A Differential Evolution with Two Mutation Strategies and a Selection Based on an Improved Constraint-Handling Technique for Bilevel Programming Problems

Table 13

Comparison of function evaluations (FE) for the upper level (UL) and lower level (LL) by the proposed differential evolution algorithm (DE) and the nested bilevel evolutionary algorithm (NBEA) in [28] for 20-dimensional test problems of the second set. A dash denotes that a feasible solution could not be obtained for the test problem. A notation of (𝑥) denotes that the algorithm terminated far away (Δ𝐹≥0.1) from the optimal solution.

Problem	Algorithm	Best FE		Median FE		Worst FE
		UL	LL	UL	LL	UL	LL
SMD1	DE	3,600	537,221	3,960	585,595	4,520	649,558
	NBEA	3,210	3,105,178	5,248	5,262,456	7,378	6,868,944
SMD2	DE	3,280	613,759	3,720	696,895	4,120	734,711
	NBEA	3,326	2,166,384	4,052	4,102,678	7,076	5,803,812
SMD3	DE	3,520	890,998	3,920	981,330	4,240	1,064,627
	NBEA	3,220	3,696,032	4,282	4,814,112	6,314	7,015,724
SMD4	DE	3,120	1,140,213	3,400	1,261,720	3,960	1,462,526
	NBEA	2,454	2,017,734	3,110	2,755,534	5,296	4,364,876
SMD5	DE	3,600	1,175,815	3,960	1,308,906	4,520	1,516,729
	NBEA	4,488	4,574,482	7,004	8,800,232	9,290	12,064,566
SMD6	DE	3,680	338,737	4,080	375,752	4,560	421,805
	NBEA	3,530	5,026,522	6,962	8,448,154	9,978(𝑥)	12,448,922(𝑥)
SMD7	DE	3,840	633,706	4,560	747,605	8,040	1,161,633
	NBEA	—	—	—	—	—	—
SMD8	DE	5,560	2,738,987	6,060	3,003,491	6,640	3,253,497
	NBEA	—	—	—	—	—	—
SMD9	DE	5,760	1,240,993	6,840	1,504,952	7,240	1,589,831
	NBEA	—	—	—	—	—	—
SMD10	DE	—	—	—	—	—	—
	NBEA	—	—	—	—	—	—
SMD11	DE	—	—	—	—	—	—
	NBEA	—	—	—	—	—	—
SMD12	DE	—	—	—	—	—	—
	NBEA	—	—	—	—	—	—