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European Journal of Operational Research 152 (2004) 1–13
www.elsevier.com/locate/dsw
Invited Review
A concise survey of scheduling with time-dependentprocessing times
T.C.E. Cheng a,*, Q. Ding a, B.M.T. Lin b
a Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kongb Department of Information Management, National Chi Nan University, Pu-Li, Nan-Tou County, Taiwan
Received 22 April 2002; accepted 4 December 2002
Abstract
We consider a class of machine scheduling problems in which the processing time of a task is dependent on its
starting time in a schedule. On reviewing the literature on this topic, we provide a framework to illustrate how models
for this class of problems have been generalized from the classical scheduling theory. A complexity boundary is pre-
sented for each model and related existing results are consolidated. We also introduce some enumerative solution al-
gorithms and heuristics and analyze their performance. Finally, we suggest a few interesting areas for future research.
� 2003 Elsevier B.V. All rights reserved.
Keywords: Survey; Scheduling; Sequencing; Time dependence; Computational complexity
1. Introduction
Machine scheduling problems with time-de-
pendent processing times have received increas-
ing attention in recent years. For many years,
most scheduling research has focused on problems
with deterministic parameters. As mentioned in
Rinnooy Kan (1976), the traditional restrictiveassumptions may correspond to a somewhat over-
simplified picture of reality, though they can take
great advantage of computational convenience. In
real-life applications, many systems exhibit dy-
* Corresponding author. Tel.: +852-2766-5215; fax: +852-
2364-5245.
E-mail addresses: [email protected] (T.C.E. Cheng),
[email protected] (B.M.T. Lin).
0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv
doi:10.1016/S0377-2217(02)00909-8
namic behaviors characterized by a set of dynamic
parameters. This fact is commonly recognized in
control theory, systems engineering and many
other areas. But scheduling problems with dy-
namic parameters have been studied only in a few
papers. It should, however, be noted that a con-
siderable body of literature has existed for sto-
chastic scheduling that deals with schedulingproblems in an environment of uncertainty, see,
Mohring and Rademacher (1985), Righter (1994)
and Pinedo (1995).
Based on some scheduling problems with dy-
namic parameters considered by Gupta et al.
(1987) or some earlier Russian papers (e.g. Tanaev
et al., 1994), Gupta and Gupta (1988) introduced
an interesting scheduling model in which the pro-cessing time of a task is a polynomial function of
its starting time. From a modeling perspective,
ed.
2 T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13
however, the makespan scheduling problem withquadratic time-dependent processing times is al-
ready very intricate. For this reason, most subse-
quent research along this line has concentrated on
problems with linear or piecewise linear time-
dependent processing times.
This model reflects some real-life situations in
which the expected processing time of a task in-
creases/decreases linearly or piecewise linearly withits starting time. Examples can be found in finan-
cial management, steel production, resource allo-
cation and national defense, where any delay in
tackling a task may result in an increasing/de-
creasing effort (time, cost, etc.) to accomplish the
task. The reader is referred to Kunnathur and
Gupta (1990), Mosheiov (1994, 1996a) and Sun-
dararaghavan and Kunnathur (1994) for a list ofapplications. Moreover, it seems that in other
cases, for example, fire fighting, learning effect and
maintenance scheduling, a linear or piecewise lin-
ear function is a close approximation of the actual
nonlinear phenomenon.
Research on time-dependent problems has
spawned a new area in the scheduling field. It has
uncovered many new properties neglected in theclassical scheduling theory and led to efficient
methodological approaches to algorithm design
and NP-complete reduction. For example, tech-
niques based on reductions from a multiplicative
type NP-complete problem, such as SUBSETSUBSET
PRODUCTPRODUCT, are crucial to the NP-completeness
proofs for many time-dependent scheduling prob-
lems. Regarding the development of polynomialtime algorithms, a very interesting phenomenon is
related to the existence of algorithms with time
complexities of Oðn5Þ or Oðn6 log nÞ, which is notso common in the deterministic scheduling litera-
ture. Thus, research on these problems is signifi-
cant in both practical and theoretical senses.
Alidaee and Womer (1999) presented a review
on scheduling problems with time-dependent pro-cessing times. Our study aims not only at survey-
ing recent developments in this line of research but
also at investigating several unsolved problems.
Based upon state-of-the-art status of research on
scheduling problems with time-dependent pro-
cessing times, we discuss the relationships of dif-
ferent models and explicate how they are
generated from a basic linear model. A complexityboundary is presented for each model and existing
and new results are consolidated. For the intrac-
table problems, we also introduce some enumera-
tive solution algorithms and heuristics and analyze
their performance. Finally, we give some insights
into scheduling problems of this type, which reveal
several potential future directions of research for
this exciting field of study.In Section 2, we introduce a notation-and-
model system for the scheduling problem with
time-dependent processing times. In Section 3, we
present a set of complexity results for each model.
In Section 4, we illustrate a series of polynomial
and pseudo-polynomial algorithms. In Section 5,
we discuss some enumerative and heuristic solu-
tion algorithms in the literature. Some concludingremarks and suggestions for future research are
given in Section 6.
2. Notation and models
Since most of the time-dependent scheduling
problems are a natural generalization of their
classical counterpart, we adopt the notation, defi-
nitions and assumptions prevalent in classical
scheduling theory, see the survey of Graham et al.
(1976).Research on time-dependent problems has
mainly dealt with the single machine model, with
only a few exceptions dealing with the parallel
machine and flow shop situations. The objective
has been confined to the minimization of a handful
of traditional regular performance measures, such
as makespan, flow time, maximum lateness and
number of tardy tasks. The time-dependent func-tion used to model the processing time of a task is
usually a linear or piecewise linear function of the
starting time of the task in a schedule. We give
below a formal statement for the basic linear
model.
A task system consisting of n independent tasksis denoted by TS ¼ ðfTig; faig; fbig; frig; fdigÞ.Each task Ti is associated with a release time ri anda due date di, and is characterized by a normalprocessing time ai P 0 and a processing rate bi.The actual processing time of Ti depends on itsstart time si and is given by pi ¼ ai � bisi.
T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13 3
Gupta et al. (1987), Gupta and Gupta (1988),Browne and Yechiali (1990), Gawiejnowicz and
Pankowska (1995) and some earlier Russian pa-
pers (e.g. Tanaev et al., 1994) proposed the model
pi ¼ ai þ bisi from different perspectives. The
model reflects real-life situations, such as searching
for an object under worsening weather or perfor-
mance of medical treatments under deteriorating
health conditions, where any delay may incur extraefforts to accomplish the task (Mosheiov, 1994).
Motivated by a military application concerning
aerial treats, Ho et al. (1993) proposed the model
pi ¼ ai � bisi with deadline di. Some special casesof the model pi ¼ ai � bisi have also been investi-gated in the literature.
Mosheiov (1994) first considered the special
case with ai ¼ 0, i.e., the model pi ¼ bisi. For thecases with bi ¼ b or ai ¼ a, the problems with adeadline or release time restriction have been
studied by Cheng and Ding (1998a,b, 1999, 2000,
2003). Next we introduce some piecewise linear
models.
In some situations, if a task starts after a time
di, its processing time deteriorates with its startingtime. Note that the parameter di is not the classicaldeadline or due date, but a kind of deteriorating
(decreasing) break point. Practical examples arise
from jobbing production where jobs are produced
at a normal or overtime cost depending on whe-
ther the job is started before a specified time point,
i.e., the break point. Kunnathur and Gupta (1990)
proposed a model with piecewise increasing pro-
cessing times, denoted by pi ¼ maxfai; ai þbiðsi � diÞg. Sundararaghavan and Kunnathur
(1994) and Mosheiov (1995) independently intro-
duced another model with step deteriorating pro-
cessing times, denoted by pi ¼ ai or pi ¼ ai þ bi,where pi ¼ ai, if si 6 di; pi ¼ ai þ bi, otherwise.When the parameter di is considered, only themakespan and flow time problems have been
considered in the literature.To approximate the learning effect, Cheng et al.
(2003) studied a model with piecewise linear de-
creasing processing times, denoted by pi ¼ai � biminfsi; dig. In this model, if a task startsbefore or at di, its processing time decreases withits starting time linearly; otherwise, the decrease of
its processing time is a constant bidi. Bachman et al.
(2002b) investigated the model pi ¼ ai � bisi, illus-trating with an example where a worker gains
knowledge and skills when assembling a large
quantity of similar products.
For the models pi ¼ ai � bisi and pi ¼ ai �biminfsi; dig, if bi > 1, then some performancemeasures become nonregular. If ai < bidi, then theactual processing time may reduce to 0. To avoid
these unrealistic and uninteresting cases, we as-sume that 0 < bi < 1 and ai > bidi for these twomodels. For the case without the deadline restric-
tion, we assume ai > biP
aj. For the modelpi ¼ bisi, if si ¼ 0, then every actual processingtime is 0, a trivial case. Thus, we assume that si P efor the model pi ¼ bisi, where e is a given smallpositive number.
For the model pi ¼ ai or pi ¼ ai þ bi, all pa-rameters are assumed to be integers. For the other
models, the normal processing times ai are as-sumed to be integers. Since the deteriorating rates
bi are rational numbers in most practical cases andthe actual processing times are always greatly af-
fected by some exponential terms of the corre-
sponding deteriorating rates, the other parameters
are allowed to be positive rational numbers.Discussions and justifications for these as-
sumptions are provided in the remainder of this
paper when necessary. Table 1 presents a list of the
different models that have appeared in the litera-
ture, along with the corresponding references. Fig.
1 illustrates the time-dependent function of each
model.
3. NP-complete problems
The NP-complete results for the class of clas-
sical scheduling problems have been well re-
searched and surveyed in several papers and
books, for example, Rinnooy Kan (1976), Graham
et al. (1976) and Lawler et al. (1993). However, thetime-dependent scheduling problems present quite
different boundaries for the computational com-
plexities of the problems. In this section, we in-
troduce the new scheduling results, starting from
the simple models and gradually progressing to
their generalizations.
Table 1
Time-dependent scheduling models and references
Model References
pi ¼ ai � bisi Gupta et al. (1987), Gupta and Gupta (1988), Browne and Yechiali (1990), Mosheiov
(1991, 1994, 1995, 1996a, 2002), Ho et al. (1993), Woeginger (1995), Chen (1995, 1996),
Gawiejnowicz and Pankowska (1995), Kononov (1997); Cheng and Ding (1998a,b,
1999, 2000, 2003); Hsieh and Bricker (1997); Bachman and Janiak (1997, 2000),
Kononov and Gawiejnowicz (2001); Ng et al. (2002), Bachman et al. (2002a)
pi ¼ maxfai; ai þ biðsi � diÞg Kunnathur and Gupta (1990); Sundararaghavan and Kunnathur (1990), Cai et al.
(1998), Kubiak and van de Velde (1998)
pi ¼ ai or pi ¼ ai þ bI Sundararaghavan and Kunnathur (1994), Mosheiov (1995, 1998), Cheng and Ding
(2001), Jeng and Lin (2002)
pi ¼ ai � biminfsi; dig Cheng et al. (2003)
Fig. 1. Time-dependent functions.
4 T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13
In the literature, studies involving multiple
machines or shops center around the two models
pi ¼ bisi and pi ¼ ai þ bisi. For the model pi ¼ bisi,Mosheiov (1994, 1998), Chen (1996) and Mos-
heiov (1996a, 2002) reduced the SUBSET PRODUCTSUBSET PRODUCT
problem (Garey and Johnson, 1979) to P2jpi ¼bisijCmax, P2jpi ¼ bisij
PCi and F 3jpi ¼ bisijCmax
respectively, where the respective parameters were
explained at the beginning of Section 2 and the
three-field notation is adopted from Graham et al.
(1976). Kononov (1997) independently established
the ordinary NP-completeness of P2jpi ¼ bisijCmaxand P2jpi ¼ bisij
PCi. As mentioned in Chen
(1997), SUBSET PRODUCTSUBSET PRODUCT is NP-complete in the
ordinary sense, not strongly NP-complete as used
in the above three papers. See also Johnson (1981)for a correction of the complexity results of SUB-SUB-
SET PRODUCTSET PRODUCT. Thus, these three problems are
only NP-complete in the ordinary sense and the
respective original reductions need some modifi-
cation for their correctness. For the open shop and
job shop problem, Mosheiov (2002) showed that
O3jpi ¼ bisijCmax and J2jpi ¼ bisijCmax are NP-complete. The above results demonstrate a com-plexity structure similar to that of classical shop
floor scheduling problems where pi ¼ ai.For the model pi ¼ ai þ bisi on multiple
machines, Kononov and Gawiejnowicz (2001) pre-
sented a strong NP-completeness proof for F 3jpi ¼ai þ bisijCmax by a reduction from 3-PARTITIONPARTITION,
which is known to be strongly NP-complete
(Garey and Johnson, 1979). They also presented areduction from PARTITIONPARTITION for the model pi ¼ai þ bisi with two machines. Moreover, by a re-duction from SUBSET PRODUCTSUBSET PRODUCT, they showed that
makespan minimization on a three-machine open
shop with the model pi ¼ ai þ bisi remains NP-complete even if all jobs have the same deteriora-
tion rate on the third machine.
For the model pi ¼ ai � bisi with bi ¼ b and dion a single machine, Cheng and Ding (1998a) re-
duced PARTITIONPARTITION to 1jpi ¼ ai � bsi; di 2 fD1;D2gjCmax and Cheng and Ding (1999) reduced 3-PAR-PAR-
TITIONTITION to 1jpi ¼ ai � bsi; dijCmax. Naturally, thecorresponding maximum lateness problems are
T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13 5
also NP-complete and strongly NP-complete,respectively.
Given a schedule, one convenient approach to
analyze it is to interchange the positions of one
pair of adjacent tasks and compare the properties
(such as the makespan and flow time) of the
original and the new schedules. Such an approach
of analysis is called the adjacent jobs interchange
argument. The use of this approach to facilitate theNP-completeness analysis of a problem is outlined
in the following example, which reduces PARTI-PARTI-
TIONTION to 1jpi ¼ ai � bsi; di ¼ DjP
Ci.
Given an instance I of PARTITIONPARTITION with a list
H ¼ fh1; h2; . . . ; hng and B ¼ 12
Phi, construct an
instance II of 1jpi ¼ ai � bsi; di ¼ DjP
Ci, where
TS consists of 2n tasks fT1;1; T1;2; T2;1; T2;2; . . . ;Tn;1;Tn;2g, ai;1¼ i2Bþhi, ai;2¼ i2B and b¼1=29n3B.D and G are given bounds based on makespan andflow time formulas plus reasonable error toler-
ances, respectively. Defining Ri¼fTi;1; Ti;2g, we canshow that there exists a solution in the form
ðR1;...;RnÞ, if there exists a solution to II. Given aschedule in the form of ðR1;...RnÞ, only swap theorder of one pair of tasks Ri¼fTi;1;Ti;2g. From theadjacent jobs interchange argument, the makespanof the new schedule increases/decreases about bhi,while the flow time decreases/increases about hi.Thus, we obtain a situation where I has a solution
if and only if II has a solution. Details of this
approach can be found in the reduction for
1jp¼ai�biminfsi;DgjP
Ci presented in Cheng
et al. (2003). Thus, we obtain the following theo-
rem immediately.
Theorem 1. The problem 1jpi ¼ ai � bsi; di ¼ DjPCi is NP-complete.
For the model pi ¼ ai � bisi with bi ¼ b and ri,Cheng and Ding (1998b) showed that 1=pi ¼ai � bsi; ri ¼ 0;R=Cmax is NP-complete and 1=pi ¼ai � bsi; ri=Cmax is strongly NP-complete by re-ducing 1=pi ¼ ai � bsi; di ¼ D1;D2=Cmax and 1=pi ¼ai � bsi; di=Cmax to them, respectively.For the model pi ¼ ai þ bisi on a single ma-
chine, Kononov (1997) and Bachman and Janiak
(2000) independently showed that 1jpi ¼ ai þbisijLmax is NP-complete. For the model pi ¼ ai �bisi with ai ¼ a, using an approach similar to the
case with bi ¼ b, Cheng and Ding (2003) gavea reduction to show that 1jp ¼ 1� bisi; di 2 fD1;D2gjCmax is NP-complete and 1jpi ¼ 1� bisi; dijCmaxis strongly NP-complete. Applying the same ap-
proach, they showed that the three problems
1jpi ¼ 1þ bisi; di ¼ fD1;D2gjCmax, 1jpi ¼ 1þ bisi;ri ¼ f0;RgjCmax and 1jpi ¼ 1� bisi; ri ¼ f0;RgjCmax are NP-complete; and 1jpi ¼ 1þ bisi; dijCmax,1jpi ¼ 1þ bisi; rijCmax and 1jpi ¼ 1� bisi; rijCmaxare strongly NP-complete. Moreover, the corre-
sponding flow time, maximum lateness and num-
ber of tardy task problems are NP-complete in the
ordinary sense or strongly NP-complete. If no
deadline restriction is imposed, the complexities
of 1jpi ¼ 1þ bisijP
Ci and 1jpi ¼ 1� bisijP
Ci
are open, although they seem to be NP-complete.
For the general model pi ¼ ai þ bisi with di ¼ D,using the same approach in Theorem 1, we obtain
the following theorem.
Theorem 2. The problem 1jpi ¼ ai þ bisi; dijP
Ci isNP-complete.
Now we introduce the NP-completeness results
for the piecewise linear models. Generally, thecomplexity boundaries of these models are delin-
eated based on the cases with di ¼ D.For the model pi ¼ maxfai; ai þ biðsi � DÞg,
Kubiak and van de Velde (1998) showed that
1jpi ¼ maxfai; ai þ biðsi � DÞgjCmax is NP-com-
plete in the ordinary sense. For the model pi ¼ aior pi ¼ ai þ bi, Mosheiov (1995) and Cheng andDing (2001) showed that 1jpi ¼ ai or pi ¼ ai þbi; di ¼ DjCmax is NP-complete in the ordinarysense. Jeng and Lin (2002) proposed a pseudo-
polynomial time dynamic programming algorithm
and a branch-and-bound algorithm for 1jpi ¼ aior pi ¼ ai þ bi; dijCmax.For the corresponding flow time problems,
Cheng and Ding (2001) reduced PARTITIONPARTITION to
1jpi ¼ ai or pi ¼ ai þ bi; di ¼ DjP
Ci as follows.Given an instance I of PARTITIONPARTITION with a list
H ¼ fh1; h2; . . . ; hng and B ¼ 12
Phi, they con-
structed an instance II of 1jpi ¼ ai or pi ¼ ai þbi; di ¼ Dj
PCi, which has an optimal schedule
in the form of ðTk1;1; Tk2;2; . . . ; Tkn;nÞðT0ÞðT3�k1;1;T3�k2;2; . . . T3�kn;nÞ, for 16 i6 n and ki ¼ 1 or 2. Inthis structure, each operation of exchanging a pair
6 T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13
of tasks T1;i and T2;i corresponds to moving hi be-tween ðTk1;1; Tk2;2; . . . ; Tkn;nÞ and ðT3�k1;1; T3�k2;2; . . . ;T3�kn;nÞ. If the sum of hi chosen in ðTk1;1; Tk2;2; . . . ;Tkn;nÞ is larger than B, then the starting time of T0cannot meet D and result in a large increasing
processing time from T0. If the sum of hi chosenin ðTk1;1; Tk2;2; . . . ; Tkn;nÞ is smaller than B, thenthe total increasing processing time in ðT3�k1;1;T3�k2;2; . . . ; T3�kn;nÞ is larger than a given threshold.Thus, each solution for II corresponds to a solu-
tion for I.
Now we transform II to an instance II0 of
1jpi ¼ maxfai; ai þ biðsi � DÞgjP
Ci by defining
a00 ¼ a0, a0i ¼ ai, b00 ¼ b0 and b0i ¼ bi=a0. Let a0 be anumber larger than ½4n
Pi6¼0 ðai þ biÞ�2. If the in-
creasing processing time of a task (except T0) in II0
is larger than 0, then it is dominated by the termb0ia0 ¼ bi. That is, II
0 is almost equivalent to II.
Thus, we obtain the following theorem.
Theorem 3. The problem 1jpi ¼ maxfai; ai þbiðsi � DÞgj
PCi is NP-complete.
For the model pi ¼ ai � biminðsi; diÞ with
di ¼ D, Cheng et al. (2003) showed that 1jpi ¼ai � biminfsi;DgjCmax is NP-complete in the or-dinary sense and 1jpi ¼ ai � biminfsi;Dgj
PCi
is NP-complete.
A complete classification of the NP-complete-
ness results of the scheduling problems with time-
dependent processing times that have been studied
in the literature is given in Table 2.
4. Polynomially and pseudo-polynomially solvable
problems
In the light of the complexity boundaries of
the time-dependent problems, based on the results
reported in the last section, we attempt to de-
velop polynomial or pseudo-polynomial solutionalgorithms for those tractable and semi-tractable
problems in this section. Now we introduce
the solvable cases, beginning with the simplest
models and gradually progressing to their gener-
alizations.
First, we consider the model pi ¼ bis. Mosheiov(1994) presented several solvable cases as follows:
The makespan Cmax ¼ eQ16 i6 n ð1þ biÞ is se-
quence independent. Nondecreasing order of
bi=ðiþ biÞwi minimizes 1jpi ¼ bisjP
wiCi, where wi
is the weight. The earliest due date (EDD) rule
minimizes 1jpi ¼ bisi; dijLmax. The scheduling rulefor 1jdij
PUi can be adapted to 1jpi ¼
bisi; dijP
Ui. Mosheiov (1996a, 2002) showed that
Johnson�s rule for F 2jjCmax can be adapted toF 2jpi ¼ bisijCmax. There are many problems thatcan be solved by comparing the results for the
corresponding classical cases pi ¼ ai. For example,adapting Smith�s backward scheduling rule for1jdij
PCi to 1jpi ¼ bisi; dij
PCi, we obtain the
following theorem immediately.
Theorem 4. The problem 1jpi ¼ bisi; dijP
Ci ispolynomially solvable.
The second interesting case is the model
pi ¼ ai � bisi with bi ¼ b. Mosheiov (1996b) con-sidered the case to minimize the sum of weighted
completion times with weights proportional to the
basic processing times, i.e., wi ¼ dai, where d is agiven constant. He showed that the considered
problem can be solved in Oðn log nÞ time by a K-shaped policy with respect to the initial processing
rates. Cheng and Ding (2000) gave a generalized
Smith�s rule to solve 1jpi ¼ ai þ bsi; dijCmax inOðn5Þ time. They also showed that 1jpi ¼ai þ bsi; dij
PCi is equivalent to 1jpi ¼ ai þ
bsi; dijCmax. By an application of this schedulingapproach, 1jpi ¼ ai þ bsi; dijLmax can be solved inOðn6 log nÞ time. Moreover, Cheng and Ding(1998b) presented another equivalence relation-
ship between 1jpi ¼ ai þ bsi; dijCmax and 1jpi ¼ai � bsi; rijCmax and gave a dynamic programmingalgorithm to solve 1jpi ¼ ai � bsi; rijCmax in
Oðn6 log nÞ time. Ng et al. (2002) provided
Oðn log nÞ algorithms for 1jpi ¼ ai � bsijP
Ci and
1jpi ¼ ai � kaisijP
Ci, and a pseudo-polynomial
time dynamic programming algorithm for 1jpi ¼ai � bsi; dij
PCi. They also suggested a corre-
spondence between the complexities of the prob-
lem with an increasing linear model and the
problem with a decreasing linear model. Bachman
and Janiak (1997) showed that the EDD rule
yields optimal solutions to 1jpi ¼ ai þ kaisi; dijLmax.
Table 2
NP-complete results of time-dependent scheduling problems
Problem Complexity Reference
1jpi ¼ 1þ bisijP
Ci Open problem
1jpi ¼ 1� bisijP
Ci Open problem
1jpi ¼ ai þ bsi; dijP
Ui Open problem
1jpi ¼ 1 orpi ¼ 1þ bisi; di ¼ Dj
PwiCi
Open problem
F 3jpi ¼ bisijCmax NP-complete Mosheiov (1996a, 2002)
F 3jpi ¼ ai þ bisijCmax Strongly NP-complete Kononov and Gawiejnowicz (2001)
O3jpi ¼ bisijCmax NP-complete Kononov and Gawiejnowicz (2001), Mosheiov
(2002)
P2jpi ¼ bisijCmax NP-complete Kononov (1997); Mosheiov (1998)
P2jpi ¼ bisijP
Ci NP-complete Chen (1996); Kononov (1997)
J2jpi ¼ bisijCmax NP-complete Mosheiov (2002)
1jpi ¼ ai þ bsi; ri 2 f0;RgjCmax NP-complete Cheng and Ding (1998b)
1jpi ¼ ai þ bsi; rijCmax Strongly NP-complete Cheng and Ding (1998b)
1jpi ¼ ai � bsi; di ¼ DjP
Ci NP-complete Theorem 1
1jpi ¼ ai � bsi; di 2 fD1;D2gjCmax NP-complete Cheng and Ding (1998a)
1jpi ¼ ai � bsi; dijCmax Strongly NP-complete Cheng and Ding (1999)
1jpi ¼ 1þ bisi; di 2 fD1;D2gjCmax NP-complete Cheng and Ding (2003)
1jpi ¼ 1þ bisi; dijCmax Strongly NP-complete Cheng and Ding (2003)
1jpi ¼ 1þ bisi; ri 2 f0;RgjCmax NP-complete Cheng and Ding (1998b)
1jpi ¼ 1þ bisi; rijCmax Strongly NP-complete Cheng and Ding (1998b)
1jpi ¼ 1� bisi; di 2 fD1;D2gjCmax NP-complete Cheng and Ding (2003)
1jpi ¼ 1� bisi; dijCmax Strongly NP-complete Cheng and Ding (2003)
1jpi ¼ 1� bisi; ri 2 f0;RgjCmax NP-complete Cheng and Ding (1998b)
1jpi ¼ 1� bisi; rijCmax Strongly NP-complete Cheng and Ding (1998b)
1jpi ¼ ai þ bisijLmax NP-complete Kononov (1997), Bachman and Janiak (2000)
1jpi ¼ ai þ bisijP
wiCi NP-complete Bachman et al. (2002b)
1jpi ¼ ai � bisijP
wiCi NP-complete Bachman et al. (2002a)
1jpi ¼ ai þ bisi; di ¼ DjP
Ci NP-complete Theorem 2
1jpi ¼ maxfai; ai þ biðsi � DÞjCmax NP-complete in the ordinary sense Kubiak and van de Velde (1998), Cai et al. (1998)
1jpi ¼ maxfai; ai þ biðsi � DÞjP
Ci NP-complete Theorem 3
1jpi ¼ ai or pi ¼ ai þ bi; di ¼ DjCmax NP-complete in the ordinary sense Mosheiov (1995), Cheng and Ding (2001)
1jpi ¼ ai or pi ¼ ai þ bi; dijCmax NP-complete in the ordinary sense Jeng and Lin (2002)
1jpi ¼ ai or pi ¼ ai þ bi; di ¼ DjP
Ci NP-complete Cheng and Ding (2001)
1jpi ¼ ai � biminfsi;DgjCmax NP-complete in the ordinary sense Cheng et al. (2003)
1jpi ¼ ai � biminfsi;DgjP
Ci NP-complete Cheng et al. (2003)
T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13 7
The complexity of 1jpi ¼ ai þ bsi; dijP
Ci is
open, but there exists a polynomial algorithm
for the case with a fixed number of distinct dead-
lines, i.e., di ¼ D1; . . . ;Dm. For 1jpi ¼ ai þ bsi; di 2fD1; . . . ;Dmgj
PUi, each schedule can be divided
into two partial schedules: on-time portion andtardy portion. For two tasks with identical dead-lines, from the adjacent jobs interchange argument
(see Section 3), there exists an optimal schedule
such that the task with a smaller ai precedes an-other on entering the on-time portion. If the
number of tasks in the on-time portion and their
corresponding distinct deadlines are given, i.e., if a
distribution is given, then the task set in the on-time portion is determined. The scheduling of the
on-time portion is a corresponding makespanproblem and can be solved in polynomial time by
the generalized Smith�s rule proposed in Chengand Ding (2000). The tardy portion is sequence
independent. Hence, there exists a unique potential
optimal schedule for each distribution. Moreover,
a final optimal schedule can be obtained by enu-
merating such a distribution. Thus, we obtain the
following theorem.
8 T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13
Theorem 5. The problem 1jpi ¼ ai þ bsi; di 2fD1; . . . ;Dmgj
PUi is polynomially solvable.
For the model pi ¼ ai � bisi with ai ¼ a, Chengand Ding (2003) gave an algorithm that solves
1jpi ¼ a� bisi; bi 2 fB1;B2g; dijCmax optimally in
Oðn log nÞ.For the general pi ¼ ai � bisi model, polynomial
algorithms only exist for the case with di ¼ D andri ¼ 0. Gupta and Gupta (1988) and Browne andYechiali (1990) showed that the nondecreasing
order of ai=bi is optimal for 1jpi ¼ ai þ bisijCmax.Gawiejnowicz and Pankowska (1995) gave a more
detailed analysis of the cases with ai ¼ 0, bi ¼ 0 orai 6¼ 0 and bi 6¼ 0. Similarly, Ho et al. (1993)showed that the nonincreasing order of ai=bi isoptimal for 1jpi ¼ ai � bisi; di ¼ DjCmax.For 1jpi ¼ ai � bisi; di ¼ Dj
PUi, Woeginger
(1995) and Chen (1995) gave a dynamic pro-
gramming algorithm to solve it in Oðn3Þ and Oðn2Þtime, respectively. The algorithm in Chen (1995)
constructs a schedule iteratively by assigning an
unscheduled task with the largest ratio of ai=bieither to the position immediately following thecurrent last on-time task, or to the position fol-
lowing the current last task. Using the same ap-
proach, we can construct a counterpart algorithm
for 1jpi ¼ ai þ bisi; di ¼ DjP
Ui. Thus, the follow-
ing theorem is established.
Theorem 6. The problem 1jpi ¼ ai þ bisi; di ¼Dj
PUi can be solved in Oðn2Þ time.
For the piecewise linear models with di ¼ D,polynomial algorithms only exist for the case in
which the task system can be divided into a fixed
number of task chains. Cheng and Ding (2001)
introduced a method to divide a task system of
1jpi ¼ ai, or pi ¼ ai þ bi; di ¼ DjP
Ci into a series
of task chains as follows: Re-index the task systemin the order of nondecreasing ai, breaking ties inthe order of nonincreasing ai þ bi. Let the first taskbe the head of a chain. Put the next task, denoted
as Tn, at the end of the chain, if it has agreeableparameters with the last task, denoted as Tl, in thechain, i.e., al6 an and al þ bl P an þ bn. Repeatthis operation until all tasks are checked. For the
remaining task set, repeat the procedure to estab-
lish a new chain until no task remains. If the tasksystem can be divided into a fixed number of such
task chains, we use ch6M as an additional
descriptor in the problem description.
Similar to the approach used in Theorem 5,
Cheng and Ding (2001) gave an optimal algorithm
for 1jpi ¼ ai or pi ¼ ai þ bi; di ¼ DjP
Ci. The al-
gorithm is polynomial for the case with ch6M ,i.e., the number of task chains is fixed. Hence,1jpi ¼ ai or pi ¼ ai þ bi; di ¼ D; ch6M j
PCi is
polynomially solvable. Moreover, they adapted
the approach to the corresponding makespan and
weighted total completion time problems. Simi-
larly, adapting this approach to the models pi ¼maxfai; ai þ biðsi �DÞg and pi ¼ ai � biminfsi;Dg,we obtain the following theorem.
Theorem 7. If an instance can be divided into afixed number of task chains, then 1jpi ¼maxfai;ai þ biðsi �DÞg; ch6M jCmax and 1jpi ¼ai � bi minfsi;Dg, ch6M jCmax are polynomiallysolvable.
Since we have not been able to determine the
complexity of either the tardy partial schedule of1jpi ¼ maxf1; 1þ biðsi � DÞgj
PCi or the on-time
partial schedule of 1jpi ¼ 1� biminfsi;DgjP
Ci,
the above results cannot be applied to these two
problems. For the case with bi ¼ b, using the ad-jacent jobs interchange argument (see Section 3),
Kunnathur and Gupta (1990) showed that the
shortest ai first rule can solve 1jpi ¼ maxfai; ai þbiðsi � DÞgj
PCi to optimality. Similarly, Cheng
et al. (2003) showed that the largest ai first rule isoptimal for 1jpi ¼ ai � biminfsi;Dgj
PCi.
By now we have covered all solvable cases for
the time-dependent model considered in this
paper. It is obvious that the number of solvable
cases is quickly shrinking when the scheduling
model is generalized. Now we continue our dis-
cussion into pseudo-polynomial algorithms.The actual processing time is usually sequence
dependent in a time-dependent problem. This
feature makes it very difficult to find a pseudo-
polynomial algorithm. However, Kubiak and van
de Velde (1998) proposed a pseudo-polynomial
algorithm for 1jpi ¼ maxfai; ai þ biðsi � DÞgjCmax,which can be stated as follows: Fix any task Ti at
T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13 9
any position such that Ti is processed at time D.Re-index the other tasks in nondecreasing order of
ai=bi. Use a dynamic program to allot the tasks
into two partial sequences: an early sequence be-
fore the fixed Ti; a tardy sequence, otherwise.There are only two choices for each task––either
put it in the early sequence in a sequence inde-
pendent way, or put it at the end of the tardy
sequence. This is somewhat similar to theKNAPSACKKNAPSACK problem. Thus, 1jpi ¼ maxfai; ai þbiðsi � DÞgjCmax is pseudo-polynomially solvable.Applying this approach, Cheng et al. (2003)
and Cheng and Ding (2001) constructed a simi-
lar pseudo-polynomial algorithm for 1jpi ¼ ai �biminfsi � DgjCmax and 1jpi ¼ ai or pi ¼ ai þ bi;di ¼ DjCmax, respectively.A complete classification of the solvable cases
of the scheduling problems with time-dependent
processing times is presented in Table 3. The re-
sults in Tables 2 and 3 give sharp boundaries de-
Table 3
P-Solvable cases of time-dependent scheduling problems
Problem Com
F 2jpi ¼ bisijCmax Oðn lO2jpi ¼ bisijCmax OðnÞ1jpi ¼ bisijCmax OðnÞ1jpi ¼ bisij
PwiCi Oðn l
1jpi ¼ bisi; dijP
Ci Oðn l1jpi ¼ bisi; dijLmax Oðn l1jpi ¼ bisi; dij
PUi Oðn l
1jpi ¼ ai þ bsi; dijCmax Oðn51jpi ¼ ai þ bsi; dij
PCi Oðn5
1jpi ¼ ai þ bsi; dijLmax Oðn61jpi ¼ ai þ kaisijLmax Oðn l1jpi ¼ ai þ bsi; di 2 fD1; . . . ;Dmgj
PUi Oðnm
1jpi ¼ ai � bsi; rijCmax Oðn61jpi ¼ ai þ bisijCmax Oðn l1jpi ¼ ai þ bsij
PwiCi Oðn l
1jpi ¼ ai � bsijP
Ci Oðn l1jpi ¼ ai � kaisij
PCi Oðn l
1jpi ¼ ai þ bisi; di ¼ DjP
Ui Oðn21jpi ¼ aI � bisi; di ¼ DjCmax Oðn l1jpi ¼ ai � bisi; di ¼ Dj
PUi Oðn2
1jpi ¼ a� bisi; di; bi 2 fB1;B2gjCmax Oðn l1jpi ¼ maxfai; ai þ biðsi � DÞg; ch6M jCmax OðnM1jpi ¼ maxfai; ai þ bðsi � DÞg; ch6M j
PCi Oðn l
1jpi ¼ ai or pi ¼ ai þ bi; di ¼ D; ch6M jCmax OðnM1jpi ¼ ai or pi ¼ ai þ bi; di ¼ D; ch6M j
PCi OðnM
1jpi ¼ ai � biminfsi; dig; ch6M jCmax OðnM1jpi ¼ ai � bminfsi; dig; ch6M j
PCi Oðn l
lineating the complexities of the consideredproblems.
5. Enumerative and heuristic algorithms
There are a number of enumerative and heu-
ristic solution algorithms for some intractable
problems reported in the literature. Mosheiov(1996a,b, 1998) and Chen (1996) presented heu-
ristic algorithms for the model pi ¼ bisi. Guptaet al. (1987), Gupta and Gupta (1988), Mosheiov
(1991) and Hsu and Lin (2002) investigated heu-
ristic rules, dynamic programming procedures and
branch and bound algorithms for the model
pi ¼ ai þ bisi. Kunnathur and Gupta (1990) con-sidered heuristic rules and a branch and bound al-gorithm for the model pi¼maxfai;aiþbiðsi�diÞg.For makespan minimization with the stepwise
linear model, Kovalyov and Kubiak (1998)
plexity Reference
og nÞ Mosheiov (1996a, 2002)
Mosheiov (2002)
Mosheiov (1994)
og nÞ Mosheiov (1994)
og nÞ Theorem 4
og nÞ Mosheiov (1994)
og nÞ Mosheiov (1994)
Þ Cheng and Ding (2000)
Þ Cheng and Ding (2000)
log nÞ Cheng and Ding (2000)
og nÞ Bachman and Janiak (1997)þ5Þ Theorem 5
log nÞ Cheng and Ding (1998b)
og nÞ Gupta and Gupta (1988)
og nÞ Mosheiov (1996b)
og nÞ Ng et al. (2002)
og nÞ Ng et al. (2002)
Þ Theorem 6
og nÞ Ho et al. (1993)
Þ Chen (1995)
og nÞ Cheng and Ding (2003)
log nÞ Theorem 7
og nÞ Kunnathur and Gupta (1990)
log nÞ Cheng and Ding (2001)
log nÞ Cheng and Ding (2001)
log nÞ Theorem 7
og nÞ Cheng et al. (2003)
10 T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13
developed a fully polynomial approximationscheme. For the 1jpi¼maxfai;aiþbiðsi�DÞgjCmaxproblem, after indicating its NP-hardness, Cai
et al. (1998) proposed a fully polynomial approxi-
mation scheme. Sundararaghavan and Kunnathur
(1994), Mosheiov (1995) and Cheng and Ding
(2001) presented heuristics for the model pi¼ai orpi¼aiþbi. Gawiejnowicz et al. (2002a) presented agreedy heuristic for the pi¼1þbisi model to min-imize the total completion time. Gawiejnowicz
et al. (2002b) considered the same model to simul-
taneously minimize the makespan and total com-
pletion time. Some heuristics for the more general
models have also been presented in the literature,
see Gupta et al. (1987), Gupta and Gupta (1988),
and Mosheiov (1995).
As mentioned in Section 2, there exists aframework for the time-dependent scheduling
problems considered in this paper. Many NP-
complete problems with time-dependent process-
ing times have a corresponding solvable classical
scheduling problem as their counterparts. The
corresponding scheduling rule in the classical the-
ory becomes a natural heuristic for the time-
dependent case. The classical theory also offersabundant dominant properties, which are often
applicable for the time-dependent models, espe-
cially for tasks with agreeable parameters. This is
why most heuristic algorithms presented in the
literature perform very well in general. Gupta and
Gupta (1988), Kunnathur and Gupta (1990), and
Mosheiov (1995) presented a great deal of work
on numerical evaluation of heuristics.Due to the inherent exponential characteristics
of the actual processing times in the time-depen-
dent models, the worst-case bounds for the heu-
ristic algorithms are often difficult to determine.
Chen (1996) showed that there is no polynomial
heuristic rule with a constant worst-case ratio for
Pmjpi ¼ bisijP
Ci, unless NP ¼ P . He also showedthat the heuristic rule used for P2jpi ¼ bisij
PCi is
unbounded. Kononov and Gawiejnowicz (2001)
showed that for the F 3jpi ¼ bisijCmax problem,there is no polynomial approximation algorithm
with a constant worst-case ratio, unless NP ¼ P .Hsieh and Bricker (1997) proposed three heuristics
for Pmjpi ¼ bisijCmax and Pmjpi ¼ ai þ bisijCmax.They proved that the heuristic for Pmjpi ¼ bisijCmax
is asymptotically optimal, i.e., the worst-case ratioapproaches 1 when n is large. Mosheiov (1995,1998) presented a similar analysis of the proposed
heuristics for P2jpi ¼ bisijCmax and Pmjpi ¼ ai orpi ¼ ai þ bijCmax, respectively. The worst-case ra-tios are also unbounded. Cheng and Ding (2001)
showed that the heuristic switching technique in-
troduced by Sundararaghavan and Kunnathur
(1994) for 1jpi ¼ ai or pi ¼ ai þ bi; di ¼ DjP
wiCi
is not an optimal algorithm.
Finally, we expand the scope of application of
an interesting heuristic rule in the following. For
1jpi ¼ aþ bisijP
Ci, Mosheiov (1991) showed that
there exists an optimal schedule that is V-shaped
with respect to deterioration rates, i.e., the tasks
appearing before the task with the smallest bi aresequenced in nonincreasing order of bi, and theones after it are sequenced in nondecreasing order
of bi. He also defined a perfectly symmetric V-shaped sequence and showed that a perfectly
symmetric V-shaped sequence is optimal, if it can
be constructed from the task system. Since
1jpi ¼ a� bisijP
Ci has a structure converse to
that of 1jpi ¼ aþ bisijP
Ci, the above properties
can be adapted to 1jpi ¼ a� bisijP
Ci. Thus, weobtain the following theorem.
Theorem 8. For the problem 1jpi ¼ a� bisijP
Ci,the optimal sequence has a K-shaped property. Aperfectly symmetric K-shaped sequence is optimal, ifit can be constructed from the task system.
6. Conclusions
In this paper, we consider a class of machine
scheduling problems in which the processing time
of a task is dependent on its starting time in a
schedule. On reviewing the literature on this topic,
we provide a framework to illustrate how the
models have been generalized from the classicalscheduling theory. A complexity boundary is pre-
sented for each model and many existing results
are consolidated. We also introduce some enu-
merative and heuristic solution algorithms and
analyze their performance. In view of the work
that we have surveyed, certain topics emerge as
fertile ground for future research.
T.C.E. Cheng et al. / European Journal of Operational Research 152 (2004) 1–13 11
The complexity reductions for many consideredmodels remain a challenge. Many problems were
posed as open problems in the literature and some
of them are still open. In Table 2, four problems
are listed as open problems and, for more than half
of the problems, neither strongly NP-complete
reductions nor pseudo-polynomial algorithms
have been provided. Since each of them possesses a
very intricate geometric structure, we conjecturethat most of them should be strongly NP-com-
plete. However, it remains a great challenge for
researchers to come up with ways to settle these
issues.
There are a series of polynomial, heuristic and
enumerative algorithms proposed in the literature.
Most of them perform very well on average, but
unbounded in the worst case. Hence, whether thereexists a polynomial approximation scheme for the
considered problems becomes an interesting re-
search direction. On the other hand, we can pre-
dict the complexity of an instance by checking the
agreeable extension of its parameters, then
choosing a suitable algorithm to obtain a solution
by a trade-off consideration between computa-
tional complexity and accuracy of the solution.This kind of combinational algorithms is a mini-
expert system and should yield much better results
on the average. This approach also can be used
to deal with the multi-machine systems or other
more general models.
There are two common assumptions used in the
classical scheduling theory, namely preemptive
scheduling and unit processing time scheduling.The function pi ¼ ai � bisi can be transformed topi ¼ aið1� cisiÞ. Assume that ai are allowed to bepreemptive. Thus, we can consider preemptive
scheduling for the time-dependent models. The
model pi ¼ ai � bisi with ai ¼ a and bi ¼ b is ageneralized form of the classical scheduling prob-
lem with unit processing times. Thus, it will be an
interesting branch of the time-dependent prob-lems.
We have mainly considered some basic models
with typical performance measures in this paper.
Clearly, at a later stage of research both the time-
dependent processing time function and the per-
formance measures may be expanded to include
other considerations such as stochastic parameters.
Besides investigating the linear and piece-linearprocessing time problems, there are some papers
studying other kinds of time-dependent problems.
For examples, a group of flow shop problems with
state-dependent processing times were first intro-
duced by Sriskandarajah and Goyal (1989), con-
tinued by Sriskandarajah and Wagneur (1991) and
Wagneur and Sriskandarajah (1993a,b), as well as
extended by Finke and Jiang (1997). Alidaee(1991) considered a few single machine scheduling
problems with nonlinear cost functions. Janiak
and Kovalyov (1996) considered a class of resource
dependent scheduling problems. Gawiejnowicz
(1996) considered the scheduling problem with
speed dependent processor. Cheng and Wang
(2000) considered a class of scheduling problems in
which the processing time of a task depends on thenumber of tasks before it. Biskup (1999) and
Mosheiov (2001) investigated the scheduling prob-
lems with learning considerations in a nonlinear
form.
In fact, in almost every classical scheduling
area, there exists the time dependence phenome-
non. Thus, a beginning has just been made on the
study of time-dependent scheduling problems.
Acknowledgements
This research was supported in part by The
Hong Kong Polytechnic University under grant
number G-S818. We are grateful for three anon-
ymous referees for their constructive commentson an earlier version of this paper.
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