Threat evaluation is extremely important to decision makers in many situations, such as military application and physical protection systems. In this paper, a new threat assessment model based on interval number to deal with the intrinsic uncertainty and imprecision in combat environment is proposed. Both objective and subjective factors are taken into consideration in the proposed model. For the objective factors, the genetic algorithm (GA) is used to search out an optimal interval number representing all the attribute values of each object. In addition, for the subjective factors, the interval Analytic Hierarchy Process (AHP) is adopted to determine each object’s threat weight according to the experience of commanders/experts. Then a discounting method is proposed to integrate the objective and subjective factors. At last, the ideal of Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is applied to obtain the threat ranking of all the objects. A real application is used to illustrate the effectiveness of the proposed model.
In the real world, it is always complex and difficult to make a decision and to deal with threat assessment in most situations, because the problems we are faced with are always intrinsic imprecision and uncertainty which is typically represented by the threat assessment for multiobject and multiattribute with uncertain information. Hence, it is meaningful to develop a reasonable model to solve this problem. The threat assessment of multiobject and multiattribute has been researched by many researchers. Generally speaking, threat assessment can be seen as a decisionmaking problem under uncertain environment. We first introduce some related works about decision making and threat assessment.
Decision making has been the popular topic of research in many fields since the concept was proposed. Saaty [
A threat is defined as any act, entity, event, or phenomenon with the potential to harm a person or thing. In other words, a threat is a source of potential harm. Sometimes the word hazard or risk is used as a synonym for threat. The term “threat assessment” can be broadly interpreted as evaluation of impending danger or harm by a person, group, circumstance, or set of conditions. It is a pattern of activities involving detection and analysis of threat stimuli and the situations in which the threat is encountered. Threat or risk assessment serving as the foundation of regulatory decision making whether to take actions to reduce the loss has been researched by many researchers. Pedrycz et al. [
Evaluation for aerial objects is an important part in the aerial defensive operation, which can be described as the assessment of multiobject and multiattribute under uncertain environment. Due to the limitation of detection means or secrecy, the effective information we can obtain from the threat objects is little. Generally, the interval number can be used to represent the uncertain or imprecise information. In this paper, a new model is proposed to rank the threat objects based on interval numbers, which takes the subjective and objective factors into consideration. Two examples are used to illustrate the effectiveness of the proposed threat evaluation model.
This paper is organized as follows. Section
As mentioned above, there are many math tools to handle uncertain information, such as fuzzy sets theory [
The interval approach is originally developed by Moore [
Interval number is defined as an ordered pair
Let
The problem of a multiobjective threat assessment can be described with the mathematical model. The object set can be represented with
The attribute of each object is composed of two types, namely, “cost type” and “benefit type.” Let
Let
It can be proved that
Although only the lower and upper bound values of the two interval numbers appear in (
The simple genetic algorithm (Algorithm
For
Assume that a decision maker provides interval judgements instead of precise judgements for a pairwise comparison matrix. For example, it could be judged that
Let
Assume that a MCDM problem has
(1) Normalize the decision matrix
(2) Calculate the weighted normalized decision matrix
(3) Determine the ideal and negativeideal solutions:
(4) Calculate the Euclidean distances of each alternative from the ideal solution and the negativeideal solution, respectively:
(5) Calculate the relative closeness of each alternative to the idea solution. The relative closeness of the alternative
(6) Rank the alternatives according to the relative closeness to the ideal solution. The bigger the
The mathematic model of threat assessment is detailed in this section. As shown in Figure
The threat evaluation model.
Due to the imprecision of the sensors in the complex situation, sometimes, we should make use of interval numbers to represent the attributive values of the threat targets. In order to measure the different threat degree, the problem arises regarding how best we can aggregate this interval number into a general interval number. The main idea of our proposed method can be shown as follows: this paper integrates the fuzzy numbers through the distance among interval numbers to find a special interval number
Now the procedure of aggregating interval numbers (objective factors) is made by GA in detail as follows.
After conducting several Roulette Wheel tests, assume that chromosome
The crossover operator in this paper adopts the strategy of a single cutting crossover. This method considers the two flanks of the cutting into two substrings; then the right substrings should be exchanged with each other to get two new individuals. If the crossover probability
Mutation operator is to change some gene of chromosome with a tiny possibility. If the mutation probability
Three interval numbers are as follows:
Assume that the importance of each interval number is the same. First, we suppose that the integrated interval number of the three fuzzy numbers is represented with
Hence, the integration of three interval numbers is mapped into solving the minimum of the formula
Assume that the accuracy of encoding
For the subjective factors, the interval AHP (interval eigenvector) is applied to deal with the object’s threat weight according to experience of the commanders/experts. A numerical example is used to describe the process as follows.
Suppose that we get a pairwise comparison matrix through interval judgements which is denoted as
Then the interval comparison matrix
Then the principal eigenvalue of
In this part, the method of discounting will be used to integrate the objective and subjective factors. Through the two procedures above, for each threat object, the integrative interval number describing the objective factor can be integrated by the convergent process of genetic algorithm. Hence, assume that there are
Sometimes, the commanders need to evaluate the threat degree of each target according to their previous experience in order to control the situation. Hence, through constructing the interval decision matrix and calculating the interval eigenvector, the subjective interval numbers representing the evaluation of experts can be described as
Then the fusion weight of all the threat targets
In this part, the method of TOPSIS will be made use of to deal with the ranking of the threat objects according to their different threat degree. The classical TOPSIS method is a technique for order preference by similarity to ideal solution. The ideal solution (also called positive ideal solution) is a solution that maximizes the benefit criteria/attributes and minimizes the cost criteria/attributes, whereas the negative ideal solution (also called antiideal solution) maximizes the cost criteria/attributes and minimizes the benefit criteria/attributes. The socalled benefit criteria/attributes are those for maximization, while the cost criteria/attributes are those for minimization. The best alternative is the one which is closest to the ideal solution and farthest from the negative ideal solution.
Because the normalized interval numbers all distribute in
Then the distance between the ideal solution
Assume that an interval number
In this part, a numerical example [
Interval decision matrix.

 


[0.75, 1.24] 


[1.83, 2.11] 


[4.90, 5.73] 

First, the interval decision matrix can be normalized through (
Normalized decision matrix.

 


[0.55, 1.40]  [0.41, 0.50] 

[0.33, 0.57]  [0.54, 0.60] 

[0.12, 0.21]  [0.64, 0.73] 
Second, through the proposed objective aggregating method with GA, the aggregated results can be converged as in Table
Aggregated decision matrix.
 


[0.47, 0.96] 

[0.43, 0.59] 

[0.31, 0.52] 
At last, the priority of the three alternatives can be determined by TOPSIS, and the result can be shown as in Table
Priority of decision matrix.



RC  


[0.47, 0.96]  0.53  0.10  0.84 

[0.43, 0.59]  0.26  0.24  0.52 

[0.31, 0.52]  0.18  0.35  0.34 
This numerical example illustrates the effectiveness of the proposed model without considering the subjective factors. The aggregation of the objective factors is always optimum with GA. In the following part, another numerical example is presented to illustrate the effectiveness of the proposed threat model with considering both objective factors and subjective factors.
In this part, a real application in aerial object threat will be introduced to describe the operational procedure of threat assessment. The threat objects and their attributes can be shown as in Table
The attributive value of threat objects.
TT  ST  TA  SA  

Object 1 ( 
E  1500 

9 
Object 2 ( 
C 



Object 3 ( 
C 



Object 4 ( 
D  1600 

8 
Object 5 ( 
E 



In Table
Different numerical value of TT.
Target style  Threat degree  Details 

A  0.8  Refuelling plane and so forth 
B  0.6  Transport plane and so forth 
C  0.9  Strike aircraft and so forth 
D  0.8  Helicopter 
E  1.0  Missile and so forth 
F  0.2  Fictional goal and so forth 
In order to eliminate the influence resulting from different measures, the normalization can be made through (
Normalized attribute value of threat objects.
TT  ST  TA  SA  


[0.48, 0.48]  [0.62, 0.64]  [0.71, 0.60]  [0.52, 0.50] 

[0.44, 0.44]  [0.16, 0.19]  [0.37, 0.18]  [0.47, 0.41] 

[0.44, 0.44]  [0.17, 0.22]  [0.37, 0.16]  [0.43, 0.38] 

[0.48, 0.48]  [0.67, 0.68]  [0.62, 0.54]  [0.58, 0.56] 

[0.39, 0.39]  [0.25, 0.30]  [0.37, 0.23]  [0.26, 0.23] 
Through the objective fusion method by GA, the integrated results can be converged as in Table
Integrated result of attributes of each corresponding object.
Fusion data  minSD  


[0.58, 0.60]  0.0114 

[0.28, 0.31]  0.0364 

[0.25, 0.38]  0.0311 

[0.57, 0.60]  0.0101 

[0.30, 0.31]  0.0318 
In the following part, we will take the commanders’ experience into consideration. The interval decision by experts for the five threat objects is described as
As for the subjective factors, the method of interval eigenvector is used to generate the interval threat weights of all targets. The fusion process of subjective factors is like Example
Hence, at present, the integration between objective factors and subjective factors can be made by the method of discounting. It can be denoted as in Table
Fusion between objective and subjective weight (O(
O( 
S( 
I( 



[0.58, 0.60]  [0.41, 0.46]  [0.51, 0.54] 

[0.28, 0.31]  [0.15, 0.19]  [0.23, 0.26] 

[0.25, 0.38]  [0.19, 0.23]  [0.23, 0.32] 

[0.57, 0.60]  [0.11, 0.13]  [0.39, 0.41] 

[0.30, 0.31]  [0.06, 0.07]  [0.20, 0.21] 
According to the idea of TOPSIS, we can get the ideal distance
Relative closeness (RC) of all targets:
I( 


RC  


[0.51, 0.54]  0.3194  0.1894  0.3722 

[0.23, 0.26]  0.5701  0.0601  0.0954 

[0.23, 0.32]  0.5293  0.0753  0.1245 

[0.39, 0.41]  0.3613  0.1593  0.3060 

[0.20, 0.21]  0.6257  0.0437  0.0653 
The threat degree of all threat objects.
As presented above, the proposed threat model can not only be applied to the objective issue, but can also be easily extended to considering the subjective issue from knowledge of the domain experts. When the objective discounting factor
Threat assessment is very important in many fields. This paper proposes a new model of threat assessment based on interval number. Both subjective and objective factors are taken into consideration in this threat assessment model. GA is used to determine the objective factors. In addition, interval AHP is applied to determine subjective factors. Then a method of discounting is proposed to integrate the subjective weight and objective weight. At last, the idea of TOPSIS is adopted to rank the objects according to their threat degree. A numerical example and a real application are used to illustrate the effectiveness of the proposed threat model.
The author declares that there is no conflict of interests regarding the publication of this paper.
The author greatly appreciates the reviewers’ valuable suggestions. This work is partially supported by National High Technology Research and Development Program of China (863 Program) (Grant no. 2013AA013801), National Natural Science Foundation of China (Grant no. 61174022), Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20131102130002), R&D Program of China (2012BAH07B01), and the Open Funding Project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant no. BUAAVR14KF02).