### 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 5: Results of the capacitated facility locations problems on the AP1000

### Table 1. Verifying optimality by preprocessing

2001

"... In PAGE 16: ...Table 1. Verifying optimality by preprocessing Table1 shows a few problem instances from the MIPLIB where setting up a simplex tableau for the optimal integral solution and application of Algorithm 13 suffice to prove optimality. The table shows the number of columns and rows of the problem and the time spent in Algorithm 13.... ..."

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### Table 17. Verifying optimality by preprocessing

"... In PAGE 32: ...6. Table17 shows a few problem instances from the MIPLIB [BCMS98] where setting up a simplex tableau for the optimal integral solution and applying the preprocessing algorithm (Table 13) suffice to prove optimality. The table shows the number of columns and rows of the problem and the time spent in the preprocessing stage.... ..."

### Table 17. Verifying optimality by preprocessing

"... In PAGE 32: ...6. Table17 shows a few problem instances from the MIPLIB [BCMS98] where setting up a simplex tableau for the optimal integral solution and applying the preprocessing algorithm (Table 13) suffice to prove optimality. The table shows the number of columns and rows of the problem and the time spent in the preprocessing stage.... ..."

### Table 1: Dwell Task Parameters

2004

"... In PAGE 18: ...1 Experiment Setup Although the online template generation algorithm can handle all possible dwells, for the purpose of comparison we restricted the dwell types to those employed in past work. The workload parameters are listed in Table1 . The three element tuples for execution time and power consumption denote the time and average power consumption for the sending, round-trip delay and receiving phases of a dwell task.... ..."

Cited by 3

### Table 1: Node counts and time for instances of multi-commodity network ow problems CPLEX CPLEX + CUTS

2007

"... In PAGE 126: ... CPLEX branch-and-bound was used to solve the two mixed integer programming formulations. Table1 1: Comparison of two formulations: lower and upper bounds were returned at the end of 300s of computation time (P1) (P2) Prob LB UB LB UB E10 10 0.00 0.... In PAGE 127: ... This shows that as an integer programming formulation, with no additional cuts or heuristics added, formulation (P 2) performs better than formulation (P 1). Table1 2: Comparison of two formulations: Node counts and solve times (P1) (P2) Prob Node Count Time Node Count Time E10 10 240 1.5 56 0.... In PAGE 128: ... The time limit was 300s, so if optimal solution is not found in the allotted time for a problem the corresponding entry for solve time is 300s and node count entry is the number of nodes explored in 300s. Table1 3: Comparison of two formulations with cutting planes and heuristics: lower and upper Bounds after 300s of computation time (P1) (P2) Prob LB UB LB UB E10 10 0.00 0.... In PAGE 128: ...00 0.00 Entries in bold represent that optimal solution was found in 300 second Looking at the results from Table1 3, we can see that, with the help of cuts and heuristics, formulation (P 1) was able to provide better results than (P 2). More problems were solved to optimality and for except one, the bounds provided for the problems not solved to optimality in allotted time by formulation (P 1) were stronger than formulation (P 2).... In PAGE 129: ...Table1 4: Comparison of two formulations with cutting Planes and heuristics: node counts and computation times (P1) (P2) Prob Node Count Time Node Count Time E10 10 0 4.01 0 0.... ..."

### Table 4 Number of solved pricing problems.

2004

"... In PAGE 25: ...ave been solved to proven optimality. The run-time is 160.68s on average, which is the fastest of our four B amp;P variants. In Table4 , the total number of pricing problems solved in each class and their sums are given for the B amp;P approaches. Furthermore, the bar charts shown in Fig.... In PAGE 29: ...Table4 we can observe that when using FFBC only (BPNoR) the number of solved pricing problems is lower than the one of BP where CPLEX(restricted 3-stage 2DKP) is used. Pricing using a more sophisticated heuristic, in this case exactly solving restricted 3-stage 2DKP, can therefore improve the overall results, see also Table 3.... In PAGE 29: ... These pricing problems can be denoted as easy ones. Looking at absolute numbers shows that CPLEX(restricted 3-stage 2DKP) successfully solved 21 500 pricing problems, which approximately corresponds to the increase of solved pricing problems when comparing BPNoR to BP in Table4 . The bar charts showing the relative success rates of the pric- ing algorithms indicate that the absolute number of easy pricing problems roughly remained the same.... ..."

Cited by 3

### Table 4 Number of solved pricing problems.

2004

"... In PAGE 28: ...nstances have been solved to proven optimality. The run-time is 160.68s on average, which is the fastest of our four B amp;P variants. In Table4 , the total number of pricing problems solved in each class as well as their sums are given for the B amp;P approaches. Furthermore, the bar charts shown in Fig.... In PAGE 28: ...lass. The origin of the charts were shifted to 0.5 because for almost all the variants, the greedy FFBC algorithm solved more than half of the pricing problems. In Table4 , we can observe that, when using FFBC only (BPNoR), the number of solved pricing problems is lower than the one of BP where CPLEX(restricted 3-stage 2DKP) is used. Pricing using a more sophisticated heuristic, in this case exactly solving restricted 3-stage 2DKP, can therefore improve the overall results, see also Table 3.... In PAGE 30: ... These pricing problems can be denoted as easy ones. Looking at absolute numbers shows that CPLEX(restricted 3-stage 2DKP) successfully solved 21 500 pricing problems, which approximately corresponds to the increase of solved pricing problems when BPNoR is compared to BP in Table4 . The bar charts showing the relative success rates of the pric- ing algorithms indicate that the absolute number of easy pricing problems roughly remained the same.... ..."

Cited by 3

### Table 3. We give some basic statistics on the distribution of the computation times to solve instances of the three problem sets. We indicate the number of jobs (n), the average time (averaged over the 125 instances) to solve the benchmark set (tavg) and its standard deviation ( t), the average time to solve the easiest and the hardest instance (tmin and tmax, respectively), and the quantils of the average time to solve a given percentage of the instances. Qx indicates the average time to solve x%ofthe benchmark instances.

2001

"... In PAGE 9: ... On each instance 25 trials were performed with a large computation time limit, which was enough that each instance could be solved to the best-known solutions, which we conjecture to be optimal, in each single trial. Of the 125 instances with 100 jobs (those with 40 or 50 jobs are very easily solved, see Table3 ), only 15 took an average time to optimal larger than 10 seconds; only 7 of these longer than 20 seconds. The large majority of the benchmark instances was either solved with a single local search or within very few seconds.... ..."

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