A Novel Portfolio Based on Interval-Valued Intuitionistic Fuzzy AHP with Improved Combination Weight Method and New Score Function

: The classical Analytic Hierarchy Process (AHP) requires an exact value to compare the relative importance of two attributes, but experts often can not obtain an accurate assessment of every attribute in the decision-making process, there are always some uncertainty and hesitation. Compared with classical AHP, our new defined interval-valued intuitionistic fuzzy AHP has accurately descripted the vagueness and uncertainty. In decision matrix, the real numbers are substituted by fuzzy numbers. In addition, each expert will make different evaluations according to different experiences for each attribute in the subjective weighting method, which neglects objective factors and then generates some deviations in some cases. This paper provides two ways to make up for this disadvantage. On the one hand, by combining the interval-valued intuitionistic fuzzy AHP with entropy weight, an improved combination weighting method is proposed, which can overcome the limitations of unilateral weighted method only considering the objective or subjective factors. On the other hand, a new score function is presented by adjusting the parameters, which can overcome the invalidity of some existing score functions. In theory, some theorems and properties for the new score functions are given with strictly mathematical proof to validate its rationality and effectiveness. In application, a novel fuzzy portfolio is proposed based on the improved combination weighted method and new score function. A numerical example shows that these results of our new score function are consistent with those of most existing score functions, which verifies that our model is feasible and effective.


Introduction
The portfolio theory was first proposed by Markowitz [1] in 1952. He used the average rate of historical return to measure the expected return level of investments and the variance of the rate of return to measure the investment risk. On this basis, the mean-variance model was established to explain how to achieve the best balance between returns and risks through the selection of securities portfolios. The central problem of portfolio theory is how to choose the combination of return and risk in the decision of securities investment. In this way, the expected benefits can be maximized at a given level of expected risk, or the expected risks can be minimized at a given level of expected return. Makovitz assumed that the income distribution was symmetric and used the variance to depict the risk. But in actual cases, variance is not always accurate to describe the risk, and the distribution of income is not necessarily symmetric. So, many scholars put forward their own different views on this. Hogan and Warren [2], Harlow [3] proposed that the lower half variance can depict the risk more accurately and discussed he mean half variance model. Konno and Suzuki [4] studied the mean variance-skewness model which is valuable in the case of asymmetric return distribution. Konno and Yamazaki [5] used expected absolute deviation to describe risks and established a linear programming model for portfolio selection, which is called mean absolute deviation model. Athayde and Flores [6] considered the asset allocation under the condition of asymmetric distribution. Jondeau and Rockinger [7] considered the non-normal distribution and time-varying characteristics of the rate of return. In their paper, Taylor expansion was performed on the final expected income and the first four high order moments are taken, and the first order condition was used to optimize the asset allocation. Li [8] constructed an asymmetry robust mean absolute deviation (ARMAD) model that takes the asymmetry distribution of returns into consideration. Deng and Yuan [32] constructed a hybrid multi-objective portfolio model which considers fuzzy return status, systematic risk, non-systematic risk and entropy.
The concept of entropy originating from thermodynamics reflects the degree of chaos in a system. The smaller the corresponding entropy value of the system, the more stable the system. Zadeh [9] put forward the concept of fuzzy entropy for the first time in 1965. Then many scholars had offered different definitions of interval intuitionistic fuzzy entropy. In the case of attributes was completely unknown, most scholars used entropy weight method to determine weights. Burillo and Bustince [30] proposed the notions of entropy of to measure the degree of intuitionism of interval-valued intuitionistic fuzzy sets and intuitionistic fuzzy sets. Szimidt and Kacprzyk [31] proposed a non-probabilistic-type entropy measure with a geometric interpretation of intuitionistic fuzzy sets. Deng and Liu [33] used improved entropy method to calculated the weight of each indicator in order to conduct quantitative analysis on 20 indicator variables which can be divided into four digital economic types in Guangdong Province from 2015 to 2018.
Analytic hierarchy process [10] is a multi-criteria decision-making method combining qualitative and quantitative analysis, which is practical in the case of complex target structure and lack of necessary data. Because the fuzziness of expert judgment is not considered when evaluating the weight distribution of various factors, Atanassov [11,12] successively proposed the intuitionistic fuzzy set and the interval-valued intuitionistic fuzzy set to effectively solve the double fuzzy situation in life. Sadiq [13] applied the intuitionistic fuzzy set to the AHP and then construct the intuitionistic fuzzy AHP method.
The sort function is a mean to measure the intuitionistic fuzzy number. In order to compare the advantages and disadvantages of two interval-valued intuitionistic fuzzy numbers, Xu [14] defined the score function and the accurate function of interval-valued intuitionistic fuzzy numbers as the sort function, and gave the corresponding sorting rules. However, Xu's ordering rules are invalid for some interval-valued intuitionistic fuzzy numbers. Therefore, many scholars put forward new sort functions from different perspectives. Ye [23] proposed the accurate function from the perspective of hesitation in 2009. Nayagam and Sivaraman et al. [24,25], Gao [26], Kang [27] and Wang [28,29] proposed some different sort functions. But there are still cases of sorting failure.
The inherent statistical rules and authoritative values among index data should be considered when assigning weights to indexes. Many scholars have studied different combination weighting methods to make up for the limitations of a single weighting method. Wang [16] used combination weighting method and the fuzzy multi-criteria model to select the optimal cool storage system for air conditioning. In the evaluation process, the optimal weighting method combined the subjective knowledge of decision-maker and the objectivity of numerical data to obtain the comprehensive weights of criteria and avoided the subjective one-sidedness of weights. To get the subjective and objective weights, Yi [17] used fuzzy analytic hierarchy process method and improved criteria importance through inter-criteria correlation, and he applied the least square method to obtain the combined weights, which reduced the influence of artificial experience. Hu [18] established a credit evaluation model based on the combination weighting method, considering the information volume, volatility, and difference of the road transportation enterprises data and using normalized constraints of maximum variance to determine the combination weights. The model fully considered the degree of difference between the indicators and made up for the deviation of the single weighting method. Tan [19] used an improved analytic hierarchy process AHP and the entropy method to make the suitability evaluation of underground space. The method ensured rationality of the evaluation results to the greatest extent, thereby providing a certain guiding significance for the development of underground space. Wu [20] used the coefficient of variation method and entropy weight method to determine the combined weight of the evaluation indicators, and realized the optimization of the green building programs in the South Sichuan Economic Zone.
As one kind of decision problem, portfolio selection also needs to take the subjective knowledge of investor into account. In addition, to better describe the uncertainty of financial market, using interval-valued fuzzy can obtain more detailed information of securities. For this purpose, we study the interval-valued fuzzy portfolio model based on combination weighting method and score function. The main contributions of this paper are as follows. (1) Combining the interval-valued intuitionistic fuzzy AHP with entropy weight method, we get the combination weighting method. It overcomes the defects of unilateral empowerment law; (2) A new score function is obtained by adjusting the parameters. It can overcome the invalidity of the previous score function for some special interval-valued intuitionistic fuzzy numbers; (3) We propose an interval fuzzy portfolio model based on combination weighting method and new score function, and the theoretical theorem and proof are given. In practical application, a numerical example is given to verify the feasibility and effectiveness of the model. The rest of this paper is arranged as follows. In Section 2, the basic theory is introduced. In Section 3, the interval-valued intuitionistic fuzzy analytic hierarchy process is introduced. In Section 4, we introduce ten kinds of score functions and accurate functions and their limitations, and construct a new score function. In Section 5, a novel portfolio model with improved interval-valued intuitionistic AHP and score function is constructed, and the feasibility of the model is verified by a numerical example. In Section 6, we summarize the work of this paper.

Some existing definitions and properties
Definition 1 [12] Suppose   int 0,1 is the collection of closed subsets of the interval-valued number [0,1] .
X is a given theoretical field, is called an interval-valued intuitionistic fuzzy set on the theoretical domain X .
is the non-membership of the element x : The hesitating degree of the element is denoted by where Particularly, when we have the following relationships is a set of interval-valued intuitionistic fuzzy numbers, then which is called interval-valued intuitionistic fuzzy weighted arithmetic average operator; which is called interval-valued intuitionistic fuzzy weighted geometric average operator.
Our above proof process is inspired by Atanassov K.T. and Gargov G [12].

Proof:
(1) If we have then we can prove that 1 2    is the interval-valued intuitionistic fuzzy number.
we have By (5), we have Correspondingly, we obtain By given conditions, it is obvious that we have 1 2 (3) On the one hand, since we have On the other hand, we obtain a a a a a a a b b b a a a a a a a a a b b b

a a a a a a a a a a
so, we have On the one hand, since we have On the other hand, we obtain we have (34) we can obtain   Similarly, we also can prove Remark 1 It should be noted that the following equation does not hold: Obviously, we can see that

A new division definition and related proof
Since subtraction comes from multiplication and division from addition, 1 we have the following two properties.

Proof:
(1) According to Definition 5 and Theorem 1, we can get (2) According to Definition 5 and Theorem 1, we can also get

The ideas and steps of our new interval-valued intuitionistic fuzzy AHP
With the help of the idea of intuitionistic fuzzy AHP of Xu [21], we can construct interval-valued intuitionistic AHP, the basic ideas and specific steps are as follows.

The basic ideas
Our inspiration comes from Xu [21]'s research on intuitive AHP. He extended the classical AHP and the fuzzy AHP to the context of IFS, where the scale of pairwise comparisons of the decision maker was represented by intuitionistic fuzzy numbers. He established a perfect multiplicative consistent interval-valued intuitionistic fuzzy judgment matrix to check whether the preference relation is consistent or not. The weight vector of the intuitionistic preference relation could be derived through a normalizing rank summation method. Based on the weight vector and score function, the score of each index was obtained and finally obtained the normalized weight vector. Then we try to extend the AHP to the context of interval-valued intuitionistic fuzzy set, which is powerful in describing vagueness and uncertainty.

The specific steps
Step 1: Establish an interval-valued intuitionistic fuzzy judgment matrix.
First, we establish an interval-valued intuitionistic fuzzy judgment matrix .
The consistency of R is acceptable if R and R meet the following conditions: Step 3: Calculate the weights of the indexes by introducing parameters. For the interval-valued intuitionistic fuzzy number can be converted to Since  [21] to obtain the weights as follows: Then we convert the interval 1 1 Next, we can show that  Step 4: Sort the indexes.
We sort the indicators using the score function. Set the score function as follows.
We calculate the normalized weights of each indicator according to Formula (56). index, and then we can change an interval-valued intuitionistic fuzzy number into an intuitionistic fuzzy number. Schematic diagram of the IVIFAHP is as follows: Step 5: Calculate the normalized weights of each indicator according to Formula (56).
Step 4: Sort the indicators using the score function.

No
Step 1: Establish an interval intuitionistic fuzzy judgment matrix Step 3: Calculate the weights of the indexes by introducing parameters.
Repair the process.

The limitation and improvement analysis of score functions
In the previous section, we have already introduced the construction of interval-valued intuitionistic fuzzy hierarchical analysis. In this chapter, we will introduce the existing score functions and accurate functions and give a new score function.

Score functions, accurate functions and their limitation analysis
The existing score functions and accurate functions have specific definitions, but they have limitations for some data. We have carefully combed and explored the existing references and obtained the following seven score functions and accurate functions. After a rigorous review and analysis of the existing references, we believe that the sort function is a means to compare the merits of the two intuitionistic fuzzy numbers, so it is necessary to consider this criterion. In this paper, the limitations of these seven score functions and Step 2: Consistent or not accurate functions are analyzed as follows.

The 1 st kind of score function, accurate function and their limitation analysis
Xu [14] defined the score function and accurate function and gave the ordering rules in 2007.

Definition 6 [14] Suppose
is an interval-valued intuitionistic fuzzy numbers, its score function   S  and accurate function   h  can be defined as follows We can't judge whether 1  or 2  is better.

The 2 nd kind of accurate function and its limitation analysis
After pointing out the limitations of Xu [14], Ye [23] proposed the following accurate function from the perspective of hesitation in 2009.

Definition 7 [23] Suppose
is an interval-valued intuitionistic fuzzy number, its accurate function   h  can be defined as follows: Limitation analysis: . This is clearly not practical.
(2) This function does not make full use of the change information of the upper and lower bounds of membership and non-membership. When the midpoint of the membership degree and the non-membership degree of two interval-valued intuitionistic fuzzy numbers are equal, their exact functions will be equal, and this sort method will fail.

The 3 rd kind of accurate function and its limitation analysis
In 2011, Nayagam and Sivaraman et al. [24,25] proposed two accurate functions based on Xu [14] and Ye [23].

Limitation analysis:
(1) For the first accurate function, . This is not consistent with common sense.
(2) For the second accurate function, the presence of  strengthens the subjective evaluation of the decision maker, and the determination of its value is also a problem. (3) For two accurate functions, when , , a b d are fixed, we seek the partial derivatives for c, then we can find the partial derivative is less than 0, indicating that the accurate function value increases as the lower bound c of the non-membership interval-value decreases. It is obviously questionable.

The 4 th kind of accurate function and its limitation analysis
In 2014, Gao et al. [26] proposed the following accurate functions.

Definition 9 [26] Suppose
is an interval-valued intuitionistic fuzzy number, its accurate function   h  can be defined as follows. can not be judged in this case.

The 5 th kind of accurate function and its limitation analysis
In 2015, Kang et al. [27] modified the accurate function and gave the following three definitions.

Definition 10 [27] Suppose
is an interval-valued intuitionistic fuzzy number, its accurate function   h  can be defined as follows, where 1  and 2  are weights determined by the intention of the decision maker.
Limitation analysis: (1) When , , a b d are fixed, we seek the partial derivatives for c, then we can find the partial derivative is less than 0, indicating that the accurate function value increases as the lower bound c of the non-membership interval-value decreases. which is obviously questionable.
(2) For the three accurate functions, the  strengthens the subjective evaluation of the decision maker, and the determination of its value is also a problem.
Then 1  and 2  can not be judged in this case.

The 6 th kind of accurate function and its limitations
New score functions were presented by Wang et al. [28] in 2017.
Definition 11 [28] Suppose is an interval-valued intuitionistic fuzzy number, its score function   S  can be defined as follows: (65) Limitation analysis: For some special interval-valued intuitionistic fuzzy numbers, this method will fail.  and 2  can not be judged in this case.

The 7 th kind of score function and accurate function and their limitations
In 2018, Wang et al. [29] proposed new score and accurate functions.

Definition 12 [29] Suppose
is an interval-valued intuitionistic fuzzy number, its score function   S  and accurate function   h  can be defined as follows:

Limitation analysis:
For some special interval-valued intuitionistic fuzzy numbers, this method will fail. For example,

Our new score function, theorems and proofs
In the above cases that the sorting function above has the possibility of failure, we give the new score function drawn from the score function of intuitionistic fuzzy sets of Li [6]. Then the proof of relevant theorems is given.
is an interval-valued intuitionistic fuzzy number, its score function   S  can be defined as follows: The formula considers membership, non-membership and hesitancy of the interval-valued intuitionistic fuzzy set. Let 88 then this formula satisfies the following theorems.
we have According to Theorem 3, we have Similarly, we also can prove    

Remark 5
The score function proposed by Li (2015) is based on the intuitionistic fuzzy set. Here we extend it to the interval-valued intuitionistic fuzzy set. The original score function is Finally, we obtain the score function defined above (77)

Comparison of our new score function with existing sort functions
We get the following table by comparing the existing score and accurate functions with the new score function. The results show that the merits of certain interval-valued intuitionistic fuzzy numbers can not be judged through the existing sort functions, while the new sort function can obtain reasonable sorting results. To solve these problems, we propose a multi-attribute decision making method based on the improved interval-valued intuitionistic fuzzy AHP and new score function. Its specific steps are as follows.
Step 1: According to the interval-valued intuitionistic fuzzy judgment matrix (   (79) Step 6: Sort the alternative portfolios.

Numerical example
To better choose the portfolio, we usually evaluate the portfolio with income, risk, Sharp ratio, and this paper uses income, risk, Sharp ratio as attributes, then we consider an investor who wants to buy a portfolio.
There are two kinds of portfolios 1 2 { , } A A A  and three attributes which are rate of return 1 C , risk 2 C and Sharpe ratio 3 C . After the data processing, we can obtain the interval-valued intuitionistic fuzzy judgment matrix ( ) ij n m R r     as shown in Table 3. After data processing, we can obtain the interval-valued intuitionistic fuzzy decision matrix as shown in Table 4. Step 6: We sort the alternative portfolios according to the score function value   i S A . We can get that 1 2 A A  , so the portfolio 1 A is selected: Through calculation, the sort function value of each portfolio is obtained. The sorting results are finally given, as shown in Table 5. Our new score function achieves the same result with these six formulae. From these line charts, we can find that The line chart for portfolio 1 A is basically above the line chart for portfolio 2 A (except for Formula (57)), which intuitively shows that portfolio 1 A is superior to portfolio 2 A .
.  We draw bar charts and line charts of the normalized scores for two portfolios. From these bar charts, we can see the normalized sores of the two portfolios obtained by each sort function. Only the normalized sore of 1 A obtained by Formula (57) is larger than that of 2 A ( ) 0.5014 0.4986  . The normalized sores of 1 A obtained by the other six formulae are larger than that of 2 A . Our new score function achieves the same result with these six formulae. From these line charts, we can find that the normalized sore of 1 A is basically bigger than 0.5 and the normalized sore of 2 A is basically smaller than 0.5 (except for Formula (57)), which intuitively shows that portfolio 1 A is superior to portfolio 2 A .
From the figures above, we can see the sorting results obtained by the existing 7 sort functions. Of these, the better is 1 A . And the sorting result we get with the new score function is 1 2 A A  . Combined with the analysis in Section4, it is doubtless that the result of the new score function is better in application.

Conclusion
With the help of Xu [21]'s intuitionistic fuzzy AHP, we apply AHP to the context of interval-valued intuitionistic fuzzy sets. Because the interval-valued intuitionistic fuzzy set is powerful in describing fuzziness and uncertainty, the interval-valued intuitionistic fuzzy AHP can describe the decision-making process more accurately, which makes the interval-valued intuitionistic fuzzy AHP have more advantages than AHP and FAHP. It can solve the problem that the classical AHP ignores the fuzziness of expert judgment. In addition, we combine interval-valued intuitionistic fuzzy AHP and entropy weight method to get the combined weight method, which overcomes the limitations of unilateral authorization method. By the score function obtained by adjusting the parameters, we solve the problem that the previous sort functions are invalid for some interval intuitionistic fuzzy numbers. Finally, we propose a novel portfolio with the improved intuitionistic AHP and the new score function. The feasibility and validity of the model are proved by applying it to the portfolio decision problem through an example. These result of our proposed new score function is consistent with those of most existing sort functions. Therefore, our proposed score function is not only effective in practice, but also able to overcome some shortcomings of other score functions in theory. The research results of this paper will provide more theoretical and practical reference for investment decision makers. In the future research, we are interested in further theoretical discussion of entropy weight method and the application of new analytic hierarchy process combined with dual hesitation fuzzy sets.

Author Contributions
Xue Deng and Jianxin Yang were responsible for the overall understanding of the structure of the article, the main idea of the article, the conclusion analysis of the article and the revision of the full text.
Fengting Geng performed the data analyses and wrote the manuscript.

Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.