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

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

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

    Citation:

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

    $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $C_6$
    $X_1$ $\langle[0.2, 0.3], [0.4, 0.5]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.4, 0.5], [0.2, 0.4]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.1, 0.3], [0.5, 0.6]\rangle$ $\langle[0.5, 0.7], [0.2, 0.3]\rangle$
    $X_2$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.6, 0.7], [0.1, 0.2]\rangle$ $\langle[0.3, 0.4], [0.5, 0.6]\rangle$ $\langle[0.4, 0.7], [0.1, 0.2]\rangle$
    $X_3$ $\langle[0.4, 0.5], [0.3, 0.4]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.5, 0.6], [0.3, 0.4]\rangle$ $\langle[0.6, 0.7], [0.1, 0.3]\rangle$ $\langle[0.4, 0.5], [0.3, 0.4]\rangle$ $\langle[0.3, 0.5], [0.1, 0.3]\rangle$
    $X_4$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$ $\langle[0.3, 0.4], [0.1, 0.2]\rangle$ $\langle[0.5, 0.6], [0.1, 0.3]\rangle$ $\langle[0.7, 0.8], [0.1, 0.2]\rangle$
    $X_5$ $\langle[0.5, 0.6], [0.3, 0.4]\rangle$ $\langle[0.3, 0.4], [0.3, 0.5]\rangle$ $\langle[0.6, 0.7], [0.1, 0.3]\rangle$ $\langle[0.6, 0.8], [0.1, 0.2]\rangle$ $\langle[0.6, 0.7], [0.2, 0.3]\rangle$ $\langle[0.5, 0.6], [0.2, 0.4]\rangle$
     | Show Table
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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 $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_5 \succ X_2 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$
     | Show Table
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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 $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$ $X_4 \succ X_2 \succ X_5 \succ X_3 \succ X_1$
     | Show Table
    DownLoad: CSV

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

    $\gamma$ By IIFHIWG By IIFHIHWG
    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 $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$ $X_4\succ X_2\succ X_5 \succ X_3 \succ X_1$
     | Show Table
    DownLoad: CSV
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