X–Y band and modified (s, S) policy

This paper considers the stochastic, single-item, periodic review inventory problem. Most importantly we assume a finite production capacity per period and a production cost function containing a fixed (as well as a variable) component. With stationary data, a convex expected holding and shortage cost function, we show that generally the modified (s, S) policy is not optimal to the finite horizon problems. The optimal policy does, however, show a systematic pattern which we call the X–Y band structure. This X–Y band policy is interpreted as follows: whenever the inventory level drops below X, order up to capacity; when the inventory level is above Y, do nothing; if the inventory level is between X and Y, however, the ordering pattern is different from problem to problem. Although the X and Y bounds may change from period to period, we prove the existence of a pair of finite X and Y values that can apply for all the periods (i.e., bounds on individual bounds). One calculation for such X and Y bounds that are tight in some cases is also provided. By exploring the X–Y band structure, we can drastically reduce the computation effort for finding the optimal policies.