Circular Intuitionistic Fuzzy Median Ranking Model with a Novel Scoring Mechanism for Multiple Criteria Decision Analytics

Figures & data

Figure 1. Geometrical visualization of intuitionistic fuzzy and C-IF numbers.

Figure 2. The framework of the C-IF scoring-mediated median ranking model.

Figure 3. Hierarchical configuration and C-IF data for the multiple-expert supplier appraisement.

Table 1. Detailed outcomes of interval-valued estimates and C-IF scoring functions ($\mathit{\rho }=0.5$).

	Results for Supplier #1 (${\alpha }_1$):			Results for Supplier #2 (${\alpha }_2$):			Results for Supplier #3 (${\alpha }_3$):
${\mathfrak{z}}_j$	$\left[m_{1j}^{-},m_{1j}^{+}\right]$	$\left[n_{1j}^{-},n_{1j}^{+}\right]$	$S^{\rho }\left(C_{1\mathit{j}}\right)$	$\left[m_{2j}^{-},m_{2j}^{+}\right]$	$\left[n_{2j}^{-},n_{2j}^{+}\right]$	$S^{\rho }\left(C_{2\mathit{j}}\right)$	$\left[m_{3j}^{-},m_{3j}^{+}\right]$	$\left[n_{3j}^{-},n_{3j}^{+}\right]$	$S^{\rho }\left(C_{3\mathit{j}}\right)$
${\mathfrak{z}}_1$	[0.1544, 0.5376]	[0.3364, 0.7196]	0.3162	[0.5087, 0.7393]	[0.1387, 0.3693]	0.6736	[0.5473, 0.6887]	[0.2063, 0.3477]	0.6729
${\mathfrak{z}}_2$	[0.3386, 0.5394]	[0.3386, 0.5394]	0.4724	[0.6500, 0.6500]	[0.2500, 0.2500]	0.7150	[0.5406, 0.8094]	[0.0726, 0.3414]	0.7186
${\mathfrak{z}}_3$	[0.1544, 0.5376]	[0.3364, 0.7196]	0.3162	[0.4898, 0.6482]	[0.2478, 0.4062]	0.6156	[0.5484, 0.7096]	[0.1864, 0.3476]	0.6814
${\mathfrak{z}}_4$	[0.2610, 0.5410]	[0.3400, 0.6200]	0.4095	[0.6457, 0.7503]	[0.1417, 0.2463]	0.7679	[0.4891, 0.7409]	[0.1411, 0.3929]	0.6559
${\mathfrak{z}}_5$	[0.5087, 0.6473]	[0.2467, 0.3853]	0.6297	[0.6459, 0.7901]	[0.1029, 0.2471]	0.7844	[0.5473, 0.6887]	[0.2063, 0.3477]	0.6729
${\mathfrak{z}}_6$	[0.5473, 0.6887]	[0.2063, 0.3477]	0.6729	[0.6345, 0.8395]	[0.0375, 0.2425]	0.8066	[0.6353, 0.9167]	[0.0000, 0.2367]	0.8215
${\mathfrak{z}}_7$	[0.5087, 0.6473]	[0.2467, 0.3853]	0.6297	[0.4891, 0.7409]	[0.1411, 0.3929]	0.6559	[0.6477, 0.8103]	[0.0847, 0.2473]	0.7923
${\mathfrak{z}}_8$	[0.3329, 0.8151]	[0.0519, 0.5341]	0.5341	[0.5471, 0.6489]	[0.2451, 0.3469]	0.6562	[0.4898, 0.6482]	[0.2478, 0.4062]	0.6156
${\mathfrak{z}}_9$	[0.3364, 0.7196]	[0.1544, 0.5376]	0.5211	[0.4898, 0.6482]	[0.2478, 0.4062]	0.6156	[0.5471, 0.6489]	[0.2451, 0.3469]	0.6562

Table 2. Calculation outcomes related to comprehensive C-IF scoring functions.

${\alpha }_i$	${\mathfrak{z}}_j$	${\mathfrak{C}}_{ij}^{0.5}$	${\widetilde{{\mathit{I}}_{\mathfrak{C}}}}_{\mathit{ij}}^{0.5}$	$\mathfrak{S}\left({\mathfrak{C}}_{ij}^{0.5}\right)$
${\alpha }_1$	${\mathfrak{z}}_1$	(0.2359, 0.7240; 0.1580)	([0.1242, 0.3476], [0.6123, 0.8357])	0.2204
	${\mathfrak{z}}_2$	(0.1706, 0.7788; 0.1120)	([0.0914, 0.2498], [0.6996, 0.8580])	0.1667
	${\mathfrak{z}}_3$	(0.1751, 0.7871; 0.1580)	([0.0634, 0.2868], [0.6754, 0.8988])	0.1567
	${\mathfrak{z}}_4$	(0.2195, 0.7404; 0.1740)	([0.0965, 0.3425], [0.6174, 0.8634])	0.1980
	${\mathfrak{z}}_5$	(0.4131, 0.5311; 0.1740)	([0.2901, 0.5361], [0.4080, 0.6541])	0.4059
	${\mathfrak{z}}_6$	(0.2455, 0.6883; 0.1610)	([0.1316, 0.3593], [0.5745, 0.8021])	0.2358
	${\mathfrak{z}}_7$	(0.2185, 0.7189; 0.0740)	([0.1662, 0.2708], [0.6665, 0.7712])	0.2267
	${\mathfrak{z}}_8$	(0.3684, 0.5795; 0.0740)	([0.3161, 0.4207], [0.5272, 0.6318])	0.3821
	${\mathfrak{z}}_9$	(0.2732, 0.6770; 0.0000)	([0.2732, 0.2732], [0.6770, 0.6770])	0.2868
${\alpha }_2$	${\mathfrak{z}}_1$	(0.4363, 0.5025; 0.1580)	([0.3245, 0.5480], [0.3908, 0.6142])	0.4380
	${\mathfrak{z}}_2$	(0.2466, 0.6849; 0.1120)	([0.1674, 0.3258], [0.6057, 0.7641])	0.2509
	${\mathfrak{z}}_3$	(0.3126, 0.6274; 0.1580)	([0.2009, 0.4243], [0.5157, 0.7392])	0.3064
	${\mathfrak{z}}_4$	(0.3717, 0.5691; 0.1740)	([0.2486, 0.4947], [0.4461, 0.6922])	0.3634
	${\mathfrak{z}}_5$	(0.4851, 0.4546; 0.1740)	([0.3621, 0.6082], [0.3315, 0.5776])	0.4841
	${\mathfrak{z}}_6$	(0.2865, 0.6390; 0.1610)	([0.1727, 0.4004], [0.5252, 0.7529])	0.2819
	${\mathfrak{z}}_7$	(0.2265, 0.7090; 0.0740)	([0.1742, 0.2788], [0.6567, 0.7614])	0.2356
	${\mathfrak{z}}_8$	(0.4314, 0.5115; 0.0740)	([0.3791, 0.4837], [0.4592, 0.5638])	0.4506
	${\mathfrak{z}}_9$	(0.3141, 0.6307; 0.0000)	([0.3141, 0.3141], [0.6307, 0.6307])	0.3315
${\alpha }_3$	${\mathfrak{z}}_1$	(0.4359, 0.5029; 0.1580)	([0.3242, 0.5477], [0.3911, 0.6146])	0.4377
	${\mathfrak{z}}_2$	(0.2476, 0.6836; 0.1120)	([0.1685, 0.3268], [0.6044, 0.7628])	0.2521
	${\mathfrak{z}}_3$	(0.3396, 0.5969; 0.1580)	([0.2278, 0.4513], [0.4852, 0.7087])	0.3362
	${\mathfrak{z}}_4$	(0.3276, 0.6179; 0.1740)	([0.2046, 0.4506], [0.4949, 0.7410])	0.3152
	${\mathfrak{z}}_5$	(0.4342, 0.5085; 0.1740)	([0.3111, 0.5572], [0.3854, 0.6315])	0.4288
	${\mathfrak{z}}_6$	(0.2910, 0.6338; 0.1610)	([0.1771, 0.4048], [0.5199, 0.7476])	0.2869
	${\mathfrak{z}}_7$	(0.2667, 0.6601; 0.0740)	([0.2144, 0.3191], [0.6078, 0.7124])	0.2808
	${\mathfrak{z}}_8$	(0.4112, 0.5331; 0.0740)	([0.3589, 0.4635], [0.4808, 0.5855])	0.4286
	${\mathfrak{z}}_9$	(0.3310, 0.6118; 0.0000)	([0.3310, 0.3310], [0.6118, 0.6118])	0.3499

Figure 4. Comparative patterns of C-IF scoring outcomes across various $\mathit{\rho }$ values.

Figure 5. Comparative patterns of comprehensive C-IF scoring outcomes across various $\mathit{\rho }$ values.

Figure 6. Comparisons of the comprehensive disagreement metric $d_{\mathit{ik}}$ across various $\mathit{\rho }$ values.

Table 3. Assessment of the final sorting outcomes generated by different methods.

Solution outcome derived from Otay and Kahraman’s (2022) approach			Ultimate supplier rank order
Technique	Indicator	Inclination	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
Integrated C-IF AHP	Group utility measure	Optimism	1st	2nd	3rd
& VIKOR approach		Pessimism	3rd	1st	2nd
		Comprehensiveness	3rd	1st	2nd
	Individual regret measure	Optimism	1st	2nd	3rd
		Pessimism	3rd	1st	2nd
		Comprehensiveness	3rd	1st	2nd
	Trade-off index	Optimism	1st	2nd	3rd
		Pessimism	3rd	1st	2nd
		Comprehensiveness	3rd	1st	2nd
Solution outcome derived from the developed approach in the current paper			Ultimate supplier rank order
Technique	Indicator	Parameter setting	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
C-IF scoring-mediated	Comprehensive C-IF scoring	$\mathit{\rho }\in \left\{0,0.1,\cdots ,0.4\right\}$	3rd	2nd	1st
median ranking model	function/disagreement metric	$\mathit{\rho }\in \left\{0.5,0.6,\cdots ,1\right\}$	3rd	1st	2nd

Otay, İ., and C. Kahraman. 2022. A novel circular intuitionistic fuzzy AHP-VIKOR methodology: An application to a multi-expert supplier evaluation problem. Pamukkale University Journal of Engineering Sciences 28 (1):194–207. doi:10.5505/pajes.2021.90023.

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Data availability statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.