Scheduling parallel machines by the dynamic newsboy problem

Abstract

The newsboy problem is a well-known operations research model. Its various extensions have been applied to scheduling and evaluating advanced orders in manufacturing, retail and service industries. This paper focuses on a dynamic, continuous-time generalization of the single period newsboy problem. Similar to the classical newsboy problem, the model may represent the inventory of an item that becomes obsolete quickly, spoils quickly, or has a future that is uncertain beyond a single period. The problem is characterized by a number of newsboys (machines) whose operations are organized and controlled in parallel. The objective is to minimize shortage and surplus costs occurring at the end of the period as in the classical newsboy problem, as well as intermediate production and surplus costs that are incurred at each time point along the period. We prove that this continuous-time problem can be reduced to a number of discrete-time problems which are determined by loose, balanced and pressing production conditions. As a result, a polynomial-time combinatorial algorithm is derived in order to find globally optimal solutions. Scope and purpose The classical, single-period newsboy problem is to find a product order quantity that either maximizes the expected profit or minimizes the expected costs of overestimating and underestimating probabilistic demand. The basic point of the classical newsboy problem is that while a decision has to be made at the beginning of a period of time there is no way to either get or use information (or updates) on the demand realization before it is accumulated, i.e., before the end of the period. This very point is adopted in the paper. The importance and applicability of such a model are widely discussed in literature. Specifically, the model may represent the inventory of an item that becomes obsolete quickly, spoils quickly, or has a future that is uncertain beyond a single period. Furthermore, the decision on inventory in the classical newsboy problem is determined in terms of the total amount to be acquired or produced over the entire planning period. In contrast, we suggest a dynamic continuous-time extension that enables us to make a decision at each point of time and take into account all associated costs during the planning horizon.