Location and Allocation Decisionin supply chain manangement

Expert Systems with Applications 40 (2013) 551–562

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Expert Systems with Applications

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Location and allocation decisions for multi-echelon supply chain network –A multi-objective evolutionary approach

B. Latha Shankar a, S. Basavarajappa b, Jason C.H. Chen c,⇑, Rajeshwar S. Kadadevaramath a

a Dept. of Industrial Engg. & Management, SIT, Tumkur, Indiab Mechanical Department, UBDT, Davangere, Indiac Graduate School of Business, Gonzaga University, Spokane, WA, USA

a r t i c l e i n f o a b s t r a c t

Keywords:Four-echelon supply chain architectureEvolutionary approachNon-dominated sorting algorithmMOHPSOSwarm intelligence

0957-4174/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.eswa.2012.07.065

⇑ Corresponding author.E-mail address: [email protected] (J.C.H. C

This paper aims at multi-objective optimization of single-product for four-echelon supply chain architec-ture consisting of suppliers, production plants, distribution centers (DCs) and customer zones (CZs). Thekey design decisions considered are: the number and location of plants in the system, the flow of rawmaterials from suppliers to plants, the quantity of products to be shipped from plants to DCs, fromDCs to CZs so as to minimize the combined facility location and shipment costs subject to a requirementthat maximum customer demands be met. To optimize these two objectives simultaneously, four-eche-lon network model is mathematically represented considering the associated constraints, capacity, pro-duction and shipment costs and solved using swarm intelligence based Multi-objective Hybrid ParticleSwarm Optimization (MOHPSO) algorithm. This evolutionary based algorithm incorporates non-domi-nated sorting algorithm into particle swarm optimization so as to allow this heuristic to optimize twoobjective functions simultaneously. This can be used as decision support system for location of facilities,allocation of demand points and monitoring of material flow for four-echelon supply chain network.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Supply chain (SC) is an integrated system of facilities and activ-ities that synchronizes inter-related business functions of materialprocurement, material transformation to intermediates and finalproducts and distribution of these products to customers. Supplychain management is a set of approaches utilized to efficientlyintegrate suppliers, manufacturers, warehouses, and stores, so thatmerchandise is produced and distributed at the right quantities, tothe right locations, and at the right time, in order to minimize sys-tem-wide costs while satisfying service level requirements(Simchi-Levi, Kaminsky, & Simchi-Levi, 2000). Above definition re-veals that there are many independent entities in a supply chaineach of which tries to maximize its own inherent objective func-tions in business transactions. This is a complicated problem astoo many factors are involved and needs more than one objectiveto be satisfied simultaneously. Such a problem is called multi-objective optimization problem and has many Pareto solutions.The final decision is made taking the total balance over all criteriainto account. This balancing over criteria is called trade-off.

Since today, the success measures for the companies arethought as lower costs, shorter production time, shorter lead time,

ll rights reserved.

hen).

less stock, larger product range, more reliable delivery time, bettercustomer services, higher quality, and providing the efficient coor-dination between demand, supply and production, the trade-offbetween cost investment and service levels may change over time.Hence the supply chain performance needs to be evaluated contin-uously and supply chain managers should make timely and rightdecisions (Shen, 2007).

The key issues in supply chain management can broadly be di-vided into three main categories: (i) supply chain design (ii) supplychain planning and (iii) supply chain control. In the supply chaindesign phase, strategic decisions, such as facility location decisionsand technology selection decisions play major roles. It is veryimportant to design an efficient supply chain to facilitate themovements of goods. These strategic decisions lead to costly, timeconsuming investment as the facilities located today, are expectedto remain in operation for long time. Environmental changes dur-ing the facility’s lifetime can drastically alter the appeal of a partic-ular site, turning today’s optimal location into tomorrow’sinvestment blunder. Determining the best locations for new facili-ties is thus an important strategic challenge (Owen & Daskin,1998). Once the supply chain configuration is determined, the fo-cus shifts to decisions at the tactical and operational levels, suchas inventory management decisions on raw materials, intermedi-ate products, and end products and distribution decisions withinthe supply chain (Chopra & Meindl, 2005).

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552 B. Latha Shankar et al. / Expert Systems with Applications 40 (2013) 551–562

In traditional supply chain management, the focus of the de-signs of supply chain network is usually on single objective, mini-mum cost or maximum profit. But the design, planning andscheduling projects are usually involving trade-offs among differ-ent incompatible goals such as fair profit distribution among allmembers, customer service levels, fill rates, safe inventory levels,volume flexibility, etc. (Chen & Lee, 2004). Hence real supplychains are to be optimized simultaneously considering more thanone objective. Many of the problems that occur in supply chainoptimization are combinatorial in nature and picking a set of opti-mal solutions in the case of multi-objective formulations requires aalgorithm that can efficiently search the entire objective spaceusing small amounts of computation time. Literature shows thatevolutionary algorithms perform well in this respect and give goodoptimal results when applied to many combinatorial problems.This work proposes the utility of non-dominated sorting particleswarm optimization algorithm for simultaneous optimization oftwo objectives, minimizing total supply chain cost and maximizingfill rate for a four-echelon supply chain architecture so as to arriveat an efficient supply chain design and optimal transportation/shipment plan which can be used as decision support system.

2. Prior related work

Since this work considers location decisions, allocation deci-sions, multi-objective optimization using particle swarm optimiza-tion algorithm, this section deals with prior work related to allthese areas. Supply chain management has vast scope and encom-passes the decisions about (1) where to produce, what to produce,and how much to produce at each site, (2) what quantity of goodsto hold in inventory at each stage of the process, (3) where to lo-cate plants and distribution centers etc. Out of these, location deci-sions may be the most critical and most difficult of the decisionsneeded to realize an efficient supply chain as their effect is longlasting. Both the cost of whole supply chain and the level of cus-tomer service provided by the system are significantly affectedby the number, size and locations of facilities, as well as by theallocation decisions as to which customers to be served from whichupstream supplier. Consequently, a significant amount of researchhas been devoted to the development of efficient supply chain de-sign. The early study of location theory begin in 1909 with AlfredWeber’s work on positioning a single warehouse so as to minimizethe total distance between it and several customers (Weber, 1929).After that considerable work in location theory is done by Hakimi(1964), who work on locating switching centers in a communica-tions network and police stations in a highway system. Later manyresearchers work on basic facility location problem formulationsrecognized as static and deterministic which take constant, knownquantities as inputs and derive a single solution to be implementedat one point in time. These fundamental location problems are cat-egorized into median problems (Hakimi, 1964; ReVelle, 1986), cov-ering problems (Daskin, 1995; Church & ReVelle, 1976), centerproblems (Daskin, 1995), etc. Later focus is shifted to location-allo-cation problems which simultaneously locate facilities and dictateflows between facilities and demands. These problems are re-viewed by Scott (1971). Additional variety of problems includemodels with multiple commodities, unreliable supply, etc. War-szawski and Peer (1973) and Warszawski (1973) are among thefirst to study the multi-commodity location problem. These modelsconsider fixed location costs and linear transportation costs, andassume that each warehouse can be assigned at most one com-modity. Geoffrion and Graves (1974) consider the capacitated ver-sion of the multi-commodity location problem and present amodel to solve the problem of designing a distribution system withoptimal location of the intermediate distribution facilities between

plants and customers. The risks arising from the use of heuristics indistribution planning are discussed early on by Geoffrion and Van-roy (1979). They present three examples in the area of distributionplanning demonstrating the failure of common sense methods tocome up with the best solution.

In literature, another set of problems considered is called fixedcharge facility location problems which consider fixed charge asso-ciated with locating at each potential facility site. There are twotypes of problems capacitated and uncapacitated plant locationproblems. Uncapacitated and capacitated plant location modelsare extensively dealt in Mirchandani and Francis (1990) and ReV-elle, Eiselt, and Daskin (2008) and capacitated plant location mod-els in Sridharan (1995). Also basic facility location problems aregiven a new orientation with integrated approach. This is due tothe realization of fact that location decisions taken without consid-ering inventory and shipment costs can lead to sub-optimality.Hence facility location models are developed considering locationassociated costs as well as production, inventory and distributioncosts. Integrated decision making models in particular focus atcoordination of any two of the three important supply chain deci-sions. Based on the factors considered they are catagorized into (1)location–routing (LR) models; (2) inventory–routing (IR) models;and (3) location–inventory (Li) models. These problems are exten-sively reviewed by Shen (2007).

Dynamic facility location models next evolved make an attemptto capture many of the characteristics of real-world location prob-lems. Ballou (1968) first use dynamic programming to determineoptimal location and relocation strategy for the planning period.Wesolowsky (1973) examines another, unconstrained, version ofthe single facility location problem over a finite planning horizonwith explicit facility relocation costs. Scott (1971) develop multipledynamic facility location-allocation problem. Wesolowsky andTruscott (1976) present an integer programming model to extendthe analysis of multi-period node location-allocation problems,allowing facilities to be relocated in response to predicted changesin demand. Erlenkotter (1981) compares the performance of sev-eral heuristic solution approaches on a single problem formulation.He examines a dynamic, fixed charge, capacitated, cost minimiza-tion problem with discrete time intervals.

In traditional supply chain management, minimizing costs ormaximizing profit is the primitive objective in most of the supplychain network design models (Cohen & Lee, 1989; Tsiakis, Shah,& Pantelides, 2001). But for a supply chain, producing products atminimum cost is not the only objective, satisfying customers isalso equally important. Later some researchers start incorporatingmore than one competing objectives such as improving customerservice and reducing cost in their models. Different methodologiesfound in literature for treating multiobjective optimization prob-lems are the weighted-sum method, the e-constraint method andthe goal-programming method, fuzzy method, etc. (Azapagic &Clift, 1999; Chen & Lee, 2004; Cheng-Liang, Wang, & Wen-Cheng,2003; Zhou, Cheng, & Hua, 2000).

Sabri and Beamon (2000) develop a model for supply chainmanagement by combining strategic and operational design andplanning decisions and solve it using an iterative procedure. Theyadopt multi-objective decision analysis and optimize simulta-neously cost, fill rate and flexibility. Nozick and Turnquist (2001)present an optimization model which minimizes cost and maxi-mizes service. They use a linear function to approximate the safetystock inventory cost function, which is then embedded in a fixed-charge facility location model. For a review of other multi-objec-tive location models, publication by Shen, Coullard, and Daskin(2003) can be referred. Chen and Lee (2004) propose a modelwhich simultaneously optimize conflicting objectives such as eachparticipant’s profit, the average customer service level, and theaverage safe inventory level. Guilléna, Melea, Bagajewiczb,

B. Latha Shankar et al. / Expert Systems with Applications 40 (2013) 551–562 553

Espuñaa, and Puigjanera (2004) formulate a multiobjective sto-chastic Mixed Integer Linear Programming model for supply chaindesign, which is solved by using the standard e-constraint method,and branch and bound techniques. This formulation takes into ac-count not only Supply Chain profit and customer satisfaction level,but also uncertainty by means of the concept of financial risk,which is defined as the probability of not meeting a certain profitaspiration level. Shen (2006) addresses profit-maximizing supplychain design model where in a company can choose whether tosatisfy a customer’s demand.

Particle swarm optimization (PSO) is one of the evolutionarycomputation techniques and is a population-based search algo-rithm which is initialized with a population of random solutionscalled particles. Each particle in PSO is also associated with a veloc-ity. Particles fly through the search space with velocities which aredynamically adjusted according to their historical behaviors. Dueto this the particles have a tendency to fly towards the betterand better search area over the course of search process. This tech-nique is originally proposed by Kennedy and Russell Eberhart as asimulation of social behavior, and it is initially introduced as anoptimization method in 1995 (Kennedy & Eberhart, 1995). PSOcan be easily implemented and it is computationally in expensive,since its memory and CPU speed requirements are low. The maindifficulty in Multi-objective Particle Swarm Optimization (MOPSO)is to pick suitable global best (gbest) and personal best (pbest) tomove the particles through search space. In general, a good MOPSOmethod must obtain solutions with (a) a good convergence, and (b)a diversity spread along the Pareto-optimal front (Coello, 1999).

Parsopoulos and Vrahatis (2002) are one among to study firstthe performance of the PSO in multi-objective optimization prob-lems and find Pareto front in weighted aggregation cases. Hu andEberhart (2002) present a MOPSO that uses dynamic neighborhoodstrategy to obtain the best local guide for each particle in multi-objective optimization problems. Coello and Lechuga (2002) pro-pose a MOPSO where the objective space is divided to hyper cubes.Fieldsend and Singh (2002) propose a MOPSO which uses a domi-nated tree for the search of the best local solution. Mostaghim andTeich (2003) propose a sigma method and ‘‘turbulence’’ operator toimprove the convergence and diversity for multi-objective optimi-zation. Hu, Eberhart, and Shi (2003) adopt a secondary populationcalled extended memory to improve their previous dynamic neigh-borhood PSO approach presented in (2002). Mostaghim and Teich(2004) propose a new method which divides population of the cov-ering MOPSO into sub-swarms. The sub-swarms try to cover thegaps between the non-dominated solutions found in the initialrun. In later research in the field of MOPSO, different approacheshave been introduced to identify the local best solution based onmemorizing all the non-dominated solutions. These approacheswhen implemented on standard test problems show that keepingthe particle archive improves significantly the effectiveness ofthe technique (Jürgen & Sanaz, 2006). Mahnam, Yadollahpour, Fa-mil-Dardashti, and Hejazi (2009) develop an inventory model foran assembly supply chain network where in the performance ofsupply chain is assessed by two criteria total cost and fill rate usinga fuzzy expert system. To solve this bi-criteria model, hybridizationof multi-objective particle swarm optimization and simulationoptimization are considered. Results indicate the efficiency of pro-posed approach in performance measurement. Guo, Li, Mileham,and Owen (2009) apply effectively a variant of PSO to integrateprocess planning and scheduling. Recently Guo and Hou (2010)develop a simulation model to describe the operations of athird-party logistics provider. They develop an MOPSO algorithmcombining with the simulation model to identify non-dominatedsolutions that constitute the trade-off curve between fill rate andtotal inventory. Marinakis and Marinaki (2010) propose a newhybrid algorithmic nature inspired approach based on particle

swarm optimization (PSO), Greedy Randomized Adaptive SearchProcedure (GRASP) and Expanding Neighborhood Search (ENS)Strategy and apply it to the Probabilistic Traveling SalesmanProblem (PTSP). They test the algorithm on numerous benchmarkproblems and show satisfactory results.

Tsou, Yang, Chen, and Lee (2011) present a bi-objective modelto represent a fixed order system under lost sales. They considercycle stock investment and service level to be simultaneously opti-mized. They utilize a solver based on MOPSO to find the inventorymanagement policies and show that combining local search andMOPSO will hopefully improve the chance of finding bettertrade-offs in inventory cost and customer service. Study of Cheng-ming (2011) focus on swarm intelligence where particles adjusttheir flying time adaptively through generations. This insures es-cape from local minima. They call this algorithm as Improved Dis-crete Particle Swarm optimization and apply it to optimizecapacitated vehicle routing problem. An integrated optimizationapproach using an artificial neural network and a bidirectional par-ticle swarm is proposed by Navalertporn and Afzulpurkar (2011).They apply the approach to the real-life ceramic tile pressing pro-cess where multiple responses are considered and their experi-mental results indicate superior performance of the proposedapproach over the conventional ones.

The work of Mukhopadhyay and Banerjee (2012) emphasizesthe stochastic nature of chaotic carrier by embedding the meritsof it in a Swarm Intelligence optimization technique. The outcomeis a modified improved version of particle swarm optimization andthey apply this for global optimization of the unknown parametersof a laser system derived from Maxwell–Bloch’s Equations. Theircomparative analysis reveals that the proposed version is effec-tively tackling the problem of premature convergence with en-hanced convergence speed. An optimization hybrid swarmalgorithm namely particle-bee algorithm is developed by Lienand Cheng (2012). They apply their algorithm to the constructionsite layout (CSL) design problems and show that the above saidalgorithm can be efficiently employed to solve CSL problems withhigh dimensionality. Mousa, El-Shorbagy, and Abd-El-Wahed(2012) propose a hybrid multi-objective evolutionary algorithmcombining two heuristic optimization techniques, genetic algo-rithm (GA) and particle swarm optimization (PSO). They apply hy-brid algorithm to various kinds of multi-objective (MO) benchmarkproblems to stress the importance of hybridization algorithms ingenerating Pareto optimal sets for multi-objective optimizationproblems. Prasanna Venkatesan and Kumanan (2012) propose dis-crete particle swarm algorithm (MODPS) to optimize the supplychain network with the objectives of minimization of supply chaincost, minimization of demand fulfillment lead time and maximiza-tion of volume flexibility. In the first stage the performance of twoglobal guide selection techniques are evaluated and in the secondstage proposed (MODPS) is compared with non-dominated sortinggenetic algorithm-II. The results indicate that the proposedapproach is effective in producing high-quality Pareto-optimalsolutions. Che (2012) present a modified PSO method formulti-echelon unbalanced supply chain planning. The experimental resultsinfer that the modified method they develop obtain a better qualitysolution compared to classical GA and PSO. Kadadevaramath, Chen,Latha Shankar, and Rameshkumar (2012) has proposed modelingand optimization of a three echelon supply chain network usingthe particle swarm optimization to address the demand uncer-tainty and constraints posed by the every echelon in the supplychain design operations.

The above literature survey indicates that very little researchhas been carried out to implement MOPSO algorithm in supplychain network optimization. The purpose of this paper is to formu-late and analyze a strategic plant location-allocation model for sin-gle product four-echelon supply chain network with respect two


conflicting objectives; minimizing total supply chain cost consist-ing of production, transportation and distribution cost and maxi-mizing fill rate using MOHPSO algorithm. The proposed strategyshould lead to a final SC design which would represent the desiredcompromise among the different objectives from the decision-ma-ker’s perspective.

3. Modeling and mathematical formulation

This section discusses the objectives of the network optimiza-tion model, modeling of the supply chain network (SCN) and themathematical programming of the designed model. During supplychain network design, in one of the phases, managers must decideon location and capacity allocation for each facility which can bemade jointly. The demand allocation decisions can be altered onregular basis as costs change and markets evolve (Chopra &Meindl, 2005). This work considers a general supply chain networkconsisting of four different level enterprises. The first-level enter-prise is the CZ from which the products are sold to customers sub-ject to a given lower bound of customer service. The second-levelenterprise is the DC from which the products are transported toCZ. The third level enterprise is the plant or manufacturer andfourth level is the supplier. Revenues come from sale of productsand costs arise from facilities, labor, transportation, material andinventories. In order to overcome difficulty in mathematical for-mulation and convergence, a minimum target for the demand sat-isfaction, which must be attained in all the scenarios, isincorporated as a constraint within the existing formulation.

Assumptions

– Product produced needs three raw materials.– Capacities of suppliers for different raw materials, cost of mak-

ing each raw material at each supplier, cost of transportationper raw material per supplier–plant pair are known in advance.

– Potential location sites for plants and DCs are known.– Production, inventory costs per product and distribution costs

per plant–DC pair are known and fixed.– Inventory costs per product and cost of transportation per DC–

CZ pair are known and fixed.– Minimum fill rate is to be maintained.

The overall problem can thus be stated as follows:Given

� Number of suppliers and their capacities for each raw material.� Number of potential plants, DCs and their capacities.� Cost of making of each raw material at each supplier and trans-

portation cost per supplier–plant pair.� Production and inventory costs in each plant.� Distribution costs per product per plant–DC pair.� Product throughput cost in each DC and transportation cost per

DC–CZ pair.� Number of CZs and their demands.� Minimum fill rate (fraction of demand satisfied) to be

maintained.

To determine

� Quantity of each raw material to be shipped from each supplierto a plant.� Number and location of plants.� Flow of products from located plants to DCs.� Number and location of DCs.� Allocation of CZs to DCs.

Objective functions

� To minimize total cost of supply chain which includes rawmaterial making and transportation, plant location, productionand inventory costs in plant, distribution cost from plant toDC, throughput cost in DC, distribution cost from DC to CZ, etc.� To maximize the fill rate.

Product availability reflects a firm’s ability to fill a customerorder out of available inventory. A stock-out results if a customerorder arrives when product is not available. Product fill rate (fr)and cycle service level (CSL) are the ways to measure productavailability. Product fill rate is the fraction of product demand thatis satisfied from product in inventory. CSL is the fraction of replen-ishment cycles that end with all the customer demand being met(Chopra & Meindl, 2005). This work considers a constrainedbi-objective problem formulation for a pull based supply chainand proposes the utility of using hybrid MOPSO algorithm for thesimultaneous optimization of both cost and product fill rate.

4. Problem description

This work considers four-echelon capacitated plant location-allocation network model as shown in Fig. 1. The notations usedin this network model are listed in Table 1 with their meaning.

The model defines decision variables as shown in Table 2.The problem is then formulated as the following mixed integer

model:

Objective 1 : MinXn

i¼1

fiyi þXt

e¼1

feye þXn

i¼1

Xl

h¼1

Xp

c¼1

cchixhci

þXn

i¼1

Xt

e¼1

ciexie þXt

e¼1

Xm

j¼1

cejxej ð1Þ

Objective 2 : Max

Pte¼1

Pmj¼1xejPm

j¼1Djð2Þ

Subject to ð3ÞXn

i¼1

xhci 6 Sch for h¼ 1; . . . :l; c ¼ 1; ::p ð4Þ

Xt

e¼1

xej 6 Dj for j¼ 1; . . . ;m ð5Þ

Xt

e¼1

xie 6 Kiyi for i¼ 1; . . . :n ð6Þ

Xm

j¼1

xej 6 Keye for e¼ 1; . . . ; t ð7Þ

Xl

h¼1

xhci �Xt

e¼1

xie P 0 for i¼ 1; . . . ;n;

c ¼ 1; . . . ;p ð8ÞXn

i¼1

xie �Xm

j¼1

xej P 0 for e¼ 1; . . . ; t ð9Þ

0:806

Pte¼1

Pmj¼1xejPm

j¼1Dj6 1 ð10Þ

yi; ye 2 f0;1g for i¼ 1 . . . . . . n; e¼ 1; . . . ; t ð11Þ

The objective function (1) minimizes the total cost (fixed + variable)of setting up and operating the network and objective function (2)maximizes fill rate.

The constraint in Eq. (4) specifies that the total quantity shippedfrom a supplier cannot exceed the supplier’s capacity. The

Fig. 1. Four-Echelon supply chain network of 3 suppliers, 5 plants, 6 dcs and 7 CZs.

Table 1Notations and their meaning used in the mathematical formulation.

Notationsused

Meaning

m Number of customer zones (markets) or demand points(j = 1,2 . . . ,m)

t Number of warehouse (DC) locations (e = 1,2, . . . , t)n Number of potential plant locations (i = 1,2 . . . ,n)l Number of suppliers (h = 1,2, . . ., l)p Number of components (c = 1,2, . . . ,p)Dj Average demand from market j/periodKi Potential capacity of plant iKe Potential capacity of warehouse eSch Supply capacity at supplier h for component cfi Annualised fixed cost of keeping plant i open/yearfe Annualised fixed cost of keeping warehouse e open/yearcchi Cost of making and shipping a component c from supply source

h to plant i/unitcie Cost of producing, stocking and shipping one unit from plant i

to warehouse e/unitcej Cost of throughput and shipping one unit from warehouse e to

customer zone j/unitCSch Cost of making a component c by supplier h/unitSTCchi Transportation cost of a component c from supplier h to plant i/

unitci Manufacturing cost at plant i/unitICi Inventory cost at plant i/unitIci Inventory of component c at plant iPTCie Plant transportation cost from plant i to warehouse e/unit

Table 2Decision variables used in the mathematical formulation.

Notations used Meaning

yi 1, if plant i is open, 0 otherwise.ye 1, if warehouse e is open, 0 otherwise.xhci Quantity of component c shipped from supplier h to plant ixie Quantity shipped from plant i to warehouse exej Quantity shipped from warehouse e to customer zone j


constraint in Eq. (5) requires that the demand at each regionalmarket be satisfied to the maximum extent. The constraint in Eq.(6) states that no plant can supply more than its capacity. Capacityis 0 if the plant is closed and Ki if it is open. Similarly Eq. (7) states

that no warehouse can supply more than its capacity. Theconstraint in Eq. (8) states that the quantity shipped out of a plantcannot exceed the quantity of raw material received. Theconstraint in Eq. (9) states that the quantity transported from awarehouse cannot exceed the quantity received. The constraintin Eq. (10) states that fill rate can vary from 80% to 100%. Theconstraint in Eq. (11) enforces that each plant is either open(yi = 1) or closed (yi = 0).

The solution has the decision variables which characterize thenetwork configuration. They include binary variables that repre-sent the existence of plants and warehouses (DCs) of the SC andthe continuous ones that represent the flows of materials from var-ious suppliers to various plants from plants to different ware-houses and the allocation of regional demand to these warehouses.

5. Methodology

5.1. Introduction to particle swarm optimization algorithm

Particle swarm optimization (PSO) algorithm is a promisingnew population-based optimization technique developed by


Kennedy and Eberhart (1995). This algorithm models a set of po-tential problem solutions as a swarm of particles moving aboutin a virtual search space. PSO is based on the behavior of commu-nities that have both social and individual communication, similarto birds searching for food. The bird would find food through socialcooperation with other birds around it (within its neighborhood).

In PSO algorithm, each individual (particle) represents a solu-tion in a n-dimensional space. Each particle also has knowledgeof its previous best experience and knows the global best experi-ence (solution) found by the entire swarm. Each particle updatesits direction using the following equations:

v ij ¼ w � v ij þ c1 � r1 � ðpij � xijÞ þ c2 � r2 � ðpgj � xijÞxij ¼ xij þ v ij

where w is the inertia factor influencing the local and global abili-ties of the algorithm, vij is the velocity of the particle i in the jthdimension, c1 and c2 are weights affecting the cognitive and socialfactors, respectively. r1 and r2 are uniform random variables be-tween 0 and 1, and two random values are generated indepen-dently. pij stands for the best value found by particle i (pbest) andpg denotes the global best found by the entire swarm (gbest). Afterthe velocity is updated, the new position i in its jth dimension is cal-culated. This process is repeated for every dimension and for all theparticles in the swarm. Eventually the swarm as a whole, like a flockof birds collectively foraging for food, is likely to move close to anoptimum of the fitness function. PSO algorithm is becoming verypopular due to its simplicity of implementation and ability toquickly converge to a reasonably good solution. This is achievedby adjusting easily few parameters in PSO algorithm to have moreglobal search ability at the beginning of the run and have more localsearch ability near the end of the run.

5.2. Particle swarm model for binary decision

This model proposes that the probability of an individual’sdeciding yes or no, true or false or making some other binary deci-sion, is a function of personal and social factors and is given by,

PðxidðtÞ ¼ 1Þ ¼ f ðxidðt � 1Þ;v idðt � 1Þ; pid;pgdÞ

where

� P(xid(t) = 1) is the probability that individual i will choose 1.� (xid(t)) is the current state of individual i.� t is the current time step, and t � 1 is the previous step.� vid(t � 1) is a measure of the individual’s current probability of

deciding 1.� pid is the best state found so far, for example, it is 1 if the indi-

vidual’s best success occurred when xid was 1 and 0 if it was 0. Itis referred as pbest.� pgd is the neighborhood best, again 1 if the best success attained

by any member of the neighborhood was when it was in the 1state and 0 otherwise. It is referred as gbest.

The parameter vid(t), will determine the probability threshold. Ifvid(t) is higher, the individual is more likely to choose 1, and lowervalues favor the 0 choice. Such a threshold stays in the range[0.0,1.0]. The following sigmoid function is used to determinethe probability threshold.

sðv idÞ ¼1

1þ expð�v idÞ

In any situation, whether individual learning (pbest) is stronger orsocial-influence (gbest) is stronger is unknown, both will beweighed by random numbers, so that sometimes the effect of oneand sometimes the other will be stronger. The symbol u is used

to represent a positive random number drawn from a uniform dis-tribution with a predefined upper limit so that the two u limits sumto 4.0. Thus the formula for binary decision is

v idðtÞ ¼ v idðt � 1Þ þu1ðpid � xidðt � 1ÞÞ þu2ðpgd � xidðt � 1ÞÞIf qid < sðv idðtÞÞ then xidðtÞ ¼ 1; else xidðtÞ ¼ 0

Then qid is a vector of random numbers, drawn from a uniform dis-tribution between 0.0 and 1.0. The constant parameter Vmax is oftenset at ±.0, so that there is always at least a chance ofs(Vmax) ffi 0.0180 that a bit will change state (Kennedy & Eberhart,2001).

5.3. The particle swarm in continuous numbers

The position of a particle i is assigned the algebraic vector sym-bol ~xi. There can be any number of particles and each vector can beof any dimension. Change of position of a particle is called ~v i thevelocity. Velocity is a vector of numbers that are added to the posi-tion co-ordinates in order to move the particle from one time stepto another. As the system is dynamic, position of each individual ischanging. The direction of movement is a function of current posi-tion and velocity, the location of individual’s previous best successcalled pbest and best position found by any member in the swarmcalled gbest of the neighborhood (Kennedy & Eberhart, 2001).

5.4. The hybrid PSO

A hybrid swarm optimizer combines both binary and real val-ued parameters in one search. This hybrid optimizer is used forsuch problems which deal with both continuous and binary vari-ables. It simply operates on binary inputs with binary particleswarm algorithm and treat the continuous variables with real val-ued particle swarm. Binary PSO algorithm is used to take the loca-tion decisions (whether to locate a facility at a given candidatesite), while the allocation decisions are obtained by continuousPSO algorithm (Kennedy & Eberhart, 2001).

5.5. Multi-objective optimization using MOHPSO

The general Multi-Objective Optimization can be defined asthe process of simultaneously optimizing two or more conflictingobjectives subjected to certain constraints. Since multi-objectiveoptimization problem has several objective functions, the solu-tion method aims at finding several ‘‘trade-off’’solutions ratherthan a single solution. The concept of Pareto optimality in mul-ti-objective optimization is the one proposed by Vilfredo Paretoin 1986.

A point x⁄ 2X is Pareto optimal if "x 2X and I = {1,2, . . . ,k}either:

"i 2 I(fi(x⁄) 6 fi(x)) and, there is at least one i e I such thatfi(x⁄) < fi(x).This definition says that x⁄ is Pareto optimal if there exists no

feasible x which would decrease some criteria without causing asimultaneous increase in at least one other criterion (Coello &Lechuga, 2002).

Multi-objective optimization algorithm makes use of domina-tion concept to arrive at optimal solutions. In this algorithm, twosolutions are compared on the basis of whether one dominatesthe other solution or not. If there are M objective functions thena solution x is said to dominate the other solution y, if both the fol-lowing conditions are true:

1. The solution x is not worse than y in all objectives.2. The solution x is strictly better than y in at-least one objective.

Table 3Supplier capacities, plant capacities and fixed costs.

Vendor capacities Plants capacities and fixed costs Warehouses Capacity

Vendors Components Plants XCapacity Fixed Cost

C1 C2 C3

V1 36 62 50 P1 18 7650 WH1 15V2 40 65 55 P2 24 3500 WH2 12V3 42 70 60 P3 37 500 WH3 14

P4 22 4100 WH4 13P5 41 2200 WH5 12

WH6 16

Table 5Manufacturing cost, inventory cost, transportation cost matrix per unit for second stage of SCN.

Plants Warehouses Manufacturing cost at plants Inventory cost at plants

WH1 WH 2 WH 3 WH 4 WH 5 WH 6

P1 7 12 15 17 18 20 3900 50P2 12 10 11 13 15 17 2010 45P3 8 10 14 15 18 21 1945 55P4 10 12 13 14 13 18 1855 48P5 8 10 11 15 11 12 1975 52

Table 6Handling cost and transportation cost matrix per unit for third stage of SCN.

Warehouses Customer zones Handling cost at warehouse

CZ1 CZ2 CZ3 CZ4 CZ5 CZ6 CZ7

WH1 8 3 9 6 7 3 4 55WH2 5 8 7 6 3 2 8 50WH3 9 3 8 6 7 5 4 60WH4 3 9 2 2 5 4 8 54WH5 7 6 3 9 4 9 4 55WH6 5 6 7 8 3 2 9 45Monthly demand Dj 3 5 4 6 4 5 3

Table 4Cost of making and shipping cost matrix per unit for first stage of SCN.

Vendors Components Cost of making the component at vendor’s place Supplier transportation costsPlants

1 2 3 4 5

V1 C1 300 10 13 8 11 15C2 115 6 7 5 8 4C3 90 3 4 5 4 5

V2 C1 320 17 14 12 12 15C2 120 6 5 7 5 7C3 85 6 6 5 6 4

V3 C1 290 13 12 14 11 9C2 125 6 5 3 4 5C3 75 3 6 3 2 3


If any of the above conditions is violated, the solution x does notdominate the solution y. Since multi-objective optimization hasmore than one objective to be optimized simultaneously, therecannot be a single optimum solution which simultaneously opti-mize all objectives. The resulting outcome is a set of optimal solu-tions with a varying degree of objective values. This set of solutionsis called non-dominated set. The following two conditions must betrue for a non-dominated set:

1. Any two solutions of non-dominated set must be non-domi-nated with respect to each other.

2. Any solution not belonging to non-dominated set is dominatedby at least one member of non-dominated set.

Such a non-dominated set is also called as Pareto-optimal set.Because minimization of total SC cost and maximization of fill

rate cannot be achieved at the same time as they are conflictingin nature, there exists a trade-off between them. This type of sys-tem clearly represents a multiobjective optimization situationwhich is a procedure looking for a compromise policy, the result,called a Pareto-optimal or noninferior solution, consists of a num-ber of options. Hence the Pareto set solutions and their corre-sponding decision variables should be provided, from which thedecision-maker can select a solution to satisfy the industrial need.

Many researchers have suggested many classical multi-objective optimization methods such as weighted sum approach,e-constraint method, weighted metric method, value function

Fig. 2. Feasible solutions where in series 2 represents Pareto-front with six non-dominated points showing trade-off between SC cost and fill rate and series 1represents other inferior points.

Fig. 3. Convergence behavior of HPSO.

Table 7Pareto-front objective function values.

Scenario Total fixedcost

Totalvariable cost

Total SCcost

Demandsatisfied

Actualdemand

1 30,050 112,256 142,306 30 302 27,150 114,235 141,385 29 303 29,750 110,879 140,629 28 304 15,600 104,304 119,904 27 305 26,950 90,868 117,818 26 306 20,900 82,390 103,290 25 307 17,950 82,530 100,480 24 30

Table 8The optimal material flows from suppliers to plants corresponding to Pareto frontpoints.

Supplier plants Component 1 Component 2 Component 3

V1 V2 V3 V1 V2 V3 V1 V2 V3

Scenario 1P1 0 0 8 10 0 0 0 12 0P2 5 0 11 10 12 0 0 22 0P3 15 0 0 18 0 0 0 12 0P4 11 0 0 0 12 0 15 0 0P5 0 0 0 0 0 0 0 0 0

Scenario 2P1 0 1 4 8 0 0 8 0 0P2 0 18 0 16 0 0 9 12 0P3 10 9 0 0 17 0 0 0 20P4 0 0 0 0 0 0 0 0 0P5 0 11 0 0 10 0 0 9 0

Scenario 3P1 7 0 10 13 5 0 5 0 14P2 0 0 6 0 8 0 0 0 9P3 0 0 0 0 0 0 0 0 0P4 8 0 0 10 0 0 7 0 0P5 0 0 0 0 0 0 0 0 0

Scenario 4P1 0 0 0 0 0 0 0 0 0P2 0 0 0 0 0 0 0 0 0P3 23 0 0 25 0 0 21 0 0P4 0 0 15 0 16 0 0 0 18P5 0 0 17 20 0 0 0 0 22

Scenario 5P1 0 11 0 0 0 12 8 3 0P2 0 6 0 0 6 0 0 0 7P3 0 0 0 0 0 0 0 0 0P4 10 0 0 9 0 2 12 0 0P5 0 0 0 0 0 0 0 0 0

Scenario 6P1 0 0 0 0 0 0 0 0 0P2 0 7 11 0 0 14 19 0 0P3 6 5 0 10 0 0 12 0 0P4 0 0 0 0 0 0 0 0 0P5 0 0 9 0 12 0 0 7 0

Scenario 7P1 0 0 9 0 0 10 0 0 11P2 0 0 0 0 0 0 0 0 0P3 17 0 0 18 0 0 0 20 0P4 0 0 0 0 0 0 0 0 0P5 0 0 0 0 0 0 0 0 0


method, goal programming method etc. But all these methods con-vert a multi-objective optimization problem into a single-objectiveoptimization problem. These methods are subjective and only onePareto-optimal solution can be expected in one simulation run.Since then there has been increasing interest to use evolutionaryalgorithms in solving multi-objective optimization because of theirability to find multiple optimal solutions in single run and com-plexity of location-allocation problems.

5.6. Selecting the global best and personal best

In MOHPSO, each particle has to change its position as guidedby two leaders pbest and gbest which must be selected from theupdated set of non-dominated solutions stored in the archive.The main difficulty in MOHPSO is to pick suitable global best(gbest) and personal best (pbest) to move the particles throughsearch space to attain a good convergence and diversity alongthe Pareto-optimal front.

In this study, method of selection of pbest and gbest used is in-spired by NSGA-II. For selecting pbest a method called Prandom isemployed, according to which a single pbest is maintained. Pbest isreplaced if new value < pbest, otherwise, if new value is found tobe mutually non-dominating with Pbest, one of the two israndomly selected to be the new pbest. ( Eveson, Fieldsend, &Singh, 2002). To select gbest from archive, this work makes use

of crowding distance measure and niche count (Deb, 2001). Thenon-dominated solutions from the last generations are kept inthe archive. The archive is an external population, in which non-dominated solutions are kept after each flight cycle. Such anarchive will allow the entrance of a solution only if: (a) it is non-dominated with respect to the contents of the archive or (b) itdominates any of the solutions within the archive (Coello &Lechuga, 2002). Finding a relatively large set of Pareto-optimaltrade-off solutions is possible by running the MOHPSO for manygenerations.

Table 9Optimum location and distribution of products from plants to DCs corresponding toPareto front points.

Supply plant D1 D2 D3 D4 D5 D6 Optimal locationof plants

Scenario 1P1 0 0 0 2 0 4 1P2 1 3 1 7 0 0 1P3 8 0 0 0 0 2 1P4 0 2 2 2 0 0 1P5 0 0 0 0 0 0 0

Scenario 2P1 1 0 0 1 0 1 1P2 2 0 0 6 4 2 1P3 2 0 9 0 1 2 1P4 0 0 0 0 0 0 0P5 0 0 0 3 0 4 1

Scenario 3P1 0 4 1 8 4 0 1P2 0 0 2 0 0 4 1P3 0 0 0 0 0 0 0P4 0 2 2 3 0 0 1P5 0 0 0 0 0 0 0

Scenario 4P1 0 0 0 0 0 0 0P2 0 0 0 0 0 0 0P3 2 0 0 5 5 2 1P4 2 0 0 4 2 2 1P5 2 0 0 0 1 9 1

Scenario 5P1 1 3 4 0 3 0 1P2 0 0 0 0 0 6 1P3 0 0 0 0 0 0 0P4 0 0 3 0 4 3 1P5 0 0 0 0 0 0 0

Scenario 6P1 0 0 0 0 0 0 0P2 0 3 7 0 4 0 1P3 5 4 0 0 0 0 1P4 0 0 0 0 0 0 0P5 0 0 0 6 1 0 1

Scenario 7P1 3 4 0 1 0 1 1P2 0 0 0 0 0 0 0P3 0 5 0 5 0 6 1P4 0 0 0 0 0 0 0P5 0 0 0 0 0 0 0


Niche count nci for ith solution is calculated using sharing func-tion sh(dij) as follows:

nci ¼XN

j¼1

shðdijÞ and shðdÞ ¼ 1� drshare

� �a; if d 6 rshare

0; otherwise

8<:

It provides an estimate of extent of crowding near a solution. Heredij is the distance between the ith and jth solutions. Its value is al-ways greater than or equal to one because sh(dii) = 1. rshare is calcu-lated imagining that the n-dimensional hypersphere of radius rmust be divided among q optima equally. After calculating nc foreach solution in the archive the gbest is selected from the archivewhose nc is the smallest. This ensures diversity being maintainedas one which is comparatively less crowded is selected (Deb,2001). The MOHPSO algorithm used in this work includes followingsteps:

� Generate randomly n particles constituting swarm, initialisetheir initial velocities, such that the particle velocity in the kthdimension is limited by some maximum value, vkmax.� Set iteration counter j = 0.� Calculate fitness values of each particle, using two objectives.

� Initialize pbest and gbest values� Do

{– Calculate new velocity using pbest and gbest values.– Update position for each particle using new velocity.– Calculate new fitness value.– Update pbest if the new fitness value is better than current

one.– Determine gbest if the new fitness value is better than cur-

rent one.– Identify the particles that give non-dominated solutions

with respect to present Gbest and store them in an archivecalled nonDomList. The solutions that are not non-domi-nating with respect to present gbest are deleted so as tolimit the list.

– Calculate the crowding distance for each particle in theSwarm for niching.

– Sort all particles in the current non-dominated front indescending order, and then choose best with least nichecount for next iteration.

– j = j + 1} till j <= maximum iteration or till termination condition is

reached.

6. Numerical example

In order to illustrate the capabilities of the proposed model andalgorithm, a hypothetical case study of four echelon supply chainnetwork connecting 3 suppliers, 5 plants, 6 DCs and 7 CZs as shownin Fig. 1 is considered. As per the problem, the company has man-ufacturing plants located in five different geographical locationswith total capacity of 142 which serve markets in seven differentplaces with total demand of 30 through six warehouses with totalcapacity of 82. In the first stage of SCN, each vendor supplies threecomponents which are required for producing one product. Sup-plier capacities, plant capacities, warehouse capacities and fixedcosts per month at each plant are given in Table 3. Supplier costof making a component and supplier transportation cost per unitshipped are given in Table 4. Variable inventory, production andtransportation costs per unit shipped and demand at differentCZs are given in Table 5. Handling cost and transportation cost ma-trix per unit for third stage of SCN is given in Table 6. This problemis analyzed with respect to two conflicting objectives, minimizingtotal cost and maximizing fill rate, with the condition that mini-mum fill rate to be attained is 80% of actual demand (24). Thereare totally 128 decision variables to be optimized and 48 con-straints. This problem results in seven Pareto solutions. The finaldecision is made among them taking the total balance over all cri-teria into account. This is a problem of value judgment of decisionmaker. Also these decisions will be revisited every year as demandand costs change.

6.1. The proposed MOPSO implementation

The proposed MOHPSO based approach was implemented usingC language and the developed software program was executed on a1.8-GHz Pentium 4 personal computer. The random seeds weregenerated by a random number generator incorporated in the pro-gram. Initially, several simulation runs have been done with differ-ent values of the PSO key parameters such as the maximumallowable velocity. Other parameters selected are: number of par-ticles n = 20, decrement constant = 0.8, c1 = 1.05, c2 = 0.05. The time

Table 10Optimum distribution of products from DCs to CZs corresponding to Pareto front points.

Supply DC CZ1 CZ2 CZ3 CZ4 CZ5 CZ6 CZ7 Optimal location of DCs

Scenario 1DC1 0 0 2 3 0 1 2 1DC2 0 0 1 1 1 0 1 1DC3 1 2 0 0 0 0 0 1DC4 1 2 1 2 1 2 0 1DC5 0 0 0 0 0 0 0 0DC6 1 1 0 0 2 2 0 1Total quantity supplied 3 5 4 6 4 5 3 30Demand 3 5 4 6 4 5 3 30




Scenario 5DC1 0 0 0 0 0 0 0 0DC2 0 0 0 3 0 0 0 1DC3 1 1 1 0 2 1 1 1DC4 0 0 0 0 0 0 0 0DC5 1 2 3 0 0 0 1 1DC6 0 1 0 3 1 4 0 1Total Quantity Supplied 2 4 4 6 3 5 2 26Demand 3 5 4 6 4 5 3 30

Scenario 6DC1 0 0 2 1 0 0 2 1DC2 0 2 1 2 0 1 0 1DC3 1 0 0 2 0 3 1 1DC4 0 0 0 1 3 0 0 1DC5 1 2 0 0 0 0 0 1DC6 0 0 0 0 0 0 0 0Total Quantity Supplied 2 4 3 6 3 4 3 25Demand 3 5 4 6 4 5 3 30



required to obtain solutions ranges from 363 to 1772 s dependingon PSO key parameters values.

7. Results and discussion

Fig. 2 depicts the obtained optimum feasible solutions. In thisPareto front is represented by series2 and all other inferior pointsby series 1. The convergence behavior of MOHPSO is shown in


Fig. 3 for total demand of 27 for both best and worst cases. Thesolution space has seven non-dominated optimum decision points,whose corresponding objective values are given in Table 7 and cor-responding decision variables representing optimum materialflows from suppliers to plants for seven different scenarios aregiven in Table 8, optimum location and distribution of goods from5 plants to 6 DCs are tabulated in Table 9 and optimum locationand flow of goods from 6 DCs to 7 CZs are tabulated in Table 10.These points clearly demonstrate tradeoffs in objective functionstotal SC cost to be minimized and fill rate to be maximized fromwhich an appropriate solution can be compromisingly chosen asper the requirements of the corresponding industrial environment.

8. Conclusion

In this paper, an analytical model is formulated for the locationand allocation of facilities of four-echelon supply chain network forthe optimal facility location and capacity allocation decisions.Fixed location and variable material cost, production, inventoryand transportation costs are considered while making strategicdecisions. Two objective functions of minimizing total SC costand maximizing fill rate are considered. A heuristic based hybridMOPSO is used as optimizer. This algorithm is mainly aimed atcharacterizing the Pareto Optimal front by computing well-distrib-uted non-dominated solutions. These solutions represent trade-offsolutions which facilitate decision makers to develop managementpolicies under a changing environment. Further this can be used asdecision support system for strategic supply chain design andmonitoring of material flow. Above explained model and algorithmfurther can consider safety stock costs, risk related and reliabilitycosts. Also the model can optimize more than two conflictingobjectives simultaneously.

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