Optimal Production Scheduling and Employment Smoothing with Deterministic Demands

In this paper we study a model that minimizes the sum of production, employment smoothing, and inventory costs subject to a schedule of known demand requirements over a finite time horizon. The three instrumental variables are work force producing at regular-time, work force producing on overtime, and the total work force. Overtime is limited to be not more than a fixed multiple of regular time. The idle portion of the work force and the levels of inventory are resultant variables. We postulate the following shape characteristics for the cost functions production costs are convex-like, smoothing costs are V-shaped, and holding costs are increasing, both the production and holding cost functions need not be stationary. In this paper, we provide upper and lower bounds on the cumulative regular-time plus overtime work force for any sequence of demand requirements. We also give the form of an optimal policy when demands are monotone (either increasing or decreasing). Finally, we derive the asymptotic behavior of optimal policies when demands are monotone and the planning horizon becomes arbitrarily long. All of these results, which convey information about the numerical values of optimal policies, given specific demands and an initial level of inventory, depend only on the shape characteristics of the cost functions. Algorithmic techniques are discussed elsewhere [Lippman, S. A., A. J. Rolfe, H. M. Wagner, J. S. C. Yuan. Algorithms for optimal production scheduling and employment smoothing. Opns Res. To appear.], [Yuan, J. S. C. 1967. Algorithms and multi-product model in production scheduling and employment smoothing. Technical Report 22 (NSF GS-552), Stanford University, August.].