Theoretical developments and application of variational principle in a production inventory problem with interval uncertainty

Abstract

The goal of this work is two folded: (i) theoretical development of optimality condition of a variational problem with interval uncertainty; (ii) application of the established results in a production inventory model with interval uncertainty. To serve the purpose of first fold, the definitions of optimizers of an interval valued variational problem (IVVP) are proposed by using interval order relations. Then both the necessary and sufficient optimality conditions of an IVVP with single as well as several variables are derived. In this derivation, the obtained necessary conditions are named as c-r Euler equations and sufficient conditions are named c-r as Legendre conditions. To achieve the purpose of second fold, a production inventory model is formulated by considering interval valued time dependent production and demand rates. Then the corresponding IVVP of the proposed model is obtained and its optimal policy is studied with the help of established c-r Euler’s equations and c-r Legendre conditions. Finally, to illustrate the optimal policy of the proposed model, a numerical example is considered and optimality of its obtained results are verified graphically.