The selected benchmark problems to be solved by the ASSF approach are Dejong-2D, Dejong-8D, Ackley-2D and Ackley-8D function optimization problems. The PSO swarm sizes were 10 and 30 and the PSO maximum iterations were 100 and 300 for 2D and 8D problems, respectively. More than one solution was determined by the ASSF approach because the best solution with the best objective value reached by the approach may not be the one having the best original objective function. Therefore, the results reported in Table 1 presents the best solutions among 10 runs of the SPSO algorithm. In this table, the approximate optimum function ⁠, the original fitness function and the related error have been reported.

Table 1

Results of the function minimization problems using the ASSF approach for different meta-models

.	Optimum point located .	Approximate obj. function .	Original obj. function .	MSE .	Number of evaluations of the original function (NFE) .
Dejong-2D
SPSO	[0 0]	–	0	–	1,000
SVM	[0 0]	−0.01	0	0.01	341
ANN	[0 0]	0.0419	0	−0.0419	627
Kriging	[0 0]	−8.84 × 10−7	0	−8.84 × 10−7	346
Polynomial	[0 0]	−4.34 × 10−15	0	4.34 × 10−15	306
Dejong-8D
SPSO	[00000000]	–	0	–	9,000
SVM	[00000000]	−0.0408	0	0.0408	1,159
ANN	[00000000]	8.14 × 10−4	0	−0.0114	2,021
Kriging	[-0.0636–0.029–0.0596–0.1245 0.0166 0.0023 0.006–0.0270]	−5.2683 × 10−4	0.8854	0.8859	2,346
Polynomial	[00000000]	−2.329 × 10−3	0	−2.329 × 10−3	1,201
Ackley-2D
SPSO	[0 0]	–	0	–	1,000
SVM	[0 0]	0.1672	0	−0.1672	256
ANN	[0 0]	0.159	0	−0.159	494
Kriging	[0 0]	0.0896	0	−0.0896	667
Polynomial	[0 0]	0.9987	0	−0.998	2,893
Ackley-8D
SPSO	[00000000]	–	0	–	9,000
SVM	[00000000]	0.00275	0	0.00275	1,956
ANN	[00000000]	0.2025	0	−0.2025	6,215
Kriging	[−0.0175 0.0779–0.0046–0.006 0.05 0.0776–0.0125 0.1071]	0.07981	0.07869	−0.012	8,345
Polynomial	[−0.2548–0.2732-0.0562–0.4618 0.1307–0.1127 0.3673–0.1004]	4.7988	2.7613	−2.0376	9,689

.	Optimum point located .	Approximate obj. function .	Original obj. function .	MSE .	Number of evaluations of the original function (NFE) .
Dejong-2D
SPSO	[0 0]	–	0	–	1,000
SVM	[0 0]	−0.01	0	0.01	341
ANN	[0 0]	0.0419	0	−0.0419	627
Kriging	[0 0]	−8.84 × 10−7	0	−8.84 × 10−7	346
Polynomial	[0 0]	−4.34 × 10−15	0	4.34 × 10−15	306
Dejong-8D
SPSO	[00000000]	–	0	–	9,000
SVM	[00000000]	−0.0408	0	0.0408	1,159
ANN	[00000000]	8.14 × 10−4	0	−0.0114	2,021
Kriging	[-0.0636–0.029–0.0596–0.1245 0.0166 0.0023 0.006–0.0270]	−5.2683 × 10−4	0.8854	0.8859	2,346
Polynomial	[00000000]	−2.329 × 10−3	0	−2.329 × 10−3	1,201
Ackley-2D
SPSO	[0 0]	–	0	–	1,000
SVM	[0 0]	0.1672	0	−0.1672	256
ANN	[0 0]	0.159	0	−0.159	494
Kriging	[0 0]	0.0896	0	−0.0896	667
Polynomial	[0 0]	0.9987	0	−0.998	2,893
Ackley-8D
SPSO	[00000000]	–	0	–	9,000
SVM	[00000000]	0.00275	0	0.00275	1,956
ANN	[00000000]	0.2025	0	−0.2025	6,215
Kriging	[−0.0175 0.0779–0.0046–0.006 0.05 0.0776–0.0125 0.1071]	0.07981	0.07869	−0.012	8,345
Polynomial	[−0.2548–0.2732-0.0562–0.4618 0.1307–0.1127 0.3673–0.1004]	4.7988	2.7613	−2.0376	9,689

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