Interval TOPSIS with a novel interval number comprehensive weight for threat evaluation on uncertain information

Abstract

Threat evaluation (TE) is essential in battlefield situation awareness and military decision-making. The current processing methods for uncertain information are not effective enough for their excessive subjectivity and difficulty to obtain detailed information about enemy weapons. In order to optimize TE on uncertain information, an approach based on interval Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) and the interval SD-G1 (SD standard deviation) method is proposed in this article. By interval SD-G1 method, interval number comprehensive weights can be calculated by combining subjective and objective weights. Specifically, the subjective weight is calculated by interval G1 method, which is an extension of G1 method into interval numbers. And the objective weight is calculated by interval SD method, which is an extension of SD method with the mean and SD of the interval array defined in this paper. Sample evaluation results show that with the interval SD-G1 method, weights of target threat attributes can be better calculated, and the approach combining interval TOPSIS and interval SD-G1 can lead to more reasonable results. Additionally, the mean and SD of interval arrays can provide a reference for other fields such as interval analysis and decision-making.

Keywords

Interval number multiple attribute decision making (MADM)interval TOPSIS comprehensive weight threat evaluation

1 Introduction

TE is a process to calculate the extent of a target’s threat according to its motion, type, equipment, operational intention etc. The extent of targets’ threat is the key information for battlefield situation awareness and the foundation of firepower allocation [1] There are two main types of TE methods, those based on machine learning and rule-based evaluation methods.

TE methods based on machine learning work by learning the nonlinear relationship between attributes and extents of the threats by machine learning algorithms [2 –4]. These methods predominantly involve neural network (NN) [5] and support vector machine (SVM) [6]. NN based methods are advantageous for their strong capability for nonlinear fitting, while they may suffer from lack of reliable training sets and the need for a great number of training samples. In contrast, SVM based methods do not need a large number of samples for training. But still, the acquisition of reliable training sets is a problem for SVM-based TE, at the same time, a long training time is required when faced with a large amount of training samples. Therefore, in order to improve the effectiveness of TE, researchers combine the advantages of machine learning algorithms with other theoretical methods. Yue proposed a GNNM model [5]. It’s a predictive model that combines the advantages of small sample prediction with gray system theory and the advantages of self-learning of NN. Yang conducted threat assessment based on BP-BN [7]. It uses the powerful reasoning ability of Bayesian networks to analyze qualitative indicators to obtain threat values.

TE methods based on rule-based evaluation are engaged with clear calculation rules for the threat extents and rankings, including Bayesian network (BN), fuzzy evaluation (FE) and multiple attribute decision making [8] (MADM). BN based methods in TE [9] have been demonstrated to be effective for dealing with uncertain information, but the determination of prior probabilities in BN-based TE depends on subjective expert experience [5]. FE in TE [10 –12] is good at processing qualitative indicators, but the determination of membership functions and evaluation rules still depends on subjective expert experience. By contrast, MADM in TE is not affected by the specific types or attribute ranges, and it is a strongly versatile method and is easy for calculations. In addition, MADM can reduce the subjectivity during evaluation by an objective weighting method. Therefore, it can be deduced that MADM is more general, more flexible and simpler than other TE methods.

There are many kinds of uncertain information in TE [9 , 14], such as expert experience, measurement errors of target parameters, qualitative indices, etc. However, the methods mentioned previously are not sufficient to effectively process these information. Specifically, the machine learning-based methods can not directly deal with uncertain information. They need to integrate with other processing methods, such as grey system theory [5], Bayesian network [9], to fulfill this job. As for rule-based evaluation methods, BN deals with uncertain information by priori probabilities, and FE tackles such information by fuzzy theories such as intuitionistic fuzzy sets (IFS) [11] and GIFSS [10]. The priori probabilities in BN and the membership function in fuzzy theory are the key to process uncertain information in their individual framework, but they are determined by subjective judgments in current TE methods.

In addition, MADM is integrated with other methods to deal with uncertain information in TE, such as IFS [13], HFS (hesitant fuzzy sets) [15 –17] and interval number theory [20]. Therefore, a more reliable, more objective and simpler method is needed to deal with uncertain information in TE.

In MADM, the weighting methods brings an important impact on the decision results. The weighting methods in interval MADM can be categorized as subjective [21, 22], objective and comprehensive methods [23]. In subjective weighting methods, the preferences of decision makers can be reflected, but their experience and knowledge restrict the weight rationality. By contrast, the results of objective weighting methods are completely determined by the actual and objective attributes, but they are easily affected by outliers, which becomes more serious when processing uncertain information. By contrast, comprehensive weighting methods combine the other two methods together to maximize the advantages of both. In practical application, the weights can be assigned as either real numbers [23] or interval numbers [20].

It has been demonstrated that interval number theory [24] is suitable to process uncertain information in TE. It only requires two parameters for calculation, which are the upper and lower bounds. No additional subjective information is involved when modeling uncertain information as interval numbers, which utilizes the known information to the utmost extent and minimizes the deviation between the model and reality. TOPSIS [25] is an alternative ranking algorithm for MADM proposed by Hwang and Yoon [18]. Since uncertain information is not considered in classical TOPSIS, TOPSIS has been modified and optimized by integrating with other uncertainty theories, giving birth to algorithms like fuzzy TOPSIS [19], interval TOPSIS [20], AHP-TOPSIS [26]. In this paper, an interval MADM model based on interval TOPSIS and interval comprehensive weighting method is established to deal with uncertain information in TE. The comprehensive weights of threat attributes are calculated as interval numbers by the interval SD-G1 method. The results of the comparative analysis based on the evaluation index system indicate that the proposed method has the feasibility for practical applications.

2 Preliminaries

2.1 Interval Numbers and its operations [5 , 27]

An interval number is denoted as

$\tilde{x} = [x^{L}, x^{U}] = {x | x^{L} ⩽ x ⩽ x^{U}; x^{L}, x^{U} \in R}$ (1) where x^L is the lower bound and x^U is the upper bound. The interval number $\tilde{x}$ is a closed interval of real numbers. Suppose that $\tilde{x}, \tilde{y}$ are two interval numbers. Arithmetic operations on interval numbers has been studied in [20]. And the distance between $\tilde{x}, \tilde{y}$ is defined as

$d (\tilde{x}, \tilde{y}) = \sqrt[p]{{| x^{L} - y^{L} |}^{p} + {| x^{U} - y^{U} |}^{p}}$ (2) where p is the distance parameter, and p ⩾ 1. p = 2 is used in the calculation in this article

2.2 Ranking of interval numbers

Possibility degrees based ranking method is most widely used to determine the order of interval numbers.

Suppose that ${\tilde{x}}_{i} = [x_{i}^{L}, x_{i}^{U}]$ and ${\tilde{x}}_{j} = [x_{j}^{L}, x_{j}^{U}]$ are two interval numbers. According to [29], the possibility degree that ${\tilde{x}}_{i}$ is no less than ${\tilde{x}}_{j}$ is defined as:

$\begin{matrix} p_{ij} = p ({\tilde{x}}_{i} ⩾ {\tilde{x}}_{j}) \\ = max {1 - max (\frac{x_{j}^{U} - x_{i}^{L}}{x_{i}^{U} - x_{i}^{L} + x_{j}^{U} - x_{j}^{L}}, 0), 0} \end{matrix}$ (3)

However, such ranking method would sometimes bring unreasonable ranking orders. In order to solve this problem, another ranking strategy based on the Boolean matrix has been proposed [30]. The actual steps for this strategy are as follows.

In order to rank a group of interval numbers ${{\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n}}$ , the possibility degrees of any two interval numbers ${\tilde{x}}_{i}, {\tilde{x}}_{j} (i, j = 1, 2, \dots, n)$ are calculated to create a possibility degree matrix P = {p_ij} _n×n.

A Boolean matrix Q = {q_ij} _n×n, named as the ranking matrix, is constructed from P, where

$q_{ij} = {\begin{matrix} 1, p_{ij} ⩾ 0.5 \\ 0, p_{ij} < 0.5 \end{matrix}$ (4)

Calculate $λ_{i}^{'} = \sum_{j = 1}^{n} q_{ij}$ and obtain the ranking vector $λ^{'} = (λ_{1}^{'}, \dots, λ_{n}^{'})$ .

Rank the interval numbers according to the element order of $λ_{i}^{'} (i = 1, 2, \dots, n)$ in the ranking vector.

3 MADM model for TE with uncertain information

The purpose of TE is to calculate the extent of target threat based on the observed target attributes. A TE problem with uncertain information can be modeled as a MADM one with interval numbers.

Suppose that m threat targets are observed, and the threat target set T is expressed as

$\begin{matrix} T = {T_{1}, T_{2}, \dots, T_{m}} \end{matrix}$

The threat assessment system consists of n attributes for evaluation, and the index set A is expressed as $\begin{matrix} A = {A_{1}, A_{2}, \dots, A_{n}} \end{matrix}$

A threat variable X_ij is defined to show the threat extent of target T_i on attribute A_j, and the weight w_j is defined to indicate the importance of A_j in TE. In order to process uncertain information, X_ij and w_j are expressed as interval numbers.

The structure of an interval MADM model for TE with uncertain information is illustrated in Fig. 1. TE based on interval MADM is carried out in the following steps.

Fig. 1

Interval MADM model for TE with uncertain information.

Threats with uncertain information are quantified as interval numbers based on the evaluation index system.

By taking T as the decision scheme set and A as the decision index set, TE is transformed into a MADM problem.

A decision matrix is established with both target and attribute dimensions based on the threats expressed as interval numbers.

The extent of target threat is calculated by interval MADM based on the decision matrix.

4 Proposed TE method based on interval TOPSIS

TE is a complex operation. When the early warning detection system finds enemy threats, multiple observation platforms will be activated to record their attributes and obtain threat information about the target, followed by subsequent TE accordingly. TE results can be used as a reference for operational decision-making or firepower allocation. Specifically, the TE process based on our proposed method is described in Fig. 2.

Fig. 2

Overview of the TE process based on our proposed method.

As can be seen from Fig. 2, this process mainly includes four steps, the details of the proposed TE method are as follows.

(1) Decision matrix establishment

The observed threat attributes about the target are quantified into interval numbers by the evaluation index system. Then an interval attribute decision matrix $M_{0} = {{\tilde{X}}_{ij}}_{m \times n}$ is established based on target and attribute dimensions.

(2) Decision matrix normalization

In this paper, vector normalization is used to normalize $M_{0} = {{\tilde{X}}_{ij}}_{m \times n}$ [31]. Attributes are classified into benefit and cost ones. The benefit attributes are positively correlated to the threat extent, while the cost ones are negatively correlated to the threat.

According to [31], If an attribute A_j is a benefit attribute, then

${\tilde{X}}_{ij} = [X_{ij}^{L}, X_{ij}^{U}] = [\frac{X_{ij}^{L}}{\sqrt{\sum_{i = 1}^{m} (X_{ij}^{U})^{2}}}, \frac{X_{ij}^{U}}{\sqrt{\sum_{i = 1}^{m} (X_{ij}^{L})^{2}}}]$ (5)

If an attribute A_j is a cost attribute, then

$\begin{matrix} {\tilde{X}}_{ij} = & [X_{ij}^{L}, X_{ij}^{U}] \\ = [\frac{1 / X_{ij}^{U}}{\sqrt{\sum_{i = 1}^{m} {(1 / X_{ij}^{L})}^{2}}}, \frac{1 / X_{ij}^{L}}{\sqrt{\sum_{i = 1}^{m} {(1 / X_{ij}^{U})}^{2}}}] \end{matrix}$ (6)

In this way, a normalized decision matrix $M = {{\tilde{X}}_{ij}}_{m \times n}$ is obtained.

(3) Threat attribute weighting

The attribute weights are calculated based on the normalized decision matrix $M = {{\tilde{X}}_{ij}}_{m \times n}$ . The weight for A_j is the interval number ${\tilde{w}}_{j} = [w_{j}^{L}, w_{j}^{U}]$ , which takes uncertainties into consideration. Consequently, the attribute weights are listed in an interval weight vector $V^{w} = ({\tilde{w}}_{1}, {\tilde{w}}_{2} \dots, {\tilde{w}}_{n})$ . In this paper, the interval SD-G1 method is proposed to calculate the attribute weights, which will be introduced in section 5. The normalized decision matrix M can be weighted based on V ^w to obtain a weighted normalized decision matrix $M_{W} = {{\tilde{X}}_{ij}^{w}}_{m \times n}$ , where

${\tilde{X}}_{ij}^{w} = {\tilde{w}}_{j} {\tilde{X}}_{ij} = [w_{j}^{L}, w_{j}^{U}] \cdot [X_{ij}^{L}, X_{ij}^{U}]$ (7)

(4) Evaluation by interval TOPSIS

➀ Determination of ideal solutions. The positive ideal solution S⁺ is composed of maximum threat attributes ${\tilde{X}}_{j}^{+}$ , while the negative ideal solution S^– consists of minimum threat attributes ${\tilde{X}}_{j}^{-}$ .

$S^{+} = {{\tilde{X}}_{1}^{+}, {\tilde{X}}_{2}^{+}, \dots, {\tilde{X}}_{n}^{+}}$ (8)

$S^{-} = {{\tilde{X}}_{1}^{-}, {\tilde{X}}_{2}^{-}, \dots, {\tilde{X}}_{n}^{-}}$ (9) where

$\begin{matrix} \begin{matrix} {\tilde{X}}_{j}^{+} = [X_{j}^{L +}, X_{j}^{U +}] \\ = [max {X_{ij}^{L}, i = 1, \dots, m}, max {X_{ij}^{U}, i = 1, \dots, m}] \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} {\tilde{X}}_{j}^{-} = [X_{j}^{L -}, X_{j}^{U -}] \\ = [min {X_{ij}^{L}, i = 1, \dots, m}, min {X_{ij}^{U}, i = 1, \dots, m}] \end{matrix} \end{matrix}$

➁ Calculate the distances between each target and the ideal solutions.

The distance between a target T_i and S⁺ is labeled as $D_{i}^{+}$ , and its distance to S^- is denoted as $D_{i}^{-}$ .

According to Equation (2), $D_{i}^{+}$ and $D_{i}^{-}$ are calculated as: $\begin{matrix} D_{i}^{+} & = \sum_{j = 1}^{n} d ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{+}) \\ = \sum_{j = 1}^{n} \sqrt{{| X_{ij}^{L} - X_{j}^{L +} |}^{2} + {| X_{ij}^{U} - X_{j}^{U +} |}^{2}} \end{matrix}$ (10)

$\begin{matrix} D_{i}^{-} & = \sum_{j = 1}^{n} d ({\tilde{X}}_{ij}, {\tilde{X}}_{j}^{-}) \\ = \sum_{j = 1}^{n} \sqrt{{| X_{ij}^{L} - X_{j}^{L -} |}^{2} + {| X_{ij}^{U} - X_{j}^{U -} |}^{2}} \end{matrix}$ (11)

➂ Calculate the relative closeness C_i between each target and S⁺.

$C_{i} = \frac{D_{i}^{-}}{D_{i}^{+} + D_{i}^{-}}$ (12)

C_i is positively related to the threat extent of target T_i. Therefore, C_i is denoted as the target threat value TV_i in the following sections [1 , 20].

➃ Rank the targets according to their threat values.

TE results can be obtained by ranking threats, where sort (C_i) (i = 1, ⋯ , m).

The proposed TE method contains these four steps. Among them, the details about the threat attribute weighting and the interval SD-G1 method are introduced in section 5.

5 Weight determination

The comprehensive weights of threat attributes are calculated as interval numbers by the interval SD-G1 method. Specifically, SD is an objective weighting method, while G1 is a subjective weighting method. Interval SD-G1 method is proposed as an extension and integration of SD and G1 methods in this paper.

5.1 Determination of subjective weights

The subjective weights in TE with uncertain information can be calculated by the interval G1 method. The G1 method is an enhancement over the AHP method proposed by Guo [32]. With the increasing of threat attributes, calculations in AHP consistency tests become more complex, and experts’ judgments on weight importance turn less accurate. On the other hand, consistency is not a problem of concern in the G1 method, which evades complicated calculations. Furthermore, in the G1 method, experts judge the importance of weights more easily by ranking and relative comparisons. Therefore, the G1 method is suitable for TE.

Since both the threat attributes and their weights are interval numbers, the interval G1 method has been proposed by extending the G1 method into interval numbers, so as to enable weight calculation even with uncertain information. The actual steps of the interval G1 method are as follows [21, 22].

Determine the order relationship among the attributes.

Decision makers sort out attributes based on their importance. The relationship between A_i and A_k in an attribute set A = {A₁, A₂, ⋯ , A_n} is recorded as A_i > A_k when A_i is not less important than A_k under a certain evaluation criterion. And A_i = A_k represents that A_i is as important as A_k. After such order relationship is set for all elements in A,

A_{sort} = {A_{1}^{s}, A_{2}^{s}, \dots, A_{n}^{s}}

is obtained as a set that satisfies the order relationship

A_{1}^{s} > A_{2}^{s} > \dots > A_{n}^{s}

. The weight of attribute

A_{j}^{s}

is recorded as

w_{j}^{s}

Determine the relative importance between adjacent attributes [21].

The weights in this paper are interval numbers, so the weight of attribute $A_{k}^{s}$ is expressed as ${\tilde{w}}_{k}^{s} = [w_{k}^{sL}, w_{k}^{sU}]$ . The relative importance r_k between adjacent attributes $A_{k - 1}^{s}$ and $A_{k}^{s}$ is defined as

${\tilde{r}}_{k} = [r_{k}^{L}, r_{k}^{U}] = [\frac{w_{k - 1}^{sL}}{w_{k}^{sL}}, \frac{w_{k - 1}^{sU}}{w_{k}^{sU}}] k = n, n - 1, \dots, 2$ (13)

In reality, ${\tilde{r}}_{k}$ is obtained by comparing the importance between adjacent attributes by experts. Its numbers can be obtained by reference [21].

It should be noted that although ${\tilde{r}}_{k}$ is expressed as interval numbers, $r_{k}^{L}$ and $r_{k}^{U}$ are not directly related and they do not necessarily satisfy $r_{k}^{L} ⩽ r_{k}^{U}$ . Furthermore, ${\tilde{r}}_{k}$ can be regarded as a generalized interval number, which is calculated following the criterion of interval number operations.

Calculate the weight for each attribute.

According to the traditional G1 method [32], and interval number arithmetic rules, the equations to calculate the subjective weight of each attribute are

$\begin{matrix} {\tilde{w}}_{n}^{s} = [w_{n}^{sL}, w_{n}^{sU}] = {(1 + \sum_{k = 2}^{n} \prod_{q = k}^{n} r_{q})}^{- 1} \\ = {(1 + \sum_{k = 2}^{n} \prod_{q = k}^{n} [r_{q}^{L}, r_{q}^{U}])}^{- 1} \\ = \frac{[1, 1]}{[1, 1] + \sum_{k = 2}^{n} \prod_{q = k}^{n} [r_{q}^{L}, r_{q}^{U}]} \end{matrix}$ (14)

$\begin{matrix} {\tilde{w}}_{k}^{s} = [w_{k}^{sL}, w_{k}^{sU}] = [r_{k}^{L} w_{k - 1}^{sL}, r_{k}^{U} w_{k - 1}^{sU}] \\ k = n, n - 1, \dots, 2 \end{matrix}$ (15)

Corresponding to the above weight results according to the order relationship, a subjective weight vector is obtained as $\begin{matrix} W_{G 1} & = ({\tilde{w}}_{1}^{G 1}, \dots, {\tilde{w}}_{n}^{G 1}) \\ = ([w_{1}^{G 1 L}, w_{1}^{G 1 U}], \dots, [w_{n}^{G 1 L}, w_{n}^{G 1 U}]) \end{matrix}$

5.2 Determination of objective weights

The objective weights of threat attributes can be calculated by the interval SD method, which is the extension of the SD method with the means and SD numbers of interval arrays defined later in this paper.

5.2.1 The SD method

Suppose an attribute X_ij is a real number. Its objective weight calculated by the SD method is

$w_{j} = \frac{σ_{j}}{\sum_{i = 1}^{m} σ_{j}}$ (16) where

$σ_{j} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} (X_{ij} - {\bar{X}}_{j})^{2}}$ (17)

${\bar{X}}_{j} = \frac{1}{m} \sum_{i = 1}^{m} X_{ij}$ (18)

SD measures the deviation of the attributes in a set from their average, and it is positively correlated with the differentiation among the attributes. Additionally, a larger SD is more helpful for decision-making. In addition, comparing with other objective weighting methods such as entropy method [33] and Criteria Importance Through Intercriteria Correlation (CRITIC) [34], the calculation of SD method is more convenient.

In this paper, the attributes are interval numbers, so the SD method needs to be enhanced to involve interval numbers. Since square and square root operations are needed to calculate SD, it is necessary to give the definition of these operations for interval numbers to calculate the SD of interval arrays.

5.2.2 The square and square root of interval numbers

The theory of interval functions has been studied in [28]. Based on the theory of interval power functions, the square and square root of interval numbers are calculated in this section.

(1) The square of interval numbers

Suppose that $\tilde{x} = [x^{L}, x^{U}]$ and $\tilde{y} = [y^{L}, y^{U}]$ are two interval numbers. The rules for multiplication [20] between interval numbers are as follows: $\begin{matrix} \begin{matrix} \tilde{x} \tilde{y} = [x^{L}, x^{U}] \cdot [y^{L}, y^{U}] \\ = [min {x^{L} y^{L}, x^{L} y^{U}, x^{U} y^{L}, x^{U} y^{U}}, \\ max {x^{L} y^{L}, x^{L} y^{U}, x^{U} y^{L}, x^{U} y^{U}}] \end{matrix} \end{matrix}$

If $\tilde{x}$ and $\tilde{y}$ are positive interval numbers, then $\begin{matrix} \tilde{x} \tilde{y} = [x^{L}, x^{U}] \cdot [y^{L}, y^{U}] = [x^{L} y^{L}, x^{U} y^{U}] . \end{matrix}$

So, its power function ${(\tilde{x})}^{n} = [x^{L}, x^{U}]^{n}$ is defined as

$\begin{matrix} {(\tilde{x})}^{n} = [x^{L}, x^{U}]^{n} \\ = {\begin{matrix} [(x^{L})^{n}, (x^{U})^{n}], n odd or x^{L} ⩾ 0 (or both) \\ [(x^{U})^{n}, (x^{L})^{n}], n even and x^{U} ⩽ 0 \\ [0, max {{| x^{L} |}^{n}, {| x^{U} |}^{n}}, n even and x^{L} < 0 < x^{U} \end{matrix} \end{matrix}$ (19)

Therefore, when n = 2, the square of an interval number can be calculated as

$\begin{matrix} {(\tilde{x})}^{2} = [x^{L}, x^{U}]^{2} \\ = {\begin{matrix} [(x^{U})^{2}, (x^{L})^{2}], x^{U} ⩽ 0 \\ [0, max {(x^{L})^{2}, (x^{U})^{2}}], x^{L} < 0 < x^{U} \end{matrix} \end{matrix}$ (20)

In this paper, Equation (20) is unified by the operation of maximum and minimum.

$\begin{matrix} {(\tilde{x})}^{2} = [x^{L}, x^{U}]^{2} \\ = [(min {max {0, x^{L}}, x^{U}})^{2}, max {(x^{L})^{2}, (x^{U})^{2}}] \end{matrix}$ (21)

(2) The square root of interval numbers

Suppose a positive interval number $\tilde{x} = [x^{L}, x^{U}]$ . When n ⩾ 0, its power function ${(\tilde{x})}^{n} = [x^{L}, x^{U}]^{n}$ is defined as

${(\tilde{x})}^{n} = {[x^{L}, x^{U}]}^{n} = [{(x^{L})}^{n}, {(x^{U})}^{n}]$ (22)

The square root operation requires the interval number to be positive. When n = 2, the square root $\sqrt{\tilde{x}} = {(\tilde{x})}^{\frac{1}{2}} = [x^{L}, x^{U}]^{\frac{1}{2}}$ can be expressed as

$\sqrt{\tilde{x}} = {[x^{L}, x^{U}]}^{\frac{1}{2}} = [{(x^{L})}^{\frac{1}{2}}, {(x^{U})}^{\frac{1}{2}}] = [\sqrt{x^{L}}, \sqrt{x^{U}}]$ (23)

5.2.3 The Mean and SD of interval arrays

Based on Equations (21) and (23), the mean and SD of an interval array X are proposed in this section.

(1) The mean of interval arrays

The mean of an interval array X is defined as $\bar{X} = [{\bar{X}}^{L}, {\bar{X}}^{U}] = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{x}}_{i}$ . According to the arithmetic operations of interval numbers [20], X’s mean is calculated as

$\begin{matrix} \bar{X} = \frac{1}{m} \sum_{i = 1}^{m} [x_{i}^{L}, x_{i}^{U}] = [\frac{1}{m} \sum_{i = 1}^{m} x_{i}^{L}, \frac{1}{m} \sum_{i = 1}^{m} x_{i}^{U}] \\ = [{\bar{X}}^{L}, {\bar{X}}^{U}] \end{matrix}$ (24)

When $x_{i}^{L} = x_{i}^{U} (i = 1, \dots, m)$ , X degenerates into a real array, and

$\bar{X} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{x}}_{i} = \frac{1}{m} \sum_{i = 1}^{m} x_{i}^{L} = \frac{1}{m} \sum_{i = 1}^{m} x_{i}^{U}$ (25) which shows that the mean of a real array can be regarded as a special case in the more general concept of the mean of interval arrays.

(2) The SD of interval arrays

The SD of an interval array X is defined as ${\tilde{σ}}_{X} = [σ_{X}^{L}, σ_{X}^{U}] = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {({\tilde{x}}_{i} - \bar{X})}^{2}}$ . Based on the arithmetic operations of interval numbers in [20], and the definitions of the square and square root of interval numbers in Section 5.2.2, X’s SD can be calculated as

$\begin{matrix} {\tilde{σ}}_{X} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {({\tilde{x}}_{i} - \bar{X})}^{2}} \\ = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {([x_{i}^{L}, x_{i}^{U}] - [{\bar{X}}^{L}, {\bar{X}}^{U}])}^{2}} \\ = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {([x_{i}^{L} - {\bar{X}}^{U}, x_{i}^{U} - {\bar{X}}^{L}])}^{2}} \\ = [\sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(min {max {0, x_{i}^{L} - {\bar{X}}^{U}}, (x_{i}^{U} - {\bar{X}}^{L})})}^{2}}, \\ \sqrt{\frac{1}{m} \sum_{i = 1}^{m} max {{(x_{i}^{L} - {\bar{X}}^{U})}^{2}, {(x_{i}^{U} - {\bar{X}}^{L})}^{2}}}] \\ = [σ_{X}^{L}, σ_{X}^{U}] \end{matrix}$ (26)

When $x_{i}^{L} = x_{i}^{U} (i = 1, \dots, m)$ , X degenerates into a real array, and

$\begin{matrix} σ_{X} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {({\tilde{x}}_{i} - \bar{X})}^{2}} \\ = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {([x_{i}^{L}, x_{i}^{U}] - [{\bar{X}}^{L}, {\bar{X}}^{U}])}^{2}} \\ = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(x_{i}^{U} - {\bar{X}}^{U})}^{2}} \end{matrix}$ (27) which shows the SD of a real array can be regarded as a special case of the SD of an interval array.

5.2.4 Interval SD method

Based on the calculation equations given in Section 5.2.3, the equation to find out the objective weights when the attributes are interval numbers is obtained and named as the interval SD method.

Suppose an attribute ${\tilde{X}}_{ij} = [X_{ij}^{L}, X_{ij}^{U}] (i = 1, \dots, m; j = 1, \dots, n)$ . The mean ${\bar{X}}_{j} = [{\bar{X}}_{j}^{L}, {\bar{X}}_{j}^{U}]$ and SD ${\tilde{σ}}_{j} = [σ_{j}^{L}, σ_{j}^{U}]$ of attribute A_j are

$\begin{matrix} {\bar{X}}_{j} = \frac{1}{m} \sum_{i = 1}^{m} {\tilde{X}}_{ij} = \frac{1}{m} \sum_{i = 1}^{m} [X_{ij}^{L}, X_{ij}^{U}] \\ = [\frac{1}{m} \sum_{i = 1}^{m} X_{ij}^{L}, \frac{1}{m} \sum_{i = 1}^{m} X_{ij}^{U}] = [{\bar{X}}_{j}^{L}, {\bar{X}}_{j}^{U}] \end{matrix}$ (28)

$\begin{matrix} {\tilde{σ}}_{j} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {({\tilde{X}}_{ij} - {\bar{X}}_{j})}^{2}} \\ = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {([X_{ij}^{L}, X_{ij}^{U}] - [{\bar{X}}_{j}^{L}, {\bar{X}}_{j}^{U}])}^{2}} \\ = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {([X_{ij}^{L} - {\bar{X}}_{j}^{U}, X_{ij}^{U} - {\bar{X}}_{j}^{L}])}^{2}} \\ = [\sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(min {max {0, X_{ij}^{L} - {\bar{X}}_{j}^{U}}, (X_{ij}^{U} - {\bar{X}}_{j}^{L})})}^{2}}, \\ \sqrt{\frac{1}{m} \sum_{i = 1}^{m} max {{(X_{ij}^{L} - {\bar{X}}_{j}^{U})}^{2}, {(X_{ij}^{U} - {\bar{X}}_{j}^{L})}^{2}}}] \\ = [σ_{j}^{L}, σ_{j}^{U}] \end{matrix}$ (29)

The objective weight of attribute A_j is calculated by

$w_{sum} = \sum_{j = 1}^{n} σ_{j}^{L} + \sum_{j = 1}^{n} σ_{j}^{U}$ (30)

${\tilde{w}}_{j} = [w_{j}^{L}, w_{j}^{U}] = [\frac{σ_{j}^{L}}{w_{sum}}, \frac{σ_{j}^{U}}{w_{sum}}]$ (31)

Consequently, the objective weight vector is $W_{σ} = ({\tilde{w}}_{1}^{σ}, \dots, {\tilde{w}}_{n}^{σ}) = ([w_{1}^{σ L}, w_{1}^{σ U}], \dots, [w_{n}^{σ L}, w_{n}^{σ U}])$ .

5.3 Calculation of the comprehensive weights

The interval G1 method can calculate the weights of threat attributes as interval numbers, and its calculation, as well as the judgment of weight importance, is relatively easy. However, as is shown in the Fig. 3, the order relationship among all attributes and the relative importance between adjacent attributes are completely dependent on experts’ subjective judgments [21]. Therefore, the interval G1 method is flawed for its excessive subjectivity.

Fig. 3

The flaw of the interval G1 method.

As the foundation of weight calculation, determinations about order relationship and relative importance are vulnerable to subjective factors. When a strong subjectivity is involved in weight calculation, the results will become unstable.

In order to determine the attribute weights more accurately based on the advantages of the interval G1 method, this paper combines subjective and objective weights together to calculate comprehensive weights, weakening the impact from subjective factors. The idea of comprehensive weights is shown in Fig. 4.

Fig. 4

The idea of comprehensive weights.

In this paper, objective weights are introduced by the interval SD method to overcome the subjectivity problem in the interval G1 method, creating the interval SD-G1 method as a consequence. Specifically, the interval SD-G1 method is performed as follows.

The objective weight vector ${\tilde{W}}_{σ}$ of A = {A₁, A₂, ⋯ , A_n} is determined by the interval SD method.

The weights of ${\tilde{W}}_{σ}$ are sorted by interval number ranking based on a Boolean matrix. And the ranking relationship of these is taken as the order relationship $A_{1}^{s} > A_{2}^{s} > \dots > A_{n}^{s}$ in the interval G1 method.

The weight vector ${\tilde{W}}_{G 1}$ is obtained by the interval G1 method, which is taken as the comprehensive weight vector $\tilde{W}$ .

6 Evaluation example

6.1 Evaluation index system

This evaluation example is a TE problem of air targets. The first step of TE is to establish an evaluation index system. In this example, three aspects are considered during system establishment, including overall combat capability, target position and motion of air targets [5 , 13]. The index system is established shown in Fig. 5.

Fig. 5

Evaluation index system of air target.

Overall combat capability. Target type (A₁) is an important index to measure the targets’ overall combat capability and their threat. In this paper, target types are mainly considered as one of the five representative target aircrafts, namely missile, fighter, bomber, attack helicopter and early warning aircraft. In addition, jamming ability (A₂) is another important indicator of combat capability. A higher jamming ability can provide an advantage in electromagnetic air combats.

A₁ and A₂ are quantified by Miller’s quantitative theory [35], where levels of extremely strong, strong, moderate, weak and extremely weak are assigned numbers of 0.9, 0.7, 0.5, 0.3 and 0.1, respectively. The details are shown in Table 1.

Target position is determined by target deviation (A₃) and target distance (A₄), which are both cost attributes. A₃ is an important index to measure the approach intention of both sides. And A₄ is an important index related to target weapon and equipment action distance.

Target motion includes target velocity (A₅) and maneuvering ability (A₆). Specifically, target velocity is a benefit attribute. It is a movement index that reflects the target’s intention of combat. And, Maneuvering ability is a qualitative index expressed as linguistic values, which is quantified in the same way as A1 and A2 (Table 1). The more maneuverable the enemy target is, the better it can track and hit friendly targets, as well as evade attacks.

Table 1

Quantification of qualitative indexes

Target type (A1)	Jamming ability (A2)	Maneuvering ability (A6)	Quantification
Missile	Strongest	Strongest	0.9
Battleplane	Strong	Strong	0.7
Bomber	Average	Average	0.5
Armed Helicopter	Weak	Weak	0.3
Early Warning Aircraft	Weakest	Weakest	0.1

In all the threat attributes mentioned above, A₁, A₂ and A₆ are constant. These constant attributes indicate the targets’ threats regardless of battlefield situations, which are called static threats in this paper. However, A₃, A₄ and A₅ are variable attributes. They reflect the targets’ movements, actions and direct threats in real battlefields, which are dynamic threats.

6.2 Parameters of the air target

Suppose there are 5 air targets in the combat airspace, labeled as T₁-T₅. Based on the air battle data from Longfei Yue and Gaige Wang [5, 37], target parameters from 2 observation platforms O_a and O_b are shown in Tables 2 and 3. Due to uncertainties caused by data transmission delays, as well as measurement errors and reliability problems of the sensors, data obtained from 2 observation platforms are different.

Table 2
Air target parameters from O_a<sub>

Target Type Jamming ability Deviation (°) Distance (km) Velocity (m/s) Maneuvering ability

T ₁ Bomber Average 100.5 290 160 Average

T ₂ Armed Helicopter Weak 9.0 65 100 Strong

T ₃ Early Warning Aircraft Strongest 90.0 200 145 Weak

T ₄ Battleplane Strong 20.0 150 380 Strong

T ₅ Missile Weakest 10.0 80 480 Strongest

Target	Type	Jamming ability	Deviation (°)	Distance (km)	Velocity (m/s)	Maneuvering ability
T ₁	Bomber	Average	100.5	290	160	Average
T ₂	Armed Helicopter	Weak	9.0	65	100	Strong
T ₃	Early Warning Aircraft	Strongest	90.0	200	145	Weak
T ₄	Battleplane	Strong	20.0	150	380	Strong
T ₅	Missile	Weakest	10.0	80	480	Strongest

Table 3

Air target parameters from O<sub>b

Target	Type	Jamming ability	Deviation (°)	Distance (km)	Velocity (m/s)	Maneuvering ability
T ₁	Bomber	Average	99.5	295	155	Average
T ₂	Armed Helicopter	Weak	10.0	55	115	Strong
T ₃	Early Warning Aircraft	Strongest	88.0	210	140	Weak
T ₄	Battleplane	Strong	19.0	160	340	Strong
T ₅	Missile	Weakest	9.0	70	500	Strongest

6.3 Evaluation process

(1) Decision matrix establishment

Based on the target parameters in Tables 2–3 and the quantification results in Table 1, a decision matrix M₀ is obtained.

$\begin{matrix} M_{0} = (\begin{matrix} 0.5 0.5 [9950, 10050] [290, 295] [155, 160] 0.5 \\ 0 . 30 . 3 [900, 1000] [55, 65] [100, 115] 0 . 7 \\ 0 . 10 . 9 [8800, 9000] [200, 210] [140, 145] 0 . 3 \\ 0 . 70 . 7 [1900, 2000] [150, 160] [340, 380] 0 . 7 \\ 0 . 90 . 1 [900, 1000] [70, 80] [480, 500] 0 . 9 \end{matrix}) \end{matrix}$

(2) Decision matrix normalization

The normalized decision matrix M is obtained by Equations (5) and (6).

$\begin{matrix} M_{0} = (\begin{matrix} 0 . 5 & 0 . 5 & [9950, 10050] & [290, 295] & [155, 160] & 0 . 5 \\ 0 . 3 & 0 . 3 & [900, 1000] & [55, 65] & [100, 115] & 0 . 7 \\ 0 . 1 & 0 . 9 & [8800, 9000] & [200, 210] & [140, 145] & 0 . 3 \\ 0 . 7 & 0 . 7 & [1900, 2000] & [150, 160] & [340, 380] & 0 . 7 \\ 0 . 9 & 0 . 1 & [900, 1000] & [70, 80] & [480, 500] & 0 . 9 \end{matrix}) \end{matrix}$

(3) Threat attribute value weighting

Comprehensive weights for the threats are calculated by the interval SD-G1 method mentioned in previous sections.

First, the objective weight vector W_σ is calculated by the interval SD method based on the normalized decision matrix M. (Equation (31)) $\begin{matrix} \begin{matrix} W_{σ} = (0 . 16820 . 1682 [0 . 1588, 0 . 2566] \\ [0 . 0976, 0 . 2520] [0 . 1366, 0 . 2123] 0 . 1067) \end{matrix} \end{matrix}$

Then, the comprehensive weight vector W_c is calculated by the interval G1 method based on W_σ through Equation (14). $\begin{matrix} \begin{matrix} W_{c} = ([0 . 0831, 0 . 2882] [0 . 0831, 0 . 2882] \\ [0 . 0785, 0 . 4398] [0 . 0483, 0 . 4319] \\ [0 . 0675, 0 . 3639] [0 . 05280 . 1829]) \end{matrix} \end{matrix}$

Finally, the weighted normalized decision matrix M_w is obtained based on the normalized decision matrix weighted by W_c.

(4) Evaluation by interval TOPSIS

The distance D⁺ between targets and the positive ideal solutions (Equation (10)), the distance D^– between targets and the negative ideal solutions (Equation (11)) and their relative closeness C are shown in Fig. 6.

Fig. 6

Variables in interval TOPSIS.

$\begin{matrix} M_{w} = (\begin{matrix} [0 . 0324, 0 . 1122] [0 . 0324, 0 . 1122] [0 . 0047, 0 . 0293] [0 . 0066, 0 . 0690] [0 . 0155, 0 . 0921] [0 . 0181, 0 . 0627] \\ [0 . 0194, 0 . 0673] [0 . 0194, 0 . 0673] [0 . 0472, 0 . 3242] [0 . 0299, 0 . 3637] [0 . 0100, 0 . 0662] [0 . 0253, 0 . 0877] \\ [0 . 0065, 0 . 0224] [0 . 0582, 0 . 2019] [0 . 0052, 0 . 0332] [0 . 0093, 0 . 1000] [0 . 0140, 0 . 0835] [0 . 0108, 0 . 0376] \\ [0 . 0453, 0 . 1571] [0 . 0453, 0 . 1571] [0 . 0236, 0 . 1536] [0 . 0122, 0 . 1334] [0 . 0341, 0 . 2188] [0 . 0253, 0 . 0877] \\ [0 . 0582, 0 . 2019] [0 . 0065, 0 . 1571] [0 . 0472, 0 . 3242] [0 . 0243, 0 . 2858] [0 . 0481, 0 . 2878] [0 . 0325, 0 . 1128] \end{matrix}) \end{matrix}$

In this paper, the relative closeness C_i is regarded as the threat value TV_i. And the TE result of T₅ > T₂ > T₄ > T₃ > T₁ could be obtained by ranking all TV_i in Fig. 6. T₅ (missile) is found to exhibit the highest comprehensive threat. The evaluation results are consistent with the actual situation of air combat [1 , 11]. Consequently, based on the assessment results, our fighting groups need to make operational decisions and firepower allocations accordingly.

6.4 Comparative analysis of weighting methods

6.4.1 Comparison with non-comprehensive weights

In order to verify the effectiveness of the interval SD-G1 method, it is compared with non-weighting, G1 subjective weighting and interval SD objective weighting strategies. The weights generated from the G1, interval SD and interval SD-G1 methods are as follows

The weights from G1 subjective weighting [21, 22] is: $\begin{matrix} W_{G 1} = (0.2652 0 . 1324 0 . 1204 0 . 0926 \\ 0.1854 0.2040) \end{matrix}$

From Equation (31), the weights from interval SD objective weighting is: $\begin{matrix} \begin{matrix} W_{σ} = (0 . 16820 . 1682 [0 . 1588, 0 . 2566] \\ [0 . 0976, 0 . 2520] [0 . 1366, 0 . 2123] 0 . 1067) \end{matrix} \end{matrix}$

The weights from the interval SD-G1 method is:

\begin{matrix} \begin{matrix} W_{c} = ([0 . 0831, 0 . 2882] [0 . 0831, 0 . 2882] \\ [0 . 0785, 0 . 4398] [0 . 0483, 0 . 4319] \\ [0 . 0675, 0 . 3639] [0 . 0528, 0 . 1829]) \end{matrix} \end{matrix}

The TE results obtained from these weighting methods are shown in Table 4.

Table 4
Comparative results obtained from non-comprehensive and comprehensive weights

Weighting method Non-weighting G1 Interval SD Interval SD-G1

TV₁ 0.2310 0.3510 0.1891 0.1884

TV₂ 0.5130 0.5210 0.5668 0.5819

TV₃ 0.1978 0.1900 0.1885 0.1886

TV₄ 0.5610 0.600 0.5598 0.5330

TV₅ 0.7991 0.8101 0.8011 0.7914

Ranking Results $\begin{matrix} T_{5} > T_{4} > T_{2} \\ > T_{1} > T_{3} \end{matrix}$ $\begin{matrix} T_{5} > T_{4} > T_{2} > \\ T_{1} > T_{3} \end{matrix}$ $\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{1} > T_{3} \end{matrix}$ $\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{3} > T_{1} \end{matrix}$

Weighting method	Non-weighting	G1	Interval SD	Interval SD-G1
TV₁	0.2310	0.3510	0.1891	0.1884
TV₂	0.5130	0.5210	0.5668	0.5819
TV₃	0.1978	0.1900	0.1885	0.1886
TV₄	0.5610	0.600	0.5598	0.5330
TV₅	0.7991	0.8101	0.8011	0.7914
Ranking Results	$\begin{matrix} T_{5} > T_{4} > T_{2} \\ > T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{4} > T_{2} > \\ T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{3} > T_{1} \end{matrix}$

As is shown in Table 4, the difference lay in the rankings of T₁, T₃, and T₂, T₄. The most intrinsic reason behind such difference is differed importance evaluation results for threat attributes as a consequence of varied weighting methods.

In the non-weighting strategy, the weight for all attributes is 1, and the importance of different threat attributes is evaluated as the same. For the G1 method, threat attributes’ weights are given as, A₁ > A₆ > A₅ > A₂ > A₃ > A₄ and the ranks of these attributes in interval SD and interval SD-G1 methods are both A₃ > A₄ > A₅ > A₂ = A₁ > A₆. Judging from the TE results, T₂ and T₄ are ranked correctly by interval SD and interval SD-G1 methods, suggesting these two strategies can assign weights to threat attributes that resemble to the actual situations more closely than the other two. In addition, it has also demonstrated that the ranks of T₂ and T₄ are mainly affected by dynamic threat attributes A₃, A₄ and A₅, since both non-weighting and G1 strategies do not use interval numbers to represent uncertainties.

In order to further compare the weighting results from interval SD and interval SD-G1 strategies, the weights of each attribute are sorted and shown in Fig. The lower bounds, upper bounds and midpoints of the weights correspond to the bottom, top and horizontal line in the middle of the bar.

As is shown in Fig. 7, the weight results for dynamic threats A₃, A₄ and A₅ in two strategies are interval numbers, whose relative size relationships are close. But the weight results for static threat attributes A₁, A₂ and A₆ are obviously different between two methods. A₁, A₂ and A₆ are quantified as real numbers in the interval SD method, but they are expressed as interval numbers in the interval SD-G1 method as a result of interval G1 processing. In addition, the results from the interval SD method are close to the midpoint of those from the interval SD-G1 method. Nevertheless, the interval SD-G1 method can express uncertain information in static threats A₁, A₂ and A₆ as interval numbers by interval G1 processing.

Fig. 7

Weights of interval SD and interval SD-G1 methods.

When comparing between T₁ and T₃, the main parameter difference lay in the static threats: A₁, A₂ and A₆. Furthermore, interval number weights from the interval SD-G1 method make the difference more significant in TE. Therefore, the interval SD-G1 strategy is more favorable to calculate the weights of real attributes. The differences of expressing weights as interval numbers or real numbers will be discussed in section 6.4.2.

In conclusion, compared with non-comprehensive weighting strategies, the interval SD-G1 method could produce better evaluation results on the importance of threat attributes and provide more reasonable weights, which is helpful for higher TE accuracy. In addition, the ranks of T₁ and T₃ are mainly affected by static threats: A₁, A₂ and A₆, while those of T₂ and T₄ are mainly affected by dynamic threats: A₃, A₄ and A₅.

6.4.2 Comparison with real number weights

In order to verify the idea that weights expressed as interval numbers can lead to better TE outcomes than real numbers, those obtained from the interval SD-G1 method are compared with other real number weights.

The degradation operation takes the midpoint of an interval number X_ij as its degraded value, which is labeled as DEGR (X_ij). For an interval array X, DEGR (X) represents the degradation of each interval number in X. For a weighting method M, DEGR (M) represents the degradation of the weights of each attribute. Degradation is capable to transform an interval number weight into a real number one, which is helpful to compare and analyze the influence brought by loss in uncertain information on the evaluations for weight importance and TE.

The interval number weights for comparison come from the interval SD-G1 and interval SD methods, and the real number weights are obtained from DEGR (interval SD-G1), DEGR (interval SD) and non-weighting.

The weight vector of DEGR (interval SD-G1) is:

$\begin{matrix} W_{σ}^{D} = (0 . 1682 0 . 1682 0 . 2077 0 . 1748 0 . 1744 0 . 1067) \end{matrix}$

The weight vector of DEGR (interval SD) is:

$\begin{matrix} W_{c}^{D} = (0 . 1857 0 . 1857 0 . 2592 0 . 2401 0 . 2157 0 . 1178) \end{matrix}$

TE results obtained from the weighting methods above are shown in Fig. 8 and Table 5 TE differences lay in the ranks of T₁, T₂, T₃ and T₄. Such differences in ranks are caused by varied weights.

Fig. 8

Threat value (TV) compared with real number weights.

Table 5

Comparative results with real number weights

Weighting method	Non-weighting	DEGR (interval SD)	Interval SD	DEGR (interval SD-G1)	Interval SD-G1
TV₁	0.2310	0.2401	0.1891	0.1887	0.1884
TV₂	0.5130	0.5529	0.5668	0.5714	0.5819
TV₃	0.1978	0.1902	0.1885	0.1885	0.1886
TV₄	0.5610	0.5825	0.5598	0.5560	0.5330
TV₅	0.7991	0.8021	0.8011	0.8102	0.7914
Ranking Results	$\begin{matrix} T_{5} > T_{4} > T_{2} \\ > T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{4} > T_{2} > \\ T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{1} > T_{3} \end{matrix}$	$\begin{matrix} T_{5} > T_{2} > T_{4} > \\ T_{3} > T_{1} \end{matrix}$

TE results from non-weighting strategy and DEGR (interval SD) method are found to be identical. Specifically, SD of the weights obtained from DEGR (interval SD) is 0.3000, suggesting little difference among weights.

When comparing DEGR (interval SD) and interval SD methods, the difference between them is the ranks of T₂ and T₄. It has been demonstrated in 6.4.1 that these ranks are mainly affected by dynamic threat attributes A₃, A₄ and A₅. Specifically, in DEGR (interval SD), the weights for dynamic threat attributes are expressed as real numbers. As the degraded result of the interval SD method, DEGR (interval SD) fails to rank T₂ and T₄ correctly. Therefore, the loss of uncertain information caused by degradation makes the evaluation on the importance of dynamic threat attributes inaccurate, and the interval number weights from the interval SD method are more favored to cope with uncertain information.

Among the interval SD, DEGR (interval SD-G1) and the interval SD-G1 strategies, the attribute ranks in the former two methods are identical. Analysis in previous sections has shown that the ranks of T₁ and T₃ are mainly affected by the static threat attributes, and interval SD-G1 method can better evaluate the importance of these attributes compared to real number weights from interval SD.

In conclusion, compared with real number weights, interval number weights can express and help cope with uncertainties, and evaluation results on attribute importance are more accurate with them.

6.5 Comparative analysis of MADM methods

In order to verify the advantages of the proposed method, some existing MADM methods are selected to assess the five targets.

SAW [36] and Original TOPSIS are non-interval MADM methods. Specifically, the threat values in SAW are calculated as ${TV}_{i} = \sum_{j - 1}^{n} w_{j} {\hat{X}}_{ij}$ , and in Original TOPSIS, they become TV_i = C_i, where all variables are real numbers. When only real numbers are involved, the evaluation results of SD-G1 weighting are consistent with those from the SD method only, and the proof for this is given in Appendix A. Consequently, the SD method is used to calculate weights for comparison with SAW and TOPSIS.

Besides, Sahin [13] dealed with uncertain information based on IVIFS (Interval-Valued Intuitionistic Fuzzy Set). He combines the characteristics of IVIFS and TOPSIS to solve the MADM problem. And Beskese [15] developed a MADM tool integrating HFS with TOPSIS. Table 6 presents the TE results of the five MADM methods based on air target parameters in Tables 2 and 3.

Table 6
Threat value (TV) compared with non-interval MADM methods

Method Evaluation Parameters TV₁ TV₂ TV₃ TV₄ TV₅ Ranking Results

SAW parameters from O_a 0.411 0.530 0.1934 0.8200 0.6200 $\begin{matrix} T_{4} ≻ T_{5} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$

parameters from O_b 0.3900 0.5210 0.1900 0.8159 0.6050 $\begin{matrix} T_{4} ≻ T_{5} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$

Original TOPSIS parameters from O_a 0.2980 0.6523 0.1788 0.7131 0.8820 $\begin{matrix} T_{5} ≻ T_{4} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$

parameters from O_b 0.3857 0.5100 0.2103 0.6944 0.7697 $\begin{matrix} T_{5} ≻ T_{4} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$

IVIFS-TOPSIS O_a &O_b 0.3978 0.7211 0.2657 0.5207 0.8772 $\begin{matrix} T_{5} ≻ T_{2} ≻ T_{4} ≻ \\ T_{3} ≻ T_{1} \end{matrix}$

HFS-TOPSIS O_a &O_b 0.1883 0.5563 0.1908 0.6983 0.8859 $\begin{matrix} T_{5} ≻ T_{4} ≻ T_{2} ≻ \\ T_{3} ≻ T_{1} \end{matrix}$

Proposed Method O_a &O_b 0.1884 0.5819 0.1886 0.5330 0.7914 $\begin{matrix} T_{5} ≻ T_{2} ≻ T_{4} ≻ \\ T_{3} ≻ T_{1} \end{matrix}$

Method	Evaluation Parameters	TV₁	TV₂	TV₃	TV₄	TV₅	Ranking Results
SAW	parameters from O_a	0.411	0.530	0.1934	0.8200	0.6200	$\begin{matrix} T_{4} ≻ T_{5} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$
	parameters from O_b	0.3900	0.5210	0.1900	0.8159	0.6050	$\begin{matrix} T_{4} ≻ T_{5} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$
Original TOPSIS	parameters from O_a	0.2980	0.6523	0.1788	0.7131	0.8820	$\begin{matrix} T_{5} ≻ T_{4} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$
	parameters from O_b	0.3857	0.5100	0.2103	0.6944	0.7697	$\begin{matrix} T_{5} ≻ T_{4} ≻ T_{2} \\ ≻ T_{1} ≻ T_{3} \end{matrix}$
IVIFS-TOPSIS	O_a &O_b	0.3978	0.7211	0.2657	0.5207	0.8772	$\begin{matrix} T_{5} ≻ T_{2} ≻ T_{4} ≻ \\ T_{3} ≻ T_{1} \end{matrix}$
HFS-TOPSIS	O_a &O_b	0.1883	0.5563	0.1908	0.6983	0.8859	$\begin{matrix} T_{5} ≻ T_{4} ≻ T_{2} ≻ \\ T_{3} ≻ T_{1} \end{matrix}$
Proposed Method	O_a &O_b	0.1884	0.5819	0.1886	0.5330	0.7914	$\begin{matrix} T_{5} ≻ T_{2} ≻ T_{4} ≻ \\ T_{3} ≻ T_{1} \end{matrix}$

In the comparison of SAW and Original TOPSIS, Original TOPSIS can rank the top threat target T₅, whose effect is better than SAW. The reason is that the threat value of TOPSIS is relative closeness, which comprehensively considers the threats of different attributes, while the threat value of SAW is the sum of weighted threat attribute values. But the threat variables of Original TOPSIS are real numbers, which cannot express uncertain information correctly. IVIFS-TOPSIS provides a very good TE result. Although IVIFS-TOPSIS and HFS-TOPSIS can describe the uncertain information very well, it is greatly influenced by subjective factors of experts. Our proposed method is more reliable and effective because it combines the interval value theory and the synthetic weight method.

7 Conclusions

TE problems with uncertain information are addressed in this paper, an interval MADM model based on interval TOPSIS and interval comprehensive weighting method is established to deal with information uncertainties in TE.

Interval number theory can help deal with uncertain information obtained from multiple observation platforms. In addition, threats from different attributes can be comprehensively considered via TOPSIS. The proposed TE method based on interval TOPSIS integrates the advantages of both, which can produce more reasonable ranking results. Specifically, the interval SD-G1 method is proposed to calculate the comprehensive weights of threat attributes, which integrates the objective weights from the interval SD method and the subjective weights from the interval G1 method. The feasibility and advantages of the proposed method are demonstrated in an evaluation example: (1) Compared with non-comprehensive weighting strategies, the interval SD-G1 method can produce more promising evaluation results on the importance of the threat attributes, which is helpful for accurate TE. (2) Compared with real number weights and other MADM methods, interval number weights could better represent and assist in the operations on uncertain information, producing more accurate evaluations on the importance of threat attributes. Finally, based on the assessment results, it can be used as a reference for operational decision-making or firepower allocation accordingly.

Compared with the weighting methods of certain MADM, there is less research on weighting methods of interval MADM. Referring to the idea of this paper, some weighting methods can be extended with interval numbers in the future. Specifically, the mean and SD of interval arrays defined in this paper are of reference value for other fields, such as interval analysis, MADM problems and decision-making. In addition, interval number theory is used to represent and cope with information uncertainties, which is mainly feasible in the situations where detailed information about enemy weapons is insufficient. However, when information is adequate, it is more suitable to establish an accurate random model or fuzzy model for TE. Therefore, the evaluation outcomes from different models under varied degrees of information adequacy about the TE remain a good research point in the future, so as to optimize the evaluation results.

8 Appendix A

When only real numbers are involved, based on the idea of the interval SD-G1 method, the weights of the SD method are taken as the reference value for the G1 method to determine the order relationships and relative importance. Weight calculations are performed as follows.

Suppose an objective weight vector W^SD of A = {A₁, A₂, ⋯ , A_n} is calculated by the SD method.

The weights of W^SD are sorted by their sizes. Suppose the order relationship is $A_{1}^{s} > A_{2}^{s} > \dots > A_{n}^{s}$ , and the objective weight of attribute $A_{j}^{s}$ is $w_{j}^{s}$ .

Calculate the subjective weight by the G1 method.

The relative importance r_k is calculated by

$r_{k} = \frac{w_{k - 1}^{s}}{w_{k}^{s}} k = n, n - 1, \dots, 2$ (32)

Then calculate the weight of $A_{n}^{s}$ . $\begin{matrix} w_{n}^{s} & = {(1 + \sum_{k = 2}^{n} \prod_{q = k}^{n} r_{q})}^{- 1} \\ = {[1 + \prod_{q = 2}^{n} r_{q} + \prod_{q = 3}^{n} r_{q} + \dots + \prod_{q = n}^{n} r_{q}]}^{- 1} \end{matrix}$

$\begin{matrix} = {[1 + (r_{2} r_{3} \dots r_{n}) + (r_{3} r_{4} \dots r_{n}) + \dots + r_{n}]}^{- 1} \\ = [1 + (\frac{w_{1}^{s}}{w_{2}^{s}} \frac{w_{2}^{s}}{w_{3}^{s}} \dots \frac{w_{n - 1}^{s}}{w_{n}^{s}}) \\ {+ (\frac{w_{2}^{s}}{w_{3}^{s}} \frac{w_{3}^{s}}{w_{4}^{s}} \dots \frac{w_{n - 1}^{s}}{w_{n}^{s}}) + \dots + \frac{w_{n - 1}^{s}}{w_{n}^{s}}]}^{- 1} \\ = {(1 + \frac{w_{1}^{s}}{w_{n}^{s}} + \frac{w_{2}^{s}}{w_{n}^{s}} + \dots + \frac{w_{n - 1}^{s}}{w_{n}^{s}})}^{- 1} \\ = {(1 + \frac{1}{w_{n}^{s}} \sum_{k = 1}^{n - 1} w_{k}^{s})}^{- 1} \\ = {[1 + \frac{1}{w_{n}^{s}} (1 - w_{n}^{s})]}^{- 1} = {(\frac{1}{w_{n}^{s}})}^{- 1} = w_{n}^{s} \end{matrix}$ (33)

Finally calculate the weight of each attribute.

$w_{k - 1}^{s} = r_{k} w_{k}^{s} = \frac{w_{k - 1}^{s}}{w_{k}^{s}} w_{k}^{s} = w_{k - 1}^{s} k = n, n - 1, \dots, 2$ (34)

Therefore, the weights calculated by the SD-G1 method are consistent with those obtained from the SD method when no interval numbers are involved.

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Interval TOPSIS with a novel interval number comprehensive weight for threat evaluation on uncertain information

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Interval Numbers and its operations [5, 27]

5.1 Determination of subjective weights

5.2.1 The SD method

6.1 Evaluation index system

6.4.1 Comparison with non-comprehensive weights

8 Appendix A

References

2.1 Interval Numbers and its operations [5 , 27]