A Multiswarm Intelligence Algorithm for Expensive Bound Constrained Optimization Problems

Table 2

The experimental results of (a) MSIA, (b) ACO, and (c) FA for CEC′13 benchmark functions in 10 dimensions [54].

Problem	Optimum	Best	Average	Standard deviation	Maximum	Algorithms
	−1400	–1.40000000e + 003	–1.28813052e + 003	3.83901313e + 002	2.67500831e + 003	a
		–1.40000000e + 003	–1.15749008e + 003	1.07195386e + 003	4.55466927e + 003	b
		–1.40000000e + 003	–1.39443074e + 003	1.05911384e + 002	1.33204120e + 003	c
	–1300	–1.30000000e + 003	1.00558869e + 002	3.34248688e + 003	1.01984633e + 003	a
		–1.31053585e + 003	1.20311183e + 002	3.71650087e + 003	1.50229253e + 003	b
		–1.30240122e + 003	1.00650874e + 001	3.34011105e + 002	1.03514064e + 003	c
	–1200	–1.20000000e + 003	1.84411229e + 003	6.23354197e + 003	9.99110936e + 004	a
		–1.31510766e + 002	2.31193243e + 004	6.50115943e + 003	2.48403247e + 005	b
		–1.29995459e + 003	1.18540350e + 003	1.61050797e + 003	7.63499302e + 004	c
	–1100	–1.10000000e + 003	1.12780473e + 003	6.26400664e − 003	1.16009458e + 003	a
		4.11467285e + 004	1.11467285e + 004	2.47506343e + 5	4.11467285e + 004	b
		–1.10000000e + 003	1.65098334e + 003	1.45403194e + 003	1.86066330e + 004	c
	–1000	–1.00000000e + 003	1.65169743e + 003	6.32203970e + 003	1.14679794e + 005	a
		–1.00000000e + 003	–8.90435476e + 002	5.25039922e + 002	3.99696948e + 003	b
		–1.00000000e + 003	–9.99017949e + 002	1.99880005e + 001	–4.21267907e + 002	c
	–900	–1.40000000e + 003	1.74778253e + 003	5.98501382e + 003	1.07202661e + 005	a
		–8.94435996e + 002	–8.72043240e + 002	9.43596267e + 001	–4.29479950e + 001	b
		–8.99971565e + 002	–8.99359770e + 002	6.00877221e + 000	–7.22793586e + 002	c
	–800	–1.40000000e + 003	1.74757754e + 003	6.32949555e + 003	1.06572975e + 005	a
		–7.99256478e + 002	–7.53886898e + 002	6.62008126e + 001	–4.39710456e + 002	b
		–8.00000000e + 002	–7.99490674e + 002	6.06214778e + 000	–6.93335202e + 002	c
	–700	–1.40000000e + 003	1.77042499e + 003	6.21341629e + 003	1.10342739e + 005	a
		–1.40000000e + 003	–1.17255767e + 003	1.00591243e + 003	5.03660790e + 003	b
		–1.40000000e + 003	–1.39333836e + 003	1.37246794e + 002	2.48848299e + 003	c
	–600	–1.40000000e + 003	1.63539741e + 003	5.79958004e + 003	1.02497124e + 005	a
		–6.79760366e + 002	–6.79685333e + 002	3.41686898e − 002	–6.79222141e + 002	b
		–6.79892570e + 002	–6.79846262e + 002	4.32771150e − 002	–6.79577360e + 002	c
	–500	–1.40000000e + 003	1.81847280e + 003	6.63718254e + 003	1.02569292e + 005	a
		–5.92651099e + 002	–5.91792871e + 002	3.09488316e − 001	–5.88815678e + 002	b
		–5.99358333e + 002	–5.99287888e + 002	3.50839046e − 001	–5.91626684e + 002	c
	–400	–1.40000000e + 003	1.82639194e + 003	6.40390942e + 003	9.39110015e + 004	a
		–4.99597235e + 002	–4.18769343e + 002	2.03331451e + 002	3.02821266e + 002	b
		–4.99953263e + 002	–4.99163416e + 002	1.17780480e + 001	–1.66277831e + 002	c
	–300	–1.40000000e + 003	2.04024682e + 003	6.43914792e + 003	1.08141375e + 005	a
		–3.74840231e + 002	–3.59247046e + 002	1.98618374e + 001	–2.62805243e + 002	b
		–4.00000000e + 002	–3.99487961e + 002	4.70209205e + 000	–3.18882349e + 002	c
	–200	–1.40000000e + 003	1.61912036e + 003	5.97109732e + 00	9.13305274e + 004	a
		–2.81939856e + 002	–2.60638895e + 002	2.09195180e + 001	–1.65501622e + 002	b
		–2.96020164e + 002	–2.95516805e + 002	4.98604369e + 000	–1.99035788e + 002	c
	–100	–1.41000000e + 003	–1.38285100e + 003	1.26343726e + 002	2.37763536e + 003	a
		–1.80661472e + 002	–1.53623927e + 002	2.15069087e + 001	–9.27553100e + 000	b
		–1.99005041e + 002	–1.98527872e + 002	4.31448924e + 000	–1.04821070e + 002	c