### Table 2: Minimizing the makespan for the given number of prototypes. Data m #Variables #Constraints Total Time (sec) Makespan (day)

"... In PAGE 6: ...Further, it has good constraint propagation techniques and efficient searching algorithms. The numbers of variables and constraints presented in Table2 correspond to the model formulated in OPL. However, these numbers are not directly related to the difficulty of the optimization problem.... In PAGE 6: ...0 GHz, and 1 GB main memory. Table2 shows the computational results. For each data set and a given number of prototypes, we minimize the makespan.... ..."

### Table 4: Summary of the test runs, Case 1. The makespan lengths are given in seconds. Legend: av.ms = avarage length of the makespan of all permutations, max.ms = worst makespan, min.ms = minimal makespan, LJF-ms = Longest Job First makespan, SJF-ms = Shortest Job First makespan, LJF di = di erence of LJF makespan and the optimal makespan, SJF di = di erence of SJF makespan and the optimal makespan.

1996

"... In PAGE 14: ...For a complete description of the assays, refer to Appendix C. Case 1 Table4 ( rst column) shows six problem instances each of which consist- ing of seven protocols. Because we can write the seven protocols in 7!(= 5040) orderings, it is possible to solve the problem exactly by considering all per- mutations (columns 2 to 4).... ..."

### Table 4: Model Characteristics for Parallel Flowshop Problems Binaries CPU secs Optimality Gap (%)

2003

"... In PAGE 20: ... 6.3 Computational Results Unlike the case of the flowshop problem discussed in Section 5, the MILP models for the multi- stage flowshop with parallel units do not exhibit good computational performance as the problem size increases (see Table4 ). This is primarily due to the large number of binary variables that are required for modeling this problem.... In PAGE 20: ... This is primarily due to the large number of binary variables that are required for modeling this problem. Although the number of binary variables can be reduced by taking in to account the forbidden assignments and postulating an appropriate number of slots for the various units, it increases dramatically with problem size as can be seen from Table4 . The solution times reported are for makespan minimization.... ..."

Cited by 3

### Table 3b. Solution Statistics for Model 2 (Minimization)

1999

"... In PAGE 4: ...6 Table 2. Problem Statistics Model 1 Model 2 Pt Rows Cols 0/1 Vars Rows Cols 0/1 Vars 1 4398 4568 4568 4398 4568 170 2 4546 4738 4738 4546 4738 192 3 3030 3128 3128 3030 3128 98 4 2774 2921 2921 2774 2921 147 5 5732 5957 5957 5732 5957 225 6 5728 5978 5978 5728 5978 250 7 2538 2658 2658 2538 2658 120 8 3506 3695 3695 3506 3695 189 9 2616 2777 2777 2616 2777 161 10 1680 1758 1758 1680 1758 78 11 5628 5848 5848 5628 5848 220 12 3484 3644 3644 3484 3644 160 13 3700 3833 3833 3700 3833 133 14 4220 4436 4436 4220 4436 216 15 2234 2330 2330 2234 2330 96 16 3823 3949 3949 3823 3949 126 17 4222 4362 4362 4222 4362 140 18 2612 2747 2747 2612 2747 135 19 2400 2484 2484 2400 2484 84 20 2298 2406 2406 2298 2406 108 Table3 a. Solution Statistics for Model 1 (Maximization) Pt Initial First Heuristic Best Best LP Obj.... In PAGE 5: ...) list the elapsed time when the heuristic procedure is first called and the objective value corresponding to the feasible integer solution returned by the heuristic. For Table3 a, the columns Best LP Obj. and Best IP Obj.... In PAGE 5: ... report, respectively, the LP objective bound corresponding to the best node in the remaining branch-and-bound tree and the incumbent objective value corresponding to the best integer feasible solution upon termination of the solution process (10,000 CPU seconds). In Table3 b, the columns Optimal IP Obj., bb nodes, and Elapsed Time report, respectively, the optimal IP objective value, the total number of branch-and-bound tree nodes solved, and the total elapsed time for the solution process.... ..."

### Table 6: Best results obtained for data sets of Taillard when treated as continuous flow-shop problem instances with total processing time objective.

2001

"... In PAGE 13: ... So, for n = 20 we compare to the optimal results, for n =50and n = 100 to lower bounds, and for n = 200 to the best results obtained during all experiments. The best objective function value obtained for each problem instance is given in Table6 in the Appendix. Table 1 shows the results of the application of construction methods.... ..."

Cited by 4

### Table 4: Complexity boundary involving scheduling problems and the makespan and mean #0Dow time

1989

"... In PAGE 2: ...1. Table4 presents the known boundary for unit execution time scheduling problems involving various precedence constraints with respect to the makespan and mean #0Dow time objective functions. Note that the results obtained in this section are either minimal NP-complete results or maximal polynomial-time solvable results.... ..."

### Table 2. A schedule for the benchmark problem in Table 1 with the makespan = 293 In the next section, we introduce the three evolutionary based heuristics we implemented and ran with OSSP benchmark problems taken from a well known source of test problems. 3 Genetic Algorithms for OSSP In this work, we use three genetic algorithms. We start our discussion by pre- senting the Permutation GA and then turn our attention to the Hybrid GA.

"... In PAGE 3: ... Most benchmark problems in the literature of scheduling have this property. A schedule to the problem instance of Table 1 is given in Table2 . We note that the operations are not scheduled in their order of appearance in Table 1.... In PAGE 3: ... We note that the operations are not scheduled in their order of appearance in Table 1. Thus, operation O32, for instance, is scheduled at time 78 while operation O31 is scheduled at time 226, as can be seen in Table2 . Operation O22 is the last one to nish, with \end time quot; equal to 293, thus, the makespan of the schedule given in Table 2 is 293, which happens to be the optimal solution for this problem.... In PAGE 3: ... Thus, operation O32, for instance, is scheduled at time 78 while operation O31 is scheduled at time 226, as can be seen in Table 2. Operation O22 is the last one to nish, with \end time quot; equal to 293, thus, the makespan of the schedule given in Table2 is 293, which happens to be the optimal solution for this problem. Machines Job J1 Job J2 Job J3 Job J4 M1 85 23 39 55 M2 85 74 56 78 M3 3 96 92 11 M4 67 45 70 75 Table 1.... ..."

### Table 1. Example of the constraint manager behavior. 2. Show the proposed system performance for a set of scheduling problems randomly established. In order to evaluate the quality of the scheduling solutions obtained by the CROSS system, a set of 8 problems was randomly generated. Table 2 shows the results obtained in terms of the maximum completion times (makespan) achieved by the proposed system and the optimal values determined using a branch and bound algorithm. This table includes also the corresponding mean completion times. For the results presented in Table 2, the solutions obtained by the CROSS system are less than 10% percent from the optimal values.

1996

"... In PAGE 4: ... Illustrate the constraint manager behavior. Table1 shows the results obtained for a 5 jobs, 3 resources problem on two di erent situations: i) the initial due dates are too wide, and ii) the initial due dates are too narrow. Although the nal due dates be quite di erent, the two mean completion times are nevertheless similar.... ..."

Cited by 1

### Table 3: Summary of the IPR Algorithms Algorithm Model Time Makespan

1997

"... In PAGE 2: ... In our experiments, IPR consistently obtained smaller makespans than all of the previous algorithms with which we compared it. A summary of our results is contained in Table3 , where OPT(I) denotes the makespan of the optimal schedule. The diagram shows the relationships between the models; an ar- row from a to b indicates a is more restricted than b.... In PAGE 2: ... In this paper we propose a set of scheduling algorithms which utilize one level of backtracking. These algorithms obtain improved schedules without significantly increasing the scheduling time (see Table3 ). We call these algorithms IPR, for immediate predecessor rescheduling.... ..."

Cited by 2

### TABLE 1. ILLUSTRATIVE TABU SEARCH APPLICATIONS Scheduling Flow-Time Cell Manufacturing Heterogeneous Processor Scheduling Workforce Planning

in Tabu Search

1997

Cited by 1