Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process

Harish Garg

doi:10.3934/jimo.2017047

Article Contents

2018, Volume 14, Issue 1: 283-308. Doi: 10.3934/jimo.2017047

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Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process

Harish Garg

School of Mathematics, Thapar University Patiala-147004, Punjab, India

Received: June 30, 2016

Revised: September 30, 2016

Early access: May 2017

Published: January 2018

The author would like to thank the Editor-in-Chief and referees for providing very helpful comments and suggestions.

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Abstract

The objective of this manuscript is to present some new interactive geometric aggregation operators for the interval-valued intuitionistic fuzzy numbers (IVIFNs). In order to achieve it, firstly the shortcomings of the existing operators have been highlighted and then resolved it by defining new operational laws based on the pairs of hesitation degree between the membership functions. By using these improved laws, some geometric aggregation operators, namely interval-valued intuitionistic fuzzy Hamacher interactive weighted and hybrid geometric labeled as IIFHIWG and IIFHIHWG operators, respectively have been proposed. Furthermore, desirable properties corresponding to these operators have been stated. Finally, a decision-making method based on the proposed operator has been illustrated to demonstrate the approach. A computed result is compared with the existing results.

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Mathematics Subject Classification: Primary: 90B50, 62A86, 03E72, 68T35.

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Table 1. Information about each alternative in the form of the IVIFNs

	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$C_6$
$X_1$	$\langle[0.2, 0.3], [0.4, 0.5]\rangle$	$\langle[0.6, 0.7], [0.2, 0.3]\rangle$	$\langle[0.4, 0.5], [0.2, 0.4]\rangle$	$\langle[0.7, 0.8], [0.1, 0.2]\rangle$	$\langle[0.1, 0.3], [0.5, 0.6]\rangle$	$\langle[0.5, 0.7], [0.2, 0.3]\rangle$
$X_2$	$\langle[0.6, 0.7], [0.2, 0.3]\rangle$	$\langle[0.5, 0.6], [0.1, 0.3]\rangle$	$\langle[0.6, 0.7], [0.2, 0.3]\rangle$	$\langle[0.6, 0.7], [0.1, 0.2]\rangle$	$\langle[0.3, 0.4], [0.5, 0.6]\rangle$	$\langle[0.4, 0.7], [0.1, 0.2]\rangle$
$X_3$	$\langle[0.4, 0.5], [0.3, 0.4]\rangle$	$\langle[0.7, 0.8], [0.1, 0.2]\rangle$	$\langle[0.5, 0.6], [0.3, 0.4]\rangle$	$\langle[0.6, 0.7], [0.1, 0.3]\rangle$	$\langle[0.4, 0.5], [0.3, 0.4]\rangle$	$\langle[0.3, 0.5], [0.1, 0.3]\rangle$
$X_4$	$\langle[0.6, 0.7], [0.2, 0.3]\rangle$	$\langle[0.5, 0.6], [0.1, 0.3]\rangle$	$\langle[0.7, 0.8], [0.1, 0.2]\rangle$	$\langle[0.3, 0.4], [0.1, 0.2]\rangle$	$\langle[0.5, 0.6], [0.1, 0.3]\rangle$	$\langle[0.7, 0.8], [0.1, 0.2]\rangle$
$X_5$	$\langle[0.5, 0.6], [0.3, 0.4]\rangle$	$\langle[0.3, 0.4], [0.3, 0.5]\rangle$	$\langle[0.6, 0.7], [0.1, 0.3]\rangle$	$\langle[0.6, 0.8], [0.1, 0.2]\rangle$	$\langle[0.6, 0.7], [0.2, 0.3]\rangle$	$\langle[0.5, 0.6], [0.2, 0.4]\rangle$

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Table 2. Effect of the parameter $\gamma$ on the ranking of the alternatives by IIFHIWG and the existing operators

	$\gamma=1$		$\gamma=2$		$\gamma=3$
	Wei and Wang [26]	Proposed	Wang and Liu [24]	Proposed	Liu [20]	Proposed
	Score value		Score value		Score value
$X_1$	0.0548	0.1346	0.0727	0.1454	0.0822	0.1517
$X_2$	0.2874	0.3174	0.2998	0.3310	0.3065	0.3388
$X_3$	0.2139	0.2713	0.2205	0.2760	0.2245	0.2793
$X_4$	0.4463	0.4997	0.4535	0.5013	0.4576	0.5024
$X_5$	0.2985	0.3119	0.3047	0.3166	0.3083	0.3197
ranking	$X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$

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Table 3. Effect of the parameter $\gamma$ on the ranking of the alternatives by using IIFHIHWG and the existing operators

	$\gamma=1$		$\gamma=2$		$\gamma=3$
	Wei and Wang [26]	Proposed	Wang and Liu [24]	Proposed	Liu [20]	Proposed
	Score value		Score value		Score value
$X_1$	0.1221	0.2080	0.1434	0.2163	0.1558	0.2151
$X_2$	0.3304	0.3674	0.3443	0.3734	0.3522	0.3795
$X_3$	0.2535	0.3019	0.2630	0.3068	0.2692	0.3113
$X_4$	0.3705	0.4853	0.3815	0.4815	0.3880	0.4828
$X_5$	0.3141	0.3414	0.3164	0.3510	0.3203	0.3500
ranking	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$	$X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$

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Table 4. Ordering of the attributes for different $\gamma$

$\gamma$		By IIFHIWG		By IIFHIHWG
$\gamma$		Aggregated IVIFN	Score values	Aggregated IVIFN	Score values
0.1	$X_1$	$\big\langle[0.3771, 0.5753], [0.2996, 0.4247]\big\rangle$	0.1140	$\big\langle[0.4562, 0.6029], [0.2818, 0.3971]\big\rangle$	0.1901
	$X_2$	$\big\langle[0.5042, 0.6545], [0.2324, 0.3455]\big\rangle$	0.2904	$\big\langle[0.5016, 0.6889], [0.1971, 0.3111]\big\rangle$	0.3411
	$X_3$	$\big\langle[0.4719, 0.6441], [0.2310, 0.3559]\big\rangle$	0.2646	$\big\langle[0.4808, 0.6605], [0.2187, 0.3395]\big\rangle$	0.2916
	$X_4$	$\big\langle[0.6130, 0.7519], [0.1218, 0.2481]\big\rangle$	0.4975	$\big\langle[0.5404, 0.7670], [0.1097, 0.2330]\big\rangle$	0.4824
	$X_5$	$\big\langle[0.5306, 0.6405], [0.2027, 0.3595]\big\rangle$	0.3045	$\big\langle[0.5338, 0.6603], [0.1876, 0.3397]\big\rangle$	0.3334
Ranking		$X_4\succ X_5 \succ X_2 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
0.5	$X_1$	$\big\langle[0.3805, 0.5819], [0.2933, 0.4181]\big\rangle$	0.1255	$\big\langle[0.4597 0.6086], [0.2764 0.3914]\big\rangle$	0.2003
	$X_2$	$\big\langle[0.5092, 0.6634], [0.2249, 0.3366]\big\rangle$	0.3056	$\big\langle[0.5062 0.6977], [0.1897 0.3023]\big\rangle$	0.3560
	$X_3$	$\big\langle[0.4734, 0.6455], [0.2285, 0.3545]\big\rangle$	0.2679	$\big\langle[0.4826 0.6633], [0.2157 0.3367]\big\rangle$	0.2967
	$X_4$	$\big\langle[0.6133, 0.7526], [0.1214, 0.2474]\big\rangle$	0.4986	$\big\langle[0.5406 0.7681], [0.1093 0.2319]\big\rangle$	0.4838
	$X_5$	$\big\langle[0.5318, 0.6429], [0.2009, 0.3571]\big\rangle$	0.3084	$\big\langle[0.5352 0.6639], [0.1855 0.3361]\big\rangle$	0.3388
Ranking		$X_4\succ X_5 \succ X_2 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
1	$X_1$	$\big\langle[0.3834, 0.5868], [0.2878, 0.4132]\big\rangle$	0.1346	$\big\langle[0.4626, 0.6125], [0.2717, 0.3875]\big\rangle$	0.2080
	$X_2$	$\big\langle[0.5134, 0.6699], [0.2185, 0.3301]\big\rangle$	0.3174	$\big\langle[0.5100, 0.7041], [0.1835, 0.2959]\big\rangle$	0.3674
	$X_3$	$\big\langle[0.4750, 0.6467], [0.2260, 0.3533]\big\rangle$	0.2713	$\big\langle[0.4845, 0.6659], [0.2126, 0.3341]\big\rangle$	0.3019
	$X_4$	$\big\langle[0.6136, 0.7533], [0.1210, 0.2467]\big\rangle$	0.4997	$\big\langle[0.5409, 0.7693], [0.1089, 0.2307]\big\rangle$	0.4853
	$X_5$	$\big\langle[0.5330, 0.6449], [0.1991, 0.3551]\big\rangle$	0.3119	$\big\langle[0.5141, 0.6751], [0.1813, 0.3249]\big\rangle$	0.3414
Ranking		$X_4\succ X_2 \succ X_5 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
2	$X_1$	$\big\langle[0.3872, 0.5923], [0.2809, 0.4077]\big\rangle$	0.1454	$\big\langle[0.4663, 0.6161], [0.2659, 0.3839]\big\rangle$	0.2163
	$X_2$	$\big\langle[0.5186, 0.6770], [0.2106, 0.3230]\big\rangle$	0.3310	$\big\langle[0.5268, 0.7023], [0.1847, 0.2977]\big\rangle$	0.3734
	$X_3$	$\big\langle[0.4774, 0.6484], [0.2221, 0.3516]\big\rangle$	0.2760	$\big\langle[0.4890, 0.6646], [0.2047, 0.3354]\big\rangle$	0.3068
	$X_4$	$\big\langle[0.6141, 0.7543], [0.1202, 0.2457]\big\rangle$	0.5013	$\big\langle[0.5556, 0.7623], [0.1172, 0.2377]\big\rangle$	0.4815
	$X_5$	$\big\langle[0.5348, 0.6473], [0.1963, 0.3527]\big\rangle$	0.3166	$\big\langle[0.5166, 0.6813], [0.1772, 0.3187]\big\rangle$	0.3510
Ranking		$X_4\succ X_2 \succ X_5 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
5	$X_1$	$\big\langle[0.3922, 0.5987], [0.2716 0.4013]\big\rangle$	0.1590	$\big\langle[0.4324, 0.6341], [0.2537, 0.3659]\big\rangle$	0.2235
	$X_2$	$\big\langle[0.5256, 0.6849], [0.1999 0.3151]\big\rangle$	0.3478	$\big\langle[0.5195, 0.7130], [0.1776, 0.2870]\big\rangle$	0.3840
	$X_3$	$\big\langle[0.4815, 0.6506], [0.2153 0.3494]\big\rangle$	0.2837	$\big\langle[0.4939, 0.6691], [0.1969, 0.3309]\big\rangle$	0.3176
	$X_4$	$\big\langle[0.6150, 0.7558], [0.1189 0.2442]\big\rangle$	0.5039	$\big\langle[0.5565, 0.7642], [0.1158, 0.2358]\big\rangle$	0.4846
	$X_5$	$\big\langle[0.5379, 0.6504], [0.1917 0.3496]\big\rangle$	0.3235	$\big\langle[0.5521, 0.6726], [0.1906, 0.3274]\big\rangle$	0.3534
Ranking		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
10	$X_1$	$\big\langle[0.3952, 0.6020], [0.2659, 0.3980]\big\rangle$	0.1667	$\big\langle[0.4360, 0.6387], [0.2474, 0.3613]\big\rangle$	0.2330
	$X_2$	$\big\langle[0.5300, 0.6889], [0.1932, 0.3111]\big\rangle$	0.3573	$\big\langle[0.5220, 0.7368], [0.1572, 0.2632]\big\rangle$	0.4192
	$X_3$	$\big\langle[0.4847, 0.6519], [0.2102, 0.3481]\big\rangle$	0.2891	$\big\langle[0.4767, 0.6968], [0.1744, 0.3032]\big\rangle$	0.3479
	$X_4$	$\big\langle[0.6158, 0.7568], [0.1178, 0.2432]\big\rangle$	0.5058	$\big\langle[0.5573, 0.7657], [0.1147, 0.2343]\big\rangle$	0.4870
	$X_5$	$\big\langle[0.5403, 0.6522], [0.1882, 0.3478]\big\rangle$	0.3282	$\big\langle[0.5608, 0.6594], [0.1933, 0.3406]\big\rangle$	0.3432
Ranking		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_3 \succ X_5 \succ X_1$
25	$X_1$	$\big\langle[0.3979, 0.6046], [0.2610, 0.3954]\big\rangle$	0.1730	$\big\langle[0.4217, 0.6566], [0.2234, 0.3434]\big\rangle$	0.2558
	$X_2$	$\big\langle[0.5338, 0.6920], [0.1874, 0.3080]\big\rangle$	0.3652	$\big\langle[0.5570, 0.7047], [0.1689, 0.2953]\big\rangle$	0.3988
	$X_3$	$\big\langle[0.4878, 0.6530], [0.2051, 0.3470]\big\rangle$	0.2943	$\big\langle[0.5176, 0.6687], [0.1871, 0.3313]\big\rangle$	0.3339
	$X_4$	$\big\langle[0.6165, 0.7576], [0.1167, 0.2424]\big\rangle$	0.5075	$\big\langle[0.6057, 0.7394], [0.1297, 0.2606]\big\rangle$	0.4774
	$X_5$	$\big\langle[0.5426, 0.6536], [0.1847, 0.3464]\big\rangle$	0.3325	$\big\langle[0.5600, 0.6525], [0.1944, 0.3475]\big\rangle$	0.3353
Ranking		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$		$X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$

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