Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 6652706 |

Ruojing Zhao, Fengbao Yang, Linna Ji, Yongqiang Bai, "Dynamic Air Target Threat Assessment Based on Interval-Valued Intuitionistic Fuzzy Sets, Game Theory, and Evidential Reasoning Methodology", Mathematical Problems in Engineering, vol. 2021, Article ID 6652706, 13 pages, 2021.

Dynamic Air Target Threat Assessment Based on Interval-Valued Intuitionistic Fuzzy Sets, Game Theory, and Evidential Reasoning Methodology

Academic Editor: Emilio Insfran Pelozo
Received15 Oct 2020
Revised08 May 2021
Accepted21 May 2021
Published09 Jun 2021


In order to reduce the uncertainty of target threat assessment results and improve exact target assessment in the complicated and changeable air combat environment, a novel method based on the combination of interval-valued intuitionistic fuzzy sets (IVIFSs), game theory, and evidential reasoning methodology is proposed in this paper. First, the imprecise and fuzzy information of battlefield air target is expressed by IVIFS. Second, the optimal index weight is determined by the interval intuitionistic fuzzy entropy and game theory. And the time series weight is calculated by the inverse Poisson distribution method. Then, the target evaluation information at different times is dynamically fused through an evidential reasoning algorithm. Finally, the accuracy function is applied to obtain the threat ranking of all the targets. A case of the threat assessment of air targets is provided to demonstrate the implementation process of the method proposed in this paper. Simulation experiments show that in a rapidly evolving combat environment, this algorithm can effectively reduce the uncertainty of target threat assessment results. It provides us with a useful way for target threat assessment based on interval-valued intuitionistic fuzzy sets, game theory, and evidential reasoning methodology.

1. Introduction

With the development of weaponry information technology, the battlefield environment becomes more complicated and diversified. How to effectively use battlefield target information to assist commanders in making scientific decisions has always been the focus of battlefield research [1, 2]. Since air combat plays a vital role in the trend of the modern battlefield, the efficient threat assessment of incoming targets can be used as a reference for reasonably allocating defensive firepower [3]. Hence, it is very necessary to carry out research on air threat assessment methods since it can provide an effective auxiliary decision-making method for the field of air defense operations.

Target threat assessment aims to analyze the operational capabilities of incoming enemy targets and obtain a quantitative description of the degree of the enemy target threat [4]. To solve the target threat assessment problem, air target threat assessment methods have been proposed. These methods mainly include Bayesian networks [57], neural networks [8, 9], rough sets [10], the cloud model [11], and so on [12, 13]. These methods are suitable for different battlefield environments and have different characteristics. However, these methods express the target information by crisp numerical numbers and do not fully consider the fuzziness and uncertainty of battlefield target information.

In recent years, the intuitionistic fuzzy set (IFS) [14] has been defined as a union of the membership and nonmembership functions and the degree of hesitation. IFSs can describe the fuzziness and uncertainty of target information and have been widely used in target threat assessment [1518]. In [15], IFSs are used to construct an intuitionistic fuzzy target information matrix and conduct target threat assessment. Due to the complex and changeable modern battlefield environment, the threat membership of battlefield targets cannot be given an exact value, resulting in a certain degree of limitations in the application of IFSs in the field of target threat assessment. Atanassov and Gargov [19] further extended the IFS and proposed the interval-valued intuitionistic fuzzy set (IVIFS), using interval numbers instead of exact values to represent membership and nonmembership. IVIFS can better describe the uncertainty and fuzziness of the decision-making process, which has attracted the attention of a large number of scholars. The research of interval intuitionistic fuzzy sets mainly focuses on the entropy [20, 21], intuitionistic fuzzy divergence [22, 23], knowledge measure [24], ranking method [25], and similarity [26] of interval-valued intuitionistic fuzzy sets. These theoretical research studies promote the application of IVIFS in decision-making fields, such as alternative fuel technology options [27], hydrogen production evaluation process [28], supplier selection problem [29], car-sharing problems [30], stock investment selection [31], and hotel location [32]. Among the exiting studies, the assessment of air targets based on IVIFS was seldom reported.

How to fuse IVIFS information effectively is the key to solving the decision-making problem with IVIFS. In [33], induced intuitionistic fuzzy ordered weighted geometry is developed to solve venture capital project selection problems. In [34], interval-valued intuitionistic fuzzy ordered weighted aggregation is proposed to solve the supplier optimization problem. Although the aggregator in the above method solves the information fusion problem in related fields, the shortcoming is that the uncertainty of the evaluation result in the information fusion process is increased by ignoring hesitation information. Fortunately, the evidential reasoning methodology (ERM) is an effective means of fusing information [3543]. In [37], the evidential reasoning algorithm is adopted to aggregate the weighted evidentiary sources and is suitable for dealing with conflicts in information fusion. In [38], ERM is adopted to aggregate the decision matrix, which improves the rationality of the decision process. In [42], the ERM is used to fuse intuitionistic fuzzy sets, instead of the aggregator. The ERM has a strong ability to aggregate intuitionistic fuzzy sets. As seen from the above study, the ERM has a good data fusion capability which can avoid information loss and retain hesitation intuitionistic fuzzy information.

Meanwhile, existing target threat assessment methods [2, 4] and [1012] only use the target measurement information at the current time for the evaluation, which means that it is difficult to comprehensively analyze the target threat level for a period of times after discarding the observation data before the current time. Without fully considering the dynamic changes of the target attributes, the results of these methods are one-sided and unreasonable to a certain extent.

To sum up, IVIFS is suitable for dealing with ambiguity and inaccurate battlefield target information, but it only uses the current target measurement information to evaluate. In order to meet the needs of the dynamic battlefield, a novel target threat assessment method based on IVIFS, game theory, and the ERM is proposed. The proposed method can effectively reduce the uncertainty of the target threat assessment and adapt to the requirements of a dynamic battlefield. Moreover, the optimal index weights are calculated through the combination interval intuitionistic fuzzy entropy and game theory method, which is used to eliminate the data sensitivity of the index weight and makes the assessment result more reliable.

The rest of this paper is organized as follows. Section 2 presents the basic theory of the IVIFS and ERM. Section 3 introduces the calculation method of the optimal attribute weight and the time series weight. A novel air threat assessment method based on the IVIFS, game theory, and the ERM is presented in Section 4. Section 5 presents an example analysis and comparison discussion. The conclusions are drawn in Section 6.

2. Preliminaries

2.1. Interval-Valued Intuitionistic Fuzzy Sets

Definition 1. In [19], the interval-valued intuitionistic fuzzy set (IVIFS) is proposed by Atanassov and Gargov. The domain X is a finite nonempty set and IVIFS A can be described as follows:where the intervals and , respectively, denote the degrees of membership and nonmembership of element of IVIFS . and , where , , , and . In addition, indicates the degree of uncertainty (degree of hesitation) for element that belongs to , where and . When and , the IVIFS degenerates into an intuitionistic fuzzy set (IFS).

Definition 2. In [43], Ye presents the accuracy function to transform an interval-valued intuitionistic fuzzy value into a real value , and it is given as follows:

2.2. Evidential Reasoning Methodology (ERM)

In the following, a brief review is given of the ERM method for multiattribute decision analysis under uncertain environments [38]. Let be a set of alternatives, be a set of attributes, and be a set of weights. where ,,. Assume that there are p evaluation grades for assessing the attributes of alternatives. assess the attributes of alternatives. Let be a degree of belief that attribute of alternative is assessed theevaluation grade , where and . Let denote the evaluation value of attribute with respect to alternative , and it is defined as follows:where is an evaluation grade, , and . The assessments of the attributes of the alternatives are represented by a decision matrix D, which is shown as follows:where and . Based on the decision matrix D, we can aggregate the evaluation values of the attributes of each alternative , where , and the details are as follows.

First, the degree of belief of the evaluation grade of an attribute for the alternative is transformed into the basic probability mass where

, , and . Let be the remaining probability mass of attribute for an alternative , which is shown as follows:where is the first part of the remaining probability mass, which is caused by the weight, and is the second part of the remaining probability mass that has not yet been assigned individual grades, which is due to the incompleteness in the assessment .

Next, the basic probability mass matrix is obtained:where , , and .

Compute the combined probability mass of each attribute of alternative as follows:

Let and , where

Based on (9) and (10), after aggregating the evaluation values of the attributes of alternative , we get the degree of belief of alternative regarding evaluation grade , which is shown as follows: is the degree of belief produced by an unknown information, where , , and

3. Weight Determination

3.1. Determination of Assessment Index Weight

When using IVIFSs to assess target threats, the assessment index weight is included in the calculation of the IVIFS data fusion process; thus, the weight value of each index affects the final evaluation result. The impact of each attribute index on the final degree of the threat is different; therefore, attribute indicators should be given different weights. In this paper, the interval-valued intuitionistic fuzzy entropy and game theory are used to determine the weights of the attributes.

3.1.1. Determination of Assessment Index Weights
Step 1. Determine the IVIFS matrix of the assessment index at time , where and , and is simplified as .Step 2. Wei et al. [44] proposed an effective method based on the intuitionistic fuzzy entropy, which is defined as follows:Step 3. The weight of attribute at time is calculated as
3.1.2. Determine the Optimal Attribute Weight Using Game Theory

The index weight determination method based on the interval-valued intuitionistic fuzzy entropy is sensitive to the data. Game theory is used to eliminate the sensitivity of the attribute weights at different times to obtain the optimal attribute weight.Step 1. Construct possible attribute weight sets.An attribute weight set is obtained using the interval-valued intuitionistic fuzzy entropy at different times , and then a set of basic attribute weight vectors is constructed. Let any linear combination of attribute weight vectors bewhere is the linkage combination coefficient and is a possible attribute weight vector based on the basic attribute weights. The entire collection represents a set of possible attribute weight vectors.Step 2. Find the optimal attribute weight vector .

 The optimal attribute weight vector is calculated using the game strategy model. In the selection process, the linkage coefficient should be optimized based on the objective of minimizing the deviation of and , and the optimization model is derived as follows:

By solving the model, the optimal attribute weight results among the attribute weights can be obtained. According to the differential properties of the matrix, the optimal first derivative condition of equation (15) is as follows:

The matrix form of the linear equation system corresponding to equation (16) is as follows:

According to equation (17), is obtained, and the normalization is as follows:

Finally, the optimal attribute weight is obtained as follows:

3.2. Determination of Time Series Weights

The impacts of different time points in air combat target data on threat assessment are different. To make a comprehensive and objective evaluation of the dynamic air combat situation, calculating the weights of different time series is a prerequisite. The air combat process is further along at the current time, and thus the measured target threat assessment information is less crucial. Thus, the inverse Poisson distribution method is adopted to calculate the time series weight vector . of times can be calculated as follows:where , , and .

4. Proposed Dynamic Target Threat Assessment Method

4.1. Threat Assessment Index System

Considering the combat capabilities and attack intentions of incoming targets, the air target assessment system is shown in Figure 1.

4.1.1. Target Combat Capabilities
(1)Target Type. The degrees of the threats of different types of targets are completely different, and the degrees of the threats in descending order are as follows: missile, fighter, bomber, Armed helicopter, and early warning aircraft.(2)Maneuvering Ability. The stronger the targe’s maneuverability, the lower the hit rate of air defense weapons against it and the greater the degree of the threat.
4.1.2. Target Attack Intentions
(1)Target Velocity. The faster the target’s flight speed, the shorter the reaction time for the commander, the greater the lethality of this target to the target we want to protect, and the higher the degree of the threat.(2)Target Height. The higher the altitude of the air target is, the more time we can take to form corresponding measures against the target, and thus the lower its threat. The lower the target height is, the higher the attack rate, and the higher the corresponding threat.(3)Route Shortcut. The air route shortcut of the target refers to the vertical distance from the point protected by our side to the projected extension line of the instantaneous velocity of the air raid target on the horizontal plane. The smaller the air route shortcut, the higher the degree of the threat of the target.(4)Target Distance. Target distance refers to the distance between the target and the boundary of the air defense area, reflecting the enemy’s attack attempt and the possibility of achieving an attack. The smaller the target distance, the higher the degree of the target threat and the greater the corresponding degree of membership.(5)Target Arrival Time. The target arrival time refers to the time when the target reaches the near boundary of the air defense fire launch zone. The smaller the value, the shorter the preparation time of the weapon system and tactical decision and the greater the threat to us.
4.2. Dynamic Threat Assessment Method Based on IVIFS, Game Theory, and ERM

To provide an effective auxiliary decision-making method, which can help the commander to make the correct decision, a novel dynamic threat assessment method based on the IVIFS, game theory, and the ERM is proposed. The process for the proposed method is shown in Figure 2.Step 1. The determination of the initial decision matrix air targets set includes m targets to be evaluated, and the attribute set consists of attributes. Denote the decision matrix at time as follows:Step 2. Calculate the index weight at different times and obtain the optimal index weight as follows:Step 3. Aggregate the initial decision matrix based on the ERM and the optimal index weights to the dynamic decision matrix.Step 3.1. LetIn equation (23), denotes the degrees of belief of the decision matrix at time with respect to attribute of target regarding the evaluation grades and , respectively. represents completely satisfying the fuzzy concept “threaten,” represents not satisfying the fuzzy concept “threaten,” and represents the evaluation grade of indeterminacy (uncertainty).Step 3.2. Transform the degree of belief at time into the basic probability mass , which is shown as follows:Step 3.3. Compute the combined probability massNote thatThe combined probability mass can be generated by combining the basic probability masses.whereStep 3.4. Aggregate the evaluated values of the attributes of target to get the degree of belief of target regarding the evaluation grade and the degree of belief produced by unknown information, respectively, as follows:where   Step 3.5. The aggregated value of the evaluated values of the attributes of alternative is represented by an IVIFS . The dynamic decision matrix is obtained as equation (24).Step 4. Calculate the time series weight .Step 5. Aggregate the dynamic decision matrix based on the ERM and the time series weights to the final decision matrix .Step 5.1. Transform the degree of belief into the basic probability mass , as follows:Step 5.2. Compute the combined probability mass.Note that andThe combined probability mass can be generated by combining the basic probability masses as described as follows:whereStep 5.3. Aggregate the evaluated values of the attributes of target to get the degree of belief of target regarding the evaluation grade and the degree of belief produced by unknown information, respectively, as follows:whereStep 5.4. The aggregated value of the evaluated values of the attributes of alternative is represented by an IVIFS . The final decision matrix is obtained as follows:Step 6. Calculate the accuracy function value of each target according to equation (2). The higher the score of the accuracy function, the greater the target threat. Thus, the evaluation ranking result is acquired.

5. Numerical Examples and Discussion

5.1. Numerical Examples
Step 1. The initial decision matrix is determined as follows.Our side finds five air targets through detection equipment. The air target set corresponds to {bomber, fighter, missile, early warning aircraft, Armed helicopter}. The attribute set consists of seven attributes, which correspond to {target type, target arrival time, target height, target distance, route shortcut, target velocity, maneuvering ability}. The IVIFS matrix of the air target at three time points , , and is as shown in Tables 13, respectively.Step 2. According to Table 1 to Table 3, the attribute weights and the optimal index weight at each time are as Table 4.Step 3. Based on the IVIFS assessment information and the optimal index weight, the dynamic decision matrix is obtained as Table 5.Step 4. The time series weight is obtained when as follows:Step 5. Based on dynamic decision matrix and the time series weight, the final decision matrix is obtained as follows:Step 6. The degrees of the target threats are sorted by the accuracy function.

According to , the ranking result of the target threat assessment is obtained: . According to the initial decision matrix data shown in Tables 13, the target threat assessment results are consistent with reality.


<[0.75, 0.85]<[0.48, 0.58]<[0.70, 0.75]<[0.50, 0.55]<[0.40, 0.45]<[0.55, 0.60]<[0.55, 0.65]
[0.05, 0.10]>[0.10, 0.20]>[0.05, 0.12]>[0.05, 0.15]>[0.30, 0.40]>[0.20, 0.30]>[0.05, 0.15]>

<[0.65, 0.75],<[0.45, 0.55]<[0.65, 0.72]<[0.45, 0.55]<[0.65, 0.70]<[0.55, 0.60]<[0.55, 0.60]
[0.15, 0.25]>[0.20, 0.30]>[0.12, 0.21]>[0.15, 0.25]>[0.15, 0.25]>[0.22, 0.37]>[0.10, 0.20]>

<[0.85, 0.95]<[0.30, 0.40]<[0.55, 0.65]<[0.5, 0.6]<[0.70, 0.75]<[0.80, 0.85]<[0.70, 0.80]
[0.01, 0.05]>[0.40, 0.50]>[0.20, 0.30]>[0.04, 0.13]>[0.05, 0.15]>[0.01, 0.12]>[0.02, 0.12]>

<[0.25, 0.35]<[0.35, 0.45]<[0.45, 0.55]<[0.60, 0.70]<[0.45, 0.55]<[0.35, 0.45]<[0.25, 0.35]
[0.30, 0.40]>[0.10, 0.30]>[0.35, 0.40]>[0.05, 0.15]>[0.1, 0.20]>[0.10, 0.20]>[0.30, 0.40]>

<[0.45, 0.55]<[0.75, 0.8]<[0.80, 0.85]<[0.80, 0.85]<[0.30, 0.35]<[0.35, 0.45]<[0.65, 0.70]
[0.30, 0.40]>[0.08, 0.15]>[0.01, 0.06]>[0.00, 0.10]>[0.20, 0.3]>[0.35, 0.50]>[0.08, 0.20]>


<[0.75, 0.85]<[0.58, 0.68]<[0.75, 0.80]<[0.55, 0.60]<[0.45, 0.50]<[0.60, 0.65]<[0.60, 0.70]
[0.05, 0.10]>[0.15, 0.20]>[0.05, 0.15]>[0.25, 0.35]>[0.35, 0.45]>[0.20, 0.30]>[0.20, 0.30]>

<[0.65, 0.75]<[0.45, 0.55]<[0.60, 0.65]<[0.40, 0.50]<[0.60, 0.70]<[0.60, 0.65]<[0.60, 0.65]
[0.15, 0.25]>[0.20, 0.30]>[0.08, 0.12]>[0.10, 0.19]>0.10, 0.15]>[0.22, 0.30]>[0.10, 0.20]>

<[0.85, 0.95]<[0.35, 0.45]<[0.60, 0.70]<[0.55, 0.60]<[0.80, 0.85]<[0.85, 0.90]<[0.75, 0.85]
[0.01, 0.05]>[0.40, 0.50]>[0.10, 0.20]>[0.10, 0.20]>[0.10, 0.14]>[0.01, 0.04]>[0.08, 0.13]>

<[0.25, 0.35]<[0.45, 0.55]<[0.45, 0.55]<[0.65, 0.70]<[0.55, 0.60]<[0.35, 0.45]<[0.25, 0.35]
[0.30, 0.40]>[0.15, 0.20]>[0.25, 0.35]>[0.10, 0.20]>[0.15, 0.25]>[0.20, 0.40]>[0.30, 0.50]>

<[0.45, 0.55]<[0.80, 0.85]<[0.80, 0.85]<[0.85, 0.90]<[0.35, 0.40]<[0.45, 0.55]<[0.65, 0.75]
[0.30, 0.40]>[0.10, 0.15]>[0.00, 0.04]>[0.00, 0.03]>[0.10, 0.20]>[0.15, 0.20]>[0.05, 0.15]>


<[0.75, 0.85]<[0.60, 0.70]<[0.80, 0.85]<[0.60, 0.65]<[0.55, 0.60]<[0.60, 0.68]<[0.65, 0.70]
[0.05, 0.10]>[0.10, 0.15]>[0.05, 0.15]>[0.10, 0.20]>[0.20, 0.35]>[0.20, 0.30]>[0.10, 0.20]>

<[0.65, 0.75]<[0.40, 0.45]<[0.55, 0.60]<[0.35, 0.45]<[0.50, 0.60]<[0.65, 0.70]<[0.65, 0.70]
[0.15, 0.25]>[0.20, 0.28]>[0.10, 0.15]>[0.10, 0.20]>[0.10, 0.14]>[0.10, 0.20]>[0.08, 0.13]>

<[0.85, 0.95]<[0.45, 0.55]<[0.65, 0.75]<[0.75, 0.85]<[0.85, 0.95]<[0.90, 0.95]<[0.85, 0.90]
[0.01, 0.05]>[0.30, 0.40]>[0.15, 0.23]>[0.04, 0.14]>[0.00, 0.02]>[0.01, 0.12]>[0.02, 0.07]>

<[0.25, 0.35]<[0.55, 0.65]<[0.50, 0.55]<[0.70, 0.75]<[0.60, 0.65]<[0.40, 0.47]<[0.25, 0.35]
[0.30, 0.40]>[0.01, 0.03]>[0.25, 0.33]>[0.10, 0.20]>[0.15, 0.20]>[0.10, 0.20]>[0.25, 0.35]>

<[0.45, 0.55]<[0.85, 0.90]<[0.85, 0.90]<[0.85, 0.90]<[0.40, 0.45]<[0.55, 0.60]<[0.75, 0.85]
[0.30, 0.40]>[0.02, 0.08]>[0.01, 0.04]>[0.01, 0.04]>[0.35, 0.40]>[0.30, 0.40]>[0.02, 0.08]>




<[0.5901, 0.6739]<[0.6479, 0.7300]<[0.6812, 0.7643]
[0.0861, 0.1607]>[0.1438, 0.2191]>[0.0855, 0.1623]>

<[0.5889, 0.6750]<[0.5778, 0.6601]<[0.5548, 0.6366]
[0.1294, 0.2229]>[0.1108, 0.1798]>[0.0981, 0.1624]>

<[0.6512, 0.7552]<[0.7032, 0.7970]<[0.7793, 0.8816]
[0.0785, 0.1526]>[0.0829, 0.1358]>[0.0528, 0.0955]>

<[0.4011, 0.5069]<[0.4363, 0.5251]<[0.4800, 0.5596]
[0.1826, 0.2803]>[0.1956, 0.3145]>[0.1559, 0.2428]>

<[0.6265, 0.7021]<[0.6549, 0.7393]<[0.7120, 0.7888]
[0.1117, 0.1973]>[0.0746, 0.1279]>[0.1046, 0.1527]>

5.2. Discussion and Comparative Analysis
5.2.1. Discussion of Experimental Results

In the dynamic target threat assessment process, the time factor has a great influence on the target threat assessment results. Table 6 shows the sorting results at each different moment.

TimeRanking result