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A Hesitant Fermatean Fuzzy CoCoSo Method for Group Decision-Making and an Application to Blockchain Platform Evaluation

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Abstract

The existing score functions of Fermatean fuzzy number (FFN) show some defects. In addition, the evaluation information given by decision-makers in real decision-making processes may be several discrete values rather than single value, which cannot be represented by the classical Fermatean fuzzy set (FFS). The combined compromise solution (CoCoSo) method can get a compromise solution by incorporating multi-aggregation strategy; however, the final aggregation operator adopted in the CoCoSo method may cause irrational results. To bridge the above gaps, in this study, we first determine a new score function of FFNs. Then, we define the hesitant Fermatean fuzzy sets based on hesitant fuzzy sets and FFSs to process uncertain information flexibly. Next, we develop an ensemble ranking approach to overcome the limitation of the original CoCoSo method regarding aggregation bias. Afterwards, we propose a hesitant Fermatean fuzzy CoCoSo method for multiple criteria group decision-making. Last, we verify its effectiveness and practicability through a case study on the selection of blockchain platform. At the same time, we perform sensitivity analysis to check the robustness of the method, and emphasize its advantages through comparative analysis.

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Acknowledgements

This paper was supported by the Project of Chongqing Technology and Business University (2152009).

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Correspondence to Huchang Liao.

Appendix

Appendix

1.1 The Proof of Proposition 1

Proof

According to Eq. (9), \({{\frac{{\partial S_{L} \left( F \right)}}{\partial \alpha } = 3\alpha_{F}^{2} \left[ {\frac{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} + 2\left( {1 - \beta_{F}^{3} } \right)}}{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} }}} \right]} \mathord{\left/ {\vphantom {{\frac{{\partial S_{L} \left( F \right)}}{\partial \alpha } = 3\alpha_{F}^{2} \left[ {\frac{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} + 2\left( {1 - \beta_{F}^{3} } \right)}}{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} }}} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}\). Since \(\alpha_{F}\),\(\beta_{F}^{{}} \in \left[ {0,1} \right]\) and \(0 \le \left( {\alpha_{F}^{3} + \beta_{F}^{3} } \right) \le 1\), it follows \(\frac{{\partial S_{{{\text{proposed}}}} \left( F \right)}}{\partial \alpha } \ge 0\).

Analogously, \({{\frac{{\partial S_{L} \left( F \right)}}{\partial \beta } = - 3\beta_{F}^{2} \left[ {\frac{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} + 2\left( {1 - \alpha_{F}^{3} } \right)}}{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} }}} \right]} \mathord{\left/ {\vphantom {{\frac{{\partial S_{L} \left( F \right)}}{\partial \beta } = - 3\beta_{F}^{2} \left[ {\frac{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} + 2\left( {1 - \alpha_{F}^{3} } \right)}}{{\left( {2 - \alpha_{F}^{3} - \beta_{F}^{3} } \right)^{2} }}} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}\; \le 0\).

Based on the above results, we can conclude that \(S_{L} \left( F \right)\) monotonically increases with the increase of \(\alpha\) and monotonically decreases with the increase of \(\beta\). This completes the proof of Proposition 1.

1.2 The Proof of Proposition 2

Proof

Based on Proposition 1, we can find that if only consider \(\alpha_{F}\) or \(\beta_{F}\), \(S_{L} \left( F \right)\) has the maximum value (i.e., F = (1,0)) or minimum value (i.e., F = (0,1)), respectively. In other words, \(S_{L} \left( F \right)_{\max } = 1\) and \(S_{L} \left( F \right)_{\min } = - 1\). This completes the proof of Proposition 2.

1.3 The Proof of Theorem 3

Proof

(1) \(f_{1} \,{ \boxplus }\,f_{2} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{a_{1}^{3} + a_{2}^{3} - a_{1}^{3} a_{2}^{3} }}} \right\},\left\{ {b_{1} b_{2} } \right\}} \right\}} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{a_{2}^{3} + a_{1}^{3} - a_{2}^{3} a_{1}^{3} }}} \right\},\left\{ {b_{2} b_{1} } \right\}} \right\}} = f_{2} \,{ \boxplus }\,f_{1}\).

(2) \(f_{1} \,{ \boxtimes }\,f_{2} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {a_{1} a_{2} } \right\},\left\{ {\sqrt[3]{{b_{1}^{3} + b_{2}^{3} - b_{1}^{3} b_{2}^{3} }}} \right\}} \right\}} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {a_{2} a_{1} } \right\},\left\{ {\sqrt[3]{{b_{2}^{3} + b_{1}^{3} - b_{2}^{3} b_{1}^{3} }}} \right\}} \right\}} = f_{2} \,{ \boxtimes }\,f_{1}\).

(3) Since \(f_{1} \,{ \boxplus }\,f_{2} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{a_{1}^{3} + a_{2}^{3} - a_{1}^{3} a_{2}^{3} }}} \right\},\left\{ {b_{1} b_{2} } \right\}} \right\}}\) and \(kf = \bigcup\limits_{a \in \alpha ,b \in \beta } {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - a^{3} } \right)^{k} }}} \right\},\left\{ {b^{k} } \right\}} \right\}}\), for a real number \(k > 0\), we get

$$k\left( {f_{1} \,{ \boxplus }\,f_{2} } \right) = \bigcup\limits_{\begin{array}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{array} } {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - \left( {\sqrt[3]{{a_{1}^{3} + a_{2}^{3} - a_{1}^{3} a_{2}^{3} }}} \right)^{3} } \right)^{k} }}} \right\},\left\{ {\left( {b_{1} b_{2} } \right)^{k} } \right\}} \right\}} = \bigcup\limits_{\begin{array}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{array} } {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - a_{1}^{3} - a_{2}^{3} + a_{1}^{3} a_{2}^{3} } \right)^{k} }}} \right\},\left\{ {\left( {b_{1} b_{2} } \right)^{k} } \right\}} \right\}} = \bigcup\limits_{\begin{array}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{array} } {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {\left( {1 - a_{1}^{3} } \right)\left( {1 - a_{2}^{3} } \right)} \right)^{k} }}} \right\},\left\{ {\left( {b_{1} b_{2} } \right)^{k} } \right\}} \right\}} = \bigcup\limits_{\begin{array}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{array} } {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {\left( {1 - a_{1}^{3} } \right)\left( {1 - a_{2}^{3} } \right)} \right)^{k} }}} \right\},\left\{ {\left( {b_{1} b_{2} } \right)^{k} } \right\}} \right\}} \,{ \boxplus }\,\bigcup\limits_{{a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} }} {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - a_{2}^{3} } \right)^{k} }}} \right\},\left\{ {b_{2}^{k} } \right\}} \right\}} = kf_{1} \,{ \boxplus }\,kf_{2}$$

(4) Since \(f_{1} \,{ \boxtimes }\,f_{2} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {a_{1} a_{2} } \right\},\left\{ {\sqrt[3]{{b_{1}^{3} + b_{2}^{3} - b_{1}^{3} b_{2}^{3} }}} \right\}} \right\}}\) and \(f^{k} = \bigcup\limits_{a \in \alpha ,b \in \beta } {\left\{ {\left\{ {a^{k} } \right\}\left\{ {\sqrt[3]{{1 - \left( {1 - b^{3} } \right)^{k} }}} \right\}} \right\}}\), for a real number \(k > 0\), we can get

$$\left( {f_{1} \,{ \boxtimes }\,f_{2} } \right)^{k} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\left( {a_{1} a_{2} } \right)^{k} } \right\},\left\{ {\sqrt[3]{{1 - \left( {1 - a_{1}^{3} - a_{2}^{3} + b_{1}^{3} b_{2}^{3} } \right)^{k} }}} \right\}} \right\}} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\left( {a_{1} a_{2} } \right)^{k} } \right\},\left\{ {\sqrt[3]{{1 - \left( {\left( {1 - a_{1}^{3} } \right)\left( {1 - a_{2}^{3} } \right)} \right)^{k} }}} \right\}} \right\}} = \bigcup\limits_{{a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} }} {\left\{ {\left\{ {a_{1}^{k} } \right\},\left\{ {\sqrt[3]{{1 - \left( {1 - b_{1}^{3} } \right)^{k} }}} \right\}} \right\}} \,{ \boxtimes }\,\bigcup\limits_{{a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} }} {\left\{ {\left\{ {a_{2}^{k} } \right\},\left\{ {\sqrt[3]{{1 - \left( {1 - b_{2}^{3} } \right)^{k} }}} \right\}} \right\}} = f_{1}^{k} \,{ \boxtimes }\,f_{2}^{k} .$$

1.4 The proof of Theorem 4

Proof

We prove Eq. (11) by mathematical induction on \(n\).

When \(n = 2\), we have \(f_{1} = \tilde{F}\left( {\alpha_{1} ,\beta_{1} } \right)\) and \(f_{2} = \tilde{F}\left( {\alpha_{2} ,\beta_{2} } \right)\). By the operations of HFFEs, we get

$${\text{HFFWA}}\left( {f_{1} ,f_{2} } \right) = \tilde{w}_{1} f_{1} \,{ \boxplus }\,\tilde{w}_{2} f_{2} = \bigcup\nolimits_{{a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} }} {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - \left( {a_{1} } \right)^{3} } \right)^{{\tilde{w}_{1} }} }}} \right\},\left\{ {\left( {b_{1} } \right)^{{\tilde{w}_{1} }} } \right\}} \right\}} \,{ \boxplus }\,\bigcup\nolimits_{{a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} }} {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - \left( {a_{2} } \right)^{3} } \right)^{{\tilde{w}_{2} }} }}} \right\},\left\{ {\left( {b_{2} } \right)^{{\tilde{w}_{2} }} } \right\}} \right\}} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,b_{1} \in \beta_{1} , \\ a_{2} \in \alpha_{2} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{\begin{gathered} 1 - \left( {1 - a_{1}^{3} } \right)^{{\tilde{w}_{1} }} + 1 - \left( {1 - a_{2}^{3} } \right)^{{\tilde{w}_{1} }} \hfill \\ - \left( {1 - \left( {1 - a_{1}^{3} } \right)^{{\tilde{w}_{1} }} } \right)\left( {1 - \left( {1 - a_{2}^{3} } \right)^{{\tilde{w}_{2} }} } \right) \hfill \\ \end{gathered} }} \right\},\left\{ {b_{1}^{{\tilde{w}_{1} }} b_{2}^{{\tilde{w}_{2} }} } \right\}} \right\}} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,a_{2} \in \alpha_{2} \\ b_{1} \in \beta_{1} ,b_{2} \in \beta_{2} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{1 - \prod\limits_{i = 1}^{2} {\left( {1 - \left( {a_{i} } \right)^{3} } \right)^{{\tilde{w}_{i} }} } }}} \right\},\left\{ {\prod\limits_{i = 1}^{n} {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } } \right\}} \right\}}$$

Thus, the result is true for \(n = 2\).

Suppose that the result given in Eq. (11) is true for \(n = k\), i.e.,

$${\text{HFFWA}}\left( {f_{1} ,f_{2} , \ldots ,f_{k} } \right) = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,a_{2} \in \alpha_{2} , \ldots ,a_{k} \in \alpha_{k} \\ b_{1} \in \beta_{1} ,b_{2} \in \beta_{2} , \ldots ,b_{k} \in \beta_{k} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{1 - \prod\limits_{i = 1}^{k} {\left( {1 - \left( {a_{i} } \right)^{3} } \right)^{{\tilde{w}_{i} }} } }}} \right\},\left\{ {\prod\limits_{i = 1}^{k} {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } } \right\}} \right\}}$$

When \(n = k + 1\), we have

$${\text{HFFWA}}\left( {f_{1} ,f_{2} , \ldots ,f_{k + 1} } \right) = \left( {\tilde{w}_{1} f_{1} \,{ \boxplus }\,\tilde{w}_{2} f_{2} \,{ \boxplus }\, \ldots \,{ \boxplus }\,\tilde{w}_{k} f_{k} } \right)\,{ \boxplus }\,\tilde{w}_{k + 1} f_{k + 1} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,a_{2} \in \alpha_{2} , \ldots ,a_{k} \in \alpha_{k} \\ b_{1} \in \beta_{1} ,b_{2} \in \beta_{2} , \ldots ,b_{k} \in \beta_{k} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{1 - \prod\limits_{i = 1}^{k} {\left( {1 - \left( {a_{i} } \right)^{3} } \right)^{{\tilde{w}_{i} }} } }}} \right\},\left\{ {\prod\limits_{i = 1}^{k} {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } } \right\}} \right\}} \,{ \boxplus }\,\bigcup\limits_{{a_{k + 1} \in \alpha_{k + 1} ,b_{k + 1} \in \beta_{k + 1} }} {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - \left( {a_{k + 1} } \right)^{3} } \right)^{{\tilde{w}_{k + 1} }} }}} \right\},\left\{ {\left( {b_{k + 1} } \right)^{{\tilde{w}_{k + 1} }} } \right\}} \right\}} = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,a_{2} \in \alpha_{2} , \ldots ,a_{k + 1} \in \alpha_{k + 1} \\ b_{1} \in \beta_{1} ,b_{2} \in \beta_{2} , \ldots ,b_{k + 1} \in \beta_{k + 1} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{1 - \prod\limits_{i = 1}^{k + 1} {\left( {1 - \left( {a_{i} } \right)^{3} } \right)^{{\tilde{w}_{i} }} } }}} \right\},\left\{ {\prod\limits_{i = 1}^{k + 1} {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } } \right\}} \right\}}$$

Hence, the result holds for \(n = k + 1\). According to the principle of mathematical induction, the result given in Eq. (11) holds for all positive integer \(n\). This completes the proof of Theorem 4.

1.5 The Proof of Property 2 (Idempotency) of the HFFWA Operator

Proof

Since all HFFEs are identical, i.e., \(f_{i} = f = \tilde{F}\left( {\alpha ,\beta } \right)\) for all \(i\), using Eq. (11), we have.

$$HFFWA\left( {f_{1} ,f_{2} , \ldots ,f_{n} } \right) = \bigcup\limits_{\begin{subarray}{l} a_{1} \in \alpha_{1} ,a_{2} \in \alpha_{2} , \ldots ,a_{n} \in \alpha_{n} \\ b_{1} \in \beta_{1} ,b_{2} \in \beta_{2} , \ldots ,b_{n} \in \beta_{n} \end{subarray} } {\left\{ {\left\{ {\sqrt[3]{{1 - \prod\limits_{i = 1}^{n} {\left( {1 - \left( {a_{i} } \right)^{3} } \right)^{{\tilde{w}_{i} }} } }}} \right\},\left\{ {\prod\limits_{i = 1}^{n} {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } } \right\}} \right\}} = \bigcup\limits_{a \in \alpha ,b \in \beta } {\left\{ {\left\{ {\sqrt[3]{{1 - \prod\limits_{i = 1}^{n} {\left( {1 - \left( a \right)^{3} } \right)^{{\tilde{w}_{i} }} } }}} \right\},\left\{ {\prod\limits_{i = 1}^{n} {\left( b \right)^{{\tilde{w}_{i} }} } } \right\}} \right\}} = \bigcup\limits_{\varphi \in \alpha ,\phi \in \beta } {\left\{ {\left\{ {\sqrt[3]{{1 - \left( {1 - \left( a \right)^{3} } \right)^{{\sum\limits_{i = 1}^{n} {\tilde{w}_{i} } }} }}} \right\},\left\{ {b^{{\sum\limits_{i = 1}^{n} {\tilde{w}_{i} } }} } \right\}} \right\}} = \bigcup\limits_{a \in \alpha ,b \in \beta } {\left\{ {\left\{ a \right\},\left\{ b \right\}} \right\}} = f$$

This completes the proof of Property 2 (Idempotency) of the HFFWA operator.

1.6 The Proof of Property 3 (Boundedness) of the HFFWA Operator

Proof

It is clear that

$$\bigcup\limits_{{a_{i} \in \alpha_{i} }} {\min_{i} \left\{ {a_{i} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\left\{ {a_{i} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\max_{i} \left\{ {a_{i} } \right\}}$$
(23)
$$\bigcup\limits_{{b_{i} \in \beta_{i} }} {\min_{i} \left\{ {b_{i} } \right\}} \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\left\{ {b_{i} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\max_{i} \left\{ {a_{i} } \right\}}$$
(24)

According to Eq. (23), we have

$$\bigcup\limits_{{a_{i} \in \alpha_{i} }} {\min_{i} \left\{ {a_{i} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\left\{ {a_{i} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\max_{i} \left\{ {a_{i} } \right\}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} }}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\prod\limits_{i = 1}^{n} {\left( {1 - \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\prod\limits_{i = 1}^{n} {\left( {1 - \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\sum\limits_{i = 1}^{n} {\tilde{w}_{i} } }} }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\sum\limits_{i = 1}^{n} {\tilde{w}_{i} } }} }}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\left( {1 - \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)}}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{ - 1 + \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{ - \prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{ - 1 + \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - 1 + \min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - \prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - 1 + \max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - \prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{\max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}}}$$
$$\Leftrightarrow \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\min_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\sqrt[3]{{1 - \prod\limits_{i = 1}^{n} {\left( {1 - \left\{ {\left( {a_{i} } \right)^{3} } \right\}} \right)^{{\tilde{w}_{i} }} } }}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\max_{i} \left\{ {\left( {a_{i} } \right)^{3} } \right\}}$$

In addition, according to Eq. (24), we have

$$\bigcup\limits_{{b_{i} \in \beta_{i} }} {\min_{i} \left\{ {b_{i} } \right\}} \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\left\{ {b_{i} } \right\}} \le \bigcup\limits_{{a_{i} \in \alpha_{i} }} {\max_{i} \left\{ {a_{i} } \right\}} \Leftrightarrow \bigcup\limits_{{b_{i} \in \beta_{i} }} {\min_{i} \left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\max_{i} \left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} \Leftrightarrow \bigcup\limits_{{b_{i} \in \beta_{i} }} {\prod\limits_{i = 1}^{n} {\min_{i} \left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} } \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\prod\limits_{i = 1}^{n} {\left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} } \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\prod\limits_{i = 1}^{n} {\max_{i} \left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} } \Leftrightarrow \bigcup\limits_{{b_{i} \in \beta_{i} }} {\min_{i} \left\{ {\left( {b_{i} } \right)^{{\sum\limits_{i = 1}^{n} {\tilde{w}_{i} } }} } \right\}} \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\prod\limits_{i = 1}^{n} {\left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} } \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\max_{i} \left\{ {\left( {b_{i} } \right)^{{\sum\limits_{i = 1}^{n} {\tilde{w}_{i} } }} } \right\}} \Leftrightarrow \bigcup\limits_{{b_{i} \in \beta_{i} }} {\min_{i} \left\{ {b_{i} } \right\}} \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\prod\limits_{i = 1}^{n} {\left\{ {\left( {b_{i} } \right)^{{\tilde{w}_{i} }} } \right\}} } \le \bigcup\limits_{{b_{i} \in \beta_{i} }} {\max_{i} \left\{ {b_{i} } \right\}} .$$

Thus, we have \({\text{HFFWA}}\left( {f_{1} ,f_{2} , \ldots ,f_{n} } \right) \ge f^{ - }\) and \({\text{HFFWA}}\left( {f_{1} ,f_{2} , \ldots ,f_{n} } \right) \le f^{ + }\). Therefore, \(f^{ - } \le {\text{HFFWA}}\left( {f_{1} ,f_{2} , \ldots ,f_{n} } \right) \le f^{ + }\). This completes the proof of Property 3 (Boundedness) of the HFFWA operator.

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Lai, H., Liao, H., Long, Y. et al. A Hesitant Fermatean Fuzzy CoCoSo Method for Group Decision-Making and an Application to Blockchain Platform Evaluation. Int. J. Fuzzy Syst. 24, 2643–2661 (2022). https://doi.org/10.1007/s40815-022-01319-7

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