Applied Mathematics and Computation 218 (2011) 2069–2073

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Applied Mathematics and Computation

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

Scheduling with deteriorating jobs and learning effects

Dar-Li Yang, Wen-Hung Kuo ⇑Department of Information Management, National Formosa University, Yunlin 632, Taiwan

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

Keywords:SchedulingDeteriorating jobsLearning effectMakespanTardy job

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.07.023

⇑ Corresponding author.E-mail address: whkuo@nfu.edu.tw (W.-H. Kuo).

This paper studies a single machine scheduling problem simultaneously with deterioratingjobs and learning effects. The objectives are to minimize the makespan and the number oftardy jobs, respectively. Two polynomial time algorithms are proposed to solve these prob-lems optimally.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Scheduling problems with deteriorating jobs have received increasing attention in recent years. In these problems, theactual processing time of a job is modeled as an increasing function of its starting time due to deterioration effects. Kunna-thur and Gupta [1] and Mosheiov [2] presented several real-life situations where deteriorating jobs might occur such as steelproduction, resource allocation, fire fighting, maintenance or cleaning, etc., in which any delay in processing a job may resultin an increasing effort to accomplish the job. For the single-machine scheduling problems, Browne and Yechiali [3] studiedthe problem with a deterioration model in which the actual processing time (pi) of job i is pi = ai + aiti, where ai, ai and ti arethe original processing time, deterioration rate and starting time of job i, respectively. They showed that sequencing the jobsin non-decreasing order of ai/ai minimizes the makespan. Mosheiov [4] introduced the above model into the total comple-tion time minimization scheduling problem. He showed that there exists an optimal schedule (V-shaped schedule) if the ori-ginal processing times of all jobs are equal (i.e. pi = a + aiti). Bachman and Janiak [5] showed that the maximum latenessminimization problem with the linear deterioration model (pi = ai + aiti) is NP-complete. Bachman et al. [6] also showed thatthe total weighted completion time minimization problem with the same deterioration model (pi = ai + aiti) is NP-hard. Mos-heiov [7] incorporated a simple model (pi = aiti) and showed that the problems of minimizing such objectives as the make-span, total completion time, total weighted completion time, total lateness, number of tardy jobs, maximum lateness, andmaximum tardiness are all polynomially solvable. Mosheiov [8] further considered another simple linear deterioration mod-el (pi = ai + ati) and showed that the problem to minimize the total weighted completion time is also polynomially solvable.For research results on other scheduling models considering deterioration effects and under different machine environ-ments, the reader may refer to the review papers of Alidaee and Womer [9], and Cheng et al. [10].

On the other hand, there is also a growing interest in the literature to study scheduling problems with a learning effect.Biskup [11] was the first to analyze the learning effect in single machine scheduling problems. He showed that single ma-chine scheduling problems with a learning effect still remain polynomially solvable if the objective is to minimize the devi-ation from a common due date or to minimize the total completion time. Mosheiov [12] applied similar solution techniquesto several other single machine problems. Lee et al. [13] considered the learning effect in a bi-criterion single machine sched-uling problem. The objective is to minimize the linear combination of total completion time and maximum tardiness. Theycreated some dominance properties and applied them to enhance the performance of a proposed algorithm. Mosheiov and

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2070 D.-L. Yang, W.-H. Kuo / Applied Mathematics and Computation 218 (2011) 2069–2073

Sidney [14] further considered a more general learning effect model in which the learning effects of some jobs are betterthan those of others in a sequence, i.e. the learning effects are job-dependent. They showed that some scheduling problemswith the job-dependent learning effect remain polynomially solvable. A survey on this kind of the scheduling research couldbe found in Bachman and Janiak [15].

However, the phenomenon of a job simultaneously with learning and deterioration effects can be found in many real-lifesituations [16]. For example, in steel mills, if the temperature of an ingot, while waiting to enter the rolling machine, dropsbelow a specified level, the ingot must be preheated up to the temperature required for rolling. It may result in additionaltime for the ingot rolling. On the other hand, the productivity of an operator can be improved through repeating the sameoperating processes. Thus, there exist both deterioration and learning effects of a job in such a situation. To the best of ourknowledge, Lee [17] first considered a single machine scheduling problem with deteriorating jobs and learning effect. Theobjectives are to minimize makespan and total completion time. He introduced the polynomial solutions for both objectivesunder the effects of simple linear deterioration and job-independent learning curves. Wang [16] considered another linearcombination model in which the actual processing time of job i scheduled in rth position is pi = ai(a(t) + bra) where a(t) is anincreasing function and weight parameter b is greater than or equal to zero. It is shown that the makespan, the total com-pletion time, and the sum of the squares of completion times minimization problems all can be optimally solved by the SPTrule. Wang and Cheng [18] considered a model in which the actual processing time of job Ji is pir = ai(b + ct)ra if it is started attime t and scheduled in position r in a sequence where ai is a deterioration rate, b > 0 and c > 0 are constants, and a is thelearning index. They introduced polynomial-time solutions to minimize the makespan, the total completion time, thesum of weighted completion times in the single-machine scheduling problem. They also provided some polynomial-timesolutions to minimize the makespan and the total completion time under certain conditions in the two-machine flowshopscheduling problem. Wang and Cheng [19] considered another model in which the actual processing time of job Ji is pir = (-p0 + ait)ra if it is started at time t and scheduled in position r in a sequence where ai is a deterioration rate, p0 is the commonbasic processing time, and a is the learning index. They gave some polynomially solvable cases to the single-machine min-imum makespan scheduling problem. Cheng et al. [20] studied a model in which the actual processing time of job Ji is

pir ¼ pip0þPr�1

l�1p½l�

p0þPn

l¼1pl

� �a1

ra2 if it is scheduled in position r in a sequence where p[l] is the normal processing time of the job sched-

uled in the lth position in a sequence, p0 > 0 is a given parameter, and a1 and a2 denote the deterioration rates and learningindices. They respectively proposed polynomial-time optimal solutions to minimize the makespan and the total completiontime and gave some polynomial-time solutions to minimize the sum of weighted completion times and the maximum late-ness under certain conditions in the single-machine scheduling problem. They also presented polynomial-time solutions tominimize the makespan and the total completion time for some special cases in permutation flowshop scheduling problem.

In this paper, we consider a more general model than that of Lee [17] in a single-machine scheduling problem. The objec-tives are to minimize the makespan and to minimize the number of tardy jobs when all jobs share a common due-date,respectively.

The remaining part of the paper is organized as follows. In Section 2, we formulate the model. In Section 3 and 4, the min-imum makespan problem and the minimum number of tardy jobs problem are respectively studied. Finally, conclusions aregiven in Section 5.

2. Problem description

To model the effect of deterioration, we follow Mosheiov [7] by incorporating a simple model (pi = aiti). The learning effectis modeled in a general learning effect form of the log-linear curve (see Mosheiov and Sidney [14]). The learning effects ofsome jobs are better than those of others in a sequence, i.e. the learning effects are job-dependent. In order to study the ef-fects of deteriorating jobs and learning simultaneously, we combine the above models to constitute our model. Formally, themodel is stated as follows:

There are n jobs to be processed on a single machine. All jobs are non-preemptive and available for processing at timet0 > 0. Moreover, all jobs share a common due-date d. The machine can handle at most one job at a time and cannot standidle until the last job assigned to it has finished processing. Let pir(t) be the actual processing time of job Ji (i = 1,2, . . . ,n) if it isstarted at time t and scheduled in position r in a sequence. Lee [17] considered the model where processing times is pir(t) = ai-

tra, where ai is the deterioration rate of job Ji and a is the learning index, given as the (base 2) logarithm of the learning rate.Mosheiov and Sidney [14] considered the job-dependent learning model where learning curves ai may be different for alljobs. In this paper, we extend the model presented by Lee to job-dependent learning environment, that is

pirðtÞ ¼ aitrai ; ð1Þ

where t P t0 is the starting time for job Ji.In addition, for a given schedule q, let Ci = Ci(q) denote the completion time of job Ji and the makespan. In addition, letPUi denote the number of tardy jobs where Ui = 1 if Ci 6 d and Ui = 0 if Ci > d. Thus, using the three-field notation introduced

by Graham et al. [21], the corresponding scheduling problems are denoted by 1=pirðtÞ ¼ aitrai=Cmax and 1=pirðtÞ ¼ aitrai=P

Ui,respectively.

D.-L. Yang, W.-H. Kuo / Applied Mathematics and Computation 218 (2011) 2069–2073 2071

3. Minimization of the makespan

In this section, a polynomial time algorithm is proposed to optimally solve the problem. Let S = (J[1], J[2], . . . , J[n]) denote asequence of these n jobs. Then, the actual processing time for the job scheduled in the first position in sequence S isp11ðt0Þ ¼ a½1�t01a½1� and its completion time is C½1� ¼ t0 þ a½1�t01a1½� ¼ t0ð1þ a½1�1a½1� Þ. Similarly, we have

Table 1The pos

J1

J2

J3

J4

J5

⁄ The o

C2 ¼ C½1� þ a½2�C ½1�2a½2� ¼ C½1�ð1þ a½2�2a½2� Þ ¼ t0ð1þ a½1�1a½1� Þð1þ a½2�2a½2� Þ;C ½3� ¼ C ½2� þ a½3�C½2�3a½3� ¼ C ½2�ð1þ a½3�3a½3� Þ ¼ t0ð1þ a½1�1a½1� Þð1þ a½2�2a½2� Þð1þ a½3�3a½3� Þ:

Finally, the completion time of the last job is

C ½n� ¼ C ½n�1� þ a½n�C ½n�1�na½n� ¼ C ½n�1�ð1þ a½n�na½n� Þ ¼ t0

Yn

r¼1

ð1þ a½r�ra½r� Þ: ð2Þ

By taking logarithms to the base e, Eq. (2) becomes as follows.

ln C ½n� ¼ ln t0 þXn

r¼1

lnð1þ a½r�ra½r� Þ: ð3Þ

Thus, to minimize the makespan (or C[n]) is equivalent to minimizing lnC[n]. Clearly, to minimize lnC[n] is independent onlnt0. Thus, the problem to minimize lnC[n] can be formulated as a weighted-bipartite matching problem. We define a bipar-tite graph G(S,T,E), where the nodes in set S represent jobs, and the nodes in set T represent the job position in the sequence.The edges in set E connect each node in S with all nodes in T. Then a polynomial time algorithm for the 1=pirðtÞ ¼ aitrai=Cmax

problem is given as follows.

Algorithm 1

Step 1. Take log of both sides of Eq. (2). Then we have ln C½n� ¼ ln t0 þPn

r¼1 lnð1þ a½r�ra½r� Þ.Step 2. For each edge (i,r) in set E, i 2 S and r 2 T, the corresponding positional weight is wir ¼ lnð1þ airai Þ. Find the match-

ing with minimum weight. Then the matching is an optimal solution.

Theorem 1. Algorithm 1 will find an optimal solution for 1=pirðtÞ ¼ aitrai=Cmax with time complexity O(n3).

Proof. As discussed above, the 1=pirðtÞ ¼ aitrai=Cmax problem can be formulated as a weighted-bipartite matching problem.Hence, Algorithm 1 can find an optimal solution of the problem. The time complexity of Step 1 is O(1). Step 2 can be solvedby Hungarian method in O(n3). Therefore, the complexity of Algorithm 1 is O(n3). h

In the following, we provide a five-job example to demonstrate the application of Algorithm 1.

Example 1. n = 5, t0 = 1, a1 = 0.2, a2 = 0.6, a3 = 0.4, a4 = 0.8, a5 = 0.7, a1 = �0.6, a2 = �0.1, a3 = �0.3, a4 = �0.4, a5 = �0.2.Based on the data in Example 1, the makespan is calculated as follows.

Cmax ¼ t0ð1þ a½1�1a½1� Þð1þ a½2�2a½2� Þð1þ a½3�3a½3� Þð1þ a½4�4a½4� Þð1þ a½5�5a½5� Þ; ð4Þ

where a[r](r = 1,2, . . . ,5) is the deterioration rate of a job scheduled in position r and a[r] is its corresponding learning index.

Step 1. Take log of both sides of Eq. (4). We obtain ln Cmax ¼ ln t0 þP5

r¼1 lnð1þ a½r�ra½r� Þ.Step 2. Calculate the positional weight wir ¼ lnð1þ airai Þ of job i when scheduled in position r (see Table 1). By solving the

weighted-bipartite matching problem, we obtain the optimal job-sequence is (2,1,3,5,4). Then the optimal value ofthe makespan is calculated as follows.

Cmax ¼ t0ð1þ a21a2 Þð1þ a12a1 Þð1þ a33a3 Þð1þ a54a5 Þð1þ a45a4 Þ ¼ 5:069:

itional weight wir ¼ lnð1þ airai Þ of job i in position r in Example 1.

Position 1 Position 2 Position 3 Position 4 Position 5

0.182 0.124⁄ 0.098 0.083 0.0730.470 ⁄ 0.444 0.430 0.420 0.4120.336 0.281 0.253⁄ 0.234 0.2210.588 0.474 0.416 0.378 0.351⁄

0.531 0.476 0.446 0.426⁄ 0.410

ptimal job-sequence is (2,1,3,5,4).

2072 D.-L. Yang, W.-H. Kuo / Applied Mathematics and Computation 218 (2011) 2069–2073

4. Minimization of the number of tardy jobs

Here, the scheduling problem of minimizing the number of tardy jobs is equivalent to that of maximizing the number ofon-time jobs. Hence, if the maximum number (k) of jobs of which the makespan is less than or equal to a common due-date(d) is found, then the problem is solved. Based on the above discussion and Algorithm 1, we have the following theorem.

Algorithm 2

Step 1. Take log of both sides of Eq. (2). Then we have ln C½n� ¼ ln t0 þPn

r¼1 lnð1þ a½r�ra½r� Þ.Step 2. Set k1 = 0, k2 = n and k = d(k1 + k2)/2e.Step 3. Let r = k. For each edge (i,r) in set E, i 2 S and r 2 T, the corresponding positional weight is wir ¼ lnð1þ airai Þ. Find the

matching with the minimum value ofPk

r¼1wir . IfPk

r¼1wir < ðln d� ln t0Þ, the makspan is less than the common due-date (d). Go to Step 4. Otherwise, go to Step 5.

Step 4. Set k1 = k and then k = d(k1 + k2)/2e. If k = k1, the minimum number of tardy jobs is (n � k). STOP. Otherwise, go toStep 3.

Step 5. Set k2 = k and then k = d(k1 + k2)/2e. If k = k2, the minimum number of tardy jobs is (n � k + 1). STOP. Otherwise, go toStep 3.

Remark. Similar to Mosheiov and Sidney [22], in Step 3 of Algorithm 2, the minimum value ofPk

r¼1wir can also be found bythe following assignment problem.

Table 2The iter

Itera

J1

J2

J3

J4

J5

Itera

J1

J2

J3

J4

J5

Itera

J1

J2

J3

J4

J5

minXn

i¼1

Xk

r¼1

wirxir;

s:t:Xn

i¼1

xir ¼ 1; r ¼ 1;2; . . . ; k;

Xn

r¼1

xir 6 1; i ¼ 1;2; . . . ; n;

xir ¼ 0 or 1; i; r ¼ 1;2; . . . ; n;

where wir ¼ lnð1þ airai Þ.Among them, the first set of constraints guarantees that each of the first k positions is assigned a single job. Since not all n

jobs need to be assigned, there are inequalities in the second set of constraints.

Theorem 2. The 1=pirðtÞ ¼ aitrai=P

Ui problem can be solved optimally in O(n3logn).

ation process of the algorithm in Example 2.

tion 1: The optimal sequence of on-time jobs is (2,1,3). Jobs 4 and 5 are tardyPosition 1 Position 2 Position 3 Position 4 Position 50.182 0.124⁄ 0.098 0.083 0.0730.470⁄ 0.444 0.430 0.420 0.4120.336 0.281 0.253⁄ 0.234 0.2210.588 0.474 0.416 0.378 0.3510.531 0.476 0.446 0.426 0.410

tion 2: The optimal sequence of on-time jobs is (2,1,3,4). Job 5 is tardyPosition 1 Position 2 Position 3 Position 4 Position 50.182 0.124⁄ 0.098 0.083 0.0730.470⁄ 0.444 0.430 0.420 0.4120.336 0.281 0.253⁄ 0.234 0.2210.588 0.474 0.416 0.378⁄ 0.3510.531 0.476 0.446 0.426 0.410

tion 3: The minimum value ofPk

r¼1wir ¼ 1:624 > ln d� ln t0 ¼ 1:504. StopPosition 1 Position 2 Position 3 Position 4 Position 50.182 0.124⁄ 0.098 0.083 0.0730.470⁄ 0.444 0.430 0.420 0.4120.336 0.281 0.253⁄ 0.234 0.2210.588 0.474 0.416 0.378 0.351⁄

0.531 0.476 0.446 0.426⁄ 0.410

D.-L. Yang, W.-H. Kuo / Applied Mathematics and Computation 218 (2011) 2069–2073 2073

Proof. Algorithm 2 is a standard binary search for the maximum number (k) of jobs of which the makespan is less than orequal to a common due-date (d). For a given k, the makespan can be found by Algorithm 1. The complexity of a standardbinary search for the optimal k value is O(logn) and the time complexity of Algorithm 1 is O(n3). Hence, the total complexityis O(n3logn). h

In the following, we use the same data in example 1 to demonstrate the application of Algorithm 2.

Example 2. n = 5, t0 = 1, a1 = 0.2, a2 = 0.6, a3 = 0.4, a4 = 0.8, a5 = 0.7, a1 = �0.6, a2 = �0.1, a3 = �0.3, a4 = �0.4, a5 = �0.2. Thecommon due-date is assumed to be d = 4.5.

Since n = 5, in the first iteration, set k1 = 0 and k2 = 5. Therefore, we first consider k = d(k1 + k2)/2e = 3. The minimum valueofPk

r¼1wir is 0.847 which is smaller than lnd � ln t0 = 1.504.So, in the second iteration, set k1 = 3 and then k = d(k1 + k2)/2e = d(3 + 5)/2e = 4. The minimum value of

Pkr¼1wir is 1.225

which is also smaller than ln d � ln t0 = 1.504.In the third iteration, set k1 = 4 k = d(4 + 5)/2e = 5. The minimum value of

Pkr¼1wir is 1.624 which is greater than

lnd � lnt0 = 1.504. Thus, set k2 = 5 and k = d(4 + 5)/2e = 5. Because k = k2, the minimum number of tardy jobs is(n � k + 1) = (5 � 5 + 1) = 1. The iteration process of the algorithm is shown in Table 2.

5. Conclusions

This paper considers single machine scheduling problems with deteriorating jobs and learning effects. The objectives areto minimize the makespan and the number of tardy jobs, respectively. Although the concept of deteriorating jobs and thelearning effects have been extensively studied, they have seldom been considered simultaneously. A polynomial time algo-rithm is proposed to optimally solve each problem.

In our study, the job-dependent learning effect of a job is assumed. However, in some other situations, a learning effectmay be time-dependent [23]. Therefore, it is interesting for future research to investigate the effects of different learning anddeterioration in the context of other scheduling settings, including multi-machine and job-shop scheduling.

Acknowledgements

We are grateful to the Editor and the referees for their helpful comments on an earlier version of this paper. This researchis supported in part by the National Science Council of Taiwan, Republic of China, under Grant No. NSC-97-2221-E150-056-MY2.

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