Intuitionistic fuzzy multi-attribute decision making with ideal-point-based method and correlation measure

Abstract

In this paper we develop a new measure for calculating the correlation coefficients between intuitionistic fuzzy sets (IFSs), and prove its desirable axiomatic properties. Then we present an algorithm for multi-attribute decision making. The algorithm first defines the intuitionistic fuzzy ideal solution (IFIS) and the intuitionistic fuzzy negative ideal solution (IFNIS). Then based on the IFIS, the IFNIS and the developed correlation coefficient, the algorithm gives a simple and straightforward method to rank and select the given alternatives. Two numerical examples are used to illustrate our algorithm. Finally we extend the correlation coefficient and the algorithm to interval-valued intuitionistic fuzzy environment.

Keywords

Intuitionistic fuzzy set interval-valued intuitionistic fuzzy set correlation coefficient ideal point

1 Introduction

Multi-attribute decision making [11] is an important content in the theory of decision making analysis, which has been widely used in many fields, such as engineering, management and military. Up to now, there have been plenty of achievements in the theory and methods about multi-attribute decision making whose attribute values are real numbers [18]. However, because of the decision makers’ insufficient knowledge and judgments, the subjective characteristics of the alternatives are generally uncertain. For example, in some decision-making processes, a decision maker may consider that an attribute of an alternative satisfies him to a certain degree but it is possible that he/she is not sure about it, or he/she has hesitation in expressing his/her preference over the alternative. Fuzzy set [31] is hard to capture the hesitation. However, we usually need to deal with such vagueness and uncertainty inherent in subjective assessments. Intuitionistic fuzzy set (IFS) [1, 2], each element of which is assigned a membership degree and a non-membership degree, can offer a possibility for handling these sorts of data and information involving the subjective characteristics of human beings in the decision-making processes. IFS is an extension of fuzzy set [31]. Later, Gau and Buehrer [12] introduced the concept of vague set, which is equivalent to IFS [6]. Atanassov and Gargov [3] further generalized IFS into the notion of interval-valued intuitionistic fuzzy set (IVIFS). Up to now, the IFS theory has been applied to various fields, including decision making [26 –28], medical diagnosis [10], pattern recognition [15–16], clustering analysis [27], etc. In recent years, decision making theory has also been in vigorous development in intuitionistic fuzzy environments. Many papers [7 , 30] have presented a lot of methods to solve multi-attribute decision making problems with intuitionistic fuzzy information. Chen and Tan [7] present some functions to measure the degree of suitability of each alternative with respect to a set of criteria presented by vague sets. Hong and Choi [17] provided some new functions giving additional information about alternatives to measure the degree of accuracy in the grades of membership of each alternative with respect to a set of criteria represented by vague sets. The techniques proposed in these two papers can provide a useful way to help the decision-maker to make his/her decision. Xu and Yager [30], and Xu [24] developed some new aggregation operators, which extend the weighted geometric operator, the ordered weighted geometric operator, the weighted averaging operator and the ordered weighted averaging operator to accommodate the environment in which the given arguments are intuitionistic fuzzy numbers (IFNs) and applied them to multi-attribute decision making based on IFNs. Szmidt and Kacprzyk [23] reconsidered the ranking of alternatives in an intuitionistic fuzzy multi-criteria decision making problem. Hwang and Yoon [18] proposed the famous TOPSIS (technique for order preference by similarity to ideal solution) to deal with multi-attribute decision making problems. In the TOPSIS method, the chosen alternative should have the shortest distance from the positive ideal-solution and the farthest distance from the negative ideal-solution. Because the traditional TOPSIS method aiming at the multi-attribute decision making problem whose attribute values are real number, many scholars have studied how to use the TOPSIS method under the intuitionistic fuzzy or interval-valued intuitionistic fuzzy situation in recent years. Xu [25] proposed intuitionistic preference relation (IPR) and applied it to group decision making. Paternain et al. [21] constructed the IPR straightforward from fuzzy preference relations and then presented a method to sort the alternatives based on the TOPSIS principle and intuitionistic fuzzy correlation measure. Boran et al. [4] utilized the normalized Euclidean distance Szmidt & Kacprzyk [22] and the TOPSIS method to rank the alternatives with intuitionistic fuzzy information. Not using the distance measure, Chen [9] defined the inclusion comparison possibilities to compute closeness coefficient for ranking the alternatives based on the TOPSIS method under interval-valued intuitionistic fuzzy environment. Kavita et al. [19] used interval-valued intuitionistic fuzzy Euclidean distance to calculate the relative closeness of each alternative to positive ideal solution and then choose the best alternative(s). Liu and Guan [20] presented a novel multi-attribute decision making approach which combines gray relational grade and the TOPSIS method to solve the multiple attribute decision making problem in which the attribute values are vague. The computation of the gray relational grade is a bit complicated leading to so many calculating works and the approach cannot be applied to interval-valued intuitionistic fuzzy environment. Considering all the methods above, we find that they solely tackled one kind of information, either intuitionistic fuzzy or interval-valued intuitionistic fuzzy one. Furthermore, the correlation measure used in the method proposed by Paternain et al. [21] did not consider the indeterminacy degree of intuitionistic fuzzy information. Szmidt and Kacprzyk [22] showed that neglecting any of the three parameters (the membership degree, the non-membership degree and the hesitancy degree) may lead to inappropriate results, so we should consider all the three parameters when calculating the distance, correlation coefficients or the inclusion comparison possibilities between IFSs. However, the existing intuitionistic fuzzy correlation measures or interval-valued intuitionistic fuzzy ones usually cannot meet these requirements. For example, Gerstenkorn and Manko [13], Bustince and Burillo [5] introduced the correlation coefficients of IFSs and IVIFSs respectively, only considering the membership degree and non-membership degree of the information not considering the hesitant degree. Aiming at these problems, in our paper, we will first propose a new intuitionistic fuzzy correlation coefficient taking into account all the three parameters of IFS and then combine it with the traditional TOPSIS method to solve intuitionistic fuzzy multiple attribute decision making problems. The method takes less effort to calculate the correlation coefficient and will be extended to interval-valued intuitionistic fuzzy environment. Moreover, it is straightforward and can be performed on computer easily.

To do that, the rest of the paper is organized as follows. Section 2 reviews the concept of IFS and introduces the intuitionistic fuzzy correlation coefficient. Based on the TOPSIS method, Section 3 introduces the ideal-point-based method for intuitionistic fuzzy multi-attribute decision making using a numerical example to demonstrate the feasibility and practicability of the proposed method. Moreover, we give another example to show that it makes sense to consider all the three parameters (the membership degree, the non-membership degree and the hesitancy degree) of IFS. In Section 4, the correlation coefficient and the ideal-point-based method are extended to interval-valued intuitionistic fuzzy environment, and we conclude the paper in Section 5.

2 Intuitionistic fuzzy correlation measures

2.1 The basic concepts about IFSs

Atanassov [2] extended fuzzy sets (FSs) [31] to intuitionistic fuzzy sets (IFSs) by adding a hesitation degree as A ={ < x, μ_A (x) , ν_A (x) > |x ∈ X }, where the function μ_A : X → [0, 1] defines the degree of membership, and ν_A : X → [0, 1] defines the degree of non-membership of x to A, respectively, with the condition 0 ≤ μ_A (x) + ν_A (x) ≤1. π_A (x) =1 - μ_A (x) - ν_A (x) is usually called the degree of hesitation of x to A. Since it assigns to each element a membership degree, a non-membership degree and a hesitancy degree, IFS is more flexible in dealing with fuzziness and uncertainty than FS.

For computational convenience, Xu [24] called α = (μ_α, ν_α) an intuitionistic fuzzy value (IFV) or intuitionistic fuzzy number (IFN). The IFV α = (μ_α, ν_α) has a physical interpretation, for example, if (μ_α, ν_α) = (0.3, 0.2), then it can be interpreted as “the vote for resolution is 3 in favor, 2 against, and 5 abstentions” Gau and Buehrer [12].

2.2 The correlation coefficient of IFSs

TOPSIS is called technique for order preference by similarity to ideal solution, so we should first seek for some sort of measure that can characterize the degree of the relationship between two alternatives (IFSs), such as

correlation measures to be prepared for the intuitionistic fuzzy multi-attribute decision making based on ideal-point-based method.

Because of its importance, the correlation measure has attracted more and more attention from researchers. We first review an axiomatic definition of correlation measure of IFSs, which has been widely employed in the literature on intuitionistic fuzzy theory. Let X be fixed, for better description, let φ (X) be the set of all IFSs on X, and A, B ∈ φ (X).

Definition 1. [13] Let ρ be a mapping ρ : (φ (x)) ² → [0, 1], then the correlation coefficient between two IFSs A and B is defined as ρ (A, B), which has the following properties: 1) 0 ≤ ρ (A, B) ≤1; 2) ρ (A, B) =1 if A = B; and 3) ρ (A, B) = ρ (B, A). Gerstenkorn and Manko [13] presented a correlation measure of IFSs, which has the following form: $ρ_{1} (A, B) = \frac{\sum_{j = 1}^{n} μ_{A} (x_{j}) \cdot μ_{B} (x_{j}) + ν_{A} (x_{j}) \cdot ν_{B} (x_{j})}{\sqrt{\sum_{j = 1}^{n} (μ_{A}^{2} (x_{j}) + ν_{A}^{2} (x_{j})) \cdot \sum_{j = 1}^{n} (μ_{B}^{2} (x_{j}) + ν_{B}^{2} (x_{j}))}}$ (1)

Hong and Hwang [14] extended Equation (1) to the following equation where the set X is infinite: $ρ_{2} (A, B) = \frac{\int_{X} (μ_{A} (x) \cdot μ_{B} (x) + ν_{A} (x) \cdot ν_{B} (x)) dx}{\sqrt{\int_{X} (μ_{A}^{2} (x) + ν_{A}^{2} (x)) dx \cdot \int_{X} (μ_{B}^{2} (x) + ν_{B}^{2} (x)) dx}}$ (2)

We can see that the two correlation coefficients of IFSs above only consider the first two parameters (the membership degree and the non-membership degree) and do not take the third parameter (the indeterminacy degree) into consideration. Szmidt and Kacprzyk [22] pointed out that omitting any one of the three parameters may lead to incorrect results, and therefore, we should pay attention to the three parameters altogether when computing the correlation coefficients between IFSs. Guided by this idea, we will provide a new correlation measure between IFSs, considering all the three parameters.

Let a set X ={ x₁, x₂, …, x_n } be a universe of discourse, and let A, B ∈ φ (X), we define a new correlation coefficient between the IFSs A and B as follows: $\begin{matrix} ρ_{3} (A, B) = \\ \frac{\sqrt{\sum_{i = 1}^{n} ({(μ_{A} (x_{i}) + μ_{B} (x_{i}))}^{2} + {(ν_{A} (x_{i}) + ν_{B} (x_{i}))}^{2} + {(2 - π_{A} (x_{i}) - π_{B} (x_{i}))}^{2})}}{\sqrt{\sum_{i = 1}^{n} (μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i}) + {(1 - π_{A} (x_{i}))}^{2})} + \sqrt{\sum_{i = 1}^{n} (μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i}) + {(1 - π_{B} (x_{i}))}^{2})}} \end{matrix}$ (3)

Equation (3) considers all the three parameters (the membership degree, the non-membership degree and the indeterminacy degree). Furthermore, we will prove that Equation (3) satisfies the three conditions in Definition 1.

Proof. Since A, B ∈ φ (X), then

$\begin{matrix} 0 \leq μ_{A} (x_{i}) \leq 1, 0 \leq ν_{A} (x_{i}) \leq 1, \\ 0 \leq π_{A} (x_{i}) \leq 1, for all x_{i} \in X \end{matrix}$ (4)

$\begin{matrix} 0 \leq μ_{B} (x_{i}) \leq 1, 0 \leq ν_{B} (x_{i}) \leq 1, \\ 0 \leq π_{B} (x_{i}) \leq 1, for all x_{i} \in X \end{matrix}$ (5)by Equation (3), we get ρ₃ (A, B) ≥0.

For the inequality ρ₃ (A, B) ≤1, we can prove it by the well-known triangle inequality: $∥ a + b ∥ \leq ∥ a ∥ + ∥ b ∥$ (6)

With equality if and only if one of the two vectors a = (a₁, a₂, …, a_n) and b = (b₁, b₂, …, b_n) are equal to the zero vector or there is a real number η (η > 0) such that a = ηb.

Because μ_A (x_i), ν_A (x_i) and π_A (x_i) cannot be all zero, it follows from Equation (3) immediately that ρ₃ (A, B) ≤1 with equality if and only if there is a real number η (η > 0) such that

$\begin{matrix} μ_{A} (x_{i}) & = η μ_{B} (x_{i}), ν_{A} (x_{i}) = η ν_{B} (x_{i}), \\ 1 - π_{A} (x_{i}) & = η (1 - π_{B} (x_{i})) \end{matrix}$ (7)

If A = B, i.e., $μ_{A} (x_{i}) = μ_{B} (x_{i}), ν_{A} (x_{i}) = ν_{B} (x_{i}), π_{A} (x_{i}) = π_{B} (x_{i})$ (8)for all x_i ∈ X, then by Equation (7), we know that η = 1, and thus, ρ₃ (A, B) = 1. In addition, we get by Equation (3) that

$\begin{matrix} ρ_{3} (A, B) = \frac{\sqrt{\sum_{i = 1}^{n} ({(μ_{A} (x_{i}) + μ_{B} (x_{i}))}^{2} + {(ν_{A} (x_{i}) + ν_{B} (x_{i}))}^{2} + {(2 - π_{A} (x_{i}) - π_{B} (x_{i}))}^{2})}}{\sqrt{\sum_{i = 1}^{n} (μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i}) + {(1 - π_{A} (x_{i}))}^{2})} + \sqrt{\sum_{i = 1}^{n} (μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i}) + {(1 - π_{B} (x_{i}))}^{2})}} \\ = \frac{\sqrt{\sum_{i = 1}^{n} ({(μ_{B} (x_{i}) + μ_{A} (x_{i}))}^{2} + {(ν_{B} (x_{i}) + ν_{A} (x_{i}))}^{2} + {(2 - π_{B} (x_{i}) - π_{A} (x_{i}))}^{2})}}{\sqrt{\sum_{i = 1}^{n} (μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i}) + {(1 - π_{B} (x_{i}))}^{2})} + \sqrt{\sum_{i = 1}^{n} (μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i}) + {(1 - π_{A} (x_{i}))}^{2})}} = ρ_{3} (B, A) \end{matrix}$ (9)

Hence, all the conditions in Definition 1 hold.

Furthermore, during the proof of the condition 2) in Definition 1, we can further know that ρ₃ (A, B) = 1 if and only if A = ηB, for some positive real number η.

In many situations, such as multi-attribute decision making, the weights of the attributes are always different, so we should take them into account, and thus extend ρ₃ (A, B) to the following form:

$ρ_{4} (A, B) = \frac{\sqrt{\sum_{i = 1}^{n} w_{i} ({(μ_{A} (x_{i}) + μ_{B} (x_{i}))}^{2} + {(ν_{A} (x_{i}) + ν_{B} (x_{i}))}^{2} + {(2 - π_{A} (x_{i}) - π_{B} (x_{i}))}^{2})}}{\sqrt{\sum_{i = 1}^{n} w_{i} (μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i}) + {(1 - π_{A} (x_{i}))}^{2})} + \sqrt{\sum_{i = 1}^{n} w_{i} (μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i}) + {(1 - π_{B} (x_{i}))}^{2})}}$ (10)where w = (w₁, w₂, …, w_n) ^T is the weight vector of x_j (j = 1, 2, …, n) with w_j ≥ 0, j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ . Similar to Eq. (3), Eq. (10)) also satisfies all the conditions of Definition 1.

If the universe of discourse, X, is continuous and the weight of the element x ∈ X = [a, b] is w (x), where w (x) ≥ 0 and $\int_{a}^{b} w (x) dx = 1$ , then Equation (10) is transformed into the following form: $\begin{matrix} ρ_{5} (A, B) \\ = \frac{\sqrt{\int_{a}^{b} w (x) ({(μ_{A} (x) + μ_{B} (x))}^{2} + {(ν_{A} (x) + ν_{B} (x))}^{2} + {(2 - π_{A} (x) - π_{B} (x))}^{2}) dx}}{\sqrt{\int_{a}^{b} w (x) (μ_{A}^{2} (x) + ν_{A}^{2} (x) + {(1 - π_{A} (x))}^{2}) dx} + \sqrt{\int_{a}^{b} w (x) (μ_{B}^{2} (x) + ν_{B}^{2} (x) + {(1 - π_{B} (x))}^{2}) dx}} \end{matrix}$ (11)

If all the elements have the same importance, i.e., $w (x) = \frac{1}{b - a} \in [0, 1]$ (in this case, b - a ≥ 1), for any x ∈ [a, b], then Equation (11) is replaced by $\begin{matrix} ρ_{6} (A, B) \\ = \frac{\sqrt{\int_{a}^{b} ({(μ_{A} (x) + μ_{B} (x))}^{2} + {(ν_{A} (x) + ν_{B} (x))}^{2} + {(2 - π_{A} (x) - π_{B} (x))}^{2}) dx}}{\sqrt{\int_{a}^{b} (μ_{A}^{2} (x) + ν_{A}^{2} (x) + {(1 - π_{A} (x))}^{2}) dx} + \sqrt{\int_{a}^{b} (μ_{B}^{2} (x) + ν_{B}^{2} (x) + {(1 - π_{B} (x))}^{2}) dx}} \end{matrix}$ (12)

3 Intuitionistic fuzzy multi-attribute decision making based on ideal-point-based method

In this section, we will focus on investigating multi-attribute decision making technique under intuitionistic fuzzy environments.

3.1 The description of intuitionistic fuzzy multi-attribute decision making problem

Let A = {A₁, A₂, ⋯ , A_n} be a discrete set of alternatives, X = {x₁, x₂, ⋯ , x_m} be the set of attributes, and w = (w₁, w₂, …, w_m) ^T be the weight vector of attributes, with w_j ≥ 0, j = 1, 2, …, m and $\sum_{j = 1}^{m} w_{j} = 1$ . Let R = (r_ij) _n×m be the decision matrix, where each r_ij denotes the attribute value provided by the decision maker for the ith alternative A_i with respect to the jth attribute X_j. In the considered problem, the decision maker expresses his/her preference information over different attributes with IFVs.

Below we give an ideal-point-based method for intuitionistic fuzzy multi-attribute decision making to rank and select the best alternative(s).

3.2 Intuitionistic fuzzy multi-attribute decision making based on ideal-point-based method

Step 1. Define the intuitionistic fuzzy ideal solution (IFIS): $A^{+} = (α_{1}^{+}, α_{2}^{+},, α_{m}^{+})$ (13)and the intuitionistic fuzzy negative ideal solution (IFNIS): $A^{-} = (α_{1}^{-}, α_{2}^{-},, α_{m}^{-})$ (14)with $α_{i}^{+} = (1, 0), α_{i}^{-} = (0, 1), i = 1, 2, \dots, m$ (15)

Step 2. Calculate the correlation coefficient between the alternative A_i and the IFIS A⁺, and the correlation coefficient between the alternative A_i and the IFNIS A^-, respectively: $s_{i}^{+} = ρ_{4} (A_{i}, A^{+}), i = 1, 2, \dots, n$ (16)and $s_{i}^{-} = ρ_{4} (A_{i}, A^{-}), i = 1, 2, \dots, n$ (17)

If the weights of the attributes are equal, then we can choose $s_{i}^{+} = ρ_{3} (A_{i}, A^{+}), i = 1, 2, \dots, n$ (18)and $s_{i}^{-} = ρ_{3} (A_{i}, A^{-}), i = 1, 2, \dots, n .$ (19)Step 3. Calculate the closeness coefficients of each alternative: $c_{i} = \frac{s_{i}^{+}}{s_{i}^{-} + s_{i}^{+}}, i = 1, 2, \dots, n$ (20)

Step 4. Rank all the alternatives A_i (i = 1, 2, ⋯ , n) according to the closeness coefficients c_i (i = 1, 2, ⋯ , n), the greater the value c_i, the better the alternative A_i.

Step 5. End.

3.3 Numerical examples

Example 1. In this section, an intuitionistic fuzzy multi-attribute decision making problem concerning with the prioritization of a set of investment programs is discussed to illustrate the method. The example is adapted from Liu [20].

A high and new technology venture capital firm needs to prioritize five investment programs according to their various risks. In order to sort these investment programs A_i (i = 1, 2, ⋯ , 5) with respect to their various risks, a team of financial experts has been set up to provide their assessment information on A_i (i = 1, 2, ⋯ , 5). The attributes (risks) which are considered here in assessment of A_i (i = 1, 2, ⋯ , 5) are: 1) x₁ is the environmental risk; 2) x₂ is the market risk; 3) x₃ is the technical risk; 4) x₄ is the production risk; 5) x₅ is the management risk; 6) x₆ is the financial risk. The team of financial experts evaluates the investment programs A_i (i = 1, 2, ⋯ , 5) according to the attributes x_j (j = 1, 2, ⋯ , 6), and gives the intuitionistic fuzzy decision making information in Table 1:

Suppose that the weights of the attributes x_j (j = 1, 2, ⋯ , 6) are equal, now we use the method developed in this paper for intuitionistic fuzzy multi-attribute decision making to prioritize these investment programs A_i (i = 1, 2, ⋯ , 5):

Step 1. Define the intuitionistic fuzzy ideal solution (IFIS): $A^{+} = ((1, 0), (1, 0), (1, 0), (1, 0), (1, 0), (1, 0))$ (21)and the intuitionistic fuzzy negative ideal solution (IFNIS): $A^{-} = ((0, 1), (0, 1), (0, 1), (0, 1), (0, 1), (0, 1))$ (22)

Step 2. Calculate the correlation coefficient between the alternative A_i and the IFIS A⁺, and the correlation coefficient between the alternative A_i and the IFNIS A^-, by Equations (18) and (19), respectively: $\begin{matrix} s_{1}^{+} = 0.9827, s_{2}^{+} = 0.9768, s_{3}^{+} = 0.9806, \\ s_{4}^{+} = 0.9843, s_{5}^{+} = 0.9882 \\ s_{1}^{-} = 0.9207, s_{2}^{-} = 0.9134, s_{3}^{-} = 0.9379, \\ s_{4}^{-} = 0.9279, s_{5}^{-} = 0.9236 \end{matrix}$

Step 3. Calculate the closeness coefficient of each alternative by using Equation (20):

$\begin{matrix} c_{1} & = 0.5163, c_{2} = 0.5168, c_{3} = 0.5111, \\ c_{4} & = 0.5147, c_{5} = 0.5169 \end{matrix}$

Step 4. Rank all the alternatives A_i (i = 1, 2, ⋯ , 5) according to the closeness coefficient c_i (i = 1, 2, ⋯ , 5), then we get $A_{5} ≻ A_{2} ≻ A_{1} ≻ A_{4} ≻ A_{3}$

In the following, we shall give another example adapted from Chen [8] to show that in the proposed intuitionistic fuzzy TOPSIS method, if the correlation coefficient does not consider the hesitant degree of IFS, the decision information will be lost and we will get the incorrect ranking result. In this example, the attribute weights are different.

Example 2. The purchasing manager of a small enterprise want to choose a supplier considering four attributes involving x₁: financial factors (e.g., economic performance, financial stability), x₂: performance (e.g., delivery, quality, price), x₃: technology (e.g., manufacturing capability, design capability, ability to cope with technology changes), and x₄: organizational culture and strategy (e.g., feeling of trust, internal and external integration of suppliers, compatibility across levels and functions of the buyer and supplier). The set of attributes is expressed by X ={ x₁ , x₂, x₃, x₄ }, whose weight vector is w = (0.34, 0.23, 0.22, 0.21) ^T. A set of six suppliers is available, denoted by A ={ A₁ , A₂, …, A₆ }. The characteristics of the suppliers A_i (i = 1, 2, ⋯ , 6) according to the attributes in X are given by the following decision matrix (Table 2):

To compare the sorting results of the alternatives, we first use the following correlation coefficient considering all the three parameters of IFS: $\begin{matrix} ρ_{4} (A, B) = \frac{\sqrt{\sum_{i = 1}^{n} w_{i} ({(μ_{A} (x_{i}) + μ_{B} (x_{i}))}^{2} + {(ν_{A} (x_{i}) + ν_{B} (x_{i}))}^{2} + {(2 - π_{A} (x_{i}) - π_{B} (x_{i}))}^{2})}}{\sqrt{\sum_{i = 1}^{n} w_{i} (μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i}) + {(1 - π_{A} (x_{i}))}^{2})} + \sqrt{\sum_{i = 1}^{n} w_{i} (μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i}) + {(1 - π_{B} (x_{i}))}^{2})}} \end{matrix}$

Step 1. Find intuitionistic fuzzy ideal solution (IFIS): $A^{+} = ((1, 0), (1, 0), (1, 0), (1, 0))$ and intuitionistic fuzzy negative ideal solution (IFNIS): $A^{-} = ((0, 1), (0, 1), (0, 1), (0, 1))$

Step 2. Compute the correlation coefficient between the supplier A_i and the IFIS A⁺, and the one between the supplier A_i and the IFNIS A^-, by Equations (18) and (19), respectively: $\begin{matrix} s_{1}^{+} = 0.963289, s_{2}^{+} = 0.966857, s_{3}^{+} = 0.967071, \\ s_{4}^{+} = 0.953108, s_{5}^{+} = 0.959345, s_{6}^{+} = 0.962207 \\ s_{1}^{-} = 0.954298, s_{2}^{-} = 0.947267, s_{3}^{-} = 0.948162, \\ s_{4}^{-} = 0.961849, s_{5}^{-} = 0.947139, s_{6}^{-} = 0.928497 . \end{matrix}$

Step 3. Calculate the closeness coefficient of each alternative by using Equation (20): $\begin{matrix} c_{1} = 0.5023, c_{2} = 0.5051, c_{3} = 0.5409, \\ c_{4} = 0.4977, c_{5} = 0.5032, c_{6} = 0.5089 . \end{matrix}$ Step 4. Rank all the alternatives A_i (i = 1, 2, ⋯ , 6) according to the closeness coefficient c_i (i = 1, 2, ⋯ , 6), then we get $A_{3} ≻ A_{6} ≻ A_{2} ≻ A_{5} ≻ A_{1} ≻ A_{4}$

If we don’t consider the hesitant degree of IFS and use the following correlation coefficient: $ρ_{4}^{'} (A, B) = \frac{\sqrt{\sum_{i = 1}^{n} w_{i} ({(μ_{A} (x_{i}) + μ_{B} (x_{i}))}^{2} + {(ν_{A} (x_{i}) + ν_{B} (x_{i}))}^{2})}}{\sqrt{\sum_{i = 1}^{n} w_{i} (μ_{A}^{2} (x_{i}) + ν_{A}^{2} (x_{i}))} + \sqrt{\sum_{i = 1}^{n} w_{i} (μ_{B}^{2} (x_{i}) + ν_{B}^{2} (x_{i}))}}$

Repeating the above steps, we will get the closeness coefficient of each alternative: $\begin{matrix} c_{1} = 0.5057, c_{2} = 0.5125, c_{3} = 0.5123, \\ c_{4} = 0.4943, c_{5} = 0.5078, c_{6} = 0.5214 . \end{matrix}$ and thus A₆ ≻ A₂ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₄.

Obviously, the rankings of the alternatives are different. The main difference lies in the ordering of the alternatives A₂, A₃, A₆. In the first method, A₃ ≻ A₆ ≻ A₂, while in the second method, A₆ ≻ A₂ ≻ A₃. In other words, the sorts of the two pairs of alternatives A₃, A₆ and A₂, A₃ in the two methods are opposite. The main reason of the above different results is that the former considers all the parameters: the membership degree, the non-membership degree and the hesitant degree of the IFS, while the latter omits the hesitant degree of it. In the practical decision making process, not considering the hesitant degree of the IFV will lose the original decision information and thus will affect the reasonability of the final decision making results. For interval-valued intuitionistic fuzzy case, the analogous work can be done for comparisons.

4 Interval-valued intuitionistic fuzzy multi-attribute decision making based on ideal-point-based method

4.1 The basic concepts about IVIFS

Atanassov and Gargov [3] considered that sometimes it is more suitable to characterize the membership degrees and the non-membership degrees for certain elements of A by using fuzzy ranges than by exactly real numbers. Hence, the notion of interval-valued intuitionistic fuzzy set (IVIFS) came into being. They defined an IVIFS $\tilde{A}$ over X an object having the form: $\tilde{A} = {< x, {\tilde{μ}}_{\tilde{A}} (x), {\tilde{ν}}_{\tilde{A}} (x) > | x \in X}$ , where ${\tilde{μ}}_{\tilde{A}} (x) \subset [0, 1]$ and ${\tilde{ν}}_{\tilde{A}} (x) \subset [0, 1]$ are intervals, and sup ${\tilde{μ}}_{\tilde{A}} (x) +$ sup ${\tilde{ν}}_{\tilde{A}} (x) \leq 1$ , for every x ∈ X. Especially, if each of the intervals ${\tilde{μ}}_{\tilde{A}} (x)$ and ${\tilde{ν}}_{\tilde{A}} (x)$ contains exactly one element, i.e., if $μ_{\tilde{A}} (x) = inf {\tilde{μ}}_{\tilde{A}} (x) = sup {\tilde{μ}}_{\tilde{A}} (x)$ , $ν_{\tilde{A}} (x) = inf {\tilde{ν}}_{\tilde{A}} (x) = sup {\tilde{ν}}_{\tilde{A}} (x)$ , for every x ∈ X, then, the given IVIFS $\tilde{A}$ reduces to an ordinary IFS.

4.2 The correlation coefficient of IVIFSs

Let $\tilde{Φ} (X)$ be the set of all IVIFSs over X, we define the concept of correlation coefficient between two IVIFSs as follows:

Definition 2. Let $\dot{ρ}$ be a mapping $\dot{ρ} : {(\tilde{Φ} (x))}^{2} \to [0, 1]$ , then the correlation coefficient between two IVIFSs $\tilde{A}$ and $\tilde{B}$ is defined as $\dot{ρ} (\tilde{A}, \tilde{B})$ , which satisfies the following conditions:

$0 \leq \dot{ρ} (\tilde{A}, \tilde{B}) \leq 1$ ;

$\dot{ρ} (\tilde{A}, \tilde{B}) = 1$ if $\tilde{A} = \tilde{B}$ ;

$\dot{ρ} (\tilde{A}, \tilde{B}) = \dot{ρ} (\tilde{B}, \tilde{A})$ .

In the case where X = {x₁, x₂, ⋯ , x_n} is a discrete universe of discourse, we extend ρ₄ (A, B) to IVIFSs to calculate the correlation coefficient between two IVIFSs $\tilde{A}$ and $\tilde{B}$ as below:

${\dot{ρ}}_{7} (\tilde{A}, \tilde{B}) = \frac{\sqrt{\sum_{j = 1}^{n} f_{\tilde{A}, \tilde{B}} (x_{j})}}{\sqrt{\sum_{j = 1}^{n} g_{\tilde{A}} (x_{j})} + \sqrt{\sum_{j = 1}^{n} g_{\tilde{B}} (x_{j})}}$ (23)where $\begin{matrix} g_{\tilde{A}} (x_{j}) & = {({\tilde{μ}}_{\tilde{A}}^{L} (x_{j}))}^{2} + {({\tilde{ν}}_{\tilde{A}}^{L} (x_{j}))}^{2} + {(1 - {\tilde{π}}_{\tilde{A}}^{L} (x_{j}))}^{2} \\ + {({\tilde{μ}}_{\tilde{A}}^{U} (x_{j}))}^{2} + {({\tilde{ν}}_{\tilde{A}}^{U} (x_{j}))}^{2} + {(1 - {\tilde{π}}_{\tilde{A}}^{U} (x_{j}))}^{2} \end{matrix}$ (24) $\begin{matrix} g_{\tilde{B}} (x_{j}) & = {({\tilde{μ}}_{\tilde{B}}^{L} (x_{j}))}^{2} + {({\tilde{ν}}_{\tilde{B}}^{L} (x_{j}))}^{2} + {(1 - {\tilde{π}}_{\tilde{B}}^{L} (x_{j}))}^{2} \\ + {({\tilde{μ}}_{\tilde{B}}^{U} (x_{j}))}^{2} + {({\tilde{ν}}_{\tilde{B}}^{U} (x_{j}))}^{2} + {(1 - {\tilde{π}}_{\tilde{B}}^{U} (x_{j}))}^{2} \end{matrix}$ (25) $\begin{matrix} f_{\tilde{A}, \tilde{B}} (x_{j}) & = {({\tilde{μ}}_{\tilde{A}}^{L} (x_{j}) + {\tilde{μ}}_{\tilde{B}}^{L} (x_{j}))}^{2} + {({\tilde{ν}}_{\tilde{A}}^{L} (x_{j}) + {\tilde{ν}}_{\tilde{B}}^{L} (x_{j}))}^{2} \\ + {(2 - {\tilde{π}}_{\tilde{A}}^{L} (x_{j}) - {\tilde{π}}_{\tilde{B}}^{L} (x_{j}))}^{2} + {({\tilde{μ}}_{\tilde{A}}^{U} (x_{j}) + {\tilde{μ}}_{\tilde{B}}^{U} (x_{j}))}^{2} \\ + {({\tilde{ν}}_{\tilde{A}}^{U} (x_{j}) + {\tilde{ν}}_{\tilde{B}}^{U} (x_{j}))}^{2} + {(2 - {\tilde{π}}_{\tilde{A}}^{U} (x_{j}) - {\tilde{π}}_{\tilde{B}}^{U} (x_{j}))}^{2} \end{matrix}$ (26)

If we need to consider the weight of the element x ∈ X, then Equation (23) can be extended to its weighted counterpart: ${\dot{ρ}}_{8} (\tilde{A}, \tilde{B}) = \frac{\sqrt{\sum_{j = 1}^{n} w_{j} f_{\tilde{A}, \tilde{B}} (x_{j})}}{\sqrt{\sum_{j = 1}^{n} w_{j} g_{\tilde{A}} (x_{j})} + \sqrt{\sum_{j = 1}^{n} w_{j} g_{\tilde{B}} (x_{j})}}$ (27)

where w = (w₁, w₂, …, w_n) ^T is the weight vector of x_i (i = 1, 2, …, n), with w_j ≥ 0, j = 1, 2, …, n and $\sum_{j = 1}^{n} w_{j} = 1$ .

If $w_{1} = w_{2} = \dots = w_{n} = \frac{1}{n}$ , then Equation (27) reduces to Equation (23).

In the following, we prove that Equation (27) satisfies all the conditions of Definition 2.

Proof. Since $\tilde{A}, \tilde{B} \in \tilde{Φ} (X)$ , then $\begin{matrix} 0 \leq {\tilde{μ}}_{\tilde{A}}^{L} (x_{j}) \leq {\tilde{μ}}_{\tilde{A}}^{U} (x_{j}) \leq 1, 0 \leq {\tilde{ν}}_{\tilde{A}}^{L} (x_{j}) \leq {\tilde{ν}}_{\tilde{A}}^{U} (x_{j}) \leq 1, \\ 0 \leq {\tilde{π}}_{\tilde{A}}^{L} (x_{j}) \leq {\tilde{π}}_{\tilde{A}}^{U} (x_{j}) \leq 1, for all x_{j} \in X \end{matrix}$ (28) $\begin{matrix} 0 \leq {\tilde{μ}}_{\tilde{B}}^{L} (x_{j}) \leq {\tilde{μ}}_{\tilde{B}}^{U} (x_{j}) \leq 1, 0 \leq {\tilde{ν}}_{\tilde{B}}^{L} (x_{j}) \leq {\tilde{ν}}_{\tilde{B}}^{U} (x_{j}) \leq 1, \\ 0 \leq {\tilde{π}}_{\tilde{B}}^{L} (x_{j}) \leq {\tilde{π}}_{\tilde{B}}^{U} (x_{j}) \leq 1, for all x_{j} \in X \end{matrix}$ (29) and thus, by Equation (27), we get ${\dot{ρ}}_{8} (\tilde{A}, \tilde{B}) \geq 0$ . According to the famous triangle inequality Equation (6), we have

$\begin{matrix} \sqrt{\sum_{j = 1}^{n} w_{j} f_{\tilde{A}, \tilde{B}} (x_{j})} & \leq \sqrt{(\sum_{j = 1}^{n} w_{j} g_{\tilde{A}} (x_{j}))} \\ + \sqrt{(\sum_{j = 1}^{n} w_{j} g_{\tilde{B}} (x_{j}))} \end{matrix}$ (30)and thus ${\dot{ρ}}_{8} (\tilde{A}, \tilde{B}) \leq 1$ with equality if and only if there is a real number η (η > 0) such that $\begin{matrix} μ_{\tilde{A}}^{L} (x_{i}) = η μ_{\tilde{B}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i}) = η μ_{\tilde{B}}^{U} (x_{i}), \\ ν_{\tilde{A}}^{L} (x_{i}) = η ν_{\tilde{B}}^{L} (x_{i}), ν_{\overset{⌢}{A}}^{U} (x_{i}) = η ν_{\tilde{B}}^{U} (x_{i}), \end{matrix}$ and

$\begin{matrix} 1 - π_{\tilde{A}}^{L} (x_{i}) & = η (1 - π_{\tilde{B}}^{L} (x_{i})), 1 - π_{\tilde{A}}^{U} (x_{i}) \\ = η (1 - π_{\tilde{B}}^{U} (x_{i})) for all x_{i} \in X \end{matrix}$ (31)

If $\tilde{A} = \tilde{B}$ , i.e., $\begin{matrix} μ_{\tilde{A}}^{L} (x_{i}) = μ_{\tilde{B}}^{L} (x_{i}), μ_{\tilde{A}}^{U} (x_{i}) = μ_{\tilde{B}}^{U} (x_{i}), \\ ν_{\overset{⌣}{A}}^{L} (x_{i}) = ν_{\tilde{B}}^{L} (x_{i}), ν_{\tilde{A}}^{U} (x_{i}) = ν_{\tilde{B}}^{U} (x_{i}), \\ π_{\tilde{A}}^{L} (x_{i}) = π_{\tilde{B}}^{L} (x_{i}), π_{\tilde{A}}^{U} (x_{i}) = π_{\tilde{B}}^{U} (x_{i}), for all x_{i} \in X \end{matrix}$ (32)then by Equation (31), we know η = 1, and thus, ${\dot{ρ}}_{8} (\tilde{A}, \tilde{B}) = 1$ . In addition, we get by Equation (27) that $\begin{matrix} {\dot{ρ}}_{8} (\tilde{A}, \tilde{B}) = \frac{\sqrt{\sum_{j = 1}^{n} w_{j} f_{\tilde{A}, \tilde{B}} (x_{j})}}{\sqrt{\sum_{j = 1}^{n} w_{j} g_{\tilde{A}} (x_{j})} + \sqrt{\sum_{j = 1}^{n} w_{j} g_{\tilde{B}} (x_{j})}} \\ = \frac{\sqrt{\sum_{j = 1}^{n} w_{j} f_{\tilde{B}, \tilde{A}} (x_{j})}}{\sqrt{\sum_{j = 1}^{n} w_{j} g_{\tilde{B}} (x_{j})} + \sqrt{\sum_{j = 1}^{n} w_{j} g_{\tilde{A}} (x_{j})}} = {\dot{ρ}}_{8} (\tilde{B}, \tilde{A}) \end{matrix}$ (33)

Hence, all the conditions in Definition 2 hold.

If the universe of discourse, X, is continuous and the weight of the element x ∈ X = [a, b] is w (x), where w (x) ≥ 0 and $\int_{a}^{b} w (x) dx = 1$ , then we get the continuous form of Equation (27): ${\dot{ρ}}_{9} (\tilde{A}, \tilde{B}) = \frac{\sqrt{\int_{a}^{b} w (x) f_{\tilde{A}, \tilde{B}} (x) dx}}{\sqrt{\int_{a}^{b} w (x) g_{\tilde{A}} (x) dx} + \sqrt{\int_{a}^{b} w (x) g_{\tilde{B}} (x) dx}}$ (34)where $\begin{matrix} g_{\tilde{A}} (x) & = {({\tilde{μ}}_{\tilde{A}}^{L} (x))}^{2} + {({\tilde{ν}}_{\tilde{A}}^{L} (x))}^{2} + {(1 - {\tilde{π}}_{\tilde{A}}^{L} (x))}^{2} \\ + {({\tilde{μ}}_{\tilde{A}}^{U} (x))}^{2} + {({\tilde{ν}}_{\tilde{A}}^{U} (x))}^{2} + {(1 - {\tilde{π}}_{\tilde{A}}^{U} (x))}^{2} \end{matrix}$ (35) $\begin{matrix} g_{\tilde{B}} (x) & = {({\tilde{μ}}_{\tilde{B}}^{L} (x))}^{2} + {({\tilde{ν}}_{\tilde{B}}^{L} (x))}^{2} + {(1 - {\tilde{π}}_{\tilde{B}}^{L} (x))}^{2} \\ + {({\tilde{μ}}_{\tilde{B}}^{U} (x))}^{2} + {({\tilde{ν}}_{\tilde{B}}^{U} (x))}^{2} + {(1 - {\tilde{π}}_{\tilde{B}}^{U} (x))}^{2} \end{matrix}$ (36) $\begin{matrix} f_{\tilde{A}, \tilde{B}} (x_{j}) & = {({\tilde{μ}}_{\tilde{A}}^{L} (x) + {\tilde{μ}}_{\tilde{B}}^{L} (x))}^{2} + {({\tilde{ν}}_{\tilde{A}}^{L} (x) + {\tilde{ν}}_{\tilde{B}}^{L} (x))}^{2} \\ + {(2 - {\tilde{π}}_{\tilde{A}}^{L} (x) - {\tilde{π}}_{\tilde{B}}^{L} (x))}^{2} + {({\tilde{μ}}_{\tilde{A}}^{U} (x) + {\tilde{μ}}_{\tilde{B}}^{U} (x))}^{2} \\ + {({\tilde{ν}}_{\tilde{A}}^{U} (x) + {\tilde{ν}}_{\tilde{B}}^{U} (x))}^{2} + {(2 - {\tilde{π}}_{\tilde{A}}^{U} (x) - {\tilde{π}}_{\tilde{B}}^{U} (x))}^{2} \end{matrix}$ (37)

If all the elements have the same importance, then Equation (34) reduces to ${\dot{ρ}}_{10} (\tilde{A}, \tilde{B}) = \frac{\sqrt{\int_{a}^{b} f_{\tilde{A}, \tilde{B}} (x) dx}}{\sqrt{\int_{a}^{b} g_{\tilde{A}} (x) dx} + \sqrt{\int_{a}^{b} g_{\tilde{B}} (x) dx}}$ (38)

4.3 The description of interval-valued intuitionistic fuzzy multi-attribute decision making problem

Let $\tilde{A} = {{\tilde{A}}_{1}, {\tilde{A}}_{2}, \dots, {\tilde{A}}_{n}}$ be a discrete set of alternatives, X = {x₁, x₂, ⋯ , x_m} be the set of attributes, and w = {w₁, w₂, ⋯ , w_m} ^T be the weight vector of attributes, with w_j ≥ 0, j = 1, 2, …, m and $\sum_{j = 1}^{m} w_{j} = 1$ . Let $\dot{R} = ({\tilde{r}}_{ij})_{n \times m}$ be the decision matrix, where each ${\tilde{r}}_{ij}$ denotes the attribute value provided by the decision maker for the ith alternative ${\tilde{A}}_{i}$ with respect to the jth attribute x_j. In this case, the preferences expressed by the decision maker over different attributes are IVIFVs.

Below we give an ideal-point-based method for interval-valued intuitionistic fuzzy multi-attribute decision making to rank the alternatives.

4.4 Interval-valued intuitionistic fuzzy multi-attribute decision making based on ideal-point-based method

Step 1. Define the interval-valued intuitionistic fuzzy ideal solution (IVIFIS): ${\tilde{A}}^{+} = ({\tilde{α}}_{1}^{+}, {\tilde{α}}_{2}^{+}, \dots, {\tilde{α}}_{m}^{+})$ (39)and the interval-valued intuitionistic fuzzy negative ideal solution (A-IVIFNIS): ${\tilde{A}}^{-} = ({\tilde{α}}_{1}^{-}, {\tilde{α}}_{2}^{-}, \dots, {\tilde{α}}_{m}^{-})$ (40)with

$\begin{matrix} {\tilde{α}}_{i}^{+} & = ([1, 1], [0, 0]), {\tilde{α}}_{i}^{-} = ([0, 0], [1, 1]), \\ i & = 1, 2, \dots, m \end{matrix}$ (41)

Step 2. Calculate the correlation coefficient between the alternative ${\tilde{A}}_{i}$ and the IVIFIS ${\tilde{A}}^{+}$ , and the correlation coefficient between the alternative ${\tilde{A}}_{i}$ and the IVIFNIS ${\tilde{A}}^{-}$ , respectively: $s_{i}^{+} = {\dot{ρ}}_{8} ({\tilde{A}}_{i}, {\tilde{A}}^{+}), i = 1, 2, \dots, n$ (42)and $s_{i}^{-} = {\dot{ρ}}_{8} ({\tilde{A}}_{i}, {\tilde{A}}^{-}), i = 1, 2, \dots, n$ (43)

If the weights of the attributes are equal, we can choose $s_{i}^{+} = {\dot{ρ}}_{7} ({\tilde{A}}_{i}, {\tilde{A}}^{+}), i = 1, 2, \dots, n$ (44)and $s_{i}^{-} = {\dot{ρ}}_{7} ({\tilde{A}}_{i}, {\tilde{A}}^{-}), i = 1, 2, \dots, n$ (45)

Step 3. Calculate the closeness coefficients of each alternative: $c_{i} = \frac{s_{i}^{+}}{s_{i}^{-} + s_{i}^{+}}, i = 1, 2, \dots, n$ (46)

Step 4. Rank all the alternatives ${\tilde{A}}_{i} (i = 1, 2, \dots, n)$ according to the closeness coefficients c_i (i = 1, 2, …, n), the greater the value c_i, the better the alternative ${\tilde{A}}_{i}$ .

Step 5. End.

4.5 Numerical example

Example 3. If the team of financial experts evaluates the investment programs ${\tilde{A}}_{i} (i = 1, 2, \dots, 5)$ according to the attributes x_j (j = 1, 2, …, 6), and gives the interval-valued intuitionistic fuzzy decision making information in Table 3:

Suppose that the weights of the attributes x_j (j = 1, 2, …, 6) are equal, now we use the ideal-point-based method for interval-valued intuitionistic fuzzy multi-attribute decision to prioritize these investment programs ${\tilde{A}}_{i} (i = 1, 2, \dots, 5)$ :

Step 1. Define the interval-valued intuitionistic fuzzy ideal solution (IVIFIS): ${\tilde{A}}^{+} = (([1, 1], [0, 0]), ([1, 1], [0, 0]), \dots, ([1, 1], [0, 0]))$ (47)and the interval-valued intuitionistic fuzzy negative ideal solution (IVIFNIS): ${\tilde{A}}^{-} = (([0, 0], [1, 1]), ([0, 0], [1, 1]), \dots, ([0, 0], [1, 1]))$ (48)

Step 2. Calculate the correlation coefficient between the alternative ${\tilde{A}}_{i}$ and the IVIFIS ${\tilde{A}}^{+}$ , and the correlation coefficient between the alternative ${\tilde{A}}_{i}$ and the IVIFNIS ${\tilde{A}}^{-}$ , by Equations (44) and (45), respectively: $\begin{matrix} s_{1}^{+} = 0.9567, s_{2}^{+} = 0.9754, s_{3}^{+} = 0.9791, \\ s_{4}^{+} = 0.9821, s_{5}^{+} = 0.9840 \\ s_{1}^{-} = 0.8943, s_{2}^{-} = 0.9214, s_{3}^{-} = 0.9335, \\ s_{4}^{-} = 0.9259, s_{5}^{-} = 0.9220 \end{matrix}$ (49)

Step 3. Calculate the closeness coefficient of each alternative by using Equation (46):

$\begin{matrix} c_{1} & = 0.5169, c_{2} = 0.5142, c_{3} = 0.5119, \\ c_{4} & = 0.5147, c_{5} = 0.5163 \end{matrix}$ (50)

Step 4. Rank all the alternatives ${\tilde{A}}_{i} (i = 1, 2, \dots, 5)$ according to the closeness coefficient c_i (i = 1, 2, …, 5), then we get ${\tilde{A}}_{1} ≻ {\tilde{A}}_{5} ≻ {\tilde{A}}_{4} ≻ {\tilde{A}}_{2} ≻ {\tilde{A}}_{3}$ (51)

Step 5. End.

Remark: Comparing the result derived by the interval-valued intuitionistic fuzzy ideal-point-based method (Example 3) with that derived by the intuitionistic fuzzy ideal-point-based method (Example 1), we find that the position of the first alternative changes a lot. Making a mathematical analysis of Equation.(23), we can see that most of the attribute values’ membership degrees and non-membership degrees of the first alternative A₁ in Example 1 of Section 3.4 are the maximal values in the corresponding attribute values’ membership intervals and non-membership intervals of the first alternative ${\tilde{A}}_{1}$ in the example of Section 4.5, respectively, which leads to the change of the closeness coefficient of the first alternative.

5 Concluding remarks

The TOPSIS method has been investigated in the existing literatures and used in a wide range of applications since its first appearance. Most of the existing TOPSIS methods are concerned with real number information, but not too many papers discussed the TOPSIS methods dealing with intuitionistic fuzzy (or interval-valued intuitionistic fuzzy) information. In this paper, we have exerted our effort on intuitionistic fuzzy TOPSIS method where the decision making information is expressed by IFSs. We have first constructed an intuitionistic fuzzy correlation measure to compute the correlation coefficient between every two IFSs. Then based on the intuitionistic fuzzy correlation measure, the intuitionistic fuzzy ideal solution (IFIS) and the intuitionistic fuzzy negative ideal solution (IFNIS), we have calculated the closeness coefficient for each alternative, by which all the alternatives can be ranked. Afterwards, we have applied the developed method to solve a problem of prioritization for investment programs. We have also pointed out that when we calculate the correlation coefficient between every pair of IFSs, the hesitant degree of IFSs should be considered. Otherwise, we will get the improper decision making results. Furthermore we have extended the method to interval-valued intuitionistic fuzzy environment.

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 61273209), and the Central University Basic Scientific Research Business Expenses Project (No. skgt201501).

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