Applied Mathematical Modelling 35 (2011) 5933–5935

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Applied Mathematical Modelling

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

Erratum

Erratum to ‘‘Single-machine scheduling problems with bothdeteriorating jobs and learning effects’’ [Appl. Math. Modell. 34 (2010)2831–2839]

Yunqiang Yin a,b, Dehua Xu a,b,⇑, Xiaokun Huang c

a State Key Laboratory Breeding Base of Nuclear Resources and Environment, East China Institute of Technology, Nanchang, Jiangxi 330013, Chinab School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, Chinac Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

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

Article history:Received 13 January 2011Received in revised form 20 April 2011Accepted 12 May 2011

Keywords:SchedulingSingle machineMakespanMean finish time

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.05.025

DOI of original article: 10.1016/j.apm.2009.12.017⇑ Corresponding author at: School of Mathematics

+86 794 8258307.E-mail addresses: yunqiangyin@gmail.com (Y. Yi

The aim of this paper is to show by counterexamples that Theorems 3–10 and Corollaries2–5 in Wang et al. [Appl. Math. Model. 34 (2010) 2831–2839] are incorrect.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

As we observe, Theorems 3–10 and Corollaries 2–5 in Wang et al. [1] are incorrect. In this note, we point out these incor-rect results by three counterexamples.

We shall follow the notations and terminologies given in Wang et al. [1]. We are given n independent and non-preemp-tive jobs J = {J1, J2, . . . , Jn}. All the jobs will be processed on a single machine starting at time 0 without overlapping and idletime between them. Associated with each job Jj (j = 1,2, . . . ,n) there is a normal processing time pj and p[r] is defined as thenormal processing time of a job if scheduled in the rth position in a sequence. If job Jj is scheduled in position r in a process-ing sequence, then its actual processing time is given by

pj½r� ¼ pj

p0 þP r�1

l¼1 p½l�p0 þ

Pnl¼1pl

!a1

ra2 ð1Þ

where p0 is a given parameter, a1 (a2) is a given constant representing a rate of change, which is common for all jobs, andP0i¼1p½i� :¼ 0. Here a1 6 0 (a2 P 0) in the case of deterioration and a1 P 0 (a2 6 0) in the case of learning. For convenience,

we denote the scheduling model with Eq. (1) by LDE.

. All rights reserved.

and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China. Tel./fax:

n), schedulingresearch@gmail.com, dhxu@ecit.edu.cn (D. Xu).

5934 Erratum / Applied Mathematical Modelling 35 (2011) 5933–5935

Wang et al. [1] gave the following results.

Theorem 10 (Wang et al. [1], Theorem 3). For the problem 1jLDE,0 6 a1 6 1, a2 P 0jCmax, an optimal schedule can be obtained bysequencing the jobs in non-increasing order of pj (i.e., the LPT rule).

Theorem 20 (Wang et al. [1], Theorem 4). For the problem 1jLDE;0 6 a1 6 1; a2 P 0jPn

l¼1Cl, an optimal schedule exists which isV-shaped with respect to the job normal processing times.

Corollary 10 (Wang et al. [1], Corollary 2). For the problem 1jLDE; 0 6 a1 6 1; a2 P 0jaCmax þ bPn

l¼1Cl, where a P 0 andb P 0, an optimal schedule exists which is V-shaped with respect to the job normal processing times.

Theorem 30 (Wang et al. [1], Theorem 5). For the problem 1jLDE,a1 P 1, a2 P 0jCmax, an optimal schedule exists which is V-shaped with respect to the job normal processing times.

Theorem 40 (Wang et al. [1], Theorem 6). For the problem 1jLDE; a1 P 1; a2 P 0jPn

l¼1Cl, an optimal schedule exists which is V-shaped with respect to the job normal processing times.

Corollary 20 (Wang et al. [1], Corollary 3). For the problem 1jLDE; a1 P 1; a2 P 0jaCmax þ bPn

l¼1Cl, where a P 0 and b P 0, anoptimal schedule exists which is V-shaped with respect to the job normal processing times.

Theorem 50 (Wang et al. [1], Theorem 7). For the problem 1jLDE, 0 6 a1 6 1, a2 6 0jCmax, an optimal schedule exists which is V-shaped with respect to the job normal processing times.

Theorem 60 (Wang et al. [1], Theorem 8). For the problem 1jLDE; 0 6 a1 6 1; a2 6 0jPn

l¼1Cl, an optimal schedule exists whichis V-shaped with respect to the job normal processing times.

Corollary 30 (Wang et al. [1], Corollary 4). For the problem 1jLDE; 0 6 a1 6 1; a2 6 0jaCmax þ bPn

l¼1Cl, where a P 0 andb P 0, an optimal schedule exists which is V-shaped with respect to the job normal processing times.

Theorem 70 (Wang et al. [1], Theorem 9). For the problem 1jLDE, a1 6 0, a2 6 0jCmax, an optimal schedule exists which is V-shaped with respect to the job normal processing times.

Theorem 80 (Wang et al. [1], Theorem 10). For the problem 1jLDE; a1 6 0; a2 6 0jPn

l¼1Cl, an optimal schedule exists which isV-shaped with respect to the job normal processing times.

Corollary 40 (Wang et al. [1], Corollary 5). For the problem 1jLDE; a1 6 0; a2 6 0jaCmax þ bPn

l¼1Cl, where a P 0 and b P 0, anoptimal schedule exists which is V-shaped with respect to the job normal processing times.

2. Counterexamples

The following counterexamples show that Theorems 3–10 and Corollaries 2–5 in Wang et al. [1] are incorrect.

Counterexample 1. Let n = 3, p0 = 1, a1 = 0.6 and a2 = 0. The normal processing times of the three jobs are given as follows:p1 = 1, p2 = 150, and p3 = 151. Then the optimal schedule for the problems 1jLDE, 0 6 a1 6 1,a2 P 0jCmax and 1jLDE,0 6 a1 6 1, a2 6 0jCmax is (J1, J3, J2) with the optimal objective value of 107.0078, which is also the optimal schedule forthe problems 1jLDE;0 6 a1 6 1; a2 P 0j

Pnl¼1Cl and 1jLDE;0 6 a1 6 1; a2 6 0j

Pnl¼1Cl with the optimal objective value of

114.4983. Note that this sequence is neither V-shaped with respect to the normal processing times: (1, 151, 150), norobtained in the LPT rule. Hence Theorems 3, 4, 7 and 8 in Wang et al. [1] are incorrect. Since Theorems 4 and 8 is arespecial cases of Corollaries 2 and 4 in Wang et al. [1], respectively, we conclude that Corollaries 2 and 4 in Wang et al.[1] are also incorrect.

Counterexample 2. Let n = 3, p0 = 100, a1 = 2 and a2 = 2. The normal processing times of the three jobs are given as fol-lows: p1 = 1, p2 = 300, and p3 = 301. Then the optimal schedule for the problem 1jLDE, a1 P 1, a2 P 0jCmax is (J2, J3, J1) withthe optimal objective value of 405.9679, which is also the optimal schedule for the problem 1jLDE; a1 P 1; a2 P 0j

Pnl¼1Cl

Erratum / Applied Mathematical Modelling 35 (2011) 5933–5935 5935

with the optimal objective value of 809.0491. Note that this sequence is not V-shaped with respect to the normal process-ing times: (300, 301, 1). Hence Theorems 5 and 6 in Wang et al. [1] are incorrect. Since Theorem 6 is a special case ofCorollary 3 in Wang et al. [1], we conclude that Corollary 3 in Wang et al. [1] is also incorrect.

Counterexample 3. Let n = 3, p0 = 1, a1 = �0.5 and a2 = 2. The normal processing times of the three jobs are given as fol-lows: p1 = 1, p2 = 100, and p3 = 101. Then the optimal schedule for the problem 1jLDE, a1 6 0, a2 P 0jCmax is (J2, J3, J1) withthe optimal objective value of 2006.6, which is also the optimal schedule for the problem 1jLDE; a1 6 0; a2 P 0j

Pnl¼1Cl

with the optimal objective value of 5428.9. Note that this sequence is not V-shaped with respect to the normal processingtimes: (100,101,1). Hence Theorems 9 and 10 in Wang et al. [1] are incorrect. Since Theorem 10 is a special case of Cor-ollary 5 in Wang et al. [1], we conclude that Corollary 5 in Wang et al. [1] is also incorrect.

Acknowledgements

This research supported in part by the Natural Science Foundation of Jiangxi (2010GQS0003), the Science Foundation ofEducation Committee of Jiangxi for Young Scholars (GJJ11143, GJJ11144), and the Doctor Foundation of East China Instituteof Technology.

Reference

[1] J.-B. Wang, D. Wang, G.-D. Zhang, Single-machine scheduling problems with both deteriorating jobs and learning effects, Appl. Math. Modell. 34 (2010)2831–2839.