*5.1. Example Analysis*

Credit scoring can help financial institutions reduce financial risks and non-performing loans. Generally, financial institutions assess the credit score of borrowers based on basic information, such as age, profession, education, income, capital gains, residence and borrowing frequency. Recently, a financial institution wants to invest an amount of money in borrowers. The institution initially selects

five borrowers as candidates. In addition, the institution makes a decision by analyzing the following four parameters: Highly educated, higher borrowing frequency, higher income and higher capital gains. Subsequently, the institution assembles a team composed of three decision makers to make the investment decision. Suppose that *U* = {*<sup>u</sup>*1, *u*2, *u*3, *u*4, *<sup>u</sup>*5} is the set of candidates, *E* = {*<sup>e</sup>*1,*e*2,*e*3,*e*4} is the parameter set, and *DM* = {*Z*1,*Z*2,*Z*3} is the set of decision makers. Let the neutrosophic soft sets (*F*(*t*), *E*) (*t* = 1, 2, 3) be the alternative evaluation values with respect to the parameters given by decision makers as follows.

(*F*(1), *E*) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *<sup>F</sup>*(1)(*<sup>e</sup>*1) = -< *u*1 0.60,0.35,0.80 >,< *u*2 0.70,0.50,0.60 >,< *u*3 0.80,0.40,0.70 >,< *u*4 0.65,0.50,0.50 >,< *u*5 0.75,0.30,0.60 > *<sup>F</sup>*(1)(*<sup>e</sup>*2) = -< *u*1 0.50,0.80,0.20 >,< *u*2 0.60,0.30,0.70 >,< *u*3 0.70,0.35,0.80 >,< *u*4 0.80,0.30,0.70 >,< *u*5 0.80,0.20,0.55 > *<sup>F</sup>*(1)1(*<sup>e</sup>*3) = -< *u*1 0.60,0.50,0.80 >,< *u*2 0.70,0.50,0.20 >,< *u*3 0.80,0.60,0.30 >,< *u*4 0.70,0.40,0.70 >,< *u*5 0.85,0.30,0.60 > *<sup>F</sup>*(1)(*<sup>e</sup>*4) = -< *u*1 0.50,0.80,0.60 >,< *u*2 0.40,0.70,0.30 >,< *u*3 0.60,0.40,0.70 >,< *u*4 0.60,0.35,0.80 >,< *u*5 0.70,0.30,0.40 > ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (*F*(2), *E*) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *<sup>F</sup>*(2)(*<sup>e</sup>*1) = -< *u*1 0.60,0.35,0.80 >,< *u*2 0.70,0.50,0.60 >,< *u*3 0.80,0.40,0.70 >,< *u*4 0.65,0.50,0.50 >,< *u*5 0.75,0.30,0.60 > *<sup>F</sup>*(2)(*<sup>e</sup>*2) = -< *u*1 0.50,0.80,0.20 >,< *u*2 0.60,0.30,0.70 >,< *u*3 0.70,0.35,0.80 >,< *u*4 0.80,0.30,0.70 >,< *u*5 0.80,0.20,0.55 > *<sup>F</sup>*(2)(*<sup>e</sup>*3) = -< *u*1 0.60,0.50,0.80 >,< *u*2 0.70,0.50,0.20 >,< *u*3 0.80,0.60,0.30 >,< *u*4 0.70,0.40,0.70 >,< *u*5 0.85,0.30,0.60 > *<sup>F</sup>*(2)(*<sup>e</sup>*4) = -< *u*1 0.50,0.80,0.60 >,< *u*2 0.40,0.70,0.30 >,< *u*3 0.60,0.40,0.70 >,< *u*4 0.60,0.35,0.80 >,< *u*5 0.70,0.30,0.40 > ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (*F*(3), *E*) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *<sup>F</sup>*(3)(*<sup>e</sup>*1) = -< *u*1 0.60,0.35,0.80 >,< *u*2 0.70,0.50,0.60 >,< *u*3 0.80,0.40,0.70 >,< *u*4 0.65,0.50,0.50 >,< *u*5 0.75,0.30,0.60 > *<sup>F</sup>*(3)(*<sup>e</sup>*2) = -< *u*1 0.50,0.80,0.20 >,< *u*2 0.60,0.30,0.70 >,< *u*3 0.70,0.35,0.80 >,< *u*4 0.80,0.30,0.70 >,< *u*5 0.80,0.20,0.55 > *<sup>F</sup>*(3)(*<sup>e</sup>*3) = -< *u*1 0.60,0.50,0.80 >,< *u*2 0.70,0.50,0.20 >,< *u*3 0.80,0.60,0.30 >,< *u*4 0.70,0.40,0.70 >,< *u*5 0.85,0.30,0.60 > *<sup>F</sup>*(3)(*<sup>e</sup>*4) = -< *u*1 0.50,0.80,0.60 >,< *u*2 0.40,0.70,0.30 >,< *u*3 0.60,0.40,0.70 >,< *u*4 0.60,0.35,0.80 >,< *u*5 0.70,0.30,0.40 > ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

The neutrisophic set *D* represents the neutrosophic weights of decision makers, and the neutrisophic soft set (*<sup>F</sup>*,*<sup>Z</sup>*) stands for neutrosophic subjective weights of parameters. They are valued as follows:

$$D = \{, , \}$$

$$F(Z, Z) = \left\{ \begin{array}{l} F(Z\_1) = \left\{ <\frac{c\_1}{0.40, 0.60, 0.50} >, <\frac{c\_2}{0.35, 0.70, 0.60} >, <\frac{c\_3}{0.40, 0.60, 0.55} >, <\frac{c\_4}{0.40, 0.60, 0.75} >\right\} \\ F(Z\_2) = \left\{ <\frac{c\_1}{0.70, 0.45, 0.30} >, <\frac{c\_2}{0.50, 0.80, 0.60} >, <\frac{c\_3}{0.70, 0.55, 0.40} >, <\frac{c\_4}{0.70, 0.40, 0.65} >\right\} \\ F(Z\_3) = \left\{ <\frac{c\_1}{0.65, 0.70, 0.40} >, <\frac{c\_2}{0.60, 0.55, 0.75} >, <\frac{c\_3}{0.40, 0.65, 0.70} >, <\frac{c\_4}{0.25, 0.60, 0.50} >\right\} \end{array} \right\}$$

Step 1: Input the neutrosophic soft sets (*F*(*t*), *E*)(*t* = 1, 2, <sup>3</sup>), (*<sup>F</sup>*, *Z*) and the neutrosophic set *D*. Step2: There is no need to normalize the neutrosophic soft sets (*F*(*t*), *E*)(*t* = 1, 2, 3) of alternatives, becausetheparametersadoptedinthisstudyarebenefitparameters.

 Step 3: Compute the determinacy degree vector of decision makers based on Equation (8) as follows:

$$\psi\_t = \{0.3478, 0.4130, 0.2391\}$$

Step 4: Construct the prospect decision matrix based on Equation (20).

$$V\_{\vec{\mathbb{q}}} = \begin{pmatrix} 0.3878 & 0.2846 & 0.3574 & 0.2274 \\ 0.3035 & 0.3751 & 0.3571 & 0.2712 \\ 0.4536 & 0.3834 & 0.3226 & 0.3180 \\ 0.3345 & 0.3294 & 0.3120 & 0.3776 \\ 0.3482 & 0.4482 & 0.4055 & 0.3481 \end{pmatrix}.$$

Step 5: Determine the comprehensive weight vector *j* = ( 1, 2, ... , *n*) for the parameters as Equation (18), and the neutrosophic subjective weights are aggregated by the weighted geometric aggregation rule as Equation (11).

$$\phi\_{j} = (0.2991, 0.2260, 0.2898, 0.1851)$$

Step 6: Obtain the comprehensive prospect value *Vi* by Equation (23).

$$V\_1 = 0.3269, V\_2 = 0.3292, V\_3 = 0.3746, V\_4 = 0.3348, V\_5 = 0.3874.$$

Step 7: Make a decision by ranking the comprehensive prospect value of the five candidates.

$$\mathbf{x\_5 \succ x\_3 \succ x\_4 \succ x\_2 \succ x\_1}$$

Therefore, we can see that the optimal candidate is *x*5. *x*3, *x*4 are suboptimal, and *x*2, *x*1 are the worst.

Furthermore, we also utilize the weighted average aggregation rule to compute the subjective weights of parameters. In addition, the computational procedure is shown as follows.

Step 1–4: Be consistent with the above steps 1–4.

Step 5: Determine the comprehensive weight vector *j* = ( 1, 2, ... , *n*) for the parameters as Equation (18), and the neutrosophic subjective weights are aggregated by the weighted average aggregation rule.

$$
\alpha\_{\bar{\jmath}} = (0.2903, 0.2127, 0.2523, 0.2447).
$$

Step 6: Obtain the comprehensive prospect value *Vi* by Equation (23).

$$V\_1 = 0.3254, V\_2 = 0.3295, V\_3 = 0.3744, V\_4 = 0.3348, V\_5 = 0.3876.$$

Step 7: Make a decision by ranking the five candidates.

$$\mathbf{x\_5 \succ x\_3 \succ x\_4 \succ x\_2 \succ x\_1.}$$

So the best optimal is still *x*5, the following are *x*3, *x*4, and the worst are *x*2, *x*1.

Obviously, we can see that the ranking orders obtained by two aggregation rules of the neutrosophic soft set are the same.
