Single machine scheduling under market uncertainty

Abstract

This paper considers single machine scheduling problems where job processing times are known and deterministic but where the reward received upon completion of a job changes stochastically over time according to Brownian motion. The objectives of maximizing expected net-present-value (NPV), minimizing the variance of NPV and maximizing the probability of achieving a minimum benchmark NPV are considered. For non-preemptive static list policies complexity results and branch and bound procedures are presented. The branch and bound procedures are shown to be effective for problem instances with 20–25 jobs. For the problem of maximizing NPV with non-preemptive dynamic policies the optimal static schedule is shown through empirical testing to be as good as a greedy heuristic and to be near optimal when the variance is not large.

Introduction

The need to respond to changes in the market environment (i.e. prices of inputs and outputs) has received much attention in the manufacturing literature in the areas of production planning, purchasing and inventory management. Models which deal with production planning have focused on the concept of dynamically changing product mix to maximize some performance measure subject to changes in the market environment (see [28], [3], [29], [5] for examples). Purchasing and inventory models have focused on valuing contracts, the ability to hold inventory, the ability to outsource and the flexibility offered by many other real options (see [20], [11], [13], [12], [17] for examples). Bengtsson [4] offers a survey of real option applications to manufacturing flexibility where as [24], [8] provide general surveys of manufacturing flexibility. The work in this area which most closely relates to single machine scheduling is provided by [15] who considers a single machine with the ability to produce a single product in two different modes. Depending on an uncertain output price and switching costs, stochastic dynamic programming is used to determine the switching policy and the value of having a machine with the flexibility to switch between processing modes. The model presented in this work differs significantly from [15] in that the machine operator does not decide what mode in which to produce identical units but rather must decide the sequence in which to process jobs with independent underlying risk. This new approach is more typical of the machine scheduling framework and the problems faced by managers with scheduling responsibilities. Such sequencing responsibilities are prevalent throughout both the manufacturing and service industries. In manufacturing settings jobs must be sequenced on individual pieces of equipment and a single job may require processing on multiple pieces of equipment across multiple departments or even multiple plants. In the service industry customers’ needs must be sequenced as is done in the transportation industry when scheduling dispatches and deliveries and as is done in the health services industry when scheduling visits and operations for patients. The reader is directed to [18], [19] for more information on scheduling in practice.

Despite the importance with which market variability has been treated in the manufacturing literature, no work which considers this type of uncertainty has yet been done in the area of machine scheduling. Rather, stochastic models in the machine scheduling literature have focused primarily on the operational uncertainties in job processing times and machine availability. The main results in stochastic machine scheduling are reviewed in [19]. The models studied in this manuscript do not consider this type of operational uncertainty but allow for deterministic processing times and uninterrupted machine availability. However, the reward received upon completion of each job is stochastic and changes over time according to Brownian motion. The jobs are then to be sequenced on a single machine so as to maximize some function of the NPV of rewards. The NPV has not received nearly as much attention in machine scheduling as in production and project scheduling. The main dissent being that the processing time of a job is not of significant length for the time value of money to be worth considering. Though it is true that a single item will likely be processed in minutes in a classical single machine environment, most shops do not operate as classical single machine environments. Rather, a particular item may need to be routed through several machines and/or facilities before completion, requiring days or even weeks to complete. Scheduling in such environments necessarily requires scheduling the individual machines which they contain. In addition, the jobs and machine need not be used to represent jobs and a machine in the classical sense. The jobs can represent batches of similar items or more generic and time consuming tasks that need to be performed by a team, a facility or an entire corporation, as is the case in [15]. Hence, though the NPV objectives are not of particular interest for a classical single machine environment they are of interest as a starting point for many other more general environments. For example, NPV objectives similar to those studied in this paper have been studied in the project scheduling literature in environments without resource constraints (see Section 3.1). Of course, most practical projects are performed under resource constraints. The machines studied in machine scheduling theory can be considered equivalent to the renewable resources studied in project scheduling theory. It follows that this paper provides a necessary step in addressing resource constrained project scheduling problems with similar NPV objectives.

Machine scheduling problems can be classified using the three-field α∣β∣γ notation where α describes the machine environment, β contains processing characteristics and γ represents the objective function. The models studied in this manuscript can be considered a generalization of the single machine scheduling problem in which jobs are to be scheduled on a single machine to minimize the discounted total weighted completion of jobs. Following the version of the three-field α∣β∣γ notation presented in [19], this problem is denoted as $1\Vert \sum w_j(1-{\mathrm{e}}^{-{\mathit{rC}}_j})$. Here, r represents the discount rate, C_j represents the completion time of the jth job and w_j represents the weight of the jth job. The idea of incorporating discounting into single machine scheduling objectives was originally proposed by [21] and further developed by [22], [23], [26]. Though [21] provides an optimal priority rule for $1\Vert \sum w_j(1-{\mathrm{e}}^{-{\mathit{rC}}_j})$, a negative result from [23] establishes that no such priority rule exists for the models developed in this work. Rothkopf [22] considers the problem with uncertain processing times and more than one processor where as [26] consider the problem with tree-like precedence constraints. The models studied in this manuscript can also be considered special cases of problem $1\Vert \sum f_j(C_j)$, the general non-linear costs single machine scheduling problem. Alidaee [2], [1] present heuristics when the functions f_j are differentiable and regular (non-decreasing in completion time). Also, [16] and [6] present optimal procedures when the functions f_j are regular. Unfortunately, the new objectives are not regular (see the proof of Theorem 2 in Appendix A) and the results of these works are not applicable here. Held and Karp [9] present an exponential time dynamic programming algorithm for $1\Vert \sum f_j(C_j)$ which provides the optimal solution if either the functions f_j are regular or if only non-delay schedules are used (as is the case in this paper). This same DP also appears in [16], [19]. However, in Section 3.3 the branch and bound procedures presented in this work are shown to significantly outperform the dynamic programming algorithm presented there.

This paper considers three versions of a new type of stochastic single machine scheduling problem where processing times are deterministic but rewards depend on a stochastic process. The new problems are formalized and the relationships between the three objectives are discussed in Section 2. Section 3 presents branch and bound procedures for obtaining optimal non-preemptive static list schedules and reports on the performance of the new procedures on a randomly generated data set. Section 4 presents a heuristic for the problem of maximizing expected NPV under non-preemptive dynamic schedules and reports on the performance of the heuristic versus an upper bound and versus the optimal static schedule. Finally, Section 5 concludes the paper with a summary of results and possible areas of future research.