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International Journal of Production Research

ISSN: 0020-7543 (Print) 1366-588X (Online) Journal homepage: http://www.tandfonline.com/loi/tprs20

An integrated inventory optimization model forfacility location-allocation problem

R.P. Manatkar, Kondapaneni Karthik, Sri Krishna Kumar & Manoj KumarTiwari

To cite this article: R.P. Manatkar, Kondapaneni Karthik, Sri Krishna Kumar & Manoj KumarTiwari (2015): An integrated inventory optimization model for facility location-allocationproblem, International Journal of Production Research, DOI: 10.1080/00207543.2015.1120903

To link to this article: http://dx.doi.org/10.1080/00207543.2015.1120903

Published online: 21 Dec 2015.

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An integrated inventory optimization model for facility location-allocation problem

R.P. Manatkar, Kondapaneni Karthik, Sri Krishna Kumar and Manoj Kumar Tiwari*

Department of Industrial and Systems Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India

(Received 25 May 2015; accepted 6 November 2015)

This paper presents an integrated inventory distribution optimisation model for multiple products in a multi-echelonsupply chain environment. Inventory, transportation and location decisions are considered. The objective is to offer prac-tical guideline to the steel retail supply chain practitioners in choosing the correct distribution centre, finding out inven-tory level at individual inventory keeping points (retailers and distribution centres) point thereby helping them inreducing overall distribution cost. The framework presented endorses systems approach and suggests near-optimalapproach to calculating inventory for an individual distributor and his retailers. Two algorithms are used to solve thisproblem, a novel hybrid Multi-objective Self-learning particle swarm optimiser and Non-dominated sorting genetic algo-rithm-II. The model and solution methods are tested on real data-sets obtained from organisations in the steel retail envi-ronment. The actual data on inventory holding, ordering and transportation costs of distributors and retailers are used asinputs. The decisions like choosing correct set of Distribution centres, keeping optimal regular and safety stock inventorylevels are arrived at by applying practical constraints in the supply chain. Model developed assists in effective andefficient distribution of the products manufactured from the optimal location at minimal cost.

Keywords: inventory distribution; facility location-allocation problem; supply chain network design; multi-objectiveoptimisation; swarm optimisation

1. Introduction

Innovative strategies and technologies are being implemented in the field of Supply Chain Management by enterprisesto tackle new challenges faced due to changing demands of customers. Efficiency and responsiveness are the two newstrategies for supply chain that have emerged in recent years. Efficiency aims at delivering products at reduced costs,while responsiveness, on the other hand, is the ability of the organisation to respond quickly to customer demands andsave costs (Liao and Hsieh 2010). Amongst other drivers, an organisation’s responsiveness depends upon managementof inventory at plant warehouses, regional warehouses, distributor warehouse, transportation and retailers. The comprisedinventories on hand can cost between 20 and 40% of the value. Therefore, careful management of inventory becomespredominant for economic reasons. Despite many strategies such as just-in-time, quick response and collaborative prac-tices are being practised by the manufacturers, distributors and retailers, still there is a big scope for improvement, sincequestion of interest to various players in the supply chain is How much inventory is the right inventory? Higher cus-tomer service level demands large inventory thereby inflating the costs and lower inventory may result in sales loss.Hence, to make decent profit margins and remain competitive, firms are looking for measures to reduce their costs bymaintaining desired service level through better inventory management practices.

Managing inventory in a distribution network consists of three critical tasks. Primarily, determine the location ofDistribution centres (DCs) (from here warehouses and distribution centres are considered same) from those locations.Second, allocate group of customers to each DC and thirdly, determine the amount of inventory at each location in theSupply chain. In a stochastic environment, inventory at any facility is categorised into two types – regular stock andsafety stock. Regular stock is carried by any facility to serve its nominal demand, while the safety stock is to mitigatethe risk of stock outs due to the unpredictable nature of supply and demand. Therefore, it is essential to maintainadequate regular and safety stock at each facility of the supply chain network (SCN).

Supply chain consists of six drivers: location, transportation, information, sourcing, pricing and inventory. These dri-vers have direct or indirect impact on each other as well as on the whole SCN. Any decision on these drivers will havean overall impact on the performance and operational efficiency of the SCN (Bashiri and Tabrizi 2010). Determiningstock point location is common in supply chain practices; however, identifying the exact location based on integrated

*Corresponding author. Email: mkt09@hotmail.com

© 2015 Taylor & Francis

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inventory model (integrating location sourcing and transportation decisions) is difficult. As the market is becoming moreand more competitive, companies need to be more aggressive in achieving their service goals at the lowest possible cost.For that reason, firms ought to design the SCN considering their inventory, location, sourcing and transportation in aholistic manner.

The model developed in this paper is on Multi-objective integrated allocation-inventory problem (MOIAIP).Theobjective is to lessen inventory carrying and transportation cost. This includes inventory holding cost, transportation costand the decisions regarding maximum inventory to be kept at any echelon. Specifically, it is helping in making anintegrated decision regarding:

• The optimal assignment of a group of retailers to multiple DCs.• The level of safety stock to be maintained at each facility by upholding given service level.• The level of regular stock to be maintained at each facility.• The maximum inventory at any echelon in the system.

2. Literature review

The literature on strategic issues like facility location and design of distribution network focuses on developing modelsfor spotting the best possible locations of the facilities, identify the optimal number of DCs and assignment of retailer’sto DCs. This mainly includes fixed transportation and facility costs, but the shortage costs and operational inventory areusually overlooked (Melkote and Daskin 2001). Daskin and Owen (2003) did the analysis on the facility locationmodelling, while Daskin (1995) clearly depicted FLPs. FLPs can sometimes be modelled as SCN design problems.Ambrosino and Scutellà (2005) have studied complex distribution network design problems, which involve warehous-ing, facility location, inventory and transportation decisions. Facility location modelling comes with issues andchallenges in designing a SCN. These are clearly shown by Melnyk, Narasimhan, and DeCampos (2014).

Recently, several researchers have put their effort on integrating inventory management theory in the supply chaindesign field. For multiple retailers, multiple supplier distribution networks a near optimal inventory policy was presentedby Ganeshan (1999). In their model, inventory at retailers and DC are synchronised to minimise the total logistics coststaking into account the service level requirements. A linear safety stock function was formulated by Nozick and Turn-quist (2001). They proposed Lagrangian-based scheme to solve uncapacitated FLP. On the similar line, a DC locationmodel was introduced by Daskin, Coullard, and Shen (2002) that incorporates safety stock and working inventory costat the DCs. Shen, Coullard, and Daskin (2003) as well as Daskin, Coullard, and Shen (2002) proposed a single-echelonset-covering problem to build an optimal SCN. He demonstrated that if the demand faced by the DC is Poisson distribu-tion, then the problem can be tackled effectively. Economic order quantity model and safety stock model are incorpo-rated into FLP by Miranda and Garrido (2004). Gebennini, Gamberini, and Manzini (2009) had proposed a model forsafety stock optimisation and control service level for the dynamic location–allocation. Mizgier, Jüttner, and Wagner(2013) proposed a new methodology to identify the bottlenecks in SCN design in order to make an informed decisionby the firms. Some of the researchers Shu (2010) and Li et al. (2013) proposed approximate algorithms to solve SCNdesign problems.

In recent years, FLPs are extended to multi-echelon SCN design. Teo and Shu (2004) proposed a two-echelon DC-retailer network for the set-covering model under deterministic demand. Later this model was extended to non-standarddemands by Shu, Teo, and Shen (2005). A two-echelon model that incorporates location-specific transportation costswith location decisions was proposed by Romeijn, Edwin, and Teo (2007). This model determines the number of DCsto open, location of the DCs and the way to serve from DCs to the retailers under a single-sourcing policy. Their workput forward inventory policies to reduce the location, transportation costs for two-echelon inventory model. Askin,Baffo, and Xia (2014) have modelled the distribution network of a logistics firm for multiple sources, multiple productsand multiple retailer environments. FLPs can be classified crisply into two types capacitated and uncapacitated. Many ofthe research works in FLPs is uncapacitated, but without considering capacities at the entities in the supply chain theproblem is always incomplete. Silva and de la Figuera (2007) presented a capacitated FLP with constrained backloggingprobabilities taking stochastic nature of demand.

With the increased complexity with multi-modal and non-linear problems, approximate algorithms have been findingincreased usage for solving this kind of problems. Integration of inventory, safety stock, allocation decisions etc., intoFLPs makes these non-linear and complex. This led many researchers to find application of EA in FLPs. Guner andSevkli (2008) have proposed a discrete particle swarm optimisation (PSO) approach to solve an uncapacitated FLP. Lee,Moon, and Park (2010) proposed a model integrating vehicle-routing decisions with FLP and solved it using a hybridEA approach based on the swarm intelligence to determine the optimal location of DCs. This paper used hybrid EA

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approach based on the fast elitist Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to determine the optimal loca-tion of DCs. Yapicioglu, Smith, and Dozier (2007) solved a semi-desirable FLP using a single-objective PSO and multi-objective PSO and found that usage of multi-objective approach was efficient. On similar lines using a multi-objectiveoptimisation approach, Bashiri and Tabrizi (2010) tackled a problem of locating a DC in a single DC, multiple retailerssystem. They have used PSO approach to finding the location of the DC.

In this paper, we adopted a similar research work proposed by Bashiri and Tabrizi (2010), however, they proposedmodel for a single DC multiple retailer environments. We extended the model to multiple DCs, multiple products andmultiple retailer environments. Even though model developed by Bashiri and Tabrizi (2010) curtailed maximum inven-tory at retailers, in our model, we are curtailing maximum inventory at DCs. Bashiri and Tabrizi (2010) solved theirmulti-objective problem by integrating two objectives and solved it using basic PSO. In our extended model with rretailers, d DCs and p products, the solution space is (d)rp. When it comes to higher dimensional problems, the solutionspace will be huge. With this huge solution space and non-linear nature of the problem, the algorithm will get struck inan infinite loop making our problem NP hard. Our allocation problem is analogous to classic halting problem as shownin Figure 1. To solve this type of NP-hard problem, we employed a novel hybrid EA, Multi-Objective Self-LearningParticle Swarm Optimisation (MOSLPSO) to estimate the Pareto front.

The rest of this work is structured as follows. Section 3 describes the problem and gives the overview of the mathe-matical model developed for MOIAIP. Section 4 discusses the MOSLPSO and NSGA-II algorithms with steps to solveMOIAIP. Section 5 presents the industrial cases for implementing the feasibility of applying the proposed MOIAIPmodel in the real situation and discusses the significant findings. Section 6 illustrates results and discussions. Section 7,concludes, recommends and gives suggestions for future work.

3. Problem environment

Manufacturing plants, in general, route their product through DCs. We considered a problem of designing distributionnetwork which encompasses a two-echelon environment with single manufacturing plant, multiple DCs distributedgeographically across a territory distributing multiple products to a set of retailers. The product shipment betweenmanufacturing plants and retailers is facilitated by DCs acting as intermediate facilities, as depicted in Figure 2. Thesetypes of network commonly exist for most of the manufacturing organisation in which positions of DCs are fixed.Important aspects concerning this network architecture are the development of an ideal network in which all the retail-ers are optimally allocated to the existing DCs in a way that inventory holding cost, total transportation cost is min-imised and optimal amount of inventory at all places. The Distributor wishes to redesign their SCN in such a waythat it supports the inventory replenishment activities of its retailers under stochastic demand at specified servicelevels and at the lowest possible cost. The nominal demands and standard deviation of the demands are available foreach retailer. The problem is MOIAIP, which deals with the modelling, redesigning, planning and controllingtwo-echelon supply chains.

Figure 1. proof of NP – hard complexity.

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3.1 Assumptions

The model is developed with following general basic assumptions:

(1) All products of each retailer will be served exactly by one warehouse.(2) There is no pipeline inventory.(3) DCs are uncapacitated and demands of all retailers are uncertain.

Based on assumptions and problem stated above, the proposed mathematical model with notations is illustratedbelow in Table 1.

3.2 Multi-objective non-linear integer programming model

As described earlier, we are considering the situation of multiple DCs, multiple products and multiple retailers.

Objective functions:

Min Z

Pi2I

Pj2J

Pp2P

Crwijpl

ripd

rwij X

rwijp þP

j2J

Pp2P

Cwmjp dwmj ðP

i2IlripX

rwijp Þ þ

Pi2I

Ari n

ri þ

Pj2J

Awj n

wj

þPi2I

Pp2P

hriplrip

2nriþP

j2J

Pp2P

Pi2I

hwjplripX

rwijp

2nwj

þPi2I

Pj2J

Pp2P

hripkripr

ripX

rwijp

ffiffiffiffiffiffiffiffiffiffiffifficripd

rwij

p þPj2J

Pp2P

hwjpkwjp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficwjpd

wmj

Pi2I

ðrripÞ2X rwijp

r

8>>>>><>>>>>:

(1)

MinW (2)

Subjected to Constraints: Xj2J

Xp2P

X rwijp ¼ 1 8i 2 I (3)

Xi2I

Xp2P

lripXrwijp

2nwjþXp2P

kwjp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficwjpd

wj

Xi2I

ðrripÞ2X rwijp

r� W 8 j 2 J (4)

Figure 2. Two-echelon distribution network problem.

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Derivation of ordering frequency for DC:

The total annual operational cost (TC) of ordering inventory from the plant at the DC j is given by

TC ¼Xp2P

Xj2J

Crwijpd

rwij l

ripX

rwijp þ Ar

i nri þ

Xp2P

hriplrip

2nri(5)

Ari �

Xp2P

hriplrip

2ðnri Þ2¼ 0 (6)

Solving for nri , we obtain,

nri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXp2P

hriplrip

2Ari

s(7)

TC ¼Xp2P

Cwmjp dwmj

Xi2I

lripXrwijp þ Aw

j nwj þ

Xp2P

Xi2I

hwjplwipX

rwijp

2nwj(8)

Where nri is the (unknown) number of orders per month for retailer i, and other parameters have their usual meaning asdescribed earlier. Equation (5) represents the total annual cost incurred by the DC to the retailers, which include trans-portation cost, ordering cost and inventory carrying cost. In order to find the optimal number of orders annually, we takethe derivative of Equation (5) with respect to nri and equate it to zero, we obtain Equation (6). Solving the Equation (6),

Table 1. Notations for mathematical model.

Notation Definition of sets and indexes

I Set of retailers, indexed by iJ Set of warehouses, indexed by jP Set of products manufactured by the plant, indexed by pDefinition of parametersAri Fixed administrative and handling cost of placing an order by retailer i for all the products required by him, ∀i ∊ I

Awj Fixed administrative and handling cost of placing an order by distributor for warehouse j for all the products required by

it, ∀j ∊ JCwrijp Cost of transporting one unit of product p from warehouse j to retailer i for one unit of distance,

8i 2 I ; 8j 2 JðiÞ; 8p 2 PðiÞ \ PðjÞCmwjp Cost of transporting one unit of product p from plant to warehouse j for one unit of distance, 8j 2 J ; p 2 PðjÞ

drwij Euclidean distance between retailer i and warehouse j, 8i 2 I ; 8 j 2 J ðiÞdwmj Euclidean distance between the plant and warehouse j, ∀j ∊ Jhrip Inventory holding cost per unit per year at retailer i for product p, ∀i ∊ I, ∀p ∊ P(i)hwjp Inventory holding cost per unit per year at warehouse j for product p, ∀j ∊ J, ∀p ∊ P( j)krip Parameter corresponding to service level in standard normal distribution at retailer i for product p. 8 i 2 I ; 8p 2 Pkwjp Parameter corresponding to service level in standard normal distribution at warehouses j for product p.

8 i 2 IðiÞ; 8p 2 PðiÞnri Number of shipments per year from the warehouse to the retailer i for all the products required by him, ∀i ∊ Inwj Number of shipments per year from the plant to the warehouse j for all the products required by warehouse, ∀j ∊ JWd Number of working days in a monthcrip Parameter that facilitate the conversion of distance between the retailer i and the warehouse which serves it into LEAD

TIME in inventory per day. Here we use crip ¼ lrip=Wd .8 i 2 IðiÞ; 8p 2 PðiÞcwjp Parameter that facilitate the conversion of distance between the plant and the warehouse j to Lead time in inventory per

day. Here we use cwjp ¼P

i¼IðjÞ\IðpÞ lripX

rwijp =Wd ,8j 2 J ; 8p 2 PðjÞ

lrip Average demand at retailer i for product p, ∀i ∊ I, ∀p ∊ P(i)rrip Standard deviation of demand at retailer i for product p, ∀i ∊ I, ∀p ∊ P(i)

Definition of variables

X rwijp

1; if warehouse j supplies product p to retailer i0; Otherwise

�W Maximum amount Inventory at warehouses

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we will get optimal number of orders for retailer i in Equation (7). On similar lines, we derive optimal number of ordersfor warehouses by taking the derivative of total annual cost in Equation (8) with respect to nwj . By taking the derivativeof Equation (8) and equating it to zero (Equation (9)) and solving it we get optimal number of orders for warehouse j(Equation (10)). Equations (7) and (10) has decision variable X rw

ijp and are now substituted in Objective function 1(Equation (1)).

Awj �

Pp2P

Pi2I h

wjpl

ripX

wijp

2ðnwj Þ2¼ 0 (9)

Solving for nwj , we obtain,

nwj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPp2P

Pi2I

hwjplrjpX

rwijp

2Awj

vuut(10)

The first two terms in Equation (1) are the cost of transportation i.e. transportation cost between the DC and retailersand the cost from plant to DCs for each product. The individual component of this cost comprises of three parts:amount of demand, distance shipped and decision variable which are then multiplied by unit cost of transportation fromorigin to destination. Third term and fourth term represents ordering cost for retailers and DCs. Fifth and sixth term rep-resents regular stock holding cost for retailers and DCs. The total ordering cost and holding cost for both retailers andDCs of all the above costs are from the Roundy’s popular 98% formulation (Roundy 1986). The last two cost terms arethe modified inventory cost for safety stock (Romeijn, Edwin, and Teo 2007) for retailer and distributor. In fact, safetystock mainly depends on the distance from the source for material replenishment. The same is not visible in the formu-lation by Romeijn, Edwin, and Teo (2007) and Qi and Shen (2007). The amount of safety stock depends on the leadtime, and the lead time depends upon the distance between source and destination of the product. This process is shownas modifications in safety stock. Equation (2) is the second objective function; W constraints the maximum inventorycarried by any DC in SCN under consideration. Equation (3) shows the constraint that retailer will have Single productsource; demand of all products of a retailer are served by a single warehouse. Equation (4) Constraints the inventorylevel at each DC. The first term in Equation (4) is the average inventory to satisfy the mean demand at the warehouse.The second term is the safety stock to maintain the required service level. Safety stock is the product of required servicelevel parameter (kwjp) and the total variation in the demand at the warehouse.

4. Solution methodology

The problem modelled in Section 3 is a multi-objective formulation and due to the inclusion of safety stock, it hasbecome non-linear and complex. In situations like this, EAs have been found effective in finding near optimal solutionsin less time. This encouraged us to employ an EA technique to solve problem at hand.

Over the years, NSGA-II has become a benchmark algorithm to solve multi-objective problems. NSGA-II works onthe similar lines of GA and having an additional feature of non-dominated sorting to rank solution with multiple objec-tives. But, it is well known that GA has the tendency to converge pre-maturely to local optima. This prompted authorsto replace the working procedure of GA in NSGA-II with Self-Learning Particle swarm optimisation (SLPSO) procedureto develop a new hybrid multi-objective EA; MOSLPSO, to tackle this problem. Flow chart of MOSLPSO is shown inFigure 3. In following parts of this section, MOSLPSO and NSGA-II procedures are clearly depicted.

4.1 Multi-objective self-learning particle swarm optimisation

By observing the communication of information in bird flocking and fish schooling, Eberhart and Kennedy (1995)developed PSO. PSO is a population (swarm)-based stochastic search technique. PSO is an iterative improvement basedmeta-heuristic, wherein each iteration, a set of possible candidate solutions called swarm updated using Equations (11)and (12). Each of the candidate solutions in the swarm is called a particle which is represented by two vectors; position(xp) and velocity (vp). Velocity of the particle is updated by guiding it partially towards its local best position (pbest)and partially towards global best position (gbest) as in Equation (11). Position of the particle is updated by simply add-ing velocity vector to old position vector as in Equation (12). x̂p and v̂p are updated position and velocity vectorsrespectively.

v̂p ¼ x� vp þ g1 � rp � ðpbest � xpÞ þ g2 � rp � ðgbest � xpÞ: (11)

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Figure 3. Flowchart of proposed MOSLPSO.

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x̂p ¼ xp þ vp (12)

In recent years, PSO has been extensively considered by researchers and applied to numerous practical complex prob-lems which yielded challenging results, yet it suffers from some drawbacks. In general, PSO algorithms use a particularlearning style (Equation (11)) for all the particles that enforce all the particles to use only one kind of thinking. Thissingle-learning strategy may lack intelligence of a particular particle; consequently, PSO algorithm may tend to beincapable to tackle problems with high complexity (Li, Yang, and Nguyen 2012). Furthermore, in many complex multi-model problemata solving the algorithm tends to local optima with a large numbers of local optima (Liang et al. 2006).Li, Yang, and Nguyen (2012) took away these drawbacks of PSO, and invented SLPSO algorithm by proposing fourlearning sources produced by four operators that guide particle (Equation (13)) to learn from the particle’s current bestsolution, (Equation (14)) jumping out of a local optimum, (Equation (15)) exploit a local optimum, and (Equation (16))explore new promising areas.

SLPSO is based on swarm intelligence that helps to find the near optimal solution to the problem. Li, Yang, andNguyen (2012) proposed four operators to update the velocity and position vectors. The four operators correspond tofour learning equations, respectively, as follow.

Operator 1: involvement of particle best fitness (pbest) in a learning source

v̂p ¼ x� vp þ g� rp � ðpbest � xpÞ: (13)

Operator 2: learning from a random position nearby, jumping out of local optima.

x̂p ¼ xp þ vavg � Nð0; 1Þ: (14)

Operator 3: involvement of the best position of a random particle (better than its fitness) in a learning source

v̂p ¼ x� vp þ g� rp � ðpbestrand � xpÞ: (15)

Operator 4: the abest position (super particle) in a learning source

v̂p ¼ x� vp þ g� rp � ðabest � xpÞ (16)

In this subsection, subscript p represents the pth particle; x̂p and xp represent the current and the previous position vec-tors of the particle p, respectively; v̂p and vp are the velocity vectors of the current and the previous iterations; ω ∊(0, 1) is the inertia weight that decide velocity preservation criterion from previous iteration; η is the acceleration coeffi-cient; rp is a random number generated uniformly from the interval [0, 1]; pbestp is the best position found by the indi-vidual particle p so far; pbestrand is the pbest of a random particle that is better than to pbestp; abest is the archivedposition of the global best (gbest) particle so far; N(0, 1) generates a random number from the standard normal distribu-tion with mean 0 and variance 1; and vavg ¼

PNp¼1 vp

��� ���=N is the average speed of all particles, where N is the popula-tion size.

These four operators produce four different learning sources that independently increase the search efficiency of eachparticle. Operator 1 exploits the local optimum solution, whereas Operator 2 is mutation operator that can be used inescaping the local optima; Operator 3 enables a particle to explore the non-searched areas with high probability, andOperator 4 enables particles converge to the current global best position (Li, Yang, and Nguyen 2012).

In this paper, we are extending SLPSO for Multi-objective optimisation; MOSLPSO, using the non-domination sort-ing (see NSGA-II; Deb et al. 2002) operation from NSGA-II. We used crowding distance operator and ranking of frontsin order to distinguish among solutions. For more detailed understanding of how final ranks of solutions are calculated(see NSGA-II; Deb et al. 2002). The steps of Algorithm are shown below. Only non-dominating sorting procedure wasadded to SLPSO to make it MOSLPSO. Rest of the steps of algorithm is same a SLPSO. In SLPSO, initially all thefour operators are given same percentage in roulette wheel. As the algorithm progresses, the probabilities of selectingeach operator will be updated. For more detailed understanding of SLPSO and updating of parameters, please see(Li, Yang, and Nguyen 2012). Flowchart of MOSLPSO is shown in Figure 3.

Step 1: Initialize swarm of particles and establish parameters for each particle.Step 2: Evaluate fitness for each particle p.Step 3: Set current generation counter t = 1.Step 4: For each particle p do

• Select one of the Operators using roulette wheel selection.○ Update velocity using selected operator i.○ Update position using selected operator i.

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○ Evaluate fitness for particle p.○ Update particle’s parameters.- UpdateGk

i .- Updategki .- Update progress value Pp

i of ith Operator for pth particle.- Update pbest of particle.- Update abest of particle.- Update mk.○ If Update criterion for selection ratios is met.- Update selection ratios.- Set mk = 0.- Set Gk

i = 0; gki = 0; Ppi = 0;

Step 5: if iteration number is one, store first non-dominated front else update first Non-dominated front.Step 6: Update current iteration counter t++.Step 7: Update algorithm parameters.Step 8: If a termination criterion is met stop the algorithm and take results, else go to Step 4.

Gki is the statistic that contains information regarding selection of Operator i for particle k.

gki is the statistic that contains information regarding successful learning times using Operator i for particle k.mk is the number of successive unsuccessful leaning for particle k.Ppi is the progress value for particle p using Operator i.

Termination condition: Pre-specified maximum number of Generations or acceptable quality of solutions is attained.

4.2 Non-dominating sorting genetic algorithm

Over the years, NSGA-II is considered as one of the earliest multi-objective EA based on Genetic Algorithm. NSGA-IIis also a population-based search technique for multi-objective optimisation. In order to find multiple optimal solutionsfor the multi-objective optimisation, which form a non-dominating front (Deb et al. 2002) developed NSGA-II. Theyused crowding distance operator and ranking of fronts in order to distinguish among solutions.

The steps followed for getting the results through NSGA-II are as follow:Step 1: Initialize population (i.e. population, P(0))Step 2: Evaluate fitness of each chromosome in the population (P(0)).Step 3: Assign fronts and crowding distance to each individual chromosome in the population(P(0)) (non-dominated

sorting).Step 4: Update the population set (P(0)) as follows,

(a) Set iteration counter t → t + 1,(b) Create a new population set (NP(t)).

(1) Select two individuals from the population (P(0)), parent 1(p1) and parent 2(p2).(2) Combine these two parents (p1 and p2) using crossover operator and produce two children (c1 and c2).(3) Apply mutation operator on each child (c1 and c2).(4) Insert children (c1 and c2) into (NP(t)).(5) Continue this process until size of new population (NP(t)) is equal to size of population(P(t)) .

(c) Combine old population (P(t – 1)) and new population (NP(t)) and do non-dominate sorting.(d) Select the best half from the combined population based on fronts and crowding distance as the new regular

population set (P(t)).

Step 5: If termination criterion is not met go to step 4, else terminate program and extract required results.Termination condition: Pre-specified maximum number of Generations or acceptable quality of solutions is attained.

5. Case study

Real-world data are considered from India’s leading manufacturer of GI wire, barbed wire and chain links. Barbed wireand chain link is the product of GI wire. This company sells GI wire and its products through a network of independentdistributors. The problem faced by the distributors includes decisions such as number of DCs required, retailer allocation

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to DC, optimal inventory at DC and optimal inventory at the retail outlets. Addressing all these issues will help thedistributor to reduce overall distribution cost including transportation and inventory holding cost (for safety and regularstocks).

5.1 Case study with scenario 1

Case scenario 1 analyses the problem with the single distributor with three DCs. The data were taken from one of thedistributors of this organisation. To ensure the confidentiality of the data, a real distributor facing the problem is referredas ‘D’. D purchases and distributes GI wires, barbed wire, chain link items to a total of over hundred and eighty-ninecustomers (e.g. retailers including institutional customers and converters) across his territory. D presently holds threeDCs, represented as DC 1, DC 2 and DC 3. D currently faces stochastic demand of retailers. It maintains regular andsafety stocks at each of his DC and supplies products from any one of the DC to retailers based on judgement. D wantsto know which DCs to be maintained to serve retailers i.e. D wants to see the effect of various options, such as closingor opening of the DC. Further, D wants to know which retailer should be served from which DC. Moreover, D wants tofind out what should be the optimal inventory (Regular stock and Safety stock) at each of his DC and also at each ofhis retailer so that a given service level can be achieved. The decision on above will help him in minimising the sys-tem-wide distribution and inventory cost.

The present system of D was optimised i.e. DC at three district places denoted as DC 1, 2 and 3 respectively andstudied the effect of possible alternative DC combinations. Other than 3 DC locations above, one more potential loca-tion was provided by management team to explore as DC 4. The authors are unable to provide input data-set in thispaper because of the excessive size of Tables, but we are able to take part of the data-set to show a numerical examplein the following sections. Input data for the model include distance between retailers and DC locations, Inventory carry-ing cost, service level, price, demand and standard deviation of each product at retailer.

5.2 Case study with scenario 2

Similar to the earlier case, the organisation under study has 30 distributor territories with more than 6000 retailers in theretail chain. The problem size is much bigger as the retailers are spread across the geographical territories of India andthere exists a gap in retailer reach in individual territory. Challenge is to identify the gap and locate appropriate DC orDC to tap the changing market demand to serve customers better and remain cost competitive.

The territory under study had six distributors with eight DCs. Other than these eight DC locations, another potentiallocation for DC was also taken for study as suggested by the top management. Each distributor has one or more thanone DC and is allotted a well-defined territory. The company manager wishes to divide the particular region into optimalnumber of territories. The study was conducted at nine potential DC locations.

6. Results and discussions

The implementation of the evolutionary multi-objective optimisation technique involved coding and testing the inte-grated allocation–inventory distribution network mathematical module and the MOSLPSO and NSGA-II algorithmsusing MATLAB (2012b). Following process is adopted to solve the problem:

(1) Devising a mathematical model that is constrained with a single product source for each retailer with samesource for all other products and retailers demanded at that site.

(2) Setting the chromosome (NSGA-II) or particle MOSLPSO structure to represent a candidate solution of(MOIAIP) for m customers and r DCs allocation problem. Each chromosome has (m + 1) genes. The first mgenes take values ranging from 1 to r. Each integer in the vector takes the value ranging from 1 to r. The integervalues in these genes represent DC allocated to that particular retailer. The (m + 1)th gene represent themaximum inventory level at any facility. Table 2 illustrates the particle representation of MOIAIP with eightcustomers served by two DCs.

Table 2. Schema of MOIAIP with two warehouses serving eight customers.

Customer 1 2 3 4 5 6 7 8 W (Inventory)Ware House 2 2 1 2 1 1 1 2 102.6 (mt)

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(3) Define the particle evolution criterion that is based on fitness value of all the objectives considered in the prob-lem.

(4) Generate initial swarm of particles. The structure of particle is specified in step 2. This step employs velocityupdating operator and particle updating operator. The particle representation for MOIAIP is set of integers (DCsid) for first m vectors and last vector is real coded. The particle and its velocity updating operators work asdescribed in Section 4.2. For NSGA-II, uniform crossover and random mutation for updating of chromosomes(see Figure 4).

6.1 Results of case study scenario I

Coding for Case Study Scenario1 was done in MATLAB (2012b), Intel core duo processor, windows basic for proposedMOSLPSO and NSGA-II algorithms by setting the parameters as shown in Table 3. The parameter values for both thealgorithms are obtained using trial and error method by means of incremental method (changing one parameter at atime). Since, we are dealing with allocation problem therefore number of DCs must be known prior. Now for N-DCs,there can be (2N – 1 – N) ways of serving the customer. In this paper, we have ruled out the possibility of the singleDC. For a P-DCs situation, each retailer can be served in P ways; for Y number of retailers, there are PY

(P � P � P � P � P ðupto Y timesÞ). The number of ways customers are served is the total search space (complexity).There are total of 11 possible combination of DCs are possible as shown in Table 4. Total search for each of 11(24 – 1– 4) possible scenarios and the number of function evaluations for each of the DC combinations to get converged areshown in Table 5. With the increased complexity of the problem, algorithm needs more function evaluations to reachthe near global optimum.

After doing experimentation with all the 11 DC combinations, final results are summarised in Table 6. Table 6shows the points in the final near optimal Pareto front for case study 1. The occurrence of multiple points in thePareto front for each combination is due to slightly different allocation of retailers to DCs which resulted in higheror lesser maximum inventory. Results indicate that only four combinations of DCs (DC1-DC4, DC1-DC2-DC4,DC1-DC3-DC4 and DC1-DC2-DC3-DC4) are making into final Pareto front. One can perceive the importance ofDC4 as it is involved in all four DC combinations that are part of final Pareto front. The points in the Pareto fronthelp the Distributor to take strategic decisions like which DCs to keep and how much maximum storage capacity aDC must be given. The question of which DCs to keep and which DCs must be removed is a strategic one. Thisdepends upon several external factors like how much storage capacity available at the new potential DCs? howmuch additional storage capacity available at present DCs? Future market trends etc. This is the main reason

Figure 4. Crossover and mutation operations for NSGA-II.

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behind considering a multi objective formulation. The solutions in the Pareto front (Figure 5) help the distributor toexplore different possibilities and find the optimal combination considering all the factors influencing it.

The solutions of the Pareto front also help us to explore the possibilities of decreasing or increasing the maximumcapacity of all the DCs in a selected combination. For example, the first and second points in the Pareto front belong tosame combination of DCs (DC1-DC4). We can see that with a slightly different allocation of retailers resulted in adecrease of 18 tonnes of inventory (15.65% increase) with just an increase of just Rs. 24,558 in total operational cost(2.88% increase).

At present, distributor of the organisation is maintaining DC1, DC2 and DC3. Distributor present total cost ofthe SCN is Rs. 1,345,781. Once the network is optimised, all the points in Pareto front (Table 6) have better costswith different varying maximum inventories. These solutions help in making an effective yet efficient decision. Min-imum total operational cost for all experimental DCs combinations are given in Table 7. Present DC combination(case id 7) is optimised but results show that other combinations give better results (Table 7). Actual cost of caseId 10 before optimisation is (Rs. 1057,892). Optimised results show an improvement of Rs. 73,305 (6.93%decrease).

In order to validate the performance of MOSLPSO, It has been compared NSGA-II which is considered as bench-mark algorithm in multi-objective optimization. Figure 6 shows the comparison of Pareto fronts of both algorithms. ThePareto front is obtained after 10 repeated experimental trails. It can be clearly seen that NSGA-II is able to generateonly part of the Pareto front obtained by MOSLPSO. In all the 10 trials experimented, solutions of MOSLPSO arefound to be non-dominated which made us to conclude that algorithm is converged.

Table 3. Parameter values of MOSLPSO and NSGA-II for case scenario 1.

Parameter values for MOSLPSO Parameter values for NSGA-II

Acceleration coefficient 0.55 Crossover probability 0.9Probability 0.8 Mutation probability 0.1Velocity preserver 0.5

Table 4. Experimented DC combinations for case study I.

Case Id DC combination Case Id DC combination

1 DC1-DC2 7 DC1-DC2-DC32 DC1-DC3 8 DC1-DC2-DC43 DC1-DC4 9 DC1-DC3-DC44 DC2-DC3 10 DC2-DC3-DC45 DC2-DC4 11 DC1-DC2-DC3-DC46 DC3-DC4

Table 5. Complexity of the cases and function evaluations required for scenario 1.

Case (warehousecombination)

Complexity (totalsearch space)

Number of functionevaluations Case

Complexity (totalsearch space)

Number of functionevaluations

1-2 2196 30,000 1-2-3 3196 35,0001-3 2196 30,000 1-2-4 3196 35,0001-4 2196 30,000 1-3-4 3196 35,0002-3 2196 30,000 2-3-4 3196 35,0002-4 2196 30,000 1-2-

3-44196 45,000

3-4 2196 30,000

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Table 6. Pareto front points of case study I.

Sr no. Total operational cost Maximum curtailed inventory DC combination

1 827638.7723 115 1-42 852196.8064 97 1-43 893473.5912 83 1-44 902289.9048 82 1-45 911225.3085 81 1-2-46 916957.2897 79 1-2-47 926566.3727 78 1-2-48 939950.1895 77 1-2-49 947675.3574 75 1-2-410 966853.1557 70 1-3-411 983786.9702 65 1-3-412 992439.6332 63 1-3-413 1012587.372 62 1-3-414 1017282.842 60 1-3-415 1030234.18 57 1-2-3-416 1041635.651 53 1-2-3-417 1052389.502 52 1-2-3-418 1075368.19 51 1-2-3-419 1078197.886 50 1-2-3-420 1112861.314 49 1-2-3-4

40

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800000 850000 900000 950000 1000000 1050000 1100000 1150000

Max

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tons

)

Total Operational cost (Rs)

Pareto front for case 1

Figure 5. Final Pareto front for case study I for MOSLPSO.

45

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800000 850000 900000 950000 1000000 1050000 1100000 1150000

Max

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(t)

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MOSLPSO

NSGA-II

Figure 6. Comparison of pareto fronts of MOSLPSO and NSGA-II.

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6.2 Results of case study scenario II

Case scenario 2 is a case of 9-DCs and 496-retailers environment. In this case, distributor wants to explore the possibil-ity of 9th DC. There are 502 (29 – 1 – 9) possible multiple DC combinations. However, we are not interested in explor-ing all the alternatives possible. With the inputs from the organisation from where the data are collected, 10 potentialDC combinations are identified (as shown in Table 8). The experiments were conducted for only these 10 combinationslisted in Table 8. Summary of the results are shown in Table 9. Minimum total operational cost for all experimentalDCs combinations are given in Table 10. Present DC combination (case id 10) is optimised but results show that othercombinations give better results (see Table 10). Actual cost of case id 10 before optimisation is (Rs. 2,250,145).Optimised results show an improvement of Rs. 162,665 (7.23% decrease). Final Pareto front of MOSLPSO is shown inFigure 7.

Table 9 shows the final Pareto front after experimentation with case study 2; Combinations with case ids 3 and5 are only part of Pareto front. Though, remaining cases are optimised, combinations 3 and 5 dominate solutions ofother cases (see Table 10). It can be inferred from the results that present DCs combination (case id 10) is not theright combination and needs to be revised to either combination 3 or combination 5. Different solutions correspondto combinations 3 and 5 are part of Pareto front which helps organisation to take informed decision regarding thecorrect combination of DCs with optimal allocation of retailers. On the similar lines of case study 1, results of casestudy 2 are also compared with NSGA-II. With increased complexity of the problem, unlike case study 1, solutionsof MOSLPSO and NSGA-II are not close. Solutions of NSGA-II found to be far away from solutions of MOS-LPSO (see Figure 8). In order to compare both the algorithms, first we find the number of objective function evalu-ations for MOSLPSO to be converged and then repeated the experimentation with NSGA-II with same number offunction evaluations. In all the 10 repeated experimental NSGA-II results never matched with the results of MOS-LPSO.

Table 7. Minimum total operational costs for all combinations of case study I.

Dc combination Total operational cost (Rs.) Maximum curtailed inventory (t)

1-2 892223.8 1001-3 953429.9 861-4 827638.8 1152-3 953419.7 1052-4 993039.6 803-4 982892.8 761-2-3 984587.3 801-2-4 911225.3 811-3-4 966853.2 702-3-4 1,048,902 601-2-3-4 1,030,234 57

Table 8. Experimented DC combinations for case study 2.

Case id DC combination

1 DCs(1-2-3-4-5-6-7-8-9)2 DCs(2-3-4-5-6-7-8-9)3 DCs(1-3-4-5-6-7-8-9)4 DCs(1-2-4-5-6-7-8-9)5 DCs(1-2-3-5-6-7-8-9)6 DCs(1-2-3-4-6-7-8-9)7 DCs(1-2-3-4-5-7-8-9)8 DCs(1-2-3-4-5-6-8-9)9 DCs(1-2-3-4-5-6-7-9)10 DCs(1-2-3-4-5-6-7-8)

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6.3 Numerical results

In this part of this section, we will demonstrate results for a smaller dimension of problem with 20 retailers, 2 productsand 4 DCs. Part of the data-set of case study scenario 1 is taken as an input to this numerical example. Input data-setfor this numerical include mean and standard deviation of demands, ordering, holding and transportation costs and dis-tances between retailers and DCs as shown in Table 11. After doing experimentation for all the combinations shown inTable 4 (same as case study scenario 1), final Pareto front solutions are shown in Table 12 and Figure 9. Table 13 givesthe optimal assignment of retailers to DCs for the first point in Pareto front. The present combination (DC1-DC2-DC3)is optimised (Rs. 92,671, 8 tonnes) but it is not included in the Pareto front because we have better solutions in termsof both the objectives in different combinations. It can be suggested to distributor to change the combination of DCs tobe operated.

Table 9. Final pareto front points of case study II.

Sr. no. Total operational cost (Rs.) Maximum curtailed inventory (tonnes) Case id

1 2069519.98 36.9890573 32 2069550.34 36.9665772 33 2069563.6 36.9569153 34 2076251.81 34.4024336 35 2077279.3 34.3895343 36 2077464.73 34.3468795 37 2077498.68 34.3219464 38 2077614.78 34.3087279 39 2079453.73 33.3625514 310 2080899.89 32.6170919 311 2081754.51 32.4499772 312 2082282.04 32.3556846 313 2098080.59 31.1470353 314 2099255.26 31.0075908 315 2099415.24 30.968622 316 2103740.36 30.541123 317 2104777.47 30.4250844 318 2106828.1 30.3700667 319 2110451.07 30.1286675 320 2110732.71 30.0580162 321 2130800.71 29.7156286 522 2132313.23 29.6853488 523 2132370.71 29.6717765 524 2132539.19 29.6639151 525 2132700.54 29.6417295 5

Table 10. Minimum total operational costs for all experimented combinations.

DC combinations Total operational cost (Rs.) Minimum curtailed inventory (t)

DCs(1-2-3-4-5-6-7-8-9) 2,086,032 37.2314DCs(2-3-4-5-6-7-8-9) 2,085,700 35.79813DCs(1-3-4-5-6-7-8-9) 2,069,520 36.98906DCs(1-2-4-5-6-7-8-9) 2,153,462 32.4154DCs(1-2-3-5-6-7-8-9) 2,130,801 29.71563DCs(1-2-3-4-6-7-8-9) 2,078,592 38.4221DCs(1-2-3-4-5-7-8-9) 2,099,981 34.2146DCs(1-2-3-4-5-6-8-9) 2,101,618 33.5734DCs(1-2-3-4-5-6-7-9) 2,084,376 38.4253DCs(1-2-3-4-5-6-7-8) 2,087,480 35.2167

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35

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39

2060000 2080000 2100000 2120000 2140000Max

imum

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ory

(t)Total operational cost (Rs.)

MOSLPSO

MOSLPSO

Figure 7. Pareto front of MOSLPSO for case study II.

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2000000 2100000 2200000 2300000 2400000

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(t)

Total operational cost (Rs.)

MOSLPSO

NSGA-II

Figure 8. Comparison of pareto fronts for case study II.

Table 11. Input data for the model.

Retailer Product id hj Aj Cj kj μj rj

Distance to DC

1 2 3 4

m = 1 1 2 517 3 3.1 5.17 2.27 386 306 392 237m = 2 1 2 18 6 3.1 0.18 0.42 450 92 471 273m = 3 2 2 225 3 3.1 2.25 1.5 386 306 392 237m = 4 2 2 90 6 3.1 0.9 0.95 182 293 180 64m = 5 2 2 18 3 3.1 0.18 0.42 229 170 226 80m = 6 1 2 213 3 3.1 2.13 1.46 345 46 384 196m = 7 1 2 217 6 3.1 2.17 1.47 370 25 363 218m = 8 1 2 113 3 3.1 1.13 1.06 380 36 405 184m = 9 1 2 113 3 3.1 1.13 1.06 370 25 363 218m = 10 1 2 4 3 3.1 0.04 0.2 345 46 384 196m = 11 2 2 200 6 3.1 2 1.41 277 172 274 128m = 12 1 2 69 3 3.1 0.69 0.83 410 45 404 255m = 13 1 2 100 6 3.1 1 1 380 306 385 232m = 14 2 2 208 3 3.1 2.08 1.44 87 122 119 235m = 15 1 2 3 6 3.1 0.03 0.17 172 32 260 229m = 16 2 2 2 3 3.1 0.02 0.14 343 75 338 192m = 17 2 2 317 6 3.1 3.17 1.78 116 495 134 264m = 18 1 2 4 5 3.1 0.04 0.2 370 25 363 218m = 19 1 2 225 10 3.1 2.25 1.5 410 45 404 255m = 20 2 2 483 3 3.1 4.83 2.2 257 169 240 108

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7. Conclusion and future work

In this paper, we have developed a model for integrated inventory distribution optimisation problem for multi-product ina multi-echelon supply chain environment. The model includes inventory analysis at DCs and retailers, demand analysisat DCs and location decisions of DCs. The model developed in the paper will help to make cost effective decisionsregarding optimal inventory at the retailers and DC locations. This paper also introduces a novel hybrid multi-objectiveMOSLPSO by combining the single-objective SLPSO with non-dominated sorting procedure of NSGA-II. The model

Table 12. Pareto front solutions for case III.

Sr. no. Total operational cost Maximum curtailed inventory DC combination

1 80446.6393 10 1-42 80634.96624 9 1-43 81300.33489 8 1-44 87386.40116 7 1-2-45 99133.53721 6 1-3

4

5

6

7

8

9

10

11

78000 83000 88000 93000 98000 103000

Max

imum

Cur

taile

din

vent

ory

(ton

s)

Total operational cost (Rs)

Pareto front for case II

Figure 9. Final Pareto front for case III using MOSLPSO.

Table 13. Optimal allocation of retailers to DCs for case III.

Retailer id Product id Optimally allocated customers

r = 1 1 4r = 2 1 4r = 3 2 4r = 4 2 1r = 5 2 4r = 6 1 4r = 7 1 4r = 8 1 4r = 9 1 4r = 10 1 1r = 11 2 1r = 12 1 4r = 13 1 4r = 14 2 1r = 15 1 1r = 16 2 4r = 17 2 1r = 18 1 4r = 19 1 4r = 20 2 4

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was applied in one of a galvanised iron wire firm. Multi-objective model was formulated and was solved using MOS-LPSO and NSGA -II. In both case studies 1 and 2, present network of DCs are optimised but, experimental resultsshows that other DC combinations yield lower possible cost with better curtailed inventory level. In case study scenario1, MOSLPSO outperformed NSGA-II, MOSLPSO is able to generate complete front, while NSGA-II has been able togenerate only part of the Pareto front. In case scenario 2 also, MOSLPSO outperformed NSGA-II. The model providesan important managerial implication regarding what-if analysis; in particular, the model is flexible to include change innumber of retailers, DCs or number of products. In both the case studies, MOSLPSO suggested that the present combi-nations of DCs are not the right choice and must be changed. Future design can also be done with this analysis. Futureresearch can address some of these issues: the model considers manufacturing plant and DC to be uncapacitated. Exten-sions of the model can include multiple manufacturing plants and relax the uncapacitated constraint. Future models canalso aim at demonstrating the performance of the proposed algorithm.

AbbreviationsDC(s) Distribution Centre(s).EA Evolutionary Algorithm.FLP Facility Location Problem.MOSLPSO Multi-Objective Self-Learning Particle Swarm Optimiser.NSGA-II Non-dominating Sorting Genetic Algorithm.PSO Particle swarm optimisation.SCN Supply chain network.SLPSO Self-Learning Particle Swarm Optimiser.

Disclosure statement

No potential conflict of interest was reported by the authors.

References

Ambrosino, D., and M. G. Scutellà. 2005. “Distribution Network Design: New Problems and Related Models.” European Journal ofOperational Research 165 (3): 610–624.

Askin, R. G., I. Baffo, and M. Xia. 2014. “Multi-commodity Warehouse Location and Distribution Planning with InventoryConsideration.” International Journal of Production Research 52 (7): 1897–1910.

Bashiri, M., and M. M. Tabrizi. 2010. “Supply Chain Design: A Holistic Approach.” Expert Systems with Applications 37 (1)688–693.

Daskin, M. S. 1995. Network and Discrete Location: Models, Algorithms and Applications. New York: Wiley.Daskin, M., and S. Owen. 2003. “Handbook of Transportation Science.” In International Series in Operations Research &

Management Science Vol. 56, edited by Randolph W. Hall. Boston, MA: Kluwer Academic.Daskin, M., C. Coullard, and Z. J. Shen. 2002. “An Inventory-location Model: Formulation, Solution Algorithm and Computational

Results.” Annals of Operations Research 110: 83–106.Deb, K., A. Pratap, S. Agarwal, and T. Meyarivan. 2002. “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II.” IEEE

Transactions on Evolutionary Computation 6 (2): 182–197.Eberhart, R., and J. Kennedy. 1995. “A New Optimizer Using Particle Swarm Theory.” Proceedings of the Sixth International

Symposium on Micro Machine and Human Science, Nagoya (1): 39–43.Ganeshan, R. 1999. “Managing Supply Chain Inventories: A Multiple Retailer, One Warehouse, Multiple Supplier Model.”

International Journal of Production Economics 59: 341–354.Gebennini, E., R. Gamberini, and R. Manzini. 2009. “An Integrated Production–distribution Model for the Dynamic Location and

Allocation Problem with Safety Stock Optimization.” International Journal of Production Economics 122 (1): 286–304.Guner, A. R., and M. Sevkli. 2008. “A Discrete Particle Swarm Optimization Algorithm for Uncapacitated Facility Location Prob-

lem.” Journal of Artificial Evolution and Applications. 2008: 10.Lee, J.-H., I.-K. Moon, and J.-H. Park. 2010. “Multi-level Supply Chain Network Design with Routing.” International Journal of

Production Research 48 (13): 3957–3976.Li, C., S. Yang, and T. T. Nguyen. 2012. “A Self-learning Particle Swarm Optimizer for Global Optimization Problems.” IEEE Trans-

actions on Systems, Man, and Cybernetics, Part B: Cybernetics 42 (3): 627–646.Li, Y., J. Shu, X. Wang, N. Xiu, D. Xu, and J. Zhang. 2013. “Approximation Algorithms for Integrated Distribution Network Design

Problems.” INFORMS Journal on Computing 25 (3): 572–584.

18 R.P. Manatkar et al.

Dow

nloa

ded

by [

115.

85.7

5.13

1] a

t 04:

42 2

2 D

ecem

ber

2015

Liang, J. J., A. K. Qin, P. N. Suganthan, and S. Baskar. 2006. “Comprehensive Learning Particle Swarm Optimizer for GlobalOptimization of Multimodal Functions” Evolutionary Computation, IEEE Transactions 10 (3): 281–295.

Liao, S. H., and C. L. Hsieh. 2010. “Integrated Location-inventory Retail Supply Chain Design: A Multi-objective EvolutionaryApproach.” Simulated Evolution and Learning 151: 533–542.

Melkote, S. and Mark S. Daskin. 2001. “An Integrated Model of Facility Location and Transportation Network Design.” Transporta-tion Research Part A: Policy and Practice 35 (6): 515–538.

Melnyk, S. A., R. Narasimhan, and H. A. DeCampos. 2014. “Supply Chain Design: Issues, Challenges, Frameworks and Solutions.”International Journal of Production Research . 52 (7): 1887–1896. doi:10.1080/00207543.2013.787175.

Miranda, P. A., and R. A. Garrido. 2004. “Incorporating Inventory Control Decisions into a Strategic Distribution Network DesignModel with Stochastic Demand.” Transportation Research Part E: Logistics and Transportation Review 40 (3): 183–207.

Mizgier, K. J., M. P. Jüttner, and S. M. Wagner. 2013. “Bottleneck Identification in Supply Chain Networks.” International Journalof Production Research 51 (5): 1477–1490.

Nozick, L. K., and M. A. Turnquist. 2001. “A Two-echelon Inventory Allocation and Distribution Center Location Analysis.”Transportation Research Part E: Logistics and Transportation Review 37 (6): 425–441.

Qi, L., and Z. M. Shen. 2007. “A Supply Chain Design Model with Unreliable Supply.” Naval Research Logistics 54 (8): 829–844.Romeijn, H., J. Shu Edwin, and C. Teo. 2007. “Designing Two-echelon Supply Networks.” European Journal of Operational

Research 178 (2): 449–462.Roundy, R. 1986. A 98%-effective Lot Sizing Rule for a Multi-product, Multi-stage Production/Inventory System. Mathematics of

operations research 11(4): 699–727.Shen, Z. J., C. Coullard, and M. Daskin. 2003. “A Joint Location-inventory Model.” A Joint Location-Inventory Model.

Transportation Science 37: 40–55.Shu, J. 2010. “An Efficient Greedy Heuristic for Warehouse-retailer Network Design Optimization.” Transportation Science 44 (2):

183–192.Shu, J., C. P. Teo, and Z. J. Shen. 2005. “Stochastic Transportation-inventory Network Design Problem.” Operations Research 53 (1):

48–60.Silva, F. J. F., and D. S. de la Figuera. 2007. “A Capacitated Facility Location Problem with Constrained Backlogging Probabilities.”

International Journal of Production Research 45 (21): 5117–5134.Teo, C. P., and J. Shu. 2004. “Warehouse-retailer Network Design Problem.” Operations Research 52 (3): 396–408.Yapicioglu, H., A. E. Smith, and G. Dozier. 2007. “Solving the Semi-desirable Facility Location Problem Using Bi-objective Particle

Swarm.” European Journal of Operational Research 177 (2): 733–749.

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