A New Method Based on PROMETHEE and TODIM for Multi-Attribute Decision-Making with Single-Valued Neutrosophic Sets

Single-valued neutrosophic sets (SVNSs) can effectively describe the multi-attribute decision-making (MADM) problems which are characterized by incompleteness and uncertainty. Aiming at the MADM problem of SVNSs, a series of methods are proposed to solve the problem, such as the TODIM and PROMETHEE methods. The main idea of the TODIM method is to establish a relative superiority function of scheme relative to other schemes based on the value function of prospect theory, and the ranking of alternatives is determined according to the obtained superiority. In the PROMETHEE method, the decision maker selects the preference function for each attribute according to their preference, and then calculates the priority index, inflow, outflow and net flow according to the difference of the attribute values of scheme, so as to determine the ranking of alternatives. In this paper, a new method based on PROMETHEE and TODIM is proposed to solve the MADM problem under the single-valued neutrosophic environment. Based on the calculation formula of inflow and outflow in PROMETHEE method, and the calculation formula of overall dominance in the TODIM method, a new integrated formula is obtained.


Introduction
In real life, most decision-making problems involve multiple schemes and multiple attributes, which are called multi-attribute decision-making (MADM) problems. How to sort and select the alternatives under given conditions is the key to solve this problem. In traditional MADM methods, alternatives are evaluated with crisp values generally. However, due to the complexity of objective things and the subjectivity of humans, MADM problems are often accompanied by uncertainty, so the decision information given is often fuzzy or linguistic. In order to describe fuzzy concepts, Zadeh [1] proposed the concept of the fuzzy set (FS) and obtained in-depth research. The FS is a tool to express fuzzy information. On this basis, Atanassov [2] defined the intuitionistic fuzzy set by adding a parameter, that is, non-membership degree. Intuitionistic fuzzy sets consider both membership degree and non-membership degree. Although FS theory has been widely developed and popularized, it cannot handle all types of uncertainties in real life, such as uncertain and inconsistent information. Therefore, Smarandache [3] proposed the concept of a neutrosophic set. The neutrosophic set is an extension of the intuitionistic fuzzy set, which contains a truth-membership function, an indeterminacy-membership function, and a falsity-membership function. After that, some scholars It can be seen from the above that the PROMETHEE and TODIM methods are very common in MADM problems. In this paper, we combine PROMETHEE with the TODIM method. In previous research, there have been combinations of the two methods-for example, Peng et al. [48] proposed TOPSIS and TODIM methods based on two-dimensional linguistic variables for the MADM problems with evaluation information as two-dimensional linguistic variables. Arya and Kumar [49] developed an algorithm for picture fuzzy set using TODIM and VIKOR methods. Liu et al. [50] extended the TODIM and TOPSIS methods to the corresponding distance measure under the Fermatean fuzzy linguistic environment.
In the TODIM method, any two alternatives need to be compared and also lead to high computational complexity. In addition, in the PROMETHEE method, the calculation of the inflows and outflow of the alternatives lack persuasiveness and readability. Both of these methods have some shortcomings, and in order to solve these problems, a new method is proposed based on them. This method does not require you to calculate the distance between the alternatives, and it also does not need you to compare any two alternatives, which reduces a huge amount of calculation. In addition, using the overall dominance and overall disadvantage to calculate the inflow and outflow, we can increase the credibility of the alternative ranking. In this paper, firstly, the SVNSs theory, PROMETHEE and TODIM method are introduced; then, a new method based on PROMETHEE and TODIM is proposed. Through a numerical example, the calculation results of the new method are compared with those of the PROMETHEE and TODIM methods. Finally, the conclusion proves the feasibility of the new method.

Neutrosophic Sets
Definition 1. Let X be a space of points (objects) with a generic element in X denoted by x, aneutrosophic set A in X is characterized by a truth-membership function T A (x), a indeterminacy-membership function I A (x), and a false-membership function Definition 2. Let X be a space of points (objects) with a generic element in X denoted by x, the complement of a neutrosophic set A in X is denoted by A C and is defined as Definition 3. Let X be a space of points (objects) with a generic element in X denoted by x, a neutrosophic set A is contained in another neutrosophic set B, A ⊆ B if and only if infT

Single-Valued Neutrosophic Sets
Definition 4. Let X be a space of points (objects), a single-valued neutrosophic set (SVNS) A in X is characterized as the following: where the truth-membership function T A (x), indeterminacy-membership function I A (x) and false-membership function Definition 5. Let A and B be two single-valued neutrosophic numbers (SVNNs), then the normalized Hamming distance between them is: (1) Definition 6. Let A = (T A , I A , F A ) be a SVNN, a score function S(A) is: Definition 7. Let A = (T A , I A , F A ) be a SVNN, an accuracy function H(A) is: where H(A) is the degree of accuracy of SVNN A. The larger the value of H(A) is, the higher the degree of accuracy of the SVNN A.
For the convenience of research, Brans et al. proposed six preference functions, as shown in Table 1. q, p and s are three parameters of the preference function: q is the indifference threshold, p is the absolute preference threshold, and s is a value between p and q.

PROMETHEE Method
Let the alternatives be A = (A 1 , . . . , A m ), and the attributes be G = (G 1 , G 2 , . . . , G n ). Let the weights of the attributes be W = (w 1 , w 2 , . . . , w n ), where 0 ≤ w j ≤ 1, n j=1 w j = 1. Let a ij , i = 1, 2, . . . , m, j = 1, 2, . . . , n, be the attribute value of the alternative A i with attribute G j , the A = (a ij ) m×n = T ij , I ij , F ij m×n is a SVNNs matrix, where T ij , I ij and F ij are membership degree, indeterminacy-membership degree and non-membership degree. The following is the calculation procedure of PROMETHEE method.
Step 1: Standardize the decision information. That is, normalizing A = (a ij ) m×n into B = (b ij ) m×n . If the decision is a cost factor, the decision information should be changed by its complementary set, while if it is an efficient factor, it should not be changed.
Step 2: Construct a preference function P j (B i , B r ) of alternative B i relative to B r under the attribute G j by There are 6 typical preference functions in Table 1. Decision makers choose preference functions according to their own preferences. In certain circumstances, decision makers can also construct preference functions by themselves. d is the priority function parameter and the difference between the criterion values of scheme B i and B r .
Step 3: We define the priority index π(B i , B r ) of the scheme B i relative to B r by Step 4: Calculate the inflow φ + (B i ), outflow φ − (B i ) and net flow φ(B i ) of the object, as following where φ + (B i ) indicates the degree to which scheme B i exceeds other schemes, and φ − (B i ) indicates the degree to which it is exceeded.
Step 5: Rank the alternatives according to the net flow value of the object.
, that means there is no difference between B i and B r .

TODIM Method
The TODIM method [51,52] describes the dominance of each alternative over others by constructing a function of multi-attribute values. The MADM problem is given with single-valued neutrosophic information.
This question is the same as that in Section 3.1, where A and G are constant. The following is the calculation procedure of TODIM method.
Step 1: Standardize the decision information. That is, normalizing A = (a ij ) m×n into B = (b ij ) m×n . If the decision is a cost factor, the decision information should be changed by its complementary set, while an efficient factor, it should not be changed.
Step 2: Calculate w jr , which is the relative weight of G j to G r , where where w j is the weight of the attribute of G j and w r = max w j j = 1, 2, . . . , n . There are many methods to determine the weight [53], and the subjective assigned method is used here.
Step 3: Find out the dominance degree of B i over every alternative B t by σ( where the parameter θ is the attenuation factor of the losses, and d(b ij , b tj ) is the distance between the Step 4: Work out the overall dominance of B i by the following function Step 5: Rank all alternatives according to the value of ξ i . The greater the value of ξ i , the better the alternative is.

A New Method Based on PROMETHEE and TODIM
This question is the same as that in Section 3.1, where A and G are constant. The following is the calculation procedure of new method.
Step 1: Standardize the decision information. That is, normalizing A = (a ij ) m×n into B = (b ij ) m×n . If the decision is a cost factor, the decision information should be changed by its complementary set, while an efficient factor, it should not be changed.
Step 2: Construct a preference function P j (B i , B r ) of scheme B i relative to B r under the attribute G j by There are 6 typical preference functions, and decision makers choose preference functions according to their own preferences. In certain circumstances, decision makers can also construct preference functions by themselves. The value of the preference function is from 0 to 1; the smaller the function value, the smaller the difference between B i and B r ; when the function value is 0, there is no difference between B i and B r . The closer the value is to 1, the higher the degree of B i being better than B r is, and when the function value is 1, B i is strictly better than B r . d is the priority function parameter and the difference between the criterion values of scheme B i and B r . Here, the score function value is used to calculate the criterion value of the scheme.
Step 3: Calculate w jr , which is the relative weight of G j to G r , where where w j is the weight of the attribute of G j and w r = max w j j = 1, 2, . . . , n .
Step 4: We define the priority index π(B i , B r ) of the scheme B i relative to B r by π(B i , B r ) = n j=1 w jr P j (B i , B r ) n j=1 w jr (15) Step 5: Calculate the inflow φ + (B i ), outflow φ − (B i ) and net flow φ(B i ) of the object, as follows Step 6: Rank all alternatives according to the value of φ(B i ). The larger the value of φ(B i ), the better the alternative is.

Numerical Example
We give an example to illustrate the feasibility and rationality of the proposed method. Suppose there are four enterprises A = (A 1 , A 2 , A 3 , A 4 ) and four attributes G = (G 1 , G 2 , G 3 , G 4 ). Here, the decision information is given in the form of single-valued neutrosophic number. Suppose a ij is given for the alternative A i under the attribute G j , i = 1,2,3,4 and j = 1,2,3,4. The first attribute is the cost factor, and the next three are efficient factors. The weight of each attribute is given by w = {0.3, 0.2, 0.1, 0.4}. Now we need to select the better one from the four enterprises. The evaluation results of the scheme are given by experts, as shown in the following matrix.
Attribute G 1 is the cost factor, and the next three, G 2 , G 3 , G 4 , are efficient factors. Therefore, the standardized decision matrix shows in the following.

Comparative Analysis
In order to verify the effectiveness and rationality of the new method, the new method is compared with the PROMETHEE and TODIM based on single-valued neutrosophic sets. The TODIM method fully considers the risk aversion attitude of the decision maker on the basis of the prospect theory, and can adjust the parameters to reflect the risk preference of the decision maker, which is more in line with actual decision-making needs, the parameter θ is the attenuation factor of the losses.
For the convenience of comparison, the alternatives and weight coefficients are also the same as that in Section 5.1. Next, we discuss the use of TODIM method to sort the alternative under different parameter values. The results are shown in Table 3.
It can be seen from Table 3 that the ranking of the alternative is unchanged as the θ value changes. In the new method, we continue to discuss the changes of the values of the parameters p and q to the results and compared the results as shown in Table 4.
It can be seen from Table 4 that reasonable selection of parameters p and q has no effect on the results.
Due to how the parameter value changes, the result is consistent; usually, the value of θ is taken as 2.5, and the value of p and q are taken as 0 and 1 respectively. Suppose θ = 2.5, p = 0, q = 1 and then the comparison results of the three methods are as Table 5: Table 5. Ranking of methods.

Method
Ranking Optimal Choice From the above analysis, through the calculation of the same example, similar sorting results are obtained, and the optimal scheme is A 2 . The new method is proposed based on the calculation formula of inflow and outflow in the PROMETHEE method and the calculation formula of overall dominance in the TODIM method. The TODIM method can reasonably depict the decision maker's behaviors under risk. The main reason for the difference is that the new method and the PROMETHEE method depend on the selection of input parameters and preference functions. Although the three methods' ranking results are slightly different, each of the three methods has its own advantages. This proves that the new method proposed is reasonable and effective.

Conclusions
The single-valued neutrosophic sets (SVNSs) are useful tools to depict the uncertainty of the MADM. In this paper, a new decision-making method based on PROMETHEE and TODIM is proposed for MADM problems. In the new method, based on the calculation formula of inflow and outflow in PROMETHEE method, and the calculation formula of overall dominance in the TODIM method, a new integrated formula is obtained. It does not require you to calculate the distance between the alternatives, and it also does not need you to compare any two alternatives, which reduces the amount of calculation by a huge degree. In addition, using the overall dominance and overall disadvantage to calculate the inflow and outflow, we can increase the credibility of the alternative ranking. Due to the complexity of objective things and human subjectivity, MADM problems are often uncertain, so the decision information often given is unclear. Therefore, this method is more applicable when the information in MADM problems in real life is unclear or there is a large amount of data. Finally, a numerical example is given under the background of SVNSs, and the results are compared with the PROMETHEE and TODIM methods to verify the effectiveness and practicability of the new method, which improves and enriches the theory of MADM.