### Table 3: Results on a set of hard job-shop scheduling problems

1999

"... In PAGE 17: ... The goal for each problem is to ful l all operations with a given time bound. Table3 shows the timings required for the di erent algorithms to solve these hard problems. For each case, the problem size (m n) is shown, where m denotes the number of jobs in each problem and n stands for the number of tasks contained in each job.... ..."

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### Table 1: An instance of job-shop scheduling problem. (Machine, Processing Time) Jobs

1995

"... In PAGE 2: ... We can represent the disjunctive graph as follows: G = (O; A; E), where O is the vertex set, A is the arc set and E is the edge set. Figure 1 presents a feasible schedule for the instance in Table1 . Figure 2 shows the disjunctive graph for the same instance.... In PAGE 24: ...Table1 0: Approximation algorithms; one trial; initial schedule: shifting bottleneck heuris- tic. initial schedule: shifting bottleneck heuristic LI SA LSLO LSRW LSMC MT10 952 949 944 937 941 2:76 100 1:08 104 6:59 104 1:01 105 9:86 104 ABZ5 1270 1246 1239 1236 1234 2:40 100 1:09 104 6:12 104 8:96 104 9:22 104 LA19 863 843 842 842 842 2:98 100 1:08 104 5:21 104 7:69 104 8:09 104 LA20 918 902 902 902 902 3:03 100 9:78 102 5:21 104 8:29 104 8:38 104 ORB1 1176 1101 1078 1083 1059 2:73 100 1:07 104 6:42 104 1:04 105 1:00 105 LA21 1128 1059 1048 1053 1047 4:53 100 2:47 104 1:43 105 2:40 105 2:49 105 ABZ8 736 679 682 681 680 2:97 101 1:03 105 5:56 105 5:01 105 5:80 105 LA27 1353 1287 1242 1254 1254 8:38 100 4:68 104 2:63 105 4:90 105 4:87 105 CAR5 8873 8308 7720 7702 7720 3:66 10?1 4:08 103 2:67 104 3:85 104 3:83 104 LA39 1301 1244 1239 1239 1240 1:78 101 5:59 103 2:68 105 4:89 105 2:33 105 ulated annealing with the same parameters as above, except for SIZEFACTOR=2 and TIT=1000.... In PAGE 25: ...Table1 1: Approximation algorithms; one trial; initial schedule: dispatching rule (most work remaining).initial schedule: shifting bottleneck heuristic LI SA LSLO LSRW LSMC MT10 1130 944 945 941 940 2:83 10?1 1:06 104 8:16 104 9:96 104 5:60 104 ABZ5 1322 1242 1263 1236 1234 2:16 10?1 1:08 104 7:51 104 9:38 104 5:27 104 LA19 993 842 842 842 842 2:66 10?1 1:08 104 6:29 104 7:16 104 4:00 104 LA20 1055 907 902 902 902 6:16 10?1 1:09 104 6:32 104 7:96 104 4:07 104 ORB1 1411 1098 1078 1074 1089 6:67 10?1 1:07 104 7:96 104 9:52 104 5:88 104 LA21 1255 1065 1048 1058 1053 7:33 10?1 2:19 104 1:96 105 2:53 105 1:57 105 ABZ8 901 695 688 682 683 2:30 100 8:73 104 6:03 105 6:04 104 6:08 105 LA27 1518 1260 1255 1255 1256 1:81 100 6:86 103 2:71 105 4:87 105 3:20 105 CAR5 11058 7832 7720 7727 7702 3:00 10?1 4:29 103 3:08 104 3:93 104 3:93 104 LA39 1560 1264 1242 1246 1242 1:73 100 4:71 104 2:44 105 4:93 105 3:09 105 Table 12: Approximation algorithms; several trials; initial schedule: shifting bottleneck heuristic.... In PAGE 25: ...work remaining).initial schedule: shifting bottleneck heuristic LI SA LSLO LSRW LSMC MT10 1130 944 945 941 940 2:83 10?1 1:06 104 8:16 104 9:96 104 5:60 104 ABZ5 1322 1242 1263 1236 1234 2:16 10?1 1:08 104 7:51 104 9:38 104 5:27 104 LA19 993 842 842 842 842 2:66 10?1 1:08 104 6:29 104 7:16 104 4:00 104 LA20 1055 907 902 902 902 6:16 10?1 1:09 104 6:32 104 7:96 104 4:07 104 ORB1 1411 1098 1078 1074 1089 6:67 10?1 1:07 104 7:96 104 9:52 104 5:88 104 LA21 1255 1065 1048 1058 1053 7:33 10?1 2:19 104 1:96 105 2:53 105 1:57 105 ABZ8 901 695 688 682 683 2:30 100 8:73 104 6:03 105 6:04 104 6:08 105 LA27 1518 1260 1255 1255 1256 1:81 100 6:86 103 2:71 105 4:87 105 3:20 105 CAR5 11058 7832 7720 7727 7702 3:00 10?1 4:29 103 3:08 104 3:93 104 3:93 104 LA39 1560 1264 1242 1246 1242 1:73 100 4:71 104 2:44 105 4:93 105 3:09 105 Table1 2: Approximation algorithms; several trials; initial schedule: shifting bottleneck heuristic. SA LSLO LSRW LSMC MT10 951 937 938 939 ABZ5 1239 1239 1238 1236 LA19 843 842 842 842 LA20 907 902 902 902 ORB1 1085 1079 1078 1079 LA21 1054 1049 1055 1056 ABZ8 679 680 680 674 LA27 1264 1258 1257 1256 CAR5 7824 7821 7702 7749 LA39 1250 1242 1247 1240... In PAGE 26: ...Table1 3: Approximation algorithms; several trials; initial schedule: simplest heuristic. Li SA LSMC MT10 1021 948 944 ABZ5 1269 1239 1236 LA19 874 842 842 LA20 914 902 902 ORB1 1203 1086 1086 LA21 1131 1054 1049 ABZ8 752 682 681 LA27 1379 1254 1254 CAR5 7749 7832 7731 LA39 1366 1248 1242 annealing method this was not the case, in 7 out of 10, the solution obtained in Table 11 was better than the one in Table 10, for the SA column.... In PAGE 26: ...Li SA LSMC MT10 1021 948 944 ABZ5 1269 1239 1236 LA19 874 842 842 LA20 914 902 902 ORB1 1203 1086 1086 LA21 1131 1054 1049 ABZ8 752 682 681 LA27 1379 1254 1254 CAR5 7749 7832 7731 LA39 1366 1248 1242 annealing method this was not the case, in 7 out of 10, the solution obtained in Table 11 was better than the one in Table1 0, for the SA column. However, for the large-step optimization methods, the use of a good initial solution gives some help, but it is not as signi catively di erent as for the local improvement.... In PAGE 28: ...Table1 4: Summary of the results. n m C maxjLB SBH SBH(new) Slug(AC) TB SA LSO MT10 930 952 940 930 935 944 937 ABZ5 1234 1270 1258 1245 1236 1238 1234 LA19 842 863 878 848 842 842 842 LA20 902 918 922 911 902 902 902 ORB1 1059 1176 1121 1070 1064 1068 1059 LA21 1040 1128 1071 1053 1048 1053 1047 ABZ8 635 738 708 687 678 679 674 LA27 1235 1353 1272 1269 1242 1254 1242 CAR5 7702 8873 7720 7702 LA39 1233 1301 1278 1242 1244 1239 outperformed the remaining methods, obtaining several optimal schedules.... ..."

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### Table 1. Types of job-shop scheduling problems

"... In PAGE 5: ... Design of the data test We have defined four types of job-shop scheduling problems. Each type has a different number of jobs, operations and resources (see Table1 ). We have randomly generated 50 problems of each type.... ..."

### Table 2: Job Shop Problem

"... In PAGE 5: ... Therefore, the job shop problem is modified into a flexible flow shop prob- lem using dummy nodes to fill in the space in the matrix where a particular step is not included in a particular job route. Table2 and 3 show an example how a complex job shop problem can be modified into a flexible flow shop problem. Each step has an associated TG type and a recipe name.... ..."

### Table 1. Comparison of SSG and SG algorithms for job shop scheduling problems

1999

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### Table 1: Performance Measures for Job-shop Scheduling and Control

2001

"... In PAGE 2: ... Our distributed, evolutionary approach to scheduling avoids these problems by removing the requirement for a truly optimal solution, requiring instead only a towards-optimal (but practical and useful) solution. The goal is to optimize (often) conflicting local and global performance measures (French, 1983), as outlined in columns one and two of Table1 . A secondary goal is to minimize the variance in the global stability measures in column three, in order to maximize the stability of the system.... ..."

Cited by 1

### Table 2. Results of the our proposed multi-objective approach after 1-hour runtime

2007

"... In PAGE 13: ...99 and the num ber of iterations within SA to be 1,000,000. Table2 lists the re- sults of using different evaluation functions on the obtained solutions. For the weighted-sum objective function, we use the sam e set of weight values as in formula (29), and list the num - ber of archived non-dom inated solutions (see colum n 2) and the best solution under this evaluation function (see colum n 3).... In PAGE 14: ...Table 2. Results of the our proposed multi-objective approach after 1-hour runtime A ccording to the results in Table2 , we can see that our proposed approach is very prom ising in solving the m ulti-objective nurse scheduling problem . In terms of the solution quality evaluated by the sam e objective function, our approach performs similar to the IP-based VNS, and significantly improve the best results of the hybrid genetic algorithm and the hybrid VNS by 25.... ..."

### Table 5: Fuzzy Job Shop Scheduling

1998

"... In PAGE 6: ...1 Job Shop Scheduling A number of papers on fuzzy job shop scheduling have been published. A summary of the direction of research on fuzzy job shop scheduling is found in Table5 . McCahon and Lee (1990) study the job sequencing problem when job processing times are represented with fuzzy numbers.... ..."

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### Table 3. Performances on the job shop scheduling problem

"... In PAGE 13: ... We consider two relatively small instances, known as FT06 (36 tasks, 6 jobs, 6 processors, solvable with deadline 55), and LA02 (50 tasks, 10 jobs, 5 processors, solvable with deadline 655). As shown in Table3 the rst instance is solved easily, and the proof of the optimality of the deadline (i.... In PAGE 13: ... We remark that, without the optimizations described in Section 3.2, none of the instances of Table3 was compiled by Spec2SAT in less than one day. 5 Conclusions, Related, and Future Work We have presented a novel approach for the execution of speci cations of prob- lems in NP, based on the translation into SAT.... ..."

### Table 1: Results on 10x10 job-shop problems for the Oz Scheduler

1997

"... In PAGE 4: ... The propagators using edge- nding can be stated as before, and one only adds the new propagators. 4 Evaluation In this section we evaluate the performance of Oz for 10x10 job-shop problem instances of [1] in Table1 . For all problems the optimal solution (starting with no information) has to be found and the optimality has to be proved.... ..."

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