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A New Risk-Based Fuzzy Cognitive Model and Its Application to Decision-Making

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Abstract

Cognitive information in real-world decision-making problems is usually associated with all sorts of ambiguities and uncertainties. Fuzzy sets have been proposed as a general workaround for such information representation. Notwithstanding, there are cases in which the fuzzy sets and fuzzy numbers have some degree of uncertainty when available data either come from unreliable sources or refer to events in the future. These situations result in some unreliability of the obtained fuzzy information. For the modeling of the possible future-event effects on the fuzzy information credibility, the present research presents a novel risk-based fuzzy cognitive methodology by investigating all possible cases to risk modeling of fuzzy sets and the governing mathematical equations. The new fuzzy cognitive model is used to develop a multi-criteria decision-making method based on a fuzzy TOPSIS method so-called RFC-TOPSIS, and the proposed approach was tested on a case study of failure modes and effects analysis problem. Based on the results, robust outcomes were obtained when the proposed methodology was used, highlighting the flexibility and the efficiency of the proposed methodology. The present concept can be used to deal with any problems, where membership function is associated with some risks and errors due to risk factors.

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Authors and Affiliations

  1. Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Hamidreza Seiti

  2. Department of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran

    Ashkan Hafezalkotob

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  1. Hamidreza Seiti
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  2. Ashkan Hafezalkotob
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Correspondence to Hamidreza Seiti.

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Appendix

Appendix

  • Example 1. Letting the risks associated with membership function and set elements of a reference set X and the fuzzy set \( \tilde{A} \) as rμ− = 0.3 and rx− = 0.2,respectively, the pessimistic set \( {\tilde{A}}_{Pb} \) can be obtained as follows.

$$ {\displaystyle \begin{array}{l}X=\left\{1,2,3,4\right\}\\ {}A=\left\{\left(1,0.2\right),\left(2,0.4\right),\left(3,0.5\right),\left(4,0.7\right)\right\}\\ {}{\tilde{A}}_P=\left\{\left(1,\left[0.14,0.2\right]\right),\left(2,\left[0.28,0.4\right]\right),\left(3,\left[0.35,0.5\right]\right),\left(4,\left[0.49,0.7\right]\right)\right\}\end{array}} $$

  • Example 2. Taking the X and fuzzy set \( \tilde{A} \) of Example 1 and assuming the risks of membership function as r− = 0.3 and r+ = 0.2, the pessimistic-optimistic set \( {\tilde{A}}_{P- OT}(x) \) can be written as follows.

$$ {\displaystyle \begin{array}{l}X=\left\{1,2,3,4\right\}\\ {}\tilde{A}=\left\{\left(1,0.2\right),\left(2,0.4\right),\left(3,0.5\right),\left(4,0.7\right)\right\}\\ {}{\tilde{\tilde{A}}}_{P- OT}(x)=\left\{\left(1,\left(0.14,0.2,0.24\right)\right),\left(2,\left(0.28,0.4,0.48\right)\right),\left(3,\left(0.35,0.5,0.6\right)\right),\left(4,\left(0.49,0.7,0.84\right)\right)\right\}\end{array}} $$

  • Example 3. Suppose GTrFN \( \tilde{A}=\left(1,2,3,4;\mathrm{0.4,0.6}\right) \) and rL− = 0.2, rR− = 0.3, rL+ = 0.3 and rR+ = 0.2 , \( {\tilde{\tilde{A}}}^{P- OT} \) is determined as follows:

$$ {\displaystyle \begin{array}{l}\tilde{A}=\left(1,2,3,4;0.4,0.6\right),\kern0.33em {r_L}^{-}=0.2,{r_R}^{-}=0.3,{r_L}^{+}=0.3\kern0.66em {r_R}^{+}=0.2\\ {}{\tilde{\tilde{A}}}^{P- OT}=\left(\left(1,2,3,4;0.32,0.42\right),\left(1,2,3,4;0.4,0.6\right),\left(1,2,3,4;0.52,0.72\right)\right)\end{array}} $$

  • Example 4. By letting ={1}, A = {(1,0.2)}, B = {(1,0.5)}, RA = {0.3, 0.4, 0.2, 0.1}, and RB = {0.2, 0.1, 0.2, 0.1},then we have

$$ {\displaystyle \begin{array}{l}A=\left\{\left(1,0.2\right)\right\},\kern0.33em {r}_A^{-}=0.3,{r}_A^{+}=0.3,A{R_A}^{-}=0.2\kern0.33em and\kern0.58em A{R}_A^{+}=0.1\kern0.33em \to RF{C}_A(x)=\left\{\left(1,\left(0.152,0.2,0.254\right)\right)\right\}\\ {}B=\left(1,0.5\right),\kern0.33em {r}_B^{-}=0.2,\kern0.33em ?{r}_B^{+}=0.1,A{R}_B^{-}=0.2\kern0.33em and\kern0.33em A{R}_A^{+}=0.1\kern0.33em \to RF{C}_B(x)=\left(1,\left(0.42,0.5,0.545\right)\right)\end{array}} $$
$$ {\displaystyle \begin{array}{l}\ {\mathrm{RFC}}_A(x)\cup {\mathrm{RFC}}_B(x)=\left\{\left(1,\left(0.42,0.5,0.545\right)\right)\right\}\\ {}\ {\mathrm{RFC}}_A(x)\cap {\mathrm{RFC}}_B(x)=\left\{\left(1,\left(0.152,0.2,0.254\right)\right)\right\}\\ {}\ {\mathrm{RFC}}_A(x)\oplus {\mathrm{RFC}}_B(x)=\left\{\left(1,\left(0.5082,0.6,0.6606\right)\right)\right\}\\ {}\ {\mathrm{RFC}}_A(x)\otimes {\mathrm{RFC}}_B(x)=\left\{\left(1,\left(0.0638,0.1,0.13843\right)\right)\right\}\end{array}} $$

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Seiti, H., Hafezalkotob, A. A New Risk-Based Fuzzy Cognitive Model and Its Application to Decision-Making. Cogn Comput 12, 309–326 (2020). https://doi.org/10.1007/s12559-019-09701-8

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