## 1 Introduction

*et al.*, 2019; Zhang

*et al.*, 2017; Zhang Z.

*et al.*, 2019). However, picture fuzzy set is still irreplaceable and unique in investigating the issues of MADM and MAGDM. Specifically, PFSs have four expressions of membership degree including the positive subordinate degree, neutral subordinate degree, negative subordinate degree and refusal subordinate degree, which is a very detailed breakdown of decision makers’ attitudes, corresponding to four descriptions (affirmative, adiaphorous, averse and refusal) when decision makers make a decision (Liang

*et al.*, 2018). Because of its excellent characteristics, many scholars use picture fuzzy sets to study decision problems. Ma

*et al.*(2019) gave complex fuzzy sets and extended the range of membership function values. Obviously, compared with other kinds of fuzzy sets, PFSs can delineate the conduct of decision-makers in more detail and are closer to human thinking and cognition of fuzzy things. Therefore, it has more advantages in solving multi-attribute decision making (MADM). Liang

*et al.*(2018) evaluated the cleaner production for gold mines with picture fuzzy information. Meksavang

*et al.*(2019) researched for the selection of sustainable suppliers. Wang

*et al.*(2018) studied the risk evaluation of construction project with PFSs. Khan

*et al.*(2019) invented logarithmic aggregation operators of Picture Fuzzy Numbers to solve MADM problems. Ju

*et al.*(2019) used extended GRP method to study the location of charging stations for electric vehicles under picture fuzzy environment. Sindhu

*et al.*(2019) developed a linear programming model with PFSs. Liu and Zhang (2018) put forward picture fuzzy linguistic set and some aggregation operator based on picture fuzzy information, such as, A-PFLWAA. Wei (2017) also gave some picture fuzzy aggregation operators.

*et al.*(2015) in 2015) and so on. Among them, the TODIM method is distinctive, which makes use of piecewise function to denote the distance between two schemes. What’s more, it is more authentic to take the different attitudes of decision makers towards gains and losses on decision making into consideration by introducing parameter in the process of evaluation. Liang

*et al.*(2020) utilized TODIM to introduce the risk appetite on three-way decisions. Wu Y.N.

*et al.*(2019) investigated the investment selection of meeting the requirements of rooftop distributed photovoltaic projects for industrial and commercial households with TODIM. Biswas and Sarkar (2019) put forward a kind of methodology based on TODIM. Liang

*et al.*(2019) proposed a mixture TODIM method to assess the risk level of the targets. Zhang Y.X.

*et al.*(2019) explored water safety evaluation on the strength of the TODIM method. Zhang Y.X.

*et al.*(2019) integrated maximizing deviation, FANP and TODIM method. Renet al. (2017) studied TODIM under probabilistic dual hesitant fuzzy environment. Zhu

*et al.*(2019) used grey relational analysis to count the dominance degree. Yuan

*et al.*(2019) got through the ranking of risk level of CFPP investment with TODIM. Liu

*et al.*(2019) generalized TODIM and TOPSIS methods to distance measure. Wu and Zhang (2019) used TODIM under intuitionistic fuzzy environment to obtain the results of product ranking. Wei (2018) accomplished the TODIM in picture fuzzy environment. Mishra and Rani (2018) designed TODIM technique to solve problems in interval-valued IFSs. Zhang

*et al.*(2019) utilized sentiment analysis as well as classical TODIM to evaluate and rank products online in an intuitionistic fuzzy environment. Yu

*et al.*(2017) combined the classical TODIM method with unbalanced hesitant fuzzy linguistic term sets to analyse multi-criteria group decision making. Liang Y.Y.

*et al.*(2019) ameliorated the conventional TODIM with a weight determination method which was based on incomplete weight information. Wang

*et al.*(2019) applied a novel function to TODIM. Liu and Teng (2019) acquired weights by means of probabilistic linguistic information, which extend the TODIM. Tian

*et al.*(2019) developed the traditional TODIM by using Cumulative Prospect Theory (CPT).

## 2 Preliminary Knowledge

### 2.1 Picture Fuzzy Sets and Picture Fuzzy Numbers

##### Definition 1 *(See* Garg, 2017*).*

*O*, Picture Fuzzy Set (PFS) is defined by

##### (1)

\[ L(\mathrm{o})=\big\{\big\langle o,{\alpha _{L}}(o),{\beta _{L}}(o),{\varphi _{L}}(o)\big\rangle \hspace{0.1667em}\big|\hspace{0.1667em}o\in O\big\},\]*L*, ${\varphi _{L}}(o)\in [0,1]$ is well-known as degree of negative membership of

*L*, ${\beta _{L}}(o)\in [0,1]$ is well-known as degree of neutral membership of

*L*. In the meantime, ${\alpha _{L}}(o)$, ${\beta _{L}}(o)$ and ${\varphi _{L}}(o)$ satisfy the relation of “$0\leqslant {\alpha _{L}}(o)+{\beta _{L}}(o)+{\varphi _{L}}(o)\leqslant 1$, $\forall o\in O$”. Then the refusal membership of

*o*in

*L*can be calculated by Eq. (2)

##### Definition 2 *(See* Meksavang *et al.*, 2019*).*

- (1) $\delta \oplus \varepsilon =({\alpha _{\delta }}+{\alpha _{\varepsilon }}-{\alpha _{\delta }}{\alpha _{\varepsilon }},{\beta _{\delta }}{\beta _{\varepsilon }},{\varphi _{\delta }}{\varphi _{\varepsilon }})$;
- (2) $\delta \otimes \varepsilon =({\alpha _{\delta }}{\alpha _{\varepsilon }},{\beta _{\delta }}+{\beta _{\varepsilon }}-{\beta _{\delta }}{\beta _{\varepsilon }},{\varphi _{\delta }}+{\varphi _{\varepsilon }}-{\varphi _{\delta }}{\varphi _{\varepsilon }})$;
- (3) $\omega \cdot \delta =(1-{(1-{\alpha _{\delta }})^{\omega }},{({\beta _{\delta }})^{\omega }},{({\varphi _{\delta }})^{\omega }})$, $\omega >0$;
- (4) ${\delta ^{\omega }}=({({\alpha _{\delta }})^{\omega }},1-{(1-{\beta _{\delta }})^{\omega }},1-{(1-{\varphi _{\delta }})^{\omega }})$, $\omega >0$;
- (5) $\overline{\delta }=({\varphi _{\delta }},{\beta _{\delta }},{\alpha _{\delta }})$.

##### Definition 3 *(See* Wei, 2018*).*

##### Definition 4 *(See* Liang *et al.*, 2018*).*

##### Definition 5 *(See* Meksavang *et al.*, 2019*).*

##### (5)

\[\begin{aligned}{}d(\delta ,\varepsilon )& =\frac{1}{4}(|{\alpha _{\delta }}-{\alpha _{\varepsilon }}|+|{\beta _{\delta }}-{\beta _{\varepsilon }}|+|{\varphi _{\delta }}-{\varphi _{\varepsilon }}|+|{z_{\delta }}-{z_{\varepsilon }}|)\\ {} & \hspace{1em}+\frac{1}{2}\max (|{\alpha _{\delta }}-{\alpha _{\varepsilon }}|+|{\beta _{\delta }}-{\beta _{\varepsilon }}|+|{\varphi _{\delta }}-{\varphi _{\varepsilon }}|+|{z_{\delta }}-{z_{\varepsilon }}|).\end{aligned}\]### 2.2 Extended TODIM Method Based on Cumulative Prospect Theory

*N*, the alternatives and attributes are displayed in accordance with decision maker’s view.

**Step 1.**The converted probability of the alternative ${V_{i}}$ to ${V_{k}}$ will be computed according to (6) or (7), where $i,k\in f$ and $i\ne k$.

##### (7)

\[ {\eta _{ikr}^{-}}({\lambda _{r}})={\lambda _{r}^{\xi }}/{\big({\lambda _{r}^{\xi }}+{(1-{\lambda _{r}})^{\xi }}\big)^{\frac{1}{\xi }}},\]*ζ*and

*ξ*are the parameters describing the curvature of the weighting function.

**Step 2.**Eq. (8) is used to determine the relative weight ${\eta _{ikr}^{\ast }}({\lambda _{r}})$ of the alternative ${V_{i}}$ to ${V_{k}}$.

##### (8)

\[ {\eta _{ikr}^{\ast }}({\lambda _{r}})={\eta _{ikr}}({\lambda _{r}})\big/\max \big\{{\eta _{ikr}}({\lambda _{p}})\hspace{0.1667em}\big|\hspace{0.1667em}p\in g\big\},\hspace{1em}r\in g,\hspace{2.5pt}\forall (i,k),\]*r*th attribute for the alternative ${V_{i}}$, which is equal to ${\eta _{ikr}^{+}}({\lambda _{r}})$ when ${n_{ir}}\geqslant {n_{kr}}$, or equal to ${\eta _{ikr}^{-}}({\lambda _{r}})$ according to (7).

**Step 3.**Figure out the relative prospect dominance of alternative ${V_{i}}$ to ${V_{k}}$ underneath the attribute

*r*with (9):

##### (9)

\[ {\vartheta _{r}}({V_{i}},{V_{k}})=\left\{\begin{array}{l@{\hskip4.0pt}l}{\eta _{ikr}^{\ast }}({\lambda _{r}})\cdot {({n_{ir}}-{n_{kr}})^{\beta }}/{\textstyle\textstyle\sum _{r=1}^{g}}{\eta _{ikr}^{\ast }}({\lambda _{r}}),\hspace{1em}& \text{if}\hspace{2.5pt}{n_{ir}}>{n_{kr}},\\ {} 0,\hspace{1em}& \text{if}\hspace{2.5pt}{n_{ir}}={n_{kr}},\\ {} -\theta ({\textstyle\textstyle\sum _{r=1}^{g}}{\eta _{ikr}^{\ast }}({\lambda _{r}}))\cdot {({n_{kr}}-{n_{ir}})^{\alpha }}/{\eta _{ikr}^{\ast }}({\lambda _{r}}),\hspace{1em}& \text{if}\hspace{2.5pt}{n_{ir}}<{n_{kr}},\end{array}\right.\]*α*,

*β*and

*θ*are the parameters.

**Step 4.**Determine the dominance degree of the alternative ${V_{i}}$ over the others, which is calculated as Eq. (10).

**Step 6.**Rank the overall dominance degree $\psi ({V_{i}})$, $i\in f$. The alternative with the bigger $\psi ({V_{i}})$ value is considered the better choice.

## 3 Extended TODIM for Picture Fuzzy MAGDM Based on Cumulative Prospect Theory

*n*decision makers, integrated into the set of decision makers $M=\{{M_{1}},{M_{2}},\dots ,{M_{n}}\}$, whose weight vector is $\chi =({\chi _{1}},{\chi _{2}},\dots ,{\chi _{n}})$, ${\chi _{t}}\geqslant 0$, $(t=1,2,\dots ,n)$, ${\textstyle\sum _{t=1}^{n}}{\chi _{t}}=1$.

*t*th decider, where ${\alpha _{sr}^{t}}$ indicates the positive subordinate degree of the

*t*th decision maker, ${\beta _{sr}^{t}}$ expresses the neutral subordinate degree of the

*t*th decision maker, ${\varphi _{sr}^{t}}$ expresses the negative subordinate degree of the

*t*th decision maker, ${\alpha _{sr}^{t}},{\beta _{sr}^{t}},{\varphi _{sr}^{t}}\in [0,1]$ and $0\leqslant {\alpha _{sr}^{t}}+{\beta _{sr}^{t}}+{\varphi _{sr}^{t}}\leqslant 1$, $s=1,2,\dots ,f$, $r=1,2,\dots ,g$, $t=1,2,\dots ,n$.

**Step 1.**Transform the cost attributes into the benefit attributes by using Eq. (12).

##### (12)

\[\begin{aligned}{}& {U^{t}}={\big({u_{sr}^{t}}\big)_{f\times g}},\hspace{1em}s=1,2,\dots ,f,\hspace{2.5pt}r=1,2,\dots ,g,\hspace{2.5pt}t=1,2,\dots ,n,\\ {} & {u_{sr}^{t}}=\big({\mu _{sr}^{t}},{\nu _{sr}^{t}},{\rho _{sr}^{t}}\big)=\left\{\begin{array}{l@{\hskip4.0pt}l}{n_{sr}^{t}}=({\alpha _{sr}^{t}},{\beta _{sr}^{t}},{\varphi _{sr}^{t}}),\hspace{1em}& {D_{r}}\hspace{2.5pt}\text{is a benefit attribute},\\ {} {\overline{n}_{sr}^{t}}=({\varphi _{sr}^{t}},{\beta _{sr}^{t}},{\alpha _{sr}^{t}}),\hspace{1em}& {D_{r}}\hspace{2.5pt}\text{is a cost attribute}.\end{array}\right.\end{aligned}\]**Step 2.**Calculate score matrix ${C^{t}}={({c_{sr}^{t}})_{f\times g}}$ for each normalized decision maker using Eq. (13), and integrate these score matrices for different decision maker into one group score matrix $Y={({y_{sr}})_{f\times g}}$ using Eq. (14).

##### Fig. 1

**Step 3.**Use Eq. (15) to normalize the group score matrix and obtain the normalized matrix $X={({x_{sr}})_{f\times g}}$.

**Step 4.**Utilize the Entropy Weight Method to obtain the original weight attributes $\lambda =({\lambda _{1}},{\lambda _{2}},\dots ,{\lambda _{n}})$, ${\lambda _{r}}\geqslant 0$, which is calculated as (16) and (17):

**Step 5.**The converted probability of the alternative ${V_{i}}$ to ${V_{k}}$ will be computed according to (18) or (19), where $i,k\in f$ and $i\ne k$.

##### (19)

\[ {\eta _{ikr}^{-}}({\lambda _{r}})={\lambda _{r}^{\xi }}\big/{\big({\lambda _{r}^{\xi }}+{(1-{\lambda _{r}})^{\xi }}\big)^{\frac{1}{\xi }}},\]*ζ*and

*ξ*are the parameters describing the curvature of the weighting function.

**Step 6.**Eq. (20) is used to determine the relative weight ${\eta _{ikr}^{\ast }}({\lambda _{r}})$ of the alternative ${V_{i}}$ to ${V_{k}}$.

##### (20)

\[ {\eta _{ikr}^{\ast }}({\lambda _{r}})={\eta _{ikr}}({\lambda _{r}})\big/\max \big\{{\eta _{ikr}}({\lambda _{p}})\big|p\in g\big\},\hspace{1em}r\in g,\hspace{2.5pt}\forall (i,k),\]*r*th attribute for the alternative ${V_{i}}$, which is equal to ${\eta _{ikr}^{+}}({\lambda _{r}})$ when ${x_{ir}}\geqslant {x_{kr}}$, or equal to ${\eta _{ikr}^{-}}({\lambda _{r}})$ according to Eq. (19).

**Step 7.**Determine the dominance degree of the alternative ${V_{i}}$ over the others, which is calculated as Eq. (21):

##### (21)

\[ \pi ({V_{i}})={\sum \limits_{k=1}^{f}}{\sum \limits_{r=1}^{g}}{\vartheta _{r}}({V_{i}},{V_{k}}),\hspace{1em}i=1,2,\dots ,f,\]##### (22)

\[ {\vartheta _{r}}({V_{i}},{V_{k}})=\left\{\begin{array}{l@{\hskip4.0pt}l}{\eta _{ikr}^{\ast }}({\lambda _{r}})\cdot {({x_{ir}}-{x_{kr}})^{\beta }}/{\textstyle\textstyle\sum _{r=1}^{g}}{\eta _{ikr}^{\ast }}({\lambda _{r}}),\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ir}}>{x_{kr}},\\ {} 0,\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ir}}={x_{kr}},\\ {} -\theta ({\textstyle\textstyle\sum _{r=1}^{g}}{\eta _{ikr}^{\ast }}({\lambda _{r}}))\cdot {({x_{kr}}-{x_{ir}})^{\alpha }}/{\eta _{ikr}^{\ast }}({\lambda _{r}}),\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ir}}<{x_{kr}}.\end{array}\right.\]*r*, and

*α*,

*β*and

*θ*are the parameters.

**Step 9.**Rank the overall dominance degree $\psi ({V_{i}})$, $i\in f$. The alternative with the bigger $\psi ({V_{i}})$ value is considered a better choice.

## 4 Numerical Instance

### 4.1 Numerical Example for Picture Fuzzy MAGDM

*et al.*, 2020; Lu

*et al.*, 2019; Wang P.

*et al.*, 2019; Wei

*et al.*, 2019a). Therefore, it is crucial for a supermarket to select the best location that may result in prodigious effectiveness. Now, for the management of a supermarket, there are four site locations ${V_{i}}$ ($i=1,2,3,4$) from which to choose. And the management adopts five attributes to assess these four alternatives: (1) ${D_{1}}$ is the shop rent, (2) ${D_{2}}$ is the population density, (3) ${D_{3}}$ is the consumption level, (4) ${D_{4}}$ is the intensity of competitive rivalry, (5) ${D_{5}}$ is the transportation convenience. Among them, ${D_{1}}$ and ${D_{4}}$ are cost attributes, and the others are benefit attributes. The four store location plans are going to be evaluated by three experts (whose weighting vector is $\chi ={({\chi _{1}},{\chi _{2}},{\chi _{3}})^{T}}={(0.43,0.22,0.35)^{T}}$) using PFNs under five attributes. Then, the picture fuzzy decision matrices which are given by the three experts are shown below.

##### Table 1

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.28,0.06,0.54)$ | $(0.86,0.03,0.11)$ | $(0.55,0.13,0.26)$ | $(0.24,0.08,0.63)$ | $(0.56,0.18,0.22)$ |

${V_{2}}$ | $(0.26,0.19,0.48)$ | $(0.53,0.12,0.03)$ | $(0.52,0.18,0.21)$ | $(0.21,0.16,0.53)$ | $(0.75,0.05,0.12)$ |

${V_{3}}$ | $(0.15,0.02,0.72)$ | $(0.74,0.06,0.05)$ | $(0.70,0.04,0.13)$ | $(0.17,0.04,0.74)$ | $(0.61,0.21,0.08)$ |

${V_{4}}$ | $(0.22,0.13,0.58)$ | $(0.69,0.21,0.08)$ | $(0.64,0.16,0.08)$ | $(0.16,0.24,0.59)$ | $(0.46,0.31,0.13)$ |

##### Table 2

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.28,0.06,0.62)$ | $(0.87,0.03,0.09)$ | $(0.71,0.05,0.20)$ | $(0.33,0.14,0.49)$ | $(0.58,0.14,0.16)$ |

${V_{2}}$ | $(0.22,0.16,0.52)$ | $(0.47,0.14,0.26)$ | $(0.62,0.13,0.25)$ | $(0.12,0.22,0.48)$ | $(0.65,0.13,0.15)$ |

${V_{3}}$ | $(0.19,0.11,0.67)$ | $(0.69,0.05,0.21)$ | $(0.84,0.02,0.13)$ | $(0.09,0.23,0.62)$ | $(0.63,0.05,0.25)$ |

${V_{4}}$ | $(0.02,0.08,0.74)$ | $(0.62,0.12,0.04)$ | $(0.58,0.26,0.14)$ | $(0.24,0.02,0.71)$ | $(0.52,0.17,0.24)$ |

##### Table 3

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.14,0.28,0.52)$ | $(0.68,0.02,0.16)$ | $(0.67,0.06,0.23)$ | $(0.21,0.09,0.66)$ | $(0.47,0.11,0.26)$ |

${V_{2}}$ | $(0.13,0.15,0.68)$ | $(0.58,0.24,0.16)$ | $(0.43,0.38,0.12)$ | $(0.15,0.26,0.48)$ | $(0.65,0.08,0.16)$ |

${V_{3}}$ | $(0.11,0.03,0.77)$ | $(0.73,0.09,0.13)$ | $(0.65,0.12,0.22)$ | $(0.16,0.08,0.72)$ | $(0.59,0.04,0.35)$ |

${V_{4}}$ | $(0.21,0.09,0.66)$ | $(0.56,0.07,0.26)$ | $(0.49,0.21,0.26)$ | $(0.28,0.03,0.54)$ | $(0.53,0.22,0.23)$ |

##### Table 4

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.54,0.06,0.28)$ | $(0.86,0.03,0.11)$ | $(0.55,0.13,0.26)$ | $(0.63,0.08,0.24)$ | $(0.56,0.18,0.22)$ |

${V_{2}}$ | $(0.48,0.19,0.26)$ | $(0.53,0.12,0.03)$ | $(0.52,0.18,0.21)$ | $(0.53,0.16,0.21)$ | $(0.75,0.05,0.12)$ |

${V_{3}}$ | $(0.72,0.02,0.15)$ | $(0.74,0.06,0.05)$ | $(0.70,0.04,0.13)$ | $(0.74,0.04,0.17)$ | $(0.61,0.21,0.08)$ |

${V_{4}}$ | $(0.58,0.13,0.22)$ | $(0.69,0.21,0.08)$ | $(0.64,0.16,0.08)$ | $(0.59,0.24,0.16)$ | $(0.46,0.31,0.13)$ |

##### Table 5

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.62,0.06,0.28)$ | $(0.87,0.03,0.09)$ | $(0.71,0.05,0.20)$ | $(0.49,0.14,0.33)$ | $(0.58,0.14,0.16)$ |

${V_{2}}$ | $(0.52,0.16,0.22)$ | $(0.47,0.14,0.26)$ | $(0.62,0.13,0.25)$ | $(0.48,0.22,0.12)$ | $(0.65,0.13,0.15)$ |

${V_{3}}$ | $(0.67,0.11,0.19)$ | $(0.69,0.05,0.21)$ | $(0.84,0.02,0.13)$ | $(0.62,0.23,0.09)$ | $(0.63,0.05,0.25)$ |

${V_{4}}$ | $(0.74,0.08,0.02)$ | $(0.62,0.12,0.04)$ | $(0.58,0.26,0.14)$ | $(0.71,0.02,0.24)$ | $(0.52,0.17,0.24)$ |

##### Table 6

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.52,0.28,0.14)$ | $(0.68,0.02,0.16)$ | $(0.67,0.06,0.23)$ | $(0.66,0.09,0.21)$ | $(0.47,0.11,0.26)$ |

${V_{2}}$ | $(0.68,0.15,0.13)$ | $(0.58,0.24,0.16)$ | $(0.43,0.38,0.12)$ | $(0.48,0.26,0.15)$ | $(0.65,0.08,0.16)$ |

${V_{3}}$ | $(0.77,0.03,0.11)$ | $(0.73,0.09,0.13)$ | $(0.65,0.12,0.22)$ | $(0.72,0.08,0.16)$ | $(0.59,0.04,0.35)$ |

${V_{4}}$ | $(0.66,0.09,0.21)$ | $(0.56,0.07,0.26)$ | $(0.49,0.21,0.26)$ | $(0.54,0.03,0.28)$ | $(0.53,0.22,0.23)$ |

**Step 2.**Calculate score matrix ${C^{t}}={({c_{sr}^{t}})_{4\times 5}}$ for each normalized decision maker using Eq. (13), and it is shown in Table 7 to Table 9. Then, integrate these score matrices for different decision maker into one group score matrix $Y={({y_{sr}})_{4\times 5}}$ using Eq. (14). The result is shown in Table 10.

##### Table 7

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | 0.6300 | 0.8750 | 0.6450 | 0.6950 | 0.6700 |

${V_{2}}$ | 0.6100 | 0.7500 | 0.6550 | 0.6600 | 0.8150 |

${V_{3}}$ | 0.7850 | 0.8450 | 0.7850 | 0.7850 | 0.7650 |

${V_{4}}$ | 0.6800 | 0.8050 | 0.7800 | 0.7150 | 0.6650 |

##### Table 8

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | 0.6700 | 0.8900 | 0.7550 | 0.5800 | 0.7100 |

${V_{2}}$ | 0.6500 | 0.6050 | 0.6850 | 0.6800 | 0.7500 |

${V_{3}}$ | 0.7400 | 0.7400 | 0.8550 | 0.7650 | 0.6900 |

${V_{4}}$ | 0.8600 | 0.7900 | 0.7200 | 0.7350 | 0.6400 |

##### Table 9

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | 0.6900 | 0.7600 | 0.7200 | 0.7250 | 0.6050 |

${V_{2}}$ | 0.7750 | 0.7100 | 0.6550 | 0.6650 | 0.7450 |

${V_{3}}$ | 0.8300 | 0.8000 | 0.7150 | 0.7800 | 0.6200 |

${V_{4}}$ | 0.7250 | 0.6500 | 0.6150 | 0.6300 | 0.6500 |

##### Table 10

*Y*.

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | 0.6598 | 0.8381 | 0.6955 | 0.6802 | 0.6561 |

${V_{2}}$ | 0.6766 | 0.7041 | 0.6616 | 0.6662 | 0.7762 |

${V_{3}}$ | 0.7909 | 0.8062 | 0.7759 | 0.7789 | 0.6978 |

${V_{4}}$ | 0.7354 | 0.7475 | 0.7091 | 0.6897 | 0.6543 |

**Step 3.**Use Eq. (15) to normalize the group score matrix and obtain the normalized matrix $X={({x_{sr}})_{4\times 5}}$ shown in Table 11.

**Step 4.**Utilize Eq. (16) and (17) to obtain the original weighting vector of attributes $\lambda =(0.2368,0.2054,0.1546,0.1765,0.2266)$.

##### Table 11

*X*.

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | 0.2305 | 0.2707 | 0.2447 | 0.2416 | 0.2356 |

${V_{2}}$ | 0.2363 | 0.2274 | 0.2328 | 0.2367 | 0.2788 |

${V_{3}}$ | 0.2763 | 0.2604 | 0.2730 | 0.2767 | 0.2506 |

${V_{4}}$ | 0.2569 | 0.2414 | 0.2495 | 0.2450 | 0.2350 |

**Step 5.**Compute the converted probability of the alternative ${V_{i}}$ to ${V_{k}}$ according to (18) or (19). The result is shown in Tables 12 to 15 ($\zeta =0.61$, $\xi =0.69$, which derive from the experiment of Kahneman, 1992).

##### Table 12

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{12}}$ | 0.2842 | 0.2642 | 0.2303 | 0.2455 | 0.2768 |

${\eta _{13}}$ | 0.2842 | 0.2642 | 0.2207 | 0.2387 | 0.2768 |

${\eta _{14}}$ | 0.2842 | 0.2642 | 0.2207 | 0.2387 | 0.2771 |

##### Table 13

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{21}}$ | 0.2831 | 0.2611 | 0.2207 | 0.2387 | 0.2771 |

${\eta _{23}}$ | 0.2842 | 0.2611 | 0.2207 | 0.2387 | 0.2771 |

${\eta _{24}}$ | 0.2842 | 0.2611 | 0.2207 | 0.2387 | 0.2771 |

##### Table 14

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{31}}$ | 0.2831 | 0.2611 | 0.2303 | 0.2455 | 0.2771 |

${\eta _{32}}$ | 0.2831 | 0.2642 | 0.2303 | 0.2455 | 0.2768 |

${\eta _{34}}$ | 0.2831 | 0.2642 | 0.2303 | 0.2455 | 0.2771 |

##### Table 15

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{41}}$ | 0.2831 | 0.2611 | 0.2303 | 0.2455 | 0.2768 |

${\eta _{42}}$ | 0.2831 | 0.2642 | 0.2303 | 0.2455 | 0.2768 |

${\eta _{43}}$ | 0.2842 | 0.2611 | 0.2207 | 0.2387 | 0.2768 |

**Step 6.**Determine the relative weight ${\eta _{ikr}^{\ast }}({\lambda _{r}})$ of the alternative ${V_{i}}$ to ${V_{k}}$ by using Eq. (20). The result is shown in Tables 16 to 19.

##### Table 16

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{12}^{\ast }}$ | 1.0000 | 0.9296 | 0.8103 | 0.8639 | 0.9742 |

${\eta _{13}^{\ast }}$ | 1.0000 | 0.9296 | 0.7765 | 0.8400 | 0.9742 |

${\eta _{14}^{\ast }}$ | 1.0000 | 0.9296 | 0.7765 | 0.8400 | 0.9751 |

##### Table 17

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{21}^{\ast }}$ | 1.0000 | 0.9223 | 0.7794 | 0.8431 | 0.9788 |

${\eta _{23}^{\ast }}$ | 1.0000 | 0.9189 | 0.7765 | 0.8400 | 0.9751 |

${\eta _{24}^{\ast }}$ | 1.0000 | 0.9189 | 0.7765 | 0.8400 | 0.9751 |

##### Table 18

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{31}^{\ast }}$ | 1.0000 | 0.9223 | 0.8133 | 0.8671 | 0.9788 |

${\eta _{32}^{\ast }}$ | 1.0000 | 0.9330 | 0.8133 | 0.8671 | 0.9779 |

${\eta _{34}^{\ast }}$ | 1.0000 | 0.9330 | 0.8133 | 0.8671 | 0.9788 |

##### Table 19

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\eta _{41}^{\ast }}$ | 1.0000 | 0.9223 | 0.8133 | 0.8671 | 0.9779 |

${\eta _{42}^{\ast }}$ | 1.0000 | 0.9330 | 0.8133 | 0.8671 | 0.9779 |

${\eta _{43}^{\ast }}$ | 1.0000 | 0.9189 | 0.7765 | 0.8400 | 0.9742 |

**Step 7.**Determine the relative prospect dominance ${\vartheta _{r}}({V_{i}},{V_{k}})$ and the dominance degree of the alternative ${V_{i}}$ ($i=1,2,3,4$) over the others, which are calculated as in Eq. (22) and (21), respectively. The result is shown in Tables 20 to 23. ($\alpha =0.88$, $\beta =0.88$, $\theta =2.25$, which derive from the experiment of Kahneman, 1992).

##### Table 20

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{12}}$ | −0.1117 | 0.0128 | 0.0036 | 0.0018 | −0.6653 |

${\vartheta _{13}}$ | −0.6741 | 0.0037 | −0.5687 | −0.6344 | −0.2589 |

${\vartheta _{14}}$ | −0.4153 | 0.0092 | −0.1190 | −0.0805 | 0.0003 |

##### Table 21

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{21}}$ | 0.0024 | −0.6960 | −0.2647 | −0.1138 | 0.0136 |

${\vartheta _{23}}$ | −0.5964 | −0.5483 | −0.7729 | −0.7117 | 0.0093 |

${\vartheta _{24}}$ | −0.3323 | −0.2581 | −0.3566 | −0.1791 | 0.0138 |

##### Table 22

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{31}}$ | 0.0145 | −0.1994 | 0.0077 | 0.0099 | 0.0053 |

${\vartheta _{32}}$ | 0.0128 | 0.0101 | 0.0105 | 0.0111 | −0.4568 |

${\vartheta _{34}}$ | 0.0068 | 0.0062 | 0.0065 | 0.0091 | 0.0055 |

##### Table 23

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{41}}$ | 0.0089 | −0.4996 | 0.0016 | 0.0013 | −0.0164 |

${\vartheta _{42}}$ | 0.0071 | 0.0047 | 0.0048 | 0.0028 | −0.6735 |

${\vartheta _{43}}$ | −0.3158 | −0.3370 | −0.4820 | −0.5792 | −0.2680 |

**Step 8.**Acquire the overall dominance degree of the alternative ${V_{i}}$ ($i=1,2,3,4$) from Eq. (23).

### 4.2 Comparative Analysis

*et al.*, 2019), the picture fuzzy weighted cross-entropy method (Wei, 2016), the projection model (Wei

*et al.*, 2018), the MULTIMOORA method (Lin

*et al.*, 2020) and the EDAS method (Li

*et al.*, 2019) to test the efficiency of the improved TODIM, respectively.

#### 4.2.1 Comparison with the Classical TODIM

**Step 1.**We can use the same method to process data, like Section 4.1 from step 1 to step 4, obtaining the weighting vector of attributes $\lambda =(0.2368,0.2054,0.1546,0.1765,0.2266)$ and normalized Group score matrix

*X*shown in Table 11.

**Step 2.**Determine the dominance degree using Eq. (24). And the result is shown in Tables 24 to 27.

##### (24)

\[ {\vartheta _{r}}({V_{i}},{V_{k}})=\left\{\begin{array}{l@{\hskip4.0pt}l}\sqrt{{\lambda _{r}}\cdot ({x_{ir}}-{x_{kr}})/{\textstyle\textstyle\sum _{r=1}^{g}}{\lambda _{r}}},\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ir}}>{x_{kr}},\\ {} 0,\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ir}}={x_{kr}},\\ {} -\theta \sqrt{({\textstyle\textstyle\sum _{r=1}^{g}}{\lambda _{r}})\cdot ({x_{kr}}-{x_{ir}})/{\lambda _{r}}},\hspace{1em}& \text{if}\hspace{2.5pt}{x_{ir}}<{x_{kr}}.\end{array}\right.\]##### Table 24

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{12}}$ | −0.7268 | 0.0459 | 0.0209 | 0.0144 | −2.0175 |

${\vartheta _{13}}$ | −2.0330 | 0.0224 | −1.9784 | −2.0601 | −1.1886 |

${\vartheta _{14}}$ | −1.5436 | 0.0377 | −0.8134 | −0.6376 | 0.0059 |

##### Table 25

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{21}}$ | 0.0181 | −2.1221 | −1.2833 | −0.7775 | 0.0481 |

${\vartheta _{23}}$ | −1.8986 | −1.8523 | −2.3581 | −2.2019 | 0.0389 |

${\vartheta _{24}}$ | −1.3618 | −1.2072 | −1.5194 | −1.0055 | 0.0485 |

##### Table 26

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{31}}$ | 0.0507 | −1.0356 | 0.0322 | 0.0383 | 0.0284 |

${\vartheta _{32}}$ | 0.0473 | 0.0400 | 0.0384 | 0.0409 | −1.6303 |

${\vartheta _{34}}$ | 0.0330 | 0.0304 | 0.0293 | 0.0364 | 0.0290 |

##### Table 27

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${\vartheta _{41}}$ | 0.0385 | −1.7453 | 0.0132 | 0.0118 | −0.2469 |

${\vartheta _{42}}$ | 0.0339 | 0.0261 | 0.0247 | 0.0187 | −2.0326 |

${\vartheta _{43}}$ | −1.3230 | −1.4048 | −1.8034 | −1.9590 | −1.2140 |

**Step 3.**Acquire the overall dominance degree of the alternative ${V_{i}}$ ($i=1,2,3,4$) from Eq. (23).

**Step 4.**Rank the overall dominance degree $\psi ({V_{i}})$, $i=1,2,3,4$.

#### 4.2.2 Comparison with the VIKOR Method

*et al.*, 2019) method, the collective picture fuzzy evaluation matrix is shown in Table 28. The positive ideal in different attributes is as follows: ${u_{1}^{+}}=(0.7290,0.0335,0.1418)$, ${u_{2}^{+}}=(0.8160,0.0260,0.1200)$, ${u_{3}^{+}}=(0.7243,0.0504,0.1563)$, ${u_{4}^{+}}=(0.7099,0.0749,0.1447)$, ${u_{5}^{+}}=(0.6971,0.0727,0.1394)$ and the negative is: ${u_{1}^{-}}=(0.5523,0.1029,0.2197)$, ${u_{2}^{-}}=(0.5360,0.1582,0.0867)$, ${u_{3}^{-}}=(0.5158,0.2177,0.1794)$, ${u_{4}^{-}}=(0.5021,0.2034,0.1650)$, ${u_{5}^{-}}=(0.4988,0.2409,0.1817)$. The distance from each scheme to the perfect point is shown in Table 29. And the values ${S_{i}}$ ($i=1,2,3,4$) and ${R_{i}}$ ($i=1,2,3,4$) are as follows: ${S_{1}}=0.6600$, ${S_{2}}=1.0086$, ${S_{3}}=0.2474$, ${S_{4}}=0.8842$; ${R_{1}}=0.2368$, ${R_{2}}=0.3642$, ${R_{3}}=0.1326$, ${R_{4}}=0.2699$. The ${Q_{i}}$ ($i=1,2,3,4$) index is as follows: ${Q_{1}}=0.4959$, ${Q_{2}}=1$, ${Q_{3}}=0$, ${Q_{4}}=0.7147$, ${Q_{3}}<{Q_{1}}<{Q_{4}}<{Q_{2}}$. Therefore, there is ${V_{3}}>{V_{1}}>{V_{4}}>{V_{2}}$. The alternative ${V_{3}}$ is the optimal one.

##### Table 28

*U*.

${D_{1}}$ | ${D_{2}}$ | ${D_{3}}$ | ${D_{4}}$ | ${D_{5}}$ | |

${V_{1}}$ | $(0.5523,0.1029,0.2197)$ | $(0.8160,0.0260,0.1200)$ | $(0.6335,0.0804,0.2351)$ | $(0.6145,0.0943,0.2457)$ | $(0.5352,0.1434,0.2175)$ |

${V_{2}}$ | $(0.5689,0.1684,0.1966)$ | $(0.5360,0.1582,0.0867)$ | $(0.5158,0.2177,0.1794)$ | $(0.5021,0.2034,0.1650)$ | $(0.6971,0.0727,0.1394)$ |

${V_{3}}$ | $(0.7290,0.0335,0.1418)$ | $(0.7261,0.0664,0.0958)$ | $(0.7243,0.0504,0.1563)$ | $(0.7099,0.0749,0.1447)$ | $(0.6077,0.0857,0.1723)$ |

${V_{4}}$ | $(0.6490,0.1027,0.1277)$ | $(0.6335,0.1264,0.1038)$ | $(0.5793,0.1958,0.1367)$ | $(0.6045,0.0671,0.2128)$ | $(0.4988,0.2409,0.1817)$ |

#### 4.2.3 Comparison with the Picture Fuzzy Weighted Cross-Entropy Method

#### 4.2.4 Comparison with the Projection Model

*et al.*, 2018). In accordance to the identical principle with picture fuzzy weighted cross-entropy method,

#### 4.2.5 Comparison with the MULTIMOORA Method

*et al.*, 2020) is to dispose of the fundamental information in different way. Finally, ranking value matrix and the corresponding ranking order matrix come into being. The ranking value matrix is

#### 4.2.6 Comparison with the EDAS Method

#### 4.2.7 Contrastive Analysis

##### Table 31

Method | The ranking result |

Classical TODIM | ${V_{3}}>{V_{4}}>{V_{1}}>{V_{2}}$ |

VIKOR | ${V_{3}}>{V_{1}}>{V_{4}}>{V_{2}}$ |

Picture fuzzy weighted cross-entropy | ${V_{3}}>{V_{1}}>{V_{4}}>{V_{2}}$ |

Projection model | ${V_{3}}>{V_{1}}>{V_{4}}>{V_{2}}$ |

MULTIMOORA | ${V_{3}}>{V_{4}}>{V_{1}}>{V_{2}}$ |

EDAS | ${V_{3}}>{V_{1}}>{V_{4}}>{V_{2}}$ |

Improved TODIM | ${V_{3}}>{V_{4}}>{V_{1}}>{V_{2}}$ |

## 5 Conclusions

*et al.*, 2019; He

*et al.*, 2019; Li and Lu, 2019; Lu and Wei, 2019; Wang J.

*et al.*, 2019; Wang, 2019; Wei

*et al.*, 2019b; Wu

*et al.*, 2019a, 2019b). And we will also continue to explore the application of this proposed TODIM method in other fields (Lu

*et al.*, 2020; Wang

*et al.*, 2020; Wei Y.

*et al.*, 2020) and seek more scientific methods to solve the multi-attribute group decision problems (Liu and Liu, 2019; Liu and Wang, 2014; Liu

*et al.*, 2018).