Some new results on rough interval linear programming problems and their application to scheduling and fixed-charge transportation problems

Abstract

This paper focuses on linear programming problems in a rough interval environment. By introducing four linear programming problems, an attempt is being made to propose some results on optimal value of a linear programming problem with rough interval parameters. To obtain optimal solutions of a linear programming problem with rough interval data, constraints of the four proposed linear problems are applied. In this regard, firstly, the largest and the smallest feasible spaces for a linear constraint set with rough interval coefficients and parameters are introduced. Then, a rough interval for optimal value of such problems is obtained. Further, an upper approximation interval and a lower approximation interval as the optimal solutions of linear programming problems with rough interval parameters are achieved. Moreover, two solution concepts, surely and possibly solutions, are defined. Some numerical examples demonstrate the validity of the results. In particular, a scheduling problem and a fixed-charge transportation problem (FCTP) under rough interval uncertainty are investigated.