Table 5.

Statistical lower bounds, upper bounds and the estimators for quality evaluation.

Sample size	Repetition	Objective	Lower bound		Upper bound		Gap estimators
			${\overline{f}}_{Q,R}$	${\sigma }_{{\overline{f}}_{Q,R}}^2$	${\hat{f}}_{Q^{’}}\left(\hat{x}\right)$	${\sigma }_{{\hat{f}}_{Q^{’}}\left(\hat{x}\right)}^2$	$ga{p}_{Q,R,Q^{’}}\left(\hat{x}\right)$	%	${\sigma }_{gap}^2$
$Q$ = 20	$R$ = 10	Cost	43,186,456	175,394	43,250,895	809,434	64,439	0.15	828,219
		Risk	96,467,552	1,512,030	97,711,236	8,528,440	1,243,684	1.29	8,661,439
$Q$ = 40	$R$ = 10	Cost	43,181,245	130,564	43,250,895	809,434	69,650	0.16	819,896
		Risk	97,021,695	1,016,312	97,402,577	8,596,540	380,882	0.39	8,656,408
$Q$ = 60	$R$ = 10	Cost	43,252,268	86,316	43,250,895	809,434	−1373	0	814,023
		Risk	97,723,873	651,248	97,513,812	8,681,283	−210,061	−0.22	8,705,676