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Applied Mathematical Modelling 37 (2013) 2388–2393
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Applied Mathematical Modelling
journal homepage: www.elsevier .com/locate /apm
Single-machine scheduling with convex resource dependent processingtimes and deteriorating jobs
Xue-Ru Wang a,⇑, Jian-Jun Wang b,c
a School of Science, Shenyang Aerospace University, Shenyang 110136, Chinab Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, Chinac College of Business, East Carolina University, Greenville, NC 27858, USA
a r t i c l e i n f o
Article history:Received 20 February 2012Received in revised form 8 May 2012Accepted 18 May 2012Available online 1 June 2012
Keywords:SchedulingSingle-machineDeteriorating jobsResource allocation
0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.05.025
⇑ Corresponding author.E-mail address: [email protected] (X.-R. Wang)
a b s t r a c t
In this study, we consider scheduling problems with convex resource dependent process-ing times and deteriorating jobs, in which the processing time of a job is a function of itsstarting time and its convex resource allocation. The objective is to find the optimalsequence of jobs and the optimal convex resource allocation separately. This paper focuson the single-machine problems with objectives of minimizing a cost function containingmakespan, total completion time, total absolute differences in completion times and totalresource cost, and a cost function containing makespan, total waiting time, total absolutedifferences in waiting times and total resource cost. It shows that the problems remainpolynomially solvable under the proposed model.
� 2012 Elsevier Inc. All rights reserved.
1. Introduction
In classical scheduling theory, it is assumed that the job processing times fixed and constant values [1]. In practice, how-ever, we often encounter settings in which job processing times may be subject to change due to the phenomenon of dete-rioration. Job deterioration appears, for instance, in the steel production where the temperature of an ingot drops below acertain level while waiting to enter a rolling machine, which requires reheating of the ingot before rolling. Similar situationswill also occur in scheduling maintenance tasks, national defense or cleaning assignments, where any delay in processing ajob is penalized by incurring additional time for accomplishing the job. Extensive surveys of different scheduling models andproblems involving deteriorating jobs can be found in Alidaee and Womer [2], Cheng et al. [3], and Gawiejnowicz [4]. Morerecent papers that have considered scheduling problems with deteriorating jobs include Rachaniotis and Pappis [5], Wanget al. [6,7], Wang and Guo [8], Wang and Wang [9], Zhao and Tang [10], Moslehi and Jafari [11], Pappis and Rachaniotis [12],Rachaniotis and Pappis [13], Voutsinas and Pappis [14], Huang et al. [15], Wang et al. [16,17], Bai et al. [18], Wei et al. [19],Wang and Wang [20], Wang et al. [21,22].
On the other hand, scheduling problems with controllable processing times in which the actual job processing time isassumed to be a function of the amount of resource allocated have been extensively studied. A more recent survey of differ-ent scheduling models and problems involving controllable processing times was given by Shabtay and Steiner [23]. Morerecent papers that have considered scheduling problems with controllable processing times include Wang and Xia [24],Shabtay and Steiner [25], Tseng et al. [26], Wang et al. [27], Zhu et al. [28], and Wang and Wang [29]. Moreover, very re-cently, based on the realistic situation, Wei et al. [19] first discussed scheduling problems considering linear resource depen-dent processing times and deteriorating jobs concurrently. In this paper, we consider single-machine scheduling problemswith convex resource dependent processing times and deteriorating jobs at the same time. The rest of this paper is organized
. All rights reserved.
.
X.-R. Wang, J.-J. Wang / Applied Mathematical Modelling 37 (2013) 2388–2393 2389
as follows. Notations and assumptions are given in Section 2. In Sections 3 and 4, we show that the problems can be solved inpolynomial time, respectively. In Section 5, conclusions are presented.
2. Problem formulation
We consider the problem of scheduling n jobs J1; J2; . . . ; Jn on a continuously available machine. All the jobs are availablefor processing at some time 0. The machine can handle one job at a time and job preemption is not allowed. Let pj be theactual processing time of job Jj. Shabtay [30], and Shabtay and Kaspi [31] considered the following single machine model,
i.e., the actual processing time of job Jj is pj ¼aj
uj
� �k, where k is a positive constant, aj P 0 is the normal (basic) processing
time of the job Jj (which represents the workload of job Jj) and uj > 0 is the amount of a non-renewable resource allocatedto job Jj. Cheng et al. [32] considered the following single machine model, i.e., the actual processing time of job Jj ispj ¼ aj þ bt, where b P 0 is the common deterioration rate for all the jobs and t P 0 is its start time. In this paper, we proposea new model that stems from Shabtay [30], Shabtay and Kaspi [31] and Cheng et al. [32]. Specially, we consider the followingtime and convex resource dependent processing times model
pj ¼aj
uj
� �k
þ bt; uj > 0; ð1Þ
where k is a positive constant, aj P 0 is the normal (basic) processing time of the job Jj (which represents the workload of jobJj), b P 0 is the common deterioration rate for all the jobs, t P 0 is its start time, and uj > 0 is the amount of a non-renewableresource allocated to job Jj.
For a given sequence p ¼ ½J1; J2; . . . ; Jn�; Cj ¼ CjðpÞ represents the completion time for job Jj. Let Cmax, TC=Pn
j¼1Cj,TW=
Pnj¼1Wj; TADC ¼
Pni¼1
Pnj¼ijCi � Cjj and TADW ¼
Pni¼1
Pnj¼ijWi �Wjj be the makespan of all jobs, the total completion
times, the total waiting times, the total absolute differences in completion times, and the total absolute differences in wait-ing times, where Wj ¼ Cj � pj be the waiting time of job Jj. The objective is to determine the optimal resource allocations andthe optimal sequence of jobs in the machine so that the corresponding value of the following cost functions can beminimized:
f1ðp;uÞ ¼ d1Cmax þ d2TCþ d3TADCþ d4
Xn
j¼1
Gjuj; ð2Þ
f2ðp;uÞ ¼ d1Cmax þ d2TWþ d3TADWþ d4
Xn
j¼1
Gjuj; ð3Þ
where weights d1 P 0; d2 P 0; d3 P 0 and d4 P 0 are given constants (the decision-maker selects the weightsd1; d2; d3; d4) and Gj is the per time unit cost associated with the resource allocation. In the remaining part of the paper,all the problems considered will be denoted using the three-field notation schema introduced by Graham et al. [33].
3. Problem 1jpj ¼ajuj
� �kþ btjd1Cmax þ d2TCþ d3TADCþ d4+
nj¼1Gjuj
Let p½r� and a½r� denote the actual processing time and the normal processing time of a job when it is scheduled in position rin a sequence, respectively. Then the completion times of jobs can be expressed as follows:
C ½1� ¼a½1�u½1�
� �k
C ½2� ¼a½1�u½1�
� �k
þ a½2�u½2�
� �k
þ ba½1�u½1�
� �k
¼ a½2�u½2�
� �k
þ ð1þ bÞ a½1�u½1�
� �k
. . .
C ½j� ¼Xj
l¼1
ð1þ bÞj�l a½l�u½l�
� �k
. . .
C ½n� ¼Xn
l¼1
ð1þ bÞn�l a½l�u½l�
� �k
:
ð4Þ
And the actual processing time of job J½j� can be expressed as follows:
p½j� ¼a½j�u½j�
� �k
þ bC ½j�1� ¼a½j�u½j�
� �k
þ bXj�1
l¼1
ð1þ bÞj�1�l a½l�u½l�
� �k !
: ð5Þ
For the model (2), if we substitute, C½j� ¼Pj
l¼1p½l�; Cmax ¼Pn
j¼1p½j�, TC ¼Pn
j¼1C½j� and TADC ¼Pn
j¼1ðj� 1Þðn� jþ 1Þp½j� [34]into (2) and simplify, we have
2390 X.-R. Wang, J.-J. Wang / Applied Mathematical Modelling 37 (2013) 2388–2393
f1ðp;uÞ ¼ d1
Xn
j¼1
p½j� þ d2
Xn
j¼1
ðn� jþ 1Þp½j� þ d3
Xn
j¼1
ðj� 1Þðn� jþ 1Þp½j� þ d4
Xn
j¼1
G½j�u½j�
¼Xn
j¼1
½d1 þ d2ðnþ 1� jÞ þ d3ðj� 1Þðn� jþ 1Þ�p½j� þ d4
Xn
j¼1
G½j�u½j� ¼Xn
j¼1
xjp½j� þ d4
Xn
j¼1
G½j�u½j�;
where xj ¼ d1 þ d2ðnþ 1� jÞ þ d3ðj� 1Þðn� jþ 1Þ.
For the model (2), if we substitute, p½j� ¼a½j�u½j�
� �kþ b
Pj�1l¼1ð1þ bÞj�1�l a½l�
u½l�
� �k� �
, Eq. (2) can be rewritten as ! !
f1ðp;uÞ ¼Xn
j¼1
xjp½j� þ d4
Xn
j¼1
G½j�u½j� ¼Xn
j¼1
xja½j�u½j�
� �k
þ bXj�1
l¼1
ð1þ bÞj�1�l a½l�u½l�
� �k
þ d4
Xn
j¼1
G½j�u½j�
¼ x1 þ bx2 þ bð1þ bÞx3 þ � � � þ bð1þ bÞn�2xn
� � a½1�u½1�
� �k
x2 þ bx3 þ bð1þ bÞx4 þ � � � þ bð1þ bÞn�3xn
� � a½2�u½2�
� �k
þ x3 þ bx4 þ bð1þ bÞx5 þ � � � þ bð1þ bÞn�4xn
� � a½3�u½3�
� �k
þ � � � þ ðxn�1 þ bxnÞa½n�1�
u½n�1�
� �k
þxna½n�u½n�
� �k
þ d4
Xn
j¼1
G½j�u½j�
¼Xn
j¼1
Xja½j�u½j�
� �k
þ d4
Xn
j¼1
G½j�u½j�; ð6Þ
where
X1 ¼ x1 þ bx2 þ bð1þ bÞx3 þ � � � þ bð1þ bÞn�2xn
X2 ¼ x2 þ bx3 þ bð1þ bÞx4 þ � � � þ bð1þ bÞn�3xn
X3 ¼ x3 þ bx4 þ bð1þ bÞx5 þ � � � þ bð1þ bÞn�4xn
. . .
Xn�1 ¼ xn�1 þ bxn
Xn ¼ xn:
ð7Þ
In the following lemma, we determine the optimal resource allocation, denoted by u�ðpÞ, as a function of the jobsequence.
Lemma 1. The optimal resource allocation as a function of the job sequence, u�ðpÞ, is:
u�½j� ¼kXj
d4G½j�
� � 1kþ1
� a½j�� � k
kþ1; ð8Þ
for the objective function d1Cmax þ d2TCþ d3TADCþ d4Pn
j¼1Gjuj, where Xj are given by Eq. (7).
Proof. By taking the derivative of the objective given by (6) with respect to u½j�; j ¼ 1;2; . . . ;n, equating it to zero and solvingit for u½j�, we obtain (8). Since the objective function is a convex function, (8) provides necessary and sufficient conditions foroptimality. h
By substituting (8) into (6), we obtain a new unified expression for the cost function for the objective functiond1Cmax þ d2TCþ d3TADCþ d4
Pnj¼1Gjuj under an optimal resource allocation and as a function of the job sequence:
f1ðp;u�ðpÞÞ ¼ k�k
kþ1 þ k1
kþ1
� �� d4ð Þ
kkþ1 �
Xn
j¼1
h½j�/j; ð9Þ
where
h½j� ¼ G½j�a½j�� � k
kþ1 ð10Þ
and/j ¼ Xj� � 1
kþ1 ð11Þ
for the objective function d1Cmax þ d2TCþ d3TADCþ d4Pn
j¼1Gjuj, where Xj are given by Eq. (7).In order to find the job sequence that minimizes f1ðp;u�ðpÞÞ, we have to optimally match the positional penalties /j with
the job-dependent costs hj. The optimal matching is obtained by applying the following lemma.
Lemma 2 (Hardy et al. [35]). The optimal job sequence is obtained by matching the smallest /j value to the job with the largest hj
value, the second smallest /j value to the job with the second largest hj value, and so on. The index of the /j matched with hj
specifies the position of job j in the optimal sequence for j ¼ 1;2; . . . ;n.
Table 1Date of Example 1.
r 1 2 3 4 5 6 7
xr 8 13 16 17 16 13 8
Xr 135.9063 93.9375 64.6250 43.7500 28.5 17 8
/r 11.6579 9.6921 8.0390 6.6144 5.3385 4.1231 2.8284
X.-R. Wang, J.-J. Wang / Applied Mathematical Modelling 37 (2013) 2388–2393 2391
The results of our analysis are summarized in the following optimization algorithm that solves the problem
1jpj ¼aj
uj
� �kþ btjd1Cmax þ d2TCþ d3TADCþ d4
Pnj¼1Gjuj.
Algorithm 1
Step 1. For each objective function, calculate /j and hj for j ¼ 1;2; . . . ;n by Eqs. ()(9)–(11).Step 2. Sequence the jobs according to Lemma 2, and denote the resulting optimal sequence by p� ¼ J½1�; J½2�; . . . ; J½n�
h i.
Step 3. Calculate the optimal resources allocation u�½j�ðp�Þ by using Eq. (8).
Theorem 1. For the scheduling problem 1jpj ¼aj
uj
� �kþ btjd1Cmax þ d2TCþ d3TADCþ d4
Pnj¼1Gjuj, an optimal solution can be
obtained by Algorithm 1 in Oðn log nÞ time.
Proof. The correctness of the algorithm follows from Lemmas 2 and 2. Steps 1 and 3 can be performed in linear time andStep 2 requires Oðn log nÞ time. Thus the overall computational complexity of the algorithm is Oðn log nÞ, which is equal tothe computational complexity of Step 2. h
The following example illustrates the working of Algorithm 1 for the problem 1jpj ¼aj
uj
� �kþ btjd1Cmax þ d2TCþ
d3TADCþ d4Pn
j¼1Gjuj.
Example 1. Data: n ¼ 7; a1 ¼ 13; a2 ¼ 4; a3 ¼ 7; a4 ¼ 16; a5 ¼ 10; a6 ¼ 3; a7 ¼ 9; G1 ¼ 2; G2 ¼ 3; G3 ¼ 1; G4 ¼ 4;G5 ¼ 2; G6 ¼ 5; G7 ¼ 6; b ¼ 0:5; k ¼ 1; d1 ¼ d2 ¼ d3 ¼ d4 ¼ 1. The input of the positional weights are given in Table 1.
From Eq. (10), we have, h1 ¼ 5:0990; h2 ¼ 3:4641; h3 ¼ 2:6458; h4 ¼ 8; h5 ¼ 4:4721; h6 ¼ 3:8730; h7 ¼ 7:3485. Stem fromAlgorithm 1, the optimal schedule is ½J3; J2; J6; J5; J1; J7; J4�, and the optimal resource allocation u�3 ¼ 30:8439; u�2 ¼11:1915; u�6 ¼ 6:2270; u�5 ¼ 14:7902; u�1 ¼ 13:6107; u�7 ¼ 5:0498; u�4 ¼ 5:6569. The total cost d1Cmax þ d2TCþ d3TADCþd4Pn
j¼1Gjuj is 205.2810.
4. Problem 1jpj ¼ajuj
� �kþ btjd1Cmax þ d2TWþ d3TADWþ d4+
nj¼1Gjuj
As in Section 3, for the objective d1Cmax þ d2TWþ d3TADWþ d4Pn
j¼1Gjuj, if we substitute, W ½j� ¼Pj�1
l¼1p½l�;Cmax ¼Pn
j¼1p½j�,
TW ¼Pn
j¼1W ½j�, TADW ¼Pn
j¼1jðn� jÞp½j� (Bagchi [36]), and p½j� ¼a½j�u½j�
� �kþ b
Pj�1l¼1ð1þ bÞj�1�l a½l�
u½l�
� �k� �
into d1Cmax þ d2TWþ
d3TADWþ d4Pn
j¼1Gjuj and simplify, we have
f2ðp;uÞ ¼ d1
Xn
j¼1
p½j� þ d2
Xn
j¼1
ðn� jÞp½j� þ d3
Xn
j¼1
jðn� jÞp½j� þ d4
Xn
j¼1
G½j�u½j� ¼Xn
j¼1
mjp½j� þ d4
Xn
j¼1
G½j�u½j�
¼Xn
j¼1
Wja½j�u½j�
� �k
þ d4
Xn
j¼1
G½j�u½j�; ð12Þ
where mj ¼ d1 þ d2ðn� jÞ þ d3jðn� jÞ, and
W1 ¼ m1 þ bm2 þ bð1þ bÞm3 þ � � � þ bð1þ bÞn�2mn
W2 ¼ m2 þ bm3 þ bð1þ bÞm4 þ � � � þ bð1þ bÞn�3mn
W3 ¼ m3 þ bm4 þ bð1þ bÞm5 þ � � � þ bð1þ bÞn�4mn
. . .
Wn�1 ¼ mn�1 þ bmn
Wn ¼ mn:
ð13Þ
As in Section 3, In the following lemma, we determine the optimal resource allocation, denoted by u�ðpÞ, as a function of thejob sequence.
Table 2Date of Example 2.
r 1 2 3 4 5 6 7
mr 13 16 17 16 13 8 1
Wr 97.7344 67.1563 45.4375 29.6250 17.7500 8.5 1
ur 9.8861 8.1949 6.7407 5.4429 4.2131 2.9155 1
2392 X.-R. Wang, J.-J. Wang / Applied Mathematical Modelling 37 (2013) 2388–2393
Lemma 3. The optimal resource allocation as a function of the job sequence, u�ðpÞ, is:
u�½j� ¼kWj
d4G½j�
� � 1kþ1
� a½j�� � k
kþ1; ð14Þ
for the objective function d1Cmax þ d2TWþ d3TADWþ d4Pn
j¼1Gjuj, where Wj are given by Eq. (13).By substituting (14) into (12), we obtain a new unified expression for the cost function for the objective function
d1Cmax þ d2TWþ d3TADWþ d4Pn
j¼1Gjuj under an optimal resource allocation and as a function of the job sequence:
f2ðp;u�ðpÞÞ ¼ k�k
kþ1 þ k1
kþ1
� �� d4ð Þ
kkþ1 �
Xn
j¼1
g½j�uj; ð15Þ
where
g½j� ¼ G½j�a½j�� � k
kþ1 ð16Þ
and
uj ¼ Wj� � 1
kþ1 ð17Þ
for the objective function d1Cmax þ d2TWþ d3TADWþ d4Pn
j¼1Gjuj, where Wj are given by Eq. (13).The results of our analysis are summarized in the following optimization algorithm that solves the problem 1jpj ¼
aj
uj
� �kþ
btjd1Cmax þ d2TWþ d3TADWþ d4Pn
j¼1Gjuj.
Algorithm 2
Step 1. For each objective function, calculate gj and uj for j ¼ 1;2; . . . ;n by Eqs. (15)–(17).Step 2. Sequence the jobs according to Lemma 2, and denote the resulting optimal sequence by p� ¼ ½J½1�; J½2�; . . . ; J½n��.Step 3. Calculate the optimal resources allocation u�½j�ðp�Þ by using Eq. (14).
Theorem 2. For the scheduling problem 1jpj ¼aj
uj
� �kþ btjd1Cmax þ d2TWþ d3TADWþ d4
Pnj¼1Gjuj, an optimal solution can be
obtained by Algorithm 2 in Oðn log nÞ time.The following example illustrates the working of Algorithm 2 for the problem 1jpj ¼
aj
uj
� �kþ btjd1Cmaxþ
d2TWþ d3TADWþ d4Pn
j¼1Gjuj.
Example 2. The data is the same as Example 1. The input of the positional weights are given in Table 2. From Eq. (16), wehave, g1 ¼ 5:0990; g2 ¼ 3:4641; g3 ¼ 2:6458; g4 ¼ 8; g5 ¼ 4:4721; g6 ¼ 3:8730; g7 ¼ 7:3485. Stem from Algorithm 2, theoptimal schedule is ½J3; J2; J6; J5; J1; J7; J4�, and the optimal resource allocation u�3 ¼ 26:1561; u�2 ¼ 9:4626; u�6 ¼ 5:2214; u�5 ¼12:1707; u�1 ¼ 10:7413; u�7 ¼ 3:5707; u�4 ¼ 2. The total cost d1Cmax þ d2TWþ d3TADWþ d4
Pnj¼1Gjuj is 155.8997.
5. Conclusions
The problem of scheduling n jobs with convex resource dependent processing times and deteriorating jobs has been stud-ied. The objective function is to minimize a cost function containing makespan, total completion (waiting) time, total abso-lute differences in completion (waiting) times and total resource cost. We have solved the problem in polynomial time. Infuture research, we plan to explore more realistic settings such as multi machines, and different objective functions.
Acknowledgements
The authors are grateful for two anonymous referees for their helpful comments on earlier version of the article. This re-search was supported by the National Natural Science Foundation of China (Grant No. 11001181). Jian-Jun Wang was also
X.-R. Wang, J.-J. Wang / Applied Mathematical Modelling 37 (2013) 2388–2393 2393
supported by the National Natural Science Foundation of China (70902033) and the Fundamental Research Funds for theCentral Universities (DUT11SX10).
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