An application of control theory for imperfect production problem with carbon emission investment policy in interval environment

The theory of optimal control plays an important role to solve several real-life problems. However, it is often seen that uncertainty is a major challenge in formulating the appropriate model corresponding to real world problems. To overcome this challenge, in this work, the theory of optimal control is developed in interval environment. Accordingly, the necessary and sufficient optimality conditions for the existence of maximizer of interval valued optimal control problem (IVOCP) are proposed. To validate the proposed theory, an imperfect production inventory model is developed in both crisp and interval environments in which customers' demand is dependent on the selling price, specific absorption rate (SAR) of products and continuous time. In the proposed model, the production rate is considered as a function of time. In addition, manufacturer invests to reduce the carbon emission and to increase the environmental sustainability of the produced items. As the production rate is taken as a function of time, optimal control theory is used to study the optimal policy of the corresponding problem. To solve the optimal control problem for the proposed model in interval environment, sufficient optimality conditions of IVOCP are used. Thereafter, two numerical examples are considered and solved with the help of different meta-heuristic algorithms. Finally, to observe the impact of various system parameters to the optimal policy of the system, sensitivity analyses are carried out for the example in interval environment and impacts are shown graphically. (C) 2022 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.