13
Invited Review A concise survey of scheduling with time-dependent processing 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 Kong b 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. Ó 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 restrictive assumptions 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- 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 scheduling problems 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, * 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 Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00909-8 European Journal of Operational Research 152 (2004) 1–13 www.elsevier.com/locate/dsw

A concise survey of scheduling with time-dependent processing times

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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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