### 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 7.3: Evaluation of composite Job-Shop Scheduling algorithms

### 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.... ..."

Cited by 4

### 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.... ..."

Cited by 5

### Table 3. Comparison of the algorithms on bi-objective instances with 50 cities Pareto GA IM MOGLS

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

1999

Cited by 24

### Table 2. Job Shop Scheduling Results with the BAS and the Petri Net algorithms.

2006

"... In PAGE 10: ...the Petri-Net models consisted of 220 places and 200 transitions. A summary of the results is shown in Table2 . Notice that both algorithms show competitive results in terms of solution quality and CPU times.... ..."

### Table continues *JIT = Just-in-time

2002

Cited by 5

### Table 1: New or improved approximation results for minsum shop scheduling problems. (*) Does not use a makespan approximation algorithm; see Sections 2.2 and 2.3 for details. (y) A simple single-machine relaxation in [21] leads to an 8/3-approximation algorithm for the closely related problem J2jrj; pmtnj PwjCj with job release dates and a weighted sum of job completion times objective, and to a 2-approximation for the problem J2jrj; pmtnj PCj with unweighted objective.

2000

"... In PAGE 2: ...um objective. On the other hand, Hoogeveen et al. [12] have shown that the problems Fj j P Cj and Oj j P Cj are MAX-SNP-hard, hence there is no hope for obtain- ing approximation guarantees arbitrarily close to 1 for these problems, unless P = NP. Using known makespan approximation algorithms, we obtain the new approximation bounds for weighted sum of operations (or job) completion times objectives given in Table1 (at the end of this paper). In this table, = maxj j denotes the largest number of operations per job.... In PAGE 2: ... [28]. The polylogarithmic performance guarantee in the rst row of Table1 dominates1 earlier O(m) results for: the classical job-shop problem Jjj P Cj with mean ow time objective (Gonzalez and Sahni [8]); the multiprocessor ow-shop problem F(P)jrjj P wjCj with weighted sum of job completion times objective (Schulz [24]); and the open shop problem Ojj P Cj with mean ow time objective (Achugbue and Chin [1]). 1Except for instances where is very large, namely, where m = o((log(m )= loglog(m ))2)), discussed under the name of \very long jobs quot; in [21].... ..."

Cited by 3

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

1999

Cited by 24