Sustainable concept selection using modified fuzzy TOPSIS: a case study

Figures & data

Figure 1 Hierarchical model used in the study.

Table 1 Linguistic variables for ratings.

Linguistic variable	Fuzzy number
Very poor (VP)	(0, 0, 1, 2)
Poor (P)	(1, 2, 2, 3)
Medium poor (MP)	(2, 3, 4, 5)
Fair (F)	(4, 5, 5, 6)
Medium good (MG)	(5, 6, 7, 8)
Good (G)	(7, 8, 8, 9)
Very good (VG)	(8, 9, 10, 10)

Table 2 Linguistic variables for performance criteria weights.

Linguistic variable	Fuzzy number
Very low (VL)	(0, 0, 0.1, 0.2)
Low (L)	(0.1, 0.2, 0.2, 0.3)
Medium low (ML)	(0.2, 0.3, 0.4, 0.5)
Medium (M)	(0.4, 0.5, 0.5, 0.6)
Medium high (MH)	(0.5, 0.6, 0.7, 0.8)
High (H)	(0.7, 0.8, 0.8, 0.9)
Very high (VH)	(0.8, 0.9, 1, 1)

Table 3 Performance ratings of five concepts with respect to 12 sustainability criteria.

		DMs
Concept	Criteria	D 1	D 2	D 3
A 1	C 1	G	MG	MG
	C 2	MG	MG	G
	C 3	G	MG	G
	C 4	MG	G	MG
	C 5	G	MG	G
	C 6	MG	G	G
	C 7	G	MG	G
	C 8	MG	G	MG
	C 9	G	MG	G
	C 10	MG	G	MG
	C 11	G	MG	G
	C 12	MG	G	MG
A 2	C 1	G	MG	G
	C 2	G	MG	G
	C 3	G	MG	G
	C 4	MG	G	MG
	C 5	G	MG	G
	C 6	MG	G	MG
	C 7	MG	G	G
	C 8	G	MG	MG
	C 9	G	MG	MG
	C 10	MG	G	G
	C 11	G	MG	MG
	C 12	G	MG	G
A 3	C 1	VG	G	VG
	C 2	G	VG	VG
	C 3	G	VG	G
	C 4	G	VG	G
	C 5	VG	G	VG
	C 6	G	VG	G
	C 7	VG	G	VG
	C 8	G	VG	G
	C 9	G	VG	VG
	C 10	VG	G	G
	C 11	G	VG	VG
	C 12	G	VG	VG
A 4	C 1	G	VG	G
	C 2	G	G	VG
	C 3	G	VG	G
	C 4	G	VG	G
	C 5	G	MG	G
	C 6	MG	G	MG
	C 7	G	MG	MG
	C 8	G	MG	VG
	C 9	VG	G	G
	C 10	G	VG	G
	C 11	G	MG	G
	C 12	G	MG	VG
A 5	C 1	VG	G	G
	C 2	G	VG	G
	C 3	G	MG	G
	C 4	G	MG	G
	C 5	G	MG	MG
	C 6	MG	G	MG
	C 7	G	MG	MG
	C 8	G	MG	MG
	C 9	G	MG	VG
	C 10	VG	G	MG
	C 11	G	VG	G
	C 12	G	MG	G

Table 4 Importance weights of the criteria by the three decision-makers.

	DM
Criteria	D 1	D 2	D 3
C 1	VH	H	VH
C 2	H	VH	H
C 3	MH	H	MH
C 4	VH	H	MH
C 5	MH	M	MH
C 6	H	MH	MH
C 7	MH	H	MH
C 8	M	H	MH
C 9	VH	H	H
C 10	H	MH	MH
C 11	H	VH	H
C 12	MH	H	VH

Table 5 Fuzzy decision matrix and fuzzy weight for each criterion.

	A 1	A 2	A 3	A 4	A 5	Weight
C 1	(5,6.67,7.33,9)	(5,7.33,7.67,9)	(7,8.67,9.33,10)	(7,8.33,8.67,10)	(7,8.33,8.67,10)	(0.7,0.87,0.93,1)
C 2	(5,6.67,7.33,9)	(5,7.33,7.67,9)	(7,8.67,9.33,10)	(7,8.33,8.67,10)	(7,8.33,8.67,10)	(0.7,0.83,0.87,1)
C 3	(5,7.33,7.67,9)	(5,7.33,7.67,9)	(7,8.33,8.67,10)	(7,8.33,8.67,10)	(5,7.33,7.67,9)	(0.5,0.67,0.73,0.9)
C 4	(5,6.67,7.33,9)	(5,6.67,7.33,9)	(7,8.33,8.67,10)	(7,8.33,8.67,10)	(5,7.33,7.67,9)	(0.5,0.77,0.83,1)
C 5	(5,7.33,7.67,9)	(5,7.33,7.67,9)	(7,8.67,9.33,10)	(5,7.33,7.67,9)	(5,6.67,7.33,9)	(0.4,0.57,0.63,0.8)
C 6	(5,7.33,7.67,9)	(5,6.67,7.33,9)	(7,8.33,8.67,10)	(5,6.67,7.33,9)	(5,6.67,7.33,9)	(0.5,0.67,0.73,0.9)
C 7	(5,7.33,7.67,9)	(5,7.33,7.67,9)	(7,8.67,9.33,10)	(5,6.67,7.33,9)	(5,6.67,7.33,9)	(0.5,0.67,0.73,0.9)
C 8	(5,6.67,7.33,9)	(5,6.67,7.33,9)	(7,8.33,8.67,10)	(5,7.67,8.33,10)	(5,6.67,7.33,9)	(0.4,0.63,0.67,0.9)
C 9	(5,7.33,7.67,9)	(5,6.67,7.33,9)	(7,8.67,9.33,10)	(7,8.33,8.67,10)	(5,7.67,8.33,10)	(0.7,0.83,0.87,1)
C 10	(5,6.67,7.33,9)	(5,7.33,7.67,9)	(7,8.33,8.67,10)	(7,8.33,8.67,10)	(5,7.67,8.33,10)	(0.5,0.67,0.73,0.9)
C 11	(5,7.33,7.67,9)	(5,6.67,7.33,9)	(7,8.67,9.33,10)	(5,7.33,7.67,9)	(7,8.33,8.67,10)	(0.7,0.83,0.87,1)
C 12	(5,6.67,7.33,9)	(5,7.33,7.67,9)	(7,8.67,9.33,10)	(5,7.67,8.33,10)	(5,7.33,7.67,9)	(0.5,0.77,0.83,1)

Table 6 Normalised fuzzy decision matrix.

	A 1	A 2	A 3	A 4	A 5
C 1	(0.5,0.67,0.73,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.87,0.93,1)	(0.7,0.83,0.87,1)	(0.7,0.83,0.87,1)
C 2	(0.5,0.67,0.73,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.87,0.93,1)	(0.7,0.83,0.87,1)	(0.7,0.83,0.87,1)
C 3	(0.5,0.73,0.77,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.83,0.87,1)	(0.7,0.83,0.87,1)	(0.5,0.73,0.77,0.9)
C 4	(0.5,0.67,0.73,0.9)	(0.5,0.67,0.73,0.9)	(0.7,0.83,0.87,1)	(0.7,0.83,0.87,1)	(0.5,0.73,0.77,0.9)
C 5	(0.5,0.73,0.77,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.87,0.93,1)	(0.5,0.73,0.77,0.9)	(0.5,0.67,0.73,0.9)
C 6	(0.5,0.73,0.77,0.9)	(0.5,0.67,0.73,0.9)	(0.7,0.83,0.87,1)	(0.5,0.67,0.73,0.9)	(0.5,0.67,0.73,0.9)
C 7	(0.5,0.73,0.77,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.87,0.93,1)	(0.5,0.67,0.73,0.9)	(0.5,0.67,0.73,0.9)
C 8	(0.5,0.67,0.73,0.9)	(0.5,0.67,0.73,0.9)	(0.7,0.83,0.87,1)	(0.5,0.77,0.83,1)	(0.5,0.67,0.73,0.9)
C 9	(0.5,0.73,0.77,0.9)	(0.5,0.67,0.73,0.9)	(0.7,0.87,0.93,1)	(0.7,0.83,0.87,1)	(0.5,0.77,0.83,1)
C 10	(0.5,0.67,0.73,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.83,0.87,1)	(0.7,0.83,0.87,1)	(0.5,0.77,0.83,1)
C 11	(0.5,0.73,0.77,0.9)	(0.5,0.67,0.73,0.9)	(0.7,0.87,0.93,1)	(0.5,0.73,0.77,0.9)	(0.7,0.83,0.87,1)
C 12	(0.5,0.67,0.73,0.9)	(0.5,0.73,0.77,0.9)	(0.7,0.87,0.93,1)	(0.5,0.77,0.83,1)	(0.5,0.73,0.77,0.9)

Table 7 Weighted normalised fuzzy decision matrix.

	A 1	A 2	A 3	A 4	A 5
C 1	(0.35,0.58,0.68,0.9)	(0.35,0.64,0.72,0.9)	(0.49,0.76,0.87,1)	(0.49,0.72,0.81,1)	(0.49,0.72,0.87,1)
C 2	(0.35,0.56,0.64,0.9)	(0.35,0.61,0.67,0.9)	(0.49,0.72,0.81,1)	(0.49,0.69,0.76,1)	(0.49,0.69,0.76,1)
C 3	(0.25,0.49,0.56,0.81)	(0.25,0.49,0.56,0.81)	(0.35.0.56.0.64.0.9)	(0.35,0.64,0.72,1)	(0.25,0.49,0.56,0.81)
C 4	(0.25,0.52,0.61,0.9)	(0.25,0.52,0.61,0.9)	(0.35,0.64,0.72,1)	(0.35,0.64,0.72,1)	(0.25,0.56,0.64,0.9)
C 5	(0.2,0.42,0.49,0.72)	(0.2,0.42,0.49,0.72)	(0.28,0.5,0.59,0.8)	(0.2,0.42,0.49,0.72)	(0.2,0.38,0.46,0.72)
C 6	(0.25,0.49,0.56,0.81)	(0.25,0.45,0.53,0.81)	(0.35,0.56,0.64,0.9)	(0.25,0.45,0.53,0.81)	(0.25,0.45,0.53,0.81)
C 7	(0.25,0.45,0.53,0.81)	(0.25,0.49,0.56,0.81)	(0.35,0.58,0.68,0.9)	(0.25,0.45,0.53,0.81)	(0.25,0.45,0.53,0.81)
C 8	(0.2,0.46,0.52,0.81)	(0.2,0.42,0.49,0.81)	(0.28,0.52,0.58,0.9)	(0.2,0.49,0.56,0.9)	(0.2,0.42,0.49,0.81)
C 9	(0.35,0.56,0.64,0.9)	(0.35,0.56,0.64,0.9)	(0.49,0.72,0.81,1)	(0.49,0.69,0.761)	(0.35,0.64,0.72,1)
C 10	(0.25,0.49,0.56,0.81)	(0.25,0.49,0.56,0.81)	(0.35,0.56,0.64,0.9)	(0.35,0.56,0.64,0.9)	(0.25,0.52,0.61,0.9)
C 11	(0.35,0.56,0.64,0.9)	(0.35,0.56,0.64,0.9)	(0.49,0.72,0.81,1)	(0.35,0.61,0.67,0.9)	(0.49,0.69,0.76,1)
C 12	(0.25,0.56,0.64,0.9)	(0.25,0.56,0.64,0.9)	(0.35,0.67,0.77,1)	(0.25,0.59,0.69,1)	(0.25,0.56,0.64,0.9)

Table 8 Ideal and anti-ideal solution for each concept.

	I ^+	I ^−
C 1	(0.49,0.76,0.87,1)	(0.35,0.58,0.68,0.9)
C 2	(0.49,0.72,0.81,1	(0.35,0.56,0.64,0.9)
C 3	(0.35,0.56,0.64,0.9)	(0.25,0.49,0.56,0.81)
C 4	(0.35,0.64,0.72,1)	(0.25,0.52,0.61,0.9)
C 5	(0.28,0.5,0.56,0.8)	(0.2,0.38,0.46,0.72)
C 6	(0.35,0.56,0.64,0.9)	(0.25,0.45,0.53,0.81)
C 7	(0.35,0.58,0.68,0.9)	(0.25,0.45,0.53,0.81)
C 8	(0.28,0.52,0.58,0.9)	(0.2,0.42,0.49,0.81)
C 9	(0.49,0.72,0.81,1)	(0.35,0.56,0.64,0.9)
C 10	(0.35,0.56,0.64,0.9)	(0.25,0.49,0.56,0.81)
C 11	(0.49,0.72,0.81,1)	(0.35,0.56,0.64,0.9)
C 12	(0.35,0.67,0.77,1)	(0.25,0.56,0.64,0.9)

Table 9 Distance of each concept from ideal and anti-ideal solutions.

Concept
A 1	1.965	0.1067
A 2	1.9385	0.1414
A 3	0	2.0571
A 4	0.9088	1.1867
A 5	1.4858	0.7188

Table 10 Closeness index.

Concept	Closeness index C i ^*
A 1	0.05
A 2	0.07
A 3	1
A 4	0.57
A 5	0.35