Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 754826, 9 pages

http://dx.doi.org/10.1155/2013/754826

Research Article

Competitive Two-Agent Scheduling with Learning Effect and Release Times on a Single Machine

Der-Chiang Li¹ and Peng-Hsiang Hsu1,2

¹Department of Industrial and Information Management, National Cheng Kung University, No. 1, University Road, Tainan 70101, Taiwan

²Department of Business Administration, Kang-Ning Junior College, Taipei, Taiwan

Received 16 May 2013; Accepted 2 July 2013

Academic Editor: Yunqiang Yin

Copyright © 2013 Der-Chiang Li and Peng-Hsiang Hsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The learning effect has gained much attention in the scheduling research recently, where many researchers have focused their problems on only one optimization. This study further addresses the scheduling problem in which two agents compete to perform their own jobs with release times on a common single machine with learning effect. The aim is to minimize the total weighted completion time of the first agent, subject to an upper bound on the maximum lateness of the second agent. We propose a branch-and-bound approach with several useful dominance properties and an effective lower bound for searching the optimal solution and three simulated-annealing algorithms for the near-optimal solutions. The computational results show that the proposed algorithms perform effectively and efficiently.

1. Introduction

In traditional scheduling problems, most studies assumed that each of the operations of all job processing times is known and fixed. But in some circumstances, the jobs processing times are affected by learning effect. The ‘‘learning effect” is the phenomenon that unit costs reduce as firms produce more of a product and gain knowledge or experience. Biskup [1] and Cheng and Wang [2] first brought the phenomenon of learning effect into scheduling field. Afterwards the learning effect is getting much to pay attention in the scheduling research in the last decade such as Mosheiov [3], Mosheiov and Sidney [4], Bachman and Janiak [5], Lee and Wu [6], Kuo and Yang [7], and Koulamas and Kyparisis [8]. Biskup [9] provided the most recent learning effects survey paper in scheduling research. More recent studies involving learning effects were by Wang et al. [10], Janiak and Rudek [11], Yin et al. [12], Toksari and Güner [13], Wang et al. [14], Lee et al. [15], J.-B. Wang and C. Wang [16], Wu et al. [17], Zhang et al. [18], Yin et al. [19], and Yang et al. [20], Yang et al. [21], Wang et al. [22], J.-B. Wang and J.-J. Wang [23], Cheng et al. [24], and so forth.

In addition, most research assumed that all jobs met a single criterion. But jobs might come from different agents. There might be multiple agents who compete on the same resources, and each agent has its own objective. This concept was first initiated and considered into scheduling field by Baker and Smith [25] and Agnetis et al. [26]. After that time, many researchers focused on multiagent in scheduling field. However, little research has been done on scheduling problem with learning effect and multiagent. Liu et al. [27] studied the optimal polynomial time algorithms to solve a single-machine scheduling problem with two-agent and position-dependent processing time aging and learning effect. The objective is to find a schedule that minimizes the total completion time of the first agent with a maximum cost limit of the second agent. Cheng et al. [28] investigated a two-agent single-machine scheduling problem with a truncated sum-of-processing times-based learning effect and developed algorithms to minimize the total weighted completion time of the jobs of the first agent, subject to the restriction that no tardy job is allowed for the second agent. Li and Hsu [29] investigated the job scheduling problem of two agents competing for the usage of a common single machine with learning effect. The objective is to minimize the total weighted completion time of both agents with the restriction that the makespan of either agent cannot exceed an upper bound. Wu et al. [30] address a two-agent single-machine scheduling problem with the co-existing sum-of-processing times-based deteriorating and learning effects. The goal is to minimize the total weighted completion time of the jobs of the first agent given that no tardy job is allowed for the second agent.

The previous both scheduling issues were yet relatively unexplored. In this paper, we therefore study a two-agent scheduling problem with position-based learning competing on a common single machine and further consider each job a different release time. The objective is to minimize the total weighted completion time of first agents, subject to the constraint that the maximum lateness of second agent the jobs cannot exceed an upper bound. In the classical scheduling notation, the problem can be notated by a triplet as . As shown by Agnetis et al. [26], it is a Binary NP-hard problem, even without release time and learning effect ().

The rest of this paper is organized as follows: the problem formulation is introduced in the next section. The branch-and-bound and simulated-annealing algorithms are employed to find the optimal solution and the near-optimal solutions, respectively. Dominance properties and a lower bound are developed to be used in the branch-and-bound algorithm in Section 3. The details of simulated-annealing algorithm are described in Section 4, and the computational experiment results are given in Section 5. In the last section, some conclusions and extensions are presented.

2. Problem Formulation

In this section, we describe a formal definition of the model as there are jobs from two competing agents ( = agent or agent ) to be scheduled. The number of jobs in the two sets is recorded as and , such that . The processing time and weight of job are known and denoted as and , respectively. Based on the learning effect, the actual processing time of job changes with position and learning ratio , that is , where and .

Furthermore, we consider the problem of scheduling a set of independent jobs with integer release times () on single machine. The goal is to minimize the weighted sum of completion times of the jobs from agent , subject to the constraint that the maximum lateness of the jobs from agent is not more than a given upper bound .

3. Branch-and-Bound Algorithm

The computational complexity of this problem does not consider ready time and learning effect (), which is showed a Binary NP-hard problem by Agnetis et al. [26]. Therefore, the addressed problem here () is more complex than the problem (). We basically try to employ the branch-and-bound algorithm to gain the optimal solution, and for speeding up the searching process, several dominance properties and a lower bound were presented in the following.

3.1. Dominance Property

Suppose that there are two contiguous jobs ( and ) in sequence , where and denote the scheduled and unscheduled partial sequence, respectively. One can perform the contiguous jobs and interchanges to obtain another sequence . To show that dominates , it is sufficient to ensure or . Before proving the proposed properties, we let denote the completion time of the last job in the scheduled partial sequence to determine the following properties.

Property 1. If , , , and , then dominates .

Proof. By the definition, the completion times of jobs and in and are, respectively,

Since and . Moreover,

By and , we have

hence , as required.

The proofs of Properties 2–4 are omitted since they are similar to that of Property 1.

Property 2. If ,, and , then dominates .

Property 3. If , , and , then dominates .

Property 4. If ,, and , then dominates .

In addition, let be a sequence of the jobs where is the scheduled part with jobs, and is the unscheduled part with jobs. The following property is found for determining the sequence feasibility by the unscheduled . Moreover, is the completion time of the last job in .

Property 5. If there is a in such that , then sequence () is not a feasible solution.

Proof. Since , lateness of the unscheduled jobs must exceed the given bound . So () is not a feasible sequence.

The next property is for assigning the unscheduled in ()th position.

Property 6. If all the unscheduled jobs belong to and there exists an such that , where for all jobs and , then job may be assigned to the ()th position.

3.2. Lower Bound

In addition to the previous properties, we hatch up a lower bound to speed up the building of searching trees in the branch-and-bound algorithm. Assume that PS and US are the two partial scheduling sequences. PS is the scheduled jobs, and US is the remaining unscheduled jobs in which is agent jobs and is agent jobs, where . The lower bound is obtained by scheduling agent jobs first and then scheduling agent jobs in any order. To elaborate this, let be the completion time of the last job in PS; then the completion time for the ()th job is

Similarly, the completion time for the job is

Hence the lower bound of the partial sequence PS can thus be found as follows:

4. Simulated-Annealing Algorithm

The simulated-annealing algorithm is one of the meta-heuristic methods to solve large-scaled combinatorial minimization problems [28, 31–34]. It was first described by Kirkpatrick et al. [35] based upon the research of Metropolis et al. [36]. The major advantage of this approach is that it avoids getting trapped in local minima for global optimization by controlling the parameter which influences the probability of accepting a worse solution in the iterative process. Here, we employ SA algorithm to obtain near-optimal solutions as described in the following.

Step 1 (initial feasible solution). An initial feasible sequence was generated by putting agent in front of agent for considering the conditionality objective. Thus, we employ the earliest due date (EDD) rule for agent first, and for agent , four different rules are employed as EDD rule, shortest processing time (SPT) rule, shortest release time (SRT) rule, and weighted shortest processing time (WSPT) rule; they were denoted by SA₁, SA₂, SA₃, and SA₄, respectively.

Step 2 (adjusting the solution). To improve on the initial schedule, we shift the neighborhood job schedules. The exchange sequence strategy procedures were to choose two different locations randomly and irregularly select one of the three resources (pairwise interchanges, extraction, and forward/backward shifted reinsertions) to ameliorate the quality of the SA.

Step 3 (acceptance probability). If there exists a new schedule that improves the value of the objective, it replaces the previous schedule. Besides, the SA algorithms prevent it to get stuck to local minima with an acceptance probability. The acceptance probability, given in the next equation, is based on the exponential distribution:

where is the control parameter and is the variation of the objective value. If , the new sequence is accepted, otherwise new sequence will be rejected. Ben-Arieh and Maimon [37] suggested that in the th iteration is

where is an experimental constant. After preliminary trials, was used in our experiment.

4.1. Stopping Condition Iterations

The SA algorithms were terminated after 300 iterations in our preliminary experiments, where is the number of jobs.

5. Computational Experiments

A computational experiment was conducted to assess the performance of the proposed branch-and-bound algorithm and the accuracy of the SA algorithms. All the algorithms were coded in Compaq Visual Fortran version 6.6 and run on an Intel(R) Core(TM) i7-2600 CPU @ 3.40 GHz with 4 GB RAM operating system under Windows 7 environment. The experimental design followed Chu's [38] and Fisher [39] framework, where the normal job processing times were generated randomly from a uniform distribution over the integers between 1 and 100; the release times were from a uniform distribution over the integers (0, 50.5), where was the job size and was a control variable; the due dates were from a uniform distribution over the range of integers to , where ,, and are the sum of the processing times of all the jobs , the tardiness factor, and the due date range, respectively. Furthermore, the bound was fixed at 0 and each agent has half of the total number of jobs.

5.1. The Accuracy of SAs for Small Job Size

First, to assess the accuracy of SA algorithms, the error percentage was calculated as

where was the solution obtained from the SA algorithm and OP was the optimal solution of the objective function obtained from the branch-and-bound algorithm. In the first simulation experiment, the job size , tardiness factor (), the due date range (), and learning effect (le) were fixed at  = . The values of were taken as 0.2,,,,,,,,, and 3.0. For each situation, 100 replications were randomly generated. The mean and maximum of the error percentages were recorded in Table 1. The results showed that SA₁, SA₂, SA₃, and SA₄ are not affected by the variation of . It was observed that the mean error percentage of the SA algorithm was less than 0.1352%. In order to diminish some peculiar worst case of SAs, we combined the four SAs to obtain SA₅ which was the smallest value of the objective function obtained from the SAs. The mean error percentage of the SA₅ was less than 0.0429%.

Table 1: The performance of the simulated-annealing algorithms at .

λ
SA₁
SA₂
SA₃
SA₄
SA₅

Mean
Max
Mean
Max
Mean
Max
Mean
Max
Mean
Max

0.20
0.0252
0.6711
0.0326
1.2731
0.0294
1.2731
0.0239
1.2731
0.0020
0.2014

0.40
0.0539
1.9013
0.0724
1.9013
0.0911
1.9013
0.1352
5.8067
0.0151
1.4364

0.60
0.0433
0.9016
0.0954
5.5725
0.0803
4.4405
0.0919
4.6113
0.0103
0.3001

0.80
0.0595
1.8690
0.0412
0.4131
0.0465
1.6664
0.0543
1.7206
0.0062
0.1241

1.00
0.0710
3.7936
0.1013
3.7936
0.0994
3.7936
0.0907
3.7936
0.0429
3.7936

1.25
0.0423
0.2853
0.1068
5.7572
0.0433
0.4495
0.0371
0.6049
0.0045
0.0665

1.50
0.0359
0.4688
0.0414
0.2745
0.0331
0.3646
0.0331
0.2364
0.0048
0.0651

1.75
0.0671
3.3128
0.0414
0.3602
0.0671
3.3128
0.0413
0.3486
0.0057
0.1112

2.00
0.0358
0.3843
0.0329
0.3610
0.0504
0.3250
0.0314
0.3368
0.0091
0.2887

3.00
0.0355
0.2558
0.0420
0.2762
0.0321
0.2738
0.0316
0.3117
0.0066
0.1416

In another simulation experiment, variables were fixed at . The values of () were taken as , , , , , and . For each situation, 100 replications were randomly generated. The mean and maximum of the percentage errors were also recorded in Table 2. The results were similar to the former experiment. The mean error percentage of the SA₅ was less than 0.0151%. By the two experiment results, it was recommended to combine SAs into SA₅.

Table 2: The performance of the simulated-annealing algorithms at .

τ
γ
SA₁
SA₂
SA₃
SA₄
SA₅

Mean
Max
Mean
Max
Mean
Max
Mean
Max
Mean
Max

0.25
0.25
0.0252
0.6711
0.0326
1.2731
0.0294
1.2731
0.0239
1.2731
0.0020
0.2014

0.50
0.0539
1.9013
0.0724
1.9013
0.0911
1.9013
0.1352
5.8067
0.0151
1.4364

0.75
0.0433
0.9016
0.0954
5.5725
0.0803
4.4405
0.0919
4.6113
0.0103
0.3001

0.50
0.25
0.0671
3.3128
0.0414
0.3602
0.0671
3.3128
0.0413
0.3486
0.0057
0.1112

0.50
0.0358
0.3843
0.0329
0.3610
0.0504
0.3250
0.0314
0.3368
0.0091
0.2887

0.75
0.0355
0.2558
0.0420
0.2762
0.0321
0.2738
0.0316
0.3117
0.0066
0.1416

5.2. Performance of the Branch-and-Bound Algorithm for Small Job Size

To evaluate the performance of the branch-and-bound algorithm, the mean and maximum numbers of nodes were recorded as well as the computation times (in seconds) to calculate the mean and maximum computation times. In this simulation experiment, the parameters were set as the job size ( and 14), release times control value (,,,,,,,,, and 3.0), tardiness factor ( and 0.5), the due date range ( and 0.75), and the learning effect (le = 70% and 90%). For each situation, 100 replications were randomly generated to yield a total of 16000 instances. Table 3 summarized the performance of the branch-and-bound and SA algorithms. It indicated as follows.

Table 3: The performance of the branch-and-bound and simulated-annealing algorithms in and .

le

Branch-and-bound algorithm

Branch-and-bound algorithm

Node
Node
NSI
SA₅
Node
Mean
NSI
SA₅

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

0.2
0.25
0.25
70
7953180
57938701
60.47
492.31
100
0.0012
0.0524
234798074
490877044
2081.69
4386.73
73
0.0201
0.8312

90
4740856
27074133
44.13
259.24
100
0.1540
8.7447
130221904
456315410
1437.82
5207.04
93
0.0044
0.2050

0.75
70
7266869
68305607
56.12
577.05
100
0.0008
0.0672
181473222
499210262
1646.57
4692.44
81
0.0107
0.8222

90
1669247
17432807
15.03
161.55
100
0.0157
0.6238
74978873
422464059
830.62
4819.28
97
0.2485
20.4500

0.5
0.25
70
4885377
30357870
37.28
260.68
100
0.0117
1.1747
185730807
467124530
1815.88
4535.62
83
0.0136
0.8498

90
420242
4048018
4.20
40.48
100
0.6183
14.4734
13870486
163643109
173.24
2134.31
100
0.5811
15.4566

0.75
70
2168845
35858649
16.14
291.38
100
0.0736
2.7652
95878107
488272101
956.07
5041.92
96
0.1735
6.8935

90
553857
8010627
5.23
77.77
100
0.2046
11.3446
14254450
139078335
162.34
1740.41
100
0.4214
23.7347

0.4
0.25
0.25
70
7095766
25477833
52.13
195.94
100
0.0032
0.0972
224196847
494946864
2117.99
5289.59
69
0.0057
0.1331

90
1437139
8454985
13.72
88.31
100
0.0661
5.5203
53829459
368620154
640.41
4722.94
99
0.1643
6.3549

0.75
70
3266325
34176162
23.38
240.77
100
0.0077
0.4400
139740047
458655160
1351.59
5074.59
92
0.0066
0.2338

90
899555
7207133
8.27
66.58
100
0.3497
10.3461
25911608
163977599
293.97
2093.06
99
0.4423
18.5009

0.5
0.25
70
1547759
9729009
11.64
73.84
100
0.0688
2.7329
70707385
375376204
715.40
3852.09
99
0.0359
1.6350

90
90231
626524
0.89
6.21
100
0.4160
13.1596
4073837
54407366
51.53
721.13
100
0.7376
21.6059

0.75
70
1351836
10319114
9.67
75.99
100
0.0049
0.1158
53013074
403629962
540.22
4223.36
98
0.0106
0.3034

90
197252
2130612
1.83
20.01
100
0.3840
14.2713
5725593
47250298
64.77
564.13
100
0.2255
4.8593

0.6
0.25
0.25
70
3862510
16167618
27.75
122.24
100
0.0071
0.1072
171949416
449957311
1674.82
4512.44
85
0.0149
0.1546

90
639113
2425615
6.18
23.70
100
0.1761
12.3540
29277986
233944079
356.17
2956.13
100
0.1433
3.3191

0.75
70
2091778
15560184
14.85
119.36
100
0.0055
0.0945
111408553
497252899
1019.55
4674.72
99
0.0284
1.4289

90
435481
4364092
3.89
42.06
100
0.1133
8.4358
13830333
162070288
152.22
1924.88
100
0.1332
3.4404

0.5
0.25
70
706061
5309118
5.16
40.39
100
0.0324
2.8857
35336302
316279317
357.65
3325.72
100
0.0107
0.4723

90
44338
273034
0.41
2.31
100
0.4392
13.6008
1421321
15206208
16.72
195.00
100
0.3227
11.0501

0.75
70
1087937
14668547
7.95
108.73
100
0.0208
1.0697
27409801
442248409
265.04
4704.04
99
0.0083
0.1406

90
115643
2408792
1.08
23.57
100
0.1092
8.6434
2959763
25796932
32.88
302.44
100
0.1744
5.1725

0.8
0.25
0.25
70
2228168
11825715
16.03
95.74
100
0.0094
0.1390
115343373
461244567
1167.23
4819.64
93
0.0171
0.1883

90
330955
1919421
3.23
21.20
100
0.2375
16.7504
11697110
120462234
139.35
1375.25
100
0.2476
4.8784

0.75
70
1860095
41493678
13.49
321.69
100
0.0076
0.0942
67104475
432826308
646.54
4682.28
98
0.0127
0.1003

90
362757
3964085
3.36
39.09
100
0.0609
5.0439
14443132
373521531
169.66
4464.73
100
0.0286
0.9466

0.5
0.25
70
524558
3799338
3.96
29.16
100
0.0061
0.2385
9415219
80038607
95.53
833.19
100
0.0324
2.5209

90
36666
577706
0.34
5.74
100
0.2641
6.3281
823660
5949505
10.63
83.68
100
0.2739
9.5707

0.75
70
403985
5962730
2.83
43.99
100
0.0062
0.0928
22181919
380549351
222.19
4369.47
100
0.0098
0.1969

90
72633
1101780
0.67
10.53
100
0.1546
4.0201
1523540
30875624
18.57
395.32
100
0.1881
8.0855

1.00
0.25
0.25
70
1830472
16744441
13.43
121.90
100
0.0095
0.1007
59443784
330818627
564.87
3015.11
98
0.0133
0.1125

90
215190
2035038
2.08
20.64
100
0.1032
4.0359
4466746
28799550
51.63
350.30
100
0.0403
2.8493

0.75
70
1129202
18263919
7.96
128.09
100
0.0065
0.1021
37106372
459794226
335.53
4123.83
99
0.0139
0.1279

90
228020
2337809
2.06
21.34
100
0.0061
0.1508
8313461
106599701
93.90
1371.25
100
0.0125
0.4674

0.5
0.25
70
214900
3407180
1.58
25.43
100
0.0068
0.4455
6115145
60562961
57.14
570.64
100
0.0087
0.0862

90
29404
276253
0.27
2.81
100
0.0589
4.6160
629722
5126101
7.19
58.67
100
0.1737
5.3500

0.75
70
301944
4639599
2.11
34.49
100
0.0064
0.2709
11218748
269394563
104.08
2817.75
100
0.0090
0.0958

90
69288
1953549
0.65
19.42
100
0.0102
0.9705
1357448
13996545
15.29
171.25
100
0.1471
4.7189

1.25
0.25
0.25
70
1089215
11257092
8.21
95.29
100
0.0090
0.0845
32149788
369070833
326.03
3946.48
100
0.0146
0.1287

90
121436
1254725
1.16
13.21
100
0.0505
2.4323
5560244
114005664
70.15
1528.11
100
0.0049
0.0518

0.75
70
442342
2908517
3.06
20.06
100
0.0079
0.0759
21335243
266345547
194.90
2685.73
99
0.0127
0.1111

90
135323
1799997
1.23
17.18
100
0.0272
1.8397
2880368
22008680
30.82
231.41
100
0.0050
0.0754

0.5
0.25
70
100924
1251332
0.75
9.97
100
0.0065
0.1067
8748002
445990918
84.10
4346.73
100
0.0054
0.0692

90
19309
331734
0.18
3.17
100
0.1289
3.6122
415265
4518256
4.64
49.28
100
0.0714
6.6160

0.75
70
133234
4014823
0.93
27.98
100
0.0073
0.2137
5501857
53869685
49.83
514.11
100
0.0113
0.0758

90
28751
250665
0.27
2.36
100
0.0037
0.2513
730907
10862579
7.95
117.42
100
0.0049
0.0849

1.50
0.25
0.25
70
664098
9051000
4.84
69.75
100
0.0105
0.1576
29590335
325690474
299.12
3285.10
99
0.0171
0.1274

90
101999
951368
0.96
8.89
100
0.0099
0.4903
2879670
35619654
35.16
391.36
100
0.0198
1.2207

0.75
70
460782
6318122
3.27
44.57
100
0.0118
0.0894
14381778
329708866
135.18
3225.48
100
0.0127
0.1105

90
88765
959182
0.80
8.91
100
0.0040
0.0662
2681801
98041164
29.74
1127.96
100
0.0064
0.0727

0.5
0.25
70
90684
718580
0.66
5.07
100
0.0055
0.1499
1217974
26068657
11.74
278.99
100
0.0073
0.0970

90
17842
97492
0.16
0.98
100
0.0534
3.6591
396960
6700590
4.53
77.83
100
0.0851
4.8506

0.75
70
69694
549161
0.50
3.90
100
0.0041
0.1045
2519744
27555493
22.84
251.41
100
0.0066
0.1271

90
25354
255908
0.23
2.43
100
0.0138
1.1731
675377
11025856
7.61
135.48
100
0.0389
2.0427

1.75
0.25
0.25
70
346213
12937042
2.56
102.62
100
0.0091
0.0727
10231727
183248629
98.28
1629.76
100
0.0139
0.1443

90
72082
962963
0.67
8.94
100
0.0192
1.5155
1524548
12214703
18.99
144.91
100
0.0159
0.7236

0.75
70
155745
1597658
1.12
11.67
100
0.0048
0.0759
8349472
233610726
82.41
2322.54
100
0.0112
0.0788

90
54346
556872
0.49
4.99
100
0.0045
0.0677
1193311
14839106
13.61
164.88
100
0.0060
0.0762

0.5
0.25
70
33618
401940
0.26
3.17
100
0.0050
0.1512
1037401
29751952
10.05
286.85
100
0.0061
0.0863

90
22010
588097
0.21
5.80
100
0.0165
1.3437
277896
2630825
3.28
30.73
100
0.0337
2.5780

0.75
70
60324
938322
0.43
6.44
100
0.0057
0.0852
1345847
17163412
12.75
169.89
100
0.0076
0.0834

90
19976
223194
0.18
1.97
100
0.0138
1.1915
450183
4753419
5.18
56.54
100
0.0053
0.1587

2.00
0.25
0.25
70
263896
5361429
1.92
42.43
100
0.0076
0.1149
7966404
294171852
80.90
3238.28
100
0.0122
0.1022

90
41536
303945
0.37
2.50
100
0.0069
0.1556
1257091
30578774
14.30
353.53
100
0.0257
1.3044

0.75
70
174184
1725739
1.25
13.27
100
0.0057
0.0808
3265403
32146416
29.53
315.98
100
0.0128
0.1015

90
48914
669835
0.42
5.74
100
0.0039
0.0705
1195686
15899030
13.15
187.01
100
0.0127
0.5283

0.5
0.25
70
34243
391787
0.25
3.09
100
0.0043
0.0473
673277
19665442
6.18
181.05
100
0.0090
0.1181

90
11611
154462
0.11
1.69
100
0.1822
16.9507
187134
2683667
2.05
30.06
100
0.0214
1.5299

0.75
70
34586
471487
0.24
3.35
100
0.0056
0.0758
1271906
25609588
11.54
234.23
100
0.0098
0.0715

90
6815
55855
0.06
0.48
100
0.0028
0.0490
225415
3858777
2.49
45.84
100
0.0081
0.3787

3.00
0.25
0.25
70
84306
3685077
0.64
29.47
100
0.0096
0.0978
1998031
52564160
19.32
556.56
100
0.0080
0.0852

90
26775
344199
0.24
3.65
100
0.0074
0.1978
781780
14099992
9.08
148.56
100
0.0063
0.0869

0.75
70
78675
2304815
0.57
18.53
100
0.0071
0.1111
1449705
38705731
13.90
405.38
100
0.0104
0.0937

90
23334
592065
0.20
5.27
100
0.0074
0.0876
318573
3935413
3.55
45.32
100
0.0085
0.0990

0.5
0.25
70
15785
616369
0.11
4.66
100
0.0028
0.0627
420907
2543861
3.95
26.30
100
0.0046
0.0764

90
3863
49539
0.03
0.50
100
0.0025
0.0815
108292
614015
1.14
6.43
100
0.1739
2.4366

0.75
70
11775
162353
0.08
1.18
100
0.0045
0.0894
276874
4007313
2.53
37.96
100
0.0074
0.0642

90
8133
270480
0.07
2.57
100
0.0033
0.0798
108638
1842254
1.21
21.81
100
0.0018
0.0236

First, the mean and maximum numbers of nodes increased as the job size became larger, as became smaller, as became smaller, as became smaller, or as learning effect became strong. The CPU times were observed to increase exponentially as the job size increased. Some of instances could not be solved in the reasonable time. We recorded the number of solvable instances (NSI) if the numbers of nodes were less than 10⁸. There were 15,848 solvable instances for all instances. When  = , the least NSI was 73 which implied that the aforesaid parameters affected the nodes. Second, the accuracy of SA algorithms was assessed, and the results presented in Table 3 indicated that the mean error percentage of SA₅ was less than 0.7376%. The performance of error percentage of SA₅ trend was not subject to the parameter influenced.

5.3. The Performance of the SA Algorithms for Large Job Size

On evaluating the accuracy of SA algorithms for large job size, we carried out the jobs size ( and 80) simulation experiments. The other parameters were taken as (,, and 0.6), ( and 0.5), (,, and 0.75), and (le = 70%, 80%, and 90%). For each situation, 100 replications were randomly generated. For evaluating the accuracy of SA algorithms, the mean relative deviance percentage was calculated as

where was the solution obtained from SA algorithms and SA₅ was the smallest value of the objective function obtained from the th SA algorithms. We get the total average for each algorithm experimental results as depicted in Figures 1–5. In Figure 4, parameters could slightly affect the performance of SAs algorithm. It was observed that there was no significant difference among the performance of SA algorithms. Besides, the CPU times of the SA algorithms were not recorded since they were completed within one second. Thus, the SA₅ is recommended for solving the large job size method.

Figure 1: The performance of SA algorithm with respect to .

Figure 2: The performance of SA algorithm with respect to .

Figure 3: The performance of SA algorithm with respect to .

Figure 4: The performance of SA algorithm with respect to .

Figure 5: The performance of SA algorithm with respect to le.

6. Conclusions

In this paper, we study the two-agent single-machine scheduling problems in which jobs with arbitrary release times in learning effect condition. The objective is minimizing the total weighted completion time of one agent, subject to an upper bound on the maximum lateness of the second agent. To solve the problem, a branch-and-bound algorithm incorporated with several dominance properties and a lower bound is developed. In addition, simulated-annealing algorithms are also developed to test the accuracy. Computational results show that the branch-and-bound algorithm can find the optimum up to 14 jobs in a reasonable time, and the simulated-annealing algorithm is effective and efficient in obtaining near-optimal solutions. Suggested further research includes considering other optimization criteria or multiobjective optimization to further verify the proposed approach.

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