Flowshop scheduling with a general exponential learning effect

Computers & Operations Research 43 (2014) 292–308

Contents lists available at ScienceDirect

Computers & Operations Research journal homepage: www.elsevier.com/locate/caor

Flowshop scheduling with a general exponential learning effect Ji-Bo Wang a, Jian-Jun Wang b,n a b

School of Science, Shenyang Aerospace University, Shenyang 110136, People's Republic of China Faculty of Management and Economics, Dalian University of Technology, Dalian 116024, People's Republic of China

art ic l e i nf o

a b s t r a c t

Available online 20 October 2013

This paper investigates flowshop scheduling problems with a general exponential learning effect, i.e., the actual processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The objective is to minimize the makespan, the total (weighted) completion time, the total weighted discounted completion time, and the sum of the quadratic job completion times, respectively. Several simple heuristic algorithms are proposed in this paper by using the optimal schedules for the corresponding single machine problems. The tight worst-case bound of these heuristic algorithms is also given. Two well-known heuristics are also proposed for the flowshop scheduling with a general exponential learning effect. & 2013 Published by Elsevier Ltd.

Keywords: Scheduling Flowshop Learning effect Heuristic algorithm Worst-case analysis

1. Introduction For most scheduling problems, it is assumed that the processing time of a job are fixed parameters [19]. However, this assumption is not appropriate for many modern industrial processes where the processing time of a job may be shorter if it is scheduled later in a sequence. For example, in many industry, economy, management, services sectors, both machines and workers can improve their performance by repeating the production operations. Therefore, the actual processing time of a job is shorter if it is scheduled later in a sequence. Such approach is called the learning effect [1,3]. Extensive surveys of different scheduling models and problems involving jobs with learning effects can be found in Biskup [4] and Janiak and Rudek [10]. Scheduling models concerning flowshop environments are encountered in many modern manufacturing processes [5,11,14,17,19]. Kuo et al. [12], Lee and Wu [15], Li et al. [16], Rudek [20], Sun et al. [23], Wang and Wang [25,26], Wang and Xia [27], Wang et al. [28], Wu and Lee [29], Xu et al. [30] considered different flowshop scheduling models with learning effects. Kuo et al. [12] and Li et al. [16] considered the following model, i.e., if job Jj is scheduled in position r on machine Mi in a sequence, its actual processing time a 1 is pijr ¼ pij ð1 þ∑rl ¼ 1 pi½l Þ , where pij denote the normal processing time of Jj on machine Mi, a r 0 is the learning index. For some regular objective functions, they gave heuristics and analyzed the worst-case bound of these heuristics. Lee and Wu [15] considered

n

Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (J.-J. Wang). 0305-0548/$ - see front matter & 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.cor.2013.09.001

the following model, i.e., if job Jj is scheduled in position r on machine Mi in a sequence, its actual processing time is a 1 pijr ¼ pij ðqðrÞ þ ∑rl ¼ 1 βl pi½r  l Þ , where q(r) denote a non-decreasing function of the job position, a r 0 is the learning index, and β1 ; β2 ; …; βn are a sequence of numbers with 0 r β1 rβ2 r⋯ r βn . For two special cases, they proved that the makespan and total completion time minimization can be solved in polynomial time. Rudek [20] considered a makespan minimization flowshop scheduling problem with stepwise learning curves, i.e., job processing times are described by step functions dependent on a jobs position in a sequence. He gave some results about computational complexity and presented solution algorithms for the general version of the considered problem. Wang and Wang [25] considered the following model, i.e., pijr ¼ pij gðrÞ where g : ½1; þ 1Þ-ð0; 1 is a non-increasing function with gð1Þ ¼ 1 and gðrÞ Z gðr þ 1Þ for each r. Wang and Xia [27], and Sun et al. [23] considered the following model, i.e., pijr ¼ pij αr  1 where 0 o α r 1 is the learning ratio. Wang and Xia [27], Xu et al. [30], and Sun et al. [23] considered the models pijr ¼ pij ðδ  θrÞ and pijr ¼ pij r a where a r 0 is the learning ratio, δ Z 0, θ Z 0 and δ θðn  1Þ 4 0. For the makespan minimization and total flow time minimization, Wang and Xia [27] proposed heuristic algorithms. For the (discounted) total weighted completion time minimization, and the sum of the quadratic job completion times minimization, Xu et al. [30] proposed heuristic algorithms. Wu and Lee [29] considered the following model, i.e., pijr ¼ pij r a where a r 0 is the learning ratio. For the total completion times minimization problem. They presented the branch-and-bound and several well-known heuristics. Wang et al. [28] considered the following model, i.e., pijr ¼ pij maxfr a ; ρ;g where a r 0 is the learning ratio and ρ is a truncation parameter with 0 r ρ r 1. For some objective functions, they presented the heuristics.

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

In this paper we consider flowshop scheduling with a general exponential learning effect [2]. The objective is to minimize the makespan, the total (weighted) completion time, the total weighted discounted completion time, and the sum of the quadratic job completion times. Several heuristic algorithms with worst-case bound for each criteria are presented. The paper is organized as follows. Some general notations and assumptions are given in Section 2. In Section 3, some heuristic algorithms with tight worst-case bound for some regular objective functions are proposed. In Section 4, we present two well-known heuristic algorithms. Computational results are described in Section 5, followed by some concluding remarks in Section 6.

2. Notations and assumptions The paper deals with the flowshop scheduling problem formulated as follows. A set of n jobs J ¼ fJ 1 ; J 2 ; …; J n g should be processed on m machines M 1 ; M 2 ; …; M m . Each job Jj, j ¼ 1; 2; …; n, consists of a chain of operations ðO1j ; O2j ; …; Omj Þ, and operation Oij has to be processed on machine M i ; i ¼ 1; 2; …; m. Also all machines process the jobs in the same order (a permutation schedule), and each machine can process only a job at a time. Let ωij be the weight of the jth position in a sequence on machine Mi. As in Bai et al. [2], in this paper, we consider a general exponential learning effect model, i.e., the actual processing time pijr of operation Oij

293

scheduled in position r is r1

r 1

pijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb

;

i ¼ 1; 2; …; m; r; j ¼ 1; 2; …; n;

ð1Þ

where pij denote the normal processing time of operation Oij (incurred when the operation is scheduled first in a sequence), α Z 0; β Z 0, 0 o a r1 and 0 o b r 1 are parameters obtained empirically, 0 o ωi1 r ωi2 r⋯ r ωin , and α þ β ¼ 1. For a permutation S, let C ij ¼ C ij ðSÞ and C j ¼ C mj represent the completion time of Oij and Jj, respectively. We consider the problem to minimize the makespan ðC max ¼ maxfC j jj ¼ 1; 2; …; ngÞ, the total completion time ð∑C j Þ, the total weighted completion time (∑wj C j , where wj 40 is a weight associated with job Jj), the total weighted discounted completion time (∑wj ð1  e  γC j Þ, where γ A ð0; 1Þ is the discount factor [19], and the sum of the quadratic job completion times (∑C 2j , [24]). According to the three-field notation introduced by Graham et al. [9], the corresponding problems can be denoted as follows: r1 r 1 FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb jρ, where ρ A fC max ; ∑C j ; ∑wj C j ;  γC j ∑wj ð1  e Þ; ∑C 2j g.

3. Worst-case behavior for the general case First, several lemmas for the single machine scheduling problems with a general exponential learning effect are given.

Table 1 Computational results of the heuristics for C max and m¼ 3. n

8

a

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m ðαaP max þ βÞ

C max ðSPTÞ C max ðSn Þ

m

C max ðARBÞ C max ðSn Þ

ðαaP max þ βÞb

n1

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.0976 1.0673 1.0887 1.0626 1.0584 1.1207 1.0405 1.0491 1.1128 1.0540 1.0777 1.1000 1.0661 1.0641 1.0913 1.0733 1.0716 1.1011 1.0681 1.1073 1.0887 1.0636 1.0951 1.0761 1.0820 1.0799 1.0566 1.0858 1.0560 1.1233 1.0449 1.1254 1.0991 1.0779 1.1163 1.1321

1.2064 1.2582 1.2131 1.2018 1.1753 1.3570 1.2078 1.1839 1.3849 1.1536 1.3584 1.2136 1.1392 1.1964 1.3063 1.2066 1.2157 1.2535 1.1700 1.1958 1.1830 1.1569 1.1932 1.2369 1.1937 1.1782 1.1927 1.1715 1.1037 1.2717 1.0881 1.2379 1.2116 1.1491 1.2136 1.2954

6.000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000

6.000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000

1.9330 1.8662 1.3311 2.0520 1.7609 1.4581 1.8684 1.6470 1.3372 1.9196 1.8384 1.3508 2.0448 1.8354 1.4166 1.8550 1.7908 1.3151 1.8553 2.0721 1.3381 2.0115 1.8530 1.3936 2.1179 1.6534 1.3153 2.3748 2.0401 1.2946 1.9063 1.8536 1.3465 1.7688 1.6652 1.4481

4.0390 3.4996 1.8476 3.7287 2.6226 1.9520 3.5978 2.4572 1.7815 4.2882 2.9107 1.9395 4.4472 3.5572 2.1702 3.1345 2.7974 1.5953 3.0633 3.5097 1.9949 3.1504 3.2374 1.9692 3.8566 2.8229 1.7562 3.7332 2.8876 1.4824 2.9116 2.8312 1.6154 2.2135 1.7946 1.6908

384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870

384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870

294

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308 r1

Lemma 1 (Bai et al. [2]). For the problem 1jpjr ¼ pj ðαa∑l ¼ 1 ωl p½l þ r1 βÞb jC max , an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of pj (i.e., the shortest processing time first (SPT) rule). r1

Lemma 2 (Bai et al. [2]). The problem 1jpjr ¼ pj ðαa∑l ¼ 1 ωl p½l þ βÞ r1 b j∑C j can be solved by the shortest processing time first (SPT) rule (i.e., sequencing the jobs in non-decreasing order of pj). r1

Lemma 3 (Bai et al. [2]). The problem 1jpjr ¼ pj ðαa∑l ¼ 1 ωl p½l þ βÞ r1 b j∑C 2j can be solved by the shortest processing time first (SPT) rule (i.e., sequencing the jobs in non-decreasing order of pj). It is well known that most of the flowshop scheduling problems are known to be NP-hard except for two-machine makespan minimization problem [11], hence we turn our attention to obtain schedules whose performance approximates that of optimal schedules. Let Sn ¼ ðJ ½1 ; J ½2 ; …; J ½n Þ be the optimal schedule, where ½j denote the job when it is scheduled in the jth position. For Sn we have C 1½j ¼ p1½1 þp1½2 ðαa

1 ∑rl  ω p ¼ 1 11 1½l

þ βÞb þ ⋯ þp1½j ðαa

∑jl ¼11 ω1l p1½l

þ βÞb

j1

r1

j1

hence C ½j ðSn Þ Z

1 j l1 ∑ L ðαaP max þβÞb ; m l ¼ 1 ½l

ð2Þ

where P max ¼ maxf∑nl¼ 1 ωil pil  pi min ji ¼ 1; 2; …; mg and pi min ¼ minfωij pij jj ¼ 1; 2; …; ng. r1 r1 Since for the problem 1jpjr ¼ pj ðαa∑l ¼ 1 ωl p½l þ βÞb jC max , SPT rule generates an optimal solution (Lemma 1), so we use the SPT rule (in the non-decreasing order of Lj ¼ ∑m approxi ¼ 1 pij ) as an r1 imate algorithm to the problem FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞ r1 b jC max . r1

Theorem 1. Let S be an SPT schedule for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ r1 βÞb jC max , we have the tight bound C max ðSÞ m ; r ðαaP max þ βÞ C max ðSn Þ where P max ¼ maxf∑nl¼ 1 ωil pil  pi min ji ¼ 1; 2; …; mg and pi min ¼ minfωij pij jj ¼ 1; 2; …; ng. Proof. Let Cj(S) be the completion time of job Jj by using the SPT schedule S (i.e., L1 r L2 r ⋯ r Ln ), we have

j1

C 2½j Zp2½1 þ p2½2 ðαa∑l ¼ 1 ω21 p2½l þ βÞb þ ⋯ þ p2½j ðαa∑l ¼ 1 ω2l p2½l þ βÞb …

2

C j ðSÞ r L1 þ L2 ðαapminfi1g þ βÞbþ L3 ðαapminfi1 þ i2g þ βÞb 1 ∑rl  ω p ¼ 1 m1 m½l

C m½j Zpm½1 þ pm½2 ðαa

∑jl ¼11 ωml pm½l

þ βÞb þ ⋯þ pm½j ðαa

j1

þ βÞb

j1

;

þ ⋯ þLj ðαapminfi1 þ i2 þ ⋯ þ i;j  1g þ βÞb

;

ð3Þ

Table 2 Computational results of the heuristics for C max and m¼ 5. n

8

a

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m ðαaP max þ βÞ

C max ðSPTÞ C max ðSn Þ

m

C max ðARBÞ C max ðSn Þ

n1

ðαaP max þ βÞb

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.0422 1.0410 1.0765 1.0424 1.0529 1.0682 1.0461 1.0590 1.0577 1.0535 1.0486 1.0848 1.0487 1.0515 1.0742 1.0589 1.0678 1.1080 1.0459 1.0797 1.0918 1.0547 1.0770 1.0850 1.0335 1.0769 1.0980 1.0496 1.0893 1.0915 1.0813 1.0682 1.0650 1.0449 1.0403 1.1028

1.1436 1.1310 1.2462 1.1918 1.1238 1.1734 1.1422 1.2603 1.2355 1.1018 1.0886 1.2064 1.2014 1.1079 1.2466 1.1889 1.2295 1.2351 1.1041 1.2737 1.1631 1.1004 1.2055 1.3150 1.0955 1.1739 1.2386 1.0859 1.1135 1.2861 1.1662 1.1352 1.1136 1.0786 1.0961 1.1747

10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000

10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000

1.5665 1.6351 1.3021 1.6828 1.4138 1.3092 1.6584 1.3870 1.2978 1.6675 1.5890 1.3355 1.5402 1.6166 1.2907 1.6261 1.5323 1.3616 1.6221 1.5605 1.3734 1.5489 1.4535 1.3592 1.9833 1.3662 1.3937 1.6611 1.6796 1.3249 1.4586 1.5665 1.3769 1.5929 1.5508 1.5240

3.5009 3.0188 1.8187 2.9303 2.1962 2.1043 3.2125 2.0863 1.7953 2.4985 2.1926 1.8141 2.4433 2.8585 1.6813 2.8637 2.0639 1.8023 2.8807 2.0296 1.9634 2.6785 1.8579 1.7447 3.8690 2.1976 1.7582 2.4679 2.0361 1.4927 1.8048 1.9297 1.5155 1.9264 1.7340 1.7186

640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117

640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

where pminfi1 þ i2 þ ⋯ þ i;j  1g ¼ minfωi1 pi1 þ ωi2 pi2 þ ⋯ þ i ¼ 1; 2; …; mg and so n

C max ðSÞ r ∑ Ll b l¼1

l1

ωi;j  1 pi;j  1 j

:

ð4Þ

rule (any busy schedule [8]) as an approximate algorithm for the r1 r1 problem FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb jC max . Theorem 2. Let S be an ARB schedule for FPmjpijr ¼ pij ðαa 1 r1 ∑rl  ω p ¼ 1 il i½l þβÞb jC max , we have the tight bound C max ðSÞ m r ; n1 C max ðSn Þ ðαaP max þβÞb

For Sn, from (2), we have C max ðSn Þ Z

295

n n 1 1 l1 l1 ðαaP max þ βÞ ∑ L½l b ¼ ðαaPmax þ βÞ ∑ Ll b ; m m l¼1 l¼1

ð5Þ

l1

as the term ∑nl¼ 1 L½l b is minimized by the increasing order of Ll (Lemma 1). From Stem form (4) and (5), we have C max ðSÞ= C max ðSn Þ rm=ðαaPmax þ βÞ. The following example can be confirmed to show that the bound m=ðαaP max þ βÞ is tight. There are m machines and m jobs J 1 ; J 2 ; …; J m with processing times pii ¼ k; pij ¼ ε ðε≪kÞ ia j ¼ 1; 2; …; m. Learning takes place by the 100%-learning curve, i.e., no learning is taking place, thus a ¼1, b¼1, and m=ðαaP max þ βÞ ¼ m. Since Lj ¼ ∑m i ¼ 1 pij ¼ k þ ðm  1Þε for all jobs, our heuristic may schedule the jobs in any order. Suppose we execute the jobs in the order of S ¼ ðJ 1 ; J 2 ; …; J m Þ; the resulting schedule has cost equal to C max ðSÞ ¼ mk þ ðm  1Þε. An optimal schedule has the jobs executed in the order Sn ¼ ðJ m ; J m  1 ; …; J 1 Þ, it has cost C max ðSn Þ ¼ k þ 2ðm  1Þε. The ratio goes to m as ε≪k. □ For the problem 1 J C max , an optimal schedule can be obtained by the any schedule (ARB rule), hence, we can also use the ARB

where P max ¼ maxf∑nl¼ 1 ωil pil  pi min ji ¼ 1; 2; …; mg and pi min ¼ minfωij pij jj ¼ 1; 2; …; ng. Proof. From Theorem 1, we have n

l1

C max ðSÞ r ∑ Ll b l¼1

n

r ∑ Ll : l¼1

For Sn, from (2), we have C max ðSn Þ Z

n 1 n 1 l1 n1 ∑ L½l ðαaP max þ βÞb Z ðαaPmax þ βÞb ∑ Ll : ml¼1 m l¼1

Consequently, we have C max ðSÞ m r : n1 C max ðSn Þ ðαaP max þβÞb The following example can be confirmed to show that the bound n1 m=ðαaP max þ βÞb is tight. Learning takes place by the 100%n1 learning curve, thus a ¼1, b ¼1, and m=ðαaP max þβÞb ¼ m.

Table 3 Computational results of the heuristics for ∑C j and m¼ 3. n

8

a

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m ðαaP max þ βÞ

∑C j ðSPTÞ ∑C j ðSn Þ

n

∑C j ðARBÞ ∑C j ðSn Þ

ðαaP max þ βÞb

n1

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.0491 1.0583 1.0664 1.0600 1.0696 1.0651 1.0545 1.0684 1.0603 1.0439 1.0815 1.0907 1.0864 1.0604 1.0736 1.0524 1.0770 1.0586 1.0537 1.0908 1.0742 1.0784 1.0780 1.0827 1.0767 1.0911 1.1015 1.1023 1.1454 1.0852 1.0449 1.0655 1.0764 1.0508 1.1027 1.0993

1.1436 1.2003 1.2003 1.2106 1.2492 1.2857 1.2291 1.1898 1.2424 1.1250 1.2618 1.2258 1.1790 1.1905 1.2108 1.1847 1.3116 1.1752 1.1335 1.2393 1.2106 1.2100 1.1891 1.2968 1.3001 1.1886 1.2560 1.2082 1.2442 1.2119 1.1230 1.0970 1.1570 1.1112 1.2131 1.2133

6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000

6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000

1.8119 1.7786 1.5868 2.2054 1.7798 1.5247 1.6768 1.6327 1.4411 1.7466 1.9507 1.5107 1.8147 1.6403 1.5317 1.8611 2.0217 1.5643 2.1060 1.7611 1.7310 2.3482 2.0516 1.5337 2.1081 1.5773 1.6526 2.5654 1.8885 1.6225 3.6307 1.7104 1.5728 2.3664 1.8176 1.5078

3.4644 3.0440 2.3487 5.8352 2.6659 2.8982 3.7874 2.5837 2.2014 3.1148 3.2418 1.9005 2.6988 2.6911 2.9264 4.7215 3.7608 2.1215 5.0926 2.5507 2.5520 4.0260 3.6615 2.3151 3.8048 2.4975 2.5480 4.1609 2.6371 2.1181 8.6067 2.6672 1.9421 3.6416 2.6365 2.0222

896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 10240.0000 495.6186 51.6235 10240.0000 495.6186 51.6235 10240.0000 495.6186 51.6235

896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 10240.0000 495.6186 51.6235 10240.0000 495.6186 51.6235 10240.0000 495.6186 51.6235

296

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

The bound m of the ARB rule for FPm J C max is tight (see [8]), hence r1 r 1 the bound for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb jC max is also tight. □ From above analysis and Lemma 2, we can use the SPT (in order of non-decreasing Lj ) rule as an approximate algorithm for r1 r 1 FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑C j . r1

Theorem 3. Let S be an SPT schedule for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ r1 βÞb j∑C j , we have the tight bound ∑nj¼ 1 C j ðSÞ

r

∑nj¼ 1 C j ðSn Þ

m ; ðαaPmax þ βÞ

l1

as the term ∑nj¼ 1 ∑jl ¼ 1 Ll b

is minimized by the increasing order

of Lj (Lemma 2). Consequently, we have ∑nj¼ 1 C j ðSÞ=∑nj¼ 1 C j ðSn Þ r m=ðαaP max þ βÞ. The following example can be confirmed to show that the bound m=ðαaP max þ βÞ is tight. Learning takes place by the 100%-learning curve, thus a ¼1, b¼ 1, and m=ð1 þ P max Þa ¼ m. The bound m of the SPT rule for FPm J ∑nj¼ 1 C j is tight (see [8]), hence the bound for the r1 r1 problem FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑C j is also tight. □ From Theorem 2, we can also use the ARB rule as an approxr1 r1 imate algorithm for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þβÞb j∑C j .

where P max ¼ maxf∑nl¼ 1 ωil pil pi min ji ¼ 1; 2; …; mg and pi min ¼ minfωij pij jj ¼ 1; 2; …; ng.

Theorem 4. Let S be an ARB schedule for r1 r1 pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑C j , we have the tight bound

Proof. Similar to the proof of Theorem 1. From Theorem 1, we

∑nj¼ 1 C j ðSÞ

l1

have C j ðSÞ r ∑jl ¼ 1 Ll b n

n

j

l1

∑ C j ðSÞ r ∑ ∑ Ll b

j¼1

j¼1l¼1

∑nj¼ 1 C j ðSn Þ

: Then :

n1

þ βÞb

;

Proof. From Theorem 1, we have

j n 1 l1 ∑ C j ðS Þ Z ðαaPmax þ βÞ ∑ ∑ L½l b m j¼1 j¼1l¼1 n

Z

n ðαaP max

where P max ¼ maxf∑nl¼ 1 ωil pil  pi min ji ¼ 1; 2; …; mg and pi min ¼ minfωij pij jj ¼ 1; 2; …; ng.

From (2), we have n

r

FPmjpijr ¼

n

n

j

n

∑ C j ðSÞ r ∑ ∑ Ll r n ∑ Lj :

j¼1

j n 1 l1 ðαaPmax þ βÞ ∑ ∑ Ll b ; m j¼1l¼1

j¼1l¼1

j¼1

Table 4 Computational results of the heuristics for ∑C j and m¼5. n

8

a

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m ðαaP max þ βÞ

∑C j ðSPTÞ ∑C j ðSn Þ

n

∑C j ðARBÞ ∑C j ðSn Þ

ðαaP max þ βÞb

n1

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.0282 1.0506 1.0498 1.0483 1.0433 1.0500 1.0463 1.0450 1.0747 1.0377 1.0529 1.0712 1.0458 1.0513 1.0751 1.0494 1.0567 1.0535 1.0579 1.0612 1.0651 1.0483 1.0635 1.0798 1.0536 1.0579 1.0691 1.0611 1.0953 1.0845 1.0451 1.0951 1.0772 1.0741 1.0462 1.0865

1.0705 1.2000 1.1784 1.1798 1.1589 1.1431 1.1241 1.1277 1.2579 1.0797 1.1524 1.1652 1.1015 1.1252 1.1628 1.1739 1.1537 1.1419 1.1111 1.1738 1.2369 1.1050 1.1509 1.2758 1.1672 1.1365 1.1540 1.0962 1.1420 1.1198 1.0698 1.1729 1.1117 1.1670 1.0784 1.1750

10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000

10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000

1.7181 1.5083 1.4169 1.4736 1.4599 1.4567 1.5906 1.5589 1.4054 1.6653 1.6743 1.4259 1.6136 1.5695 1.4382 1.5960 1.6574 1.5061 1.5882 1.5261 1.4787 1.7847 1.4926 1.5058 1.5970 1.7733 1.4427 1.8250 1.4878 1.5844 1.4938 1.4918 1.3161 1.7229 1.4484 1.8359

3.8512 2.0021 2.1191 2.1248 2.1412 1.7692 2.8258 3.5695 1.8723 2.5752 3.0391 2.1276 2.3949 2.8037 2.0169 2.6775 2.5050 2.3358 2.3527 2.9930 2.1801 3.2114 2.4814 2.0201 2.6089 4.0363 2.1433 2.8372 2.1958 1.9140 1.9941 2.2947 1.4971 2.0497 1.8956 2.2500

896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 10,240.0000 495.6186 51.6235 10,240.0000 495.6186 51.6235 10,240.0000 495.6186 51.6235

896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 896.0000 118.9980 26.3435 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 2048.0000 194.2825 33.4520 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 4608.0000 312.2397 41.8150 10,240.0000 495.6186 51.6235 10,240.0000 495.6186 51.6235 10,240.0000 495.6186 51.6235

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308 n1

From (2), we have C j ðSn Þ Z Lj ðαaPmax þ βÞb n

∑ C j ðSn Þ Z ðαaP max þβÞb

n1

j¼1

where P max ¼ maxf∑nl¼ 1 ωil pil  pi min ji ¼ 1; 2; …; mg and pi min ¼ minfωij pij jj ¼ 1; 2; …; ng.

, hence

n

∑ Lj :

Proof. Let Cj(S) be the completion time of job Jj using WSPT schedule S (i.e., L1 =w1 r L2 =w2 r ⋯ r Ln =wn ). From Theorem 1, we have

j¼1

Consequently, we have ∑nj¼ 1 C j ðSÞ n

∑nj¼ 1 C j ðS Þ

r

From Smith [21], we can use the WSPT (in order of nondecreasing Lj =wj ) rule as an approximate algorithm for r 1

r1

5. Let

S r 1

r1

pij ðαa∑l ¼ 1 ωil pi½l þ βÞb ∑nj¼ 1 wj C j ðSÞ ∑nj¼ 1 wj C j ðSn Þ

n

j

j¼1

j¼1

l¼1

∑ wj C j ðSÞ r ∑ wj ∑ Ll :

: n1 ðαaPmax þ βÞb

The following example can be confirmed to show that the bound n1 is tight. Learning takes place by the 100%n=ðαaP max þ βÞb n1 learning curve, thus a ¼1, b¼1, and n=ðαaPmax þ βÞb ¼ n. n The bound n of the ARB rule for FPm J ∑j ¼ 1 C j is tight (see [8]), r1 hence the bound for the problem FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞ r 1 b j∑C j is also tight. □

Theorem

n

n

FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb

r

be

a

WSPT

m n1

From (2), we have n

∑ wj C j ðSn Þ Z

j¼1

for

n

j

j¼1

l¼1

∑ w½j ∑ L½l

n1

n

j

j¼1

l¼1

∑ w j ∑ Ll ;

∑nj¼ 1 wj ∑jl ¼ 1 Ll

is minimized by the increasing order as the term of Lj =wj (Lemma 3), hence ∑nj¼ 1 wj C j ðSn Þ

schedule

n1

ðαaPmax þ βÞb m

ðαaP max þβÞb Z m

∑nj¼ 1 wj C j ðSÞ

j∑wj C j .

r

m n1

ðαaP max þ βÞb

:

FPmjpijr ¼

j∑wj C j , we have the tight bound

ðαaP max þ βÞb

297

The following example can be confirmed to show that the bound n1 m=ðαaP max þ βÞb is tight. Learning takes place by the 100%n1 learning curve, thus a ¼1, b ¼1, and m=ðαaP max þβÞb ¼ m. The bound m of the WSPT rule for FPm J ∑wj C j is tight (see [22]),

;

Table 5 Computational results of the heuristics for ∑wj C j and m¼ 3. n

8

a

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m

∑wj C j ðWSPTÞ ∑wj C j ðSn Þ

ðαaP max þ βÞb

1þ ðn  1Þðw=wÞ

∑wj C j ðARBÞ ∑wj C j ðSn Þ

n1

n1

ðαaP max þ βÞb

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.2157 1.1368 1.1365 1.3286 1.1608 1.1937 1.2669 1.1664 1.1763 1.2421 1.2501 1.1386 1.2453 1.3276 1.1852 1.2644 1.1813 1.1494 1.3702 1.2217 1.1438 1.1813 1.2843 1.1521 1.3098 1.2778 1.1604 1.1997 1.1542 1.2029 1.1964 1.1111 1.1051 1.2373 1.3138 1.1914

1.7290 1.4898 1.4669 3.2126 1.5493 1.4067 2.0462 1.7721 1.4524 1.7151 1.7174 1.3859 1.8482 2.6678 1.6899 2.0457 1.8616 1.5542 2.6232 2.1409 1.6209 1.7409 2.0777 1.5206 2.0469 2.2071 1.5487 1.4451 1.3923 1.2559 1.3668 1.1724 1.3759 1.6869 1.8671 1.3457

384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870

384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870

1.8106 1.8365 1.6291 2.0954 1.7653 1.5123 1.8851 1.7820 1.4300 1.8969 1.6672 1.4862 1.7138 1.9145 1.5692 1.9345 1.5208 1.6230 2.5938 1.9561 1.7195 2.0754 1.9223 1.7459 1.8652 1.9657 1.7114 1.9296 2.3971 1.4816 2.0734 1.8746 1.7844 2.2558 1.9147 1.3980

4.7935 3.7331 2.4313 5.1415 3.8172 2.5054 3.1080 3.4476 2.2191 3.7994 2.5016 2.3046 3.6148 4.5078 2.2570 3.6997 2.2342 2.3044 6.5154 3.3315 2.3631 5.1303 2.7947 2.6970 3.1239 3.5451 2.6022 2.4201 3.1565 1.9269 2.9958 3.3567 2.2287 3.9580 2.8637 1.6850

9612.8000 1011.4833 356.0132 14,758.4000 1047.1827 214.8874 10,304.0000 1842.7696 276.9828 34,214.4000 1979.2531 541.2956 33,945.6000 4308.2146 655.4506 27,404.8000 1868.7549 605.6907 97,177.6000 6987.2316 852.0974 78,950.4000 7681.0978 543.5954 65,843.2000 4919.5107 861.3897 424,960.0000 13,074.4198 1222.4444 283,033.6000 3261.1707 1138.8143 426,803.2000 2725.9025 1175.9832

69,248.0000 3382.9442 2239.1949 37,760.0000 5218.9139 545.6862 30,080.0000 8686.8567 1019.8686 161,536.0000 8184.1507 2609.2580 165,120.0000 15,834.0245 2843.4222 88,064.0000 5464.1956 2784.8811 369,152.0000 26401.6052 3498.5243 201,216.0000 27,511.7910 1788.7541 373,248.0000 13,634.4689 3201.1730 913,408.0000 39,748.6153 4326.0490 848,896.0000 8078.5839 2142.3751 21,248.0000 3618.0161 4186.6655

298

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308 r1

hence the bound for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þβÞb tight. □

r1

j∑wj C j is also

Hence ∑nj¼ 1 wj C j ðSÞ

Smutnicki [22] proposed the ARB schedule as an approximate algorithm for FPm J ∑wj C j , hence, we can also use the ARB rule r1 as an approximate algorithm for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞ r1 b j∑wj C j . r1

Theorem 6. Let S be an ARB schedule for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ r1 βÞb j∑wj C j , we have

∑nj¼ 1 wj C j ðSn Þ

1 þ ðn  1Þðw=wÞ

r

n1

ðαaP max þ βÞb

:

□

Lemma 4 (Xu et al. [30]). 1  e  γα Z αð1  e  γ Þ if 0 r γ r 1 and 0 r α r 1. Lemma 5. ∑nj¼ 1 ð1  e  γLj Þ Z 1  e  γ∑j ¼ 1 Lj . n

pimin ¼ where P max ¼ maxf∑nl¼ 1 ωil pil  pimin ji ¼ 1; 2; …; mg, minfωij pij jj ¼ 1; 2; …; ng, w ¼ minj A J wj and w ¼ maxj A J wj .

Proof (by induction). By direct calculation, we have ð1  e  γL1 Þ ¼ 1  e  γL1 , ð1 e  γL1 Þð1  e  γL2 Þ Z0 3 1  e  γL1  e  γL2 þ e  γðL1 þ L2 Þ Z 0 3 ð1 e  γL1 Þ þ ð1 e  γL2 Þ Z 1 e  γðL1 þ L2 Þ Suppose Lemma 7 holds for n¼ k, i.e., ∑kj ¼ 1 ð1  e  γLj Þ Z k 1  e  γ∑j ¼ 1 Lj . Consider n ¼ k þ 1

Proof. From Theorem 5, we have

ð1  e  γ∑j ¼ 1 Lj Þð1  e  γLk þ 1 Þ Z0

∑nj¼ 1 wj C j ðSÞ ∑nj¼ 1 wj C j ðSn Þ

r

1 þ ðn  1Þðw=wÞ n1

ðαaP max þ βÞb

;

k

3 1  e  γ∑j ¼ 1 Lj  e  γLk þ 1 þ e k

n

j

n

n

n

n

∑ wj C j ðSÞ r ∑ wj ∑ Ll r ∑ wj ∑ Ll r ðw þ ðn  1ÞwÞ ∑ Ll :

j¼1

j¼1

j¼1

l¼1

l¼1

3 ð1  e

l¼1

 γ∑kj ¼ 1 Lj

þ1  γ∑kj ¼ L 1 j

Z0

Þ þ ð1  e  γLk þ 1 Þ Z 1  e

þ1  γ∑kj ¼ L 1 j

þ1  γ∑kj ¼ L þ1  γLj 1 j. ÞZ1e ∑kj ¼ 1 ð1  e

n

From (2), C j ðS Þ Z Lj ðαa n

P max

n1

þ βÞb

n1

∑ wj C j ðSn Þ Z ðαaP max þ βÞb

j¼1

hence we have proof of Lemma 7.

, hence

n

∑ w½j L½j Z ðαaP max þβÞb

j¼1

n1

n

w ∑ Ll : l¼1

;

This completes the

□

From Pinedo [19], the WDSPT (in order of non-decreasing ð1  e  γLj Þ=wj e  γLj ) rule can be used as an approximate algorithm r1 r1 for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑wj ð1  e  γC j Þ.

Table 6 Computational results of the heuristics for ∑wj C j and m¼5. n

8

a

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m

∑wj C j ðWSPTÞ ∑wj C j ðSn Þ

1 þ ðn 1Þðw=wÞ

∑wj C j ðARBÞ ∑wj C j ðSn Þ

n1

ðαaP max þ βÞb

n1

ðαaP max þ βÞb

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.2718 1.1745 1.1690 1.2009 1.1657 1.1322 1.3316 1.1688 1.1156 1.2450 1.2779 1.2022 1.2422 1.2743 1.1823 1.3108 1.2208 1.1362 1.2730 1.3307 1.2300 1.2519 1.3391 1.2200 1.2658 1.1892 1.1947 1.3509 1.2292 1.2251 1.0373 1.1691 1.2677 1.3146 1.1862 1.3272

2.0737 1.4853 1.6034 1.9025 1.5414 1.5005 2.2172 1.4467 1.4267 2.3298 1.7057 1.5767 1.8176 2.2139 1.5261 1.9598 1.9873 1.4305 1.7850 1.7810 1.4726 2.0341 2.1710 2.1404 2.1194 1.5987 1.6592 1.8008 1.5484 1.5465 1.0682 1.3921 1.6000 1.5017 1.6144 1.4727

640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117

640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117

1.6925 1.5496 1.4430 1.5511 1.6904 1.3695 1.5426 1.3251 1.4187 2.0144 1.5829 1.4853 1.7750 1.5518 1.5648 1.5849 1.4989 1.4933 1.5777 1.7180 1.5383 1.6627 1.6684 1.5675 1.7874 1.5865 1.4906 1.7811 1.6750 1.5419 1.7819 1.8891 1.5744 1.8014 1.8320 1.7178

2.4877 2.7453 2.3531 2.3395 3.0438 2.1631 3.3732 2.2266 2.0095 4.2340 3.0060 2.0465 3.3184 2.3785 2.0036 2.5334 2.2661 2.0683 2.2729 3.4178 2.4756 3.2672 2.4017 2.0184 3.9016 2.4046 2.0251 2.1374 2.1513 2.2702 3.3739 2.3866 1.7769 2.4204 2.9947 1.9181

6195.2000 2750.5546 427.1405 11,417.6000 1939.6680 597.6204 14,566.4000 2643.4564 607.7815 41,292.8000 5319.6979 588.1284 30,720.0000 2548.7437 514.9521 34,035.2000 1231.2654 760.8245 122,572.8000 5002.7746 1344.5856 37,990.4000 9318.6218 924.5768 66,867.2000 9207.6032 915.2846 212,992.0000 12,182.3063 1110.9376 65,536.0000 11,647.0381 1482.6268 72,908.8000 11,736.2495 1092.3532

23,936.0000 9298.8466 1968.2335 65,408.0000 9910.8365 2148.8745 69,248.0000 9604.8415 2239.1949 174,080.0000 16,854.0076 2638.5285 95,232.0000 8354.1479 2901.9632 175,872.0000 3254.2320 2667.7990 397,824.0000 13,356.9225 3647.2000 127,488.0000 25,846.5123 3572.8621 328,192.0000 27,234.2445 3461.3554 756,736.0000 41,532.8424 2095.9139 93,184.0000 42,424.9560 4233.1267 111,616.0000 19,230.0034 2095.9139

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

Theorem 7. Let S be a WDSPT schedule for FPmjpijr ¼ r1 r 1 pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑wj ð1  e  γC j Þ, we have the tight bound ∑nj¼ 1 wj ð1  e  γC j ðSÞ Þ n ∑nj¼ 1 wj ð1  e  γC j ðS Þ Þ

r

m ðαaP max þβÞb

n1

; and

pimin ¼

n

j¼1

j

∑ wj ð1  e  γC j ðSÞ Þ r ∑ wj ð1  e  γ∑l ¼ 1 Ll Þ;

where the inequality derived from 1  e  γC j is an increasing function on Cj. For the optimal schedule Sn, we have n

∑ wj ð1  e

 γC j ðSn Þ

j¼1

n

Þ Z ∑ w½j ð1  e

ðfrom 1  e

n1

Z

n1

 γðαaP max þ βÞb

=m∑jl ¼ 1 L½l

ðαaPmax þ βÞb m

Þ

j

∑ w½j ð1  e  γ∑l ¼ 1 L½l Þ

∑nj¼ 1 wj ð1  e  γC j ðSÞ Þ n

ðfrom Lemma 4Þ

r

m ðαaPmax

n1

þ βÞb

:

From Pinedo [19], we have that the WDSPT schedule converges to the WSPT schedule as γ-0. From Theorem 5, we know r1 that the bound is tight for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ r1  γC j βÞb j∑wj ð1 e Þ. □ From Theorem 6, we can also use the ARB rule as an approxr1 imate algorithm for the problem FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ r1  γC j βÞb j∑wj ð1 e Þ.

∑nj¼ 1 wj ð1  e  γC j ðSÞ Þ n ∑nj¼ 1 wj ð1  e  γC j ðS Þ Þ

is an increasing function on C j Þ

n

j

Theorem 8. Let S be an ARB schedule r1 r1 pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑wj ð1 e  γC j Þ, we have

j¼1

 γC j

n

∑ wj ð1  e  γ∑l ¼ 1 Ll Þ

j¼1

∑nj¼ 1 wj ð1  e  γC j ðS Þ Þ

Proof. Similar to the proof of Theorem 5. Let Cj(S) be the completion time of job Jj using WDSPT schedule S, i.e., ð1  e  γL1 Þ= w1 e  γL1 r ð1  e  γL2 Þ=w2 e  γL2 r ⋯ r ð1  e  γLn Þ=wn e  γLn , we have n

ðαaP max þ βÞb m

Hence

where P max ¼ maxf∑nl¼ 1 ωil pil  pimin ji ¼ 1; 2; …; mg minfωij pij jj ¼ 1; 2; …; ng.

j¼1

n1

Z

299

r

1 þ ðn  1Þðw=wÞ ðαaP max þβÞb

n1

for

FPmjpijr ¼

;

pi min ¼ where P max ¼ maxf∑nl¼ 1 ωil pil  pi min ji ¼ 1; 2; …; mg, minfωij pij jj ¼ 1; 2; …; ng, w ¼ minj A J wj and w ¼ maxj A J wj .

ðfrom Lemma 6Þ

j¼1

Table 7 Computational results of the heuristics for ∑wj ð1  e  γC j Þ and m¼ 3. n

a

b

∑wj ð1 e  γC j ðWDSPTÞ Þ n

∑wj ð1  e  γC j ðS Þ Þ

8

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m

∑wj ð1 e  γC j ðARBÞ Þ

n1

ðαaP max þ βÞb

n

1 þ ðn  1Þðw=wÞ n1

∑wj ð1  e  γC j ðS Þ Þ

ðαaP max þ βÞb

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870

384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 384.0000 50.9992 11.2901 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 768.0000 72.8559 12.5445 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 1536.0000 104.0799 13.9383 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870 3072.0000 148.6856 15.4870

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

14,758.4000 2097.7654 387.6253 18,022.4000 1934.5681 398.9154 7116.8000 1322.5782 314.2400 27,494.4000 2183.2497 349.5737 29,555.2000 4741.7075 577.8837 34,662.4000 2939.7372 538.3685 38,604.8000 6071.3284 578.9059 130,560.0000 4905.6333 313.1481 73,420.8000 2310.5741 924.5768 133,734.4000 7097.2590 702.0795 229,580.8000 7810.9498 1064.4765 139,264.0000 12,003.8836 841.4630

66,176.0000 10,114.8331 1923.0733 73,088.0000 9298.8466 1019.8686 23,936.0000 8176.8651 1990.8137 145,408.0000 5294.1984 1262.8140 175,872.0000 16,854.0076 2814.1516 177,664.0000 8184.1507 2638.5285 139,776.0000 26,956.6981 3610.0311 381,440.0000 13,079.3760 1677.2474 385,536.0000 5308.0757 3684.3689 212,992.0000 16,999.7195 1910.0693 452,608.0000 21,460.2873 4372.5101 249,856.0000 36,626.2178 3257.4426

300

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

Theorem 9. Let S be an SPT schedule for r1 r1 pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑C 2j , we have the tight bound

Proof. From Theorem 6, we have n

∑ wj ð1  e  γC j ðSÞ Þ

∑nj¼ 1 C 2j ðSÞ

j¼1

n

r ∑ wj ð1  e

 γ∑jl ¼ 1 Ll

j¼1

Þ r ðw þðn 1ÞwÞð1  e

 γ∑nl ¼ 1 Ll

∑nj¼ 1 C 2j ðSn Þ

Þ:

r

!2

m ðαaP max þβÞb

;

n1

where P max ¼ maxf∑nl¼ 1 ωil pil pimin ji ¼ 1; 2; …; mg minfωij pij jj ¼ 1; 2; …; ng.

n

n

n

∑ wj ð1  e  γC j ðS Þ Þ Z ∑ w½j ð1  e  γðαa

j¼1

P max

n1

þ βÞb

L½j

Þ

n1

n

n1

n

∑nj¼ 1 wj ð1  e  γC j ðS Þ Þ

r

wð1  e  γ∑l ¼ 1 Ll Þ:

n1

pimin ¼

j¼1

j¼1

j

!2

∑ Ll

;

l¼1

n

1 þ ðn  1Þðw=wÞ ðαaP max þ βÞb

n

∑ C 2j ðSÞ r ∑

and

The last inequality is obtained by Lemma 7. Consequently, we have ∑nj¼ 1 wj ð1  e  γC j ðSÞ Þ

n

∑ w½j ð1  e  γL½j Þ

j¼1

Z ðαaPmax þ βÞb

and

Proof. Similar to the proof of Theorem 1, we have

j¼1

Z ðαaPmax þ βÞb

FPmjpijr ¼

:

n

n

j¼1

j¼1

∑ C 2j ðSn Þ Z ∑

□

From Lemma 3, the SPT rule can be used as an approximate r1 r 1 j∑C 2j . algorithm for FPmjpijr ¼ pij ðαa∑l ¼ 1 ωil pi½l þ βÞb

n1

ðαaP max þ βÞb m

ðαaP max þβÞb m

n1

Z

ðαaP max þβÞb m

n1

Z

!2

l¼1

n

∑

j¼1

!2

!2

j

∑ L½l

n

∑

j¼1

j

!2

∑ L½l

l¼1

j

∑ Ll

l¼1

!2 ;

Table 8 Computational results of the heuristics for ∑wj ð1  e  γC j Þ and m¼ 5. n

a

b

∑wj ð1 e  γC j ðWDSPTÞ Þ n

∑wj ð1  e  γC j ðS Þ Þ

8

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

m

∑wj ð1  e  γC j ðARBÞ Þ

n1

ðαaP max þ βÞb

n

1 þ ðn  1Þðw=wÞ n1

∑wj ð1 e  γC j ðS Þ Þ

ðαaP max þ βÞb

Mean

Max

Mean

Max

Mean

Max

Mean

Max

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117

640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 640.0000 84.9986 18.8168 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 1280.0000 121.4266 20.9075 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 2560.0000 173.4665 23.2306 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117 5120.0000 247.8093 25.8117

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

11,302.4000 838.0862 404.5604 13,798.4000 1465.3758 393.2704 7616.0000 1954.9677 397.7864 41,382.4000 2727.2407 563.2485 28,659.2000 3092.7347 392.0159 28,121.6000 2667.7417 541.2956 90,828.8000 5224.8117 489.7005 38,195.2000 4544.8230 865.1065 70,758.4000 7583.9565 980.3302 168,755.2000 15,037.0697 1036.5998 284,876.8000 5580.6659 878.6319 54,476.8000 18,516.3126 2179.5440

72,320.0000 3178.9475 2171.4546 67,712.0000 9604.8415 2103.7142 20,864.0000 8890.8533 2013.3938 177,664.0000 14,474.0469 2755.6106 145,408.0000 14,474.0469 1262.8140 84,480.0000 8354.1479 2638.5285 406,016.0000 26,679.1516 1788.7541 172,544.0000 15,022.2011 3647.2000 213,504.0000 26,679.1516 3349.8486 424,960.0000 43,317.0695 3861.4375 848,896.0000 9862.8110 1398.9967 93,184.0000 44,209.1831 4558.3547

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

as the term ∑nj¼ 1 ð∑jl ¼ 1 Ll Þ2 is minimized by the increasing order of Lj (Lemma 5). Hence, we have ∑nj¼ 1 C 2j ðSÞ ∑nj¼ 1 C 2j ðSn Þ

r

!2

m n1

ðαaP max þ βÞb

:

The following example can be confirmed to show that the bound n1 2 Þ is tight. Consider the case that no learning is ðm=ðαaP max þβÞb n1 2 taking place, thus a ¼1 and b¼1, i.e., ðm=ðαaP max þ βÞb Þ ¼ m2 . 2 As in Koulamas and Kyparisis [13], the bound m of the SPT rule is tight for FPm J ∑C 2j , therefore the bound for FPmjpijr ¼ r1 r 1 pij ðαa∑l ¼ 1 ωil pi½l þ βÞb j∑C 2j is also tight. □ 4. Two well-known heuristics Since the classical flowshop scheduling problems are NPcomplete except for the problem FP2 J C max [11], many studies have been devoted to constructing heuristic procedures [5,7]. In this section, we apply two well-known heuristic procedures from Nawaz et al. [18] and Laha and Sarin [14]. Now, we give the NEH and LS algorithms as follows: The Oðmn2 Þ NEH (Nawaz et al. [18]) algorithm Step 1. For each job Jj calculate Lj ¼ ∑m i ¼ 1 pij ; j ¼ 1; 2; …; n. Step 2. Arrange the jobs in descending order of Lj.

301

Step 3. Pick the two jobs from the first and second position of the list of Step 2, and find the best sequence for these two jobs by calculating total completion time (makespan, total weighted completion time, discounted total weighted completion time, sum of the quadratic job completion times, and maximum lateness) for the two possible sequences. Do not change the relative positions of these two jobs with respect to each other in the remaining steps of the algorithm. Set j¼ 3. Step 4. Pick the job in the jth position of the list generated in Step 2 and find the best sequence by placing it at all possible j positions in the partial sequence found in the previous step, without changing the relative positions to each other of the already assigned jobs. The number of enumerations at this step equals j. Step 5. If n ¼j, STOP, otherwise set j¼jþ 1 and go to Step 4. For the total flow (completion) time minimization permutation flow shop scheduling problem, Laha and Sarin [14] proposed a modification method of Framinan and Leisten [6]. Now, we give the method of Laha and Sarin [14]. The Oðmn4 Þ LS (Laha and Sarin [14]) algorithm Step 1. For each job Jj calculate Lj ¼ ∑m i ¼ 1 pij ; j ¼ 1; 2; …; n. Step 2. Sort the jobs in ascending order of Lj. Step 3. Set k ¼2. Select the first two jobs from the sorted list and select the better between the two possible sequences. Step 4. For k¼ 3 to n do the following. Insert the kth job on the sorted list into k possible positions of the ðk 1Þ-job current

Table 9 Computational results of the heuristics for ∑C 2j and m¼3. n

a

b

∑C 2j ðSPTÞ

8

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

0.5

0.7

0.9

0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

!2

m

∑C 2j ðSn Þ

ðαaP max

∑C 2j ðARBÞ ∑C 2j ðSn Þ

n1

þ βÞb

Mean

Max

Mean

Max

Mean

Max

1.1349 1.1719 1.1321 1.1208 1.1121 1.1856 1.1043 1.1765 1.1265 1.1042 1.1530 1.1549 1.1083 1.1396 1.2241 1.1458 1.1324 1.1828 1.1155 1.2129 1.2089 1.1516 1.2206 1.2085 1.0935 1.1682 1.1777 1.1197 1.1175 1.1696 1.2064 1.2639 1.2046 1.1723 1.2480 1.0841

1.4112 1.4552 1.3490 1.6031 1.2504 1.4888 1.3103 1.7305 1.4276 1.3748 1.4956 1.4405 1.3019 1.4844 1.8381 1.6302 1.3495 1.8023 1.2693 1.7196 1.7427 1.4259 1.5697 1.4560 1.2851 1.7721 1.5553 1.3351 1.2302 1.2275 1.2879 1.4888 1.4231 1.3891 1.3990 1.1555

147,456.0000 2600.9142 127.4654 147,456.0000 2600.9142 127.4654 147,456.0000 2600.9142 127.4654 589,824.0000 5307.9881 157.3647 589,824.0000 5307.9881 157.3647 589,824.0000 5307.9881 157.3647 2,359,296.0000 10,832.6288 194.2774 2,359,296.0000 10,832.6288 194.2774 2,359,296.0000 10,832.6288 194.2774 9,437,184.0000 22107.4057 239.8487 9,437,184.0000 22107.4057 239.8487 9,437,184.0000 22107.4057 239.8487

147,456.0000 2600.9142 127.4654 147,456.0000 2600.9142 127.4654 147,456.0000 2600.9142 127.4654 589,824.0000 5307.9881 157.3647 589,824.0000 5307.9881 157.3647 589,824.0000 5307.9881 157.3647 2,359,296.0000 10,832.6288 194.2774 2,359,296.0000 10,832.6288 194.2774 2,359,296.0000 10,832.6288 194.2774 9,437,184.0000 22107.4057 239.8487 9,437,184.0000 22107.4057 239.8487 9,437,184.0000 22107.4057 239.8487

4.4649 3.1214 2.2282 3.5708 3.6592 1.9346 6.5083 2.4861 2.7432 3.6613 4.7273 2.6668 5.0012 3.8553 2.3896 4.4175 4.6656 2.2006 5.7867 5.4831 2.3017 4.5196 4.0151 2.3142 4.0739 4.0420 3.1536 5.2473 5.2189 2.0602 5.9099 4.1568 3.3961 7.3689 3.8972 2.4486

12.6571 7.8149 4.3701 6.7533 8.3916 3.8593 30.2990 4.4647 5.2862 16.6603 15.3367 6.0572 16.4448 9.1338 4.9514 15.2529 14.3926 4.8263 28.0962 13.5022 4.3303 15.9987 14.2059 3.8520 10.1090 11.1736 6.5677 10.8819 15.7108 2.3270 10.3933 6.9608 5.9971 17.1242 5.9295 4.2226

302

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

sequence, thereby generating k, k-job partial sequences, and select from these a k-job partial sequence with the best total completion time (makespan, total weighted completion time, discounted total weighted completion time, and sum of the quadratic job completion times) value. Designate this as a k-job current sequence. Place each job (except for the kth job of the sorted list) of this sequence into its ðk  1Þ positions and select the best k-job sequence having the least total completion time (makespan, total weighted completion time, discounted total weighted completion time, and sum of the quadratic job completion times) value from among those generated. This becomes the next k-job current sequence. Step 5. If k ¼n, then STOP; else, go to Step 4.

5. Computational experiments The heuristic algorithms were coded in VC þ þ 6.0 and the computational experiments were programmed and tested on a Pentium 4 with 2 GB RAM personal computer. The test problems for each environment were generated as follows: (1) (2) (3) (4) (5)

pj were generated from a uniform distribution over [1, 100]; wj were generated from a uniform distribution over [1, 100]; For small-sized problems n ¼8,9,10,11, m ¼ 3,5; For large-sized problems n ¼50,100,150,200, m ¼10,20; a ¼ 0:5; 0:7; 0:9;

(6) b ¼ 0:5; 0:7; 0:9; (7) ωij ¼ 1, α ¼ β ¼ 0:5 and γ ¼ 0:9. For small-sized problems, the percentage error of the solution produced by the heuristic algorithm is calculated as CðHeurÞ= CðOptÞ, where C A fC max ; ∑C j ; ∑wj C j ; ∑wj ð1  e  γC j Þ; ∑C 2j g, Heur A fARB; SPT; WSPT; WDSPTg, CHeur is the objective value of the solution generated by the heuristic Heur and Copt is that for the optimal solution (an enumerative algorithm is used to find the optimal value of each test problem). For each problem size, 20 instances were generated. The results are summarized in Tables 1– 10. From Tables 1–4 and Tables 9 and 10, we see that the performance of the SPT rule is more effective than the ARB rule for the objective functions C max , ∑C j and ∑C 2j . From Tables 5 and 6, we see that the performance of the WSPT rule is more effective than the ARB rule for the objective function ∑wj C j . From Tables 7 and 8, we see that the performance of the WDSPT rule is almost as much as the ARB rule for the objective function ∑wj ð1  e  γC j Þ. For large-sized problems, the performance of the heuristics ARB, SPT, WSPT, WDSPT, NEH, and LS was verified by the ratios CðHeurÞ=CðARBÞ, where C A fC max ; ∑C j ; ∑wj C j ; ∑wj ð1  e  γC j Þ; ∑C 2j g, and Heur A fSPT; WSPT; WDSPT; NEH; and LSg. We did not try to optimize the running time of the heuristics, since our main goal was to evaluate the performance of the heuristics. From Tables 11–16 and Tables 19 and 20, we see that the performance of

Table 10 Computational results of the heuristics for ∑C 2j and m¼5. n

a

b

∑C 2j ðSPTÞ

8

0.5

0.7

0.9

9

0.5

0.7

0.9

10

0.5

0.7

0.9

11

0.5

0.7

0.9

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

!2

m

∑C 2j ðSn Þ

ðαaP max

∑C 2j ðARBÞ ∑C 2j ðSn Þ

n1

þ βÞb

Mean

Max

Mean

Max

Mean

Max

1.0538 1.1362 1.1170 1.0831 1.0782 1.1941 1.1124 1.1707 1.1196 1.0506 1.1022 1.1179 1.1099 1.0923 1.1321 1.0905 1.0998 1.1328 1.1220 1.1113 1.2481 1.0986 1.1479 1.1823 1.1653 1.1110 1.1572 1.0620 1.0981 1.2363 1.1315 1.0977 1.0733 1.0549 1.1081 1.1871

1.1553 1.5417 1.5360 1.4335 1.2788 1.7205 1.4939 1.8297 1.4923 1.1567 1.2482 1.3198 1.4316 1.3007 1.8100 1.3033 1.5747 1.5427 1.8508 1.4631 1.7702 1.3318 1.4533 1.7564 1.5338 1.3565 1.5774 1.0858 1.1935 1.3657 1.4774 1.2205 1.1176 1.1107 1.1474 1.2772

409,600.0000 7224.7616 354.0706 409,600.0000 7224.7616 354.0706 409,600.0000 7224.7616 354.0706 1,638,400.0000 14,744.4114 437.1242 1,638,400.0000 14744.4114 437.1242 1,638,400.0000 14744.4114 437.1242 6,553,600.0000 30,090.6355 539.6595 6,553,600.0000 30,090.6355 539.6595 6,553,600.0000 30,090.6355 539.6595 26,214,400.0000 61409.4602 666.2463 26,214,400.0000 61,409.4602 666.2463 26,214,400.0000 61,409.4602 666.2463

409,600.0000 7224.7616 354.0706 409,600.0000 7224.7616 354.0706 409,600.0000 7224.7616 354.0706 1,638,400.0000 14,744.4114 437.1242 1,638,400.0000 14744.4114 437.1242 1638400.0000 14,744.4114 437.1242 6,553,600.0000 30,090.6355 539.6595 6,553,600.0000 30,090.6355 539.6595 6553600.0000 30090.6355 539.6595 26,214,400.0000 61409.4602 666.2463 26214400.0000 61409.4602 666.2463 26,214,400.0000 61,409.4602 666.2463

3.0559 2.9759 2.2398 2.5928 2.2977 2.3484 3.3368 2.2376 2.1239 3.3400 2.8738 2.0225 2.7552 2.6448 2.0309 3.2215 2.6383 2.0683 3.2913 2.6357 1.9613 2.9676 2.9971 1.8210 2.7456 3.5327 1.9372 2.8723 4.6841 1.8204 2.5186 2.2414 2.0752 2.6155 4.3125 2.5528

9.6293 6.6395 5.9181 5.7568 5.8208 4.6612 11.5519 4.3330 4.0790 14.8613 6.2865 4.3240 10.7074 6.6085 4.9869 7.5736 5.4955 3.2033 12.2521 11.6021 3.8844 9.9612 10.8547 2.7979 7.6733 8.9828 3.1842 7.8065 6.9628 2.5290 3.6932 4.2045 3.5451 3.7437 6.3066 5.3400

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

Table 11 Computational results of the heuristics for C max and m¼ 10. n

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

C max ðSPTÞÞ C max ðARBÞ

C max ðNEHÞ C max ðARBÞ

303

Table 12 Computational results of the heuristics for C max and m¼20. C max ðLSÞ C max ðARBÞ

n

Mean

Max

Mean

Max

Mean

Max

0.6792 0.6493 0.6848 0.6946 0.6805 0.7711 0.7119 0.6363 0.7794 0.6299 0.6606 0.7279 0.6337 0.5184 0.6842 0.6195 0.6649 0.7061 0.5923 0.6226 0.6601 0.6178 0.6265 0.7018 0.5803 0.6200 0.6842 0.5216 0.5782 0.6115 0.6330 0.7096 0.7124 0.5854 0.6077 0.7306

0.8337 0.9351 0.8848 0.8926 0.8978 0.9555 0.9284 0.8491 0.9426 0.8763 0.8685 0.9161 0.8161 0.6603 0.8518 0.9557 0.8079 0.8711 0.9243 0.8315 0.8371 1.0256 0.7747 0.9278 0.6964 0.9764 0.7581 0.5835 0.6833 0.7152 0.7702 0.8558 0.8885 0.6628 0.7902 0.8616

0.6814 0.6450 0.6640 0.6989 0.6674 0.7554 0.7096 0.6379 0.7360 0.6242 0.6533 0.6986 0.6275 0.5251 0.6497 0.6247 0.6571 0.6670 0.5934 0.6269 0.6224 0.6222 0.6212 0.6666 0.5827 0.6209 0.6248 0.5259 0.5828 0.5750 0.6242 0.7152 0.6647 0.5824 0.6159 0.6836

0.8367 0.8940 0.8372 0.9044 0.9070 0.9736 0.9275 0.8592 0.8150 0.8282 0.8653 0.8411 0.7853 0.6590 0.8191 0.9611 0.8010 0.8166 0.9308 0.8313 0.7521 1.0348 0.7788 0.8583 0.6668 0.9824 0.7064 0.5841 0.6984 0.6514 0.7182 0.8696 0.8157 0.6631 0.8314 0.7590

0.6556 0.5887 0.5964 0.6669 0.6125 0.6814 0.6759 0.5702 0.6730 0.5891 0.5937 0.6274 0.5957 0.4745 0.5972 0.5826 0.6045 0.5937 0.5547 0.5554 0.5441 0.5879 0.5536 0.5848 0.5436 0.5538 0.5501 0.4979 0.5162 0.4993 0.5966 0.6473 0.5959 0.5440 0.5504 0.5938

0.8182 0.8247 0.7611 0.8826 0.8372 0.8700 0.9002 0.7809 0.7552 0.7939 0.8086 0.7651 0.7749 0.6245 0.7490 0.9215 0.7321 0.7110 0.9066 0.7215 0.6552 0.9810 0.6890 0.7534 0.6313 0.8808 0.6371 0.5647 0.6211 0.5706 0.7043 0.7805 0.7376 0.5952 0.7443 0.6490

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

C max ðSPTÞÞ C max ðARBÞ

C max ðNEHÞ C max ðARBÞ

C max ðLSÞ C max ðARBÞ

Mean

Max

Mean

Max

Mean

Max

0.7687 0.7652 0.7709 0.7574 0.7235 0.7693 0.7170 0.8225 0.8180 0.6862 0.7264 0.7587 0.7498 0.7143 0.7064 0.7102 0.6917 0.7554 0.7296 0.6909 0.7518 0.6917 0.6774 0.7404 0.6698 0.7385 0.7551 0.6480 0.7407 0.7000 0.7472 0.6876 0.7520 0.7104 0.7109 0.7098

0.9905 0.9274 0.8734 0.8599 0.8691 0.8617 0.8269 0.9703 0.9900 0.9453 0.9354 0.8235 0.8650 0.8835 0.8131 0.9027 0.7865 0.9363 0.8814 0.8756 0.9055 0.8823 0.9158 0.8038 0.8558 0.9168 0.8449 0.7089 0.8670 0.8080 0.8290 0.8069 0.8026 0.8064 0.8217 0.7916

0.7689 0.7654 0.7586 0.7594 0.7191 0.7544 0.7187 0.8204 0.7950 0.6872 0.7310 0.7382 0.7520 0.7149 0.6905 0.7123 0.6945 0.7303 0.7287 0.6913 0.7265 0.6930 0.6792 0.7144 0.6725 0.7348 0.7376 0.6495 0.7372 0.6782 0.7427 0.6844 0.7229 0.7233 0.7133 0.7026

0.9904 0.9305 0.8289 0.8598 0.8712 0.8575 0.8415 0.9663 0.9803 0.9277 0.9374 0.8262 0.8654 0.8862 0.7633 0.9065 0.7864 0.9029 0.8826 0.8836 0.8411 0.8911 0.9188 0.7925 0.8713 0.9156 0.8077 0.7125 0.8769 0.7629 0.8316 0.8017 0.7563 0.8331 0.8305 0.7966

0.7472 0.7267 0.7060 0.7396 0.6864 0.7067 0.6954 0.7748 0.7378 0.6709 0.6902 0.6795 0.7297 0.6760 0.6354 0.6912 0.6610 0.6673 0.7110 0.6474 0.6684 0.6729 0.6368 0.6491 0.6527 0.6919 0.6756 0.6354 0.6993 0.6247 0.7292 0.6446 0.6477 0.6931 0.6779 0.6275

0.9768 0.9077 0.7777 0.8355 0.8251 0.8044 0.8132 0.9250 0.8810 0.9167 0.9070 0.7565 0.8392 0.8378 0.6970 0.8814 0.7637 0.8377 0.8654 0.8072 0.7784 0.8475 0.8888 0.7160 0.8399 0.8912 0.7386 0.6939 0.8268 0.6728 0.8119 0.7481 0.6772 0.7740 0.7908 0.7114

304

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

Table 14 Computational results of the heuristics for ∑C j and m¼20.

Table 13 Computational results of the heuristics for ∑C j m¼10. n

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

∑C j ðSPTÞÞ ∑C j ðARBÞ

∑C j ðNEHÞ ∑C j ðARBÞ

n

∑C j ðLSÞ C max ðARBÞ

Mean

Max

Mean

Max

Mean

Max

0.6406 0.6417 0.7355 0.6186 0.6364 0.6730 0.6439 0.6500 0.7831 0.5677 0.5936 0.7249 0.6345 0.6386 0.6762 0.6276 0.6044 0.6274 0.6993 0.6454 0.6785 0.6072 0.5916 0.6602 0.6237 0.6600 0.6804 0.5914 0.5841 0.6851 0.5576 0.5964 0.6597 0.6159 0.6277 0.6080

0.8904 0.8496 1.0197 0.8425 0.8802 0.8100 0.9355 0.8144 0.9356 0.7674 0.7289 0.8309 0.7541 0.8839 0.8381 0.9491 0.8720 0.7932 0.9780 0.8374 0.7811 1.0006 0.7236 0.7456 0.9850 0.8511 0.8843 0.7730 0.8158 0.9149 0.6980 0.7162 0.7388 0.8441 0.8603 0.7456

0.6425 0.6441 0.7034 0.6214 0.6420 0.6461 0.6465 0.6391 0.7426 0.5688 0.5988 0.7011 0.6293 0.6418 0.6378 0.6319 0.6046 0.5914 0.6992 0.6473 0.6597 0.6075 0.5926 0.6305 0.6243 0.6834 0.6525 0.5994 0.5825 0.6562 0.5603 0.5993 0.6172 0.6216 0.6161 0.5828

0.8957 0.8644 0.9036 0.8614 0.8974 0.7828 0.9356 0.8248 0.8781 0.7611 0.7607 0.8095 0.7448 0.8836 0.7740 0.9576 0.8799 0.6701 0.9800 0.8562 0.7508 0.9908 0.7365 0.7330 0.9852 0.8472 0.8591 0.7758 0.8070 0.8124 0.7138 0.7298 0.7230 0.8993 0.8155 0.7073

0.6090 0.5905 0.6371 0.5939 0.5888 0.5788 0.6137 0.5880 0.6780 0.5421 0.5441 0.6256 0.6018 0.5808 0.5564 0.5891 0.5451 0.5213 0.6568 0.5795 0.5943 0.5712 0.5331 0.5525 0.5945 0.6036 0.5670 0.5587 0.5218 0.5739 0.5288 0.5352 0.5467 0.5793 0.5539 0.5133

0.8636 0.7949 0.8232 0.8232 0.8360 0.7044 0.8920 0.7734 0.8031 0.7415 0.6920 0.7315 0.7137 0.8057 0.6663 0.8573 0.7759 0.6164 0.9399 0.7440 0.6716 0.9355 0.6797 0.6490 0.9530 0.7700 0.7596 0.7291 0.7086 0.7308 0.6767 0.6518 0.6466 0.7878 0.7089 0.6527

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

∑C j ðSPTÞÞ ∑C j ðARBÞ

∑C j ðNEHÞ ∑C j ðARBÞ

∑C j ðLSÞ C max ðARBÞ

Mean

Max

Mean

Max

Mean

Max

0.7446 0.7198 0.7791 0.7427 0.7313 0.7935 0.6895 0.7678 0.7966 0.6870 0.7169 0.7349 0.7356 0.8222 0.7252 0.7831 0.7233 0.8284 0.7309 0.6942 0.7432 0.7194 0.6905 0.6626 0.6904 0.7307 0.7524 0.7369 0.7700 0.6778 0.6493 0.7494 0.6575 0.6288 0.7200 0.6877

0.9037 0.9859 0.9008 0.9431 0.9770 0.9558 0.7889 0.9994 0.8878 0.8039 0.9005 0.9940 0.8430 1.0248 0.8275 1.0164 0.9221 0.9514 0.9459 0.8080 0.8804 0.8678 0.7868 0.7536 0.8570 0.9209 0.8912 0.7995 0.9102 0.7618 0.7366 0.9167 0.7592 0.6903 0.7570 0.8869

0.7420 0.7170 0.7713 0.7443 0.7330 0.7801 0.6846 0.7700 0.7783 0.6894 0.7151 0.7289 0.7353 0.8183 0.7183 0.7858 0.7254 0.7915 0.7328 0.6957 0.7291 0.7226 0.6915 0.6506 0.6911 0.7316 0.7418 0.7401 0.7725 0.6609 0.6497 0.7497 0.6389 0.6303 0.7213 0.6817

0.9024 0.9518 0.9022 0.9455 1.0001 0.8816 0.7893 1.0038 0.8514 0.8086 0.9044 0.9611 0.8436 1.0210 0.8149 1.0168 0.9237 0.9426 0.9551 0.8158 0.8622 0.8691 0.7898 0.7228 0.8581 0.9248 0.8257 0.7996 0.8940 0.7293 0.7378 0.9141 0.7243 0.6922 0.7695 0.8447

0.7228 0.6830 0.7248 0.7272 0.6942 0.7226 0.6699 0.7316 0.7273 0.6727 0.6800 0.6790 0.7124 0.7770 0.6591 0.7638 0.6899 0.7388 0.7120 0.6562 0.6646 0.6985 0.6497 0.5877 0.6717 0.6932 0.6718 0.7237 0.7305 0.5996 0.6357 0.7058 0.5860 0.6091 0.6806 0.6166

0.8820 0.9230 0.8459 0.9340 0.9318 0.8345 0.7791 0.9460 0.8132 0.7942 0.8668 0.8809 0.8099 0.9711 0.7726 0.9717 0.8800 0.8980 0.9295 0.7573 0.7954 0.8314 0.7408 0.6441 0.8353 0.8758 0.7644 0.7819 0.8585 0.6731 0.7277 0.8512 0.6674 0.6769 0.7116 0.7950

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

Table 15 Computational results of the heuristics for ∑wj C j and m¼ 10. n

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

305

Table 16 Computational results of the heuristics for ∑wj C j and m ¼20.

∑wj C j ðWSPTÞÞ ∑wj C j ðARBÞ

∑wj C j ðNEHÞ ∑wj C j ðARBÞ

∑wj C j ðLSÞ ∑wj C j ðARBÞ

Mean

Max

Mean

Max

Mean

Max

0.7625 0.7326 0.7880 0.7401 0.7523 0.7694 0.6727 0.7778 0.7713 0.7491 0.6621 0.8007 0.7228 0.6984 0.7199 0.7098 0.7850 0.7189 0.7313 0.6878 0.7501 0.7417 0.7245 0.6879 0.7771 0.7240 0.7472 0.7343 0.6436 0.6914 0.6344 0.6396 0.6715 0.6470 0.5904 0.7037

1.1486 0.9228 0.9976 0.9791 1.0544 0.9205 0.9198 0.9508 0.9127 1.0496 0.8593 0.9524 0.9300 1.0663 0.9197 1.0707 1.1197 0.9046 1.3308 0.9696 0.9215 0.9776 1.0997 0.8908 1.2550 0.9744 0.9621 0.9753 0.7729 0.8599 0.6753 0.8588 0.7446 0.8847 0.6849 0.8111

0.6566 0.5808 0.6837 0.6155 0.6700 0.6813 0.5964 0.7244 0.6584 0.6384 0.5534 0.7149 0.6162 0.6341 0.6513 0.6007 0.6250 0.6494 0.6205 0.5682 0.6501 0.6306 0.5956 0.6106 0.6851 0.6462 0.6338 0.6430 0.5430 0.5946 0.6241 0.5132 0.5686 0.5634 0.4894 0.6287

1.0012 0.7557 0.7542 0.8401 0.9584 0.8639 0.7050 0.8770 0.8092 0.8758 0.6960 0.8623 0.7823 0.9857 0.8600 0.7746 0.8448 0.8390 0.8091 0.7455 0.8171 0.7692 0.7966 0.7659 0.9568 0.7467 0.7897 0.8282 0.7183 0.7293 0.6812 0.6443 0.6465 0.8425 0.5644 0.7987

0.6249 0.5298 0.6128 0.5830 0.6200 0.6046 0.5589 0.6684 0.5923 0.6098 0.5029 0.6510 0.5887 0.5795 0.5886 0.5671 0.5645 0.5798 0.5731 0.5153 0.5729 0.5946 0.5351 0.5286 0.6475 0.5734 0.5482 0.6029 0.4970 0.5153 0.5886 0.4552 0.5063 0.5283 0.4416 0.5475

0.9516 0.6952 0.6700 0.7846 0.8768 0.7735 0.6723 0.8042 0.7417 0.8414 0.6134 0.7960 0.7757 0.9295 0.7579 0.7514 0.7395 0.7704 0.7141 0.6733 0.7231 0.7053 0.6749 0.6308 0.9303 0.7058 0.6664 0.7808 0.6632 0.5887 0.6545 0.5795 0.5854 0.7649 0.5411 0.6555

n

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

∑wj C j ðWSPTÞÞ ∑wj C j ðARBÞ

∑wj C j ðNEHÞ ∑wj C j ðARBÞ

∑wj C j ðLSÞ ∑wj C j ðARBÞ

Mean

Max

Mean

Max

Mean

Max

0.8412 0.8451 0.8359 0.8307 0.8097 0.8522 0.8126 0.9026 0.8573 0.8394 0.8749 0.7778 0.8159 0.7655 0.7838 0.8069 0.8220 0.8116 0.7492 0.7569 0.9074 0.8241 0.8571 0.7156 0.7690 0.7354 0.7614 0.7712 0.8291 0.8211 0.8188 0.7891 0.7937 0.7023 0.7783 0.8031

1.2350 1.0661 1.0783 1.0103 1.0126 1.0377 0.9698 1.1406 1.0164 1.1266 1.0520 0.9142 1.2100 0.9531 0.9714 0.9934 1.0537 1.0267 0.9381 0.9788 1.0972 0.9326 1.0963 0.9483 0.9268 0.9923 0.8911 0.8687 1.0227 0.9488 0.9985 0.8440 0.8630 0.8773 0.8615 0.8278

0.7275 0.7395 0.7454 0.7333 0.7401 0.7868 0.6934 0.7900 0.7747 0.7091 0.7578 0.6955 0.6986 0.6909 0.6979 0.7271 0.7648 0.7438 0.6660 0.6848 0.8080 0.7108 0.7533 0.6478 0.6718 0.6737 0.7067 0.6530 0.6911 0.7169 0.6908 0.6613 0.7282 0.6554 0.7260 0.7109

0.9577 0.8875 0.9221 0.8620 0.8601 0.9693 0.9529 0.9632 0.9435 1.0000 0.9523 0.7745 0.8913 0.9003 0.8091 0.7919 0.9545 0.9098 0.7571 0.8180 0.9543 0.8642 0.8525 0.7414 0.8612 0.8304 0.8292 0.7466 0.8415 0.8635 0.8588 0.7290 0.7550 0.8808 0.7918 0.7570

0.7071 0.7062 0.6819 0.7171 0.7071 0.7324 0.6735 0.7565 0.7195 0.6862 0.7207 0.6406 0.6812 0.6585 0.6484 0.7077 0.7252 0.6890 0.6490 0.6508 0.7390 0.6947 0.7127 0.5902 0.6568 0.6375 0.6502 0.6284 0.6480 0.6613 0.6689 0.6219 0.6631 0.6340 0.6847 0.6443

0.9346 0.8427 0.8417 0.8470 0.8341 0.8671 0.9257 0.9053 0.8639 0.9617 0.9126 0.6996 0.8759 0.8700 0.7512 0.7686 0.8967 0.8489 0.7450 0.7847 0.8617 0.8563 0.8028 0.6858 0.8485 0.8112 0.7911 0.7132 0.7767 0.8022 0.8307 0.7032 0.6814 0.8408 0.7527 0.6852

306

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

Table 18 Computational results of the heuristics for ρ ¼ ∑wj ð1 e  γC j Þ and m¼ 20.

Table 17 Computational results of the heuristics for ρ ¼ ∑wj ð1 e  γC j Þ and m ¼10. n

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

ρðNEHÞ ρðARBÞ

ρðWDSPTÞ ρðARBÞ

n

ρðLSÞ ρðARBÞ

Mean

Max

Mean

Max

Mean

Max

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

ρðNEHÞ ρðARBÞ

ρðWDSPTÞ ρðARBÞ

ρðLSÞ ρðARBÞ

Mean

Max

Mean

Max

Mean

Max

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

307

Table 19

Table 20

Computational results of the heuristics for ∑C 2j m¼10.

Computational results of the heuristics for ∑C 2j and m ¼20.

n

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

∑C 2j ðSPTÞÞ

∑C 2j ðNEHÞ

∑C 2j ðLSÞ

∑C 2j ðARBÞ

∑C 2j ðARBÞ

∑C 2j ðARBÞ

n

Mean

Max

Mean

Max

Mean

Max

0.4038 0.4275 0.5676 0.5152 0.5169 0.5327 0.5366 0.5732 0.5378 0.4009 0.4718 0.4617 0.4842 0.4124 0.5154 0.4281 0.3771 0.4566 0.3047 0.4465 0.4190 0.3388 0.3488 0.3983 0.4104 0.3500 0.4264 0.3375 0.2932 0.5146 0.3808 0.2576 0.4162 0.3763 0.4844 0.4971

0.8750 0.8952 0.8507 0.8869 0.7629 0.7513 0.8605 0.9628 0.7027 0.8560 0.9719 0.7520 0.9262 0.6819 0.7840 0.7856 0.6471 0.7466 0.4269 0.8245 0.8068 0.6367 0.5552 0.5174 0.6660 0.4680 0.5268 0.6452 0.4440 0.8439 0.6475 0.3820 0.4952 0.7049 0.8615 0.6728

0.4079 0.4281 0.5286 0.5202 0.4890 0.5027 0.5413 0.5866 0.4952 0.4028 0.4598 0.4230 0.4902 0.4122 0.4637 0.4296 0.3828 0.4165 0.3161 0.4587 0.3703 0.3355 0.3526 0.3688 0.4172 0.3539 0.3862 0.3434 0.2853 0.4585 0.3769 0.2713 0.3702 0.3738 0.4504 0.4253

0.8718 0.9265 0.8021 0.9038 0.6473 0.7054 0.8749 0.9740 0.6182 0.8510 0.8097 0.6551 0.9269 0.6954 0.7727 0.7531 0.6582 0.6205 0.4588 0.8376 0.5959 0.5662 0.5719 0.4668 0.6747 0.4758 0.4860 0.6556 0.4148 0.6561 0.6201 0.4191 0.4548 0.7043 0.8564 0.5236

0.3674 0.3702 0.4346 0.4750 0.4223 0.4109 0.4772 0.4790 0.3995 0.3626 0.3836 0.3465 0.4279 0.3406 0.3863 0.3742 0.3172 0.3316 0.2743 0.3522 0.2930 0.3002 0.2783 0.2745 0.3623 0.2834 0.3069 0.2982 0.2314 0.3482 0.3233 0.2140 0.2846 0.3239 0.3772 0.3409

0.8074 0.8101 0.6704 0.8500 0.5865 0.6123 0.7831 0.7603 0.5262 0.7554 0.6939 0.5319 0.8247 0.5338 0.6504 0.6452 0.6167 0.5174 0.3977 0.6967 0.5072 0.5357 0.4577 0.3394 0.6163 0.4103 0.3925 0.5755 0.3292 0.4992 0.5442 0.3240 0.3425 0.5812 0.7281 0.4292

40

a

0.5

0.7

0.9

60

0.5

0.7

0.9

80

0.5

0.7

0.9

100

0.5

0.7

0.9

b

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

∑C 2j ðSPTÞÞ

∑C 2j ðNEHÞ

∑C 2j ðLSÞ

∑C 2j ðARBÞ

∑C 2j ðARBÞ

∑C 2j ðARBÞ

Mean

Max

Mean

Max

Mean

Max

0.5491 0.6312 0.5943 0.5845 0.5517 0.6061 0.5026 0.5561 0.6064 0.5455 0.5006 0.6496 0.5135 0.5435 0.5632 0.6519 0.5511 0.5401 0.4737 0.4643 0.5641 0.4918 0.5007 0.6197 0.5298 0.5864 0.5493 0.4976 0.5042 0.5669 0.5420 0.4198 0.4929 0.4445 0.5153 0.6289

0.7421 0.9655 0.7180 0.9092 0.8348 0.9086 0.8765 0.8743 0.7238 0.9477 0.7262 0.8851 0.9514 0.8302 0.7645 1.0066 0.8692 0.7613 0.7352 0.6375 0.8286 0.6949 0.7607 0.9137 0.9762 0.9791 0.7818 0.5629 0.6023 0.7752 0.6792 0.5489 0.6834 0.6479 0.8283 0.9065

0.5473 0.6260 0.5853 0.5886 0.5541 0.6029 0.5030 0.5560 0.5813 0.5440 0.5020 0.6318 0.5159 0.5456 0.5255 0.6472 0.5490 0.5300 0.4745 0.4634 0.5281 0.4915 0.5019 0.5733 0.5252 0.5765 0.5271 0.5029 0.5101 0.5350 0.5462 0.4200 0.4658 0.4468 0.5175 0.5787

0.7431 0.9694 0.6934 0.9090 0.8471 0.9433 0.8777 0.8865 0.7354 0.9409 0.7180 0.8507 0.9475 0.8318 0.7221 0.9564 0.8856 0.7271 0.7338 0.6170 0.7746 0.6968 0.7423 0.8462 0.9649 0.9463 0.7264 0.5783 0.6198 0.7155 0.6929 0.5551 0.6818 0.6526 0.8355 0.8081

0.5182 0.5786 0.5085 0.5598 0.5041 0.5095 0.4782 0.5076 0.5041 0.5183 0.4528 0.5301 0.4939 0.5037 0.4541 0.6231 0.4812 0.4567 0.4462 0.4225 0.4469 0.4681 0.4506 0.4768 0.4832 0.5072 0.4389 0.4751 0.4427 0.4561 0.5200 0.3762 0.3751 0.4208 0.4634 0.4731

0.7121 0.8659 0.6294 0.8548 0.7554 0.7620 0.8637 0.7847 0.6357 0.8900 0.6403 0.7336 0.9273 0.7743 0.6601 0.9350 0.7603 0.6464 0.6792 0.5760 0.6676 0.6545 0.6724 0.7341 0.8238 0.8431 0.6425 0.5394 0.5447 0.6005 0.6570 0.5114 0.5284 0.6131 0.7960 0.6682

308

J.-B. Wang, J.-J. Wang / Computers & Operations Research 43 (2014) 292–308

the heuristic LS is more effective than the other heuristics for the objective functions C max , ∑C j , ∑wj C j and ∑C 2j . From Tables 17 and 18, we see that the performance of the WDSPT, NEH, and LS is almost as much as the ARB rule for the objective function ∑wj ð1  e  γC j Þ. 6. Conclusions In this paper, several heuristic algorithms for the flowshop scheduling with a general exponential learning effect have been developed with five objective functions: the makespan, the total completion time, the total weighted completion time, the total weighted discounted completion time, and the sum of the quadratic job completion times. We analyzed the worst-case bound of these heuristic algorithms. Future research may focus on extending the results to the case with multiple machines or considering a metaheuristic approach involving a genetic or Tabu search algorithm. Acknowledgments The authors are grateful for two anonymous referees for their helpful comments on earlier version of the paper. This research was supported by the National Natural Science Foundation of China (Grant nos. 11001181, 71271039, and 70902033), the Program for Liaoning Excellent Talents in University (Grant no. LJQ2011014), and the open project of The State Key Laboratory for Manufacturing Systems Engineering (Grant no. sklms201306). References [1] Badiru AB. Computational survey of univariate and multivariate learning curve models. IEEE Trans Eng Manage 1992;39:176–88. [2] Bai J, Wang M-Z, Wang J-B. Single machine scheduling with a general exponential learning effect. Appl Math Model 2012;36:829–35. [3] Biskup D. Single-machine scheduling with learning considerations. Eur J Oper Res 1999;115:173–8. [4] Biskup D. A state-of-the-art review on scheduling with learning effects. Eur J Oper Res 2008;188:315–29. [5] Framinan JM, Gupta JND, Leisten R. A review and classification of heuristics for permutation flow-shop scheduling with makespan objective. J Oper Res Soc 2004;55:1243–55. [6] Framinan JM, Leisten R. An efficient constructive heuristic for flowtime minimization in permutation flow shops. OMEGA: Int J Manage Sci 2003;31:311–7.

[7] Framinan JM, Leisten M, Ruiz-Usano R. Comparison of heuristics for flow time minimization in permutation flow shop. Comput Oper Res 2005;32:1237–54. [8] Gonzalez T, Sahni S. Flowshop and jobshop schedules: complexity and approximation. Oper Res 1978;26:36–52. [9] Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG. Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 1979;5:287–326. [10] Janiak A, Rudek R. Experience based approach to scheduling problems with the learning effect. IEEE Trans Syst Man Cybern Part A 2009;39(2):344–57. [11] Johnson SM. Optimal two-and-three-stage production schedules. Nav Res Logist 1954;1:61–8. [12] Kuo W-H, Hsu C-J, Yang D-L. Worst-case and numerical analysis of heuristic algorithms for flowshop scheduling problems with a time-dependent learning effect. Inf Sci 2012;184:282–97. [13] Koulamas C, Kyparisis GJ. Algorithms with performance guarantees for flow shops with regular objective functions. IIE Trans 2005;37:1107–11. [14] Laha D, Sarin SC. A heuristic to minimize total flowtime in permutation flowshop. Omega 2009;37:734–9. [15] Lee W-C, Wu C-C. Some single-machine and m-machine flowshop scheduling problems with learning considerations. Inf Sci 2009;179:3885–92. [16] Li G, Wang X-Y, Wang J-B, Sun L-Y. Worst case analysis of flow shop scheduling problems with a time-dependent learning effect. Int J Product Econ 2013;142:98–104. [17] Nagarajan V, Sviridenko M. Tight bounds for permutation flow shop scheduling. Math Oper Res 2009;34:417–27. [18] Nawaz M, Enscore Jr EE, Ham I. A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. OMEGA: Int J Manage Sci 1983;11:91–5. [19] Pinedo M. Scheduling: theory, algorithms and systems. Englewood Cliffs, New Jersey: Prentice Hall; 2002. [20] Rudek R. Computational complexity and solution algorithms for flowshop scheduling problems with the learning effect. Comput Ind Eng 2011;61:20–31. [21] Smith WE. Various optimizers for single-stage production. Nav Res Logist 1956;3:59–66. [22] Smutnicki C. Some results of worst-case analysis for flow shop scheduling. Eur J Oper Res 1998;109:66–87. [23] Sun L-H, Cui K, Chen J-H, Wang J, He X-C. Some results of the worst-case analysis for flow shop scheduling with a learning effect. Ann Oper Res, doi:10. 1007/s10479-013-1368-6. [24] Townsend W. The single machine problem with quadratic penalty function of completion times: a branch-and-bound solution. Manage Sci 1978;24:530–4. [25] Wang J-B, Wang M-Z. Worst-case behavior of simple sequencing rules in flow shop scheduling with general position-dependent learning effects. Ann Oper Res 2011;191:155–69. [26] Wang J-B, Wang M-Z. Worst-case analysis for flow shop scheduling problems with an exponential learning effect. J Oper Res Soc 2012;63:130–7. [27] Wang J-B, Xia Z-Q. Flow shop scheduling with a learning effect. J Oper Res Soc 2005;56:1325–30. [28] Wang X-Y, Zhou Z, Zhang X, Ji P, Wang J-B. Several flow shop scheduling problems with truncated position-based learning effect. Comput Oper Res 2013;40:2906–29. [29] Wu C-C, Lee W-C. A note on the total completion time problem in a permutation flowshop with a learning effect. Eur J Oper Res 2009;192:343–7. [30] Xu Z, Sun L, Gong J. Worst-case analysis for flow shop scheduling with a learning effect. Int J Product Econ 2008;113:748–53.