Research Article Multi-Item Multiperiodic Inventory Control Problem with Variable Demand and Discounts: A Particle Swarm Optimization Algorithm

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1 e Scientific World Journal, Article ID 16047, 16 pages Research Article Multi-Ite Multiperiodic Inventory Control Proble with Variable Deand and Discounts: A Particle Swar Optiization Algorith Seyed Mohsen Mousavi, 1 S. T. A. Niaki, Ardeshir Bahreininejad, 1 and Siti Nuraya Musa 1 1 Departent of Mechanical Engineering, Faculty of Engineering, University of Malaya, 5060 Kuala Lupur, Malaysia Departent of Industrial Engineering, Sharif University of Technology, P.O. Box Azadi Avenue, Tehran , Iran Correspondence should be addressed to Ardeshir Bahreininejad; bahreininejad@u.edu.y Received 1 April 014; Accepted 8 May 014; Published 0 June 014 Acadeic Editor: Leopoldo Eduardo Cardenas-Barron Copyright 014 Seyed Mohsen Mousavi et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. A ulti-ite ultiperiod inventory control odel is developed for known-deterinistic variable deands under liited available budget. Assuing the order quantity is ore than the shortage quantity in each period, the shortage in cobination of backorder and lost sale is considered. The orders are placed in batch sizes and the decision variables are assued integer. Moreover, all unit discounts for a nuber of products and increental quantity discount for soe other ites are considered. While the objectives are to iniize both the total inventory cost and the required storage space, the odel is forulated into a fuzzy ulticriteria decision aking (FMCDM) fraework and is shown to be a ixed integer nonlinear prograing type. In order to solve the odel, a ultiobjective particle swar optiization (MOPSO) approach is applied. A set of coproise solution including optiu and near optiu ones via MOPSO has been derived for soe nuerical illustration, where the results are copared with those obtained using a weighting approach. To assess the efficiency of the proposed MOPSO, the odel is solved using ulti-objective genetic algorith (MOGA) as well. A large nuber of nuerical exaples are generated at the end, where graphical and statistical approaches show ore efficiency of MOPSO copared with MOGA. 1. Introduction and Literature Review Most real-world probles in industries and coerce are studied as an optiization proble involving a single objective. The assuption that organizations always seek to axiize (or iniize) their profit (or their cost) rather than aking trade-offs aong ultiple objectives has been censured for a long tie. Generally, classical inventory odels are developed under the basic assuption that the anageent purchases or produces a single product. However, in any real-life conditions, this assuption does not hold. Instead, any firs, enterprises, or vendors are otivated to store a nuber of products in their shops for ore profitable business affairs. Another cause of their otivation is to attract the custoers to purchase several ites in one showroo or shop. This work proposes a ultiperiod inventory odel for seasonal and fashion ites. The ultiperiodic inventory control probles have been investigated in depth in different research. Chiang [1]investigateda periodicreview inventory odel in which the period is partly long. The iportant aspect of his study was to introduce eergency orders to preventshortages.heeployedadynaicprograing approach to odel the proble. Mohebbi and Posner [] investigated an inventory syste with periodic review, ultiple replenishent, and ultilevel delivery. Assuing a Poisson process for the deand, shortages were allowed in this research, and the lost sale policy could be eployed. Lee and Kang [] developedaodelforanaginginventoryof a product in ultiple periods. Their odel was first derived for one ite and then was extended for several products. Mousavi et al. [4] proposed a ultiproduct ultiperiod inventory control proble under tie value of oney and inflation where total storage space and budget were liited. They solved the proble using two etaheuristic algoriths,

2 The Scientific World Journal that is, genetic algorith and siulated annealing. Mirzapour Al-e-hashe and Rekik [5] presented a ultiproduct ultiperiod inventory routing proble, where ultiple constrained vehicles distributed products fro ultiple supplierstoasingleplanttoeetthegivendeandofeachproduct over a finite planning horizon. Janakiraan et al. [6] analyzed the ultiperiodic newsvendor proble and proposed soe new results. The quantity discount is of increasing attention due to its practical iportance in purchasing and control of ites. Usually, one derives the better arginal cost of purchase/production by taking advantages of the chances of cost savings through bulk purchase/production. Furtherore, in supply chain environents, quantity discounts can be considered an inventory coordination echanis between buyers and suppliers [7].In the literature of quantity discounts, Benton [8] considered an inventory syste with quantity discount with ultiple price breaks and alternative purchasing and lot-sizing policy. Abad [9, 10] proposed odels for joint price and lot size deterination when supplier offers either increental (IQD) or all unit (AUD) quantity discounts. K. Maiti and M. Maiti [11] developed a odel for a ulti-ite inventory control syste of breakable ites with AUD and IQD (and a cobination of the two policies) and proposed genetic algorith to solve the odel. Sana and Chaudhuri [1] extended an EOQ odel by relaxation of thepreassuptionsrelatedtopayents,allowingdelayon delivery and discounts. They used a ixed integer nonlinear prograing technique to odel the proble. Taleizadeh et al. [1] considered a genetic algorith to optiize ultiproduct ulticonstraint inventory control systes with stochastic replenishent intervals and discount. Recently, several works such as the ones in [4, 14 16] havealsospotteddiscountsin inventory control probles. Huang and Lin [17] addressed an integrated odel that scheduled ulti-ite replenishent with uncertain deand to deterine delivery routes and truck loads. In this study, the products are purchased in different periods under AUD and IQD policies. Metaheuristic algoriths have been suggested to solve soe of the existing developed inventory probles in the literature.soeofthesealgorithsaretabusearch[18, 19, 1], genetic algoriths (GA) [0, ], siulating annealing (SA) [,, 4], evolutionary algorith [4, 5], threshold accepting [0], neural networks [6], ant colony optiization [7], fuzzy siulation [5], and harony search [6, 8 41]. Inspired by social behavior of bird flocking or fish schooling, particle swar optiization (PSO) is also a populationbased stochastic optiization etaheuristic developed by Kennedy and Eberhart [4]. Recently, researchers have eployed this effective technique to find optial or near optial solutions of their inventory control probles. For exaple, Taleizadeh et al. [4] eployed PSO to solve their integer nonlinear prograing odel of a constraint joint single buyer-single vendor inventory proble with changeable lead tie and (r, Q) policy in supply chains with stochastic deand. Chen and Dye [44] solved an inventory proble with deteriorating products and variable deands using a PSO algorith. Further, Taleizadeh et al. [7] odeled a chance constraint supply chain proble with uniforly distributed stochastic deand, where an Ant ColonyBeeandaPSOalgorithwereutilizedtosolvethe proble. Instead of optiizing a single objective, soe researchers tried to find Pareto front solutions for their ultiple objective inventory planning probles that usually consist of ultiple conflicting objectives. Agrell [45] proposed a ulticriteria fraework for inventory control proble, in which the solution procedure was an interactive ethod with preferences extracted gradually in decision analysis process to deterine batch size and security stock. Roy and Maiti [8] presented a ultiobjective inventory odel of deteriorating ites with stock-dependent deand under liited iprecise storage area and total cost budget. Tsou [46] developedaultiobjective reorder point and order size syste and proposed a ultiobjective PSO (MOPSO) to generate Pareto front solutions. He eployed TOPSIS to sort the nondoinated solutions. The objectives therein were to axiize the profit and to iniize the wastage cost where the profit goal, wastage cost, and storage area were fuzzy in nature. One of the successful applications of PSO to MOOPs is the seinal work of Coello Coello and Lechuga [47]. Yaghin et al. [9] first addressed an inventory-arketing syste to deterine the production lot size, arketing expenditure, and selling prices in which the odel was forulated as a fuzzy nonlinear ultiobjective progra. Then, they converted the odel to a classical single-objective one by a fuzzy goal prograing ethod where an efficient solution procedure using PSO was provided to solve the resulting nonlinear proble. In their study, MOPSO is not only a viable alternative to solve MOOPs, but also the only one, copared with the nondoinated sorting genetic algorith-ii (NSGA-II) [48], the Pareto archive evolutionary strategy (PAES) [49], and the icrogenetic algorith [50] for ultiobjective optiization probles [51]. Table 1 shows the literature review of the works reviewed in this work where DOE is an abbreviation of ter design of experients. In this research, the contribution of the proble is considering a new biobjective ulti-ite ultiperiodic inventory control odel where soe ites are purchased under AUD and the other ites are bought fro IQD. The deands vary in different periods, the budget is liited, the orders are placed in batch sizes, and shortages in cobination of backorder and lost sale are considered. The goal is to find the optiu inventory levels of the ites in each period such that the total inventory cost and the total required warehouse space are iniized siultaneously. Since it is not easy for theanagerstoallocatethecrispvaluestotheweightsof the objectives in a decision aking process, considering these weightsasfuzzynuberswillbetakenasanadvantage. In order to be ore understanding of the proble, we try to explain the odel with taking an exaple in the real world. We consider a copany which produces soe kinds of fashion clothes including trousers, t-shirt, and shirt in a certain period. The custoers (wholesales) of this copany with different deand rates ake the orders and receive their products in the prespecific boxes, each one consisting of a known nuber of these clothes. Moreover, due to soe unforeseen atters, such as production liitation, the

3 The Scientific World Journal Table 1: Literature review of the related works. References Multiproduct Multiperiod Fuzzy ultiobjective Discount policy Solving ethodology Shortages DOE [1] BandB [] Level-crossing [] IQD Nuerical ethods [4] IQD GA [5] CPLEX [6] [7] IQD and AUD Siulation [8] AUD Siulation [9] IQD Nuerical ethods [10] IQD Nuerical ethods [11] IQDandAUD GA [1] AUD Nuerical ethods [1] IQDandAUD GA [14] IQD Yager ranking [15] IQD Excel acro [16] TOPSISandGA [17] ACO [18] TabusearchandLagrangian [19] Tabusearch [0] IQD Goal prograing and GA [1] IQD GA and fuzzy siulation [] GA [] SAandGA [4] TOPSISandGA [5] IQD fuzzy siulation [6] Harony search [7] BeecolonyandPSO [8] Fuzzy prograing algorith [9] Fuzzy ethod This research PSO and GA Taguchi copanies are not responsive to all of the deands in a period and hence soe custoers ust wait until the next period to receive their orders. Furtherore, it is assued the copany is going to extend the production part and therefore the owner has a plan to build and optiize a new storage subject to the available space. The reainder of the paper is organized as follows. In Section,theproblealongwithitsassuptionsisdefined. In Section, the defined proble of Section is odeled. To do this, the paraeters and the variables of the proble are first introduced. A MOPSO algorith is presented in Section 4 to solve the odel. Section 5 contains a nuerical exaple for a proble with 5 ites and periods, for which a ultiobjective genetic algorith (MOGA) is also applied as benchark for coparisons. Finally, conclusion and recoendations for future research coes in Section 6.. Proble Definition, Assuptions, and Notations Consider a biobjective ulti-ite ultiperiod inventory control proble, in which an AUD policy is used for soe ites and an IQD policy for soe other ites. The inventory control proble of this research is siilar to the seasonal ites proble where the planning horizon starts in a period (or season) and finish in a certain period (or season). The total available budget in the planning horizon is liited and fixed. Due to existing ordering liitations or production constraints, the order quantities of all ites in different periods cannot be ore than their predeterined upper bounds. The deands of the products are constant and distinct, and, in case of shortage, a fraction is considered backorder and a fraction lost sale. The costs associated with the inventory control syste are holding, backorder, lost sale, and purchasing costs. Moreover, due to current anagerial decision adaptations on production policies (i.e., setting up a new anufacturing line, extending the warehouse, or building a new storage area), iniizing the total storage space is required as well as iniizing the total inventory costs. Therefore, the goal is to identify the inventory levels of the ites in each period such that the two objective functions, total inventory costs and total storage space, are iniized. In order to siplify the odeling, the following assuptionsaresettotheprobleathand. (1) The deand rate of an ite is independent of the others and is constant in a period. However, it can be different in different periods.

4 4 The Scientific World Journal ()Atostoneordercanbeplacedinaperiod.This order can include or exclude an ite. () The ites are delivered in a special container. Thus, the order quantities ust be a ultiple of a fixed-sized batch. (4) The vendor uses an AUD policy for soe ites and an IQD policy for others. (5) A fraction of the shortages is considered backorder and a fraction lost sale. (6) The initial inventory level of all ites is zero. (7) The budget is liited. (8) The planning horizon is finite and known. In the planning horizon, there are N periods of equal duration. (9) The order quantity on an ite in a period is greater than or equal to its shortage quantity in the previous period (i.e., Q i,j+1 b i,j defined below). In order to odel the proble at hand, in what coes next we first define the variables and the paraeters. Then, the proble is forulated in Section. For i = 1,,..., and j = 1,,...,N 1 and k = 1,,...,Kthevariablesandtheparaetersoftheodelare defined as follows: N: nuber of replenishent cycles during the planning horizon, :nuberofites, K: nuber of price break points, S i : required storage space per unit of the ith product, T j : total tie elapsed up to and including the jth replenishent cycle, T i,j : jth period in which the inventory level of ite i is zero (a decision variable), B i :batchsizeoftheith product, V i,j :nuberofthepacketsfortheith product order in period j (a decision variable), D i,j : deand of the ith product in period j, Q i,j :purchasequantityofitei in period j (a decision variable where Q i,j =B i V i,j ), A i : ordering cost per replenishent of product i (If an order is placed for one or ore ites in period j, this cost appears in that period), b i,j : shortage quantity of the ith product in period j (a decision variable), X i,j : the beginning positive inventory level of the ith product in period j (in j = 1, the beginning positive inventory level of all ites is zero) (a decision variable), I i,j : inventory position of the ith product in period j (it is X i,j+1 +Q i,j+1,ifi i,j 0,otherwiseequalsb i,j ), I i (t): the inventory level of the ith ite at tie t, H i : unit inventory holding cost for ite i, q i,k : kth discount point for the ith product (q i,1 =0), i,k :discountrateofitei in kth price break point ( i,1 =0), P i : purchasing cost per unit of the ith product, P i,k : purchasing cost per unit of the ith product at the kth price break point, U i,j,k :abinaryvariable,set1ifitei is purchased at price break point k in period j,and0otherwise, W i,j :abinaryvariable,set1ifapurchaseofitei is ade in period j,and0otherwise, L i,j :abinaryvariable,set1ifashortageforitei occurs in period j,and0otherwise, β i : percentage of unsatisfied deands of the ith product, that is, back ordered, π i,j : backorder cost per unit deand of the ith product in period j, π i,j :shortagecostperunitoftheith product in period j,thatis,lost, Z 1 : total inventory cost, Z :totalstoragespace, TB:totalavailablebudget, M 1 : an upper bound for order quantity of the ith ite in period j, M : an upper bound for order quantities of all ites in each period (the truck capacity), TMF: objective function (the weighted cobination of the total inventory cost and the total storage space), w 1 : a weight associated with the total inventory cost (0 w 1 1), w : a weight associated with the total storage space (0 w 1).. Proble Forulation A graphical representation of the inventory control proble at hand with 5 periods for ite i is given in Figure 1 to obtain the inventory costs. At the beginning of the priary period (T 0 ), it is assued the starting inventory level of ite i is zero and that the order quantity has been received and is available. In the following periods, shortages can either occur or not. If shortage occurs, the corresponding binary variable is1,otherwiseitiszero.inthelattercase,theinventorylevels at the beginning of each period ay be positive..1. The Objective Functions. The first objective function of the proble, the total inventory cost, is obtained as Z 1 = Total Inventory Cost = Total Ordering Cost + Total Holding Cost + Total Shortage Cost + TotalPurchasingCost, where each part is derived as follows. (1)

5 The Scientific World Journal 5 Q D i, i,1 Q i,1 Qi, Q i,4 X i, T 0 T1 T 1 β T T i b i,1 T T 4 T 5 b i,1 b i, (1 β i )b i,1 Figure 1: Soe possible situations for the inventory of ite i in 5 periods. The ordering cost of an ite in a period occurs when an orderisplacedforitinthatperiod.usingabinaryvariable W i,j,whereitis1ifanorderfortheith product in period j is placed and zero otherwise, and knowing that orders can be placed in periods 1 to the total ordering cost is obtained as Therefore, the total holding cost is obtained in Total Holding Cost = ( X i,j +Q i,j +X i,j+1 ) (6) Total Ordering Cost = A i W i,j. () Since it is assued a shortage ay occur for a product in a period or not, the holding cost derivation is not as straightforward as the ordering cost derivation. Taking advantage of a binary variable L i,j,whereitis1ifashortage for ite i in period j occurs and otherwise zero, and using Figure 1, the holding cost for ite i in the tie interval T j 1 t T j (1 L i,j )+T i,j L i,j is obtained as T j (1 L i,j )+T i,j L i,j H i I i (t) dt, () T j 1 where I i (t) is the inventory level of the ith ite at tie t. In (), if a shortage for ite i occurs, L i,j becoes 1 and the ter T j (1 L i,j )+T i,j L i,j becoes T i,j.otherwise,l i,j =0 and T j (1 L i,j )+T i,j L i,j =T j.infigure 1,thetrapezoidalarea above the horizontal tieline in each period when ultiplied by the unit inventory holding cost of an ite, H i,represents the holding cost of the ite in that period. In other word, since (T j (1 L i,j )+T i,j L i,j T j 1 )H i. The total shortage cost consists of two parts: the total backorder cost and the total lost sale cost. In Figure 1, the trapezoidal area underneath the horizontal tieline in each period (shown for the priary period) when ultiplied by the backorder cost per unit deand of the ith product in period j, π i,j, is equal to the backorder cost of the ite in that period. Therefore, the total backorder cost will be Total Backorder Cost = ( π i,jb i,j (T j T i,j )β i). Furtherore, since (1 β i ) represents the percentage deands of the ith product, that is, lost sale, the total lost sale becoes Total Lost Sale Cost = ( π i,jb i,j (T j T i,j )(1 β i)) in which T j T i,j =b i,j/d i,j. ThetotalpurchasecostalsoconsistsoftwoAUDandIQD costs. The purchasing offered by AUD policy is odeled by (7) (8) I i,j+1 =I i,j +Q i,j D i,j (4) and if I i,j+1 0then I i,j+1 =X i,j+1,otherwisei i,j+1 =b i,j,() becoes T j (1 L i,j )+T i,j L i,j H i I i (t) dt T j 1 = X i,j +Q i,j D i,j (T j (1 L i,j )+T i,j L i,j T j 1 )H i. (5) P i,1 ; 0 < Q i,j q i, { P i, ; q i, <Q i,j q i, P i =. { { P i,k ; q i,k <Q i,j. Hence, the purchasing cost of this policy is obtained as AUD Purchasing Cost K = P i,k Q i,j U i,j,k. k=1 (9) (10)

6 6 The Scientific World Journal P ijk P ijk 1 P ij P ij1 q ij1 q ij q ijk q ijk 1 Figure : AUD policy for purchasing the products in different periods. P i,1 q i, +P i, (q i, q i, )(1 i, ) + + P i,k (Q i,j q i,k )(1 i,k ) P i,1 q i, +P i, (Q i,j q i, )(1 i, ) P i,1 Q i,j P ij1 P ij P ijk 1 P ijk q ij1 q ij q ijk 1 qijk Figure : IQD policy for purchasing the products in different periods. A graphical representation of the AUD policy eployed to purchase the products in different periods is shown in Figure. In this Figure, the relation between the price break points and the purchasing costs is deonstrated clearly. Moreover, U i,j,k is a binary variable, set 1 if the ith ite is purchased with price break k in period j and 0 otherwise. In the IQD policy, the purchasing cost per unit of the ith product depends on its order quantity. Therefore, for each price break point we have P i,1 Q i,j ; 0 < Q i,j q i, P i,1 q i, +P i, (Q i,j q i, )(1 i, ); q i, <Q i,j q i,. P i,1 q i, +P i, (q i, q i, )(1 i, ) + +P i,k (Q i,j q i,k )(1 i,k ); q i,k <Q i,j. (11) Hence, the total purchasing cost under the IQD policy is obtained as IQD Purchasing Cost = {(Q i,j q i,k )P i U i,j,k (1 i,k ) K 1 + k=1 (q i,k+1 q i,k )P i (1 i,k )}. (1) Figure graphically depicts the IQD policy for each product in different periods. Thus, the first objective function of the proble at hand becoes Z 1 = A i W i,j + ( X i,j +Q i,j +X i,j+1 ) (T j (1 L i,j )+T i,j L i,j T j 1 )H i ( π i,jb i,j (T j T i,j )β i) ( π i,jb i,j (T j T i,j )(1 β i)) K k=1 Q i,j P i,k U i,j,k {(Q i,j q i,k )P i U i,j,k (1 i,k ) K 1 + k=1 (q i,k+1 q i,k )P i (1 i,k )}. (1) The second objective of the proble is to iniize the total required storage space. Since in each period, order quantities Q i,j enter the storage and the beginning inventory

7 The Scientific World Journal 7 of a period is the reainder inventory of the previous period, X i,j, the second objective function of the proble is odeled by Z = (X i,j +Q i,j )S i. (14) Finally, the fitness function is defined as the weighted cobination of the total inventory cost and the required storage space as TMF =w 1 Z 1 +w Z. (15).. The Constraints. In real-world inventory planning probles, due to existing constraints on either supplying or producing goods (e.g., budget, labor, production, carrying equipent, and the like), objectives are not et siply. This section presents forulations for soe real-world constraints. The first liitation is given in (4), where it relates the beginning inventory of the ites in a period to the beginning inventory of the ites in the previous period plus the order quantity of the previous period inus the deand of the previous period. The second liitation is due to delivering the ites in packets of batches. Since Q i,j represents the purchase quantity of ite i in period j; denoting the batch size by B i and the nuber of packets by V i,j,wehave Q i,j =B i V i,j. (16) Furtherore, since Q i,j canonlybepurchasedbasedonone price break point, the following constraint ust hold: K k=1 U i,j,k =1. (17) The prerequisite of using this strategy is that the lowest q i,k in theaudtableustbezero(i.e.,q i,1 =0). Since the total available budget is TB, the unit purchasing cost of the product is P i, and the order quantity is Q i,j,the budget constraint will be Q i,j P i TB. (18) In real-world environents, the order quantity of an ite in a period ay be liited. Defining M 1 an upper bound for this quantity, for i = 1,,..., and j = 1,,..., we have Q i,j M 1. (19) Moreover, due to transportation contract and the truck capacity, the nuber of product orders and the total order quantities in a period are liited as well. Hence, for j = 1,,...,,wehave Q i,j W i,j M, (0) where if an order occurs for ite i in period j, W i,j = 1, otherwise W i,j =0.Further,M is an upper bound on the total nuber of orders and the total order quantities in a period. As a result, the coplete atheatical odel of the proble is subject to Z 1 = Z = Min TMF =w 1 Z 1 +w Z (1) A i W i,j + ( X i,j +Q i,j +X i,j+1 ) (T j (1 L i,j )+T i,j L i,j T j 1 )H i ( π i,jb i,j (T j T i,j )β i) ( π i,jb i,j (T j T i,j )(1 β i)) K k=1 Q i,j P i,k U i,j,k {(Q i,j q i,k )P i U i,j,k (1 i,k ) K 1 + k=1 (X i,j +Q i,j )S i, (,,...,), (q i,k+1 q i,k )P i (1 i,k )}, I i,j+1 =I i,j +Q i,j D i,j ; (,,...,), I i,j+1 ={ X i,j+1; I i,j+1 0 b i,j ; I i,j+1 <0; (,,...,), (,,...,), (,,...,), Q i,j =B i V i,j ; K k=1 (,,...,), (,,...,), U i,j,k =1; (,,...,), Q i,j P i TB Q i,j M 1 ;

8 8 The Scientific World Journal (,,...,), (,,...,), W i,j {0, 1} ; Q i,j W i,j M ; (,,...,) (,,...,) U i,j,k {0, 1} ; (,,...,), (k=1,,...,k) (,,...,) Q i,j+1 b i,j. () In ost inventory-planning odels that have been developed so far, researchers have iposed soe unrealistic assuptions such that the objective function of the odel becoes concave and the odel can easily be solved by soe atheatical approaches like the Lagrangian or the derivative ethods. However, since the odel in (), which is obtained based on assuptions that are ore realistic, is an integer nonlinear prograing ixed with binary variables, reaching an analytical solution (if any) to the proble is difficult. In addition, efficient treatent of integer nonlinear optiization is one of the ost difficult probles in practical optiization [5]. As a result, in the next section a etaheuristic algorith is proposed to solve the odel in (). 4. The Proposed Multiobjective Particle Swar Optiization Algorith Many researchers have successfully used etaheuristic ethods to solve coplicated optiization probles in different fields of scientific and engineering disciplines; aong the, the particle swar optiization (PSO) algorith is one of the ost efficient ethods. That is why this approach is taken in this research to solve the odel in (). The structure of the proposed MOPSO that is based on the PSO algorith for the ultiobjective inventory planning proble at hand is given as follows Generating and Initializing the Particles Positions and Velocities. PSO is initialized by a group of rando particles (solutions) called generation and then searches for optia by updating generations. The initial population is constructed by randoly generated R particles (siilar to the chroosoes of a genetic algorith). In a d-diensional search space, let x i k = {x i k,1,xi k,,...,xi k,d } and Vi k = {V i k,1, Vi k,,...,vi k,d } be, respectively, the position and the velocity of particle i at tie k. Then,() is applied to generate initial particles, in which x in and x ax arethelowerandtheupperboundson the design variable values and RAND is a rando nuber between 0 and 1. Consider x i 0 =x in + RAND (x ax x in ) V i 0 =x in + RAND (x ax x in ). () An iportant note for the generating and initializing step of the PSO is that solutions ust be feasible and ust satisfy the constraints. As a result, if a solution vector does not satisfy a constraint, the related vector solution will be penalized by a big penalty on its fitness. 4.. Selecting the Best Position and Velocity. For every particle, denotethebestsolution(fitness)thathasbeenachievedsofar as pbest i k ={pbesti k,1,pbesti k,,...,pbesti k,d }, (4) gbest i k ={gbesti k,1,gbesti k,,...,gbesti k,d }, (5) where pbest i k in (5) isthebestpositionalreadyfoundby particle i until tie k and gbest i k in (4) isthebestposition already found by a neighbor until tie k. 4.. Velocity and Position Update. The new velocities and positions of the particles for the next fitness evaluation are calculated using [5, 54] V i k+1,d =w Vi k,d +C 1 r 1 (pbest i k,d xi k,d ) +C r (gbest i k,d xi k,d ), x i k+1,d =xi k,d + Vi k+1,d, (6) where r 1 and r are rando nubers between 0 and 1, coefficients C 1 and C are given acceleration constants towards pbest and gbest, respectively, and w is the inertia weight. Introducing a linearly decreasing inertia weight into the original PSO significantly iproves its perforance through the paraeter study of inertia weight [55, 56]. Moreover, the linear distribution of the inertia weight is expressed as follows [55]: w=w ax w ax w in iteration, (7) iter ax where iter ax is the axiu nuber of iterations and iteration is the current nuber of iteration. Equation (7) presents how the inertia weight is updated, considering w ax and w in aretheinitialandthefinalweights,respectively.the paraeters w ax = 0.9 and w in = 0.4 that were previously investigated by Naka et al. [55] and Shi and Eberhart [56]are used in this research as well Stopping Criterion. Achieving a predeterined solution, steady-state ean, and standard deviations of a solution in several consecutive generations, stopping the algorith at a certain coputer CPU tie, or stopping when a axiu nuber of iterations is reached are usual stopping rules that have been used so far in different research works. In current research, the PSO algorith stops when the axiu nuber of iterations is reached. Pseudocode 1 shows the pseudocode of the proposed MOPSO algorith. Moreover, since the proble and hence

9 The Scientific World Journal 9 for to Pop initialize position (i) initialize velocity (i) if position (i)and velocity(i) be a feasible candidate solution penalty = 0 else penalty = a positive nuber end if end for w = [0.4, 0.9] do while Iter <=Gen for to Pop Calculate new velocity of the particle Calculate new position of the particle pbest(iter) = in(pbest(i)) end for gbest(iter) = in(gbest) w=w ax ((w ax w in )/iter ax) iter odifying the velocity and position of the particle end while Pseudocode 1: The pseudocode of MOPSO algorith. Begin Set P C, P,PopandGen S 0 initialize Population (S) evaluate Population (S) while (non terinating condition) repeat S S+1 select Population (S) fro Population with roulette wheel (S 1) unifor crossover one point utation evaluate Population (S) end repeat print optiu result end Pseudocode : The pseudocode of MOGA algorith. the odel is new and there is no other available algorith to copare the results, a ultiobjective genetic algorith (MOGA) is developed in this research for validation and bencharking. MOGA was coded using roulette wheel in selection operator, population size of 40, unifor crossover with probability of 0.64, one-point rando utation with probability 0., and a axiu nuber of 500 iterations. Pseudocode shows the pseudocode of the proposed MOGA algorith. The coputer progras of the MOPSO and MOGA algoriths were developed in MATLAB software and are executed on a coputer with.50 GHz of core CPU and.00 GB of RAM. Furtherore, all the graphical and statistical analyses are perfored in MINITAB 15. In the next section, soe nuerical exaples are given to illustrate the application of the proposed MOPSO algorith in real-world environents and to evaluate and copare its perforances with the ones obtained by a MOGA ethod. 5. Nuerical Illustrations The decision variables in the inventory odel () areq i,j, X i,j, V i,j,andb i,j. We note that the deterination of the order quantity of the ites in different periods, that is, Q i,j,results in the deterination of the other decision variables as well. Hence, we first randoly generate Q i,j, that is, odeled by the particles position and velocity. Equation (8) shows a pictorial representation of the atrix Q for a proble with 4itesin4periods,whererowsandcolunscorrespondto theitesandtheperiods,respectively. The structure of a particle Q 4,4 = [ [ ]. (8) [ ]

10 10 The Scientific World Journal Table : Different probles and their optial TMF values obtained by the two algoriths. Proble N M 1 TB Objective values MOPSO MOGA MOPSO MOGA C 1 C Pop Gen P C P Pop Gen Mean St. Dev Table shows partial data for 40 different probles with different sizes along with their near optial solutions obtained by MOPSO and MOGA. In these probles, the nuber of ites varies between 1 and 0 and the nuber of periods takes values between and 15. In addition, the total available budgets and the upper bounds for the order quantities (M 1 ) are given in Table 1 for each proble. In order to illustrate how the results are obtained, consider a typical proble with 5 ites and periods (the seventh row in Table ),for which the coplete input data is given in Table. The paraeters of the MOPSO and MOGA algoriths are set by Taguchi ethod where C 1,C the nuber of populations (Pop) and nuber of generations (Gen) are the paraeters of MOPSO and crossover probability and their level values are shown in Table 4.Furtherore,the rest of MOPSO s paraeters are set as w in = 0.4, w ax = 0.9 and the tie periods T j = for j = 0, 1,,. The above paraeter settings are obtained perforing intensive runs. Furtherore, the aount of V i,j will be obtained autoatically after gaining the order quantity Q i,j.

11 The Scientific World Journal 11 Table : The general data for a proble with 5 ites and periods. Product D i,1 D i, π i,1 π i, π i,1 π i, B i H i A i β i S i μ 1 0 W a W b W c Figure 4: The triangular fuzzy nubers. Variable Table 4: The paraeters of the two algoriths and their levels. Algoriths Factors Levels [1,, ] C 1 (A) [1.5,,.5] MOPSO C (B) [1.5,,.5] Pop(C) [0, 0, 40] Gen(D) [100, 00, 500] P C (A) [0.5, 0.6, 0.7] MOGA P (B) [0.08, 0.1, 0.] Pop(C) [0, 40, 50] Gen(D) [00, 00, 500] Theweightsassociatedwiththeobjectivesareastriangular fuzzy nuber w =[ w a, w b, w c ] shown in Figure 4 where ebership function of variable x is given by 0 x < w a x w a w { w μ (x) = b w a <x< w b a w c x w { w c w b <x< w c b { 0 w c <x. (9) Now, in order to get crisp interval by α-cut operation, interval w α canbeobtainedasfollows( α [0, 1]): We have w (α) a w a =α, w b w a w (α) a =( w b w a )α+ w a ; w (α) c Therefore, w α =[ w (α) a, w(α) c ] =[( w b w a )α+ w a, w (α) c = w c ( w c w b )α. w c w (α) c w c w b =α. (0) = w c ( w c w b )α], where w 1 = [0., 0.5, 0.7], w = [0., 0., 0.6],andα = 0.5. (1) () Table 5: The Taguchi L 9 design along with objective values of the algoriths. Run nuber A B C D MOPSO MOGA Table 6: The optial levels of the algoriths paraeters for proble 7 of Table. Algoriths Factors Optial levels C 1 MOPSO C.5 Pop 40 Gen 00 P C 0.6 MOGA P 0. Pop 50 Gen 00 To perfor Taguchi approach in this paper, a L 9 design is utilized, based on which the results for proble 7 described in Table are shown in Table 5 as an exaple. The optial values of the levels of the algoriths paraeters shown in Table 5 are represented by Table 6. Figure 5 depicts the ean S/N ratio plot each level of the factors of MOPSO and MOGA forproble7in Table. Tables 7 and 8 show the best result obtained by MOPSO andmogafortheproblewith5itesandperiods (proble 7), respectively, including the aounts of decision variablesandtheoptialobjectivevalues.inthesetables,

12 1 The Scientific World Journal Main effects plot of MOPSO ean for SN ratios Mean of SN ratios Mean of SN ratios (A) (B) Mean of SN ratios Mean of SN ratios (C) (D) Signal-to-noise: saller is better (a) Main effects plot of MOGA ean for SN ratios Mean of SN ratios Mean of SN ratios (A) (B) Mean of SN ratios Mean of SN ratios (C) (D) Signal-to-noise: saller is better (b) Figure 5: The ean S/N ratio plot for paraeter levels of MOPSO and MOGA in proble 7 of Table.

13 The Scientific World Journal 1 Table 7: The best result of the MOPSO algorith. Product Q i,1 Q i, X i, X i, V i,1 V i, b i,1 b i, TMF Table8:ThebestresultoftheMOGAalgorith. Product Q i,1 Q i, X i, X i, V i,1 V i, b i,1 b i, TMF Table 9: The ANOVA analysis of the perforances. Source DF SS MS F Pvalue Factor E E Error 78.11E E + 11 Total 79.1E + 1 TMF is the best value of the biobjective inventory planning proble, which is given in the last two coluns of Table. Siilarly, the best TMF for the other probles is obtained and is suarized in Table. To copare the perforances of the MOPSO and MOGA, several statistical and graphical approaches are eployed. A one-way ANOVA analysis of the eans of the algoriths in confidence 0.95% is used to copare and evaluate the objective values of the generated 40 probles. Table 9 shows the ANOVA analysis of the results of the two algoriths that deonstrates no significant difference between both algoriths. Moreover, the ean and standard deviation (Std. Dev) of the objective values of the 0 generated probles shows that the MOPSO has the better perforance in ters of the objective values in coparison with the MOGA. In addition, a pictorial presentation of theperforancesofthetwoalgorithsshownbyfigure 6 displays that the MOPSO is ore efficient than the MOGA algorith in the large nuber of the probles. Figure 7 depicts the boxplot and the individual value plot and Figure 8 showstheresidualplotsforthealgoriths. AcoparisonoftheresultsinTable shows that the MOPSO algorith perfors better than the MOGA in ters of the fitness functions values. 6. Conclusion and Recoendations for Future Research In this paper, a biobjective ulti-ite ultiperiod inventory planning proble with total available budget under all unit discount for soe ites and increental quantity discount Objective value Variable MOPSO MOGA Figure 6: The pictorial representation of the perforances of the algoriths. for other ites was considered. The orders were assued to be placed in batch sizes and the order quantities at the end period were zeros. Shortages were allowed and contain backorder and lost sale. It was assued that the beginning inventory level in priary period was zeros and the order quantity in each period was ore than the shortage quantity in the previous period. Due to adopting decisions related to a certain departent of production planning (extending warehouse or building a new anufacturing line), the anager decided to build a new warehouse for the ordered ites. The objectives were to iniize both the total inventory costs

14 14 The Scientific World Journal 10 5 Boxplot of MOPSO and MOGA 10 5 Individual value plot of MOPSO and MOGA Data 10 Data MOPSO MOGA MOPSO (a) (b) Figure 7: The boxplot and the individual value plot of the perforances of the algoriths. MOGA Residual Plots for MOPSO and MOGA Noral probability plot of the residuals Residuals versus the fitted values (%) Residual Residual Fitted value Histogra of the residuals Frequency Residual Figure 8: The residual plots of the algoriths. and the total required storage space, for which a weighted cobination was defined as the objective function. The ai of the study was to deterine the optial order quantity and the shortage quantity of each product in each period such that the objective function is iniized and the constraints hold.thedevelopedodeloftheproblewasshowntobean integer nonlinear prograing ixed with binary variables. To solve the odel, both a ultiobjective particle swar optiization and a ultiobjective genetic algorith were applied. The results showed that for the 10 specific probles the MOPSO perfors better than the MOGA in ters of the fitness function values. Soe recoendations for future works are to expand the odel to cover a supply chain environent, to consider fuzzy or stochastic deands, and/or to take into account the inflation and the tie value of the oney. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

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