A new approach for multiple attribute group decision-making based on interval neutrosophic sets

Abstract

The performance of interval neutrosophic sets (INSs) aggregated by resent operator is sensitive to special value (e.g. zero value). This paper proposes a novel algorithm for aggregating INSs and applies it to multiple attribute group decision-making. First, INSs are projected as point sets of a 2D space, thus the interval numbers of INSs and the 2D space points correspond one-to-one. Second, the plant growth simulation algorithm is employed to aggregate each point set and calculate the optimal rally points of each point set, therefore, the collective matrix is established by the optimal rally point set. Third, the proposed method achieved in the implementation of multiple attribute group decision-making with interval neutrosophic information. The simulation shows that the proposed aggregation method is insensitive to special value and achieve better performance than compared approach.

Keywords

Interval neutrosophic set multiple attribute group decision-making plant growth simulation algorithm optimal rally point collective matrix

1 Introduction

Fuzzy information is widely used to express decision-maker (DM) preference with the increasing complexity of the decision-making environment, the decreasing professional degree of DM, and the increasing demand for real-time decision-making. Zadeh [1] developed the theory of fuzzy sets (FS), which is an effective tool for representing fuzzy information. Furthermore, Atanassov [2 –4] introduced intuitionistic FSs (IFS) and interval-valued IFS (IVIFS) to improved the limitations of FSs. Many scholars have investigated multiple attribute group decision-making (MAGDM) with interval-valued intuitionistic fuzzy information, and several effective operators have been proposed for aggregating the IVIFS [5 –9]. Yue [10] investigated a group decision-making (GDM) method with interval data and developed an approach to aggregate interval data into interval-valued intuitionistic fuzzy information. Liu [11] developed an approach to determine the integrated weights of experts, based on interval-valued preference matrices. They used the generalized Fermat point to determine the weights of experts, which provided good simulation results. Wan [12] discussed the additive consistency of interval-valued intuitionistic fuzzy preference relations and designed a likelihood comparison algorithm for ranking IVIFSs. Wang [13] developed the linear programming (LP) methodology and the extended the technique for order of preference by similarity to ideal solution method for multiple-attribute decision-making based on IVIFSs, where the LP methodology is used to obtain optimal attribute weights. However, because of time pressure, area knowledge limitations, and decision environment complexity, DMs always put forward preferences with uncertainty. For instance, when a DM gives the preference for a contingency plan at a disaster site, he or she may say that the membership degree is 0.5, the non-membership degree is 0.4, and the degree of indeterminacy is 0.3. Obviously, FSs, IFSs, IVIFSs cannot express the DM preference because the sum of three degrees is greater than 1. Therefore, more suitable theories have been developed.

Smarandache [14] developed neutrosophic logic, neutrosophic sets (NS), and the three dimensions (i.e., truth, falsehood, and indeterminacy) for 3D space, which, as described by neutrosophic logic, are subsets to ] ^-0, 1⁺ [. Truth-membership degree, indeterminacy-membership degree, and falsity-membership degree in neutrosophic sets are mutually independent. Thus, the DM statement example can be distributed as, d (0.5, 0.4, 0.3). NSs have been widely employed in many areas, such as medical diagnostics, image processing, and investment selection [15 –20]. Broumi [21] developed an extended Hausdorff distance between two NSs based on the Hausdorff metric and defined a new series of similarity measures to calculate the similarity between neutrosophic sets. Ye [22] developed three vector similarity measures between simplified NSs as a generalization of the Jaccard, Dice, and cosine similarity measures between two vectors. Peng [23] presented three approacges for MADM under single-valued neutrosophic environment and demonstrated their effectiveness and feasibility by two numerical examples. Eisa [24] described “image” as an NS and proposed a new image segmentation technique and an adaptive threshold, which proved the advantage of NS in representing uncertain information. However, NSs cannot be employed to express information in science and engineering problems because they are based on philosophical thinking. To resolve this, several scholars extended NSs to develop new concepts. Liu [25] proposed the normal neutrosophic Bonferroni mean operator and the normal neutrosophic geometric Bonferroni mean operator for aggregating the information expressed by the normal neutrosophic sets. Peng [26] introduced multi-valued neutrosophic sets and defined the operations of multi-valued neutrosophic numbers. Zhang [27] investigated a correlation coefficient measure of NSs and IFSs, and then developed a new correlation coefficient measure for INSs. According to the new measure, he established a procedure to solve multi-criteria decision-making problems. Şahin [28] developed cross-entropy for INSs by referring to the cross-entropy for FSs and SVNSs. Thus, the proposed cross-entropy is established in MAGDM, for which criteria alternatives are characterized by INSs. Ye [29] developed an improved a cross-entropy measure for SVNSs and extended it to INSs, overcoming the drawbacks of the cross-entropy measure defined by Ye [30]. Liu [31] proposed Muirhead mean (MM) operator, interval neutrosophic MM operator, interval neutrosophic weighted MM operator, and interval neutrosophic dual weighted MM operator to deal with multiple attribute DM problems under the interval neutrosophic environment.

The above operators can aggregate INSs. However, aggregation operators have some drawbacks: (1) some employ much continued product in the process of calculation of correlation coefficient, cross-entropy, and collective matrix. Thus, the aggregation result will produce great deviation when the value of alternative characterize contains zero; and (2) the operator operation process is complicated. In view of different practical problems, it is necessary to recalculate the indices between INSs. The efficiency and accuracy of some operators will be reduced when they deal with MAGDM, because they have several DM preference matrices. Compared to traditional operators, intelligent algorithms have the advantages of insensitivity to special values, strong universality, and high accuracy [9, 11].

A novel method for aggregating INFs is proposed in this paper. The INNs in INFs are projected onto a 2D coordinate plane. The optimal rally points (i.e., the point whose sum of Euclidean distances to other given points is minimal) of each set are aggregated by the PGSA. Further, the collective INS is established by the optimal rally point set. The correlation coefficient between collective INS and other given INSs are calculated by projection theory. According to the above process, an accurate INSs sorting can be efficiently obtained. The main advantages of the proposed method are: (1) because the INSs are aggregated by calculated the distance between points, the proposed method is unaffected by special values (e.g., 0); and (2) because the proposed method transforms the correlation coefficient between INSs into the distance between points. INSs of different practical problems can be directly aggregated after projecting onto a 2D coordinate plane without the use of other operators to calculate the correlation coefficient and cross-entropy between INSs.

This paper is arranged in the following manner. Section 2 briefly describes the research problem statement and contributions of this paper. Section 3 introduces the theory of INS, optimal rally point, and plant growth simulation algorithm (PGSA). The aggregation method for INS and its application in MAGDM are then introduced, in detail, in Section 4. In Section 5, an illustrative example is provided to demonstrate the application and effectiveness of the developed method. Conclusions and future work are provided in Section 6.

2 Preliminaries

In this section, basic concepts and definitions of INSs, optimal rally point, PGSA and projection theory are introduced.

2.1 Interval neutrosophic sets

Definition 1. [32] Let X be a space of points (i.e., objects), with a generic element in X denoted by x. An INS, A, in X is defined with the form, $A = {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 | x \in X},$

Where

T_A (x) = [inf T_A (x) , sup T_A (x)] ⊆ [0, 1],

I_A (x) = [inf I_A (x) , sup I_A (x)] ⊆ [0, 1],

and F_A (x) = [inf F_A (x) , sup F_A (x)] ⊆ [0, 1]

denotes the truth-membership function, the indeterminacy-membership function, and the falsity-membership function of the element x ∈ X to the set A, respectively. Because the three functions are independent of each other, for each point, x, in X, we have 0 ⩽ T_A (x) + I_A (x) + F_A (x) ⩽ 3. For convenience, let

$T_{A} (x) = [T_{A}^{L} (x), T_{A}^{U} (x)]$ ,

$I_{A} (x) = [I_{A}^{L} (x), I_{A}^{U} (x)]$ ,

$F_{A} (x) = [F_{A}^{L} (x), F_{A}^{U} (x)]$ .

Then, A ={ 〈 x, T_A (x) , I_A (x) , F_A (x) 〉 |x ∈ X }

Definition 2. [32] Let INS (X) denote the family of all INSs in universe, X. Assume A, B ∈ INS (X) such that

A ={ 〈 x, T_A (x) , I_A (x) , F_A (x) 〉 |x ∈ X }, and

B ={ 〈 x, T_B (x) , I_B (x) , F_B (x) 〉 |x ∈ X }.

Then, some operations can be defined as follows. $\begin{matrix} A \cup B = {〈 x, [max {T_{A}^{L} (x), T_{B}^{L} (x)}, max {T_{A}^{U} (x), T_{B}^{U} (x)}], \\ [min {I_{A}^{L} (x), I_{B}^{L} (x)}, min {I_{A}^{U} (x), I_{B}^{U} (x)}], \\ [min {F_{A}^{L} (x), F_{B}^{L} (x)}, min {F_{A}^{U} (x), F_{B}^{U} (x)}] 〉 | x \in X}, \end{matrix}$ (1) $\begin{matrix} A \cap B = {〈 x, [min {T_{A}^{L} (x), T_{B}^{L} (x)}, min {T_{A}^{U} (x), T_{B}^{U} (x)}], \\ [max {I_{A}^{L} (x), I_{B}^{L} (x)}, max {I_{A}^{U} (x), I_{B}^{U} (x)}], \\ [max {F_{A}^{L} (x), F_{B}^{L} (x)}, max {F_{A}^{U} (x), F_{B}^{U} (x)}] 〉 | x \in X} \end{matrix}$ (2) $\begin{matrix} A^{c} = {〈 x, [T_{A}^{L} (x), T_{A}^{U} (x)], \\ [1 - I_{A}^{U} (x), 1 - I_{A}^{L} (x)], \\ [F_{A}^{L} (x), F_{A}^{U} (x)] 〉 | x \in X} \end{matrix}$ (3)

A ⊆ B,

Iff

$T_{A}^{L} (x) ⩽ T_{B}^{L} (x), T_{A}^{U} (x) ⩽ T_{B}^{U} (x)$ ,

$I_{A}^{L} (x) ⩾ I_{B}^{L} (x), I_{A}^{U} (x) ⩾ I_{B}^{U} (x)$ and

$F_{A}^{L} (x) ⩾ F_{B}^{L} (x)$ ,

$F_{A}^{U} (x) ⩾ F_{B}^{U} (x)$ for all

x ∈ X. A = B, iff A ⊆ B and B ⊆ A.

Definition 3. [33] Let

$x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}], [F_{1}^{L}, F_{1}^{U}])$ and

$y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$

be two INNs and k > 0. The operations of INNs can be defined as follows. $\begin{matrix} x \oplus y = ([T_{1}^{L} + T_{2}^{L} - T_{1}^{L} T_{2}^{L}, T_{1}^{U} + T_{2}^{U} - T_{1}^{U} T_{2}^{U}], \\ [I_{1}^{L} I_{2}^{L}, I_{1}^{U} I_{2}^{U}], [F_{1}^{L} F_{2}^{L}, F_{1}^{U} F_{2}^{U}]) \\ [I_{1}^{L} I_{2}^{L}, I_{1}^{U} I_{2}^{U}], [F_{1}^{L} F_{2}^{L}, F_{1}^{U} F_{2}^{U}]) \end{matrix}$ (4) $\begin{matrix} x \otimes y = ([T_{1}^{L} T_{2}^{L}, T_{1}^{U} T_{2}^{U}], \\ [I_{1}^{L} + I_{2}^{L} - I_{1}^{L} I_{2}^{L}, I_{1}^{U} + I_{2}^{U} - I_{1}^{U} I_{2}^{U}], \\ [F_{1}^{L} + F_{2}^{L} - F_{1}^{L} F_{2}^{L}, F_{1}^{U} + F_{2}^{U} - F_{1}^{U} F_{2}^{U}]) \end{matrix}$ (5) $\begin{matrix} kx = ([1 - {(1 - T_{1}^{L})}^{k}, 1 - {(1 - T_{1}^{U})}^{k}], \\ [{(I_{1}^{L})}^{K}, {(I_{1}^{U})}^{K}], [{(F_{1}^{L})}^{K}, {(F_{1}^{U})}^{K}]) \end{matrix}$ (6) $\begin{matrix} x^{k} = ([{(T_{1}^{L})}^{k}, {(T_{1}^{U})}^{k}], \\ [1 - {(1 - I_{1}^{L})}^{k}, 1 - {(1 - I_{1}^{U})}^{k}], \\ [1 - {(1 - F_{1}^{L})}^{k}, 1 - {(1 - F_{1}^{U})}^{k}]) \end{matrix}$ (7)

Theorem 1. [33] Let

$x = ([T_{1}^{L}, T_{1}^{U}], [I_{1}^{L}, I_{1}^{U}], [F_{1}^{L}, F_{1}^{U}])$ ,

$y = ([T_{2}^{L}, T_{2}^{U}], [I_{2}^{L}, I_{2}^{U}], [F_{2}^{L}, F_{2}^{U}])$ , and

$z = ([T_{3}^{L}, T_{3}^{U}], [I_{3}^{L}, I_{3}^{U}], [F_{3}^{L}, F_{3}^{U}])$

be three INNs. Thus, operational rules of INNs have properties as follows.

x ⊕ y = y ⊕ x,

x ⊗ y = y ⊗ x,

λ (x ⊕ y) = λx ⊕ λy, λ > 0,

(x ⊗ y) ^λ = x^λ ⊗ y^λ, λ > 0,

λ₁z ⊕ λ₂z = (λ₁ + λ₂) z, λ₁ > 0, λ₂ > 0, and

z^{λ
₁} ⊗ z^{λ
₂} = z^λ₁+λ₂, λ₁ > 0, λ₂ > 0.

Definition 4. [28] Let A and B be two INSs in the universe discourse, X ={ x₁, x₂, ⋯ , x_n }. Then, the distance measure between them can be defined as follows.

The Hamming distance is $\begin{matrix} d_{H} (A, B) = \frac{1}{6} \sum_{i = 1}^{n} [ \\ | T_{A}^{L} (x_{i}) - T_{B}^{L} (x_{i}) | + | T_{A}^{U} (x_{i}) - T_{B}^{U} (x_{i}) | \\ + | I_{A}^{L} (x_{i}) - I_{B}^{L} (x_{i}) | + | I_{A}^{U} (x_{i}) - I_{B}^{U} (x_{i}) | \\ + | F_{A}^{L} (x_{i}) - F_{B}^{L} (x_{i}) | + | F_{A}^{U} (x_{i}) - F_{B}^{U} (x_{i}) |] . \end{matrix}$ (8)

The normalized Hamming distance is $\begin{matrix} d_{nH} (A, B) = \frac{1}{6 n} \sum_{i = 1}^{n} [ \\ | T_{A}^{L} (x_{i}) - T_{B}^{L} (x_{i}) | + | T_{A}^{U} (x_{i}) - T_{B}^{U} (x_{i}) | \\ + | I_{A}^{L} (x_{i}) - I_{B}^{L} (x_{i}) | + | I_{A}^{U} (x_{i}) - I_{B}^{U} (x_{i}) | \\ + | F_{A}^{L} (x_{i}) - F_{B}^{L} (x_{i}) | + | F_{A}^{U} (x_{i}) - F_{B}^{U} (x_{i}) |] . \end{matrix}$ (9)

Definition 5. [33] Let A and B be two INSs in the universe discourse, X ={ x₁, x₂, ⋯ , x_n }. Then, the correlation coefficient of A and B is defined by Broumi: $K (A, B) = \frac{C (A, B)}{\sqrt{E (A) \cdot E (B)}},$ (10) where the correlation of two INSs A and B is given by $\begin{matrix} C (A, B) = \frac{1}{2} \sum_{i = 1}^{n} [ \\ T_{A}^{L} (x_{i}) \cdot T_{B}^{L} (x_{i}) + T_{A}^{U} (x_{i}) \cdot T_{B}^{U} (x_{i}) \\ + I_{A}^{L} (x_{i}) \cdot I_{B}^{L} (x_{i}) + I_{A}^{U} (x_{i}) \cdot I_{B}^{U} (x_{i}) \\ + F_{A}^{L} (x_{i}) \cdot F_{B}^{L} (x_{i}) + F_{A}^{U} (x_{i}) \cdot F_{B}^{U} (x_{i})] \end{matrix}$ (11) and the informational intuitional energies of two INSs A and B are defined as: $\begin{matrix} E (A) = \sum_{i = 1}^{n} \frac{1}{2} [{(T_{A}^{L} (x_{i}))}^{2} + {(T_{A}^{U} (x_{i}))}^{2} + {(I_{A}^{L} (x_{i}))}^{2} \\ + {(I_{A}^{U} (x_{i}))}^{2} + {(F_{A}^{L} (x_{i}))}^{2} + {(F_{A}^{U} (x_{i}))}^{2}], \end{matrix}$ (12)

$\begin{matrix} E (B) = \sum_{i = 1}^{n} \frac{1}{2} [{(T_{B}^{L} (x_{i}))}^{2} + {(T_{B}^{U} (x_{i}))}^{2} + {(I_{B}^{L} (x_{i}))}^{2} \\ + {(I_{B}^{U} (x_{i}))}^{2} + {(F_{B}^{L} (x_{i}))}^{2} + {(F_{B}^{U} (x_{i}))}^{2}] \end{matrix}$ (13)

2.2 Optimal rally point

Definition 6. There are n (n > 3) weighted points in a bounded closed box in a 2D plane, whose corresponding positive weights are ξ_i ∈ [0, 1] (1 ⩽ i ⩽ n), and $\sum_{i = 1}^{n} ξ_{i} = 1$ . If a point, P^*, exists, the Euclidean distances to the other given points meet the following condition. $\begin{matrix} D = min \sum_{i = 1}^{n} | P^{*} P_{i} | \\ = min (\sqrt{(x^{*} - a_{1})^{2} + (y^{*} - b_{1})^{2}} \\ + \dots + \sqrt{(x^{*} - a_{n})^{2} + (y^{*} - b_{n})^{2}}) \end{matrix}$ (14)

Then, P^* can be defined as the optimal rally point. See Fig. 1.

Fig.1

Optimal rally point.

The optimal aggregation point is the point with the minimum sum of weighted distances between all given points. In practical problems, it can be regarded as the point with the closest relationship with all given points.

As the number of points on the plane increases, the difficulty faced in solving the problem increases exponentially. The PGSA is employed to solve this problem in this research.

2.3 Plant growth simulation algorithm

The PGSA is a heuristic algorithm based on the plant growth mechanism first proposed by the Chinese scholar, Li [34]. It is widely employed in many fields, such as decision-making, site selection, and network planning [9 , 35–37]. The PGSA is used to aggregate INNs in this study.

For a plant, the branches closer to the light source have higher morphactin concentration. Thus, they have more opportunities to grow new branches. A new branch that has a certain angle, θ, with the original branch, will grow out from the seed that has the right to grow. Lindenmayer summed up these rules and proposed a formal descriptive grammar for plant growth, called the “L-system” (see Fig. 2; θ = 45°).

Fig.2

L-system.

According to this property, we can design an algorithm to find the global optimal solution of some problems. The continuous growth of plant branches behaves like iterative algorithm steps. Gradual iteration makes the objective function converge. Thus, the plant is “finished” when the results are not changed after many iterations.

The entire growing space of a plant is in the feasible domain of the PGSA probability model. The point closest to the light source is the global optimum point, and the production process of the optimal rally point is like that of the plant growth. To avoid falling into the local optimum, we have two optimizations to the algorithm process. (1) Each iteration searches for the entire growth space. Thus, nodes with low morphactin concentration also can be selected as new growth points, rather than being ignored directly. (2) The morphactin concentration of each growth node will be updated every iteration to improve the efficiency and accuracy of the algorithm. The specific algorithm steps are as follows.

One point, x⁰, is randomly selected as the root, and the trunk, M, grows from it. Suppose there are t nodes, S_M1, S_M2, …, S_Mt, on the trunk having growth hormone concentrations C_M1, C_M2, …, C_Mt. C_Mi (1 ⩽ i ⩽ t) is given by

$C_{Mi} = \frac{f (x^{0}) - f (S_{Mi})}{\sum_{i = 1}^{t} (f (x^{0}) - f (S_{Mi}))} (1 ⩽ i ⩽ t)$ (15) where f (x⁰) is the backlight function for describing the environment of point, x⁰. The shorter the distance between x⁰ and the light source, the smaller the value of f (x⁰). The function value decreases as the illumination of the growth node increases.

$\sum_{i = 1}^{t} C_{Mi} = 1$ is clearly derived from Equation (15). Then, a special roulette (see Fig. 3) is established according to this feature. Thus, this roulette can be employed to select a new growth node.

The node, S_Mi occupies its own area on the roulette. A random number, δ, is selected in the interval [0,1]. This is a method like throwing a ball onto a state map. It will land in one of S_M1, S_M2, …, S_Mt. The corresponding node, S_Mk (1 ⩽ k ⩽ t), which is the preferential growth node, will take priority to grow a new branch in the next step.

We assume that a new branch, m, grows from S_Mk, which has r nodes, namely, S_m1, S_m2, …, S_mr. The growth hormone concentrations of the nodes on branch m are C_m1, C_m2, …, C_mr. According to the principle of plant growth, the morphactin concentration of each node in the plant is updated after each new round of branch growth. After branch, m, has grown, the concentrations of the nodes on trunk, M, except S_Mk and branch m, need to be recalculated. Meanwhile, C_Mi and C_mj (1 ⩽ i ⩽ t, 1 ⩽ j ⩽ r, i ≠ k) can be calculated by ${\begin{matrix} C_{Mi} = \frac{f (x^{0}) - f (S_{Mi})}{\sum_{i = 1}^{t} (f (x^{0}) - f (S_{Mi})) + \sum_{j = 1}^{r} (f (x^{0}) - f (S_{nj}))} \\ C_{mj} = \frac{f (x^{0}) - f (S_{nj})}{\sum_{i = 1}^{t} (f (x^{0}) - f (S_{Mi})) + \sum_{j = 1}^{r} (f (x^{0}) - f (S_{nj}))} \end{matrix}$ (16)

$\sum_{i = 1, i \neq k}^{t} C_{Mi} + \sum_{j = 1}^{r} C_{mj} = 1$ can evidently be derived from Equation (16). A new roulette is established like in the previous step. All the nodes on trunk, M, except S_Mk and branch m, respectively, occupy their own range on the roulette. One of them is selected as the new growth point for a new branch.

Fig.3

Special roulette.

The growth process is repeated until the new branch reaches the optimal point. The PGSA cannot easily fall into local optimum, because the morphactin concentrations of all nodes are updated during each growth step.

2.4 Projection method for scoring INSs

Xu [38] proposed several definitions (i.e., Definitions 5–7) for uncertain multiple attribute decision-making, as follows.

Definition 5. Let α = (α₁, α₂, …, α_n) and β = (β₁, β₂, …, β_n) be two vectors. Then, ${Prj}_{β} (α) = | α | cos (α, β) = | α | \frac{α β}{| α | | β |} = \frac{α β}{| β |}$ (17) is called the projection of the vector, α, on the vector, β.

Generally, Prj (α) shows the approaching degree of the vector, α, to the vector, β, whose value rises with an increase in the two vectors’ approaching degree.

Definition 6. Let ${\ddot{μ}}_{j} = ({\dot{μ}}_{j {min}^{L}, {\dot{μ}}_{j {max}^{U}}})$ . Then, $\ddot{μ} = ({\ddot{μ}}_{1}, {\ddot{μ}}_{2}, \dots, {\ddot{μ}}_{j})$ is called the positive idea vector of the membership degree values of all alternatives.

Definition 7. Let ${\ddot{ν}}_{j} = ({\dot{ν}}_{j {max}^{L}, {\dot{ν}}_{j {min}^{U}}})$ . Then, $\ddot{ν} = ({\ddot{ν}}_{1}, {\ddot{ν}}_{2}, \dots, {\ddot{ν}}_{j})$ is called the positive idea vector of the non-membership degree values of all alternatives.

Regarding the above definitions, we propose the projection method for scoring INSs.

Definition 8. Let INS(X) denote the family of all the INSs in universe, X. Assume A₁, A₂, ⋯ , A_n ∈ INS (X), where $\begin{matrix} A_{n} = {〈 x, [T_{A_{n}}^{L} (x), T_{A_{n}}^{U} (x)], [I_{A_{n}}^{L} (x), I_{A_{n}}^{U} (x)], \\ [F_{A_{n}}^{L} (x), F_{A_{n}}^{U} (x)] 〉 | x \in X} . \end{matrix}$

Let

$\ddot{T} (x) = ({\dot{T}}_{n min}^{L} (x), {\dot{T}}_{n max}^{U} (x))$ ,

$\ddot{I} (x) = ({\dot{I}}_{n max}^{L} (x), {\dot{I}}_{n min}^{U} (x))$ and

$\ddot{F} (x) = ({\dot{F}}_{n max}^{L} (x), {\dot{F}}_{n min}^{U} (x))$ .

Then, $\dot{T}, \dot{I}$ and $\dot{F}$ care called the positive idea vector of the truth-membership degree indeterminacy-membership degree, and falsity-membership degree of all INSs, ∈INS (X).

Definition 9. Let $\dot{A}$ = {〈x, $\dot{T} (x)$ , $\dot{I} (x)$ , $\dot{F} (x) 〉 |$ x ∈ X} be the positive idea vector of INSs ∈ INS (X). Then, the projection of the truth-membership degree values of each INS, ${Prj}_{\ddot{T}} (A_{n})$ . can be calculated as follows. $\begin{matrix} {Prj}_{\ddot{T}} (A_{n}) = \frac{T_{A_{n}} (x) \cdot \ddot{T} (x)}{| \ddot{T} (x) |} \\ = \frac{T_{A_{n}}^{L} (x) \cdot {\dot{T}}_{n min}^{L} (x) + T_{A_{n}}^{U} (x) \cdot {\dot{T}}_{n max}^{U} (x)}{\sqrt{{({\dot{T}}_{n min}^{L} (x))}^{2} + {({\dot{T}}_{n max}^{U} (x))}^{2}}} \end{matrix}$ (18) $\begin{matrix} {Prj}_{\ddot{F}} (A_{n}) = \frac{F_{A_{n}} (x) \cdot \ddot{F} (x)}{| \ddot{F} (x) |} \\ = \frac{F_{A_{n}}^{L} (x) \cdot {\dot{F}}_{n max}^{L} (x) + F_{A_{n}}^{U} (x) \cdot {\dot{F}}_{n min}^{U} (x)}{\sqrt{{({\dot{F}}_{n max}^{L} (x))}^{2} + {({\dot{F}}_{n min}^{U} (x))}^{2}}} . \end{matrix}$ (19)

Obviously, the larger the value of ${Prj}_{\dot{T}} (A_{n})$ , ${Prj}_{\dot{I}} (A_{n})$ and ${Prj}_{\dot{F}} (A_{n})$ , the higher is the degree of the n^th INS approaching the positive idea vector. $\dot{T} (x)$ .

Finally, the score of INS, A_n, can be evaluated as follows.

$S (A_{n}) = {Prj}_{\dot{T}} (A_{n}) + {Prj}_{\dot{I}} (A_{n}) - {Prj}_{\dot{F}} (A_{n})$ (20)

3 Aggregation method for INSs and its application in MAGDM

The process of the proposed aggregation method is divided into three main steps: (1) project the INSs onto the 2D coordinate; (2) aggregate point sets and calculate the optimal rally point; (3) establish the collective INS and sort the INSs. These processes and their application in MAGDM are described in detail in this section.

3.1 Projection process of INSs

Let A₁, A₂, …, A_n be n INSs in universe, X. Store them in a special matrix, called INSs matrix (see Fig. 4).

Fig.4

INSs matrix.

Three special 2D coordinate systems are established in view of the three criteria of the matrix. The lower limit and the upper limit of the INN are used as the axis of the abscissa and the axis of the ordinate. Thus, all INNs in the INSs matrix are projected as interval neutrosophic points (INP) in their respective coordinate systems (See Fig. 5).

Fig.5

INSs coordinate systems.

3.2 Aggregation of INPs

The optimal rally point of each INP set is calculated by employing the PGSA. The detailed algorithm steps for optimal rally INN of truth-membership are shown next. The aggregation processes of indeterminacy-membership degree and falsity-membership degree are similar to the aggregation process of truth-membership degree.

Suppose Ω is the bounded closed box in R^N having length, l. There are n truth-membership degree points, $({\tilde{T}}_{A_{1}}, {\tilde{T}}_{A_{2}}, {\tilde{T}}_{A 3}, \dots, {\tilde{T}}_{Ax}) \in Ω$ . The backlight function, f (γ_i), is continuous on Ω. The sum of weighted Euclidean distances between γ_i and truth-membership degree points is determined as the backlight function of γ_i. The corresponding positive weights of these points are ξ₁, ξ₂, …, ξ_n. The aggregation algorithm is carried out in the following steps.

Step 1. Determine γ⁰ ∈ Ω as the initial growth point and λ = l/ - 1000 as the initial step size. Select k growth points, $γ_{i}^{0} \in Ω (1 ⩽ i ⩽ k)$ , whose growth hormone concentration, $C_{γ_{i}^{0}}$ , can be calculated as follows. $C_{γ_{i}^{0}} = \frac{f (γ_{i}^{0})}{\sum_{i = 1}^{k} f (γ_{i}^{0})}$ (21)

Thus, a special roulette can be established according to the respective growth hormone concentrations for all growth points. The new growth point, γ^0*, can then be selected randomly. Set γ⁰ = γ^0*, Γ_min = γ⁰, F_min = f (γ⁰).

Step 2. Establish L-system (θ = 22 . 5°) with γ⁰ as the thought center. This is the first layer of the “branch.” Then, select k growth points, $γ_{i}^{1} (1 ⩽ i ⩽ k)$ , from L-system, randomly.

Step 3. Update the growth hormone concentration of all “nodes.” If $f (γ^{0}) ⩽ f (γ_{i}^{1})$ , then $C_{γ_{i}^{1}} = 0$ . Otherwise, the following formula is used to calculate the new growth hormone concentration of $γ_{i}^{1}$ . $C_{γ_{i}^{1}} = \frac{f (γ^{0}) - f (γ_{i}^{1})}{\sum_{i = 1}^{k} f (γ^{0}) - f (γ_{i}^{1})}$ (22)Step 4. Establish the special roulette according to the calculated growth hormone concentrations, and select γ^1* as the new growth point randomly. Set γ¹ = γ^1*, Γ_min = γ¹, F_min = f (γ¹).

Step 5. Establish L-system (θ = 22 . 5°) with γ¹ as the thought center. This is the second layer of “branch.” Then, select k growth points, $γ_{i}^{2} (1 ⩽ i ⩽ k)$ , from L-system, randomly.

Step 6. Update the growth hormone concentration of all “nodes” on the first and second layers of “branch” to avoid the global optimization of the algorithm.

If $f (γ^{1}) ⩽ f (γ_{i}^{1})$ , then $C_{γ_{i}^{1}} = 0$ . Otherwise, the following formula is used to calculate the new growth hormone concentration of $γ_{i}^{1}$ $C_{γ_{i}^{1}} = \frac{f (γ^{1}) - f (γ_{i}^{1})}{\sum_{i = 1}^{k} [f (γ^{1}) - f (γ_{i}^{1})] + \sum_{i = 1}^{k} [f (γ^{1}) - f (γ_{i}^{2})]}$ (23)

If $f (γ^{1}) ⩽ f (γ_{i}^{2})$ , then $C_{γ_{i}^{2}} = 0$ . Otherwise, the following formula is used to calculate the new growth hormone concentration of $γ_{i}^{2}$ . $C_{γ_{i}^{2}} = \frac{f (γ^{1}) - f (γ_{i}^{2})}{\sum_{i = 1}^{k} [f (γ^{1}) - f (γ_{i}^{1})] + \sum_{i = 1}^{k} [f (γ^{1}) - f (γ_{i}^{2})]}$ (24)

Step 7. Select the new growth point, γ^2*, by employing the special roulette. Set γ² = γ^2*, Γ_min = γ², F_min = f (γ²).

Step 8. Repeat Steps 5 through 7. The algorithm ends when the number of iterations reaches the default value, or F_min remains unchanged. The γ^* = Γ_min is the optimal rally point for truth-membership degree point set. The flowchart of the algorithm is shown in Fig. 6.

Fig.6

The flowchart of the proposed algorithm.

The optimal rally points, η^* and κ^*, for indeterminacy-membership degree point set and falsity-membership degree point set, can be also selected by employing the above steps. Thus, A^* = [γ^*, η^*, κ^*] = [(γ^L, γ^U) , (η^L, η^U) , (κ^L, κ^U)] is the optimal rally INN for A₁, A₂, ⋯ A_n (See Fig. 7).

Fig.7

Optimal rally INP.

3.3 Application in MAGDM based on INS

MAGDM is widely employed to solve various practical decision-making problems, such as emergency management, peer review, and business plan formulation. DMs’ preference information can be abstracted as an interval neutrosophic matrix, because INNs express fuzzy information accurately. Assume there are k DMs to evaluate n alternatives with m properties. ξ₁, ξ₂, ξ₃, …, ξ_k are the weighted vector of DMs, and ζ₁, ζ₂, ζ₃, …, ζ_m are the corresponding positive weights of properties. The DMs’ preference matrices can be expressed as follows.

${[D_{k}]}_{n \times m} = (\begin{matrix} ([T_{A_{11}}^{(k) L}, T_{A_{11}}^{(k) U}], [I_{A_{11}}^{(k) L}, I_{A_{11}}^{(k) U}], [F_{A_{11}}^{(k) L}, F_{A_{11}}^{(k) U}]) \dots ([T_{A_{1 m}}^{(k) L}, T_{A_{1 m}}^{(k) U}], [I_{A_{1 m}}^{(k) L}, I_{A_{1 m}}^{(k) U}], [F_{A_{1 m}}^{(k) L}, F_{A_{1 m}}^{(k) U}]) \\ ([T_{A_{21}}^{(k) L}, T_{A_{21}}^{(k) U}], [I_{A_{21}}^{(k) L}, I_{A_{21}}^{(k) U}], [F_{A_{21}}^{(k) L}, F_{A_{21}}^{(k) U}]) \dots ([T_{A_{2 m}}^{(k) L}, T_{A_{2 m}}^{(k) U}], [I_{A_{2 m}}^{(k) L}, I_{A_{2 m}}^{(k) U}], [F_{A_{2 m}}^{(k) L}, F_{A_{2 m}}^{(k) U}]) \\ \dots \dots \dots \\ ([T_{A_{n 1}}^{(k) L}, T_{A_{n 1}}^{(k) U}], [I_{A_{n 1}}^{(k) L}, I_{A_{n 1}}^{(k) U}], [F_{A_{n 1}}^{(k) L}, F_{A_{n 1}}^{(k) U}]) \dots ([T_{A_{n m}}^{(k) L}, T_{A_{n m}}^{(k) U}], [I_{A_{n m}}^{(k) L}, I_{A_{n m}}^{(k) U}], [F_{A_{n m}}^{(k) L}, F_{A_{n m}}^{(k) U}]) \end{matrix})$

Establish attribute preference matrices:

$\begin{matrix} {[P_{ij}]}_{1 \times k} = (([T_{A_{ij}}^{(1) L}, T_{A_{ij}}^{(1) U}], [I_{A_{ij}}^{(1) L}, I_{A_{ij}}^{(1) U}], [F_{A_{ij}}^{(1) L}, F_{A_{ij}}^{(1) U}]), \\ \dots, ([T_{A_{ij}}^{(k) L}, T_{A_{ij}}^{(k) U}], [I_{A_{ij}}^{(k) L}, I_{A_{ij}}^{(k) U}], [F_{A_{ij}}^{(k) L}, F_{A_{ij}}^{(k) U}])) \end{matrix}$

to store the DMs’ preferences for the ith property of the jth alternative, (1 ⩽ i ⩽ n, 1 ⩽ j ⩽ m). The attribute preference matrices are then projected as INPs on the 2D coordinate systems by employing the method of 3.1. Then, employ the method of 3.2 to select the optimal rally INPs, $[{\dot{P}}_{ij}] = ([γ_{A_{ij}}^{L}, γ_{A_{ij}}^{U}], [η_{A_{ij}}^{L}, η_{A_{ij}}^{U}], [κ_{A_{ij}}^{L}, κ_{A_{ij}}^{U}])$ , for all attributes of alternatives. Thus, the optimal collective preference matrix expressed by INNs can be describe as follows.

$[\dot{D}] = {[{\dot{P}}_{ij}]}_{n \times m} = (\begin{matrix} ([γ_{A_{11}}^{L}, γ_{A_{11}}^{U}], [η_{A_{11}}^{L}, η_{A_{11}}^{U}], [κ_{A_{11}}^{L}, κ_{A_{11}}^{U}]) & \dots & ([γ_{A_{1 m}}^{L}, γ_{A_{1 m}}^{U}], [η_{A_{1 m}}^{L}, η_{A_{1 m}}^{U}], [κ_{A_{1 m}}^{L}, κ_{A_{1 m}}^{U}]) \\ ([γ_{A_{21}}^{L}, γ_{A_{21}}^{U}], [η_{A_{21}}^{L}, η_{A_{21}}^{U}], [κ_{A_{21}}^{L}, κ_{A_{21}}^{U}]) & \dots & ([γ_{A_{2 m}}^{L}, γ_{A_{2 m}}^{U}], [η_{A_{2 m}}^{L}, η_{A_{2 m}}^{U}], [κ_{A_{2 m}}^{L}, κ_{A_{2 m}}^{U}]) \\ \dots & \dots & \dots \\ ([γ_{A_{n 1}}^{L}, γ_{A_{n 1}}^{U}], [η_{A_{n 1}}^{L}, η_{A_{n 1}}^{U}], [κ_{A_{n 1}}^{L}, κ_{A_{n 1}}^{U}]) & \dots & ([γ_{A_{nm}}^{L}, γ_{A_{nm}}^{U}], [η_{A_{nm}}^{L}, η_{A_{nm}}^{U}], [κ_{A_{nm}}^{L}, κ_{A_{nm}}^{U}]) \end{matrix})$

The MATLAB simulation of the aggregation process is shown in Fig. 8.

Fig.8

Aggregation process for INSs.

Finally, sort the alternatives by employing the alternative score calculated by the projection theory introduced in 2.4. the positive idea vector of $[\dot{D}]$ should be established by employing the following formulas, because each attribute has its own weight in practical problem.

${Prj}_{\ddot{γ}} (A_{p}) = \frac{\sum_{q = 1}^{m} (γ_{pq}^{L} \cdot {\dot{γ}}_{p min}^{L} \cdot ζ_{q} + γ_{pq}^{U} \cdot {\dot{γ}}_{p max}^{U} \cdot ζ_{q})}{\sqrt{\sum_{q = 1}^{m} [{({\dot{γ}}_{n min}^{L})}^{2} \cdot ζ_{q} + {({\dot{γ}}_{n max}^{U})}^{2} \cdot ζ_{q}]}}$ (25)

${Prj}_{\ddot{η}} (A_{p}) = \frac{\sum_{q = 1}^{m} (η_{pq}^{L} \cdot {\dot{η}}_{p max}^{L} \cdot ζ_{q} + η_{pq}^{U} \cdot {\dot{η}}_{p min}^{U} \cdot ζ_{q})}{\sqrt{\sum_{q = 1}^{m} [{({\dot{η}}_{n max}^{L})}^{2} \cdot ζ_{q} + {({\dot{η}}_{n min}^{U})}^{2} \cdot ζ_{q}]}}$ (26)

${Prj}_{\ddot{κ}} (A_{p}) = \frac{\sum_{q = 1}^{m} (κ_{pq}^{L} \cdot {\dot{κ}}_{p max}^{L} \cdot ζ_{q} + κ_{pq}^{U} \cdot {\dot{κ}}_{p min}^{U} \cdot ζ_{q})}{\sqrt{\sum_{q = 1}^{m} [{({\dot{κ}}_{n max}^{L})}^{2} \cdot ζ_{q} + {({\dot{κ}}_{n min}^{U})}^{2} \cdot ζ_{q}]}}$ (27)

The score of the pth alternative can be calculated as follows. $S (A_{p}) = {Prj}_{\dot{γ}} (A_{p}) + {Prj}_{\dot{η}} (A_{p}) - {Prj}_{\dot{κ}} (A_{p})$ (28)

4 Illustrative example

This paper solves the example problem proposed by Ye [29] by employing our proposed method. At the end of this section, a detailed comparison of the experimental results is given.

Three DMs, D^k (k = 1, 2, 3), evaluate four alternatives, A_n (n = 1, 2, 3, 4), with respect to three properties, C_m (m = 1, 2, 3). The weight vector of the properties is W = (0.35, 0.25, 0.4) ^T and the weight vector of DMs is given as V = (0.37, 0.33, 0.3) ^T. The DM preferences described as INNs and their preference matrices are shown as follows. $\begin{matrix} D^{1} = (\begin{matrix} ([0.4, 0.5], [0.2, 0.3], [0.3, 0.4]) & ([0.4, 0.5], [0.1, 0.2], [0.3, 0.4]) & ([0.3, 0.4], [0.1, 0.2], [0.5, 0.6]) \\ ([0.5, 0.7], [0.1, 0.2], [0.2, 0.3]) & ([0.6, 0.7], [0.1, 0.2], [0.1, 0.2]) & ([0.5, 0.6], [0.1, 0.2], [0.1, 0.2]) \\ ([0.3, 0.4], [0.1, 0.2], [0.3, 0.4]) & ([0.4, 0.5], [0.1, 0.3], [0.3, 0.4]) & ([0.5, 0.6], [0.2, 0.3], [0.0, 0.1]) \\ ([0.6, 0.7], [0.0, 0.1], [0.1, 0.2]) & ([0.5, 0.7], [0.0, 0.1], [0.1, 0.2]) & ([0.3, 0.4], [0.1, 0.2], [0.1, 0.2]) \end{matrix}) \\ D^{2} = (\begin{matrix} ([0.4, 0.5], [0.2, 0.3], [0.3, 0.4]) & ([0.4, 0.5], [0.0, 0.1], [0.2, 0.3]) & ([0.2, 0.3], [0.0, 0.1], [0.6, 0.7]) \\ ([0.6, 0.7], [0.1, 0.2], [0.2, 0.3]) & ([0.6, 0.7], [0.1, 0.2], [0.2, 0.3]) & ([0.5, 0.7], [0.1, 0.2], [0.1, 0.2]) \\ ([0.4, 0.5], [0.1, 0.2], [0.3, 0.4]) & ([0.5, 0.6], [0.2, 0.3], [0.3, 0.4]) & ([0.5, 0.6], [0.1, 0.2], [0.2, 0.3]) \\ ([0.6, 0.8], [0.0, 0.1], [0.1, 0.2]) & ([0.6, 0.7], [0.1, 0.2], [0.2, 0.3]) & ([0.3, 0.4], [0.1, 0.2], [0.1, 0.2]) \end{matrix}) \\ D^{3} = (\begin{matrix} ([0.4, 0.5], [0.1, 0.3], [0.2, 0.3]) & ([0.5, 0.6], [0.1, 0.2], [0.3, 0.4]) & ([0.1, 0.3], [0.0, 0.1], [0.5, 0.6]) \\ ([0.5, 0.6], [0.1, 0.2], [0.2, 0.3]) & ([0.7, 0.8], [0.1, 0.2], [0.1, 0.2]) & ([0.7, 0.8], [0.1, 0.2], [0.0, 0.1]) \\ ([0.4, 0.5], [0.0, 0.1], [0.2, 0.3]) & ([0.5, 0.6], [0.0, 0.1], [0.3, 0.4]) & ([0.6, 0.7], [0.1, 0.2], [0.0, 0.1]) \\ ([0.6, 0.7], [0.0, 0.1], [0.1, 0.2]) & ([0.5, 0.6], [0.1, 0.2], [0.1, 0.2]) & ([0.4, 0.5], [0.0, 0.1], [0.0, 0.1]) \end{matrix}) \end{matrix}$

The collective matrix, $\dot{D}$ , as shown below, can be established by employing the method introduced in 3.1–3.3.

$\dot{D} = (\begin{matrix} ([0.4000, 0.5000], [0.1664, 0.3021], [0.2753, 0.3719]) & ([0.4320, 0.5322], [0.0694, 0.1710], [0.3640, 0.2039]) & ([0.2039, 0.3407], [0.0357, 0.1360], [0.5322, 0.6335]) \\ ([0.5324, 0.6693], [0.1000, 0.2000], [0.2000, 0.3000]) & ([0.6304, 0.7284], [0.1000, 0.2000], [0.1336, 0.2331]) & ([0.5589, 0.6918], [0.1000, 0.2000], [0.0729, 0.1687]) \\ ([0.3638, 0.4589], [0.0700, 0.1699], [0.2753, 0.3719]) & ([0.4661, 0.5642], [0.1095, 0.2465], [0.3000, 0.4000]) & ([0.5276, 0.6343], [0.1402, 0.2397], [0.0667, 0.1616]) \\ ([0.6043, 0.7344], [0.0000, 0.1000], [0.1000, 0.2000]) & ([0.5331, 0.6760], [0.1623, 0.1336], [0.1336, 0.2331]) & ([0.3307, 0.4312], [0.0714, 0.1707], [0.0694, 0.1754]) \end{matrix})$

According to the projection theory, to establish the positive idea vector, $\ddot{D}$ ,

$\ddot{D} = {(\begin{matrix} ([0.3638, 0.7344], [0.1664, 0.1000], [0.2753, 0.2000]) \\ ([0.4320, 0.7284], [0.1095, 0.1623], [0.3000, 0.2331]) \\ ([0.2039, 0.6918], [0.1402, 0.1360], [0.5322, 0.1616]) \end{matrix})}^{T}$

Finally, the scores of each alternative can be calculated by Equations (25–28). $\begin{matrix} S (A_{1}) = 0.035942, S (A_{2}) = 0.419571, \\ S (A_{3}) = 0.320757, S (A_{4}) = 0.350617 \end{matrix}$

Thus, A₂ ≻ A₄ ≻ A₃ ≻ A₁. Therefore, we see that the alternative, A₂, is the best choice among all alternatives.

Ye got the result of A₂ ≻ A₄ ≻ A₃ ≻ A₁ in [29]. This paper carries out several discussions. The integrated matrix established by Yu’s method is shown as follows. ${\dot{D}}_{Yu} = (\begin{matrix} ([0.4000, 0.5000], [0.1625, 0.3000], [0.2656, 0.3669]) & ([0.4319, 0.5324], [0.0000, 0.1591], [0.2624, 0.3638]) & ([0.2112, 0.3388], [0.0000, 0.1292], [0.5310, 0.6313]) \\ ([0.5355, 0.6730], [0.1000, 0.2000], [0.2000, 0.3000]) & ([0.6331, 0.7344], [0.1000, 0.2000], [0.1257, 0.2286]) & ([0.5710, 0.7045], [0.1000, 0.2000], [0.0000, 0.1625]) \\ ([0.3648, 0.4651], [0.0000, 0.1625], [0.2656, 0.3669]) & ([0.4651, 0.5656], [0.0000, 0.2158], [0.3000, 0.4000]) & ([0.5324, 0.6331], [0.1292, 0.2324], [0.0000, 0.1437]) \\ ([0.6000, 0.7000], [0.0000, 0.1000], [0.1000, 0.2000]) & ([0.5355, 0.6730], [0.0000, 0.1548], [0.1257, 0.2286]) & ([0.3316, 0.4319], [0.0000, 0.1625], [0.0000, 0.1625]) \end{matrix})$

Compared to the collective matrix, differences in black numbers are very small. Most are the differences in percentile, even in the thousandths. The collective established by Yu contains a lot of zeroes. There is a large error in these values. The zeroes annotated in blue is not a problem, because three experts give this attribute a zero preference. However, the zeroes annotated in red are inaccurate. Many continuous products are employed in the establishment of the collective matrix, by Yu’s method, which leads to the neglect of preferences of the other DMs when a DM’s preference is zero. The quality of the aggregation matrix directly leads to the decline in the precision of the final sort. The normalized Hamming distance between the collective matrices established by two methods, and the DMs preferences matrices are calculated by Equation (9), $d_{nH}^{PGSA} = 0.092708$ and $d_{nH}^{{Ye}^{'} s} = 0.111329$ .

The correlation between the collective matrices established by two methods, and the DMs preferences matrices are calculated by Equation (11), Fig. 9 shows the simulation curve of MATLAB.

Obviously, the result obtained by the proposed method is more accurate.

Fig.9

Experimental comparison.

In order to prove the practicability of the proposed method, we have increased the number of DMs in the above calculation to 5 and the weight vector of DMs is given as V = (0.12, 0.23, 0.15, 0.28, 0.22) ^T. The preference matrices of new DMs as shown as follows:

$\begin{matrix} D^{4} = (\begin{matrix} ([0.6, 0.8], [0.2, 0.5], [0.1, 0.2]) & ([0.4, 0.5], [0.0, 0.2], [0.4, 0.6]) & ([0.0, 0.3], [0.1, 0.3], [0.3, 0.5]) \\ ([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) & ([0.5, 0.6], [0.1, 0.3], [0.0, 0.2]) & ([0.5, 0.6], [0.0, 0.2], [0.1, 0.2]) \\ ([0.2, 0.5], [0.1, 0.2], [0.2, 0.3]) & ([0.4, 0.5], [0.1, 0.3], [0.2, 0.4]) & ([0.3, 0.5], [0.2, 0.3], [0.2, 0.3]) \\ ([0.4, 0.5], [0.0, 0.2], [0.1, 0.2]) & ([0.3, 0.5], [0.0, 0.1], [0.2, 0.3]) & ([0.3, 0.4], [0.1, 0.2], [0.0, 0.1]) \end{matrix}) \\ D^{5} = (\begin{matrix} ([0.6, 0.7], [0.3, 0.4], [0.0, 0.2]) & ([0.3, 0.5], [0.0, 0.2], [0.2, 0.4]) & ([0.0, 0.2], [0.0, 0.1], [0.7, 0.8]) \\ ([0.4, 0.7], [0.1, 0.2], [0.1, 0.3]) & ([0.5, 0.6], [0.1, 0.2], [0.1, 0.3]) & ([0.6, 0.8], [0.1, 0.2], [0.1, 0.2]) \\ ([0.5, 0.6], [0.0, 0.1], [0.1, 0.2]) & ([0.3, 0.5], [0.0, 0.2], [0.1, 0.2]) & ([0.7, 0.9], [0.1, 0.2], [0.0, 0.1]) \\ ([0.7, 0.8], [0.1, 0.2], [0.2, 0.3]) & ([0.4, 0.6], [0.1, 0.3], [0.2, 0.3]) & ([0.2, 0.4], [0.0, 0.1], [0.1, 0.2]) \end{matrix}) \end{matrix}$ The collective matrix, ${\dot{D}}^{*}$ , as shown below. $\begin{matrix} {\dot{D}}^{*} = (\begin{matrix} ([0.5000, 0.6249], [0.1849, 0.3759], [0.1665, 0.2868]) ([0.3920, 0.5138], [0.0220, 0.1778], [0.2799, 0.4333]) \\ ([0.4468, 0.6013], [0.1000, 0.2000], [0.1763, 0.2998]) ([0.5650, 0.6641], [0.1051, 0.2302], [0.0934, 0.2446]) \\ ([0.3517, 0.5094], [0.0605, 0.1663], [0.2087, 0.3126]) ([0.4115, 0.5375], [0.0846, 0.2509], [0.2353, 0.3600]) \\ ([0.5628, 0.6871], [0.0222, 0.1537], [0.1215, 0.2211]) ([0.4477, 0.6094], [0.0608, 0.1812], [0.1779, 0.2769]) \end{matrix} \\ \begin{matrix} ([0.1018, 0.2907], [0.0424, 0.1671], [0.5113, 0.6381]) \\ ([0.5504, 0.6984], [0.0714, 0.1978], [0.0800, 0.1809]) \\ ([0.5023, 0.6543], [0.1391, 0.2443], [0.1056, 0.1970]) \\ ([0.2964, 0.4179], [0.0610, 0.1626], [0.0540, 0.1575]) \end{matrix} \end{matrix}$ establish the positive idea vector, ${\ddot{D}}^{*}$ , $\begin{matrix} {\ddot{D}}^{*} = {(\begin{matrix} ([0.352, 0.687], [0.185, 0.154], [0.209, 0.221]) \\ ([0.392, 0.664], [0.105, 0.178], [0.280, 0.245]) \\ ([0.102, 0.698], [0.139, 0.163], [0.511, 0.158]) \end{matrix})}^{T} \end{matrix}$ the scores of each alternative are shown as follows. $\begin{matrix} S (A_{1}) = 0.143005, S (A_{2}) = 0.454649, \\ S (A_{3}) = 0.362536, S (A_{4}) = 0.37876 \end{matrix}$ The performance of proposed method is the same as that of the group decision making (GDM). The proposed method will be expanded to 3D space for solving the problem presented by Herrera firstly and modified by Xu [39].

The DM preference matrices are shown as follows. $\begin{matrix} D_{A}^{1} = (\begin{matrix} \begin{matrix} (0.3, 0.4) & (0.4, 0.6) & (0.0, 0.3) \end{matrix} \\ \begin{matrix} (0.6, 0.8) & (0.1, 0.2) & (0.0, 0.3) \end{matrix} \\ \begin{matrix} (0.5, 0.8) & (0.1, 0.2) & (0.0, 0.4) \end{matrix} \end{matrix}) \end{matrix}$ $D_{A}^{2} = (\begin{matrix} \begin{matrix} (0.4, 0.5) & (0.3, 0.4) & (0.0, 0.3) \end{matrix} \\ \begin{matrix} (0.6, 0.8) & (0.1, 0.2) & (0.0, 0.3) \end{matrix} \\ \begin{matrix} (0.5, 0.6) & (0.3, 0.4) & (0.0, 0.2) \end{matrix} \end{matrix})$ $D_{A}^{3} = (\begin{matrix} \begin{matrix} (0.4, 0.6) & (0.3, 0.4) & (0.0, 0.3) \end{matrix} \\ \begin{matrix} (0.5, 0.8) & (0.1, 0.2) & (0.0, 0.4) \end{matrix} \\ \begin{matrix} (0.5, 0.6) & (0.0, 0.1) & (0.3, 0.5) \end{matrix} \end{matrix})$ The collective matrix aggregated by the PGSA is as follows. ${\dot{D}}^{*} = (\begin{matrix} \begin{matrix} (0.3501, 0.4810) & (0.3490, 0.4732) & (0.4569, 0.3008) \\ (0.5513, 0.7736) & (0.1000, 0.2263) & (0.0000, 0.3487) \\ (0.4389, 0.6262) & (0.1324, 0.2329) & (0.1409, 0.4287) \end{matrix} \end{matrix})$ the scores of each alternative are shown as follows. $\begin{matrix} S (A_{1}) = 1.821623, S (A_{2}) = 1.724433, \\ S (A_{3}) = 0.986398 \end{matrix}$ Thus A₁ ≻ A₂ ≻ A₃.

The collective matrices aggregated by IIFHA operator and IIFHG operator are $\begin{matrix} {\dot{D}}_{IIFHA}^{*} = (\begin{matrix} (0.348, 0.471) & (0.338, 0.473) & (0.056, 0.314) \\ (0.549, 0.790) & (0.100, 0.210) & (0.000, 0.351) \\ (0.470, 0.630) & (0.000, 0.120) & (0.250, 0.530) \end{matrix}) \\ {\dot{D}}_{IIFHG}^{*} = (\begin{matrix} (0.350, 0.486) & (0.348, 0.470) & (0.043, 0.302) \\ (0.552, 0.771) & (0.100, 0.229) & (0.000, 0.348) \\ (0.446, 0.643) & (0.144, 0.245) & (0.113, 0.410) \end{matrix}) \end{matrix}$ The results of IIFHA and IIFHG are the same as the proposed method, but the Euclidean distance between the collective matrices and preference matrices of IIFHA and IIFHG is larger. Therefore, the performance of the proposed method is better.

5 Conclusion

This paper proposed a new approach for aggregating INSs and investigated its application in MAGDM. The greatest advantage of the proposed method is as follows:

The typical operators are sensitive to special value, but proposed algorithm no longer exist this disadvantage.

The typical mathematical method needs to explore various of operators for different fuzzy sets. The proposed algorithm has stronger generality of fuzzy sets with interval-valued numbers.

In the future, we will explore the application of fuzzy set aggregation in site selection problem and extend the proposed algorithm to 3D space. For example, site selection problem of Radar station in mountainous region. Also, we consider to recommend this work to Multi-objective aggregation for dealing with more complex problems. For instance, the location problem of multiple logistics stations and scheme evaluation under different demands.

References

Zadeh

L.A.

, Fuzzy sets, Inf Control 8(3) (1965), 338–356.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov

K.T.

, Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 64(2) (1994), 159–174.

Atanassov

K.T.

, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31(3) (1989), 343–349.

Chu

, Liu

, Wang

, Chin

, A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations, Computers & Industrial Engineering 101 (2016), 227–242.

Miguel

, Bustince

, Fernandez

, Indurin

, Construction of admissible linear orders for interval- valued atanassov intuitionistic fuzzy sets with an application to decision making, Information Fusion 27 (2016), 189–197.

Joshi

, Kumar

, Interval-valued intuitionistic hesitant fuzzy choquet integral based topsis method for multi-criteria group decision making, European Journal of Operational Research 248(1) (2016), 183–191.

Liao

H.C.

, Xu

Z.S.

, Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency, IEEE Transactions on Fuzzy Systems 22(6) (2014), 1669–1681.

Qiu

J.D.

, Li

, A new approach for multiple attribute group decision making with interval-valued intuitionistic fuzzy information, Applied Soft Computing 61 (2017), 111–121.

10.

Yue

Z.L.

, A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information, Applied Mathematical Modelling 38(2) (2014), 683–698.

11.

Liu

, Li

, An approach to determining the integrated weights of decision makers based on interval number group decision matrices, Knowledge-Based Systems 90(C) (2015), 92–98.

12.

Wan

S.P.

, Wang

, Dong

J.Y.

, Additive consistent interval-valued atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making, European Journal of Operational Research 263(2) (2017), 571–582.

13.

Wang

C.Y.

, Chen

S.M.

, A new multiple attribute decision making method based on interval-valued intuitionistic fuzzy sets, linear programming methodology, and the TOPSIS method, Ninth International Conference on Advanced Computational Intelligence (ICACI), 2017, pp. 260–263.

14.

Smarandache

, A unifying field in logics: Neutrosophic logics, Multiple-Valued Logic 8(3) (1999), 489–503.

15.

, Wu

, Zhou

L.G.

, et al., Some new Hamacher aggregation operators under single-valued neutrosophic 2-tuple linguistic environment and their applications to multi-attribute group decision making, Computers & Industrial Engineering 116 (2018), 144–162.

16.

Liu

, Liu

, The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making[J], International Journal of Machine Learning & Cybernetics 9(2) (2018), 347–358.

17.

Peng

, Dai

J.G.

, Algorithms for interval neutrosophic multiple attribute decision making based on MABAC, similarity measure and EDAS, International Journal for Uncertainty Quantification 7(5) (2017), 395–421.

18.

Peng

, Dai

J.G.

, A bibliometric analysis of neutrosophic set: Two decades review from 1998–2017, Artificial Intelligence Review (2018). doi: 10.1007/s10462-018-9652-0

19.

Peng

, Dai

J.G.

, Approaches to single-valued neutrosophic MADM basedon MABAC, TOPSIS and new similarity measure with score function, Neural Computing and Applications 29(10) (2018), 939–954.

20.

, Multiple attribute group decision making based on interval neutrosophic uncertain linguistic variables, International Journal of Machine Learning & Cybernetics 8(3) (2015), 1–12.

21.

Broumi

, Smarandache

, Several similarity measures of neutrosophic sets, Neutrosophic Sets & Systems 1(1), 54–62.

22.

, Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making, International Journal of Fuzzy Systems 16(2) (2014), 204–211.

23.

Peng

, Liu

, Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set, Journal of Intelligent & Fuzzy Systems 32(1) (2017), 955–968.

24.

Eisa

, A new approach for enhancing image retrieval using neutrosophic sets, International Journal of Computer Applications 95(8) (2014), 12–20.

25.

Liu

, Teng

, Multiple attribute decision making method based on normal neutrosophic generalized weighted power averaging operator, International Journal of Machine Learning & Cybernetics (2015), 1–13.

26.

Peng

J.J.

, Wang

J.Q.

, Wu

X.H.

, et al., Multivalued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decision-making problems, International Journal of Intelligent Systems 8(2) (2015), 345–363.

27.

Zhang

H.Y.

, Ji

, Wang

J.Q.

, et al., An improved weighted correlation coefficient based on integrated weight for interval neutrosophic sets and its application in multi-criteria decision-making problems, International Journal of Computational Intelligence Systems 8(6) (2015), 1027–1043.

28.

Şahin

, Cross-entropy measure on interval neutrosophic sets and its applications in multicriteria decision making, Neural Computing & Applications 28(5) (2017), 1177–1187.

29.

, Improved cross entropy measures of single valued neutrosophic sets and interval neutrosophic sets and their multicriteria decision making methods, Cybernetics & Information Technologies 15(4) (2015), 13–26.

30.

, Single valued neutrosophic cross-entropy for multicriteria decision making problems, Applied Mathematical Modelling 38(3) (2014), 1170–1175.

31.

Liu

P.D.

, Zhang

L.L.

, Liu

, et al., Multi-valued neutrosophic number bonferroni mean operators with their applications in multiple attribute group decision making, International Journal of Information Technology & Decision Making 15(5) (2016), 1181–1210.

32.

Wang

H.B.

, Smarandache

, Zhang

Y.Q.

, et al., Single valued neutrosophic sets, Multispace Multistruct 4 (2010), 410–413.

33.

Broumi

, Correlation coefficient of interval neutrosophic set, Applied Mechanics & Materials 436(4) (2013), 511–517.

34.

, Wang

C.F.

, Wang

W.B.

, et al., A global optimization bionics algorithm for solving integer programming – plant growth simulation algorithm, Systems Engineering-Theory & Practice 25(1) (2005), 76–85.

35.

Tang

, Liu

, Ni

, A novel wireless sensor network localization approach: Localization based on plant growth simulation algorithm, Elektronika Ir Elektrotechnika 19(8) (2013), 97–100.

36.

Rao

R.S.

, Narasimham

S.V.L.

, Ramalingaraju

, A novel wireless sensor network localization ap proach: Localization based on plant growth simulation algorithm, International Journal of Electrical Power & Energy Systems 33(5) (2011), 1133–1139.

37.

S.L.

, Yu

S.Z.

, A fuzzy k-coverage approach for rfid network planning using plant growth simulation algorithm, Journal of Network and Computer Applications archive 39(1) (2014), 280–291.

38.

Z.S.

, On method for uncertain multiple attribute decision making problems with uncertain multiplicative preference information on alternatives, Fuzzy Optimization & Decision Making 4(2) (2005), 131–139.

39.

Z.S.

, Intuitionistic Fuzzy Information Aggregation: Theory and Application. Beijing. China: SciencePress, 2008, pp. 45–67.