Air multi-target threat assessment method based on improved GGIFSS

Abstract

In order to accurately assess the threat of air multi-target in the complicated and changeable air combat environment, an assessment method based on improved group generalized intuitionistic fuzzy soft set (I-GGIFSS) is proposed in this paper. Firstly, considering the characteristics of air target and the influence factors of threat assessment, a reasonable threat assessment system is established, and the appropriate assessment index is determined. Secondly, the generalized parameter matrix provided by many experts is introduced into the generalized intuitionistic fuzzy soft set (GIFSS) to form the group generalized intuitionistic fuzzy soft set (GGIFSS) to compensate for the knowledge limitation and assessment error of a single expert in traditional GIFSS. Finally, subjective weight is determined by group AHP (GAHP) and objective weight is determined by intuitionistic fuzzy entropy (IFE), then subjective weight and objective weight are combined based on relative entropy theory to determine reasonable index weight and expert weight, thus I-GGIFSS is obtained. The validity and superiority of I-GGIFSS are verified by the calculation and comparison of an example.

Keywords

Threat dynamic assessment group generalized intuitionistic fuzzy soft set group AHP intuitionistic fuzzy entropy relative entropy

1 Introduction

In the process of air defense command and decision-making, threat assessment of air multi-target is one of the key links, which provides a prerequisite for fire distribution and tactical decision-making, and can improve the ability of air defense weapons to deal with multi-target attack [1]. As the air combat environment becomes more complicated and changeable, how to accurately assess the threat of air target determines the operational effectiveness of the air defense weapon system, and then determines the success or failure of air defense operation. Therefore, it is of great significance to carry out research on air threat assessment methods.

At present, the methods of target threat assessment mainly include Bayesian networks [2], neural networks [3], rough sets [4], intuitionistic fuzzy set (IFS) [5, 6], and so on. These methods have their own characteristics and are suitable for different operational environments. In recent years, IFS has been used in threat assessment and some research results have been obtained. Lei et al. [7] established a system reasoning model based on IFS to solve the problem of joint air defense threat assessment. Zhang et al. [1] proposed a method to solve the multi-target threat assessment problem, which the target attribute matrix is determined by IFS and the target attribute weight is determined by the intuitionistic fuzzy entropy (IFE). Xu et al. [6] proposed a multi-attribute decision-making method for air target threat assessment based on IFS.

Although these methods based on IFS have been widely used in threat assessment, there are still difficulties in accurately determining target membership degree. This is related to the incompleteness and uncertainty of the information as well as the limitations of professional knowledge and personal preferences of decision-makers [8]. In addition, IFS lacks parameterization tools. In order to solve such problems, Molodtsov [8] proposed the concept of soft set (SS). Since then, remarkable achievements have been made in various fields based on SS [9 –11]. Inspired by IFS and SS, Maji et al. [12] proposed the concept of intuitionistic fuzzy soft set (IFSS), which achieved satisfactory results in multi-attribute decision-making. Agarwal et al. [13] proposed the theory of generalized intuitionistic fuzzy soft set (GIFSS). On the basis of describing the information by IFS, a generalized parameter was introduced to express the expert’s judgment on the validity of the information provided. GIFSS had been successfully applied in medical diagnosis and other aspects [14, 15].

GIFSS has a good effect on dealing with uncertain and inaccurate information, and is suitable for threat assessment of air target with greater measurement uncertainty. However, in a complex threat assessment environment, the generalized parameter provided by a single expert is difficult to take into account all important aspects of the decision problem. Therefore, it is necessary to introduce group generalized intuitionistic fuzzy soft set (GGIFSS) which contains the generalized parameter matrix provided by multiple experts into threat assessment. GGIFSS of assessment target consists of the GIFSS of assessment index and the generalized parameter set provided by experts. The ranking of the assessment target depends on the calculation result of the aggregation operator, and the index weight and the expert weight are involved in the calculation of the aggregation operator, so the weight value affects the final assessment result. However, in GGIFSS, index weight and expert weight are based on subjective cognition or human experience, and there are great uncertainties and errors. Therefore, an improved GGIFSS (I-GGIFSS) method for threat assessment of air target is proposed. The rationale for improving GGIFSS is: firstly, subjective weight is determined by group AHP (GAHP) and objective weight is determined by IFE, and then subjective weight and objective weight are combined by relative entropy theory. Thus the problem of index weight and expert weight is reasonably solved, which makes the assessment result is more accurate and effective. Through an example analysis and comparison, the effectiveness and superiority of the proposed method are verified.

2 Assessment index

The assessment index system is an important part of the air target threat assessment model. A reasonable selection of the index is a prerequisite for correct assessment. Threat assessment of air target needs to consider the influence of multiple factors, so it is necessary to select representative factors that can reflect the threat level of air target from different perspectives when determining assessment index. This paper takes the assessment of air target threat as the goal, and takes the overall target characteristics, the target position characteristics and the target motion characteristics as the selection criteria, and then determines seven assessment indexes: target type, jamming ability, target height, target distance, route shortcut, target velocity and maneuvering ability. The assessment index system of the air target threat is shown in Fig. 1.

Fig. 1

Air target threat assessment index system.

However, in actual air combat, in addition to the above seven indexes, other factors also have important reference value for threat assessment of air target, for instance, geographical factor such as the location of defensive areas, natural factor such as weather and climate, as well as human factor such as higher authorities and tactical arrangements. Considering that these factors are difficult to be expressed quantitatively by determining the index, it is necessary to introduce GGIFSS, and to propose the generalized parameter set for various factors by a number of experts, so that the assessment result is more accurate and reasonable.

3 Preliminaries

In this section, some basis concepts related to IFS, SS, IFSS, GIFSS, and GGIFSS are defined. These concepts are the basis for the calculation of the assessment method proposed in this paper and will be used in the following sections.

Definition 1. Let U ={ x₁, x₂, ⋯ , x_n } be a nonempty domain, the IFSA is defined as [16] $A = {〈 x_{i}, μ_{A} (x_{i}), ν_{A} (x_{i}) 〉 | x_{i} \in U}$ (1) where μ_A (x_i) and ν_A (x_i) denote the membership degree and non-membership degree of x_i ∈ U to A respectively, such that for ∀x_i ∈ U: 0 ≤ μ_A (x_i) , ν_A (x_i) ≤ 1, and 0 ≤ μ_A (x_i) + ν_A (x_i) ≤ 1.

In addition, let π_A (x_i) be the hesitation degree of x_i to A $π_{A} (x_{i}) = 1 - μ_{A} (x_{i}) - ν_{A} (x_{i})$ (2)

Definition 2. Let any intuitionistic fuzzy number (IFN) be α = (μ_α, ν_α), its score function and exact function [13] are as follows: $E_{α} = μ_{α}^{2} - ν_{α}^{2}$ (3) $F_{α} = μ_{α}^{2} + ν_{α}^{2}$ (4)

Definition 3. Let two IFNs be α = (μ_α, ν_α) and β = (μ_β, ν_β), the ranking rules of IFN [13] are as follows:

$\begin{matrix} * \begin{matrix} \end{matrix} If E_{α} < E_{β}, \begin{matrix} \end{matrix} then α < β; \\ * \begin{matrix} \end{matrix} If E_{α} < E_{β}, \begin{matrix} \end{matrix} then \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} If F_{α} = F_{β}, \begin{matrix} \end{matrix} then α = β; \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} If F_{α} < F_{β}, \begin{matrix} \end{matrix} then α < β . \end{matrix}$ (5)

Definition 4. Let U ={ x₁, x₂, ⋯ , x_n } be a nonempty domain, α (x_i) = (μ_α (x_i) , ν_α (x_i)) (i = 1, 2, ⋯ , n) be a set of IFNs, and ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of IFN α (x_i), with ω_i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1$ , then the intuitionistic fuzzy weighted averaging (IFWA) operator [17] is

$\begin{matrix} IFWA (α (x_{1}), α (x_{2}), \dots, α (x_{n})) = \\ (ω_{1} α (x_{1}) \oplus ω_{2} α (x_{2}) \oplus \dots \oplus ω_{n} α (x_{n})) = \\ (1 - \prod_{i = 1}^{n} {(1 - μ_{α} (x_{i}))}^{ω_{i}}, \prod_{i = 1}^{n} {(ν_{α} (x_{i}))}^{ω_{i}}) \end{matrix}$ (6)

Definition 5. Let U be a nonempty domain, IF^U represents the set of all intuitionistic fuzzy subsets on U. Moreover, let E be a set of parameters, and A ∈ E, then the pair (P, A) is an IFSS on U, where P is a mapping: A → IF^U [18].

Definition 6. Let U be a set of elements, and E be a set of parameters, then the pair (U, E) is a SS. Moreover, let A ⊆ E, Q is a mapping: A → IF^U, IF^U represents the set of all intuitionistic fuzzy subsets on U, and the generalized parameter g is an intuitionistic fuzzy subset of E. Then Q_g is the GIFSS based on SS (U, E) [14], and can be expressed as $Q_{g} = (Q (e), g (e)), Q (e) \in {IF}^{U}, g (e) \in IF$ (7) where Q_g is a mapping: A → IF^U × IF, Q (e) is the membership degree of elements in IFSS, g (e) is the possibility degree of U element membership degree in Q (e).

Definition 7. Let U be a set of elements, and E be a set of parameters, then the order pair (U, E) is a SS. Moreover, let A ⊆ E, Q is mapping: A → IF^U, IF^U represents the set of all intuitionistic fuzzy subsets on U, and the generalized parameter G = {δ₁, δ₂, ⋯ , δ_n} is the set of all intuitionistic fuzzy subsets of E. The weight set ω_δ = {ω₁, ω₂, ⋯ , ω_n} is the weight corresponding to G. Then Q_G is the GGIFSS based on SS (U, E) [13], and can be expressed as

$\begin{matrix} s . t . Q (e) \in {IF}^{U}, G (e) \in IF; \\ Q_{G} = (Q (e), G (e)) = \\ (Q (e), {δ_{1} (e) | ω_{1}, δ_{2} (e) | ω_{2}, \dots, δ_{n} (e) | ω_{n}}) \end{matrix}$ (8) where Q_G is a mapping: A → IF^U × IF, Q (e) is the membership degree of elements in IFSS, G (e) = {δ₁ (e) |ω₁, δ₂ (e) |ω₂, ⋯ , δ_n (e) |ω_n } is the possibility degree of U element membership degree in Q (e).

Definition 8. Let Q_{G
_i} = (Q_i (e) , G_i (e)) be a set of GIFSS, and $Q (e) = {a_{i 1}, a_{i 2}, \dots, a_{im}}$ (9) $G (e) = {q_{i 1}, q_{i 2}, \dots, q_{il}}$ (10)

Their weight vectors are $ω = (ω_{1}, ω_{2}, \dots, ω_{j}, \dots, ω_{m})$ (11)

$ρ = (ρ_{1}, ρ_{2}, \dots, ρ_{k}, \dots, ρ_{l})$ (12)

Then the aggregation operator of GGIFSS can be expressed as [5]

$\begin{matrix} Z_{Q_{i}} (a_{i 1}, a_{i 2}, \dots, a_{im}) \\ = IFWA (q_{i 1}, q_{i 2}, \dots, q_{il}) \\ \otimes IFWA (a_{i 1}, a_{i 2}, \dots, a_{im}) \\ = [(1 - \prod_{k = 1}^{l} (1 - μ_{q_{ij}})^{ρ_{k}} - \prod_{j = 1}^{m} (1 - μ_{a_{ij}})^{ω_{j}} \\ + \prod_{k = 1}^{l} (1 - μ_{q_{ij}})^{ρ_{k}} \prod_{j = 1}^{m} (1 - μ_{a_{ij}})^{ω_{j}}),] \\ (\prod_{k = 1}^{l} (ν_{q_{ij}})^{ρ_{k}} + \prod_{j = 1}^{m} (ν_{a_{ij}})^{ω_{j}} \\ - \prod_{k = 1}^{l} (ν_{q_{ij}})^{ρ_{k}} \prod_{j = 1}^{m} (ν_{a_{ij}})^{ω_{j}})] (13) \end{matrix}$ (13)

It can be seen that index weight ω and expert weight ρ are involved in the calculation of aggregation operator. The weight value affects the calculation result of aggregation operator, and then affects the assessment result. In traditional GGIFSS, the weight value is not very reasonable because of the less factors considered in the weighting method. So it is necessary to adopt a reasonable weighting method, which is introduced in the next section.

4 Weight determination

The weighting methods are mainly divided into three categories: subjective weighting method, objective weighting method, and combination weighting method [19]. The subjective weighting method can fully reflect the opinions of the decision-makers, but it is easy to be influenced by the experience and knowledge of the decision-makers, which makes the decision-making process and results more subjective. The objective weighting method is based on the information contained in the objective data, which has a strong theoretical basis, but the data fluctuation has a great influence on the calculation result, and neglects the influence of the decision-maker’s subjective intention. Combination weighting method is a compromise method which combines subjective preference of decision with objective information contained in the attribute. It can reflect the influence of subjective and objective factors on the decision to some extent, and greatly reduce the loss of information caused by single weighting method, so the index weight and expert weight are determined by the combination weighting method in this paper.

In this section, GAHP is first used to determine the subjective weight, and then IFE is used to determine the objective weight. Finally, the subjective weight and objective weight are combined based on the relative entropy theory to determine the reasonable index weight and expert weight.

4.1 Determination of subjective weight

The AHP proposed by Thomas L. Saaty is an effective method for multi-objective and multi-factor decision-making, and often used to calculate the subjective weight in decision-making problems. The GAHP is a method for different decision-makers to use AHP to analyze decision-making problem, and then assemble the information of each decision-maker to form a more objective group decision result. Compared with the AHP, the GAHP can determine more reasonable subjective weight, so that the assessment results can be more accurate.

Assume that the subjective weight is α = (α₁, α₂, ⋯ , α_j, ⋯ , α_m), the decision-maker group is E = {e₁, ⋯ , e_l}, and the judgment matrix given by decision-maker e_k is $A_{k} = {(a_{ij}^{k})}_{m \times m}$ . The judgment matrix weight λ_k of the decision-maker e_k is determined by the following two aspects.

4.1.1 Consistency level of the judgment matrix

In order to prevent the deviation between the assessment result from being too large, the consistency level of the judgment matrix needs to be tested. The consistency test formula is ${CR}_{k} = \frac{λ_{max}^{k} - l}{(l - 1) RI}$ (14) where $λ_{max}^{k}$ is the maximum eigenvalue of the judgment matrix A_k, n is the order of the pairwise comparison factor, and RI is the average random consistency index.

The smaller the consistency level, the higher the credibility of the judgment matrix, and the greater the weight of the judgment matrix of the relevant decision-makers. The greater the consistency level, the more serious the logic conflict of the judgment matrix, and the smaller the weight of the judgment matrix of the relevant decision-maker. Therefore, the weight of the decision-maker judgment matrix based on the consistency level of judgment matrix is

$λ_{k}^{c} = \frac{1 / - e^{{CR}_{k}}}{\sum_{i = 1}^{m} (1 / - e^{{CR}_{k}})} \begin{matrix} , & k = 1, 2, \dots, l \end{matrix}$ (15)

4.1.2 Deviation between individual judgment and group judgment

The consistency level of the judgment matrix only reflects the thinking logic of the decision-maker itself, and the weight of the decision-maker judgment matrix also needs to consider the difference between the decision-maker individual judgment and the group judgment. In order to facilitate the analysis of the difference of the judgment matrix, the following definition is given.

Definition 9. Let A = (a_ij) _m×m be the m-order judgment matrix, B = (b_ij) _m×m be the scale matrix of matrix A [20], where

$\begin{matrix} b_{ij} = {\begin{matrix} a_{ij} - 1 \begin{matrix} , & a_{ij} \geq 1 \end{matrix} \\ (a_{ij} - 1) / - a_{ij} \begin{matrix} , & a_{ij} < 1 \end{matrix} \end{matrix}, \\ \begin{matrix} i, j = 1, 2, \dots, m \end{matrix} \end{matrix}$ (16)

If the deviation between individual judgment and group judgment of a decision-maker is small, it means that the decision-maker’s opinion is generally supported by the group, so the decision-maker should have a higher weight. Therefore, the decision-maker judgment matrix weight $λ_{k}^{d}$ based on the deviation between individual judgment and group judgment is as follows:

$\begin{matrix} λ_{k}^{d} = \frac{1 / - (\sum_{i = 1}^{l} {∥ B_{k} - B_{i} ∥}_{F})}{\sum_{i = 1}^{l} (1 / - (\sum_{j = 1}^{l} {∥ B_{i} - B_{j} ∥}_{F}))}, \\ \begin{matrix} k = 1, 2, \dots, l \end{matrix} \end{matrix}$ (17)

In the above, the decision-maker judgment matrix weight $λ_{k}^{c}$ and $λ_{k}^{d}$ are obtained respectively, and then the decision-maker judgement matrix comprehensive weight λ_k can be obtained as $λ_{k} = γ λ_{k}^{c} + (1 - γ) λ_{k}^{d}$ (18) where γ is an adjustment factor, and the smaller γ is, the more attention is paid to the consistency of the group opinion.

Then, the comprehensive judgment matrix is defined and the relevant theorem is given.

Definition 10. Let $A_{k} = (a_{ij}^{k})_{m \times m} (k = 1, 2, \dots,$ m) be the judgment matrix given by decision-maker e_k, λ_k denotes the comprehensive weight of the judgment matrix of decision-maker e_k, satisfying λ_k ≥ 0 and $\sum_{k = 1}^{l} λ_{k} = 1$ . Then the Hadamard convex combination of judgement matrix A₁, A₂, ⋯ , A_m (that is, weighted geometric average comprehensive judgment matrix) [20] is

$\begin{matrix} \bar{A} = {({\bar{a}}_{ij})}_{m \times m} = {(\prod_{k = 1}^{l} {(a_{ij}^{k})}^{λ^{k}})}_{m \times m}, \\ \begin{matrix} i, j = 1, 2, \dots, m \end{matrix} \end{matrix}$ (19)

Theorem 1. If the judgment matrix A₁, A₂, ⋯ , A_m is uniformly acceptable, then the Hadamard convex combination of A₁, A₂, ⋯ , A_m is also uniformly acceptable.

The subjective weight α can be obtained by using the eigenvalue method for the determined comprehensive judgment matrix. In summary, the determination steps of α are as follows:

Step 1: Determine the judgment matrix A_k = ${(a_{ij}^{k})}_{m \times m} (k = 1, 2, \dots, m)$ of n decision-makers;

Step 2: The consistency test of judgment matrix is carried out, and the judgment matrix weight $λ_{k}^{c}$ is determined based on the consistency level of judgment matrix.

Step 3: The scale matrix B = (b_ij) _m×m of the judgment matrix is determined, and the other judgment matrix weight $λ_{k}^{d}$ is determined based on the deviation between individual judgment and group judgment.

Step 4: Determine the decision-maker judgement matrix comprehensive weight λ_k

Step 5: Determine the weighted geometric average comprehensive judgment matrix $\bar{A} = {({\bar{a}}_{ij})}_{m \times m}$ of judgement matrix A₁, A₂, ⋯ , A_m

Step 6: By using the eigenvalue method for the $\bar{A} = {({\bar{a}}_{ij})}_{m \times m}$ , the subjective weight can be obtained as

$\begin{matrix} α = (α_{1}, α_{2}, \dots, α_{j}, \dots, α_{m}), \\ \begin{matrix} j = 1, 2, \dots, m \end{matrix} \end{matrix}$ (20)

4.2 Determination of objective weight

Considering that IFE uses probability theory as a mathematical tool to measure information, it has the advantages of overcoming the influence of intuition and fuzziness on uncertain information and measuring the uncertain information of IFS. Therefore, this paper uses IFE to reasonably determine the objective weight.

Assume that the objective weight is β = (β₁, β₂, ⋯ , β_j, ⋯ , β_m), there are many definitions about the calculation of IFE. This paper adopts the calculation method in reference [21], which is defined as follows.

Definition 11. Let U be a nonempty domain, A ={ 〈 x_i, μ_A (x_i) , ν_A (x_i) 〉 |x_i ∈ U } be an IFS defined on domain U, and then the IFE is

$\begin{matrix} E (A) = \frac{1}{n} \sum_{i = 1}^{n} \\ [\frac{1 - {| μ_{A} (x_{i}) - ν_{A} (x_{i}) |}^{2} + π_{A}^{2} (x_{i})}{2}] \end{matrix}$ (21)

The calculation steps of objective weight β are as follows:

Step 1: Determine the IFS matrix.

$F = (f_{ij})_{n \times m} = [\begin{matrix} f_{11} & f_{12} & \dots & f_{1 m} \\ f_{21} & f_{22} & \dots & f_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ f_{n 1} & f_{n 2} & \dots & f_{nm} \end{matrix}]$ (22) where f_ij ={ (μ_ij, ν_ij, π_ij) }, and i = 1, 2, ⋯ , n ; j = 1, 2, ⋯, m.

Step 2: Calculate the IFE.

$E_{j} = \frac{1}{n} \sum_{i = 1}^{n} [\frac{1 - | μ_{ij} - ν_{ij} | + π_{ij}^{2}}{2}]$ (23)

Then, the IFE matrix E = (E_j) _1×m is determined.

Step 3: Construct the index weight nonlinear programming model

${\begin{matrix} min \sum_{j = 1}^{m} (β_{j})^{2} \cdot E_{j} \\ s . t . \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} \sum_{j = 1}^{m} β_{j} = 1 \end{matrix}$ (24) where β_j is the weight of the j index, and β_j ≥ 0.

Step 4: Calculate the attribute weight value β_j. The Lagrange function is established as

$L (β, λ) = \sum_{j = 1}^{m} (β_{j})^{2} \cdot E_{j} + 2 λ (\sum_{j = 1}^{m} β_{j} - 1)$ (25)

Then calculate the partial derivative for β_j and λ, respectively.

${\begin{matrix} \frac{\partial L (β, λ)}{\partial β_{j}} = 2 β_{j} \cdot E_{j} + 2 λ = 0 \\ \frac{\partial L (β, λ)}{\partial λ} = 2 (\sum_{j = 1}^{m} β_{j} - 1) = 0 \end{matrix}$ (26)

Then the attribute weight value is calculated as follows

$β_{j} = \frac{(E_{j})^{- 1}}{\sum_{j = 1}^{m} (E_{j})^{- 1}}$ (27)

Step 5: Determine the objective weight

$\begin{matrix} β = (β_{1}, β_{2}, \dots, β_{j}, \dots, β_{m}), \\ \begin{matrix} j = 1, 2, \dots, m \end{matrix} \end{matrix}$ (28)

4.3 Combination of the subjective weight and objective weight

The relative entropy theory is used to combine the subjective weight and objective weight. First, the definition and properties of relative entropy are given as follows [22].

Definition 12. Let X = (x₁, x₂, ⋯ , x_i, ⋯ , x_n) and Y = (y₁, y₂, ⋯ , y_i, ⋯ , y_n) be two discrete probability distributions, satisfying x_i, y_i ≥ 0 (i = 1, 2, ⋯ , n) and $\sum_{i = 1}^{n} x_{i} = \sum_{i = 1}^{n} y_{i} = 1$ , then the relative entropy of X relative to Y is

$h (X, Y) = \sum_{i = 1}^{n} x_{i} log \frac{x_{i}}{y_{i}}$ (29)

Theorem 2. The relative entropy satisfies the following properties:

$\sum_{i = 1}^{n} x_{i} log \frac{x_{i}}{y_{i}} \geq 0;$

The necessary and sufficient condition for $\sum_{i = 1}^{n} x_{i} log \frac{x_{i}}{y_{i}} = 0$ isx_i = y_i.

The relative entropy h (X, Y) can be used to measure the closeness degree between X and Y. When X and Y are the weight vectors obtained by two different methods, the closeness degree between the two different weights can be obtained.

In order to obtain the index weight, α and β need to be firstly aggregated to obtain the aggregation weight d = { d₁, d₂, ⋯, d_j, ⋯ , d_m }. The problem of determining aggregation weight can be transformed into the following optimization problems.

${\begin{matrix} min h (d) = \sum_{j = 1}^{m} d_{j} log \frac{d_{j}}{α_{j}} + \sum_{j = 1}^{m} d_{j} log \frac{d_{j}}{β_{j}} \\ s . t . \begin{matrix} \sum_{j = 1}^{m} d_{j} = 1, d_{j} > 0 \end{matrix} \end{matrix}$ (30)

For the above optimization problem, there is a global optimal solution $d^{*} = {d_{1}^{*}, d_{2}^{*}, \dots,$ $d_{j}^{*}, \dots, d_{m}^{*}}$

$d_{j}^{*} = \frac{{(α_{j})}^{\frac{1}{2}} {(β_{j})}^{\frac{1}{2}}}{\sum_{j = 1}^{m} ({(α_{j})}^{\frac{1}{2}} {(β_{j})}^{\frac{1}{2}})}$ (31)

Based on the relative entropy theory, the steps of combining α and β are as follows:

Step 1: Determine α and β.

Step 2: Construct an optimization model, as shown in Eq. (30)

Step 3: Calculate the aggregation weight d^*;

Step 4: Calculate the closeness degree $h (α, d_{i}^{*})$ of the subjective weight and the aggregation weight, and calculate the closeness degree $h (β, d_{i}^{*})$ of the objective weight and the aggregation weight.

Step 5: According to the closeness degree, the reliability of α and β are calculated respectively

${\begin{matrix} ψ_{α} = \frac{h (α, d^{*})}{h (α, d^{*}) + h (β, d^{*})} \\ ψ_{β} = \frac{h (β, d^{*})}{h (β, d^{*}) + h (β, d^{*})} \end{matrix}$ (32)

Step 6: Assume that the subjective weight and objective weight of the index are α_ω and β_ω respectively, and the subjective weight and objective weight of the expert are α_ρ and β_ρ respectively. According to the reliability of the subjective weight and objective weight, the combination index weight and expert weight can be obtained as

${\begin{matrix} ω = ψ_{α_{ω}} α_{ω} + ψ_{β_{ω}} β_{ω} \\ ρ = ψ_{α_{ρ}} α_{ρ} + ψ_{β_{ρ}} β_{ρ} \end{matrix}$ (33)

5 Assessment process of air multi-target threat based on I-GGIFSS

In order to accurately assess the threat of air multi-target in the complicated and changeable air combat environment, an air multi-target threat assessment method based on I-GGIFSS is proposed. The specific assessment process is as follows:

Step 1: Determine the air target threat membership matrix at the current time. According to Table 1, the membership degree of each air target assessment index is calculated, and then the air target threat membership matrix N_n×m can be obtained.

Table 1
Calculation method of membership degree of each index

Index Calculation method of membership degree

Target type Missile Battleplane Bomber Armed Helicopter Early Warning Aircraft

0.9 0.7 0.5 0.3 0.1

Jamming ability Strongest Strong Average Weak Weakest

0.9 0.7 0.5 0.3 0.1

Target height μ_h = (max { h_i } - h_i)/  - (max { h_i } - min { h_i })

Target distance μ_l = (max { l_i } - l_i)/  - (max { l_i } - min { l_i })

Route shortcut μ_s = (max { s_i } - s_i)/  - (max { s_i } - min { s_i })

Target velocity μ_v = (v_i - min { v_i })/  - (max { v_i } - min { v_i })

Maneuvering ability Strongest Strong Average Weak Weakest

0.9 0.7 0.5 0.3 0.1

Index	Calculation method of membership degree
Target type	Missile	Battleplane	Bomber	Armed Helicopter	Early Warning Aircraft
	0.9	0.7	0.5	0.3	0.1
Jamming ability	Strongest	Strong	Average	Weak	Weakest
	0.9	0.7	0.5	0.3	0.1
Target height	μ_h = (max { h_i } - h_i)/ - (max { h_i } - min { h_i })
Target distance	μ_l = (max { l_i } - l_i)/ - (max { l_i } - min { l_i })
Route shortcut	μ_s = (max { s_i } - s_i)/ - (max { s_i } - min { s_i })
Target velocity	μ_v = (v_i - min { v_i })/ - (max { v_i } - min { v_i })
Maneuvering ability	Strongest	Strong	Average	Weak	Weakest
	0.9	0.7	0.5	0.3	0.1

Step 2: Calculate the hesitation value and determine the hesitation matrix. Hesitation expresses the potential threat motive. Comparing the results of two real-time samplings, it can reflect the changing trend of air target at a certain time. Let t be the previous time and t + Δt be the later time. The measured value at time t is r_t, and at time t + Δt is r_t+Δt. Then the rate of change measured can be calculated as $k = \frac{r_{t + Δ t} - r_{t}}{r}$ (34)

The initial hesitation degree of the index can be calculated as $π = 0 . 5 (1 - μ)$ (35)

For an economic index, the hesitation value is calculated as $π = (0 . 5 + k) (1 - μ)$ (36)

For a cost index, the hesitation value is calculated as $π = (0 . 5 - k) (1 - μ)$ (37)

Among the seven indexes, the target type, jamming ability, target velocity and maneuvering ability are economic index. While the target height, target distance, and route shortcut are cost index.

Step 3: Determine the IFS matrix. The non-membership matrix is determined by the membership matrix and hesitation matrix, and then the intuitionistic fuzzy set matrix M_n×m can be determined.

Step 4: Determine the generalized parametric set matrix G_n×l provided by l experts.

Step 5: Based on the IFS matrix and generalized parameter set matrix, the GGIFSS matrix Q_n×(m+l) is obtained.

Step 6: Determine α_ω and α_ρ according to the GAHP method respectively.

Step 7: Determine β_ω and β_ρ according to the IFE respectively.

Step 8: Determine ω and ρ according to the relative entropy theory respectively.

Step 9: Calculate the aggregation operator Z_{Q
_i} of each air target.

Step 10: Calculate the score function value and exact function value of the aggregation operator, and rank the each target according to Eq. (5), thus the assessment result is obtained.

6 Assessment example

Our side found five air targets through radar and other detection equipment. The elements of set T ={ T₁, T₂, ⋯ , T₅ } correspond to target 1 to target 5, and the elements of set C ={ C₁, C₂, ⋯ , C₇ } are seven assessment indexes. The air target index parameters of the previous time t₁ and the current time t₂ are shown in Tables 2 and 3, respectively.

Table 2
Parameters of the air target index at t₁

Target Type Jamming ability Height (m) Distance (km) Route shortcut (km) Velocity (m/s) Maneuvering ability

T ₁ Missile Weakest 1000 80 7 480 Strongest

T ₂ Battleplane Strong 2200 210 12 380 Strong

T ₃ Bomber Average 3600 230 13 250 Average

T ₄ Early Warning Aircraft Strongest 8000 85 8 150 Weak

T ₅ Armed Helicopter Average 1100 65 5 100 Strong

Target	Type	Jamming ability	Height (m)	Distance (km)	Route shortcut (km)	Velocity (m/s)	Maneuvering ability
T ₁	Missile	Weakest	1000	80	7	480	Strongest
T ₂	Battleplane	Strong	2200	210	12	380	Strong
T ₃	Bomber	Average	3600	230	13	250	Average
T ₄	Early Warning Aircraft	Strongest	8000	85	8	150	Weak
T ₅	Armed Helicopter	Average	1100	65	5	100	Strong

Table 3

Parameters of the air target index at t₂

Target	Type	Jamming ability	Height (m)	Distance (km)	Route shortcut (km)	Velocity (m/s)	Maneuvering ability
T ₁	Missile	Weakest	800	70	6	500	Strongest
T ₂	Battleplane	Strong	1900	260	15	340	Strong
T ₃	Bomber	Average	3400	195	11	260	Average
T ₄	Early Warning Aircraft	Strongest	6500	100	9	170	Weak
T ₅	Armed Helicopter	Weak	1000	55	4	115	Strongest

6.1 Assessment process

The membership matrix of air target threat at t₂ can be obtained according to Table 1, and the hesitation matrix of air target threat at t₂ can be obtained according Equations (27) to (30). According to Eq. (2), the non-membership matrix of threat assessment can be obtained, and then the IFS M_5×7 can be sorted out, as shown in Table 4.

Table 4
IFS matrix of the air target threat at t₂

Target C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇

T ₁ (0.9,0.05,0.05) (0.1,0.45,0.45) (1,0,0) (0.927,0.027,0.046) (0.818,0.065,0.117) (1,0,0) (0.9,0.05,0.05)

T ₂ (0.7,0.15,0.15) (0.7,0.15,0.15) (0.807,0.070,0.123) (0,0.738,0.262) (0,0.750,0.250) (0.584,0.252,0.164) (0.7,0.15,0.15)

T ₃ (0.5,0.25,0.25) (0.5,0.25,0.25) (0.544,0.203,0.253) (0.317,0.238,0.445) (0.364,0.220,0.416) (0.378,0.286,0.336) (0.5,0.25,0.25)

T ₄ (0.1,0.45,0.45) (0.9,0.05,0.05) (0,0.312,0.688) (0.780,0.149,0.071) (0.545,0.284,0.171) (0.143,0.314,0.543) (0.3,0.35,0.35)

T ₅ (0.3,0.35,0.35) (0.3,0.63,0.07) (0.965,0.014,0.021) (1,0,0) (1,0,0) (0,0.350,0.650) (0.9,0.05,0.05)

Target	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇
T ₁	(0.9,0.05,0.05)	(0.1,0.45,0.45)	(1,0,0)	(0.927,0.027,0.046)	(0.818,0.065,0.117)	(1,0,0)	(0.9,0.05,0.05)
T ₂	(0.7,0.15,0.15)	(0.7,0.15,0.15)	(0.807,0.070,0.123)	(0,0.738,0.262)	(0,0.750,0.250)	(0.584,0.252,0.164)	(0.7,0.15,0.15)
T ₃	(0.5,0.25,0.25)	(0.5,0.25,0.25)	(0.544,0.203,0.253)	(0.317,0.238,0.445)	(0.364,0.220,0.416)	(0.378,0.286,0.336)	(0.5,0.25,0.25)
T ₄	(0.1,0.45,0.45)	(0.9,0.05,0.05)	(0,0.312,0.688)	(0.780,0.149,0.071)	(0.545,0.284,0.171)	(0.143,0.314,0.543)	(0.3,0.35,0.35)
T ₅	(0.3,0.35,0.35)	(0.3,0.63,0.07)	(0.965,0.014,0.021)	(1,0,0)	(1,0,0)	(0,0.350,0.650)	(0.9,0.05,0.05)

The expert group participating in the provision of the generalized parameter set is E ={ e₁, e₂, e₃ }, and the generalized parameter set matrix G_5×3 is shown in Table 5. Then, GGIFSS matrix Q_5×10 = (M_5×7, G_5×3) is obtained.

Table 5

Generalized parameter set matrix

Target	e ₁	e ₂	e ₃
T ₁	(0.9,0.07,0.03)	(0.9,0.05,0.05)	(0.9,0.01,0.09)
T ₂	(0.8,0.12,0.08)	(0.8,0.16,0.04)	(0.8,0.05,0.15)
T ₃	(0.7,0.1,0.2)	(0.7,0.13,0.17)	(0.9,0.04,0.06)
T ₄	(0.8,0.04,0.16)	(0.7,0.12,0.18)	(0.6,0.25,0.15)
T ₅	(0.7,0.21,0.09)	(0.6,0.32,0.08)	(0.7,0.28,0.02)

Next, according to the calculation steps in Section 4, the index weight and expert weight are calculated.

The subjective weight and objective weigh of the index is $\begin{matrix} \begin{matrix} α_{ω} = (0.2283, 0.1117, 0.0709, 0.1614, \\ \begin{matrix} \end{matrix} 0.2577, 0.1411, 0.0559) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} β_{ω} = (0.1147, 0.1211, 0.1666, 0.1915, \\ \begin{matrix} \end{matrix} 0.1574, 0.1050, 0.1437) \end{matrix} \end{matrix}$

The reliability of the subjective weight of the index is ψ_{α
_ω} = 0.3998, the reliability of the objective weight of the index is ψ_{β
_ω} = 0.6002.

Then the index weight is $\begin{matrix} \begin{matrix} ω = (0.1829, 0.1155, 0.1092, 0.1734, \\ \begin{matrix} \end{matrix} 0.2176, 0.1276, 0.0910) \end{matrix} \end{matrix}$

The subjective weight and objective weight of the expert is $\begin{matrix} α_{ρ} & = & (0.4830, 0.2594, 0.2576) \\ β_{ρ} & = & (0.3457, 0.2973, 0.3571) \end{matrix}$

The reliability of the subjective weight of the expert is ψ_{α
_ρ} = 0.4943, the reliability of the objective weight of the expert is ψ_{β
_ρ} = 0.5057.

Then the expert weight is $ρ = (0.4151, 0.2781, 0.3068)$

According to (12), the aggregation operator calculation results of the five targets are as follows: $\begin{matrix} Z_{Q 1} & = & [0.9, 0.0351] Z_{Q 2} = [0.426, 0.3399] \\ Z_{Q 3} & = & [0.3462, 0.2963] Z_{Q 4} = [0.3887, 0.3040] \\ Z_{Q 5} & = & [0.6750, 0.2579] \end{matrix}$

Calculate the score function value and the exact function value of the aggregation operator of the five targets respectively. The calculation results are shown in the Fig. 2.

Fig. 2

Calculation results of the score function value and the exact function value under I-GGIFSS.

According to the ranking rules, the ranking result of five targets is as follows. $T_{1} > T_{5} > T_{2} > T_{4} > T_{3}$

6.2 Comparative analysis

In order to verify the superiority of I-GIFSS, other methods are selected to assess the five targets, and the assessment results are compared and analyzed.

1) Comparison method 1. Threat assessment method based on IFS.

According to the calculation process in reference [17], the aggregation operator calculation results of the five targets are as follows: $\begin{matrix} Z_{I 1} & = & [1, 0] Z_{I 2} = [0.5325, 0.2671] \\ Z_{I 3} & = & [0.4406, 0.2341] Z_{I 4} = [0.5376, 0.2307] \\ Z_{I 5} & = & [1, 0] \end{matrix}$

2) Comparison method 2. Threat assessment method based on GIFSS.

In order to verify the validity of GGIFSS, the assessment result of the air target threat is calculated when the generalized parameter is proposed by only one expert. The method based on expert 1, expert 2 and expert 3 is GIFSS-1, GIFSS-2 and GIFSS-3, respectively.

The aggregation operator of GIFSS-1 is $\begin{matrix} Z_{e 11} & = & [0.90, 0.07] Z_{e 12} = [0.4260, 0.3550] \\ Z_{e 13} & = & [0.3084, 0.3107] Z_{e 14} = [0.4301, 0.2615] \\ Z_{e 15} & = & [0.70, 0.21] \end{matrix}$

The aggregation operator of GIFSS-2 is $\begin{matrix} Z_{e 21} & = & [0.90, 0.05] Z_{e 22} = [0.4260, 0.3843] \\ Z_{e 23} & = & [0.3084, 0.3337] Z_{e 24} = [0.3763, 0.3230] \\ Z_{e 25} & = & [0.60, 0.32] \end{matrix}$

The aggregation operator of GIFSS-3 is $\begin{matrix} Z_{e 31} & = & [0.90, 0.01] Z_{e 32} = [0.4260, 0.3037] \\ Z_{e 33} & = & [0.3965, 0.2647] Z_{e 34} = [0.3225, 0.4230] \\ Z_{e 35} & = & [0.70, 0.28] \end{matrix}$

3) Comparison method 3. Threat assessment method based on GGIFSS.

In order to verify the correctness of the weight determination method, the assessment result of air target threat is calculated when the index weight and expert weight is subjective weight or objective weight respectively. The method based on the subjective weight is GGIFSS-1, and the method based on the objective weight is GGIFSS-2.

The aggregation operator of GGIFSS-1 is $\begin{matrix} Z_{α 1} = [0.9000, 0.0389] Z_{α 2} = [0.4094, 0.3583] \\ Z_{α 3} = [0.3386, 0.2971] Z_{α 4} = [0.3944, 0.2991] \\ Z_{α 5} = [0.6768, 0.2523] \end{matrix}$ The aggregation operator of GGIFSS-2 is $\begin{matrix} Z_{β 1} = [0.9000, 0.0316] Z_{β 2} = [0.4488, 0.3161] \\ Z_{β 3} = [0.3545, 0.2972] Z_{β 4} = [0.1958, 0.3028] \\ Z_{β 5} = [0.6733, 0.2637] \end{matrix}$

The score function value and the exact function value of the aggregation operator of the five targets obtained by the above methods are calculated respectively, and the calculation results are shown in Fig. 3.

Fig. 3

Calculation results of the score function value and the exact function value under above comparison methods.

According to the Fig. 3, the ranking results of the proposed method and various comparison methods are shown in the Table 6.

Table 6

Ranking results of the proposed method and various comparison methods

	Method	Ranking result
Proposed method	I-GGIFSS	T₁ > T₅ > T₂ > T₄ > T₃
Comparison method 1	IFS	T₁ = T₅ > T₄ > T₂ > T₃
Comparison method 2	GIFSS-1	T₁ > T₅ > T₄ > T₂ > T₃
	GIFSS-2	T₁ > T₅ > T₄ > T₂ > T₃
	GIFSS-3	T₁ > T₅ > T₂ > T₃ > T₄
Comparison methods 3	GGIFSS-1	T₁ > T₅ > T₄ > T₂ > T₃
	GGIFSS-2	T₁ > T₅ > T₂ > T₃ > T₄

As can be seen from the Table 6, the ranking results of T₁ and T₅ are the same under each method (except IFS), the difference is that the ranking results of T₂, T₃ and T₄ are not the same. Compared with other targets, T₁ and T₅ are closer in distance, lower in height, stronger in maneuverability, so the threat is larger. T₁ is a missile, which is faster than T₅ and has a strong destructive capability. Thus, the actual threat degree is ranked as T₁>T₅. Compared with T₄, T₂ is lower in height, lager in velocity, stronger in maneuverability, and T₂ is a battleplane that can carry missiles, so the actual threat degree is ranked as T₂>T₄. Compared with T₃, T₄ is closer in distance, shorter in route shortcut, stronger in jamming ability, while T₃ is far away from the target and does not reach the optimal attack area, so the actual threat degree is ranked as T₄> T₃. In summary, the actual threat degree is ranked as T₁>T₅ > T₂ >T₄ >T₃. By comparing the result obtained by the proposed method with the results obtained by the three comparison methods, it can be seen that the assessment result of the proposed method is most consistent with the actual situation, which shows the validity of I-GGIFSS.

IFS is based on assessment index for threat assessment without considering other influencing factors. In a complex operational environment, the threat assessment result can be biased due to the uncertainties in the information obtained. In contrast, by introducing expert correction, GIFSS can effectively adjust the stiffness of assessment index and reduce the error caused by the uncertainty of the information obtained. However, because the knowledge background and personal preference of different expert is different, the assessment result is different when the generalized parameter provided by different expert is used. GGIFSS-1 and GGIFSS-2 use subjective weight and objective weight respectively, the weight information only reflects the influence of subjective factor or objective factor on decision-making, which results in deviation of the assessment result. Through the above analysis, the superiority of I-GGIFSS compared with other methods is verified.

Through the above comparison and analysis of the ranking results, it can be seen that I-GGIFSS overcomes the problem of unreasonable weight determination in GGIFSS, so it can assess the threat of air targets more accurately. Compared with IFS, GIFSS and GGIFSS, I-GGIFSS is more suitable for threat assessment of air target in complex and dynamic air combat environment, and the assessment performance is the best.

7 Conclusions

In the complicated and changeable air combat environment, how to accurately assess the threat of air multi-target is the key factor to determine the success or failure of air defense operation. In this paper, I- GGIFSS method for threat assessment of air multi-target is proposed. The main conclusions are as follows:

Due to the knowledge limitations of a single expert, the assessment result of GIFSS is different when the generalized parameter is provided by different expert. Compared with GIFSS, GGIFSS adopts the generalized parameters provided by many experts, so it takes into account the influence factors of threat assessment in a more comprehensive way, and makes the assessment results more reliable.

When different weighting methods are used to obtain index weight and expert weight, the assessment result of GGIFSS is different, which indicates that the weight value affects the accuracy of assessment result. I-GGIFSS proposed in this paper takes into account the influence of subjective factor and objective factor on the weight by using the combination weighting method, so it reasonably determines the index weight and expert weight, and makes the assessment result is more accurate.

In summary, I-GGIFSS can reliably and accurately assess air multi-target threat in complicated and changeable air combat environment. In addition, GGIFSS can be improved from many aspects, not limited to the reasonable weight determination, thus, the proposed method also has some limitations. How to improve GGIFSS from different aspects so that it can more effectively assess the threat of air targets is the next step of our work.

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