Research on the optimal aggregation method of fuzzy preference information based on spatial Steiner-Weber point

Abstract

In view of the present situation that most aggregation methods of fuzzy preference information are extended or mixed by classical aggregation operators, which leads to the aggregation accuracy is not high. The purpose of this paper is to develop a novel method for spatial aggregation of fuzzy preference information. Thus we map the fuzzy preference information to a set of three-dimensional coordinate and construct the spatial aggregation model based on Steiner-Weber point. Then, the plant growth simulation algorithm (PGSA) algorithm is used to find the spatial aggregation point. According to the comparison and analysis of the numerical example, the aggregation matrix established by our method is closer to the group preference matrices. Therefore, the optimal aggregation point obtained by using the optimal aggregation method based on spatial Steiner-Weber point can best represent the comprehensive opinion of the decision makers.

Keywords

Fuzzy preference information Steiner-Weber point spatial aggregation model aggregation operator PGSA

1 Introduction

Multiple attribute group decision making plays an important role in many fields of government, industry, management, education and engineering [1 –3]. It provides effective support for the aggregation of group preferences by evaluating and synthesizing individual opinions of group members. However, due to the ambiguity of decision makers’ understanding, the incompleteness of decision information, and the differences of decision makers’ personal preferences, it is often difficult for decision makers to reach consensus. The existing methods often have the uncertainty of decision data and decision environment, especially in new decision problems. How to propose an efficient and reasonable method to aggregate the preferences provided by different experts into a group of representative opinions has become an urgent problem in the process of MAGDM.

Aggregation method is an important research content of MAGDM. For the aggregation method of group preference, researchers at home and abroad have put forward many methods. Some researchers have developed many aggregation operators for the aggregation of group preference. The geometric average (GA) is considered to be the most widely used method to aggregate the preferences of decision makers [4] [5]. Compared with the WA and OWA operators [6], Xu [7] provided in-depth analysis of the consistency in GDM, which developed a theoretical foundation for the weighted geometric mean method (WGMM). In order to solve the multiplicative preference information, Xu and Yager [8] presented the power-geometric (PG) operator and three extended operators. However, far too little attention has been paid to the fuzzy information in the aggregation process.

As a solution, fuzzy sets were applied in MAGDM to deal with the uncertainty of data and decision making process [9] [10]. A number of researchers focused on fuzzy sets. Kahraman et al. [11] presented a hesitant fuzzy linguistic method. In the meaning of fuzzy logic, Matzenauer et al. [12] extended the research on fuzzy operators and the main properties. Seiti et al. [13] developed the R-numbers methodology to construct the model about the risk of fuzzy sets. In order to estimate the security of Hospital Management System Software, Kumar et al. [14] employed a comprehensive method of fuzzy logic, ANP and TOPSIS. Rehman and Ali [15] introduced some novel multigranulation fuzzy rough sets and studied their related properties. Deli [16] focused on MADM problems with generalized trapezoidal hesitant fuzzy information. However, the insufficiency of fuzzy sets in classification accuracy has not been well resolved. To solve this problem, Sun et al. [17] introduced a novel fuzzy MAGDM method based on maximal consistent block by using rough set and fuzzy set. At present, fuzzy sets are mainly divided into intuitionistic fuzzy sets, Pythagorean fuzzy sets, and spherical fuzzy sets.

A number of scholars have sought to focus on intuitionistic fuzzy sets. Based on the extension of T-conorm and Schweizer-Sklar T-norm to intuitionistic fuzzy numbers, Wang and Liu [18] developed two Maclaurin symmetric mean aggregation operators. Liu et al. [19] developed a novel MAGDM method for intuitionistic fuzzy with weighted partitioned Maclaurin symmetric mean operators. In order to aggregate interval-valued intuitionistic fuzzy numbers, Kumar and Chen [20] proposed a novel multiattribute decision making method on the basis of the set pair analysis and connection numbers. Akram et al. [21] introduced four aggregation operators to aggregate complex intuitionistic fuzzy information. Sirbiladze et al. [22] constructed two operators named As-IP-IFOWA and As-IP-IFOWG in intuitionistic fuzzy values. On the basis of complex intuitionistic fuzzy preference information, Rani and Garg [23] presented single and GDM methods. Alcantud et al. [24] developed a method to aggregate infinite sequences of intuitionistic fuzzy set. Garg [25] defined several weighted exponential operators presented for MAGDM problem under the intuitionistic multiplicative environment. For two nonlinear chaotic systems, Hamdy et al. [26] presented control and synchronization based on the intuitionistic fuzzy control scheme. Kala and Deepa [27] developed spatial rough intuitionistic fuzzy C-means approach to overcome the problems in medical image segmentation. On the basis of interval-valued intuitionistic fuzzy sets and DEA, Davoudabadi et al. [28] proposed a novel method in renewable energy evaluation.

Based on the power aggregating operators, Ak and Gul [29] developed a new methodology based on extension of Pythagorean fuzzy sets and AHP–TOPSIS aggregation method. Aydin et al. [30] proposed three harmonic aggregation operators for Pythagorean fuzzy numbers. Zhang et al. [31] developed several new similarity measures of the Pythagorean fuzzy set. Haktanır [32] offered two novel aggregation operators named interval valued Pythagorean fuzzy weighted geometric and interval valued Pythagorean fuzzy weighted averaging, which can actually represent the subjective evaluation of decision makers. For Pythagorean fuzzy information, Garg [33] defined some new aggregation operator with fuzzy weighted, ordered weighted and mixed neutral average, which can deal with the degree of membership and non-membership in a neutral manner. In order to overcome the shortcomings of proposed models by previous studies, Hendiani et al. [34] presented a novel soft computing Pythagorean fuzzy distance to ideal solution. On the basis of Einstein t-norms and t-conorms, Sarkar and Biswas [35] developed a novel linguistic Pythagorean fuzzy aggregation operator.

Intuitionistic fuzzy sets and Pythagorean fuzzy sets have been widely studied and applied in different research fields, but their scope of expressing information is limited. To solve this problem, the spherical fuzzy sets were applied to describe the vagueness [36] [37]. Kutlu Gündoğdu and Kahraman [38] developed a novel theory with some operators, and proposed the spherical fuzzy TOPSIS method. Based on the linguistic fuzzy and spherical fuzzy set, Jin et al. [39] proposed the linguistic spherical fuzzy aggregation operators. For aggregating the spherical fuzzy information, Ashraf et al. [40] introduced some spherical fuzzy triangular norms and conforms. On the basis of T-spherical fuzzy numbers, Ullah et al. [41] presented the Hamacher aggregation operators. Mathew et al. [42] elaborated a new framework combined AHP and TOPSIS with the spherical fuzzy set. For spherical fuzzy sets, Aydogdu and Gul [43] proposed a new entropy measure and proved that it can provide the required properties. In the spherical fuzzy environment, Seyfi-Shishavan et al. [44] proposed a novel beneficial method for solving MCGDM problems. In order to overcome the shortcomings of the traditional risk priority number calculation, Gul and Ak [45] developed an improved failure modes and effects analysis model based on the interval-valued spherical fuzzy information. Akram et al. [46] established complex spherical fuzzy model and developed four aggregation operators.

Some scholars attempted to improve the aggregation method by the aggregation algorithm. Altuzarra et al. [47] proposed the Bayesian priorization procedure to the aggregation of group preference. In a peer-to-peer dynamic decision making environment, Blagojevic et al. [48] presented a heuristic aggregation method based on simulated annealing (SA) algorithm. Kohler et al. [49] developed a novel particle swarm optimization algorithm named PSO+, which can solve the problems of linear and nonlinear constraints. Based on Dempster’s combination rule and probability theory, Fu et al. [50] presented evidence reasoning (ER) algorithm for combining belief distribution on criteria. These aggregation algorithms run slowly in solving practical problems and need to be simplified mathematically. However, a major problem with these aggregation algorithms needs to be simplified mathematically.

A number of researchers have attempted to discuss the PGSA algorithm to improve the aggregation algorithm. Li Tong et al. [51] proposed the PGSA algorithm, which was an effective global optimization algorithm for solving integer programming. Li et al. [52] combined PGSA with GDM to find the aggregation point. Liu and Li [53] attempted to determine the comprehensive weight about interval number by PGSA. For aggregating the interval-valued intuitionistic fuzzy information, Qiu and Li [54] presented a novel approach for MAGDM. Many scholars at home and abroad have applied PGSA to their respective research fields [55 –58].

Previous research has indicated that the efficient aggregation operators and the aggregation algorithm can improve the accuracy of group preference aggregation methods to some extent. However, there are still some problems worthy of further exploration and research. Firstly, the existing aggregation operators of fuzzy information are mainly expanded or improved based on GA operator, OWA operator, PG operator and other aggregation operators. The aggregation results of fuzzy preference information cannot accurately reflect group opinions to some extent. If the experts’ preference information is far away from the mean, there will be a large deviation in the aggregation result. It is necessary to perfect the aggregation point by Pareto improvements. Secondly, the existing aggregation models can improve the accuracy of aggregation to a certain extent, but the biggest problem with these methods is that the obtained aggregation points are not the optimal aggregation points, which means that the experts’ group opinions have not reached a complete consensus. How to construct the spatial optimal aggregation model from the perspective of Steiner-Weber point is worthy of further study. Thirdly, many researchers usually use the barycenter method and the method of partial derivative to solve the plane Steiner-Weber point problem. How to solve the spatial Steiner-Weber point problem is not available for reference. In order to solve this problem, we adopt the PGSA algorithm to solve the Steiner-Weber point problem. However, few writers have been able to draw on any systematic research into the spatial aggregation method of fuzzy preference information based on Steiner-Weber point and PGSA algorithm. Hence, a further study with more focuses on the spatial aggregation of fuzzy preference information is therefore suggested.

The purpose of this paper is to develop a novel approach for the spatial accurate aggregation of fuzzy preference information. Thus, we adopt the spatial mapping rules to map the fuzzy preference information to a set of spatial three-dimensional coordinate systems. Then, a three-dimensional aggregation model is constructed based on spatial Steiner-Weber point. Finally, the PGSA algorithm is used to find the spatial optimal aggregation point, which has the minimum spatial weighted Euclidean distance to all decision makers’ preferences. The main contributions of this study can be characterized by three aspects. Firstly, as a new perspective for solving MAGDM problem, this paper proposes the optimal aggregation method based on Steiner-Weber points for fuzzy preference information. Based on the optimal aggregation model constructed in this paper, the evaluation spatial preference scores of three-dimensional attribute indicators given by experts are aggregated into a group’s evaluation aggregation point (group consensus point). This model extends the traditional planar Steiner point problem to three-dimensional space. Secondly, we construct the three-dimensional aggregation model based on Steiner-Weber points as a prototype and developed on this basis. We theoretically proved the proposed model in this paper as an optimal model, which can make up for the disadvantages of existing aggregation method. Thirdly, the proposed model extends Steiner points from a plane to a three-dimensional space. How to solve the spatial Steiner-Weber point problem is not available for reference. Therefore, this paper introduces an intelligent optimization algorithm PGSA to solve the spatial Steiner-Weber point.

The remaining part of the paper proceeds as follows: Section 2 displays the preliminary knowledge, including definition of optimal aggregation based on Steiner-Weber point, and the principle of PGSA algorithm. Section 3 elaborates the preference aggregation of decision makers based on spatial Steiner-Weber points. In section 4, we employ numerical analysis to illustrate the effectiveness and rationality of our approach. Finally, the conclusion gives a brief summary and suggestions for further research is identified.

2 Preliminary knowledge

2.1 The principle of the optimal aggregation model

The prototype of the optimal aggregation model proposed in this paper is based on Steiner-Weber points. This problem can be traced back to the Fermat Point Problem first proposed by French mathematician Fermat in 1643. Suppose there are three points e₁, e₂ and e₃ on the plane that is not on the same straight line. If there is a point e^*, the sum of the distances from point e^* to the three points e₁, e₂ and e₃ is the smallest. This point e^* is called the “Fermat point”. Since Torricelli first solved the problem with elementary geometry, this point is often called the “Fermat-Torricelli” point. Later, some scholars extended the known points to more points: e₁, e₂, . . . e_p (p > 3). If we can find a point E^* that makes $D = \sum_{i = 1}^{p} | e_{i} E^{*} |$ is the minimum. Then the point E^* is called the generalized “Fermat point” of e₁, e₂, . . . e_p.

As far back as 1837, Steiner extended the above problem to Steiner point with weight [59]. Given n points on the plane, if there is a point P on the plane that makes $D = \sum_{i = 1}^{n} w_{i} | {PP}_{i} |$ the minimum. Then the point P is called “Steiner point”. Steiner’s problem is now summarized as the “Steiner-Weber point” problem. However, it is much more difficult to solve the “Steiner-Weber point” problem than the “Fermat-Torricelli point” problem. Some scholars have conducted in-depth research on this problem. Daniel and Adel [60] gave the properties of the optimal solution of the Steiner-Weber location problem on unit weighted sphere. Gorner and Kanzow [61] used Newton method to solve Fermat-Steiner problem, and verified the effectiveness of the method by numerical comparison. How to solve the spatial Steiner-Weber point problem is not available for reference.

Inspired by the above Steiner-Weber point problem, this paper proposes the spatial optimal aggregation model of fuzzy preference information based on Steiner-Weber points. Firstly, the multi-attribute fuzzy preference information of experts is mapped into three-dimensional coordinate system by using spatial mapping rules. Then, the optimal aggregation model based on the extended spatial Steiner-Weber point problem is established. PGSA algorithm is used to find the spatial optimal aggregation point. This method is helpful to improve the accuracy of the aggregation and make the experts involved in the evaluation reach the optimal consensus.

2.2 Definition of optimal aggregation based on Steiner-Weber point

In this paper, the Steiner-Weber point is applied to the process of information aggregation, and the definition of optimal aggregation of plane and space is proposed.

Definition 1. [54] There are n points R_i (a_i, b_i) with distance weight in the point set R = { R₁, R₂, …, R_n } (n ⩾ 3) of the closed region in the plane, whose corresponding weights δ_i ∈ [0, 1], and $\sum_{i = 1}^{n} δ_{i} = 1$ . If there is a point R^* (x^*, y^*), whose weighted Euclidean distance D_{R
^*} to all the points in R is minimum. If it satisfies the following condition:

$D_{R^{*}} = min \sum_{i = 1}^{n} (δ_{i} \sqrt{{(x^{*} - a_{i})}^{2} + {(y^{*} - b_{i})}^{2}})$ (1)

Then the point R^* (x^*, y^*) is called the optimal aggregation point in the plane of point set R (as shown in Fig. 1).

Fig. 1

The optimal aggregation point in the plane.

Definition 2. There are m points S_i (x_i, y_i, z_i) (i = 1, 2, ⋯ , m) with distance weight in the point set S = { S¹, S², …, S^m } (m ⩾ 3) of the closed region in 3-dimensional space, the weight of Sⁱ is ηⁱ ∈ [0, 1], and $\sum_{i = 1}^{m} η^{i} = 1$ . If there is a point S^* (x^*, y^*, z^*), whose weighted Euclidean distance D_{S
^*} to all the points in S is minimum. If it satisfies the following condition:

$\begin{matrix} D_{S^{*}} = min \sum_{i = 1}^{n} \\ (ζ_{i} \sqrt{{(x_{i} - x^{*})}^{2} + {(y_{i} - y^{*})}^{2} + {(z_{i} - z^{*})}^{2}}) \end{matrix}$ (2)

Then the point S^* (x^*, y^*, z^*) is called the spatial optimal aggregation point of the point set S (as shown in Fig. 2).

2.3 Plant growth simulation algorithm

Fig. 2

The optimal aggregation point in 3-dimensional space.

The PGSA is a heuristic intelligent optimization algorithm, which was proposed by Li Tong et al. in 2005 [51]. Compared with these intelligent algorithms such as MTACO, BA and GA, Guney et al. [58] pointed out that PGSA has two main advantages. First of all, PGSA can determine the direction by the morphological concentration of the branches, and its running speed is fast. Secondly, PGSA has the advantage that the objective function and constraints can be handled separately, which avoids the construction of new objective function.

2.3.1 The principle of PGSA

The principle of PGSA algorithm is to simulate the phototropic theory of real plants. It takes the feasible region of integer programming as the growth environment of plants, regard the global optimal solution as the light source, and establishes the dynamic mechanism for the rapid growth of branches and leaves to the global optimal solution under different light intensity.

As an artificial self-organizing growth model, the focus of PGSA is to establish the plant system deductive mode based on growth rules and the probabilistic growth model based on plant phototropism theory. The dynamic process formed by the combination of the plant system deductive mode and the probabilistic growth model. It is to realize the process of simulating the plant from the initial state to the complete form of the final state (without the growth of new branches) in the feasible region space of integer programming. The path of PGSA to find the optimal solution is shown in Fig. 3.

2.3.2 The probabilistic model of PGSA

Fig. 3

The path of PGSA to find the optimal solution in a bounded closed area O.

The deduction method of the PGSA is described by using the following steps. The numerical space of the research problem is taken as the field of plant growth, and the point of sun irradiation is regarded as the global optimal point. When the plant grows all the way to the position of the light source, the plant stops growing.

The entire growth space of plants is defined as a feasible region in the probability model of PGSA. Assuming that M represents the length of the branch, there are T growing points on the trunk are S_M = (S_M1, S_M2, . . . S_MT), and the morphological element concentration of each growing point is P_M = (P_M1, P_M2, . . . P_MT). Suppose the unit length of the branch is m (m < M), there are r growing points defined S_m = (S_m1, S_m2, . . . S_mr) on it. The morphological element concentration of each growing point is p_M = (p_m1, p_m2, . . . p_mr). The morphological element concentration of the growing points on the trunk and branches can be calculated by the following formula:

$P_{M i} = \frac{f (x_{0}) - f (S_{M i})}{\sum_{i_{1} = 1}^{T} [f (x_{0}) - f (S_{M i})] + \sum_{j = 1}^{r} [f (x_{0}) - f (S_{m j})]}$ (3)

$P_{M j} = \frac{f (x_{0}) - f (S_{M j})}{\sum_{i_{1} = 1}^{T} [f (x_{0}) - f (S_{M i})] + \sum_{j = 1}^{r} [f (x_{0}) - f (S_{m j})]}$ (4)

The point x₀ is a random point on the feasible region, and f^* (x) is the backlight function. If the light intensity increases, the value of f^* (x) will decrease. From the above formula, we can derive $\sum_{i = 1}^{K} p_{Mi} + \sum_{j = 1}^{q} p_{mj} = 1$ . Suppose there are K + 1 growing points on the trunk and branches, we can obtain the morphological concentration of the growing points is p₁, p₂, . . . , p_K+1 from Equation (4). According to the randomness process of the PGSA algorithm, a value in the interval [0, 1] is randomly generated as a new growing point for the next growth. The values of T and r will change as the plants grow. New growing points will be added to the original set of growing points until no new branches are produced.

3 Fuzzy preference aggregation of decision makers based on spatial Steiner-Weber points

3.1 Spatial mapping of the fuzzy preference information

In multi-attribute group decision making, the experts’ fuzzy preference information is mapped to a set of 3-dimensional points. Let A = {A₁, A₂, ⋯ , A_n} be a finite scheme set, C = {C₁, C₂, ⋯ , C_m} be an attribute set, E = {E₁, E₂, ⋯ , E_q} be an expert set for evaluation. The evaluation vector of the expert E_l for the scheme A_i is ${\vec{x}}_{i}^{l} = [x_{ij}^{l}]_{1 \times m} = [x_{i 1}^{l}, x_{i 2}^{l}, \dots, x_{im}^{l}], j = 1, 2, \dots, m$ .

Each of these attributes represents a dimension, that is, m attributes represent m dimensions. The evaluation vector ${\vec{x}}_{i}^{l}$ of the scheme A_i by the expert E_l can be mapped to the point $x_{i}^{l}$ in 3-dimensional space through the relevant conversion rules, where the point $x_{i}^{l}$ coordinate is $(x_{i 1}^{l}, x_{i 2}^{l}, \dots, x_{im}^{l})$ .

$[x_{ij}^{l}]_{1 \times m} \to (x_{i 1}^{l}, x_{i 2}^{l}, \dots, x_{im}^{l}) \in R^{3}$ (5)

From Equation (5), R³ is called 3-dimensional Euclidean space (i = 1, 2, ⋯ , n ; j = 1, 2, ⋯ , m ; l = 1, 2, ⋯ , q).

3.2 Construction of spatial optimal aggregation model of fuzzy preference information

For the same scheme A_i (i = 1, 2, ⋯ , n), q experts have a corresponding evaluation vector. The spatial mapping rules in Equation (5) correspond to q points $x_{i}^{1}, x_{i}^{2}, \dots, x_{i}^{q}$ in 3-dimensional space. The purpose of spatial aggregation based on Steiner-Weber point is to find the point $x_{i}^{*}$ so that weighted Euclidean distance $D = \sum_{l = 1}^{q} v_{l} | x_{i}^{l} x_{i}^{*} |$ is minimized. The point $x_{i}^{*}$ is the aggregation of q experts’ evaluation vectors for the scheme A_i, which can reflect the overall opinions of q experts on the same scheme. Similarly, for n schemes, we can obtain the aggregation matrix $R^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*})^{'}$ composed of n spatial aggregation points.

Theorem 1. Suppose that there are m points P_i (x_i, y_i, z_i) , (i = 1, 2, ⋯ , m) with distance weight in the point set P = { P¹, P², …, P^m } (m ⩾ 3) of the closed region in 3-dimensional space. The weight vector of q experts is w = (w₁, w₂, ⋯ , w_i), in which $0 ⩽ w_{l} ⩽ 1 (l = 1, 2, \dots, i), \sum_{w = 1}^{i} w_{l} = 1$ . If the sum of the weighted Euclidean distances from the point Qⁿ (xⁿ, yⁿ, zⁿ) to other points should satisfy the following conditions:

$\begin{matrix} D_{Q^{n}} = min \sum_{i = 1}^{n} \\ (w_{i} \sqrt{{(x_{i} - x^{n})}^{2} + {(y_{i} - y^{n})}^{2} + {(z_{i} - z^{n})}^{2}}) \end{matrix}$ (6)

The point Qⁿ (xⁿ, yⁿ, zⁿ) is called the spatial optimal aggregation point, which satisfies the Pareto optimality and represents the willingness of group decision making. If we can find a global minimum objective function, then this method is the optimal, which can make the results of group decision making reach the optimum to some extent.

Proof. Let X be a bounded closed box in 3-dimensional space. There are n known vector groups p = (p₁, p₂, ⋯ p_n) ∈ X_Ip₁ ∈ [x₁, y₁, z₁], p₂ ∈ [x₂, y₂, z₂] , . . . , p_n ∈ [x_n, y_n, z_n] (where X_I represents all of the integer points in X, x₁, y₁, z₁, x₂, y₂, z₂, . . . , x_n, y_n, z_n are integers). It is assumed that f (x) is a continuous function defined on X. We can use the PGSA algorithm to obtain the value of min f (x). According to the characteristics of the Steiner-Weber problem, it is assumed that the growing point grows in the four directions of east, west, north and south, and new branches are produced constantly. The new branch rotates at angle of 90 degrees. The length of the branch can be set to l/1000 (l is the length of the bounded closed box). The step length is represented by f.

Suppose that the aggregation point obtained from the i-th iteration of the PGSA algorithm is Qⁿ (xⁿ, yⁿ, zⁿ), and the aggregation point obtained from the (i + 1)-th iteration is Q^n +1 (xⁿ⁺¹, yⁿ⁺¹, zⁿ⁺¹), then the point Q^n +1 can be represented by the point Qⁿ as Q^n +1 (w₁xⁿ + f₁, w₂yⁿ + f₂, w₃zⁿ + f₃).

Then the spatial optimal aggregation model objective function of point Qⁿ and point Qⁿ⁺¹ can be expressed as:

$\begin{matrix} f (Q^{n}) = \sum_{i = 1}^{n} (w_{i} \sqrt{{(x_{i} - x^{n})}^{2} + {(y_{i} - y^{n})}^{2} + {(z_{i} - z^{n})}^{2}}) \end{matrix}$ $f (Q^{n + 1}) = \sum_{i = 1}^{n} (w_{i} \sqrt{{(x_{i} - x^{n + 1})}^{2} + {(y_{i} - y^{n + 1})}^{2} + {(z_{i} - z^{n + 1})}^{2}})$ $= \sum_{i = 1}^{n} (w_{i} \sqrt{{[x_{i} - (w_{1} x^{n} + f_{1})]}^{2} + {[y_{i} - (w_{2} y^{n} + f_{2})]}^{2} + {[z_{i} - (w_{3} z^{n} + f_{3})]}^{2}})$ $f (Q^{n + 1}) - f (Q^{n}) = \sum_{i = 1}^{n} (w_{i} \sqrt{{(x_{i} - w_{1} x^{n} - f_{1})}^{2} + {(y_{i} - w_{2} y^{n} - f_{2})}^{2} + {(z_{i} - w_{3} z^{n} - f_{3})}^{2}})$ $- \sum_{i = 1}^{n} (w_{i} \sqrt{{(x_{i} - x^{n})}^{2} + {(y_{i} - y^{n})}^{2} + {(z_{i} - z^{n})}^{2}})$

To compare the values of f (Qⁿ) and f (Q^n +1), we only need to compare the values of (x_i - xⁿ) and (x_i - w₁xⁿ - f₁), (y_i - yⁿ) and (y_i - w₂yⁿ - f₂), (z_i - zⁿ) and (z_i - w₃zⁿ - f₃).

Assume that (x_i - xⁿ) > (x_i - w₁xⁿ - f₁), then $x^{n} < (w_{1} x^{n} + f_{1}), x^{n} < \frac{f_{1}}{1 - w_{1}}$

If x > 1, then w₁ → 1; If 0 < x < 1, then 0 < w₁ < 1. This is consistent with the hypothesis. Therefore, we can get (x_i - xⁿ) > (x_i - w₁xⁿ - f₁).

According to the above proof process, similarly, we can get (y_i - yⁿ) > (y_i - w₂yⁿ - f₂), (z_i - zⁿ) > (z_i - w₃zⁿ - f₃).

Then, we can get f (Q^n +1) - f (Qⁿ) < 0.

That is, we can obtain f (Q^n +1) < f (Qⁿ).

The PGSA algorithm converges to one point, and the function has a minimal solution. Therefore, if the optimal aggregation point Qⁿ exists, the sum of the Euclidean distances from Qⁿ to all other points should satisfy the following condition: $\begin{matrix} D_{Q^{n}} = min \sum_{i = 1}^{n} \\ (w_{i} \sqrt{{(x_{i} - x^{n})}^{2} + {(y_{i} - y^{n})}^{2} + {(z_{i} - z^{n})}^{2}}) \end{matrix}$

In the research of group preference aggregation, the experts’ preferences are mapped to a set of points in the 3-dimensional space and the distance between the two points reflects the difference of the experts’ preferences. The aggregation matrix is constructed by calculating the optimal aggregation points from the experts’ preferences. However, with the increase of the set of spatial points, how to solve the optimal aggregation point has become an NP-hard problem. In this paper, we use the spatial optimal aggregation model and the PGSA algorithm to solve this problem.

3.3 Core steps for searching spatial optimal aggregation point by PGSA algorithm

In order to obtain the optimal aggregation point, the key process of PGSA can be described in five aspects. First, set the initial growth point x⁰ ∈ X_I, X is the bounded closed box in Rⁿ. The initial growing point x⁰ is distributed randomly and uniformly in the bounded closed box. Second, calculate the morphological element concentration of each growing point. We can use Equation (7) to calculate P_j:

$P_{j} = \frac{\sum_{i = 1}^{n} (1 / | x_{j} X_{i} |)}{\sum_{j = 1}^{v} \sum_{i = 1}^{n} (1 / | x_{j} X_{i} |)}, j = 1, 2, . . ., v$ (7) where j is the serial number of growing points, and v is the number of growing points. Third, according to the calculation results of step 2, the probability space of each growth point in [0, 1] is selected, and the growth point x_j in this iteration is selected by random number. Fourth, determine the step length as f (can be set to l/1000). According to the L-system with α = 90°. Then, we can replace x_j with the best point of the new growing point. Fifth, if there is no new growing point generated after loop iteration, we can and local optimal solution. Then the optimal aggregation point is obtained, and the algorithm ends; otherwise, return to step 2 and continue to iterate until the global optimal solution is found. The program block diagram is shown in Fig. 4.

4 An approach for solving MAGDM problem based on spatial optimal aggregation model

Fig. 4

The block diagram for obtaining the optimal aggregation point by PGSA.

In the above chapters, we have discussed the spatial optimal aggregation model and the efficient algorithm (PGSA) in detail. Let A = {A₁, A₂, ⋯ , A_n} be a scheme set, C = {C₁, C₂, ⋯ , C_m} be an attribute set, and E = {E₁, E₂, ⋯ , E_q} be an expert set for evaluation. The attribute weight is W = (w₁, w₂, …, w_m), the expert weight is V = (v₁, v₂, …, v_q), and 0 ⩽ w_j, v_l ⩽ 1, $\sum_{j = 1}^{m} w_{j} = \sum_{l = 1}^{q} v_{l} = 1$ , j = 1, 2, …, m, l = 1, 2, ⋯ , q. In MAGDM, each expert evaluate each attribute of each scheme. We can express the fuzzy decision matrix by G = (δ_ij) _n×m = 〈o_ij, p_ij, q_ij〉_n×m, where o_ij denotes the degree to which the alternative meets the criteria considered by decision maker, p_ij denotes the degree to which the alternative is neutral to the criteria considered by decision maker, q_ij denotes the degree to which the alternative does not meet the criteria considered by decision maker.

An approach is developed here to solve MAGDM problem with the spatial optimal aggregation model and the PGSA algorithm. The core steps of this methods are as follows:

Step 1 According to the MAGDM problem, express the preference matrix of each expert. $G_{n \times m}^{l} = (\begin{matrix} 〈 o_{11}, p_{11}, q_{11} 〉 & 〈 o_{12}, p_{12}, q_{12} 〉 & \dots & 〈 o_{1 m}, p_{1 m}, q_{1 m} 〉 \\ 〈 o_{21}, p_{21}, q_{21} 〉 & 〈 o_{22}, p_{22}, q_{22} 〉 & \dots & 〈 o_{2 m}, p_{2 m}, q_{2 m} 〉 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 〈 o_{n 1}, p_{n 1}, q_{n 1} 〉 & 〈 o_{n 2}, p_{n 2}, q_{n 2} 〉 & \dots & 〈 o_{nm}, p_{nm}, q_{nm} 〉 \end{matrix})$

Step 2 Normalize the preference matrix $G_{n \times m}^{l}$ . If the attribute stands for a benefit type, we can obtain the normalized matrix $G_{ij}^{l} = (o_{ij}, p_{ij}, q_{ij})$ . If the attribute stands for cost attribute, we can get the normalized matrix $G_{ij}^{l} = (q_{ij}, p_{ij}, o_{ij})$ .

Step 3 According to the transformation rules of formula (5), map the expert preference matrices $G_{n \times m}^{l}$ to 3-dimensional Euclidean spaces respectively.

Step 4 Aiming at the fuzzy preference information contained in the decision matrix, use formula (6) to construct the spatial optimal aggregation model. Based on the spatial Steiner-Weber point, the purpose of aggregating experts’ preferences is to find a point $x_{i}^{*}$ that minimizes $D = \sum_{l = 1}^{r} v_{l} | x_{i}^{l} x_{i}^{*} |$ .

Step 5 We employ the PGSA algorithm to aggregate all the experts’ opinions. The obtained point $x_{i}^{*}$ is the spatial optimal aggregation point, which can reflect the overall opinions of l experts on the same alternative. The optimal aggregation matrix $G_{n \times m}^{*}$ is composed of these optimal aggregation points.

Step 6 The expert’s comprehensive evaluation score sc (δ_i) for each alternative is calculated by the formula from Ashraf [62]. Let δ_h = { o_h, p_h, q_h } , (h = 1, 2, 3, . . . , n), then we can obtain the score value of δ_h by the formula $sc (δ_{h}) = \frac{1}{3} (2 + o_{h} - p_{h} - q_{h})$ . Rank the alternatives based on the maximum score value and select the optimal one.

5 Numerical analysis

In the above chapters, we have discussed the spatial optimal aggregation model and the efficient algorithm (PGSA) in detail. In this section, we select one of the current hot issues of energy consumption for an example analysis. Because a country’s energy capacity depends on its economic growth, every country hopes to become a powerful economy by using the least energy. Renewable energy source (RES) selection is the most important factor for any country to survive in the competition. In order to illustrate the proposed method in this paper, we adopt the numerical analysis example offered by Ashraf et al. [62]. We give RES selection of Pakistan for numerical analysis. Due to the depth development of traditional energy such as coal, oil, natural gas, et al., there is a huge energy crisis in the world. Pakistan seeks high demand for energy due to its diverse population and technological progress.

The five alternatives were represented as S₁ (solar) , S₂ (wind) , S₃ (coal) , S₄ (natural gas), and S₅ (petroleum). The four attributes were denoted as f₁ (energy efficiency), f₂ (complexity of techno - logy), f₃ (job creation), and f₄ (CO2 emission). The weights of the three experts are (0.314, 0.355, 0.331) ^T, and the criteria weights are (0.256, 0.248, 0.245, 0.251) ^T. Suppose there are three evaluation tables given by three experts for selecting a proper RES, as show in Tables 1 –3.

Table 1
Spherical fuzzy information of expert 1

f ₁ f ₂ f ₃ f ₄

S ₁ (0.84,0.34,0.40) (0.43,0.39,0.78) (0.67,0.50,0.30) (0.31,0.21,0.71)

S ₂ (0.60,0.11,0.53) (0.23,0.35,0.59) (0.72,0.31,0.41) (0.11,0.25,0.82)

S ₃ (0.79,0.19,0.39) (0.11,0.21,0.91) (0.71,0.41,0.13) (0.34,0.25,0.51)

S ₄ (0.63,0.51,0.13) (0.49,0.33,0.42) (0.61,0.43,0.45) (0.49,0.37,0.59)

S ₅ (0.57,0.36,0.29) (0.50,0.15,0.60) (0.70,0.32,0.40) (0.33,0.44,0.65)

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.84,0.34,0.40)	(0.43,0.39,0.78)	(0.67,0.50,0.30)	(0.31,0.21,0.71)
S ₂	(0.60,0.11,0.53)	(0.23,0.35,0.59)	(0.72,0.31,0.41)	(0.11,0.25,0.82)
S ₃	(0.79,0.19,0.39)	(0.11,0.21,0.91)	(0.71,0.41,0.13)	(0.34,0.25,0.51)
S ₄	(0.63,0.51,0.13)	(0.49,0.33,0.42)	(0.61,0.43,0.45)	(0.49,0.37,0.59)
S ₅	(0.57,0.36,0.29)	(0.50,0.15,0.60)	(0.70,0.32,0.40)	(0.33,0.44,0.65)

Table 2

Spherical fuzzy information of expert 2

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.61,0.15,0.53)	(0.16,0.35,0.62)	(0.61,0.35,0.47)	(0.55,0.17,0.74)
S ₂	(0.66,0.11,0.51)	(0.43,0.23,0.77)	(0.93,0.08,0.09)	(0.02,0.06,0.99)
S ₃	(0.88,0.09,0.07)	(0.05,0.06,0.89)	(0.56,0.17,0.44)	(0.43,0.13,0.61)
S ₄	(0.59,0.32,0.34)	(0.24,0.48,0.51)	(0.68,0.53,0.39)	(0.34,0.21,0.61)
S ₅	(0.71,0.31,0.24)	(0.35,0.41,0.69)	(0.73,0.44,0.21)	(0.22,0.49,0.74)

Table 3

Spherical fuzzy information of expert 3

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.85,0.25,0.15)	(0.14,0.23,0.88)	(0.78,0.38,0.18)	(0.29,0.39,0.83)
S ₂	(0.94,0.04,0.07)	(0.39,0.19,0.61)	(0.63,0.18,0.35)	(0.48,0.49,0.56)
S ₃	(0.73,0.13,0.46)	(0.19,0.39,0.88)	(0.87,0.35,0.18)	(0.41,0.13,0.81)
S ₄	(0.82,0.12,0.43)	(0.55,0.21,0.63)	(0.53,0.33,0.47)	(0.46,0.23,0.51)
S ₅	(0.61,0.33,0.29)	(0.28,0.41,0.63)	(0.74,0.34,0.14)	(0.37,0.32,0.65)

In his major study, Ashraf et al. [62] used the spherical fuzzy (SF) weighted averaging (SFWA) operator to aggregate the spherical fuzzy information for the first time. Then, he adopted the SF symmetric weighted averaging (SFSWA), the SF symmetric ordered weighted averaging (SFSOWA) and the SF symmetric hybrid weighted averaging (SFSHWA) operators to aggregate the spherical fuzzy information for the second time. To illustrate the advantages of the proposed method in this paper, we adopted the proposed method to aggregate the spherical fuzzy information for the first time and compared it with the SFWA operator. Then, we used the proposed method to aggregate the spherical fuzzy information for the second time, and compared it with the SFSWA, SFSOWA and SFSHWA operators.

Step 1 According to the MAGDM problem, express the preference matrix of each expert, as show in Tables 1 –3.

Step 2 Normalize the preference matrix. The experts believe that attribute f₁ and attribute f₃ are benefit type, attribute f₂ and attribute f₄ are cost attributes. The evaluation information is normalized, and the results are shown in Tables 4 –6.

Table 4

Normalized spherical fuzzy information of expert 1

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.84,0.34,0.40)	(0.78,0.39,0.43)	(0.67,0.50,0.30)	(0.71,0.21,0.31)
S ₂	(0.60,0.11,0.53)	(0.59,0.35,0.23)	(0.72,0.31,0.41)	(0.82,0.25,0.11)
S ₃	(0.79,0.19,0.39)	(0.91,0.21,0.11)	(0.71,0.41,0.13)	(0.51,0.25,0.34)
S ₄	(0.63,0.51,0.13)	(0.42,0.33,0.49)	(0.61,0.43,0.45)	(0.59,0.37,0.49)
S ₅	(0.57,0.36,0.29)	(0.60,0.15,0.50)	(0.70,0.32,0.40)	(0.65,0.44,0.33)

Table 5

Normalized spherical fuzzy information of expert 2

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.61,0.15,0.53)	(0.62,0.35,0.16)	(0.61,0.35,0.47)	(0.74,0.17,0.55)
S ₂	(0.66,0.11,0.51)	(0.77,0.23,0.43)	(0.93,0.08,0.09)	(0.99,0.06,0.02)
S ₃	(0.88,0.09,0.07)	(0.89,0.06,0.05)	(0.56,0.17,0.44)	(0.61,0.13,0.43)
S ₄	(0.59,0.32,0.34)	(0.51,0.48,0.24)	(0.68,0.53,0.39)	(0.61,0.21,0.34)
S ₅	(0.71,0.31,0.24)	(0.69,0.41,0.35)	(0.73,0.44,0.21)	(0.74,0.49,0.22)

Table 6

Normalized spherical fuzzy information of expert 3

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.85,0.25,0.15)	(0.88,0.23,0.14)	(0.78,0.38,0.18)	(0.83,0.39,0.29)
S ₂	(0.94,0.04,0.07)	(0.61,0.19,0.39)	(0.63,0.18,0.35)	(0.56,0.49,0.48)
S ₃	(0.73,0.13,0.46)	(0.88,0.39,0.19)	(0.87,0.35,0.18)	(0.81,0.13,0.41)
S ₄	(0.82,0.12,0.43)	(0.63,0.21,0.55)	(0.53,0.33,0.47)	(0.51,0.23,0.46)
S ₅	(0.61,0.33,0.29)	(0.63,0.41,0.28)	(0.74,0.34,0.14)	(0.65,0.32,0.37)

Step 3 According to the spatial mapping rule of fuzzy preference information proposed in 3.1, we can map normalized spherical fuzzy information into the 3-dimensional spatial point.

Step 4 Construct the spatial optimal aggregation model based on the spatial Steiner-Weber point to find a point $x_{i}^{*}$ that minimizes $D = \sum_{l = 1}^{r} v_{l} | x_{i}^{l} x_{i}^{*} |$ .

Step 5 We adopt the proposed method in this paper to obtain the aggregation result of the 3-dimensional preference points of each expert, as shown in Table 7. We can also obtain the aggregation matrix of Ashraf’s method (see Table 8). Then the matrix in Table 7 is aggregated again by our method. From Table 9, we can get the collective preference values obtained by SFSWA operator, SFSOWA operator, SFSHWA operator, and the proposed method in this paper.

Table 7

Aggregation point obtained by the proposed method in this paper

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.803,0.290,0.373)	(0.737,0.323,0.219)	(0.681,0.449,0.307)	(0.740,0.239,0.353)
S ₂	(0.660,0.110,0.510)	(0.647,0.223,0.377)	(0.692,0.206,0.337)	(0.820,0.250,0.110)
S ₃	(0.790,0.190,0.390)	(0.910,0.210,0.110)	(0.752,0.354,0.192)	(0.610,0.130,0.430)
S ₄	(0.590,0.320,0.340)	(0.478,0.332,0.458)	(0.610,0.430,0.450)	(0.553,0.254,0.435)
S ₅	(0.611,0.313,0.219)	(0.675,0.398,0.344)	(0.729,0.407,0.214)	(0.652,0.438,0.328)

Table 8

Aggregation point obtained by SFWA operator

	f ₁	f ₂	f ₃	f ₄
S ₁	(0.788,0.229,0.319)	(0.785,0.315,0.208)	(0.696,0.402,0.297)	(0.767,0.239,0.371)
S ₂	(0.807,0.078,0.279)	(0.674,0.246,0.342)	(0.818,0.160,0.227)	(0.919,0.188,0.097)
S ₃	(0.814,0.128,0.223)	(0.893,0.165,0.099)	(0.748,0.284,0.223)	(0.677,0.159,0.393)
S ₄	(0.702,0.267,0.271)	(0.533,0.324,0.395)	(0.615,0.424,0.433)	(0.573,0.258,0.421)
S ₅	(0.639,0.331,0.271)	(0.644,0.298,0.363)	(0.724,0.365,0.224)	(0.685,0.411,0.296)

Table 9

Collective preference values obtained by four aggregation methods

	SFSWA operator	SFSOWA operator	SFSHWA operator	This paper
S ₁	(0.76020,0.30345,0.30469)	(0.75976,0.30508,0.30344)	(0.77225,0.27258,0.31515)	(0.75240,0.30120,0.32620)
S ₂	(0.80980,0.17791,0.25206)	(0.81003,0.17877,0.25161)	(0.82220,0.15240,0.24468)	(0.69200,0.20600,0.33700)
S ₃	(0.78763,0.19270,0.25719)	(0.78831,0.19293,0.25657)	(0.79913,0.19697,0.22875)	(0.76850,0.21430,0.31710)
S ₄	(0.60993,0.32436,0.38416)	(0.61028,0.32522,0.38432)	(0.79913,0.19697,0.22875)	(0.55300,0.32030,0.41900)
S ₅	(0.67353,0.35372,0.29288)	(0.67446,0.35417,0.29215)	(0.66306,0.33768,0.29619)	(0.66670,0.40390,0.30640)

Step 6 Calculate the alternative scores sc (S_i) (i = 1, 2, 3, 4, 5): sc (S₁) =0.7083, sc (S₂) =0.7163, sc (S₃) =0.7457, sc (S₄) =0.6046, sc (S₅) =0.6521. Thus, the ranking orders are S₃ ≻ S₂ ≻ S₁ ≻ S₅ ≻ S₄, and the optimal alternative is S₃.

The proposed method was used to aggregate the spherical fuzzy information for the first time, and compared with the SFWA operator. Then, the proposed method was adopted to aggregate the spherical fuzzy information for the second time, and compared it with the SFSWA, the SFSOWA and the SFSHWA operators. The advantages of our method are illustrated by discussing the different results obtained by the proposed method and these aggregation operators.

In order to visually demonstrate expert preference points and different aggregation points, the aggregation process can be expressed in the 3-dimensional coordinate system through the MATLAB simulation, as shown in Figs. 5 24. The black origin represents the experts preference points, the red “pentastar point” in the figure represents the aggregation point obtained by proposed method in this paper, and the blue “diamond point” represents the aggregation point obtained by SFWA aggregation operator. When the expert preference point appears, this signifies that the aggregation points obtained in this paper coincide with the preference point.

Fig. 5

The aggregation process of $(x_{i 1}^{1}, x_{i 2}^{1}, x_{i 3}^{1})$ .

Fig. 6

The aggregation process of $(x_{i 1}^{2}, x_{i 2}^{2}, x_{i 3}^{2})$ .

Fig. 7

The aggregation process of $(x_{i 1}^{3}, x_{i 2}^{3}, x_{i 3}^{3})$ .

Fig. 8

The aggregation process of $(x_{i 1}^{4}, x_{i 2}^{4}, x_{i 3}^{4})$ .

Fig. 9

The aggregation process of $(x_{i 1}^{5}, x_{i 2}^{5}, x_{i 3}^{5})$ .

Fig. 10

The aggregation process of $(x_{i 1}^{6}, x_{i 2}^{6}, x_{i 3}^{6})$ .

Fig. 11

The aggregation process of $(x_{i 1}^{7}, x_{i 2}^{7}, x_{i 3}^{7})$ .

Fig. 12

The aggregation process of $(x_{i 1}^{8}, x_{i 2}^{8}, x_{i 3}^{8})$ .

Fig. 13

The aggregation process of $(x_{i 1}^{9}, x_{i 2}^{9}, x_{i 3}^{9})$ .

Fig. 14

The aggregation process of $(x_{i 1}^{10}, x_{i 2}^{10}, x_{i 3}^{10})$ .

Fig. 15

The aggregation process of $(x_{i 1}^{11}, x_{i 2}^{11}, x_{i 3}^{11})$ .

Fig. 16

The aggregation process of $(x_{i 1}^{12}, x_{i 2}^{12}, x_{i 3}^{12})$ .

Fig. 17

The aggregation process of $(x_{i 1}^{13}, x_{i 2}^{13}, x_{i 3}^{13})$ .

Fig. 18

The aggregation process of $(x_{i 1}^{14}, x_{i 2}^{14}, x_{i 3}^{14})$ .

Fig. 19

The aggregation process of $(x_{i 1}^{15}, x_{i 2}^{15}, x_{i 3}^{15})$ .

Fig. 20

The aggregation process of $(x_{i 1}^{16}, x_{i 2}^{16}, x_{i 3}^{16})$ .

Fig. 21

The aggregation process of $(x_{i 1}^{17}, x_{i 2}^{17}, x_{i 3}^{17})$ .

Fig. 22

The aggregation process of $(x_{i 1}^{18}, x_{i 2}^{18}, x_{i 3}^{18})$ .

Fig. 23

The aggregation process of $(x_{i 1}^{19}, x_{i 2}^{19}, x_{i 3}^{19})$ .

Fig. 24

The aggregation process of $(x_{i 1}^{20}, x_{i 2}^{20}, x_{i 3}^{20})$ .

We obtain the optimal aggregation point of three experts on the spherical fuzzy information based on PGSA. From Table 7, we obtain the aggregation point by the proposed method. Then, we calculate the sum of weighted Euclidean distances from the spatial aggregation point obtained to all the experts’ preference points by SFWA operator and this paper. Let D₁ be the sum of weighted Euclidean distances from the aggregation point obtained by SFWA aggregation operator to experts’ preference points. Let D₂ be the sum of weighted Euclidean distances from the aggregation point obtained by our method to experts’ preference points. Let ΔD be the difference between D₁ and D₂. The correlation comparison analysis results can be compared in Table 10.

Table 10

The comparison between SFWA operator and this paper

	SFWA operator	This paper	Total path comparison	Difference ΔD
$(x_{i 1}^{1}, x_{i 2}^{1}, x_{i 3}^{1})$	D₁=0.208055	D₂=0.199224	D₂ < D₁	ΔD=0.008831
$(x_{i 1}^{2}, x_{i 2}^{2}, x_{i 3}^{2})$	D₁=0.283472	D₂=0.194036	D₂ < D₁	ΔD=0.089436
$(x_{i 1}^{3}, x_{i 2}^{3}, x_{i 3}^{3})$	D₁=0.200344	D₂=0.159642	D₂ < D₁	ΔD=0.040702
$(x_{i 1}^{4}, x_{i 2}^{4}, x_{i 3}^{4})$	D₁=0.223041	D₂=0.195000	D₂ < D₁	ΔD=0.028041
$(x_{i 1}^{5}, x_{i 2}^{5}, x_{i 3}^{5})$	D₁=0.064223	D₂=0.056020	D₂ < D₁	ΔD=0.008203
$(x_{i 1}^{6}, x_{i 2}^{6}, x_{i 3}^{6})$	D₁=0.183673	D₂=0.180531	D₂ < D₁	ΔD=0.003142
$(x_{i 1}^{7}, x_{i 2}^{7}, x_{i 3}^{7})$	D₁=0.133665	D₂=0.128146	D₂ < D₁	ΔD=0.005519
$(x_{i 1}^{8}, x_{i 2}^{8}, x_{i 3}^{8})$	D₁=0.137093	D₂=0.123741	D₂ < D₁	ΔD=0.013379
$(x_{i 1}^{9}, x_{i 2}^{9}, x_{i 3}^{9})$	D₁=0.196210	D₂=0.186376	D₂ < D₁	ΔD=0.009834
$(x_{i 1}^{10}, x_{i 2}^{10}, x_{i 3}^{10})$	D₁=0.154420	D₂=0.128307	D₂ < D₁	ΔD=0.026113
$(x_{i 1}^{11}, x_{i 2}^{11}, x_{i 3}^{11})$	D₁=0.151104	D₂=0.146758	D₂ < D₁	ΔD=0.004346
$(x_{i 1}^{12}, x_{i 2}^{12}, x_{i 3}^{12})$	D₁=0.224017	D₂=0.193243	D₂ < D₁	ΔD=0.030774
$(x_{i 1}^{13}, x_{i 2}^{13}, x_{i 3}^{13})$	D₁=0.208334	D₂=0.197731	D₂ < D₁	ΔD=0.010603
$(x_{i 1}^{14}, x_{i 2}^{14}, x_{i 3}^{14})$	D₁=0.096280	D₂=0.091188	D₂ < D₁	ΔD=0.005092
$(x_{i 1}^{15}, x_{i 2}^{15}, x_{i 3}^{15})$	D₁=0.114195	D₂=0.110165	D₂ < D₁	ΔD=0.004030
$(x_{i 1}^{16}, x_{i 2}^{16}, x_{i 3}^{16})$	D₁=0.156955	D₂=0.154724	D₂ < D₁	ΔD=0.002231
$(x_{i 1}^{17}, x_{i 2}^{17}, x_{i 3}^{17})$	D₁=0.296079	D₂=0.265439	D₂ < D₁	ΔD=0.030640
$(x_{i 1}^{18}, x_{i 2}^{18}, x_{i 3}^{18})$	D₁=0.136457	D₂=0.123137	D₂ < D₁	ΔD=0.013320
$(x_{i 1}^{19}, x_{i 2}^{19}, x_{i 3}^{19})$	D₁=0.103781	D₂=0.102550	D₂ < D₁	ΔD=0.001231
$(x_{i 1}^{20}, x_{i 2}^{20}, x_{i 3}^{20})$	D₁=0.101877	D₂=0.095340	D₂ < D₁	ΔD=0.006537

From Table 10, we can see that that D₁ is greater than D₂. The sum of the weighted Euclidean distances by the proposed method in this paper is minimal, which indicates that the aggregation point obtained by our method is superior to that obtained by SFWA operator. According to Theory 1, the smaller the sum of the distances from the aggregation point to all known points, the higher the aggregation accuracy. Through the analysis and comparison of Table 10, it is found that D₂ is the minimum value. This result shows that the method proposed in this paper has the highest aggregation accuracy. The greater the values of ΔD, it indicates that the method proposed in this paper is more accurate than other methods. Therefore, the aggregation matrix established by our method is closer to the group preference matrices, which can best represent the decision makers’ comprehensive opinion. In order to intuitively show the advantages of the method presented in this paper, the difference between SFWA operator and the proposed method is shown in Fig. 25. From the polyline, we can obtain that the accuracy of the method proposed in this paper is higher than that of the SFWA operator. Our method has a corresponding improvement on the spatial aggregation point 1 to spatial aggregation point 20. This shows that the method proposed in this paper has greatly improved these spatial aggregation points.

Fig. 25

Difference between SFWA operator and this paper.

From Table 9, we obtain the aggregation result by four different aggregation methods. From Table 11, we can see that score and ranking orders of four aggregation methods. The ranking orders are S₃ ≻ S₂ ≻ S₁ ≻ S₅ ≻ S₄, and the optimal alternative is S₃. We obtain the different sort and optimal alternative. Because the aggregation model proposed in this paper is an optimization model based on Steiner-Weber point. We proposed Theorem 1 and proved it. If we can find the global minimum objective function, the spatial aggregation model is optimal which can make the group decision result optimal to a certain extent. The idea of “Steiner-Weber” is introduced into the construction of aggregation model, which is the optimal aggregation model. We employ the PGSA to solve the aggregation points, and use the optimal aggregation model to calculate the sum of weighted Euclidean distance to see whether it is optimal. The PGSA algorithm converges to one point, and the function has a minimal solution. According to the comparison and analysis of the numerical example, the aggregation matrix established by our method is closer to the group preference matrices. We get the optimal aggregation point, which is the optimal consensus point of expert group evaluation. If it deviates from this aggregation point, there will be incomplete consensus. Therefore, this research proposed a new method to aggregate fuzzy preference information. The spatial optimal aggregation model and PGSA algorithm are used to find the optimal alternative scheme from a group of alternatives given by the decision makers. Thus, the MAGDM method based on spatial optimal aggregation proposed in this paper provides another perspective for finding the best alternative in the decision support system.

Table 11

Score and ranking orders of four aggregation methods

	sc (S₁)	sc (S₂)	sc (S₃)	sc (S₄)	sc (S₅)	Ranking
SFSWA operator	0.7173	0.7932	0.7792	0.6338	0.6756	S₂ ≻ S₃ ≻ S₁ ≻ S₅ ≻ S₄
SFSOWA operator	0.7170	0.7932	0.7796	0.6335	0.6760	S₂ ≻ S₃ ≻ S₁ ≻ S₅ ≻ S₄
SFSHWA operator	0.5120	0.5610	0.5550	0.4700	0.4950	S₂ ≻ S₃ ≻ S₁ ≻ S₅ ≻ S₄
This paper	0.7083	0.7163	0.7457	0.6046	0.6521	S₃ ≻ S₂ ≻ S₁ ≻ S₅ ≻ S₄

6 Conclusion

In this paper, we proposed an optimal aggregation method of fuzzy preference information based on spatial Steiner-Weber points, which aggregates several fuzzy preference relations into a set fuzzy preference relationship. As an analytical method, it is very simple to use, and we mainly analyze it from four parts. Firstly, we express the fuzzy preference matrix and normalize it. Secondly, use spatial mapping rules to map the fuzzy preference information to a set of spatial three-dimensional coordinate systems. Thirdly, a spatial aggregation model is constructed based on spatial Steiner-Weber point. Finally, the PGSA algorithm is used to find the spatial optimal aggregation point, which has the minimum spatial weighted Euclidean distance to all decision makers’ preferences.

By comparing with other aggregation operators (such as SFWA, SFSWA, SFSWA and SFSHWA operators), the spatial weighted Euclidean distance from the aggregation point obtained by the proposed method to all decision makers’ preference points is minimal. The optimal aggregation matrix is composed of these optimal aggregation points, which can accurately reflect decision makers’ comprehensive opinions.

This paper contributes to providing a novel method for spatial aggregation of fuzzy preference information. We believe that our approach offers an effective method for aggregating multiple fuzzy preference information. In our opinion, for the three-dimensional spatial aggregation problem, the proposed method provides some advantages over other fuzzy preference aggregation methods in this field. This study provides important enlightenment for the new approach of spatial optimal aggregation of MAGDM.

In the future, the proposed method will be used to solve some practical MAGDM problems, such as air quality evaluation, medical diagnosis, emergency decision making, etc. In addition, we will extend this method to a diversified environment, combining decision-making theory with social network, and consensus reaching in group decision making [63 –65] to achieve more objective and scientific decision making.

Footnotes

Acknowledgments

The authors would like to thank editors and anonymous reviewers for their helpful comments and suggestions. The research reported in this paper was partially supported by the National Natural Science Foundation of China (No. 71871106) and the Fundamental Research Funds for the Central Universities (Nos. JUSRP1809ZD; 2019JDZD06; JUSRP321016). The work was also sponsored by the Major Projects of Educational Science Fund of Jiangsu Province in 13th Five-Year Plan (No. A/2016/01); the Key Project of Philosophy and Social Science Research in Universities of Jiangsu Province (No. 2018SJZDI051); the Major Projects of Philosophy and Social Science Research of Guizhou Province (No. 21GZZB32); Project of Chinese Academic Degrees and Graduate Education (2020ZDB2). Key project of Philosophy and Social Science Research of Wuxi City (WXSK21-A-06).

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