Available online at www.sciencedirect.com

Applied Mathematical Modelling 33 (2009) 1141–1150

www.elsevier.com/locate/apm

A bicriteria parallel machine scheduling with a learning effectof setup and removal times

Tamer Eren *

Kırıkkale University, Faculty of Engineering, Department of Industrial Engineering, 71451 Kırıkkale, Turkey

Received 1 April 2006; received in revised form 1 January 2008; accepted 7 January 2008Available online 24 January 2008

Abstract

In studies on scheduling problems, generally setup times and removal times of jobs have been neglected or by includingthose into processing times. However, in some production systems, setup times and removal times are very important suchthat they should be considered independent from processing times. Since, in general jobs are done according to automaticmachine processes in production systems processing times do not differ according to process sequence. But, since humanfactor becomes influential when setup times and removal times are taken into consideration, setup times will be decreasingby repeating setup processes frequently. This fact is defined with learning effect in scheduling literature. In this study, abicriteria m-identical parallel machines scheduling problem with a learning effect of setup times and removal times is con-sidered. The objective function of the problem is minimization of the weighted sum of total completion time and total tar-diness. A mathematical programming model is developed for the problem which belongs to NP-hard class. Results ofcomputational tests show that the proposed model is effective in solving problems with up to 15 jobs and five machines.We also proposed three heuristic approaches for solving large jobs problems. According to the best of our knowledge, nowork exists on the minimization of the weighted sum of total completion time and total tardiness with a learning effect ofsetup times and removal times.� 2008 Elsevier Inc. All rights reserved.

Keywords: Parallel machine scheduling; Bicriteria; Learning effect; Setup times; Removal times; Mathematical programming model

1. Introduction

While studying the parallel machine scheduling problem, it is generally assumed that the changeover times,where changeover time is defined as the time required changing from one job to another on a given machine,are either included in the processing times, or negligible, and are therefore ignored. However, in many prac-tical situations changeover times are separable and can be classified into two components, namely setup timeand removal time. The setup time is the time span required to prepare machine j for processing job i, e.g., bymounting the necessary tools, jigs and fixtures. The removal time is the time span needed to restore the initial

0307-904X/$ - see front matter � 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2008.01.010

* Tel.: +90 318 3573576x1008; fax: +90 318 3572459.E-mail address: teren@kku.edu.tr.

1142 T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150

state of machine j after completion of job i, e.g., by removing the tools, jigs and fixtures [1]. In studies onscheduling problems, generally setup times and removal times of jobs have been neglected or by includingthose into processing times. However, as setup times and removal times may be too important to be neglectedin some production systems, it may also be necessary to consider processing times independent from setuptimes and removal times. Since, in general jobs are done according to automatic machine processes in produc-tion systems processing times do not differ according to process sequence. But, since human factor becomesinfluential when setup times and removal times are taken into consideration, setup times will be decreasing byrepeating setup processes frequently. This fact is defined with learning effect in scheduling literature [2]. Thelearning effect has been widely employed in management science since its discovery by Wright [3] over half acentury ago. However, there have been very few studies in the general context of production scheduling. Bis-kup [4] was the first to investigate the learning effect in a scheduling setting. Biskup [4] studied single-machineproblems and considered the objective of (i) minimizing completion time, and (ii) minimizing the weightedsum of completion time deviations from a common due date and the sum of job completion times. Later,Mosheiv [5] showed that the single-machine makespan minimization problem remained polynomial solvablewhen the learning effect was taken into consideration. Mosheiv [6] considered completion time minimizationon identical machines in parallel and showed that this problem has a polynomial time solution. Mosheiv andSidney [7] extended the setting to account for the possibility that learning in the production process is fasterfor some jobs than for others. They showed that the makespan and total completion time minimization prob-lems on a single-machine, a due date assignment problem, and total completion time minimization on unre-lated parallel machines remain polynomial solvable. Eren and Guner [8] worked on the well-known single-machine total tardiness problem with learning effects. Lee et al. [9] analyzed the single-machine bicriteria prob-lem with a learning effect. The objective was to find a sequence that minimizes a linear combination of the totalcompletion time and the maximum tardiness. Eren and Guner [10] worked on a different bicriteria single-machine scheduling problem, namely jointly minimizing the sum of completion times and the total tardiness,1=LE=a

PC þ b

PT . The authors developed a mathematical programming and heuristics to solve this prob-

lem apart from these articles in the single-machine there are other studies with the learning effect: Cheng andWang [11], Kuo and Yang [12], Lee [13], Biskup and Simons [14], Mosheiov and Sidney [15]. Eren and Guner[16] analyzed a scheduling problem with job-dependent learning effect in a two-machine flowshop with make-span by performance measure. They showed that Johnson algorithm cannot guarantee the best results in thesituation with job-dependent learning effect. They also proposed a mixed integer programming model for thisproblem. In addition, studies about a learning effect in a two-machine flowshop scheduling with total comple-tion times has been considered by Lee and Wu [17] and Eren and Guner [18]. Eren and Guner [19] analyzed thebicriteria flowshop problem with a learning effect. The objective was to find a sequence that minimizes a linearcombination of the total completion time and the makespan. In this paper a bicriteria parallel machine sched-uling problem with a learning effect of setup times and removal times is considered. The objective function ofthe problem is to find a sequence that minimizes a weighted sum of total completion time and total tardiness.

Total completion time and total tardiness are widely used performance measures in scheduling literature.Panwalker and Iskander [20] pointed out that the completion time and the tardiness are the most prominentmeasures among the scheduling objectives in industrial applications. Minimizing the completion time involvesmaintaining the work-in-process inventory at a low level. Minimizing the tardiness involves reducing the pen-alties incurred for late jobs. The former represents internal efficiency while the latter represents externalefficiency.

This bicriteria problem is an NP-hard problem since its simpler case of the single criterion problem on onemachine case ð1==

PT Þ is already an NP-hard [21]. According to the best of our knowledge, no work exists on

the minimization of the weighted sum of total completion time and total tardiness with a learning effect ofsetup times and removal times.

In this paper a bicriteria m-identical parallel machine scheduling problem with a learning effect of setuptimes and removal times are considered. To get exact solution of the problem, a mathematical programmingmodel is proposed. We also proposed three heuristic approaches for solving large jobs problems.

The rest of the study is organized as follows. In Section 2, the problem is described and in Section 3, theproblem and the proposed a mathematical programming model are described. Three heuristic methods thatare used to solve large size problems are presented in the Section 4. The experimental results are given in

T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150 1143

the Section 5. Finally, Section 6 provides conclusions and evaluations of the study and suggests somedirections for future researches.

2. Problem description

We consider n jobs available at time 0 and m-identical machines in parallel. As in Mosheiov [6], let ni

denote the number of jobs assigned to machine i, i = 1,2, . . . ,m. ðPm

i¼1ni ¼ nÞ. For a given vectorV = (n1,n2, . . . ,nm), i.e. when the number of jobs per machine is known, the positional weights are uniquelyspecified for each machine. n jobs are to be processed on m-parallel identical machines. The normal setupand removal times for job j on Mi are denoted Sji(=Sj) and Rji(=Rj). Furthermore, we assume that bothmachines have the same learning effect. That is if Sjir and Rjir are the actual setup and removal times ofjob j scheduled in position r in a sequence, then Sjir = Sjir

a and Rjir = Rjira (where a 6 0 is the learning

effect, given is the logarithm to the base 2 of the learning rate) the objective is to find a schedule thatminimizes the weighted sum of total completion time and total tardiness on m-identical parallel machineswith a learning effect of setup times and removal times (the problem is denoted as P m=SLE

j ;RLE

j =aP

C þ bP

T , where LE is learning effect).Assumptions made in this paper are:

1. Setup and removal times are known and is not included in the processing time.2. Machine preemption is disallowed, each operation, once started, must be completed before another oper-

ation may be started on the same machines.3. Machines are stable and remain available throughout the scheduling period.4. No job may be processed on more than one machine simultaneously.5. A machine may only process one job at a time.

3. A proposed mathematical programming model

In this proposed model, there are n2 + 6n variables and 8n constraints (for assigned to machine), where ndenotes the number of jobs. The parameters and variables in the model are described below and then the pro-posed model is given.

3.1. Parameters

n the number of jobs j = 1, 2, . . . ,n

m the number of machine i = 1, 2, . . . ,m

a the weight for total completion time a P 0

b the weight for total tardiness a + b = 1

pj the processing time of job j, pj = pji, j = 1, 2, . . . ,n

Sj the setup time of job j, Sj = Sji, j = 1, 2, . . . ,n

Rj the removal time of job j, Rj = Rji, j = 1, 2, . . . ,n

dj the due date of job j, j = 1, 2, . . . ,n

ni the number of jobs assigned to machine iPm

i¼1ni ¼ n i = 1, 2, . . . ,m

3.2. Decision variable

�

X jir

1 job j is scheduled on machine i in position r

0 otherwisei ¼ 1; 2; . . . ;m j; r ¼ 1; 2; . . . ; n:

1144 T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150

3.3. Mathematical programming model

Objective function:

Min aX

C þ bX

T :

Constraints:

Xn

j¼1

X jir ¼ 1 i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð1Þ

Xm

i¼1

Xni

r¼1

X jir ¼ 1 j ¼ 1; 2; . . . ; n; ð2Þ

p ir½ � ¼Xn

j¼1

X jirpj i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð3Þ

S ir½ � ¼Xn

j¼1

X jirSjir i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð4Þ

R ir½ � ¼Xn

j¼1

X jirRjir i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð5Þ

Cir P Ci;r�1 þ p ir½ � i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð6Þ

d ir½ � ¼Xn

j¼1

X jirdj i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð7Þ

T ir P Cir � d ir½ � i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni; ð8Þ

X jir ¼ 0 or 1 j ¼ 1; 2; . . . ; n i ¼ 1; 2; . . . ;m r ¼ 1; 2; . . . ; ni:

Constraint (1) specifies that only one job be scheduled at the rth job priority. Constraint (2) defines that eachjob be scheduled only once. Constraints (3)–(8) denotes the rth of ith machine position jobs processing time,setup time, removal time, completion time, due date and tardiness time, respectively. All variables are positiveand integer.

4. Heuristic methods

The considered bicriteria m-identical parallel machines scheduling problem with a learning effect of setuptimes and removal times can be solved optimally for small size problems with up to five machines and 15 jobsby the proposed mathematical programming model. Three heuristics methods are developed for solving largesize problems. Steps of special heuristic are given below.

4.1. An improvement algorithm

Step 1. Obtain an initial sequence.Step 2. Set k = 2. Pick the first two jobs from the rearranged jobs list and schedule them in order to minimi-

zation of the weighted sum of total completion time and total tardiness as if there are only two jobs.Set the better one as the current solution.

Step 3. Increment k by 1. Generate k candidate sequences by inserting the first job in the remaining job listinto each slot of the current solution. Among these candidates, select the best one with the least par-tial minimization of the weighted sum of total completion time and total tardiness. Update theselected partial solution as the new current solution.

Step 4. If k = ni, a schedule (the current solution) has been found and stop. Otherwise, go to step 3.

T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150 1145

Special Heuristic 1, 2 and 3 (SH1, SH2, SH3) are obtained by using SSPRT (Shortest sum of Setup,Processing and Removal Times), SSRT (Shortest sum of Setup and Removal Times) and EDD (EarliestDue Date) sequence, respectively, in Step 1.

5. Experimental results

In this study, all experimental tests were conducted on a personal computer with Pentium IV/2 512 RAM.The mathematical programming model is used to find the optimal solutions of the considered problem usingHyper LINDO/PC 6.01 software package. Special heuristic methods used in this paper were coded on C++Builder 5. The experimental design is similar to Hsu and Lin [22]’s, Eren and Guner [23–25]’s and Eren [26]processing times on machine is generated from a uniform distribution in the ranges [1,100]. The setup andremoval times are randomly generated from another uniform distribution on the integers between [0,24].The due dates are randomly generated from another uniform distribution on the integers between [0 � Cmax]

ðCmax ¼Pn

r¼1p½r�r

a

m rank : minnr¼1p½r�Þ. It is seen that in manufacturing systems and especially in the assembly

lines the learning rate fits to the 80% learning curve. In this paper we also used the same rate as a learningrate. The experimental set is given in Table 1. As seen from Table 1, totally 120 problems are solved [27].

n = 15 for allocation vector V = (n1, n2, . . . ,nm) sets are shown in Table 2. When n = 15 and m = 2 alloca-tion vectors should be considered: (14,1), (13,2), (12,3), (11,4), (10,5), (9,6) and (8,7).

Averagely CPU times of problem sets are shown in Figs. 1–4. As can be seen in Figs. 1–4 the consideredbicriteria problem with a learning effect of setup and removal times can be solved up to five machines 15 jobsby the proposed mathematical programming model (for a = 0.50 and b = 0.50). Since the average CPU timesfor (a = 0.25 and b = 0.75) and (a = 0.75 and b = 0.25) weights were very similar to the (a = 0.50 and

Table 1The parameters of problem

Parameter Alternatives Values

Weights (a,b) 3 (0.25,0.75); (0.50,0.50); (0.75,0.25)Number of jobs, n 1 15Machines, m 4 2, 3, 4, 5Processing time, pj 1 �U[1, 100]Setup times, removal times Sj, Rj 1 �U[0, 24]Due date, dj 1 �U[0,Cmax]Learning rate (LE) 1 80%Number of solution problem 10 10

Total problem 3 � 1 � 4 � 1 � 1 � 1 � 1 � 1 � 10 = 120

Table 2n = 15 for Allocation vector V = (n1, n2, . . . ,nm)

m V = (n1, n2, . . . ,nm) Alternatives

2 (n1, n2) 7 (14,1); (13,2); (12,3); (11,4); (10,5); (9,6); (8,7)3 (n1, n2, n3) 19 (13,1,1); (12,2,1); (11,3,1); (11,2,2); (10,4,1); (10,3,2); (9,5,1);

(9,4,2); (9,3,3); (8,6,1); (8,5,2); (8,4,3); (7,7,1); (7,6,2); (7,5,3);(7,4,4); (6,6,3); (6,5,4); (5,5,5)

4 (n1, n2, n3, n4) 27 (12,1,1,1); (11,2,1,1); (10,3,1,1); (10,2,2,1); (9,4,1,1); (9,3,2,1);(9,2,2,2); (8,5,1,1); (8,4,2,1); (8,3,3,1); (8,3,2,2); (7,6,1,1); (7,5,2,1);(7,4,3,1); (7,4,2,2); (7,3,3,2); (6,6,2,1); (6,5,3,1); (6,5,2,2); (6,4,4,1);(6,4,3,2); (6,3,3,3); (5,5,4,1); (5,5,3,2); (5,4,4,2); (5,4,3,3); (4,4,4,3)

5 (n1, n2, n3, n4, n5) 29 (11,1,1,1,1); (10,2,1,1,1); (9,3,1,1,1); (9,2,2,1,1); (8,4,1,1,1); (8,3,2,1,1);(8,2,2,2,1); (7,5,1,1,1); (7,4,2,1,1); (7,3,3,1,1); (7,3,2,2,1); (7,2,2,2,2);(6,6,1,1,1); (6,5,2,1,1); (6,4,3,1,1); (6,4,2,2,1); (6,3,3,2,1); (6,3,2,2,2);(5,5,3,1,1); (5,5,2,2,1); (5,4,4,1,1); (5,4,3,2,1); (5,4,2,2,2); (5,3,3,2,2);(4,4,4,2,1); (4,4,3,3,1); (4,4,3,2,2); (4,3,3,3,2); (3,3,3,3,3)

0 20 40 60 80

100

120

(14,1)

(13,2)

(12,3)

(11,4)

(10,5)

(9,6)

(8,7)

(n1,n2)

CPU times (s)

0.250.50

0.75

Fig.

1.C

PU

times

(s)fo

r(n,m

)=

(15,2).

0 50

100

150

200

250

300

350

400

(13,1,1)

(12,2,1)

(11,3,1)

(11,2,2)

(10,4,1)

(10,3,2)

(9,5,1)

(9,4,2)

(9,3,3)

(8,6,1)

(8,5,2)

(8,4,3)

(7,7,1)

(7,6,2)

(7,5,3)

(7,4,4)

(6,6,3)

(6,5,4)

(5,5,5)

(n1,n2,n3)

CPU times (s)

0.250.50

0.75

Fig.

2.C

PU

times

(s)fo

r(n,m

)=

(15,3).

0

200

400

600

800

1000

1200

(12,1,1,1)

(11,2,1,1)

(10,3,1,1)

(10,2,2,1)

(9,4,1,1)

(9,3,2,1)

(9,2,2,2)

(8,5,1,1)

(8,4,2,1)

(8,3,3,1)

(8,3,2,2)

(7,6,1,1)

(7,5,2,1)

(7,4,3,1)

(7,4,2,2)

(7,3,3,2)

(6,6,2,1)

(6,5,3,1)

(6,5,2,2)

(6,4,4,1)

(6,4,3,2)

(6,3,3,3)

(5,5,4,1)

(5,5,3,2)

(5,4,4,2)

(5,4,3,3)

(4,4,4,3)

(n1,n2,n3,n4)

CPU times (s)

0.250.50

0.75

Fig.

3.C

PU

times

(s)fo

r(n,m

)=

(15,4).

1146T

.E

ren/A

pp

liedM

ath

ema

tical

Mo

dellin

g3

3(

20

09

)1

14

1–

11

50

0

200

400

600

800

1000

1200

1400

1600

(11,

1,1,

1,1)

(10,

2,1,

1,1)

(9,3

,1,1

,1)

(9,2

,2,1

,1)

(8,4

,1,1

,1)

(8,3

,2,1

,1)

(8,2

,2,2

,1)

(7,5

,1,1

,1)

(7,4

,2,1

,1)

(7,3

,3,1

,1)

(7,3

,2,2

,1)

(7,2

,2,2

,2)

(6,6

,1,1

,1)

(6,5

,2,1

,1)

(6,4

,3,1

,1)

(6,4

,2,2

,1)

(6,3

,3,2

,1)

(6,3

,2,2

,2)

(5,5

,3,1

,1)

(5,5

,2,2

,1)

(5,4

,4,1

,1)

(5,4

,3,2

,1)

(5,4

,2,2

,2)

(5,3

,3,2

,2)

(4,4

,4,2

,1)

(4,4

,3,3

,1)

(4,4

,3,2

,2)

(4,3

,3,3

,2)

(3,3

,3,3

,3)

(n1,n2,n3,n4,n5)

CP

U ti

mes

(s)

0.25 0.50 0.75

Fig. 4. CPU times (s) for (n,m) = (15,5).

T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150 1147

b = 0.50) (a = 0.50 and b = 0.50) weight, their CPU times figures are not given. According to weight valuesproblem solutions get difficult as the weight value of total tardiness increases.

The optimal solutions of the considered problem can be found up to 15 jobs and five machines. To solvelarge size problems in a short time and to find the optimal or near optimal solutions, heuristics should be used.For the problem considered, three heuristic methods were used. Heuristic error is calculated as follows:

Error ¼ Heuristic solution �Optimal solution

Optimal solution:

The results of the heuristic methods were compared with the results of the optimal solutions obtained by themathematical programming model, and the average errors of these heuristics are given in Figs. 5–8. As seenfrom Figs. 5–8, the SH1 gives fairly good results in terms of error and it is quite effective in this bicriteria m-parallel machine problem. SH1 gives the best results of three heuristics, SH2 also gives a good performanceand its solution error is averagely 1%. On the other hand the solution error of the SH3 random search is 2%.

The results of the large size problems up to 500 jobs and 10 machines were also computed using heuristicmethods. Since the optimal results of these problems have not been known, the results of the heuristics werecompared with the best result in order to define their performances. In this comparison, the error, formulatedbelow, was used as a performance measure:

Error ¼ Heuristic solution � Best heuristic solution

Best heuristic solution:

0.00

0.01

0.02

0.03

0.04

0.05

(14,

1)

(13,

2)

(12,

3)

(11,

4)

(10,

5)

(9,6

)

(8,7

)

(n1,n2)

CP

U ti

mes

(s)

SH1 SH2 SH3

Fig. 5. Heuristics error for (n,m) = (15,2).

Figs.

9–11th

eaverage

error

of

heu

risticsacco

rdin

gto

the

weigh

tvalu

es.Th

eS

H1

giveth

eb

estresu

ltso

fth

ep

rob

lemfo

rall

weigh

tvalu

es.S

H1

givesth

eb

estresu

ltso

fth

reeh

euristics

for

a=

0.25an

db

=0.75,

SH

2also

givesa

goo

dp

erform

ance

and

itsso

lutio

nerro

ris

averagely2.5%

.O

nth

eo

ther

han

dth

eso

lutio

nerro

ro

fth

eS

H3

heu

risticis

3.5%.

Fo

ra

=0.50

and

b=

0.50,w

hile

SH

1again

givesth

eb

estresu

lts,th

eS

H2

results

app

roach

verym

uch

toS

H1,

and

SH

3gives

the

wo

rstp

erform

ance

of

three

heu

ristics.

0.00

0.01

0.02

0.03

0.04

0.05

(13,1,1)

(12,2,1)

(11,3,1)

(11,2,2)

(10,4,1)

(10,3,2)

(9,5,1)

(9,4,2)

(9,3,3)

(8,6,1)

(8,5,2)

(8,4,3)

(7,7,1)

(7,6,2)

(7,5,3)

(7,4,4)

(6,6,3)

(6,5,4)

(5,5,5)

(n1,n2,n3)

CPU times (s)

SH

1S

H2

SH

3

Fig.

6.H

euristics

error

for

(n,m)

=(15,3).

0.00

0.01

0.02

0.03

0.04

0.05

(12,1,1,1)

(11,2,1,1)

(10,3,1,1)

(10,2,2,1)

(9,4,1,1)

(9,3,2,1)

(9,2,2,2)

(8,5,1,1)

(8,4,2,1)

(8,3,3,1)

(8,3,2,2)

(7,6,1,1)

(7,5,2,1)

(7,4,3,1)

(7,4,2,2)

(7,3,3,2)

(6,6,2,1)

(6,5,3,1)

(6,5,2,2)

(6,4,4,1)

(6,4,3,2)

(6,3,3,3)

(5,5,4,1)

(5,5,3,2)

(5,4,4,2)

(5,4,3,3)

(4,4,4,3)

(n1,n2,n3,n4)

CPU times (s)

SH1

SH2

SH3

Fig.

7.H

euristics

error

for

(n,m)

=(15,4).

0.00

0.01

0.02

0.03

0.0

0.04 5

(11,1,1,1,1)

(10,2,1,1,1)

(9,3,1,1,1)

(9,2,2,1,1)

(8,4,1,1,1)

(8,3,2,1,1)

(8,2,2,2,1)

(7,5,1,1,1)

(7,4,2,1,1)

(7,3,3,1,1)

(7,3,2,2,1)

(7,2,2,2,2)

(6,6,1,1,1)

(6,5,2,1,1)

(6,4,3,1,1)

(6,4,2,2,1)

(6,3,3,2,1)

(6,3,2,2,2)

(5,5,3,1,1)

(5,5,2,2,1)

(5,4,4,1,1)

(5,4,3,2,1)

(5,4,2,2,2)

(5,3,3,2,2)

(4,4,4,2,1)

(4,4,3,3,1)

(4,4,3,2,2)

(4,3,3,3,2)

(3,3,3,3,3)

(n1,n2,n3,n4,n5)

CPU times (s)

SH1

SH2

SH3

Fig.

8.H

euristics

error

for

(n,m)

=(15,5).

1148T

.E

ren/A

pp

liedM

ath

ema

tical

Mo

dellin

g3

3(

20

09

)1

14

1–

11

50

0.00

0.01

0.02

0.03

0.04

0.05

100 200 300 400 500

number of jobs (n)

Err

or h

euri

stic

SH1 SH2 SH3

Fig. 9. Heuristics error for (n,m) = (100 6 n 6 500,10) and (a,b) = (0.25,0.75) problem.

0.00

0.01

0.02

0.03

0.04

0.05

100 200 300 400 500

number of jobs (n)

Err

or h

euri

stic

SH1 SH2 SH3

Fig. 10. Heuristics error for (n,m) = (100 6 n 6 500,10) and (a,b) = (0.50,0.50) problem.

0.00

0.01

0.02

0.03

0.04

0.05

100 200 300 400 500

number of jobs (n)

Err

or h

euri

stic

SH1 SH2 SH3

Fig. 11. Heuristics error for (n,m) = (100 6 n 6 500,10) and (a,b) = (0.75,0.25) problem.

T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150 1149

For a = 0.75 and b = 0.25, while SH1 gives the best result again, the SH2 results approach very much toSH1 results. SH3 results are worse than the previous weights levels.

1150 T. Eren / Applied Mathematical Modelling 33 (2009) 1141–1150

6. Conclusions

In this paper, a bicriteria m-identical parallel machines scheduling problem with a learning effect of setuptimes and removal times is considered. The objective function of the problem is minimization of the weightedsum of total completion time and total tardiness. A mathematical programming model is developed for theproblem which belongs to NP-hard class. Results of computational tests show that the proposed model iseffective in solving problems with up to 15 jobs and five machines.

To solve the large sizes problems up to 500 jobs and 10 machines, special heuristics methods were used. Theperformances of heuristics about the solution error were evaluated with the mathematical programming modelresults for small size problems and each other for large size problems. According to results, the special heu-ristic for all weight values was the more effective than others.

Other performance criteria with a learning effect in parallel machines can be also considered for future studies.

References

[1] C. Rajendran, H. Ziegler, Heuristics for scheduling in a flowshop with setup and processing and removal times separated, ProductionPlanning & Control 8 (6) (1997) 568–576.

[2] T. Eren, E. Guner, Setup and removal times with learning effect in single machine scheduling problems, Trakya University Journal ofScience 8 (1) (2007) 07–13 (in Turkish).

[3] T.P. Wright, Factors affecting the cost of airplanes, Journal of the Aeronautical Sciences 3 (1936) 122–128.[4] D. Biskup, Single machine scheduling with learning considerations, European Journal of Operational Research 115 (1999) 173–178.[5] G. Mosheiov, Scheduling problems with learning effect, European Journal of Operational Research 132 (2001) 687–693.[6] G. Mosheiov, Parallel machine scheduling with learning effect, Journal of the Operational Research Society 52 (2001) 1165–1169.[7] G. Mosheiov, J.B. Sidney, Scheduling with general job-dependent learning curves, European Journal of Operational Research 147

(2003) 665–670.[8] T. Eren, E. Guner, Minimizing total tardiness in a scheduling problem with a learning effect, Applied Mathematical Modelling 31 (7)

(2007) 1351–1361.[9] W.-C. Lee, C.-C. Wu, H.-J. Sung, A bi-criterion single-machine scheduling problem with learning considerations, Acta Informatica 40

(2004) 303–315.[10] T. Eren, E. Guner, A bicriteria scheduling with a learning effect: total completion time and total tardiness, Infor, accepted for

publication.[11] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Annals of Operations Research 98 (2000) 273–290.[12] W.-H. Kuo, D.-L. Yang, Minimizing the makespan in a single machine scheduling problem with a time-based learning effect,

Information Processing Letters 97 (2006) 64–67.[13] W.-C. Lee, A note on deteriorating jobs and learning in single-machine scheduling problems, International Journal of Business and

Economics 3 (2004) 83–89.[14] D. Biskup, D. Simons, Common due date scheduling with autonomous and induced learning, European Journal of Operational

Research 159 (2004) 606–616.[15] G. Mosheiov, J.B. Sidney, Note on scheduling with general learning curves to minimize the number of tardy jobs, Journal of the

Operational Research Society 56 (2005) 110–112.[16] T. Eren, E. Guner, Flowshop scheduling with general job-dependent learning effect, K.H.O, Journal of Defense Science 2 (2003) 1–11,

In Turkish.[17] W.-C. Lee, C.-C. Wu, Minimizing total completion time in a two-machine flowshop with a learning effect, International Journal of

Production Economics 88 (2004) 85–93.[18] T. Eren, E. Guner, Minimizing mean flow time in a flowshop scheduling with learning effect, Journal of the Faculty of Engineering

and Architecture of Gazi University 19 (2004) 119–124 (in Turkish).[19] T. Eren, E. Guner, A bicriteria flowshop scheduling with a learning effect, Applied Mathematical Modelling 32 (2007) 1719–1733.[20] S.S. Panwalker, W. Iskander, A survey of scheduling rule, Operations Research 25 (1977) 45–61.[21] J. Du, J.Y.T. Leung, Minimizing total tardiness on one machine is NP-hard, Mathematics of Operations Research 15 (1990) 483–495.[22] Y.S. Hsu, B.N.T. Lin, Minimizing of maximum lateness under linear deterioration, Omega 31 (2003) 459–469.[23] T. Eren, E. Guner, A bicriteria scheduling with sequence-dependent setup times, Applied Mathematics and Computation 179 (1)

(2006) 378–385.[24] T. Eren, E. Guner, A bicriteria flowshop scheduling problem with setup times, Applied Mathematics and Computation 183 (2) (2006)

1292–1300.[25] T. Eren, E. Guner, Setup times with a learning effect in flowshop scheduling problem, Journal of the Faculty of Engineering and

Architecture of Gazi University 22 (2) (2007) 353–362.[26] T. Eren, A multicriteria flowshop scheduling problem with setup times, Journal of Materials Processing Technology 186 (1–3) (2007) 60–65.[27] http://www.kku.edu.tr/akademik/goster.php?sicilno=0816.