Order-based neighborhoods for project scheduling
with nonregular objective functions
K. Neumann *, C. Schwindt, J. Zimmermann
Institut f€ur Wirtschaftstheorie und Operations Research, University of Karlsruhe, 76128 Karlsruhe, Germany
Abstract
This paper is concerned with project scheduling where scarce resources are taken into account and some nonregular
objective function is to be minimized. The activities of the project are linked by general temporal constraints. For a
variety of nonregular objective functions, the search for optimal solutions can be restricted to special types of schedules
which correspond to specific points of the feasible region. Each of those schedules can be associated with some strict
order in the activity set. We study three neighborhoods on the set of spanning forests and spanning trees, respectively,
of order networks arising from such strict orders in the activity set. The first neighborhood is dedicated to objective
functions for which any local minimizer on an order-induced subset of the feasible region is a global minimizer on this
subset as well. The second and the third neighborhoods cope with the case of objective functions which are minimized
by vertices and minimal points, respectively, of order-induced subsets of the feasible region. For all neighborhoods,
weak optimal connectivity of the corresponding neighborhood graph is shown.

2002 Elsevier Science B.V. All rights reserved.
Keywords: Project scheduling; Resource allocation; General temporal constraints; Nonregular objective functions; Order-based
neighborhoods
1. Introduction
We deal with resource allocation problems in
project scheduling where the temporal constraints
are given by minimum and maximum time lags
between project activities and some nonregular
objective function is to be minimized. For such
problems with special nonregular objective func-
tions of types net present value, resource invest-
ment, and resource levelling, some exact and
heuristic solution procedures have been proposed.
For the net present value objective, we refer to De
Reyck and Herroelen (1998), Herroelen et al.
(1997), Icmeli and Ereng€uc
(1994, 1996), Neu-
mann and Zimmermann (2000, 2003), Russell
(1970), Schwindt and Zimmermann (2001), and
Zimmermann (2001). Methods for resource in-
vestment and resource levelling problems have
been devised by Bandelloni et al. (1994), Demeu-
lemeester (1995), Easa (1989), Neumann and
Zimmermann (1999a,b, 2000), and N€ubel (1999).
A first structural analysis of the feasible region
S of resource-constrained project scheduling
*Corresponding author. Tel.: +49-721-608-3807; fax: +49-
721-608-6616.
E-mail addresses: neumann@wior.uni-karlsruhe.de (K. Neu-
mann), schwindt@wior.uni-karlsruhe.de (C. Schwindt), juergen.
zimmermann@tu-clausthal.de (J. Zimmermann).
0377-2217/03/$ - see front matter
2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0377-2217(02)00765-8
European Journal of Operational Research 149 (2003) 325–343
www.elsevier.com/locate/dsw

problems with general temporal constraints pro-
viding an order-based characterization of S has
been offered by Bartusch et al. (1988). In Neumann
et al. (2000) it has been shown that special points
of S (for example, minimal or extreme points of
S or of subsets of S induced by strict orders in
the set of activities) represent possible optimal
solutions to project scheduling problems with
regular or specific nonregular objective functions.
This paper is based upon the latter results. In
Section 2, we give a precise formulation of the
basic project scheduling problem. In Section 3, we
summarize some structural properties of the fea-
sible region S, strict orders in the set of activities,
and corresponding order networks. Some of those
results can already be found in Bartusch et al.
(1988) and Neumann et al. (2000). Special points
of S and corresponding types of schedules which
represent possible optimal solutions to project
scheduling problems with specific nonregular ob-
jective functions are discussed in Section 4. In
Section 5, we show that the different types of
schedules introduced can be represented by span-
ning forests of order networks. Sections 6 and 7
are concerned with different neighborhoods on the
set of spanning forests associated with different
schedule types. Neighborhoods constitute the es-
sential building block of local search algorithms
such as hill climbing, tabu search, simulated an-
nealing, or threshold accepting. It is shown that
the proposed neighborhoods allow local search
algorithms to reach optimal solutions indepen-
dently of the (possibly even infeasible) initial so-
lution chosen, where appropriate schedule sets are
explored.
In project scheduling literature, various types of
neighborhoods have been discussed. A great deal
of effort has been devoted to the project scheduling
problem where the project duration is to be mini-
mized subject to renewable-resource constraints
and precedence relationships among activities.
Neighborhoods are based on representations of
schedules. Following the classification of Kolisch
and Hartmann (1999), we distinguish between the
activity list representation, random key repre-
sentation, shift vector representation, schedule
scheme representation, and start time vector rep-
resentation of schedules. Table 1 classifies recent
references discussing local search algorithms for
different resource-constrained project scheduling
problems according to the schedule representation
used and objective function considered. Bold-face
entries indicate that the corresponding neighbor-
hood is either explicitly dedicated to the case of
general temporal constraints (Franck et al., 2001;
Neumann and Zimmermann, 2000) or can be
adapted to cope with this type of constraints (Baar
et al., 1998; Icmeli and Ereng€uc
, 1994). Though
the papers by Icmeli and Ereng€uc
(1994) and Zhu
and Padman (1999) are concerned with the net
present value problem, the proposed neighbor-
hoods are suited for dealing with general objective
functions as well.
Sampson and Weiss (1993) discuss a neigh-
borhood which is based on modifying shift vec-
tors. For each activity, a shift vector specifies the
Table 1
Local search algorithms from literature classified according to schedule representation and objective function
Representation Objective function
Project duration Net present value Levelling General
Activity list Baar et al. (1998), Bouleimen and
Lecocq (1998), Cho and Kim
(1997), Franck et al. (2001),
Pinson et al. (1994)
Mayer (1998),
Yang et al. (1995)
Random key Cho and Kim (1997), Lee and
Kim (1996), Leon and Balakrish-
nan (1995), Naphade et al. (1997)
Shift vector Sampson and Weiss (1993)
Schedule scheme Baar et al. (1998)
Start time vector Neumann and Zim-
mermann (2000)
Icmeli and Ereng€uc¸ (1994),
Zhu and Padman (1999)
326 K. Neumann et al. / European Journal of Operational Research 149 (2003) 325–343