J. Christopher Beck

Problem difficulty for tabu search in job-shop scheduling

Tabu search algorithms are among the most effective approaches for solving the job-shop scheduling problem (JSP). Yet, we have little understanding of why these algorithms work so well, and under what conditions. We develop a model of problem difficulty for tabu search in the JSP, borrowing from similar models developed for SAT and other NP-complete problems. We show that the mean distance between random local optima and the nearest optimal solution is highly correlated with the cost of locating optimal solutions to typical, random JSPs. Additionally, this model accounts for the cost of locating sub-optimal solutions, and provides an explanation for differences in the relative difficulty of square versus rectangular JSPs. We also identify two important limitations of our model. First, model accuracy is inversely correlated with problem difficulty, and is exceptionally poor for rare, very high-cost problem instances. Second, the model is significantly less accurate for structured, non-random JSPs. Our results are also likely to be useful in future research on difficulty models of local search in SAT, as local search cost in both SAT and the JSP is largely dictated by the same search space features. Similarly, our research represents the first attempt to quantitatively model the cost of tabu search for any NP-complete problem, and may possibly be leveraged in an effort to understand tabu search in problems other than job-shop scheduling.

APPLYING MACHINE LEARNING TO LOW-KNOWLEDGE CONTROL OF OPTIMIZATION ALGORITHMS

This paper addresses the question of allocating computational resources among a set of algorithms to achieve the best performance on scheduling problems. Our primary motivation in addressing this problem is to reduce the expertise needed to apply optimization technology. Therefore, we investigate algorithm control techniques that make decisions based only on observations of the improvement in solution quality achieved by each algorithm. We call our approach “low knowledge” since it does not rely on complex prediction models, either of the problem domain or of algorithm behavior. We show that a low‐knowledge approach results in a system that achieves significantly better performance than all of the pure algorithms without requiring additional human expertise. Furthermore the low‐knowledge approach achieves performance equivalent to a perfect high‐knowledge classification approach.

A theoretic and practical framework for scheduling in a stochastic environment

There are many systems and techniques that address stochastic planning and scheduling problems, based on distinct and sometimes opposite approaches, especially in terms of how generation and execution of the plan, or the schedule, are combined, and if and when knowledge about the uncertainties is taken into account. In many real-life problems, it appears that many of these approaches are needed and should be combined, which to our knowledge has never been done. In this paper, we propose a typology that distinguishes between proactive, progressive, and revision approaches. Then, focusing on scheduling and schedule execution, a theoretic model integrating those three approaches is defined. This model serves as a general template to implement a system that will fit specific application needs: we introduce and discuss our experimental prototypes which validate our model in part, and suggest how this framework could be extended to more general planning systems.

Constraint-directed techniques for scheduling alternative activities

Abstract In this paper, we expand the scope of constraint-directed scheduling techniques to deal with the case where the scheduling problem includes alternative activities. That is, not only does the scheduling problem consist of determining when an activity is to execute, but also determining which set of alternative activities is to execute at all. Such problems encompass both alternative resource problems and alternative process plan problems. We formulate a constraint-based representation of alternative activities to model problems containing such choices. We then extend existing constraint-directed scheduling heuristic commitment techniques and propagators to reason directly about the fact that an activity does not necessarily have to exist in a final schedule. Experimental results show that an algorithm using a novel texture-based heuristic commitment technique together with extended edge-finding propagators achieves the best overall performance of the techniques tested.

Proactive algorithms for job shop scheduling with probabilistic durations

Most classical scheduling formulations assume a fixed and known duration for each activity. In this paper, we weaken this assumption, requiring instead that each duration can be represented by an independent random variable with a known mean and variance. The best solutions are ones which have a high probability of achieving a good makespan. We first create a theoretical framework, formally showing how Monte Carlo simulation can be combined with deterministic scheduling algorithms to solve this problem. We propose an associated deterministic scheduling problem whose solution is proved, under certain conditions, to be a lower bound for the probabilistic problem. We then propose and investigate a number of techniques for solving such problems based on combinations of Monte Carlo simulation, solutions to the associated deterministic problem, and either constraint programming or tabu search. Our empirical results demonstrate that a combination of the use of the associated deterministic problem and Monte Carlo simulation results in algorithms that scale best both in terms of problem size and uncertainty. Further experiments point to the correlation between the quality of the deterministic solution and the quality of the probabilistic solution as a major factor responsible for this success.

Dynamic problem structure analysis as a basis for constraint-directed scheduling heuristics

Abstract While the exploitation of problem structure by heuristic search techniques has a long history in AI (Simon, 1973), many of the advances in constraint-directed scheduling technology in the 1990s have resulted from the creation of powerful propagation techniques. In this paper, we return to the hypothesis that understanding of problem structure plays a critical role in successful heuristic search even in the presence of powerful propagators. In particular, we examine three heuristic commitment techniques and show that the two techniques based on dynamic problem structure analysis achieve superior performance across all experiments. More interestingly, we demonstrate that the heuristic commitment technique that exploits dynamic resource-level non-uniformities achieves superior overall performance when those non-uniformities are present in the problem instances.

A Hybrid Approach to Scheduling with Earliness and Tardiness Costs

A hybrid technique using constraint programming and linear programming is applied to the problem of scheduling with earliness and tardiness costs. The linear model maintains a set of relaxed optimal start times which are used to guide the constraint programming search heuristic. In addition, the constraint programming problem model employs the strong constraint propagation techniques responsible for many of the advances in constraint programming for scheduling in the past few years. Empirical results validate our approach and show, in particular, that creating and solving a subproblem containing only the activities with direct impact on the cost function and then using this solution in the main search, significantly increases the number of problems that can be solved to optimality while significantly decreasing the search time.

A generic framework for constraint-directed search and scheduling

This article introduces a generic framework for constraint-directed search. The research literature in constraint-directed scheduling is placed within the framework both to provide insight into, and examples of, the framework and to allow a new perspective on the scheduling literature. We show how a number of algorithms from the constraint-directed scheduling research can be conceptualized within the framework. This conceptualization allows us to identify and compare variations of components of our framework and provides new perspective on open research issues. We discuss the prospects for an overall comparison of scheduling strategies and show that firm conclusions vis-a-vis such a comparison are not supported by the literature. Our principal conclusion is the need for an empirical model of both the characteristics of scheduling problems and the solution techniques themselves. Our framework is offered as a tool for the development of such an understanding of constraint-directed scheduling and, more generally, constraint-directed search.

Combining Constraint Programming and Local Search for Job-Shop Scheduling

Since their introduction, local search algorithms have consistently represented the state of the art in solution techniques for the classical job-shop scheduling problem. This dominance is despite the availability of powerful search and inference techniques for scheduling problems developed by the constraint programming community. In this paper, we introduce a simple hybrid algorithm for job-shop scheduling that leverages both the fast, broad search capabilities of modern tabu search algorithms and the scheduling-specific inference capabilities of constraint programming. The hybrid algorithm significantly improves the performance of a state-of-the-art tabu search algorithm for the job-shop problem and represents the first instance in which a constraint programming algorithm obtains performance competitive with the best local search algorithms. Furthermore, the variability in solution quality obtained by the hybrid is significantly lower than that of pure local search algorithms. Beyond performance demonstration, we perform a series of experiments that provide insights into the roles of the two component algorithms in the overall performance of the hybrid.

Scheduling with uncertain durations: Modeling β-robust scheduling with constraints

Many real-world scheduling problems are subject to change, and scheduling solutions should be robust to those changes. We consider a single-machine scheduling problem where the processing time of each activity is characterized by a normally distributed random variable, with flowtime as the main solution criterion. The objective is to find the @b-robust schedule-the schedule that minimizes the risk of the flowtime exceeding a threshold. We show how to represent this problem as a constraint model, explicitly representing the uncertainty and robustness as input parameters and objectives, and enabling the uncertainty to propagate using constraint propagation. Specifically, we develop three models (primal, dual and hybrid), and we show the effect of dominance rules on the search space.