Studying the Changes of an Optimal Trajectory

Abstract

This paper provides an extension to an optimal control problem using the negative logarithm of deterioration and spoilage function as total cost. This function must be minimized at the end of planning period depending on the alternative quadratic exponential form. The co-state variable has negative values along the optimal trajectory according to the Pontryagin Minimum Principle (PMP). The different values of this co-state variable are investigated using initial values for the optimal control rates, separately. The controlled system according to each value is presented. Studying the behavior of optimal inventory levels, the optimal production rates, and the optimal spoilage function, it is our optimal solution along the optimal trajectory. The effectiveness of increasing and decreasing the co-state values on the optimal trajectory especially at the end of planning period is investigated. Also, the sensitivity analysis that reflects the effect of changes of different parameters (the deterioration and spoilage parameters values, and the initial values of inventory levels and production rates) on the optimal solution is explained with many different cases. Finally, we compared, numerically, the results for using these different co-state values with the results obtained when this value is negative.