Pricing and dynamic service policy for an imperfect production system: Extended Pontryagin’s maximum principle for interval control problems

Abstract

Pontryagin’s maximum principle is one of the most important topics of control theory. It plays a crucial role in solving variational problems in an alternative way. In this paper, Pontryagin’s maximum principle is extended in order to solve interval valued control problems (IVCPs). To serve this purpose, firstly all the formal definitions (viz. IVCP, interval ranking and optimizer of IVCP etc.) are presumed. Further, the necessary and sufficient condition, i.e., Pontryagin’s principles are extended in interval environment in order to solve an IVCP. The second part of this work presents an application of the proposed extension of Pontryagin’s maximum principle for IVCPs. Then, an imperfect production inventory problem is considered by assuming flexibility of all the involved parameters in interval environment. The proposed production problem mainly addresses a joint optimal policy for pricing, dynamic service investment and reworked/salvaged under interval flexibility. The IVCP related to the proposed model is solved using extended Pontryagin’s maximum principle and the interval optimization problem corresponding to the model is solved using improved c-r optimization technique. Further, a numerical example is taken and solved using the sparrow search algorithm (SSA) in order to illustrate the proposed theoretical results and validate the proposed model. Finally, sensitivity analyses are carried out w. r. to various system parameters in order to evaluate some implications to the manager.