Information Sciences 180 (2010) 3814–3816

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Information Sciences

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

Note on ‘‘Single-machine and flowshop scheduling with a generallearning effect model” and ‘‘Some single-machine and m-machineflowshop scheduling problems with learning considerations”

Wen-Hung Kuo *, Dar-Li YangDepartment of Information Management, National Formosa University, Yun-Lin, Taiwan 632, ROC

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

Article history:Received 26 August 2009Received in revised form 23 February 2010Accepted 21 May 2010

Keywords:SchedulingLearning effectSingle-machineFlowshop

0020-0255/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.ins.2010.05.026

* Corresponding author. Tel.: +886 5 6315733.E-mail address: whkuo@nfu.edu.tw (W.-H. Kuo).

In this note, we show that the main results in the two papers [C.C. Wu, W.C. Lee, Single-machine and flowshop scheduling with a general learning effect model, Computers andIndustrial Engineering 56 (2009) 1553–1558, W.C. Lee, C.C. Wu, Some single-machineand m-machine flowshop scheduling problems with learning considerations, InformationSciences 179 (2009) 3885–3892] are incorrect.

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1. Introduction

Recently, Wu and Lee [2] and Lee and Wu [1] studied some scheduling problems with general learning effect models,respectively. In the proposed learning effect models, they considered both the human and the machine learning effectssimultaneously. The model in Wu and Lee [2] is described as follows. There are n jobs ready to be processed on a single ma-chine. Each job j has a due-date dj. The actual processing time of job j when scheduled in the rth position is as follows:

pj½r� ¼ pjqa1r c0 þ

Xr�1

l¼1

br�lp½l�

!a2

ð1Þ

where pj is the normal processing time of job j, p[l] is the normal processing time of a job when scheduled in the lth positionin a sequence, c0 > 0 is a constant, qr is a non-decreasing function of the job position, bi is a non-decreasing sequence of coef-ficients, a1 6 0 and a2 6 0 are the learning indices.

On the other hand, in the model of Lee and Wu [1], the actual processing time of job j when scheduled in the rth positionis as follows:

pj½r� ¼ pjðqðrÞ þ br�1p½1� þ � � � þ b1p½r�1�Þa ¼ pj qðrÞ þ

Xr�1

l¼1

br�lp½l�

!a

ð2Þ

where q(r) is a non-decreasing function of the job position, b1,b2, . . .,bn are numbers with 0 6 b1 6 b2 6 � � � 6 bn and a is thelearning index with a < 0.

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W.-H. Kuo, D.-L. Yang / Information Sciences 180 (2010) 3814–3816 3815

For convenience, we denote the learning model with Eq. (1) by LE(1) and that with Eq. (2) by LE(2). Then, using the con-ventional notation, the corresponding problems in Wu and Lee [2] are denoted by 1=LEð1Þ=Cmax;1=LEð1Þ=

PCj;1=LEð1Þ=

PwjCj

and 1=LEð1Þ=P

Ti, and those in Lee and Wu [1] are denoted by 1=LEð2Þ=Cmax;1=LEð2Þ=P

Cj;1=LEð2Þ=P

wjCj and 1/LE(2)/Lmax.

2. Counter examples

In the following, we show that Properties 1–4 in Wu and Lee [2] are not correct by Counter-example 1.Counter-example 1: Let n = 3, p1 = 1, p2 = 2, p3 = 100, d1 = 1, d2 = 2, d3 = 30, b1 = 2, b2 = 3, a1 = 0, a2 = �0.5, c0 = 1 and qr = r.If the jobs are arranged to be processed according to the SPT rule, the sequence of the jobs is J1, J2 and J3. Then

p1½1� ¼ p1ð1Þ0ð1Þ�0:5 ¼ 1

p2½2� ¼ p2ð2Þa1 ð1þ b1p½1�Þ

a2 ¼ ð2Þð2Þ0ð1þ 2� 1Þ�0:5 ¼ 1:155

p3½3� ¼ p3ð3Þa1 ð1þ b2p½1� þ b1p½2�Þ

a2 ¼ ð100Þð3Þ0ð1þ 3� 1þ 2� 2Þ�0:5 ¼ 35:355:

The makespan Cmax and the total completion timeP

Cj are respectively calculated as follows:

Cmax ¼ 1þ 1:155þ 35:355 ¼ 37:51

andP

Cj ¼ 1� 3þ 1:155� 2þ 35:355 ¼ 40:665.However, if the sequence of the jobs is J2, J1 and J3, then we have

p2½1� ¼ p2ð1Þ0ð1Þ�0:5 ¼ 2

p1½2� ¼ p1ð2Þa1 ð1þ b1p½1�Þ

a2 ¼ ð1Þð2Þ0ð1þ 2� 2Þ�0:5 ¼ 0:447

and p3½3� ¼ p3ð3Þa1 ð1þ b2p½1� þ b1p½2�Þ

a2 ¼ ð100Þð3Þ0ð1þ 3� 2þ 2� 1Þ�0:5 ¼ 33:333.Similarly, the makespan Cmax and the total completion time

PCj are respectively calculated as follows:

Cmax ¼ 2þ 0:447þ 33:333 ¼ 35:78

andP

Cj ¼ 2� 3þ 0:447� 2þ 33:333 ¼ 40:227.Apparently, the SPT sequence is not the optimal sequence for the two problems 1/LE(1)/Cmax and 1=LEð1Þ=

PCj. Hence,

Properties 1 and 2 are not correct. In addition, since the problem 1=LEð1Þ=P

Cj is a special case of the problem1=LEð1Þ=

PwjCj, Property 3 is also not correct.

Next, If the jobs are sequenced in non-decreasing order of di, the sequence of the jobs is J1, J2 and J3. The total tardiness iscalculated as follows:

T1 ¼maxð0;C1 � d1Þ ¼ maxð0;1� 1Þ ¼ 0T2 ¼maxð0;C2 � d2Þ ¼ maxð0; ð1þ 1:155Þ � 2Þ ¼ 0:155T3 ¼maxð0;C3 � d3Þ ¼ maxð0; ð1þ 1:155þ 35:355Þ � 30Þ ¼ 7:510X

Ti ¼ 0þ 0:155þ 7:510 ¼ 7:665:

However, if the sequence of the jobs is J2, J1 and J3, then we have

T2 ¼maxð0;C2 � d2Þ ¼ maxð0;2� 2Þ ¼ 0T1 ¼maxð0;C1 � d1Þ ¼ maxð0; ð2þ 0:447Þ � 1Þ ¼ 1:447T3 ¼maxð0;C3 � d3Þ ¼ maxð0; ð2þ 0:447þ 33:333Þ � 30Þ ¼ 5:780X

Ti ¼ 0þ 1:447þ 5:780 ¼ 7:227:

From the above result, we can see that Property 4 is also not correct. Finally, since the proofs of Properties 5–11 in the flow-shop scheduling problem are based on Property 1, they are also incorrect.

Similarly, we show that Theorems 1–4 in Lee and Wu [1] are not correct by Counter-example 2.Counter-example 2: Let n = 3, p1 = 1, p2 = 2, p3 = 100, d1 = 1, d2 = 2, d3 = 30, b1 = 2, b2 = 3, a = �0.5 and q(r) = 1.When the above data are used in the learning model proposed by Lee and Wu [1], the learning model is equivalent to that

proposed by Wu and Lee [2]. That is, Eq. (2) is equal to Eq. (1). Hence, it is very straightforward that Theorems 1–3 are incor-rect. Besides, in such a situation, the lateness of each job is equal to its tardiness. From the results of Counter-example 1, themaximum lateness of the sequence (J1 ? J2 ? J3) is 7.510 and that for the sequence (J2 ? J1 ? J3) is 5.780. Thus, Theorem 4 isalso incorrect. Similarly, since the proofs of Theorems 5–8 in the flowshop scheduling problem are based on Theorem 1, theyare also incorrect.

3. Conclusion

Although the results of these papers are not correct, the authors introduced two new learning models in which they con-sidered both the machine and human learning effects simultaneously. Such learning effects happen in many realistic situ-

3816 W.-H. Kuo, D.-L. Yang / Information Sciences 180 (2010) 3814–3816

ations. Hence, it is worthwhile to further discuss the complexity of the scheduling problems with the proposed learningmodels or to find the sufficient conditions to ensure that the properties or theorems in these papers still hold.

Acknowledgement

We are grateful to Editor-in-Chief and the referees for their valuable comments on earlier versions of this paper. This re-search was supported in part by the National Science Council of Taiwan, Republic of China, under Grant No. NSC-98-2221-E-150-032.

References

[1] W.C. Lee, C.C. Wu, Some single-machine and m-machine flowshop scheduling problems with learning considerations, Information Sciences 179 (2009)3885–3892.

[2] C.C. Wu, W.C. Lee, Single-machine and flowshop scheduling with a general learning effect model, Computer and Industrial Engineering 56 (2009) 1553–1558.