The m-polar fuzzy ELECTRE-I integrated AHP approach for selection of non-traditional machining processes

Figures & data

Table 1. Expert system developed by various authors for NTM selection

Sr. No.	Name of expert system	Working on an expert system	Name of author
1	Computer-aided procedure	It uses a 16-digit classification code to rank the NTM process.	(Cogun, 1993)
2	Multi-attribute-based selection procedure	They have employed AHP and TOPSIS to rank feasible NTM processes according to machining application.	(Yurdakul & Çoǧun, 2003)
3	AHP-based expert system	They have considered criteria and sub-criteria priority values related to a particular NTM process.	(Chakraborty & Dey, 2006)
4	Quality function deployment–based expert system	Estimated overall scores from various product and process characteristics and their weights.	(Chakraborty & Dey, 2007)
5	n-tier management information system	Considered parameter selection and optimization in NTM processes using the architecture of MIS.	(Chakrabarti et al., 2007)
6	Combination of TOPSIS and AHP methods	AHP method to calculate weights of parameters and TOPSIS method for ranking alternatives.	(N. D. Chakladar & Chakraborty, 2008)
7	Diagraph-based method	Used digraph method to select the NTM process	(N. Das Chakladar et al., 2009)
8	Web-based, knowledge-based system	They have employed three-tier web architecture to implement user modules to select and expert modules to update the knowledge base.	(Edison Chandrasselan et al., 2008)
9	Two-phase decision model	The first phase uses the Charnes, Cooper, and Rhodes model of data envelopment analysis to select the most efficient process for shape and material. The second phase uses the weighted overall efficiency ranking method to rank alternatives.	(Sadhu & Chakraborty, 2011)
10	Analytical network process–based approach	Analytical network process to consider interdependency and feedback relationship among various criteria and graphic user interface to automate the method.	(Das & Chakraborty, 2011)
11	MOORA method	They have multi-objective optimization based on ratio analysis (MOORA) to choose a process for various machining applications.	(Chakraborty, 2011)
12	Fuzzy-based decision support model	They have used a fuzzy and crisp environment for NTM process selection by assessing the potential of distinct NTM processes.	(Temuçin et al., 2014)
13	PROMETHEE- GAIA method	They have used the PROMETHEE- GAIA method, a visual aid to the decision engineers.	(Karande & Chakraborty, 2012)
14	QFD-based decision-making model	They have created an automated QFD-based decision-making model with the help of Visual BASIC 6.0.	(Prasad & Chakraborty, 2014)
15	Fuzzy axiomatic design principle	They have applied the fuzzy axiomatic design principle to select suitable NTM processes.	(Khandekar & Chakraborty, 2016)
16	Fuzzy AHP and QFD	Fuzzy AHP evaluates the relative importance of various NTM processes and QFD to calculate the overall score of the NTM process.	(Roy et al., 1968)
17	Case-based reasoning approach	They have used a memory model to represent, index, and organize past cases and develop new ones.	(Boral & Chakraborty, 2016)
18	Integrated factor relationship (FARE) and MABAC methods	Used integrated FARE and MABAC methods for NTM process selection.	(Chatterjee et al., 2017)
19	Decision guidance framework	They have designed and developed a decision guidance framework in VISUAL BASICS 6.0 to select suitable NTM processes for intricate and complex shapes.	(Prasad & Chakraborty, 2018)

Table 2. Decision matrix: performance of various NTM processes to different attributes (N. D. Chakladar & Chakraborty, 2008)

Sr.No.	Parameters
	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
1	1.0	10.00	500.0	2	4	2	3	1	4	1
2	2.5	0.22	0.8	1	4	2	2	3	4	1
3	2.5	0.24	0.5	1	4	2	2	3	4	1
4	3.0	100.00	400.0	5	2	3	1	3	5	4
5	3.0	0.40	15.0	3	3	2	1	3	5	1
6	3.5	2.70	800.0	3	4	4	4	3	5	1
7	3.5	2.50	600.0	3	4	4	4	3	5	1
8	2.5	0.20	1.6	4	5	2	1	3	4	1
9	2.0	1.40	0.1	3	5	2	1	1	4	1

Table 3. Normalised decision matrix

Sr.	Parameters
	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
o1	0.1229	0.0994	0.4210	0.2195	0.3345	0.2481	0.4121	0.1240	0.2981	0.2041
o2	0.3071	0.0022	0.0007	0.1098	0.3345	0.2481	0.2747	0.3721	0.2981	0.2041
o3	0.3071	0.0024	0.0004	0.1098	0.3345	0.2481	0.2747	0.3721	0.2981	0.2041
o4	0.3686	0.9943	0.3368	0.5488	0.1672	0.3721	0.1374	0.3721	0.3727	0.8165
o5	0.3686	0.0040	0.0126	0.3293	0.2509	0.2481	0.1374	0.3721	0.3727	0.2041
o6	0.4300	0.0268	0.6737	0.3293	0.3345	0.4961	0.5494	0.3721	0.3727	0.2041
o7	0.4300	0.0249	0.5053	0.3293	0.3345	0.4961	0.5494	0.3721	0.3727	0.2041
o8	0.3071	0.0020	0.0013	0.4391	0.4181	0.2481	0.1374	0.3721	0.2981	0.2041
o9	0.2457	0.0139	0.0001	0.3293	0.4181	0.2481	0.1374	0.1240	0.2981	0.2041

Table 4. Pairwise comparison matrix

	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
u1	1	2	1/3	1/2	3	4	5	6	1/5	1/4
u2	1/2	1	1/3	1/2	2	3	4	5	1/5	1/4
u3	3	3	1	2	5	6	7	8	1/2	1/3
u4	2	2	1/2	1	4	5	6	7	1/4	1/3
u5	1/3	1/2	1/5	1/4	1	2	3	4	1/7	1/6
u6	1/4	1/3	1/6	1/5	1/2	1	2	3	1/8	1/7
u7	1/5	1/4	1/7	1/6	1/3	1/2	1	2	1/9	1/8
u8	1/6	1/5	1/8	1/7	1/4	1/3	1/2	1	1/9	1/9
u9	5	5	2	4	7	8	9	9	1	2
u10	4	4	3	3	6	7	8	9	1/2	1

Table 5. Weighted normalised matrix

Sr.	Parameters
	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
o1	0.0096	0.0061	0.0646	0.0236	0.0128	0.0067	0.0080	0.0018	0.0825	0.0457
o2	0.0240	0.0001	0.0001	0.0118	0.0128	0.0067	0.0054	0.0054	0.0825	0.0457
o3	0.0240	0.0001	0.0001	0.0118	0.0128	0.0067	0.0054	0.0054	0.0825	0.0457
o4	0.0289	0.0607	0.0517	0.0589	0.0064	0.0101	0.0027	0.0054	0.1031	0.1827
o5	0.0289	0.0002	0.0019	0.0353	0.0096	0.0067	0.0027	0.0054	0.1031	0.0457
o6	0.0337	0.0016	0.1034	0.0353	0.0128	0.0134	0.0107	0.0054	0.1031	0.0457
o7	0.0337	0.0015	0.0776	0.0353	0.0128	0.0134	0.0107	0.0054	0.1031	0.0457
o8	0.0240	0.0001	0.0002	0.0471	0.0160	0.0067	0.0027	0.0054	0.0825	0.0457
o9	0.0192	0.0009	0.0001	0.0353	0.0160	0.0067	0.0027	0.0018	0.0825	0.0457

Table 6. Concordance set

j	1	2	3	4	5	6	7	8	9
C1j	-	{2,3,4,5,6,7,9,10}	{2,3,4,5,6,7,9,10}	{3,5,7}	{2,3,5,6,7,10}	{2,5,10}	{2,5,10}	{2,3,6,7,9,10}	{2,3,6,7,8,9,10}
C2j	{1,5,6,8,9,10}	-	{1,3,4,5,6,7,8,9,10}	{5,7,8}	{5,6,7,8,10}	{5,8,10}	{5,8,10}	{1,2,6,7,8,9,10}	{1,3,6,7,8,9,10}
C3j	{1,5,6,8,9,10}	{1,2,4,5,6,7,8,9,10}	-	{5,7,8}	{5,6,7,8,10}	{5,8,10}	{5,8,10}	{1,2,6,7,8,9,10}	{1,3,6,7,8,9,10}
C4j	{1,2,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	-	{1,2,3,4,6,7,8,9,10}	{2,4,8,9,10}	{2,4,8,9,10}	{1,2,3,4,6,7,8,9,10}	{1,2,3,4,6,7,8,9,10}
C5j	{1,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	{1,5,7,8,9}	-	{4,8,9,10}	{4,8,9,10}	{1,2,3,6,7,8,9,10}	{1,3,4,6,7,8,9,10}
C6j	{1,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,3,5,6,7,8,9}	{1,2,3,4,5,6,7,8,9,10}	-	{1,2,3,4,5,6,7,8,9,10}	{1,2,3,6,7,8,9,10}	{1,2,3,4,6,7,8,9,10}
C7j	{1,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,3,5,6,7,8,9}	{1,2,3,4,5,6,7,8,9,10}	{1,4,5,6,7,8,9,10}	-	{1,2,3,6,7,8,9,10}	{1,2,3,4,6,7,8,9,10}
C8j	{1,4,5,6,8,9,10}	{1,3,4,5,6,8,9,10}	{1,3,4,5,6,8,9,10}	{5,7,8}	{4,5,6,7,8,10}	{4,5,8,10}	{4,5,8,10}	-	{1,3,4,5,6,7,8,9,10}
C9j	{1,4,5,6,8,9,10}	{2,4,5,6,9,10}	{2,4,5,6,9,10}	{5,7}	{2,4,5,6,7,10}	{4,5,10}	{4,5,10}	{2,5,6,7,9,10}	-

Figure 1. Console of the soft solution with instructions given for the input and output expected after the implementation of the mFS ELECTRE-I integrated approach.

Figure 2. Actual input is given to the console in the form of normalised decision matrix and weight obtained from the AHP method.

Figure 3. Results obtained for the given input data.

Figure 4. Directed graph is plotted on the console with C++ code for example-I.

Table 7. Discordance set

j	1	2	3	4	5	6	7	8	9
A1j	-	{1,8}	{1,8}	{1,2,4,6,8,9,10}	{1,4,8,9}	{1,3,4,6,7,8,9}	{1,3,4,6,7,8,9}	{1,4,5,8}	{1,4,5}
A2j	{2,3,4,7}	-	{2}	{1,2,3,4,6,9,10}	{1,2,3,4,9}	{1,2,3,4,6,7,9}	{1,2,3,4,6,7,9}	{3,4,5}	{2,4,5}
A3j	{2,3,4,7}	{3}	-	{1,2,3,4,6,9,10}	{1,2,3,4,9}	{1,2,3,4,6,7,9}	{1,2,3,4,6,7,9}	{3,4,5}	{2,4,5}
A4j	{3,5,7}	{5,7}	{5,7}	-	{5}	{1,3,5,6,7}	{1,3,5,6,7}	{4,5}	{2,5}
A5j	{2,3,5,7}	{5,7}	{5,7}	{2,3,4,6,10}	-	{1,2,3,5,6,7}	{1,2,3,5,6,7}	{4,5}	{2,5}
A6j	{2}	{}	{}	{2,4,10}	{}	-	{}	{4,5}	{5}
A7j	{2}	{}	{}	{2,4,10}	{}	{2,3}	-	{4,5}	{5}
A8j	{2,3,7}	{2,7}	{2,7}	{1,2,3,4,6,9,10}	{1,2,3,9}	{1,2,3,6,7,9}	{1,2,3,6,7,9}	-	{2}
A9j	{2,3,7}	{1,3,7,8}	{1,3,7,8}	{1,2,3,4,6,8,9,10}	{1,3,8,9}	{1,2,3,6,7,8,9}	{1,2,3,6,7,8,9}	{1,3,4,8}	-

Figure 5. Colour-directed graph, for example-I.

Table 8. Outranking relations between NTM processes

Comparison of NTM processes	Cpq	Apq	cpq	apq	l	e	f	Ranking
(1,2)	{2,3,4,5,6,7,9,10}	{1,8}	0.9071	0.223555	1	1	1	1–2
(1,3)	{2,3,4,5,6,7,9,10}	{1,8}	0.9071	0.223395	1	1	1	1–3
(1,4)	{3,5,7}	{1,2,4,6,8,9,10}	0.2113	1	0	0	0	Incomparable
(1,5)	{2,3,5,6,7,10}	{1,4,8,9}	0.5232	0.329152	0	1	0	Incomparable
(1,6)	{2,5,10}	{1,3,4,6,7,8,9}	0.3231	1	0	0	0	Incomparable
(1,7)	{2,5,10}	{1,3,4,6,7,8,9}	0.3231	1	0	0	0	Incomparable
(1,8)	{2,3,6,7,9,10}	{1,4,5,8}	0.7615	0.36575	1	1	1	1–8
(1,9)	{2,3,6,7,8,9,10}	{1,4,5}	0.7761	0.182354	1	1	1	1–9
(2,1)	{1,5,6,8,9,10}	{2,3,4,7}	0.6586	1	0	0	0	Incomparable
(2,3)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.265364	1	1	1	2–3
(2,4)	{5,7,8}	{1,2,3,4,6,9,10}	0.0724	1	0	0	0	Incomparable
(2,5)	{5,6,7,8,10}	{1,2,3,4,9}	0.3232	1	0	0	0	Incomparable
(2,6)	{5,8,10}	{1,2,3,4,6,7,9}	0.2766	1	0	0	0	Incomparable
(2,7)	{5,8,10}	{1,2,3,4,6,7,9}	0.2766	1	0	0	0	Incomparable
(2,8)	{1,2,6,7,8,9,10}	{3,4,5}	0.7009	1	1	0	0	Incomparable
(2,9)	{1,3,6,7,8,9,10}	{2,4,5}	0.7933	1	1	0	0	Incomparable
(3,1)	{1,5,6,8,9,10}	{2,3,4,7}	0.6586	1	0	0	0	Incomparable
(3,2)	{1,2,4,5,6,7,8,9,10}	{3}	0.8465	1	1	0	0	Incomparable
(3,4)	{5,7,8}	{1,2,3,4,6,9,10}	0.0724	1	0	0	0	Incomparable
(3,5)	{5,6,7,8,10}	{1,2,3,4,9}	0.3232	1	0	0	0	Incomparable
(3,6)	{5,8,10}	{1,2,3,4,6,7,9}	0.2766	1	0	0	0	Incomparable
(3,7)	{5,8,10}	{1,2,3,4,6,7,9}	0.2766	1	0	0	0	Incomparable
(3,8)	{1,2,6,7,8,9,10}	{3,4,5}	0.7009	1	1	0	0	Incomparable
(3,9)	{1,3,6,7,8,9,10}	{2,4,5}	0.7933	1	1	0	0	Incomparable
(4,1)	{1,2,4,6,8,9,10}	{3,5,7}	0.7887	0.0943451	1	1	1	4–1
(4,2)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.0467728	1	1	1	4–2
(4,3)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.0467728	1	1	1	4–3
(4,5)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.0234004	1	1	1	4–5
(4,6)	{2,4,8,9,10}	{1,3,5,6,7}	0.6833	0.377492	1	1	1	4–6
(4,7)	{2,4,8,9,10}	{1,3,5,6,7}	0.6833	0.188802	1	1	1	4–7
(4,8)	{1,2,3,4,6,7,8,9,10}	{4,5}	0.9617	0.0701453	1	1	1	4–8
(4,9)	{1,2,3,4,6,7,8,9,10}	{2,5}	0.9617	0.0701453	1	1	1	4–9
(5,1)	{1,4,6,8,9,10}	{2,3,5,7}	0.7276	1	1	0	0	Incomparable
(5,2)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.135947	1	1	1	5–2
(5,3)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.135947	1	1	1	5–3
(5,4)	{1,5,7,8,9}	{2,3,4,6,10}	0.4273	1	0	0	0	Incomparable
(5,6)	{4,8,9,10}	{1,2,3,5,6,7}	0.6222	1	0	0	0	Incomparable
(5,7)	{4,8,9,10}	{1,2,3,5,6,7}	0.6222	1	0	0	0	Incomparable
(5,8)	{1,2,3,6,7,8,9,10}	{4,5}	0.8544	0.570967	1	0	0	Incomparable
(5,9)	{1,3,4,6,7,8,9,10}	{2,5}	0.9006	0.310344	1	1	1	5–9
(6,1)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.114357	1	1	1	6–1
(6,2)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–2
(6,3)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–3
(6,4)	{1,3,5,6,7,8,9}	{2,4,10}	0.6079	1	0	0	0	Incomparable
(6,5)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–5
(6,7)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–7
(6,8)	{1,2,3,6,7,8,9,10}	{4,5}	0.8544	0.114147	1	1	1	6–8
(6,9)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.0309667	1	1	1	6–9
(7,1)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.189302	1	1	1	7–1
(7,2)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	7–2
(7,3)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	7–3
(7,4)	{1,3,5,6,7,8,9}	{2,4,10}	0.6079	1	0	0	0	Incomparable
(7,5)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	7–5
(7,6)	{1,4,5,6,7,8,9,10}	{2,3}	0.7854	1	1	0	0	Incomparable
(7,8)	{1,2,3,6,7,8,9,10}	{4,5}	0.8544	0.152287	1	1	1	7–8
(7,9)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.0412889	1	1	1	7–9
(8,1)	{1,4,5,6,8,9,10}	{2,3,7}	0.7659	1	1	0	0	Incomparable
(8,2)	{1,3,4,5,6,8,9,10}	{2,7}	0.9194	0.0757729	1	1	1	8–2
(8,3)	{1,3,4,5,6,8,9,10}	{2,7}	0.9194	0.0757729	1	1	1	8–3
(8,4)	{5,7,8}	{1,2,3,4,6,9,10}	0.0724	1	0	0	0	Incomparable
(8,5)	{4,5,6,7,8,10}	{1,2,3,9}	0.4305	1	0	0	0	Incomparable
(8,6)	{4,5,8,10}	{1,2,3,6,7,9}	0.3839	1	0	0	0	Incomparable
(8,7)	{4,5,8,10}	{1,2,3,6,7,9}	0.3839	1	0	0	0	Incomparable
(8,9)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.0617143	1	1	1	8–9
(9,1)	{1,4,5,6,8,9,10}	{2,3,7}	0.7659	1	1	0	0	Incomparable
(9,2)	{2,4,5,6,9,10}	{1,3,7,8}	0.7341	0.204125	1	1	1	9–2
(9,3)	{2,4,5,6,9,10}	{1,3,7,8}	0.7341	0.204125	1	1	1	9–3
(9,4)	{5,7}	{1,2,3,4,6,8,9,10}	0.0578	1	0	0	0	Incomparable
(9,5)	{2,4,5,6,7,10}	{1,3,8,9}	0.477	1	0	0	0	Incomparable
(9,6)	{4,5,10}	{1,2,3,6,7,8,9}	0.3693	1	0	0	0	Incomparable
(9,7)	{4,5,10}	{1,2,3,6,7,8,9}	0.3693	1	0	0	0	Incomparable
(9,8)	{2,5,6,7,9,10}	{1,3,4,8}	0.6463	1	0	0	0	Incomparable

Table 9. Criteria weight variation for single-dimensional weight sensitivity analysis

u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
0.0783	0.0611	0.1535	0.1073	0.0383	0.0271	0.0195	0.0146	0.2766	0.2237
Criteria weight variation with weight additive constraint
0.109	0.0918	0.1842	0.138	0.069	0.0578	0.0502	0.0453	0	0.2544
0.09792	0.08072	0.17312	0.12692	0.05792	0.04672	0.03912	0.03422	0.1	0.24332
0.0868	0.0696	0.162	0.1158	0.0468	0.0356	0.028	0.0231	0.2	0.2322
0.0757	0.0585	0.1509	0.1047	0.0357	0.0245	0.0169	0.012	0.3	0.2296
0.0663	0.0491	0.1415	0.0953	0.0263	0.0151	0.0075	0	0.3846	0.2117

Table 10. Alternatives with their ranks for criteria weight variation

Alternative	1	2	3	4	5
o1	4	4	4	4	4
o2	7	7	8	8	8
o3	8	8	9	9	9
o4	2	2	2	1	1
o5	6	6	5	5	5
o6	1	1	1	2	2
o7	3	3	3	3	3
o8	5	5	6	6	6
o9	9	9	7	7	7

Figure 6. Criteria weight variation for “Material” criteria v/s rank of alternatives.

Table 11. Decision matrix: performance of various NTM processes to different attributes (N. D. Chakladar & Chakraborty, 2008)

Sr.	Process	Parameters
		u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
1	o1	1.0	10.00	500.0	2	4	2	3	1	4	1
2	o2	2.5	0.22	0.8	1	4	2	2	3	3	1
3	o3	2.5	0.24	0.5	1	4	2	2	3	3	1
4	o4	3.0	100.00	400.0	5	2	3	1	3	5	4
5	o5	3.0	0.40	15.0	3	3	2	1	3	5	4
6	o6	3.5	2.70	800.0	3	4	4	4	3	4	5
7	o7	3.5	2.50	600.0	3	4	4	4	3	4	5
8	o8	2.5	0.20	1.6	4	5	2	1	3	4	1
9	o9	2.0	1.40	0.1	3	5	2	1	1	4	1

Table 12. Normalised decision matrix

Sr.	Process	Parameters
		u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
1	o1	0.1229	0.0994	0.4210	0.2195	0.3345	0.2481	0.4121	0.1240	0.3288	0.1072
2	o2	0.3071	0.0022	0.0007	0.1098	0.3345	0.2481	0.2747	0.3721	0.2466	0.1072
3	o3	0.3071	0.0024	0.0004	0.1098	0.3345	0.2481	0.2747	0.3721	0.2466	0.1072
4	o4	0.3686	0.9943	0.3368	0.5488	0.1672	0.3721	0.1374	0.3721	0.4110	0.4288
5	o5	0.3686	0.0040	0.0126	0.3293	0.2509	0.2481	0.1374	0.3721	0.4110	0.4288
6	o6	0.4300	0.0268	0.6737	0.3293	0.3345	0.4961	0.5494	0.3721	0.3288	0.5361
7	o7	0.4300	0.0249	0.5053	0.3293	0.3345	0.4961	0.5494	0.3721	0.3288	0.5361
8	o8	0.3071	0.0020	0.0013	0.4391	0.4181	0.2481	0.1374	0.3721	0.3288	0.1072
9	o9	0.2457	0.0139	0.0001	0.3293	0.4181	0.2481	0.1374	0.1240	0.3288	0.1072

Table 13. Pairwise comparison matrix

	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
u1	1	2	1/3	1/2	3	4	5	6	1/5	1/4
u2	1/2	1	1/3	1/2	2	3	4	5	1/5	1/4
u3	3	3	1	2	5	6	7	8	1/2	1/3
u4	2	2	1/2	1	4	5	6	7	1/4	1/3
u5	1/3	1/2	1/5	1/4	1	2	3	4	1/7	1/6
u6	1/4	1/3	1/6	1/5	1/2	1	2	3	1/8	1/7
u7	1/5	1/4	1/7	1/6	1/3	1/2	1	2	1/9	1/8
u8	1/6	1/5	1/8	1/7	1/4	1/3	1/2	1	1/9	1/9
u9	5	5	2	4	7	8	9	9	1	2
u10	4	4	3	3	6	7	8	9	1/2	1

Table 14. Weighted normalised matrix

Sr.	Parameters
	u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
1	0.0096	0.0061	0.0646	0.0236	0.0128	0.0067	0.0080	0.0018	0.0909	0.0240
2	0.0240	0.0001	0.0001	0.0118	0.0128	0.0067	0.0054	0.0054	0.0682	0.0240
3	0.0240	0.0001	0.0001	0.0118	0.0128	0.0067	0.0054	0.0054	0.0682	0.0240
4	0.0289	0.0607	0.0517	0.0589	0.0064	0.0101	0.0027	0.0054	0.1137	0.0959
5	0.0289	0.0002	0.0019	0.0353	0.0096	0.0067	0.0027	0.0054	0.1137	0.0959
6	0.0337	0.0016	0.1034	0.0353	0.0128	0.0134	0.0107	0.0054	0.0909	0.1199
7	0.0337	0.0015	0.0776	0.0353	0.0128	0.0134	0.0107	0.0054	0.0909	0.1199
8	0.0240	0.0001	0.0002	0.0471	0.0160	0.0067	0.0027	0.0054	0.0909	0.0240
9	0.0192	0.0009	0.0001	0.0353	0.0160	0.0067	0.0027	0.0018	0.0909	0.0240

Table 15. Concordance set

j	1	2	3	4	5	6	7	8	9
C1j	-	{2,3,4,5,6,7,9,10}	{2,3,4,5,6,7,9,10}	{3,5,7}	{2,3,5,6,7}	{2,5,9}	{2,5,9}	{2,3,6,7,9,10}	{2,3,6,7,8,9,10}
C2j	{1,5,6,8,10}	-	{1,3,4,5,6,7,8,9,10}	{5,7,8}	{5,6,7,8}	{5,8}	{5,8}	{1,2,6,7,8,10}	{1,3,6,7,8,10}
C3j	{1,5,6,8,10}	{1,2,4,5,6,7,8,9,10}	-	{5,7,8}	{5,6,7,8}	{5,8}	{5,8}	{1,2,6,7,8,10}	{1,3,6,7,8,10}
C4j	{1,2,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	-	{1,2,3,4,6,7,8,9,10}	{2,4,8,9}	{2,4,8,9}	{1,2,3,4,6,7,8,9,10}	{1,2,3,4,6,7,8,9,10}
C5j	{1,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	{1,2,3,4,6,8,9,10}	{1,5,7,8,9,10}	-	{4,8,9}	{4,8,9}	{1,2,3,6,7,8,9,10}	{1,3,4,6,7,8,9,10}
C6j	{1,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,3,5,6,7,8,10}	{1,2,3,4,5,6,7,8,10}	-	{1,2,3,4,5,6,7,8,9,10}	{1,2,3,6,7,8,9,10}	{1,2,3,4,6,7,8,9,10}
C7j	{1,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,2,3,4,5,6,7,8,9,10}	{1,3,5,6,7,8,10}	{1,2,3,4,5,6,7,8,10}	{1,4,5,6,7,8,9,10}	-	{1,2,3,6,7,8,9,10}	{1,2,3,4,6,7,8,9,10}
C8j	{1,4,5,6,8,9,10}	{1,3,4,5,6,8,9,10}	{1,3,4,5,6,8,9,10}	{5,7,8}	{4,5,6,7,8}	{4,5,8,9}	{4,5,8,9}	-	{1,3,4,5,6,7,8,9,10}
C9j	{1,4,5,6,8,9,10}	{2,4,5,6,9,10}	{2,4,5,6,9,10}	{5,7}	{2,4,5,6,7}	{4,5,9}	{4,5,9}	{2,5,6,7,9,10}	-

Table 16. Discordance set

j	1	2	3	4	5	6	7	8	9
A1j	-	{1,8}	{1,8}	{1,2,4,6,8,9,10}	{1,4,8,9,10}	{1,3,4,6,7,8,10}	{1,3,4,6,7,8,10}	{1,4,5,8}	{1,4,5}
A2j	{2,3,4,7,9}	-	{2}	{1,2,3,4,6,9,10}	{1,2,3,4,9,10}	{1,2,3,4,6,7,9,10}	{1,2,3,4,6,7,9,10}	{3,4,5,9}	{2,4,5,9}
A3j	{2,3,4,7,9}	{3}	-	{1,2,3,4,6,9,10}	{1,2,3,4,9,10}	{1,2,3,4,6,7,9,10}	{1,2,3,4,6,7,9,10}	{3,4,5,9}	{2,4,5,9}
A4j	{3,5,7}	{5,7}	{5,7}	-	{5}	{1,3,5,6,7,10}	{1,3,5,6,7,10}	{5}	{5}
A5j	{2,3,5,7}	{5,7}	{5,7}	{2,3,4,6}	-	{1,2,3,5,6,7,10}	{1,2,3,5,6,7,10}	{4,5}	{2,5}
A6j	{2}	{}	{}	{2,4,9}	{9}	-	{}	{4,5}	{5}
A7j	{2}	{}	{}	{2,4,9}	{9}	{2,3}	-	{4,5}	{5}
A8j	{2,3,7}	{2,7}	{2,7}	{1,2,3,4,6,9,10}	{1,2,3,9,10}	{1,2,3,6,7,10}	{1,2,3,6,7,10}	-	{2}
A9j	{2,3,7}	{1,3,7,8}	{1,3,7,8}	{1,2,3,4,6,8,9,10}	{1,3,8,9,10}	{1,2,3,6,7,8,10}	{1,2,3,6,7,8,10}	{1,3,4,8}	-

Figure 7. Actual input is given to the console in the form of a normalised decision matrix and weight obtained from the AHP method for example-II.

Figure 8. Results obtained for the given input data.

Figure 9. Directed graph is plotted on the console with C++ code for example-II.

Figure 10. Colour-directed graph, for example-II.

Table 17. Outranking relations between NTM processes

Comparison of NTMP’s	Cpq	Apq	cpq	apq	l	e	f	Ranking
(1,2)	{2,3,4,5,6,7,9,10}	{1,8}	0.9071	0.223555	1	1	1	1–2
(1,3)	{2,3,4,5,6,7,9,10}	{1,8}	0.9071	0.223395	1	1	1	1–3
(1,4)	{3,5,7}	{1,2,4,6,8,9,10}	0.2113	1	0	0	0	Incomparable
(1,5)	{2,3,5,6,7}	{1,4,8,9,10}	0.5232	1	0	0	0	Incomparable
(1,6)	{2,5,9}	{1,3,4,6,7,8,10}	0.3231	1	0	0	0	Incomparable
(1,7)	{2,5,9}	{1,3,4,6,7,8,10}	0.3231	1	0	0	0	Incomparable
(1,8)	{2,3,6,7,9,10}	{1,4,5,8}	0.7615	0.36575	1	1	1	1–8
(1,9)	{2,3,6,7,8,9,10}	{1,4,5}	0.7761	0.182354	1	1	1	1–9
(2,1)	{1,5,6,8,10}	{2,3,4,7,9}	0.382	1	0	0	0	Incomparable
(2,3)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.265364	1	1	1	2–3
(2,4)	{5,7,8}	{1,2,3,4,6,9,10}	0.0724	1	0	0	0	Incomparable
(2,5)	{5,6,7,8}	{1,2,3,4,9,10}	0.0995	1	0	0	0	Incomparable
(2,6)	{5,8}	{1,2,3,4,6,7,9,10}	0.0529	1	0	0	0	Incomparable
(2,7)	{5,8}	{1,2,3,4,6,7,9,10}	0.0529	1	0	0	0	Incomparable
(2,8)	{1,2,6,7,8,10}	{3,4,5,9}	0.4243	1	0	0	0	Incomparable
(2,9)	{1,3,6,7,8,10}	{2,4,5,9}	0.5167	1	0	0	0	Incomparable
(3,1)	{1,5,6,8,10}	{2,3,4,7,9}	0.382	1	1	0	0	Incomparable
(3,2)	{1,2,4,5,6,7,8,9,10}	{3}	0.8465	1	1	0	0	Incomparable
(3,4)	{5,7,8}	{1,2,3,4,6,9,10}	0.0724	1	1	0	0	Incomparable
(3,5)	{5,6,7,8}	{1,2,3,4,9,10}	0.0995	1	1	0	0	Incomparable
(3,6)	{5,8}	{1,2,3,4,6,7,9,10}	0.0529	1	0	0	0	Incomparable
(3,7)	{5,8}	{1,2,3,4,6,7,9,10}	0.0529	1	0	0	0	Incomparable
(3,8)	{1,2,6,7,8,10}	{3,4,5,9}	0.4243	1	1	0	0	Incomparable
(3,9)	{1,3,6,7,8,10}	{2,4,5,9}	0.5167	1	1	0	0	Incomparable
(4,1)	{1,2,4,6,8,9,10}	{3,5,7}	0.7887	0.1796	1	1	1	4–1
(4,2)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.0890	1	1	1	4–2
(4,3)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.0890	1	1	1	4–3
(4,5)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.0529	1	1	1	4–5
(4,6)	{2,4,8,9}	{1,3,5,6,7,10}	0.4596	0.8748	0	0	0	Incomparable
(4,7)	{2,4,8,9}	{1,3,5,6,7,10}	0.4596	0.4366	0	1	0	Incomparable
(4,8)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.1335	1	1	1	4–8
(4,9)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.1335	1	1	1	4–9
(5,1)	{1,4,6,8,9,10}	{2,3,5,7}	0.7276	0.8713	1	0	0	Incomparable
(5,2)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.0445	1	1	1	5–2
(5,3)	{1,2,3,4,6,8,9,10}	{5,7}	0.9422	0.0445	1	1	1	5–3
(5,4)	{1,5,7,8,9,10}	{2,3,4,6}	0.651	1	1	0	0	Incomparable
(5,6)	{4,8,9}	{1,2,3,5,6,7,10}	0.3985	1	0	0	0	Incomparable
(5,7)	{4,8,9}	{1,2,3,5,6,7,10}	0.3985	1	0	0	0	Incomparable
(5,8)	{1,2,3,6,7,8,9,10}	{4,5}	0.8544	0.1637	1	1	1	5–8
(5,9)	{1,3,4,6,7,8,9,10}	{2,5}	0.9006	0.0890	1	1	1	5–9
(6,1)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.0462	1	1	1	6–1
(6,2)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–2
(6,3)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–3
(6,4)	{1,3,5,6,7,8,10}	{2,4,9}	0.555	1	0	0	0	Incomparable
(6,5)	{1,2,3,4,5,6,7,8,10}	{9}	0.7234	0.2240	1	1	1	6–5
(6,7)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	6–7
(6,8)	{1,2,3,6,7,8,9,10}	{4,5}	0.8544	0.114147	1	1	1	6–8
(6,9)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.0309667	1	1	1	6–9
(7,1)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.0474	1	1	1	7–1
(7,2)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	7–2
(7,3)	{1,2,3,4,5,6,7,8,9,10}	{}	1	0	1	1	1	7–3
(7,4)	{1,3,5,6,7,8,10}	{2,4,9}	0.555	1	0	0	0	Incomparable
(7,5)	{1,2,3,4,5,6,7,8,10}	{9}	0.7234	0.3006	1	1	1	7–5
(7,6)	{1,4,5,6,7,8,9,10}	{2,3}	0.7854	1	1	0	0	Incomparable
(7,8)	{1,2,3,6,7,8,9,10}	{4,5}	0.8544	0.1227	1	1	1	7–8
(7,9)	{1,2,3,4,6,7,8,9,10}	{5}	0.9617	0.0333	1	1	1	7–9
(8,1)	{1,4,5,6,8,9,10}	{2,3,7}	0.7659	1	1	0	0	Incomparable
(8,2)	{1,3,4,5,6,8,9,10}	{2,7}	0.9194	0.0757729	1	1	1	8–2
(8,3)	{1,3,4,5,6,8,9,10}	{2,7}	0.9194	0.0757729	1	1	1	8–3
(8,4)	{5,7,8}	{1,2,3,4,6,9,10}	0.0724	1	0	0	0	Incomparable
(8,5)	{4,5,6,7,8}	{1,2,3,9,10}	0.2068	1	0	0	0	Incomparable
(8,6)	{4,5,8,9}	{1,2,3,6,7,10}	0.4368	1	0	0	0	Incomparable
(8,7)	{4,5,8,9}	{1,2,3,6,7,10}	0.4368	1	0	0	0	Incomparable
(8,9)	{1,3,4,5,6,7,8,9,10}	{2}	0.9389	0.0617143	1	1	1	8–9
(9,1)	{1,4,5,6,8,9,10}	{2,3,7}	0.7659	1	1	0	0	Incomparable
(9,2)	{2,4,5,6,9,10}	{1,3,7,8}	0.7341	0.204125	1	1	1	9–2
(9,3)	{2,4,5,6,9,10}	{1,3,7,8}	0.7341	0.204125	1	1	1	9–3
(9,4)	{5,7}	{1,2,3,4,6,8,9,10}	0.0578	1	0	0	0	Incomparable
(9,5)	{2,4,5,6,7}	{1,3,8,9,10}	0.2533	1	0	0	0	Incomparable
(9,6)	{4,5,9}	{1,2,3,6,7,8,10}	0.4222	1	0	0	0	Incomparable
(9,7)	{4,5,9}	{1,2,3,6,7,8,10}	0.4222	1	0	0	0	Incomparable
(9,8)	{2,5,6,7,9,10}	{1,3,4,8}	0.6463	1	1	0	0	Incomparable

Table 18. Criteria weight variation for single-dimensional weight sensitivity analysis

u1	u2	u3	u4	u5	u6	u7	u8	u9	u10
0.0783	0.0611	0.1535	0.1073	0.0383	0.0271	0.0195	0.0146	0.2766	0.2237
Criteria weight variation with weight additive constraint
0.109	0.0918	0.1842	0.138	0.069	0.0578	0.0502	0.0453	0	0.2544
0.09792	0.08072	0.17312	0.12692	0.05792	0.04672	0.03912	0.03422	0.1	0.24332
0.0868	0.0696	0.162	0.1158	0.0468	0.0356	0.028	0.0231	0.2	0.2322
0.0757	0.0585	0.1509	0.1047	0.0357	0.0245	0.0169	0.012	0.3	0.2296
0.0663	0.0491	0.1415	0.0953	0.0263	0.0151	0.0075	0	0.3846	0.2117

Table 19. Alternatives with their ranks for criteria weight variation

Alternative	1	2	3	4	5
o1	4	4	4	4	4
o2	8	8	8	8	8
o3	9	9	9	9	9
o4	2	2	2	2	1
o5	5	5	5	5	5
o6	1	1	1	1	2
o7	3	3	3	3	3
o8	6	6	6	6	6
o9	7	7	7	7	7

Figure 11. Criteria weight variation for “Material” criteria v/s rank of alternatives.

Table 20. Validation of results by comparing with the results of N. D. Chakladar & Chakraborty (2008)

	Example-I		Example-II
Ranks	TOPSIS-AHP method (N. D. Chakladar & Chakraborty, 2008)	m-polar ELECTRE-I integrated AHP	TOPSIS-AHP method (N. D. Chakladar & Chakraborty, 2008)	m-polar ELECTRE-I integrated AHP
1	ECM	ECM	EDM	EDM
2	EDM	EDM	ECM	ECM
3	WEDM	WEDM	WEDM	WEDM
4	USM	USM	CHM	CHM
5	EBM	EBM	USM	USM
6	CHM	CHM	EBM	EBM
7	LBM	LBM	LBM	LBM
8	WJM	WJM	WJM	WJM
9	AJM	AJM	AJM	AJM

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