Thomas E. Morton

Filtered beam search in scheduling

Beam search is a technique for searching decision trees, particularly where the solution space is vast. The technique involves systematically developing a small number of solutions in parallel so as to attempt to maximize the probability of finding a good solution with minimal search effort. In this paper, we systematically study the performance behaviour of beam search with other heuristic methods for scheduling, and the effects of using different evaluation functions to guide the search. We also develop a new variation of beam search, called filtered beam search which is computationally simple yet produces high quality solutions.

Myopic Heuristics for the Random Yield Problem

We consider a single item periodic review inventory problem with random yield and stochastic demand. The yield is proportional to the quantity ordered, with the multiplicative factor being a random variable. The demands are stochastic and are independent across the periods, but they need not be stationary. The holding, penalty, and ordering costs are linear. Any unsatisfied demands are backlogged. Two cases for the ordering cost are considered: The ordering cost can be proportional to either the quantity ordered (e.g., in house production) or the quantity received (e.g., delivery by an external supplier). Random yield problems have been addressed previously in the literature, but no constructive solutions or algorithms are presented except for simple heuristics that are far from optimal. In this paper, we present a novel analysis of the problem in terms of the inventory position at the end of a period. This analysis provides interesting insights into the problem and leads to easily implementable and highly accurate myopic heuristics. A detailed computational study is done to evaluate the heuristics. The study is done for the infinite horizon case, with stationary yields and demands and for the finite horizon case with a 26-period seasonal demand pattern. The best of our heuristics has worst-case errors of 3.0% and 5.0% and average errors of 0.6% and 1.2% for the infinite and finite horizon cases, respectively.

Resource-constrained multi-project scheduling with tardy costs: Comparing myopic, bottleneck, and resource pricing heuristics

Abstract This paper addresses the problem of scheduling multiple resource-constrained projects with the objective of minimizing weighted tardiness costs. Extending our earlier heuristic scheduling work for production shops, we develop an efficient and effective means of generating low cost schedules for multiple projects requiring multiple resources. A ‘cost-benefit’ scheduling policy with resource pricing is developed which balances the marginal cost of delaying the start of an eligible activity with the marginal benefit of such a delay. A central part of this policy is the heuristic estimation of implicit resource prices, which form the basis for calculating marginal delay costs. The resulting policies are tested against a number of dispatch scheduling rules taken from the project scheduling literature, and against several new scheduling rules, with encouraging results for both the weighted tardiness problem and for the special case of weighted project delay.

Order acceptance with weighted tardiness

Over the past decade the strategic importance of order acceptance has been widely recognized in practice as well as academic research. This paper examines order acceptance decisions when capacity is limited, customers receive a discount for late delivery, but early delivery is neither penalized nor rewarded. We model a manufacturing facility that considers a pool of orders, and chooses for processing the subset that results in the highest profit. We present several solution methods, beginning with a straightforward application of an approach which separates sequencing and job acceptance. We then develop an optimal branch-and-bound procedure that uses a linear (integer) relaxation for bounding and performs the sequencing and job acceptance decisions jointly. We develop a variety of fast and high-quality heuristics based on this approach. For small problems, beam search runs almost 20 times faster than the benchmark, with a high degree of accuracy, and a branch-and-bound heuristic using Vogels method for bounding is over 100 times faster with very high accuracy. For larger problems, a myopic heuristic based on the relaxation runs 2000 times faster than the beam-search benchmark, with comparable accuracy.

Selecting jobs for a heavily loaded shop with lateness penalties

Abstract One way of adjusting the workload in a manufacturing facility when available jobs exceed current capacity is to select a subset of jobs with the objective of maximizing total net profit for the firm, that is revenue contributed by jobs processed less a penalty for lateness. This motivates us to formulate the objective function in terms of revenues minus costs, in order to reflect the trade-offs faced by the firm itself, which views costs mainly in the context of the reduction of its profits. We present a model that uses weighted lateness as a criterion for time-related penalties. Taking advantage of two special characteristics of the problem, we develop an optimal algorithm and two heuristic procedures. Computational results show that the myopic heuristic demonstrates near-optimal performance in a fraction of the processing time of the optimal benchmark.

The Near-Myopic Nature of the Lagged-Proportional-Cost Inventory Problem with Lost Sales

A considerable literature has been devoted to periodic-review inventory problems with proportional ordering costs and known independent demand distributions, because of the optimality under broad conditions of “base-stock” policies. An important exception is the case when order delivery is lagged and excess demand is simply lost, which is more complicated. By further restricting holding and penalty costs to be proportional, the author has extended preliminary work by Karlin and Scarf to obtain good bounds for both the optimal order policy and the associated minimum-expected-discounted-cost function for the stationary problem. These bounds possess a “newsboy” interpretation primarily in terms of costs to be incurred specifically for the review period (the period marked by the arrival of this order). In fact, the upper ordering bound is precisely the “myopic” policy in Veinotts sense. (However, the myopic policy is not base stock.) Experimental results for the one- and two-period lag problems strongly supp...

The Effects of Learning on Optimal Lot Sizes: Further Developments on the Single Product Case

Abstract This paper reviews the previous research on the problem of determining optimal lot sizes when production costs decrease on a log-linear function. Problems due to simplifying assumptions of the previous work are discussed and modifications of previous models are presented to address more realistic conditions. Using a dynamic programming approach, the new results for two cases of learning, total transmission of learning (from lot to lot) and partial transmission, indicate that optimal lot sizes are increasing in the long-run and long transient states exist, contrary to conclusions of prior papers. The effect of forgetting indicates that operations managers should consider longer production runs for many processes. The revised model formulation presented in this study is also capable of handling demand characteristics of variance, growth and seasonality, significantly extending the application possibilities of the previous models which were limited to a continuous, deterministic demand pattern.

Universal Planning Horizons for Generalized Convex Production Scheduling

We study an infinite-horizon analog of the well-known finite-horizon convex production scheduling model with positive or negative demands, production, and inventory, and with constraints on inventory and production. We develop a forward algorithm for this problem allowing successively tighter bounds on the optimal first-period production to be given as successive demand forecasts are added to the problem. We show under very general conditions that the algorithm will always lead to an exact or asymptotic planning horizon. Planning horizon results due to others form special cases. Applications of the generalized production scheduling model include capacity expansion problems.

A Simple Heuristic for Computing Nonstationary ( S, S ) Policies

Nonstationary inventory problems with set-up costs, proportional ordering costs, and stochastic demands occur in a large number of industrial, distribution, and service contexts. It is well known that nonstationary (s, S) policies are optimal for such problems. In this paper, we propose a simple, myopic heuristic for computing the policies. The heuristic involves approximating the future problem at each period by a stationary one and obtaining the solution to the corresponding stationary problem. We numerically compare our heuristic with an earlier myopic heuristic and the optimal dynamic programming solution procedure. Over all problems tested, the new heuristic averaged 1.7% error, compared with 2.0% error for the old procedure, and is on average 399 times as fast as the D.P. and 2062 as fast as the old heuristic. Moreover, our heuristic, owing to its myopic nature, requires the demand data only a few periods into the future, while the dynamic programming solution needs the demand data for the entire time horizon-which are typically not available in most practical situations.

Infinite-Horizon Dynamic Programming Models—A Planning-Horizon Formulation

Two major areas of research in dynamic programming are optimality criteria for infinite-horizon models with divergent total costs and forward algorithm planning-horizon procedures. A fundamental observation for both problems is that the relative cost of two possible initial actions for a given initial state may be quite insensitive to structural information in all but the first few periods of a multiperiod model. Recently Lundin and Morton have developed a unified machinery for the dynamic lot size model that provides a general optimality criterion for the infinite horizon problem and complete planning-horizon procedures. Here those ideas are extended to provide a formal infinite-horizon/planning-horizon framework for a reasonably general form of the dynamic programming problem. Several examples and illustrations are provided. The approach provides a good vehicle for investigating the non-stationary stochastic inventory problem; this work will appear elsewhere.