*3.2. PULIFPR*

**Definition 4.** *Let U* = (*u*(*p*)*ij*)*n*×*n be a matrix on the object set X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*} *for the LTS S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>2</sup>*τ*]}*, where <sup>u</sup>*(*p*)*ij* = {(*sukij* ,*svkij*), *pkij*} *is a PULIFS, sukij* = [*sukij* ,*sukij* ] *represents the preference of DMs for xi over xj, svkij* = [*svkij* ,*svkij* ] *represents the non-preference of DMs for xi over xj, and <sup>s</sup>πkij* = [*sπkij* ,*sπkij* ] *indicates the hesitancy (vagueness) degree to the preference of DMs for xi over xj. πkij* = 2*τ* − *ukij* − *vkij*, *πkij* = 2*τ* − *ukij* − *vkij, k* = 1, 2, ··· , #*u*(*p*)*ij, and* #*u*(*p*)*ij is the number of PULIFE in <sup>u</sup>*(*p*)*ij. U is called a PULIFPR, if it satisfies the following conditions:*

*(1) pkij*,= *pkji*, *pkii*= 1*;*

*(2) sukij*= *svkji*,*svkij*= *sukji*

$$\mu(3)\qquad\mu(\stackrel{\circ}{p})\_{ii} = \{ ([\stackrel{\circ}{s}\_{\tau}, s\_{\tau}], ^{\prime}[s\_{\tau}, s\_{\tau}]), 1 \} = s\_{\tau};$$

*;*

*(4)* #*u*(*p*)*ij* = #*u*(*p*)*ji;*

*for all i*, *j* = 1, 2, ··· , *n with i* = *j, and* ∑#*u*(*p*)*ij k*=1 *pkij* ≤ 1, 0 ≤ *ukij* + *vkij* ≤ <sup>2</sup>*τ*,*sukij* ⊆ [*<sup>s</sup>*0,*s*2*τ*],*svkij* ⊆ [*<sup>s</sup>*0,*s*2*τ*]*. In particular, when* #*u*(*p*)*ij* = 1 *and* (*sukij* ,*svkij*) ∈ {([*<sup>s</sup>*0,*s*0], [*<sup>s</sup>*2*τ*,*s*2*τ*]),([*<sup>s</sup>*2*τ*,*s*2*τ*], [*<sup>s</sup>*0,*s*0])}*, it means that the preference information given by the decision maker is certain and extreme for xi over xj. However, in a complex decision-making environment, the decision maker often does not give such a judgment with extreme certainty, so this paper only considers the case of* (*sukij* ,*svkij*) ∈ { / ([*<sup>s</sup>*0,*s*0], [*<sup>s</sup>*2*τ*,*s*2*τ*]),([*<sup>s</sup>*2*τ*,*s*2*τ*], [*<sup>s</sup>*0,*s*0])}*. In addition,* ∑#*u*(*p*)*ij k*=1 *pkij* = 0 *means that the decision maker cannot give preference information for xi over xj. Therefore, in order to ensure the completeness of information, we assume that* ∑#*u*(*p*)*ij k*=1 *pkij* > 0*.*

In GDM, it is often necessary to aggregate individual preference information into group preference information. However, due to the knowledge and experience gaps between individuals, the determination of individual weight in the aggregation process is particularly important. Therefore, in order to determine a reasonable individual weight, we first introduce the definition of the distance measure of PULIFPRs

#### *3.3. The Distance Measure of PULIFSs*

Considering that different PULIFS may have different numbers of PULIFE, it may be too complicated to give the distance measurement directly. Therefore, before giving the distance measure of PULIFS, we need to convert PULIFS. According to the partition of uncertain space of PULIFS in Figure 1, we transform the information expressed by PULIFS into two parts: non-fuzzy uncertain information and fuzzy uncertain information. Inspired by the conversion method of probabilistic interval-valued intuitionistic hesitant fuzzy set (PIVIHFS) proposed by Zhai et al. [28], we present the conversion function as follows

**Definition 5.** *Let u*(*p*) = {([*suk* ,*suk* ], [*svk* ,*svk* ]), *p<sup>k</sup>*} *be a PULIFS associated with S, then its non-fuzzy uncertain information transformation function f is defined as*

$$f(u(p)) = \sum\_{k=1}^{\#u(p)} p^k \times \frac{I(s\_{\underline{\mathfrak{u}}^k}) - I(s\_{\overline{\mathfrak{p}}^k}) + I(s\_{\overline{\mathfrak{T}}^k}) - I(s\_{\underline{\mathfrak{p}}^k}) + 4\pi}{8\pi} \tag{5}$$

*and its fuzzy uncertain information transformation function g is defined as*

$$g(u(p)) = \sum\_{k=1}^{\#u(p)} p^k \times \frac{I(s\_{\frac{\pi^k}{\#}}) + I(s\_{\overline{\pi^k}})}{4\pi} \tag{6}$$

*where* #*u*(*p*) *is the number of PULIFE in <sup>u</sup>*(*p*)*, <sup>I</sup>*(·) *is the subscript function of the linguistic term, that is <sup>I</sup>*(*st*) = *t. Moreover, f*(*u*(*p*)) *represents the non-fuzzy information part of PULIFS, <sup>I</sup>*(*suk* ) − *<sup>I</sup>*(*svk* ) *and* *<sup>I</sup>*(*suk* ) − *<sup>I</sup>*(*svk* ) *in Equation (5) can be respectively interpreted as the pessimistic and optimistic attitude values of DMs. On the contrary, g*(*u*(*p*)) *denotes the fuzzy information part of PULIFS, which can be interpreted as the average of information that DMs fail to grasp or ignore.*

**Remark 1.** *On the premise that the original meaning expressed by PULIFS is not lost, we used Equations (5) and (6) to transform the qualitative non-fuzzy and fuzzy information into the specific values in [0,1], so as to simplify the calculation of distance measure. For convenience, we used v* = (*f* , *g*) *to represent the converted PULIFS and call it the conversion set (CS). Thus, for each PULIFPR U* = (*u*(*p*)*ij*)*n*×*n, there is a transformation matrix V* = (*vij*)*n*×*n, where vij* = (*fij*, *gij*)*. Now, we give the definition of the distance measure of PULIFSs.*

**Definition 6.** *Let <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *be two PULIFSs associated with S, v*1 = (*f*1, *g*1) *and v*2 = (*f*2, *g*2) *be the corresponding CSs of <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*)*, then the Hamming distance between <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *is:*

$$d(\mu\_1(p), \mu\_2(p)) = d(v\_1, v\_2) = \frac{1}{2}(|f\_1 - f\_2| + |g\_1 - g\_2|)\tag{7}$$

*the Euclidean distance between <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *is:*

$$d(\mu\_1(p), \mu\_2(p)) = d(v\_1, v\_2) = [\frac{(f\_1 - f\_2)^2 + (g\_1 - g\_2)^2}{2}]^{\frac{1}{2}} \tag{8}$$

*It is obvious that the given distance measure satisfies the following properties:*


For convenience, this paper only takes hamming distance for discussion, and based on the relationship between distance measure and similarity degree, we further give the similarity degree of PULIFSs.

$$s\left(u\_1(p), u\_2(p)\right) = 1 - d(u\_1(p), u\_2(p)) = 1 - \frac{1}{2}(|f\_1 - f\_2| + |g\_1 - g\_2|)\tag{9}$$

Lets give a simple example to show the distance calculation between PULIFSs *<sup>u</sup>*1(*p*) and *<sup>u</sup>*2(*p*).

**Example 1.** *Let LTS S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>8</sup>]}*, and the two PULIFSs are shown below: <sup>u</sup>*1(*p*) = {([*<sup>s</sup>*1,*s*2], [*<sup>s</sup>*4,*s*5]), 0.2,([*<sup>s</sup>*0,*s*2], [*<sup>s</sup>*3,*s*5]), 0.3,([*<sup>s</sup>*2,*s*3], [*<sup>s</sup>*4,*s*5]), 0.5} *<sup>u</sup>*2(*p*) = {([*<sup>s</sup>*4,*s*6], [*<sup>s</sup>*0,*s*1]), 0.45,([*<sup>s</sup>*3,*s*5], [*<sup>s</sup>*1,*s*2]), 0.5} *the values of non-fuzzy function and fuzzy function corresponding to <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *can be easily obtained from Equations (5) and (6) are as follows f*1 = 0.2 × 1−5+2−4+16 32 + 0.3 × 0−5+2−3+16 32 + 0.5 × 2−5+3−4+16 32 = 0.3438*, g*1 = 0.2 × 8−2−5+8−1−4 16 + 0.3 × 8−2−5+8−0−3 16 + 0.5 × 8−3−5+8−2−4 16 = 0.2250*, f*2 = 0.45 × 4−1+6−0+16 32 + 0.5 × 3−2+5−1+16 32 = 0.6797*, g*2 = 0.45 × 8−6−1+8−4−0 16 + 0.5 × 8−5−2+8−3−1 16 = 0.2969*, then the distance between <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *is: d*(*<sup>u</sup>*1(*p*), *<sup>u</sup>*2(*p*)) = 12 (| *f*1 − *f*2| + |*g*1 − *g*2|) = 12 (|0.3438 − 0.6797| + |0.225 − 0.2969|) = 0.2039*, and the corresponding similarity degree is <sup>s</sup>*(*<sup>u</sup>*1(*p*), *<sup>u</sup>*2(*p*)) = 1 − *d*(*<sup>u</sup>*1(*p*), *<sup>u</sup>*2(*p*)) = 0.7961*.*

**Remark 2.** *From the above example, it is not difficult to find that compared with the distance measure defined in general literature, the distance measure proposed in this paper does not need to normalize the initial set, which allows different sets to have different elements and allows for the absence of probability information* (0 < ∑#*u*(*p*) *k*=1 *pk* ≤ 1)*. In addition, under the premise that the original information is not lost, multiple elements in PULIFS are integrated into two parts of fuzzy information and non-fuzzy information, which greatly simplifies the calculation between sets.*

*Symmetry* **2019**, *11*, 234

> Based on the distance between PULIFSs, we give the distance measure of PULIFPRs.

**Definition 7.** *Let Ul* = (*u*(*p*)*lij*)*n*×*n and Um* = (*u*(*p*)*mij*)*n*×*n be two PULIFPRs, and their corresponding transformation matrices are Vl* = (*vlij*)*n*×*n and Vm* = (*vmij*)*n*×*n, where vlij* = (*f lij*, *<sup>g</sup>lij*), *vmij* = (*f mij* , *gmij* )*. Similar to Equation (7), the hamming distance between individual PULIFPRs Ul and Um is defined as:*

$$d(\mathcal{U}^l, \mathcal{U}^m) = d(V^l, V^m) = \frac{1}{n \times (n-1)} \sum\_{i$$

*where f lij* = ∑#*u*(*p*)*lij k*=1 *pkij* × *<sup>I</sup>*(*sukij*)−*<sup>I</sup>*(*svkij*)+*I*(*sukij*)−*<sup>I</sup>*(*svkij*)+<sup>4</sup>*<sup>τ</sup>* 8*τ and glij* = ∑#*u*(*p*)*lij k*=1 *pkij* × *<sup>I</sup>*(*<sup>s</sup>πkij*)+*I*(*<sup>s</sup>πkij*) 4*τ* 

Then the similarity degree between *Ul* and *Um* is defined as:

$$s(\mathcal{U}^l, \mathcal{U}^m) = 1 - d(\mathcal{U}^l, \mathcal{U}^m) \tag{11}$$

*.*

Next, we have used the distance measure and similarity degree between individual PULIFPRs to present the aggregation process of GDM.

#### *3.4. Deriving Individual Weights and Aggregating Individual PULIFPRs*

For GDM problems, without loss of generality, we supposed there are *q* DMs *D* = {*d*1, *d*2, ··· , *dq*} who are invited to compare *n* alternatives *X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*}, and *Ul* = (*u*(*p*)*lij*)*n*×*n* be the individual PULIFPR provided by the DMs *dl*,(*l* = 1, ··· , *q*). Then, based on the similarity degree of PULIFPRs given by the DMs, we defined the confidence degree of the *l*-th decision maker *dl* as:

$$\text{cs}\_{l} = \sum\_{m=1, m \neq l}^{q} \text{s}(\mathcal{U}^{l}, \mathcal{U}^{m}) (l = 1, 2, \dots, \cdot, q) \tag{12}$$

Obviously, the higher the confidence degree of a decision maker, the higher the overall similarity between the decision maker and other DMs, and the greater the importance of the decision maker in GDM. Therefore, we regarded the normalized confidence degree *csNl* as the weight of individual in GDM, where *csNl* = *csl* <sup>∑</sup>*ql*=<sup>1</sup> *csl* . Let the weight of the *l*-th decision maker be *wl* = *csNl* , then <sup>∑</sup>*ql*=<sup>1</sup> *wl* = 1 and 0 ≤ *wl* ≤ 1,(*l* = 1, 2, ··· , *q*).

In order to aggregate individual PULIFPRs into a collective one, the basic operational laws between PULIFSs *<sup>u</sup>*1(*p*) = {([*suk*1 ,*suk*1 ], [*svk*1 ,*svk*1 ]), *<sup>p</sup>k*1} and *<sup>u</sup>*2(*p*) = {([*suk*2 ,*suk*2 ], [*svk*2 ,*svk*2 ]), *<sup>p</sup>k*2} is given as follows:

$$u\_1(p)\bigoplus u\_{\overline{\mathbb{L}}}(p) = \bigcup\_{k \in \{1, \cdots, \#u\_1(p)\}} \{ (\langle [s\_{\underline{\mathbb{L}}^k\_1 + \underline{\mathbb{L}}^k\_1} s\_{\overline{\mathbb{L}}^k\_1 + \overline{\mathbb{L}}^k\_1}], [s\_{\underline{\mathbb{L}}^k\_1 + \underline{\mathbb{L}}^k\_1} s\_{\overline{\mathbb{L}}^k\_1 + \overline{\mathbb{L}}^k\_1}]), \frac{p\_1^k + p\_2^k}{2} \rangle \} \tag{13}$$

$$\lambda \mu\_1(p) = \bigcup\_{k \in \{1, \dots, \#\mu\_1(p)\}} \{ ([\mathbf{s}\_{\lambda \underline{\mathbf{u}}^k \prime} s\_{\lambda \overline{\mathbf{u}}^k\_1}] \prime [\mathbf{s}\_{\lambda \underline{\mathbf{u}}^k\_1 \prime} s\_{\lambda \overline{\mathbf{v}}^k\_1}]) \} \; p\_1^k \} \tag{14}$$

where #*u*1(*p*) = #*u*2(*p*), when #*u*1(*p*) = #*u*2(*p*), we normalized it by the following method:

If #*u*1(*p*) = #*u*2(*p*), #*u*1(*p*) > #*u*2(*p*), then we have added #*u*1(*p*) − #*u*2(*p*) PULIFEs to *<sup>u</sup>*2(*p*) so that the PULIFSs *<sup>u</sup>*1(*p*) and *<sup>u</sup>*2(*p*) have the same number of elements. The added uncertain linguistic intuitionistic variables (ULIVs) are the smallest one(s) in *<sup>u</sup>*2(*p*), and the probabilities of the added ULIVs are zero. In addition, the comparison method of two PULIFE *ek* = ([*suk* ,*suk* ], [*svk* ,*svk* ]), *p<sup>k</sup>* and *el* = ([*sul* ,*sul* ], [*svl* ,*svl* ]), *pl* in PULIFS is as follows Let 

$$\begin{aligned} f\_i &= p^i \times [I(\mathbf{s\_{\underline{n}^i}}) - I(\mathbf{s\_{\overline{n}^i}}) + I(\mathbf{s\_{\overline{n}^i}}) - I(\mathbf{s\_{\overline{n}^i}})], \\ g\_i &= p^i \times [I(\mathbf{s\_{\overline{n}^i}}) + I(\mathbf{s\_{\overline{n}^i}})], (i = k, l). \end{aligned} \tag{15}$$

(1) If *gk* > *gl*, then *ek* < *el*; (2) If *gk* = *gl*, then

> (*a*) If *fk* > *fl*, then *ek* > *el*; (*b*) If *fk* = *fl*, then *ek* = *el*.

The larger the PULIFE, the larger its corresponding ULIV. Based on this, we give the definition of probabilistic uncertain linguistic intuitionistic weighted average (PULIWA) operator.

**Definition 8.** *Given q PULIFSs ui*(*p*) = {([*suki* ,*suki* ], [*svki* ,*svki* ]), *pki* },(*<sup>i</sup>* = 1, 2, ··· , *q*), *k* = 1, 2 ··· , #*ui*(*p*)*, the weight vector W* = (*<sup>w</sup>*1, *w*2, ··· , *wq*), *wi* ∈ [0, 1], ∑*qi*=<sup>1</sup> *wi* = 1*, then we called*

$$\begin{split} PWA(u\_1(p), \dots, u\_q(p)) &= \bigoplus\_{i=1}^q w\_i u\_i(p) \\ &= \bigcup\_{k=1,2,\dots,\emptyset u\_i(p)} \{ (\left[ \mathbf{s}\_{\sum\_{i=1}^q w\_i \mathbf{z}\_i^{k\_i}, \sum\_{i=1}^q w\_i \mathbf{z}\_i^{k\_i} \right], \left[ \mathbf{s}\_{\sum\_{i=1}^q w\_i \mathbf{z}\_i^{k\_i}, \sum\_{i=1}^q w\_i \mathbf{z}\_i^{k\_i} \right]), \sum\_{i=1}^q p\_i^k} \} \end{split} \tag{16}$$

*the PULIWA operator.*

**Example 2.** *Continuing with Example 1, assuming that the weight values of both PULIFS <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *are 0.5. Since* #*u*2(*p*) = 2 < #*u*1(*p*) = 3*, we can easily know from Equation (15) that g*21 = 0.45 · [8 − 1 − 6 + 8 − 4 − 0] = 2.25 *and g*22 = 0.5 · [8 − 5 − 2 + 8 − 3 − 1] = 2.5 *in <sup>u</sup>*2(*p*)*. Therefore, g*21 < *g*22,*e*21 > *e*22*, the normalized <sup>u</sup>*2(*p*) = {([*<sup>s</sup>*4,*s*6], [*<sup>s</sup>*0,*s*1]), 0.45,([*<sup>s</sup>*3,*s*5], [*<sup>s</sup>*1,*s*2]), 0.5, ([*<sup>s</sup>*3,*s*5], [*<sup>s</sup>*1,*s*2]), <sup>0</sup>}*, and the PULIWA operator PWA*(*<sup>u</sup>*1(*p*), *<sup>u</sup>*2(*p*)) = {([*<sup>s</sup>*2.5,*s*4], [*<sup>s</sup>*2,*s*3]), 0.325, ([*<sup>s</sup>*1.5,*s*3.5], [*<sup>s</sup>*2,*s*3.5]), 0.4,([*<sup>s</sup>*2.5,*s*4], [*<sup>s</sup>*2.5,*s*3.5]), 0.5} *of <sup>u</sup>*1(*p*) *and <sup>u</sup>*2(*p*) *can be obtained by using Equation (16).*

Obviously, it is easy to aggregate individual PR into collective PR by using Equation (16). Therefore, the process of obtaining priority weights is given in the following discussion based on the consistency of collective PULIFPR *U* ˜ .

#### **4. Consistency Analysis of PULIFPR and Acquisition of Its Priority Weight**

#### *4.1. Consistency Analysis of PULIFPR*

At present, the research on the consistency of PR is mainly divided into two categories: multiplicative consistency and additive consistency. Without loss of generality, this paper discusses PULIFPR consistency based on multiplicative consistency. Therefore, before giving the definition of PULIFPR consistency, lets review the multiplicative consistency of fuzzy preference relations (FPRs).

**Definition 9.** *[29]. For the FPR R* = (*rij*)*n*×*n*,(*<sup>i</sup>*, *j* = 1, 2, ··· , *<sup>n</sup>*),*rij* ∈ [0, 1]*, if we have*

$$r\_{ij} = \frac{w\_i}{w\_i + w\_j}.\tag{17}$$

*for all i*, *j* = 1, 2, ··· , *n, and which satisfies: 1) rii* = 0.5*; 2) rij* + *rji* = 1*; 3)* ∑*ni*=<sup>1</sup> *wi* = 1*. then we called the FPR R is multiplicative consistent, where rij is the preference degree of the objectives xi over xj, and wi* ∈ [0, 1]*, w* = (*<sup>w</sup>*1, *w*2, ··· , *wn*) *is the priority vector of R.*

Inspired by this, we presented the following definition of consistency by combining the preferences, non-preferences and vagueness information expressed by PULIFPR.

**Definition 10.** *Let U* ˜ = (*u*(*p*)*ij*)*n*×*n be a PULIFPR on the object set X* = {*<sup>x</sup>*1, *x*2, ··· , *xn*} *for the LTS S* = {*<sup>s</sup>α*|*<sup>α</sup>* ∈ [0, <sup>2</sup>*τ*]}*, its corresponding transformation matrix is V* = (*vij*)*n*×*n, where vij* = (*fij*, *gij*) *is a CS. Based on this, we can extract the FPR H* = (*hij*)*n*×*n from the PULIFPR U, where* ˜

$$h\_{ij} = \begin{cases} \theta\_{i\bar{j}} f\_{i\bar{j}} + (1 - \theta\_{i\bar{j}}) g\_{i\bar{j}\nu} & \text{if } \ i < j \\ 0.5, & \text{if } \ i = j \\ 1 - h\_{\bar{j}i\nu} & \text{if } \ i > j \end{cases} \tag{18}$$

*If we have*

$$h\_{i\bar{j}} = \frac{w\_i}{w\_{\bar{i}} + w\_{\bar{j}}} \tag{19}$$

*for all i*, *j* = 1, 2, ··· , *n, then we called PULIFPR U* ˜ *multiplicative consistent, where θij* ∈ [0, 1] *represents the importance of non-fuzzy information fij extracted from <sup>u</sup>*(*p*)*ij , and w* = (*<sup>w</sup>*1, *w*2, ··· , *wn*) *is the priority vector of U, satisfying w* ˜ *i* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *wi* = 1*.*

**Remark 3.** *From Equations (5) and (6), it is not difficult to see that the values of non-fuzzy information fij and fuzzy information gij extracted from PULIFPR are all located in [0,1]. Furthermore, it is easy to know that fii* = 0.5 *and gii* = 0 *by the nature of PULIFPR. Therefore, the FPR H* = (*hij*)*n*×*n is generated by combining the non-fuzzy and fuzzy information of the DMs, and the consistency of PULIFPR is transformed into the consistency of the FPR. However, this definition of consistency only considers the fuzzy and non-fuzzy space of PULIFPR in general. In order to make full use of the decision-making information expressed by PULIFPR, we have considered the decision maker's risk attitude and further discuss its consistency with the preference and non-preference information in the non-fuzzy space.*

**Definition 11.** *Based on Definitions 4 and 5, we set aij* = ∑#*u*(*p*) *k*=1 *pk* × *<sup>I</sup>*(*sukij*)−*<sup>I</sup>*(*svkij*)+<sup>2</sup>*<sup>τ</sup>* 4*τ as the maximum preference (the most optimistic judgment) of DMs for xi over xj, while bij* = ∑#*u*(*p*) *k*=1 *pk* × *<sup>I</sup>*(*sukij*)−*<sup>I</sup>*(*svkij*)+<sup>2</sup>*<sup>τ</sup>* 4*τ as the minimum preference (the most pessimistic judgment) of DMs for xi over xj. So similarly, we can extract a FPR D* = (*dij*)*n*×*n from the PULIFPR U, where* ˜

$$d\_{ij} = \begin{cases} t\_{ij}a\_{ij} + (1 - t\_{ij})b\_{ij\prime} & \text{If } i < j \\ 0.5, & \text{If } i = j \\ 1 - d\_{ji\prime} & \text{If } i > j \end{cases} \tag{20}$$

*If we have*

$$d\_{ij} = \frac{w\_i'}{w\_i' + w\_j'} \tag{21}$$

*for all i*, *j* = 1, 2, ··· , *n, then we also called PULIFPR U* ˜ *multiplicative consistent, where tij* ∈ [0, 1] *indicates the degree of optimism of the DMs, the bigger the values of tij, the higher DM's optimistic degree, and w* = (*w* 1, *w* 2, ··· , *w n*) *is the priority vector of U, satisfying w* ˜ *i* ∈ [0, 1] *and* ∑*ni*=<sup>1</sup> *w i* = 1*.*

Based on the two consistency definitions given above, we give the method to obtain the priority weight of collective PULIFPR.

#### *4.2. Determine the Priority Weights of PULIFPR through the GPM*

The consistency definition given by Equation (19) integrates the fuzzy uncertainty and non-fuzzy uncertainty of the information given by the decision-maker. However, in actual decision-making, we hope that the fuzzy uncertainty degree of information expressed by PULIFPR is as small as possible, so as to make the ranking result as reasonable and accurate as possible. Therefore, the higher the value of parameter *<sup>θ</sup>ij*, which indicates the importance of non-fuzzy uncertainty information, the more reasonable the result will be. Based on this principle, we establish the following GPM to obtain the priority weight of the PULIFPR.

$$\max \theta = \sum\_{i$$

$$\text{s.t.} \begin{cases} h\_{ij} = \frac{w\_i}{w\_i + w\_j}, & \text{(1)}\\ \sum\_{i \neq j} w\_i > w\_j - 0.5, & \text{(2)}\\ \sum\_{i=1}^n w\_i = 1, w\_i \ge 0, & \text{(3)}\\ 0 \le \theta\_{ij} \le 1, (i, j = 1, 2, \dots, n; i < j). \end{cases} \tag{22}$$

In Equation (22), the constraint condition (1) guarantees the consistency of the FPR *H* extracted from PULIFPR *U* ˜ , and the constraint condition (2) avoids the occurrence of extreme judgment caused by individual subjective preference, thus guaranteeing the objectivity of decision-making process. In addition, the literature [30] shows that the consistency of the FPR only needs to discuss the upper triangular part of it. So to simplify the calculation, we have *i* < *j*.

If the feasible region of Equation (22) is nonempty, the optimal solutions *θij* and priority weight vector *wi*,(*<sup>i</sup>* = 1, 2, ··· , *n*) can be obtained by solving it. However, it does not guarantee that there will always be nonempty feasible regions. Therefore, when the feasible region is empty, we expand the feasible region of the model by appropriately increasing the fuzzy uncertain information value *gij* and reducing the non-fuzzy uncertain information value *fij*, and the expanded model is as follows

$$\begin{aligned} \max \theta &= \sum\_{i w\_j - 0.5, \\ \sum\_{i=1}^n w\_i = 1, w\_i \ge 0, \ \phi\_{ij}, \psi\_{ij} \ge 0, \\ 0 \le \theta\_{ij} \le 1, (i, j = 1, 2, \dots, n; i < j). \end{cases} \end{aligned} \tag{23}$$

where *φij* and *ψij* are the deviation variables, satisfying *φij* ≥ 0, *ψij* ≥ 0. Then, by solving (23), the optimal solutions *θij* and priority weight vector *wi*,(*<sup>i</sup>* = 1, 2, ··· , *n*) can be obtained.

Similarly, it is easy to know from Definition 11 that the bigger the value of *tij*, the higher the degree of optimism of the decision maker. Therefore, we combine the two extreme attitudes of the decision maker, the most optimistic and the most pessimistic, and respectively present the following GPM.

$$\max t = \sum\_{i$$

$$\begin{cases} d\_{ij} = \frac{w\_i^+}{w\_i^+ + w\_j^+}, \\ \sum\_{i \ne j} w\_i^+ > w\_j^+ - 0.5, \\ \sum\_{i < j} t\_{ij} < \frac{n \times (n-1)}{2} - 1, \\ \sum\_{i=1}^n w\_i^+ = 1, w\_i^+ \ge 0, \\ 0 \le t\_{ij} \le 1, (i, j = 1, 2, \dots, n; i < j) \end{cases} \tag{24}$$
 
$$\min t = \sum\_{i < j} t\_{ij}$$

$$\begin{cases} d\_{lj} = \frac{w\_i^-}{w\_i^- + w\_j^-}, \\ \sum\_{i \neq j} w\_i^- > w\_j^- - 0.5, \\ \sum\_{i < j} t\_{ij} > 1, \\ \sum\_{i=1}^n w\_i^- = 1, w\_i^- \ge 0, \\ 0 \le t\_{ij} \le 1, (i, j = 1, 2, \cdots, n; i < j) \end{cases} \tag{25}$$

where *w*<sup>+</sup> = ( *w*<sup>+</sup> 1 , *w*<sup>+</sup> 2 , ··· , *w*<sup>+</sup> *n* ) represents the most optimistic weight vector and *w*<sup>−</sup> = (*w*<sup>−</sup> 1 , *w*<sup>−</sup> 2 , ··· , *w*<sup>−</sup> *n* ) represents the most pessimistic weight vector. It is noted that different from Model (22), Equations (24) and (25) have added restriction condition ∑*<sup>i</sup>*<*<sup>j</sup> tij* < *<sup>n</sup>*<sup>×</sup>(*<sup>n</sup>*−<sup>1</sup>) 2 − 1 and ∑*<sup>i</sup>*<*<sup>j</sup> tij* > 1 respectively, which ensures that the decision maker does not show overly optimistic or pessimistic judgment information when in a rational state. Similarly, for Model (23), when the feasible regions of Equations (24) and (25) are empty, we give the expansion model as follows

$$\max t = \sum\_{i
$$\begin{cases} t\_{ij}(a\_{ij} - a\_{ij}) + (1 - t\_{ij})(b\_{ij} + \beta\_{ij}) = \frac{w\_{i}^{+}}{w\_{i}^{+} + w\_{j}^{+}},\\ \sum\_{i \neq j} w\_{i}^{+} > w\_{i}^{+} - 0.5, \\ \sum\_{i
$$\begin{cases} 0 \le t\_{ij} \le 1, (i, j = 1, 2, \cdots, n; i < j) \\\\ \min t = \sum\_{i
$$\begin{cases} t\_{ij}(a\_{ij} + a\_{ij}) + (1 - t\_{ij})(b\_{ij} - \beta\_{ij}) = \frac{w\_{i}^{-}}{w\_{i}^{+} + w\_{j}^{-}},\\ \sum\_{i w\_{i}^{-} - 0.5, \end{cases}$$
$$
$$
$$

$$\begin{cases} \text{u}\_{l}(w\_{l} + w\_{l}) + (1 - w\_{l})(w\_{l} - p\_{l}) - w\_{w\_{l}} \cdot \text{w} \\ \sum\_{i \neq j} w\_{i}^{-} > w\_{j}^{-} - 0.5, \\ \sum\_{i < j} t\_{ij} > 1, \\ \sum\_{i=1}^{n} w\_{i}^{-} = 1, w\_{i}^{-} \ge 0, \ a\_{ij}, \beta\_{ij} \ge 0. \\ 0 \le t\_{ij} \le 1, (i, j = 1, 2, \dots, n; i < j) \end{cases} \tag{27}$$

where *<sup>α</sup>ij* and *βij* are the deviation variables, satisfying *<sup>α</sup>ij* ≥ 0, *βij* ≥ 0.

By solving Equations (26) and (27), the optimal weight vectors *w*<sup>+</sup> *i* and *w*<sup>−</sup> *i* (*i* = 1, 2, ··· , *n*) can be obtained respectively. Combining *w*<sup>+</sup> *i* with *w*<sup>−</sup> *i* , the compromise weight vector can be obtained as follows:

$$w\_i' = \lambda w\_i^+ + (1 - \lambda)w\_i^-, i = 1, 2, \cdots, n \tag{28}$$

where *λ* ∈ [0, 1] represents the risk attitude of DMs. If 0 ≤ *λ* < 0.5, DMs are risk averse; If *λ* = 0.5, DMs are risk neutral; If 0.5 < *λ* ≤ 1, DMs are risk taking.

Considering that the decision maker pays more attention to the final result in the actual decision, we take the average value of priority weight obtained under the two consistency definitions as the final ranking weight, namely *<sup>w</sup>*¯*i* = *wi*+*w i* 2 , *i* = 1, 2, ··· , *n*.

**Remark 4.** *Compared with general programming models for solving priority weights, the main advantages of the programming models presented in this paper are as follows:*

*(1) At present, most of the programming models proposed in many literatures only consider the principle of minimum consistency deviation, such as literatures [27,31–34]. In this paper, the consistency of the newly proposed PULIFPR is considered comprehensively from the three aspects of fuzzy and non-fuzzy uncertain information and DM's risk attitude. Therefore, the rationality of decision result is greatly improved.*

*(2) Currently, most of the research on PR needs to test its consistency, and some literatures that needs to test the consistency of acceptable PR fails to provide a reasonable test method, such as the research on triangular FPR by Wang [35], and the research on interval-valued intuitionistic FPR by Wan et al. [36]. In this paper, the priority weight of consistent PULIFPR can be obtained directly through the proposed programming models without considering the consistency test, which greatly simplifies the DM process.*

#### *4.3. A New Algorithm for Solving GDM with PULIFPR*

Summarizing above analyses, a new method for GDM with PULIFPR is developed as follows: Step 1. Calculate the distance measure *<sup>d</sup>*(*Ul*, *Um*) and similar measure *<sup>s</sup>*(*Ul*, *Um*)(*l*, *m* =

1, 2, ··· , *q*, *l* = *m*) between individual PULIFPRs by Equations (10) and (11).

Step 2. Use Equation (12) to calculate the confidence degree *csl* and determine the individual weight *w<sup>l</sup>*(*<sup>l</sup>* = 1, 2, ··· , *q*).

Step 3. Aggregating individual PULIFPR *U* into collective PULIFPR *U* ˜ by Equation (16).

Step 4. When feasible regions of Models (22), (24) and (25) are nonempty, priority weights *wi*, *<sup>w</sup>*+*i* and *<sup>w</sup>*<sup>−</sup>*i* can be solved respectively. Otherwise, priority weights *wi*, *<sup>w</sup>*+*i* and *<sup>w</sup>*<sup>−</sup>*i* shall be obtained by solving Equations (23), (26) and (27).

Step 5. Determining the risk parameter value *λ*, and then the compromise weight *w i* is obtained by Equation (28).

Step 6. Combining *wi* and *w i*, the comprehensive weight *<sup>w</sup>*¯*i* = *wi*+*w i* 2is obtained.

Step 7. According to the comprehensive weight value to compare the alternatives, the best alternative has the bigger value.

The graphical process of solving GDM using PULIFPR is shown in Figure 2.

**Figure 2.** Process of group decision-making (GDM) with PULIFPRs.

#### **5. Case Application and Comparative Analysis**

In order to demonstrate the effectiveness and practicability of the proposed method, this section is mainly divided into two parts. The first part discusses the application of the proposed method in the world VR industry conference 2018. The second part gives the comparative analysis between the proposed method and other methods.

#### *5.1. Application in VR Project Selection*

In recent years, virtual reality (VR) technology has received unprecedented attention from all sectors of society, and it is regarded as the portal of the next generation general computing platform and Internet along with augmented reality (AR) and mixed reality (MR). In addition, as an important force leading a new round of industrial reform in the world, it plays an important role in promoting new economic development. Therefore, in order to explore the key and common problems in the development of VR, as well as the industrial development trend and solutions, the 2018 world VR industry conference was successfully held in nanchang, jiangxi province on 19 October. As one of the important activities of the conference, the industrial counterpart conference was successfully held in nanchang on 20 October.

However, in order to ensure the successful holding of the industrial docking conference, it is particularly important for the organizers to have extensive and in-depth communication with the investors in the early stage of the conference. On the one hand, it can enable investors to have a deep and sufficient understanding of each VR project in our province so that investors can select the best cooperation project. On the other hand, it is convenient for every VR industry company in our province to select the best partner or investor. Finally, the cooperation agreements reached at the industry conference are guaranteed. Therefore, the communication and mutual selection process is an important preparation work in the early stage of the conference.

Due to the complexity of VR technology, VR project selection is a very challenging task for investors. It requires investors to make a comprehensive analysis and judgment on the competitive advantage, profitability, viability and development potential of VR project from the perspectives of simulation technology and computer graphics, man-machine interface technology, sensor technology and network technology,etc. Therefore, the project selection process is often a GDM. Without loss of generality, in order to demonstrate the GDM process using the proposed method, we take the four important projects selected by Microsoft as an example. The four projects are Touch display integration project *x*1, Optoelectronic project *x*2, Network security industry center project *x*3 and Intelligent VR visual equipment project *x*4 respectively.

In view of the complexity of VR project, Microsoft sent two investment teams (*<sup>e</sup>*1,*<sup>e</sup>*2) to inspect the project content and one technical team (*e*3) to inspect the company's technical equipment. Due to the wide range of knowledge involved in VR project and the complexity of factors to be considered by DMs, the decision team can only give judgment information from positive and negative aspects based on the LTS *S* = {*<sup>s</sup>*0: extremely poor, *s*1: very poor, *s*2: poor, *s*3: slightly poor, *s*4: fair, *s*5: slightly good, *s*6: good, *s*7: very good, *s*8: extremely good}.

For example, by analyzing and comparing projects *x*1 and *x*2, the decision team *e*1 gave the following judgment information:

$$\mu(p)\_{12} = \{ \langle ([\mathbf{s}\_{4\prime}, \mathbf{s}\_6], [\mathbf{s}\_1, \mathbf{s}\_1]), \mathbf{0.45} \rangle, \langle ([\mathbf{s}\_3, \mathbf{s}\_5], [\mathbf{s}\_1, \mathbf{s}\_2]), \mathbf{0.5} \rangle \}$$

where [*<sup>s</sup>*4,*s*6] indicates that the DM's preference degree for *x*1 over *x*2 is between fair and good, [*<sup>s</sup>*1,*s*1] expresses that the DM's non-preference degree for *x*1 over *x*2 is very poor. The probability 0.45 indicates that 45% of the people in investment teams *e*1 give interval intuitionistic judgment information ([*<sup>s</sup>*4,*s*6], [*<sup>s</sup>*1,*s*1]). Similarly, PULIFE ([*<sup>s</sup>*3,*s*5], [*<sup>s</sup>*1,*s*2]), 0.5 indicated that 50% of the people in investment team *e*1 gave the interval intuitionistic judgment information as ([*<sup>s</sup>*3,*s*5], [*<sup>s</sup>*1,*s*2]). In addition, 5% of the people failed to give any judgment information.

Now, we regard the three teams *e*1,*e*2 and *e*3 sent by Microsoft as three individuals, and take the four projects of jiangxi province, *x*1, *x*2, *x*3, and *x*4 as the alternatives. The preference information given by the three teams in the form of PULIFPR is as follows

*U*1 = (*<sup>u</sup>*1(*p*)*ij*)<sup>4</sup>×4,(*<sup>i</sup>*, *j* = 1, 2, 3, 4) where *<sup>u</sup>*1(*p*)11 = *<sup>u</sup>*1(*p*)22 = *<sup>u</sup>*1(*p*)33 = *<sup>u</sup>*1(*p*)44 = *s*4; *<sup>u</sup>*1(*p*)12 = {([*<sup>s</sup>*1,*s*3], [*<sup>s</sup>*4,*s*5]), 0.3,([*<sup>s</sup>*1,*s*2], [*<sup>s</sup>*5,*s*6]), 0.6} *<sup>u</sup>*1(*p*)13 = {([*<sup>s</sup>*4,*s*6], [*<sup>s</sup>*0,*s*1]), 0.8,([*<sup>s</sup>*5,*s*6], [*<sup>s</sup>*1,*s*2]), 0.2} *<sup>u</sup>*1(*p*)14 = {([*<sup>s</sup>*3,*s*4], [*<sup>s</sup>*4,*s*4]), 0.6,([*<sup>s</sup>*2,*s*3], [*<sup>s</sup>*4,*s*5]), 0.2} *<sup>u</sup>*1(*p*)23 = {([*<sup>s</sup>*0,*s*2], [*<sup>s</sup>*6,*s*6]), 0.7,([*<sup>s</sup>*1,*s*3], [*<sup>s</sup>*5,*s*5]), 0.3} *<sup>u</sup>*1(*p*)24 = {([*<sup>s</sup>*5,*s*6], [*<sup>s</sup>*0,*s*1]), 0.2,([*<sup>s</sup>*6,*s*7], [*<sup>s</sup>*0,*s*1]), 0.8} *<sup>u</sup>*1(*p*)34 = {([*<sup>s</sup>*0,*s*1], [*<sup>s</sup>*5,*s*6]), 0.9,([*<sup>s</sup>*0,*s*1], [*<sup>s</sup>*7,*s*7]), 0.1} *U*2 = (*<sup>u</sup>*2(*p*)*ij*)<sup>4</sup>×4,(*<sup>i</sup>*, *j* = 1, 2, 3, 4) where *<sup>u</sup>*2(*p*)11 = *<sup>u</sup>*2(*p*)22 = *<sup>u</sup>*2(*p*)33 = *<sup>u</sup>*2(*p*)44 = *s*4; *<sup>u</sup>*2(*p*)12 = {([*<sup>s</sup>*2,*s*3], [*<sup>s</sup>*4,*s*4]), 0.4,([*<sup>s</sup>*3,*s*4], [*<sup>s</sup>*4,*s*5]), 0.6} *<sup>u</sup>*2(*p*)13 = {([*<sup>s</sup>*1,*s*2], [*<sup>s</sup>*5,*s*5]), 0.3,([*<sup>s</sup>*2,*s*3], [*<sup>s</sup>*4,*s*5]), 0.5} *<sup>u</sup>*2(*p*)14 = {([*<sup>s</sup>*5,*s*6], [*<sup>s</sup>*1,*s*1]), 0.3,([*<sup>s</sup>*6,*s*6], [*<sup>s</sup>*1,*s*2]), 0.6} *<sup>u</sup>*2(*p*)23 = {([*<sup>s</sup>*6,*s*7], [*<sup>s</sup>*1,*s*1]), 0.4,([*<sup>s</sup>*5,*s*6], [*<sup>s</sup>*1,*s*2]), 0.5} *<sup>u</sup>*2(*p*)24 = {([*<sup>s</sup>*0,*s*2], [*<sup>s</sup>*5,*s*6]), 0.7,([*<sup>s</sup>*1,*s*2], [*<sup>s</sup>*6,*s*6]), 0.3} *<sup>u</sup>*2(*p*)34 = {([*<sup>s</sup>*3,*s*5], [*<sup>s</sup>*2,*s*2]), 0.4,([*<sup>s</sup>*2,*s*5], [*<sup>s</sup>*1,*s*2]), 0.4} *U*3 = (*<sup>u</sup>*3(*p*)*ij*)<sup>4</sup>×4,(*<sup>i</sup>*, *j* = 1, 2, 3, 4) where *<sup>u</sup>*3(*p*)11 = *<sup>u</sup>*1(*p*)22 = *<sup>u</sup>*1(*p*)33 = *<sup>u</sup>*1(*p*)44 = *s*4; *<sup>u</sup>*3(*p*)12 = {([*<sup>s</sup>*2,*s*4], [*<sup>s</sup>*3,*s*4]), 0.2,([*<sup>s</sup>*4,*s*5], [*<sup>s</sup>*3,*s*3]), 0.5} *<sup>u</sup>*3(*p*)13 = {([*<sup>s</sup>*1,*s*2], [*<sup>s</sup>*5,*s*6]), 0.8,([*<sup>s</sup>*2,*s*3], [*<sup>s</sup>*5,*s*5]), 0.2} *<sup>u</sup>*3(*p*)14 = {([*<sup>s</sup>*7,*s*8], [*<sup>s</sup>*0,*s*0]), 0.3,([*<sup>s</sup>*6,*s*7], [*<sup>s</sup>*1,*s*1]), 0.6} *<sup>u</sup>*3(*p*)23 = {([*<sup>s</sup>*0,*s*1], [*<sup>s</sup>*6,*s*6]), 0.6,([*<sup>s</sup>*1,*s*1], [*<sup>s</sup>*6,*s*7]), 0.3} *<sup>u</sup>*3(*p*)24 = {([*<sup>s</sup>*4,*s*5], [*<sup>s</sup>*1,*s*2]), 0.7,([*<sup>s</sup>*5,*s*5], [*<sup>s</sup>*2,*s*3]), 0.3} *<sup>u</sup>*3(*p*)34={([*<sup>s</sup>*7,*s*7],[*<sup>s</sup>*0,*s*1]),0.3,([*<sup>s</sup>*5,*s*6],[*<sup>s</sup>*1,*s*2]),0.6}

Since the upper triangle of PULIFPR has a one-to-one correspondence with the lower triangle, we only give the upper triangle of the preference relation. According to Section 4.3, we can solve the GDM problem about project selection as follows:

**Step 1**: According to Equation (10), the distance measure between *U*1 and *U*2 is *d*(*<sup>U</sup>*1, *<sup>U</sup>*2) = 1 4×3 (| *f* 112 − *f* 212| + |*g*112 − *g*212| + | *f* 113 − *f* 213| + |*g*113 − *g*213| + | *f* 114 − *f* 214| + |*g*114 − *g*214| + | *f* 123 − *f* 223| + |*g*123 − *g*223| + | *f* 124 − *f* 224| + |*g*124 − *g*224| + | *f* 134 − *f* 234| + |*g*134 − *g*234|) = 112 (|0.2531 − 0.425| + |0.775 − 0.2719| + |0.3563 − 0.7031| + |0.225 − 0.7125| + |0.8625 − 0.2188| + |0.1781 − 0.5| + |0.1313 − 0.075| + |0.275 − 0.1188| + |0.0625 − 0.0938| + |0.125 − 0.0875| + |0.15 − 0.15| + |0.2313 − 0.25|) = 0.2313, Similarly, we can calculate *d*(*<sup>U</sup>*1, *<sup>U</sup>*3) = 0.1930 and *d*(*<sup>U</sup>*2, *<sup>U</sup>*3) = 0.1398 respectively, so the corresponding similarity degree is *<sup>s</sup>*(*<sup>U</sup>*1, *<sup>U</sup>*2) = 0.7687,*s*(*<sup>U</sup>*1, *<sup>U</sup>*3) = 0.8070 and *<sup>s</sup>*(*<sup>U</sup>*2, *<sup>U</sup>*3) = 0.8602 respectively.

**Step 2**: According to Equation (12), the confidence degree of the three teams can be calculated as *cs*1 = *<sup>s</sup>*(*<sup>U</sup>*1, *<sup>U</sup>*2) + *<sup>s</sup>*(*<sup>U</sup>*1, *<sup>U</sup>*3) = 1.5758, *cs*2 = 1.6289 and *cs*3 = 1.6672, so the weight of each team can be further determined as *w*<sup>1</sup> = *cs*1 *cs*1+*cs*2+*cs*3= 0.3234, *w*<sup>2</sup> = 0.3344, and *w*<sup>3</sup> = 0.3422.

**Step 3**: By using Equation (16), the collective PULIFPR *U* ˜ can be obtained as follows *U* ˜ = (*u*˜(*p*)*ij*)<sup>4</sup>×4,(*<sup>i</sup>*, *j* = 1, 2, 3, 4)

where

*u* ˜(*p*)11 = *<sup>u</sup>*˜(*p*)22 = *<sup>u</sup>*˜(*p*)33 = *<sup>u</sup>*˜(*p*)44 = *s*4; *u* ˜(*p*)12 = {([*<sup>s</sup>*1.6766,*s*3.3442], [*<sup>s</sup>*3.6578,*s*4.3234]), 0.3,([*<sup>s</sup>*2.6953,*s*3.6953], [*<sup>s</sup>*3.0717,*s*4.6390]), 0.5667}; *u* ˜(*p*)13 = {([*<sup>s</sup>*1.9703,*s*3.2938], [*<sup>s</sup>*3.3828,*s*4.0484]), 0.6333,([*<sup>s</sup>*2.9703,*s*3.9703], [*<sup>s</sup>*3.3719,*s*4.0297]), 0.3}; *u* ˜(*p*)14 = {([*<sup>s</sup>*5.0375,*s*6.0375], [*<sup>s</sup>*1.6281,*s*1.6281]), 0.4,([*<sup>s</sup>*4.7062,*s*5.3719], [*<sup>s</sup>*1.9703,*s*2.6281]), 0.4667}; *u* ˜(*p*)23 = {([*<sup>s</sup>*2.0061,*s*3.3295], [*<sup>s</sup>*4.3283,*s*4.3283]), 0.5667,([*<sup>s</sup>*2.3374,*s*3.3186], [*<sup>s</sup>*4.0048,*s*4.6814]), 0.3667}; *u* ˜(*p*)24 = {([*<sup>s</sup>*2.9860,*s*4.3204], [*<sup>s</sup>*2.0140,*s*3.0140]), 0.5333,([*<sup>s</sup>*3.9860,*s*4.6438], [*<sup>s</sup>*2.6905,*s*3.3562]), 0.4667}; *u* ˜(*p*)34= {([*<sup>s</sup>*3.3985,*s*4.3906], [*<sup>s</sup>*2.2859,*s*2.9516]), 0.5333,([*<sup>s</sup>*2.3797,*s*4.0484], [*<sup>s</sup>*2.9407,*s*3.6172]), 0.3667}.

**Step 4**: The non-fuzzy uncertain information values *fij*(*<sup>i</sup>*, *j* = 1, 2, 3, 4, *i* < *j*) and the fuzzy uncertain information values *gij*(*<sup>i</sup>*, *j* = 1, 2, 3, 4, *i* < *j*) of PULIFPR *U*˜ are calculated respectively. By substituting them into Equation (22), the following model can be obtained

$$\max \theta = \theta\_{12} + \theta\_{13} + \theta\_{14} + \theta\_{23} + \theta\_{24} + \theta\_{34}$$

$$\begin{cases} \left[0.3822\theta\_{12} + 0.1235(1 - \theta\_{12})\right](w\_1 + w\_2) - w\_1 = 0, \\ \left[0.4199\theta\_{13} + 0.1619(1 - \theta\_{13})\right](w\_1 + w\_3) - w\_1 = 0, \\ \left[0.6110\theta\_{14} + 0.0803(1 - \theta\_{14})\right](w\_1 + w\_4) - w\_1 = 0, \\ \left[0.3731\theta\_{23} + 0.1091(1 - \theta\_{23})\right](w\_2 + w\_3) - w\_2 = 0, \\ \left[0.5756\theta\_{24} + 0.1608(1 - \theta\_{24})\right](w\_2 + w\_4) - w\_2 = 0, \\ \left[0.4191\theta\_{34} + 0.1682(1 - \theta\_{34})\right](w\_3 + w\_4) - w\_3 = 0, \\ w\_1 + w\_2 + w\_3 > w\_4 - 0.5, \ w\_1 + w\_2 + w\_4 > w\_3 - 0.5, \\ w\_1 + w\_3 + w\_4 > w\_2 - 0.5, \ w\_2 + w\_3 + w\_4 > w\_1 - 0.5, \\ w\_1 + w\_2 + w\_3 + w\_4 = 1, \ w\_1, w\_2, w\_3, w\_4 \ge 0, \\ 0 \le \theta\_{12}\theta\_{13}, \theta\_{14}\theta\_{23}, \theta\_{24}\theta\_{34} \le 1. \end{cases} \tag{29}$$

By solving this model, priority weights and parameter values can be obtained as *w*1 = 0.1227, *w*2 = 0.1984, *w*3 = 0.3333, *w*4 = 0.3456, *θ*13 = 0.4162, *θ*14 = 0.3425, *θ*24 = 0.4916, *θ*12 = *θ*23 = *θ*34 = 1.

**Step 5**: By calculating the optimistic judgment values *aij* = ∑#*u*(*p*) *k*=1 *pk* × *<sup>I</sup>*(*sukij*)−*<sup>I</sup>*(*svkij*)+<sup>2</sup>*<sup>τ</sup>*4*τ* and pessimistic judgment values *bij* = ∑#*u*(*p*) *k*=1 *pk* × *<sup>I</sup>*(*sukij*)−*<sup>I</sup>*(*svkij*)+<sup>2</sup>*<sup>τ</sup>* 4*τ* ,(*<sup>i</sup>*, *j* = 1, 2, 3, 4, *i* < *j*) and substituting them into Equations (24) and (25) respectively to solve the weight. But their feasible regions are all empty. Therefore, substitute the values of *aij* and *bij* into Equations (26) and (27) respectively, then the priority weights can be obtained as follows

*<sup>w</sup>*+1 = 0.2149, *<sup>w</sup>*+2 = 0.2631, *<sup>w</sup>*+3 = 0.3700, *<sup>w</sup>*+4 = 0.1520, *w*− =0.1395,*<sup>w</sup>*<sup>−</sup>2=0.3034, *<sup>w</sup>*<sup>−</sup>3=0.2432,*<sup>w</sup>*<sup>−</sup>4=0.3139.

1 Without loss of generality, assume that the value of risk parameter *λ* determined by Microsoft is 0.5. Then the priority weights can be obtained as *w* 1 = 0.5*w*+1 + (1 − 0.5)*w*<sup>−</sup>1 = 0.1783, *w* 2 = 0.2823, *w* 3= 0.3058, *w* 4= 0.2336.

**Step 6**: Combining the results obtained in steps 4 and 5, the comprehensive ranking weight of *U* ˜ can be obtained as *w*¯1 = 0.1505, *w*¯2 = 0.2404, *w*¯3 = 0.3195, *w*¯4 = 0.2896. Therefore, the final ranking result is *w*¯3 > *w*¯4 > *w*¯2 > *w*¯1 , namely, *x*3 is the best candidate partner of Microsoft.

In addition, the sorting results for different risk parameter values *λ* are shown in Table 2.


**Table 2.** Ranking orders of alternatives with different parameter values *λ*.

It can be seen from Table 2 that different sorting results may occur for different risk parameter values *λ*. When 0 < *λ* < 0.3, the sorting result is *x*4 > *x*3 > *x*2 > *x*1, and when 0.3 ≤ *λ* ≤ 0.9, the sorting result is *x*3 > *x*4 > *x*2 > *x*1. This fully demonstrates the importance of DM's risk attitude in GDM and the rationality of the method proposed in this paper. In addition, to further reflect the impact of risk parameter value *λ* on GDM. We give the variation trend diagram of the compromise weight *w i*and the comprehensive weight *w*¯ (see Figure 3).

**Figure 3.** The variation trend of weights *w i* and *<sup>w</sup>*¯*i* based on different parameter values *λ*.

It can be seen intuitively from Figure 3 that the variation trend of the compromise weight *w i* and the comprehensive weight *<sup>w</sup>*¯*i* with the increase of *λ*. Furthermore, by comparing *w i* and *<sup>w</sup>*¯*i* , it is easy to see that project *x*4 is greatly influenced by *λ* when only taking into account DM's risk attitude (as the value of *λ* increases, the value of *w* 4 decreases from the maximum to the minimum), while its comprehensive weight *w*¯4 is less affected by *λ*. This further illustrates the necessity and rationality of comprehensive consideration of risk attitude, fuzzy and non-fuzzy uncertain information in GDM problems.
