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Int. J. Production Economics

Int. J. Production Economics 118 (2009) 424–429

0925-52

doi:10.1

� Cor

Systems

Univers

fax: +86

E-m

(G. Li),

(Z. Xu).

journal homepage: www.elsevier.com/locate/ijpe

Single machine scheduling of deteriorating jobs to minimize totalabsolute differences in completion times

Yongqiang Li a,b,�, Gang Li a, Linyan Sun a, Zhiyong Xu a

a The Management School of Xi’an Jiaotong University, The State Key Lab for Manufacturing Systems Engineering, Xi’an 710049, Chinab School of Economics and Management, Xidian University, Xi’an 710071, China

a r t i c l e i n f o

Article history:

Received 27 December 2007

Accepted 12 November 2008Available online 24 December 2008

Keywords:

Scheduling

Single machine

Deteriorating jobs

The total absolute deviation of completion

times (TADC)

73/$ - see front matter & 2009 Elsevier B.V. A

016/j.ijpe.2008.11.011

responding author at: The State Key Lab

Engineering, The Management School

ity, Xi’an 710049, China. Tel.: +86 1331929266

29 82664643.

ail addresses: qlyong@xidian.edu.cn (Y. Li), gle

lysun@mail.xjtu.edu.cn (L. Sun), zhiyongxu@m

a b s t r a c t

This paper investigates a single machine scheduling problem with deteriorating jobs. By

a deteriorating job, we mean that the processing time is an increasing function of its

execution starting time. Job deterioration is described by a function which is

proportional to a linear function of time. The objective is to find a schedule that

minimizes total absolute differences in completion times (TADC). We show that the

optimal schedule is V-shaped, i.e., jobs are arranged in descending order of their

deterioration rates if they are placed before the job with the smallest deterioration rate,

but in ascending order of their deterioration rates if placed after it. We also prove some

other properties of an optimal schedule, and propose two heuristic algorithms that are

tested against a lower bound. We also provide computational results to evaluate the

performance of the heuristic algorithms.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

In many scheduling environments, it is assumed thatthe processing times of jobs are an increasing function oftheir starting times. This phenomenon, known as deterior-

ating jobs, has been extensively studied in the last decadein different scheduling models and problems (see recentreviews Alidaee and Womer, 1998; Cheng et al., 2004a).

Scheduling deteriorating jobs was first considered byBrowne and Yechiali (1990) who assumed that the jobprocessing time is a non-decreasing, start-time dependentlinear function. They showed that the single machineexpected makespan minimization problem could be

ll rights reserved.

for Manufacturing

of Xi’an Jiaotong

5;

e@mail.xjtu.edu.cn

ail.xjtu.edu.cn

solved in polynomial time. Mosheiov (1991) consideredthe problem that all the jobs are characterized by acommon positive basic processing time. Using this basicassumption, Mosheiov proved that the optimal scheduleto minimize flowtime is symmetric and has a V-shapedproperty with respect to the increasing rates. Mosheiov(1994) considered the following objective functions:makespan, total flow time, sum of weighted completiontimes, total lateness, maximum lateness and maximumtardiness, and the number of tardy jobs. When the valuesof the basic processing time equal zero, all these problemscan be solved polynomially. Wu and Lee (2008) investi-gated two single machine group-scheduling problemswhere the group setup times and the job processing timesare both increasing functions of their starting times. Theyproved that the makespan minimization problem remainspolynomially solvable when the deterioration is present,and the sum of completion times problem is polynomiallysolvable when the numbers of jobs in each group areequal. Sundararaghavan and Kunnathur (1994) consideredthe single machine scheduling problem in which theprocessing time is a binary function of a common start

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Y. Li et al. / Int. J. Production Economics 118 (2009) 424–429 425

time due date. The jobs have processing time penalties forstarting after the due date, and the objective was tominimize the sum of the weighted completion times.Three special cases of this problem can be solvedoptimally. Bachman and Janiak (2000) showed thatthe maximum lateness minimization problem underthe linear deterioration assumption is NP-hard, and twoheuristic algorithms were presented as a consequence.Bachman et al. (2002) considered the problem of mini-mizing the total weighted completion time with generallinear deterioration. They proved that the problem is NP-hard. Wang and Xia (2005) considered the schedulingproblems under a special type of linear decreasingdeterioration. They presented optimal algorithms forsingle machine scheduling of minimizing the makespan,maximum lateness, maximum cost and number of latejobs, respectively. For the two-machine flow shop sche-duling problem to minimize the makespan, they provedthat the optimal schedule can be obtained by Johnson’srule. If the processing times of the operations are equal foreach job, they proved that the flow shop schedulingproblems could be transformed into single machinescheduling problems. Wang et al. (2008) consideredthe single machine scheduling problems with deteriorat-ing jobs, and the jobs are related by a series-parallelgraph. They proved that for the general linear problemto minimize the makespan, polynomial algorithms exist.They also proved that for the proportional linear problemto minimize the total weighted completion time, poly-nomial algorithms exist, too. Wu et al. (2007) consideredthe problem of minimizing the total weighted completiontime introduced by Bachman et al. (2002). They gave threeheuristic algorithms and a branch-and-bound algorithmthat incorporates two lower bounds. They also studiedthe effects of normal processing times and deteriorationrates.

Chen (1996), Hsieh and Bricker (1997), and Mosheiov(1998) considered scheduling deteriorating jobs in amulti-machine setting. They assumed that the jobsdeteriorate linearly and they are processed on parallelidentical machines. Chen (1996) considered minimizingthe flow time, while Hsieh and Bricker (1997), andMosheiov (1998) studied makespan minimization. Thereader is referred to the survey by Cheng et al. (2004a)for more details on single machine and multi-machinescheduling with deteriorating jobs.

Kononov and Gawiejnowicz (2001) considered themakespan minimization problem. They showed thatunder linear deterioration the two-machine flow shopproblem is strongly NP-hard, and the two-machine openshop problem is ordinary NP-hard. They also showed thatin the three-machine flow shop with simple deterioration,there does not exist a polynomial time approximationalgorithm with a worst-case ratio bounded by a constant.Finally, they proved that the three-machine open shopproblem with simple linear deterioration is ordinaryNP-hard. Kang and Ng (2007) studied the NP-hardproblem of scheduling n deteriorating jobs on m identicalparallel machines to minimize the makespan. Theyassumed that each job’s processing time is a linear non-decreasing function of its start time and proposed a fully

polynomial time approximation scheme for the problem.Mosheiov (2002) considered the computational complex-ity of flow shop, open shop and job shop problems withsimple linear deterioration of minimizing the makespan.He introduced a polynomial time algorithm for thetwo-machine flow shop and two-machine open shopproblems, respectively. He also proved that the three-machine flow shop, the three-machine open shop and thetwo-machine job shop problems are all NP-hard. Wangand Xia (2006a, b) considered flow shop schedulingproblems with job processing times dependent on theirstarting times. In these problems there are some dom-inating relationships between the machines. They showedthat for the problems to minimize makespan or minimizeweighted sum of completion time, polynomial algorithmsstill exist. However, when the objective is to minimizemaximum lateness, the solutions of a classical versionmay not hold. Wang et al. (2006) considered a two-machine flow shop scheduling problem with a simplelinear deterioration. Several dominance conditions andtwo lower bounds for the problem to minimize totalcompletion time were implemented in the proposedbranch-and-bound algorithm to search for the optimalsolution. They also provided a heuristic algorithm toovercome the inefficiency of the branch-and-bound algo-rithm for large-sized problems.

Most of the scheduling of the frontal papers withdeteriorating jobs literature examines regular measures ofthe performance, which are non-decreasing functions ofjob completion times. Yet in certain situations one is moreinterested in performance measures that are non-regular.To the best of our knowledge, there exist only a fewresearch results on scheduling models considering non-regular performance measures. Cheng et al. (2004b, 2005)considered single machine scheduling problem withlinear job-independent increasing (decreasing) deteriora-tion jobs. The problem was to determine an optimalcombination of the due date and schedule so as tominimize the sum of due date, earliness and tardinesspenalties. They gave a polynomial time algorithm to solvethis problem. Oron (2008) considered a single machinescheduling with simple linear deterioration. The objectivefunction was to minimize the total absolute deviationof completion times (TADC). They proved some propertiesof an optimal schedule, and introduced two heuristicalgorithms to solve this problem.

In this paper we consider a single machine schedulingproblem with proportional deterioration to minimize anon-regular performance measure, i.e., the total absolutedifferences in completion times (TADC). This objectivewas proposed by Kanet (1981) as an alternative measureof completion time variation.

The rest of this paper is organized as follows. In thenext section we give the problem description. In Section 3we introduce the V-type and some other properties. InSection 4 we analyse the complexity of the problem, andoffer two efficient polynomial time heuristic algorithmsand a lower bound. In Section 5 we present computationalexperiments to evaluate the performance of the heuristicalgorithms. Concluding remarks are given in the lastsection.

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2. Formulation of the problem

We now formulate the problem of single machinescheduling with deteriorating jobs to minimize the totalabsolute deviation of completion times (TADC). We aregiven a set of n jobs N ¼ fJ1; J2; . . . ; Jng that have to beprocessed on a single machine. All the jobs are available attime t0X0. The machine can handle one job at a time andpreemption is not allowed. The machine is assumed to becontinuously available from time zero onwards only.The problem is to schedule the jobs in such a way thatthe variation in flow time (time-in-system) is minimal.This type of problem has applications in any service ormanufacturing setting whenever it is deemed desirable toprovide jobs (customers) the same treatment; i.e., eachcustomer spends approximately the same time in thesystem or waits for service the same time as every othercustomer (Kanet, 1981). As in Kononov and Gawiejnowicz(2001), we assume that the actual processing time pj ofjob Jj if its starting time is Sj is given by pj ¼ bj(a+bSj),where bj denotes the deterioration rate of job Jj. For agiven schedule p ¼ [J1,J2,y,Jn], Cj ¼ Cj(p) represents thecompletion time of job Jj. The objective is to find aschedule that minimizes the total absolute deviation ofcompletion times (TADC): TADC ¼

Pnj¼1

Pni¼jjCj � Cij,

where |Ci�Cj| is the absolute difference of pair-wisecomparisons of job completion times, and the n jobs arenumbered in order of completion. Using the three-fieldnotation for scheduling problem classification, the pro-blem can be denoted as 1|pj ¼ bj(a+bSj)|TADC.

3. Properties of the optimal schedule

Lemma 1. For a given schedule p ¼ [J1,J2,y,Jn] of

1|pj ¼ bj(a+bSj)|TADC, if all jobs start at time t0X0, then

the actual processing time pj of job Jj is equal to

pj ¼ b t0 þa

b

� �bj

Yj�1

i¼1

ð1þ bbiÞ, (1)

whereQ0

i¼1ð1þ bbiÞ ¼ 1.

Proof. For a given schedule p ¼ [J1,J2,y,Jn], the startingtime Sj of job Jj is Sj ¼ Cj�1 ¼ ðt0 þ a=bÞ

Qj¼1i¼1ð1þ bbiÞ � a=b

(Kononov and Gawiejnowicz, 2001), hence,

pj ¼ bjðaþ bSjÞ ¼ bjðaþ bCj�1Þ

¼ bj aþ b t0 þa

b

� �Yj�1

i¼1

ð1þ bbiÞ �a

b

! !

¼ b t0 þa

b

� �bj

Yj�1

i¼1

ð1þ bbiÞ

This completes the lemma. &

For a given schedule p ¼ [J1,J2,y,Jn], from TADC ¼Pnj¼1ðj� 1Þðn� jþ 1Þpj (Kanet, 1981), and Lemma 1,

we have

TADC ¼Xn

j¼1

ðj� 1Þðn� jþ 1Þpj

¼ b t0 þa

b

� �Xn

j¼1

ðj� 1Þðn� jþ 1Þbj

Yj�1

i¼1

ð1þ bbiÞ. (2)

Similar to Oron (2008), we have Propositions 1–5 andTheorem 1.

Proposition 1. For problem 1|pj ¼ bj(a+bSj)|TADC, let

l ¼ arg minj2Nfbj; j ¼ 1;2; . . . ;ng. If nX3, then, within the

job set N, Jl is scheduled neither first nor the last in the

optimal schedule.

Proof. For problem 1|pj ¼ bj(a+bSj)|TADC, consider anyschedule with job Jl placed first. For convenience, letp1 ¼ [Jl,J[2],J[3],y,J[n]], where J[i] denotes the job scheduledin the ith position (C[i] and b[i] are defined accordingly). p2

is the schedule obtained by interchanging the first twojobs, i.e., p2 ¼ [J[2],Jl,J[3],y,J[n]]. Then

TADCðp1Þ � TADCðp2Þ ¼ b t0 þa

b

� �ðn� 1Þðb½2� � blÞ. (3)

Since b[2]Xbl, (3) is non-negative and therefore p2 is abetter policy.

Similarly, let p01 ¼ ½J½1�; J½2�; . . . ; J½n�1�; Jl�. p02 is the schedule

obtained by interchanging the last two jobs, i.e.,

p02 ¼ ½J½1�; J½2�; . . . ; Jl; J½n�1��. Then

TADCðp01Þ � TADCðp02Þ

¼ b t0 þa

b

� �ðn� 3Þðb½n�1� � blÞ

Yn�2

i¼1

ð1þ bb½i�Þ. (4)

Since b[n�1]Xbl and nX3, (4) is non-negative and there-

fore p02 is a better policy. This completes the proof. &

Proposition 2. A schedule containing three consecutive

jobs, Ji�1, Ji and Ji+1, such that bi4bi�1 and bi4bi+1 is not

optimal.

Proof. We show that an interchange between Ji and Ji�1

or between Ji and Ji+1 reduces the value of TADC.Let p1 ¼ ½J½1�; J½2�; . . . ; J½i�2�; Ji�1; Ji; Jiþ1; J½iþ2�; . . . ; J½n��, p2 ¼

½J½1�; J½2�; . . . ; J½i�2�; ; Ji; Ji�1; Jiþ1; J½iþ2�; . . . ; J½n�� and p3 ¼ ½J½1�; J½2�;

. . . ; J½i�2�; Ji�1; Jiþ1; Ji; J½iþ2�; . . . ; Jn�.

Then

TADCðp1Þ � TADCðp2Þ

¼ b t0 þa

b

� �ðnþ 3� 2iÞðbi � bi�1Þ

Yi�2

j¼1

ð1þ bb½j�Þ, (5)

TADCðp1Þ � TADCðp3Þ

¼ b t0 þa

b

� �ð2i� n� 1Þðbi � biþ1Þð1þ bbi�1Þ

�Yi�2

j¼1

ð1þ bb½j�Þ. (6)

We show that at least one of the schedules is better than

the original one, i.e., that either TADC(p1)�TADC(p2)40 or

TADC(p1)�TADC(p3)40.

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Y. Li et al. / Int. J. Production Economics 118 (2009) 424–429 427

Under the condition that bi4bi�1 and bi4bi+1, if the

term n+3�2i in (5) is negative, we have TADC(p1)�

TADC(p2)o0; it implies that i4(n+3)/2. Under the same

condition, TADC(p1)�TADC(p3)o0 only if the term 2i�n�1

in (6) is negative, it implies that io(n+1)/2, a contra-

diction. We conclude that the schedule p1 is never

optimal. &

Theorem 1. (V-shape). The optimal schedule has a V-shape,i.e., jobs are arranged in descending order of their deteriora-

tion rates if they are placed before the job with the smallest

bl, but in ascending order of their deterioration rates if placed

after it.

Proof. It is straightforward from Propositions 1 and 2. &

In the following we give some propositions about theposition of the smallest b-value job in an optimalschedule.

Proposition 3. If the number of jobs n is even, then an

optimal schedule consists of sequencing the smallest b-value

job, Jl, in the n/2+1 position.

Proof. Using schedules p1, p2 and p3 in Proposition 2, ifbiobi�1 and biobi+1, then Eqs. (5) and (6) have to be non-positive. This follows from the fact that the optimalschedule has a V-shape.

From (5) p0, we have n+3�2iX0, hence ip(n+3)/2.

Similarly, from (6) p0, we have 2i�n�1 and iX(n+1)/2.

Thus, ðnþ 1Þ=2pipðnþ 3Þ=2. Because the number of

jobs n is even, we have i ¼ n/2+1. &

Proposition 4. If the number of jobs n is odd, then an

optimal schedule consists of sequencing the smallest b-value

job, Jl, in position (n+1)/2 or (n+3)/2.

Proof. Using the same conditions on Eqs. (5) and (6) weobtain that Jl should be scheduled in the ith positionwhere ðnþ 1Þ=2pipðnþ 3Þ=2. Consequently, Jl can bescheduled in one of two positions (n+1)/2 and (n+3)/2.Note that an interchange of jobs scheduled in these twopositions is no change in the objective function TADC. Letp1 ¼ ½J½1�; J½2�; . . . ; J½ðnþ1Þ=2�; J½ðnþ3Þ=2�; . . . ; J½n��,p2 ¼ ½J½1�; J½2�; . . . ; J½ðnþ3Þ=2�; J½ðnþ1Þ=2�; . . . ; J½n��; then

TADCðp1Þ � TADCðp2Þ

¼ b t0 þa

b

� � nþ 1

2� 1

� �n�

nþ 1

2þ 1

� �

� b½ðnþ1Þ=2�

Yðnþ1Þ=2�1

j¼1

ð1þ bb½j�Þ þ b t0 þa

b

� � nþ 3

2� 1

� �

� n�nþ 3

2þ 1

� �b½ðnþ3Þ=�ð1þ bb½ðnþ1Þ=2�Þ

Yðnþ1Þ=2�1

j¼1

ð1þ bb½j�Þ

� b t0 þa

b

� � nþ 1

2� 1

� �n�

nþ 1

2þ 1

� �

� b½ðnþ3Þ=2�

Yðnþ1Þ=2�1

j¼1

ð1þ bb½j�Þ � b t0 þa

b

� � nþ 3

2� 1

� �

� n�nþ 3

2þ 1

� �b½ðnþ1Þ=2�ð1þ bb½ðnþ3Þ=2�Þ

Yðnþ1Þ=2�1

j¼1

ð1þ bb½j�Þ

¼ b t0 þa

b

� � nþ 1

2� 1

� �n�

nþ 1

2þ 1

� ��

�nþ 3

2� 1

� �n�

nþ 3

2þ 1

� ��

� ðb½ðnþ1Þ=2� � b½ðnþ3Þ=2�ÞYðn�1Þ=2

j¼1

ð1þ bb½j�Þ

Since ((n+1/2�1)(n�(n+1)/2+1)�((n+3)/2�1)(n�(n+3)/2+1) ¼ 0, hence TADC(p1)�TADC(p2) ¼ 0. We can obtain thatthe smallest b-value job can be scheduled in position(n+1)/2 or (n+3)/2. &

For convenience, we choose to schedule the smallest b-value job in position (n+3)/2 if n is odd. From Proposition4, we know that when the number of jobs is odd, theoptimal solution is not unique.

Proposition 5. In an optimal schedule, the job scheduled i

positions after the smallest b-value job will have a larger b-

value than the job scheduled i positions prior to the smallest

b-value job, i ¼ 1;2; . . . ; bn=2� 1c.

Proof. Consider p1 to be an optimal schedule of job set{J1,J2,y,Jn} and p2 (p3) to be the schedule obtained from p1

by interchanging jobs J[n/2+1�i] (J[(n+3)/2�i]) and J[n/2+1+i]

(J[(n+3)/2+i]) when n is even (odd).

When n is even, we have

TADCðp1Þ � TADCðp2Þ

¼ b t0 þa

b

� � n

2� i

� � n

2þ i

� �b½n=2�iþ1�

Yn=2�i

j¼1

ð1þ bb½j�Þ

þ b t0 þa

b

� � n

2� iþ 1

� � n

2þ i� 1

� �

� b½n=2�iþ2�ð1þ bb½n=2�iþ1�ÞYn=2�i

j¼1

ð1þ bb½j�Þ þ � � � þ b t0 þa

b

� �

�n

2þ i

� � n

2� i

� �b½n=2þiþ1�

Yn=2þi

k¼n=2�iþ1

ð1þ bb½k�ÞYn=2�i

j¼1

ð1þ bb½j�Þ

� b t0 þa

b

� � n

2� i

� � n

2þ i

� �b½n=2þiþ1�

Yn=2�i

j¼1

ð1þ bb½j�Þ

� b t0 þa

b

� � n

2� iþ 1

� � n

2þ i� 1

� �

� b½n=2�iþ2�ð1þ bb½n=2þiþ1�ÞYn=2�i

j¼1

ð1þ bb½j�Þ � � � � � b t0 þa

b

� �

�n

2þ i

� � n

2� i

� �b½n=2�iþ1�ð1þ bb½n=2þiþ1�Þ

�Yn=2þi

k¼n=2�iþ2

ð1þ bb½k�ÞYn=2�i

j¼1

ð1þ bb½j�Þ

¼ b t0 þa

b

� �ðb½n=2þ1�i� � b½n=2þ1þi�Þ

Yn=2�i

j¼1

ð1þ bb½j�Þ

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Y. Li et al. / Int. J. Production Economics 118 (2009) 424–429428

�n

2� i

� � n

2þ i

� �þ

Xn=2þi

k¼n=2�iþ2

ðk� 1Þðn� kþ 1Þbb½k�

24

�Yk�1

j¼n=2�iþ2

ð1þ bb½j�Þ �n

2þ i

� � n

2� i

� � Yn=2þi

k¼n=2�iþ2

ð1þ bb½k�Þ

35.

(7)

Obviously, the term bðt0 þ a=bÞQn=2�i

j¼1 ð1þ bbjÞ is alwayspositive, and

n

2� i

� � n

2þ i

� �þ

Xn=2þi

k¼n=2�iþ2

ðk� 1Þðn� kþ 1Þbb½k�

�Yk�1

j¼n=2�iþ2

ð1þ bb½j�Þ �n

2þ i

� � n

2� i

� � Yn=2þi

k¼n=2�iþ2

ð1þ bb½k�Þ

4n

2� i

� � n

2þ i

� �1þ

Xn=2þi

k¼n=2�iþ2

bb½k�Yk�1

j¼n=2�iþ2

ð1þ bb½j�Þ

0@

�Yn=2þi

k¼n=2�iþ2

ð1þ bb½k�Þ

1A ¼ 0.

Thus, under the assumption that p1 is an optimalschedule, (7) is non-positive if and only if b½n=2þ1�i�pb½n=2þ1þi�.

When n is old, the proof is similar to the above. &

4. Complexity analysis and heuristic algorithms

Similar to Oron (2008), from Propositions 3–5, we canobtain the complexity of the problem 1|pj ¼ bj(a+bSj)|-TADC.

Proposition 6. For the problem 1|pj ¼ bj(a+bSj)|TADC, the

computational time required to find the optimal schedule is

O((n)�3/22n).

Proof. From the proof of Oron (2008), the result can beeasily obtained. &

From Proposition 6, we know that the computationaltime of the problem 1|pj ¼ bj(a+bSj)|TADC remains expo-nential in the number of jobs. Thus, the heuristicalgorithms are the main tool for solving the problemcontaining a large number of jobs. Therefore, in orderto evaluate any proposed heuristic algorithms, we firstintroduce a lower bound LB.

Proposition 7. For the problem 1|pj ¼ bj(a+bSj)|TADC, when

n is even, a lower bound is equal to

LB1 ¼ b t0 þa

b

� �Xn=2

j¼1

ð2j� n� 1ÞYj

i¼1

ð1þ bbnþ2�2jÞ

þ b t0 þa

b

� � Xn

j¼n=2þ1

ð2j� n� 1ÞYj

i¼1

ð1þ bbiÞ, (8)

where b1pb2p � � �pbn.

Proof. Similar to the proof of Oron (2008). &

Proposition 8. For the problem 1|pj ¼ bj(a+bSj)|TADC, when

n is even, a lower bound is equal to

LB2 ¼ b t0 þa

b

� � Xn=2þ1

j¼1

ðj� 1Þðn� jþ 1Þbn=2þ2�j

Yj�1

i¼1

ð1þ bbn=2þ2�iÞ

þ b t0 þa

b

� � Xn

j¼n=2þ2

ðj� 1Þðn� jþ 1Þb2j�n�1

�Yj�1

i¼1

ð1þ bbiÞ, (9)

where b1pb2p � � �pbn.

Proof. Similar to the proof of Oron (2008). &

Proposition 9. For the problem 1|pj ¼ bj(a+bSj)|TADC, when

n is odd, a lower bound is equal to

LB2 ¼ b t0 þa

b

� � Xðnþ3Þ=2

j¼1

ðj� 1Þðn� jþ 1Þbðnþ3Þ=2þ1�j

�Yj�1

i¼1

ð1þ bbðnþ3Þ=2þ1�iÞ

þ b t0 þa

b

� � Xn

j¼ðnþ5Þ=2

ðj� 1Þðn� jþ 1Þb2j�n�2

�Yj�1

i¼1

ð1þ bbiÞ, (10)

where b1pb2p � � �pbn.

Proof. Similar to the proof of Oron (2008). &

In order to make the lower bound tighter, we choose themaximum value of Eqs. (8), (9) or (10) as a lower boundfor the problem 1|pj ¼ bj(a+bSj)|TADC. That is,

LB ¼ maxfLB1; LB2g.

As in Oron (2008), we introduce two simple heuristicalgorithms to solve the problem 1|pj ¼ bj(a+bSj)|TADC.

As previously, we assume that the job set N ¼

{J1,J2,y,Jn} is ordered in non-decreasing order of bj, i.e.,b1pb2p � � �pbn.

Heuristic 1 (H1). If n is even proceed to Step 1 and if n

is odd proceed to Step 10.Step 1: Assign job Jn/2+2�i to position i, i ¼ 1,2,y,n/2+1.Step 2: Assign job Ji to position i, i ¼ n=2þ 2;

n=2þ 3; . . . ;n. The resulting permutation is of the form½Jn=2þ1; Jn=2; . . . ; J3; J2; J1; Jn=2þ2; Jn=2þ3; . . . ; Jn�.

Step 10: Assign job Jðnþ5Þ=2�i to position i, i ¼ 1,2,y,(n+3)/2.

Step 20: Assign job Ji to position i, i ¼ ðnþ 5Þ=2;ðnþ 7Þ=2; . . . ;n. The resulting permutation is of the form½Jðnþ3Þ=2; Jðnþ1Þ=2; . . . ; J3; J2; J1; Jðnþ5Þ=2; Jðnþ7Þ=2; . . . ; Jn�.

Heuristic 2 (H2). If n is even proceed to Step 1 and if n

is odd proceed to Step 10.Step 1: Assign job Jn+2�2i to position i, i ¼ 1,2,y,n/2.Step 2: Assign job J2i�1 to position n/2+i, i ¼ 1,2,y,n/2.

The resulting permutation is of the form ½Jn; Jn�2;

Jn�4; . . . ; J2; J1; J3; . . . ; Jn�3; Jn�1�.Step 10: Assign job Jn+2�2i to position i, i ¼ 1,2,y,n+1/2.Step 20: Assign job J2i to position (n+1)/2+i, i ¼ 1,2,

y,n�1/2. The resulting permutation is of the form½Jn; Jn�2; . . . ; J5; J3; J1; J2; J4; . . . ; Jn�3; Jn�1�.

Obviously, the running time of both heuristic algo-rithms is O(n log n).

ARTICLE IN PRESS

Table 1Optimality gaps for heuristic algorithms H1 and H2.

Problem H1 H2

Mean (%) Best case (%) Worst case (%) Mean (%) Best case (%) Worst case (%)

20 jobs 1.169 7.53�10�2 6.663 5.613 1.993 14.638

50 jobs 8.91�10�3 7.76�10�4 0.154 2.214 0.723 7.879

100 jobs 6.61�10�7 2.63�10�8 2.68�10�5 1.375 0.281 4.771

Y. Li et al. / Int. J. Production Economics 118 (2009) 424–429 429

5. Computational experiments

Computational experiments were conducted to evalu-ate the effectiveness of the heuristic algorithms. Theheuristic algorithms were coded in VC++ 6.0 and thecomputational experiments were run on a Pentium4 personal computer with a RAM size of 512 MB.The deterioration rates were generated from a uniformdistribution over (0.05, 1), i.e., bj�(0.05,1). The heuristicalgorithms were tested for three sizes of the problem(n ¼ 20, 50 and 100). As a consequence, 3 experimentalconditions were examined and 200 instances randomlygenerated for each condition. A total of 600 problemswere tested. For all the tests, the values t0 ¼ 0, a ¼ 1 andb ¼ 0.1 were used. For the heuristic algorithm, the worst,average and best percentage deviations of the heuristicsolution from the lower bound, i.e., (Heuristic-LB)/LB, arereported. From Table 1 we see that the performance of theheuristic algorithms is effective in obtaining near-optimalsolutions for large problems.

6. Conclusions

In this paper we considered a single machine schedul-ing problem with proportional deterioration. We consid-ered a non-regular objective function, i.e., minimizing thetotal deviation of job completion time (TADC). Severalimportant properties of an optimal schedule were proved:the optimal schedule has a V-shape with respect to thedeterioration rates, the positions of the smallest b-valuewere found, and a relationship between jobs scheduledprior and after the smallest b-value was introduced.Two heuristic algorithms were also proposed, which wasshown by computational experiments to be effective andefficient in obtaining near-optimal solutions. Furtherresearch may focus on determining the complexity ofthe problem, considering the general deteriorating model,or studying the other non-regular objective functions.

Acknowledgements

The authors wish to thank the anonymous referees fortheir valuable suggestions.

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