سیستم کنترل فازی با نرم افزار برای مشکلات برنامه ریزی تولید

A considerable part of the literature on fuzzy sets is devoted to the field of fuzzy control system. In this paper, an alternative control system is introduced to describe a dynamic system with fuzzy white noise. In order to find optimal ways to control such a system, fuzzy optimal control theory is further developed. Specifically, a linear quadratic model is formulated and solved as a fuzzy optimal control problem. The formulation and solution of this model provide an economic interpretation of a production planning model both in the finite horizon and in the infinite horizon.

Classical control systems are expressed by differential equations. However, uncertainty is inherent in most dynamic systems. In order to characterize a system with white noise, stochastic control systems were introduced via Ito’s stochastic differential equation. Stochastic control theory is developed to find optimal ways to control such a system, and has been widely applied to physical, biological, economic and management problems. It is well known that fuzziness is another kind of uncertainty present in real systems. Fuzzy set theory [17] has been applied for modeling control systems in different ways such as in Zadeh [18], [19] and [20]. Traditionally, these models use If-Then rules and logical connectives to establish relations between the system variables. Depending on the form of the propositions and on the structure of the rule, different types of rule-based fuzzy models can be distinguished. See Sousa and Kaymak [14] and references therein for a review of these models, and several new developments [3] and [4]. In the past, two basic types of fuzzy control systems have been studied. They are the Mamdani fuzzy control system [10] and the Takagi–Sugeno fuzzy control system [15]. Each of these fuzzy control systems can be used to produce rule-based models for deterministic systems. An alternative approach, the Liu fuzzy control system, introduced in this paper, is that of a fuzzy control system driven by a Liu process [7]. Unlike the Mamdani and Takagi–Sugeno systems, the Liu fuzzy control system is not deterministic. Instead, it is characterized by a fuzzy differential equation. Based on this, a linear quadratic model is proposed and the corresponding fuzzy optimal control problem is solved. Finally, the system is applied to model production planning problems. The remainder of this paper is organized as follows: Section 2 recalls some basic concepts of Liu fuzzy control system. The proposed fuzzy optimal control problem is introduced and some results are given in Section 3. The linear quadratic model is formulated and the general solution procedure is obtained in Section 4. In Section 5, a type of production planning model in the finite horizon is formulated and solved, and some economic interpretations are given. Section 6 considered the infinite horizon case of production planning model. The related works are shown in Section 7, and the comparisons with stochastic production planning model and the limitations of our approach are discussed in Section 8. Finally, some conclusions of our work are listed.

Following the Mamdani and Takagi–Sugeno fuzzy control systems, the concept of Liu fuzzy control system was introduced. Associated with some functional, the fuzzy optimal control problem was formulated to model dynamic optimization problems. In particular, a linear quadratic model was proposed and one solution was obtained. As an application, both finite and infinite horizon versions of a production planning problem were investigated and solved. Finally, concrete interpretations are given to explain the solutions obtained. This paper has produced novel theoretical results; however, further developments will contribute to enrich the fuzzy optimal control theory and enhance its applicability. Specifically, the following issues remain to be considered in future investigations: (1) The mathematical properties of the Liu fuzzy control system; (2) Maximum principle of fuzzy optimal control theory; (3) Other special models of fuzzy optimal control problem; (4) More complex production planning problems will be proposed and solved by using fuzzy optimal control theory.