Adaptive Control of a Reverse Logistic Inventory Model with Uncertain Deteriorations and Disposal Rates

An adaptive control of a reverse logistic inventory system with unknown deterioration and disposal rates is considered. An adaptive control approach with a feedback is applied to track the inventory levels toward their goal levels. Also, the updating rules of both deterioration and disposal rates are derived from the conditions of asymptotic stability of the reference model. Important characteristics of the adaptive inventory system are discussed. The adaptive controlled system is modeled by a nonlinear system of differential equations. Finally, the numerical solution of the controlled system is discussed and displayed graphically. 1. Introduction Applications of optimal control theory to management science, and especially to production planning, are proving to be quite fruitful; see Sethi and Thompson [1] and El-Gohary et al. [2]. The optimal control problem characterizes the system optimal trajectory in time. The application of optimal control theory supplies nice information about the system optimal path and decision rules in time, as for the following examples.(i)Tadj et al. [3] applied the optimal control theory to an inventory system with ameliorating and deteriorating items. They derived the optimal inventory level and optimal production rate for different cases of both amelioration and deterioration rates.(ii)El-Gohary and Elsyed [4] discussed the problem of optimal control of multi-item inventory models with different types of deterioration. They derived inventory levels which minimize the total holding cost of the system.(iii)Alshamrani and El-Gohary [5] applied Pontryagin principle to a two-item inventory model with different types of deteriorating items. They minimized the total cost which includes the sum of the holding costs of inventory items, the holding costs of one item due to the presence of the other, and the production cost.(iv)Foul et al. [6] studied the adaptive control of a continuous-time model of a production inventory system in which a manufacturing firm produces a single product, selling some, and stocking the remaining. The model reference adaptive control with feedback is applied to track the output of the system toward the inventory goal level.(v)El-Gohary and Yassen [7] used the adaptive control and synchronization procedures to the coupled dynamo system with unknown parameters. Based on the Liapunov stability technique, an adaptive control laws are derived so that the coupled dynamo system is asymptotically stable and the two identical dynamo systems are asymptotically synchronized. Also the updating rules of the unknown