An Efficient Branch-and-Bound Algorithm for Globally Solving Minimax Linear Fractional Programming Problem

Table 1

Numerical comparisons among the algorithm presented in the works of Feng et al. [8], Jiao et al. [12], Wang et al. [24], and our algorithm on Examples 1–8.

Example	Refs.	Optimal value	Optimal solution	Iter	CPU time
1	Feng et al. [8]	0.57310	(1.0157, 0.5905, 1.4037)	1897	88.40531
	Jiao and Liu [12]	0.57310	(1.0157, 0.5905, 1.4037)	22	1.05172
	Wang et al. [24]	0.57310	(1.0157, 0.5905, 1.4037)	22	1.38462
	Ours	0.57335	(1.0157, 0.5902, 1.4041)	1	0.20335
2	Feng et al. [8]	0.67136	(1.5000, 1.5000)	123	3.69388
	Jiao and Liu [12]	0.67136	(1.5000, 1.5000)	16	0.52438
	Wang et al. [24]	0.67136	(1.5000, 1.5000)	20	1.11778
	Ours	0.67136	(1.5000, 1.5000)	1	0.05756
3	Feng et al. [8]	1.34783	(1.0167, 0.5500, 1.4500)	428	14.23580
	Jiao and Liu [12]	1.34783	(1.0167, 0.5500, 1.4500)	22	1.01556
	Wang et al. [24]	1.34783	(1.0167, 0.5500, 1.4500)	19	1.26376
	Ours	1.34783	(1.0167, 0.5500, 1.4500)	17	0.89739
4	Feng et al. [8]	2.40000	(1.0167, 0.5500, 1.4500)	494	17.10538
	Jiao and Liu [12]	2.40000	(1.0167, 0.5500, 1.4500)	23	0.93779
	Wang et al. [24]	2.40000	(1.0167, 0.5500, 1.4500)	19	1.33390
	Ours	2.40000	(1.0167, 0.5500, 1.4500)	43	2.13518
5	Feng et al. [8]	1.16157	(1.0000, 0.5500, 1.4500)	313	10.15298
	Jiao and Liu [12]	1.16157	(1.0000, 0.5500, 1.4500)	21	0.85291
	Wang et al. [24]	1.16157	(1.0000, 0.5500, 1.4500)	17	1.18794
	Ours	1.16157	(1.0000, 0.5500, 1.4500)	19	0.88201
6	Feng et al. [8]	0.98971	(1.3452, 0.5000, 1.9464)	2272	113.77662
	Jiao and Liu [12]	0.98971	(1.3452, 0.5000, 1.9465)	40	1.84189
	Wang et al. [24]	0.98971	(1.3452, 0.5000, 1.9465)	39	2.04739
	Ours	0.99279	(1.3500, 0.5000, 1.9417)	91	4.06822
7	Feng et al. [8]	1.11789	(1.5054, 0.3500, 1.5500)	2076	96.72385
	Jiao and Liu [12]	1.11789	(1.5054, 0.3500, 1.5500)	39	1.58835
	Wang et al. [24]	1.11789	(1.5054, 0.3500, 1.5500)	41	2.08152
	Ours	1.12533	(1.4844, 0.3500, 1.5500)	98	4.67947
8	Feng et al. [8]	1.11838	(1.7538, 0.3500, 1.5500)	10326	477.60639
	Jiao and Liu [12]	1.11838	(1.7538, 0.3500, 1.5500)	46	1.77758
	Wang et al. [24]	1.11838	(1.7538, 0.3500, 1.5500)	60	2.60233
	Ours	1.13750	(1.4546, 0.3500, 1.3967)	149	6.75431