Group Decision-Making with Linguistic Cognition from a Reliability Perspective


To deal reliably with the cognitive uncertainty experienced by decision-makers when facing problems involving linguistic group decision-making, we investigate a new research perspective: cognitive familiarity is regarded as a measure of cognitive reliability. The linguistic variables examined in this work are quantified with the use of several granulation optimization models that include consideration of cognitive reliability. Three types of linguistic variables are used for describing the alternative grades, attribute weights, and levels of cognitive familiarity associated with the experts involved. Three interrelated optimization models are built to quantify these linguistic variables. An information entropy model first determines cognitive familiarity, which is applied as a measure of cognitive reliability. Using group consistency and cognitive reliability, two proposed optimization models then successively establish the attribute weights and alternative grades. The final element is a proposed new selection method based on the aggregated values for the alternative grades and cognitive reliability. An illustrative example clarifies the steps in the proposed method, which produces a ranking of three alternatives as a final decision. The validity and advantages of the proposed method are verified through a comparison with existing approaches. The proposed method can be employed for effectively resolving decision-making uncertainty through the improvement of the group consistency and cognitive reliability of the experts. Sensitivity analysis also reveals that cognitive reliability has a strong impact on decision-making and should thus be considered during fusion processes.

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This work was funded by the National Natural Science Foundation of China (nos. 71171112 and 71502073), the Key Project of National Social Science Foundation of China (no. 14AZD049), the Scientific Innovation Research of College Graduates Jiangsu Province (no. KYZZ150094), the Foundation of the Ministry of Education (no. 14YJC630120), and the Anhui Provincial Natural Science Foundation (no. 1708085MG168).

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Appendix A: Calculation of F pp(x) and F qp(x)

Five situations are considered for the computation of Fpp(x) and Fqp(x):

  1. 1.

    If \(0=\hat {g}_{p}^{G}<\hat {g}_{q}^{G}<M/2\), then Fpp(x) = 0 and

    $$\begin{array}{@{}rcl@{}} \setlength{\abovecaptionskip}{0cm} \setlength{\belowcaptionskip}{15cm} F_{qp}(x)&=&\displaystyle{{\int}_{0}^{\hat{g}_{q}^{G}}}\left[\frac{\delta_{q}\left( x-\hat{g}_{q}^{G}\right)}{M-\hat{g}_{q}^{G}} +\delta_{q}\right]dx\\&&+\displaystyle{{\int}_{\hat{g}_{q}^{G}}^{M}}\frac{\delta_{q}(M-x)}{M-\hat{g}_{q}^{G}}dx =\frac{\delta_{q}\left[M^{2}-2\left( \hat{g}_{q}^{G}\right)^{2}\right]}{2\left( M-\hat{g}_{q}^{G}\right)}. \end{array} $$
  2. 2.

    If \(0=\hat {g}_{p}^{G}<M/2\leq \hat {g}_{q}^{G}\), then Fpp(x) = 0 and

    $$\begin{array}{@{}rcl@{}} F_{qp}(x)&=&\displaystyle{{\int}_{0}^{\hat{g}_{q}^{G}}}\frac{\delta_{q}x}{\hat{g}_{q}^{G}}dx+ \displaystyle{{\int}_{\hat{g}_{q}^{G}}^{M}}\left( \frac{-\delta_{q}x}{\hat{g}_{q}^{G}}+ 2\delta_{q}\right)dx \\&=&\frac{\delta_{q}\left[4M\hat{g}_{q}^{G}-M^{2}-2\left( \hat{g}_{q}^{G}\right)^{2}\right]}{2\hat{g}_{q}^{G}}. \end{array} $$
  3. 3.

    If \(0<\hat {g}_{p}^{G}<\hat {g}_{q}^{G}<M/2\), we have

    $$\begin{array}{@{}rcl@{}} F_{pp}(x)&=&\displaystyle{{\int}_{\hat{g}_{p}^{G}}^{M}}\frac{\delta_{p}(M-x)}{M-\hat{g}_{p}^{G}}dx =\delta_{p}\left[\frac{Mx}{M-\hat{g}_{p}^{G}}|_{\hat{g}_{p}^{G}}^{M}-\frac{x^{2}}{2\left( M-\hat{g}_{p}^{G}\right)} |_{\hat{g}_{p}^{G}}^{M}\right]=\frac{\delta_{p}\left( M-\hat{g}_{p}^{G}\right)}{2},\\ F_{qp}(x)&=&\displaystyle{{\int}_{\hat{g}_{p}^{G}}^{\hat{g}_{q}^{G}}}\left[\frac{\delta_{q}\left( x-\hat{g}_{q}^{G}\right)} {M-\hat{g}_{q}^{G}}+\delta_{q}\right]dx+\displaystyle{{\int}_{\hat{g}_{q}^{G}}^{M}}\frac{\delta_{q}(M-x)} {M-\hat{g}_{q}^{G}}dx\\ &=&\delta_{q}\left[\frac{x^{2}}{2\left( M-\hat{g}_{q}^{G}\right)}+\frac{\left( M-2\hat{g}_{q}^{G}\right)x}{M-\hat{g}_{q}^{G}}\right] |_{\hat{g}_{p}^{G}}^{\hat{g}_{q}^{G}}+\delta_{q}\left[\frac{Mx}{M-\hat{g}_{q}^{G}}-\frac{x^{2}}{2\left( M- \hat{g}_{q}^{G}\right)}\right]|_{\hat{g}_{q}^{G}}^{M}\\ &=&\frac{\delta_{q}\left[M^{2}-2M\hat{g}_{p}^{G}-\left( \hat{g}_{p}^{G}\right)^{2}+ 4\hat{g}_{p}^{G}\hat{g}_{q}^{G}- 2\left( \hat{g}_{q}^{G}\right)^{2}\right]}{2\left( M-\hat{g}_{q}^{G}\right)}. \end{array} $$
  4. 4.

    If \(0<\hat {g}_{p}^{G}<M/2\leq \hat {g}_{q}^{G}\), we have

    $$\begin{array}{@{}rcl@{}} F_{pp}(x)&=&\displaystyle{{\int}_{\hat{g}_{p}^{G}}^{M}}\frac{\delta_{p}(M-x)}{M-\hat{g}_{p}^{G}}dx =\frac{\delta_{p}\left( M-\hat{g}_{p}^{G}\right)}{2},\\ F_{qp}(x)&=&\displaystyle{{\int}_{\hat{g}_{p}^{G}}^{\hat{g}_{q}^{G}}}\frac{\delta_{q}x}{\hat{g}_{q}^{G}}dx+ \displaystyle{{\int}_{\hat{g}_{q}^{G}}^{M}}\left( \frac{-\delta_{q}x}{\hat{g}_{q}^{G}}+ 2\delta_{q}\right)dx\\ &=&\frac{\delta_{q}\left[4M\hat{g}_{q}^{G}-M^{2}-2\left( \hat{g}_{q}^{G}\right)^{2}-\left( \hat{g}_{p}^{G}\right)^{2}\right]}{2\hat{g}_{q}^{G}}. \end{array} $$
  5. 5.

    If \(M/2<\hat {g}_{p}^{G}<\hat {g}_{q}^{G}\), we have

    $$\begin{array}{@{}rcl@{}} F_{pp}(x)&=&\displaystyle{{\int}_{\hat{g}_{p}^{G}}^{M}}\left( \frac{-\delta_{p}x}{\hat{g}_{p}^{G}}+ 2\delta_{p}\right)dx \\ &=&\frac{\delta_{p}\left[4M\hat{g}_{p}^{G}-M^{2}-3\left( \hat{g}_{p}^{G}\right)^{2}\right]}{2\hat{g}_{p}^{G}},\\ F_{qp}(x)&=&\displaystyle{{\int}_{\hat{g}_{p}^{G}}^{\hat{g}_{q}^{G}}}\frac{\delta_{q}x}{\hat{g}_{q}^{G}}dx+ \displaystyle{{\int}_{\hat{g}_{q}^{G}}^{M}}\left( \frac{-\delta_{q}x}{\hat{g}_{q}^{G}}+ 2\delta_{q}\right)dx \\&=&\frac{\delta_{q}\left[4M\hat{g}_{q}^{G}-M^{2}-2\left( \hat{g}_{q}^{G}\right)^{2}-\left( \hat{g}_{p}^{G}\right)^{2}\right]}{2\hat{g}_{q}^{G}}. \end{array} $$

Appendix B: Initial Linguistic Terms Given by the Experts

$$\setlength{\arraycolsep}{1pt} e^{1}=\left[ \begin{array}{cccccccc} (g_{6},c_{5},h_{6}), & (g_{7},c_{5},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{4},c_{5},h_{7}), & (g_{6},c_{4},h_{6}), & (g_{6},c_{5},h_{5}), & (g_{7},c_{5},h_{7}),\\ (g_{5},c_{2},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{6},c_{3},h_{5}), & (g_{6},c_{2},h_{7}), & (g_{5},c_{4},h_{6}), & (g_{4},c_{5},h_{6}), & (g_{7},c_{4},h_{7}), & (g_{6},c_{3},h_{6});\\ (g_{5},c_{5},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{5},c_{5},h_{7}), & (g_{6},c_{3},h_{6}), & (g_{6},c_{4},h_{5}), & (g_{7},c_{5},h_{7}),\\ (g_{6},c_{3},h_{6}), & (g_{7},c_{5},h_{6}), & (g_{5},c_{3},h_{5}), & (g_{6},c_{2},h_{7}), & (g_{4},c_{3},h_{6}), & (g_{5},c_{5},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{6},c_{3},h_{6});\\ (g_{6},c_{5},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{6},c_{4},h_{6}), & (g_{6},c_{5},h_{5}), & (g_{7},c_{4},h_{7}),\\ (g_{6},c_{3},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{5},c_{2},h_{5}), & (g_{6},c_{2},h_{7}), & (g_{5},c_{3},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{7},c_{5},h_{7}), & (g_{6},c_{4},h_{6}). \end{array} \right]; $$
$$\setlength{\arraycolsep}{2pt} e^{2}=\left[ \begin{array}{cccccccc} (g_{5},c_{5},h_{6}), & (g_{6},c_{5},h_{5}), & (g_{5},c_{5},h_{7}), & (g_{4},c_{4},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{4},c_{4},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{4},c_{5},h_{6}),\\ (g_{4},c_{4},h_{7}), & (g_{5},c_{5},h_{6}), & (g_{4},c_{3},h_{6}), & (g_{5},c_{3},h_{5}), & (g_{5},c_{4},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{5},c_{5},h_{6}), & (g_{5},c_{4},h_{7});\\ (g_{4},c_{5},h_{6}), & (g_{6},c_{4},h_{5}), & (g_{4},c_{5},h_{7}), & (g_{4},c_{4},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{5},c_{5},h_{6}), & (g_{4},c_{5},h_{6}),\\ (g_{5},c_{4},h_{7}), & (g_{5},c_{4},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{5},c_{3},h_{5}), & (g_{4},c_{5},h_{6}), & (g_{5},c_{4},h_{7}), & (g_{6},c_{5},h_{6}), & (g_{4},c_{3},h_{7});\\ (g_{6},c_{5},h_{6}), & (g_{7},c_{5},h_{5}), & (g_{5},c_{5},h_{7}), & (g_{6},c_{5},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{7},c_{4},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{5},c_{5},h_{6}),\\ (g_{6},c_{4},h_{7}), & (g_{5},c_{5},h_{6}), & (g_{6},c_{3},h_{6}), & (g_{5},c_{2},h_{5}), & (g_{5},c_{5},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{6},c_{5},h_{6}), & (g_{5},c_{4},h_{7}). \end{array} \right]; $$
$$\setlength{\arraycolsep}{1pt} e^{3}=\left[ \begin{array}{cccccccc} (g_{5},c_{5},h_{6}), & (g_{6},c_{5},h_{5}), & (g_{6},c_{5},h_{6}), & (g_{4},c_{5},h_{6}), & (g_{5},c_{5},h_{6}), & (g_{4},c_{5},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{4},c_{5},h_{6}),\\ (g_{4},c_{5},h_{7}), & (g_{5},c_{5},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{6},c_{4},h_{6}), & (g_{6},c_{4},h_{7}), & (g_{5},c_{5},h_{7}), & (g_{5},c_{4},h_{5}), & (g_{5},c_{5},h_{7});\\ (g_{6},c_{5},h_{6}), & (g_{6},c_{4},h_{5}), & (g_{5},c_{4},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{4},c_{5},h_{6}), & (g_{3},c_{4},h_{6}), & (g_{5},c_{5},h_{7}), & (g_{3},c_{5},h_{6}),\\ (g_{4},c_{4},h_{7}), & (g_{4},c_{5},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{5},c_{2},h_{6}), & (g_{4},c_{4},h_{7}), & (g_{4},c_{5},h_{7}), & (g_{6},c_{4},h_{5}), & (g_{5},c_{5},h_{7});\\ (g_{6},c_{5},h_{6}), & (g_{6},c_{5},h_{5}), & (g_{5},c_{5},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{5},c_{5},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{5},c_{5},h_{6}),\\ (g_{6},c_{4},h_{7}), & (g_{6},c_{5},h_{6}), & (g_{6},c_{5},h_{6}), & (g_{5},c_{4},h_{6}), & (g_{6},c_{5},h_{7}), & (g_{5},c_{3},h_{7}), & (g_{6},c_{5},h_{5}), & (g_{5},c_{5},h_{7}). \end{array} \right]. $$

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Ma, Z., Zhu, J., Ponnambalam, K. et al. Group Decision-Making with Linguistic Cognition from a Reliability Perspective. Cogn Comput 11, 172–192 (2019).

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  • Decision analysis
  • Cognitive reliability
  • Group consistency
  • Cognitive familiarity