Stork, Frederik

Uetz, Marc

English

Research Papers in Economics

FACHBEREICH 3
MATHEMATIK
Resource-Constrained
Project Scheduling: From a
Lagrangian Relaxation to
Competitive Solutions
by
Frederik Stork Marc Uetz
No. 661/2000Resource-Constrained Project Scheduling: From a
Lagrangian Relaxation to Competitive Solutions
Frederik Stork¤ Marc Uetz¤
February 7, 2000
Abstract
List scheduling belongs to the classical and widely used algorithms for schedul-
ing problems, but for resource-constrained project scheduling problems most
standard priority lists do not capture enough of the problem structure, often
resulting in poor performance. We use a well-known Lagrangian relaxation to
¯rst compute schedules which do not necessarily respect the resource constraints.
We then apply list scheduling in the order of so-called ®-completion times of jobs.
Embedded into a standard subgradient optimization, our computational results
show that the schedules compare to those obtained by state-of-the-art local search
algorithms. In contrast to purely primal heuristics, however, the Lagrangian re-
laxation also provides powerful lower bounds, thus the deviation between lower
and upper bounds can be drastically reduced by this approach.
1 Introduction
Resource-constrained project scheduling. Within resource-constrained project
scheduling, jobs have to be scheduled subject to both temporal and resource constraints
in order to minimize a given objective function. Temporal constraints are given by
precedence relations between pairs of jobs indicating that the start of a job must not
occur before the completion of its predecessors. While in process, every job requires
a certain amount of renewable resources (e.g., machines and/or personnel), and the
availability of these resources is limited. A schedule is called time- (resource-) feasible if
it respects the temporal (resource) constraints. The objective addressed in this abstract
is to ¯nd a time- and resource-feasible schedule minimizing the project makespan, which
is the time required to complete all jobs. (PSjprecjCmax in the notation of [2].)
¤Technische UniversitÄ at Berlin, Fachbereich Mathematik, Sekr. MA 6{1, Stra¼e des 17. Juni 136,
D-10623 Berlin, Germany. Email: fstork, uetzg@math.tu{berlin.de
1From a Lagrangian relaxation to a series of infeasible schedules. In order to
compute lower bounds on the optimal objective value for resource-constrained project
scheduling problems, Christo¯des et al.[3] proposed a Lagrangian relaxation where
resource constraints are dualized, and the corresponding Lagrangian multiplier problem
is solved by subgradient optimization. It has recently been shown by MÄ ohring et al.[9]
that each subproblem which has to be solved within the subgradient optimization is
equivalent to a minimum-cut problem in a simple auxiliary graph, and can thus be
solved e±ciently. What is important for the thread of the present paper is that in each
iteration of the subgradient optimization procedure, a time-feasible but not necessarily
resource-feasible schedule is computed. Moreover, during its course the subgradient
optimization tends to reduce these resource infeasibilities. The intuition behind our
approach is to exploit the series of these resource-infeasible schedules in order to obtain
priority lists which capture a fair amount of the problem structure.
2 From infeasible to feasible schedules
List scheduling. In the literature, two list scheduling algorithms are distinguished,
the parallel and the serial scheme. In both cases a priority list L of the jobs is given
which determines the order in which the jobs are considered. A job is usually called
available at a time t if all its predecessors have already been completed by t.
² Parallel List Scheduling proceeds over time (starting at time t = 0). At any
time t, as many available jobs as possible are scheduled according to the order
given by L. If no more job can be scheduled at time t, t is augmented to the next
completion time of a job.
² Serial List Scheduling proceeds job by job. In the order given by L, each job
is scheduled as early as possible with respect to the jobs scheduled so far. To
make sure that each job is available at the time it is considered, the priority list
has to be compatible with the precedence constraints.
List scheduling by ®-completion times. One possible way of computing a feasible
schedule on the basis of a resource-infeasible one is to apply list scheduling in order
of non-decreasing start times of jobs. However, motivated by several approximation
results in machine scheduling which make use of so-called ®-points of jobs [10, 5, 4],
this approach can be re¯ned as follows. Let a time-feasible but resource-infeasible
schedule S =( S1 :::;S n) be given, where n is the number of jobs and Sj denotes the
start time of job j. Then, for 0 · ® · 1, let Sj + ®p j be the ®-completion time of job
j, where pj denotes the processing time of job j. For a given ® (and a given schedule
S), the ordering according to non-decreasing ®-completion times of the jobs can now
be used as a priority list for parallel or serial list scheduling. Note that, since the
original schedule S was time-feasible, the ordering is compatible with the precedence
constraints for all values of ®. It is not di±cult to see that the number of di®erent
2priority lists one can obtain using di®erent values of ® is not more than n ¢ m, where
n is the number of jobs and m is the maximum number of jobs processed in parallel in
schedule S. We call these values of ® representative.
The algorithm. In each iteration of the subgradient optimization algorithm, a time-
feasible schedule is computed (by solving a minimum cut problem; see [9]). Using the
priority lists obtained as orderings according to ®-completion times for all representa-
tive values of ®, we apply both parallel and serial list scheduling. As a folklore trick,
we also apply parallel list scheduling backwards, i.e., we schedule the jobs in decreasing
time according to the reverse order of ®-completion times. We ¯nally output the best
schedule found.
3 Computational results
Test set. We have conducted experiments using the well known ProGen test set [8],
consisting of instances with 60, 90, and 120 jobs (480, 480, and 600 instances, respec-
tively). In order to exclude \trivial" instances, we ¯rst computed feasible schedules
using a set of 10 standard priority lists from the literature (see, e.g., [7]). We consid-
ered only those instances where the minimal makespan obtained with these priority
lists was above the critical path lower bound. The maximal number of iterations in the
subgradient optimization was set to 50, which means that we have evaluated maximally
50 infeasible schedules per instance. The number of representative values of ® (i.e., the
number of di®erent priority lists) for each of these schedules was roughly n=4.
Results. The ¯rst two columns of Table 1 show the number of jobs per instance
(jobs), and the number of instances we considered (inst.). The next columns display
the average and maximum computation times per instance in seconds (CPU), and
the average number of iterations in the subgradient optimization (it.). Note that
the computation time refers to the whole subgradient optimization procedure. We
further display the average deviations of the obtained solutions from the critical path
lower bound (dv.LBcp), the lower bound obtained with the Lagrangian relaxation
(dv.LBlg), and the best known solutions (dv.UBbest). The latter are maintained in
[1], and have been obtained by di®erent, partly time-intensive algorithms including
branch-and-bound as well as various local search procedures. Finally, we display the
number of instances that have been solved optimally with our procedure by computing
matching lower and upper bounds (opt.), as well as the number of instances where a
solution was found which matches the currently best known solution (best).
Conclusions. The quality of the solutions we obtain by using Lagrangian relaxation,
resource-infeasible schedules and list scheduling according to ®-completion times com-
pares to the most powerful heuristic algorithms that have been recently summarized
in [7] and [6]. The solutions, particularly the quality of the lower bounds, can be fur-
ther improved by allowing more iterations within the subgradient optimization, clearly
at the expense of higher computation times. Most important, however, is that the
3jobs inst. av.CPU mx.CPU it. dv.LBcp dv.LBlg dv.UBbest opt. best
60 266 5.8 18.4 39 22.6 % 12.1 % 2.2 % 88 129
90 253 14.0 39.5 39 22.7 % 11.5 % 2.3 % 102 119
120 563 33.7 147.2 42 38.9 % 16.3 % 3.1 % 107 134
Table 1: Results obtained on a Sun Ultrasparc with 200 MHz clock-pulse. The code
was compiled with the EGCS C++ compiler v. 1.1.2 running under Solaris.
approach provides powerful lower and upper bounds at the same time. Compared to
purely primal heuristics which rely on the critical path lower bound only, the deviation
between lower and upper bounds can thus be drastically reduced.
References
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5Reports from the group
\Combinatorial Optimization and Graph
Algorithms"
of the Department of Mathematics, TU Berlin
661/2000 Frederik Stork and Marc Uetz: Resource-Constrained Project Scheduling:
From a Lagrangian Relaxation to Competitive Solutions
658/1999 Olaf Jahn, Rolf H. MÄ ohring, and Andreas S. Schulz: Optimal Routing of
Tra±c Flows with Length Restrictions in Networks with Congestion
655/1999 Michel X. Goemans and Martin Skutella: Cooperative facility location
games
654/1999 Michel X. Goemans, Maurice Queyranne, Andreas S. Schulz, Martin
Skutella, and Yaoguang Wang: Single Machine Scheduling with Release Dates
653/1999 Andreas S. Schulz and Martin Skutella: Scheduling unrelated machines by
randomized rounding
646/1999 Rolf H. MÄ ohring, Martin Skutella, and Frederik Stork: Forcing Relations
for AND/OR Precedence Constraints
640/1999 Foto Afrati, Evripidis Bampis, Chandra Chekuri, David Karger, Claire
Kenyon, Sanjeev Khanna, Ioannis Milis, Maurice Queyranne, Martin Skutella,
Cli® Stein, and Maxim Sviridenko: Approximation Schemes for Minimizing Av-
erage Weighted Completion Time with Release Dates
639/1999 Andreas S. Schulz and Martin Skutella: The Power of ®-Points in Preemp-
tive Single Machine Scheduling
634/1999 Karsten Weihe, Ulrik Brandes, Annegret Liebers, Matthias MÄ uller{
Hannemann, Dorothea Wagner and Thomas Willhalm: Empirical Design of Ge-
ometric Algorithms
633/1999 Matthias MÄ uller{Hannemann and Karsten Weihe: On the Discrete Core
of Quadrilateral Mesh Re¯nement
632/1999 Matthias MÄ uller{Hannemann: Shelling Hexahedral Complexes for Mesh
Generation in CAD631/1999 Matthias MÄ uller{Hannemann and Alexander Schwartz: Implementing
Weighted b-Matching Algorithms: Insights from a Computational Study
629/1999 Martin Skutella: Convex Quadratic Programming Relaxations for Network
Scheduling Problems
628/1999 Martin Skutella and Gerhard J. Woeginger: A PTAS for minimizing the
total weighted completion time on identical parallel machines
627/1998 Jens Gustedt: Specifying Characteristics of Digital Filters with FilterPro
620/1998 Rolf H. MÄ ohring, Andreas S. Schulz, Frederik Stork, and Marc Uetz: Re-
source Constrained Project Scheduling: Computing Lower Bounds by Solving
Minimum Cut Problems
619/1998 Rolf H. MÄ ohring, Martin Oellrich, and Andreas S. Schulz: E±cient Algo-
rithms for the Minimum-Cost Embedding of Reliable Virtual Private Networks
into Telecommunication Networks
618/1998 Friedrich Eisenbrand and Andreas S. Schulz: Bounds on the Chv¶ atal Rank
of Polytopes in the 0/1-Cube
617/1998 Andreas S. Schulz and Robert Weismantel: An Oracle-Polynomial Time
Augmentation Algorithm for Integer Proramming
616/1998 Alexander Bockmayr, Friedrich Eisenbrand, Mark Hartmann, and Andreas
S. Schulz: On the Chv¶ atal Rank of Polytopes in the 0/1 Cube
615/1998 Ekkehard KÄ ohler and Matthias Kriesell: Edge-Dominating Trails in AT-free
Graphs
613/1998 Frederik Stork: A branch and bound algorithm for minimizing expected
makespan in stochastic project networks with resource constraints
612/1998 Rolf H. MÄ ohring and Frederik Stork: Linear preselective policies for stochas-
tic project scheduling
609/1998 Arfst Ludwig, Rolf H. MÄ ohring, and Frederik Stork: A computational study
on bounding the makespan distribution in stochastic project networks
605/1998 Friedrich Eisenbrand: A Note on the Membership Problem for the Elemen-
tary Closure of a Polyhedron
596/1998 Andreas Fest, Rolf H. MÄ ohring, Frederik Stork, and Marc Uetz: Resource
Constrained Project Scheduling with Time Windows: A Branching Scheme Based
on Dynamic Release Dates
595/1998 Rolf H. MÄ ohring Andreas S. Schulz, and Marc Uetz: Approximation in
Stochastic Scheduling: The Power of LP-based Priority Policies591/1998 Matthias MÄ uller{Hannemann and Alexander Schwartz: Implementing
Weighted b-Matching Algorithms: Towards a Flexible Software Design
590/1998 Stefan Felsner and Jens Gustedt and Michel Morvan: Interval Reductions
and Extensions of Orders: Bijections to Chains in Lattices
584/1998 Alix Munier, Maurice Queyranne, and Andreas S. Schulz: Approximation
Bounds for a General Class of Precedence Constrained Parallel Machine Schedul-
ing Problems
577/1998 Martin Skutella: Semide¯nite Relaxations for Parallel Machine Scheduling
Reports may be requested from: Hannelore Vogt-MÄ oller
Fachbereich Mathematik, MA 6{1
TU Berlin
Stra¼e des 17. Juni 136
D-10623 Berlin { Germany
e-mail: moeller@math.TU-Berlin.DE
Reports are also available in various formats from
http://www.math.tu-berlin.de/coga/publications/techreports/
and via anonymous ftp as
ftp://ftp.math.tu-berlin.de/pub/Preprints/combi/Report-number-year.ps

[1] ftp://ftp.bwl.uni-kiel.de/pub/operations-research/psplib/HTML/ index.html,

Heuristic algorithms for solving the resourceconstrained project scheduling problem: Classi¯cation and computational analysis,

Improved approximation algorithms for scheduling with release dates,

Minimizing average completion time in the presence of release dates,

Project scheduling with resource constraints: A branch and bound approach,

PSPLIB - A project scheduling problem library,

Resource-constrained project scheduling: Notation, classi¯cation, models, and methods,

Resourceconstrained project scheduling: Computing lower bounds by solving minimum cut problems,

Scheduling to minimize average completion time: O®-line and on-line algorithms,

Self-adapting genetic algorithms with an application to project scheduling,

Resource-constrained project scheduling: from a lagrangian relaxation to competitive solutions.

Resource-constrained project scheduling: from a lagrangian relaxation to competitive solutions.

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