896 we say that 119883 le 119884

if 119909119894

le 119910119894 119894 = 1 119896 and that 119883 dominates 119884 (or presented

by 119883 ≺ 119884) if 119883 le 119884 and 119883 = 119884

Definition 2 We say 119883 isin 120594 sub 119877119896 is nondominated with

respect to 120594 if another 1198831015840

isin 120594 does not exist such that119891(1198831015840) ≺ 119891(119883)

Selected non-dominated solutions are stored in an exter-nal archive Archive updates in each iteration of the algo-rithm When the running algorithm is terminated externalarchive presents final results

Recently the multi-objective gravitational search algo-rithm (MOGSA) [43ndash45] has been developed to solve multi-objective problems by using the UniformMutation Operator[46] and an Elitist Policy [47] as well-known operators ingenetic algorithm (GA) into GSA MOGSA also uses anexternal archive to reserve the non-dominated solutionsand update it the same as Simple Multi-Objective PSO(SMOPSO) [47] The first operator selects one of the objectrsquosdimensions randomly and adds a signed random term to itThe total value should be in the permitted range otherwiseit will be truncated to its nearest feasible value The non-dominated solutions found are stored by the Elitist Policyin an archive that has a grid structure The grid structure iscreated as follows each dimension in the objective space isdivided into 2

119899119894 equal divisions where 119894 denotes dimensionindex and hence for a 119896th objective optimisation problemtherewill be 2

(sum119896

119894=1119899119894) different segments As long as the archive

is not full new non-dominated solutions are added to itAs soon as the number of elements in the archive meets itslimit one of the elements should be omitted from it Thestrategy to remove an element from the archive is to findthe most crowded hypercube and randomly select one of theelements in it to extract For this purpose objective functionsand solution space are divided into hyper-rectangles When

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

2 Problem Modeling

In this section the mathematical formulation of the designmodel for a supply chain network will be presented indetail In automotive industries the general structure ofthe proposed supply chain logistic network is illustrated inFigure 1 In the forward direction the suppliers are respon-sible for providing all the parts needed by the car manu-facturing facilities Suppliers have been grouped into threecategories as tier-1 suppliersThe first group of tier-1 suppliersis responsible for supplyingmetal-based parts such as wheelsthe main body and springs The second group of suppliersis responsible for supplying electric parts such as wiringharnesses instrument panels and engine control units Thelast tier-1 group supplies plastics- and latex-based partssuch as tires interior parts and dashboard panels The newproducts are conveyed fromplants to retailers via distributioncentres to meet customer demand All assumptions andlimitations sets parameters and variables are defined andfollow a model-detailed description

21 Assumptions and Limitations

(i) The model is deterministic so that demand is fixedand known and must be satisfied

(ii) The model selects plants warehouses and distribu-tions from a given set of potential locations More-over the model also selects the supplier and retailerfrom a set of fixed locations

(iii) Suppliers plants warehouse and distributors have acapacity limit for the product

(iv) Cost is independent of product load but is varied forany kind of transportation vehicles such as the truckroad classes

(v) Carbon emission is dependent on product load(capacity of truck type) and also is varied for any kindof transportation vehicles

(vi) All truck types in transportation are turned off whenunloading product

(vii) Theflow is only allowed to be transferred between twosequential echelons

(1) Indices and Sets

119868 = 1 119873V set of fixed location of metal-partssuppliersrsquo tier-1119869 = 1 119873

119892 set of fixed location of electric-parts

suppliersrsquo tier-1119873 = 1 119873

119889 set of fixed location of plastic-parts

suppliersrsquo tier-1119870 = 1 119873

119891 set of potential available for plants

119871 = 1 119873119908

set of potential available for ware-houses and distributors119872 = 1 119873

119903 set of fixed location of retailers

(2) Parameters

120576119905 maximum capacity in truck type 119905 (tone) 119905 isin 119879

120591119905 transportation cost mode 119905 per km ($km) 119905 isin 119879

120593119905 carbon dioxide emission by truck type 119905 (gram

km) 119905 isin 119879120597V119891119894119896 distance between metal-parts suppliersrsquo tier-1 119894 to

plant 119896 (km) that 119894 isin 119868 119896 isin 119870

120597119892119891

119895119896 distance between electric-parts suppliersrsquo tier-1 119895

to plant 119896 (km) that 119894 isin 119868 119896 isin 119870

120597119889119891

119899119896 distance between plastic-parts suppliersrsquo tier-1 119899

to plant 119896 (km) that 119899 isin 119873 119896 isin 119870

120597119891119908

119896119897 distance between plant 119896 to distributor 119897 (km)

that 119896 isin 119870 119897 isin 119871120597119908119903

119897119898 distance between warehouse and distributor 119897 to

retailer 119898 (km) that 119897 isin 119871 119896 isin 119870120596119901 weight of product (tone)

120588119898 rate of metal parts contribution in the product (0-

1)120588119890 rate of electrical parts contribution in the product

(0-1)120588119901 rate of plastic parts contribution in the product (0-

1)120578119898 product demand retailer 119898

120583V119894 maximum capacity metal-parts suppliersrsquo tier-1 119894

(tone) 119894 isin 119868120583119892

119895 maximum capacity electric-parts suppliersrsquo tier-1

119895 isin 119869120583119889

119899 maximum capacity plastic-parts suppliersrsquo tier-1 119899

(tone) 119899 isin 119873120583119891

119896 maximumproduct capacity for plant 119896 that 119896 isin 119870

120583119908

119897 maximum product capacity for warehouse and

distributor 119897 that 119897 isin 119871120574119891

119896 unit Production and material handling cost of

product in plant 119896 ($) that 119896 isin 119870120574119908

119897 unit operation cost of product in distributor 119897 ($)

that 119897 isin 119871120574119903

119898 unit operation cost of product in retailer119898 ($) that

119898 isin 119872120572119891

119896 fixed cost of opening plant 119896 ($) that 119896 isin 119870

120572119908

119897 fixed cost of opening distribution center 119897 ($) that

119897 isin 119871

(3) Decision Variables

1198831

119894119896119905 quantity of product produced in plant 119896 using

metal parts of metal-parts suppliersrsquo tier-1 119894 by trucktype 119905 119896 isin 119870 119894 isin 119868 119905 isin 1198791198832

119895119896119905 quantity of product produced in plant 119896 using

electrical parts of electric-parts suppliersrsquo tier-1 119895 bytruck type 119905 119896 isin 119870 119895 isin 119869 119905 isin 119879

4 Mathematical Problems in Engineering

Supplier group-1

Plant Distribution center Retailers

Potential available facilitiesSet of available facilities

Fossil fuel

Supplier group-3

Supplier group-2

CO2 emission

Figure 1 General structure of automotive supply chain logistic network

1198833

119899119896119905 quantity of product produced in plant 119896 using

plastic parts of plastic-parts suppliersrsquo tier-1 119899 by trucktype 119905 119899 isin 119873 119896 isin 119870 119905 isin 1198791198834

119896119897119905 quantity of product shipped from plant 119896 todis-

tribution center 119897 by truck type 119905 119896 isin 119870 119897 isin 119871 119905 isin 1198791198835

119897119898119905 quantity of product shipped fromdistribution

center 119897 to retailer 119898 by truck type 119905 119897 isin 119871 119898 isin 119872119905 isin 119879

1198841

119896=

1 if factory 119896 is opened 119896 isin 119870

0 otherwise

1198842

119897=

1 if warehouse 119897 is opened 119897 isin 119871

0 otherwise

(1)

Objective Functions With regards to the above descriptionthe first objective function that minimizes the total cost ofsupply chain network is as follows

Total Cost = Transportation Cost + Operation Cost

+ Initial Facility Cost(2)

In objective function (3) the first five terms define thetotal transportation cost of the products between each

pair suppliers plants and distributors Next two terms arereferred to operation cost in plants and distributors Fixedsetup cost for opening the facilities (factories distributor cen-ters resp) can be defined by the last two terms

Total Cost = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120591119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119890

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120591119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120591119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120591119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120591119905

+ sum

119905

sum

119897

sum

119896

1199094

119896119897119905sdot 120574119891

119896+ sum

119905

sum

119898

sum

119897

1199095

119897119898119905sdot 120574119908

119897

+ sum

119896isin119870

120572119891

119896sdot 1198841

119896+ sum

119897isin119871

120572119908

119897sdot 1198842

119897

(3)

Mathematical Problems in Engineering 5

The second objective function that should be minimizedis total CO

2emission in supply chain network due to trans-

portation between all facilities

Carbon Emission = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120593119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119899

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120593119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120593119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120593119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120593119905

(4)

ConstrainsThis section is a representation of the constraintsof the proposed model

sum

119905

sum

119897

1199095

119897119898119905= 120578119898

forall119898 isin 119872 (5)

120596119901

sdot 120588119898

sdot sum

119906

sum

119905

sum

119894

1199091

119894119896119905+ 120596119901

sdot 120588119890

sdot sum

119905

sum

119895

1199092

119895119896119905

+ 120596119901

sdot 120588119901

sum

119905

sum

119899

1199093

119899119896119905= sum

119905

sum

119897

120596119901

sdot 1199094

119896119897119905forall119896 isin 119870

(6)

sum

119905

sum

119896

1199094

119897119896119905119906= sum

119905

sum

119898

1199095

119897119898119905forall119897 isin 119871 (7)

120588119898

sdot sum

119905

sum

119896

1199091

119894119896119905le 120583

V119894

forall119894 isin 119868 (8)

120588119890

sdot sum

119905

sum

119896

1199092

119895119896119905le 120583119889

119895forall119895 isin 119869 (9)

120588119901

sdot sum

119905

sum

119896

1199093

119899119896119905le 120583119889

119895forall119899 isin 119873 (10)

sum

119905

sum

119897

1199094

119896119897119905119906le 120583119891

119896sdot 1198841

119896forall119896 isin 119870 (11)

sum

119905

sum

119898

1199095

119897119898119905119906le 120583119908

119897sdot 1198842

119897forall119897 isin 119871 (12)

1198831

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905ge 0 forall119894 119895 119896 119897 119898 119905 (13)

1198841

119896 1198842

119897isin 0 1 forall119896 119897 (14)

Equation (5) states that for each product the flow exitingthe distribution centers must satisfy the demand of all cus-tomers Equation (6) shows that for each product the sum ofthe flow entering each plant from all suppliers is equal to theflow leaving the facility to the distribution centers Equation(7) insures that the sum of the product flow entering each

distribution centre from all plants is equal to the flow leavingthis distribution centre to retailers Equation (8) ensures thatfor each product the sum of the flow exiting eachmetal-partssupplier to all plants does not exceed the supplierrsquos capacityEquation (9) ensures that for each product the sum of theflow exiting each electric-parts supplier to all plants doesnot exceed the supplierrsquos capacity Equation (10) ensures thatfor each product the sum of the flow exiting each electric-parts supplier to all plants does not exceed the supplierrsquoscapacity Equation (11) states that the sum of the flow exitingeach plant to all distribution counters does not exceed theplantrsquos capacity And finally (12) shows that the sum of theflow exiting from each distribution centre to all customersdoes not exceed the distributorrsquos capacity Constraints (13)and (14) impose binary and nonnegativity restrictions on thecorresponding decision variables respectively

3 Solving Method

In this section we first introduce the Gravitational SearchAlgorithm (GSA) and then explain the extended GSA whichis used to solve multi-objective optimisation known as themulti-objective gravitational search algorithm (MOGSA)

31 GSA One of the novel meta-heuristic stochastic opti-misation algorithms inspired by the law of gravity and massinteractions is called the Gravitational Search Algorithm(GSA) [38] In GSA the individuals are called agents and area collection of objects that interact with each other based onNewtonian gravity and the laws of motion The agents shareinformation using gravitational force to guide the searchtowards the best location in the search spaceThe high perfor-mance and the global search ability of GSA in solving variousnonlinear functions infers from the results of experimentsundertaken previously [38ndash42] At GSA searcher agents are acollection of objects All these objects attract one another bygravitational force and this force causes a global movementof all objects toward the objects with heavier masses asFigure 2 shows Hence objects cooperate using a direct formof communication through gravitational force The heavyobjects that correspond to good solutions move more slowlythan the lighter objects which guarantee the exploitationability of the algorithm

The position of the objects corresponds to a solution ofthe problem and itsmass is determined using amap of fitnessfunction By changing the velocities over time the agents arelikely to move towards the global optima In this algorithmthere are objects The position of the 119894th agent is defined by

119883119894

= (1199091

119894 119909

119889

119894 119909

119899

119894) 119894 = 1 2 119873 (15)

where 119909119889

119894presents the position of the 119894th agent in the 119889th

dimension Based on theNewton gravitation theory the forceacting on the 119894th object from the 119895th object is

119865119889

119894119895(119905) = 119866 (119905)

119872119894(119905) times 119872

119895(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905)) (16)

119866 (119905) = 1198660

lowast exp(minus119886119905

119879) (17)

6 Mathematical Problems in Engineering

Figure 2 Each object accelerates toward the result force that attractsit from the other objects

where 119872119894is the mass of object 119894 119872

119895is the mass of object 119895

and 119866(119905) is the gravitational constant at time 119905 Notably thegravitational constant 119866(119905) is important in determining theperformance of GSA 119866(119905) is defined as a decreasing function(17) of time which is set to 119866

0at the beginning and decreased

exponentially to zero as time progressesTo compute the acceleration of an agent total forces from

a set of heavier masses that apply on it should be consideredbased on law of gravity (18) which is followed by calculationof agent acceleration using law of motion (19) Afterwardthe next velocity of an agent is calculated as a fraction ofits current velocity added to its acceleration (20) Then itsposition could be calculated using

119865119889

119894(119905) = sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905) 119872119894(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

(18)

119886119889

119894(119905) =

119865119889

119894(119905)

119872119894(119905)

= sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905)

119877119894119895

(119905) + 120576

times (119909119889

119895(119905) minus 119909

119889

119894(119905))

(19)

V119889119894

(119905 + 1) = rand119894

times V119889119894

(119905) + 119886119889

119894(119905) (20)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1) (21)

where rand119894and rand

119895are two uniform randoms in the

interval [0 1] 120576 is a small value and 119877119894119895

(119905) is the Euclidiandistance between two agents 119894 and 119895 defined as 119877

119894119895(119905) =

119883119894(119905) 119883119895(119905)2 119896best is the set of first 119870 agents with the best

fitness value and biggest mass 119896best is a function of timeinitialized to 119870

0at the beginning and decreasing with time

Here 1198700is set to 119873 (total number of agents) and is decreased

linearly to 1Based on [39] the mass of each agent is calculated after

computing current populationrsquos fitness as follows

119902119894(119905) =

fit119894(119905) minus worst (119905)

best (119905) minus worst (119905)

119872119894(119905) =

119902119894(119905)

sum119904

119895=1119902119895

(119905)

(22)

where 119872119894(119905) and fit

119894(119905) represent the mass and the fitness

value of the agent 119894 at 119905 and worst(119905) and best(119905) are definedas follows (for a minimization problem)

best (119905) = min119895isin1119904

fit119895

(119905)

worst (119905) = max119895isin1119904

fit119895

(119905)

(23)

Figure 3 illustrates the flow chart of the GSA

32 MOGSA Suppose that 119883 = (1199091 1199092 119909

119899) is the vector

of decision variable 119891119894

119877119899

rarr 119877 119894 = 1 119896 are theobjective functions and 119892

119894 ℎ119894

119877119899

rarr 119877 119894 = 1 119898 119895 =

1 119901 are the constraints of the problem Pareto domain-based algorithmMOGSA that is used in this paper presentsthis problem and definition as follows

(i) Minimize 119891(119883) = [1198911(119883) 119891

2(119883) 119891

119896(119883)]

(ii) Subject to 119892119894(119883) le 0 119894 = 1 2 119898

(iii) ℎ119895(119883) = 0 119895 = 1 2 119901

Definition 1 Given two vectors 119883 119884 isin 119877119896 we say that 119883 le 119884

if 119909119894

le 119910119894 119894 = 1 119896 and that 119883 dominates 119884 (or presented

by 119883 ≺ 119884) if 119883 le 119884 and 119883 = 119884

Definition 2 We say 119883 isin 120594 sub 119877119896 is nondominated with

respect to 120594 if another 1198831015840

isin 120594 does not exist such that119891(1198831015840) ≺ 119891(119883)

Selected non-dominated solutions are stored in an exter-nal archive Archive updates in each iteration of the algo-rithm When the running algorithm is terminated externalarchive presents final results

Recently the multi-objective gravitational search algo-rithm (MOGSA) [43ndash45] has been developed to solve multi-objective problems by using the UniformMutation Operator[46] and an Elitist Policy [47] as well-known operators ingenetic algorithm (GA) into GSA MOGSA also uses anexternal archive to reserve the non-dominated solutionsand update it the same as Simple Multi-Objective PSO(SMOPSO) [47] The first operator selects one of the objectrsquosdimensions randomly and adds a signed random term to itThe total value should be in the permitted range otherwiseit will be truncated to its nearest feasible value The non-dominated solutions found are stored by the Elitist Policyin an archive that has a grid structure The grid structure iscreated as follows each dimension in the objective space isdivided into 2

119899119894 equal divisions where 119894 denotes dimensionindex and hence for a 119896th objective optimisation problemtherewill be 2

(sum119896

119894=1119899119894) different segments As long as the archive

is not full new non-dominated solutions are added to itAs soon as the number of elements in the archive meets itslimit one of the elements should be omitted from it Thestrategy to remove an element from the archive is to findthe most crowded hypercube and randomly select one of theelements in it to extract For this purpose objective functionsand solution space are divided into hyper-rectangles When

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Supplier group-1

Plant Distribution center Retailers

Potential available facilitiesSet of available facilities

Fossil fuel

Supplier group-3

Supplier group-2

CO2 emission

Figure 1 General structure of automotive supply chain logistic network

1198833

119899119896119905 quantity of product produced in plant 119896 using

plastic parts of plastic-parts suppliersrsquo tier-1 119899 by trucktype 119905 119899 isin 119873 119896 isin 119870 119905 isin 1198791198834

119896119897119905 quantity of product shipped from plant 119896 todis-

tribution center 119897 by truck type 119905 119896 isin 119870 119897 isin 119871 119905 isin 1198791198835

119897119898119905 quantity of product shipped fromdistribution

center 119897 to retailer 119898 by truck type 119905 119897 isin 119871 119898 isin 119872119905 isin 119879

1198841

119896=

1 if factory 119896 is opened 119896 isin 119870

0 otherwise

1198842

119897=

1 if warehouse 119897 is opened 119897 isin 119871

0 otherwise

(1)

Objective Functions With regards to the above descriptionthe first objective function that minimizes the total cost ofsupply chain network is as follows

Total Cost = Transportation Cost + Operation Cost

+ Initial Facility Cost(2)

In objective function (3) the first five terms define thetotal transportation cost of the products between each

pair suppliers plants and distributors Next two terms arereferred to operation cost in plants and distributors Fixedsetup cost for opening the facilities (factories distributor cen-ters resp) can be defined by the last two terms

Total Cost = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120591119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119890

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120591119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120591119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120591119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120591119905

+ sum

119905

sum

119897

sum

119896

1199094

119896119897119905sdot 120574119891

119896+ sum

119905

sum

119898

sum

119897

1199095

119897119898119905sdot 120574119908

119897

+ sum

119896isin119870

120572119891

119896sdot 1198841

119896+ sum

119897isin119871

120572119908

119897sdot 1198842

119897

(3)

Mathematical Problems in Engineering 5

The second objective function that should be minimizedis total CO

2emission in supply chain network due to trans-

portation between all facilities

Carbon Emission = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120593119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119899

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120593119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120593119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120593119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120593119905

(4)

ConstrainsThis section is a representation of the constraintsof the proposed model

sum

119905

sum

119897

1199095

119897119898119905= 120578119898

forall119898 isin 119872 (5)

120596119901

sdot 120588119898

sdot sum

119906

sum

119905

sum

119894

1199091

119894119896119905+ 120596119901

sdot 120588119890

sdot sum

119905

sum

119895

1199092

119895119896119905

+ 120596119901

sdot 120588119901

sum

119905

sum

119899

1199093

119899119896119905= sum

119905

sum

119897

120596119901

sdot 1199094

119896119897119905forall119896 isin 119870

(6)

sum

119905

sum

119896

1199094

119897119896119905119906= sum

119905

sum

119898

1199095

119897119898119905forall119897 isin 119871 (7)

120588119898

sdot sum

119905

sum

119896

1199091

119894119896119905le 120583

V119894

forall119894 isin 119868 (8)

120588119890

sdot sum

119905

sum

119896

1199092

119895119896119905le 120583119889

119895forall119895 isin 119869 (9)

120588119901

sdot sum

119905

sum

119896

1199093

119899119896119905le 120583119889

119895forall119899 isin 119873 (10)

sum

119905

sum

119897

1199094

119896119897119905119906le 120583119891

119896sdot 1198841

119896forall119896 isin 119870 (11)

sum

119905

sum

119898

1199095

119897119898119905119906le 120583119908

119897sdot 1198842

119897forall119897 isin 119871 (12)

1198831

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905ge 0 forall119894 119895 119896 119897 119898 119905 (13)

1198841

119896 1198842

119897isin 0 1 forall119896 119897 (14)

Equation (5) states that for each product the flow exitingthe distribution centers must satisfy the demand of all cus-tomers Equation (6) shows that for each product the sum ofthe flow entering each plant from all suppliers is equal to theflow leaving the facility to the distribution centers Equation(7) insures that the sum of the product flow entering each

distribution centre from all plants is equal to the flow leavingthis distribution centre to retailers Equation (8) ensures thatfor each product the sum of the flow exiting eachmetal-partssupplier to all plants does not exceed the supplierrsquos capacityEquation (9) ensures that for each product the sum of theflow exiting each electric-parts supplier to all plants doesnot exceed the supplierrsquos capacity Equation (10) ensures thatfor each product the sum of the flow exiting each electric-parts supplier to all plants does not exceed the supplierrsquoscapacity Equation (11) states that the sum of the flow exitingeach plant to all distribution counters does not exceed theplantrsquos capacity And finally (12) shows that the sum of theflow exiting from each distribution centre to all customersdoes not exceed the distributorrsquos capacity Constraints (13)and (14) impose binary and nonnegativity restrictions on thecorresponding decision variables respectively

3 Solving Method

In this section we first introduce the Gravitational SearchAlgorithm (GSA) and then explain the extended GSA whichis used to solve multi-objective optimisation known as themulti-objective gravitational search algorithm (MOGSA)

31 GSA One of the novel meta-heuristic stochastic opti-misation algorithms inspired by the law of gravity and massinteractions is called the Gravitational Search Algorithm(GSA) [38] In GSA the individuals are called agents and area collection of objects that interact with each other based onNewtonian gravity and the laws of motion The agents shareinformation using gravitational force to guide the searchtowards the best location in the search spaceThe high perfor-mance and the global search ability of GSA in solving variousnonlinear functions infers from the results of experimentsundertaken previously [38ndash42] At GSA searcher agents are acollection of objects All these objects attract one another bygravitational force and this force causes a global movementof all objects toward the objects with heavier masses asFigure 2 shows Hence objects cooperate using a direct formof communication through gravitational force The heavyobjects that correspond to good solutions move more slowlythan the lighter objects which guarantee the exploitationability of the algorithm

The position of the objects corresponds to a solution ofthe problem and itsmass is determined using amap of fitnessfunction By changing the velocities over time the agents arelikely to move towards the global optima In this algorithmthere are objects The position of the 119894th agent is defined by

119883119894

= (1199091

119894 119909

119889

119894 119909

119899

119894) 119894 = 1 2 119873 (15)

where 119909119889

119894presents the position of the 119894th agent in the 119889th

dimension Based on theNewton gravitation theory the forceacting on the 119894th object from the 119895th object is

119865119889

119894119895(119905) = 119866 (119905)

119872119894(119905) times 119872

119895(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905)) (16)

119866 (119905) = 1198660

lowast exp(minus119886119905

119879) (17)

6 Mathematical Problems in Engineering

Figure 2 Each object accelerates toward the result force that attractsit from the other objects

where 119872119894is the mass of object 119894 119872

119895is the mass of object 119895

and 119866(119905) is the gravitational constant at time 119905 Notably thegravitational constant 119866(119905) is important in determining theperformance of GSA 119866(119905) is defined as a decreasing function(17) of time which is set to 119866

0at the beginning and decreased

exponentially to zero as time progressesTo compute the acceleration of an agent total forces from

a set of heavier masses that apply on it should be consideredbased on law of gravity (18) which is followed by calculationof agent acceleration using law of motion (19) Afterwardthe next velocity of an agent is calculated as a fraction ofits current velocity added to its acceleration (20) Then itsposition could be calculated using

119865119889

119894(119905) = sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905) 119872119894(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

(18)

119886119889

119894(119905) =

119865119889

119894(119905)

119872119894(119905)

= sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905)

119877119894119895

(119905) + 120576

times (119909119889

119895(119905) minus 119909

119889

119894(119905))

(19)

V119889119894

(119905 + 1) = rand119894

times V119889119894

(119905) + 119886119889

119894(119905) (20)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1) (21)

where rand119894and rand

119895are two uniform randoms in the

interval [0 1] 120576 is a small value and 119877119894119895

(119905) is the Euclidiandistance between two agents 119894 and 119895 defined as 119877

119894119895(119905) =

119883119894(119905) 119883119895(119905)2 119896best is the set of first 119870 agents with the best

fitness value and biggest mass 119896best is a function of timeinitialized to 119870

0at the beginning and decreasing with time

Here 1198700is set to 119873 (total number of agents) and is decreased

linearly to 1Based on [39] the mass of each agent is calculated after

computing current populationrsquos fitness as follows

119902119894(119905) =

fit119894(119905) minus worst (119905)

best (119905) minus worst (119905)

119872119894(119905) =

119902119894(119905)

sum119904

119895=1119902119895

(119905)

(22)

where 119872119894(119905) and fit

119894(119905) represent the mass and the fitness

value of the agent 119894 at 119905 and worst(119905) and best(119905) are definedas follows (for a minimization problem)

best (119905) = min119895isin1119904

fit119895

(119905)

worst (119905) = max119895isin1119904

fit119895

(119905)

(23)

Figure 3 illustrates the flow chart of the GSA

32 MOGSA Suppose that 119883 = (1199091 1199092 119909

119899) is the vector

of decision variable 119891119894

119877119899

rarr 119877 119894 = 1 119896 are theobjective functions and 119892

119894 ℎ119894

119877119899

rarr 119877 119894 = 1 119898 119895 =

1 119901 are the constraints of the problem Pareto domain-based algorithmMOGSA that is used in this paper presentsthis problem and definition as follows

(i) Minimize 119891(119883) = [1198911(119883) 119891

2(119883) 119891

119896(119883)]

(ii) Subject to 119892119894(119883) le 0 119894 = 1 2 119898

(iii) ℎ119895(119883) = 0 119895 = 1 2 119901

Definition 1 Given two vectors 119883 119884 isin 119877119896 we say that 119883 le 119884

if 119909119894

le 119910119894 119894 = 1 119896 and that 119883 dominates 119884 (or presented

by 119883 ≺ 119884) if 119883 le 119884 and 119883 = 119884

Definition 2 We say 119883 isin 120594 sub 119877119896 is nondominated with

respect to 120594 if another 1198831015840

isin 120594 does not exist such that119891(1198831015840) ≺ 119891(119883)

Selected non-dominated solutions are stored in an exter-nal archive Archive updates in each iteration of the algo-rithm When the running algorithm is terminated externalarchive presents final results

Recently the multi-objective gravitational search algo-rithm (MOGSA) [43ndash45] has been developed to solve multi-objective problems by using the UniformMutation Operator[46] and an Elitist Policy [47] as well-known operators ingenetic algorithm (GA) into GSA MOGSA also uses anexternal archive to reserve the non-dominated solutionsand update it the same as Simple Multi-Objective PSO(SMOPSO) [47] The first operator selects one of the objectrsquosdimensions randomly and adds a signed random term to itThe total value should be in the permitted range otherwiseit will be truncated to its nearest feasible value The non-dominated solutions found are stored by the Elitist Policyin an archive that has a grid structure The grid structure iscreated as follows each dimension in the objective space isdivided into 2

119899119894 equal divisions where 119894 denotes dimensionindex and hence for a 119896th objective optimisation problemtherewill be 2

(sum119896

119894=1119899119894) different segments As long as the archive

is not full new non-dominated solutions are added to itAs soon as the number of elements in the archive meets itslimit one of the elements should be omitted from it Thestrategy to remove an element from the archive is to findthe most crowded hypercube and randomly select one of theelements in it to extract For this purpose objective functionsand solution space are divided into hyper-rectangles When

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The second objective function that should be minimizedis total CO

2emission in supply chain network due to trans-

portation between all facilities

Carbon Emission = sum

119905

sum

119896

sum

119894

(

120596119901

sdot 120588119898

sdot 1199091

119894119896119905

120576119905

) sdot 120597V119891119894119896

sdot 120593119905

+ sum

119905

sum

119896

sum

119895

(

120596119901

sdot 120588119899

sdot 1199092

119895119896119905

120576119905

) sdot 120597119892119891

119895119896sdot 120593119905

+ sum

119905

sum

119896

sum

119899

(

120596119901

sdot 120588119901

sdot 1199093

119899119896119905

120576119905

) sdot 120597119889119891

119899119896sdot 120593119905

+ sum

119905

sum

119897

sum

119896

(

120596119901

sdot 1199094

119896119897119905

120576119905

) sdot 120597119891119908

119896119897sdot 120593119905

+ sum

119905

sum

119898

sum

119897

(

120596119901

sdot 1199095

119897119898119905

120576119905

) sdot 120597119908119903

119897119898sdot 120593119905

(4)

ConstrainsThis section is a representation of the constraintsof the proposed model

sum

119905

sum

119897

1199095

119897119898119905= 120578119898

forall119898 isin 119872 (5)

120596119901

sdot 120588119898

sdot sum

119906

sum

119905

sum

119894

1199091

119894119896119905+ 120596119901

sdot 120588119890

sdot sum

119905

sum

119895

1199092

119895119896119905

+ 120596119901

sdot 120588119901

sum

119905

sum

119899

1199093

119899119896119905= sum

119905

sum

119897

120596119901

sdot 1199094

119896119897119905forall119896 isin 119870

(6)

sum

119905

sum

119896

1199094

119897119896119905119906= sum

119905

sum

119898

1199095

119897119898119905forall119897 isin 119871 (7)

120588119898

sdot sum

119905

sum

119896

1199091

119894119896119905le 120583

V119894

forall119894 isin 119868 (8)

120588119890

sdot sum

119905

sum

119896

1199092

119895119896119905le 120583119889

119895forall119895 isin 119869 (9)

120588119901

sdot sum

119905

sum

119896

1199093

119899119896119905le 120583119889

119895forall119899 isin 119873 (10)

sum

119905

sum

119897

1199094

119896119897119905119906le 120583119891

119896sdot 1198841

119896forall119896 isin 119870 (11)

sum

119905

sum

119898

1199095

119897119898119905119906le 120583119908

119897sdot 1198842

119897forall119897 isin 119871 (12)

1198831

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905ge 0 forall119894 119895 119896 119897 119898 119905 (13)

1198841

119896 1198842

119897isin 0 1 forall119896 119897 (14)

Equation (5) states that for each product the flow exitingthe distribution centers must satisfy the demand of all cus-tomers Equation (6) shows that for each product the sum ofthe flow entering each plant from all suppliers is equal to theflow leaving the facility to the distribution centers Equation(7) insures that the sum of the product flow entering each

distribution centre from all plants is equal to the flow leavingthis distribution centre to retailers Equation (8) ensures thatfor each product the sum of the flow exiting eachmetal-partssupplier to all plants does not exceed the supplierrsquos capacityEquation (9) ensures that for each product the sum of theflow exiting each electric-parts supplier to all plants doesnot exceed the supplierrsquos capacity Equation (10) ensures thatfor each product the sum of the flow exiting each electric-parts supplier to all plants does not exceed the supplierrsquoscapacity Equation (11) states that the sum of the flow exitingeach plant to all distribution counters does not exceed theplantrsquos capacity And finally (12) shows that the sum of theflow exiting from each distribution centre to all customersdoes not exceed the distributorrsquos capacity Constraints (13)and (14) impose binary and nonnegativity restrictions on thecorresponding decision variables respectively

3 Solving Method

In this section we first introduce the Gravitational SearchAlgorithm (GSA) and then explain the extended GSA whichis used to solve multi-objective optimisation known as themulti-objective gravitational search algorithm (MOGSA)

31 GSA One of the novel meta-heuristic stochastic opti-misation algorithms inspired by the law of gravity and massinteractions is called the Gravitational Search Algorithm(GSA) [38] In GSA the individuals are called agents and area collection of objects that interact with each other based onNewtonian gravity and the laws of motion The agents shareinformation using gravitational force to guide the searchtowards the best location in the search spaceThe high perfor-mance and the global search ability of GSA in solving variousnonlinear functions infers from the results of experimentsundertaken previously [38ndash42] At GSA searcher agents are acollection of objects All these objects attract one another bygravitational force and this force causes a global movementof all objects toward the objects with heavier masses asFigure 2 shows Hence objects cooperate using a direct formof communication through gravitational force The heavyobjects that correspond to good solutions move more slowlythan the lighter objects which guarantee the exploitationability of the algorithm

The position of the objects corresponds to a solution ofthe problem and itsmass is determined using amap of fitnessfunction By changing the velocities over time the agents arelikely to move towards the global optima In this algorithmthere are objects The position of the 119894th agent is defined by

119883119894

= (1199091

119894 119909

119889

119894 119909

119899

119894) 119894 = 1 2 119873 (15)

where 119909119889

119894presents the position of the 119894th agent in the 119889th

dimension Based on theNewton gravitation theory the forceacting on the 119894th object from the 119895th object is

119865119889

119894119895(119905) = 119866 (119905)

119872119894(119905) times 119872

119895(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905)) (16)

119866 (119905) = 1198660

lowast exp(minus119886119905

119879) (17)

6 Mathematical Problems in Engineering

Figure 2 Each object accelerates toward the result force that attractsit from the other objects

where 119872119894is the mass of object 119894 119872

119895is the mass of object 119895

and 119866(119905) is the gravitational constant at time 119905 Notably thegravitational constant 119866(119905) is important in determining theperformance of GSA 119866(119905) is defined as a decreasing function(17) of time which is set to 119866

0at the beginning and decreased

exponentially to zero as time progressesTo compute the acceleration of an agent total forces from

a set of heavier masses that apply on it should be consideredbased on law of gravity (18) which is followed by calculationof agent acceleration using law of motion (19) Afterwardthe next velocity of an agent is calculated as a fraction ofits current velocity added to its acceleration (20) Then itsposition could be calculated using

119865119889

119894(119905) = sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905) 119872119894(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

(18)

119886119889

119894(119905) =

119865119889

119894(119905)

119872119894(119905)

= sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905)

119877119894119895

(119905) + 120576

times (119909119889

119895(119905) minus 119909

119889

119894(119905))

(19)

V119889119894

(119905 + 1) = rand119894

times V119889119894

(119905) + 119886119889

119894(119905) (20)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1) (21)

where rand119894and rand

119895are two uniform randoms in the

interval [0 1] 120576 is a small value and 119877119894119895

(119905) is the Euclidiandistance between two agents 119894 and 119895 defined as 119877

119894119895(119905) =

119883119894(119905) 119883119895(119905)2 119896best is the set of first 119870 agents with the best

fitness value and biggest mass 119896best is a function of timeinitialized to 119870

0at the beginning and decreasing with time

Here 1198700is set to 119873 (total number of agents) and is decreased

linearly to 1Based on [39] the mass of each agent is calculated after

computing current populationrsquos fitness as follows

119902119894(119905) =

fit119894(119905) minus worst (119905)

best (119905) minus worst (119905)

119872119894(119905) =

119902119894(119905)

sum119904

119895=1119902119895

(119905)

(22)

where 119872119894(119905) and fit

119894(119905) represent the mass and the fitness

value of the agent 119894 at 119905 and worst(119905) and best(119905) are definedas follows (for a minimization problem)

best (119905) = min119895isin1119904

fit119895

(119905)

worst (119905) = max119895isin1119904

fit119895

(119905)

(23)

Figure 3 illustrates the flow chart of the GSA

32 MOGSA Suppose that 119883 = (1199091 1199092 119909

119899) is the vector

of decision variable 119891119894

119877119899

rarr 119877 119894 = 1 119896 are theobjective functions and 119892

119894 ℎ119894

119877119899

rarr 119877 119894 = 1 119898 119895 =

1 119901 are the constraints of the problem Pareto domain-based algorithmMOGSA that is used in this paper presentsthis problem and definition as follows

(i) Minimize 119891(119883) = [1198911(119883) 119891

2(119883) 119891

119896(119883)]

(ii) Subject to 119892119894(119883) le 0 119894 = 1 2 119898

(iii) ℎ119895(119883) = 0 119895 = 1 2 119901

Definition 1 Given two vectors 119883 119884 isin 119877119896 we say that 119883 le 119884

if 119909119894

le 119910119894 119894 = 1 119896 and that 119883 dominates 119884 (or presented

by 119883 ≺ 119884) if 119883 le 119884 and 119883 = 119884

Definition 2 We say 119883 isin 120594 sub 119877119896 is nondominated with

respect to 120594 if another 1198831015840

isin 120594 does not exist such that119891(1198831015840) ≺ 119891(119883)

Selected non-dominated solutions are stored in an exter-nal archive Archive updates in each iteration of the algo-rithm When the running algorithm is terminated externalarchive presents final results

Recently the multi-objective gravitational search algo-rithm (MOGSA) [43ndash45] has been developed to solve multi-objective problems by using the UniformMutation Operator[46] and an Elitist Policy [47] as well-known operators ingenetic algorithm (GA) into GSA MOGSA also uses anexternal archive to reserve the non-dominated solutionsand update it the same as Simple Multi-Objective PSO(SMOPSO) [47] The first operator selects one of the objectrsquosdimensions randomly and adds a signed random term to itThe total value should be in the permitted range otherwiseit will be truncated to its nearest feasible value The non-dominated solutions found are stored by the Elitist Policyin an archive that has a grid structure The grid structure iscreated as follows each dimension in the objective space isdivided into 2

119899119894 equal divisions where 119894 denotes dimensionindex and hence for a 119896th objective optimisation problemtherewill be 2

(sum119896

119894=1119899119894) different segments As long as the archive

is not full new non-dominated solutions are added to itAs soon as the number of elements in the archive meets itslimit one of the elements should be omitted from it Thestrategy to remove an element from the archive is to findthe most crowded hypercube and randomly select one of theelements in it to extract For this purpose objective functionsand solution space are divided into hyper-rectangles When

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Figure 2 Each object accelerates toward the result force that attractsit from the other objects

where 119872119894is the mass of object 119894 119872

119895is the mass of object 119895

and 119866(119905) is the gravitational constant at time 119905 Notably thegravitational constant 119866(119905) is important in determining theperformance of GSA 119866(119905) is defined as a decreasing function(17) of time which is set to 119866

0at the beginning and decreased

exponentially to zero as time progressesTo compute the acceleration of an agent total forces from

a set of heavier masses that apply on it should be consideredbased on law of gravity (18) which is followed by calculationof agent acceleration using law of motion (19) Afterwardthe next velocity of an agent is calculated as a fraction ofits current velocity added to its acceleration (20) Then itsposition could be calculated using

119865119889

119894(119905) = sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905) 119872119894(119905)

119877119894119895

(119905) + 120576(119909119889

119895(119905) minus 119909

119889

119894(119905))

(18)

119886119889

119894(119905) =

119865119889

119894(119905)

119872119894(119905)

= sum

119895isin 119896best119895 = 119894rand119895119866 (119905)

119872119895

(119905)

119877119894119895

(119905) + 120576

times (119909119889

119895(119905) minus 119909

119889

119894(119905))

(19)

V119889119894

(119905 + 1) = rand119894

times V119889119894

(119905) + 119886119889

119894(119905) (20)

119909119889

119894(119905 + 1) = 119909

119889

119894(119905) + V119889

119894(119905 + 1) (21)

where rand119894and rand

119895are two uniform randoms in the

interval [0 1] 120576 is a small value and 119877119894119895

(119905) is the Euclidiandistance between two agents 119894 and 119895 defined as 119877

119894119895(119905) =

119883119894(119905) 119883119895(119905)2 119896best is the set of first 119870 agents with the best

fitness value and biggest mass 119896best is a function of timeinitialized to 119870

0at the beginning and decreasing with time

Here 1198700is set to 119873 (total number of agents) and is decreased

linearly to 1Based on [39] the mass of each agent is calculated after

computing current populationrsquos fitness as follows

119902119894(119905) =

fit119894(119905) minus worst (119905)

best (119905) minus worst (119905)

119872119894(119905) =

119902119894(119905)

sum119904

119895=1119902119895

(119905)

(22)

where 119872119894(119905) and fit

119894(119905) represent the mass and the fitness

value of the agent 119894 at 119905 and worst(119905) and best(119905) are definedas follows (for a minimization problem)

best (119905) = min119895isin1119904

fit119895

(119905)

worst (119905) = max119895isin1119904

fit119895

(119905)

(23)

Figure 3 illustrates the flow chart of the GSA

32 MOGSA Suppose that 119883 = (1199091 1199092 119909

119899) is the vector

of decision variable 119891119894

119877119899

rarr 119877 119894 = 1 119896 are theobjective functions and 119892

119894 ℎ119894

119877119899

rarr 119877 119894 = 1 119898 119895 =

1 119901 are the constraints of the problem Pareto domain-based algorithmMOGSA that is used in this paper presentsthis problem and definition as follows

(i) Minimize 119891(119883) = [1198911(119883) 119891

2(119883) 119891

119896(119883)]

(ii) Subject to 119892119894(119883) le 0 119894 = 1 2 119898

(iii) ℎ119895(119883) = 0 119895 = 1 2 119901

Definition 1 Given two vectors 119883 119884 isin 119877119896 we say that 119883 le 119884

if 119909119894

le 119910119894 119894 = 1 119896 and that 119883 dominates 119884 (or presented

by 119883 ≺ 119884) if 119883 le 119884 and 119883 = 119884

Definition 2 We say 119883 isin 120594 sub 119877119896 is nondominated with

respect to 120594 if another 1198831015840

isin 120594 does not exist such that119891(1198831015840) ≺ 119891(119883)

Selected non-dominated solutions are stored in an exter-nal archive Archive updates in each iteration of the algo-rithm When the running algorithm is terminated externalarchive presents final results

Recently the multi-objective gravitational search algo-rithm (MOGSA) [43ndash45] has been developed to solve multi-objective problems by using the UniformMutation Operator[46] and an Elitist Policy [47] as well-known operators ingenetic algorithm (GA) into GSA MOGSA also uses anexternal archive to reserve the non-dominated solutionsand update it the same as Simple Multi-Objective PSO(SMOPSO) [47] The first operator selects one of the objectrsquosdimensions randomly and adds a signed random term to itThe total value should be in the permitted range otherwiseit will be truncated to its nearest feasible value The non-dominated solutions found are stored by the Elitist Policyin an archive that has a grid structure The grid structure iscreated as follows each dimension in the objective space isdivided into 2

119899119894 equal divisions where 119894 denotes dimensionindex and hence for a 119896th objective optimisation problemtherewill be 2

(sum119896

119894=1119899119894) different segments As long as the archive

is not full new non-dominated solutions are added to itAs soon as the number of elements in the archive meets itslimit one of the elements should be omitted from it Thestrategy to remove an element from the archive is to findthe most crowded hypercube and randomly select one of theelements in it to extract For this purpose objective functionsand solution space are divided into hyper-rectangles When

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Start

Initialize no of particles m and initial positions X(0)

Compute masses based on the fitness function

Compute the gravitational constant G(t)

Compute distance between agents total force and acceleration of each agent

End

Initialize algorithm parameters G0 a

Compute new velocity and new position xi (t) for each agent

Is fitness functionsatisfied or t = Tmax

(number of iteration)

Figure 3 General flowchart of GSA

archive overflow is reached one particle from the mostcrowded hyper-rectangle is randomly selected and removedIn MOGSA the distance between each particle to its nearestneighbor is calculated and its mass is updated Distributionof archived particles is done by using the niching techniqueIn this algorithm the archived particles apply gravitationalforce The list below shows the pseudocode of MOGSA thatis similar to Hassanzadeh and Rouhani [43]

(1) Set the population size 1198660 119886 and 120576

(2) Initialize population of solutions(3) Evaluate population (with regards to objective func-

tions CO2and Cost)

(4) Update gravitational constant(5) Calculate the masses(6) Calculate force and acceleration between objects(7) Update velocity and position per each object(8) Select set of best solutions (Pareto front) Update

external archive

(9) Generate new population

(10) If the fitness function is satisfied or 119905 = 119879max go to 4otherwise continued

(11) Figure set of Pareto solutions and end

4 Experimental Results and Discussion

In this section a multi-objective mixed integer linear pro-gramming supply chain logistics network model is applied toa real world companyThe case study is conducted to illustratethe efficiency and consistency of the proposed method tosolve multi-objective optimization in supply chain networksThis case is motivated by the Pars Khodro Company whichis located in Tehran Iran and is one the largest car manu-facturers in the Middle East The company has been underpressure to satisfy recently enacted environmental legislationOur purpose is to provide some strategic viewpoints and rec-ommendations for the supply chain network designwhere theCO2emission issue is being considered In this regard we set

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

up a bi-objective optimization model and conducted a caseanalysis based on a network consisting of the following thethree groups of tier-1 suppliers (metal-based parts electricalparts and plastic parts) which include 17 22 and 17 tier-1suppliers in each group respectively the 9 potential locationsfor establishing the manufacturer the 23 potential locationsfor establishing the distribution centres the 47 retailers thethree trucks type including a new 12-ton truck an 8-tontruck and an old 12-ton truck Figure 4 illustrates some ofthe potential networks in the supply chain logistics networkThe formulation of this case contains 31 binary variables 1792continuous variables and 302 constraints

In Table 1 the parameters of the problem are given Thetransportation costs andCO

2emissions for all truck types are

defined as corresponding to every kilometer between facili-ties on each layer of the supply chain network Moreover thefixed costs and capacity of the same facilities and the demandof each retailer are discriminated

For MOGSA implementation as a solving method mate-rial and product flows between facilities in automotive supplychains such as suppliers manufacturers distributors andretailers as objects with some respective masses should beconsidered And each particle is as a vector with 7 dimen-sions 119872 = [119883

1

119894119896119905 1198832

119895119896119905 1198833

119899119896119905 1198834

119896119897119905 1198835

119897119898119905 1198841

119896 1198842

119897] In this case

we set MOGSA regarding to the parameter setting that wasmentioned in Table 2 The problem is solved by the MOGSAmethod and it is implemented using MATLAB software Allthe experiments are conducted using a laptop with an IntelCore 3 Duo 219GHz and 1GB RAM

Since the Pareto frontier can provide the decision makerwith a portfolio of alternative ldquooptimal solutionsrdquo the solu-tions to the assignment problem estimated previously werethe marginal points as shown in Figure 5 The minimizationof total logistics cost including freight transportation systemcosts operation costs and the initial cost of opening andrunning all the facilities is shown on the upper left side andthe minimisation of CO

2emissions is shown on the lower

right side It clearly demonstrates the trade-off between thetotal cost and the total CO

2emissions It coincides with our

intuition that a lower CO2emission can only be reached

by applying more investment Figure 5 shows 20 solutionsbut these are only a subset of the Pareto optimal solutionsHowever the algorithm can estimate less than 20 solutionsbecause there might be unfeasible solutions in the iterationsThe relationship between cost and CO

2emissions in the

entire network is not exactly linearThe linearity of the Paretooptimal solution is not necessary even if all the objectivefunctions are linear Specifically the changed cost amountsare not necessarily proportional as theCO

2emissions allowed

are decreased Also we can find that the absolute value of theslope of the curve decreases and the curves became flatter asthe total cost increases

Solution selection is depended on how much total costthe company would like to invest for environmental issueFor example if the company prefers to produce minimumCO2emission (180 tones) should pay about 990560 thousand

US dollar for total logistic cost If the company is notunder government and customer pressure to decrease CO

2

Potential position manufacturer

Fix position of retailer

Potential position distribution center Fix position metal parts supplier tier-1 Fix position electrical parts supplier tier-1 Fix position plastic parts supplier tier-1

Potential logistics network

Figure 4 Potential logistics network in the case study supply chain

Table 1 Value of parameters in the numerical experiments

Parameter Value Parameter Value Parameter Value1205761 1205763

12 120588119890

03 1205931

6001205762

6 120588119898

05 1205932

4001205911

02 1205781

minus 12057847

12000 1205933

8001205912 1205913

015 120574119908

1minus 120574119908

2380 120572

119891

1minus 120572119891

950000000

120574119891

11000 120574

119903

1minus 120574119903

4765 120572

119908

1minus 120572119908

235000000

120574119891

21200 120574

119891

3minus 120574119891

9900 120596

1199012

Table 2 Parameters values of MOGSA algorithm

119866constant 8Mutation prop 05Number of divisions in each dimension [3 2 2]

emission it can select the cheaper solution such as the firstsolution In this case total logistic cost would be 290260thousand US dollar but more than 680 tones CO

2emission is

emitted in environment Decision maker compare the Paretosolution set with the real situation for the company regardingthe total cost and environmental policy andfinally choose oneof the solutions to trade-off these limitations

5 Conclusion

The quantitative relationship between CO2emissions and

logistics costs has been gaining importance in the logisticsfield because of global warming as well as rapidly increasingfuel costsThis study is an attempt to estimate the relationship

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

18002300280033003800430048005300580063006800

Total cost (k $)lowastPareto optimal solutions

CO2

(times102

kg)

600

000

109

056

090

091

278

424

472

911

168

055

262

263

755

813

253

335

148

401

646

610

845

058

643

724

940

974

640

064

938

582

536

584

034

824

332

891

532

648

1

Figure 5 Pareto optimal solution to the case study

by using multi-objective linear programming mathematicalmodelling in the automotive supply chain network Thismodel will have an important application in the regional orglobal automotive supply chain network design with greenconsiderations The model is a multi-objective model thatconsists of minimising the total cost and the CO

2emissionmdash

the type of pollution human kind is most responsible forThe MOGSA method is used as a new swarm intelligencealgorithm to solve the model and obtain a Pareto optimal setAfter that the model was tested by a real case study to illus-trate the model and MOGSA The Pareto optimal solutionsgiven by the model provide a portfolio of configuration fordecisionmakers and the computation experiments show thatthe model can serve as an effective tool in designing a greensupply chain network The results show that the trade-offcurves have almost a linear relationship in that logistics costsshould be higher as a greater reduction of CO

2emissions is

requiredThequantity of the relationship varies ranging from290 to 990 billionUSdollars in terms of theCO

2emission lev-

els Also the absolute value of the slope of the curve decreasesand the curves became flatter as the total cost increases

The research can be extended to minimise the lead timeor ensuring just-in-time arrival that is not taken into accountin this study Furthermore since considering reverse logisticsin the supply chain network is attracting great interest amongresearchers and it can be added to the network model-ling authors are starting to extend the model as a closedloop Accordingly nonlinear programming would be a betteroption as a modelling method to find a more accuratesolution which should be investigated more in the futureAlso comparing the MOGSA result with the other multi-objective optimization algorithm such as NondominatedSorting Genetic Algorithm (NSGA) or Multi-objective Par-ticle Swarm Optimization (MOPSO) can be one of the futureresearch

Acknowledgments

Thiswork received financial support from theMinistry of Sci-ence Technology and Innovation ofMalaysia under Research

Project no 5527085 and was supported by the Institute ofAdvanced Technology of University Putra Malaysia and isgratefully acknowledged

References

[1] X Xi X Ding D Fu L Zhou and K Liu ldquoRegional Δ14Cpatterns and fossil fuel derived CO

2distribution in the Beijing

area using annual plantsrdquo Chinese Science Bulletin vol 56 no16 pp 1721ndash1726 2011

[2] S C Davis S W Diegel R G Boundy Engineering ORNLDivision TS and Program VT Transportation Energy DataBook Oak Ridge National Laboratory 29th edition 2010

[3] S Elhedhli and RMerrick ldquoGreen supply chain network designto reduce carbon emissionsrdquo Transportation Research D vol 17no 5 pp 370ndash379 2012

[4] J F Cordeau F Pasin andMM Solomon ldquoAn integratedmod-el for logistics network designrdquo Annals of Operations Researchvol 144 pp 59ndash82 2006

[5] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[6] P Georgiadis D Vlachos and E Iakovou ldquoA system dynamicsmodeling framework for the strategic supply chain manage-ment of food chainsrdquo Journal of Food Engineering vol 70 no3 pp 351ndash364 2005

[7] Y Ge J B Yang N Proudlove andM Spring ldquoSystem dynam-ics modelling for supply chain management a case study ona supermarket chain in the UKrdquo International Transactions inOperational Research vol 11 no 5 pp 495ndash509 2004

[8] F Du and G W Evans ldquoA bi-objective reverse logistics net-work analysis for post-sale servicerdquo Computers and OperationsResearch vol 35 no 8 pp 2617ndash2634 2008

[9] B M Beamon and C Fernandes ldquoSupply-chain network con-figuration for product recoveryrdquo Production Planning and Con-trol vol 15 no 3 pp 270ndash281 2004

[10] S Chopra ldquoDesigning the distribution network in a supplychainrdquo Transportation Research E vol 39 no 2 pp 123ndash1402003

[11] D Peidro JMula R Poler and FC Lario ldquoQuantitativemodelsfor supply chain planning under uncertaintyrdquo InternationalJournal of Advanced Manufacturing Technology vol 43 no 3-4 pp 400ndash420 2009

[12] N S KimM Janic and B vanWee ldquoTrade-off between carbondioxide emissions and logistics costs based on multiobjectiveoptimizationrdquo Transportation Research Record no 2139 pp107ndash116 2009

[13] F Schultmann M Zumkeller and O Rentz ldquoModeling reverselogistic tasks within closed-loop supply chains an examplefrom the automotive industryrdquo European Journal of OperationalResearch vol 171 no 3 pp 1033ndash1050 2006

[14] R J Merrick and J H Bookbinder ldquoEnvironmental assessmentof shipment release policiesrdquo International Journal of PhysicalDistribution and Logistics Management vol 40 no 10 pp 748ndash762 2010

[15] E L Plambeck ldquoReducing greenhouse gas emissions throughoperations and supply chain managementrdquo Energy Economicsvol 34 supplement 1 pp S64ndashS74 2012

[16] C Reich-Weiser and D A Dornfeld ldquoA discussion of green-house gas emission tradeoffs and water scarcity within the sup-ply chainrdquo Journal of Manufacturing Systems vol 28 no 1 pp23ndash27 2009

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[17] S Melkote andM S Daskin ldquoCapacitated facility locationnet-work design problemsrdquo European Journal of OperationalResearch vol 129 no 3 pp 481ndash495 2001

[18] T Aslam P Hedenstierna A H C Ng L Wang and K DebldquoMulti-objective optimisation in manufacturing supply chainsystems design a comprehensive survey and new directionsrdquoinMulti-Objective EvolutionaryOptimisation for Product Designand Manufacturing pp 35ndash70 Springer London UK 2011

[19] R Sun X Wang and G Zhao ldquoAn ant colony optimizationapproach tomulti-objective supply chainmodelrdquo in Proceedingsof the 2nd IEEE International Conference on Secure SystemIntegration and Reliability Improvement (SSIRI rsquo08) pp 193ndash194Yokohama Japan July 2008

[20] R K Pati P Vrat and P Kumar ldquoA goal programming modelfor paper recycling systemrdquo Omega vol 36 no 3 pp 405ndash4172008

[21] Y Bouzembrak H Allaoui G Goncalves and H BouchrihaldquoA multi-objective green supply chain network designrdquo inProceedings of the 4th International Conference on Logistics(LOGISTIQUA rsquo2011) pp 357ndash361 Hammamet Tunisia June2011

[22] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquo EuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[23] V V Kumar F T S Chan N Mishra and V Kumar ldquoEnvi-ronmental integrated closed loop logistics model an artificialbee colony approachrdquo in Proceedings of the 8th InternationalConference on Supply Chain Management and InformationSystems (SCMIS rsquo10) pp 1ndash7 October 2010

[24] M S Pishvaee K Kianfar and B Karimi ldquoReverse logistics net-work design using simulated annealingrdquo International Journalof AdvancedManufacturing Technology vol 47 no 1ndash4 pp 269ndash281 2010

[25] MDorigoM Birattari andT Stutzle ldquoAnt colony optimizationartificial ants as a computational intelligence techniquerdquo IEEEComputational Intelligence Magazine vol 1 no 4 pp 28ndash392006

[26] M Dorigo and G Di Caro ldquoAnt colony optimization a newmeta-heuristicrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) vol 2 Washington DC USA 1999

[27] M Dorigo and T Stutzle ldquoThe ant colony optimization meta-heuristic algorithms applications and advancesrdquo inHandbookof Metaheuristics pp 250ndash285 2003

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Wash USA December1995

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 2007

[30] S Das and B K Panigrahi ldquoMulti-objective evolutionary algo-rithmsrdquo in Encyclopedia of Artificial Intelligence vol 3 pp 1145ndash1151 2009

[31] S Mondal A Bhattacharya and S H N Dey ldquoMulti-objectiveeconomic emission load dispatch solution using gravitationalsearch algorithm and considering wind power penetrationrdquoInternational Journal of Electrical Power andEnergy Systems vol44 no 1 pp 282ndash292 2013

[32] F Altiparmak M Gen L Lin and T Paksoy ldquoA geneticalgorithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[33] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[34] L Alcada-Almeida J Coutinho-Rodrigues and J Current ldquoAmultiobjective modeling approach to locating incineratorsrdquoSocio-Economic Planning Sciences vol 43 no 2 pp 111ndash1202009

[35] T Paksoya E Ozceylana and G WWeberb ldquoA multi objectivemodel for optimization of a green supply chain networkrdquoGlobalJournal of Technology and Optimization vol 2 pp 84ndash96 2011

[36] F Wang X Lai and N Shi ldquoA multi-objective optimization forgreen supply chain network designrdquo Decision Support Systemsvol 51 no 2 pp 262ndash269 2011

[37] RDekker J Bloemhof and IMallidis ldquoOperations research forgreen logisticsmdashan overview of aspects issues contributionsand challengesrdquo European Journal of Operational Research vol219 no 3 pp 671ndash679 2012

[38] E Rashedi HNezamabadi-pour and S Saryazdi ldquoGSA a grav-itational search algorithmrdquo Information Sciences vol 179 no 13pp 2232ndash2248 2009

[39] E Rashedi H Nezamabadi-Pour and S Saryazdi ldquoFilter mod-eling using gravitational search algorithmrdquoEngineeringApplica-tions of Artificial Intelligence vol 24 no 1 pp 117ndash122 2011

[40] M Khajehzadeh M R Taha A El-Shafie and M Eslami ldquoAmodified gravitational search algorithm for slope stability anal-ysisrdquo Engineering Applications of Artificial Intelligence vol 25no 8 pp 1589ndash1597 2012

[41] M Ojha K Deep A Nagar M Pant and J C Bansal ldquoOpti-mizing supply chain management using gravitational searchalgorithm and multi agent systemrdquo in Proceedings of the In-ternational Conference on Soft Computing for Problem Solving(SocProS rsquo11) December 20ndash22 2011 vol 130 of Advances inIntelligent and Soft Computing pp 481ndash491 Springer BerlinGermany 2012

[42] C Li and J Zhou ldquoParameters identification of hydraulic tur-bine governing system using improved gravitational searchalgorithmrdquo Energy Conversion and Management vol 52 no 1pp 374ndash381 2011

[43] H R Hassanzadeh andM Rouhani ldquoAmulti-objective gravita-tional search algorithmrdquo in Proceedings of the 2nd InternationalConference on Computational Intelligence Communication Sys-tems and Networks (CICSyN rsquo10) pp 7ndash12 Liverpool UK July2010

[44] T Ganesan I Elamvazuthi K Z K Shaari and P VasantldquoSwarm intelligence and gravitational search algorithm formulti-objective optimization of synthesis gas productionrdquoApplied Energy vol 103 pp 368ndash374 2013

[45] M Arsuaga-Ros M A Vega-Rodrguez and F Prieto-CastrilloldquoMeta-schedulers for grid computing based on multi-objectiveswarm algorithmsrdquo Applied Soft Computing Journal vol 13 no4 pp 1567ndash1582 2013

[46] T Back D B Fogel and Z Michalewicz Handbook of Evolu-tionary Computation IOP Publishers Bristol UK 1997

[47] L Cagnina S Esquivel and C A C Coello ldquoA particleswarm optimizer for multi-objective optimizationrdquo Journal ofComputer Science and Technology vol 5 pp 204ndash210 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of