Celia A. Glass

The scheduling of maintenance service

Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single activity can be scheduled to any one period. Applications of such a model are the scheduling of maintenance service to machines and multi-item replenishment of stock. In this paper we assume that the cost associated with any given type of activity increases linearly with the number of periods since the last execution of this type. The problem is to find an optimal schedule specifying at which periods to execute each of the activity types in order to minimize the long-run average cost per period. We investigate properties of an optimal solution and show that there is always a cyclic optimal policy. We propose a greedy algorithm and report on computational comparison with the optimal. We also provide a heuristic, based on regular cycles for all but one activity type, with a guaranteed worse case bound.

Unrelated parallel machine scheduling using local search

Simulated annealing and taboo search are well-established local search methods for obtaining approximate solutions to a variety of combinatorial optimization problems. More recently, genetic algorithms have also been applied. However, there are few studies which compare the relative performance of these different methods on the same problem. In this paper, these techniques are applied to the problem of scheduling jobs on unrelated parallel machines to minimize the maximum completion time. Results of extensive computational tests indicate that the quality of solutions generated by a genetic algorithm is poor. However, a hybrid method in which descent is incorporated into the genetic algorithm is comparable in performance with simulated annealing and taboo search.

Lot streaming in three-stage production processes

Lot streaming is the process of splitting a given lot or job to allow the overlapping of successive operations in multi-stage production systems, thereby reducing the makespan of the corresponding schedule. This paper develops algorithms to minimize the makespan for a single job in three-stage production processes. At each stage, the job is split into s sublots. For both the flow shop and job shop problems, an algorithm is proposed which computes the minimum makespan in O(log s) time. However, for the open shop, it is shown that to evaluate the minimum makespan requires constant time. Various results are derived which are applicable when the number of stages of production exceeds three.

The nurse rostering problem: A critical appraisal of the problem structure

This paper is concerned with the problem of nurse rostering within hospitals. We analyse a class of four benchmark instances from the nurse rostering literature to provide insight into the nature of the problem. By highlighting the structure of the problem we are able to reduce the relevant solution space. A mixed integer linear programme is then able to find optimal solutions to all four instances of this class of benchmark problems, each within half an hour. Our second contribution is to extend current mathematical approaches to nurse rostering to take better account of the practical considerations. We provide a methodology for handling rostering constraints and preferences arising from the continuity from one scheduling period to the next.

Scheduling for Parallel Dedicated Machines with a Single Server

This paper examines scheduling problems in which the setup phase of each operation needs to be attended by a single server, common for all jobs and different from the processing machines. The objective in each situation is to minimize the makespan. For the processing system consisting of two parallel dedicated machines we prove that the problem of finding an optimal schedule isNP-hard in the strong sense even if all setup times are equal or if all processing times are equal. For the case of m parallel dedicated machines, a simple greedy algorithm is shown to create a schedule with the makespan that is at most twice the optimum value. For the two machine case, an improved heuristic guarantees a tight worst-case ratio of 3/2. We also describe several polynomially solvable cases of the later problem. The two-machine flow shop and the open shop problems with a single server are also shown to be NP-hard in the strong sense. However, we reduce the two-machine flow shop no-wait problem with a single server to the Gilmore{Gomory traveling salesman problem and solve it in polynomial time. c 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 304{328, 2000

A New Heuristic for Three-Machine Flow Shop Scheduling

This paper considers the problem of sequencing n jobs in a three-machine flow shop with the objective of minimizing the makespan, which is the completion time of the last job. An On log n time heuristic that is based on Johnsons algorithm is presented. It is shown to generate a schedule with length at most 5/3 times that of an optimal schedule, thereby reducing the previous best available worst-case performance ratio of 2. An application to the general flow shop is also discussed.

Scheduling Batches with Sequential Job Processing for Two-Machine Flow and Open Shops

In this paper, we study a problem of scheduling and batching on two machines in a flow-shop and open-shop environment. Each machine processes operations in batches, and the processing time of a batch is the sum of the processing times of the operations in that batch. A setup time, which depends only on the machine, is required before a batch is processed on a machine, and all jobs in a batch remain at the machine until the entire batch is processed. The aim is to make batching and sequencing decisions, which specify a partition of the jobs into batches on each machine, and a processing order of the batches on each machine, respectively, so that the makespan is minimized. The flow-shop problem is shown to be strongly NP-hard. We demonstrate that there is an optimal solution with the same batches on the two machines; we refer to these asconsistent batches. A heuristic is developed that selects the best schedule among several with one, two, or three consistent batches, and is shown to have a worst-case performance ratio of 4/3. For the open-shop, we show that the problem is NP-hard in the ordinary sense. By proving the existence of an optimal solution with one, two or three consistent batches, a close relationship is established with the problem of scheduling two or three identical parallel machines to minimize the makespan. This allows a pseudo-polynomial algorithm to be derived, and various heuristic methods to be suggested.

Scheduling maintenance services to three machines

We study a discrete problem of scheduling activities of three types under the constraintthat at most a single activity can be scheduled to any one period. Applications of such amodel are the scheduling of maintenance service to machines and multi‐item replenishmentof stock. We assume that the cost associated with any given type of activity increases linearlywith the number of periods since the last execution of this type. The problem is to specifyat which periods to execute each of the activity types in order to minimize the long‐runaverage cost per period. We analyze various forms of optimal solution which may occur,relating them to the combination of the three machine cost constants. Some cases remainunsolved by this method and for these we develop a heuristic whose worst case performanceis no more than 3.33% from the optimal.

Genetic Algorithm for graph coloring: Exploration of Galinier and Hao's Algorithm

This paper examines the best current algorithm for solving the Chromatic Number Problem, due to Galinier and Hao (Journal of Combinatorial Optimization, vol. 3, no. 4, pp. 379–397, 1999). The algorithm combines a Genetic Algorithm with Tabu Search. We show that the algorithm remains powerful even if the Tabu Search component is eliminated, and explore the reasons for its success where other Genetic Algorithms have failed. In addition we propose a generalized algorithm for the Frequency Assignment Problem.

Structural Properties of Lot Streaming in a Flow Shop

Lot streaming is the process of splitting a given lot or job to allow the overlapping of successive operations in multi-stage production systems, thereby reducing the makespan of the corresponding schedule. This paper considers the problem of finding sublot sizes to minimize the makespan for a single job in an m-machine flow shop. On each machine, the job is to be partitioned into a given number of sublots, and sublot sizes are the same on each machine. We introduce the concept of machine dominance, and propose an algorithm to reduce the problem so that only dominant machines are considered explicitly. After defining a network representation in which it is required to find the shortest critical path length, we identify the structure of critical paths for optimal sublot sizes. Knowledge of this structure considerably reduces the search for an optimal solution, which we illustrate by presenting algorithms for finding optimal sublot sizes.