Multi-objective inventory planning using MOPSO and TOPSIS

doi:10.1016/j.eswa.2007.06.009

Expert Systems with Applications

Volume 35, Issues 1–2, July–August 2008, Pages 136-142

https://doi.org/10.1016/j.eswa.2007.06.009 Get rights and content

Abstract

One of the main characteristics of today’s business tends to vary often. Under such environment, many decisions should be carefully pondered over from relevant aspects which are usually conflicting. Hence, inventory planning problems, which address how much and when to order what customers need at the least relevant cost while maintaining a desirable service level expected by customers, could be recast into a multi-objective optimization problem (MOOP). In a MOOP there are normally infinite numbers of optimal solutions in the Pareto front due to the conflicts among objectives. Unfortunately, most multi-objective inventory models have been solved by aggregation methods through a linear combination of specific weights or only one objective was optimized and the others were turned into constraints. Therefore, the challenges decision makers face are not only modeling the problem in a multi-objective context, but also the effort dedicated to build the Pareto front of MOOPs. This paper first employs the multi-objective particle swarm optimization (MOPSO) algorithm to generate the non-dominated solutions of a reorder point and order size system. A ranking method called technique for order preference by similarity to ideal solution (TOPSIS) is then used to sort the non-dominated solutions by the preference of decision makers. That is, a two-stage multi-criteria decision framework which consists of MOPSO and TOPSIS is presented to find out a compromise solution for decision makers. By varying the weights of various criteria, including minimization of the annual expected total relevant cost, minimization of the annual expected frequency of stock-out occasions, and minimization of the annual expected number of stock-outs, managers can determine the order size and safety stock simultaneously which fits their preference under different situations.

Introduction

Most real-world problems in business are modeled as an optimization problem involving a single objective. The assumption that firms always seek to maximize (or minimize) their profit (or cost) rather than making tradeoffs among multiple objectives has been criticized for a long time. For example, an inventory control system should operate at least cost while maintaining desirable service level expected by customers. The objectives of cost minimization and service level maximization are incommensurate and conflicting with each other. Hence, the stage is set for a multi-objective analysis of decision making not only in business, but also for engineering design and scientific experiments.

Up to now, most multi-objective inventory models have put their emphasis on deteriorating items. Padmanabhan and Vrat (1990) solved a multi-objective inventory of deteriorating items with stock-dependent demand by a nonlinear goal programming method. Agrell (1995) presented a decision support system for multi-criteria inventory control. The solution procedure embedded is an interactive method with preferences extracted progressively in decision analysis process to determine batch size and security stock. Roy and Maiti (1998) formulated a multi-objective inventory model of deteriorating items with stock-dependent demand under limited imprecise storage area and total cost budget. The objectives therein are to maximize the profit and to minimize the wastage cost where the profit goal, wastage cost and storage area are fuzzy in nature. The problem was solved by Fuzzy Non-Linear Programming (FNLP) and Fuzzy Additive Goal Programming (FAGP). Mahapatra and Maiti (2005) considered a multi-objective inventory model of stochastically deteriorating items and incorporated the impact of quality level into the demand and deterioration function. Mandal, Roy, and Maiti (2005) presented a multi-item multi-objective fuzzy inventory model with three constraints, actually a cost minimization objective with three fuzzy goal constraints, to find the demand, order size, and shortage level for each item. They solved the problem by geometric programming method.

All works mentioned above using a relative weight vector to scalarize multiple objectives into a single objective and solve the corresponding problem by traditional single-objective optimization techniques. This approach contradicts our intuition that single-objective optimization is a degenerate case of MOOP (Deb, 2001). Furthermore, a MOOP does not have a single solution that could optimize all objectives simultaneously. Therefore, solving MOOP is not to search for optimal solutions but for efficient solutions that can be expressed in terms of non-dominated solutions in the objective space. A solution is said to be dominant over another only if it has superior, at least no inferior, performance in all objectives. The result of preference-based approach is a compromise solution whose non-dominance is not guaranteed (Liu, Yang, & Whidborne, 2003). Lastly, but not the least, a single optimized solution could only be found in each simulation run of traditional optimization techniques. Therefore, using a population of solutions to evolve towards several non-dominated solutions in each run makes evolutionary algorithms (EA) popular in solving MOOPs.

PSO is also a population-based stochastic optimization heuristic developed by Kennedy and Eberhart (1995), which was inspired by social behavior of bird flocking or fish schooling. It has been an effective technique to search for optima of optimization problems. Although it does not use the selection operation because the members of the entire population are maintained throughout the search procedure, PSO actually shares many similarities with evolutionary computation techniques (Eberhart & Shi, 1998). Applications, parameter selection, and modified versions of PSO can be found in Eberhart and Shi (2001) and Shi and Eberhart, 1998a, Shi and Eberhart, 1998b.

One of the successful applications of PSO to MOOPs, named multi-objective PSO (MOPSO), is the seminal work of Coello-Coello and Lechuga (2002). In a subsequent study done by them, MOPSO is not only a viable alternative to solve MOOPs, but also the only one, compared with the non-dominated sorting genetic algorithm-II (NSGA-II) (Deb, Pratap, Agarwal, & Meyarivan, 2002), the Pareto archive evolutionary strategy (PAES) (Knowles & Corne, 2000), and the micro-genetic algorithm (microGA) (Coello-Coello & Pulido, 2001) for multi-objective optimization, can cover the full Pareto front MOPSO of all the test functions therein (Coello-Coello, Pulido, & Lechuga, 2004).

In addition to generating possible tradeoff solutions by considering all objectives, the effort devoted to solicit a compromise solution amenable to decision maker’s preference is equally important. An ideal approach to analyze the decision in a multi-objective context is first to find multiple tradeoff optimal solutions, so called non-dominated solutions, of a MOOP. Then, one of the obtained solutions is chosen by using preference information of decision makers. It is normally implemented as a ranking process. In a nutshell, resolution of MOOP involves two stages, one is how to find the non-dominated front in the objective space, and the other is how to rank the non-dominated solutions by subjective judgments or preference information provided by decision makers. Both stages require an iterative solution procedure in which the decision maker investigates a variety of solutions to find one or some that is most satisfactory. Therefore, this paper first applies the MOPSO to generate the non-dominated solutions of order size and safety stock in a multi-objective inventory planning model. After that, a ranking method called technique for order preference by similarity to ideal solution (TOPSIS) is used to prioritize the non-dominated solutions for decision makers (Yoon & Hwang, 1995).

The rest of this paper is organized as follows. Section 2 reviews a multi-objective inventory planning model presented by Agrell (1995). Next, some preliminary knowledge about PSO and MOOP are described in Section 3. Sections 4 A multi-criteria decision framework, 5 Experimental results report the solution procedure and the computational results, respectively. Finally, conclusions and future research directions are drawn out in Section 6.

Section snippets

Multi-objective inventory planning

The control of inventories has been a major issue in the field of industrial engineering and operational research (IE/OR) for a long time. As an essential activity for any enterprise, inventory planning tries to determine the decisions about when to order and how much should order for different control mechanisms. A common control mechanism is a continuous-review (r,Q) system in which an order of size Q is placed whenever the inventory position drops to the reorder point, r (Silver, Pyke, &

Pareto optimality concepts

Pareto optimality is the most important solution concept in MOOP. It is also a major way to find the equilibrium results in non-cooperative game theory. Before presenting the approach to plan for the control parameters in a multi-objective inventory planning context, some preliminary knowledge about Pareto optimality is briefly introduced in this section.

A MOOP problem with K objectives and M constraints can be stated as follows. $Minimize \vec{f} (\vec{x}) = [f_{1} (\vec{x}), f_{2} (\vec{x}), \dots, f_{K} (\vec{x})]^{T}$ $Subject to \vec{x} \in Ω$ $Ω = {\vec{x} | g_{j} (\vec{x}) ⩽$

A multi-criteria decision framework

A two-stage multi-criteria decision framework of the inventory planning problem is presented in Fig. 1. In the first stage, decision makers set up the parameters needed in MOPSO to generate the non-dominated front of the multi-objective inventory planning problem. Next, according to the weights reflecting the preference on each objective, TOPSIS is used to rank the non-dominated solutions generated in the first stage. Decision makers could adjust their preference to find the compromise solution

Experimental results

The pharmaceuticals data shown in Table 2 is used to demonstrate the multi-criteria decision framework for inventory planning problems. All computation is executed on a laptop computer with Pentium M-1.6 GHz CPU and 1024 MB RAM. The first job is to set up various parameters for specific problem to get a better performance. These parameters include the number of iterations, number of particles, and maximum velocity of each variable (it is denoted as a percentage of the upper bound of each decision

Conclusion

Traditional single-objective inventory planning models require the knowledge of the stock-out cost or service level. The difficulty in estimation of these parameters has been generally acknowledged. Hence the stage is set for a multi-objective analysis of the problem. Unfortunately, most multi-objective inventory models have been solved by aggregation methods through a linear combination of specific weights or only one objective was optimized and the others were turned into constraints. The

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Multi-objective inventory planning using MOPSO and TOPSIS

Abstract

Introduction

Section snippets

Multi-objective inventory planning

Pareto optimality concepts

A multi-criteria decision framework

Experimental results

Conclusion

International Journal of Production Economics

Computers and Operations Research

Fuzzy Sets and Systems

Engineering Cost and Production Economics

Computers and Operations Research

A MOPSO algorithm based exclusively on Pareto dominance concepts

Lecture Notes in Computer Science

Handling multiple objectives with particle swarm optimization

IEEE Transactions on Evolutionary Computation

Multi-objective optimization using evolutionary algorithms

A fast and elitist multiobjective genetic algorithm: NSGA-II

IEEE Transactions on Evolutionary Computation