### Table 1: Running results corresponding to one call to the sub-problem

2006

"... In PAGE 17: ...1 are based on solving the sub-problem to provable optimality. As is demonstrated anecdotally in Table1 by a representative iteration from each of the problem instances, however, the optimal solution is often found signi cantly earlier and the remainder of the time is spent on proving the optimality of this solution. Furthermore, the simplex method does not depend on pivoting in the most negative reduced cost column { the column simply needs to be strictly negative.... ..."

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### Table 11. The Details of Solving Sub-Problem 8

### Table 2: Costs Associated with the Two Sub-Problems for the Simple Example

### Table 2.1: Undefuzzified Outputs from the Sub-problems

2006

### Table 2.2: Labelled Outputs from the Sub-problems

2006

### Table 3. Detailed research findings of research sub problems. College Of Graduate

### Table 7: Size of the IP-based sub-problems

2008

"... In PAGE 15: ... This is primarily because of the growth both in number of variables and number of constraints. Table7 demonstrates the differences in size between our two test instances.... ..."

### Table 4: Running results for data instance 6 (2,320 ights) 60 iterations of the sub-problem

2006

"... In PAGE 18: ...e-solved. Thus, a second design decision is how to increase the threshold. For each of the four initial values of p, we considered two di erent constant values (10 and 50) for increasing the threshold whenever no negative reduced cost pairing was found by a known value (step-size) each time the restricted ight set resulted in no negative reduced cost pairing. Computational results for these eight scenarios appear in Table4 . This table presents the initial threshold value, the number of ights included in the rst iteration of the pricing problem as a result of this threshold, the overall improvement to the objective function from the initial set of columns to the solution after 60 iterations of the pricing problem, and statistics on the performance of... ..."

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