Applied Mathematical Modelling 36 (2012) 792–798
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Applied Mathematical Modelling
journal homepage: www.elsevier .com/locate /apm
Single-machine scheduling with time-and-resource-dependentprocessing times
Cai-Min Wei a,⇑, Ji-Bo Wang b, Ping Ji c
a Department of Mathematics, Shantou University, Shantou 515063, Chinab School of Science, Shenyang Aerospace University, Shenyang 110136, Chinac Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
a r t i c l e i n f o
Article history:Received 5 September 2009Received in revised form 24 May 2011Accepted 1 July 2011Available online 23 July 2011
Keywords:SchedulingSingle-machineDeteriorating jobsResource allocation
0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.07.005
⇑ Corresponding author.E-mail addresses: [email protected] (C.-M
a b s t r a c t
We consider single-machine scheduling problems in which the processing time of a job is afunction of its starting time and its resource allocation. The objective is to find the optimalsequence of jobs and the optimal resource allocation separately. We concentrate on twogoals separately, namely, minimizing a cost function containing makespan, total completiontime, total absolute differences in completion times and total resource cost; minimizing acost function containing makespan, total waiting time, total absolute differences in waitingtimes and total resource cost. We show that the problems remain polynomially solvableunder the proposed model.
� 2011 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,however, we often encounter settings in which job processing times may be subject to change due to the phenomenonof deterioration. Job deterioration appears, for instance, in the steel production where the temperature of an ingot dropsbelow a certain level while waiting to enter a rolling machine, which requires reheating of the ingot before rolling. Similarsituations will also occur in scheduling maintenance tasks, national defense or cleaning assignments, where any delay inprocessing a job is penalized by incurring additional time for accomplishing the job. Extensive surveys of different sched-uling models and problems involving deteriorating jobs can be found in Alidaee and Womer [2], and Cheng et al. [3]. Morerecent papers that have considered scheduling problems with deteriorating jobs include Wang and Xia [4], Gawiejnowiczet al. [5], Janiak and Kovalyov [6], Wu and Lee [7], Gawiejnowicz [8], Wang et al. [9], Lee et al. [10, 11], Li et al. [12], Tangand Liu [13], Ng et al. [14], Yang [15], Yang and Wang [16], Huang and Wang [17], Wang and Wang [18], and Wang et al.[19]. Wang and Xia [4] considered various single-machine and flow-shop scheduling problems with decreasing lineardeterioration of job processing times. Gawiejnowicz et al. [5] considered two single-machine bicriterion scheduling prob-lems with time-dependent job processing times. Janiak and Kovalyov [6] considered the problem of scheduling jobs exe-cuted by a human in a contaminated area. Wu and Lee [7] considered the two-machine flow shop total completion timeminimization problem with deteriorating jobs. They derived several dominance properties and two lower bounds to facil-itate the search for the optimal solution in the branch-and-bound algorithm. Gawiejnowicz [8] considered two single-ma-chine makespan minimization scheduling problems with proportionally deteriorating jobs. In the first problem, the
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. Wei), [email protected] (J.-B. Wang), [email protected] (P. Ji).
C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792–798 793
machine is not continuously available for processing but the number of non-availability periods, and the start time andthe end time of each period are known in advance. In the second problem, the machine is available all the time butfor each job a ready time and a deadline are defined. He showed that decision versions of these two problems are NP-com-plete in the ordinary sense or in the strong sense, depending on the number of non-availability periods or the number ofdistinct ready times and deadlines. Wang et al. [9] considered single-machine scheduling with deteriorating jobs in whichthe jobs are constrained by a series–parallel graph constraint. They proved that the problem can be solved in polynomialtime. Lee et al. [10] considered the same model of Wu and Lee [7]. But with a makespan objective function. Lee et al. [11]considered the permutation flow shop scheduling with machine-dependent job deterioration rates. The objective functionis to minimize the total completion time. They proposed a dominance rule and an efficient lower bound to speed up thesearching for the optimal solution. Li et al. [12] considered single machine scheduling of deteriorating jobs to minimizetotal absolute differences in completion times. Tang and Liu [13] considered two scheduling problems for a two-machineflowshop where a single machine is followed by a batching machine. The first problem is that there is a transporter tocarry the jobs between machines. The second problem is that there are deteriorating jobs to be processed on the singlemachine. For the first problem with minimizing the makespan, they formulate it as a mixed integer programming modeland then proved that it is strongly NP-hard. A heuristic algorithm is proposed for solving this problem and its worst caseperformance is analyzed. For the second problem, they derived the optimal algorithms with polynomial time for minimiz-ing the makespan, the total completion time and the maximum lateness, respectively. Ng et al. [14] considered a two-ma-chine flow shop scheduling problem to minimize the total completion time with proportional linear deterioration. Theyderived several dominance properties, some lower bounds, and an initial upper bound and applied them in a proposedbranch-and-bound algorithm to search for the optimal solution. Yang [15] considered some single-machine schedulingproblems with both start-time dependent learning and position dependent aging effects under deteriorating maintenanceconsideration. Yang and Wang [16] considered a two-machine flow shop scheduling problem with simple linear deterio-ration and total eighted completion time criterion. They derived several dominance properties, some lower bounds, and aninitial upper bound and applied them in a proposed branch-and-bound algorithm to search for the optimal solution.Huang and Wang [17] considered parallel identical machines scheduling problems with deteriorating jobs. They showedthat two scheduling problems remains polynomially solvable under the proposed model. Wang and Wang [18] consideredthe single-machine scheduling problems with nonlinear deterioration. They showed that even with the introduction ofnonlinear deterioration to job processing times, single machine makespan minimization problem remains polynomiallysolvable. Wang et al. [19] considered a single machine scheduling problem with simple linear deterioration. For the jobswith chain precedence constraints, they proved that the weighted sum of squared completion times minimization problemwith strong chains and weak chains can be solved in polynomial time, respectively. We refer the reader to review Gaw-iejnowicz [20] for more details on single-machine, parallel-machine and dedicated-machine scheduling problems withtime-dependent processing times.
On the other hand, the problems with fixed job processing times dependent on resources have been considered by Ja-niak [21], Nowicki and Zdrzalka [22], Panwalkar and Rajagopalan [23], Cheng and Janiak [24], Blazewicz et al. [25], Wangand Xia [26], and Tseng et al. [27]. However, to the best of our knowledge, there exist only a few research results onscheduling models considering the resource allocation and deteriorating jobs at the same time. The phenomena of re-source allocation and deteriorating jobs occurring simultaneously can be found in many real-life situations. For example,in steel production, more precisely, in the process of preheating ingots by gas to prepare them for hot rolling on theblooming mill. Before the ingots can be hot rolled, they have to achieve the required temperature. However, the preheat-ing time of the ingots depends on their starting temperature, i.e., the longer ingots wait for the start of the preheatingprocess, the lower goes their temperature and therefore the longer lasts the preheating process. The preheating timecan be shortened by the increase of the gas flow intensity, i.e., the more gas is consumed, the shorter lasts the preheatingprocess. Thus, the ingot preheating time depends on the starting moment of the preheating process and the amount of gasconsumed during it [28]. Bachman and Janiak [28] first considered single-machine scheduling with job processing timesdependent on the starting moments of job execution and on the amounts of resource allocation to the jobs. They provedthat the makespan minimization problem is NP-hard. They also gave some properties of the optimal resource allocation.Janiak and Iwanowski [29] considered the single machine scheduling problems with time and resource dependent pro-cessing times. They considered the following criteria: the makespan and the total completion time subject to a given con-straint on the total resource consumption and the total resource consumption criterion subject to a given constraint eitheron the makespan or on the total completion time, respectively. For a given schedule of jobs, they proved that an optimalresource allocation vector can be constructed in polynomial time. We proved that they operate in polynomial time. Zhaoet al. [30,31] considered single-machine scheduling with deteriorating jobs where the release times of the jobs depend onthe amounts of resource allocation. For two resource constrained scheduling problems, they gave optimal algorithms tofind the optimal resource allocations.
In this paper, we consider single machine scheduling problems with time and resource dependent processing times at thesame time. The rest of this paper is organized as follows. Notations and assumptions are given in Section 2. In Sections 3 and4, we show that the problems can be formulated as an assignment problem, respectively. In Section 5, a test example is gi-ven. In Section 6, conclusions are presented.
794 C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792–798
2. Problem formulation
We consider the problem of scheduling n jobs J1, J2, . . . , Jn on a continuously available machine. All the jobs are available forprocessing at some time 0. The machine can handle one job at a time and job preemption is not allowed. Let pj be the actualprocessing time of job Jj. In this paper, we consider the following time-and-resource dependent processing times model
pj ¼ aj þ bt � huj; ð1Þ
where aj P 0 is the normal (basic) processing time of the job Jj, b P 0 is the common deterioration rate for all the jobs, t P 0is its start time, h P 0 and uj is the amount of a non-renewable resource allocated to job Jj, with 0 6 uj 6 mj 6
aj
h and mj is theupper bound on the amount of resource that can be allocated to job Jj.
For a given sequence p = [J1, J2, . . . , Jn], Cj = Cj(p) represents the completion time for job Jj. LetCmax; TC ¼
Pnj¼1Cj;TW ¼
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 absolutedifferences in waiting times, where Wj = Cj � pj be the waiting time of job Jj. The objective is to determine the optimal re-source allocations and the optimal sequence of jobs in the machine so that the corresponding value of the following costfunctions be optimal:
f ðp;uÞ ¼ d1Cmax þ d2TCþ d3 TADCþ d4
Xn
j¼1
Gjuj; ð2Þ
f ðp;uÞ ¼ d1Cmax þ d2TWþ d3 TADWþ 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 weights d1, 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 problemsconsidered will be denoted using the three-field notation schema introduced by Graham et al. [32].
3. Problem 1jpj ¼ aj þ bt � hujjd1Cmax þ d2TCþ d3 TADCþ d4+nj¼1Gjuj
Let p[r] and a[r] denote the actual processing time and the normal processing time of a job when it is scheduled in positionr in a sequence, respectively. Then the completion times of jobs ’ be expressed as follows:
C½1� ¼ a½1� � hu½1�;
C½2� ¼ a½1� � hu½1� þ a½2� þ bða½1� � hu½1�Þ � hu½2� ¼ a½2� � hu½2� þ ð1þ bÞða½1� � hu½1�Þ;C½3� ¼ C ½2� þ a½3� þ bC½2� � hu½3� ¼ a½3� � hu½3� þ ð1þ bÞða½2� � hu½2�Þ þ ð1þ bÞ2ða½1� � hu½1�Þ;
. . .
C½j� ¼Pj
l¼1ð1þ bÞj�lða½l� � hu½l�Þ;
. . .
C½n� ¼Pnj¼1ð1þ bÞn�jða½j� � hu½j�Þ:
ð4Þ
And the actual processing time of job J[r] can be expressed as follows:
p½j� ¼ a½j� þ bC ½j�1� � hu½j� ¼ a½j� � hu½j� þ bXj�1
l¼1
ð1þ bÞj�1�lða½l� � hu½l�Þ: ð5Þ
For the model (2), if we substitute, C½j� ¼Pj
l¼1p½l�;Cmax ¼Pn
j¼1p½j�;TC ¼Pn
j¼1C½j� and TADC ¼Pn
j¼1ðj� 1Þðn� jþ 1Þp½j� [33]into (2) and simplify, we have
f ðp;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 = d1 + d2(n + 1 � j) + d3(j � 1)(n � j + 1).For the model (2), if we substitute, p½j� ¼ a½j� � hu½j� þ b
Pj�1l¼1ð1þ bÞj�1�lða½l� � hu½l�Þ, Eq. (2) can be rewritten as
C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792–798 795
f ðp;uÞ ¼Xn
j¼1
xjp½j� þ d4
Xn
j¼1
G½j�u½j� ¼Xn
j¼1
xj a½j� � hu½j� þ bXj�1
l¼1
ð1þ bÞj�1�lða½l� � hu½l�Þ !
þ d4
Xn
j¼1
G½j�u½j�
¼ x1ða½1� � hu½1�Þ þx2ða½2� � hu½2� þ bða½1� � hu½1�ÞÞ þx3ða½3� � hu½3� þ bða½2� � hu½2� þ ð1þ bÞða½1� � hu½1�ÞÞÞ
þx4ða½4� � hu½4� þ bða½3� � hu½3� þ ð1þ bÞða½2� � hu½2�Þ þ ð1þ bÞ2ða½1� � hu½1�ÞÞÞ þ � � � þxn�1ða½n�1� � hu½n�1�
þ bða½n�2� � hu½n�2� þ ð1þ bÞða½n�3� � hu½n�3�Þ þ � � � þ ð1þ bÞn�4ða½2� � hu½2�Þ þ ð1þ bÞn�3ða½1� � hu½1�ÞÞÞ þxnða½n�� hu½n� þ bða½n�1� � hu½n�1� þ ð1þ bÞða½n�2� � hu½n�2�Þ þ � � � þ ð1þ bÞn�3ða½2� � hu½2�Þ þ ð1þ bÞn�2ða½1� � hu½1�ÞÞÞ
þ d4
Xn
j¼1
G½j�u½j�
¼ x1 þ bx2 þ bð1þ bÞx3 þ � � � þ bð1þ bÞn�2xn
� �ða½1� � hu½1�Þ
þ x2 þ bx3 þ bð1þ bÞx4 þ � � � þ bð1þ bÞn�3xn
� �ða½2� � hu½2�Þ
þ x3 þ bx4 þ bð1þ bÞx5 þ � � � þ bð1þ bÞn�4xn
� �ða½3� � hu½3�Þ þ � � � þ ðxn�1 þ bxnÞða½n�1� � hu½n�1�Þ þxnða½n�
� hu½n�Þ þ d4
Xn
j¼1
G½j�u½j�
¼Xn
j¼1
Xja½j� þXn
j¼1
ðd4G½j� � hXjÞu½j�;
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:
In order to obtain the optimal resource allocations and the optimal sequence of jobs, we formulate the model (2) as anassignment problem, respectively. Let
kij ¼Xjai; if d4Gi � hXj P 0;Xjai þ ðd4Gi � hXjÞmi; if d4Gi � hXj < 0:
�ð6Þ
Furthermore, let zij be a 0/1 variable such that zij = 1 if job Ji is scheduled in position j, and zij = 0, otherwise. As in Panwalkarand Rajagopalan [23], the optimal matching of jobs to positions requires a solution for the following assignment problem:
minPni¼1
Pnj¼1
kijzij
subject toPni¼1
zij ¼ 1; i ¼ 1;2; . . . ;n;
Pnj¼1
zij ¼ 1; j ¼ 1;2; . . . ; n;
zij ¼ 0 or 1; i; j ¼ 1;2; . . . ;n:
ð7Þ
Recall that solving an assignment problem of size n requires an effort of O(n3) (using the well-known Hungarian method),hence the optimal solution can be found in polynomial time.
4. Problem 1jpj ¼ aj þ bt � hujjd1Cmax þ d2TWþ d3 TADWþ d4+nj¼1Gjuj
As in Section 3, for the model (3), if we substitute, W ½j� ¼Pj�1
l¼1p½l�;Cmax ¼Pn
j¼1p½j�; TW ¼Pn
j¼1W ½j� andTADW ¼
Pnj¼1jðn� jÞp½j� [34] into (3) and simplify, we have
f ðp;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�;
where mj = d1 + d2(n � j) + d3j(n � j).
Table 1Date of Example 1.
i 1 2 3 4 5
ai 5 4 3 1 2mi 4 3 2 1 1Gi 1 3 4 3 2
Table 2Weight of Example 1.
j 1 2 3 4 5
xj 6 9 10 9 6Xj 9.8876 11.716 11.56 9.6 6
Table 3kij of Example 1.
inj 1 2 3 4 5
1 21.8876 23.716 23.56 21.6 182 36.8876 38.716 38.56 36.6 24
kij= 3 29.6628 35.148 34.68 28.8 184 9 9 9 9 65 15.8876 17.716 17.56 15.6 12
796 C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792–798
For the model (3), if we substitute, p½j� ¼ a½j� � hu½j� þ bPj�1
l¼1ð1þ bÞj�1�lða½l� � hu½l�Þ, Eq. (3) can be rewritten as
f ðp;uÞ ¼Xn
j¼1
mjp½j� þ d4
Xn
j¼1
G½j�u½j� ¼Xn
j¼1
Wja½j� þXn
j¼1
ðd4G½j� � hWjÞu½j�;
where
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:
As in Section 3, let
hij ¼Wjai; if d4Gi � hWj P 0;Wjai þ ðd4Gi � hWjÞmi; if d4Gi � hWj < 0;
�ð9Þ
The optimal sequence is obtained, as the following assignment problem
minPni¼1
Pnj¼1
hijzij
subject toPni¼1
zij ¼ 1; i ¼ 1;2; . . . ;n;
Pnj¼1
zij ¼ 1; j ¼ 1;2; . . . ;n; zij ¼ 0 or 1; i; j ¼ 1;2; . . . ;n:
ð10Þ
5. A test example
Since the algorithm is similar for these two problems, in the numerical example we will apply the algorithm only for theproblem 1jpj ¼ aj þ bt � hujjd1Cmax þ d2TCþ d3TADCþ d4
Pnj¼1Gjuj.
C.-M. Wei et al. / Applied Mathematical Modelling 36 (2012) 792–798 797
Example 1. In the processing of preheating ingots by gas to prepare them for hot rolling on the blooming mill, the time-and-resource-dependent processing times approach takes into account the starting times and consumption of resources (gas).Consider 5 ingots (jobs) with: b = 0.1, h = 1, d1 = d2 = d3 = 1, d4 = 3 and the parameters for each job, the weights and the kij asgiven in Tables 1–3. From assignment problem (7), the optimal schedule is [J5, J4, J1, J3, J2] and the optimal total cost is101.2476.
However if we use the simple First In, First Out (FIFO) rule, the schedule is [J1, J2, J3, J4, J5] and the total cost is 116.2836,their relative error is 116:2836�101:2476
101:2476 � 100% ¼ 14:85%. If we use the smallest processing time first (SPT) rule, the schedule is[J4, J5, J3, J2, J1] and the total cost is 116.2836, their relative error is 115:996�101:2476
101:2476 � 100% ¼ 14:57%. If we use the largest pro-cessing time first (LPT) rule, the schedule is [J1, J2, J3, J5,J4] and the total cost is 116.2836, their relative error is116:8836�101:2476
101:2476 � 100% ¼ 15:44%.
Remark 1. The significance of the results of our analysis for this case is that the optimal schedule can be obtained by anassignment problem (7).
6. Conclusions
The problem of scheduling n jobs with time-and-resource-dependent processing times has been studied. The objectivefunction is to minimize a cost function containing makespan, total completion (waiting) time, total absolute differencesin completion (waiting) times and total resource cost. We have solved the problem by formulating it as an assignment prob-lem. The model can also be easily applied to the ’mirror’ scheduling problem in which the actual processing time of job isgiven by pj(t) = aj � b t � huj for t P 0 (the parameter b must be given properly to guarantee pj > 0). In future research, weplan to explore more general time-and-resource-dependent processing times models and extend the problems to multiplemachine settings.
Acknowledgements
We are grateful to the editor and an anonymous referee for their helpful comments on an earlier version of this paper. Weacknowledge the Research Grants Council (RGC) of Hong Kong for financial support under the project. This research was alsosupported by the National Natural Science Foundation of China (Grant Nos.: 11001181 and 71065001).
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