Enhancing Artificial Bee Colony Algorithm with Self-Adaptive Searching Strategy and Artificial Immune Network Operators for Global Optimization

Table 2

Benchmark functions used in experiments for testing the performances of EABC, ACO, PSO, and ABC.

Function number	Min		ACO	PSO	ABC	EABC
𝑓1(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖	0	Best	0	0	0	0
		Worst	0	0	0	0
		Mean	0	0	0	0
		SD	0	0	0	0
𝑓2(𝑥)=∑^𝐷𝑖=1𝑖𝑥^2𝑖	0	Best	0	0	0	0
		Worst	0	0	0	0
		Mean	0	0	0	0
		SD	0	0	0	0
𝑓3(𝑥)=25+∑^𝐷𝑖=1𝑥𝑖	0	Best	0	0	0	0
		Worst	0	0	0	0
		Mean	0	0	0	0
		SD	0	0	0	0
𝑓4(𝑥)=∑^𝐷𝑖=1𝑥𝑖+0.5^2	0	Best	0	0	0	0
		Worst	0	0	0	0
		Mean	0	0	0	0
		SD	0	0	0	0
𝑓5(𝑥)=∑^𝐷𝑖=1𝑖𝑥^4𝑖+random(0,1)	0	Best	0	0	0	0
		Worst	0.00289	0.00321	0.00543	0.00422
		Mean	0.00136	0.00116	0.00300	0.00196
		SD	0.00219	0.00276	0.00387	0.00208
𝑓6(𝑥)=max𝑖𝑥𝑖,1≤𝑖≤𝐷	0	Best	0	0	0	0
		Worst	0.00246	0.00305	0.00110	0.00400
		Mean	0.00180	0.00156	0.0066	0.00210
		SD	0.00039	0.00058	0.00092	0.00037
𝑓7(𝑥)=−∏^𝐷𝑖=1cos𝑥𝑖exp−∑^𝐷𝑖=1𝑥𝑖−𝜋^2	−1	Best	−1	−1	−1	−1
		Worst	−1	−1	−1	−1
		Mean	−1	−1	−1	−1
		SD	0	0	0	0
𝑓8(𝑥)=0.26∑^𝐷𝑖=1𝑥^2𝑖−0.48∏^𝐷𝑖=1𝑥𝑖	0	Best	0	0	0	0
		Worst	0	0	0	0
		Mean	0	0	0	0
		SD	0	0	0	0
𝑓9(𝑥)=(𝑥1−1)^2+∑^𝐷𝑖=2𝑖(2𝑥^2𝑖−𝑥𝑖−1)^2	0	Best	0.66667	0.6667	0	0
		Worst	0.66667	0.6667	0	0
		Mean	0.66667	0.6667	0	0
		SD	0.00001	0.00001	0	0
𝑓10(𝑥)=∑^𝐷𝑖=1100𝑥𝑖+1−𝑥^2𝑖^2+1−𝑥𝑖^2	0	Best	8.7513	10.5433	19.6788	9.1578
		Worst	32.4215	24.6711	54.2333	26.9874
		Mean	18.2039	15.0886	33.1227	17.3558
		SD	5.0361	24.1702	154.1443	11.4774
𝑓11(𝑥)=∑^𝐷𝑖=1𝑥𝑖−10cos2𝜋𝑥𝑖+10	0	Best	52.6677	43.5774	0	0
		Worst	53.2331	44.1131	0	0
		Mean	52.9226	43.9771	0	0
		SD	4.5649	11.7286	0	0
𝑓121(𝑥)=4000^𝐷𝑖∑=1𝑥𝑖−100^2−∏^𝐷𝑖=1𝑥cos^𝑖−100√𝑖+1	0	Best	0.01470	0.017112	0.008531	0
		Worst	0.01499	0.017989	0.017356	0
		Mean	0.01479	0.017391	0.011447	0
		SD	0.00296	0.020808	0.001223	0
𝑓13(𝑥)=10×𝐷+∑^𝐷𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖	0	Best	4.56781	21.44755	3.34788	0
		Worst	8.77993	39.55741	10.91447	0
		Mean	5.85411	26.3991	5.59331	0
		SD	13.1142	155.6380	10.04216	0
𝑓14(𝑥)=0.5+sin^2∑^𝐷𝑖=1𝑥^2𝑖−0.51+0.001∑^𝐷𝑖=1𝑥^2𝑖^2	0	Best	0	0	0	0
		Worst	0	0	0	0
		Mean	0	0	0	0
		SD	0	0	0	0
𝑓15(𝑥)=∑^𝐷𝑖=1−𝑥𝑖sin𝑥𝑖	−12569.5	Best	−10296	−6993.47	−11566.9	−12568.7
		Worst	−10237	−6883.33	−11498.8	−12514.3
		Mean	−10266	−6909.12	−11544.1	−12551.1
		SD	521.849	457.9577	125.4471	101.3217