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Erratum to: A note on flow shop scheduling problems with a learning effect on no-idle dominant machines [Appl. Math. Comput. 184 (2007) 945–949]
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Applied Mathematics and Computation 202 (2008) 897–898 www.elsevier.com/locate/amc
Erratum
Erratum to: A note on flow shop scheduling problems with a learning effect on no-idle dominant machines [Appl. Math. Comput. 184 (2007) 945–949] Ji-Bo Wang Department of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, People’s Republic of China
As we observe, part of the results in the paper [1] are incorrect. In this note, we point out these wrong results by a counterexample. We shall follow the notation and terminology given in Cheng et al. [1]. There are given n jobs J 1 ; J 2 ; . . . ; J n to be processed on an m machine permutation flow shop. The actual processing time of job Jj if scheduled in position r on the machine i is pijr ¼ pij ra ;
i ¼ 1; 2; . . . ; m; r; j ¼ 1; 2; . . . ; n;
ð1Þ
where pij is the normal processing time of job Jj on machine i, and a 6 0 denotes a learning index. Idle time between consecutive operations on each machine is not allowed. Let Cij be the completion time of job Jj on machine i. The objective is minimum the maximum completion time, i.e., the makespan C max ¼ maxfC ij ji ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; ng. Machine Mi is dominated by Mk, or machine Mk dominates Mi iff maxfpij jj ¼ 1; 2; . . . ; ng 6 minfpkj jj ¼ 1; 2; . . . ; ng [3]. In abbreviated notation, it is denoted as M i < M k . For the case of M 1 > M 2 > > M m ðddmÞ, Cheng et al. [1] gave the following result: Lemma 3 [1]. For the problem Fmjpijr ¼ pij ra ; no-idle; ddmjC max and a given schedule r ¼ f½1; ½2; . . . ; ½ng, the completion time of job J[j] is C ½j ¼
n1 X k¼1
p1½k k a þ
m X k¼1
pk½n na
n X
pm½k k a :
ð2Þ
k¼jþ1
In the following example, we show that the result (2) is not correct for the problem Fmjpijr ¼ pij ra ; no-idle; ddmjC max . Example 1. n ¼ 2; m ¼ 2; p11 ¼ 45; p21 ¼ 45; p12 ¼ 50; p22 ¼ 40. Learning take place by the 80%-learning curve, that is a ¼ 0:322 ([2,3]). Obviously, the condition of Example 1 conform to the case of ddm, that is M 1 > M 2.
DOI of original article: 10.1016/j.amc.2006.05.206 E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.049
898
J.-B. Wang / Applied Mathematics and Computation 202 (2008) 897–898
Fig. 1. Schedule fJ 1 ; J 2 g.
Fig. 2. Schedule fJ 2 ; J 1 g.
If the schedule is fJ 1 ; J 2 g, then according to the result of Lemma 3, C 1 ¼ p11 þ p12 20:322 ¼ 85, C 2 ¼ p11 þ p12 20:322 þ p22 20:322 ¼ 117. Obviously, the actual completion time of jobs are: C 1 ¼ p11 þ p21 ¼ 90, C 2 ¼ p11 þ p21 þ p22 20:322 ¼ 122. The reason for the incorrectness is that p21 ¼ 45 > 40 ¼ p12 20:322 (see Fig. 1). If the schedule is fJ 2 ; J 1 g, then according to the result of Lemma 3, C 2 ¼ p12 þ p11 20:322 ¼ 86, C 1 ¼ p12 þ p11 20:322 þ p21 20:322 ¼ 122. Obviously, the actual completion time of jobs are: C 2 ¼ p12 þ p22 ¼ 90, C 1 ¼ p12 þ p22 þ p21 20:322 ¼ 126. The reason for the incorrectness is that p22 ¼ 40 > 36 ¼ p11 20:322 (see Fig. 2). Hence, the result of Lemma 3 (no-idle, ddm) is incorrect. For the case no-idle, ddm is a special case of the cases no-idle, idm-ddm and no-idle, ddm-idm, hence, Lemma 4 (no-idle, idm-ddm), Lemma 5 (no-idle, ddm-idm), Theorem 7 (Fmjpijr ¼ pij ra ; no-idle; ddmjC max ), Theorem 8 (Fmjpijr ¼ pij ra ; no-idle; idm-ddmjC max ), and Theorem 9 (Fmjpijr ¼ pij ra ; no-idle; ddm-idmjC max ) are incorrect. Acknowledgement This research was supported by the Science Research Foundation of the Educational Department of Liaoning Province, China, under Grant Number 20060662. References [1] M. Cheng, S. Sun, Y. Yu, A note on flow shop scheduling problems with a learning effect on no-idle dominant machines, Applied Mathematics and Computation 184 (2007) 945–949. [2] A.B. Badiru, Computational survey of univariate and multivariate learning curve models, IEEE Transactions on Engineering Management 39 (1992) 176–188. [3] J.-B. Wang, Z.-Q. Xia, Flow-shop scheduling with a learning effect, Journal of the Operational Research Society 56 (2005) 1325–1330.
Applied Mathematics and Computation 202 (2008) 897–898 www.elsevier.com/locate/amc
Erratum
Erratum to: A note on flow shop scheduling problems with a learning effect on no-idle dominant machines [Appl. Math. Comput. 184 (2007) 945–949] Ji-Bo Wang Department of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, People’s Republic of China
As we observe, part of the results in the paper [1] are incorrect. In this note, we point out these wrong results by a counterexample. We shall follow the notation and terminology given in Cheng et al. [1]. There are given n jobs J 1 ; J 2 ; . . . ; J n to be processed on an m machine permutation flow shop. The actual processing time of job Jj if scheduled in position r on the machine i is pijr ¼ pij ra ;
i ¼ 1; 2; . . . ; m; r; j ¼ 1; 2; . . . ; n;
ð1Þ
where pij is the normal processing time of job Jj on machine i, and a 6 0 denotes a learning index. Idle time between consecutive operations on each machine is not allowed. Let Cij be the completion time of job Jj on machine i. The objective is minimum the maximum completion time, i.e., the makespan C max ¼ maxfC ij ji ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; ng. Machine Mi is dominated by Mk, or machine Mk dominates Mi iff maxfpij jj ¼ 1; 2; . . . ; ng 6 minfpkj jj ¼ 1; 2; . . . ; ng [3]. In abbreviated notation, it is denoted as M i < M k . For the case of M 1 > M 2 > > M m ðddmÞ, Cheng et al. [1] gave the following result: Lemma 3 [1]. For the problem Fmjpijr ¼ pij ra ; no-idle; ddmjC max and a given schedule r ¼ f½1; ½2; . . . ; ½ng, the completion time of job J[j] is C ½j ¼
n1 X k¼1
p1½k k a þ
m X k¼1
pk½n na
n X
pm½k k a :
ð2Þ
k¼jþ1
In the following example, we show that the result (2) is not correct for the problem Fmjpijr ¼ pij ra ; no-idle; ddmjC max . Example 1. n ¼ 2; m ¼ 2; p11 ¼ 45; p21 ¼ 45; p12 ¼ 50; p22 ¼ 40. Learning take place by the 80%-learning curve, that is a ¼ 0:322 ([2,3]). Obviously, the condition of Example 1 conform to the case of ddm, that is M 1 > M 2.
DOI of original article: 10.1016/j.amc.2006.05.206 E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.049
898
J.-B. Wang / Applied Mathematics and Computation 202 (2008) 897–898
Fig. 1. Schedule fJ 1 ; J 2 g.
Fig. 2. Schedule fJ 2 ; J 1 g.
If the schedule is fJ 1 ; J 2 g, then according to the result of Lemma 3, C 1 ¼ p11 þ p12 20:322 ¼ 85, C 2 ¼ p11 þ p12 20:322 þ p22 20:322 ¼ 117. Obviously, the actual completion time of jobs are: C 1 ¼ p11 þ p21 ¼ 90, C 2 ¼ p11 þ p21 þ p22 20:322 ¼ 122. The reason for the incorrectness is that p21 ¼ 45 > 40 ¼ p12 20:322 (see Fig. 1). If the schedule is fJ 2 ; J 1 g, then according to the result of Lemma 3, C 2 ¼ p12 þ p11 20:322 ¼ 86, C 1 ¼ p12 þ p11 20:322 þ p21 20:322 ¼ 122. Obviously, the actual completion time of jobs are: C 2 ¼ p12 þ p22 ¼ 90, C 1 ¼ p12 þ p22 þ p21 20:322 ¼ 126. The reason for the incorrectness is that p22 ¼ 40 > 36 ¼ p11 20:322 (see Fig. 2). Hence, the result of Lemma 3 (no-idle, ddm) is incorrect. For the case no-idle, ddm is a special case of the cases no-idle, idm-ddm and no-idle, ddm-idm, hence, Lemma 4 (no-idle, idm-ddm), Lemma 5 (no-idle, ddm-idm), Theorem 7 (Fmjpijr ¼ pij ra ; no-idle; ddmjC max ), Theorem 8 (Fmjpijr ¼ pij ra ; no-idle; idm-ddmjC max ), and Theorem 9 (Fmjpijr ¼ pij ra ; no-idle; ddm-idmjC max ) are incorrect. Acknowledgement This research was supported by the Science Research Foundation of the Educational Department of Liaoning Province, China, under Grant Number 20060662. References [1] M. Cheng, S. Sun, Y. Yu, A note on flow shop scheduling problems with a learning effect on no-idle dominant machines, Applied Mathematics and Computation 184 (2007) 945–949. [2] A.B. Badiru, Computational survey of univariate and multivariate learning curve models, IEEE Transactions on Engineering Management 39 (1992) 176–188. [3] J.-B. Wang, Z.-Q. Xia, Flow-shop scheduling with a learning effect, Journal of the Operational Research Society 56 (2005) 1325–1330.