Optimal control problem for an imperfect production process using fuzzy variational principle

The aim of this paper is to formulate and solve an optimal control problem under finite time horizon in fuzzy environment using fuzzy variational principle. Here an imperfect/defective item is produced to meet a time-dependent demand for a finite time period having no stock at both ends. The unit production cost is a function of production rate and also dependent on raw material cost, development costs due to durability and wear-tear cost. The cost function which consists of revenue, production and holding costs is formulated as a Fixed-Final Time and Fixed State System optimal control problem with finite time horizon. Here production rate is unknown and considered as a control variable and stock level is taken as a state variable. It is formulated to optimize the production rate so that total cost is minimum. For the fuzzy model, the production rate, stock level, inventory cost and development cost are taken as fuzzy. The models are solved by using conventional Variational Principle for crisp model and Fuzzy Variational Principle (FVP) for fuzzy model. For simulation, Mathematica-9.0 and the non-linear optimization technique Generalised Reduced Gradient Method (LINGO 11.0) have been used. The optimum results are illustrated both numerically and graphically. For the fuzzy model, the membership functions of fuzzy outputs are presented. The results of crisp model are also obtained from the fuzzy model.