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Permutation flowshop scheduling to minimize the total tardiness with

learning effects

Wen-Chiung Lee

a

, Yu-Hsiang Chung

b,n a

Department of Statistics, Feng Chia University, Taichung, Taiwan

bDepartment of Industrial & Engineering Management, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history: Received 4 January 2011 Accepted 1 August 2012 Available online 25 August 2012 Keywords: Scheduling Permutation flowshop Total tardiness Learning effect

a b s t r a c t

Scheduling with learning effects has received considerable attention recently. Often, numbers of operations have to be done on every job in many manufacturing and assembly facilities. However, it is seldom discussed in the general multiple-machine setting, especially without the assumptions of identical processing time on all the machines or dominant machines. With the current emphasis of customer service and meeting the promised delivery dates, we consider a permutation flowshop scheduling problem with learning effects where the objective is to minimize the total tardiness. A branch-and-bound algorithm and two heuristic algorithms are established to search for the optimal and near-optimal solutions. Computational experiments are also given to evaluate the performance of the algorithms.

&2012 Elsevier B.V. All rights reserved.

1. Introduction

In classical scheduling, the job processing times are assumed to be fixed and known throughout the entire process. However, this assumption might not reflect many real-life situations. For

example,Biskup (1999)pointed out that repeated processing of

similar tasks improves the worker skills; workers are able to perform setup, to deal with machine operations or software, or to

handle raw materials and components at a greater pace.Biskup

(1999) and Cheng and Wang (2000) were among the pioneers that brought the concept of learning effects into the scheduling field. Many researchers have devoted to this young but vivid area

since. Biskup (2008) provided a comprehensive review of the

scheduling models and problems with learning effects.

Recently, Wang (2007) considered some single-machine

pro-blems with the effects of learning and deterioration, and proved that the makespan and the sum of completion time problems remain polynomially solvable. He also showed that the weighted shortest processing time rule and the earliest due date rule provide the optimal schedules for the weighted sum of completion time and

the maximum lateness problems in some special cases.Janiak and

Rudek (2008)considered a scheduling problem in which each job provides a different experience to the processor. They relaxed one of the rigorous constraints, and thus each job can provide different experience to the processor in their model. They then formulated

the job processing time as a non-increasing k-stepwise function that in general is not restricted to a certain learning curve, thereby

it can accurately fit every possible shape of a learning function.Lee

and Wu (2004)investigated a two machine flowshop scheduling problem with learning consideration to minimize the total comple-tion time. They utilized the branch-and-bound algorithm incorpo-rated with several dominance properties and lower bounds to obtain the optimal solution. An accurate heuristic algorithm was

also proposed to obtain the near-optimal solution. Cheng et al.

(in press)studied a two-machine flowshop scheduling problem with a truncated learning function to minimize the makespan. They utilized a branch-and-bound and three heuristic algorithms to derive

the optimal and near-optimal solutions.Wang (2008)studied some

single-machine problems with the sum-of-processing-time-based learning effect. He showed by examples that the classical optimal rules no longer provide the optimal solutions under the proposed model. He also provided the optimal solutions for some

single-machine problems under certain conditions. Cheng et al. (2008),

Lee and Wu (2009), Yin et al. (2009) andZhang and Yan (2010)

considered a variety of models in which the actual job processing time not only depends on its scheduled position, but also depends on the sum of the processing times of jobs already processed. They provided the optimal schedules for some single machine problems.

Janiak and Rudek (2010) presented a new approach called multi-abilities learning that generalizes the existing ones and models. On this basis, they focused on the makespan problem and provided the

optimal polynomial time algorithms for some special cases.Lee et al.

(2010) investigated a single-machine problem with the learning effect and release times where the objective is to minimize the

Contents lists available atSciVerse ScienceDirect

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

Int. J. Production Economics

0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.08.014

n

Corresponding author. Tel.: þ886 3 928 395863; fax: þ 886 3 572 9101. E-mail address: yhchung.iem96g@nctu.edu.tw (Y.-H. Chung).

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makespan.Chang et al. (2009)studied a single machine scheduling problem, in which the learning/aging effect is considered. The objective is to determine the common due date and the sequence of jobs that minimizes a cost function.

Often numbers of operations have to be done on every job in

many manufacturing and assembly facilities (Pinedo, 2002; Tseng

and Lin, 2010; Wang et al., 2010; Zhao and Tang, 2012; Shabtay et al., 2012;Sun et al., 2012). However, it is seldom discussed in the general multiple-machine setting, especially without the assump-tions of identical processing time on all the machines or dominant

machines.Wang and Xia (2005)studied some permutation

flow-shop problems when the learning effect is present. They provided the worst-case bound of the shortest processing time rule for the makespan and the total completion time problems. They also showed that the makespan and the total completion time problems

remain polynomially solvable for two special cases.Xu et al. (2008)

provided heuristic algorithms for some permutation flowshop problems. They also analyzed the worst case bounds for the

proposed algorithms.Wu and Lee (2009)considered a permutation

flowshop scheduling problem to minimize the total completion time. They also analyzed the performance of the existing heuristic algorithms when the learning effect is present.

With the current emphasis of customer service and meeting the promised delivery dates, in this paper we consider a permutation flowshop scheduling problem to minimize the total tardiness with the learning effect. Although the classical problem without the consideration of learning effects has attracted the attention of numerous researchers due to its simple definition, most of the research focused on developing the heuristic or meta-heuristic

algorithms due to the complexity of the problem. Recently,Vallada

et al. (2008) provided a comprehensive review of the heuristic

algorithms. To the best of our knowledge, Kim (1995)andChung

et al. (2006) were the only authors who derived the optimal schedules. In this paper, we provide a branch-and-bound and two heuristic algorithms when the learning effect is present. The rest of the paper is organized as follows. In the next section we describe the

formulation of our problem. InSection 3, we construct a

branch-and-bound algorithm using an elimination rule and a lower branch-and-bound to

speed up the search for the optimal solution. In Section 4, two

heuristic algorithms are proposed to solve this problem. InSection 5,

a computational experiment is conducted to evaluate the efficiency of the branch-and-bound algorithm and the performance of the heuristic algorithms. A conclusion is given in the last section.

2. Problem description

There are n jobs and m machines. For each job j, there are

associated with m operations O1j, O2j, y, Omjwhere operation Oij

must be processed on machine i, i ¼ 1,2,. . .,m. Processing of

operation Oi þ 1, jcan start only after operation Oij is completed.

Moreover, we focus on the permutation flowshop case which implies the job sequence is the same in all the machines. The

normal processing time of operation Oijis denoted by pijand the

due date of job j is dj. The actual processing time pijrof operation

Oijis a function of its position in a schedule. That is,

pijr¼pijra, i ¼ 1,2,:::,m; r ¼ 1,2,:::,n,

if it is scheduled in the rth position and ao0 is the learning effect.

For a given schedule S, let CijðSÞ denote the completion time of

job j on machine i, TjðSÞ ¼ max0,CmjðSÞdjdenote the tardiness of

job j, and Ci½jðSÞ denote the completion time of the job scheduled

in the jth position on machine i. The objective of this paper is to find a schedule that minimizes the total tardiness, a widely used performance measure in scheduling literature. That is, we want to

find a schedule Sn

such thatPTjðS

n

ÞrPTjðSÞ for any schedule S.

3. A branch-and-bound algorithm

The problem under study is NP-hard since it already is even

without the learning effect (Pinedo, 2002). Thus, the

branch-and-bound algorithm might be a good way to obtain the optimal solution. In this section, we first provide a dominance property, followed by the lower bound to speed up the search process, and finally the branch-and-bound algorithm.

3.1. Dominance property

Chung et al. (2006)gave a dominance property for the classical problem. In this subsection, we modified the property to take the learning effect into consideration. Before presenting the property,

we first state a lemma fromChung et al. (2006).

Lemma 1. max0,ab Z max0,acmax0,bc for arbitrary real numbers a, b, and c.

Property 1. Suppose that S1¼ ð

s

1,

p

Þ and S2¼ ð

s

2,

p

Þ are two

sequences where

s

1and

s

2are partial sequences which contains

the same set of s jobs. If

Xs j ¼ 1 T½jð

s

2Þ Xs j ¼ 1 T½jð

s

1Þ Z ðnsÞmax0, max 1r i r mCi½sð

s

1ÞCi½sð

s

2Þ, then S1dominates S2.

Proof. By definition, the completion time of the nth job of S1on

machine m is Cm½nðS1Þ ¼ max 1r i r mCi½n1ðS1Þ þ Xm l ¼ i pl½nna¼Ci1½n1ðS1Þ þ Xm l ¼ i1 pl½nna

for some i1where 1ri1rm. Similarly, the completion time of the

nth job of S2on machine m is Cm½nðS2Þ ¼ max 1r i r mCi½n1ðS2Þ þ Xm l ¼ i pl½nna¼Ci2½n1ðS2Þ þ Xm l ¼ i2 pl½nna

for some i2where 1ri2rm. Thus, we have

Cm½nðS1ÞCm½nðS2Þ ¼ ½Ci1½n1ðS1Þ þ Xm l ¼ i1 pl½nna½Ci2½n1ðS2Þ þ Xm l ¼ i2 pl½nna r½Ci1½n1ðS1Þ þ Xm l ¼ i1 pl½nna½Ci1½n1ðS2Þ þ Xm l ¼ i1 pl½nna rCi1½n1ðS1ÞCi1½n1ðS2Þr max 1r i r mCi½n1ðS1ÞCi½n1ðS2Þ By an induction argument, we have

Cm½jðS1ÞCm½jðS2Þr max

ir i r mCi½sðS1ÞCi½sðS2Þfor j ¼ s þ1,. . .,n ð1Þ

FromLemma 1and Eq.(1), the difference between the total tardiness of S1 and S2is Xn j ¼ 1 T½jðS2Þ Xn j ¼ 1 T½jðS1Þ ¼ Xs j ¼ 1 T½jðS2Þ Xs j ¼ 1 T½jðS1Þ½ Xn j ¼ s þ 1 T½jðS1Þ Xn j ¼ s þ 1 T½jðS2Þ ¼X s j ¼ 1 T½jðS2Þ Xs j ¼ 1 T½jðS1Þ Xn j ¼ s þ 1 ðmax0,Cm½jðS1Þd½j max0,Cm½jðS2Þd½jÞ Z Xs j ¼ 1 T½jðS2Þ Xs j ¼ 1 T½jðS1Þ ðnsÞmax0, max 1r i r mCi½sðS1ÞCi½sðS2Þ

It implies that S1dominates S2and this completes the proof.

Property 1. can be simplified to the case of two adjacent jobs which is stated without proof.

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Corollary 1. Let S1¼ ð

p

,j1,j2,

p

0Þand S2¼ ð

p

,j2,j1,

p

0Þbe two

sche-dules where partial sequence

p

contains s jobs. If

Tj1ðS2Þ þTj2ðS2ÞTj1ðS1ÞTj2ðS1Þ Z ðns2Þmax0, max1r i r m½Ckj2ðS1ÞCkj1ðS2Þ,

then S1dominates S2.

3.2. A lower bound

In this subsection, a lower bound is established to facilitate the

search process of the branch-and-bound algorithm. Let

y

¼ ð

p

,

p

cÞ

denote a sequence in which

p

contains s scheduled jobs and

p

c

contains ns unscheduled jobs. Without loss of generality, we assume that the job processing times of the ns unscheduled jobs

on machine k are pkðs þ 1Þrpkðs þ 2Þr    rpkðnÞ when they are

arranged in non-decreasing order. We also assume that the due dates of the ns unscheduled jobs are dðs þ 1Þrdðs þ 2Þr    rdðnÞ

when they are arranged in non-decreasing order. By definition, the completion time of the (s þ 1)th job on machine k is

Ck½s þ 1ð

y

Þ ¼maxCk½sð

y

Þ,Ck1½s þ 1ð

y

Þ þpk½s þ 1ðs þ 1Þa

ZCk½sð

y

Þ þpk½s þ 1ðs þ1Þa:

By an induction argument, the completion time of the (s þ 1)th job on machine m is Cm½s þ 1ð

y

Þ ZCk½sð

y

Þ þ ðs þ 1Þa Xm l ¼ k pl½s þ 1: Therefore, we have Cm½s þ 1ð

y

Þ Z max 1r k r mCk½sð

y

Þ þ ðs þ 1Þ aXm l ¼ k pl½s þ 1 ð2Þ

Similarly, the completion time of the (s þ 2)th job on machine k is

Ck½s þ 2ð

y

Þ ¼maxCk½s þ 1ð

y

Þ, Ck1½s þ 2ð

y

Þ þpk½s þ 2ðs þ 2Þa

ZCk½s þ 1ð

y

Þ þpk½s þ 2ðs þ 2Þa

ZCk½sð

y

Þ þpk½s þ 1ðs þ1Þaþpk½s þ 2ðs þ 2Þa

By an induction argument, the completion time of the (s þ 2)th job on machine m is Cm½s þ 2ð

y

Þ ZCk½sð

y

Þ þpk½s þ 1ðs þ1Þaþ ðs þ 2Þa Xm l ¼ k pl½s þ 2: Thus, we have Cm½s þ 2ð

y

Þ Z max 1r k r mCk½sð

y

Þ þpk½s þ 1ðs þ 1Þ a þ ðs þ 2ÞaX m l ¼ k pl½s þ 2 ð3Þ

Using the same argument as to derive Eqs.(2) and (3), it is

obtained that the completion time of the (s þ j)th job on machine m is Cm½s þ jðyÞ Z max 1r k r mCk½sðyÞ þ Xj1 v ¼ 1 pk½s þ vðs þ vÞaþ ðs þ jÞaXm l ¼ k pl½s þ j ð4Þ

for 1rjrns. From Eq.(4), it implies that the total tardiness of

sequence

y

is Xn j ¼ 1 TjðyÞ ¼ Xs j ¼ 1 TjðyÞ þ Xns j ¼ 1 max0,Cm½s þ jðyÞd½s þ jZX s j ¼ 1 TjðyÞ þX ns j ¼ 1 max0, max 1r k r mCk½sðyÞ þ Xj1 v ¼ 1 pk½s þ vðsþ vÞa þ ðs þ jÞaXm l ¼ k pl½s þ jd½s þ jZX s j ¼ 1 TjðyÞ þ Xns j ¼ 1 max0, max 1r k r mCk½sðyÞ þX j1 v ¼ 1 pkðs þ vÞðs þvÞaþ ðs þjÞamin i Apc Xm l ¼ k plidðs þ jÞ

Thus, the lower bound on the total tardiness of sequence

y

based on s scheduled jobs is

LBð

y

Þ ¼ X s j ¼ 1 Tjð

y

Þ þ Xns j ¼ 1 max0, max 1r k r mCk½sð

y

Þ þ Xj1 v ¼ 1 pkðs þ vÞðs þ vÞa þ ðs þjÞamin i Apc Xm l ¼ k plidðs þ jÞ:

3.3. Description of the branch-and-bound algorithm

A depth-first search is adopted in the branching procedure. This method has the advantage that it only requires less storage space. In this paper, the algorithm assigns jobs in a forward manner starting from the first position. In the searching tree, we choose a branch and systematically work down the tree until we either eliminate it by virtue of the dominance property, the lower bound or reach its final node, in which case this sequence either replace the initial solution or is eliminated. The outline of the branch-and-bound algorithm is described as follows.

Step 1. {Initialization} Implement the proposed heuristic algo-rithm to obtain a sequence as the initial incumbent solution. Step 2. {Reduction} Apply Corollary 1 to eliminate the domi-nated partial sequence.

Step 3. {Branching} For the non-dominated nodes, compute the lower bound on the total tardiness of the unfathomed partial sequences or the total tardiness of the completed sequences. If the lower bound on the total tardiness for the partial sequence is greater than the initial solution, eliminate that node and all the nodes beyond it in the branch. If the value of the completed sequence is less than the initial solution, replace it as the new solution. Otherwise, eliminate it.

4. The heuristic algorithms

The computation effort can be reduced by using a heuristic solution as an upper bound prior to the application of the branch-and-bound algorithm. Furthermore, the search for the optimal solution for a problem with a large number of jobs is time consuming, but an effective heuristic can provide a time-saving

approximate solution with a small margin of error. Recently,Vallada

et al. (2008) provided a review and comprehensive evaluation of

Table 1

Data of the demonstrative examples with a ¼ 0:152. Problem 1 pij j 1 2 3 4 5 1 92 71 32 81 74 i 2 29 42 74 93 67 3 5 4 50 70 99 Due date 245 491 373 315 386 Problem 2 pij j 1 2 3 4 5 1 22 1 1 62 99 i 2 90 64 14 3 58 3 33 28 52 77 42 Due date 265 279 284 121 196 Problem 3 pij j 1 2 3 4 5 1 88 1 64 9 34 i 2 84 36 1 54 41 3 77 75 87 95 66 Due date 437 167 238 257 376

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40 different heuristics and metaheuristics for the classical m-machine permutation flowshop problem to minimize the total tardiness. They

found that ENS2 proposed byKim et al. (1996) is one of the best

heuristic among the existing algorithms. The main idea of algorithm

ENS2 is modified fromNawaz et al. (1983)by applying the EDD rule

instead, and further improving the solution by the pairwise inter-change movement. Thus, the algorithm ENS2 is used as the base for subsequently analysis.

4.1. A weight combination search algorithm

In our preliminary tests, it was observed that a job should be scheduled in an earlier position if it has a small value of the sum of the

job processing times Ppij or a small value of due date dj. This

motivated the usage of a combination of these two factors. That is, we would choose a proper weight w and arrange the jobs in a

non-decreasing order on the values of w Pm

i ¼ 1

pijþ ð1wÞdj. However, it was

very difficult to find the weight that could yield good solutions for all

the problems, as illustrated by the examples inTable 1. In problem 1,

the optimal solution was yielded when w ¼ 1, but the error percen-tages were both 23.05% whilew ¼ 0 and w ¼ 0:5. In problem 2, the optimal solution was yielded when w ¼ 0:5 and the error percentages were both 36.31% for the other two cases. In problem 3, the optimal solution was yielded when w ¼ 0, but the error percentages were both 141.93% for the other two cases. The results were recorded in

Table 2. This motivated the usage of a range of the weights, and the near-optimal solution was chosen as the best one among the solutions yielded from different weights. In our preliminary tests with 12 jobs, it was found that the quality of the solutions was quite stable after a partition of (0, 1) into 20 points. The proposed heuristic algorithm is

denoted as WSNEH þ PI, and the procedures are given as follows.

WSNEH þ PIalgorithm :

Step 1. Set w ¼ 0, Sn

¼ ð,    ,Þ with a total tardiness of 1: Step 2. Arrange jobs in the non-decreasing order of

wPm

i ¼ 1

pijþ ð1wÞdj. Let US denote the resulting sequence.

Step 3. Set k ¼ 1, select the first job in US to create a partial sequence PS.

Step 4. Update k¼kþ1. Select the kth job from US and insert it in k possible positions in the current partial sequence PS. Among k sequences, select the one with the minimum total tardiness as the current partial sequence PS.

Step 5. If k ¼ n, then replace Sn

by PS if the total tardiness of PS

is smaller than that of Sn

, else go to Step 4. Step 6. If wo1, set w ¼ wþ0:05 and go to Step 2. Step 7. Set k¼1.

Step 8. If kon, set l ¼ kþ1 and go to Step 9. Otherwise, stop

and output the sequence Sn

.

Step 9. Create a new sequence S by exchanging the jobs in positions k and l in Sn

. Replace Sn

by S if the total tardiness of S

is smaller than that of Sn

.

Step 10. If lon, then set l ¼ lþ1 and go to Step 9. Otherwise,

set k ¼ k þ 1 and go to Step 8. 4.2. The simulated annealing algorithm

The simulated annealing (SA) algorithm, proposed by

Kirkpatrick et al. (1983), was among the most popular meta-heuristic algorithms. The advantage of SA algorithm was that it could avoid getting trapped in a local optimum. In this section, the SA algorithm was utilized to derive a near-optimal solution. A brief description of the SA procedure was as follows. Given an initial sequence, a new sequence is created by a random neigh-borhood generation. The new sequence is accepted if its objective function has a smaller value than that of the original sequence; otherwise, it is accepted with some probability that decreases as the process evolves. The temperature is initially set to a high level so that a neighborhood exchange happens frequently in early iterations. It is gradually lowered using a predetermined cooling schedule so that it becomes more difficult to exchange in later iterations unless a better solution is obtained.

The most important implementations of the SA algorithm included:

(1) Initial sequence: As pointed out byVallada et al. (2008), ENS2

proposed byKim et al. (1996)was one of the best heuristic

among the existing algorithms. Thus, it was used as the initial sequence.

(2) Neighborhood generation: Neighborhood generation plays an important role in the efficiency of the SA method. Three neighborhood generation methods were used in the preliminary trials. They are the pairwise interchange (PI), the extraction and forward-shifted reinsertion (EFSR), and the extraction and backward-shifted reinsertion (EBSR) movements. It was observed that the PI movement yielded a better solution in the preliminary trials. Thus, it was used in subsequently analysis.

(3) Acceptance probability: In SA, solutions are accepted accord-ing to the magnitude of increase in the objective function and the temperature. The probability of acceptance is generated from an exponential distribution,

PðacceptÞ ¼ expð

a



D

TCÞ,

where

a

is the control parameter and

D

TC is the change in the

objective function. In addition, the method of changing

a

at

the kth iteration is obtained from Ben-Arieh and Maimon

(1992)and is given by

a

¼k

b

where

b

is an experimental factor. After some pretests, we

chose

b

¼85,000. If the total tardiness increases as a result of

a random pairwise interchange, the new sequence is accepted when PðacceptÞ 4 r, where r is a uniform random number between 0 and 1.

(4) Stopping condition: Our preliminary tests showed that the schedule is quite stable after 800n iterations, where n is the Table 2

The error percentages for the problems with different weights.

Optimal w ¼ 0 w ¼ 0:5 w ¼ 1

Schedule (total tardiness) Schedule (total tardiness) Error (%) Schedule (total tardiness) Error (%) Schedule (total tardiness) Error (%)

Problem 1 3, 4, 1, 5, 2 (11.93) 4, 1, 3, 5, 2 (14.68) 23.05 4, 1, 3, 5, 2 (14.68) 23.05 3, 4, 1, 5, 2 (11.93) 0

Problem 2 3, 4, 2, 5, 1 (54.26) 3, 2, 4, 1, 5 (73.96) 36.31 3, 4, 2, 5, 1 (54.26) 0 3, 2, 4, 1, 5 (73.96) 36.31

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number of jobs. Thus, 800n was used as the number of iterations.

5. Computational experiments

In order to evaluate the performance of the branch-and-bound and the heuristic algorithms, a computational experiment was conducted. All the proposed algorithms were coded in Fortran 90 and run on a personal computer with 2.66 GHz Intel Core 2 Quad CPU Q9400 and 3.25 GB RAM under Windows XP. The normal processing times on the machines were generated from a uniform distribution over the integers 1 to 99 as it was common in the literature. The due dates were generated from another uniform distribution over the

integers between Tð1

t

R=2Þ and Tð1

t

þR=2Þ, where R is the

due date range,

t

is the tardiness factor, and T is max

1r i r m Pn j ¼ 1 pij þ min 1r j r nð P i1 l ¼ 1 pljÞ þ min 1r j r nð Pm l ¼ i þ 1

pljÞ,which is a lower bound of the

makespan (1995).

The computational experiment consisted of three parts. In the first part, the job size was fixed at 10 and the numbers of

machines were m ¼3 and 5. The due date factors ð

t

,RÞ took the

values of (0.4, 0.8), (0.4, 1.0), (0.5, 0.8) and (0.5, 1.0) and the values of the learning effects were taken to be 90% and 80%, which

corresponded to a ¼ 0:152 and 0:322 according to Biskup’s

(1999)model. To test the efficiency of the dominance property and the lower bound separately, the branch-and-bound algorithm

with only the dominance property was denoted as BBp, the

branch-and-bound algorithm with only the lower bound was

denoted as BBL, and the branch-and-bound algorithm with both

the property and the lower bound was denoted as BBP þ L. The

results were compared with the enumeration method. The mean and maximum number of nodes and the mean and maximum CPU time (in seconds) were reported for the branch-and-bound algo-rithms, while only the mean and maximum CPU time (in seconds) were given for the enumeration method. As a consequence, there were 16 experimental conditions examined in the first part of the experiment, and 100 replications were randomly generated for

each condition. The results were presented inTable 3. It was seen

that the dominance property and the lower bound are efficient in the searching process for the optimal solution. Moreover, it was noted that the lower bound is more efficient than the dominance

property in terms of the execution time and the number of nodes. It was also noticed that the performance of the property is not influenced by the number of machines, while the lower bound is more powerful as the number of machines increases.

In the second part of the experiment, the impacts of the tardiness and the range factors were studied. The number of jobs was fixed at 12, the number of machines was fixed at 5, and the learning effect

was 90%. As in Kim et al. (1996), the values of

t

and R ranged from

0.1 to 0.5 and from 0.8 to 1.8. The combinations of

t

and R were

selected in a way that the due dates generated were nonnegative.

As a result, 20 combinations of

t

and R were examined. 100

replications were generated for each situation, and the results were

given inTable 4. When the range factor was fixed, it was seen that the

problems are harder to solve as the tardiness factor increases. On the other hand, there was no significant trend on the range factor when the tardiness factor was fixed. Thus, we would choose the last

5 values of

t

and R in subsequently experiments.

In the last part of the computational experiments, three different job sizes (n ¼14, 16, and 18) and two numbers of machines (m¼3 and 5) were tested. The values of the learning

effects were chosen to be 90% and 80%. The combination of ð

t

,RÞ

took the values of (0.4, 0.8), (0.4, 1.0), (0.4, 1.2), (0.5, 0.8), and (0.5, 1.0). As a consequence, 60 experimental situations were examined. A set of 100 instances were randomly generated for each situation,

and the results were presented in Tables 5–7. The same sets of

instances were used to test the performance of the branch-and-bound and the heuristic algorithms. For the branch-and-branch-and-bound algorithm, the mean and the maximum execution times (in seconds) as well as the mean and the maximum number of nodes

were reported. It was noted inTable 7that the branch-and-bound

algorithm was terminated if the number of nodes explored was

over 108, which was approximately 6 h in terms of the execution

time. The instances with more than 108 nodes were denoted as

unsolvable instances (USI), which were also reported inTable 7. For

the heuristic algorithms, the mean and the maximum error per-centages were noted. The error percentage of the solution produced by the heuristic algorithm is calculated as

ðVVn

Þ Vn

100%

where V is the total tardiness of the solution generated by the

heuristic method and Vn

is the total tardiness of the optimal schedule. It was noticed that there might be some instances where

Table 3

Performance of the branch-and-bound algorithms and the enumeration method (n ¼ 10).

m t R A (%) Number of nodes CPU time

BBP BBL BBP þ L BBP BBL BBP þ L Enumeration

Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max

3 0.4 0.8 90 86,445.2 306,377 1876.1 21,474 1390.5 17,501 1.23 4.16 0.07 0.63 0.05 0.58 13.57 13.81 80 58,910.3 467,754 348.8 4,985 286.5 4,195 0.80 5.69 0.02 0.19 0.01 0.19 13.49 13.69 1.0 90 99,352.7 336,606 1199.5 17,138 860.4 13,904 1.38 4.48 0.04 0.52 0.04 0.47 13.58 13.83 80 84,013.7 478,202 327.4 3,179 255.6 2,783 1.13 5.77 0.01 0.13 0.01 0.13 13.51 13.72 0.5 0.8 90 101,868.8 499,679 2463.1 17,206 1558.6 11,418 1.43 6.81 0.09 0.50 0.06 0.45 13.54 13.70 80 73,966.9 294,852 507.3 7,392 377.9 5,192 1.03 3.70 0.02 0.23 0.02 0.19 13.50 13.72 1.0 90 98,875.7 271,798 1635.0 11,577 1036.0 5,074 1.38 3.89 0.06 0.36 0.04 0.20 13.55 13.73 80 89,270.3 280,130 413.3 5,962 286.6 3,026 1.22 3.56 0.02 0.17 0.01 0.09 13.50 13.69 5 0.4 0.8 90 148,516.4 516,225 1289.7 10,438 1052.3 7,564 3.27 10.89 0.08 0.48 0.07 0.44 21.81 22.16 80 96,536.0 420,392 289.9 3,032 264.5 2,919 2.10 8.73 0.02 0.19 0.02 0.20 21.68 21.91 1.0 90 160,427.6 501,092 897.7 8,093 732.2 5,953 3.48 10.08 0.05 0.41 0.05 0.34 21.83 22.11 80 130,483.0 524,809 259.0 2,339 236.0 2,242 2.77 10.64 0.02 0.14 0.02 0.16 21.69 21.94 0.5 0.8 90 222,501.4 661,605 1965.9 14,354 1552.8 9,528 4.74 13.16 0.11 0.56 0.10 0.42 21.78 22.02 80 122,595.4 605,737 390.4 3,656 338.0 3,181 2.66 11.77 0.03 0.22 0.03 0.22 21.66 21.98 1.0 90 220,272.3 779,452 1503.3 12,731 1194.4 8,584 4.67 15.02 0.08 0.52 0.08 0.45 21.80 22.08 80 143,913.9 383,365 335.5 2,711 293.1 2,372 3.07 7.98 0.02 0.16 0.02 0.16 21.67 21.97

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the tardiness value of the branch-and-bound algorithm is zero but that of the heuristic algorithm is not. In that case, the error

percentage was not computed, and denoted as E1 in the tables.

Thus, the error percentages inTables 5–7could only be regarded as

the lower bounds of the errors. The computational times of the heuristic algorithms were not recorded since they were finished within a second. It was observed from the tables that the number of nodes and the execution time grow exponentially as the number of jobs increases. There were 5 unsolvable instances out of a total of 2000 when n ¼ 18. The most time-consuming solvable case took about 4.9 h. It was observed that the impact of the number of machines on the performance of the branch-and-bound algorithm

is not significant when the values of the other parameters were fixed. It was seen that the problems are easier to solve as the learning effect is stronger. As stated earlier, the problems are harder to solve as the tardiness factor increases. On the other hand, there was no significant trend on the range factor when the tardiness factor was fixed. As the performance of the heuristic algorithms, it was seen from the error percentage of ENS2 that the problems are harder when the learning effect is considered. It was observed that both the weight combination and the SA approaches significantly improved the quality of the near-optimal solutions. However, there was no absolutely dominance relation between the performance of

the WSNEH þ PI and the SA approaches. It was worth to mentioned

that out of the 5995 solvable instances, ENS2 has 33 instances that the tardiness value of the branch-and-bound algorithm is zero but that of the heuristic algorithm is not, WSNEH þ PIhas 7 instances, and

SA has only 2 instances. Although the error percentages of the WSNEH þ PIand the SA algorithms were still high for some instances,

it was due to the fact that the optimal tardiness values are relatively

small. For instance, the tardiness values of the WSNEH þ PIand the SA

algorithms were 5.56 and 9.56, while the optimal tardiness value was 2.055. This yielded an error percentage of 170.56% and

365.21%, respectively when n ¼ 16, m ¼ 3, ð

t

,RÞ ¼(0.4, 0.8), and

the learning effect was 90%. The overall performances of the WSNEH þ PI and the SA algorithms were quite good and they were

recommended when the learning effect was present.

6. Conclusion

In this paper, we studied an m-machine permutation flowshop problem to minimize the total tardiness with the learning effect. The computational experiments showed that the dominance rule and the lower bound facilitate the search for the optimal solution. The results also showed that the proposed branch-and-bound algorithm could solve most of the problems with up to 18 jobs in a reasonable amount of time. In addition, computational experi-ments showed that the performance of the proposed heuristics were good for instances of up to 18 jobs.

Table 4

Performance of the branch-and-bound algorithm with respect to t and R (n ¼ 12,m ¼ 5 and LE ¼90%).

t R Branch-and-bound algorithm

Number of nodes CPU time

Mean Max Mean Max

0.1 0.8 907.7 43,155 0.097 3.938 0.1 1.0 1,633.4 110,311 0.168 10.031 0.1 1.2 300.7 7,317 0.042 0.922 0.1 1.4 401.1 13,319 0.053 1.484 0.1 1.6 954.0 37,936 0.110 3.719 0.1 1.8 1,198.4 56,129 0.133 4.953 0.2 0.8 3,323.0 117,571 0.343 11.844 0.2 1.0 2,572.4 34,737 0.262 3.094 0.2 1.2 899.2 17,019 0.110 1.766 0.2 1.4 1,233.5 40,551 0.133 3.203 0.2 1.6 915.8 25,563 0.114 2.797 0.3 0.8 4,324.9 67,270 0.435 6.031 0.3 1.0 2,727.8 29,106 0.291 2.938 0.3 1.2 2,652.3 38,465 0.284 3.250 0.3 1.4 2,535.7 54,157 0.266 4.453 0.4 0.8 8,463.7 76,582 0.779 5.969 0.4 1.0 6,416.3 68,311 0.617 6.078 0.4 1.2 2,044.0 20,851 0.226 1.750 0.5 0.8 12,890.6 95,669 1.153 7.938 0.5 1.0 9,212.9 83,816 0.840 6.109 Table 5

Performance of the branch-and-bound and the heuristic algorithms (n ¼ 14 ).

m a t R Branch-and-bound algorithm Lower bound on the error percentage of the heuristics E1

Number of nodes CPU time ENS2 WSNEH þ PI SA ENS2 WSNEH þ PI SA

Mean Max Mean Max Mean Max Mean Max Mean Max

3 90% 0.4 0.8 31,665.4 745,932 2.8 59.8 4.43 66.66 0.81 30.91 0.45 32.42 2 0 0 0.4 1.0 16,479.1 320,261 1.5 23.4 5.68 178.62 0.39 14.77 0.33 25.61 1 0 0 0.4 1.2 29,930.9 918,455 2.4 71.6 2.65 35.44 0.46 10.70 0.25 9.58 0 0 0 0.5 0.8 164,189.2 3936,793 12.0 274.9 10.37 364.86 4.68 364.86 2.39 198.56 0 0 0 0.5 1.0 27,701.1 494,319 2.4 40.3 4.09 92.53 0.31 5.27 0.13 4.00 0 0 0 80% 0.4 0.8 14,133.8 1290,356 1.0 83.7 5.48 148.65 0.47 44.98 0.04 4.18 3 1 0 0.4 1.0 2,243.5 38,408 0.2 4.4 3.92 75.40 0.25 7.64 0.05 3.21 3 0 0 0.4 1.2 4,122.0 244,220 0.4 16.4 2.45 130.28 0.13 4.13 0.04 2.88 1 0 0 0.5 0.8 5,607.0 105,935 0.5 8.4 26.79 2,336.72 0.54 12.58 0.07 5.53 0 0 0 0.5 1.0 13,962.1 247,802 1.1 19.1 1.89 18.08 0.16 3.97 0.05 3.37 0 0 0 5 90% 0.4 0.8 67,788.0 2218,760 7.8 152.8 10.77 166.27 2.06 27.16 0.73 11.27 0 0 0 0.4 1.0 32,581.3 433,044 4.4 48.7 4.81 38.03 0.45 4.81 0.42 5.35 0 0 0 0.4 1.2 13,864.8 322,278 2.1 41.5 2.23 18.44 0.54 9.28 0.22 3.22 0 0 0 0.5 0.8 78,784.8 850,160 10.1 93.1 6.21 47.95 1.70 30.28 0.60 8.80 0 0 0 0.5 1.0 40,729.5 552,293 5.5 66.5 3.12 20.57 0.92 9.04 0.43 5.75 0 0 0 80% 0.4 0.8 3,740.1 50,277 0.7 8.5 135.48 12,580.30 0.85 53.30 0.86 34.55 1 0 0 0.4 1.0 2,582.3 27,599 0.5 4.6 3.88 126.19 0.30 9.09 0.10 4.85 0 0 0 0.4 1.2 2,306.7 38,603 0.4 5.2 44.53 4,037.97 22.38 2138.06 14.63 1453.15 0 0 0 0.5 0.8 7,614.9 89,364 1.1 11.5 2.48 27.15 0.48 8.31 0.29 8.02 0 0 0 0.5 1.0 4,257.0 62,313 0.7 10.0 2.72 63.46 0.39 8.43 0.30 6.90 0 0 0

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Acknowledgements

The authors are grateful to the editor and the referees, whose constructive comments have led to a substantial improvement in the presentation of the paper. This work was supported by the NSC of Taiwan, ROC, under NSC 98–2221-E-035-033-MY2. References

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Table 6

Performance of the branch-and-bound and the heuristic algorithms (n ¼ 16 ).

m a t R Branch-and-bound algorithm Lower bound on the error percentage of the heuristics E1

Number of nodes CPU time ENS2 WSNEH þ PI SA ENS2 WSNEH þ PI SA

Mean Nax Mean Max Mean Max Mean Max Mean Max

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Performance of the branch-and-bound and the heuristic algorithms (n ¼ 18 ).

m a t R Branch-and-bound algorithm Lower bound on the error percentage of the heuristics E1 USI

Number of nodes CPU time ENS2 WSNEH þ PI SA ENS2 WSNEH þ PI SA

Mean Max Mean Max Mean Max Mean Max Mean Max

3 90% 0.4 0.8 713,301.6 19,509,458 112.2 2,739.4 309.17 26,525.00 12.33 1089.64 21.63 1896.62 2 1 0 0 0.4 1.0 1,193,123.5 59,651,120 192.2 9,211.3 14.12 602.08 4.07 340.34 0.79 54.74 3 0 0 0 0.4 1.2 368,085.7n 9,941,654n 68.1n 1,931.5n 2.79 50.51 0.75 39.27 0.15 5.06 0 0 0 1 0.5 0.8 3,807,059.3n 69,093,792n 499.2n 9,345.5n 17.82 798.11 2.44 109.39 2.14 109.39 2 0 0 2 0.5 1.0 1,464,720.8n 38,858,224n 218.2n 5,040.9n 2.56 37.01 0.53 7.14 0.51 10.28 0 0 0 1 80% 0.4 0.8 99,529.5 7,693,594 18.0 1,308.8 1.53 63.88 0.00 0.00 0.02 1.72 1 0 0 0 0.4 1.0 63,843.9 2,911,862 12.3 532.1 5.26 114.00 0.24 5.34 0.18 7.79 1 1 0 0 0.4 1.2 110,169.3 8,875,314 18.1 1,338.4 1.55 23.17 0.15 6.09 0.21 7.35 0 0 0 0 0.5 0.8 198,012.8 9,168,472 33.7 1,389.6 10.11 282.92 0.34 9.75 0.76 53.80 0 0 0 0 0.5 1.0 364,668.3 29,460,512 61.1 4,730.8 3.25 70.22 0.90 31.28 0.43 17.92 0 0 0 0 5 90% 0.4 0.8 693,555.3n 9,369,027n 182.4n 2,280.3n 11.96 364.93 3.53 123.37 3.58 163.08 2 1 2 1 0.4 1.0 234,498.1 4,391,182 67.2 1,279.4 10.13 479.05 3.84 276.03 0.78 30.40 0 0 0 0 0.4 1.2 442,895.6 14,758,786 120.3 3,244.0 2.69 32.28 0.58 10.14 0.55 18.29 0 0 0 0 0.5 0.8 3,615,769.3 83,501,184 828.2 17,656.1 8.53 62.14 1.89 19.04 2.60 62.14 0 0 0 0 0.5 1.0 3,187,206.0 50,813,748 764.4 12,792.8 5.34 66.97 1.37 19.85 0.84 19.14 0 0 0 0 80% 0.4 0.8 25,074.3 753,083 7.6 242.0 10.99 191.43 0.40 20.27 0.89 48.92 3 2 0 0 0.4 1.0 97,149.0 7,736,601 24.8 1,816.3 12.24 624.45 1.17 52.76 6.80 624.45 1 0 0 0 0.4 1.2 48,316.4 2,568,162 14. 668.4 2.61 30.63 0.38 11.06 0.36 12.01 0 0 0 0 0.5 0.8 54,895.8 1,435,597 16.0 332.0 5.44 100.05 0.68 19.06 0.43 13.91 1 0 0 0 0.5 1.0 99,501.8 1,743,517 29.9 435.4 2.91 31.20 0.66 15.54 0.28 4.68 0 0 0 0 n

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數據

Table 2 . This motivated the usage of a range of the weights, and the near-optimal solution was chosen as the best one among the solutions yielded from different weights

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