Heidemarie Bräsel

Constructive heuristic algorithms for the open shop problem

In this paper we consider constructive heuristic algorithms for the open shop problem with minimization of the schedule length. By means of investigations of the structure of a feasible solution two types of heuristic algorithms are developed: construction of a rank-minimal schedule by solving successively weighted maximum cardinality matching problems and construction of an approximate schedule by applying insertion techniques combined with beam search. All presented algorithms are tested on benchmark problems from the literature. Our computational results demonstrate the excellent solution quality of our insertion algorithm, especially for greater job and machine numbers. For 29 of 30 benchmark problems with at least 10 jobs and 10 machines we improve the best known values obtained by tabu search.ZusammenfassungMit dem Ziel der Minimierung der Gesamtbearbeitungszeit werden konstruktive Heuristiken für das open shop Problem betrachtet. Durch strukturelle Untersuchungen einer zulässigen Lösung werden zwei Arten von Heuristiken entwickelt: Konstruktion eines rangminimalen Bearbeitungsplanes durch sukzessives Lösen von gewichteten Matchingproblemen mit maximaler Kardinalität und Konstruktion einer Näherungslösung durch Anwendung von Einfügungstechniken kombiniert mit beam search. Die Verfahren werden an den aus der Literatur bekannten Benchmark Beispielen getestet. Die Resultate unserer Testrechnungen demonstrieren eindrucksvoll die Qualität unseres Einfügungsalgorithmus, insbesondere für wachsende Auftrags- und Maschinenzahl. Für 29 der 30 Benchmark Beispiele mit mindestens 10 Aufträgen und 10 Maschinen wird die mit Tabusuche ermittelte Näherungslösung verbessert.

Simulated annealing and genetic algorithms for minimizing mean flow time in an open shop

This paper considers the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. It continues recent work by Brasel et al. [H. Brasel, A. Herms, M. Morig, T. Tautenhahn, T. Tusch, F. Werner, Heuristic constructive algorithms for open shop scheduling to minmize mean flow time, European J. Oper. Res., in press (doi.10.1016/j.ejor.2007.02.057)] on constructive algorithms. For this strongly NP-hard problem, we present two iterative algorithms, namely a simulated annealing and a genetic algorithm. For the simulated annealing algorithm, several neighborhoods are suggested and tested together with the control parameters of the algorithm. For the genetic algorithm, new genetic operators are suggested based on the representation of a solution by the rank matrix describing the job and machine orders. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The algorithms are compared relative to each other, and the quality of the results is also estimated partially by a lower bound for the corresponding preemptive open shop problem. For most of the problems, the genetic algorithm is superior when fixing the same number of 30000 generated solutions for each algorithm. However, in contrast to makespan minimization problems, where the focus is on problems with an equal number of jobs and machines, it turns out that problems with a larger number of jobs than machines are the hardest problems.

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/m.

Stability of a schedule minimizing mean flow time

This paper is devoted to the calculation of the stability radius of an optimal schedule for a general shop scheduling problem, where the objective is to minimize mean flow time. The stability radius denotes the largest quantity of independent variations of the processing times of the operations such that an optimal schedule of the problem remains optimal. We derive formulas for calculating the stability radius, and necessary and sufficient conditions when it is equal to zero. Moreover, computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are discussed.

On the open-shop problem with preemption and minimizing the average completion time

Abstract We consider an open-shop problem: n jobs have to be processed on m machines where each machine can process at most one job at a time and each job can be processed on at most one machine at a time. The processing times for all operations are given. Preemptions are allowed, that means each operation can be stopped and continued later on. The order of machines for job J i is called machine order of job J i , the order of jobs on machine M j is called job order on machine M j . In the case of an open-shop problem all machine orders and job orders have to be determined in such a way, that the average completion time is minimized. New models of scheduling problems with preemption are presented which base on models for problems without preemption. Because the considered problem is NP-hard, lower bounds and heuristics are developed. Computational experiments demonstrate the quality of the algorithms. For small numbers of jobs and machines the obtained results are also compared with the optimal solutions determined by an exact algorithm.

New steps in the amazing world of sequences and schedules

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timepi,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan.We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules.Furthermore, we investigate the problemO∥Cmax on an operation set with spanning tree structure.

On the set of solutions of the open shop problem

In the classical open shop problem, n jobs have to be processedon m machines, where both job orders and machine orders can be chosen arbitrarily.A feasible (i.e., acyclic) combination of all job orders and machine orders iscalled a (multi‐) sequence. We investigatea set of sequences which are structurally optimal in the sense that there is at least oneoptimal sequence in this set for each instance of processing times. Such sequences arecalled irreducible. Investigations about irreducible sequences are believed to provide apowerful tool to improve exact and heuristic algorithms. Furthermore, structural propertiesof sequences are important for problems with uncertain processing times.We prove necessary and sufficient conditions for the irreducibility of a sequence. Forseveral values of n and m, we give the numbers of allsequences, of the sequences satisfying each of these conditions and of the irreduciblesequences, respectively. It turns out that only a very small fraction of all sequences isirreducible. Thus, algorithms which work only on the set of irreducible sequences insteadof the set of all sequences can potentially perform much better than conventional algorithms.

A polynomial algorithm for an open shop problem with unit processing times and tree constraints

In this paper we consider the open shop problem with unit processing times and tree constraints (outtree) between the jobs. The complexity of this problem was open. We present a polynomial algorithm which decomposes the problem into subproblems by means of the occurrence of unavoidable idle time. Each subproblem can be solved on the base of an algorithm for the corresponding open shop problem without tree constraints.

A polynomial algorithm for the [n/m/0, tij = 1, tree/Cmax] open shop problem

Abstract In this paper we consider the open shop problem with unit processing times and tree constraints among the jobs. The objective is to mimimize the schedule length C max . The complexity of this problem was open. We present a polynomial algorithm which decomposes the problem into subproblems by means of the occurrence of unavoidable idel times. We consider two types to subproblems which can be solved by constructing special latin rectangles.

Using Simulated Annealing for Open Shop Scheduling with Sum Criteria

In this chapter, we consider the open shop scheduling problem which can be described as follows. A set of n jobs J1, J2, . . . , Jn has to be processed on a set of m machines M1,M2, . . . , Mm. The processing of job Ji on machine Mj is denoted as operation (i, j), and the sequence in which the operations of a job are processed on the machines is arbitrary. Moreover, each machine can process at most one job at a time and each job can be processed on at most one machine at a time. Such an open shop environment arises in many industrial applications. For example, consider a large aircraft garage with specialized work-centers. An airplane may require repairs on its engine and electrical circuit system. These two tasks may be carried out in any order but it is not possible to do these tasks on the same plane simultaneously. Further applications of open shop scheduling problems in automobile repair, quality control centers, semiconductor manufacturing, teacher-class assignments, examination scheduling, and satellite communications are described by Kubiak et al. (1991), Liu and Bulfin (1987) and Prins (1994). For each job Ji, i = 1, 2, . . . , n, there may be given a release date ri ≥ 0 which is the earliest possible time when the first operation of this job may start, a weight wi and a due date di ≥ 0 by which the job should be completed. The processing time of operation (i, j) is denoted as tij. It is assumed that the processing times of all operations are assumed to be given in advance. Let Ci be the completion time of job Ji, i.e. the time when the last operation of this job is completed. Traditional optimization criteria are basically partitioned into two types: either