## 1 Introduction

*et al.*, 2005; Ronald, 2015; Xu, 2007; Xu and Yager, 2006; Konwar and Debnath, 2017; Wu and Chiclana, 2014; Wang, 2017; Chen, 2011; Chen and Li, 2011; Chen, 2014, 2016; Chen and Chiou, 2015; Garg, 2016; Li, 2011; Zhao and Wei, 2013; Liu, 2017b; Zhang, 2017; Song and Wang, 2017; Ye, 2009, 2010; Wei and Zhao, 2012; Liu, 2017a). Recently, Cuong (2013) developed picture fuzzy set (PFS) and studied the properties and basic operations laws of PFS. Singh (2014) studied the correlation coefficients for PFSs. Son (2015) and Thong (2015) proposed several clustering algorithms with PFSs. Thong (2015) proposed a hybrid method between PF clustering and IF recommender systems. Wei (2016) proposed the cross-entropy for MADM problems with PFNs. Wei (2017a) investigated the picture fuzzy aggregation operators for MADM problems. Wei

*et al.*(2016b) gave the projection models for MADM with picture fuzzy information. Thong and Son (2016b) considered the improvement of FCM on the PFSs. Thong and Son (2016a) proposed the Automatic Picture Fuzzy Clustering (AFC-PFS). Son (2016) proposed a generalized picture distance measure. Son (2017) proposed the generalized picture distance measures and association measures. Son

*et al.*(2017) proposed the picture inference system (PIS). Son and Thong (2017) developed two hybrid forecast models with picture fuzzy clustering.

*et al.*, 2012; Liu

*et al.*, 2014; Wu and Chiclana, 2014). In order to show the risk and uncertainty of the MADM problems simultaneously, more and more scholars have proposed some fuzzy TODIM approach (Konwar and Debnath, 2017; Fan

*et al.*, 2013), the intuitionistic fuzzy TODIM approach (Lourenzutti and Krohling, 2013; Krohling

*et al.*, 2013), Pythagorean fuzzy TODIM approach (Ren

*et al.*, 2016), multi-hesitant fuzzy linguistic TODIM approach (Wang

*et al.*, 2016; Wei

*et al.*, 2015), interval type-2 fuzzy sets-based TODIM method (Sang and Liu, 2016), intuitionistic linguistic TODIM method (Wang and Liu, 2017) and 2-dimension uncertain linguistic TODIM method (Liu and Teng, 2016). But until now, no research extend TODIM model for PFNs. Therefore, it is necessary to investigate this issue. The purpose of this paper is to expand TODIM model to MADM with PFNs to overcome this limitation. The rest of this paper is organized as follows. In Section 2, we introduce the concepts of PFNs and classical TODIM model. In Section 3 we develop the TODIM model for MADM with PFNs. In Section 4, an illustrative example for potential evaluation of emerging technology commercialization is pointed out and some comparative analysis is conducted. In Section 5 we conclude this paper.

## 2 Preliminaries

### 2.1 Picture Fuzzy Set Sets (PFSs)

##### Definition 1 *(See* Atanassov, 1986, 1989*).*

*A*in

*X*is given by

##### (1)

\[ A=\big\{\big\langle x,{\mu _{A}}(x),{\nu _{A}}(x)\big\rangle \hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\}\]*x*to the set

*A*.

##### Definition 2 *(See* Cuong, 2013*).*

*A*on the universe

*X*is an object of the form

##### (2)

\[ A=\big\{\big\langle x,{\mu _{A}}(x),{\eta _{A}}(x),{\nu _{A}}(x)\big\rangle \hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\}\]*A*”, ${\eta _{A}}(x)\in [0,1]$ is called the “degree of neutral membership of

*A*” and ${\nu _{A}}(x)\in [0,1]$ is called the “degree of negative membership of

*A*”, and ${\mu _{A}}(x),{\eta _{A}}(x),{\nu _{A}}(x)$ satisfy the following condition: $0\leqslant {\mu _{A}}(x)+{\eta _{A}}(x)+{\nu _{A}}(x)\leqslant 1$, $\forall x\in X$. Then for $x\in X$, ${\pi _{A}}(x)=1-({\mu _{A}}(x)+{\eta _{A}}(x)+{\nu _{A}}(x))$ could be called the degree of refusal membership of

*x*in

*X*.

##### Definition 3 *(See* Abdellaoui *et al.*, 2017*).*

##### Definition 4 *(See* Wei, 2017a*).*

##### Definition 5 *(See* Wei, 2017a*).*

*α*and

*β*, respectively, and let $H(\alpha )={\mu _{\alpha }}+{\eta _{\alpha }}+{\nu _{\alpha }}$ and $H(\beta )={\mu _{\beta }}+{\eta _{\beta }}+{\nu _{\beta }}$ be the accuracy degrees of

*α*and

*β*, respectively, then if $S(\alpha )<S(\beta )$, then $\alpha <\beta $; if $S(\alpha )=S(\beta )$, then (1) if $H(\alpha )=H(\beta )$, then $\alpha =\beta $; (2) if $H(\alpha )<H(\beta )$, then $\alpha <\beta $.

##### Definition 6.

### 2.2 The TODIM Approach

- Step 1. Normalize the $A={({a_{ij}})_{m\times n}}$ into $B={({b_{ij}})_{m\times n}}$.
- Step 2. Compute the dominance degree of ${A_{i}}$ over each alternative ${A_{t}}$ for ${G_{j}}$:
##### (6)

\[ \delta ({A_{i}},{A_{t}})={\sum \limits_{j=1}^{n}}{\phi _{j}}({A_{i}},{A_{t}})\hspace{1em}(i,t=1,2,\dots ,m)\]##### (7)

\[ {\phi _{j}}({A_{j}},{A_{t}})=\left\{\begin{array}{l@{\hskip4.0pt}l}\sqrt{\frac{{w_{jr}}({b_{ij}}-{b_{tj}})}{{\textstyle\textstyle\sum _{j=1}^{n}}{w_{jr}}}},\hspace{1em}& if{b_{ij}}>{b_{tj}};\\ {} 0,\hspace{1em}& if{b_{ij}}={b_{tj}};\\ {} -\frac{1}{\theta }\sqrt{\frac{({b_{ij}}-{b_{tj}}){\textstyle\textstyle\sum _{j=1}^{n}}{w_{jr}}}{{w_{jr}}}},\hspace{1em}& if{b_{ij}}<{b_{tj}},\end{array}\right.\]*θ*depict the attenuation factor of the losses. If ${b_{ij}}-{b_{tj}}>0$, then ${\phi _{j}}({A_{i}},{A_{t}})$ represents a gain; if ${b_{ij}}-{b_{tj}}<0$, then ${\phi _{j}}({A_{i}},{A_{t}})$ signifies a loss. - Step 3. Compute the overall dominance of the alternative ${A_{i}}$ with the following formula:
##### (8)

\[\begin{array}{l}\displaystyle \phi ({A_{i}})=\frac{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})-{\min _{i}}\big\{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})\big\}}{{\max _{i}}\big\{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})\big\}-{\min _{i}}\big\{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})\big\}},\\ {} \displaystyle \hspace{1em}i=1,2,\dots ,m.\end{array}\] - Step 4. Rank and select the best alternative by the overall values $\phi ({A_{i}})$ $(i=1,2,\dots ,m)$. The alternative with the minimum value is the worst. Inversely, the maximum value is the most desirable one.

## 3 TODIM Method for Picture Fuzzy MADM Problems

##### (10)

\[ {\phi _{j}}({A_{j}},{A_{t}})=\left\{\begin{array}{l@{\hskip4.0pt}l}\sqrt{\frac{{w_{jr}}d({r_{ij}},{r_{tj}})}{{\textstyle\textstyle\sum _{j=1}^{n}}{w_{jr}}}},\hspace{1em}& if{r_{ij}}>{r_{tj}};\\ {} 0,\hspace{1em}& if{r_{ij}}={r_{tj}};\\ {} -\frac{1}{\theta }\sqrt{\frac{d({r_{ij}},{d_{tj}}){\textstyle\textstyle\sum _{j=1}^{n}}{w_{jr}}}{{w_{jr}}}},\hspace{1em}& if{r_{ij}}<{r_{tj}}.\end{array}\right.\]##### (11)

\[ d({r_{ij}},{r_{tj}})=\frac{1}{2}\big(|{\mu _{ij}}-{\mu _{tj}}|+|{\eta _{ij}}-{\eta _{tj}}|+|{\nu _{ij}}-{\nu _{tj}}|\big)\]*θ*is the attenuation factor of the losses.

##### (13)

\[ \delta ({A_{i}},{A_{j}})={\sum \limits_{j=1}^{n}}{\phi _{j}}({A_{i}},{A_{t}}),\hspace{1em}i,t=1,2,\dots ,m.\]##### (15)

\[\begin{array}{l}\displaystyle \delta ({A_{i}})=\frac{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})-{\min _{i}}\big\{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})\big\}}{{\max _{i}}\big\{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})\big\}-{\min _{i}}\big\{{\textstyle\textstyle\sum _{t=1}^{m}}\delta ({A_{i}},{A_{t}})\big\}},\\ {} \displaystyle \hspace{1em}i=1,2,\dots ,m,\end{array}\]## 4 Numerical Example and Comparative Analysis

### 4.1 Numerical Example

### 4.2 Comparative Analysis

##### Definition 7.

##### (16)

\[\begin{aligned}{}{r_{i}}=& ({\mu _{i}},{\eta _{i}},{\nu _{i}})\\ {} =& {\mathit{PFWA}_{w}}({r_{i1}},{r_{i2}},\dots ,{r_{im}})={\underset{j=1}{\overset{n}{\bigoplus }}}({w_{j}}{r_{ij}})\\ {} =& \Bigg(1-{\prod \limits_{j=1}^{n}}{(1-{\mu _{ij}})^{{w_{i}}}},{\prod \limits_{j=1}^{n}}{({\eta _{ij}})^{{w_{i}}}},{\prod \limits_{j=1}^{n}}{({\nu _{ij}})^{{w_{i}}}}\Bigg),\end{aligned}\]##### (17)

\[\begin{aligned}{}{r_{i}}=& ({\mu _{i}},{\eta _{i}},{\nu _{i}})\\ {} =& {\mathit{PFWG}_{w}}({r_{i1}},{r_{i2}},\dots ,{r_{im}})={\underset{j=1}{\overset{n}{\bigotimes }}}{({r_{ij}})^{{w_{j}}}}\\ {} =& \Bigg({\prod \limits_{j=1}^{n}}{({\mu _{ij}})^{{w_{i}}}},1-{\prod \limits_{j=1}^{n}}{(1-{\eta _{ij}})^{{w_{i}}}},1-{\prod \limits_{j=1}^{n}}{(1-{\nu _{ij}})^{{w_{i}}}}\Bigg).\end{aligned}\]##### Table 1

PFWA | PFWG | |

${A_{1}}$ | (0.6880,0.2170,0.0390) | (0.3840,0.5363,0.0521) |

${A_{2}}$ | (0.6216,0.2020,0.1120) | (0.4211,0.3729,0.1154) |

${A_{3}}$ | (0.5121,0.2218,0.1077) | (0.4042,0.3270,0.1431) |

${A_{4}}$ | (0.6547,0.2482,0.0607) | (0.2801,0.5696,0.0632) |

${A_{5}}$ | (0.5008,0.1647,0.0561) | (0.3131,0.4816,0.0858) |

##### Table 2

PFWA | PFWG | |

${A_{1}}$ | 0.8245 | 0.6664) |

${A_{2}}$ | 0.7548 | 0.6529 |

${A_{3}}$ | 0.7022 | 0.6305 |

${A_{4}}$ | 0.7970 | 0.6084 |

${A_{5}}$ | 0.7223 | 0.6137 |

##### Table 3

Ordering | |

PFWA | ${A_{1}}>{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$ |

PFWG | ${A_{1}}>{A_{2}}>{A_{3}}>{A_{4}}>{A_{1}}$ |

*et al.*, 2016b), generalized picture fuzzy distance measure (Son, 2016), similarity measures for picture fuzzy sets (Wei, 2018b) and cosine similarity measures for picture fuzzy sets (Wei, 2017c) as shown in Table 4.

##### Table 4

Ordering | |

Picture fuzzy cross-entropy (Wei, 2016) | ${A_{1}}>{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$ |

Picture fuzzy projection models (Wei et al., 2016b) | ${A_{1}}>{A_{2}}>{A_{3}}>{A_{4}}>{A_{1}}$ |

Generalized picture fuzzy distance measure (Son, 2016) | ${A_{1}}>{A_{4}}>{A_{2}}>{A_{3}}>{A_{5}}$ |

Similarity measures for picture fuzzy sets (Wei, 2018b) | ${A_{1}}>{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$ |

Cosine similarity measures for picture fuzzy sets (Wei, 2017c) | ${A_{1}}>{A_{4}}>{A_{2}}>{A_{5}}>{A_{3}}$ |

## 5 Conclusion

*et al.*, 2016; Wei

*et al.*, 2018a; Wei, 2017b; Merigo and Casanovas, 2009; Wei and Wei, 2018; Wei and Lu, 2018b; Zeng, 2017; Wei and Lu, 2017; Wei

*et al.*, 2018b; Gao

*et al.*, 2018a, 2018b; Tang and Wei, 2018; Wei and Lu, 2018a; Wei

*et al.*, 2016a; Wei and Zhang, 2018; Wei

*et al.*, 2018c; Wang

*et al.*, 2018).