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Page 1: Single machine group scheduling with decreasing time-dependent processing times subject to release dates

Applied Mathematics and Computation 234 (2014) 286–292

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Applied Mathematics and Computation

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

Single machine group scheduling with decreasingtime-dependent processing times subject to release dates

http://dx.doi.org/10.1016/j.amc.2014.01.1680096-3003/� 2014 Published by Elsevier Inc.

⇑ Corresponding author.E-mail addresses: [email protected] (Y.-Y. Lu), [email protected] (J.-B. Wang).

Yuan-Yuan Lu a,⇑, Jian-Jun Wang b, Ji-Bo Wang c

a College of Mathematics, Jilin Normal University, Siping, Jilin 136000, Chinab Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, Chinac School of Science, Shenyang Aerospace University, Shenyang 110136, China

a r t i c l e i n f o

Keywords:SchedulingSingle machineTime-dependent processing timesGroup technologyReady times

a b s t r a c t

In this paper we investigate a single machine scheduling problem with decreasing time-dependent processing times and group technology assumption. By the decreasing time-dependent processing times and group technology assumption, we mean that the groupsetup times and job processing times are both decreasing linear functions of their startingtimes. We want to minimize the makespan subject to release dates. We show that theproblem can be solved in polynomial time.

� 2014 Published by Elsevier Inc.

1. Introduction

Traditional scheduling models and problems usually involve jobs with constant, independent processing times. In prac-tice, however, we often encounter settings in which job processing times increase or decrease over time, e.g., in the model-ling of the forging process in steel plants, finance management and scheduling maintenance or learning activities. This iswhy in recent years more and more researchers are considered scheduling problems with time-dependent processingtimes. Extensive surveys of different scheduling models and problems involving jobs with start time dependent processingtimes can be found in Alidaee and Womer [1], Cheng et al. [2] and Gawiejnowicz [3]. More recent papers which have con-sidered scheduling problems with job time-dependent processing times include Lee et al. [4], Wu et al. [5], Wu and Lee [6],Wang et al. [7], Wang et al. [8], Lee et al. [9], He et al. [10], Chung et al. [11], Li et al. [12], Yang and Kuo [13], Wang and Sun[14], Wei and Wang [15], Yang and Yang [16], Yang et al. [17], Zhang and Yan [18], Zhao and Tang [19], Lee et al. [20], Zhuet al. [21], Huang and Wang [22], Zhao and Tang [23], Wang et al. [24], Yang and Wang [25], Sun et al. [26], Wang et al. [27],Wang et al. [28], Lee and Lu [29], Zhang and Luo [30], Liu et al. [31], Wu et al. [32], Wang and Wang [33–37], Xu et al. [38].

Generally, there are two types of models describing this kind of scheduling. The first type is devoted to the problems inwhich the processing time of a job is an increasing (non-decreasing) function of its starting time (deteriorating job process-ing times). The second type concerns problems in which the processing time of a job is a decreasing (non-increasing) func-tion of its starting time (shortening processing times). Wang et al. [24] considered flow shop scheduling problem withdecreasing linear deterioration, i.e., the (actual) processing time of job Jj on machine Mh is phj ¼ ahjð1� btÞ;h ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;n, where ahj is the normal (basic) processing time of job Jj on machine Mh; t is its starting time,b > 0 is a decreasing rate such that ð1� btÞ > 0. When some dominance relations between m� 1 machines can be satisfied,they showed that the makespan minimization problem can be solved in polynomial time. Wang et al. [27] considered single

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Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286–292 287

machine scheduling problem with decreasing linear deterioration, i.e., the (actual) processing time of job Jj ispj ¼ ajða� btÞ; j ¼ 1;2; . . . ;n, where a > 0; b > 0 and ða� btÞ > 0. For the total absolute differences in waiting times mini-mization problem, they proved several properties of an optimal schedule, and proposed two heuristic algorithms.

On the other hand, scheduling models and problems in a group technology (GT) environment have attracted numerousresearchers due to their frequent real-life occurrence (Potts and Van Wassenhove [39], Webster and Baker [40], Liaee andEmmons [41], Janiak et al. [42], Bozorgirad and Logendran [43], Ji et al. [44]). Group technology that groups similar productsinto families helps increase the efficiency of operations and decrease the requirement of facilities.

It is natural to study the situations where group scheduling and time dependent processing times are combined. To thebest of our knowledge, only a few results concerning scheduling problems with time dependent processing times and grouptechnology simultaneously are known. Wu et al. [5] considered a situation where group setup times and job processing timesare both described by a simple linear deterioration function, i.e., the (actual) processing time of job Jj in group Gi ispij ¼ bijt; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ; ni, where bij > 0 is the deterioration rate of job Jj in group Gi; the (actual) setup timeof group Gi is si ¼ git, where gi > 0 is the deterioration rate of the setup time for group Gi. Using the extended three-fieldnotation scheme (Graham et al. [45]), they proved that the makespan minimization problem (1jpij ¼ bijt; si ¼ git;GTjCmax)and the total completion time minimization problem (1jpij ¼ bijt; si ¼ git;GTj

PPCij) can be solved in polynomial time,

where Cij represents the completion time of job Jj in group Gi, Cmax ¼maxfCijji ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nig represent make-span of a given schedule. Wu and Lee [6] considered a situation where group setup times and job processing times are bothdescribed by a linear deterioration function, i.e., pij ¼ aij þ bt; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni, and si ¼ di þ gt; i ¼ 1;2; . . . ;m,where aij P 0 is the normal (basic) processing time of job Jj in group Gi; b > 0 is the deterioration rate of jobs, di P 0 isthe normal (basic) setup time for group Gi; g is a deterioration rate of setup times. They showed that the makespan minimi-zation problem (1jpij ¼ aij þ bt; si ¼ di þ gt;GTjCmax) remain polynomially solvable. For the sum of completion times problem,they showed that the problem remains polynomially solvable under the assumption that the numbers of jobs in each groupare equal. Wang et al. [7] considered the following model: pij ¼ aijðaþ btÞ; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni, andsi ¼ diðaþ btÞ; i ¼ 1;2; . . . ;m. They proved that the problems 1jpij ¼ aijðaþ btÞ; si ¼ diðaþ btÞ;GTjCmax and1jpij ¼ aijðaþ btÞ; si ¼ diðaþ btÞ;GTj

PPwijCij can be solved in polynomial time, where wij denote the weight of job Jj in

group Gi. Wang et al. [8] considered a situation where group setup times and job processing times are both described bya general linear deterioration function, i.e., pij ¼ aij þ bijt; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni, and si ¼ di þ git; i ¼ 1;2; . . . ;m. Theyproved that the makespan minimization problem (1jpij ¼ aij þ bijt; si ¼ di þ git;GTjCmax) can be solved in polynomial time.Wei and Wang [15] proved that the problems 1jpij ¼ bijt; si ¼ git;GTj

PwijC

2ij and 1jpij ¼ bijt; si ¼ git;GTj

PwijW

2ij can be

solved in polynomial time, where Wij ¼ Cij � pij is the waiting time of job Jj in group Gi. Yang and Yang [16] considered sched-uling problems under the effects of deterioration and learning under group technology, i.e.,pij ¼ aijrai ; pij ¼ aijð1þ

Pr�1q¼1ai½q�Þ

ai; si ¼ dit, where ai � 0 denote the learning factor of group Gi, and r denote the job position.

They showed that the makespan minimization problems can be solved in polynomial time. They also showed that the totalcompletion time minimization problem have a polynomial optimal solution under agreeable restrictions. Zhang and Yan [18]considered group scheduling with deterioration and learning effect on a single machine, i.e., pij ¼ ðaij þ btÞra1 ka2 ; si ¼ dira1 ,where a1 � 0 and a2 � 0 denote the learning effect, and r and k denote the group position and the job position. They showedthat the makespan and the total completion time minimization problems can be solved in polynomial time. They alsoshowed that the maximum lateness minimization problem have a polynomial optimal solution under agreeable conditions.Lee and Lu [29] considered the problem 1jpij ¼ bijt; si ¼ git;GTj

PwjUj, where UjðpÞ ¼ 1 if CjðpÞ > dj and 0 otherwise, where

dj denote the due date of job Jj, they proposed a branch-and-bound algorithm to solve this problem.Wang and Sun [14] considered the group scheduling with linearly decreasing time-dependent setup times and job pro-

cessing times on a single machine, i.e., pij ¼ aij � bijt; si ¼ di � git, where bij is the decreasing rate of job Jj in group Gi. Theyproved that the problem 1jpij ¼ aij � bijt; si ¼ di � git;GTjCmax can be solved in polynomial time. For a special casebij ¼ baij; gi ¼ bdi, they proved that the problem 1jpij ¼ aijð1� btÞ; si ¼ dið1� btÞ;GTj

PwijCij can be solved in polynomial

time.Wang et al. [28] considered scheduling with independent setup times, ready times, and deteriorating job processing times

under group technology assumption on a single machine. They proved that the problem 1jrij; pij ¼ bijt; si;GTjCmax have a poly-nomial optimal solution under an agreeable condition, where rij denote the ready times (release dates) of job Jj in group Gi.Xu et al. [38] considered the problem 1jrij; pij ¼ bijðaþ btÞ; si;GTjCmax. For some special cases, they proved that the problemcan be solved in polynomial time.

In this paper we consider single machine group scheduling with release dates, decreasing time-dependent setup timesand job processing times (to our knowledge for the first time) at the same time. We show that the makespan minimizationproblem can be solved in polynomial time. The remaining part of the paper is organized as follows. In the next section, aprecise formulation of the problem is given. The problem of minimizing the makespan is given in the Section 3. The last sec-tion contains some conclusions.

2. Problem formulation

The single machine group scheduling problem with group setup times can be stated as follows: There are n jobs groupedinto m groups, and these n jobs are to be processed on a single machine. A setup time is required if the machine switches

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from one group to another and all setup times of groups for processing at time t0 P 0. We also assume that the processing ofa job may not be interrupted. Let ni be the number of jobs belonging to group Gi, thus, n1 þ n2 þ . . .þ nm ¼ n. Let Jij denote thejth job in group Gi, i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni; rij > 0 denote the ready (arrival) time of job Jij. Let pij be the actual process-ing time of the jth job in the group Gi; si be the setup time of group Gi. In this paper, we consider the following model:

pij ¼ aijð1� btÞ;

where aij > 0 is the normal processing time of the jth job in the group Gi, i.e., aij is the initial processing requirement to com-plete the jth job in the group Gi if it is starting at time 0, t is its start time, and 0 < b < 1 is the decreasing rate. As in the abovemodel, we also assume that the setup time of group Gi is

si ¼ dið1� btÞ;

where di > 0 is the normal setup time of the group Gi, i.e., di is the initial setup requirement to adjust the group Gi if it isstarting at time 0. It is assumed that normal processing times and setup times satisfy the following condition:

b t0 þ rmax þXm

i¼1

Xni

j¼1

aij þXm

i¼1

di � hmin

!< 1;

where rmax ¼maxfrjjj ¼ 1;2; . . . ;ng and hmin ¼minfaij; diji ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nig. The condition ensures that all thejob processing times are positive in any feasible schedule (see also Wang and Sun [14], and Ho et al. [46] for detailedexplanations).

For a given schedule p;CijðpÞ represents the completion time of job Jij in group Gi under schedule p. The objective is tominimize the makespan (the maximum completion time of all jobs), i.e., the problem1jrij; pij ¼ aijð1� btÞ; si ¼ dið1� btÞ;GTjCmax.

3. Results

First, we consider a single machine scheduling problem with decreasing time-dependent processing times and readytimes of the jobs. The objective function is to minimize the makespan of all jobs.

Lemma 1. For the problem 1jrj; pj ¼ ajð1� btÞjCmax, an optimal schedule can be obtained by sequencing the jobs in nondecreasingorder of rj.

Proof. Suppose that p and p0 are two job schedules. The difference between p and p0 is a pairwise interchange of two adja-cent jobs Ji and Jj, i.e., p ¼ ðS1; Ji; Jj; S2Þ and p0 ¼ ðS1; Jj; Ji; S2Þ, where S1 and S2 denote a partial sequence. In addition, let Bdenote the completion time of the last job in S1. Under p, the completion times of jobs Ji and Jj are

CiðpÞ ¼ maxfB; rig þ aið1� b maxfB; rigÞ

¼max B� 1b

� �ð1� baiÞ; ri �

1b

� �ð1� baiÞ

� �þ 1

bð1Þ

and

CjðpÞ ¼ maxfCi; rjg þ ajð1� b maxfCi; rjgÞ

¼ max B� 1b

� �ð1� baiÞð1� bajÞ; ri �

1b

� �ð1� baiÞð1� bajÞ; rj �

1b

� �ð1� bajÞ

� �þ 1

b:

ð2Þ

Similarly, the completion times of jobs Jj and Ji in p0 are respectively

Cjðp0Þ ¼ max B� 1b

� �ð1� bajÞ; rj �

1b

� �ð1� bajÞ

� �þ 1

bð3Þ

and

Ciðp0Þ ¼ max B� 1b

� �ð1� bajÞð1� baiÞ; rj �

1b

� �ð1� bajÞð1� baiÞ; ri �

1b

� �ð1� baiÞ

� �þ 1

b: ð4Þ

Based on Eqs. (2) and (4), we have

Ciðp0Þ � CjðpÞ ¼max B� 1b

� �ð1� bajÞð1� baiÞ; rj �

1b

� �ð1� bajÞð1� baiÞ; ri �

1b

� �ð1� baiÞ

� �

�max B� 1b

� �ð1� baiÞð1� bajÞ; ri �

1b

� �ð1� baiÞð1� bajÞ; rj �

1b

� �ð1� bajÞ

� �P 0:

if and only if ri 6 rj.

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Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286–292 289

Hence, we have Ciðp0ÞP CjðpÞ iff ri 6 rj. We conclude that an optimal schedule can be obtained by sequencing the jobs innondecreasing order of rj. h

Now, we consider a single machine group scheduling problem with decreasing time-dependent processing times, i.e., theproblem 1jrij; pij ¼ aijð1� btÞ; si ¼ dið1� btÞ;GTjCmax. We assume that D denotes the completion time of the ði� 1Þth groupand rið1Þ 6 rið2Þ 6 � � � 6 riðniÞ is satisfied in the ith group Gi. Then, the completion time of the ith group Gi are:

CiðniÞðGiÞ¼max D�1b

� �ð1�bdiÞ

Yni

l¼1

ð1�baiðlÞÞ; rið1Þ�1b

� �Yni

l¼1

ð1�baiðlÞÞ; rið2Þ�1b

� �Yni

l¼2

ð1�baiðlÞÞ;���; riðniÞ�1b

� �ð1�baiðniÞÞ

( )

þ1b¼max D�1

b;

riBðiÞ�1b

ð1�bdiÞQBðiÞ�1

l¼1 ð1�baiðlÞÞ

( )ð1�bdiÞ

Yni

l¼1

ð1�baiðlÞÞþ1b

where BðiÞ 2 f1;2; . . . ;nig denotes the index of a job in group Gi for which the maximal value of the expressionmaxfðrið1Þ � 1

bÞQni

l¼1ð1� baiðlÞÞ; ðrið2Þ � 1bÞQni

l¼2ð1� baiðlÞÞ; . . . ; ðriðniÞ � 1bÞð1� baiðniÞÞg.

Theorem 1. For the problem 1jrij; pij ¼ aijð1� btÞ; si ¼ dið1� btÞ;GTjCmax, the optimal schedule can be constructed in thefollowing way:

1. The job sequence in each group is in nondecreasing order of ri, i.e.,

rið1Þ 6 rið2Þ 6 . . . 6 riðniÞ; i ¼ 1;2; . . . ;m:

2. The groups are arranged in nondecreasing order of

riBðiÞ � 1b

ð1� diÞQBðiÞ�1

l¼1 ð1� baiðlÞÞ;

where ðriBðiÞ � 1bÞQni

l¼BðiÞð1� baiðlÞÞ ¼ maxfðrið1Þ � 1bÞQni

l¼1ð1� baiðlÞÞ; ðrið2Þ � 1bÞQni

l¼2ð1� baiðlÞÞ; . . . ; ðriðniÞ � 1bÞð1� baiðniÞÞg, BðiÞ 2

f1;2; . . . ; nig.

Proof. The form of 1 follows from Lemma 1.Next, we consider the case in item 2. Let p and p0 be two schedules where the difference between p and p0 is a pairwise

interchange of two adjacent groups Gi and Gj, that is, p ¼ ½S1;Gi;Gj; S2�; p0 ¼ ½S1;Gj;Gi; S2�, where S1 and S2 are partialsequences. Furthermore, we assume that A denote the completion time of the last job in S1. To show p dominates p0, itsuffices to show that CjðnjÞðpÞ 6 CiðniÞðp0Þ. Under p, using Eq. (5), we obtain that the completion time of the group Gi is

CiðniÞðpÞ ¼max A� 1b;

riBðiÞ � 1b

ð1� bdiÞQBðiÞ�1

l¼1 ð1� baiðlÞÞ

( )ð1� bdiÞ

Yni

l¼1

ð1� baiðlÞÞ þ1b

and the completion time of the group Gj is

CjðnjÞðpÞ¼max CiðniÞðpÞ�1b;

rjBðjÞ �1b

ð1�bdjÞQBðjÞ�1

l¼1 ð1�bajðlÞÞ

( )ð1�bdjÞ

Ynj

l¼1

ð1�baiðlÞÞþ1b

¼max A�1b

� �ð1�bdiÞ

Yni

l¼1

ð1�baiðlÞÞ;riBðiÞ �1

b

ð1�bdiÞQBðiÞ�1

l¼1 ð1�baiðlÞÞð1�bdiÞ

Yni

l¼1

ð1�baiðlÞÞ;rjBðjÞ � 1

b

ð1�bdjÞQBðjÞ�1

l¼1 ð1�bajðlÞÞ

( )

�ð1�bdjÞYnj

l¼1

ð1�bajðlÞÞþ1b

ð6Þ

Under p0, the completion times of the groups Gj and Gi are

CjðnjÞðp0Þ ¼max A� 1

b;

rjBðjÞ � 1b

ð1� bdjÞQBðjÞ�1

l¼1 ð1� bajðlÞÞ

( )ð1� bdjÞ

Ynj

l¼1

ð1� bajðlÞÞ þ1b

and the completion time of the group Gi is

CiðniÞðp0Þ¼max CjðnjÞðp

0Þ�1b;

riBðiÞ �1b

ð1�bdiÞQBðiÞ�1

l¼1 ð1�baiðlÞÞ

( )ð1�bdiÞ

Yni

l¼1

ð1�baiðlÞÞþ1b

¼max A�1b

� �ð1�bdjÞ

Ynj

l¼1

ð1�bajðlÞÞ;rjBðjÞ �1

b

ð1�bdjÞQBðjÞ�1

l¼1 ð1�bajðlÞÞð1�bdjÞ

Ynj

l¼1

ð1�bajðlÞÞ;riBðiÞ �1

b

ð1�bdiÞQBðiÞ�1

l¼1 ð1�baiðlÞÞ

( )

�ð1�bdiÞYni

l¼1

ð1�baiðlÞÞþ1b

ð7Þ

Based on Eqs. (6) and (7), we have

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290 Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286–292

CjðnjÞðpÞ � CiðniÞðp0Þ ¼max A� 1

b

� �ð1� bdiÞ

Yni

l¼1

ð1� baiðlÞÞ;riBðiÞ � 1

b

ð1� bdiÞQBðiÞ�1

l¼1 ð1� baiðlÞÞð1� bdiÞ

(

�Yni

l¼1

ð1� baiðlÞÞ;rjBðjÞ � 1

b

ð1� bdjÞQBðjÞ�1

l¼1 ð1� bajðlÞÞ

)ð1� bdjÞ

Ynj

l¼1

ð1� bajðlÞÞ

�max A� 1b

� �ð1� bdjÞ

Ynj

l¼1

ð1� bajðlÞÞ;rjBðjÞ � 1

b

ð1� bdjÞQBðjÞ�1

l¼1 ð1� bajðlÞÞ

(

� ð1� bdjÞYnj

l¼1

ð1� bajðlÞÞ;riBðiÞ � 1

b

ð1� bdiÞQBðiÞ�1

l¼1 ð1� baiðlÞÞ

)ð1� bdiÞ

Yni

l¼1

ð1� baiðlÞÞ

6 max A� 1b

� �ð1� bdiÞð1� bdjÞ

Yni

l¼1

ð1� baiðlÞÞYnj

l¼1

ð1� bajðlÞÞ;rjBðjÞ � 1

bQBðjÞ�1l¼1 ð1� bajðlÞÞ

ð1� bdiÞYnj

l¼1

ð1� bajðlÞÞYni

l¼1

ð1� baiðlÞÞ;(

rjBðjÞ � 1bQBðjÞ�1

l¼1 ð1� bajðlÞÞ

Ynj

l¼1

ð1� bajðlÞÞ)�max A� 1

b

� �ð1� bdjÞð1� bdiÞ

Ynj

l¼1

ð1� bajðlÞÞYni

l¼1

ð1� baiðlÞÞ;(

rjBðjÞ � 1bQBðjÞ�1

l¼1 ð1� bajðlÞÞð1� bdiÞYnj

l¼1

ð1� bajðlÞÞYni

l¼1

ð1� baiðlÞÞ;riBðiÞ�1

bQBðiÞ�1

l¼1ð1�baiðlÞÞ

Yni

l¼1

ð1� baiðlÞÞ

9>>>>=>>>>;¼ 0:

if and only if

riBðiÞ � 1b

ð1� bdiÞQBðiÞ�1

l¼1 ð1� baiðlÞÞ6

rjBðjÞ � 1b

ð1� bdjÞQBðjÞ�1

l¼1 ð1� bajðlÞÞ:

Therefore, we have CjðnjÞðpÞ 6 CiðniÞðp0Þ iff riBðiÞ�1b

ð1�bdiÞQBðiÞ�1

l¼1ð1�baiðlÞÞ

6rjBðjÞ�1

b

ð1�bdjÞQBðjÞ�1

l¼1ð1�bajðlÞÞ

. This completes the proof. h

Using Theorem 1, the problem 1jrij; pij ¼ aijð1� btÞ; si ¼ dið1� btÞ;GTjCmax can be solved by the following algorithm:

Algorithm 1

Step 1. Jobs in each group scheduled in nondecreasing order of rij, i.e.,

rið1Þ 6 rið2Þ 6 � � � 6 riðniÞ; i ¼ 1;2; . . . ;m:

Step 2. Let ðriBðiÞ � 1bÞQni

l¼BðiÞð1� baiðlÞÞ ¼maxfðrið1Þ � 1bÞQni

l¼1ð1� baiðlÞÞ; ðrið2Þ � 1bÞQni

l¼2ð1� baiðlÞÞ; . . . ; ðriðniÞ � 1bÞð1� baiðniÞÞg,

BðiÞ 2 f1;2; . . . ;nig, calculate BðiÞ and riBðiÞ�1b

ð1�bdiÞQBðiÞ�1

l¼1ð1�baiðlÞÞ

; i ¼ 1;2; . . . ;m.

Step 3. Groups scheduled in nondecreasing order of

qðGiÞ ¼riBðiÞ � a

b

ð1� bdiÞQBðiÞ�1

l¼1 ð1� baiðlÞÞ:

Obviously, the complexity of obtaining the optimal job sequence within group Gi is OðnilogniÞ and that of obtaining theoptimal group sequence is OðmlogmÞ. It is easy to show that

Pmi¼1Oðni log niÞ 6 Oðn log nÞ. Hence, the complexity of Algorithm

1 is at most OðnlognÞ, where n1 þ n2 þ � � � þ nm ¼ n. In addition, we demonstrate the algorithm in the following example.

Example 1. Let n ¼ 8; m ¼ 3; b ¼ 0:01 and t0 ¼ 1. Also, G1 : fJ11; J12g; d1 ¼ 1; a11 ¼ 1; a12 ¼ 3; r11 ¼ 16; r12 ¼ 6;G2 : fJ21; J22; J23g; d2 ¼ 2; a21 ¼ 1; a22 ¼ 2; a23 ¼ 3; r21 ¼ 9; r22 ¼ 6; r23 ¼ 8; G3 : fJ31; J32; J33g; d3 ¼ 3; a31 ¼ 4; a32 ¼ 1;a33 ¼ 2; r31 ¼ 5; r32 ¼ 8; r33 ¼ 2.

Solution. According to Algorithm 1, we solve Example 1 as follows:Step 1: In group G1, the optimal job sequence is J12 ! J11. In group G2, the optimal job sequence is J22 ! J23 ! J21. In group

G3, the optimal job sequence is J33 ! J31 ! J32.

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Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286–292 291

Step 2, 3: Next, we compute the following values for each group:G1: ðr1Bð1Þ � 1

bÞQn1

l¼Bð1Þð1� baiðlÞÞ ¼maxfð6� 100Þð1� 0:03Þð1� 0:01Þ; ð16� 100Þð1� 0:01Þg ¼ �83:1600, Bð1Þ ¼ 2; qðG1Þ¼ 16�100ð1�0:01Þð1�0:01Þ ¼ �85:7055;

G2: ðr2Bð2Þ � 1bÞQn2

l¼Bð2Þð1� baiðlÞÞ ¼ maxfð6� 100Þð1� 0:03Þð1� 0:09Þð1� 0:02Þ; ð8� 100Þð1� 0:09Þð1� 0:02Þ; ð9� 100Þð1� 0:02Þg ¼ �81:3143; Bð2Þ ¼ 1; qðG2Þ ¼ 6�100

1�0:02 ¼ �95:9184;

G3: ðr3Bð3Þ � 1bÞQn3

l¼Bð3Þð1� baiðlÞÞ ¼maxfð2� 100Þð1� 0:02Þð1� 0:04Þð1� 0:01Þ; ð5� 100Þð1� 0:04Þð1� 0:01Þ; ð8� 100Þð1� 0:01Þg ¼ �90:2880; Bð3Þ ¼ 2; qðG3Þ ¼ 5�100

ð1�0:03Þð1�0:02Þ ¼ �99:9369.

Since qðG3Þ ¼ �99:9369 < qðG2Þ ¼ �95:9184 < qðG1Þ ¼ �85:7055, hence, the optimal group sequence is G3 ! G2 ! G1.Therefore, the optimal schedule is ½J33 ! J31 ! J32� ! ½J22 ! J23 ! J21� ! ½J12 ! J11�, and the optimal value of the makespanis 21.5775.

4. Conclusions

In this paper we have considered the single machine scheduling problems with decreasing time-dependent job process-ing times and decreasing time-dependent group setup times. We showed that the makespan minimization problem withready times can be solved in polynomial time. In addition, we proposed an algorithm to solve the problem. In the future re-search, it is worth to consider more general decreasing time-dependent scheduling model and to investigate multi-machinescheduling problems.

Acknowledgments

We are grateful to two anonymous referees for their helpful comments on an earlier version of this paper. This researchwas supported by the Science and Technology Development Project of Jilin province of China (Grant No. 20140520057JH),the National Natural Science Foundation of China (71271039), New Century Excellent Talents in University (NCET-13-0082), Changjiang Scholars and Innovative Research Team in University (IRT1214).

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