Proper choice of salvage term in optimal control of stochastic production inventory system

This paper modifies the existing stochastic production inventory control models with linear salvage term, constant deterioration and time-varying demand by replacing the linear salvage term with a quadratic salvage term. Analytical solutions for optimal expected production levels and optimal expected inventory levels are obtained separately for existing and modified models. It is observed that mathematical inappropriateness occurs theoretically in the use of linear salvage terms when trying for analytical solutions, but these could be avoided with the use of a quadratic one. Next, numerical analysis is done for all the cases under constant choice of parameters and different choice of demand function, viz., constant, linear, quadratic and exponential. In all the models with linear salvage terms for different types of demand rates, the graphs of production levels drop from the initial level for a certain range of time, while the graph of inventory levels increases either slowly or abruptly. This is not usual, as the production level may remain stationary, but it cannot fall down, even in the case when the inventory level increases. In case of quadratic salvage term, the graphs of production levels are either remaining stationary or increasing, as the case may be. This is quite expected. Thus, geometrical interpretations always come out unrealistic with a linear salvage term, but they remain realistic for a quadratic salvage term. This justifies the introduction of a quadratic salvage term in the stochastic inventory control model.