# Normalized Weighted Bonferroni Harmonic Mean-Based Intuitionistic Fuzzy Operators and Their Application to the Sustainable Selection of Search and Rescue Robots

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. TIFNs and the Associated Arithmetic Operations

- ${A}_{1}\oplus {A}_{2}=([{a}_{1}+{a}_{2},{b}_{1}+{b}_{2},{c}_{1}+{c}_{2}];{\omega}_{{A}_{1}}\wedge {\omega}_{{A}_{2}},{u}_{{A}_{1}}\vee {u}_{{A}_{2}}),$where “$\wedge $” and “$\vee $” stand for the min and max operators, respectively;
- ${A}_{1}\otimes {A}_{2}=([{a}_{1}{a}_{2},{b}_{1}{b}_{2},{c}_{1}{c}_{2}];{\omega}_{{A}_{1}}\wedge {\omega}_{{A}_{2}},{u}_{{A}_{1}}\vee {u}_{{A}_{2}});$
- $\lambda {A}_{1}=\left([\lambda {a}_{1},\lambda {b}_{1},\lambda {c}_{1}];{\omega}_{{A}_{1}},{u}_{{A}_{1}}\right)$;
- ${A}_{1}^{\lambda}={\left([{a}_{1}^{\lambda},{b}_{1}^{\lambda},{c}_{1}^{\lambda}];{\omega}_{{A}_{1}},{u}_{{A}_{1}}\right)}_{}.$

- Commutativity: ${A}_{1}\oplus {A}_{2}={A}_{2}\oplus {A}_{1},{A}_{1}\otimes {A}_{2}={A}_{2}\otimes {A}_{1};$
- Distributivity: $\lambda \left({A}_{1}\oplus {A}_{2}\right)=\lambda {A}_{1}\oplus \lambda {A}_{2},\lambda \left({A}_{1}\otimes {A}_{2}\right)=\lambda {A}_{1}\otimes {A}_{2}={A}_{1}\otimes \lambda {A}_{2};$
- Associativity: ${\lambda}_{1}A+{\lambda}_{2}A=\left({\lambda}_{1}+{\lambda}_{2}\right)A,{A}^{{\lambda}_{1}}\otimes {A}^{{\lambda}_{2}}={A}^{{\lambda}_{1}+{\lambda}_{2}},{\lambda}_{1}>0,{\lambda}_{2}>0.$

**Proof.**

#### 2.2. Bonferroni Mean

- $B{M}^{p,q}(0,0,\cdots ,0)=0$, i.e., aggregation of the null values renders the null value too;
- (Idempotency) $B{M}^{p,q}(a,a,\cdots ,a)=a$, i.e., aggregating a constant returns the same constant as an outcome;
- (Monotonicity) $B{M}^{p,q}({a}_{1},{a}_{2},\cdots ,{a}_{n})\ge B{M}^{p,q}({b}_{1},{b}_{2},\cdots ,{b}_{n}),$ i.e., $B{M}^{p,q}$ is monotonic in its arguments for ${a}_{i}\ge {b}_{i}$, $i=1,2,\cdots ,n$;
- (Boundedness) $\underset{i}{\mathrm{min}}\{{a}_{i}\}\le B{M}^{p,q}({a}_{1},{a}_{2},\cdots ,{a}_{n})\le \underset{i}{\mathrm{max}}\{{a}_{i}\}$, i.e., the result of aggregation is bounded from below and above by the extreme values of the arguments.

- If one sets $p=2,q=0$, then the interactions are ignored and higher values of the arguments are additionally rewarded and Equation (6) becomes the square mean:$$B{M}^{2,0}({a}_{1},{a}_{2},\cdots ,{a}_{n})={\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{a}_{i}^{2}}\right)}^{\frac{1}{2}}.$$
- If one assumes $p=1,q=0$, then interactions remain ignored and arguments do not benefit from showing higher values, with Equation (6) becoming the arithmetic average:$$B{M}^{1,0}({a}_{1},{a}_{2},\cdots ,{a}_{n})=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{a}_{i}}.$$
- If one picks the boundary condition $p\to \infty ,q=0$, then the interactions remain ignored, with the greatest importance put on the largest argument, i.e., Equation (6) boils down to the maximum operator:$$\underset{p\to \infty}{\mathrm{lim}}B{M}^{p,0}({a}_{1},{a}_{2},\cdots ,{a}_{n})=\underset{i}{\mathrm{max}}\{{a}_{i}\}.$$
- If the boundary condition is set with $p\to 0,q=0$, then the interactions among the arguments are ignored and the lowest values become the most important ones, with Equation (6) being reduced to the geometric mean operator:$$\underset{p\to 0}{\mathrm{lim}}B{M}^{p,0}({a}_{1},{a}_{2},\cdots ,{a}_{n})={\left({\displaystyle \prod _{i=1}^{n}{a}_{i}}\right)}^{\frac{1}{n}}.$$

#### 2.3. Normalized Weighted Bonferroni Harmonic Mean

#### 2.4. A Ranking Approach for TIFNs

^{L}, a

^{U}] and b = [b

^{L}, b

^{U}] be the two interval numbers, where the endpoints are represented by the ordered values so that ${a}^{L}\le {a}^{U}$ and ${b}^{L}\le {b}^{U}$. Note that if ${a}^{L}={a}^{U}$, then the interval number degenerates to a real number ${a}^{\prime}$.

^{L}, a

^{U}] and b = [b

^{L}, b

^{U}]. For these two numbers, the probability of dominance of $a$ over $b$, i.e., $a\ge b$, can be calculated as follows:

- (1)
- $0\le p(a\ge b)\le 1$.
- (2)
- $p(a\ge b)+p(a\le b)=1$.
- (3)
- $p(a\ge b)=p(a\le b)=0.5$, if $p(a\ge b)=p(a\le b)$.
- (4)
- $p(a\ge b)=0,$ if ${a}^{U}\le {b}^{L}$.
- (5)
- Assuming there exist interval numbers a, b, and c, $p(a\ge c)\ge p(b\ge c)$ if $a\ge b$.

**Step 1.**For each TIFN, compute the $(\alpha ,\beta )$-cut by using Equations (3) and (4), where parameters $\alpha $ and $\beta $ are chosen with respect to the extreme values of the membership and non-membership functions for a given set of TIFNs so that $0\le \alpha \le {\wedge}_{i=1}^{m}{\omega}_{{A}_{i}},$ ${\vee}_{i=1}^{m}{u}_{{A}_{i}}\le \beta \le 1$ and $0\le \alpha +\beta \le 1$. The resulting interval numbers representing the TIFNs are given by:

**Step 2.**Calculate the composite interval capturing both the membership and non-membership functions for a certain TIFN:

**Step 3.**Establish the preference relations matrix representing pairwise comparisons among all the alternatives:

**Step 4.**Aggregate results of the pairwise comparisons for each alternative by calculating the ranking indicator RI(A

_{i}) as follows [34]:

**Step 5.**The TIFNs are ranked with respect to the associated values of the ranking indicator RI(A

_{i}), i = 1,2, ..., m, so that higher values of the indicator imply higher ranking of the alternatives.

#### 2.5. Normalized Weighted Triangular Intuitionistic Fuzzy Bonferroni Harmonic Mean

**Idempotency.**If there exists a collection of TIFNs A

_{i}, i = 1,2,...,n, where all the elements are equal to a certain value, i.e., ${A}_{i}=A=([a,b,c],{\omega}_{A},{u}_{A})$, then the application of the NWTIFBHM operator results in that value:

**Commutativity.**Let there be a set of positive TIFNs, ${A}_{i}={([{a}_{i},{b}_{i},{c}_{i}],{\omega}_{{A}_{i}},{u}_{{A}_{i}})}_{}$, i = 1,2,...,n, and let there be a permutation of $({A}_{1},{A}_{2},\cdots ,{A}_{n})$ denoted by $({\tilde{A}}_{1},{\tilde{A}}_{2},\cdots ,{\tilde{A}}_{n})$. Then, the following relationship holds:

**Monotonicity.**Let there be the two sets of TIFNs, ${A}_{i}={([{a}_{i},{b}_{i},{c}_{i}],{\omega}_{{A}_{i}},{u}_{{A}_{i}})}_{}$ and ${\overline{A}}_{i}=([{\overline{a}}_{i},{\overline{b}}_{i},{\overline{c}}_{i}],$ ${\omega}_{{\overline{A}}_{i}},{u}_{{\overline{A}}_{i}}),$ with i = 1,2,...,n. If ${a}_{i}\ge {\overline{a}}_{i},{b}_{i}\ge {\overline{b}}_{i},$ ${c}_{i}\ge {\overline{c}}_{i},{\omega}_{{A}_{i}}\ge {\omega}_{{\overline{A}}_{i}}$ and ${u}_{{A}_{i}}\ge {u}_{{\overline{A}}_{i}}$ for all $i$. Then, the results of aggregation are also related in the same manner. Formally,

**Boundedness.**Let there be a collection of TIFNs denoted by ${A}_{i}=([{a}_{i},{b}_{i},{c}_{i}],{\omega}_{{A}_{i}},{u}_{{A}_{i}}),i=1,2,\cdots ,n$. Furthermore, let there be negative and positive ideal solutions associated with the set defined by ${A}^{-}=([{\wedge}_{i}{a}_{i},{\wedge}_{i}{b}_{i},{\wedge}_{i}{c}_{i}],{\wedge}_{i}{\omega}_{{A}_{i}},{\vee}_{i}{u}_{{A}_{i}})$ and ${A}^{+}=([{\vee}_{i}{a}_{i},{\vee}_{i}{b}_{i},{\vee}_{i}{c}_{i}],{\vee}_{i}{\omega}_{{A}_{i}},{\wedge}_{i}{u}_{{A}_{i}}),$ respectively. Then, the result of aggregation by the NWTIFBHM is bounded by those two ideal solutions as follows:

## 3. MAGDM Based on the Triangular Intuitionistic Fuzzy Information and the NWTIFBHM Operator

#### 3.1. MAGDM Framework

**Step 1.**Establish the individual decision matrices ${A}_{t}$. The weights of criteria are arranged into vector $w$. Note that the weights can be established based on objective methods (e.g., entropy) or subjective ones (e.g., pair-wise comparisons).

**Step 2.**Aggregate the ratings provided by the decision makers for each alternative and criterion. The NWTIFBHM operator given by Equation (16) can be applied (assuming $p=q=1$) for the aggregation. The resulting elements of the aggregate matrix are thus defined as:

**Step 3**Calculate the final fuzzy utility scores for each alternative considering all the criteria and experts respectively by exploiting Equation (16).

_{i}, i = 1,2, ..., n.

**Step 4.**Rank the alternatives based on the values of the ranking indicator RI(A

_{t}) by assigning the highest ranks to the alternatives featuring the highest values of RI(A

_{t}).

#### 3.2. Application for the Case of Search and Rescue Robot Selection

_{i}(i = 1,2,3,4) to be evaluated. Furthermore, the evaluation relies on expert opinions (i.e., one needs to solve an MAGDM problem). The experts provide their ratings for each alternative against the four criteria. The resulting individual decision matrices are outlined in Table 1, Table 2, Table 3 and Table 4. The group of experts is assumed not to be a completely homogenous one. Accordingly, the experts are assigned with different weights arranged into vector η = (0.20,0.30,0.35,0.15)

^{T}, where each element is associated with a corresponding expert D

_{t}(t = 1,2,3,4).

_{t}are constructed and the decision making proceeds as follows:

**Step 1.**Provide decision matrices ${A}_{t}$, $t=1,2,3,4$, and the weight vector of criteria $w={(0.22,0.20,0.28,0.30)}^{T}$.

**Step 2.**The overall utilities are obtained for the alternatives under consideration. Decision makers’ rankings of all the alternatives are calculated and the weight vector $\eta ={(0.20,0.30,0.35,0.15)}^{T}$ of decision makers and the aggregated value are given as follows:

**Step 3.**The overall utility scores are expressed in the TIFNs. Therefore, we further utilize the probabilistic ranking approach outlined in Section 2.4 The ranking indicators are obtained by assuming $\alpha =\beta =\lambda =0.5$. The following values of the ranking indicator are obtained for each alternative X

_{i}:

**Step 4.**Given the values of the ranking indicator, the following ranking is obtained: $RI({A}_{4})\text{}\text{}RI({A}_{3})\text{}\text{}RI({A}_{2})\text{}\text{}RI({A}_{1}).$ X

_{4}is identified as the most preferable (in the sense of the underlying fuzzy utility) search and rescue robot, as evidenced by the associated ranking indicator RI(A

_{4}) showing the largest value among the alternatives.

#### 3.3. Comparative Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof.**

- (1)
- when $n=2$, given (15), we can show:$$\begin{array}{l}{\oplus}_{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{2}((\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}})\otimes (\frac{{w}_{j}}{{A}_{j}^{q}}))=((\frac{{w}_{1}}{(1-{w}_{1}){A}_{1}^{p}})\otimes (\frac{{w}_{2}}{{A}_{2}^{q}}))\oplus ((\frac{{w}_{2}}{(1-{w}_{2}){A}_{2}^{p}})\otimes (\frac{{w}_{1}}{{A}_{1}^{q}}))\\ =([\frac{{w}_{1}{w}_{2}}{(1-{w}_{1}){a}_{1}^{p}{a}_{2}^{q}}+\frac{{w}_{2}{w}_{1}}{(1-{w}_{2}){a}_{2}^{p}{a}_{1}^{q}},\frac{{w}_{1}{w}_{2}}{(1-{w}_{1}){b}_{1}^{p}{b}_{2}^{q}}+\frac{{w}_{2}{w}_{1}}{(1-{w}_{2}){b}_{2}^{p}{b}_{1}^{q}},\frac{{w}_{1}{w}_{2}}{(1-{w}_{1}){c}_{1}^{p}{c}_{2}^{q}}+\frac{{w}_{2}{w}_{1}}{(1-{w}_{2}){c}_{2}^{p}{c}_{1}^{q}}];\\ {\omega}_{{A}_{1}}\wedge {\omega}_{{A}_{2}},{u}_{{A}_{1}}\vee {u}_{{A}_{2}})\\ =([{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{2}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){a}_{i}^{p}{a}_{j}^{q}}},{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{2}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){b}_{i}^{p}{b}_{j}^{q}}},{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{2}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){c}_{i}^{p}{c}_{j}^{q}}}];{\wedge}_{i=1}^{2}{\omega}_{{A}_{i}},{\vee}_{i=1}^{2}{u}_{{A}_{i}})\end{array}$$
- (2)
- assume that $n=k$ and Equation (15) holds so that$$\begin{array}{l}{\oplus}_{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{k}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{j}}{{A}_{j}^{q}}\right)\right)\\ =([{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{k}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){a}_{i}^{p}{a}_{j}^{q}}},{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{k}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){b}_{i}^{p}{b}_{j}^{q}}},{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{k}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){c}_{i}^{p}{c}_{j}^{q}}}];{\wedge}_{i=1}^{k}{\omega}_{{A}_{i}},{\vee}_{i=1}^{k}{u}_{{A}_{i}})\end{array}$$
- (3)
- subsequently, assume $n=k+1$ and by the virtue of (15), get$$\begin{array}{l}{\oplus}_{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{k+1}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{j}}{{A}_{j}^{q}}\right)\right)=\left({\oplus}_{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{k}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{j}}{{A}_{j}^{q}}\right)\right)\right)\oplus \\ \left({\oplus}_{i=1}^{k}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)\right)\oplus \left({\oplus}_{j=1}^{k}\left(\left(\frac{{w}_{k+1}}{(1-{w}_{k+1}){A}_{k+1}^{p}}\right)\otimes \left(\frac{{w}_{j}}{{A}_{j}^{q}}\right)\right)\right)\end{array}$$

- (a)
- Let $k=2$, and by the virtue of Equation (A4), one can show$$\begin{array}{l}{\oplus}_{i=1}^{2}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)=\left(\left(\frac{{w}_{1}}{(1-{w}_{1}){A}_{1}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)\oplus \left(\left(\frac{{w}_{2}}{(1-{w}_{2}){A}_{2}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)\\ =([\frac{{w}_{1}{w}_{k+1}}{(1-{w}_{1}){a}_{1}^{p}{a}_{k+1}^{q}}+\frac{{w}_{2}{w}_{k+1}}{(1-{w}_{2}){a}_{2}^{p}{a}_{k+1}^{q}},\frac{{w}_{1}{w}_{k+1}}{(1-{w}_{1}){b}_{1}^{p}{b}_{k+1}^{q}}+\frac{{w}_{2}{w}_{k+1}}{(1-{w}_{2}){b}_{2}^{p}{b}_{k+1}^{q}},\\ \frac{{w}_{1}{w}_{k+1}}{(1-{w}_{1}){c}_{1}^{p}{c}_{k+1}^{q}}+\frac{{w}_{2}{w}_{k+1}}{(1-{w}_{2}){c}_{2}^{p}{c}_{k+1}^{q}}];({\omega}_{{A}_{1}}\wedge {\omega}_{{A}_{k+1}})\wedge ({\omega}_{{A}_{2}}\wedge {\omega}_{{A}_{k+1}}),({u}_{{A}_{1}}\vee {u}_{{A}_{k+1}})\vee ({u}_{{A}_{1}}\vee {u}_{{A}_{k+1}}))\end{array}$$$$=([{\displaystyle \sum _{i=1}^{2}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){a}_{i}^{p}{a}_{j}^{q}}},{\displaystyle \sum _{i=1}^{2}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){b}_{i}^{p}{b}_{j}^{q}}},{\displaystyle \sum _{i=1}^{2}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){c}_{i}^{p}{c}_{j}^{q}}}];{\wedge}_{i=1}^{2}({\omega}_{{A}_{i}}\wedge {\omega}_{{A}_{k+1}}),{\vee}_{i=1}^{2}({u}_{{A}_{i}}\vee {u}_{{A}_{k+1}}))$$
- (b)
- Assume Equation (A4) is valid for any given $k={k}_{0}$$${\oplus}_{i=1}^{{k}_{0}}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)=\phantom{\rule{0ex}{0ex}}([{\displaystyle \sum _{i=1}^{{k}_{0}}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){a}_{i}^{p}{a}_{j}^{q}}},{\displaystyle \sum _{i=1}^{{k}_{0}}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){b}_{i}^{p}{b}_{j}^{q}}},{\displaystyle \sum _{i=1}^{{k}_{0}}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){c}_{i}^{p}{c}_{j}^{q}}}];{\wedge}_{i=1}^{{k}_{0}}({\omega}_{{A}_{i}}\wedge {\omega}_{{A}_{k+1}}),{\vee}_{i=1}^{{k}_{0}}({u}_{{A}_{i}}\vee {u}_{{A}_{k+1}}))$$
- (c)
- Subsequently, we demonstrate that the following holds for any $k={k}_{0}+1$:$$\begin{array}{l}{\oplus}_{i=1}^{{k}_{0}+1}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)={\oplus}_{i=1}^{{k}_{0}}\left(\left(\frac{{w}_{i}}{(1-{w}_{i}){A}_{i}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)\oplus \left(\left(\frac{{w}_{{k}_{0}+1}}{(1-{w}_{{k}_{0}+1}){A}_{{k}_{0}+1}^{p}}\right)\otimes \left(\frac{{w}_{k+1}}{{A}_{k+1}^{q}}\right)\right)\\ =([{\displaystyle \sum _{i=1}^{{k}_{0}}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){a}_{i}^{p}{a}_{j}^{q}}}+\frac{{w}_{{k}_{0}+1}{w}_{k+1}}{(1-{w}_{{k}_{0}+1}){a}_{{k}_{0}+1}^{p}{a}_{k+1}^{q}},{\displaystyle \sum _{i=1}^{{k}_{0}}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){b}_{i}^{p}{b}_{j}^{q}}}+\frac{{w}_{{k}_{0}+1}{w}_{k+1}}{(1-{w}_{{k}_{0}+1}){b}_{{k}_{0}+1}^{p}{b}_{k+1}^{q}},\\ \hspace{1em}\hspace{1em}{\displaystyle \sum _{i=1}^{{k}_{0}}\frac{{w}_{i}{w}_{j}}{(1-{w}_{i}){c}_{i}^{p}{c}_{j}^{q}}}+\frac{{w}_{{k}_{0}+1}{w}_{k+1}}{(1-{w}_{{k}_{0}+1}){c}_{{k}_{0}+1}^{p}{c}_{k+1}^{q}}];\\ {\wedge}_{i=1}^{{k}_{0}}({\omega}_{{A}_{i}}\wedge {\omega}_{{A}_{k+1}})\wedge ({\omega}_{{A}_{{k}_{0}+1}}\wedge {\omega}_{{A}_{k+1}}),{\vee}_{i=1}^{{k}_{0}}({u}_{{A}_{i}}\vee {u}_{{A}_{k+1}})\vee ({u}_{{A}_{{k}_{0}+1}}\vee {u}_{{A}_{k+1}}))\end{array}$$

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Alternative | C_{1} | C_{2} | C_{3} | C_{4} |
---|---|---|---|---|

X_{1} | ([0.05,0.1,0.15];0.7,0.2) | ([0.1,0.15,0.2];0.5,0.4) | ([0.1,0.2,0.25];0.6,0.4) | ([0.75,0.8,0.9];0.8,0.1) |

X_{2} | ([0.2,0.25,0.3];0.6,0.3) | ([0.8,0.85,0.95];0.8,0.2) | ([0.15,0.2,0.25];0.7,0.2) | ([0.2,0.25,0.3];0.6,0.3) |

X_{3} | ([0.1,0.2,0.3];0.5,0.4) | ([0.1,0.2,0.3];0.7,0.2) | ([0.85,0.9,0.95];0.6,0.3) | ([0.15,0.2,0.3];0.7,0.1) |

X_{4} | ([0.85,0.9,0.95];0.5,0.3) | ([0.2,0.3,0.35];0.6,0.3) | ([0.15,0.3,0.4];0.5,0.2) | ([0.1,0.25,0.35];0.8,0.1) |

Alternative | C_{1} | C_{2} | C_{3} | C_{4} |
---|---|---|---|---|

X_{1} | ([0.05,0.15,0.25];0.6,0.4) | ([0.1,0.15,0.2];0.6,0.3) | ([0.1,0.15,0.2];0.6,0.4) | ([0.85,0.9,0.95];0.6,0.3) |

X_{2} | ([0.15,0.25,0.3];0.6,0.3) | ([0.75,0.85,0.95];0.7,0.2) | ([0.15,0.2,0.25];0.7,0.2) | ([0.2,0.25,0.3];0.6,0.4) |

X_{3} | ([0.75,0.8,0.85];0.9,0.1) | ([0.1,0.2,0.25];0.5,0.3) | ([0.1,0.25,0.3];0.7,0.2) | ([0.15,0.25,0.3];0.8,0.1) |

X_{4} | ([0.1,0.3,0.4];0.6,0.2) | ([0.2,0.25,0.3];0.8,0.1) | ([0.8,0.85,0.95];0.7,0.3) | ([0.1,0.25,0.35];0.5,0.4) |

Alternative | C_{1} | C_{2} | C_{3} | C_{4} |
---|---|---|---|---|

X_{1} | ([0.8,0.85,0.9];0.9,0.1) | ([0.2,0.25,0.3];0.5,0.4) | ([0.1,0.2,0.25];0.6,0.4) | ([0.15,0.2,0.3];0.8,0.1) |

X_{2} | ([0.15,0.25,0.3];0.6,0.2) | ([0.1,0.15,0.2];0.6,0.2) | ([0.15,0.2,0.25];0.7,0.2) | ([0.8,0.85,0.95];0.8,0.2) |

X_{3} | ([0.2,0.25,0.3];0.5,0.4) | ([0.05,0.1,0.15];0.7,0.2) | ([0.85,0.9,0.95];0.6,0.25) | ([0.15,0.2,0.25];0.7,0.1) |

X_{4} | ([0.1,0.2,0.25];0.7,0.2) | ([0.75,0.8,0.9];0.6,0.2) | ([0.2,0.25,0.3];0.5,0.4) | ([0.1,0.25,0.3];0.6,0.3) |

Alternative | C_{1} | C_{2} | C_{3} | C_{4} |
---|---|---|---|---|

X_{1} | ([0.15,0.2,0.3];0.5,0.5) | ([0.25,0.3,0.35];0.4,0.4) | ([0.75,0.85,0.9];0.5,0.4) | ([0.2,0.35,0.4];0.7,0.2) |

X_{2} | ([0.85,0.9,0.95];0.8,0.1) | ([0.05,0.1,0.15];0.6,0.3) | ([0.2,0.25,0.3];0.7,0.2) | ([0.1,0.15,0.2];0.9,0.1) |

X_{3} | ([0.2,0.25,0.3];0.5,0.4) | ([0.8,0.85,0.9];0.8,0.1) | ([0.05,0.1,0.15];0.7,0.2) | ([0.25,0.3,0.35];0.5,0.4) |

X_{4} | ([0.1,0.2,0.3];0.7,0.2) | ([0.15,0.25,0.35];0.5,0.3) | ([0.25,0.3,0.35];0.6,0.3) | ([0.8,0.9,0.95];0.6,0.2) |

Alternative | D_{1} | D_{2} | D_{3} | D_{4} |
---|---|---|---|---|

X_{1} | ([0.1196,0.2204,0.2640]; 0.5,0.4) | ([0.1196,0.2304,0.2827]; 0.6,0.4) | ([0.3140,0.4742,0.5667]; 0.5,0.4) | ([0.4376,0.5420,0.6837]; 0.5,0.4) |

X_{2} | ([0.3673,0.4620,0.5584]; 0.6,0.3) | ([0.3225,0.4620,0.5584]; 0.6,0.4) | ([0.2017,0.2990,0.3778]; 0.6,0.2) | ([0.2333,0.3562,0.4641]; 0.6,0.3) |

X_{3} | ([0.2598,0.4703,0.6546]; 0.5,0.4) | ([0.2328,0.4727,0.5643]; 0.5,0.3) | ([0.2533,0.3826,0.4945]; 0.5,0.4) | ([0.2190,0.3363,0.4401]; 0.5,0.4) |

X_{4} | ([0.3815,0.6127,0.7405]; 0.5,0.3) | ([0.3420,0.5948,0.7293]; 0.5,0.4) | ([0.3058,0.4600,0.5559]; 0.5,0.4) | ([0.2360,0.3796,0.5107]; 0.5,0.3) |

Method | Ranking Order | Best Alternative |
---|---|---|

TIFWPA | ${X}_{4}\succ {X}_{2}\succ {X}_{1}\text{}\succ {X}_{3}$ | ${X}_{4}$ |

TIFWPG | ${X}_{4}\succ {X}_{2}\succ {X}_{1}\text{}\succ {X}_{3}$ | ${X}_{4}$ |

TIFWGM | ${X}_{1}\succ {X}_{4}\succ {X}_{2}\text{}\succ \text{}{X}_{3}$ | ${X}_{1}$ |

TIFWAM | ${X}_{3}\succ {X}_{4}\succ {X}_{1}\text{}\succ \text{}{X}_{2}$ | ${X}_{3}$ |

TIFWPHM | ${X}_{4}\succ {X}_{2}\succ {X}_{1}\text{}\succ {X}_{3}$ | ${X}_{4}$ |

NWTIFBHM | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ | ${X}_{4}$ |

$\mathit{\lambda}$ | Ranking Index | Ranking Order |
---|---|---|

$\lambda =0.1$ | $\begin{array}{l}RI({A}_{1})=0.1178,RI({A}_{2})=0.2004,\\ RI({A}_{3})=0.2588,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\lambda =0.4$ | $\begin{array}{l}RI({A}_{1})=0.1154,RI({A}_{2})=0.1945,\\ RI({A}_{3})=0.2671,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\lambda =0.6$ | $\begin{array}{l}RI({A}_{1})=0.1154,RI({A}_{2})=0.1923,\\ RI({A}_{3})=0.2692,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\lambda =0.9$ | $\begin{array}{l}RI({A}_{1})=0.1154,RI({A}_{2})=0.1923,\\ RI({A}_{3})=0.2692,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\mathit{\alpha}$ | Ranking Index | Ranking Order |
---|---|---|

$\alpha =0.1$ | $\begin{array}{l}RI({A}_{1})=0.1608,RI({A}_{2})=0.2020,\\ RI({A}_{3})=0.2317,RI({A}_{4})=0.3285\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\alpha =0.2$ | $\begin{array}{l}RI({A}_{1})=0.1535,RI({A}_{2})=0.1984,\\ RI({A}_{3})=0.2325,RI({A}_{4})=0.3387\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\alpha =0.3$ | $\begin{array}{l}RI({A}_{1})=0.1412,RI({A}_{2})=0.1983,\\ RI({A}_{3})=0.2374,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\alpha =0.4$ | $\begin{array}{l}RI({A}_{1})=0.1276,RI({A}_{2})=0.1978,\\ RI({A}_{3})=0.2516,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\mathit{\beta}$ | Ranking Index | Ranking Order |
---|---|---|

$\beta =0.6$ | $\begin{array}{l}RI({A}_{1})=0.1269,RI({A}_{2})=0.1971,\\ RI({A}_{3})=0.2530,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\beta =0.7$ | $\begin{array}{l}RI({A}_{1})=0.1363,RI({A}_{2})=0.1974,\\ RI({A}_{3})=0.2432,RI({A}_{4})=0.3462\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\beta =0.8$ | $\begin{array}{l}RI({A}_{1})=0.1494,RI({A}_{2})=0.1975,\\ RI({A}_{3})=0.2326,RI({A}_{4})=0.3435\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

$\beta =0.9$ | $\begin{array}{l}RI({A}_{1})=0.1574,RI({A}_{2})=0.1976,\\ RI({A}_{3})=0.2320,RI({A}_{4})=0.3361\end{array}$ | ${X}_{4}\succ {X}_{3}\succ {X}_{2}\text{}\succ \text{}{X}_{1}$ |

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## Share and Cite

**MDPI and ACS Style**

Zhou, J.; Baležentis, T.; Streimikiene, D.
Normalized Weighted Bonferroni Harmonic Mean-Based Intuitionistic Fuzzy Operators and Their Application to the Sustainable Selection of Search and Rescue Robots. *Symmetry* **2019**, *11*, 218.
https://doi.org/10.3390/sym11020218

**AMA Style**

Zhou J, Baležentis T, Streimikiene D.
Normalized Weighted Bonferroni Harmonic Mean-Based Intuitionistic Fuzzy Operators and Their Application to the Sustainable Selection of Search and Rescue Robots. *Symmetry*. 2019; 11(2):218.
https://doi.org/10.3390/sym11020218

**Chicago/Turabian Style**

Zhou, Jinming, Tomas Baležentis, and Dalia Streimikiene.
2019. "Normalized Weighted Bonferroni Harmonic Mean-Based Intuitionistic Fuzzy Operators and Their Application to the Sustainable Selection of Search and Rescue Robots" *Symmetry* 11, no. 2: 218.
https://doi.org/10.3390/sym11020218