Single-machine scheduling with deteriorating functions for job processing times

Applied Mathematical Modelling 34 (2010) 4171–4178

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Applied Mathematical Modelling

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

Single-machine scheduling with deteriorating functionsfor job processing times

T.C.E. Cheng a, Wen-Chiung Lee b, Chin-Chia Wu b,*

a Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kongb Department of Statistics, Feng Chia University, Taichung, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 January 2010Received in revised form 6 April 2010Accepted 22 April 2010Available online 28 April 2010

Keywords:SchedulingSingle machineDeteriorating jobs

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.04.014

* Corresponding author.E-mail address: [email protected] (C.-C. Wu).

In many realistic scheduling settings a job processed later consumes more time than whenit is processed earlier – this phenomenon is known as scheduling with deteriorating jobs. Inthe literature on deteriorating job scheduling problems, majority of the research assumedthat the actual job processing time of a job is a function of its starting time. In this paper weconsider a new deterioration model where the actual job processing time of a job is a func-tion of the processing times of the jobs already processed. We show that the single-machine scheduling problems to minimize the makespan and total completion timeremain polynomially solvable under the proposed model. In addition, we prove that theproblems to minimize the total weighted completion time, maximum lateness, and maxi-mum tardiness are polynomially solvable under certain agreeable conditions.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In many real-life scheduling settings a job processed later consumes more time than when it is processed earlier – thisphenomenon is known as scheduling with deteriorating jobs. For example, Gupta and Gupta [1] pointed out that in steel pro-duction, the temperature of an ingot drops below a certain level while waiting to enter a rolling machine, which requiresreheating of the ingot before rolling. Kunnathur and Gupta [2] showed that the time and effort required to control a fire in-crease if there is a delay in commencing the fire-fighting effort. In those cases, accomplishing a task might need more time astime passes. Scheduling in such settings is known as ‘‘scheduling deteriorating jobs”.

Gupta and Gupta [1] and Browne and Yechiali [3] independently initiated research on scheduling with deteriorating jobsor time-dependent processing times. Since then, machine scheduling problems with time-dependent processing times havereceived tremendous attention from the scheduling research community. Researchers constructed models where theactual processing time of a job is a function of its starting time. Depending on the processing time functions, Alidaee andWomer [4] classified scheduling models with deteriorating jobs into three different types, namely linear, piecewise linear,and non-linear functions.

In the literature most research assumed a linear processing time function for each job. For instance, Cheng and Ding [5]considered a family of scheduling problems for a set of start-time-dependent tasks with release times and linearly increas-ing/decreasing processing rates on a single machine to minimize the makespan. Ng et al. [6] investigated three schedulingproblems with deteriorating jobs to minimize the total completion time on a single machine. Cheng and Ding [7] studied thefeasibility problem of scheduling a set of start-time-dependent tasks on a single machine with known deadlines and process-ing rates and identical initial processing times. Guo and Wang [8] considered the model where jobs belong to different

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4172 T.C.E. Cheng et al. / Applied Mathematical Modelling 34 (2010) 4171–4178

groups and the actual processing time of a job is given by pij(t) = pij(a + bt), where pij is the nominal processing time of job i ingroup j and t is its starting time. They showed that the problem to minimize the makespan is polynomially solvable underthe group technology assumption. Moreover, Wang et al. [9] showed that the single-machine group-scheduling problems tominimize the makespan and total completion time are polynomially solvable under the model that the actual processingtime of a job is pij(t) = aij � bijt, where aij and bij denote the basic processing time and the deteriorating rate of job i in groupj, respectively. Kang and Ng [10] proved the NP-hard problem of scheduling n deteriorating jobs on m identical parallel ma-chines to minimize the makespan in which the actual processing time of a job is pj(t) = aj + bjt, and aj and bj are integral.Chung et al. [11] considered the single-machine problem to minimize the sum of squares of job lateness under the simplelinear deterioration assumption, i.e., pj(t) = ajt. In addition, Wu and Lee [12] considered two single-machine group-schedul-ing problems in which the group setup times and the job processing times are both increasing functions of their startingtimes, i.e., the actual job processing time is pij(t) = aij + bt and the actual group setup time is sj(t) = dj + gt, where b > 0 andg > 0. Wang et al. [13] considered the single-machine scheduling problem with job-position-based and sum-of-process-

ing-times-based processing times, i.e., the actual job processing time is pjr ¼ pjp0þPr�1

l¼1p½l�

p0þPn

l¼1pl

� �a1

ra2 , where p0 is a given param-

eter, and a1 (a2) is a given constant representing a rate of change. Wang et al. [14] considered the single-machine schedulingproblem in which the processing time of a job is an exponential function of the sum of the logarithm of the processing times

of the jobs already processed, i.e., pjrðtÞ ¼ pj aaPr�1

i¼1ln p½i� þ b

� �, where 0 < a 6 1, a P 0, and b P 0.

However, there are situations in which extra time is needed for the successful execution of a task if certain maintenanceprocedures fail to complete prior to a pre-specified deadline, and this motivates the piecewise-linear model. For example,Sundararaghavan and Kunnathur [15] proposed optimal and heuristic algorithms for the problems to minimize the make-span and the total weighted completion time, respectively. Cheng and Ding [16] considered a piecewise-linear model inwhich each task has a normal processing time that deteriorates as a step function if its starting time is beyond a given dete-riorating date. Cheng et al. [30] studied scheduling problems for a set of non-preemptive jobs on a single machine or multiplemachines without idle times where the processing time of a job is a piecewise non-increasing function of its starting time. Nget al. [17] considered a two-machine flow shop scheduling problem with deteriorating jobs in which the actual processingtime of a job is pij(t) = aij + bijt, where aij P 0 and bij P 0.

Research on non-linear models is relatively limited. Gupta and Gupta [1] introduced the problem with polynomial pro-cessing time functions and proposed branch-and-bound and heuristic algorithms to search for optimal and near-optimalsolutions for the makespan problem. Browne and Yechiali [3] introduced the makespan problem with exponential job pro-cessing times and provided insights into the optimal solutions. Voutsinas and Pappis [18] introduced a new type of modelwhere the job value deteriorates exponentially over time. Comprehensive reviews of different deteriorating job schedulingmodels and problems have been given by Alidaee and Womer [4] and Cheng et al. [19]. Recently, Cheng et al. [20] studied amodel with deteriorating jobs and learning effects. Under the proposed model, they assumed that the actual processing timeof a job depends not only on the total normal processing times of the jobs already processed but also on its scheduled posi-

tion, i.e., pj½r� ¼ pjp0þPr�1

l¼1p½l�

p0þPn

l¼1p½l�

� �a1

ra2 , where p0 > 0, a1 6 0, and a2 6 0. Sun [21] introduced a new scheduling model that consid-

ers both deteriorating jobs and learning effect simultaneously in which the actual processing time of a job depends not onlyon the processing time of the jobs already processed but also on its scheduled position, i.e., pj½r� ¼ pj 1þ

Pr�1l¼1 p½l�

� �arb, where

a P 1 and b < 0. Wang et al. [22] considered a model in which the processing times of jobs are defined as functions of theirstarting times and positions in a sequence, i.e., pj[r](t) = pj(aar�1 + b)(bt + c), where a P 0, 0 < a 6 1, b P 0, b P 0, c P 0, anda + b = 1. Lee et al. [23] proposed a sum-of-processing-time-based deterioration model in which the actual job processing

time is pj½r� ¼ pj 1þPr�1

l¼1p½l�Pn

l¼1pl

� �a

, where 0 6 a 6 1. Wang et al. [24] extended the restriction of the upper bound to the learning

rate and considered the single-machine scheduling problem with a deteriorating function where the actual processing timeof a job is a function of the jobs already processed before the job, i.e., pj½r� ¼ pj 1þ

Pr�1l¼1 p½l�

� �arb, where a P 1 and b 6 0. Wei

and Wang [25] considered the case where the job processing times are simple linear functions of their starting times, i.e., theactual processing time of job Jij is given by pij(t) = aijt, where aij > 0 is the deterioration rate of job Jij, t is the starting time, andthe actual setup time of group Gi is di(t) = bit, where di > 0 is the deterioration rate of group Gi. In addition, Lee et al. [26] con-sidered a single-machine scheduling problem with a linear deterioration assumption where the objective is to minimize thetotal weighted completion time of the jobs of the first agent with the restriction that no job of the second agent is tardy.Gawiejnowicz and Kononov [27] considered a set of independent, resumable, and proportionally deteriorating jobs to be pro-cessed on a single machine, which is not continuously available for processing. Wang and Quo [28] considered a single-ma-chine scheduling problem with the effects of learning and deterioration where the job processing times are functions of theirstarting times and positions in a sequence. Their problem is to determine an optimal combination of the due-date and sche-dule so as to minimize the sum of earliness, tardiness, and due-date costs. Recently, Gawiejnowicz [29] presented a compre-hensive discussion of different aspects of time-dependent scheduling and its applications.

It is noticed that most of the works mentioned above assumed that the actual processing time of a job is a function of itsstarting time. In this paper we consider a different model where the deterioration phenomenon is expressed as a function ofthe processing times of the jobs already processed. Specifically, we investigate several single-machine scheduling problemsunder the proposed model to minimize the makespan, total completion time, total weighted completion time, maximum

T.C.E. Cheng et al. / Applied Mathematical Modelling 34 (2010) 4171–4178 4173

lateness, and maximum tardiness. The remainder of this paper is organized into four sections. The proposed model is intro-duced in the next section. The single-machine problems under the sum-of-processing-times-based deteriorating model arediscussed in Section 3. Conclusions are given in the last section.

2. The sum-of-processing-times-based deteriorating model

In this paper we consider a job deteriorating scheduling model on a single machine specifically, the actual processing timeof job j if it is scheduled in the rth position of a sequence is

pj½r� ¼ pj 1�Pr�1

l¼1 p½l�Pnl¼1pl

!a

¼ pj

Pnl¼rp½l�Pnl¼1pl

!a

; ð1Þ

where pj is the basic processing time of job j, p[l] denotes the basic processing time of the job scheduled in the lth position ofthe sequence, and a is the deteriorating index whose value is non-positive, i.e., a 6 0. Under the sum-of-processing-times-based deteriorating model, the actual processing times cannot be computed without knowledge of the processing times,i.e., without knowing the identities of the jobs already processed. It is also observed from (1) that the longer the jobs alreadyprocessed is, the stronger the deteriorating effect is on all the subsequent jobs that are yet to be processed. The objectivesconsidered in this paper are the makespan, total completion time, total weighted completion time, maximum lateness, andmaximum tardiness.

Before presenting the results, we first derive several lemmas in the following, which will be used in the proofs of the the-orems in the sequel (The proofs are given in the Appendix).

Lemma 1. 1 � (1 � x)a � ax(1 � x)a�1 P 0 for a 6 0 and 0 6 x 6 1.

Lemma 2. k½1� ð1� xÞa� � ½1� ð1� kxÞa�P 0 for a 6 0, 0 6 x 6 1, and 1 � k � 1=x.

Lemma 3. 1 + k[1 � (1 � x)a] � ax(1 � kx)a�1 P 0 for 0 < k 6 1, 0 6 x 6 1/k, and a = �2n, where n is an integer number.

Lemma 4. k½1� ð1� xÞa� � 1k ½1� ð1� kxÞa�P 0 for 0 < k 6 1, 0 6 x 6 1/k, and a = �2n, where n is an integer number.

Lemma 5. ðk� 1Þ þ kk½1� ð1� xÞa� � 1k ½1� ð1� kkxÞa�P 0 for k P 1, 0 < k 6 1, 0 6 x 6 1=kk, and a = �2n, where n is an inte-

ger number.

3. Several single-machine problems

In this section we examine several classical single-machine scheduling problems under the proposed deteriorating model.We show that the first two problems have polynomial-time solutions and the last three problems are polynomially solvableunder certain agreeable conditions.

Suppose that S and S0 are two job schedules where the difference between S and S0 is a pairwise interchange of two adja-cent jobs i and j, i.e., S = (pijp0) and S

0= (pijp0), where p and p0 denote partial sequences. Furthermore, assume that there are

r � 1 jobs in p. Thus, jobs i and j are the rth and (r + 1)th jobs in S, while jobs j and i are scheduled in the rth and (r + 1)thpositions in S0. In addition, let B denote the completion time of the last job in p. Under S, the completion times of jobs i and jare, respectively

CiðSÞ ¼ Bþ pi 1�Pr�1


!a

ð2Þ

and

CjðSÞ ¼ Bþ pi 1�Pr�1


!a

þ pj 1�Pr�1

l¼1 p½l� þ piPnl¼1pl

!a

: ð3Þ

Similarly, the completion times of jobs j and i in S0 are, respectively

CjðS0Þ ¼ Bþ pj 1�Pr�1


!a

ð4Þ

and

CiðS0Þ ¼ Bþ pj 1�Pr�1


!a

þ pi 1�Pr�1

l¼1 p½l� þ pjPnl¼1pl

!a

: ð5Þ


3.1. Minimizing makespan

In this subsection we tackle the problem to minimize the makespan under the proposed deteriorating model. It is as-sumed that the actual job processing time of a job is an increasing function of the processing times of the jobs already pro-cessed. We show that the optimal solution for the makespan problem is obtained by sequencing jobs in the shortestprocessing time (SPT) order.

Theorem 1. For the makespan problem under the sum-of-processing-times-based deteriorating model, the optimal schedule isobtained by sequencing jobs in the SPT order.

Proof. Suppose pi 6 pj. To show that S dominates S0, it suffices to show that Cj(S) 6 Ci(S0). h

Taking the difference between (3) and (5), we have

CiðS0Þ � CjðSÞ ¼ ðpj � piÞ 1�Pr�1


!a

þ pi 1�Pr�1


!a

� pj 1�Pr�1


!a

: ð6Þ

Substituting P ¼Pn

l¼1pl; Pr ¼Pr�1

l¼1 p½l�, and t ¼ 1� PrP into (6), we obtain

CiðS0Þ � CjðSÞ ¼ ðpj � piÞta þ pi t �pj

P

� �a

� pj t � pi

P

� �a: ð7Þ

It is noted that 0 6 t 6 1 since the job processing time is non-negative. Let k ¼ pj

piand x ¼ pi

Pt, then (7) can be re-written as

CiðS0Þ � CjðSÞta ¼ pifk½1� ð1� xÞa� � ½1� ð1� kxÞa�g; ð8Þ

where a 6 0, 0 6 x 6 1, and 1 6 k 6 1=x. From Lemma 2, we have

CiðS0Þ � CjðSÞta P 0:

This implies that the jobs processed after job i under S0 must be processed at a later time than job j under S. Thus, themakespan of the jobs under S0 is strictly greater than that under S. Thus, repeating this interchange argument for all the jobsnot sequenced in the SPT order completes the proof of Theorem 1.

3.2. Minimizing total completion time

In this subsection we show that the SPT sequence yields an optimal schedule for the problem to minimize the total com-pletion times under the proposed sum-of-processing-times-based deteriorating model.

Theorem 2. For the total completion time minimization problem, an optimal schedule is obtained by sequencing jobs in the SPTorder.

Proof. Suppose that pi 6 pj. From Theorem 1, we have Cj(S) 6 Ci(S0) since pi 6 pj. To show that S dominates S0, it suffices toshow that

CiðSÞ þ CjðSÞ � CjðS0Þ þ CiðS0Þ: �

From (2)–(5), we have

fCjðS0Þ þ CiðS0Þg � fCiðSÞ þ CjðSÞg ¼ ðpj � piÞ 1�Pr�1


!a

þ ðpj � piÞ 1�Pr�1


!a

þ pi 1�Pr�1


!a

� pj 1�Pr�1


!a

: ð9Þ

The first term of (9) is non-negative since pi 6 pj. It is also noted from (6)–(8) that the sum of the last three terms is non-negative, too. So, this implies that Ci(S) + Cj(S) 6 Cj(S0) + Ci(S0). Thus, repeating this interchange argument for all the jobs notsequenced in the SPT order completes the proof of Theorem 2.

3.3. Minimizing total weighted completion time

It is well-known that sequencing jobs according to the weighted smallest processing time (WSPT) rule provides an opti-mal schedule for the classical total weighted completion time problem, i.e., sequencing jobs in non-decreasing order of pj/wj,


where wj is the weight of job j. However, the WSPT order does not yield an optimal schedule under the proposed deterio-rating model, as shown by the example below.

Example 1. n = 2, p1 = 2, p2 = 3, w1 = 3, w2 = 5, and a = �2. The WSPT sequence (2, 1) yields a total weighted completion timeof 61.5, while the sequence (1, 2) yields the optimal value of 57.67.

Although the WSPT order does not provide an optimal schedule under the proposed model, it still gives an optimal solu-tion if the processing times and the weights are agreeable, i.e., pi 6 pj implies wi P wj for all jobs i and j. Moreover, we assumethat a = �2n, where n is an integer number. The result is stated in the following theorem.

Theorem 3. For the total weighted completion time problem under the sum-of-processing-times-based deteriorating model, anoptimal schedule is obtained by sequencing jobs in non-decreasing order of pi/wi if the processing times and the weights areagreeable, i.e., pi 6 pj implies wi P wj for all jobs i and j.

Proof. Suppose pi/wi 6 pj/wj. Since pi 6 pj, it is seen from Theorem 1 that Cj(S) 6 Ci(S0). Thus, to show that S dominates S0, itsuffices to show that wiCi(S) + wjCj(S) 6 wjCj(S0) + wiCi(S0). From (2)–(5), we have h

½wjCjðS0Þ þwiCiðS0Þ� � ½wiCiðSÞ þwjCjðSÞ�

¼ wj Bþ pj 1�Pr�1


!a" #þwi Bþ pj 1�

Pr�1l¼1 p½l�Pnl¼1pl

!a

þ pi 1�Pr�1


!a" #( )

� wi Bþ pi 1�Pr�1


!a" #þwj Bþ pi 1�

Pr�1l¼1 p½l�Pnl¼1pl

!a

þ pj 1�Pr�1


!a" #( )

¼ ðwipj �wjpiÞ 1�Pr�1


!a

þwjpj 1�Pr�1


!a

� 1�Pr�1


!a" #

�wipi 1�Pr�1


!a

� 1�Pr�1


!a" #: ð10Þ

Substituting k ¼ pj=wj

pi=wiP 1; t ¼ 1�

Pr�1

l¼1p½l�Pn

l¼1pl

� �; x ¼ piPn

l¼1pl�Pr�1

l¼1p½l�

, and k ¼ wj

wiinto (10), we have

½wjCjðS0Þ þwiCiðS0Þ� � ½wiCiðSÞ þwjCjðSÞ� ¼ wjpita ðk� 1Þ þ kk½1� ð1� xÞa� � 1

k½1� ð1� kkxÞa�

� �: ð11Þ

From Lemma 5, the value of (11) is non-negative, so we have

wjCjðS0Þ þwiCiðS0ÞP wiCiðSÞ þwjCjðSÞ:

Thus, repeating this interchange argument for all the jobs not sequenced according to the WSPT rule completes the proofof Theorem 3.

3.4. Minimizing maximum lateness and maximum tardiness

If di denotes the due date of job i, then Li = Ci � di is its lateness and Ti(S) = max {Li(S), 0} is its tardiness, i = 1, 2, . . . , n.Ordering jobs according to the earliest due-date (EDD) rule provides an optimal sequence for the classical maximum latenessproblem. However, this policy is not optimal under the proposed model, as shown by the example below.

Example 2. n = 2, p1 = 10, p2 = 5, d1 = 11, d2 = 12, and a = �0.5. The EDD sequence (1, 2) yields a maximum lateness of 6.66,while the sequence (2, 1) yields the optimal value of 6.25.

Although the EDD order does not provide the optimal solution for the maximum lateness problem under the proposedmodel, it is still the optimal policy if the job processing times and due dates are agreeable, i.e., di 6 dj implies pi 6 pj forall jobs i and j. The result is stated in the following theorem.

Theorem 4. For the maximum lateness problem under the sum-of-processing-times-based-deteriorating model, an optimalschedule is obtained by sequencing jobs in non-decreasing order of di (i.e., the EDD order) if the job processing times and due datesare agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j.

Proof. Suppose di 6 dj, which implies pi 6 pj. Thus, it is seen from Theorem 1 that Cj(S) 6 Ci(S0). To show that S dominates S0, itsuffices to show that max {Li(S), Lj(S)} 6max {Lj(S0), Li(S0)}. By definition, the lateness of jobs i and j in S and jobs j and i in S

0

are


LiðSÞ ¼ CiðSÞ � di;

LjðSÞ ¼ CjðSÞ � dj;

LjðS0Þ ¼ CjðS0Þ � dj

and

LiðS0Þ ¼ CiðS0Þ � di: �

Since pi 6 pj, we have from Theorem 1 that

CjðSÞ 6 CiðS0Þ: ð12Þ

With the condition that di 6 dj, we have

LjðSÞ 6 LiðS0Þ: ð13Þ
From (12), and since job i is processed before job j in S, we have
LiðSÞ 6 LiðS0Þ: ð14Þ
From (13) and (14), we have max {Li(S), Lj(S)} 6max {Li(S0), Lj(S0)}.
Thus, repeating this interchange argument for all the jobs not sequenced according to the EDD rule completes the proof ofTheorem 4.

Corollary 1. For the maximum tardiness problem under the sum-of-processing-times-based-deteriorating model, an optimalschedule is obtained by sequencing jobs in non-decreasing order of di (i.e., the EDD order) if the job processing times and due datesare agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j.

Proof. The proof is similar to that of Theorem 4. h

4. Conclusions

The deteriorating job scheduling problem has been extensively studied under various machine settings and with respectto different performance measures in recent years. However, most research on deteriorating job scheduling problems as-sumed that the actual processing time of a job is a function of its starting time. In this paper we considered a sum-of-pro-cessing-times-based deteriorating model, where the actual processing time of a job depends on the processing times of thejobs already processed. Specifically, we investigated two single-machine scheduling problems to minimize the makespanand total completion time. We showed that both problems remain polynomially solvable when deterioration is considered.In addition, we also showed that the WSPT order and the EDD order yield the optimal schedules for the problems to min-imize the sum of weighted completion times, and maximum lateness, and maximum tardiness, respectively, under agreeableassumptions. The models including release times and extending to a general form of the processing time function are inter-esting for future studies.

Acknowledgements

We are grateful to the Editor and the referees for their constructive comments on an earlier version of our paper.

Appendix A

Lemma 1. 1 � (1 � x)a � ax(1 � x)a�1 P 0 for a 6 0 and 0 6 x 6 1.

Proof. Let h(x) = 1 � (1 � x)a � ax(1 � x)a�1. We have

h0ðxÞ ¼ að1� xÞa�1 � að1� xÞa�1 þ aða� 1Þxð1� xÞa�2 ¼ aða� 1Þxð1� xÞa�2 P 0

for a 6 0 and 0 6 x 6 1. Therefore, h(x) is a non-decreasing function for 0 6 x 6 1. Since h(0) = 0, 1 � (1 � x)a � ax(1 �x)a�1 P 0 for a 6 0 and 0 6 x 6 1. This completes the proof. h

Lemma 2. k½1� ð1� xÞa� � ½1� ð1� kxÞa�P 0 for a 6 0, 0 6 x 6 1, and 1 6 k 6 1=x.

Proof. Consider the following equation

f ðkÞ ¼ k½1� ð1� xÞa� � ½1� ð1� kxÞa�:

Taking the first and second derivatives of f ðkÞ with respect to k, we have


f 0ðkÞ ¼ 1� ð1� xÞa � axð1� kxÞa�1;

and

f 00ðkÞ ¼ aða� 1Þx2ð1� kxÞa�2:

Since a 6 0, 0 6 x 6 1, and 1 6 k 6 1=x, f 00ðkÞP 0. Thus, f 0ðkÞ is a non-decreasing function for 1 6 k 6 1=x. From Lemma 1, wehave

f 0ð1Þ ¼ 1� ð1� xÞa � axð1� xÞa�1 P 0:

Using the fact that f 0ðkÞ is a non-decreasing function for 1 6 k 6 1=x, we have

f 0ðkÞP f 0ð1ÞP 0:

This implies that f ðkÞ is also a non-decreasing function for 1 6 k 6 1=x. Since f(1) = 0, f ðkÞ ¼ k½1� ð1� xÞa� � ½1� ð1� kxÞa�P0 for a 6 0, 0 6 x 6 1, and 1 6 k 6 1=x. This completes the proof. h

Lemma 3. 1 + k[1 � (1 � x)a] � ax(1 � kx)a�1 P 0 for 0 < k 6 1, 0 6 x 6 1/k, and a = �2n, where n is an integer number.

Proof. Let f(x) = 1 + k[1 � (1 � x)a] � ax(1 � kx)a�1. Taking the first derivative of f(x) with respect to x, we have

f 0ðxÞ ¼ akð1� xÞa�1 � að1� kxÞa�1 þ aða� 1Þkxð1� kxÞa�2:

Since 0 < k 6 1, 0 6 x 6 1/k, [k(1 � x)a�1 � (1 � kx)a�1] < 0, and a = �2n, f0(x) > 0. This implies that f(x) is a non-decreasing

function for 0 6 x 6 1/k. Since f(0) = 1, f(x) > 0. This completes the proof. h

Lemma 4. k½1� ð1� xÞa� � 1k ½1� ð1� kxÞa�P 0 for 0 < k 6 1, 0 6 x 6 1/k, and a = �2n, where n is an integer number.

Proof. Consider the following function:

f ðxÞ ¼ k½1� ð1� xÞa� � 1k½1� ð1� kxÞa�:

Taking the first derivative of f(x) with respect to x, we have

f 0ðxÞ ¼ akð1� xÞa�1 � að1� kxÞa�1 ¼ a½kð1� xÞa�1 � ð1� kxÞa�1�:

Since 0 < k < 1, 0 < x < 1/k, a is an even negative integer number, and [k(1 � x)a�1 � (1 � kx)a�1] < 0, f0(x) > 0. This implies that

f(x) is a non-decreasing function for 0 6 x 6 1/k. Thus, f(x) P f(0) = 0. This completes the proof. h

Lemma 5. ðk� 1Þ þ kk½1� ð1� xÞa� � 1k ½1� ð1� kkxÞa�P 0 for k P 1, 0 < k 6 1, 0 6 x 6 1=kk, and a = �2n, where n is an inte-

ger number.

Proof. Let gðkÞ ¼ ðk� 1Þ þ kk½1� ð1� xÞa� � 1k ½1� ð1� kkxÞa�. Taking the first and second derivatives of gðkÞ with respect to

k, we have

g0ðkÞ ¼ 1þ k½1� ð1� xÞa� � axð1� kkxÞa�1

and

g00ðkÞ ¼ aða� 1Þkx2ð1� kkxÞa�2:

Since k P 1, 0 < k 6 1, 0 6 x 6 1=kk, and a = �2n, g00ðkÞP 0. This implies that g0ðkÞ is a non-decreasing function for k P 1.From Lemma 3, we have

g0ðkÞP g0ð1Þ ¼ 1þ k½1� ð1� xÞa� � ax½1� kx�a�1 P 0:

This implies that g0ðkÞP 0 and gðkÞ is a non-decreasing function for k P 1, too. Therefore, we have from Lemma 4 that

gðkÞP gð1Þ ¼ k½1� ð1� xÞa� � 1k½1� ð1� kxÞa�P 0:

The proof is completed. h


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Single-machine scheduling with deteriorating functions for job processing times