### Table 3: Proving optimality of generic MILPs with SYMPHONY (unconditional diving)

2006

"... In PAGE 30: ... To test the effect of having a good a priori upper bound, we performed a similar set of experiments in which the optimal solution value was provided a priori, so that the goal was simply to prove optimality of a known solution. The results (with a slightly smaller test set) are shown in Table3 . In this case, it is advantageous to follow an unconditional diving strategy in which the child of the current node is always preferred when there is one.... ..."

### Table III. Comparison of the minimum wakeup time obtained by the proposed algorithm and the optimal MILP formulation.

### Table 5.1: Accuracy of MILP, mFOIL (best m and m=0) and optimal theory: noise in arguments

1994

### Table 5.2: Accuracy of MILP, mFOIL (best m and m=0) and optimal theory: noise in class variable

1994

### Table 2 Results of Example 1 by GA and MILP (min PT) GA MILP Orders

"... In PAGE 5: ... (2000) for minimizing the total processing time with different problem sizes. Results are shown in Table2 . It can be found that: (1) For small-size problems, MILP can get the optimal solution of the problem within a short time.... In PAGE 16: ... The parameters and the termination condition are the same as previous. Table2 summarizes the results of Example 1 produced by GA and MILP with problem size from 5 to 24 orders. The maximum number of iteration for MILP was set to be 100,000.... In PAGE 18: ... 17 Gantt chart of a schedule for Example 1 (20 orders over 25 units) by GA (min PT) Fig. 18 Gantt chart of a schedule for Example 1 (24 orders over 25 units) by GA (min PT) From Table2 , we can see that GA performs much better than MILP both in solution quality and in search time: Solution quality: For the 5-order problem, both GA and MILP obtained the optimal solution. From the 8-order problem to the 24-order problem, GA obtained better objective value than MILP.... In PAGE 19: ... Fig. 19 is drawn from the data in Table2 and shows the ith the increasing of the problem size, the difference ge-size problem, GA performs much better than MILP. To test that stability of the proposed approach, 10 com lem size in Example 1-3.... ..."

### Table 5: Computational results for Example 1 for constant processing times. MILP STN Model MILP/CP Hybrid Scheme

2004

"... In PAGE 28: ...ith 13 time points, while instances Ms2 and Ms3 were solved for various time grids (i.e. no of time points) but no integer feasible solution was found in 36,000 CPU seconds. The computational statistics reported in Table5 , correspond to the MILP with the minimum number of time points that can represent the optimal solution found by the hybrid algorithm. The proposed algorithm, on the other hand, obtained the optimal solution for all instances of both objectives.... ..."

Cited by 7

### Table 8. Model sizes and solution time for Example 4 Solution MILP-1 Solution NLP Solution MILP-2 Outer

1914

"... In PAGE 21: ...5% requiring 5 iterations in the inner optimization. Table8 shows the computing times and the problem sizes. The total time required by the algorithm was 11.... In PAGE 37: ... Table 7. Solution steps for Example 4 Table8 . Model sizes and solution time for Example 4 Table 9: Distribution of the pollutant Hej and concentration of pollutant in organic phase Coj Table 10: Inlet Streams data for Example 5 Table 11: Cost and removal ratio data for the equipments in Example 5 Table 12.... ..."

Cited by 1

### Table 2: Clock period minimization with Algorithm RSMINP and an exact MILP-based branch-and-bound algo- rithm.

2001

"... In PAGE 21: ... To evaluate the relative speed and efficiency of our retiming and clock scheduling heuristic, we independently developed a MILP-based branch-and-bound solver. Table2 compares the runtimes and output clock periods of the two programs for a subset of our test suite. In general, the CPU requirements of the MILP-based optimizer grow very fast, due to the high computational complexity of mixed-integer linear programming.... In PAGE 21: ... With a 48-hour timeout, the MILP-solver runs out of time on most circuits, without having discovered a better solution than the heuristic scheme. The last column of Table2 shows the relative clock period improvement achieved by the MILP- based solver over our heuristic. Except for daio, the fastest circuit computed by the heuristic is as good as that of the MILP solver.... ..."

### Tables 5.1-5.3 show the comparison of MILP and mFOIL on the KRK chess end-game (see Sect 3.5.1) for di erent types of noise added. Also the accuracy of the optimal optimally augmented theory (see Algorithm 3.1) according to the evaluation function used by MILP [11] is shown. Since the performance of mFOIL depends heavily on a user de ned parameter m [6] the results for m = 0, which is the default value in mFOIL, and the accuracy for best m

1994

### Table 4: Progress of algorithm for Example 1. Iteration Master MILP Problem CP Subproblem

2004

"... In PAGE 24: ... If we allow up to five copies for all tasks, the optimal solution with an objective function value of 12 units is found in 4 iterations. The bounds and the assignments of all iterations are reported in Table4 , with the number of copies in parentheses if greater than 1. If a feasible solution has been found (i.... ..."

Cited by 7