ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 2 (2006) No. 5, pp. 312-321
Multiproduct aggregate production planning in fuzzy random environments∗ Yufu Ning1,2 , Wansheng Tang1 † , Ruiqing Zhao1 1 2
Institute of Systems Engineering, Tianjin University, Tianjin 300072, China Department of Computer Science, Dezhou University, Dezhou 253023, China (Received November 17 2005, Accepted February 27 2006)
Abstract. This paper discusses multiproduct aggregate production planning (APP) decision making problems in fuzzy random environments. To maximize the chance of obtaining the profit more than the predetermined profit over the whole planning horizon, a fuzzy random APP model is established, in which the market demand, production cost, subcontracting cost, inventory carrying cost, backorder cost, product capacity, product sales revenue, maximum labor level, maximum capital level, etc., are all characterized as fuzzy random variables. Then a hybrid optimization algorithm combining fuzzy random simulation, genetic algorithm (GA), neural network (NN) and simultaneous perturbation stochastic approximation (SPSA) algorithm is proposed to solve the model. At the end of this paper, an example is given to illustrate the effectiveness of the proposed method. Keywords: aggregate production planning (APP), fuzzy random variable, fuzzy random simulation, simultaneous perturbation stochastic approximation (SPSA)
1
Introduction
The goal of a manufacturing enterprise for making multiproduct aggregate production planning (APP) is to obtain the maximum profit or minimum cost by determining the product quantity, subcontracting quantity, labor level, etc., to meet the market demand in a long term. Since Holt et al[5] proposed the HMMS rule, a lot of researchers have developed various types of models and approaches to solve APP decision making problems. Bergstrom and Smith [1] generalized the HMMS approach to a multiproduct formulation, which was further extended by Hausman and Mcclain[4] into a stochastic programming model to cater to the randomness of product demand. Bitran and Yanassee[2] considered the problems of determining production plans over a number of time periods under stochastic demands. Fung et al[3] developed a fuzzy multiproduct aggregate production planning model whose solutions were introduced to cater to different scenarios under various decision making preferences by using parametric programming, best balance and interactive techniques. Wang and Fang[21] proposed a genetics-based imprecise approach to imitate the human decision procedure for production planning. The proposed approach could find a family of inexact solutions within an acceptable level by adopting a mutation operator to move along a weighted gradient direction. Furthermore, Wang and Fang[22] presented a fuzzy linear programming method for solving APP problems with multiple objectives where the product price, unit cost to subcontract, work force level, production capacity and market demand were fuzzy in nature. Then an interactive solution procedure was developed to provide a compromise solution. Wang and Liang[23] developed a fuzzy multi-objective linear programming model for solving the multiproduct aggregate production planning decision making problems in a fuzzy environment. The proposed model could yield an efficient compromise solution and the decision maker’s overall levels of satisfaction. Wang and Liang[24] provided an ∗
†
This work was supported by the National Natural Science Foundation of China Grant No. 70471049 and China Postdoctoral Science Foundation No. 2004035013. E-mail address:
[email protected]
Published by World Academic Press, World Academic Union
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World Journal of Modelling and Simulation, Vol. 2 (2006) No. 5, pp. 312-321
interactive possibilistic linear programming approach for solving APP problems with fuzzy demand, interrelated operating costs, and capacity. The proposed approach used the strategy of simultaneously minimizing the most possible value of the imprecise total costs, maximizing the possibility of obtaining lower total costs, and minimizing the risk of obtaining higher total costs. Therefore, the approach could yield an efficient APP compromise solution and overall degree of decision maker’s satisfaction with determined goal values. Yan et al[25] formulated a fuzzy programming model based on credibility measure for fuzzy production planning problems with fuzzy unit profit, fuzzy capacity, and fuzzy demand, and developed a fuzzy simulation-based genetic algorithm to solve the model. However, in the real APP decision making problems, randomness and fuzziness usually coexist. For example, due to lack of historical data, the demand for a kind of product may be characterized as a fuzzy variable by experts’ experience. But the demand may also be affected by some stochastic factors, such as climate, competition from other homogeneous companies, and national policies. In such cases, it is appropriate to characterize the demand as a fuzzy random variable. Fuzzy random variable is a strong tool to deal with the above problems. Since Kwakernaak[7, 8] introduced the concept of fuzzy random variable, many researchers have presented different definitions according to different requirements of measurability, such as Puri and Ralescu[18] , Kruse and Meyer[6] , Liu[9–11, 13] , and Liu and Liu[15, 16] . Liu and Liu[15] gave a new definition of fuzzy random variable and a definition of scalar expected value operator for fuzzy random variable, then they discussed some properties concerning the measurability of fuzzy random variable. The concept of chance measure of a fuzzy random event was first given by Liu[9, 10] . Liu and Liu[16] presented three kinds of mean chances of a fuzzy random event via Choquet integrals, then they provided a class of fuzzy random minimumrisk models, where the objective and constraints were all defined by the mean chances. This paper proposes an APP model with fuzzy random coefficients by the concept of mean chance of a fuzzy random event. Specifically, the market demand, production cost, subcontracting cost, inventory carrying cost, backorder cost, product capacity, product sales revenue, maximum labor level, maximum capital level, etc., are all characterized as fuzzy random variables. The objective of the model is to maximize the mean chance of obtaining the profit more than the predetermined profit over the whole planning horizon, and the constraints are defined by the mean chances. In most cases, it is impossible to obtain the exact values of these chance functions due to the complexity of uncertain factors. Therefore, it is necessary to employ fuzzy random simulation to estimate the values. To solve the model, this paper proposes a hybrid optimization algorithm based on fuzzy random simulation. At first, fuzzy random simulation is employed to generate a set of input-output data. Then an NN is trained by the data to approximate these chance functions. Finally, the NN is embedded in both GA and SPSA. GA is used to search the global optimal solution, while SPSA is used to improve the chromosomes in each generation. The remainder of this paper is organized as follows. Some concepts about fuzzy variables and fuzzy random variables are introduced in Section 2, and fuzzy random APP model is proposed in Section 3. Then a hybrid optimization algorithm is designed to solve the model in Section 4. An example is presented in Section 5.
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Fuzzy variables and fuzzy random variables
Let (Θ, P (Θ), P os) be a possibility space, where Θ is a universe, P (Θ) the power set of Θ, P os a possibility measure. Definition 1. [12, 17] A fuzzy variable is defined as a function from a possibility space to the set of real numbers. Definition 2. [14] Let ξ be a fuzzy variable on the possibility space (Θ, P (Θ), P os). Then the credibility measure of a fuzzy event {ξ ≥ r} can be represented by Cr{ξ ≥ r} =
1 (1 + P os{ξ ≥ r} − P os{ξ < r}) . 2
(1)
Definition 3. [12] Let (Θi , P (Θi ), P osi ) be possibility spaces, i = 1, 2, · · · , n, Θ = Θ1 × Θ2 × · · · × Θn and P os = P os1 ∧ P os2 ∧ · · · ∧ P osn . Then (Θ, P (Θ), P os) is called the product possibility space of (Θi , P (Θi ), P osi ), i = 1, 2, · · · , n. WJMS email for subscription:
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Definition 4. [15] A fuzzy random variable ξ is defined as a function from a probability space (Ω, A, Pr) to the set of fuzzy variables such that P os{ξ(ω) ∈ B} is a measurable function of ω for any Borel set B of
