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

Table 12

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 10-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	1,800	60,209	2,040	68,569	2,320	77,968
	NBEA	1,080	862,653	2,534	1,623,356	3,488	2,074,334
SMD2	DE	120	5,987	1,800	65,604	2,240	86,422
	NBEA	1,398	1,055,976	2,366	1,467,246	3,418	2,114,442
SMD3	DE	1,880	122,781	2,080	134,397	2,360	154,032
	NBEA	1,210	900,358	2,278	1,383,632	2,862	1,805,562
SMD4	DE	120	2,989	1,740	73,836	2,240	94,518
	NBEA	678	566,344	1,598	1,087,632	2,028	1,314,986
SMD5	DE	1,560	116,112	1,960	143,344	2,240	165,197
	NBEA	1,620	1,226,344	2,890	1,993,124	3,492	2,483,442
SMD6	DE	1,800	31,436	2,040	35,490	2,320	39,932
	NBEA	1,502	1,225,742	2,936	2,224,450	4,278(𝑥)	3,786,498(𝑥)
SMD7	DE	120	6,037	2,240	79,360	3,960	142,556
	NBEA	1,382	932,460	2,394	1,566,481	3,858(𝑥)	2,435,994(𝑥)
SMD8	DE	3,040	485,145	3,400	557,742	4,120	671,696
	NBEA	2,116	1,457,480	4,188	2,710,132	5,986(𝑥)	5,294,734(𝑥)
SMD9	DE	120	9,469	2,720	220,749	3,360	293,523
	NBEA	—	—	—	—	—	—
SMD10	DE	5,840	493,079	6,600	556,981	7,880	658,663
	NBEA	—	—	—	—	—	—
SMD11	DE	5,520	592,663	10,960	1,143,648	12,040	1,257,916
	NBEA	—	—	—	—	—	—
SMD12	DE	7,000	597,165	7,320	624,658	8,320	709,605
	NBEA	—	—	—	—	—	—