9
Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times Na Yin a,b , Liying Kang a,, Tian-Chuan Sun a , Chao Yue a , Xue-Ru Wang b a Department of Mathematics, Shanghai University, Shanghai 200444, China b School of Science, Shenyang Aerospace University, Shenyang 110136, China article info Article history: Received 5 June 2012 Received in revised form 18 February 2014 Accepted 11 March 2014 Available online xxxx Keywords: Scheduling Unrelated parallel machines Deteriorating jobs Resource allocation abstract We consider unrelated parallel machines scheduling problems involving resource depen- dent (controllable) processing times and deteriorating jobs simultaneously, i.e., the actual processing time of a job is a function of its starting time and its resource allocation. Two generally resource consumption functions, the linear and convex resource, were investi- gated. The objective is to find the optimal sequence of jobs and the optimal resource allo- cation separately. This paper focus on the objectives of minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost, and a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. If the number of unrelated parallel machines is a given constant, we show that the problems remain polynomially solvable under the proposed model. Ó 2014 Published by Elsevier Inc. 1. Introduction There are various situations in which the job processing times may be increase due to delays or waiting of jobs, such kinds of tasks are called deteriorating jobs. Job deterioration (time-dependent scheduling) appears, for instance, in the steel pro- duction where the temperature of an ingot drops below a certain level while waiting to enter a rolling machine, which re- quires reheating of the ingot before rolling. Similar situations will also occur in maintenance planning and scheduling, national defense or cleaning assignments, where any delay in processing a job is penalized by incurring additional time for accomplishing the job. An extensive survey of different scheduling theory and models involving deteriorating jobs (time-dependent scheduling) can be found in Gawiejnowicz [1]. Huang and Wang [2] considered scheduling problems with deteriorating jobs, i.e., the model p j ¼ a j þ bt, where p j ; a j and t, respectively, represent the actual processing time, normal (basic) processing time and the starting time of job J j on a single machine, and b represents the common deterioration rate. For single machine, parallel identical machines and unrelated machines, they proved that the total absolute differences in completion (waiting) times minimization can be solved in polynomial time (i.e., 1jp j ¼ a j þ btjTADCðTADWÞ; Pmjp j ¼ a j þ btjTADCðTADWÞ; Rmjp j ¼ a j þ btjTADCðTADWÞ, where C j (W j ) is the completion (waiting) time of job J j ; TADC ¼ P n i¼1 P n j¼i jC i C j j and TADW ¼ P n i¼1 P n j¼i jW i W j j). Jafari and Moslehi [3] considered the same model with Huang and Wang [2], for the number of tardy jobs minimization problem (1jp j ¼ a j þ btj P n j¼1 U j ), they proposed a branch-and-bound http://dx.doi.org/10.1016/j.apm.2014.03.022 0307-904X/Ó 2014 Published by Elsevier Inc. Corresponding author. Tel.: +86 21 66135652. E-mail addresses: [email protected] (N. Yin), [email protected] (L. Kang), [email protected] (X.-R. 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: N. Yin et al., Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.022

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Page 1: Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times

Applied Mathematical Modelling xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Applied Mathematical Modelling

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

Unrelated parallel machines scheduling with deteriorating jobsand resource dependent processing times

http://dx.doi.org/10.1016/j.apm.2014.03.0220307-904X/� 2014 Published by Elsevier Inc.

⇑ Corresponding author. Tel.: +86 21 66135652.E-mail addresses: [email protected] (N. Yin), [email protected] (L. Kang), [email protected] (X.-R. Wang).

Please cite this article in press as: N. Yin et al., Unrelated parallel machines scheduling with deteriorating jobs and resource depprocessing times, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.022

Na Yin a,b, Liying Kang a,⇑, Tian-Chuan Sun a, Chao Yue a, Xue-Ru Wang b

a Department of Mathematics, Shanghai University, Shanghai 200444, Chinab School of Science, Shenyang Aerospace University, Shenyang 110136, China

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

Article history:Received 5 June 2012Received in revised form 18 February 2014Accepted 11 March 2014Available online xxxx

Keywords:SchedulingUnrelated parallel machinesDeteriorating jobsResource allocation

We consider unrelated parallel machines scheduling problems involving resource depen-dent (controllable) processing times and deteriorating jobs simultaneously, i.e., the actualprocessing time of a job is a function of its starting time and its resource allocation. Twogenerally resource consumption functions, the linear and convex resource, were investi-gated. The objective is to find the optimal sequence of jobs and the optimal resource allo-cation separately. This paper focus on the objectives of minimizing a cost functioncontaining makespan, total completion time, total absolute differences in completion timesand total resource cost, and a cost function containing makespan, total waiting time, totalabsolute differences in waiting times and total resource cost. If the number of unrelatedparallel machines is a given constant, we show that the problems remain polynomiallysolvable under the proposed model.

� 2014 Published by Elsevier Inc.

1. Introduction

There are various situations in which the job processing times may be increase due to delays or waiting of jobs, such kindsof tasks are called deteriorating jobs. Job deterioration (time-dependent scheduling) appears, for instance, in the steel pro-duction where the temperature of an ingot drops below a certain level while waiting to enter a rolling machine, which re-quires reheating of the ingot before rolling. Similar situations will also occur in maintenance planning and scheduling,national defense or cleaning assignments, where any delay in processing a job is penalized by incurring additional timefor accomplishing the job. An extensive survey of different scheduling theory and models involving deteriorating jobs(time-dependent scheduling) can be found in Gawiejnowicz [1]. Huang and Wang [2] considered scheduling problems withdeteriorating jobs, i.e., the model pj ¼ aj þ bt, where pj; aj and t, respectively, represent the actual processing time, normal(basic) processing time and the starting time of job Jj on a single machine, and b represents the common deteriorationrate. For single machine, parallel identical machines and unrelated machines, they proved that the total absolute differencesin completion (waiting) times minimization can be solved in polynomial time (i.e., 1jpj ¼ aj þ btjTADCðTADWÞ;Pmjpj ¼ aj þ btjTADCðTADWÞ; Rmjpj ¼ aj þ btjTADCðTADWÞ, where Cj (Wj) is the completion (waiting) time of job

Jj; TADC ¼Pn

i¼1

Pnj¼ijCi � Cjj and TADW ¼

Pni¼1

Pnj¼ijWi �Wjj). Jafari and Moslehi [3] considered the same model with Huang

and Wang [2], for the number of tardy jobs minimization problem (1jpj ¼ aj þ btjPn

j¼1Uj), they proposed a branch-and-bound

endent

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2 N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

algorithm. Wang and Wang [4] considered a three-machine permutation flow shop problem with deteriorating jobs, inwhich the actual processing time of job Jj on machine Mi is pij ¼ aij þ bt, where aij is a normal (basic) processing time ofjob Jj on machine Mi. For the makespan minimization scheduling problem (i.e., F3jprmu; pij ¼ aij þ bt; b P 0jCmax), theyproposed two heuristic algorithms and a branch-and-bound algorithm.

On the other hand, scheduling problems with controllable processing times (resource-dependent processing times) inwhich the actual job processing time is assumed to be a function of the amount of resource allocated have been extensivelystudied. More recent surveys of different scheduling models and problems involving controllable processing times (resource-dependent processing times) were given by Shabtay and Steiner [5], and Edis et al. [6]. Shabtay et al. [7] considered a singlemachine group scheduling problem with a family due date assignment method. They proved that a cost function consistingof earliness, tardiness and due date assignment penalties problem can be solved in polynomial time if all jobs belonging togroup technology assumption. They also extended the results to the case of controllable processing times for both linear andconvex resource consumption functions (i.e., the models pj ¼ aj � hjuj (where hj P 0 and uj is the amount of a non-renewable

resource allocated to job Jj) and pj ¼aj

uj

� �k(k > 0 is a constant, and uj > 0 is the amount of a non-renewable resource allo-

cated to job Jj)). Xu et al. [8] considered a single machine total tardiness minimization scheduling problem with a convex

controllable processing times, i.e., the model pj ¼aj

uj

� �k. They proved that the problem 1jpj ¼

aj

uj

� �k;Pn

j¼1Tj 6 T; dj ¼

djPn

j¼1uj can be solved in Oðn2Þ time, where dj;Cj and Tj ¼ maxfCj � djg is the due date, the completion time and the tardiness

of job Jj, respectively. Wang and Wang [9] considered the problem 1jpj ¼aj

uj

� �k;Pn

j¼1wjCj 6 FjPn

j¼1uj, where wj is the weight

of the job Jj and F is the total weighted flow time of a given permutation. In order to solve this problem, they proposed aheuristic algorithm and a branch-and-bound algorithm.

It is natural to study scheduling problems combining deteriorating jobs and controllable processing times (resource allo-cation). However, to the best of our knowledge, there exist only a few papers dealing with the resource allocation and dete-riorating jobs simultaneously. Wei et al. [10] discussed single-machine scheduling problems considering linear resourcedependent processing times and deteriorating jobs concurrently, i.e., pj ¼ aj þ bt � hjuj, where aj; b; t; hj and uj all have the

same meaning as before. They proved that the problems 1jpj ¼ aj þ bt � hjujjd1Cmax þ d2TC þ d3TADC þ d4Pn

j¼1Gjuj and

1jpj ¼ aj þ bt � hjujjd1Cmax þ d2TW þ d3TADW þ d4Pn

j¼1Gjuj can be solved in polynomial time, where

TC ¼Pn

j¼1Cj; TW ¼Pn

j¼1Wj;Gj denote the per time unit cost associated with the resource allocation of job Jj andd1 P 0; d2 P 0; d3 P 0 and d4 P 0 denote the weights. Wang and Wang [11] discussed single-machine scheduling problems

considering convex resource dependent processing times and deteriorating jobs concurrently, i.e., pj ¼aj

uj

� �kþ bt;uj > 0.

They proved that the problems 1jpj ¼aj

uj

� �kþ btjd1Cmax þ d2TC þ d3TADC þ d4

Pnj¼1Gjuj and 1jpj ¼

aj

uj

� �kþ btjd1Cmaxþ

d2TW þ d3TADW þ d4Pn

j¼1Gjuj can be solved in polynomial time. Wang and Wang [12] considered the problems

1jpj ¼ aj þ bt � hjujjPðaEj þ bTj þ cdj þ GjujÞ and 1jpj ¼

aj

uj

� �kþ btj

PðaEj þ bTj þ cdj þ GjujÞ. For three popular due date

assignment methods, they proposed a polynomial time algorithm, respectively. The phenomena resource allocation andtime-dependent processing times (deteriorating jobs) occurring simultaneously can be found in steel production, i.e., inthe process of preheating ingots by gas to prepare them for hot rolling on the blooming mill. Before the ingots can be hotrolled, they have to achieve the required temperature. However, the preheating time of the ingots depends on their startingtemperature. The preheating time can be shortened by the increase of the gas flow intensity, i.e., the more gas is consumed,the shorter lasts the preheating process. Thus, the ingot preheating time depends on the starting moment of the preheatingprocess and the amount of gas consumed during it (Wei et al. [10], and Bachman and Janiak [13]). However, in the realisticproduction settings, there is rarely a single-machine environment, i.e., parallel machine scheduling problems are morepractical in industrial production environments, such as electronic industry, mechanical industry and so on. Therefore, in thispaper, we consider unrelated parallel machines scheduling with linear (convex) resource dependent processing times anddeteriorating jobs at the same time. The rest of this paper is organized as follows. Notations and assumptions are given inSection 2. In Sections 3 and 4, we show that the problems can be solved in polynomial time for linear and convexresource dependent processing times, respectively. Section 5 are devoted to examples. In Section 6, conclusions arepresented.

2. Problem formulation

Given n independent and non-preemptive jobs J ¼ fJ1; J2; . . . ; Jng and m unrelated parallel machines M ¼ fM1;M2; . . . ;Mmg(each job could be processed by a free machine) with associated the actual processing time pij ði ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;nÞ.All the jobs are available for processing at time zero. In this research, we consider the following models:

A time and linear resource dependent processing times model (Wei et al. [10]):

Pleaseproces

pij ¼ aij þ bt � hijuij; ð1Þ

cite this article in press as: N. Yin et al., Unrelated parallel machines scheduling with deteriorating jobs and resource dependentsing times, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.022

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N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 3

where aij P 0 is the normal (basic) processing time of the job Jj on machine Mi; hij P 0 and uij is the amount of a non-renew-able resource allocated to job Jj on machine Mi, with 0 6 uij 6 mij 6

aij

hijand mij is the upper bound on the amount of resource

that can be allocated to job Jj on machine Mi.A time and convex resource dependent processing times model (Wang and Wang [11]):

Pleaseproces

pij ¼aij

uij

� �k

þ bt; uij > 0; ð2Þ

where uij > 0 is the amount of a non-renewable resource allocated to job Jj on machine Mi.For a given sequence, suppose Mi processed ni ði ¼ 1;2; . . . ;mÞ ðn1 þ n2 þ � � �nm ¼ nÞ jobs, let i½j� denote the jth position on

machine Mi. Let Ji½j� denote the jth job on machine Mi Ci½j� (Wi½j�) denote the completion (waiting) time of job Ji½j�, where

Wi½j� ¼ Ci½j� � pi½j�; pi½j�; ai½j�; hi½j�;ui½j�, and Gi½j� are defined similarly. For convenience, we denote Cimax; TCi ¼

Pnij¼1Cij; TWi ¼Pni

j¼1Wij; TADCi ¼Pni

l¼1

Pnij¼ljCil � Cijj and TADWi ¼

Pnil¼1

Pnij¼ljWil �Wijj as the makespan, the total completion time, the total

waiting time, the total absolute deviation of job completion time, and the total absolute deviation of job waiting time on

machine Mi, respectively. ThenPm

i¼1Cimax;

Pmi¼1TCi;

Pmi¼1TWi,

Pmi¼1TADCi and

Pmi¼1TADWi denote the total load, the total com-

pletion time, the total waiting time, the total absolute deviation of job completion time, and the total absolute deviation ofjob waiting time on all machines, respectively.

Let ðn1;n2; . . . ;nmÞ denote the allocation vector of job numbers. The problem is to determine the sets N1;N2; . . . ;Nm, whereNij j ¼ ni ði ¼ 1;2; . . . ;mÞ, the sequence of job on each machine and the resource allocation u ¼ ðu1;u2; . . . ;unÞ to minimize

objective functions

f1ðp;uÞ ¼ d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TCi þ d3

Xm

i¼1

TADCi þ d4

Xm

i¼1

Xni

j¼1

Gijuij; ð3Þ

f2ðp;uÞ ¼ d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TWi þ d3

Xm

i¼1

TADWi þ d4

Xm

i¼1

Xni

j¼1

Gijuij; ð4Þ

where Gij is the per time unit cost associated with the resource allocation of job Jj on machine Mi. Then, following the stan-dard notation used in the scheduling literature (Gawiejnowicz [1]), we denote our scheduling problems as

Rm pij ¼ aij þ bt � hijuij

�� ��d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TCi þ d3

Xm

i¼1

TADCi þ d4

Xm

i¼1

Xni

j¼1

Gijuij;

Rm pij ¼ aij þ bt � hijuij

�� ��d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TWi þ d3

Xm

i¼1

TADWi þ d4

Xm

i¼1

Xni

j¼1

Gijuij;

Rm pij ¼aij

uij

� �k

þ bt

����������d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TCi þ d3

Xm

i¼1

TADCi þ d4

Xm

i¼1

Xni

j¼1

Gijuij and

Rm pij ¼aij

uij

� �k

þ bt

����������d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TWi þ d3

Xm

i¼1

TADWi þ d4

Xm

i¼1

Xni

j¼1

Gijuij:

3. Solution with a time and linear resource dependent processing times model

3.1. Problem Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCi þ d4

Pmi¼1

Pnij¼1Gijuij

When ðn1;n2; . . . ;nmÞ and a sequence of job on each machine are given, then from Wei et al. [10], we have

Ci½j� ¼Xj

l¼1

ð1þ bÞj�lðai½l� � hi½l�ui½l�Þ ð5Þ

and

pi½j� ¼ ai½j� � hi½j�ui½j� þ bXj�1

l¼1

ð1þ bÞj�1�lðai½l� � hi½l�ui½l�Þ: ð6Þ

For the model (3), if we substitute, Ci½j� ¼Pj

l¼1pi½l�; Cimax ¼

Pnij¼1pi½j�; TCi ¼

Pnij¼1Ci½j�, TADCi ¼

Pnij¼1ðj� 1Þðni � jþ 1Þpi½j�

(Kanet [14]) and pi½j� ¼ ai½j� � hi½j�ui½j� þ bPj�1

l¼1ð1þ bÞj�1�lðai½l� � hi½l�ui½l�Þ into (3), we have

cite this article in press as: N. Yin et al., Unrelated parallel machines scheduling with deteriorating jobs and resource dependentsing times, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.022

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4 N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

Pleaseproces

f1ðp;uÞ ¼ d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TCi þ d3

Xm

i¼1

TADCi þ d4

Xm

i¼1

Xni

j¼1

Gijuij

¼Xm

i¼1

Xni

j¼1

½d1 þ d2ðni þ 1� jÞ þ d3ðj� 1Þðni � jþ 1Þ�pi½j� þ d4

Xm

i¼1

Xn

j¼1

Gi½j�ui½j�

¼Xm

i¼1

Xni

j¼1

xij ai½j� � hi½j�ui½j� þ bXj�1

l¼1

ð1þ bÞj�1�lðai½l� � hi½l�ui½l�Þ" #

þ d4

Xm

i¼1

Xn

j¼1

Gi½j�ui½j�

¼Xm

i¼1

Xni

j¼1

Xijai½j� þXm

i¼1

Xni

j¼1

ðd4Gi½j� � hi½j�XijÞui½j�; ð7Þ

where

xij ¼ d1 þ d2ðni þ 1� jÞ þ d3ðj� 1Þðni � jþ 1ÞXi1 ¼ xi1 þ bxi2 þ bð1þ bÞxi3 þ � � � þ bð1þ bÞni�2xini

Xi2 ¼ xi2 þ bxi3 þ bð1þ bÞxi4 þ � � � þ bð1þ bÞni�3xini

Xi3 ¼ xi3 þ bxi4 þ bð1þ bÞxi5 þ � � � þ bð1þ bÞni�4xini

� � �Xi;ni�1 ¼ xi;ni�1 þ bxini

Xini¼ xini

:

In order to obtain the optimal resource allocations and the optimal job sequence, for a given ðn1;n2; . . . ;nmÞ vector, we

formulate the problem Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCi þ d4

Pmi¼1

Pnij¼1Gijuij as an assignment

problem. Let xijr be a 0/1 variable such that xijr ¼ 1 if job Jj (j ¼ 1;2; . . . ;n) is assigned to machine Mi (i ¼ 1;2; . . . ;m) at posi-tion r (r ¼ 1;2; . . . ;ni), and xjir ¼ 0, otherwise. For a given ðn1;n2; . . . ;nmÞ vector, from (7), the problem

Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCi þd4

Pmi¼1

Pnij¼1Gijuij can be solved by the following assignment

problem:

minXm

i¼1

Xni

r¼1

Xn

j¼1

kijrxijr ð8Þ

subject to

Xm

i¼1

Xni

r¼1

xijr ¼ 1; j ¼ 1;2; . . . ;n;

Xn

j¼1xijr ¼ 1; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni;

xijr ¼ 0 or 1; j ¼ 1;2; . . . ; n; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni:

where

kijr ¼Xiraij; if d4Gij � hijXir P 0;Xiraij þ ðd4Gij � hijXirÞmij; if d4Gij � hijXir < 0:

�ð9Þ

The remaining question is how many ðn1;n2; . . . ;nmÞ vectors exist. Note that ni may be 0;1;2; . . . ; n for i ¼ 1;2; . . . ;m. So ifwe know that the numbers of jobs on the first m� 1 machines, the number of jobs on the last machine is then uniquelydetermined due to the fact that

Pmi¼1ni ¼ n. Therefore, an upper bound on the number ðn1;n2; . . . ;nmÞ is ðnþ 1Þm�1. Thus,

we conclude that the following Theorem holds.

Theorem 1. When m is a given constant, the problem Rm pij ¼ aij þ bt � hijuij�� ��d1

Pmi¼1Ci

max þ d2Pm

i¼1TCi þ d3Pm

i¼1TADCiþd4Pm

i¼1Pni

j¼1Gijuij can be solved in polynomial time Oðnmþ2Þ.

Proof. As discussed above, to solve the problem Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCiþ

d4Pm

i¼1

Pnij¼1Gijuij, a polynomial number (i.e., ðnþ 1Þm�1) of assignment problems need to be solved. Each assignment problem

is solved in Oðn3Þ time (using the well-known Hungarian method). Hence, the time complexity of the problem

Rmjpij ¼ aij þ bt � hijuijjd1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCi þ d4

Pmi¼1

Pnij¼1Gijuij can be solved in Oðnmþ2Þ time. h

cite this article in press as: N. Yin et al., Unrelated parallel machines scheduling with deteriorating jobs and resource dependentsing times, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.022

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N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 5

3.2. Problem Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWi þ d4

Pmi¼1

Pnij¼1Gijuij

As in Section 3.1, for the problem Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWi þ d4

Pmi¼1

Pnij¼1Gijuij, if

we substitute, Wi½j� ¼Pj�1

l¼1pi½l�;Cimax ¼

Pnij¼1pi½j�; TWi ¼

Pnij¼1Wi½j�, TADWi ¼

Pnij¼1jðni � jÞpi½j� (Bagchi [15]) and pi½j� ¼ ai½j��

hi½j�ui½j� þ bPj�1

l¼1ð1þ bÞj�1�lðai½l� � hi½l�ui½l�Þ into (4), we have

Pleaseproces

f2ðp;uÞ ¼ d1

Xm

i¼1

Cimax þ d2

Xm

i¼1

TWi þ d3

Xm

i¼1

TADWi þ d4

Xm

i¼1

Xni

j¼1

Gijuij ¼Xm

i¼1

Xni

j¼1

Wijai½j� þXm

i¼1

Xni

j¼1

ðd4Gi½j� � hi½j�WijÞui½j�; ð10Þ

where

mij ¼ d1 þ d2ðni � jÞ þ d3jðni � jÞWi1 ¼ mi1 þ bmi2 þ bð1þ bÞmi3 þ � � � þ bð1þ bÞni�2mini

Wi2 ¼ mi2 þ bmi3 þ bð1þ bÞmi4 þ � � � þ bð1þ bÞni�3mini

Wi3 ¼ mi3 þ bmi4 þ bð1þ bÞmi5 þ � � � þ bð1þ bÞni�4mini

� � �Wi;ni�1 ¼ mi;ni�1 þ bmini

Wini¼ mini

:

For a given ðn1;n2; . . . ; nmÞ vector, from (10), the problem Rm pij ¼ aij þ bt � hijuij

�� ��d1Pm

i¼1Cimax þd2

Pmi¼1TWiþ

d3Pm

i¼1TADWi þ d4Pm

i¼1

Pnij¼1Gijuij can be solved by the following assignment problem:

minXm

i¼1

Xni

r¼1

Xn

j¼1

~kijrxijr ð11Þ

subject to

Xm

i¼1

Xni

r¼1

xijr ¼ 1; j ¼ 1;2; . . . ; n;

Xn

j¼1

xijr ¼ 1; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni;

xijr ¼ 0 or 1; j ¼ 1;2; . . . ;n; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni;

where

~kijr ¼Wiraij; if d4Gij � hijWir P 0;Wiraij þ ðd4Gij � hijWirÞmij; if d4Gij � hijWir < 0:

Theorem 2. When m is a given constant, the problem Rm pij ¼ aij þ bt � hijuij�� ��d1

Pmi¼1Ci

max þ d2Pm

i¼1TWiþd3Pm

i¼1TADWi þ d4Pm

i¼1Pni

j¼1Gijuij can be solved in polynomial time Oðnmþ2Þ.

4. Solution with a time and convex resource dependent processing times model

4.1. Rm pij ¼aij

uij

� �kþ bt

��������d1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCi þ d4

Pmi¼1

Pnij¼1Gijuij

From Wang and Wang [11], we have

Ci½j� ¼Xj

l¼1

ð1þ bÞj�l ai½l�

ui½l�

� �k

ð12Þ

and

pi½j� ¼ai½j�

ui½j�

� �k

þ bXj�1

l¼1

ð1þ bÞj�1�l ai½l�

ui½l�

� �k !

: ð13Þ

f1ðp;uÞ ¼Xm

i¼1

Xni

j¼1

Xijai½j�

ui½j�

� �k

þ d4

Xm

i¼1

Xni

j¼1

Gi½j�ui½j�; ð14Þ

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Page 6: Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times

6 N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

where

Pleaseproces

xij ¼ d1 þ d2ðni þ 1� jÞ þ d3ðj� 1Þðni � jþ 1ÞXi1 ¼ xi1 þ bxi2 þ bð1þ bÞxi3 þ � � � þ bð1þ bÞni�2xini

Xi2 ¼ xi2 þ bxi3 þ bð1þ bÞxi4 þ � � � þ bð1þ bÞni�3xini

Xi3 ¼ xi3 þ bxi4 þ bð1þ bÞxi5 þ � � � þ bð1þ bÞni�4xini

� � �Xi;ni�1 ¼ xi;ni�1 þ bxini

Xini¼ xini

:

Taking the first derivative to Eq. (14) with respect to ui½j�; i ¼ 1;2; . . . ;m; j ¼ 1;2; . . . ;ni, equating it to zero and solving it,we have:

Lemma 1. The optimal resource allocation of problem Rm pij ¼aij

uij

� �kþ bt

��������d1Pm

i¼1Cimax þ d2

Pmi¼1TCi þ d3

Pmi¼1TADCiþ

d4Pm

i¼1Pni

j¼1Gijuij can be expressed as follows:

u�i½j� ¼kXij

d4Gi½j�

� � 1kþ1

� ai½j�� k

kþ1: ð15Þ

By substituting (15) into (14), we have

f1ðp;uÞ ¼ k�k

kþ1 þ k1

kþ1

� �� d4ð Þ

kkþ1 �

Xm

i¼1

Xni

j¼1

Gi½j�ai½j�� k

kþ1 Xij� 1

kþ1; ð16Þ

where Xij are given by Eq. (14).Similar to the analysis in the previous sub-section, if the vector ðn1;n2; . . . ;nmÞ is known, the optimal sequence is obtained

via the following assignment problem:

minXm

i¼1

Xni

r¼1

Xn

j¼1

hijrxijr ð17Þ

subject to

Xm

i¼1

Xni

r¼1

xijr ¼ 1; j ¼ 1;2; . . . ;n;

Xn

j¼1

xijr ¼ 1; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni;

xijr ¼ 0 or 1; j ¼ 1;2; . . . ; n; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni:

where hijr ¼ Gijaij� k

kþ1 Xirð Þ1

kþ1.

Theorem 3. When m is a given constant, the problem Rm pij ¼aij

uij

� �kþ bt

��������d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWiþ

d4Pm

i¼1Pni

j¼1Gijuij can be solved in polynomial time Oðnmþ2Þ.

4.2. Rm pij ¼aij

uij

� �kþ bt

��������d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWi þ d4

Pmi¼1

Pnij¼1Gijuij

As in Sections 3.2 and 4.1, we have

f2ðp;uÞ ¼Xm

i¼1

Xni

j¼1

Wijai½j�

ui½j�

� �k

þ d4

Xm

i¼1

Xni

j¼1

Gi½j�ui½j�; ð18Þ

where

mij ¼ d1 þ d2ðni � jÞ þ d3jðni � jÞWi1 ¼ mi1 þ bmi2 þ bð1þ bÞmi3 þ � � � þ bð1þ bÞni�2mini

Wi2 ¼ mi2 þ bmi3 þ bð1þ bÞmi4 þ � � � þ bð1þ bÞni�3mini

Wi3 ¼ mi3 þ bmi4 þ bð1þ bÞmi5 þ � � � þ bð1þ bÞni�4mini

� � �Wi;ni�1 ¼ mi;ni�1 þ bmini

Wini¼ mini

:

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Page 7: Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times

N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx 7

Lemma 2. The optimal resource allocation of problem Rm pij ¼aij

uij

� �kþ bt

��������d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWiþ

d4Pm

i¼1Pni

j¼1Gijuij can be expressed as follows:

Pleaseproces

u�i½j� ¼kWij

d4Gi½j�

� � 1kþ1

� ai½j�� k

kþ1: ð19Þ

By substituting (19) into (18), we have

f2ðp;uÞ ¼ k�k

kþ1 þ k1

kþ1

� �� d4ð Þ

kkþ1 �

Xm

i¼1

Xni

j¼1

Gi½j�ai½j�� k

kþ1 Wij� 1

kþ1; ð20Þ

where Wij are given by Eq. (18).If the vector ðn1;n2; . . . ;nmÞ is known, the optimal sequence is obtained via the following assignment problem:

minXm

i¼1

Xni

r¼1

Xn

j¼1

~hijrxijr ð21Þ

subject to

Xm

i¼1

Xni

r¼1

xijr ¼ 1; j ¼ 1;2; . . . ; n;

Xn

j¼1

xijr ¼ 1; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni;

xijr ¼ 0 or 1; j ¼ 1;2; . . . ;n; i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ;ni;

where ~hijr ¼ Gijaij� k

kþ1 Wirð Þ1

kþ1.

Theorem 4. When m is a given constant, the problem Rm pij ¼aij

uij

� �kþ bt

��������d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWiþ

d4Pm

i¼1Pni

j¼1Gijuij can be solved in polynomial time Oðnmþ2Þ.

5. Examples

Example 1. Steel ingots are to be heated by gas to prepare them for hot rolling on the blooming mill, the deteriorating jobsand resource dependent processing times takes into account the consumption of resources (gas) and the starting times ofjobs (ingots). There are several work centers, each with a number of unrelated parallel machines. For the modelpij ¼ aij þ bt � hijuij, consider 4 jobs (ingots) to be processed on machines M1 and M2 with the dates are given in Table 1. Thecost parameters are: d1 ¼ d2 ¼ d3 ¼ d4 ¼ 1, and the deterioration rate is b ¼ 0:1.

According Theorem 1, the results are:ðn1;n2Þ ¼ ð0;4Þ; x21 ¼ 5; x22 ¼ 7; x23 ¼ 7; x24 ¼ 5; X21 ¼ 7:075; X22 ¼ 8:25; X23 ¼ 7:5; X24 ¼ 5, and the cost is

118.65;ðn1;n2Þ ¼ ð1;3Þ; x11 ¼ 2; x21 ¼ 4; x22 ¼ 5; x23 ¼ 4; X11 ¼ 2; X21 ¼ 4:94; X22 ¼ 5:4; X23 ¼ 4, the cost is 69;ðn1;n2Þ ¼ ð2;2Þ; x11 ¼ 3; x12 ¼ 3; x21 ¼ 3; x22 ¼ 3; X11 ¼ 3:3; X12 ¼ 3; X21 ¼ 3:3; X22 ¼ 3, the cost is 46.5;ðn1;n2Þ ¼ ð3;1Þ; x11 ¼ 4; x12 ¼ 5; x13 ¼ 4; x21 ¼ 2; X11 ¼ 4:94; X12 ¼ 5:4; X13 ¼ 4; X21 ¼ 2, the cost is 57.16;ðn1;n2Þ ¼ ð4;0Þ; x11 ¼ 5; x12 ¼ 7; x13 ¼ 7; x14 ¼ 5; X11 ¼ 7:075; X12 ¼ 8:25; X13 ¼ 7:5; X14 ¼ 5, and the cost is

95.575.Hence for the objective d1

Pmi¼1Ci

max þ d2Pm

i¼1TCi þ d3Pm

i¼1TADCi þ d4Pm

i¼1

Pnij¼1Gijuij, an optimal schedule is ½J2; J1� on

machine M1; u12 ¼ 0; u11 ¼ 0; C12 ¼ 3; C11 ¼ 3þ 5þ 0:1 � 3 ¼ 8:3, and ½J3; J4� on machine M2; u23 ¼ 0; u24 ¼ 0; C23 ¼2; C24 ¼ 2þ 5þ 0:1 � 2 ¼ 7:2 and the optimal cost is 46.5.

Table 1Date of Example 1.

j ¼ 1 j ¼ 2 j ¼ 3 j ¼ 4

a1j 5 3 6 4a2j 8 6 2 5h1j 1 2 2 1h2j 2 1 1 2m1j 4 1 3 3m2j 3 3 2 2G1j 2 8 10 6G2j 11 9 7 10

cite this article in press as: N. Yin et al., Unrelated parallel machines scheduling with deteriorating jobs and resource dependentsing times, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.03.022

Page 8: Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times

Table 2Date of Example 2.

j ¼ 1 j ¼ 2 j ¼ 3 j ¼ 4

a1j 5 3 6 4a2j 8 6 2 5G1j 12 8 10 6G2j 11 9 7 10

8 N. Yin et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx

Similarly, for the objective d1Pm

i¼1Cimax þ d2

Pmi¼1TWi þ d3

Pmi¼1TADWi þ d4

Pmi¼1

Pnij¼1Gijuij; m11 ¼ 3; m12 ¼ 1; m21 ¼

3; m22 ¼ 1; W11 ¼ 3:1; W12 ¼ 1; W21 ¼ 3:1; W22 ¼ 1, an optimal schedule is ½J2; J1� on machine M1; u12 ¼ 0; u11 ¼ 0; C12 ¼3; C11 ¼ 8:3; W12 ¼ 0; W11 ¼ 3, and ½J3; J4� on machine M2; u23 ¼ 0; u24 ¼ 0; C23 ¼ 2; C24 ¼ 7:2; W23 ¼ 0; W24 ¼ 2 andthe optimal cost is 25.5.

Example 2. For the model pij ¼aij

uij

� �kþ bt, consider 4 jobs (ingots) to be processed on machines M1 and M2 with the dates

are given in Table 2. The cost parameters are: d1 ¼ d2 ¼ d3 ¼ d4 ¼ 1, the deterioration rate is b ¼ 0:1, and k ¼ 2.According Theorem 3, the results are:ðn1;n2Þ ¼ ð0;4Þ;x21 ¼ 5;x22 ¼ 7; x23 ¼ 7; x24 ¼ 5; X21 ¼ 7:075; X22 ¼ 8:25; X23 ¼ 7:5; X24 ¼ 5, and the cost is

188.1588;ðn1;n2Þ ¼ ð1;3Þ; x11 ¼ 2; x21 ¼ 4; x22 ¼ 5; x23 ¼ 4; X11 ¼ 2; X21 ¼ 4:94; X22 ¼ 5:4; X23 ¼ 4, the cost is 142.1085;ðn1;n2Þ ¼ ð2;2Þ; x11 ¼ 3; x12 ¼ 3; x21 ¼ 3; x22 ¼ 3; X11 ¼ 3:3; X12 ¼ 3; X21 ¼ 3:3; X22 ¼ 3, the cost is 116.3590;ðn1;n2Þ ¼ ð3;1Þ; x11 ¼ 4; x12 ¼ 5; x13 ¼ 4; x21 ¼ 2; X11 ¼ 4:94; X12 ¼ 5:4; X13 ¼ 4; X21 ¼ 2, the cost is 114.1775;ðn1;n2Þ ¼ ð4;0Þ; x11 ¼ 5; x12 ¼ 7; x13 ¼ 7; x14 ¼ 5; X11 ¼ 7:075; X12 ¼ 8:25; X13 ¼ 7:5; X14 ¼ 5, and the cost is

167.6862.Hence for the objective d1

Pmi¼1Ci

max þ d2Pm

i¼1TCi þ d3Pm

i¼1TADCi þ d4Pm

i¼1

Pnij¼1Gijuij, an optimal schedule is ½J4; J2; J1� on ma-

chine M1; u14 ¼ 2 � 4:946

� 13 � 4

23 ¼ 2:9756; u12 ¼ 2 � 5:4

8

� 13 � 3

23 ¼ 2:2989; u11 ¼ 2 � 4

12

� 13 � 5

23 ¼ 2:5544; C14 ¼ 4

2:9756

� 2 ¼ 1:8071;

C12 ¼ 1:8071þ 32:2989

� 2 þ 0:1 � 1:8071 ¼ 3:6908; C11 ¼ 3:6908þ 52:5544

� 2 þ 0:1 � 3:6908 ¼ 7:8913, and ½J3� on machine

M2; u23 ¼ 2 � 27

� 13 � 2

231:3173; C23 ¼ 2

1:3173

� 2 ¼ 2:3051 and the optimal cost is 114.1775.Similarly, for the objective d1

Pmi¼1Ci

max þ d2Pm

i¼1TWi þ d3Pm

i¼1TADWi þ d4Pm

i¼1

Pnij¼1Gijuij; m11 ¼ 3; m12 ¼ 1; m21 ¼ 3; m22 ¼

1; W11 ¼ 3:1; W12 ¼ 1; W21 ¼ 3:1; W22 ¼ 1, an optimal schedule is ½J4; J2� on machine M1; u14 ¼ 2 � 3:16

� 13 � 4

23 ¼ 2:5475;

u12 ¼ 2 � 18

� 13 � 3

23 ¼ 1:3104; C14 ¼ 4

2:5475

� 2 ¼ 2:4654; C12 ¼ 2:4654þ 31:3104

� 2 þ 0:1 � 2:4654 ¼ 7:9532; W14 ¼ 0; W12 ¼ 2:4654,

and ½J3; J1� on machine M2; u23 ¼ 2 � 3:17

� 13 � 2

23 ¼ 1:5245; u21 ¼ 2 � 1

11

� 13 � 8

23 ¼ 2:2661; C23 ¼ 2

1:5245

� 2 ¼ 1:7211; C21 ¼1:7211þ 6

2:2661

� 2 þ 0:1 � 1:7211 ¼ 8:9036; W23 ¼ 0; W21 ¼ 1:7211 and the optimal cost is 92.0492.

6. Summary and conclusions

We studied the unrelated parallel machines scheduling with deteriorating jobs and resource dependent (controllable)processing times. We assume that the processing time of a job is a (linear or convex) function of the amount of resource allo-cated to the job and its starting time. The goal is to find a job sequence and the resource allocation that jointly minimize thetotal cost of makespan, total completion (waiting) time, total absolute differences in completion (waiting) times and re-source consumption. If m is a given constant, we showed that the problems can be solved in polynomial time Oðnmþ2Þ. Infuture research, we plan to consider flow shop scheduling (an potential application can be found in steel mills), availableconstraints and release time of this problem. Other challenging possible extensions are either to (i) Parero-optimal schedules(solutions) ðd1

Pmi¼1Ci

max þ d2Pm

i¼1TCi þ d3Pm

i¼1TADCi;Pm

i¼1

Pnij¼1GijuijÞ, or to (ii) problems subject to limited resource availabil-

ity (e.g.,Pm

i¼1

Pnij¼1Gijuij 6 U, where U is a real number).

Acknowledgments

The authors are grateful for the Editor and two anonymous referees for their helpful comments on earlier version of thearticle. This research was supported by the National Natural Science Foundation of China under Grant Nos. 11001181,11171207, 91130032.

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