Bi-criteria minimization for the permutation flowshop scheduling problem
with machine-based learning effects
q
Yu-Hsiang Chung
⇑, Lee-Ing Tong
Department of Industrial Engineering and Management, National Chiao Tung University, 1001 University Road, Hsinchu 300, Taiwan, ROC
a r t i c l e
i n f o
Article history:
Received 28 September 2011
Received in revised form 13 March 2012 Accepted 14 March 2012
Available online 23 March 2012
Keywords: Scheduling Learning effect Flowshop
Total completion time Makespan
a b s t r a c t
In traditional scheduling problems, the processing time for the given job is assumed to be a constant regardless of whether the job is scheduled earlier or later. However, the phenomenon named ‘‘learning effect’’ has extensively been studied recently, in which job processing times decline as workers gain more experience. This paper discusses a bi-criteria scheduling problem in an m-machine permutation flowshop environment with varied learning effects on different machines. The objective of this paper is to minimize the weighted sum of the total completion time and the makespan. A dominance criterion and a lower bound are proposed to accelerate the branch-and-bound algorithm for deriving the optimal solution. In addition, the near-optimal solutions are derived by adapting two well-known heuristic algorithms. The computational experiments reveal that the proposed branch-and-bound algorithm can effectively deal with problems with up to 16 jobs, and the proposed heuristic algorithms can yield accurate near-optimal solutions.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
In classical scheduling problems, all the processing times of jobs are often assumed to be fixed and known before processing the jobs (Pinedo, 2002). However, in reality, the workers acquire expe-rience after repetitiously operating similar tasks in many practical environments. As a result, the processing times of the jobs decline as the skill of the workers improves. This phenomenon is known as the ‘‘learning effect’’.
Biskup (1999)addressed a learning effect model in a single-machine scheduling problem in which the processing time of the job is a function of its position in a schedule. The problems for min-imizing the deviation from a common due date and minmin-imizing the total flow time were demonstrated to be polynomially solvable. Subsequently, the learning effect has received extensive discussion in the scheduling field (Janiak & Rudek, 2008, 2010; Mosheiov & Sidney, 2003). Biskup (2008) proposed a detailed review of scheduling problems with learning effect in which the existing models are distinguished into two distinct types: the position-based learning and the sum-of-processing-time-position-based learning types. The position-based learning effect is affected by the number of processed jobs. Meanwhile, the sum-of-processing-time-based learning effect is influenced by the total processing times of the finished jobs.
With regard to the position-based learning effect,Lee, Wu, and Hsu (2010) studied a single-machine scheduling problem with release times under learning consideration. They proposed a branch-and-bound and a heuristic algorithm to obtain the optimal and near-optimal solution for minimizing the makespan.Toksari (2011) addressed a single-machine scheduling problem with unequal release times for minimizing the makespan, in which the learning effect and the deteriorating jobs are concurrently consid-ered. Several dominance criteria and the lower bounds are estab-lished to facilitate the branch-and-bound algorithm for deriving the optimal solution. In addition,Zhu, Sun, Chu, and Liu (2011)
studied two single-machine group scheduling problems. The job processing time is a function of job position, group position and the amount of resources assigned to the group. They verified that minimizing the total weighted sum of the makespan and the total resources constrained remain polynomial solvable.Huang, Wang, and Wang (2011)investigated two resources constrained single-machine group scheduling problems in which the learning effect and deteriorating jobs are considered simultaneously. They pro-posed polynomial solutions under certain constraints to minimize the makespan and the resource consumption, respectively. More-over,Lee and Lai (2011) considered both the effect of learning and deterioration in a scheduling model. The actual job processing time is a function on the processing times of scheduled jobs and its position in the schedule. They showed that some single-machine scheduling problems remain polynomial solvable.
In terms of the sum-of-processing-time-based learning effect,
Koulamas and Kyparisis (2007) indicated that employees learn
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http://dx.doi.org/10.1016/j.cie.2012.03.009 q
This manuscript was processed by Area Editor (T.C. Edwin Cheng).
⇑Corresponding author. Tel.: +886 928 395863.
E-mail address:yhchung.iem96g@nctu.edu.tw(Y.-H. Chung).
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more when performing jobs with a longer processing time. They brought in a sum-of-job-processing-time-based learning effect scheduling model and showed that the makespan and the total completion time problems for the single machine setting, and two-machine flowshop setting with ordered job processing times remain polynomially solvable. Furthermore,Cheng, Lai, Wu, and Lee (2009) introduced a learning effect model into a single-machine scheduling problem. The actual job processing time is de-rived from the sum of the logarithm of the processing times of pro-cessed jobs. They showed that the makespan and total completion
time problems are polynomially solvable.Wang and Wang (2011)
introduced an exponential sum-of-actual-processing-time-based learning effect into a single-machine scheduling problem. The spe-cial cases of the total weighted completion time problem and the maximum lateness problem are proved to be polynomial solvable
under an adequate condition. In addition, Cheng, Cheng, Wu,
Hsu, and Wu (2011) proposed a two-agent scheduling problem with a truncated sum-of-processing-time-based learning effect on a single machine. A branch-and-bound algorithm was utilized to obtain the optimal solution for minimizing the total weighted completion time for the jobs of the first agent subject to no tardy job of the second agent.
The position-based and the sum-of-processing-time-based learning effects have been concurrently considered in recent literatures.Cheng, Wu, and Lee (2008)developed a model of the learning effect on a single machine in which the actual processing time of the job is a function of the total normal processing time of processed jobs and the position of the job in a schedule. They then proved that the makespan and total completion time problems re-main polynomially solvable.Lai and Lee (2011)addressed a general scheduling model in which the position-based and the sum-of-pro-cessing-time-based learning effects are concurrently considered. They showed that most of the models in the literatures are special cases of the model they proposed. Furthermore,Yin, Xu, Sun, and Li (2009) considered learning effect in some single-machine and m-machine flowhop scheduling problems. The actual job process-ing time is a function of the total normal processprocess-ing times of the
jobs already processed and the number of processed jobs. Lee
and Wu (2009)developed a general learning model that associates
with the position-based and sum-of-processing-time-based
learning effects. They then showed that the single-machine make-span and the total completion time problems are polynomially solvable, and proposed polynomial-time optimal solutions to minimize the makespan and total completion time under certain agreeable conditions for the flowshop setting.
In terms of the flowshop scheduling problems,Johnson (1954)
was the pioneer for discussing this topic.Chung, Flynn, and Kirca (2002)investigated an m-machine flowshop problem with a total completion time objective. Then a brand-and-bound algorithm incorporated with a creative lower bound and a dominance rule was conducted to derive the optimal solution.Chung, Flynn, and Kirca (2006) studied a total tardiness minimization scheduling problem in an m-machine flowshop environment. They sought the optimal solution by implementing a branch-and-bound algo-rithm. In addition, the learning effect has been recently introduced into flowshop scheduling problems (Lee and Wu (2004), Wu, Lee, and Wang (2007)).Chen, Wu, and Lee (2006)considered a bi-crite-ria two-machine flowshop scheduling problem with the learning effect in which the objective is to minimize the weighted sum of the total completion time and the maximum tardiness. They estab-lished a branch-and-bound algorithm and two heuristic algorithms to obtain the optimal and near-optimal solutions.Li, Hsu, Wu, and Cheng (2011)discussed a two-machine flowshop scheduling prob-lem with a truncated learning effect which considers the position of the job in a schedule and the control parameter. Then the branch-and-bound and three simulated annealing algorithms were
conducted to seek the optimal and near-optimal solutions.
Addi-tionally, Wang and Xia (2005) considered flowshop scheduling
problems with a learning effect. They demonstrated examples to prove that the Johnson’s rule is not the optimal method to minimize the makespan problem for a two-machine setting with learning consideration. Then they showed that two special cases remained polynomially solvable with makespan and total
comple-tion time objectives. Wu and Lee (2009) discussed a flowshop
scheduling problem with a learning effect for minimizing the total completion time and then utilized a branch-and-bound algorithm to obtain the optimal solution. Then they adapted four well-known heuristic algorithms to derive the near-optimal solutions and compare the proposed heuristic algorithms.
Due to the complexity of the flowshop environment, most of the researchers have devoted to discovering near-optimal solutions.
Nawaz, Enscore, and Ham (1983)investigated an m-machine flow-shop scheduling problem with the makespan objective. They sta-ted that jobs with larger total processing times have higher priority and should be scheduled earlier. Then they showed that the algorithm they proposed has remarkable performance, particu-larly in the large job-sized problems. Subsequently, Liu and Ong (2002)andRuiz and Maroto (2005)pointed out that the algorithm
Nawaz et al. (1983)proposed is preferable to other existing poly-nomial algorithms for the m-machine flowshop scheduling
prob-lem with makespan objective. Framinan, Gupta, and Leisten
(2004)presented a review and classification for the heuristic
algo-rithms with a makespan objective. Furthermore, Framinan and
Leisten (2003) considered an m-machine flowshop scheduling problem for minimizing the mean flow time. They established an efficient constructive heuristic algorithm by first utilizing the con-cept of the algorithm proposed byNawaz et al. (1983), and then implemented a general pairwise interchange movement to im-prove the solutions. Thereafter,Wu and Lee (2009)indicated that the heuristic algorithm proposed byFraminan and Leisten (2003)
is recommended to approximate the total completion time for the flowshop scheduling problem with a general position-based learning effect. In addition,Wang, Pan, and Tasgetiren (2011) pro-posed a modified global-best harmony search algorithm to obtain the near-optimal solution for dealing with a makespan scheduling
problem in a blocking permutation flowshop environment.Zhang
and Li (2011) addressed an estimation of distribution algorithm for a permutation flowshop scheduling problem with the objective of minimizing the total flowtime.
In most of the literature for the flowshop scheduling problems with learning effects, it is assumed that the learning effects are identical on all machines. Therefore, in this paper, the machine-based learning effect is introduced to the model ofBiskup (1999)
for the m-machine permutation flowshop setting, in which the learning effects are varied on different machines. In scheduling field, most of the objective functions are relative to the completion time of the tasks, and proposing approaches to minimize the com-pletion time is an important topic in many studies. Therefore, two widely used objective functions are considered simultaneously in this paper, which are total completion time and makespan. Due to the combinative performance of these two objective functions is emphasized, the objective function of proposed problem in this paper is to minimize the weighted sum of total completion time and makespan. The outline of this paper is constructing algorithms for obtaining the optimal and near-optimal solutions of proposed problem, and utilizing the computational experiment to evaluate the performance of proposed algorithms. The description of the remaining sections of this paper is structured as follows. The formulation of the problem is elaborated on Section2. Then four heuristic algorithms are proposed in Section3to obtain the near-optimal solution. A dominance criterion and a lower bound of the branch-and-bound algorithm are established in Section 4 for
optimizing the proposed problem. The computational experiments are implemented in Section 5to assess the performances of all proposed algorithms. Eventually, the conclusions are represented in Section6.
2. Problem description
The following notations are utilized throughout this paper. N: Set of jobs which contains n jobs, i.e., N = {1, 2, . . ., n}. S: Subset of N with s scheduled jobs.
U: Subset of N with n s unscheduled jobs. m: Number of machines.
Mi: ith machine, where i = 1, 2, . . ., m.
Jj: Job j, where j = 1, 2, . . ., n.
pij: Basic processing time of Jjon Mi.
pijr: Actual processing time of Jjon Miwhen Jjis scheduled at
position r.
ai: Learning index on Miwith ai< 0 for i = 1, 2, . . ., m. []: The symbol which denotes the job order in a sequence.
a
: The weight of the objective function with 0 6a
61. Ci[r](h): The completion time at the position r on Miin sequenceh.
LB: The lower bound for the objective based on the given node. The details of the proposed problem with the machine-based learning effect in an m-machine permutation flowshop environ-ment are described as follows: Assume that there is a jobs set N with n jobs to be processed on m machines. Each Jjincludes m
operations on corresponding machines which denote as Oi,j for
i = 1, 2, . . ., m and j = 1, 2, . . ., n. For the processing procedure, the starting time of Oi,jmust be the larger one of the completion times
of Oi1,jand Oi,j1. In addition, a permutation flowshop does not
al-low sequence changes between machines (Pinedo, 2002). There-fore, the sequence of jobs is identical to all the machines in this paper. Since the machine-based learning effect is considered, the actual processing time pijrof Jjscheduled at position r on Mi
de-clines based on its position, i.e.,
pijr¼ pijrai
where i = 1, 2, . . ., m, and j, r = 1, 2, . . ., n.
The aim of this paper is to seek a sequence for minimizing the weighted sum of total completion time and makespan. And then the proposed problem is defined as Fmjpijr¼ pijraij
a
Pn j¼1Cjþ
ð1
a
ÞCmax.3. Heuristic algorithms
For the m-machine permutation flowshop scheduling problem, the total completion time problem is demonstrated to be a strongly NP-hard problem for m P 2 without considering the learning effect (Lenstra, Rinnooy Kan, & Brucker, 1977). In addition, Garey, Johnson, and Sethi (1976) showed that the classical makespan problem is NP-hard for m P 3. Therefore, the proposed problem in this paper is conjectured to be a NP-hard problem. While the number of jobs increases, obtaining the optimal solution of an NP-hard scheduling problem is time consuming. Therefore, many studies are devoted to developing efficient heuristic algorithms to derive the near-optimal solution. In this paper, four heuristic algo-rithms are proposed and denoted as NEH, FL, NEH_W and FL_W. Since the objective function in this paper consists of the makespan and the total completion time, NEH and FL are respectively adapted from the heuristic algorithm proposed inNawaz et al. (1983)and
Framinan and Leisten (2003) by considering the learning effect and adjusting the objective function to the bi-criteria one proposed
in this paper. In the procedure of proposed heuristic algorithms, the jobs with larger total processing time (i.e.Pmi¼1pijfor j = 1, 2, . . ., n)
have higher priority to be selected in NEH, while smaller in FL. In addition, since the machine-based learning effects are considered in this paper, the ratios of the reduction for the actual processing time are varied on different machines. Therefore, NEH_W and FL_W are adapted from NEH and FL by utilizing the total weighted processing time (i.e.Pmi¼1wipijfor j = 1, 2, . . ., n) to determine the
priority of the jobs, in which the machines with weaker learning ef-fect have larger weight. Eventually, the procedures of NEH_W and FL_W are detailed as follows.
3.1. NEH_W algorithm
Step 1: Set sequence PS and US with empty set.
Step 2: Arrange the jobs in descending order of the total weighted normal processing time, and then schedule the jobs into US.
Step 3: Set k = 1.
Step 4: Select the first job from US into PS, and remove the job from US.
Step 5: If k = 1, go to Step 4. Otherwise, generate k sequence by respectively inserting the job into each slot of PS.
Step 6: Select the sequence with the least objective value among k candidate sequences and update the sequence as PS. Step 7: Set k = k + 1. If k 6 n, go to Step 4. Otherwise, the near-optimal sequence is set as PS.
3.2. FL_W algorithm
Step 1: Set sequence PS and US with empty set.
Step 2: Arrange the jobs in ascending order of the total weighted normal processing time, and then schedule the jobs into US. Step 3: Set k = 1.
Step 4: Select the first job from US into PS, and remove the job from US.
Step 5: If k = 1, go to Step 4. Otherwise, generate k sequence by respectively inserting the job into each slot of PS.
Step 6: Select the sequence with the least objective value among k candidate sequences and update the sequence as PS. Step 7: If k < 3, go to Step 8. Otherwise, generatekðk1Þ
2 sequences
based on PS by performing pairwise interchange procedure. Then select the sequence with the least objective value and set as PS0. If PS can be dominated by PS0in terms of the objective value, replace PS with PS0.
Step 8: Set k = k + 1. If k 6 n, go to Step 4. Otherwise, the near-optimal sequence is set as PS.
4. Branch-and-bound algorithm
In order to seek the optimal solution, the branch-and-bound algorithm is implemented in this paper. For the proposed branch-and-bound algorithm, we addressed a dominance criterion modi-fied fromChung et al. (2002)and a lower bound incorporated with the Hungarian method, to facilitate the procedure for deriving the optimal solution. In this section, the specification of the dominance criterion and the lower bound are demonstrated. Eventually, the summary of the proposed branch-and-bound algorithm is represented.
4.1. Theorem and corollary of the dominance criterion
A rule is represented in following theorem which determines the dominance from two varied sequences concluding the same jobs set. If a sequence is dominated by another one, the node based on the sequence is eliminated in the branching tree.
Theorem 4.1. There are two sequence of set N, that is h1= (
r
1,p
) andh2= (
r
2,p
), in whichr
1andr
2denote two different sequence of set S,and
p
denotes a sequence of set U. Ifa
Psj¼1
ðCm½jð
r
2Þ Cm½jðr
1ÞÞ > ða
ðn s 1Þ þ 1Þmax16i6m Ci½sð
r
1Þ Ci½sðr
2Þ
ð1Þ
then
r
1dominatesr
2.Proof. For k = 1, 2, . . ., m, we have Ck½nðh1Þ ¼ max 16i6kfCi½n1ðh1Þ þ Pk u¼i pu½nnaug Ci1½n1ðh1Þ þ Pk u¼i1 pu½nnau ð2aÞ for some 1 6 i16k Similarly, Ck½nðh2Þ Ci2½n1ðh2Þ þ Pk u¼i2 pu½nnau ð2bÞ for some 1 6 i26k(2b).
From Eqs.(2a) and (2b), we have
Ck½nðh1Þ Ck½nðh2Þ ¼ Ci1½n1ðh1Þ þ Pk u¼i1 pu½nnau " # Ci2;½n1ðh2Þ þ Pk u¼i2 pu½nnau " # 6 Ci 1½n1ðh1Þ þ Pk u¼i1 pu½nnau " # Ci1;½n1ðh2Þ þ Pk u¼i1 pu½nnau " # Then Ck½n1ðh1Þ Ck½n1ðh2Þ 6 Ci1½n1ðh1Þ Ci1½n1ðh2Þ 6max
16i6mfCi½n1ðh1Þ Ci½n1ðh2Þg By an induction, for k = m, we have
Cm½sþlðh1Þ Cm½sþlðh2Þ 6 max
16i6mfCi;½sð
r
1Þ Ci;½sðr
2Þg ð3Þ where 1 6 l 6 n s.From Eq.(3), we have
½
a
Pn j¼1 Cm½jðh1Þ þ ð1a
ÞCm½nðh1Þ ½a
P n j¼1 Cm½jðh2Þ þ ð1a
ÞCm½nðh2Þ 6a
P s j¼1 ½Cm½jðr
1Þ Cm½jðr
2Þ þ ½a
ðn s 1Þ þ 1max16i6mfCi½sð
r
1Þ Ci½sðr
2Þg ð4ÞThe value for the left side of Eq.(4)is negative by Eq.(1)and it im-plies h1dominates h2.
Therefore, we have
r
1dominatesr
2. hIn this paper, the theorem is simplified as the corollary which requires considering two adjacent jobs. The corollary is utilized in the proposed branch-and-bound algorithm and presented below.
Corollary 4.1. In set S, let
r
denote a partial sequence with s 2 jobs. In addition, the remaining jobs are scheduled in the last two positions as J1 and J2. The two sequences based onr
are presented asS1= (
r
, J1, J2) and S2= (r
, J2, J1), and CiJjðSlÞ denotes the completiontime of Jjon Miin Slfor j, l = 1, 2 and i = 1, 2, . . ., m. If
a
½CmJ2ðS2Þ þ CmJ1ðS2Þ CmJ1ðS1Þ CmJ2ðS1Þ>½
a
ðn s 1Þ þ 1max16i6mfCiJ2ðS1Þ CiJ1ðS2Þg ð5Þ
then S1dominates S2.
4.2. Calculation of the lower bound
In addition to the dominance criterion, another procedure to eliminate nodes in the branching tree is calculating the lower bound of the objective value. In this paper, we establish a lower bound to speed up the procedure of the proposed branch-and-bound algorithm. The proposed lower branch-and-bound requires O(mn3)
com-putation time, and is presented as follows.
Let h denote a sequence with s scheduled and n s unscheduled jobs of set N. For 1 6 k 6 m, the completion time of (s + 1)th job on Mkis presented as
Ck½sþ1ðhÞ ¼ maxfCk1½sþ1ðhÞ; Ck½sðhÞg þ pk½sþ1ðs þ 1Þ ak
PCk½sðhÞ þ pk½sþ1ðs þ 1Þ ak
Thus, the completion time of (s + 1)th job on Mmis presented as Cm½sþ1ðhÞ P Ck½sðhÞ þ pk½sþ1ðs þ 1Þ akþ P m i¼kþ1 pi½sþ1ðs þ 1Þ ai
Furthermore, the completion time of (s + 2)th job on Mkis presented
as Ck½sþ2ðhÞ ¼ maxfCk1½sþ2ðhÞ; Ck½sþ1ðhÞg þ pk½sþ2ðs þ 2Þ ak PCk½sðhÞ þ pk½sþ1ðs þ 1Þ akþ p k½sþ2ðs þ 2Þ ak
Thus, the completion time of (s + 2)th job on Mmis presented as Cm½sþ2ðhÞ P Ck½sðhÞ þP 2 v¼1 pk½sþvðs þ
v
Þ akþ P m i¼kþ1 pi½sþ2ðs þ 2Þai By an induction, the underestimated value of the completion time for (s + l)th job on Mmbased on Mkis presented asCk½sðhÞ þP l v¼1 pk½sþvðs þ
v
Þ akþ P m i¼kþ1 pi½sþlðs þ lÞai ð6Þ Then we havea
Pn j¼1 Cm½jðhÞ þ ð1a
ÞCm½nðhÞ Pa
P s j¼1 Cm½jðhÞ þ ½a
ðn s 1Þ þ 1Ck½sðhÞ þPns l¼1 ½a
ðn s lÞ þ 1ðs þ lÞakp k½sþl þPns l¼1 ½ða
þ IðlÞÞP m i¼kþ1 pi½sþlðs þ lÞ ai where IðlÞ ¼ 1a
; l ¼ n s 0; l – n s ð7ÞSince ½
a
ðn s lÞ þ 1ðs þ lÞak decreases as l increases, we havea
Pn j¼1 Cm½jðhÞ þ ð1a
ÞCm½nðhÞ Pa
P s j¼1 Cm½jðhÞ þ ½a
ðn s 1Þ þ 1Ck½sðhÞ þnsP l¼1 ½a
ðn s lÞ þ 1ðs þ lÞakp kðsþlÞ þnsP l¼1 ½ða
þ IðlÞÞP m i¼kþ1 pi½sþlðs þ lÞai ð8Þwhere pk(s+l)denotes the lth smallest basic processing time on Mkof
Table 1
The index set of the learning effects. m Learning indices 5 0.152 0.234 0.322 0.415 0.515 7 0.152 0.218 0.269 0.322 0.377 0.434 0.515 10 0.152 0.188 0.225 0.263 0.302 0.342 0.383 0.426 0.469 0.515 15 0.152 0.175 0.199 0.222 0.247 0.271 0.296 0.322 0.348 0.374 0.401 0.429 0.457 0.485 0.515 Ran Inc Dec SL WL 0 100 200 300 400 500 600 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 α Me a n num be r of node s
Fig. 1. The number of nodes for the branch-and-bound algorithm under differenta
(n = 10). Ran Inc Dec SL WL 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Me a n RPD α
Fig. 2. The relative deviation percentage of the learning patterns for the optimal objective value under differenta(n = 10).
Table 2
The performance of the branch-and-bound algorithm.
n m Pattern a= 0.25 a= 0.50 a= 0.75
Number of nodes Cpu times Number of nodes Cpu times m Cpu times
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
12 5 Ran 1550.43 9213 1.84 8.34 1145.36 8605 1.55 8.63 901.69 6087 1.36 7.14 Inc 3992.39 34418 2.66 19.56 2548.42 14103 1.98 10.08 1893.09 10818 1.65 7.86 Dec 429.49 3109 0.40 2.09 347.98 3010 0.37 1.98 297.70 2682 0.34 1.78 SL 1561.97 10346 1.01 4.20 1170.21 10062 0.88 4.02 969.24 7022 0.79 3.30 WL 1045.37 11976 0.86 4.97 755.21 7559 0.73 4.58 612.87 6531 0.66 4.09 7 Ran 2328.46 19105 1.32 9.09 1624.33 22124 1.08 10.16 1049.45 12119 0.88 5.38 Inc 6377.07 54718 5.68 25.11 3816.40 25625 4.19 20.97 2713.35 18196 3.45 17.97 Dec 635.63 6673 0.89 6.38 486.78 5711 0.80 5.70 363.18 3977 0.71 5.39 SL 1920.45 10152 2.03 8.16 1278.22 6932 1.67 7.67 983.96 7241 1.45 7.92 WL 1277.90 10061 1.70 10.41 871.76 6145 1.38 7.09 709.46 5993 1.22 7.05 14 5 Ran 8397.72 97712 13.43 99.28 5727.28 56055 10.91 74.48 4845.95 46274 9.82 73.30 Inc 31883.81 418776 44.54 372.50 17452.31 169405 30.27 216.80 13144.88 97435 25.12 175.41 Dec 1769.26 12080 3.22 16.92 1374.09 9108 2.94 17.28 1179.42 10186 2.76 17.73 SL 10382.89 115578 12.64 55.16 6689.93 49969 10.17 44.33 5140.81 28730 8.80 43.48 WL 3992.79 38389 10.23 84.28 2948.51 28213 8.34 67.02 2563.70 25591 7.50 62.02 7 Ran 16271.02 142636 32.37 290.52 9917.81 122468 24.74 274.36 7584.29 99124 21.15 244.52 Inc 74506.94 1980840 102.24 981.52 32558.74 322750 67.46 399.08 20916.03 124587 52.36 355.20 Dec 3317.76 87609 7.91 148.41 2486.66 74205 6.95 139.94 1767.69 47392 5.95 108.64 SL 12229.56 86179 23.97 123.91 8137.72 42532 19.40 99.19 6565.91 35383 17.09 89.53 WL 8076.68 158484 22.38 389.20 5646.80 105127 17.57 280.86 4478.99 76297 15.16 224.95 16 5 Ran 105044.01 2408452 349.62 4578.30 55620.78 600087 246.60 2274.17 39622.63 426218 201.16 1909.00 Inc 201758.16 2061604 514.57 3209.59 120521.06 1270538 350.70 2457.27 93087.06 1411099 288.07 2618.19 Dec 7520.64 107243 24.58 200.00 5155.33 41868 21.69 156.70 4148.91 37436 19.70 140.19 SL 66831.54 626475 233.56 1304.53 43107.35 360252 186.24 931.81 35647.61 289344 166.05 888.14 WL 31332.85 957031 103.05 1421.47 20896.89 575854 78.99 992.58 16706.67 402084 68.83 781.89 7 Ran 116674.80 2455027 238.17 3462.94 69780.49 1029782 167.75 2050.94 48757.11 674012 201.16 1909.00 Inc 740132.88 5188924 2252.10 16791.42 383647.16 4065895 1369.95 13054.45 263149.47 3726265 1034.44 11907.66 Dec 18756.10 145112 83.65 545.95 13486.31 101862 73.19 504.02 11616.93 89494 67.83 524.33 SL 105188.86 1447154 185.16 1820.66 62497.38 842215 130.71 864.78 44370.55 531122 108.13 666.53 WL 55492.39 1245433 287.12 6346.81 36222.31 1007826 218.60 5101.81 29516.95 783194 188.49 4129.69
In order to underestimate the final term on the right side of Eq.
(8), a Hungarian method is applied and the matrix of which is formed as follows.
a
Pm i¼kþ1 piJsþ1ðs þ 1Þ aia
P m i¼kþ1 piJsþ1ðs þ 2Þ aia
P m i¼kþ1 piJsþ1ðn 1Þ ai P m i¼kþ1 piJsþ1ðnÞ aia
Pm i¼kþ1 piJsþ2ðs þ 1Þ aia
P m i¼kþ1 piJsþ2ðs þ 2Þ aia
P m i¼kþ1 piJsþ2ðn 1Þ ai P m i¼kþ1 piJsþ2ðnÞ ai .. . .. . . . . .. . .. .a
Pm i¼kþ1 piJnðs þ 1Þ aia
P m i¼kþ1 piJnðs þ 2Þ aia
P m i¼kþ1 piJnðn 1Þ ai P m i¼kþ1 piJnðnÞ ai 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5where piJsþl is the basic processing time on Miof Js+lin set U for
1 6 l 6 n s. In the matrix, the information of a given job only can be assigned to a position from position s + 1 to n. And then the information of all jobs are sum up as Hkwhich denotes the
opti-mal value of the proposed Hungarian method. Hence, the
underes-timated value of the objective function based on Mk for h is
presented as
a
Ps j¼1 Cm½jðhÞ þ ½a
ðn s 1Þ þ 1Ck½sðhÞ þP ns l¼1 ½a
ðn s lÞ þ 1ðs þ lÞakp kðsþlÞþ Hk ð9ÞIn order to make the lower bound stricter, the lower bound is evaluated as LB ¼
a
Ps j¼1 Cm½jðhÞ þ max 16k6mf½a
ðn s 1Þ þ 1Ck½sðhÞ þ P ns l¼1 ½a
ðn s lÞ þ 1ðs þ lÞakp kðsþlÞþ Hkg ð10Þ4.3. Summary of the branch-and-bound algorithm
In this paper, the depth-first search plus forward manner is implemented in the branching procedure, i.e. the nodes are spread from (1, , , ) to (1, 2, , , ), and finally to (n, n 1, , 1). The advantages of the depth-first search are storing less numbers of dynamic nodes and seeking the bottom node rapidly to derive a feasible solution. The summary is listed as follows.
Step 1: {Initialization} Implement the heuristic algorithms to obtain an initial incumbent solution.
Step 2: {Branching} Utilize Corollary 4.1 to eliminate nodes. Step 3: {Bounding} If the lower bound exceeds the incumbent solution, eliminate the node and its offspring.
5. Computational results
Several computational experiments are implemented in this section to assess the performance of the branch-and-bound and the heuristic algorithms. All the algorithms are coded in Fortran 90 and run on a personal computer with 2.89 GHz AMD Athlon™ II X4 635 Processor and 3.25 GB RAM with Windows XP. Since the machine-based learning effect is considered, the issue for allo-cating the learning effects to the machines is discussed in this paper. In general, the stronger learning effect is assigned to the ma-chine with heavier workload, like the concept of bottlenecks. In or-der to verify this viewpoint, five learning patterns unor-der the same learning indices set are proposed to discuss the influence on the proposed algorithms. The five learning patterns are denoted as Ran, Inc, Dec, SL and WL and expressed as follows.
Ran: The learning effects are randomly assigned to the machines.
Inc: The stronger learning effects are assigned to the rear machines.
Dec: The weaker learning effects are assigned to the rear machines
SL: The stronger learning effects are assigned to the machines with the larger value ofPn
j¼1pijfor i = 1, 2, . . ., m.
WL: The weaker learning effects are assigned to the machines with the larger value ofPnj¼1pijfor i = 1, 2, . . ., m.
For all computational experiments in this paper, the basic pro-cessing times are randomly generated from a discrete uniform
dis-tribution over the integer 1–100 (i.e., pij U(1, 100) for
j = 1, 2, . . ., n and i = 1, 2, . . ., m). The sets for the learning indices under different number of machines (i.e., aifor i = 1, 2, . . ., m) are
shown in Table 1. And then the simulated results output the
number of nodes, the execution time and the objective value to evaluate the performance of proposed algorithm under different experimental parameters.
The computational experiments consist of three parts. In the first part, the influence of different
a
on the branch-and-bound algorithm is evaluated. The number of jobs and machines is respec-tively set as 10 and 5 and then 100 replications are randomly gen-erated. Consequently, a total of 100 examples are generated to be tested. In addition, 51 differenta
are given with values from 0 to 1 with an increment as 0.02, i.e.,a
= 0, 0.02, 0.04, . . ., 1. The five learning patterns and 51 differenta
are considered in each exam-ple and the results are illustrated inFigs. 1 and 2.In Fig. 1, the mean numbers of nodes for all experimental conditions are illustrated. It is observed that the problem proposed in this paper is easier to solve as
a
increases with respect to the trend of the mean number of nodes. The reason is that the corollary and the lower bound are more efficient in the branch-and-boundTable 3
The comparison among five learning patterns for the optimal objective value. n m Patten RDPO
a= 0.25 a= 0.50 n
Mean Max Mean Max Mean Max
12 5 Ran 1.089 1.297 1.078 1.275 1.073 1.267 Inc 1.049 1.292 1.046 1.275 1.044 1.267 Dec 1.125 1.356 1.109 1.324 1.103 1.310 SL 1.007 1.046 1.006 1.041 1.005 1.039 WL 1.161 1.346 1.144 1.316 1.137 1.303 7 Ran 1.070 1.195 1.063 1.167 1.060 1.163 Inc 1.025 1.119 1.024 1.105 1.024 1.103 Dec 1.125 1.237 1.109 1.217 1.103 1.212 SL 1.007 1.064 1.006 1.057 1.005 1.057 WL 1.136 1.301 1.121 1.273 1.115 1.264 14 5 Ran 1.099 1.392 1.090 1.346 1.086 1.327 Inc 1.054 1.200 1.050 1.197 1.048 1.194 Dec 1.132 1.392 1.117 1.346 1.111 1.327 SL 1.006 1.067 1.005 1.046 1.005 1.037 WL 1.171 1.398 1.155 1.351 1.148 1.332 7 Ran 1.083 1.327 1.075 1.296 1.071 1.288 Inc 1.022 1.157 1.022 1.136 1.021 1.126 Dec 1.141 1.330 1.125 1.299 1.119 1.288 SL 1.006 1.056 1.005 1.039 1.004 1.038 WL 1.157 1.328 1.141 1.295 1.135 1.282 16 5 Ran 1.095 1.271 1.086 1.247 1.083 1.239 Inc 1.068 1.301 1.063 1.286 1.061 1.279 Dec 1.136 1.413 1.122 1.376 1.117 1.365 SL 1.006 1.060 1.005 1.054 1.005 1.052 WL 1.180 1.416 1.166 1.379 1.160 1.369 7 Ran 1.084 1.276 1.075 1.250 1.072 1.242 Inc 1.031 1.183 1.029 1.167 1.028 1.158 Dec 1.128 1.411 1.116 1.371 1.111 1.357 SL 1.008 1.068 1.007 1.061 1.007 1.061 WL 1.158 1.405 1.143 1.364 1.137 1.349
algorithm with larger
a
. Furthermore, Dec is the easiest among five learning patterns for seeking the optimal solution, and Inc is the worst. In addition, the optimal objective values for five learning patterns are discussed. Then the relative deviation percentage for five learning patterns is denoted as RDPOand its mean is illustratedinFig. 2. For each example, the RDPOis calculated as k kmin
kmin 100%
where k denotes the optimal objective value under one of five given learning patterns, and kminis the minimum among all k. It is
ob-served that the optimal objective value under SL is the lowest among five learning patterns, followed by Inc, Ran and Dec, and fi-nally WL. However, there is no determined priority among five learning patterns since all mean RDPOare larger than zero.
In the second part of the computational experiments, the num-bers of jobs are set as 12, 14 and 16, and numnum-bers of machines are set as 5 and 7. Furthermore, three
a
are given as 0.25, 0.50 and 0.75 and then 100 replications are randomly generated. Hence, a total of 1800 examples are generated to be tested in which the five learn-ing patterns are considered. Then the results are listed inTables 2– 6.The mean and maximum number of nodes, and the mean and maximum CPU times (in seconds) of the branch-and-bound algo-rithm are reported inTable 2. It reveals that the number of nodes and the execution times increase significantly as the number of jobs or machines increases since the problem proposed in this pa-per is NP-hard. The optimal solution is easier to be derived for the proposed problem with a larger
a
in terms of the number of nodesand CPU times. Furthermore, the problem under Dec is the easiest among the five learning patterns to be solved, and Inc is the worst. Moreover, the branch-and-bound algorithm can deal with the problems with up to 16 jobs within a reasonable amount of time. In order to discuss the priority over five learning patterns for obtaining lower optimal objective value, the mean and maximum RDPOare recorded for all computational conditions inTable 3.
As shown inTable 3, it reveals that the optimal objective value under SL is the lowest among five learning patterns, follows by Inc, Ran and Dec, and finally WL. It implies that assigning the stronger learning effect to the machine with the heavier workload might obtain a lower optimal objective value.
For the proposed heuristic algorithms, the mean and maximum error percentages under different
a
are reported inTables 4–6. The CPU times are not presented since all heuristic algorithms for each example are executed within a second. The error percentage of the given heuristic algorithm is calculated asV V V 100% where V and V⁄
respectively denotes the objective value yielded by the heuristic algorithm, and the optimal objective value derived by the branch-and-bound algorithm. In addition, min {NEH, NEH_W} denotes the better one of NEH and NEH_W for the given example, and min {FL, FL_W} as well denotes the better one of FL and FL_W.
As shown inTables 4–6, it is observed that all heuristic algo-rithms proposed in this paper are quite accurate since the error percentages are all less than 1%. For evaluating the influence on the performance of the heuristic algorithms, several two-way
anal-Table 4
The performance of the heuristic algorithms (a= 0.25). n m Patten Error percentages
NEH NEH_W min{NEH,NEH_W} FL FL_W min{FL,FLW}
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
12 5 Ran 0.051 0.132 0.059 0.134 0.045 0.110 0.011 0.067 0.012 0.067 0.008 0.067 Inc 0.033 0.097 0.037 0.105 0.029 0.097 0.010 0.056 0.012 0.054 0.007 0.045 Dec 0.062 0.144 0.065 0.201 0.052 0.144 0.010 0.043 0.009 0.060 0.006 0.030 SL 0.055 0.138 0.057 0.138 0.046 0.126 0.015 0.060 0.017 0.049 0.011 0.049 WL 0.045 0.134 0.054 0.155 0.039 0.134 0.008 0.066 0.009 0.068 0.006 0.066 7 Ran 0.046 0.141 0.052 0.150 0.040 0.097 0.015 0.054 0.014 0.058 0.010 0.054 Inc 0.042 0.102 0.043 0.113 0.035 0.085 0.015 0.064 0.015 0.060 0.011 0.058 Dec 0.057 0.152 0.056 0.131 0.045 0.102 0.010 0.059 0.011 0.072 0.006 0.037 SL 0.051 0.140 0.051 0.115 0.041 0.110 0.014 0.049 0.016 0.066 0.011 0.039 WL 0.048 0.122 0.049 0.112 0.041 0.112 0.011 0.043 0.011 0.041 0.007 0.037 14 5 Ran 0.050 0.110 0.058 0.136 0.045 0.095 0.012 0.044 0.011 0.065 0.008 0.040 Inc 0.034 0.093 0.039 0.108 0.030 0.093 0.010 0.044 0.011 0.052 0.007 0.035 Dec 0.069 0.147 0.075 0.146 0.060 0.121 0.011 0.046 0.011 0.056 0.007 0.043 SL 0.058 0.152 0.063 0.152 0.049 0.120 0.016 0.057 0.019 0.065 0.011 0.037 WL 0.050 0.139 0.058 0.136 0.045 0.114 0.008 0.041 0.008 0.049 0.005 0.033 7 Ran 0.051 0.101 0.054 0.178 0.042 0.101 0.014 0.049 0.014 0.084 0.010 0.037 Inc 0.041 0.101 0.047 0.097 0.035 0.082 0.016 0.056 0.018 0.070 0.012 0.046 Dec 0.063 0.154 0.065 0.169 0.053 0.132 0.012 0.044 0.012 0.062 0.008 0.037 SL 0.057 0.140 0.057 0.131 0.049 0.131 0.017 0.063 0.020 0.071 0.013 0.046 WL 0.047 0.097 0.054 0.116 0.043 0.093 0.010 0.046 0.012 0.094 0.007 0.045 16 5 Ran 0.061 0.182 0.069 0.164 0.054 0.122 0.013 0.069 0.014 0.069 0.010 0.069 Inc 0.041 0.147 0.043 0.155 0.035 0.079 0.011 0.049 0.011 0.061 0.008 0.044 Dec 0.079 0.177 0.084 0.169 0.069 0.167 0.011 0.039 0.011 0.045 0.008 0.035 SL 0.069 0.163 0.074 0.162 0.059 0.129 0.019 0.087 0.021 0.066 0.014 0.049 WL 0.055 0.177 0.062 0.161 0.049 0.104 0.007 0.029 0.009 0.042 0.005 0.024 7 Ran 0.058 0.147 0.061 0.133 0.049 0.127 0.014 0.060 0.016 0.061 0.011 0.060 Inc 0.040 0.109 0.049 0.103 0.037 0.089 0.017 0.069 0.016 0.062 0.012 0.062 Dec 0.069 0.159 0.068 0.139 0.057 0.134 0.014 0.052 0.014 0.041 0.010 0.041 SL 0.066 0.181 0.064 0.168 0.054 0.132 0.020 0.068 0.024 0.069 0.015 0.048 WL 0.055 0.127 0.062 0.150 0.049 0.108 0.012 0.046 0.012 0.061 0.009 0.046
ysis of variance (ANOVA) are performed to test three hypotheses at the 0.05 level of significance. The first two null hypotheses are assumed as that the mean error percentages of the given heuristic algorithm are all identical among the three
a
settings, and five learning patterns, respectively. The last null hypothesis is assumed as that there is no interaction betweena
and learning patterns. And then the results are reported inTable 7.As shown inTable 7, it is observed that
a
does not have a signif-icant effect on the accuracy for all heuristic algorithms except NEH. Then it is shown inTables 4–6that the mean error percentage of NEH descends asa
decreases, and the reason is that the NEH is ini-tially devoted to solving the makespan problem. Furthermore, it reveals that the learning pattern has a significant effect on the accuracy for all proposed heuristic algorithms. A close observation of Tables 4–6 shows that Inc is the most accurate under NEH, NEH_W and min{NEH,NEH_W}, and Dec is the least accurate. Mean-while, SL is the most accurate under FL, FL_W and min{FL,FL_W}, and WL is the least. In addition, there is no interaction betweena
and the learning pattern for all heuristic algorithms. Moreover, it is shown that min{NEH,NEH_W} is more accurate than NEH and NEH_W, and min{FL,FL_W} is more accurate than FL and FL_W. It implies that there is no priority between two methods for selecting jobs which are utilized in the proposed heuristic algorithms. Even-tually, min{FL,FL_W} is the most accurate among all heuristic algo-rithms, followed by FL and FL_W, min{NEH,NEH_W}, and finally NEH and NEH_W.In the last part of the computational experiments, the examples with large size of jobs are generated to perform the heuristic algo-rithms proposed in this paper. Let
a
be set as 0.50 since most of the proposed heuristic algorithms are not affected bya
for thestatisti-cal analysis inTable 7. Additionally, the numbers of jobs are set as 50 and 100, and numbers of machines are set as 10 and 15. Then 100 replications are randomly generated. A total of 400 examples are generated to be tested in which five learning patterns are considered in each example. Consequently, the relative deviation percentage for all heuristic algorithms is denoted as RDPH, and its
mean and maximum values are listed inTable 8. For each example, the RDPHis calculated as
l
l
minl
min 100%where
l
denotes the near-optimal objective value for given one of all heuristic algorithms, andl
minis the minimum among alll
.As shown inTable 8that FL and FL_W are both better than min{-NEH,NEH_W} in terms of the RDPH. It implies that the heuristic
algorithm proposed byFraminan and Leisten (2003)is more proper than the algorithm proposed byNawaz et al. (1983)to obtain the near-optimal solution for the problem proposed in this paper. Fi-nally, it is observed that min{FL,FL_W} is the most accurate of all proposed heuristic algorithms because of that the RDPHare all zero.
Therefore, min{FL,FL_W} is recommended to yield the near-optimal schedule for the problem proposed in this paper.
6. Conclusions
In this paper, an m-machine permutation flowshop scheduling problem with machine-based learning effects is studied to mini-mize the weighted sum of the total completion time and the make-span. The branch-and-bound algorithm incorporated with a
Table 5
The performance of the heuristic algorithms (a= 0.50). n m Patten Error percentages
NEH NEH_W Min{NEH,NEH_W} FL FL_W Min{FL,FLW}
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
12 5 Ran 0.056 0.153 0.063 0.146 0.049 0.128 0.010 0.065 0.012 0.064 0.008 0.064 Inc 0.035 0.090 0.041 0.118 0.031 0.090 0.011 0.047 0.011 0.049 0.008 0.043 Dec 0.066 0.173 0.069 0.167 0.056 0.167 0.009 0.051 0.009 0.049 0.006 0.041 SL 0.049 0.142 0.054 0.138 0.042 0.138 0.013 0.049 0.014 0.048 0.010 0.048 WL 0.049 0.145 0.055 0.154 0.043 0.128 0.006 0.033 0.008 0.068 0.004 0.025 7 Ran 0.050 0.137 0.051 0.153 0.041 0.106 0.015 0.060 0.013 0.063 0.010 0.046 Inc 0.042 0.106 0.042 0.111 0.035 0.085 0.014 0.053 0.013 0.051 0.010 0.049 Dec 0.063 0.181 0.057 0.136 0.048 0.118 0.010 0.050 0.010 0.050 0.007 0.050 SL 0.052 0.137 0.051 0.130 0.042 0.125 0.013 0.051 0.015 0.049 0.010 0.037 WL 0.050 0.137 0.052 0.134 0.042 0.120 0.012 0.069 0.012 0.074 0.009 0.060 14 5 Ran 0.056 0.124 0.062 0.161 0.048 0.104 0.013 0.056 0.012 0.056 0.008 0.047 Inc 0.038 0.093 0.043 0.137 0.033 0.086 0.012 0.045 0.011 0.045 0.009 0.045 Dec 0.079 0.169 0.085 0.186 0.068 0.156 0.011 0.044 0.011 0.064 0.007 0.028 SL 0.058 0.129 0.066 0.151 0.051 0.122 0.016 0.064 0.019 0.082 0.012 0.055 WL 0.056 0.139 0.063 0.161 0.048 0.105 0.008 0.044 0.008 0.065 0.005 0.037 7 Ran 0.051 0.117 0.055 0.115 0.044 0.115 0.012 0.043 0.014 0.054 0.010 0.037 Inc 0.041 0.116 0.046 0.107 0.036 0.090 0.016 0.063 0.016 0.063 0.012 0.063 Dec 0.068 0.173 0.067 0.173 0.056 0.161 0.010 0.054 0.012 0.051 0.008 0.043 SL 0.058 0.138 0.056 0.144 0.047 0.138 0.016 0.049 0.018 0.069 0.013 0.049 WL 0.051 0.111 0.055 0.116 0.045 0.095 0.011 0.073 0.011 0.067 0.008 0.067 16 5 Ran 0.067 0.203 0.072 0.185 0.058 0.135 0.013 0.063 0.014 0.073 0.010 0.060 Inc 0.047 0.139 0.047 0.161 0.039 0.093 0.012 0.072 0.013 0.051 0.009 0.051 Dec 0.088 0.196 0.090 0.167 0.076 0.167 0.013 0.059 0.011 0.068 0.008 0.047 SL 0.071 0.164 0.075 0.172 0.060 0.131 0.019 0.074 0.021 0.147 0.014 0.047 WL 0.061 0.196 0.068 0.161 0.054 0.117 0.008 0.038 0.008 0.041 0.005 0.025 7 Ran 0.061 0.150 0.065 0.148 0.053 0.144 0.014 0.061 0.014 0.048 0.010 0.046 Inc 0.044 0.095 0.048 0.098 0.038 0.088 0.016 0.057 0.014 0.062 0.011 0.057 Dec 0.074 0.157 0.072 0.179 0.062 0.157 0.013 0.048 0.013 0.053 0.008 0.038 SL 0.066 0.146 0.065 0.177 0.055 0.146 0.019 0.059 0.019 0.073 0.013 0.040 WL 0.061 0.143 0.069 0.151 0.056 0.133 0.012 0.058 0.013 0.052 0.008 0.039
Table 6
The performance of the heuristic algorithms (a= 0.75). n m Patten Error percentages
NEH NEH_W Min{NEH,NEH_W} FL FL_W Min{FL,FLW}
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
12 5 Ran 0.058 0.159 0.064 0.157 0.050 0.132 0.012 0.064 0.011 0.044 0.008 0.044 Inc 0.036 0.086 0.039 0.092 0.031 0.081 0.011 0.044 0.011 0.053 0.008 0.035 Dec 0.069 0.149 0.070 0.170 0.057 0.122 0.009 0.052 0.009 0.044 0.006 0.030 SL 0.056 0.179 0.057 0.144 0.047 0.144 0.013 0.046 0.016 0.055 0.010 0.041 WL 0.052 0.146 0.058 0.187 0.045 0.114 0.006 0.050 0.007 0.059 0.005 0.038 7 Ran 0.050 0.137 0.052 0.133 0.042 0.109 0.013 0.051 0.013 0.046 0.009 0.043 Inc 0.042 0.104 0.041 0.091 0.034 0.081 0.013 0.064 0.012 0.051 0.010 0.051 Dec 0.067 0.191 0.057 0.137 0.050 0.129 0.009 0.040 0.009 0.041 0.006 0.037 SL 0.054 0.143 0.052 0.132 0.042 0.130 0.014 0.060 0.015 0.045 0.010 0.043 WL 0.053 0.132 0.051 0.145 0.043 0.130 0.011 0.057 0.011 0.047 0.008 0.044 14 5 Ran 0.057 0.131 0.064 0.131 0.050 0.109 0.013 0.054 0.012 0.050 0.008 0.045 Inc 0.039 0.103 0.043 0.105 0.034 0.083 0.012 0.040 0.013 0.059 0.009 0.040 Dec 0.081 0.180 0.082 0.199 0.068 0.159 0.011 0.048 0.011 0.053 0.007 0.044 SL 0.060 0.137 0.066 0.154 0.050 0.127 0.017 0.084 0.018 0.057 0.012 0.043 WL 0.058 0.146 0.063 0.140 0.049 0.112 0.009 0.061 0.009 0.061 0.006 0.061 7 Ran 0.051 0.126 0.056 0.152 0.044 0.126 0.013 0.047 0.013 0.050 0.010 0.042 Inc 0.043 0.120 0.045 0.103 0.036 0.103 0.017 0.060 0.016 0.058 0.013 0.056 Dec 0.068 0.180 0.068 0.178 0.056 0.167 0.010 0.045 0.011 0.067 0.007 0.045 SL 0.058 0.126 0.057 0.151 0.047 0.125 0.016 0.053 0.019 0.065 0.012 0.053 WL 0.053 0.118 0.055 0.119 0.045 0.100 0.011 0.073 0.011 0.047 0.007 0.047 16 5 Ran 0.069 0.222 0.071 0.173 0.059 0.153 0.016 0.091 0.014 0.056 0.010 0.042 Inc 0.048 0.151 0.048 0.164 0.040 0.094 0.012 0.064 0.013 0.039 0.008 0.035 Dec 0.089 0.207 0.090 0.179 0.075 0.149 0.011 0.043 0.011 0.068 0.008 0.032 SL 0.072 0.169 0.074 0.172 0.060 0.144 0.020 0.076 0.022 0.160 0.014 0.076 WL 0.064 0.207 0.071 0.167 0.056 0.126 0.010 0.076 0.010 0.042 0.006 0.037 7 Ran 0.063 0.157 0.065 0.155 0.054 0.151 0.014 0.046 0.015 0.055 0.010 0.044 Inc 0.045 0.097 0.046 0.110 0.038 0.094 0.015 0.054 0.016 0.067 0.011 0.043 Dec 0.076 0.161 0.075 0.189 0.064 0.161 0.013 0.042 0.013 0.053 0.009 0.035 SL 0.067 0.159 0.065 0.182 0.056 0.159 0.018 0.074 0.020 0.074 0.014 0.074 WL 0.063 0.150 0.070 0.161 0.057 0.139 0.014 0.055 0.014 0.054 0.010 0.035 Table 7
Two-way ANOVA of the error percentages for all heuristic algorithms.
Heuristic algorithm Source DF SS MS F p-Value
NEH a 2 0.0004311 0.0002155 5.08 0.009 Learning patterns 4 0.0089166 0.0022292 52.50 0.000 Interaction 8 0.0001122 0.0000140 0.33 0.952 Error 75 0.0031847 0.0000425 Total 89 0.0126446 NEH_W a 2 0.0001460 0.0000730 1.21 0.304 Learning patterns 4 0.0735550 0.0018389 30.46 0.000 Interaction 8 0.0000621 0.0000078 0.13 0.998 Error 75 0.0045283 0.0000604 Total 89 0.0120920 Min{NEH,NEH_W} a 2 0.0001948 0.0000974 2.43 0.095 Learning patterns 4 0.0056230 0.0014058 35.04 0.000 Interaction 8 0.0000657 0.0000082 0.20 0.989 Error 75 0.0030085 0.0000401 Total 89 0.0088921 FL a 2 0.0000008 0.0000004 0.08 0.919 Learning patterns 4 0.0004772 0.0001193 25.31 0.000 Interaction 8 0.0000074 0.0000009 0.20 0.991 Error 75 0.0003535 0.0000047 Total 89 0.0008389 FL_W a 2 0.0000075 0.0000039 0.90 0.412 Learning patterns 4 0.0007590 0.0001898 43.70 0.000 Interaction 8 0.0000071 0.0000009 0.20 0.989 Error 75 0.0003257 0.0000043 Total 89 0.0010996 Min{FL,FL_W} a 2 0.0000002 0.0000001 0.03 0.970 Learning patterns 4 0.0003397 0.0000849 33.00 0.000 Interaction 8 0.0000030 0.0000004 0.14 0.997 Error 75 0.0001930 0.0000026 Total 89 0.0005358
dominance criterion and a lower bound is proposed to seek the optimal solution, and several heuristic algorithms are established to yield the near-optimal solutions. As shown in the computational results, the proposed problem can be dealt with up to 16 jobs with-in a reasonable amount of time. When the learnwith-ing pattern is set as Inc, or if
a
is smaller, the proposed problem is harder to search for the optimal solution by implementing the proposed branch-and-bound algorithm. Furthermore, the performances of all proposed heuristic algorithms are accurate while min{FL, FL_W} is recom-mended to obtain the near-optimal solution. Finally, the issue for allocating the learning effects to the machines is discussed in this paper, and it is verified that assigning the stronger learning effects to the machines with the heavier workload might obtain the better result for the problem proposed in this paper.References
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Table 8
The comparison of the heuristic algorithms for large job-sized problem (a= 0.50).
n m Pattern RDPH
NEH NEH_W Min{NEH,NEH_W} FL FL_W Min{FL,FLW}
Mean Max Mean Max Mean Max Mean Max Mean Max Mean Max
50 10 Ran 0.072 0.143 0.073 0.133 0.064 0.106 0.004 0.053 0.004 0.030 0.000 0.008 Inc 0.043 0.078 0.047 0.083 0.039 0.074 0.003 0.035 0.003 0.035 0.000 0.000 Dec 0.095 0.137 0.075 0.129 0.073 0.128 0.004 0.037 0.003 0.026 0.000 0.000 SL 0.073 0.135 0.071 0.117 0.064 0.117 0.004 0.027 0.007 0.047 0.000 0.000 WL 0.070 0.146 0.067 0.119 0.061 0.119 0.004 0.027 0.003 0.024 0.000 0.000 15 Ran 0.062 0.127 0.060 0.114 0.053 0.108 0.006 0.038 0.003 0.028 0.000 0.000 Inc 0.037 0.077 0.040 0.072 0.033 0.068 0.005 0.029 0.004 0.035 0.000 0.004 Dec 0.078 0.135 0.061 0.123 0.059 0.123 0.004 0.021 0.003 0.022 0.000 0.000 SL 0.065 0.107 0.059 0.107 0.054 0.106 0.005 0.029 0.004 0.024 0.000 0.000 WL 0.063 0.143 0.060 0.108 0.054 0.108 0.004 0.030 0.002 0.026 0.000 0.000 100 10 Ran 0.080 0.133 0.081 0.136 0.072 0.116 0.004 0.028 0.004 0.034 0.000 0.000 Inc 0.055 0.090 0.057 0.091 0.052 0.079 0.003 0.023 0.002 0.024 0.000 0.000 Dec 0.102 0.155 0.079 0.129 0.078 0.129 0.004 0.019 0.002 0.013 0.000 0.000 SL 0.089 0.133 0.087 0.132 0.082 0.132 0.003 0.018 0.004 0.029 0.000 0.000 WL 0.075 0.137 0.077 0.130 0.069 0.117 0.004 0.020 0.002 0.024 0.000 0.000 15 Ran 0.073 0.132 0.068 0.109 0.065 0.103 0.005 0.047 0.002 0.021 0.000 0.000 Inc 0.048 0.079 0.052 0.083 0.045 0.071 0.004 0.029 0.002 0.018 0.000 0.000 Dec 0.093 0.135 0.067 0.113 0.067 0.113 0.004 0.021 0.002 0.015 0.000 0.000 SL 0.073 0.117 0.068 0.105 0.063 0.102 0.004 0.022 0.003 0.022 0.000 0.000 WL 0.071 0.120 0.068 0.121 0.064 0.105 0.005 0.029 0.002 0.025 0.000 0.000
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