کاربرد برنامه ریزی خطی چند هدفه فازی برای برنامه ریزی تولید سنگدانه

This study develops a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-product aggregate production planning (APP) decision problem in a fuzzy environment. The proposed model attempts to minimize total production costs, carrying and backordering costs and rates of changes in labor levels considering inventory level, labor levels, capacity, warehouse space and the time value of money. A numerical example demonstrates the feasibility of applying the proposed model to APP problem. Its advantages are also discussed. The proposed model yields a compromise solution and the decision maker's overall levels of satisfaction. In particular, in contrast to other APP models, several significant characteristics of the proposed model are presented.

Aggregate production planning (APP) deals with matching capacity to demand of forecasted, varying customer orders over the medium term, often from 3 to 18 months in advance. APP aims to (1) to set overall production levels for each product category to meet fluctuating or uncertain demand in the near future, and (2) to set decisions and policies concerning hiring, layoffs, overtime, backorders, subcontracting and inventory level, and thus determining appropriate resources to be used (Lai & Hwang, 1992). APP has attracted considerable attention from both practitioners and academia (Shi & Haase, 1996). In the field of planning, it falls between the broad decisions of long-range planning and the highly specific and detailed short-range planning decisions. APP is one of the most important functions in production and operations management. Other forms of family disaggregation planning involve a master production schedule, capacity requirements planning, material requirements planning, which all depend on APP in a hierarchical way. Since Holt, Modigliani, and Simon (1955) proposed the HMMS rule in 1955, researchers have developed numerous models to help to solve the APP problem, each with their own pros and cons. According to Saad (1982), all traditional models of APP problems may be classified into six categories—(1) linear programming (LP) (Charnes and Cooper, 1961 and Singhal and Adlakha, 1989), (2) linear decision rule (LDR) (Holt et al., 1955), (3) transportation method (Bowman, 1956), (4) management coefficient approach (Bowman, 1963), (5) search decision rule (SDR) (Taubert, 1968), and (6) simulation (Jones, 1967). When using any of the APP models, the goals and model inputs (resources and demand) are generally assumed to be deterministic/crisp and only APP problems with the single objective of minimizing cost over the planning period can be solved. The best APP balances the cost of building and taking inventory with the cost of the adjusting activity levels to meet fluctuating demand. However, in real-world APP problems, the input data or parameters, such as demand, resources, cost and the objective function are often imprecise/fuzzy because some information is incomplete or unobtainable. Conventional mathematical programming schemes clearly cannot solve all fuzzy programming problems. The current APP model represents information in a fuzzy environment where the objective function and parameters are incompletely defined and cannot be accurately measured. In 1976, Zimmermann (1976) first introduced fuzzy set theory into conventional LP problems. That study considered LP problems with a fuzzy goal and fuzzy constraints. Following the fuzzy decision-making method proposed by Bellman and Zadeh (1970) and using linear membership functions, that same study confirmed that there exists an equivalent LP problem. Thereafter, fuzzy linear programming (FLP) has been developed into a number of fuzzy optimization methods for solving the APP problem. Hintz and Zimmermann (1989) presented an approach based primarily on FLP and approximate reasoning to solve APP, releasing of parts and machine scheduling problems in flexible manufacturing systems. Additional references on the use of FLP to solve APP problems include Masud and Hwang, 1980, Rinks, 1982, Lee, 1990, Tang et al., 2000, Wang and Fang, 2000 and Wang and Fang, 2001. However, in practical production planning systems, the many functional areas in an organization that yield an input to the aggregate plan normally have conflicting objectives governing the use of the organization's resources. These objectives minimize costs/maximize profits, inventory investment, customer service, changes in production rates, changes in work-force levels and utilization of plant and equipment (Krajewski & Ritzman, 1999). Moreover, the solution of fuzzy multi-objective optimization problems benefits from considering the imprecision of the decision maker's (DM's) judgments such as, ‘the objective function of the annual total production costs should be substantially less than or equal to 5 millions’, or ‘the changes in labor levels should be substantially less than or equal to 200 man-hours’. Especially, these conflicting objectives are required to be optimized simultaneously by the DM in the framework of fuzzy aspiration levels. In 1978, Zimmermann (1978) first extended his FLP approach (Zimmermann, 1976) to a conventional multi-objective linear programming (MOLP) problem. For each of the objective functions of this problem, assume that the DM has a fuzzy goal such as ‘the objective functions should be essentially less than or equal to some value’. Then, the corresponding linear membership function is defined and the minimum operator proposed by Bellman and Zadeh (1970) is applied to combine all objective functions. By introducing the auxiliary variable, this problem can be transformed into the equivalent conventional LP problem and can be easily solved by the simplex method of LP. Subsequent works on fuzzy goal programming (FGP) included Leberling, 1981, Hannan, 1981, Luhandjula, 1982 and Sakawa, 1988. Therefore, the aim of this study is to develop a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-product APP decision problem in a fuzzy environment. First, a MOLP model of a multi-product APP decision problem is constructed. The model attempts to minimize total production costs, carrying and backordering costs, and rates of changes in labor levels with reference to inventory level, labor levels, capacity, warehouse space and the time value of money. Furthermore, this model is converted into an FMOLP model by integrating fuzzy sets and objective programming approaches. The rest of this paper is organized as follows. Section 2 describes the problem, details the assumptions and develops the MOLP and FMOLP mathematical models of the APP decision problem. Section 3 presents a numerical example to demonstrate the application of the proposed model. Based on this numerical example, the proposed model is implemented using seven scenarios. Section 4 discusses the advantages of the model over other APP models. The final section draws conclusions and makes relevant recommendations.

APP deals with matching supply and demand of forecasted, varying customer orders over the medium term. The aim of APP decision-making is to set overall production levels for each product category to meet fluctuating or uncertain demands in the near future, such that APP also determines the appropriate resources to be used. This study develops a FMOLP model of the multi-product APP decision problem in a fuzzy environment. The proposed model aims to minimize total production costs, carrying and backordering costs, and the rates of changes in labor levels with reference to inventory level, labor levels, capacity, warehouse space and the time value of money. The proposed model yields a compromise solution and the DM's overall levels of satisfaction, given these solved fuzzy multi-objective values. Moreover, the proposed model provides a systematic framework that facilitates the decision-making process, enabling a DM interactively to modify the membership functions of the objectives until a satisfactory solution is obtained. Consequently, the proposed model is the most practically applicable for making APP decisions. The major limitations of the proposed model concern the assumptions made for each of the decision parameters with reference to production costs, forecasted demand, maximum inventory and labor levels, maximum capacity and warehouse space available, and relevant production resources. Hence, the proposed model must be modified make it better suited to the practical application. Furthermore, future researchers may explore the fuzzy properties of decision variable, coefficients, and relevant decision parameters in APP problems. The proposed FMOLP model is based on Hannan's approach (1981), which implicitly assumes that the minimum operator is the proper representation of the human DM who combines fuzzy statements by ‘and’. Therefore, future research may also apply the averaging or other operators to solve APP decision problems in a fuzzy environment.