Performance Evaluation of Human Resources Based on Linguistic Neutrosophic Maclaurin Symmetric Mean Operators

Abstract

For corporates, the performance evaluation of human resources is always a significant strategic activity based on cognitive information. This study aims to develop a novel decision-making method with the computation of cognitive information to make assessments of human resources. First, the cognitive information is described by means of linguistic neutrosophic numbers (LNNs) to capture aspects of indeterminacy and fuzziness. Then, as the Maclaurin symmetric mean (MSM) operators can reflect the interrelations among multiple inputs, several extended MSM operators are proposed to aggregate cognitive information in the linguistic neutrosophic environments. Meanwhile, some important properties of these operators are justified. Thereafter, a linguistic neutrosophic decision-making method based on MSM operators is introduced to address qualitative evaluation problems during cognitive processes. Finally, the validity of our method is revealed by presenting a case study of selecting the best employee in a company. Moreover, the advantages of the proposed method are highlighted by the discussion of the effect of the parameter existing in aggregation operators and the comparison with other methods. The results show that the proposed method is feasible and the study can provide guidelines for the performance evaluation and management of human resources. The utilization of LNNs enriches the expression of cognitive information. Furthermore, the proposed method can be regarded as a potential choice for disposing of cognitive computation.

Appendices

Appendix A

Proof 3

(1) According to Theorem 1

, $LNMS{M}^{(s)}({A}_{1},{A}_{2},...,{A}_{m})=LNMS{M}^{(s)}(A,A,...,A)$

$$=({l}_{2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{(\frac{T}{2v})}^{s})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}},{l}_{2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{(1-\frac{I}{2v})}^{s})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}},{l}_{2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{(1-\frac{F}{2v})}^{s}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}})$$

$$\begin{aligned}&=\left({l}_{2v{\left(1-\left(1-{\left(\frac{T}{2v}\right)}^{s}\right)\right)}^\frac{1}{s}},{l}_{2v-2v{\left(1-\left(1-{\left(1-\frac{I}{2v}\right)}^{s}\right)\right)}^\frac{1}{s}},{l}_{2v-2v{\left(1-\left(1-{\left(1-\frac{F}{2v}\right)}^{s}\right)\right)}^\frac{1}{s}}\right)\\&=\left({l}_{T},{l}_{I},{l}_{F}\right)=A\end{aligned}$$

(2) As $0\le {l}_{T{a}_{i}}\le {l}_{T{b}_{i}}$ $\Rightarrow$ $T{a}_{i}\le T{b}_{i}$ $\Rightarrow$ $\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}}\le \frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{(2v)}^{s}}$ $\Rightarrow 1-\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}}\ge 1-\frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{(2v)}^{s}}$

$$\Rightarrow {\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}})\ge {\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{(2v)}^{s}})$$

$$\begin{aligned}&\Rightarrow {\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\\&\ge {\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\end{aligned}$$

$$\begin{aligned}&\Rightarrow {\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\\&\le {\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\end{aligned}$$

$$\begin{aligned}&\Rightarrow 2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\\&\le 2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-\frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{\left(2v\right)}^{s}}\right)\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s},\end{aligned}$$

Then ${l}_{2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{a}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}\le {l}_{2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-\frac{{\prod }_{j=1}^{s}T{b}_{{i}_{j}}}{{(2v)}^{s}})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}$.

Additionally, ${l}_{I{a}_{i}}\ge {l}_{I{b}_{i}}\ge 0$ $\Rightarrow$ $I{a}_{i}\ge I{b}_{i}$ $\Rightarrow$ $\frac{I{a}_{{i}_{j}}}{2v}\ge \frac{I{b}_{{i}_{j}}}{2v}$ $\Rightarrow$ ${\prod }_{j=1}^{s}(1-\frac{I{a}_{{i}_{j}}}{2v})\le {\prod }_{j=1}^{s}(1-\frac{I{b}_{{i}_{j}}}{2v})$

$$\Rightarrow 1-{\prod }_{j=1}^{s}(1-\frac{I{a}_{{i}_{j}}}{2v})\ge 1-{\prod }_{j=1}^{s}(1-\frac{I{b}_{{i}_{j}}}{2v})$$

$$\begin{aligned}&\Rightarrow {\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{I{a}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\\&\ge {\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{I{b}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\end{aligned}$$

$$\begin{aligned}&\Rightarrow {\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{I{a}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\\&\le {\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{I{b}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\end{aligned}$$

$$\begin{aligned}&\Rightarrow 2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{I{a}_{{i}_{j}}}{2v}\right))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\\&\ge 2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{I{b}_{{i}_{j}}}{2v}\right))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s},\end{aligned}$$

then ${l}_{2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{I{a}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}\ge {l}_{2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{I{b}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}$.

Similarly,${l}_{2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{F{a}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}\ge {l}_{2v-2v{\left(1-{\left({\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{F{b}_{{i}_{j}}}{2v}))\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}$.

Thus,$LNMS{M}^{(s)}({A}_{1},{A}_{2},...,{A}_{m})\le LNMS{M}^{(s)}({B}_{1},{B}_{2},...,{B}_{m})$.

Appendix B

Proof 4

(1) Based on Definition 3, ${\sum }_{j=1}^{s}{A}_{{i}_{j}}=({l}_{2v-2v{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})},{l}_{2v{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}},{l}_{2v{\prod }_{j=1}^{s}\frac{{F}_{{i}_{j}}}{2v}})$

$$\Rightarrow {\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}{\left({\sum }_{j=1}^{s}{A}_{{i}_{j}}\right)}^{\frac{1}{{C}_{m}^{s}}}$$

$$\begin{aligned}=({l}&_{2v{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}{\left(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}},{l}\\&_{2v-2v{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}{\left(1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}},{l}\\&_{2v-2v{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}{\left(1-{\prod }_{j=1}^{s}\frac{{F}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}})\end{aligned}$$

$${\Rightarrow} \frac{1}{s}({{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}({\sum }_{j=1}^{s}{A}_{{i}_{j}})}^{\frac{1}{{C}_{m}^{s}}})$$

$$\begin{aligned}=&\left({l}_{2v-2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}},{l}\right.\\&\left._{2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}},{l}\right.\\&\left._{2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{F}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}\right),\end{aligned}$$

then $LNMS{M}^{(s)}({A}_{1},{A}_{2},...,{A}_{m})$

$$\begin{aligned}=&\left({l}_{2v-2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}},{l}\right.\\&\left._{2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}},{l}\right.\\&\left._{2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{F}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}}\right).\end{aligned}$$

(2) As $0\le {T}_{{i}_{j}}\le 2v\Rightarrow 0\le 1-\frac{{T}_{{i}_{j}}}{2v}\le 1\Rightarrow 0\le 1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\le 1$

$$\begin{aligned}&\Rightarrow 0\le 1-{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v}))\\&\le 1\Rightarrow 0\le {{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}\le 1\end{aligned}$$

$$\Rightarrow 0\le {\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\le 1,$$

then $0\le 2v-2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}(1-\frac{{T}_{{i}_{j}}}{2v})\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\le 2v$; and $0\le {I}_{{i}_{j}}\le 2v$ $\Rightarrow$ $0\le \frac{{I}_{{i}_{j}}}{2v}\le 1$ $\Rightarrow$ $0\le 1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}\le 1$ $\Rightarrow 0\le {{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\le 1$ $\Rightarrow$ $0\le {\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\le 1$, then $0\le 2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{I}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\le 2v$.

Similarly, $0\le 2v{\left(1-{{\prod }_{1\le {i}_{1}<\cdots <{i}_{s}\le m}\left(1-{\prod }_{j=1}^{s}\frac{{F}_{{i}_{j}}}{2v}\right)}^{\frac{1}{{C}_{m}^{s}}}\right)}^\frac{1}{s}\le 2v$. Thus, $LNDMS{M}^{(s)}({A}_{1},{A}_{2},...,{A}_{m})$ is still a LNN.

Combined (1) and (2), Proof 3 now is completed.