Decision-making model with fuzzy preference relations based on consistency local adjustment strategy and DEA

Abstract

As one of the most useful tools, fuzzy preference relations (FPRs) can cope with the situations in which the experts are more comfortable providing their evaluation information with numerical values. Consistency-improving process and deriving the reliable priority weight vector for alternatives are two significant and challenging issues in decision making with FPRs. This paper investigates a novel decision-making model with FPRs on the basis of consistency local adjustment strategy and data envelopment analysis (DEA). Firstly, a new approach is proposed to generate the multiplicative consistent FPRs. Subsequently, a convergent consistency-improving algorithm for FPRs is developed to transform the unacceptable multiplicative consistent FPRs into the acceptable ones. In the consistency-improving process for FPRs, the local adjustment strategy is presented to employ decision-maker original evaluation information sufficiently. In order to determine the priority weight vector for alternatives, a novel fuzzy DEA model is constructed. Furthermore, a decision-making model with FPRs is designed to derive the reliable decision-making results. Finally, a numerical example of selecting the most important influence factor for fog–haze is provided, and the comparison with existing approaches is made to validate the rationality and effectiveness of the developed model.

Appendix A

Step-by-step calculations by using Ma et al.’s [39] method.

Step 1 Let $B^{(0)} = (b_{ij}^{(0)} )_{4 \times 4} = A = (a_{ij} )_{4 \times 4}$. Utilizing Eq. (4) in [39], one can obtain the following preference matrix $Q^{(0)}$ of FPR $B^{(0)} = (b_{ij}^{(0)} )_{4 \times 4}$:

$$Q^{(0)} = \left( {\begin{array}{*{20}c} 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{array} } \right),$$

and then we can get that $c^{(0)} = 3$ by using Eqs. (5) and (6) in Ma et al. [39]. According to Eq. (7) in [39], the reachable matrix $T^{(0)}$ of $B^{(0)} = (b_{ij}^{(0)} )_{4 \times 4}$ can be derived as follows:

$$T^{(1)} = \left( {\begin{array}{*{20}c} 1 & 1 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 2 \\ \end{array} } \right).$$

Based on Theorem 1 in [39], we know that FPR $B^{(0)} = (b_{ij}^{(0)} )_{4 \times 4}$ is inconsistent, then it is need to improve the consistency of FPR $B^{(0)} = (b_{ij}^{(0)} )_{4 \times 4}$.

Step 2 Construct the corresponding additive consistent FPR $R = (r_{ij} )_{4 \times 4}$ of $B^{(0)}$ by using Eq. (8) in [39] as follows:

$$R = \left( {\begin{array}{*{20}c} {0.5} & {0.3250} & {0.4750} & {0.6000} \\ {0.6750} & {0.5} & {0.6500} & {0.7750} \\ {0.5250} & {0.3500} & {0.5} & {0.6250} \\ {0.4000} & {0.2250} & {0.3750} & {0.5} \\ \end{array} } \right).$$

Based on the FPR $B^{(0)}$ and additive consistent FPR $R$, we construct a new FPR $B^{(1)} = (b_{ij}^{(1)} )_{4 \times 4}$ by using Eq. (10) as follows:

$$B^{(1)} = \left( {\begin{array}{*{20}c} {0.5} & {0.1225} & {0.5875} & {0.6900} \\ {0.8775} & {0.5} & {0.7850} & {0.4375} \\ {0.4125} & {0.2150} & {0.5} & {0.8725} \\ {0.3100} & {0.5625} & {0.1275} & {0.5} \\ \end{array} } \right).$$

After three iterations, the following adjusted FPR $B^{(3)} = (b_{ij}^{(3)} )_{4 \times 4}$ can be obtained:

$$B^{(3)} = \left( {\begin{array}{*{20}c} {0.5} & {0.1675} & {0.5625} & {0.6700} \\ {0.8325} & {0.5} & {0.7550} & {0.5125} \\ {0.4375} & {0.2450} & {0.5} & {0.8175} \\ {0.3300} & {0.4875} & {0.1825} & {0.5} \\ \end{array} } \right),$$

and we can get the reachable matrix $T^{(3)}$ of $B^{(3)}$ as follows:

$$T^{(3)} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right),$$

and then $c^{(3)} = 0$. According to Proposition 1 in [39], the adjusted FPR $B^{(3)} = (b_{ij}^{(3)} )_{4 \times 4}$ has weak transitivity.

Step 3 Apply arithmetical averaging operator to calculate the overall preference degree $b_{i}^{(3)} (i = 1,2,3,4)$ of the influence factor $x_{i} (i = 1,2,3,4)$:

$$b_{1}^{(3)} = 0.4750,b_{2}^{(3)} = 0.6500,b_{3}^{(3)} = 0.5000,b_{4}^{(3)} = 0.3750.$$

Step 4 It can be seen that $b_{2}^{(3)} > b_{3}^{(3)} > b_{1}^{(3)} > b_{4}^{(3)}$. Therefore, the importance of the influence factors is ranked as $x_{2} \succ x_{3}$$\succ x_{1} \succ x_{4}$, and the fog–haze weather’s most important influence factor is $x_{2}$.

Step-by-step calculations by using Xu et al.’s [19] method.

Step 1′ Let $P^{(0)} = A$, and then, we can construct the consistent FPR $\bar{P}^{(0)} = (\bar{p}_{ij}^{(0)} )_{4 \times 4}$ (where $\bar{p}_{ij}^{(t)} = \frac{1}{4}\sum_{k = 1}^{4} {(p_{ik}^{(t)} + p_{kj}^{(t)} )}$$- 0.5$) and $V^{(0)} = (v_{ij}^{(0)} )_{4 \times 4}$ (where $v_{ij}^{(t)} = [\hbox{min} \{ p_{ij}^{(t)} ,\bar{p}_{ij}^{(t)} \},$$\hbox{max} \{ p_{ij}^{(t)} ,\bar{p}_{ij}^{(t)} \} ]$) [19]:

$$\begin{aligned} \bar{P}^{(0)} & = \left( {\begin{array}{*{20}c} {0.5} & {0.3250} & {0.4750} & {0.6000} \\ {0.6750} & {0.5} & {0.6500} & {0.7750} \\ {0.5250} & {0.3500} & {0.5} & {0.6250} \\ {0.4000} & {0.2250} & {0.3750} & {0.5} \\ \end{array} } \right), \\ V^{(0)} & = \left( {\begin{array}{*{20}c} {0.5} & {[0.1000,0.3250]} \\ {[0.6750,0.9000]} & {0.5} \\ {[0.4000,0.5250]} & {[0.2000,0.3500]} \\ {[0.3000,0.4000]} & {[0.2250,0.6000]} \\ \end{array} } \right.\left. {\;\;\begin{array}{*{20}c} {[0.4750,0.6000]} & {[0.6000,0.7000]} \\ {[0.6500,0.8000]} & {[0.4000,0.7750]} \\ {0.5} & {[0.6250,0.9000]} \\ {[0.1000,0.3750]} & {0.5} \\ \end{array} } \right). \\ \end{aligned}$$

Step 2′ As consistency index of $P^{(0)}$ is $CI(P^{(0)} ) = 0.2291 > \overline{CI}$, then return $P^{(0)}$ to the DM, and a new LPR $P^{(1)} = (p_{ij}^{(1)} )_{4 \times 4}$ is provided by DM in accordance with his/her preference (where $p_{ij}^{(1)} \in v_{ij}^{(0)}$ and $p_{ij}^{(1)} +$$p_{ji}^{(1)} = 1$):

$$P^{(1)} = \left( {\begin{array}{*{20}c} {0.5} & {0.2} & {0.56} & {0.65} \\ {0.8} & {0.5} & {0.75} & {0.5} \\ {0.44} & {0.25} & {0.5} & {0.8} \\ {0.35} & {0.5} & {0.2} & {0.5} \\ \end{array} } \right),$$

and then, the corresponding additive consistent FPR $\bar{P}^{(1)} = (\bar{p}_{ij}^{(1)} )_{4 \times 4}$ and $V^{(1)} = (v_{ij}^{(1)} )_{4 \times 4}$ can be obtained as follows:

$$\begin{aligned} \bar{P}^{(1)} & = \left( {\begin{array}{*{20}c} {0.5} & {0.3650} & {0.4925} & {0.6025} \\ {0.6350} & {0.5} & {0.6400} & {0.7500} \\ {0.5075} & {0.3600} & {0.5} & {0.6100} \\ {0.3975} & {0.2500} & {0.3900} & {0.5} \\ \end{array} } \right), \\ V^{(2)} & = \left( {\begin{array}{*{20}c} {0.5} & {[0.2000,0.3650]} \\ {[0.6350,0.8000]} & {0.5} \\ {[0.4400,0.5075]} & {[0.2500,0.3600]} \\ {[0.3500,0.3975]} & {[0.2500,0.5000]} \\ \end{array} } \right.\left. {\;\;\begin{array}{*{20}c} {[0.4925,0.5600]} & {[0.6025,0.6500]} \\ {[0.6400,0.7500]} & {[0.5000,0.7500]} \\ {0.5} & {[0.6100,0.8000]} \\ {[0.2000,0.3900]} & {0.5} \\ \end{array} } \right). \\ \end{aligned}$$

Thus, the consistency index of $P^{(1)}$ is $CI(P^{(1)} ) = 0.1550 > \overline{CI}$. After nine iterations, the following adjusted FPR $P^{(9)} = (p_{ij}^{(9)} )_{4 \times 4}$ can be obtained:

$$P^{(9)} = \left( {\begin{array}{*{20}c} {0.5} & {0.34} & {0.51} & {0.63} \\ {0.66} & {0.5} & {0.63} & {0.76} \\ {0.49} & {0.37} & {0.5} & {0.61} \\ {0.37} & {0.24} & {0.39} & {0.5} \\ \end{array} } \right),$$

and the consistency index $CI(P^{(9)} ) = 0.051 < \overline{CI}$, which means that the adjusted FPR $P^{(9)}$ is acceptable additive consistent.

Step 3′ Apply arithmetical averaging operator to calculate the overall preference degree $p_{i}^{(9)} (i = 1,2,3,4)$ of the influence factor $x_{i} (i = 1,2,3,4)$:

$$p_{1}^{(9)} = 0.4950,\quad p_{2}^{(9)} = 0.6375,\quad p_{3}^{(9)} = 0.4925,\quad p_{4}^{(9)} = 0.3750.$$

Step 4′ Obviously, $p_{2}^{(9)} > p_{1}^{(9)} > p_{3}^{(9)} > p_{4}^{(9)}$. Therefore, the importance of the influence factors is ranked as $x_{2} \succ x_{1} \succ x_{3}$$\succ x_{4}$, and the fog–haze weather’s most important influence factor is $x_{2}$.

Step-by-step calculations by using Liu et al.’s [40] method.

Step 1″ Let $D^{(0)} = (d_{ij}^{(0)} )_{4 \times 4} = A = (a_{ij} )_{4 \times 4} ,\theta = 0.2$. By utilizing Eq. (13), we can get the additive consistency index of $D^{(0)}$ of $CI(D^{(0)} ) = 0.2479 > \overline{CI}$, which indicates FPR $D^{(0)}$ is unacceptable consistent. By Algorithm 1 in [40], the additive consistent FPR $\bar{D}^{(0)} = (\bar{d}_{ij}^{(0)} )_{4 \times 4}$ can be constructed as follows:

$$\bar{D}^{(0)} = \left( {\begin{array}{*{20}c} {0.5} & {0.4324} & {0.4842} & {0.5209} \\ {0.5676} & {0.5} & {0.5518} & {0.5885} \\ {0.5158} & {0.4482} & {0.5} & {0.5367} \\ {0.4791} & {0.4115} & {0.4633} & {0.5} \\ \end{array} } \right).$$

Step 2″ Apply Eq. (15) in [40] to adjust the consistency of FPR $D^{(0)} = (d_{ij}^{(0)} )_{4 \times 4}$ and obtain the adjusted FPR $D^{(1)} = (d_{ij}^{(1)} )_{4 \times 4}$ as follows:

$$D^{(1)} = \left( {\begin{array}{*{20}c} {0.5} & {0.1665} & {0.5768} & {0.6642} \\ {0.8335} & {0.5} & {0.7504} & {0.4377} \\ {0.4232} & {0.2496} & {0.5} & {0.8273} \\ {0.3358} & {0.5623} & {0.1727} & {0.5} \\ \end{array} } \right).$$

Step 3″ After 21 iterations, the following acceptable additive consistent FPR $D^{(21)} = (d_{ij}^{(21)} )_{4 \times 4}$ is derived as

$$D^{(21)} = \left( {\begin{array}{*{20}c} {0.5} & {0.4173} & {0.5002} & {0.5288} \\ {0.5827} & {0.5} & {0.5610} & {0.5533} \\ {0.4998} & {0.4390} & {0.5} & {0.5419} \\ {0.4712} & {0.4467} & {0.4581} & {0.5} \\ \end{array} } \right),$$

and the consistency index $CI(D^{(21)} ) = 0.0438 < \overline{CI}$.

Step 4″ The overall fuzzy preference degrees of the four influence factors can be calculated as follows:

$$d_{1}^{(21)} = 0.4866,\quad d_{2}^{(21)} = 0.5493,\quad d_{3}^{(21)} = 0.4952,\quad d_{4}^{(21)} = 04690.$$

Step 5″ Since $d_{2}^{(21)} > d_{3}^{(21)} > d_{1}^{(21)} > d_{4}^{(21)}$, the importance of the influence factors is ranked as $x_{2} \succ x_{3} \succ x_{1}$$\succ x_{4}$, and the fog–haze weather’s most important influence factor is $x_{2}$.