Parallel machine scheduling with a deteriorating maintenance activity

Applied Mathematics and Computation 217 (2011) 8093–8099

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Parallel machine scheduling with a deteriorating maintenance activity and total absolute differences penalties Ji-Bo Wang a,⇑, Cai-Min Wei b a b

School of Science, Shenyang Aerospace University, Shenyang 110136, China Department of Mathematics, Shantou University, Shantou 515063, China

a r t i c l e

i n f o

Keywords: Scheduling Parallel machine Maintenance activity Polynomial algorithm

a b s t r a c t In this paper we consider identical parallel machines scheduling problems with a deteriorating maintenance activity. In this model, each machine has a deteriorating maintenance activity, that is, delaying the maintenance increases the time required to perform it. We need to make a decision on when to schedule the rate-modifying activities and the sequence of jobs to minimize some objective function. We concentrate on two goals separately, namely, minimizing the total absolute differences in completion times (TADC) and the total absolute differences in waiting times (TADW). We show that the problems remain polynomially solvable under the proposed model. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction A common assumption in classical scheduling theory is that the machines are available at all times. However, there are many situations where machines need to be maintained and become unavailable during certain periods. Motivated by the problem commonly found in the surface-mount technology lines of electronic assembly systems, Lee and Leon [1] considered single machine scheduling with a rate-modifying activity. A rate-modifying activity is an activity that changes the production rate of the machine. Machine scheduling with a rate-modifying activity can be considered as a spacial type of scheduling with maintenance. Lee and Leon [1] studied several single machine scheduling problems in this class: minimizing makespan, flow-time, weighted flow-time and maximum lateness. Graves and Lee [2] considered problems where the machine maintenance can happen at most one or two times during the planning horizon. Qi et al. [3] considered single machine scheduling with multiple maintenance activities need to be scheduled jointly with jobs. Lee and Chen [4] discussed parallel machine scheduling where each machine must be maintained once during the planning horizon. Lee and Lin [5] investigated single machine scheduling with maintenance and repair rate-modifying activity. Whitaker [6] developed a branch-andbound procedure for single machine scheduling with rate-modifying activity to minimize total completion time. Mosheiov and Sidney [7] addressed the problems of minimizing makespan with precedence relations, minimizing makespan with learning effect, and minimizing the number of tardy jobs. Mosheiov and Oron [8] considered maintenance activity scheduling and due-date assignment simultaneously. They showed that the problem remains solvable in polynomial time. Gordon and Tarasevich [9] considered the single machine common due date assignment and scheduling problem with the possibility to perform a rate-modifying activity for changing the processing times of the jobs following this activity. The objective is to minimize the total weighted sum of earliness, tardiness and due date costs. They proved several properties of the problem which in some cases can reduce the complexity of the solution algorithm. Zhao et al. [10] considered the parallel machine scheduling problem with rate-modifying activities. For the total completion time minimization problem, they provided a polynomial algorithm to solve the problem optimally. For the total weighted completion time minimization problem, when ⇑ Corresponding author. E-mail address: [email protected] (J.-B. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.010

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jobs satisfy agreeable ratio condition, they proposed a pseudo-polynomial time dynamic programming algorithm. Sun and Li [11] considered scheduling problems with multiple maintenances and non-preemptive jobs on two identical parallel machines. Ji and Cheng [12] considered a scheduling problem with job-dependent learning effects and multiple rate-modifying activities. We show that all the cases of the total completion time minimization problem are polynomially solvable. In a recent paper, Kubzin and Strusevich [13] considered a setting where the maintenance activity is deteriorating, that is, delaying the maintenance activity increases the time required to perform it. The longer the maintenance activity is delayed, the worse the system conditions become, so that the maintenance requires more effort and time. Kubzin and Strusevich [13] studied makespan minimization on a two-machine flowshop and a two-machine openshop. Mosheiov and Sidney [14] studied single machine scheduling problems with an option to perform a deteriorating maintenance activity. They considered the following objective functions: makespan, flowtime, maximum lateness, total earliness, tardiness and due-date cost, and number of tardy jobs. They introduced polynomial time solutions for all these problems. Wang et al. [15] considered parallel identical machines scheduling problems with a deteriorating maintenance activity. They proved that the total completion time minimization problem can be solved in polynomial time. Yang et al. [16] considered the due-window assignment and scheduling problem with job-dependent aging effects and a deteriorating maintenance. The objective is to find jointly the optimal location of the maintenance operation, the optimal location and size of the due-window, and the optimal job sequence to minimize the total earliness, tardiness, and due-window related costs. They proposed polynomial time solutions for all the problems under study. Yang and Yang [17] considered a single-machine scheduling with a position-dependent aging effect described by a power function under maintenance activities and variable maintenance duration considerations simultaneously. They showed that all the studied problems can be optimally solved in polynomial time. Yang and Yang [18] considered a single-machine scheduling with aging or deteriorating effects and deteriorating maintenance activities simultaneously. The objective is to minimize the total completion time when the upper bound of the maintenance frequency is given in advance. They showed that the problem under study is polynomially solvable. Cheng et al. [19] considered the a single-machine problem of common due-window assignment and scheduling of deteriorating jobs and a maintenance activity simultaneously. They provided polynomial time solutions for the problem and some of its special cases, where the objective is to simultaneously minimize the earliness, tardiness, due-window starting time, and due-window size costs. Cheng et al. [20] considered the unrelated parallel-machine scheduling with deteriorating maintenance activities. The objective is to minimize the total completion time or the total machine load. They showed that both versions of the problem can be optimally solved in polynomial time. Yang [21] studied single-machine scheduling problems simultaneous with the phenomena of learning and aging under a deteriorating maintenance consideration, in which the processing time of a job depends on its starting time and position in a sequence. The objectives were to find the optimal solutions for minimizing the makespan, the total completion time, the total absolute deviation of completion times (TADC), and the due-window related costs problems. They showed that these problems are polynomially solvable. However, most of the scheduling literature examines regular measures of the performance, which are nondecreasing functions of job completion times. One of the most commonly occurring regular measures is the minimization of mean completion times. Its attractiveness is perhaps due to its equivalence to mean waiting time, mean lateness, and average in-process inventory. Yet in certain situations one is more interested in reducing the variability in the completion times, resulting in performance measures that are nonregular. For instance, in a service-oriented environment, one might be interested in providing as much uniform quality of service as possible based on the customers’ waiting times in system. Another example where one might be interested in a variability measure was given by Merten and Muller [22] in the context of organization of computer databases. They noted that, in an organization of computer files in the large databases, it is desirable to provide uniform response time to users. The objective then is to determine the arrangement that minimizes variation of access time to different records in the file. As measures of variation, Merten and Muller [22] considered completion time variance (CTV) and waiting time variance (WTV). These measures have also been used by Schrage [23], Eilon and Chowdhury [24], Vani and Raghavachari [25], Wang and Xia [26] and Mor and Mosheiov [27]. Although several properties of the optimal schedules for these measures have been established, no efficient procedure exists for solving these problems. Kanet [28] proposed using total absolute differences in completion times (TADC) as an alternative measure of completion time variation, and presented an efficient algorithm for minimizing this measure. While Bagchi [29] proposed using total absolute differences in waiting times (TADW) as an alternative measure of waiting time variation, and presented an efficient algorithm for minimizing this measure. Following Kubzin and Strusevich [13] and Mosheiov and Sidney [14], in this paper we extend some results of [13,14] to the parallel machine scheduling with a deteriorating maintenance activity. The underlying assumptions are those of Lee and Leon [1]: the maintenance activity is optional, it may be performed once during the production process, and it improves the production rate of the processor (which is reflected in smaller processing times of jobs scheduled after the maintenance activity). As in Kubzin and Strusevich [13] and Mosheiov and Sidney [14], we assume that the maintenance time increases with the starting time of the maintenance activity. Similar to many scheduling models with deteriorating jobs (see the recent results on scheduling with variable processing times, Cheng et al., [30], Gawiejnowicz [31], Wang [32] and Yang and Wang [33]), we assume here that the maintenance time deteriorates linearly. We study this setting for the following two non-regular goals: minimizing total absolute differences in completion times (TADC) and total absolute differences in waiting times (TADW). All problems are shown to be solved in polynomial time. The remaining part of this paper is organized as follows. In Section 2 we formulate the model. In Section 3 we consider parallel identical machines scheduling problems. The last section presents the conclusions.

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2. Problem description The problem under investigation can be described as follows: There are n independent jobs fJ 1 ; J 2 ; . . . ; J n g to be processed on m identical parallel machines fM1 ; M 2 ; . . . ; M m g. Each of them is available at time 0. The machine can handle one job at a time and preemption is not allowed. Associated with each job J j ðj ¼ 1; 2; . . . ; nÞ there is a normal processing time aj . Machine Mi has a deteriorating maintenance activity, that is, its duration time is T i ¼ t i þ di tði ¼ 1; 2; . . . ; mÞ, where ti and di are positive constants, and t is the starting time of the deteriorating maintenance activity. If job J j is processed before the rate-modifying activity, its processing time is aj , and if job J j is processed after the rate-modifying activity, its processing time is bj ðbj 6 aj Þ; j ¼ 1; 2; . . . ; n. For a given schedule, the completion (waiting) time of job J j is denoted by C j ðW j Þ. The objective functions considered in this paper are TADC: The total absolute differences in completion times, that is

TADC ¼

n X n X i¼1

jC i  C j j;

ð1Þ

j¼i

and TADW: The total absolute differences in waiting times, that is

TADW ¼

n X n X i¼1

jW i  W j j:

ð2Þ

j¼i

Using the three-field notation of Graham et al. [34] the problems can be denoted as PmjT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADC and PmjT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADW. It is easy to show that, there exist an optimal solution where the rate-modifying activity on each machine starts without interrupting the processing of some jobs. For a given sequence, suppose M i processed ni ði ¼ 1; 2; . . . ; mÞðn1 þ n2 þ . . . nm ¼ nÞ jobs, let i½j denote the jth position on machine M i . Let J i½j denote the jth job on machine M i ; C i½j denote the completion time of job J i½j and W i½j denote the waiting time of job J i½j . We say that the rate-modifying activity of machine M i is in position ki if it is scheduled immediately after the completion of job J i½ki  ði ¼ 1; 2; . . . ; mÞ. ki ¼ 0 means that we implement the rate-modifying activity before we process any job on machine M i and ki ¼ ni implies that we do not implement it on machine M i in the current planning horizon. Let ðn1 ; n2 ; . . . ; nm Þ denote the allocation vector of job numbers and ðk1 ; k2 ; . . . ; km Þ ð0 6 ki 6 ni Þ denote the allocation vector of rate-modifying activities positions for a given ðn1 ; n2 ; . . . ; nm Þ. The problem is to determine the sets N 1 ; N 2 ; . . . ; N m , where jN i j ¼ ni ði ¼ 1; 2; . . . ; mÞ, the sequence of job on each machine and a ðk1 ; k2 ; . . . ; km Þ vector to minimize objective function c; c 2 fTADC; TADWg. 3. The total absolute differences in completion (weighted) times problem 3.1. Preliminary analysis for m ¼ 1 As in Lee and Leon [1], the completion (waiting) times of jobs can be expressed as follows:

C ½j ¼ C ½j1 þ a½j ; j ¼ 1; 2; . . . ; k1 ; C ½j ¼ C ½j1 þ t1 þ d1 ðC ½k1  Þ þ b½j ; j ¼ k1 þ 1; C ½j ¼ C ½j1 þ b½j ; j ¼ k1 þ 2; . . . ; n; C ½0 ¼ 0; k1 2 Z þ ; 0 6 k1 6 n: W ½j ¼ W ½j1 þ a½j1 ; j ¼ 2; . . . ; k1 ; W ½j ¼ W ½j1 þ t 1 þ d1 ðW ½k1  þ a½k1  Þ; j ¼ k1 þ 1; W ½j ¼ W ½j1 þ b½j1 ; j ¼ k1 þ 2; . . . ; n; W ½1 ¼ 0; k1 2 Z þ ; 0 6 k1 6 n: From (1), TADC can be rewritten as

TADCðn; k1 Þ ¼

n X n X i¼1

jC i  C j j

j¼i

¼ ðC ½2  C ½1 Þ þ ðC ½3  C ½1 Þ þ   þ ðC ½n  C ½1 Þ þ ðC ½3  C ½2 Þ þ ðC ½4  C ½2 Þ þ   þ ðC ½n  C ½2 Þ þ   þ ðC ½n  C ½n1 Þ ¼ a½2 þ ða½2 þ a½3 Þ þ   þ ða½2 þ a½3 þ   ;a½k1  Þ þ ða½2 þ a½3 þ   ;a½k1  þ t1 þ d1 ða½1 þ a½2 þ  ; a½k1  Þ þ b½k1 þ1 Þ þ   þ ða½2 þ a½3 þ  ; a½k þ t1 þ d1 ða½1 þ a½2 þ  ; a½k1  Þ þ b½k1 þ1 þ   þ b½n Þa½3 þ ða½3 þ a½4 Þ þ   þ ða½3 þ a½4 þ  ; a½k1  Þ þ ða½3 þ a½4 þ   ;a½k þ t 1 þ d1 ða½1 þ a½2 þ   ;a½k1  Þ þ b½k1 þ1 Þ þ   þ ða½3 þ a½4 þ  ; a½k þ t1 þ d1 ða½1 þ a½2 þ  ; a½k1  Þ þ b½k1 þ1 þ   þ b½n Þ þ   þ b½n ¼

k1 n X X 1 ðj  1Þðn  j þ 1Þa½j þ ðn  k1 Þðn  k1 þ 1Þ½t 1 þ d1 ða½1 þ a½2 þ   ;a½k1  Þ þ ðj  1Þðn  j þ 1Þb½j : 2 j¼1 j¼k þ1 1

ð3Þ

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From (2), TADW can be rewritten as

TADWðn; k1 Þ ¼

n X n X i¼1

jW i  W j j

j¼i

¼ ðW ½2  W ½1 Þ þ ðW ½3  W ½1 Þ þ    þ ðW ½n  W ½1 Þ þ ðW ½3  W ½2 Þ þ ðW ½4  W ½2 Þ þ    þ ðW ½n  W ½2 Þ þ    þ ðW ½n  W ½n1 Þ ¼ a½1 þ ða½1 þ a½2 Þ þ    þ ða½1 þ a½2 þ    ; a½k1  Þ þ ða½1 þ a½2 þ    ; a½k1  þ t 1 þ d1 ða½1 þ a½2 þ    ; a½k1  Þ þ b½k1 þ1 Þ þ    þ ða½1 þ a½2 þ    ; a½k1  þ t1 þ d1 ða½1 þ a½2 þ    ; a½k1  Þ þ b½k1 þ1 þ    þ b½n1 Þa½2 þ ða½2 þ a½3 Þ þ    þ ða½2 þ a½3 þ    ; a½k1  Þ þ ða½2 þ a½3 þ    ; a½k1  þ t 1 þ d1 ða½1 þ a½2 þ    ; a½k1  Þ þ b½k1 þ1 Þ þ    þ ða½2 þ a½3 þ    ; a½k1  þ t1 þ d1 ða½1 þ a½2 þ    ; a½k1  Þ þ b½k1 þ1 þ    þ b½n1 Þ þ    þ b½n1 ¼

k1 X j¼1

n X 1 jðn  jÞa½j þ ðn  k1  1Þðn  k1 Þ½t 1 þ d1 ða½1 þ a½2 þ    ; a½k1  Þ þ jðn  jÞb½j : 2 j¼k þ1

ð4Þ

1

3.2. Main results for m ¼ 2 In this subsection we consider problems P2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADC and P2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADW. When the sets N 1 and N 2 , a ðk1 ; k2 Þ vector and a sequence of job on each machine are given, from (3) and (4), we have k1 X 1 ðj  1Þðn1  j þ 1Þa1½j þ ðn1  k1 Þðn1  k1 þ 1Þ½t 1 þ d1 ða1½1 þ a1½2 þ    ; a1½k1  Þ 2 j¼1

TADCððn1 ; n2 Þ; ðk1 ; k2 ÞÞ ¼

þ

n1 X

ðj  1Þðn1  j þ 1Þb1½j þ

k2 X ðj  1Þðn2  j þ 1Þa2½j j¼1

j¼k1 þ1

n2 X 1 þ ðn2  k2 Þðn2  k2 þ 1Þ½t 2 þ d2 ða2½1 þ a2½2 þ    ; a2½k2  Þ þ ðj  1Þðn2  j þ 1Þb2½j 2 j¼k þ1 2

k1 X 1 1 ¼ ½ðj  1Þðn1  j þ 1Þ þ d1 ðn1  k1 Þðn1  k1 þ 1Þa1½j þ ðn1  k1 Þðn1  k1 þ 1Þt 1 2 2 j¼1

þ

n1 X

ðj  1Þðn1  j þ 1Þb1½j þ

j¼k1 þ1

k2 X 1 ½ðj  1Þðn2  j þ 1Þ þ d2 ðn2  k2 Þðn2  k2 þ 1Þa2½j 2 j¼1

n2 X 1 þ ðn2  k2 Þðn2  k2 þ 1Þt2 þ ðj  1Þðn2  j þ 1Þb2½j : 2 j¼k þ1 2

TADWððn1 ; n2 Þ; ðk1 ; k2 ÞÞ ¼

k1 X

1 jðn1  jÞa1½j þ ðn1  k1  1Þðn1  k1 Þ½t 1 þ d1 ða1½1 þ a1½2 þ    ; a1½k1  Þ 2 j¼1

þ

n1 X

jðn1  jÞb1½j þ

j¼k1 þ1

þ

n2 X

n1 X

k1 X 1 1 ½jðn1  jÞ þ d1 ðn1  k1  1Þðn1  k1 Þa1½j þ ðn1  k1  1Þðn1  k1 Þt 1 2 2 j¼1

jðn1  jÞb1½j þ

k2 X 1 1 ½jðn2  jÞ þ d2 ðn2  k2  1Þðn2  k2 Þa2½j þ ðn2  k2  1Þðn2  k2 Þt2 2 2 j¼1

j¼k1 þ1

þ

n2 X

1 jðn2  jÞa2½j þ ðn2  k2  1Þðn2  k2 Þ½t 2 þ d2 ða2½1 þ a2½2 þ    ; a2½k2  Þ 2 j¼1

jðn2  jÞb2½j ¼

j¼k2 þ1

þ

k2 X

jðn2  jÞb2½j :

j¼k2 þ1

We need to determine the sets N 1 and N 2 , a ðk1 ; k2 Þ vector and a sequence of job on each machine to minimize TADCððn1 ; n2 Þ; ðk1 ; k2 ÞÞ and TADWððn1 ; n2 Þ; ðk1 ; k2 ÞÞ. In the following, we will show that when a ðn1 ; n2 Þ vector and a ðk1 ; k2 Þ vector are given, the problem can be formulated as a weighted-bipartite matching problem (or assignment problem). Let

Aðk1 ; k2 Þ ¼

k1 n1 X X 1 ½ðj  1Þðn1  j þ 1Þ þ d1 ðn1  k1 Þðn1  k1 þ 1Þa1½j þ ðj  1Þðn1  j þ 1Þb1½j 2 j¼1 j¼k þ1 1

k2 n2 X X 1 þ ½ðj  1Þðn2  j þ 1Þ þ d2 ðn2  k2 Þðn2  k2 þ 1Þa2½j þ ðj  1Þðn2  j þ 1Þb2½j 2 j¼1 j¼k þ1 2

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and

Bðk1 ; k2 Þ ¼

k1 n1 X X 1 ½jðn1  jÞ þ d1 ðn1  k1  1Þðn1  k1 Þa1½j þ jðn1  jÞb1½j 2 j¼1 j¼k þ1 1

þ

k2 X

n2 X 1 ½jðn2  jÞ þ d2 ðn2  k2  1Þðn2  k2 Þa2½j þ jðn2  jÞb2½j : 2 j¼1 j¼k þ1 2

Since TADCððn1 ; n2 Þ; ðk1 ; k2 ÞÞ ¼ Aðk1 ; k2 Þ þ 12 ðn1  k1 Þðn1  k1 þ 1Þt1 þ 12 ðn2  k2 Þðn2  k2 þ 1Þt2 , for a given a ðn1 ; n2 Þ vector and a ðk1 ; k2 Þ vector, minimize TADCððn1 ; n2 Þ; ðk1 ; k2 ÞÞ is equivalent to minimizing Aðk1 ; k2 Þ. Suppose job J j ðj ¼ 1; 2; . . . ; nÞ is scheduled in position r ðr ¼ 1; 2; . . . ; ni Þ on machine M i ði ¼ 1; 2Þ, then its contribution for objective is ðj  1Þðni  j þ 1Þ þ 12 di ðni  ki Þðni  ki þ 1Þaj if r 6 ki , and ðj  1Þðni  j þ 1Þbj if r > ki . We define a bipartite graph GðV; EÞ; V ¼ V 1 [ V 2 ; V 1 \ V 2 ¼ ;; and jV 1 j ¼ jV 2 j ¼ n. The vertices in set V 1 represent jobs, and the vertices in set V 2 represent positions in the sequence. The edges in set E connect each vertex in V 1 with all vertices in V 2 . Associated with edge ði; jÞ; i 2 V 1 and j 2 V 2 , there is a weight wjir .

( wjir ¼

½ðj  1Þðni  j þ 1Þ þ 12 di ðni  ki Þðni  ki þ 1Þaj ; if r 6 ki ; ðj  1Þðni  j þ 1Þbj ;

Similar to the case of TADC, for the problem P2jrmjTADW, we can define a weight

v jir ¼

(  jðni  jÞ þ 12 di ðni  ki  1Þðni  ki Þ aj ; if r 6 ki ; jðni  jÞbj ;

ð5Þ

if r > ki :

if r > ki :

v jir

for edge ði; jÞ; i 2 V 1 and j 2 V 2 .

ð6Þ

The matching with minimum weight specifies the optimal value of Aðk1 ; k2 ÞðBðk1 ; k2 ÞÞ. Hence, for a given ðn1 ; n2 Þ vector and a ðk1 ; k2 Þ vector, the problem P2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADCðP2jT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADWÞ can be solved by a weightedbipartite matching problems. We have to solve such a weighted-bipartite matching problems for each ðn1 ; n2 Þ vector and a ðk1 ; k2 Þ vector. Although two machines are identical, two vectors ðn1 ; n2 Þ and ðk1 ; k2 Þ will produce the different TADC (TADW) since t1 – t2 and/or d1 – d2 . Hence the vectors ðn1 ; n2 Þ are ð0; nÞ; ð1; n  1Þ; . . . ; ðn; 0Þ, and the number of vectors ðn1 ; n2 Þ is n þ 1. For a given vector ðn1 ; n2 Þ, the vectors ðk1 ; k2 Þ are ð0; 0Þ; ð0; 1Þ; . . . ; ð0; n2 Þ; ð1; 0Þ; ð1; 1Þ; . . . ; ð1; n2 Þ; . . . ; ðn1 ; 0Þ; ðn1 ; 1Þ . . . ; ðn1 ; n2 Þ, the number of vectors ðk1 ; k2 Þ is ðn1 þ 1Þðn2 þ 1Þ. We have to solve a weighted-bipartite matching problems for each ðn1 ; n2 Þ vector and a ðk1 ; k2 Þ vector. The number of all weighted-bipartite matching problems is ðn þ 1Þðn1 þ 1Þðn2 þ 1Þ. As in Zhao et al. [10], we now to give a polynomial time algorithm for the problem P2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADCðP2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADWÞ. Algorithm 1

Step 0. Set n1 ¼ 0. Step 1. Set n2 ¼ n  n1 . Step 1.1. Set k1 ¼ 0. Step 1.1.1. Set k2 ¼ 0. Step 1.1.2. For a given pair of ðk1 ; k2 Þ create a weighted-bipartite matching problem with weight wjir ðv jir Þ defined above. Solve the weighted-bipartite matching problem and let the corresponding total cost be Aðk1 ; k2 ÞðBðk1 ; k2 ÞÞ. Set TADC ¼ Aðk1 ; k2 Þ þ 12 ðn1  k1 Þðn1  k1 þ 1Þt 1 þ 12 ðn2  k2 Þðn2  k2 þ 1Þt2 ðTADW ¼ Bðk1 ; k2 Þ þ 12 ðn1  k1  1Þðn1  k1 Þ t 1 þ 12 ðn2  k2  1Þðn2  k2 Þt 2 Þ. Step 1.1.3. Set k2 ¼ k2 þ 1, If k2 > n2 , then go to Step 1.1.4. Otherwise go to Step 1.1.2. Step 1.1.4. Set k1 ¼ k1 þ 1, If k1 > n1 , then go to Step 2. Otherwise go to Step 1.1.1. Step 2. Set n1 ¼ n1 þ 1, If n1 > n, then go to Step 3. Otherwise go to Step 1. Step 3. The optimal solution is the best one: 



TADCðTADWÞððn1 ; n2 Þ; ðk1 ; k2 ÞÞ ¼ minfTADCðTADWÞððn1 ; n2 Þ; ðk1 ; k2 ÞÞg: Theorem 1. Algorithm 1 will find an optimal solution for the problem P2jT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADCðP2jT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADWÞ with time complexity Oðn6 Þ. Proof. As discussed above, we can convert the problem P2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADCðP2jT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADWÞ to a weighted-bipartite matching problem and find an optimal solution by Algorithm 1. Step 1.1.2 can be solved in Oðn3 Þ for each weighted-bipartite matching problem, the number of weighted bipartite matching problems is Oðn3 Þ at most. Step 1 is executed n þ 1 times. Step 3 can be solved in Oðn3 Þ. Consequently, the overall time requirement of Algorithm 1 is Oðn6 Þ. h

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3.3. Extensions to m identical parallel machines Now we discuss how the results for the case of m ¼ 2 can be extended to m identical parallel machines ðm > 2Þ. Similarly to the case of m ¼ 2, when ðn1 ; n2 ; . . . ; nm Þ ðn1 þ n2 þ . . . nm ¼ nÞ and ðk1 ; k2 ; . . . ; km Þ ð0 6 ki 6 ni Þ and a sequence of job on each machine are given, we have

TADCððn1 ; n2 ; . . . ; nm Þ; ðk1 ; k2 ; . . . ; km ÞÞ ¼

 ki  m X X 1 ðj  1Þðni  j þ 1Þ þ di ðni  ki Þðni  ki þ 1Þ ai½j 2 i¼1 j¼1 þ

ni m m X X X 1 ðni  ki Þðni  ki þ 1Þt i þ ðj  1Þðni  j þ 1Þbi½j 2 i¼1 i¼1 j¼k þ1 i

and

TADWððn1 ; n2 ; . . . ; nm Þ; ðk1 ; k2 ; . . . ; km ÞÞ ¼

ki  m X X i¼1

j¼1

 1 jðni  jÞ þ di ðni  ki  1Þðni  ki Þ ai½j 2

ni m m X X ðni  ki  1Þðni  ki Þt i X þ jðni  jÞbi½j : þ 2 i¼1 i¼1 j¼k þ1 i

Hence, for a given ðn1 ; n2 ; . . . ; nm Þ vector and a ðk1 ; k2 ; . . . ; km Þ vector, the problem PmjT i ¼ ti þ di t; pj ¼ ðaj ; bj Þ jTADCðPmjT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADWÞ can be solved by a weighted-bipartite matching problems. For a given Q ðn1 ; n2 ; . . . ; nm Þ, the number of vectors ðk1 ; k2 ; . . . ; km Þ are m i ðni þ 1Þ. The remaining question is how many ðn1 ; n2 ; . . . ; nm Þ vectors exist. Note that ni may be 0; 1; 2; . . . ; n for i ¼ 1; 2; . . . ; m. So if we know that the numbers of jobs on the first P m  1 machines, the number of jobs on the last machine is then uniquely determined due to the fact that m i¼1 ni ¼ n. Therem1 fore we conclude that an upper bound on the number ðn1 ; n2 ; . . . ; nm Þ is ðn þ 1Þ . Thus, the number of weighted-bipartite Qm matching problems is at most i ðni þ 1Þðn þ 1Þm1 . We conclude that minimizing the sum of the TADC values on parallel identical machines can be solved in polynomial time, i.e., Theorem 2. When m > 2 is given, the problem PmjT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADCPmjT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADWÞ can be solved in Oðn2mþ2 Þ. Remark. The solution procedure for minimizing the sum of the TADC (TADW) values on identical parallel machines can be easily extended to the case of unrelated machines. Using a similar model to that of Mosheiov [35], let xjir be a 0/1 variable such that xjir ¼ 1 if job J j ðj ¼ 1; 2; . . . ; nÞ is assigned to machine Mi ði ¼ 1; 2; . . . ; mÞ at position rðr ¼ 1; 2; . . . ; ni Þ, and xjir ¼ 0, otherwise. For a given ðn1 ; n2 ; . . . ; nm Þ vector and a ðk1 ; k2 ; . . . ; km Þ vector, the problem RmjT i ¼ t i þ di t; pj ¼ ðaj ; bj ÞjTADCðRmjT i ¼ ti þ di t; pj ¼ ðaj ; bj ÞjTADWÞ can be solved by the following assignment problem:

min

ni X m X n X i¼1

st

wjir ðv jir Þxjir =sij

r¼1 j¼1

ni m X X

xjir ¼ 1;

i¼1 r¼1 n X

xjir ¼ 1;

j ¼ 1; 2; . . . ; n;

i ¼ 1; 2; . . . ; m;

r ¼ 1; 2; . . . ; ni ;

j ¼ 1; 2; . . . ; n;

i ¼ 1; 2; . . . ; m;

j¼1

xjir ¼ 0 or 1;

r ¼ 1; 2; . . . ; ni ;

where sij denote the process speed of job J j ðj ¼ 1; 2; . . . ; nÞ on machine M i ði ¼ 1; 2; . . . ; mÞ, wjir and and (6).

v jir are calculated by (5)

4. Conclusions In this paper we have studied the identical parallel machines scheduling problems with total absolute differences penalties (TADC in single machine systems was introduced by Kanet [27] and Bagchi [28] was the first to consider TADW in single machine systems). They proved that the problems in single machine systems can be solved in Oðn log nÞ time. We extend this setting to the case of jobs with a deteriorating maintenance activity, i.e., each machine is subject to preventive maintenance and the length of each maintenance period depends on its starting time. We proved that the problems can be solved in polynomial time. Further research includes the investigation of other non-regular objectives, and a more general (no necessarily linear) maintenance deterioration.

J.-B. Wang, C.-M. Wei / Applied Mathematics and Computation 217 (2011) 8093–8099

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Acknowledgements We are grateful to the editor and two anonymous referees for their helpful comments on an earlier version of this paper. This research was supported by the National Natural Science Foundation of China under Grant No. 11001181. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

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