Production scheduling can be defined as the allocation of available production resources over time to best satisfy some set of criteria. Typically, the scheduling problem involves a set of tasks to be performed, and the criteria may involve both tradeoffs between early and late completion of a task, and between holding inventory for the task and frequent production changeovers. The intent of this paper is to present a broad classification for various scheduling problems, to review important theoretical developments for these problem classes, and to contrast the currently available theory with the practice of production scheduling. This paper will highlight problem areas for which there is both a significant discrepancy between practice and theory, and for which the practice corresponds closely to the theory.

A Review of Production Scheduling

A class of production planning problems is considered in which known demands have to be satisfied over a finite horizon at minimum total costs. For each period, production and storage cost functions are specified. The production costs may include set-up costs and the production levels may he subject to capacity limits. The computational complexity of the problems in this class is investigated. Several algorithms proposed for their solution are described and analyzed. It is also shown that some special cases are NP-hard and hence unlikely to be solvable in polynomial time.

Deterministic Production Planning: Algorithms and Complexity

In this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings. For the single item case new properties are introduced, classes of problems solvable by polynomial time algorithms are identified, and efficient solution procedures are given. We show that special classes are NP-hard, and that the problem with two items and independent setups is NP-hard under conditions similar to those where the single item problem is easy. Topics for further research are discussed in the last section.

Computational Complexity of the Capacitated Lot Size Problem

Planning for the expansion of production capacity is of vital importance in many applications within the private and public sectors. Examples can be found in heavy process industries, communication networks, electrical power services, and water resource systems. In all of these applications, the expansion of production capacity requires the commitment of substantial capital resources over long periods of time. Capacity expansion planning consists primarily of determining future expansion times, sizes, and locations, as well as the types of production facilities. Since the late 1950s, operations research methodology has been used to develop various models and solution approaches suitable for different applications. In this paper, we attempt to unify the existing literature on capacity expansion problems, emphasizing modeling approaches, algorithmic solutions, and relevant applications. The paper includes an extensive list of references covering a broad spectrum of capacity expansion problems.

Operations Research and Capacity Expansion Problems: A Survey

A warehouse supplies N retailers. Constant external demand occurs at each retailer, and shortages are not allowed. There are linear holding costs and fixed costs for ordering at the warehouse and at each retailer. The goal is to minimize the long-run average cost over an infinite time horizon.

We define a new class of policies for this problem whose simple structure facilitates both computation and implementation. The cost of a policy that is optimal within this class is shown to be within 2% of the cost of an optimal policy for the original problem, in the worst case. Such a policy can be computed in ON log N time.

98%-Effective Integer-Ratio Lot-Sizing for One-Warehouse Multi-Retailer Systems

An n period single-product single-facility model with known requirements and separable piecewise concave production and storage costs is considered. It is shown using network flow concepts that for arbitrary bounds on production and inventory in each period there is an optimal schedule such that if, for any two periods, production does not equal zero or its upper or lower bound, then the inventory level in some intermediate period equals zero or its upper or lower bound. An algorithm for searching such schedules for an optimal one is given where the bounds on production are -∞, 0 or ∞. A more efficient algorithm assumes further that inventory bounds satisfy “exact requirements.”

https://www.jstor.org/stable/info/2629721

Bounded Production and Inventory Models with Piecewise Concave Costs

We consider an n-period single-product inventory model with known requirements and separable concave production and storage costs. The model is a multi-echelon structure in which N facilities, labeled 1,…,N, are arranged in series. We show that if storage and production costs are respectively nondecreasing in order of facilities and nonincreasing in time, then an optimal schedule has the property that if in a given period, facility j produces, then facility j + 1 does also. This nested structure is exploited in an algorithm for finding an optimal schedule. The computational effort increases in proportion to the cube of the number of periods and linearly with the number of facilities. For the stationary infinite horizon case an algorithm yields a periodic optimal schedule.

Stephen F. Love

Papers

Bounded Production and Inventory Models with Piecewise Concave Costs

A Facilities in Series Inventory Model with Nested Schedules