Multi-item Optimal control problem with fuzzy costs and constraints using Fuzzy variational principle

An imperfect multi-item production system is considered against time dependent demands for a finite time horizon. Here production is defective. Following [Khouja and Mehrez J. Oper. Res. Soc. 45 (1994) 1405–1417], unit production cost depends on production, raw-material and maintenance costs. Produced items have same fixed life-time. Warehouse capacity is limited and used as a constraint. Available space, production, stock and different costs are assumed as crisp or imprecise. With the above considerations, crisp and fuzzy constrained optimal control problems are formulated for the minimization of total cost consisting of raw-material, production and holding costs. These models are solved using conventional and fuzzy variational principles with equality constraint condition and no-stock as end conditions. For the first time, the inequality space constraint is converted into an equality constraint introducing a pseudo state variable following Bang Bang control. [Roul et al. , J. Intell. Fuzzy Syst. 32 (2017) 565–577], as stock is mainly controlled by production, for the control problems production is taken as the control variable and stock as state variable. The reduced optimal control problem is solved by generalised reduced gradient method using Lingo-11.0. The models are illustrated numerically. For the fuzzy model, optimum results are obtained as fuzzy numbers expressed by their membership functions. From fuzzy results, crisp results are derived using α -cuts.