Evaluating the Efficiency of Decision Making Units in Fuzzy two-stage DEA Models

Figures & data

Figure 1. The generic two-stage process.

Figure 2. $\tilde{A}=\left(\underset{\_}{A}\right(r),\overline{A}(r\left)\right)$.

Figure 3. A triangular fuzzy number.

Figure 4. A trapezoidal fuzzy number.

Table 1. The data set for a simple hypotehetical fuzzy two-stage DEA example.

DMUs	$\tilde{x}$	$\tilde{z}$	$\tilde{y}$
A	(1.5,0,1.5)	(1,1,2)	(1,0.5,1)
B	(4,1,0.5)	(6,1,2)	(4,1,3)
C	(6,1,0)	(7,3,1)	(7,2,0.5)
D	(2,2,0)	(2.5,1,0.5)	(3,2,1)
E	(5,0,2)	(4,1,2)	(6,1,0.5)
F	(4,1,1)	(3.5,0.5,0.5)	(2,1,1)
G	(5.5,0.5,1.5)	(5,1,0.5)	(5,3,1)
H	(4.5,0.5,0.5)	(6,1,1)	(6,1,2)
I	(7,4,1)	(6.5,0.5,1)	(6.5,0.5,2.5)

Table 2. $Q_{\alpha }$ function of input, intermediate and output fuzzy data in example 1.

Firms	$Q_{\alpha }\left(\tilde{X}\right)$	$Q_{\alpha }\left(\tilde{Z}\right)$	$Q_{\alpha }\left(\tilde{Y}\right)$
A	3(1−α) + 0.75(1−α)^2	2(1−α) + 0.5(1−α)^2	2(1−α) + 0.25(1−α)^2
B	8(1−α) − 0.25(1−α)^2	12(1−α) + 0.5(1−α)^2	8(1−α) + (1−α)^2
C	12(1−α) − 0.5(1−α)^2	14(1−α) − (1−α)^2	14(1−α) − 0.75(1−α)^2
D	4(1−α) − (1−α)^2	5(1−α) − 0.25(1−α)^2	6(1−α) − 0.5(1−α)^2
E	10(1−α)+ (1−α)^2	8(1−α) + 0.5(1−α)^2	12(1−α)−0.25(1−α)^2
F	8(1−α)	7(1−α)	4(1−α)
G	11(1−α) + 0.5(1−α)^2	10(1−α) − 0.25(1−α)^2	10(1−α) −  (1−α)^2
H	9(1−α)	12(1−α)	12(1−α) + 0.5(1−α)^2
I	14(1−α) − 1.5(1−α)^2	13(1−α) + 0.25(1−α)^2	13(1−α)+(1−α)^2

Figure 5. Efficiency graph of DMUs under CRS assumption for different α.

Table 3. Efficiencies of DMUs in two-stage DEA model with the assumption of CRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version
A	0.269106	0.271331	0.273647	0.276060	0.278576	0.281203	0.283948	0.286821	0.289829	0.292984	0.295958	0.296296
B	0.520851	0.512999	0.505194	0.497436	0.489726	0.482062	0.474446	0.466875	0.459352	0.451875	0.445185	0.444444
C	0.516762	0.517026	0.517270	0.517494	0.517699	0.517885	0.518051	0.518198	0.518324	0.518431	0.518511	0.518519
D	0.822270	0.802339	0.783625	0.766015	0.749411	0.733727	0.718885	0.704817	0.691460	0.678760	0.667850	0.666667
E	0.479091	0.484129	0.489247	0.494447	0.499732	0.505102	0.510561	0.516111	0.521755	0.527495	0.532745	0.533333
F	0.224255	0.224077	0.223893	0.223703	0.223509	0.223308	0.223102	0.222891	0.222674	0.222451	0.222245	0.222222
G	0.351008	0.356174	0.361371	0.366598	0.371855	0.377143	0.382461	0.387810	0.393190	0.398600	0.403495	0.404040
H	0.622931	0.619946	0.616949	0.613942	0.610924	0.607895	0.604855	0.601805	0.598745	0.595674	0.592901	0.592593
I	0.502332	0.492437	0.482769	0.473319	0.464079	0.455043	0.446205	0.437557	0.429094	0.420809	0.413502	0.412698

Table 4. Ranking DMUs under CRS assumption for three different α’s.

α	A	B	C	D	E	F	G	H	I
0.1	8	4	3	1	6	9	7	2	5
0.5	8	5	3	1	4	9	7	2	6
0.9	8	5	4	1	3	9	7	2	6

Table 5. Efficiencies of DMUs in two-stage DEA model with the assumption of VRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version
A	0.800000	0.843537	0.888889	0.936170	0.985507	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000
B	0.546306	0.545116	0.544113	0.543297	0.542669	0.542228	0.541975	0.541911	0.542036	0.542351	0.542798	0.542857
C	0.629500	0.645892	0.664065	0.756693	0.883761	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000
D	1.000000	1.000000	1.000000	1.000000	1.000000	0.994636	0.986584	0.978002	0.968990	0.959633	0.950974	0.950000
E	0.481672	0.489772	0.498051	0.506513	0.515164	0.524011	0.533061	0.542320	0.551796	0.561496	0.570425	0.571429
F	0.375000	0.387500	0.400000	0.412500	0.425000	0.428743	0.428656	0.428214	0.427442	0.426364	0.425149	0.425000
G	0.368163	0.376840	0.385561	0.394329	0.403144	0.412009	0.420926	0.429896	0.438920	0.448002	0.456226	0.457143
H	0.696535	0.695823	0.694503	0.692453	0.689520	0.685499	0.674886	0.664514	0.654390	0.644523	0.635869	0.634921
I	1.000000	1.000000	0.906250	0.860750	0.826336	0.799057	0.693497	0.608231	0.569700	0.549821	0.532503	0.530612

Figure 6. Efficiency graph of DMUs under VRS assumption for different α.

Table 6. Ranking DMUs under VRS assumption for three different ’s.

α	A	B	C	D	E	F	G	H	I
0.1	2	5	4	1	6	7	8	3	1
0.5	1	5	1	2	6	7	8	4	3
0.9	1	6	1	2	4	8	7	3	5

Table 7. Triangular fuzzy input, intermediate and output data of 24 insurance companies in Taiwan.

Co.	First input $\left({\tilde{X}}_1\right)$	Second input $\left({\tilde{X}}_2\right)$	First intermediate $\left({\tilde{Z}}_1\right)$	Second intermediate $\left({\tilde{Z}}_2\right)$	First output $\left({\tilde{Y}}_1\right)$	Second output $\left({\tilde{Y}}_2\right)$
1	(1178,65,78)	(673,37,44)	(7451,410,492)	(856,47,56)	(984,54,65)	(681,37,45)
2	(1381,76,91)	(1352,74,89)	(10020,551,661)	(1812,100,120)	(1228,68,81)	(834,46,55)
3	(1177,65,78)	(592,33,39)	(4776,263,315)	(560,31,37)	(293,16,19)	(658,36,43)
4	(601,33,40)	(594,33,39)	(3174,175,209)	(371,20,24)	(248,14,16)	(177,10,12)
5	(6699,368,442)	(3351,184,221)	(37362,2027,2318)	(1753,96,116)	(7851,432,518)	(3925,216,259)
6	(2627,144,173)	(668,37,44)	(9747,536,643)	(952,52,63)	(1713,94,113)	(415,23,27)
7	(1942,89,105)	(1443,66,78)	(10685,492,577)	(643,30,35)	(2239,103,121)	(439,20,24)
8	(3789,174,205)	(1873,86,101)	(17267,794,932)	(1134,52,61)	(3899,179,211)	(622,29,34)
9	(1567,72,85)	(950,44,51)	(11473,528,620)	(546,25,29)	(1043,48,56)	(264,12,14)
10	(1303,60,70)	(1298,60,70)	(8210,378,443)	(504,23,27)	(1697,78,92)	(554,25,30)
11	(1962,90,106)	(672,31,36)	(7222,332,390)	(643,30,35)	(1486,68,80)	(18,1,1)
12	(2592,119,140)	(650,30,35)	(9434,434,509)	(1118,51,60)	(1574,72,85)	(909,42,49)
13	(2609,128,130)	(1368,67,68)	(13921,682,696)	(811,40,41)	(3609,177,180)	(223,11,11)
14	(1396,68,70)	(988,48,49)	(7396,362,370)	(465,23,23)	(1401,69,70)	(332,16,17)
15	(2184,107,109)	(651,32,33)	(10422,511,521)	(749,37,37)	(3355,164,168)	(555,27,28)
16	(1211,59,61)	(415,20,21)	(5606,275,280)	(402,20,20)	(854,42,43)	(197,10,10)
17	(1453,71,73)	(1085,53,54)	(7695,377,385)	(342,17,17)	(3144,154,157)	(371,18,19)
18	(757,37,38)	(547,27,27)	(3631,178,182)	(995,49,50)	(692,34,35)	(163,8,8)
19	(159,8,8)	(182,9,9)	(1141,58,55)	(483,25,23)	(519,26,25)	(46,2,2)
20	(145,7,7)	(53,3,3)	(316,16,15)	(131,7,6)	(355,18,17)	(26,1,1)
21	(84,4,4)	(26,1,1)	(225,11,11)	(40,2,2)	(51,3,2)	(6,0,0)
22	(15,1,1)	(10,1,0)	(52,3,2)	(14,1,1)	(82,4,4)	(4,0,0)
23	(54,3,3)	(28,1,1)	(245,12,12)	(49,2,2)	(1,0,0)	(18,1,1)
24	(163,8,8)	(235,12,11)	(476,24,23)	(644,33,31)	(142,7,7)	(16,1,1)

Table 8. Efficiency scores of 24 insurance companies in Taiwan with the assumption of CRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version	Rank	Rank from Liu [41]
1	0.699671	0.699656	0.699642	0.699627	0.699613	0.699599	0.699584	0.699570	0.699555	0.699541	0.699528	0.699526	3	3
2	0.625627	0.625638	0.625650	0.625661	0.625672	0.625683	0.625695	0.625706	0.625717	0.625728	0.625738	0.625740	5	5
3	0.690428	0.690408	0.690388	0.690369	0.690349	0.690329	0.690310	0.690290	0.690270	0.690251	0.690233	0.690231	4	4
4	0.304236	0.304244	0.304251	0.304259	0.304267	0.304275	0.304283	0.304291	0.304299	0.304307	0.304315	0.304315	15	15
5	0.791964	0.791981	0.791999	0.792016	0.792034	0.792052	0.792069	0.792087	0.792105	0.792122	0.792138	0.792140	1	1
6	0.388887	0.388934	0.388980	0.389027	0.389073	0.389120	0.389166	0.389213	0.389259	0.389306	0.389348	0.389352	12	12
7	0.277049	0.277072	0.277094	0.277116	0.277139	0.277161	0.277183	0.277206	0.277228	0.277250	0.277270	0.277273	17	17
8	0.274688	0.274710	0.274733	0.274755	0.274778	0.274800	0.274823	0.274845	0.274868	0.274890	0.274917	0.274920	18	18
9	0.223579	0.223595	0.223611	0.223627	0.223644	0.223660	0.223676	0.223692	0.223708	0.223724	0.223738	0.223740	20	20
10	0.467387	0.467406	0.467425	0.467445	0.467464	0.467483	0.467502	0.467522	0.467541	0.467560	0.467577	0.467579	9	9
11	0.159349	0.159359	0.159369	0.159379	0.159388	0.159398	0.159408	0.159418	0.159428	0.159438	0.159447	0.159448	23	23
12	0.759372	0.759404	0.759436	0.759468	0.759500	0.759532	0.759564	0.759596	0.759629	0.759661	0.759690	0.759693	2	2
13	0.206355	0.206388	0.206421	0.206454	0.206487	0.122230	0.206553	0.206586	0.206619	0.206652	0.206682	0.206685	21	21
14	0.289089	0.289101	0.289113	0.289125	0.289137	0.289149	0.289161	0.289173	0.289185	0.289197	0.289207	0.289209	16	16
15	0.610980	0.611048	0.611116	0.611184	0.611252	0.611320	0.611388	0.611456	0.611524	0.611592	0.611654	0.611661	6	6
16	0.318516	0.318559	0.318602	0.318645	0.318688	0.318731	0.318774	0.318817	0.318860	0.318903	0.318941	0.318946	14	14
17	0.360319	0.360365	0.360410	0.360456	0.360501	0.360547	0.360592	0.360638	0.360684	0.360729	0.360770	0.360775	13	13
18	0.258324	0.258345	0.258366	0.258386	0.258407	0.258428	0.258448	0.258469	0.258490	0.258511	0.258529	0.258531	19	19
19	0.411984	0.412108	0.412232	0.412356	0.412480	0.412605	0.412729	0.412853	0.412977	0.413101	0.413213	0.413226	10	11
20	0.538344	0.538417	0.538491	0.538564	0.538637	0.538711	0.538784	0.538858	0.538931	0.539005	0.539071	0.539078	8	8
21	0.190364	0.190431	0.190499	0.190566	0.190634	0.190701	0.190769	0.190837	0.190904	0.190972	0.191033	0.191040	22	22
22	0.618817	0.617519	0.616227	0.614941	0.613660	0.612385	0.611115	0.609850	0.608591	0.607338	0.606214	0.606089	7	7
23	0.404265	0.404255	0.404244	0.404234	0.404224	0.404214	0.404204	0.404194	0.404184	0.404174	0.404165	0.404164	11	10
24	0.130450	0.130481	0.130511	0.130542	0.130572	0.130603	0.130634	0.130664	0.130695	0.130725	0.130753	0.130756	24	24

Table 9. Efficiency scores of 24 insurance companies from different approaches with the assumption of CRS (α = 0).

α = 0
Co.	Our proposed approach	Kao and Liu [37]	Lozano [38]	Tavana et al. [55]	Hatami-Marbini and Saati [56]
1	0.699671	(0.493,0.906)	(0.500,0.904)	(0.679,0.9)	(0.437,0.559)
2	0.625627	(0.439,0.798)	(0.447,0.795)	(0.604,0.8)	(0.331,0.422)
3	0.690428	(0.487,0.762)	(0.549,0.861)	(0.669,0.86)	(0.406,0.501)
4	0.304236	(0.213,0.426)	(0.213,0.426)	(0.284,0.43)	(0.160,0.203)
5	0.791964	(0.562,0.957)	(0.563,1.000)	(0.771,1.00)	(0.499,0.637)
6	0.388887	(0.279,0.514)	(0.279,0.510)	(0.369,0.51)	(0.214,0.273)
7	0.277049	(0.202,0.377)	(0.203,0.375)	(0.26,0.38)	(0.217,0.250)
8	0.274688	(0.202,0.374)	(0.203,0.371)	(0.258,0.37)	(0.201,0.246)
9	0.223579	(0.162,0.296)	(0.163,0.294)	(0.206,0.29)	(0.157,0.191)
10	0.467387	(0.335,0.638)	(0.337,0.636)	(0.449,0.64)	(0.272,0.332)
11	0.159349	(0.122,0.221)	(0.116,0.217)	(0.147,0.22)	(0.081,0.099)
12	0.759372	(0.553,0.945)	(0.559,0.942)	(0.773,0.94)	(0.392,0.484)
13	0.206355	(0.153,0.280)	(0.153,0.276)	(0.192,0.28)	(0.169,0.206)
14	0.289089	(0.211,0.394)	(0.212,0.392)	(0.273,0.39)	(0.207,0.253)
15	0.610980	(0.449,0.797)	(0.460,0.794)	(0.598,0.79)	(0.383,0.466)
16	0.318516	(0.233,0.436)	(0.233,0.433)	(0.304,0.43)	(0.205,0.250)
17	0.360319	(0.263,0.488)	(0.266,0.484)	(0.344,0.48)	(0.287,0.349)
18	0.258324	(0.189,0.352)	(0.189,0.350)	(0.243,0.35)	(0.189,0.227)
19	0.411984	(0.300,0.513)	(0.309,0.497)	(0.395,0.5)	(0.280,0.307)
20	0.538344	(0.405,0.735)	(0.395,0.725)	(0.532,0.72)	(0.378,0.458)
21	0.190364	(0.148,0.273)	(0.145,0.247)	(0.189,0.25)	(0.127,0.146)
22	0.618817	(0.438,0.651)	(0.510,0.783)	(0.483,0.78)	(0.563,0.679)
23	0.404265	(0.302,0.578)	(0.291,0.556)	(0.409,0.56)	(0.234,0.285)
24	0.130450	(0.097,0.183)	(0.095,0.176)	(0.12,0.18)	(0.072,0.089)

Table 10. Efficiency scores of 24 insurance companies in Taiwan with the assumption of VRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version	Rank
1	0.735957	0.735976	0.735995	0.736002	0.736009	0.736016	0.736024	0.736031	0.736038	0.736045	0.736052	0.736053	5
2	0.710066	0.710091	0.710116	0.710141	0.710166	0.710191	0.710216	0.710241	0.710266	0.710291	0.710313	0.710316	7
3	0.690765	0.690743	0.690721	0.690699	0.690677	0.690654	0.690632	0.690610	0.690587	0.690565	0.690545	0.690543	8
4	0.311879	0.311873	0.311867	0.311860	0.311854	0.311848	0.311841	0.311835	0.311829	0.311822	0.311817	0.311816	21
5	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1
6	0.490102	0.490125	0.490148	0.490170	0.490193	0.490216	0.490239	0.490262	0.490285	0.490308	0.490329	0.490331	14
7	0.446903	0.446903	0.446904	0.446904	0.446905	0.446905	0.446906	0.446907	0.446907	0.446908	0.446908	0.446908	15
8	0.526460	0.526385	0.526310	0.526235	0.526160	0.526085	0.526009	0.525934	0.525859	0.525783	0.525715	0.525708	13
9	0.278449	0.278467	0.278484	0.278502	0.278519	0.278537	0.278555	0.278572	0.278590	0.278608	0.278624	0.278626	22
10	0.626336	0.626341	0.626345	0.626350	0.626354	0.626359	0.626364	0.626368	0.626373	0.626378	0.626382	0.626382	11
11	0.332613	0.332608	0.332603	0.332598	0.332593	0.332588	0.332582	0.332577	0.332572	0.332567	0.332562	0.332562	19
12	0.782873	0.782912	0.782952	0.782991	0.783031	0.783070	0.783110	0.783149	0.783189	0.783229	0.783264	0.783268	4
13	0.612840	0.612877	0.612915	0.612952	0.612989	0.144899	0.613064	0.613102	0.613139	0.613177	0.613210	0.613214	12
14	0.403446	0.403452	0.403459	0.403465	0.403472	0.403478	0.403484	0.403491	0.403497	0.403503	0.403509	0.403510	17
15	0.880752	0.880768	0.880785	0.880802	0.880819	0.880835	0.880852	0.880869	0.880885	0.880902	0.880917	0.880919	3
16	0.382897	0.382942	0.382987	0.383032	0.383077	0.383122	0.383167	0.383213	0.383258	0.383303	0.383343	0.383348	18
17	0.724138	0.724136	0.724135	0.724133	0.724132	0.724130	0.724129	0.724127	0.724126	0.724124	0.724123	0.724123	6
18	0.324558	0.324566	0.324574	0.324582	0.324591	0.324599	0.324607	0.324616	0.324624	0.324632	0.324640	0.324641	20
19	0.674994	0.674969	0.674944	0.674920	0.674895	0.674870	0.674846	0.674821	0.674796	0.674771	0.674749	0.674747	9
20	0.894859	0.895292	0.895726	0.896159	0.896593	0.897027	0.897461	0.897895	0.898329	0.898763	0.899154	0.899198	2
21	0.412761	0.413720	0.414680	0.415640	0.416600	0.417560	0.418520	0.419480	0.420440	0.421399	0.422263	0.422359	16
22	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1
23	0.629541	0.630058	0.630575	0.631091	0.631608	0.632125	0.632641	0.633158	0.633675	0.634192	0.634657	0.634708	10
24	0.180399	0.180393	0.180387	0.180381	0.180374	0.180368	0.180362	0.180356	0.180350	0.180343	0.180338	0.180337	23

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