Necessity constraint in two plant optimal production problem with imprecise parameters

The purpose of this paper is to present and solve a real-life problem of two plants producing the same item under fuzzy-stochastic environment. Here, an item alongwith random defective units is produced at two different plants situated in different locations under a single management. The rates of demand, production and defectiveness at these places are different. Demands of the item are primarily met locally from the respective plants but if a stock-out situation occurs in a plant, immediately some stock, from the other plant if available, is rushed to the stock-out plant. The demands are known but production rates are unknown, functions of time are taken as control variables. The available budget for the management house is imprecise. The holding, shortage and transportation costs are assumed to be imprecise and represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics and nearest interval approximation, the objective function is changed to respective multi-objective functions and thus the single-objective fuzzy problem is reduced to a crisp multi-objective decision making (MODM) problem. The MODM problem is then again transformed to a single crisp objective function with the help of weighted sum method. Using fuzzy relations, the imprecise budget constraint expressed in the form of necessity constraint is transformed into an equivalent crisp one. Then, total cost consisting of production, holding, shortage and transportation (from one plant to another) costs is expressed as an optimal control problem and solved using weighted sum method, the Kuhn–Tucker conditions, Pontryagin’s Optimal Control principle and generalized reduced gradient (GRG) technique. The model has been illustrated by numerical data. The optimum results are presented in both tabular and graphical forms.