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Picture Fuzzy Parameterized Picture Fuzzy Soft Sets and Their Application in a Performance-Based Value Assignment Problem to Salt-and-Pepper Noise Removal Filters

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Abstract

In real-world problems such as voting for an election, decisions of the electorate may be split into three types: yes, no, and abstain. The concept of picture fuzzy sets (pf-sets) has been put forward to model such problems. This study introduces a new concept, i.e., picture fuzzy parameterized picture fuzzy soft sets (pfppfs-sets), to model problems containing parameters and alternatives with picture fuzzy membership. Afterwards, it proposes a soft decision-making (SDM) method via pfppfs-sets. Next, the proposed SDM method is applied to a performance-based value assignment (PVA) problem to salt-and-pepper noise (SPN) removal filters and compared with the four SDM methods in different structures. The results manifest that pfppfs-sets and the proposed SDM method produce consistent ranking order for PVA problems to SPN removal filters.

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Abbreviations

\(\mu (x)\) :

Membership degree of x

\(\nu (x)\) :

Non-membership degree of x

\(\eta (x)\) :

Neutral membership degree of x

pf-sets:

Picture fuzzy sets

\(\left\langle \begin{array}{l} {\mu (x)}\\ {\eta (x)}\\ {\nu (x)} \end{array}\right\rangle\) :

Picture fuzzy value of x

pfs-sets:

Picture fuzzy soft sets

gpfs-sets:

Generalized picture fuzzy soft sets

fpfs-sets:

Fuzzy parametrized fuzzy soft sets

ifpifs-sets:

Intuitionistic fuzzy parametrized

intuitionistic fuzzy soft sets

ivifpivifs-sets:

Interval-valued intuitionistic fuzzy

parametrized interval-valued

intuitionistic fuzzy soft sets

pfppfs-sets:

Picture fuzzy parametrized

picture fuzzy soft sets

SDM:

Soft decision-making

PVA:

Performance-based value assignment

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Authors and Affiliations

  1. Department of Computer Engineering, Faculty of Engineering and Natural Sciences, İstanbul Rumeli University, İstanbul, 34570, Türkiye

    Samet Memiş

  2. Department of Marine Engineering, Faculty of Maritime, Bandırma Onyedi Eylül University, Balıkesir, 10200, Türkiye

    Samet Memiş

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The original online version of this article was revised due to formulas in Definition 22, Definition 23, Definition 27, Example 5 and Example 6 were incorrectly formatted and corrected in this version. In addition, the abbreviations list was incorrectly published as Table 1 and corrected in this version.

Appendix

Appendix

$$\begin{aligned} \alpha= & {} \left\{ \left( {\left\langle \begin{array}{c} 0.1\\ 0.5\\ 0.9 \end{array}\right\rangle }x_1, \left\{ {\left\langle \begin{array}{c} 0.9783\\ 0.0783\\ 0.0217 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.9796\\ 0.0796\\ 0.0204 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.9774\\ 0.0774\\ 0.0226 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.9748\\ 0.0748\\ 0.0252 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9748\\ 0.0854\\ 0.0146 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9728\\ 0.0728\\ 0.0272 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9868\\ 0.0868\\ 0.0132 \end{array}\right\rangle }u_7\right\} \right) ,\right. \\{} & {} \left( {\left\langle \begin{array}{c} 0.2\\ 0.5\\ 0.8 \end{array}\right\rangle }x_2, \left\{ {\left\langle \begin{array}{c} 0.9536\\ 0.1536\\ 0.0464 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.9584\\ 0.1584\\ 0.0416 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.9197\\ 0.1197\\ 0.0803 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.9504\\ 0.1504\\ 0.0496 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9699\\ 0.1699\\ 0.0301 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9622\\ 0.1622\\ 0.0378 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9735\\ 0.1735\\ 0.0265 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.3\\ 0.5\\ 0.7 \end{array}\right\rangle }x_3, \left\{ {\left\langle \begin{array}{c} 0.9229\\ 0.2229\\ 0.0771 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.9315\\ 0.2315\\ 0.0685 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8117\\ 0.1117\\ 0.1883 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.9248\\ 0.2248\\ 0.0752 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9516\\ 0.2516\\ 0.0484 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9484\\ 0.2484\\ 0.0516 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9581\\ 0.2581\\ 0.0419 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.4\\ 0.5\\ 0.6 \end{array}\right\rangle }x_4, \left\{ {\left\langle \begin{array}{c} 0.8838\\ 0.2838\\ 0.1162 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.8968\\ 0.2968\\ 0.1032 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.7973\\ 0.1973\\ 0.2027 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.8973\\ 0.2973\\ 0.1027 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9303\\ 0.3303\\ 0.0697 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9315\\ 0.3315\\ 0.0685 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9400\\ 0.3400\\ 0.0600 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.5\\ 0.5\\ 0.5 \end{array}\right\rangle }x_5, \left\{ {\left\langle \begin{array}{c} 0.8323\\ 0.3323\\ 0.1677 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.8520\\ 0.3520\\ 0.1480 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8399\\ 0.3399\\ 0.1601 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.8666\\ 0.3666\\ 0.1334 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9051\\ 0.4051\\ 0.0949 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9098\\ 0.4098\\ 0.0902 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9173\\ 0.4173\\ 0.0827 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.6\\ 0.5\\ 0.4 \end{array}\right\rangle }x_6, \left\{ {\left\langle \begin{array}{c} 0.7634\\ 0.3634\\ 0.2366 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.7949\\ 0.3949\\ 0.2051 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8410\\ 0.4410\\ 0.1590 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.8320\\ 0.4320\\ 0.1680 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.8748\\ 0.4748\\ 0.1252 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.8816\\ 0.4816\\ 0.1184 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.8880\\ 0.4880\\ 0.1120 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.7\\ 0.5\\ 0.3 \end{array}\right\rangle }x_7, \left\{ {\left\langle \begin{array}{c} 0.6680\\ 0.3680\\ 0.3320 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.7213\\ 0.4213\\ 0.2787 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8025\\ 0.5025\\ 0.1975 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.7910\\ 0.4910\\ 0.2090 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.8368\\ 0.5368\\ 0.1632 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.8437\\ 0.5437\\ 0.1563 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.8491\\ 0.5491\\ 0.1509 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.8\\ 0.5\\ 0.2 \end{array}\right\rangle }x_8, \left\{ {\left\langle \begin{array}{c} 0.5096\\ 0.3096\\ 0.4904 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.6265\\ 0.4265\\ 0.3735 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.7023\\ 0.5023\\ 0.2977 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.7357\\ 0.5357\\ 0.2643 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.7846\\ 0.5846\\ 0.2154 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.7904\\ 0.5904\\ 0.2096 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.7947\\ 0.5947\\ 0.2053 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left. \left( {\left\langle \begin{array}{c} 0.9\\ 0.5\\ 0.1 \end{array}\right\rangle }x_9, \left\{ {\left\langle \begin{array}{c} 0.2585\\ 0.1585\\ 0.7415 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.4966\\ 0.3966\\ 0.5034 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.3566\\ 0.2566\\ 0.6434 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.6190\\ 0.5190\\ 0.3810 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.6964\\ 0.5964\\ 0.3036 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.7028\\ 0.6028\\ 0.2972 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.7056\\ 0.6056\\ 0.2944 \end{array}\right\rangle }u_7\right\} \right) \right\} \\{} & {} \alpha_2=\left\{ \left( {\left\langle \begin{array}{c} 0.1\\ 1\\ 0.9 \end{array}\right\rangle }x_1, \left\{ {\left\langle \begin{array}{c} 0.9783\\ 1\\ 0.0217 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.9796\\ 1\\ 0.0204 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.9774\\ 1\\ 0.0226 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.9748\\ 1\\ 0.0252 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9748\\ 1\\ 0.0146 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9728\\ 1\\ 0.0272 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9868\\ 1\\ 0.0132 \end{array}\right\rangle }u_7\right\} \right) ,\right. \\{} & {} \left( {\left\langle \begin{array}{c} 0.2\\ 1\\ 0.8 \end{array}\right\rangle }x_2, \left\{ {\left\langle \begin{array}{c} 0.9536\\ 1\\ 0.0464 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.9584\\ 1\\ 0.0416 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.9197\\ 1\\ 0.0803 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.9504\\ 1\\ 0.0496 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9699\\ 1\\ 0.0301 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9622\\ 1\\ 0.0378 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9735\\ 1\\ 0.0265 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.3\\ 1\\ 0.7 \end{array}\right\rangle }x_3, \left\{ {\left\langle \begin{array}{c} 0.9229\\ 1\\ 0.0771 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.9315\\ 1\\ 0.0685 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8117\\ 1\\ 0.1883 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.9248\\ 1\\ 0.0752 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9516\\ 1\\ 0.0484 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9484\\ 1\\ 0.0516 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9581\\ 1\\ 0.0419 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.4\\ 1\\ 0.6 \end{array}\right\rangle }x_4, \left\{ {\left\langle \begin{array}{c} 0.8838\\ 1\\ 0.1162 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.8968\\ 1\\ 0.1032 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.7973\\ 1\\ 0.2027 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.8973\\ 1\\ 0.1027 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9303\\ 1\\ 0.0697 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9315\\ 1\\ 0.0685 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9400\\ 1\\ 0.0600 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.5\\ 1\\ 0.5 \end{array}\right\rangle }x_5, \left\{ {\left\langle \begin{array}{c} 0.8323\\ 1\\ 0.1677 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.8520\\ 1\\ 0.1480 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8399\\ 1\\ 0.1601 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.8666\\ 1\\ 0.1334 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.9051\\ 1\\ 0.0949 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.9098\\ 1\\ 0.0902 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.9173\\ 1\\ 0.0827 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.6\\ 1\\ 0.4 \end{array}\right\rangle }x_6, \left\{ {\left\langle \begin{array}{c} 0.7634\\ 1\\ 0.2366 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.7949\\ 1\\ 0.2051 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8410\\ 1\\ 0.1590 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.8320\\ 1\\ 0.1680 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.8748\\ 1\\ 0.1252 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.8816\\ 1\\ 0.1184 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.8880\\ 1\\ 0.1120 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.7\\ 1\\ 0.3 \end{array}\right\rangle }x_7, \left\{ {\left\langle \begin{array}{c} 0.6680\\ 1\\ 0.3320 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.7213\\ 1\\ 0.2787 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.8025\\ 1\\ 0.1975 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.7910\\ 1\\ 0.2090 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.8368\\ 1\\ 0.1632 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.8437\\ 1\\ 0.1563 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.8491\\ 1\\ 0.1509 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left( {\left\langle \begin{array}{c} 0.8\\ 1\\ 0.2 \end{array}\right\rangle }x_8, \left\{ {\left\langle \begin{array}{c} 0.5096\\ 1\\ 0.4904 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.6265\\ 1\\ 0.3735 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.7023\\ 1\\ 0.2977 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.7357\\ 1\\ 0.2643 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.7846\\ 1\\ 0.2154 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.7904\\ 1\\ 0.2096 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.7947\\ 1\\ 0.2053 \end{array}\right\rangle }u_7\right\} \right) ,\\{} & {} \left. \left( {\left\langle \begin{array}{c} 0.9\\ 1\\ 0.1 \end{array}\right\rangle }x_9, \left\{ {\left\langle \begin{array}{c} 0.2585\\ 1\\ 0.7415 \end{array}\right\rangle }u_1, {\left\langle \begin{array}{c} 0.4966\\ 1\\ 0.5034 \end{array}\right\rangle }u_2, {\left\langle \begin{array}{c} 0.3566\\ 1\\ 0.6434 \end{array}\right\rangle }u_3, {\left\langle \begin{array}{c} 0.6190\\ 1\\ 0.3810 \end{array}\right\rangle }u_4, {\left\langle \begin{array}{c} 0.6964\\ 1\\ 0.3036 \end{array}\right\rangle }u_5, {\left\langle \begin{array}{c} 0.7028\\ 1\\ 0.2972 \end{array}\right\rangle }u_6, {\left\langle \begin{array}{c} 0.7056\\ 1\\ 0.2944 \end{array}\right\rangle }u_7\right\} \right) \right\} \\{} & {} \alpha_3=\left\{ \left( ^{0.1}x_1, \left\{ ^{0.9783}u_1, ^{0.9796}u_2, ^{0.9774}u_3, ^{0.9748}u_4, ^{0.9748}u_5, ^{0.9728}u_6, ^{0.9868}u_7\right\} \right) ,\right. \\{} & {} \left( ^{0.2}x_2, \left\{ ^{0.9536}u_1, ^{0.9584}u_2, ^{0.9197}u_3, ^{0.9504}u_4, ^{0.9699}u_5, ^{0.9622}u_6, ^{0.9735}u_7\right\} \right) ,\\{} & {} \left( ^{0.3}x_3, \left\{ ^{0.9229}u_1, ^{0.9315}u_2, ^{0.8117}u_3, ^{0.9248}u_4, ^{0.9516}u_5, ^{0.9484}u_6, ^{0.9581}u_7\right\} \right) ,\\{} & {} \left( ^{0.4}x_4, \left\{ ^{0.8838}u_1, ^{0.8968}u_2, ^{0.7973}u_3, ^{0.8973}u_4, ^{0.9303}u_5, ^{0.9315}u_6, ^{0.9400}u_7\right\} \right) ,\\{} & {} \left( ^{0.5}x_5, \left\{ ^{0.8323}u_1, ^{0.8520}u_2, ^{0.8399}u_3, ^{0.8666}u_4, ^{0.9051}u_5, ^{0.9098}u_6, ^{0.9173}u_7\right\} \right) ,\\{} & {} \left( ^{0.6}x_6, \left\{ ^{0.7634}u_1, ^{0.7949}u_2, ^{0.8410}u_3, ^{0.8320}u_4, ^{0.8748}u_5, ^{0.8816}u_6, ^{0.8880}u_7\right\} \right) ,\\{} & {} \left( ^{0.7}x_7, \left\{ ^{0.6680}u_1, ^{0.7213}u_2, ^{0.8025}u_3, ^{0.7910}u_4, ^{0.8368}u_5, ^{0.8437}u_6, ^{0.8491}u_7\right\} \right) ,\\{} & {} \left( ^{0.8}x_8, \left\{ ^{0.5096}u_1, ^{0.6265}u_2, ^{0.7023}u_3, ^{0.7357}u_4, ^{0.7846}u_5, ^{0.7904}u_6, ^{0.7947}u_7\right\} \right) ,\\{} & {} \left. \left( ^{0.9}x_9, \left\{ ^{0.2585}u_1, ^{0.4966}u_2, ^{0.3566}u_3, ^{0.6190}u_4, ^{0.6964}u_5, ^{0.7028}u_6, ^{0.7056}u_7\right\} \right) \right\} \\{} & {} \alpha_4=\left\{ \left( ^{0.1}_{0.9}x_1, \left\{ ^{0.9783}_{0.0217}u_1, ^{0.9796}_{0.0204}u_2, ^{0.9774}_{0.0226}u_3, ^{0.9748}_{0.0252}u_4, ^{0.9748}_{0.0146}u_5, ^{0.9728}_{0.0272}u_6, ^{0.9868}_{0.0132}u_7\right\} \right) ,\right. \\{} & {} \left( ^{0.2}_{0.8}x_2, \left\{ ^{0.9536}_{0.0464}u_1, ^{0.9584}_{0.0416}u_2, ^{0.9197}_{0.0803}u_3, ^{0.9504}_{0.0496}u_4, ^{0.9699}_{0.0301}u_5, ^{0.9622}_{0.0378}u_6, ^{0.9735}_{0.0265}u_7\right\} \right) ,\\{} & {} \left( ^{0.3}_{0.7}x_3, \left\{ ^{0.9229}_{0.0771}u_1, ^{0.9315}_{0.0685}u_2, ^{0.8117}_{0.1883}u_3, ^{0.9248}_{0.0752}u_4, ^{0.9516}_{0.0484}u_5, ^{0.9484}_{0.0516}u_6, ^{0.9581}_{0.0419}u_7\right\} \right) ,\\{} & {} \left( ^{0.4}_{0.6}x_4, \left\{ ^{0.8838}_{0.1162}u_1, ^{0.8968}_{0.1032}u_2, ^{0.7973}_{0.2027}u_3, ^{0.8973}_{0.1027}u_4, ^{0.9303}_{0.0697}u_5, ^{0.9315}_{0.0685}u_6, ^{0.9400}_{0.0600}u_7\right\} \right) ,\\{} & {} \left( ^{0.5}_{0.5}x_5, \left\{ ^{0.8323}_{0.1677}u_1, ^{0.8520}_{0.1480}u_2, ^{0.8399}_{0.1601}u_3, ^{0.8666}_{0.1334}u_4, ^{0.9051}_{0.0949}u_5, ^{0.9098}_{0.0902}u_6, ^{0.9173}_{0.0827}u_7\right\} \right) ,\\{} & {} \left( ^{0.6}_{0.4}x_6, \left\{ ^{0.7634}_{0.2366}u_1, ^{0.7949}_{0.2051}u_2, ^{0.8410}_{0.1590}u_3, ^{0.8320}_{0.1680}u_4, ^{0.8748}_{0.1252}u_5, ^{0.8816}_{0.1184}u_6, ^{0.8880}_{0.1120}u_7\right\} \right) ,\\{} & {} \left( ^{0.7}_{0.3}x_7, \left\{ ^{0.6680}_{0.3320}u_1, ^{0.7213}_{0.2787}u_2, ^{0.8025}_{0.1975}u_3, ^{0.7910}_{0.2090}u_4, ^{0.8368}_{0.1632}u_5, ^{0.8437}_{0.1563}u_6, ^{0.8491}_{0.1509}u_7\right\} \right) ,\\{} & {} \left( ^{0.8}_{0.2}x_8, \left\{ ^{0.5096}_{0.4904}u_1, ^{0.6265}_{0.3735}u_2, ^{0.7023}_{0.2977}u_3, ^{0.7357}_{0.2643}u_4, ^{0.7846}_{0.2154}u_5, ^{0.7904}_{0.2096}u_6, ^{0.7947}_{0.2053}u_7\right\} \right) ,\\{} & {} \left. \left( ^{0.9}_{0.1}x_9, \left\{ ^{0.2585}_{0.7415}u_1, ^{0.4966}_{0.5034}u_2, ^{0.3566}_{0.6434}u_3, ^{0.6190}_{0.3810}u_4, ^{0.6964}_{0.3036}u_5, ^{0.7028}_{0.2972}u_6, ^{0.7056}_{0.2944}u_7\right\} \right) , \right\} \end{aligned}$$

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Memiş, S. Picture Fuzzy Parameterized Picture Fuzzy Soft Sets and Their Application in a Performance-Based Value Assignment Problem to Salt-and-Pepper Noise Removal Filters. Int. J. Fuzzy Syst. 25, 2860–2875 (2023). https://doi.org/10.1007/s40815-023-01547-5

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