A bicriteria flowshop scheduling with a learning effect

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Applied Mathematical Modelling 32 (2008) 1719–1733

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A bicriteria flowshop scheduling with a learning effect

Tamer Eren a,*, Ertan Guner b,1

a Kırıkkale University, Faculty of Engineering, Department of Industrial Engineering, 71450 Kırıkkale, Turkeyb Gazi University, Faculty of Engineering and Architecture, Department of Industrial Engineering, 06570 Maltepe, Ankara, Turkey

Received 1 October 2005; received in revised form 1 February 2007; accepted 4 June 2007Available online 16 June 2007

Abstract

In many situations, a worker’s ability improves as a result of repeating the same or similar tasks; this phenomenon isknown as the learning effect. In this paper the learning effect is considered in a two-machine flowshop. The objective is tofind a sequence that minimizes a weighted sum of total completion time and makespan. Total completion time and make-span are widely used performance measures in scheduling literature. To solve this scheduling problem, an integer program-ming model with n2 + 6n variables and 7n constraints where n is the number of jobs is formulated. Because of the lengthycomputing time and high computing complexity of the integer programming model, the problem with up to 30 jobs can besolved. A heuristic algorithm and a tabu search based heuristic algorithm are presented to solve large size problems. Exper-imental results show that the proposed heuristic methods can solve this problem with up to 300 jobs rapidly. According tothe best of our knowledge, no work exists on the bicriteria flowshop with a learning effect.� 2007 Elsevier Inc. All rights reserved.

Keywords: Flowshop scheduling; Bicriteria scheduling; Learning effect; Makespan; Completion time; Integer programming model;Heuristic methods

1. Introduction

In many realistic scheduling setting, the production facility (a machine, a worker) improves continuouslywith time. As a result, the processing time of a given job is shorter if it is scheduled later, rather than earlierin the sequence. This phenomenon is known as the ‘‘learning effect’’ in the literature [1]. The impact of learningon productivity in manufacturing was first observed by Wright [2] in the aircraft industry and was subse-quently discovered in many industries in both manufacturing and service sectors. However, the effect of learn-ing has not been investigated in the context of scheduling problems until recently.

Biskup [1] was the first to investigate the learning effect in a scheduling setting. Biskup [1] studied single-machine problems and considered the objective of (i) minimizing completion time, and (ii) minimizing the

0307-904X/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2007.06.009

* Corresponding author. Tel.: +90 318 3573576 1008; fax: +90 318 3572459.E-mail addresses: [email protected] (T. Eren), [email protected] (E. Guner).

1 Tel.: +90 312 2317400 2855; fax: +90 312 2308434.

mailto:[email protected]

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1720 T. Eren, E. Guner / Applied Mathematical Modelling 32 (2008) 1719–1733

weighted sum of completion time deviations from a common due date and the sum of job completion times.Later, Mosheiv [3] showed that the single-machine makespan minimization problem remained polynomialsolvable when the learning effect was taken into consideration. Mosheiv [4] considered completion time min-imization on identical machines in parallel and showed that this problem has a polynomial time solution.Mosheiv and Sidney [5] extended the setting to account for the possibility that learning in the production pro-cess is faster for some jobs than for others. They showed that the makespan and total completion time min-imization problems on a single machine, a due date assignment problem, and total completion timeminimization on unrelated parallel machines remain polynomial solvable.

Lee et al. [6] analyzed the single-machine bicriteria problem with a learning effect. The objective was to finda sequence that minimizes a linear combination of the total completion time and the maximum tardiness. Erenand Guner [7] worked on a different bicriteria single-machine scheduling problem, namely jointly minimizingthe sum of completion times and the total tadiness, 1=LE=a

PC þ b

PT . The authors developed a mathemat-

ical programming and heuristics to solve this problem. This apart from these articles in the single machinethere are other studies with the learning effect: Eren and Guner [8], Cheng and Wang [9]; Lee [10], Biskupand Simons [11], and Mosheiov and Sidney [12].

Eren and Guner [13] analyzed a scheduling problem with job-dependent learning effect in a two-machineflowshop with makespan by performance measure. They showed that Johnson algorithm cannot guaranteethe best results in the situation with job-dependent learning effect. They also proposed a mixed integer pro-gramming model for this problem. In addition, studies about a learning effect in a two-machine flowshopscheduling with total completion times has been considered by Lee and Wu [14] and Eren and Guner [15].

This paper addresses a bicriteria two-machine flowshop scheduling problem with a learning effect. The gen-eral two-machine flowshop scheduling problem can be described in the following way. Each job in a set of n

job is processed first on the first machine and then on the second machine. Job can only be processed one jobat time. Jobs are processed without interruption or preemption. Both machines are available at all times andall jobs are ready at time zero. The problem is to find the optimal schedule that minimizes the weighted sum ofthe total completion time and makespan with a learning effect.

The considered criteria in this paper are makespan and the total completion time. Johnson’s [16] rule ofminimizing makespan without the learning effect in the two-machine flowshop scheduling environment withina polynomial time, presented in 1954 has received considerable interest in related fields.

In the literature related to the completion time on the two-machine flowshop scheduling problem withoutthe learning effect, Kohler and Steiglitz [17] suggested different heuristic algorithms to obtain approximatesolutions. Later, Gonzalez and Sahni [18] showed that the two-machine flowshop problem is NP-hard whenthe objective is to minimize the total completion time. Thereafter, researchers concentrated on develop-ing branch-and-bound and/or heuristic approaches for the flowshop problem with the completion time[19–47].

Only one criterion ðCmax;P

C;CÞ was considered in the above mentioned studies. However, a decisionmaker usually needs to consider two or more criteria at the same time [48,49]. An optimal schedule undera specific criterion may be poor under another criterion. Therefore, a multicriteria two-machine flowshopscheduling problem is important. In multicriteria two-machine flowshop scheduling, the objective of minimiz-ing weighted sum of total completion time and makespan ðn=2=a

PC þ bCmaxÞ and the weighted sum of

makespan and maximum tardiness n/2/aCmax + bTmax problems were considered by a number of researchers[50–83].

The most widely used criteria in multicriteria scheduling literature were makespan and total completiontime. Therefore, in this study we address the two-machine flowshop scheduling problem with these two criteriaby taking into account a learning effect.

To increase system performance of a two-machine flowshop scheduling, the lowering of both through-put time and work in process (WIP) as much as possible is important. The scheduling criterion of totalcompletion time minimization can effectively reduce work in process [84]. This paper attempts to minimizethe weighted sum of these two scheduling criteria in a two-machine flowshop scheduling with a learningeffect.

The rest of the study is organized as follows: in Section 2, the problem and the proposed integerprogramming model are described. The heuristics that are used to solve large size problems are presented

T. Eren, E. Guner / Applied Mathematical Modelling 32 (2008) 1719–1733 1721

in the Section 3. The experimental results are given in the Section 4. Finally, Section 5 provides conclusionsand evaluations of the study and suggests some directions for future researches.

2. Problem description

Let n = {J1, J2, . . . , Jn} be the set of jobs to be scheduled and M = {M1, M2} be the two machines. Thenormal processing times for job j on M1 and M2 are denoted pj1 and pj2, respectively. Furthermore, we assumethat both machines have the same learning effect. That is if pj1r and pj2r are the actual processing times of job j

scheduled in position r in a sequence, then pj1r = pj1ra and pj2r = pj2ra (where a 6 0 is the learning effect forboth machines, 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 makespan on two-machine flowshop (the problem isdenoted as F 2=LE=a

PC þ bCmax, where LE is learning effect). This problem is NP-hard, since Gonzalez and

Sahni [18] showed that minimization of the total completion time in a two-machine flowshop that is the sim-pler variant of our problem is NP-hard even without a learning effect.

Assumptions made in this paper are:

1. Setup time is known and is included in the processing time.2. Machine preemption is not allowed, 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.

In order to solve the two-machine flowshop scheduling problem considered in this paper, an integer pro-gramming model is given. The base of this model is structured on Chou and Lee’s, [63] integer programmingmodel.

In this proposed model, there are n2 + 6n variables and 7n constraints, where n denotes the number ofjobs. The parameters and variables in the model are described below and then the proposed model isgiven.

2.1. Parameters

j the number of jobs j = 1, 2, . . . , n,i the number of machines i = 1, 2,a the weight for total completion time a P 0,b the weight for makespan b P 0, a + b = 1,pji the processing time of job j on the ith machine i = 1, 2, j = 1, 2, . . . , n,ra learning effect on rth position, r = 1, 2, . . . , n,pjir the processing time of job j, rth position on the ith machine.

pjir ¼ pjira i ¼ 1; 2; j; r ¼ 1; 2; . . . ; n:

2.2. Decision variables

Zjr if job j is scheduled at the rth position to be processed, Zjr = 1, otherwise Zjr = 0, j, r = 1, 2, . . . , n.Xr the idle time on the second machine between the starting of the rth position job and the completion of

the (r � 1)th position job, r = 1, 2, . . . , n.Yr the time between its completion at the first machine and its begin processing at the second machine for

the rth position job, r = 1, 2, . . . , n.Sr the starting time for the rth position job at the first machine, r = 1, 2, . . . , n.


2.3. Auxiliary variables

Ar the rth position learning effect jobs processing time at the first machine r = 1, 2, . . . , n.Br the rth position learning effect jobs processing time at the second machine r = 1, 2, . . . , n.Cr the completion time of the rth ranked job at the second machine (M2) r = 1, 2, . . . , n.Cmax makespan.

2.4. Proposed integer programming model

Objective function:

MinaXn

r¼1

Cr þ bCmax:

Constraints:

Ar ¼ raXn

j¼1

Zjrpj1; r ¼ 1; 2; . . . ; n; ð1Þ

Br ¼ raXn

j¼1

Zjrpj2; r ¼ 1; 2; . . . ; n; ð2Þ

Cmax ¼Xn

r¼1

X r þXn

r¼1

Br; ð3Þ

Xn

j¼1

Zjr ¼ 1; r ¼ 1; 2; . . . ; n; ð4Þ

Xn

r¼1

Zjr ¼ 1; j ¼ 1; 2; . . . ; n; ð5Þ

Sr P Sr�1 þ Ar�1; r ¼ 1; 2; . . . ; n; ð6ÞX r ¼ Sr þ Ar þ Y r � Cr�1; r ¼ 1; 2; . . . ; n; ð7ÞZjr: 0� 1;

S0 ¼ 0 and C0 ¼ 0:

All variables are positive and integer.In Eqs. (1)–(3), constraint (4) specifies that only one job be scheduled at the rth job priority. Constraint (5)

defines that each job be scheduled only once. Constraint (6) represents that the beginning processing time ofthe rth ranked job be greater than or equal to the previous jobs completion time at the first machine. Con-straints (7) indicates that the idle time on the second machine to process the rth ranked job (Xr) equals thestarting time for the rth ranked job on the first machine (Sr) plus its processing time on the first machine(Ar), plus the time between its completion on the first machine and begin processing time on the secondmachine (Yr) minus the completion time for the (r � 1)th ranked job at the second machine (Cr�1). All vari-ables should be greater than or equal to zero and Zjr is a binary integer.

3. Heuristic methods

In this section, firstly the heuristic methods used to solve the large size problems are presented and thentheir adaptations on the problem are explained. These heuristics are a random search method, the modifiedNEH algorithm [85] and a tabu search-based heuristic method. These methods will be explained in the follow-ing sections.


3.1. Random search

Random search is a method that selects a specific number of solution points (a sample size) randomly fromthe solution space. Random search evaluates the selected points (sequences) with respect to objective functionand identifies the best sequence in the sample. The best sequence is stored and this process is repeated. Whenthe new best sequence in the later iterations is better than the previous one, the previous sequence is updatedwith the new sequence. The search continues until meeting a stopping criterion. There are two parameters inthe random search method: the first is sample size and the other is stopping criteria.

In our experimental study, while the sample size is taken (n � 1), no improvement for n repetitions for theproblem is considered as a stopping criterion.

3.2. The modified NEH algorithm

The NEH algorithm [85] is developed for minimization of makespan for flowshop scheduling (Fm//Cmax).In this study, NEH algorithm is modified for the objective function which consist of the weighted sum of totalcompletion time and makespan. The steps of the modified NEH algorithm are given as follows:

Step 1. Arrange the jobs in descending order of the sum of processing times.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 makespan as if there are only two jobs. Setthe 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 makespan. Update the selectedpartial solution as the new current solution.

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

3.3. Tabu search method

Tabu search has been used widely in combinatorial optimization. The basic idea is to slightly alter a known(current) solution in a certain manner (called neighborhood structure) and take the best alteration as the newcurrent solution. Such altered solutions are called neighbors of the current solution. An operation that yields aneighbor is called a move. To avoid being trapped at local optima, the best neighbor that is worse than thecurrent solution is allowed to become the new current solution. To avoid cycling, certain moves are markedas tabu. A tabu move may be allowed if an aspiration criterion is satisfied [86]. This procedure continues untila criterion is met.

3.4. Parameters of tabu search

In this paper we use tabu search parameters as follows:

3.4.1. Initial solution

Initial solution is set according to the result of modified NEH algorithm.

3.4.2. Neighborhood search strategy

The neighborhood search strategy is set according to API (adjacent pairwise interchange) mechanism.

3.4.3. Tabu list

The length of tabu list is ð2 ffiffiffinp Þ.

3.4.4. Aspiration criterionA tabu move is accepted if it produces a solution better than the best obtained so far.

Table 1Tabu search parameters

Parameters Values

Initial solution NEHTabu list length

ffiffiffinp

Neighborhood search strategy APIStopping criterion No improvement for n repetitions

Fig. 1. Flow chart for the tabu search-based heuristic method.


3.4.5. Stopping criterion

A stopping criterion is taken as n repetitions no improvement.These parameters are shown in Table 1. Flowchart of the used tabu search method is shown in Fig. 1.

4. Experimental results

In this study, all experimental tests are conducted on a personal computer with Pentium IV/2 512 Ram. Theexact solutions for five different job sizes as n = 10, 15, 20, 25 and 30 are found. Ten replications are made foreach problem size. Experimental results are conducted at two stages:

Table 2Experimental set for small size problems

Parameters Values

Weights (a, b) (0.25, 0.75); (0.50, 0.50); (0.75, 0.25)Number of job 10, 15, 20, 25, 30Learning effect alternatives (%) 70, 80, 90Number of replications for each problem 10Total problem 3 · 5 · 3 · 10 = 450

0

1000

2000

3000

4000

5000

10 15 20 25 30

number of jobs (n)

CPU

tim

es (

s)

( )=(0.25,0.75) ( . )=(0.50, 0.5 0) ( . )=(0.75, 0.25)

0

1000

2000

3000

4000

5000

10 15 20 25 30

number of jobs (n)

CPU

tim

es (

s)

( . )=(0.25, 0.75) ( . )=(0.50, 0.50) ( . )=(0.75, 0.25)

0

1000

2000

3000

4000

5000

10 15 20 25 30

number of jobs (n)

CPU

tim

es (

s)

( . )=(0.25, 0.75) ( . )=(0.50, 0.50) ( . )=(0.75, 0.25)

α β α β α β

α β α β α β

α β α β α β

a

b

c

.

Fig. 2. Average CPU times (s) versus number of jobs: (a) 70%, (b) 80%, and (c) 90%.



1. Finding exact results for small size problems with integer model and solving the small size problems withheuristic methods and comparing the solutions with the exact results.

2. Solving the large size problems by the heuristic methods.

4.1. Exact solution results and heuristics errors

The integer programming model is used to find the optimal solutions of the considered problem usingHyper LINDO/PC 6.01 software package [87]. Processing times on machines M1 and M2 are generated from

Table 3Average CPU times (s) of heuristics for small size problems

(a, b) n LE Average CPU times (s)

Optimal NEH Tabu Random

(0.25, 0.75) 10 0.7 0.30 0.0093 0.0204 0.00890.8 0.20 0.0094 0.0187 0.01260.9 0.50 0.0061 0.0200 0.0140

15 0.7 223.71 0.0124 0.0375 0.01840.8 239.60 0.0187 0.0471 0.02070.9 233.89 0.0172 0.0531 0.0220

20 0.7 593.63 0.0265 0.0859 0.04410.8 570.25 0.0234 0.0889 0.04880.9 564.22 0.0234 0.0953 0.0648

25 0.7 1595.76 0.0348 0.1699 0.09230.8 1536.93 0.0341 0.1605 0.09860.9 1650.50 0.0304 0.1644 0.1296

30 0.7 4854.65 0.0509 0.3253 0.15430.8 4647.64 0.0503 0.3349 0.16720.9 4600.58 0.0487 0.3253 0.2323

(0.50, 0.50) 10 0.7 0.30 0.0123 0.0312 0.02190.8 0.10 0.0138 0.0326 0.02070.9 0.30 0.0141 0.0314 0.0153

15 0.7 164.17 0.0265 0.0786 0.05190.8 139.89 0.0217 0.0798 0.04180.9 163.20 0.0250 0.0859 0.0368

20 0.7 365.71 0.0452 0.1624 0.12420.8 297.00 0.0436 0.1626 0.12060.9 351.00 0.0408 0.1501 0.0827

25 0.7 1159.49 0.0330 0.1645 0.12280.8 1062.82 0.0315 0.1652 0.13120.9 1190.91 0.0361 0.1690 0.1281

30 0.7 3443.04 0.0495 0.3259 0.21950.8 3177.63 0.0473 0.3268 0.14700.9 3133.78 0.0523 0.3306 0.2628

(0.75, 0.25) 10 0.7 0.10 0.0127 0.0328 0.01870.8 0.10 0.0125 0.0392 0.02420.9 0.20 0.0124 0.0360 0.0158

15 0.7 117.17 0.0280 0.0861 0.04870.8 200.01 0.0311 0.0939 0.05180.9 176.67 0.0296 0.0997 0.0487

20 0.7 419.56 0.0514 0.1781 0.10830.8 492.36 0.0453 0.2140 0.15230.9 417.87 0.0500 0.1874 0.1199

25 0.7 1501.14 0.0346 0.1632 0.07830.8 1690.69 0.0358 0.1691 0.08700.9 1524.24 0.0385 0.1603 0.1218

30 0.7 4920.94 0.0486 0.3314 0.26430.8 4655.19 0.0516 0.3381 0.22580.9 4802.02 0.0454 0.3351 0.1947

0.00

0.02

0.04

0.06

0.08

0.10

10 15 20 25 30

number of jobs (n)

erro

r

Random

NEH

Tabu

0.00

0.02

0.04

0.06

0.08

0.10

10 15 20 25 30

number of jobs (n)

erro

r

Random

NEH

Tabu

0.00

0.02

0.04

0.06

0.08

0.10

10 15 20 25 30

number of jobs (n)

erro

r

Random

NEH

Tabu

a

c

b

Fig. 3. Average percentage error versus number of jobs for a = 0.50 and b = 0.50. (a) 70%, (b) 80%, and (c) 90%.


a uniform distribution in the ranges [1, 100]. Three different weight factors (for a and b) and three differentlearning rates are used and ten problems are generated for each problem size. The parameters of small sizeproblems in the experimental set are given in Table 2. As can be seen from Table 2, 450 small size problemsare solved.


The average CPU times (in s) of integer programming model solutions based on the number of jobs for the70%, 80% and 90% learning rates are given in Fig. 2 and Table 3. As can be seen in Fig. 2, the solution times ofproblems increase exponential as the number of jobs increase and the considered bicriteria problem can besolved optimally up to 30 jobs.

Since the optimal solutions are obtained for the problems up 30 jobs, we employ the heuristic methods men-tioned in Section 3 to find the optimal/near optimal solutions for large size problems in a reasonable time.These three heuristic methods are the random search method, the modified NEH algorithm and the tabusearch-based heuristic method. Heuristic methods are coded on C++ Builder 5.

The effectiveness of heuristics for small size problems is evaluated in terms of error which is calculated inEq. (8)

TableAverag

(a, b)

(0.25,

(0.50,

(0.75,

TableExperi

Param

WeighNumbLearniNumbTotal

Error ¼ Heuristic solution�Optimal solution

Optimal solution: ð8Þ

The results of the heuristic methods were compared with the results of the optimal solutions, and the errors ofthese heuristics are given in Fig. 3 (for a = 0.50 and b = 0.50) and in Table 4 (for all the three a and b com-binations). As can be seen Fig. 3 and Table 4, the tabu search gives considerably better results in terms of theaverage error and it is quite effective for all 450 small size problems.

4.2. The large size problem results

The results of the large size problems (40 6 n 6 300) were also computed using heuristic methods, in orderto investigate their performances. The experimental set of the large size problems is given in Table 5. As can be

4e percentage error of heuristics for small size problems

n Learning effect (LE)

70% 80% 90%

Random NEH Tabu Random NEH Tabu Random NEH Tabu

0.75) 10 0.035 0.003 0.000 0.037 0.003 0.000 0.038 0.003 0.00115 0.038 0.003 0.000 0.049 0.003 0.000 0.056 0.004 0.00120 0.040 0.003 0.000 0.040 0.003 0.001 0.048 0.004 0.00125 0.043 0.003 0.000 0.049 0.004 0.001 0.050 0.004 0.00130 0.046 0.003 0.000 0.053 0.004 0.000 0.057 0.005 0.001

0.50) 10 0.036 0.003 0.000 0.039 0.003 0.000 0.037 0.003 0.00115 0.038 0.003 0.000 0.050 0.003 0.001 0.058 0.004 0.00120 0.037 0.003 0.000 0.042 0.004 0.000 0.043 0.003 0.00025 0.039 0.003 0.000 0.048 0.004 0.001 0.048 0.004 0.00130 0.046 0.003 0.000 0.050 0.004 0.000 0.062 0.005 0.001

0.25) 10 0.037 0.003 0.000 0.040 0.003 0.000 0.040 0.003 0.00015 0.037 0.003 0.000 0.045 0.003 0.001 0.054 0.004 0.00120 0.040 0.003 0.000 0.046 0.004 0.000 0.043 0.003 0.00025 0.037 0.003 0.000 0.047 0.004 0.001 0.051 0.004 0.00130 0.048 0.003 0.000 0.055 0.004 0.000 0.068 0.005 0.001

5mental set for large size problems

eters Values

ts (a, b) (0.25, 0.75); (0.50, 0.50); (0.75, 0.25)er of job 40, 50, 100, 200, 300ng effect alternatives (%) 70, 80, 90er of replications for each problem 30problem 3 · 5 · 3 · 30 = 1350

Table 6Average CPU times (s) of heuristics for large size problems

CPU times (s)

Weight (a, b) n LE NEH Tabu Random

(0.25, 0.75) 40 0.7 0.2919 0.4337 0.35650.8 0.2070 0.2865 0.30320.9 0.2618 0.3769 0.4122

50 0.7 0.3882 0.5294 0.47670.8 0.2813 0.3144 0.31950.9 0.3540 0.4588 0.4404

100 0.7 0.9586 1.3977 1.34280.8 0.7330 1.0498 0.98550.9 0.9479 1.1825 1.0076

200 0.7 2.3419 3.5902 4.46570.8 1.7485 2.4882 2.18460.9 2.5753 2.9907 2.8732

300 0.7 6.1936 8.8761 9.18640.8 4.6205 7.0294 5.76670.9 6.9328 8.2345 8.8456

(0.50, 0.50) 40 0.7 0.2214 0.2920 0.31060.8 0.3022 0.3625 0.32090.9 0.2012 0.3124 0.2508

50 0.7 0.2880 0.3770 0.32530.8 0.4028 0.5425 0.44380.9 0.2644 0.3652 0.4145

100 0.7 0.8332 1.1313 1.34870.8 1.1149 1.2746 1.46670.9 0.7389 1.0921 1.2347

200 0.7 2.2648 2.9347 2.90550.8 2.9876 3.6912 4.16140.9 1.8685 2.4914 2.9602

300 0.7 5.5434 8.7746 9.11210.8 7.6710 11.7963 11.85520.9 5.1338 7.6182 6.6050

(0.75, 0.25) 40 0.7 0.2457 0.3686 0.29600.8 0.2172 0.2653 0.32920.9 0.2929 0.4516 0.3739

50 0.7 0.3287 0.4162 0.33530.8 0.2938 0.4634 0.43590.9 0.4070 0.5921 0.5676

100 0.7 0.8672 1.3744 1.70180.8 0.7104 1.0588 0.91100.9 1.0583 1.2901 1.3310

200 0.7 2.1277 3.2368 3.54470.8 1.8403 2.9207 2.67050.9 2.9908 4.3069 5.4380

300 0.7 5.3908 7.4071 7.53880.8 4.3581 6.3984 6.04150.9 7.7397 8.9204 8.7050


seen in Table 5, 1350 problems are solved for five different problem sizes, three different weight factors (for aand b) and three different learning rates and thirty replications are made for each problem size. The averageCPU times (in s) of heuristics solutions based on the number of jobs for the 70%, 80% and 90% learning ratesare given in Table 6.

Since the optimal results of these problems are not known, the results of the tabu and the modified NEHwere compared with the results of the random search. In this comparison, an improvement rate is formulatedin Eqs. (9) and (10), is used as a performance measure:

Table 7Heuristic improvement for large size problems

Learning effect (%) n Modified NEH improvement Tabu improvement

Mean Max Min Std Mean Max Min Std

70 40 0.1581 0.2488 0.0504 0.0738 0.3349 0.4981 0.1165 0.117450 0.2470 0.3115 0.1031 0.0655 0.2787 0.5575 0.1388 0.1493

100 0.2101 0.3477 0.1014 0.0768 0.3966 0.6357 0.1834 0.1770200 0.2288 0.3584 0.1461 0.0815 0.4089 0.6315 0.2065 0.1407300 0.2899 0.4481 0.2042 0.0853 0.5057 0.7719 0.2138 0.1900

80 40 0.1780 0.2483 0.0649 0.0659 0.2930 0.4849 0.1331 0.114850 0.1614 0.2610 0.0727 0.0684 0.3181 0.5574 0.1799 0.1408

100 0.2634 0.3893 0.1374 0.0973 0.3993 0.6187 0.1725 0.1312200 0.3089 0.4549 0.1181 0.1192 0.3416 0.5737 0.1770 0.1277300 0.3509 0.5136 0.1416 0.1199 0.4648 0.7520 0.2905 0.1643

90 40 0.1461 0.2466 0.0518 0.0693 0.2948 0.4775 0.1048 0.144650 0.2282 0.2993 0.1031 0.0711 0.3321 0.4613 0.1937 0.0872

100 0.2542 0.3875 0.1332 0.0829 0.3905 0.6237 0.1586 0.1725200 0.3065 0.4447 0.1487 0.1081 0.4281 0.6544 0.1823 0.1699300 0.2682 0.4428 0.1597 0.1026 0.4183 0.6111 0.2792 0.1069


Improvement rate of NEH ¼ Random solution�NEH solution

NEH solutionð9Þ

and

Improvement rate of tabu ¼ Random solution� Tabu solution

Tabu solution: ð10Þ

Table 7 presents the modified NEH and tabu errors compared with the error average of the random search. Itcan be seen in Table 7 that while the average improvement rate of the modified NEH was between 15–28%;16–35% and 14–30% for 70%, 80% and 90% learning rates, respectively, the average improvement rate of tabuwas between 27–50%; 29–46% and 29–42% for 70%, 80% and 90% learning rates, respectively. Based on theseresults, tabu search outperforms the modified NEH and the random search with respect to the average errorfor large size problems.

5. Conclusions

Total completion time and makespan are widely used performance criteria in the scheduling literature. Inthis paper, the bicriteria flowshop scheduling problem with the learning effect was considered. An integermodel that finds the exact solution of the problem with up to 30 jobs was proposed. Also, NEH algorithmwas modified to the problem and a tabu search-based heuristic and a random search is developed. Whenthe solution results of the heuristics were compared with optimal results, it was seen that heuristics solutionresults were fairly good. Also, problems with up to 300 jobs were solved by these heuristics and the perfor-mances of the modified NEH and tabu were compared with the random search. Other performance criteriawith a learning effect in a flowshop can be also considered for future studies.

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A bicriteria flowshop scheduling with a learning effect