Applied Mathematics and Computation 234 (2014) 286–292
Contents lists available at ScienceDirect
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
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
288 Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286–292
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.
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
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.
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).
References
[1] B. Alidaee, N.K. Womer, Scheduling with time dependent processing times: review and extensions, J. Oper. Res. Soc. 50 (1999) 711–720.[2] T.C.E. Cheng, Q. Ding, B.M.T. Lin, A concise survey of scheduling with time-dependent processing times, Eur. J. Oper. Res. 152 (2004) 1–13.[3] S. Gawiejnowicz, Time-Dependent Scheduling, Springer, Berlin, 2008.[4] W.C. Lee, Y.H. Chung, C.C. Wu, Scheduling deteriorating jobs on a single machine with release times, Comput. Ind. Eng. 54 (2008) 441–452.[5] C.C. Wu, Y.R. Shiau, W.-C. Lee, Single-machine group scheduling problems with deterioration consideration, Comput. Oper. Res. 35 (2008) 1652–1659.[6] C.C. Wu, W.C. Lee, Single-machine group-scheduling problems with deteriorating setup times and job-processing times, Int. J. Prod. Econ. 115 (2008)
128–133.[7] J.-B. Wang, L. Lin, F. Shan, Single-machine group scheduling problems with deteriorating jobs, Int. J. Adv. Manuf. Technol. 39 (2008) 808–812.[8] J.-B. Wang, W.-J. Gao, L.-Y. Wang, D. Wang, Single machine group scheduling with general linear deterioration to minimize the makespan, Int. J. Adv.
Manuf. Technol. 43 (2009) 146–150.[9] W.C. Lee, C.C. Wu, Y.H. Chung, H.C. Liu, Minimizing the total completion time in permutation flow shop with machine-dependent job deterioration
rates, Comput. Oper. Res. 36 (2009) 2111–2121.[10] C.C. He, C.C. Wu, W.C. Lee, Branch-and-bound and weight-combination search algorithms for the total completion time problem with step-
deteriorating jobs, J. Oper. Res. Soc. 60 (2009) 1759–1766.[11] Y.H. Chung, H.C. Liu, C.C. Wu, W.C. Lee, A deteriorating jobs problem with quadratic function of job lateness, Comput. Ind. Eng. 57 (2009) 1182–1186.[12] Y. Li, G. Li, L. Sun, Z. Xu, Single machine scheduling of deteriorating jobs to minimize total absolute differences in completion times, Int. J. Prod. Econ.
118 (2009) 424–429.[13] D.-L. Yang, W.-H. Kuo, Single-machine scheduling with both deterioration and learning effects, Ann. Oper. Res. 172 (2009) 315–327.[14] J.-B. Wang, L.-Y. Sun, Single-machine group scheduling with decreasing linear deterioration setup times and job processing times, Int. J. Adv. Manuf.
Technol. 49 (2010) 765–772.[15] C.-M. Wei, J.-B. Wang, Single machine quadratic penalty function scheduling with deteriorating jobs and group technology, Appl. Math. Model. 34
(2010) 3642–3647.[16] S.-J. Yang, D.-L. Yang, Single-machine group scheduling problems under the effects of deterioration and learning, Comput. Ind. Eng. 58 (2010) 754–758.[17] S.-J. Yang, D.-L. Yang, T.C.E. Cheng, Single-machine due-window assignment and scheduling with job-dependent aging effects and deteriorating
maintenance, Comput. Oper. Res. 37 (2010) 1510–1514.[18] X. Zhang, G. Yan, Single-machine group scheduling problems with deteriorated and learning effect, Appl. Math. Comput. 216 (2010) 1259–1266.[19] C.-L. Zhao, H.-Y. Tang, Scheduling deteriorating jobs under disruption, Int. J. Prod. Econ. 125 (2010) 294–299.[20] W.-C. Lee, Y.-R. Shiau, S.-K. Chen, C.-C. Wu, A two-machine flowshop scheduling problem with deteriorating jobs and blocking, Int. J. Prod. Econ. 124
(2010) 188–197.[21] V.C.Y. Zhu, L. Sun, L. Sun, X. Li, Single machine scheduling time-dependent jobs with resource-dependent ready times, Comput. Ind. Eng. 58 (2010) 84–
87.[22] X. Huang, M.-Z. Wang, Parallel identical machines scheduling with deteriorating jobs and total absolute differences penalties, Appl. Math. Model. 35
(2011) 1349–1353.[23] C.-L. Zhao, H.-Y. Tang, A note on two-machine no-wait flow shop scheduling with deteriorating jobs and machine availability constraints, Optim. Lett. 5
(2011) 183–190.[24] X.-Y. Wang, M.-Z. Wang, J.-B. Wang, Flow shop scheduling to minimize makespan with decreasing time-dependent job processing times, Comput. Ind.
Eng. 60 (2011) 840–844.[25] S.-H. Yang, J.-B. Wang, Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration, Appl.
Math. Comput. 217 (2011) 4819–4826.[26] L.-H. Sun, L.-Y. Sun, M.-Z. Wang, J.-B. Wang, Flow shop makespan minimization scheduling with deteriorating jobs under dominating machines, Int. J.
Prod. Econ. 138 (2012) 195–200.
292 Y.-Y. Lu et al. / Applied Mathematics and Computation 234 (2014) 286–292
[27] D. Wang, Y.-B. Wu, J.-B. Wang, P. Ji, Single-machine scheduling with decreasing time-dependent processing times to minimize total absolutedifferences in waiting times, J. Chin. Inst. Ind. Eng. 29 (2012) 444–453.
[28] J.-B. Wang, X. Huang, Y.-B. Wu, P. Ji, Group scheduling with independent setup times, ready times, and deteriorating job processing times, Int. J. Adv.Manuf. Technol. 60 (2012) 643–649.
[29] W.-C. Lee, Z.-S. Lu, Group scheduling with deteriorating jobs to minimize the total weighted number of late jobs, Appl. Math. Comput. 218 (2012)8750–8757.
[30] M. Zhang, C. Luo, Parallel-machine scheduling with deteriorating jobs, rejection and a fixed non-availability interval, Appl. Math. Comput. 224 (2013)405–411.
[31] P. Liu, N. Yi, X. Zhou, H. Gong, Scheduling two agents with sum-of-processing-times-based deterioration on a single machine, Appl. Math. Comput. 219(2013) 8848–8855.
[32] C.-C. Wu, S.-R. Cheng, W.-H. Wu, Y. Yin, W.-H. Wu, The single-machine total tardiness problem with unequal release times and a linear deterioration,Appl. Math. Comput. 219 (2013) 10401–10415.
[33] J.-B. Wang, M.-Z. Wang, Minimizing makespan in three-machine flow shops with deteriorating jobs, Comput. Oper. Res. 40 (2013) 547–557.[34] J.-B. Wang, M.-Z. Wang, Solution algorithms for the total weighted completion time minimization flow shop scheduling with shortening job processing
times, Int. J. Adv. Manuf. Technol. 67 (2013) 243–253.[35] J.-B. Wang, M.-Z. Wang, Minimizing makespan on three-machine flow shop scheduling with deteriorating jobs, Asia-Pacific J. Oper. Res. 30 (2013)
1350022.[36] X.-R. Wang, J.-J. Wang, Single-machine scheduling with convex resource dependent processing times and deteriorating jobs, Appl. Math. Model. 37
(2013) 2388–2393.[37] X.-Y. Wang, J.-J. Wang, Single-machine due date assignment problem with deteriorating jobs and resource-dependent processing times, Int. J. Adv.
Manuf. Technol. 67 (2013) 255–260.[38] Y.-T. Xu, Y. Zhang, X. Huang, Single-machine ready times scheduling with group technology and proportional linear deterioration, Appl. Math. Model.
38 (2014) 384–391.[39] C.N. Potts, L.N. Van Wassenhove, Integrating scheduling with batching and lot-sizing: a review of algorithms and complexity, J. Oper. Res. Soc. 43
(1992) 395–406.[40] S. Webster, K.R. Baker, Scheduling groups of jobs on a single machine, Oper. Res. 43 (1995) 692–703.[41] M.M. Liaee, H. Emmons, Scheduling families of jobs with setup times, Int. J. Prod. Econ. 51 (1997) 165–176.[42] A. Janiak, M.Y. Kovalyov, M.-C. Portmann, Single machine group scheduling with resource dependent setup and processing times, Eur. J. Oper. Res. 162
(2005) 112–121.[43] M.A. Bozorgirad, R. Logendran, Bi-criteria group scheduling in hybrid flowshops, Int. J. Prod. Econ. 145 (2013) 599–612.[44] M. Ji, K. Chen, J. Ge, T.C.E. Cheng, Group scheduling and job-dependent due window assignment based on a common flow allowance, Comput. Ind. Eng.
68 (2014) 35–41.[45] R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Ann.
Discrete Math. 5 (1979) 287–326.[46] K.I.-J. Ho, J.Y.T. Leung, W.-D. Wei, Complexity of scheduling tasks with timedependent execution times, Inf. Process. Lett. 48 (1993) 315–320.