Interval-Valued Hesitant Fuzzy Stochastic Decision-Making Method Based on Regret Theory

Abstract

Different studies show that human beings are usually limited rational, their regret aversion behavior is playing an important role in the process of stochastic decision-making. However, such psychological behavior is neglected in current studies. The interval-valued hesitant fuzzy sets can more effectively depict the uncertain information than hesitant fuzzy sets. Therefore, we propose an interval-valued hesitant fuzzy stochastic decision-making approach based on group satisfaction degree and regret theory. Firstly, based on the score and variance function, a novel group satisfaction degree is defined, which can fully reflect the overall level and group divergence. Secondly, the attribute weights optimization model based on the group satisfaction degree and the deviation of attribute values is constructed to obtain the weight vector of attributes. Then, the regret value and the rejoice value are obtained by the novel regret-rejoice function, and the alternatives are ranked according to the total psychological perception values of the decision maker. Finally, we illustrated the application of the developed method with an emergency decision-making problem. Sensitivity and comparative analyses are implemented to demonstrate the superiority, stability, and validity of the proposed method based on regret theory.

Appendices

Appendix A

Proof of Proposition 3.1.

(P.3.1.1) $0 \le \varphi (\tilde{h}) = \frac{{s(\tilde{h})}}{{1 + v(\tilde{h})}} \le s(\tilde{h}) \le 1$

(P.3.1.2) If $\tilde{h} = \left\{ {\left[ {\gamma^{L} ,\gamma^{U} } \right]} \right\}$, then $s(\tilde{h}) = \frac{1}{2}(\gamma^{L} + \gamma^{U} )$, $v(\tilde{h}) = \frac{{(\gamma^{L} - \gamma^{U} )^{2} }}{12}$. Thus, we can obtain $\varphi (\tilde{h}) = \frac{{6(\gamma^{L} + \gamma^{U} )}}{{12 + (\gamma^{L} - \gamma^{U} )^{2} }}$.

(P.3.1.3) To prove the property, we only need to prove that $v(\tilde{h}) = v(\tilde{h}^{c} )$.

Based on the complement transformation, the following equation holds:

$$\begin{aligned} v\left( {\tilde{h}^{c} } \right) & = \frac{1}{{3l_{{\tilde{h}}} }}\sum\limits_{i = 1}^{{l_{{\tilde{h}}} }} {\left[ {\left( {1 - \gamma_{i}^{U} } \right)^{2} + \left( {1 - \gamma_{i}^{L} } \right)^{2} + \left( {1 - \gamma_{i}^{U} } \right) \times \left( {1 - \gamma_{i}^{L} } \right)} \right]} - \frac{1}{{4l_{{\tilde{h}}}^{2} }}\left[ {\sum\limits_{i = 1}^{{l_{{\tilde{h}}} }} {\left( {1 - \gamma_{i}^{U} } \right) + \left( {1 - \gamma_{i}^{L} } \right)} } \right]^{2} \\ & = \frac{1}{{3l_{{\tilde{h}}} }}\sum\limits_{i = 1}^{{l_{{\tilde{h}}} }} {\left[ {3 - 3\gamma_{i}^{L} - 3\gamma_{i}^{U} + \left( {\gamma_{i}^{L} } \right)^{2} + \left( {\gamma_{i}^{U} } \right)^{2} + \gamma_{i}^{L} \times \gamma_{i}^{U} } \right]} - \frac{1}{{4l_{{\tilde{h}}}^{2} }}\left[ {\sum\limits_{i = 1}^{{l_{{\tilde{h}}} }} {\left( {2 - \gamma_{i}^{L} - \gamma_{i}^{U} } \right)} } \right]^{2} = v(\tilde{h}) \\ \end{aligned}$$

Then, $\frac{{\varphi (\tilde{h})}}{{\varphi (\tilde{h}^{c} )}} = \frac{{\frac{{s(\tilde{h})}}{{1 + v(\tilde{h})}}}}{{\frac{{s(\tilde{h}^{c} )}}{{1 + v(\tilde{h}^{c} )}}}} = \frac{{s(\tilde{h})}}{{s(\tilde{h}^{c} )}}$.

Therefore, we have proved it is valid.

The Process of solving Model 2

In order to ensure that the method for determining the attribute weights is easily understood in Model 2, the following calculating process is provided.

Model 2

$$\begin{aligned} & \text{max}\,f_{i}^{t} (w) = \frac{1}{2}\sum\limits_{j = 1}^{n} {\sum\limits_{t = 1}^{g} {\sum\limits_{i = 1}^{m} {\sum\limits_{q = 1}^{m} {\left( {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right| + \frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}}} \right)} } } w_{j} } \\ & s.t.\sum\limits_{j = 1}^{n} {w_{j}^{2} } = 1,\quad 0 \le w_{j} \le 1,\quad j = 1,2, \ldots ,n \\ \end{aligned}$$

To obtain the optimal attribute weight vector, we construct the Lagrange function to solve the model as follows:

$$L(w,\lambda ) = \frac{1}{2}\sum\limits_{j = 1}^{n} {\sum\limits_{t = 1}^{g} {\sum\limits_{i = 1}^{m} {\sum\limits_{q = 1}^{m} {\left( {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right| + \frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}}} \right)} } } w_{j} } + \frac{1}{2}\lambda \left(\sum\limits_{j = 1}^{n} {w_{j}^{2} - 1} \right)$$

where $\lambda$ is the Lagrange multiplier.

Then, $L(w,\lambda )$ is differentiated with respect to $w_{j} (j = 1,2, \cdots ,n)$ and $\lambda$, then we can obtain the following equations:

$$\frac{\partial L(w,\lambda )}{{\partial w_{j} }} = \frac{1}{2}\sum\limits_{t = 1}^{g} {\sum\limits_{i = 1}^{m} {\sum\limits_{q = 1}^{m} {\left( {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right| + \frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}}} \right)} } } + \lambda w_{j} = 0$$

$$\frac{\partial L(w,\lambda )}{\partial \lambda } = \frac{1}{2}\left( {\sum\limits_{j = 1}^{n} {w_{j}^{2} - 1} } \right) = 0$$

Then, calculating the above equations, the optimal attribute weights are obtained as follows:

$$w_{j} = \frac{{\sum\nolimits_{t = 1}^{g} {\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{q = 1}^{m} {\left( {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right| + \frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}}} \right)} } } }}{{\sqrt {\sum\nolimits_{j = 1}^{n} {\left( {\sum\nolimits_{t = 1}^{g} {\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{q = 1}^{m} {\left( {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right| + \frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}}} \right)} } } } \right)} }^{2} }}$$

Normalize $w_{j}$, have

$$w_{j} = \frac{{\left( {\sum\nolimits_{t = 1}^{g} {\sum\nolimits_{i = 1}^{m} {\frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}}} } + \sum\nolimits_{t = 1}^{g} {\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{q = 1}^{m} {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right|} } } } \right)}}{{\sum\nolimits_{j = 1}^{n} {\left( {\sum\nolimits_{t = 1}^{g} {\sum\nolimits_{i = 1}^{m} {\frac{{s(\tilde{h}_{ij}^{t} )}}{{1 + v(\tilde{h}_{ij}^{t} )}} + \sum\nolimits_{t = 1}^{g} {\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{q = 1}^{m} {\left| {d(\tilde{h}_{ij}^{t} ,\tilde{1}) - d(\tilde{h}_{qj}^{t} ,\tilde{1})} \right|} } } } } } \right)} }},\quad j = 1,2, \ldots ,n$$

Appendix B

Table 7 shows the meaning of all abbreviations in the paper to help the readers understand better.

Table 7 The list of abbreviations and parameters