Information Sciences xxx (2014) xxx–xxx

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Single machine group scheduling with time dependentprocessing times and ready times

http://dx.doi.org/10.1016/j.ins.2014.02.0340020-0255/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +86 411 84707834.E-mail addresses: wangjibo75@163.com (J.-B. Wang), wangjianjun_2008@163.com, wmeagle717@163.com (J.-J. Wang).

Please cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times antimes, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034

Ji-Bo Wang a, Jian-Jun Wang b,⇑a School of Science, Shenyang Aerospace University, Shenyang 110136, Chinab Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, China

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

Article history:Received 13 March 2009Accepted 6 February 2014Available online xxxx

Keywords:SchedulingSingle machineStart time dependent processing timeGroup technologyReady time

In this paper we investigate a single machine scheduling problem with time dependentprocessing times and group technology (GT) assumption. By time dependent processingtimes and group technology assumption, we mean that the group setup times and jobprocessing times are both increasing functions of their starting times, i.e., group setuptimes and job processing times are both described by function which is proportional to alinear function of time. We attempt to minimize the makespan with ready times ofthe jobs. We show that the problem can be solved in polynomial time when start timedependent processing times and group technology are considered simultaneously.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Traditional scheduling problems usually involve jobs with constant, independent processing times (Janiak et al. [12],Wang and Wei [30]). In practice, however, we often encounter settings in which the job processing times vary with learningeffect (Cheng et al. [5,6], Lee et al. [18,19], Rudek [21], Sun et al. [22,23], Wang et al. [31], Wu et al. [35], Yeh et al. [40], Yinet al. [41]) and deterioration effect (Alidaee and Womer [1], Cheng et al. [2] and Gawiejnowicz [7]). Scheduling problemsinvolving deterioration effect (time dependent processing times) appears, e.g., in scheduling maintenance jobs, steelproduction, national defense, emergency medicine or cleaning assignments, where any delay in processing a job is penalizedby incurring additional time for accomplishing the job. Extensive surveys of different scheduling models and problemsinvolving jobs with start time dependent processing times can be found in Alidaee and Womer [1], Cheng et al. [2] andGawiejnowicz [7]. More recent papers which have considered scheduling jobs with deterioration effects include Chenget al. [3,4], Gawiejnowicz [8], Gawiejnowicz et al. [9], Hsu et al. [11], Janiak and Kovalyov [13], Kang and Ng [15], Leeet al. [17], Wang [24], Wang et al. [25–28], Wang and Sun [29], Wei and Wang [33], Wu and Lee [34], Wu et al. [36–38], Yangand Wang [39], Yin et al. [42], and Xu et al. [43].

On the other hand, many manufacturers have implemented the concept of group technology (GT) in order to reduce setupcosts, lead times, work-in-process inventory costs, and material handling costs. In GT scheduling, it is conventional toschedule continuously all jobs from the same group. Group technology that groups similar products into families helpsincrease the efficiency of operations and decrease the requirement of facilities (Janiak et al. [14], Liaee and Emmons [16],Potts and Van Wassenhove [20], Webster and Baker [32]).

It is natural to study the situations where scheduling with group technology and start time dependent processing times(time-dependent scheduling) are combined. For the case of the setup time of each group is a fixed constant: Wang et al. [27]

d ready

2 J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxx–xxx

considered single machine group scheduling in which the actual processing time of a job is a general linear decreasing func-tion of its starting time. For the makespan minimization problem and total completion time minimization problem theyshowed that some problems can be solved in polynomial time. Wang et al. [25] and Xu et al. [43] considered the single ma-chine scheduling problems with group technology and release times. Wang et al. [25] considered a simple linear deteriora-tion function and Xu et al. [43] considered a proportional linear deterioration function.

For the case of the setup time of each group is a linear deterioration function of its starting time: Since longer setup orpreparation might be necessary as food quality deteriorates or a patient’s condition worsens, Wu et al. [37] considered a sit-uation where the setup time groups and jobs in each group deteriorate as they wait for processing, i.e., group setup times andjob processing times are both described by a simple linear deterioration function. They proved that the single machine make-span minimization problem and the total completion time minimization problem can be solved in polynomial time. Wanget al. [28] considered single machine scheduling where group setup times and processing times of jobs are both described bya proportional linear deterioration function. For the makespan and the total completion time minimization problems, theyproposed a polynomial time solution, respectively. Wu and Lee [34] considered a situation where group setup times and jobprocessing times are both described by a linear deterioration function. They showed that the makespan minimization prob-lem remain polynomially solvable. For the sum of completion times problem, they showed that the problem remains poly-nomially solvable for a special case. Wang et al. [26] considered a situation where the group setup times and processingtimes of jobs are both described by a general linear deterioration function. They proved that the single machine makespanminimization problem can be solved in polynomial time.

In reality, longer setup or preparation might be necessary as food quality deteriorates or a patient’s condition worsens. Inthis paper, we consider the single machine scheduling with ready times of the jobs under the group technology assumptionand starting time-dependent setup times. The remaining part of the paper is organized as follows. In the next section, a pre-cise formulation of the problem is given. The problem of minimization of the makespan is given in the Section 3. The lastsection contains some conclusions.

2. Problem formulation

The single machine group scheduling problem with group setup times can be stated as follows: We consider nnon-preemptive jobs to be grouped into m groups and to be processed on a single machine. The jobs in the same group areconsecutively processed and a setup time is required if the machine switches from one group to another and all setup timesof groups for processing at time t0 > 0. Let Jij denote the jth job in group Gi; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni, rij > 0 denote theready (arrival) time of job Jij, where ni denotes the number of jobs belonging to group Gi (i.e., n1 þ n2 þ . . .þ nm ¼ n). As inWang et al. [28], we consider the following proportional deterioration model, i.e., the actual processing time of job Jij is

Pleasetimes,

pij ¼ aijðaþ btÞ;

where aij is the deterioration rate of job Jij and t is its starting time. As in the above proportional model, we also assume thatthe setup time of group Gi is a proportional deterioration model, i.e., the actual setup time of group Gi is

si ¼ diðaþ btÞ;

where di is the deterioration rate of the group Gi. Our objective is to find the optimal group sequence and the optimal jobsequence within each group to minimize the maximum completion time of all jobs, i.e., the makespan.

For a given schedule p, let Cij ¼ CijðpÞ denote the completion time of job Jij, and Cmax ¼ maxfCijji ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nig represent the makespan of a given schedule. Using the three-field notation schema in scheduling problems(Graham et al. [10]) the makespan minimization problem is denoted as 1jrj; pij ¼ aijðaþ btÞ; si ¼ diðaþ btÞ; GTjCmax.

3. The makespan minimization problem

In the following section, we will prove that the single machine makespan minimization scheduling problem withdeteriorating jobs and ready times of the jobs is polynomially solvable.

Lemma 1. For the problem 1jrj; pj ¼ ajðaþ btÞjCmax, the optimal job sequence can be obtained by sequencing the jobs in thenondecreasing order of rj.

Proof. Suppose that p ¼ ½S1; Ji; Jj; S2� and p0 ¼ ½S1; Jj; Ji; S2� are two job sequences, where S1 and S2 denote a partialsequence (note that S1 and S2 may be empty) and the difference between p and p0 is a pairwise interchange of two adjacentjobs Ji and Jj. In addition, denote by A as the completion time of the last job of S1 in sequence p (p0). Then the completiontimes of jobs Ji and Jj under p are

CiðpÞ ¼maxfA; rig þ aiðaþ b maxfA; rigÞ ¼ max Aþ ab

� �ð1þ baiÞ; ri þ

ab

� �ð1þ baiÞ

n o� a

bð1Þ

cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readyInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034

J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxx–xxx 3

and

Pleasetimes,

CjðpÞ ¼maxfCi; rjg þ ajðaþ b maxfCi; rjgÞ

¼max Aþ ab

� �ð1þ baiÞð1þ bajÞ; ri þ

ab

� �ð1þ baiÞð1þ bajÞ; rj þ

ab

� �ð1þ bajÞ

n o� a

b: ð2Þ

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

Cjðp0Þ ¼max Aþ ab

� �ð1þ bajÞ; rj þ

ab

� �ð1þ bajÞ

n o� a

bð3Þ

and

Ciðp0Þ ¼max Aþ ab

� �ð1þ bajÞð1þ baiÞ; rj þ

ab

� �ð1þ bajÞð1þ baiÞ; ri þ

ab

� �ð1þ baiÞ

n o� a

b: ð4Þ

Suppose ri 6 rj, based on Eqs. (2) and (4), we have

Ciðp0Þ � CjðpÞ ¼max Aþ ab

� �ð1þ bajÞð1þ baiÞ; rj þ

ab

� �ð1þ bajÞð1þ baiÞ; ri þ

ab

� �ð1þ baiÞ

n o�max Aþ a

b

� �ð1þ baiÞð1þ bajÞ; ri þ

ab

� �ð1þ baiÞð1þ bajÞ; rj þ

ab

� �ð1þ bajÞ

n oP 0:

Hence, we have Ciðp0ÞP CjðpÞ. Repeating this interchange argument, we conclude that an optimal schedule can beobtained by sequencing the jobs in nondecreasing order of rj. h

Now we consider the problem 1jrj; pij ¼ aijðaþ btÞ; si ¼ diðaþ btÞ; GTjCmax. We assume that A denotes the completiontime of the ði� 1Þth group and rið1Þ 6 rið2Þ 6 . . . 6 riðniÞ is satisfied in the ith group Gi. Then, for the ith group Gi, we have

CiðniÞðGiÞ¼max Aþab

� �ð1þbdiÞ

Yni

l¼1

ð1þbaiðlÞÞ; rið1Þ þab

� �Yni

l¼1

ð1þbaiðlÞÞ; rið2Þ þab

� �Yni

l¼2

ð1þbaiðlÞÞ; . . . ; riðniÞ þab

� �ð1þbaiðniÞÞ

( )�a

b

¼max Aþab;

rBðiÞ þ ab

ð1þbdiÞQBðiÞ�1

l¼1 ð1þbaiðlÞÞ

( )ð1þbdiÞ

Yni

l¼1

ð1þbaiðlÞÞ�ab

ð5Þ

where ðrBðiÞ þ abÞQni

l¼BðiÞð1þ baiðlÞÞ ¼max rið1Þ þ ab

� �Qnil¼1ð1þ baiðlÞÞ; rið2Þ þ a

b

� �Qnil¼2ð1þ baiðlÞÞ; . . . ; riðniÞ þ a

b

� �ð1þ baiðniÞÞ

� �, BðiÞ 2

f1;2; . . . ; nig.

Theorem 1. For the problem 1jrj; pij ¼ aijðaþ btÞ; si ¼ diðaþ btÞ; GTjCmax, the optimal schedule satisfies the following:

(1) The job sequence in each group is in nondecreasing order of rj, i.e.,

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

(2) The groups are arranged in nondecreasing order of

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞ;

where rBðiÞ þ ab

� �Qnil¼BðiÞð1þ baiðlÞÞ ¼ max ðrið1Þ þ a

bÞQni

l¼1ð1þ baiðlÞÞ; rið2Þ þ ab

� �Qnil¼2ð1þ baiðlÞÞ; . . . ; riðniÞ þ a

b

� �ð1þ baiðniÞÞ

� �, BðiÞ 2

f1;2; . . . ; nig.

Proof. In the same group, the result of (1) can be easily obtained by 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 Aþ ab;

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞð1þ bdiÞ

Yni

l¼1ð1þ baiðlÞÞ �

ab

and the completion time of the group Gj is

CjðnjÞðpÞ ¼max CiðniÞðpÞ þab;

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞ

( )ð1þ bdjÞ

Ynj

l¼1ð1þ baiðlÞÞ �

ab

¼max Aþ ab

� �ð1þ bdiÞ

Yni

l¼1ð1þ baiðlÞÞ;

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞð1þ bdiÞ

Yni

l¼1ð1þ baiðlÞÞ;

(

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞ

)ð1þ bdjÞ

Ynj

l¼1ð1þ bajðlÞÞ �

ab

ð6Þ

cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readyInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034

4 J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxx–xxx

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

Pleasetimes,

CjðnjÞðp0Þ ¼max Aþ a

b;

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞ

( )ð1þ bdjÞ

Ynj

l¼1ð1þ bajðlÞÞ �

ab

and the completion time of the group Gi is

CiðniÞðp0Þ ¼max CjðnjÞðp

0Þ þ ab;

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞ

( )ð1þ bdiÞ

Yni

l¼1

ð1þ baiðlÞÞ �ab

¼max Aþ ab

� �ð1þ bdjÞ

Ynj

l¼1ð1þ bajðlÞÞ;

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞð1þ bdjÞ

Ynj

l¼1

ð1þ bajðlÞÞ;(

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞ

)ð1þ bdiÞ

Yni

l¼1

ð1þ baiðlÞÞ �ab

ð7Þ

Suppose

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞ6

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞ;

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

CjðnjÞðpÞ � CiðniÞðp0Þ

¼max Aþ ab

� �ð1þ bdiÞ

Yni

l¼1

ð1þ baiðlÞÞ;(

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞð1þ bdiÞ

Yni

l¼1

ð1þ baiðlÞÞ;

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞ

)ð1þ bdjÞ

Ynj

l¼1

ð1þ bajðlÞÞ �max Aþ ab

� �ð1þ bdjÞ

Ynj

l¼1ð1þ bajðlÞÞ;

n

rBðjÞ þ ab

ð1þ bdjÞQBðjÞ�1

l¼1 ð1þ bajðlÞÞð1þ bdjÞ

Ynj

l¼1ð1þ bajðlÞÞ;

rBðiÞ þ ab

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞ

)ð1þ bdiÞ

Yni

l¼1

ð1þ baiðlÞÞ

6 max Aþ ab

� �ð1þ bdiÞð1þ bdjÞ

Yni

l¼1

ð1þ baiðlÞÞYnj

l¼1ð1þ bajðlÞÞ;

(

rBðjÞ þ abQBðjÞ�1

l¼1 ð1þ bajðlÞÞð1þ bdiÞ

Ynj

l¼1ð1þ bajðlÞÞ

Yni

l¼1ð1þ baiðlÞÞ;

rBðjÞ þ abQBðjÞ�1

l¼1 ð1þ bajðlÞÞ

Ynj

l¼1

ð1þ bajðlÞÞ)

�max Aþ ab

� �ð1þ bdjÞð1þ bdiÞ

Ynj

l¼1

ð1þ bajðlÞÞYni

l¼1

ð1þ baiðlÞÞ;(

rBðjÞ þ abQBðjÞ�1

l¼1 ð1þ bajðlÞÞð1þ bdiÞ

Ynj

l¼1

ð1þ bajðlÞÞYni

l¼1

ð1þ baiðlÞÞ;rBðiÞ þ a

bQBðiÞ�1l¼1 ð1þ baiðlÞÞ

Yni

l¼1

ð1þ baiðlÞÞ)¼ 0:

Therefore,

CjðnjÞðpÞ 6 CiðniÞðp0Þ:

This completes the proof. h

Based on the result of Theorem 1, for the problem 1jrj; pij ¼ aijðaþ btÞ; si ¼ diðaþ btÞ;GTjCmax, we provide a simple algo-rithm presented in the following:

Algorithm 1.

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

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

Step 2. Let rBðiÞ þ ab

� �Qnil¼BðiÞð1þ baiðlÞÞ ¼max rið1Þ þ a

b

� �Qnil¼1ð1þ baiðlÞÞ; rið2Þ þ a

b

� �Qnil¼2ð1þ baiðlÞÞ; . . . ; riðniÞ þ a

b

� �ð1þ baiðniÞÞ

� �,

BðiÞ 2 f1;2; . . . ;nig, calculate BðiÞ and rBðiÞþab

ð1þbdiÞQBðiÞ�1

l¼1ð1þbaiðlÞÞ

; i ¼ 1;2; . . . ;m.

Step 3. Groups scheduled in nondecreasing order of

cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readyInform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034

J.-B. Wang, J.-J. Wang / Information Sciences xxx (2014) xxx–xxx 5

Pleasetimes,

qðGiÞ ¼rBðiÞ þ a

b

ð1þ bdiÞQBðiÞ�1

l¼1 ð1þ baiðlÞÞ:

Obviously, the complexity of obtaining the optimal job sequence within a certain group Gi is Oðni log niÞ and that ofobtaining the optimal group sequence is Oðm log mÞ. It is easy to show that

Pmi¼1Oðni log niÞ 6 Oðn log nÞ. Hence, the complex-

ity of Algorithm 1 is at most Oðn log nÞ. In addition, we demonstrate Algorithm 1 by the following example.

Example 1. There are eight jobs (n ¼ 8) divided into three groups (m ¼ 3) to be processed on a single machine. Leta ¼ 1; b ¼ 0:1; t0 ¼ 1 and G1 : f J11; 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 groupG3, the optimal job sequence is J33 ! J31 ! J32.Step 2, 3: Next, we compute the following values for each group:

G1 : rBð1Þ þ ab

� � Yn1

l¼Bð1Þð1þbaiðlÞÞ ¼maxfð6þ10Þð1þ0:3Þð1þ0:1Þ;ð16þ10Þð1þ0:1Þg¼28:6; Bð1Þ¼2;

qðG1Þ¼ 16þ10ð1þ0:1Þð1þ0:1Þ ¼21:4876;

G2 : rBð2Þ þ ab

� � Yn2

l¼Bð2Þð1þbaiðlÞÞ ¼maxfð6þ10Þð1þ0:3Þð1þ0:9Þð1þ0:2Þ;ð8þ10Þð1þ0:9Þð1þ0:2Þ;ð9þ10Þð1þ0:2Þg¼47:424;

Bð2Þ¼1;qðG2Þ¼ 6þ101þ0:2¼13:3333;

G3 : rBð3Þ þ ab

� � Yn3

l¼Bð3Þð1þbaiðlÞÞ ¼maxfð2þ10Þð1þ0:2Þð1þ0:4Þð1þ0:1Þ;ð5þ10Þð1þ0:4Þð1þ0:1Þ;ð8þ10Þð1þ0:1Þg¼23:1;

Bð3Þ¼2; qðG3Þ¼ 5þ10ð1þ0:3Þð1þ0:2Þ ¼9:6154:

Since qðG3Þ ¼ 9:6154 < qðG2Þ ¼ 13:3333 < qðG1Þ ¼ 21:4876, 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 makespan is 137.8516.

4. Conclusions

Group technology is important to modem manufacturing industry. Scheduling with time dependent processing times hasbeen widely studied. In this paper we have considered the scheduling problem with group technology and time dependentprocessing times. For the case of the group setup times and job processing times are proportional linear deterioration func-tions, we showed that the makespan minimization problem with ready times can be solved in polynomial time. In addition,we proposed a algorithm to solve the problem. Future research may focus on considering more general deterioration func-tion, proposing more general and practical models with group technology, or investigating multi-machine schedulingproblems.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (71271039), New Century ExcellentTalents in University (NCET-13-0082), Changjiang Scholars and Innovative Research Team in University (IRT1214), andthe open project of The State Key Laboratory for Manufacturing Systems Engineering (Xi’an Jiaotong University) (GrantNo. sklms201306).

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Please cite this article in press as: J.-B. Wang, J.-J. Wang, Single machine group scheduling with time dependent processing times and readytimes, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.034