Supply chain management which involves managing the flow of material andinformation from sources to customers has been one of the most challenging issuesfacing both the academicians and the practitioners for years. Inventory control is acrucial part of tactical decision level affecting the performance of supply chain indistribution and production. The main focus of this study is to compare theperformance of different multi-objective metaheuristic techniques to optimizeinventory parameters for single-product continuous review stochastic inventorysystem with transportation costs. The simulation-based optimization method is usedto solve the problem by combining the simulation model and metaheuristicalgorithms in order to determine the inventory policy taking into account twoconflicting objectives: customer service level and total inventory cost. We build adiscrete event simulation model to evaluate the objective function of the problem.The Metaheuristic techniques such as the genetic algorithm and particle swarmoptimization are applied to search the solution space. The results obtained by allthese proposed techniques are compared and the effectiveness of each technique hasbeen illustrated

Bernd Noche

Emre Koc

Mustafa Guller

Epoka University

“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.590A Comparison Study of Multi-Objective MetaheuristicTechniques for Continuous Review Stochastic InventorySystemMustafa GÜLLER1, Emre KOÇ2 and Bernd NOCHE31IPE Transport System and Logistics, University of Duisburg- EssenEmail: mustafa.gueller@stud.uni-due.de1IPE Transport System and Logistics, University of Duisburg- EssenEmail: emre.koc@uni-due.de1IPE Transport System and Logistics, University of Duisburg- EssenEmail: bernd.noche@uni-due.deABSTRACTSupply chain management which involves managing the flow of material andinformation from sources to customers has been one of the most challenging issuesfacing both the academicians and the practitioners for years. Inventory control is acrucial part of tactical decision level affecting the performance of supply chain indistribution and production. The main focus of this study is to compare theperformance of different multi-objective metaheuristic techniques to optimizeinventory parameters for single-product continuous review stochastic inventorysystem with transportation costs. The simulation-based optimization method is usedto solve the problem by combining the simulation model and metaheuristicalgorithms in order to determine the inventory policy taking into account twoconflicting objectives: customer service level and total inventory cost. We build adiscrete event simulation model to evaluate the objective function of the problem.The Metaheuristic techniques such as the genetic algorithm and particle swarmoptimization are applied to search the solution space. The results obtained by allthese proposed techniques are compared and the effectiveness of each technique hasbeen illustrated.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.591INTRODUCTIONOne of the most important aspects affecting the performance of a supplychain is the management of inventories. Managing inventory in complexsupply chains is typically difficult, and may have a significant impact on thecustomer service level and system-wide cost. For this reason, it has beenstudied in considerable depth. Many researchers have dedicated themselvesto deal with inventory control involving more than one objective over thepast several decades. The role of the inventory management is that try to finda way to minimize the level of their stocks without negatively impactingavailability or customer service level. Due to the complication of realinventory system, single mathematic model cannot describe stochasticinventory problems. Now research on optimization algorithm has been usedto resolve the multi-objective optimization problems.Simulation-based optimization provides a structured approach to the system designand configuration when analytical expressions are very complex and mathematicallyintractable. In recent years researchers have attempted to combine simulation andoptimization procedures to provide a complete solution [1]. Simulation-basedoptimization is an attractive combined strategy which integrates optimizationtechniques into simulation analysis where the simulation model is regarded as afunction.This paper presents a performance comparison of metaheuristics to solve theinventory problem under continuous review policy by using simulation-basedoptimization approach. The remainder of this paper is organized as follows. In thenext section, we review the related literature on inventory optimization by usingmetaheuristics techniques. Section 3 introduces the multi-objective inventorycontrol. In section 4 we explain how simulation optimization methods have beenapplied in inventory control. Section 5 shows the numerical results along withdiscussions. Finally, section 6 summarizes the conclusions.LITERATURE REVIEWOne of the most important aims of multi-objective optimization is to obtain anefficient solution taking into accounts the balance among several conflictingobjectives. One of them is the use of evolutionary algorithms to solve multi-objective problems. Evolutionary algorithm is a particularly important subset ofrandom-based solution space searching meta-heuristic methods. A literature reviewsurvey of evolutionary multi-objective optimization techniques in all the fields ofscience, engineering and technology, and also a brief general concept the analysis of“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.592past and current research performed on evolutionary multi-objective optimization ispresented in [2].Although simulation is not an optimization tool by itself, simulation-basedoptimization is an active area of research in stochastic, uncertain environments.There are different ways to optimize supply chain inventory parameters representedby means of a simulation model. A computer aided simulation model and thegenetic algorithm for optimization of the control parameters in stochastic inventoryprocess is presented [3]. A new solution procedure based on GAs to find the set ofPareto optimal solutions for multi-echelon supply chain cyclic planning andoptimization is proposed in [4]. They built a simulation model to estimate systemperformance while GAs uses the responses generated by the simulation model tocontrol the optimization process. [5] considered a multi-echelon inventory systemcomposed of five serial stages and applied simulation-based optimization to set theoptimal policies. The performance of a single product serial supply chain operatingwith a base-stock policy is studied and optimized the base stock levels in the supplychain in [6]. They proposed a simulation-based genetic algorithm to optimize thebase stock levels with the objective of minimizing the supply chain cost.An evolutionary computation to find the non-dominated control policies forstochastic multi-objective (R, Q) inventory control systems is presented by [7]. Heproposed two evolutionary optimizers, multi-objective electromagnetism-likeoptimization (MOEMO) and multi-objective particle swarm optimization (MOPSO).Computational results show that the evolutionary Pareto optimizers generate welland fast approximate the non-dominated policies in term of lot size and safety stock.Enhanced particle swarm optimization (EPSO) [8] is applied to solve a three stageinventory problem to address the outsourcing issues with different shipmentpolicies. Particle Swarm Optimization algorithm is proposed to overcome inmaintaining the optimal stock levels in the supply chain in [9]. Their proposedmethodology reduced the total supply chain cost as it undoubtedly established themost probable excess stock level and shortage level along with the consideration oflead time in supplying the stocks as well as raw materials that are required forinventory optimization.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.593Problem formulationWe are interested in solving problem of the type:)}(),...,(),(min{ 2211 xfzxfzxfz mm  (1)qixg i ,...,1,0)(  (2)kqjxhi ,...,1,0)(  (3)where x is the decision variable, fm is objective m, gi is inequality constraint and hj isequality constraint j. For the minimization multi-objective inventory controlproblem under continuous review system, a decision variable is xi = (Ri, Qi). Twoobjective functions are introduced in the optimization problem. The average totalinventory cost which includes holding cost, shortage cost and ordering cost. Thesecond objective function is to maximize the service level. When the (R, Q)inventory system is simulated for a period of n days, the total cost can be written as:TtAdptBhtIQRTC1))()((),( (4)nQRTCQRC ),(),(min  (5)DemandBackorder1max (6)where,t: time indexI(t): on-hand inventory at the end of time period tB(t): backordered demand at the end of time period th: unit holding cost per item per timep: unit shortage cost per item per timeA: ordering costelse0 timespecifiedinpointstockbyordereditemanif1idPerformance measures are the average total system cost per unit time which includesholding cost, shortage cost, ordering cost and the service level of the γ-type which isdefined as one minus the amount of cumulative backorders divided by thecumulative demand during the simulation periods [10].“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.594One of the most challenging problems in multi objective optimization is related tothe identification of the Pareto optimal set. A Solution X in objective space is said aPareto-optimal (non-dominated solution), if and only if there is no other solution Yin the search space which could dominate X. In other words, X dominates Y, if X isbetter than Y in at least one objective function and not worse with respect to allother objective functions [11]. The set including all Pareto-optimal solutions istermed the Pareto set as shown in Figure 1.Figure 9 The Pareto front of a set of solutions in a two objective spaceSIMULATION OPTIMIZATION approach based on metaheuristicsAlthough simulation technology has been widely used as one of the most powerfultechniques for analyzing and improving enterprises supply chain management andlogistics operations, simulation provides no concrete solutions to optimizationproblems. Users need to evaluate many feasible solutions in order to find a goodsolution to the problem. The aim of simulation based optimization is to integrateoptimization techniques into simulation analysis. In mathematical programming thedecision variables are assigned in an analytical function of decision variables. Thisfunction is known as the objective function. In simulation-based optimization, theperformance measure becomes one (or a function of several) of the responsesgenerated by a simulation model [1][12]. Figure 2 illustrates the general scheme ofthe simulation-based optimization procedure and interaction between the simulationmodel and the optimization model.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.595Figure 10 Simulation-Based Optimization SchemeIn the simulation optimization procedure, the control parameters which are obtainedfrom optimization tool are assigned in simulation tool to update the objectivefunctions for a pre-specified period of time. At the end of the simulation run theservice level values and related inventory costs for the stock point are sent tooptimization tool from simulation. Optimization tool then uses this information tocheck if all the constraints are satisfied and the process iterates until there is nofurther improvement in the solution or an iteration limit is reached.NUMERICAL EXAMPLEDiscrete-even simulation tool of inventory and all multi-objective optimizationalgorithms such as MOPSO (multi-objective particle swarm optimization) andNSGA-II (Non-Dominated Sorting Genetic Algorithm) have been coded by usingMicrosoft Visual C-Sharp programming language. To compare the meta-heuristicalgorithms we presented inventory problem with different coefficient of variance(CV) for daily demand. The run length of the simulation experiment is 300 days and5 replications. We first use the normal distribution with different standard deviationsby following numerical example:days.2time)-L(leadandday,perunitper5pday,perunitper1horderper1000A20and10day,perunits50 “1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.596Figures 3 and 4 show non-dominated solutions received from optimization tool forhigh and low coefficient of variance. As is illustrated from Figures 3 and 4, thePareto front of efficient solutions is well populated for both optimization techniquesand the daily inventory cost seems to vary between the values of 280 and 320 asservice level vary between 0.9 and 1. For high CV, the daily inventory cost variesbetween of 280 and 350.Figure 11 Non-dominated solutions for CV = 0.2Figure 12 Non-dominated solutions for CV = 0.4The result of different coefficient of variance is reported in Table 1. Clearly, andsupported by the results in Table 1, NSGA-II’s output approximates better to lowercost than MOPSO.Table 3 Average results of non-dominated solutions for the different standarddeviationR Q Cost Service Levelσ=10NSGA-II 109 348 291 0,962802852902953003053103153203250.88 0.9 0.92 0.94 0.96 0.98 1Daily Inventory CostService LevelParticle SwarmOptimization2802903003103203303403503600.9 0.92 0.94 0.96 0.98 1Daily Inventory CostService LevelParticle SwarmOptimization“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.597MOPSO 125 326 300 0,97σ=20NSGA-II 118 335 298 0,96MOPSO 137 337 314 0,98In many inventory models, the demand process is described by the normaldistribution, but many scholars have criticized the use of this normal approach [13].If the coefficient of variation of demand is too large (greater than 0.5), the normalapproximation will lead to be a negative value becomes too large to be ignored. Inthis study, gamma and lognormal distributions are chosen to represent the demandprocess of high coefficient of variance (CV=1).Figure 13 Non-dominated solutions for Lognormal distribution (CV=1)Figure 14 Non-dominated solutions for Gamma distribution (CV=1)2903103303503703904104304500.9 0.92 0.94 0.96 0.98 1Daily Inventory CostService LevelParticle Swarm Optimization-MOPSO2903103303503703904104300.8 0.85 0.9 0.95 1Daily Inventory CostService LevelParticle Swarm Optimization-MOPSO“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.598To study the effects of the variability of daily demand and the types of probabilitydensity function used on the cost performance of the inventory system, we usedsame mean value with previous experiment such as mean and standard deviationequals 50. Figure 5 and 6 sets of non-dominated solutions for lognormal and gammadistribution with same coefficient of variance. The result of different demanddistribution is reported in Table 2. From results, we conclude that the form of dailydemand distribution has an influence on the control parameters for high coefficientof variations.Table 4 Average results of non-dominated solutions for the different demanddistribution (CV=1)R Q Cost Service LevelLognormalNSGA-II 122 424 324 0,95MOPSO 174 386 360 0,97GammaNSGA-II 119 408 324 0,94MOPSO 156 344 348 0,94“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.599ConclusionThis work presented a comparison study of optimization approaches based onmetaheuristics to solve the stochastic inventory problem under continuous reviewpolicy. One of the objectives was to utilize simulation and optimization methods.The concept of multi-objective particle swarm optimization (MOPSO) and Non-Dominated Sorting Genetic Algorithm (NSGA-II) approaches were tested to find aset of near optimal solutions for multiple objectives by using simulation-basedoptimization. We also studied the effects of the variability of daily demand and thetypes of probability density function used on the cost performance of the inventorysystem.This paper illustrated that multi-objective metaheuristics techniques can be used tosuccessfully model a stochastic inventory problem. The proposed models can beextended in other inventory policies and different inventory input configurations.REFERENCES[1] Ammeri, A. Chabchoub, H. Hachicha, W. and Masmoudi, F. (2010) AComprehensive Literature Classification of Simulation optimization Methods.The 9th International Conference on Multiple Objective Programming andGoal Programming, Tunisia.[2] Kadadevaramath, R.S. and Mohanasudaram, K.M. (2007). Multi-ObjectiveTrade-off Analysis: State of art: Methods, Applications, and future ResearchDirections in Production and Operations Management. ManufacturingEngineering, Issue: 2, pp: 70-78.[3] Bona, K. (2004). Simulation Supported Optimization of Inventory ControlProcesses by Application of Genetic Algorithm. Proceedings of 16thEuropean Simulation Symposium.[4] Merkuryeva, G. and Napalkova, L. (2009). Supply chain cyclic planning andoptimization. Simulation-based studies in logistics: Education and appliedresearch. Berlin: Springer-Verlag, London, UK.[5] Köchel, P. and Nieländer, U. (2005). Simulation-based optimization of multi-echelon inventory systems. International Journal of Production Economics,Vol.93.94, 505-513.[6] Daniel, J. R.S. and Rajendran, C. (2005). A simulation-based geneticalgorithm for inventory optimization in a serial supply chain. InternationalTransactions In Operational Research, 101–127.[7] Tsou, C (2009). Evolutionary Pareto optimizers for continuous reviewstochastic inventory systems. European Journal of Operational Research,195, 364–371.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.600[8] Hiremath, N. C. Sahu, S. and Tiwari, M. K. (2010). Optimization for EfficientSupply Chain Using Swarm Intelligence: an Outsourcing Perspective.Proceedings of the World Congress on Engineering, Vol III.[9] Narmadha, S. Selladurai, Dr.V. and Sathish, G. (2010). Efficient InventoryOptimization of Multi Product, Multiple Suppliers with Lead Time usingPSO. International Journal of Computer Science and Information Security,Vol. 7, No. 1.[10] Diks, E.B. De Kok, A.G. and  Lagodimos, A.G. (1996). Multi-echelonsystems: A service measure perspective. European Journal of OperationalResearch 95, 241-263.[11] Yu, X., & Gen, M. (2010). Introduction to Evolutionary Algorithms.Springer-Verlag, London.[12] Mele, F.D. Guillen, G. Espuna, A. and Puigjaner, L. (2006). A Simulation-Based Optimization Framework for Parameter Optimization of Supply-ChainNetworks. Industrial & Engineering Chemistry Research, 45, 3133-3148.[13] Tyworth, J.E. and O’Neill, L. (1997). Robustness of the NormalApproximation of Lead-Time Demand in a Distribution Setting. NavalLogistics Research, 44, 165-186.

A Comparison Study of Multi-Objective Metaheuristic Techniques for Continuous Review Stochastic Inventory System

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A Comparison Study of Multi-Objective Metaheuristic Techniques for Continuous Review Stochastic Inventory System

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