Peter Brucker

Resource-constrained project scheduling : Notation, classification, models, and methods

Abstract Project scheduling is concerned with single-item or small batch production where scarce resources have to be allocated to dependent activities over time. Applications can be found in diverse industries such as construction engineering, software development, etc. Also, project scheduling is increasingly important for make-to-order companies where the capacities have been cut down in order to meet lean management concepts. Likewise, project scheduling is very attractive for researchers, because the models in this area are rich and, hence, difficult to solve. For instance, the resource-constrained project scheduling problem contains the job shop scheduling problem as a special case. So far, no classification scheme exists which is compatible with what is commonly accepted in machine scheduling. Also, a variety of symbols are used by project scheduling researchers in order to denote one and the same subject. Hence, there is a gap between machine scheduling on the one hand and project scheduling on the other with respect to both, viz. a common notation and a classification scheme. As a matter of fact, in project scheduling, an ever growing number of papers is going to be published and it becomes more and more difficult for the scientific community to keep track of what is really new and relevant. One purpose of our paper is to close this gap. That is, we provide a classification scheme, i.e. a description of the resource environment, the activity characteristics, and the objective function, respectively, which is compatible with machine scheduling and which allows to classify the most important models dealt with so far. Also, we propose a unifying notation. The second purpose of this paper is to review some of the recent developments. More specifically, we review exact and heuristic algorithms for the single-mode and the multi-mode case, for the time–cost tradeoff problem, for problems with minimum and maximum time lags, for problems with other objectives than makespan minimization and, last but not least, for problems with stochastic activity durations.

A branch and bound algorithm for the job-shop scheduling problem

Abstract A fast branch and bound algorithm for the job-shop scheduling problem has been developed. Among other hard problems it solves the 10 × 10 benchmark problem which has been open for more than 20 years. We will give a complete description of this algorithm and will present computational results.

Scheduling a batching machine

textabstractWe study the problem of scheduling a chain-reentrant shop, in which each job goes for its processing first to a machine called the primary machine, then to a number of other machines in a fixed sequence, and finally back to the primary machine for its last operation. The problem is to schedule the jobs so as to minimize the makespan. This problem is unary NP-hard for a general number of machines. We focus in particular on the two-machine case that is also at least binary NP-hard. We prove some properties that identify a specific class of optimal schedules, and then use these properties in designing an approximation algorithm and a branch-and-bound type optimization algorithm. The approximation algorithm, of which we present three versions, has a worst-case performance guarantee of f32 along with an excellent empirical performance. The optimization algorithm solves large instances quickly. Finally, we identify a few well solvable special cases and present a pseudo-polynomial algorithm for the case in which the first and the last operations of any job (on the primary machine) are identical.

Job-shop scheduling with multi-purpose machines

Consider the following generalization of the classical job-shop scheduling problem in which a set of machines is associated with each operation of a job. The operation can be processed on any of the machines in this set. For each assignment μ of operations to machines letP(μ) be the corresponding job-shop problem andf(μ) be the minimum makespan ofP(μ). How to find an assignment which minimizesf(μ)? For problems with two jobs a polynomial algorithm is derived.ZusammenfassungFolgende Verallgemeinerung des klassischen Job-Shop Scheduling Problems wird untersucht. Jeder Operation eines Jobs sei eine Menge von Maschinen zugeordnet. Wählt man für jede Operation genau eine Maschine aus dieser Menge aus, so erhält man ein klassisches Job-Shop Problem, dessen minimale Gesamtbearbeitungszeitf(μ) von dieser Zuordnung μ abhängt. Gesucht ist eine Zuordnung μ, dief(μ) minimiert. Für zwei Jobs wird ein polynomialer Algorithmus entwickelt, der dieses Problem löst.

A branch and bound algorithm for the resource-constrained project scheduling problem

A branch and bound algorithm is presented for the resource-constrained project scheduling problem (RCPSP). Given are n activities which have to be processed without preemptions. During the processing period of an activity constant amounts of renewable resources are needed where the available capacity of each resource type is limited. Furthermore, finish–start precedence relations between the activities are given. The objective is to determine a schedule with minimal makespan. The branching scheme starts from a graph representing a set of conjunctions (the classical finish–start precedence constraints) and disjunctions (induced by the resource constraints). Then it either introduces disjunctive constraints between pairs of activities or places these activities in parallel. Concepts of immediate selection are developed in connection with this branching scheme. Immediate Selection allows to add conjunctions as well as further disjunctions and parallelity relations. Computational results based on the test data of Kolisch et al. (Kolisch, R., Sprecher, A., Drexl, A., 1995. Management Science 41, 1693–1703.) and Kolisch and Sprecher (Kolisch, R., Sprecher, A., 1997. PSPLIB – A project scheduling problem library, EJOR 96, pp. 205–216.) are reported.

Review of recent development: An O( n) algorithm for quadratic knapsack problems

An algorithm is presented which solves bounded quadratic optimization problems with n variables and one linear constraint in at most O(n) steps. The algorithm is based on a parametric approach combined with well-known ideas for constructing efficient algorithms. It improves an O(n log n) algorithm which has been developed for a more restricted case of the problem.

A branch & bound method for the general-shop problem with sequence dependent setup-times

A branch & bound algorithm is presented for a very general scheduling problem withn jobs andm machines. Each job consists of a set of operations. Each operation has to be processed on a dedicated machine. There may be arbitrary precedence relations between the operations. The set of all operations is partitioned into groups. If on a machine an operation belonging to groupGg is processed immediately after an operation belonging to groupGf there is a setup ofsfg time units. We assume thatsfg=0 iff=g and that thesfg satisfy the triangle inequality. Computational results for this general problem as well as for special cases like the job-shop problem and the open-shop problem are reported.ZusammenfassungEs wird ein Branch & Bound-Algorithmus für ein sehr allgemeines Scheduling Problem mitn Jobs undm Maschinen vorgestellt. Jeder Job besteht aus einer Menge von Operationen, die auf einer ausgewählten Maschine bearbeitet werden müssen. Zwischen den Operationen sind beliebige Vorrangbeziehungen möglich. Ferner werden die Operationen in Gruppen aufgeteilt. Wenn auf einer Maschine eine Operation der GruppeGg unmittelbar nach einer Operation der GruppeGf bearbeitet wird, ist eine Rüstzeit vonsfg Zeiteinheiten notwendig. Wir setzen voraus, daßsfg=0 fürf=g und daß diesfg die Dreiecksungleichung erfüllen. Sowohl für dieses allgemeine Problem als auch für Spezialfälle wie das Job-Shop Problem und das Open-Shop Problem werden Rechenergebnisse vorgestellt.

A linear programming and constraint propagation-based lower bound for the RCPSP

Abstract A new destructive lower bound for the resource-constrained project scheduling problem (RCPSP) is presented. Given are n activities which have to be processed without preemptions. During the processing period of an activity constant amounts of renewable resources are needed where the available capacity of each resource type is limited. Furthermore, finish–start precedence relations between the activities are given. The objective is to determine a schedule with minimal makespan. The lower bound calculations are based on two methods for proving infeasibility of a given threshold value T for the makespan. The first uses constraint propagation techniques, while the second is based on a linear programming formulation. A column generation procedure is presented for the linear programming formulation and computational results are reported for an algorithm combining both concepts.

A branch & bound algorithm for the open-shop problem

A fast branch & bound method for the open-shop problem based on a disjunctive graph formulation of the problem is developed. Computational results show that the method yields excellent results. Some benchmark problems from the literature were solved to optimality for the first time.

Tabu Search Algorithms and Lower Bounds for the Resource-Constrained Project Scheduling Problem

We present two tabu search algorithms for the resource-constrained project scheduling problem. Given are n activities which have to be processed without preemptions. During the processing period of an activity constant amounts of renewable resources are needed where the available capacity of each resource type is limited. Furthermore, finish-start precedence relations between the activities are given. The objective is to determine a schedule with minimal makespan. The first tabu search approach relies on elimination of critical arcs and list-scheduling techniques. The second neighborhood is based on schedule schemes, where neighbors are generated by placing activities in parallel or deleting a parallelity relation. Furthermore, a column-generation approach for a linear programming-based lower bound is presented and computational results are reported.