Applied Mathematics and Computation 217 (2011) 7361–7364

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Applied Mathematics and Computation

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

Comments on ‘‘A bicriteria flowshop scheduling problem with setup times’’

Dehua Xu, Yunqiang Yin ⇑School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China

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

Keywords:Flowshop schedulingBicriteriaSetup timeInteger programming

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.01.101

⇑ Corresponding author.E-mail addresses: dhxu@ecit.edu.cn, schedulingr

The aim of this paper is to point out that the integer programming model proposed by Erenand Güner [T. Eren, E. Güner, A bicriteria flowshop scheduling problem with setup times,Appl. Math. Comput. 183 (2006) 1292–1300] is incorrect. We propose a new integer pro-gramming model for the same scheduling problem based on their model.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In a recent paper, Eren and Güner [1] consider a two-machine flowshop scheduling problem where n independent non-preemptive jobs 1,2, . . . ,n need to be processed. The processing time and setup time for job j on the ith machine are pji andsji, respectively. The due date of job j is dj. The objective is to find a schedule that minimizes the weighted sum of totalcompletion time and total tardiness. Let Cj and Tj denote the completion time and the tardiness of job j, respectively. Thescheduling problem under consideration is denoted by F2=sji=a

Pnj¼1Cj þ ba

Pnj¼1Tj in [1], where a and b are the weights of

the total completion time and the total tardiness, respectively; 0 < a < 1; and a + b = 1.Eren and Güner [1] first define some variables, which are as follows.

Zjk – if job j is scheduled at the kth rank to be processed, Zjk = 1, otherwise Zjk = 0, j = 1,2, . . . ,n, k = 1,2, . . . ,n.Xk – the idle time on the second machine between the starting of the kth ranked job and the completion of the (k � 1)stranked job, k = 1,2, . . . ,n.Yk – the time between its completion at the first machine and its begin processing at the second machine for the kthranked job, k = 1,2, . . . ,n.Zk – the idle time on the second machine between the completion time of the (k � 1)st ranked job on the second machineand the starting time of the kth ranked job on the first machine, Zk = max{Sk � Ck�1,2,0}, k = 1,2, . . . ,n.Sk – the starting time for the kth ranked job at the first machine, k = 1,2, . . . ,n.Tk – the tardiness for the kth ranked job,

Tk P Ck � d�k; k ¼ 1;2; . . . ; n: ð1Þ

p[ki] – the processing time of the kth ranked job at the ith machine,

p½ki� ¼Xk

j¼1

Zjkpji; i ¼ 1;2; k ¼ 1;2; . . . ; n: ð2Þ

. All rights reserved.

esearch@gmail.com (D. Xu), yunqiangyin@gmail.com (Y. Yin).

7362 D. Xu, Y. Yin / Applied Mathematics and Computation 217 (2011) 7361–7364

s[ki] – the setup time of the kth ranked job at the ith machine,

s½ki� ¼Xk

j¼1

Zjksji; i ¼ 1;2; k ¼ 1;2; . . . ; n: ð3Þ

Ck – the completion time of the kth ranked job at the second machine,

Ck ¼Xk

j¼1

Xj þXk

j¼1

p½j2�; k ¼ 1;2; . . . ;n: ð4Þ

d�½ki� – the due date of the kth ranked job,

d�½k� ¼Xn

j¼1

Zjkdj; k ¼ 1;2; . . . ; n: ð5Þ

Eren and Güner [1] then propose the following integer programming model for the considered scheduling problem.

Objective function :

Min Z ¼ aX

C þ bX

T

Constraints :

Xn

j¼1

Zjk ¼ 1; k ¼ 1;2; . . . ; n; ð6Þ

Xn

k¼1

Zjk ¼ 1; j ¼ 1;2; . . . ;n; ð7Þ

Sk P Sk�1 þ s½k�1;1� þ p½k�1;1�; k ¼ 2;3; . . . ;n; ð8ÞC12 ¼ Z1 þ s½12� þ X1 þ p½12�; ð9ÞCk2 ¼ Ck�1;2 þ Zk þ s½k2� þ Xk þ p½k2�; k ¼ 2;3; . . . ;n; ð10ÞZ1 ¼ S1 þ s½11� þ p½11� þ Y1 � s½12� � X1; ð11ÞZk ¼ Sk þ s½k1� þ p½k1� þ Yk � Ck�1;2 � s½k2� � Xk; k ¼ 2;3; . . . ;n; ð12ÞX1 ¼ S1 þ s½11� þ p½11� þ Y1 � Z1 � s½12�; ð13ÞXk ¼ Sk þ s½k1� þ p½k1� þ Yk � Ck�1;2 � Zk � s½k2�; k ¼ 2;3; . . . ;n; ð14Þ

All variables in (1)–(5) are positive and integer.

2. Comments

X1 = s[11] + p[11] and S1 = 0 should be defined when introducing variables Xk and Sk in [1], where k = 1,2, . . . ,n.Equation set (1) should be

\Tk ¼maxfCk � d�k;0g; k ¼ 1;2; . . . ; n:":

Equation set (2) should be

\p½ki� ¼Xn

j¼1

Zjkpji; i ¼ 1;2; k ¼ 1;2; . . . ;n:": ð15Þ

Equation set (3) should be

\s½ki� ¼Xn

j¼1

Zjksji; i ¼ 1;2; k ¼ 1;2; . . . ;n:": ð16Þ

Equation set (4) should be

\Ck ¼Xk

j¼1

Xj þXk

j¼1

p½j2� þXk

j¼1

s½j2�; k ¼ 1;2; . . . ; n:"; ð17Þ

since that the setup times should be included.

D. Xu, Y. Yin / Applied Mathematics and Computation 217 (2011) 7361–7364 7363

The objective function should be

\Min Z ¼ aXn

j¼1

Cj þ bXn

j¼1

Tj"

or

\Min Z ¼ aXn

k¼1

Ck þ bXn

k¼1

Tk";

since that C and T are not defined in [1].Note that the notations Ck2 (k = 1,2, . . . ,n), which are used in the integer programming model, are not defined in [1] and

that the notations Ck (k = 1,2, . . . ,n), which are defined, are not used in the integer programming model in [1]. According tothe definition of Ck, it is reasonable we think that Ck2 = Ck, where k = 1,2, . . . ,n.

According to (17), constraint sets (9) and (10) should be

\C1 ¼ X1 þ s½12� þ p½12�; "

and

\Ck ¼ Ck�1 þ Xk þ s½k2� þ p½k2�; k ¼ 2;3; . . . ; n; ";

respectively.Note that constraint sets (13) and (14) are variances of the constraint sets (11) and (12), respectively. Without loss of gen-

erality, we will focus our attention on constraint sets (13) and (14) in what follows.Constraint set (13) should be

\X1 ¼ s½11� þ p½11�; ":

According to the definition of Xk, constraint set (14) should be

\Xk ¼ Sk þ s½k1� þ p½k1� þ Yk � s½k2� � Ck�1; k ¼ 2;3; . . . ;n; ":

Recall that Yk is the time between its completion at the first machine and its begin processing at the second machine for the kthranked job, k 2 {1,2, . . . ,n}. See Fig. 1 for an illustration of this constraint set.

We also observe that the constraints in the integer programming model proposed by Eren and Güner [1] are insufficient.See, for example, the equation/inequality sets (1), (15), (16), and (5) are not included in the model.

Moreover, according to the definitions of Yk (k = 1,2, . . . ,n), we have

Yk ¼ Ck � p½k2� � Sk � s½k1� � p½k1�; k ¼ 1;2; . . . ;n: ð18Þ

For an illustration of (18) see Fig. 1.In order to avoid the case presented in Fig. 2, we redefine Yk as the time between its completion at the first machine and

its starting time of the setup at the second machine for the kth ranked job (see Fig. 3), where k 2 {1,2, . . . ,n}. In other words,

Yk ¼ Ck � p½k2� � s½k2� � Sk � s½k1� � p½k1�; k ¼ 1;2; . . . ;n:

Fig. 1. An illustration of Xk and Yk.

Fig. 2. A part of an infeasible schedule.

Fig. 3. An illustration of Xk and the redefined Yk.

7364 D. Xu, Y. Yin / Applied Mathematics and Computation 217 (2011) 7361–7364

According to the new definition of Yk, constraint set (14) becomes

\Xk ¼ Sk þ s½k1� þ p½k1� þ Yk � Ck�1; k ¼ 2;3; . . . ;n:":

The corrected and simplified integer programming model for the scheduling problem under consideration is as follows,where 0 < a < 1, a + b = 1, n and pji (j = 1,2, . . . ,n) are positive integers, and sji (i = 1,2; j = 1,2, . . . ,n) and dj (j = 1,2, . . . ,n) arenon-negative integers.

Objective function :

Min Z ¼ aXn

k¼1

Ck þ bXn

k¼1

Tk

Constraints :

Xn

j¼1

Zjk ¼ 1; k ¼ 1;2; . . . ;n;

Xn

k¼1

Zjk ¼ 1; j ¼ 1;2; . . . ; n;

p½ki� ¼Xn

j¼1

Zjkpji; i ¼ 1;2; k ¼ 1;2; . . . ;n;

s½ki� ¼Xn

j¼1

Zjksji; i ¼ 1;2; k ¼ 1;2; . . . ;n;

d�½k� ¼Xn

j¼1

Zjkdj; k ¼ 1;2; . . . ;n;

S1 ¼ 0;Sk ¼ Sk�1 þ s½k�1;1� þ p½k�1;1�; k ¼ 2;3; . . . ;n;

C1 ¼ X1 þ s½12� þ p½12�;

Ck ¼ Ck�1 þ Xk þ s½k2� þ p½k2�; k ¼ 2;3; . . . ; n;

X1 ¼ s½11� þ p½11�;

Xk ¼ Sk þ s½k1� þ p½k1� þ Yk � Ck�1; k ¼ 2;3; . . . ;n;

Tk P Ck � d�k; k ¼ 1;2; . . . ;n;

Xk 2 f0;1;2; . . .g; k ¼ 1;2; . . . ;n;

Yk 2 f0;1;2; . . .g; k ¼ 2;3; . . . ;n;

Tk 2 f0;1;2; . . .g; k ¼ 1;2; . . . ; n;

Zjk 2 f0;1g; j ¼ 1;2; . . . ;n; k ¼ 1;2; . . . ;n:

Acknowledgements

This research was supported in part by the Natural Science Foundation of Jiangxi (2010GQS0003), the Science Foundationof Education Committee of Jiangxi for Young Scholars (GJJ11143, GJJ11144), and the Doctor Foundation of East China Insti-tute of Technology.

Reference

[1] T. Eren, E. Güner, A bicriteria flowshop scheduling problem with setup times, Appl. Math. Comput. 183 (2006) 1292–1300.