11
Single-machine scheduling problems with precedence constraints and simple linear deterioration Ji-Bo Wang a,, Jian-Jun Wang b a School of Science, Shenyang Aerospace University, Shenyang 110136, China b Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, China article info Article history: Received 9 October 2010 Received in revised form 27 June 2014 Accepted 23 July 2014 Available online xxxx Keywords: Scheduling Deteriorating jobs Single machine Precedence constraint abstract This paper deals with single machine scheduling problems with simple linear deterioration in which the processing time of a job is a simple linear function of its execution starting time. The objective is to determine the optimal schedule to minimize the weighted sum of the hth (h is a positive integer number) power of waiting times. It is proved that the gen- eral problem can be solved in polynomial time. In addition, for the jobs with weak (strong) parallel chains and a series–parallel digraph precedence constraints, it is also proved that these problems can be solved in polynomial time, respectively. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Time-dependent scheduling problems have received increasing attention in recent years. Generally, there are two types of models describing such kind of process, i.e., scheduling problems with deteriorating job processing times and scheduling problems with shorting job processing times. Scheduling problems with deteriorating job processing times appear, e.g., in scheduling maintenance jobs or cleaning assignments, where any delay in processing a job is penalized by incurring addi- tional time for accomplishing the job. Scheduling problems with shorting job processing times appear, e.g., in recognizing aerial threats [1]. An extensive surveys of different scheduling models and problems involving deteriorating (shorting) job processing times can be found in Gawiejnowicz [2]. More recent papers which have considered scheduling jobs with dete- riorating (shorting) job processing times Wang et al. [3], Lee et al. [4], Wang [5], Gao et al. [6], Wei and Wang [7], Zhao and Tang [8], Moslehi and Jafari [9], Wang and Guo [10], Wang and Wang [11], Wang and Wang [12], Sun et al. [13], Wei et al. [14], Sun et al. [15], Wang et al. [16], Wang and Wang [17], Low and Lin [18], Wang and Wang [19], Xu et al. [20], Lu et al. [21], Wang and Wang [22], Yin et al. [23], Yin et al. [24], and Yin et al. [25]. Most of the works mentioned above consider the scheduling problems without precedence constraints. It is relatively unex- plored in the precedence constraints environment. In this paper we consider single machine scheduling problems with a sim- ple linear deterioration. The objective is to find a schedule in order to minimize the weighted sum of the hth (h is a positive integer number) power of waiting times (i.e., P n j¼1 w j W h j , where w j (W j ) is the weight (waiting time) of job J j ; n is a positive integer number of jobs). Minimizing this objective function would mean reducing the inventory costs of raw materials and work-in-process inventories [26]. For the classical problem of minimizing the weighted sum of squared waiting times, i.e., http://dx.doi.org/10.1016/j.apm.2014.07.028 0307-904X/Ó 2014 Elsevier Inc. All rights reserved. Corresponding author. E-mail address: [email protected] (J.-B. Wang). Applied Mathematical Modelling xxx (2014) xxx–xxx Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm Please cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin- ear deterioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

Single-machine scheduling problems with precedence constraints and simple linear deterioration

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Applied Mathematical Modelling xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Applied Mathematical Modelling

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

Single-machine scheduling problems with precedenceconstraints and simple linear deterioration

http://dx.doi.org/10.1016/j.apm.2014.07.0280307-904X/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (J.-B. Wang).

Please cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simear deterioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

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 9 October 2010Received in revised form 27 June 2014Accepted 23 July 2014Available online xxxx

Keywords:SchedulingDeteriorating jobsSingle machinePrecedence constraint

This paper deals with single machine scheduling problems with simple linear deteriorationin which the processing time of a job is a simple linear function of its execution startingtime. The objective is to determine the optimal schedule to minimize the weighted sumof the hth (h is a positive integer number) power of waiting times. It is proved that the gen-eral problem can be solved in polynomial time. In addition, for the jobs with weak (strong)parallel chains and a series–parallel digraph precedence constraints, it is also proved thatthese problems can be solved in polynomial time, respectively.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Time-dependent scheduling problems have received increasing attention in recent years. Generally, there are two typesof models describing such kind of process, i.e., scheduling problems with deteriorating job processing times and schedulingproblems with shorting job processing times. Scheduling problems with deteriorating job processing times appear, e.g., inscheduling maintenance jobs or cleaning assignments, where any delay in processing a job is penalized by incurring addi-tional time for accomplishing the job. Scheduling problems with shorting job processing times appear, e.g., in recognizingaerial threats [1]. An extensive surveys of different scheduling models and problems involving deteriorating (shorting) jobprocessing times can be found in Gawiejnowicz [2]. More recent papers which have considered scheduling jobs with dete-riorating (shorting) job processing times Wang et al. [3], Lee et al. [4], Wang [5], Gao et al. [6], Wei and Wang [7], Zhao andTang [8], Moslehi and Jafari [9], Wang and Guo [10], Wang and Wang [11], Wang and Wang [12], Sun et al. [13], Wei et al.[14], Sun et al. [15], Wang et al. [16], Wang and Wang [17], Low and Lin [18], Wang and Wang [19], Xu et al. [20], Lu et al.[21], Wang and Wang [22], Yin et al. [23], Yin et al. [24], and Yin et al. [25].

Most of the works mentioned above consider the scheduling problems without precedence constraints. It is relatively unex-plored in the precedence constraints environment. In this paper we consider single machine scheduling problems with a sim-ple linear deterioration. The objective is to find a schedule in order to minimize the weighted sum of the hth (h is a positiveinteger number) power of waiting times (i.e.,

Pnj¼1wjW

hj , where wj (Wj) is the weight (waiting time) of job Jj; n is a positive

integer number of jobs). Minimizing this objective function would mean reducing the inventory costs of raw materials andwork-in-process inventories [26]. For the classical problem of minimizing the weighted sum of squared waiting times, i.e.,

ple lin-

2 J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx

Pnj¼1wjW

2j , Szwarc and Mukhopadhyay [26] first formulated this problem and presented a powerful decomposition

mechanism. They incorporated this mechanism with new branching rules in a branch-and-bound method that efficiently han-dles tough problems. Based on our knowledge the scheduling problems with a simple linear deterioration, precedence con-straints and the objective function is to minimize the weighted sum of the hth power of waiting times has not beenconsidered so far.

The remaining of this paper is organized as follows. In Section 2, we consider the general problem (i.e., without prece-dence constraints). The problems under parallel chains and a series–parallel digraph are studied in the third section andthe fourth section, respectively. The last section contains some concluding remarks.

2. A general problem without precedence constraints

The single machine problem is to schedule n jobs N ¼ fJ1; J2; . . . ; Jng on one machine. All the jobs are available for process-ing at some time t0 > 0. Each job Jj has a deterioration rate bj and a weight wj (deterioration rates bj and weights wj are posi-tive respectively). As in Mosheiov [27], we assume that the (actual) processing time pj of job Jj is a simple linear function ofits starting time, i.e.,

Pleaseear de

pj ¼ bjt; j ¼ 1;2; . . . ;n; ð1Þ

where t P t0 > 0 is the job’s starting time.For a given schedule p ¼ ½J1; J2; . . . ; Jn�, let Cj ¼ CjðpÞ be the completion time of job Jj. The goal is to find a schedule of n jobs

in order to minimizePn

j¼1wjWhj , where Wj ¼ Cj � pj ¼ Cj�1 is the waiting time of job Jj, h is a positive integer number. Using

the three-field notation for scheduling problem, this problem can be denoted as 1jpj ¼ bjtjP

wjWhj .

Lemma 1 (Mosheiov [27]). For a given schedule p ¼ ½J1; J2; . . . ; Jn� of the 1jpj ¼ bjtjP

wjWhj problem, if the first job (i.e., J1) starts

at time t0 > 0, then job Jj’s waiting time Wj is equal to

Wj ¼ Cj�1 ¼ t0

Yj�1

i¼1

ð1þ biÞ: ð2Þ

Theorem 1. For the 1jpj ¼ bjtjP

wjWhj problem, an optimal schedule can be obtained by sequencing the jobs in non-decreasing

order of ð1þbjÞh�1

wj.

Proof. . By contradiction. Let p ¼ ½S1; Jj; Jk; S2�;p0 ¼ ½S1; Jk; Jj; S2�, where S1 and S2 are partial sequences. Assume that the sche-

dule p in which ð1þbjÞh�1

wj> ð1þbkÞ

h�1wk

is optimal. The weighted sum of the hth (h > 0) power of waiting times of the jobs in S1 and

S2 is not affected by the interchange. Let t denote the completion time of the last job in S1. The waiting times for jobs Jj and Jk

are

WjðpÞ ¼ t: ð3Þ

WkðpÞ ¼ tð1þ bjÞ: ð4Þ

Wkðp0Þ ¼ t ð5Þ

and

Wjðp0Þ ¼ tð1þ bkÞ: ð6Þ

Based on Eqs. (3)–(6), we have

wjWhj ðpÞ þwkWh

kðpÞ �wkWhkðp0Þ �wjW

hj ðp0Þ ¼ th½wkðð1þ bjÞh � 1Þ �wjðð1þ bkÞh � 1Þ� > 0:

This contradicts the optimality of p. h

Corollary 1. For the 1jpj ¼ bjtjP

wjW2j problem, an optimal schedule can be obtained by sequencing the jobs in non-decreasing

order of ð1þbjÞ2�1

wj.

Remark 1. This result is in marked contrast to the result of Szwarc and Mukhopadhyay [26], they formulated the problem1jjP

wjW2j and presented a branch and bound algorithm incorporated with a powerful decomposition mechanism to solve

it. The complexity status of problem 1jjP

wjW2j is still open [28], but the problem 1jpj ¼ bjtj

PwjW

2j can be solved in

Oðn log nÞ time.

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-terioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx 3

3. Chains precedence problem

The single machine scheduling problem with chain precedence constraints can be formulated as follows: The jobs of setN ¼ fJ1; J2; . . . ; Jng have to be processed on a single machine, and these n jobs are grouped into m chains. If job Jj must beprocessed before job Jk, then we say that job Jj has precedence over job Jk and denote it by Ji ! Jj. Let ni be the number ofjobs belonging to chain Li; Jij be the jth job in chain Li; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni;n1 þ n2 þ . . .þ nm ¼ n. As in Section 2,we assume

Pleaseear de

pij ¼ bijt; ð7Þ

where pij (bij) is the processing time (deterioration rate) of job Jij; t P t0 > 0 is the start time of job Jij.In our discussion, we shall distinguish between strong and weak chain precedence relations. In a strong chain precedence

relation, the jobs of a chain have to be done on the machine with no job from another chain inserted between them. A weakchain allows for such insertions [29, 12]. Let Cij ¼ CijðpÞ represents the completion time of job Jij in chain Li. The objective is

to find a schedule of n jobs in order to minimizePm

i¼1

Pnij¼1wijW

hij, where Wij ¼ Cij � pij ¼ Ci;j�1 is the waiting time of job Jij; h is

a positive integer number. Using the three-field notation for scheduling problem classification, the problems can be denotedas 1jpij ¼ bijt; strong chainsj

PPwijW

hij and 1jpij ¼ bijt;weak chainsj

PPwijW

hij.

For the 1jpij ¼ bijt; strong chainsjPP

wijWhij problem, we consider any two chains of jobs Li and Ljðj – iÞ, i.e.,

Li : Ji1 ! Ji2 ! . . .! Jini

and

Lj : Jj1 ! Jj2 ! . . .! Jjnj:

Theorem 2. Consider two feasible schedules a ¼ ½U; Li; Lj;V � and b ¼ ½U; Lj; Li;V �, where U and V are any other chains.Pmi¼1Pni

j¼1wijWhijðaÞ 6

Pmi¼1Pni

j¼1wijWhijðbÞ if and only if

Qnik¼1ð1þ bikÞh � 1Pni

k¼1wikQk�1

l¼1 ð1þ bilÞh6

Qnj

k¼1ð1þ bjkÞh � 1Pnj

k¼1wjkQk�1

l¼1 ð1þ bjlÞh: ð8Þ

Proof. Let the execution time of the jobs start at the same time t P t0 > 0 in both schedules a and b. Then, for a, we have

Wik ¼ tYk�1

l¼1

ð1þ ailÞ; k ¼ 1;2; . . . ; ni;

and

Wjk ¼ tYni

l¼1

ð1þ ailÞYk�1

l¼1

ð1þ ajlÞ; k ¼ 1;2; . . . ; nj:

For b, we have

Wjk ¼ tYk�1

l¼1

ð1þ ajlÞ; k ¼ 1;2; . . . ; nj

and

Wik ¼ tYnj

l¼1

ð1þ ailÞYk�1

l¼1

ð1þ ailÞ; k ¼ 1;2; . . . ; ni:

If

Qnik¼1ð1þ bikÞh � 1Pni

k¼1wikQk�1

l¼1 ð1þ bilÞh6

Qnj

k¼1ð1þ bjkÞh � 1Pnj

k¼1wjkQk�1

l¼1 ð1þ bjlÞh;

then

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-terioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

4 J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx

Pleaseear de

Xm

i¼1

Xj¼1

ni wijWhijðaÞ�

Xm

i¼1

Xni

j¼1

wijWhijðbÞ¼ th

Xnj

k¼1

wjk

Yk�1

l¼1

ð1þbjlÞhYni

k¼1

ð1þbikÞh�1

!� th

Xni

k¼1

wik

Yk�1

l¼1

ð1þbilÞhYnj

k¼1

ð1þbjkÞh�1

!60: �

From Theorem 2, we have

Theorem 3. For the problem 1jpij ¼ bijt; strong chainsjPP

wijWhij, the optimal schedule can be obtained by non-decreasing

order of RðLiÞ ¼Qni

k¼1ð1þbikÞ

h�1Pnik¼1

wik

Qk�1

l¼1ð1þbilÞ

h, i ¼ 1;2; . . . ;m.

Lemma 2. For any b > 0; d > 0 and k > 0, cd <

ab if and only if aþkc

bþkd <ab.

Lemma 3. If Ji1�!Ji2�! . . .�!Jiu�!Ji;uþ1�! . . .�!Ji;l� and

Ql�

k¼1ð1þ bikÞh � 1Pl�

k¼1wikQk�1

j¼1 ð1þ bijÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh;

then, we can obtain

Ql�

k¼uþ1ð1þ bikÞh � 1Pl�

k¼uþ1wikQk�1

j¼1 ð1þ bijÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh:

Proof. Stem from

Ql�

k¼1ð1þ bikÞh � 1Pl�

k¼1wikQk�1

j¼1 ð1þ bijÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh;

we have

Quk¼1ð1þ bikÞh � 1þ ð

Ql�

k¼uþ1ð1þ bikÞh � 1ÞQu

k¼1ð1þ bikÞhPuk¼1wik

Qk�1j¼1 ð1þ bijÞh þ

Pl�

k¼uþ1wikQk�1

j¼uþ1ð1þ bijÞhQu

k¼1ð1þ bikÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh;

From Lemma 2, we have

Ql�

k¼uþ1ð1þ bikÞh � 1Pl�

k¼uþ1wikQk�1

j¼1 ð1þ bijÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh: �

For the 1jpij ¼ bijt;weak chainsjPP

wijWhij problem, let l� be the smallest integer satisfying

R�ðLiÞ ¼Ql�

k¼1ð1þ bikÞh � 1Pl�

k¼1wikQk�1

l¼1 ð1þ bilÞh¼ min

16s6ni

Qsk¼1ð1þ bikÞh � 1Ps

k¼1wikQk�1

l¼1 ð1þ bilÞh

( ): ð9Þ

The ratio R�ðLiÞ on (9) is called the R�-factor of chain Li and job Jl� is the job that determines the R�-factor of this chain.

Theorem 4. For the 1jpij ¼ bijt; weak chainsjPP

wijWhij problem, if Jil� determines R�ðLiÞ, then there exists an optimal schedule

in which jobs Ji1; Ji2; . . . Ji;l� are executed sequentially, without idle times and such that no jobs from other chains are executedbetween the jobs of this chain.

Proof (Proof by contradiction). We assume that under the optimal schedule the processing of the subschedule Ji1; Ji2; . . . Jil� isinterrupted by another chain job Jjv ðj – iÞ. Let p ¼ ½Ji1; Ji2; . . . ; Jiu; Jjv ; Ji;uþ1; . . . ; Jil� � be a subschedule of an optimal schedule. It issufficient to show that either with subschedule p0 ¼ ½Jjv ; Ji1; Ji2; . . . ; Jiu; Ji;uþ1; . . . ; Jil� �, or with subschedule p00 ¼ ½Ji1; Ji2;

. . . ; Jiu; Ji;uþ1; . . . ; Jil� ; Jjv �, the minimizingPP

wijWhij is less than that with subschedule p. If it is not less than that with the

subsequence p0, then it has to be less than that with the subsequence p00, and vice versa. From Theorem 2, it follows thatif the minimizing

PPwijW

hij of p is less than or equal to those with p0 and p00, then

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh6ð1þ bjvÞh � 1

wjv6

Ql�

k¼uþ1ð1þ bikÞh � 1Pl�

k¼uþ1wikQk�1

j¼uþ1ð1þ bijÞh: ð10Þ

Since job Ji;l� is the job that determines the R�-factor of Ji1; Ji2; . . . Ji;l� , it follows that

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-terioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

Table 1The dat

jobs

wij

bij

J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx 5

Pleaseear de

Ql�

k¼1ð1þ bikÞh � 1Pl�

k¼1wikQk�1

j¼1 ð1þ bijÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh; ð11Þ

From (11) and Lemma 3, it follows that

Ql�

k¼uþ1ð1þ bikÞh � 1Pl�

k¼uþ1wikQk�1

j¼1 ð1þ bijÞh<

Quk¼1ð1þ bikÞh � 1Pu

k¼1wikQk�1

j¼1 ð1þ bijÞh:

It is a contradiction to (10). The same argument can be applied if the interruption of Ji1; Ji2; . . . Jil� is caused by more thanone job. h

Similar to Pinedo [30] and Gawiejnowicz [2], we can formulate the following algorithm for the 1jpij ¼ bijt; weak chainsjPP

wijWhij problem.

Algorithm 1

a of Example 1.

J11 J12 J13 J14 J21 J22 J23 J31 J32 J33 J34 J41 J42 J43 J4

3 5 7 6 1 6 2 3 4 5 6 5 6 7 30.3 0.2 0.1 0.4 0.3 0.1 0.2 0.3 0.4 0.1 0.3 0.2 0.4 0.2 0

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simpleterioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

Step 0 Let initial current chains set be N. The initial current schedule a is the empty schedule.

Step 1 Find any R�-minimal sub-chain Si for any chain Li . Let c be the one with the minimal R�-factor (ratio R�ðLiÞ from (9)) among the remainingchains. Step 2 Append c to the end of a. Step 3 Replace N by N � c. Step 4 If N is empty, stop; a is optimal. Otherwise, return to Step 1.

Remark 2. Obviously, the overall time complexity of Algorithm 1 is Oðn log nÞ.For the 1jpij ¼ bijt; strong chainsj

PPwijW

hij problem, the following example illustrates the working of Theorem 3.

Example 1. Consider the problem 1jpij ¼ bijt; strong chainsjPP

wijWhij with the following four chains:

L1 : J11 ! J12 ! J13 ! J14;

L2 : J21 ! J22 ! J23;

L3 : J31 ! J32 ! J33 ! J34

and

L4 : J41 ! J42 ! J43 ! J44:

All jobs are available at t0 ¼ 1 and h ¼ 3. The deterioration rates and weights of all jobs are given in Table 1:Applied Theorem 3 to Example 1, we have

RðL1Þ ¼Qn1

k¼1ð1þb1kÞ

h�1Pn1k¼1

w1k

Qk�1

l¼1ð1þb1lÞ

h¼ ð1þ0:3Þ3�ð1þ0:2Þ3�ð1þ0:1Þ3�ð1þ0:4Þ3�1

3þ5�ð1þ0:3Þ3þ7�ð1þ0:3Þ3�ð1þ0:2Þ3þ6�ð1þ0:3Þ3�ð1þ0:2Þ3�ð1þ0:1Þ3¼ 0:1815,

RðL2Þ ¼ ð1þ0:3Þ3�ð1þ0:1Þ3�ð1þ0:2Þ3�11þ6�ð1þ0:3Þ3þ2�ð1þ0:3Þ3�ð1þ0:1Þ3

¼ 0:2023,

RðL3Þ ¼ ð1þ0:3Þ3�ð1þ0:4Þ3�ð1þ0:1Þ3�ð1þ0:3Þ3�13þ4�ð1þ0:3Þ3þ5�ð1þ0:3Þ3�ð1þ0:4Þ3þ6�ð1þ0:3Þ3�ð1þ0:4Þ3�ð1þ0:1Þ3

¼ 0:1846,

RðL4Þ ¼ ð1þ0:2Þ3�ð1þ0:4Þ3�ð1þ0:2Þ3�ð1þ0:5Þ3�15þ6�ð1þ0:2Þ3þ7�ð1þ0:2Þ3�ð1þ0:4Þ3þ3�ð1þ0:2Þ3�ð1þ0:4Þ3�ð1þ0:2Þ3

¼ 0:3644.

Hence, the optimal sequence is p ¼ J11 ! J12 ! J13 ! J14 ! J31 ! J32 ! J33 ! J34 ! J21 ! J22 ! J23 ! J41 !½ J42 ! J43 !J44� and the optimal value of

Pmi¼1Pni

j¼1wijW3ij is 96552.82.

The following example illustrates applying Algorithm 1 to find the optimal solution for the problem1jpij ¼ bijt;weak chainsj

PPwijW

hij.

Example 2. Consider the problem 1jpij ¼ bijt; weak chainsjPP

wijWhij with the following two chains:

L1 : J11 ! J12 ! J13 ! J14

4

.5

lin-

6 J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx

and

Table 2The dat

jobs

wij

bij

Pleaseear de

L2 : J21 ! J22 ! J23:

All jobs are available at t0 ¼ 1 and h ¼ 2. The weights and deterioration rates of jobs are given in Table 2:Applied Algorithm 1 to Example 2, we can obtain:

Step 0: N ¼ fJ11; J12; J13; J14; J21; J22; J23g, a ¼ ð;Þ.Step 1:

R�ðL1Þ ¼ min16s6n1

Qsk¼1ð1þ b1kÞh � 1Ps

k¼1w1kQk�1

l¼1 ð1þ b1lÞh

( )

¼minð1þ 0:4Þ2 � 1

3;ð1þ 0:4Þ2 � ð1þ 0:2Þ2 � 1

3þ 4 � ð1þ 0:4Þ2;

ð1þ 0:4Þ2 � ð1þ 0:2Þ2 � ð1þ 0:1Þ2 � 1

3þ 4 � ð1þ 0:4Þ2 þ 7 � ð1þ 0:4Þ2 � ð1þ 0:2Þ2;

(

ð1þ 0:4Þ2 � ð1þ 0:2Þ2 � ð1þ 0:1Þ2 � ð1þ 0:3Þ2 � 1

3þ 4 � ð1þ 0:4Þ2 þ 7 � ð1þ 0:4Þ2 � ð1þ 0:2Þ2 þ 2 � ð1þ 0:4Þ2 � ð1þ 0:2Þ2 � ð1þ 0:1Þ2

)

¼minf0:3200;0:1681; 0:0789;0:1275g ¼ 0:0789

and is determined by job J13. R�ðL2Þ ¼ 0:1515 and is determined by job J22. Hence, c ¼ fJ11; J12; J13g.Step 2: a ¼ ½J11; J12; J13�.Step 3: N ¼ fJ14; J21; J22; J23g.

The new two chains are L01 : J14, and L02 : J21 ! J22 ! J23.

Step 1: R�ðL01Þ ¼ 0:345 and is determined by job J14. R�ðL02Þ ¼ 0:1515 and is determined by job J22. Hence, c ¼ fJ21; J22g.Step 2: a ¼ ½J11; J12; J13; J21; J22�.Step 3: N ¼ fJ14; J23g.

The new two chains are L001 : J14, and L002 : J23.

Step 1:R�ðL001Þ ¼ 0:345 and is determined by job J14. R�ðL002Þ ¼ 0:44 and is determined by job J23. Hence, c ¼ fJ14g.Step 2: a ¼ ½J11; J12; J13; J21; J22; J14�.Step 3: N ¼ fJ23g.

Hence, the optimal sequence is a ¼ ½J11 ! J12 ! J13 ! J21 ! J22 ! J14 ! J23� and the optimal value ofPm

i¼1Pni

j¼1wijW2ij is

141.6944.

4. A series–parallel digraph precedence constraint

In this section, we consider the problem 1jpj ¼ bjt; sp� digraphjP

wjWhj , where sp� digraph denotes the job precedence

constraints are in the form of a series–parallel digraph G ¼ ðV ;AÞ, where V and A denote the set of vertices (i.e., jobs) and theset of arcs (i.e., job precedence constraints). Now, we give some notation and terminology.

Definition 1 (Lawler [31], Gawiejnowicz [2]). The series–parallel digraph G ¼ ðV ;AÞ is defined recursively as follows:

1. A digraph consisting of a single node, e.g., G ¼ ðfJjg; ;Þ, is a series–parallel digraph.2. If G1 ¼ ðV1;A1Þ and G2 ¼ ðV2;A2Þ, where V1 \ V2 ¼ ;, are series–parallel digraphs, then:

(a) The digraph

GS ¼ ðV1 [ V2;A1 [ A2 [ ðV1 � V2ÞÞ

is a series–parallel digraph, too. GS is said to be a series composition of digraphs G1 and G2.

a of Example 2.

J11 J12 J13 J14 J21 J22 J23

3 4 7 2 5 6 10.4 0.2 0.1 0.3 0.5 0.3 0.2

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-terioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx 7

(b) The digraph

Pleaseear de

GP ¼ ðV1 [ V2;A1 [ A2Þ

is a series–parallel digraph, too. GP is said to be a parallel composition of digraphs G1 and G2.

From Definition 1, every series–parallel digraph G ¼ ðV ;AÞ can be represented by a binary decomposition tree TðGÞ inwhich the leaves denoting jobs and the internal vertexes denoting either a parallel or series composition of the twocorresponding subtrees. For a given series–parallel digraph, its decomposition tree can be constructed in OðjV j þ jAjÞ � Oðn2Þtime [32]. The internal vertexes of a decomposition tree that correspond to parallel and series compositions are labeled by Pand S, respectively. For series composition S, the left child (son) precedes the right child. Fig. 2 shows a decomposition treeTðGÞ for the digraph G in Fig. 1.

Definition 2 (Sidney [33]). A non-empty subset M # V is called a job module if for each job Jj 2 V nM, there holds one of thefollowing conditions:

(a) job Jj precedes all jobs of M,(b) job Jj follows all jobs of M,(c) there are no precedence constraints between an arbitrary job from M and job Jj.

Definition 3 (Sidney [33]). Let M be a job module. A subset I # M is an initial set of M, if for each job Jj 2 I, all the predeces-sors of Jj in M are in I, too.

Suppose that p ¼ ½Jpð1Þ; Jpð2Þ; . . . ; JpðnÞ� is any schedule of V and U ¼ fJpðlÞ; Jpðlþ1Þ; . . . ; JpðmÞg � V , let

RðU;pÞ ¼Pm

j¼lwpðjÞQj�1

i¼l ð1þ bpðiÞÞhQmi¼lð1þ bpðiÞÞh � 1

; ð12Þ

whereQm

i¼mþ1ð1þ bpðiÞÞ :¼ 1.

Definition 4 (Sidney [33]). Let M be a module. An initial set I of M is said to be R-maximal for G ¼ ðM;AÞ if qðIÞP qðVÞ forany initial set V in M.

Definition 5 (Sidney [33]). Let M be a module. An initial set I� of M is said to be R�-maximal for G ¼ ðM;AÞ if

(a) I� is R-maximal for G;(b) there is no proper subset V � M (V – I�) that is R-maximal for G.

Every module M admits at least one R-maximal initial set, possibly M itself.

Similar to the proof of Wang et al. [3], and Wang and Wang [34], we have

Theorem 5. Let M be a module of G ¼ ðV ;AÞ and I� be a R�-maximal for ðM;AÞ, then there exists an optimal schedule for V in whichthe jobs in I� precede all the other jobs in M.

Theorem 6. Let M be a module of G ¼ ðV ;AÞ and I� be a R�-maximal, then I� is a consecutive subschedule in every optimal schedulefor G ¼ ðV ;AÞ.

Fig. 1. Series–parallel digraph G.

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-terioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

8 J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx

Theorem 7. Let M be a module of G ¼ ðV ;AÞ and r be an optimal schedule for M. Then there exists an optimal schedule for V that isconsistent with r (i.e., in which the jobs in M appear in the same order as in r). Hence, from Theorems 5–7, we can generalize themethods of Lawler [31], Brucker [35], Wang et al. [3], and Wang and Wang [34] to the problem 1jpj ¼ bjt; sp� digraphj

PwjW

hj .

To describe the algorithm in more detail we need some notation. Let f be an internal vertex of the decomposition tree, and Mf be theunion of the two sets Mi and Mj. Similar to the algorithms of Lawer [31], Brucker [35], Wang et al. [3], and Wang and Wang [34],we proceed the algorithm from the bottom of the decomposition tree upward, finding an optimal sequence by using the series com-position and parallel composition.

Pea

Algorithm 2

lease cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simpler deterioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

1. FOR ALL leaves Jj of the decomposition tree DO Mj ¼ fJjg;

2. WHILE there exists an internal vertex f with two leaves as children DO

BEGIN

3. Ji:¼ leftchild(f); Jj:¼ rightchild(f); 4. IF f has label P THEN 5. Mf :¼ Mi [Mj

ELSE

6.

6.1 Find Ji 2 Mi such that RðJiÞ ¼minfRðJkÞjJk 2 Mig and Jj 2 Mj such that RðJjÞ ¼maxfRðJkÞjJk 2 Mjg. If RðJiÞ > RðJjÞ,let Mf ¼ Mi [Mj and halt. Otherwise, remove Ji from Mi; Jj from Mj and form the composite Jk ¼ ðJi; JjÞ.

6.2 6.2.1 Find Ji 2 Mi such that RðJiÞ ¼minfRðJkÞjJk 2 Mig. If RðJiÞ > RðJkÞ (RðJkÞ is computed by (12)), go to Step 6.3.1. 6.2.2 Remove Ji from Mi and form the composite job Jk ¼ ðJi; JkÞ. Return to Step 6.2.1. 6.3 6.3.1 Find Jj 2 Mj such that RðJjÞ ¼maxfRðJkÞjJk 2 Mjg. If RðJkÞ > RðJjÞ, let Mf ¼ Mi [Mj [ fJkg and halt. 6.3.2 Remove Jj from Mj and form the composite job Jk ¼ ðJk; JjÞ. Go to Step 6.2.1. END {IF}

7. Eliminate Ji and Jj and replace f by a leaf with label Mf .

END {WHILE}

8. Construct p� by concatenating all the subsequences of the single leaf in non-increasing order of R-values.

Remark 3. Obviously, if the decomposition tree TðGÞ of the series–parallel digraph G is given, the overall time complexity ofAlgorithm 2 is Oðn log nÞ [2, 35]. If the decomposition tree TðGÞ of the series–parallel digraph G is not given, Algorithm 2 muststart with the step in which the tree can be constructed in OðjV j þ jAjÞ � Oðn2Þ time [32], in this case the overall timecomplexity of Algorithm 2 increases to Oðn2Þ time [2].

The following examples illustrate applying Algorithm 2 to find the optimal solution for the problem 1 pj ¼ bjt;��

sp� digraphjP

wjWhj .

Example 3. Consider the problem 1 pj ¼ bjt; sp� digraph�� ��PwjW

hj with a digraph G of precedence constraint given in Fig. 1

(the decomposition tree TðGÞ is given in Fig. 2), with deterioration rates and weights as shown in Table 3, h ¼ 2, and t0 ¼ 1.For P1;RðJ4Þ ¼ 2

ð1þ0:6Þ2�1¼ 1:2821 < RðJ5Þ ¼ 4

ð1þ0:5Þ2�1¼ 3:2000, hence P1 : MP1 ¼ fJ5; J4g. For S2; RðJ2Þ ¼ 2

ð1þ0:4Þ2�1¼

2:0833; RðJ5; J4Þ ¼4þ2�ð1þ0:5Þ2

ð1þ0:5Þ2�ð1þ0:6Þ2�1¼ 1:7857; RðJ2Þ < maxfRðJ5Þ;RðJ5; J4Þg ¼ RðJ5Þ, hence J2 and J5 form a composite job

ðJ2; J5Þ, and S2 : MS2 ¼ fðJ2; J5Þ; J4Þg. For P3 : RðJ3Þ ¼ 3ð1þ0:7Þ2�1

¼ 1:5873 < RðJ2; J5Þ ¼2þ4�ð1þ0:4Þ2

ð1þ0:4Þ2�ð1þ0:5Þ2�1¼ 2:8856, hence

MP3 ¼ fðJ2; J5Þ; J3; J4Þg; S4 : RðJ1Þ ¼ 1ð1þ0:3Þ2�1

¼ 1:4493; MS4 ¼ fðJ2; J5Þ; J3; J1; J4g. Hence, the optimal sequence is ½J2; J5; J3; J1; J4�and the optimal value of the total weighted completion time

X

wjWhj ¼ 2þ 4� ð1þ 0:4Þ2 þ 3� ð1þ 0:4Þ2 � ð1þ 0:5Þ2 þ ð1þ 0:4Þ2 � ð1þ 0:5Þ2 � ð1þ 0:7Þ2 þ 2� ð1þ 0:4Þ2

� ð1þ 0:5Þ2 � ð1þ 0:7Þ2 � ð1þ 0:3Þ2 ¼ 78:8927:

Example 4. Consider the problem 1 pj ¼ bjt; sp� digraph�� ��PwjW

hj with a digraph G of precedence constraint given in Fig. 3

(the decomposition tree TðGÞ is given in Fig. 4), with deterioration rates and weights as shown in Table 4, h ¼ 3, and t0 ¼ 1.

For S1; RðJ2Þ ¼ 2ð1þ0:4Þ3�1

¼ 1:1468 < RðJ3Þ ¼ 3ð1þ0:3Þ3�1

¼ 2:5063, hence J2 and J3 form a composite job ðJ2; J3Þ, andS1 : MS1 ¼ fðJ2; J3Þg.

lin-

Fig. 2. Decomposition tree T(G).

Table 3Values of bj and wj

jobs J1 J2 J3 J4 J5

bj 0.3 0.4 0.7 0.6 0.5wj 1 2 3 2 4

Fig. 3. Series–parallel digraph G in Example 4.

J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx 9

For P2; RðJ4Þ ¼ 2ð1þ0:2Þ3�1

¼ 2:7473 > RðJ2; J3Þ ¼2þ3�ð1þ0:4Þ3

ð1þ0:4Þ3�ð1þ0:3Þ3�1¼ 2:0348; P2 : MP2 ¼ fJ4; ðJ2; J3Þg.

For P3 : RðJ6Þ ¼ 3ð1þ0:4Þ3�1

¼ 1:7202 > RðJ5Þ ¼ 1ð1þ0:5Þ3�1

¼ 0:4211, hence MP3 ¼ fJ6; J5g.

For S4 : minfRðJ4Þ;RðJ2; J3Þg ¼ RðJ2; J3Þ ¼ 2:0348 > maxfRðJ6Þ;RðJ5Þg ¼ RðJ6Þ ¼ 1:7202, then MS4 ¼ fJ4; ðJ2; J3Þ; J6; J5g.

For S5 : RðJ1Þ ¼ 4ð1þ0:2Þ3�1

¼ 5:4945 > maxfRðJ4Þ;RðJ2; J3Þ;RðJ6Þ;RðJ5Þg ¼ RðJ4Þ ¼ 2:7473, then MS5 ¼ fJ1; J4; ðJ2; J3Þ; J6; J5g.

Hence, the optimal sequence is ½J1; J4; J2; J3; J6; J5� and the optimal value of the total weighted completion time

Pleaseear de

XwjW

hj ¼ 4þ 2� ð1þ 0:2Þ3 þ 2� ð1þ 0:2Þ3 � ð1þ 0:2Þ3 þ 3� ð1þ 0:2Þ3 � ð1þ 0:2Þ3 � ð1þ 0:4Þ3 þ 3� ð1þ 0:2Þ3

� ð1þ 0:2Þ3 � ð1þ 0:4Þ3 � ð1þ 0:3Þ3 þ ð1þ 0:2Þ3 � ð1þ 0:2Þ3 � ð1þ 0:4Þ3 � ð1þ 0:3Þ3 � ð1þ 0:4Þ3

¼ 141:4075:

cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-terioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028

Fig. 4. Decomposition tree T(G) of Example 4.

Table 4Values of bj and wj

jobs J1 J2 J3 J4 J5 J6

bj 0.2 0.4 0.3 0.2 0.5 0.4wj 4 2 3 2 1 3

10 J.-B. Wang, J.-J. Wang / Applied Mathematical Modelling xxx (2014) xxx–xxx

5. Conclusions

In his paper we have studied the problem of scheduling deteriorating jobs under simple linear deterioration with the aimof minimizing the weighted sum of the hth power of waiting times on a single machine. We proved that the general problem(i.e., 1 pj ¼ bjt

�� ��PwjWhj ) can be solved in polynomial time. We also considered the scheduling problem with weak (strong)

parallel chains (i.e., 1 pj ¼ bjt;weak chains�� ��PwjW

hj and 1 pj ¼ bjt; strong chains

�� ��PwjWhj ) and a series–parallel digraph pre-

cedence constraints (i.e., 1 pj ¼ bjt; sp� digraph�� ��PwjW

hj ), and proved that the these problems can be solved in polynomial

time, respectively.

Acknowledgments

We are grateful to an anonymous referee for his/her constructive comments on earlier versions of our paper. Thisresearch was supported by the National Natural Science Foundation of China (Grant nos, 11001181 and 71271039), NewCentury Excellent Talents in University (NCET-13-0082), Changjiang Scholars and Innovative Research Team in Univer-sity(IRT1214), the Fundamental Research Funds for the Central Universities (DUT14YQ211).

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Please cite this article in press as: J.-B. Wang, J.-J. Wang, Single-machine scheduling problems with precedence constraints and simple lin-ear deterioration, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.07.028