An extension of Pontryagin Maximum principle in interval environment and its application to inventory problem

Abstract

The control theory is one of the most fundamental branches of engineering as it implicates solving abilities of numerous non-linear engineering design problems efficiently. Again, Pontryagin’s maximum principle is one of the most salient topics of control theory as it is involved in solving various important problems. However, in the current highly complex situation, most of such real-life problems appear to be highly uncertain/imprecise in nature. Consequently, in order to analyse such problems accurately, uncertainty/flexibility of such problems cannot be overestimated. Motivating from this fact and as a necessity, in this study, the Pontryagin’s maximum principles are extended in interval environment for interval valued control problems (IVCPs). In this context, an IVCP is defined along with different formal terminologies. Further, the necessary and sufficient optimality conditions (i.e., Pontryagin’s maximum principles) are extended for IVCPs using existing interval ranking proposed by Bhunia and Samanta (2014). Further, in the second part of this work, in order to test the effectiveness of the proposed theories, an economic order quantity (EOQ) model is developed by considering dynamic servicing strategies in interval environment. With the help of numerical example, the proposed extension of Pontryagin’s maximum principles for IVCPs is well validated.