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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
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
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
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
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
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
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
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
Xmi¼1
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
:
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
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
Xmi¼1
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
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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