Analytical solution of a modified single item continuous production inventory model under constant deterioration, goal levels and penalties

The paper tries to modify the existing single-item continuous inventory production management model with constant deterioration, goal levels and penalties under the addition of quadratic salvage term and product term in the expression of the objective function to make the model more general and meaningful. The model is solved by applying Pontryagin’s maximum principle. The solution reduces to the solution of a pair of ordinary differential equations, one for the state variable and the other one for the co-state variable. The latter equation is solved by a change of variables, where the function expressing the salvage term takes part directly. This makes the inclusion of salvage term indispensable in solving such inventory control problems. Once the co-state variable is obtained, it is used to give optimal control. Finally, using this expression of optimal control, the optimal inventory level is obtained by solving the corresponding state equation. Now the solutions of optimal control and also of the inventory are obtained under a special choice of parameters with constant, linear and quadratic demands separately. In each case, the graphical solution is shown separately, and its interpretation is given accordingly. Lastly, a proper sensitivity analysis of the model is also carried out.