Composite control of periodic-review just-in-time inventory systems with asymmetric costs

In this paper, the problem of inventory management in production-inventory systems is addressed from the control-theoretic perspective. We analyze the systems in which customers (or retailers) are served from a common distribution point. The stock at the distribution center used to satisfy an unknown, time-varying customers' demand is replenished with delay from a supply source which is subject to capacity limitations. The purpose of the control action is to keep the stock level close to zero despite unpredictable demand variations. In this way we minimize the costs of holding inventory at the distribution center imposed by a positive stock level, and reduce the shortage costs generated when customers need to wait for the ordered goods since not all of the demand can be satisfied from the readily available resources and the stock level is negative. Since, typically, the holding and shortage costs are not equal to each other (asymmetric costs), a different controller should be used depending on the sign of the stock level. For this purpose we propose a composite control structure consisting of two linear-quadratic (LQ) optimal controllers. It is shown that under the proposed policy the stock level is finite and converges to zero asymptotically with nonoscillatory ordering signal.