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Hybridizing biogeography-based optimization and integer programming for solving the travelling tournament ...

·10 5 NL 16 ·10 5 NL 14
4.55 2.92
quality 4.5 2.9

solution 4.45 2.88

2.86
8 10 12 14 16 8 10 12 14
Ft Ft

Figure 5. Influence of Ft on the travel cost value

NL 12 4 NL 10
5
·10 ·10
8.5
1.65
quality

solution 1.6 8

1.55
7.5
6 8 10 12 4 6 8
Ft Ft

Figure 6. Influence of Ft on the travel cost value.

NL 14
NL 16

6,000 2,000

Time(s) 4,000 1,000

2,000

0 0
8 10 12 14 6 8 10 12
Ft Ft

Figure 7. Influence of Ft on the runtime

benchmarks. As indicated, our method outper- CON 16 , CON 14 , CON 12 , CON 22 , and CON 4 .
forms the current best solutions reported in 12 Additionally, the Average Gap (AG) for CON
for NL 16 . Moreover our approach successfully instances is AG(CON) = 0.38%, demonstrating
that our results are highly aligned with the opti-
achieves the best-known results for NL 4 , NL 6 ,
mal solutions, showing only a 0.38% deviation.
NL 8 , NL 10 , and NL 12 .
The results demonstrate that our new ap-
To evaluate the performance of our approach
proach successfully attains the optimal solution
in solving the Traveling Tournament Problem
for instances CON 24 , CON 10 , CON 8 , CON 6 ,
(TTP),we carried out a detailed comparison
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