Radiator threat evaluation with missing data based on improved technique for order preference by similarity to ideal solution

Abstract

To solve the problem of the missing data of radiator during the aerial war, and to address the problem that traditional algorithms rely on prior knowledge and specialized systems too much, an algorithm for radiator threat evaluation with missing data based on improved Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) has been proposed. The null estimation algorithm based on Induced Ordered Weighted Averaging (IOWA) is adopted to calculate the aggregate value for predicting missing data. The attribute reduction is realized by using the Rough Sets (RS) theory, and the attribute weights are reasonably allocated with the theory of Shapley. Threat degrees can be achieved through quantization and ranking of radiators by constructing a TOPSIS decision space. Experiment results show that this algorithm can solve the incompleteness of radiator threat evaluation, and the ranking result is in line with the actual situation. Moreover, the proposed algorithm is highly automated and does not rely on prior knowledge and expert systems.

Keywords

IOWA Shapley attribute reduction TOPSIS incompleteness

1 Introduction

Radiator threat evaluation is an important research topic within the field of Electronic Countermeasures (ECM) [1]. The accuracy of radiator threat evaluation indirectly affects the outcome of the war. As the types and operation modes of radiator become ever more complex and the anti-jamming capability of targets develops, radiator threat evaluation faces great challenges.

At present, a wealth of research accomplishments have been made in the field of target threat evaluation. Bayesian Network Image was introduced to intuitively describe the reasoning process of target threat evaluation, and fuzzy relation in threat evaluation is clearly expressed through a directed acyclic graph [2, 3]. Evidence theory was used to deal with uncertain problems, and combat intentions and equipment capability of air combat targets were deeply considered to realize threat evaluation [4]. Information entropy, radar perception network, and cognitive mapping were combined with fuzzy sets theory to solve the uncertain decision problems in threat evaluation [5 –7]. XU et al. regarded target threat evaluation as a multi-attribute decision problem, and constructed an air combat target threat evaluation model based on the improved Intuitionistic Fuzzy Sets (IFS).

The above researches rely on prior knowledge and expert system support, which may not have applied to all cases. To address this challenge, Zhang et al. used intuitionistic fuzzy entropy to determine attribute weights and evaluated the target threat through a multi-criteria decision model based on VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) algorithm [10]. Zhang et al. introduced the TOPSIS algorithm, which has been widely used in various fields to calculate the nearness degrees of each target and obtained the threat degrees of targets by comparing them [11 –19]. To improve the reliability of an evaluation model, a radiator threat evaluation algorithm based on Shapley-TOPSIS was proposed, which could assign attribute weights objectively [20].

The processed data of the above research were complete. However, within the actual operational process, the radiator information system for detection and collection was usually incomplete. For the problem of target threat evaluation with missing data, ZHANG et al. introduced an improved tolerance relation rough set algorithm to build a complete set of decision rules extracting model [21]. However, the algorithm could only divide the radiator into several categories according to the threat degrees and was unable to distinguish the threat degree in detail for the targets in the categories. In addition, SUN et al. proposed a threat evaluation method based on dynamic Bayesian Networks of constraint parameter learning, and the AR(p) model was used to predict the missing data on the time series [22]. This method needed the support from target data in multiple historical moments, difficult to obtain the data in actual combat. Therefore, the method has some limitations. To achieve the detailed ranking of the radiator threat degree in the case of missing data and expand application scope, effectively overcoming the problem of incomplete data, we combine the IOWA operator [23 –28] with Shapley-TOPSIS algorithm. Then, a radiator threat evaluation model with missing data based on improved TOPSIS is constructed. The proposed algorithm can deal with incomplete information systems and has the value of the military promotion.

2 Null estimation algorithm based on IOWA

The null estimation algorithm based on IOWA considers both the correlation between objects and attributes and can adjust the weight vector according to different backgrounds, which can effectively make up the missing data of the object.

Definition 1. There is an information system (X, S, g), where, X = {x₁, x₂, …, x_m} is the object set and S = {S₁, S₂, …, S_n} is the attribute set. The g is the set of relationships between X and S, and the attribute S_j (j = 1, 2, …, n) value of object x_i (i = 1, 2, …, m) is a_ij.

Definition 2. The correlation function of attribute s_j and s_k is sim (s_j, s_k), which denotes the degree of proximity between attributes. The greater the correlation function value is, the closer the two attributes are. The expression of the correlation function is as follows [25]. $sim (s_{j}, s_{k}) = \frac{\sum_{x_{i} \in {NA}_{ik}} [(a_{ij} - \bar{V_{jk}^{j}}) (a_{ik} - \bar{V_{jk}^{k}})]}{\sqrt{\sum_{x_{i} \in {NA}_{ik}} {(a_{ij} - \bar{V_{jk}^{j}})}^{2} \sum_{x_{i} \in {NA}_{ik}} {(a_{ik} - \bar{V_{jk}^{k}})}^{2}}}$ (1) where NA_jk represents a collection of objects with attribute s_j and s_k not missing in the object set X. $\bar{V_{jk}^{j}}$ represents the mean of all values under attribute s_j in NA_jk.

Definition 3. If the correlation attribute of s_j is s_k, 〈a_ij, a_ik〉 (x_i ∈ NA_jk) is called an OWA pair. Supposing there are p elements in NA_jk, the elements are arranged into a vector $(a_{k}^{1}, a_{k}^{2}, \dots, a_{k}^{p})$ in order of attribute values from large to small, and the vector $(a_{j}^{1}, a_{j}^{2}, \dots, a_{j}^{p})$ of the attribute s_j can be formed according to OWA pairs then [24].

Definition 4. If the attribute value of s_j for radiator x_i is missing, it should be replaced by an aggregate value F_IOWA, the expression of which is as follows [24]. $F_{IOWA} = {(ω_{1}, ω_{2}, \dots ω_{p})}^{T} (a_{j}^{1}, a_{j}^{2}, \dots, a_{j}^{p})$ (2)

(ω₁, ω₂, ⋯ ω_p) is the weight vector of OWA pairs, which is determined by the following function: $ω_{j} = Q (\frac{j}{p}) - Q (\frac{j - 1}{p}), j = 1, 2, \dots, p$ (3)

Among them, $Q (x) = {\begin{matrix} 0, x < a \\ \frac{x - a}{b - a}, a ⩽ x ⩽ b \\ 1, x > b \end{matrix}$ .

The processing principle of null estimation algorithm based on IOWA:

For the missing attribute, only one null value can be predicted for each iteration.

If the value of the correlation attribute of the missing attribute changes during the iteration, the missing data of the attribute can be predicted through the next iteration.

Fig. 1

Processing flow of the null estimation algorithm based on IOWA.

The processing flow of the null estimation algorithm based on IOWA is shown in Fig. 1.

The specific steps are as follows:

For the first iteration, the correlation attributes of the attributes with missing data are determined. Firstly, the attributes with missing data and incomplete information systems are selected. The information contained in complete data-sets is fully utilized to calculate the correlation function values of each attribute and the remaining attributes, respectively. Then, the values of the correlation functions are compared, and the attribute corresponding to the maximum value is its correlation attribute.

OWA pairs are calculated. After obtaining all correlation attributes through Step 1, for each attribute with missing data, the corresponding objects when both attribute values are complete are listed in combination with its correlation attribute, and then the attribute values of correlation attributes are listed in accordance with the principle from large to small in order to form OWA pairs.

The data-set is completed. The parameter values of the weight calculation formula are determined based on practical situations, and the weights of OWA pairs are reasonably allocated. Then, the OWA pairs and weights are combined to calculate the aggregate value, which is used for replacing the null value. Finally, the first iteration is completed.

If there are attributes with missing data after the first iteration, the next iteration according to Step 1-Step 3, is undertaken until the data of the system is complete.

3 Radiator threat evaluation based on TOPSIS

The basic principle of the TOPSIS algorithm is to construct an evaluation space through positive samples and negative samples based on the standardized sample evaluation matrix. The object to be evaluated can be regarded as a point in the evaluation space and the evaluation result is provided through calculating the relative “distance” (Euclidean distance) between objects and positive and negative ideal solution.

At present, radiator threat evaluation is mainly carried out by analyzing radiator pulse parameters and platform situation. The specific steps of the evaluation algorithm based on TOPSIS are as follows.

(1) We normalize radiator data. Assuming that our reconnaissance receiving system obtains n attributes data of m radiators, the initial evaluation matrix A is as follows. $A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{mn} \end{matrix}]$ (4) where a_ij (1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n) represents the attribute value of j for radiator i. According to the normalization treatment method in Ref. [14], the cost-type index and the benefit-type index are respectively normalized. For the benefit-type index, it can be processed as Equation (5), and for the cost-type index, it can be processed as Equation (6). Then, the updated normalized evaluation matrix B is as follows. $b_{ij} = \frac{a_{ij} - min_{i} a_{ij}}{max_{i} a_{ij} - min_{i} a_{ij}}$ (5) $b_{ij} = \frac{max_{i} a_{ij} - a_{ij}}{max_{i} a_{ij} - min_{i} a_{ij}}$ (6) $B = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{22} & \dots & b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{m 1} & b_{m 2} & \dots & b_{mn} \end{matrix}]$ (7) where b_ij (1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n) represents the attribute value of j for radiator i after normalization.

(2) We construct the decision space and generate the ideal solution for the model. According to the matrix B, the positive ideal solution C⁺ corresponding to the maximum threat degree of all attributes and the negative ideal solution C^- corresponding to the minimum threat degree of all attributes are generated. The calculation formulas are as follows. $C^{+} = {max_{1 ⩽ i ⩽ m} b_{ij}} = {b_{1}^{+}, b_{2}^{+}, \dots, b_{n}^{+}}$ (8) $C^{-} = {min_{1 ⩽ i ⩽ m} b_{ij}} = {b_{1}^{-}, b_{2}^{-}, \dots, b_{n}^{-}}$ (9)

(3) We calculate the threat degrees of radiators. Based on the positive and negative ideal solutions of the model, the relative “distance” between each radiator and the positive ideal solution and the negative ideal solution can be calculated as Equation (10) and Equation (11) respectively. And then, radiator threat degree ranking is obtained based on the nearness degrees which are calculated according to Equation (12). $D_{i}^{+} = \sqrt{\sum_{j = 1}^{n} {(b_{ij} - b_{j}^{+})}^{2}}$ (10) $D_{i}^{-} = \sqrt{\sum_{j = 1}^{n} {(b_{ij} - b_{j}^{-})}^{2}}$ (11) $D = \frac{D_{i}^{-}}{D_{i}^{+} + D_{i}^{-}}$ (12)

4 Attribute weight allocation based on Shapley value dominance relation rough set

Many algorithms have emerged during the radiator threat evaluation process, including expert assignment, Analytic Hierarchy Process (AHP), IFS, etc. These algorithms have defects, which may include excessive subjectivity, having an overly dependence on prior knowledge, or being highly complex, so this work introduces Shapley value dominance relation rough set to optimize attribute weight allocation.

Definition 5. For the attribute subset V ⊆ S, if g_{s
_k} (x_i) ⩽ g_{s
_k} (x_j) when ∀_{S
_k} ∈ V, so that x_j is superior to x_i for V. And the superior set of x_i is defined as follows [24]. $H_{V}^{⩽} (x_{i}) = {x_{j} | g_{s_{k}} (x_{i}) ⩽ g_{s_{k}} (x_{j}), \forall s_{k} \in V}$ (13)

Definition 6. The amount of information of the attribute subset V is as follows. $H (V) = \sum_{i = 1}^{| X |} \frac{1}{| X |} (1 - \frac{| H_{V}^{⩽} (x_{i}) |}{| X |})$ (14) where $| H_{V}^{⩽} (x_{i}) |$ is the number of elements in $H_{V}^{⩽} (x_{i})$ . The amount of information not only has the ability to represent the uncertainty of the advantage set, but also can be used as a measure of the importance of attributes. Then, the loss of information represents the amount of information lost. For instance, the loss of information of S is as follows. $c (S) = \sum_{s_{k} \in D} H ({s_{k}}) - H (S)$ (15)

Definition 7. In the information system (X, S, g), the identity matrix is defined as follows. $Z = {H^{<} (x_{i}, x_{j}) | H^{<} (x_{i}, x_{j}) \neq ø, x_{i}, x_{j} \in X}$ (16) Where, $H^{<} (x_{i}, x_{j}) = {s_{k} | g_{k} (x_{i}) < g_{k} (x_{j}), s_{k} \in S}$ (17)

Definition 8. The attribute subset V is defined as the reduction set of S when V∩ F ≠ ø, where, F is an element in the identity matrix Z, which is composed of several attributes from S.

Definition 9. For attribute subset S = {s₁, s₂, ⋯ , s_n}, the Shapley value of the attribute s_k is as follows [23].

$\begin{matrix} φ_{s_{k}} (c) = \sum_{S^{'} \in S_{k}} (\frac{(n - | S^{'} |)! (| S^{'} | - 1)!}{n!} \\ [c (S^{'}) - c (S^{'} ∖ s_{k})]) \end{matrix}$ (18)

Where, S_k is a collection consisting of all subsets which include s_k, and S′ is one of the subsets which include s_k. S′ ∖ s_k represents that s_k is removed from S′. And the Shapley value of attribute s_k represents the contribution of s_k to the information loss of attribute subset.

Through the above analysis, we combine the amount of information and Shapley value to quantify the importance of attribute to decision making, and the weights of attributes are obtained using the following formula. $ω_{k} = \frac{H (s_{k}) - φ_{s_{k}} (c)}{\sum_{j = 1}^{n} (H (s_{k}) - φ_{s_{k}} (c))}$ (19)

Fig. 2

Processing flow of the proposed model.

5 Model of radiator threat evaluation with missing data based on improved TOPSIS

The processing flow of the algorithm of radiator threat evaluation with missing data based on improved TOPSIS is shown in Fig. 2.

The specific steps of the model are as follows:

We complete the radiator data through the null estimation algorithm based on IOWA. Given the attribute with missing data in the radiator signal pulse, the correlation function is calculated to determine the correlation attribute, and then the OWA pairs composed of non-null values of the two attributes are listed. Combined with the weight vector, the aggregate value is obtained to complete the missing data.

We construct an evaluation matrix. After that the incompleteness of the information system is solved through Step1, the initial evaluation matrix is constructed according to the radiator attribute parameters, and the difference of dimensions of different attributes is eliminated by distinguishing the benefit-type index and the cost-type index, so as to obtain the standardized evaluation matrix.

We assign attribute weights based on the Shapley value dominance relation algorithm. According to the evaluation matrix constructed by Step 2, the dominant relationship between radiator attributes is analyzed by rough set theory, with the attribute set reduced. Then, based on the principle of the attribute’s contribution to the total information loss, the Shapley value of each attribute is calculated, and the attribute weight vector of the radiator is obtained after normalization processing.

We calculate the threat degrees of radiators. Combining the evaluation matrix of Step 2 and the attribute weight vector of Step3, the TOPSIS model decision space is constructed to generate positive and negative solutions, and the relative “distance” between each evaluation object and the ideal solution is calculated to obtain the relative ranking of threat degree.

6 Experiment

To verify the validity of the algorithm and model across this work, radiator data in Ref. [29] are introduced, as shown in Table 1. s₁, s₂, s₃, s₄, s₅, s₆, s₇ respectively represent platform height (/km), platform approach speed (/Ma), platform distance (/km), angle of attack (/°), signal radio frequency (/GHz), signal pulse repetition frequency (/KHz), and signal pulse width (/μs).

Table 1
Data of radiators

Object s ₁ s ₂ s ₃ s ₄ s ₅ s ₆ s ₇

x ₁ 6.8 1.8 72 14 32 102 0.3

x ₂ 7.8 0.9 190 17 14 32 4.6

x ₃ 6.2 1.3 120 13 21 47 1.2

x ₄ 5.4 1.4 113 12 19 51 2.6

x ₅ 8.4 0.4 213 20 4.3 5.6 5.9

x ₆ 4.7 1.6 104 8 16 19 1.7

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	6.8	1.8	72	14	32	102	0.3
x ₂	7.8	0.9	190	17	14	32	4.6
x ₃	6.2	1.3	120	13	21	47	1.2
x ₄	5.4	1.4	113	12	19	51	2.6
x ₅	8.4	0.4	213	20	4.3	5.6	5.9
x ₆	4.7	1.6	104	8	16	19	1.7

To improve the validity and reliability of the experimental results and verify the universality of the algorithm in this paper, three groups of experiments are set up in this section, in which partial radiator’s information is randomly removed in Table 1 respectively.

(1) Experiment 1

Partial information in Table 1 is randomly removed to obtain incomplete radiator data as shown in Table 2.

Table 2

Incomplete data of radiators (experiment 1)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	6.8	Null	72	14	32	102	0.3
x ₂	7.8	0.9	190	17	14	Null	4.6
x ₃	Null	1.3	120	13	21	47	1.2
x ₄	5.4	1.4	113	Null	19	51	2.6
x ₅	8.4	0.4	213	20	4.3	5.6	5.9
x ₆	4.7	1.6	104	8	16	19	Null

The specific processing steps of the experiment are as follows.

(1) We predict the missing data in Table 2 through the null estimation algorithm based on IOWA. The correlation function between attributes can be calculated as Equation (1), and then, the correlation attributes of s₁, s₂, s₄, s₆, s₇ are obtained. The correlation attributes of s₁, s₂, s₄, s₆, s₇ are s₄, s₄, s₁, s₅, s₃ respectively.

The first iteration. For attribute s₁, OWA pairs are obtained according to Definition 3 as follows, which is the basis for generating the aggregate values. ${\begin{matrix} 〈 6.8, 14 〉 \\ 〈 7.8, 17 〉 \\ 〈 8.4, 20 〉 \\ 〈 4.7, 8 〉 \end{matrix}$ (20)

Then, according to Equation (3), the weights of OWA pairs are determined as follows. ${\begin{matrix} ω_{1} = 0 \\ ω_{2} = 0 \\ ω_{3} = 0.5 \\ ω_{4} = 0.5 \end{matrix} ({\begin{matrix} a = 0.5 \\ b = 1.0 \end{matrix})$ (21)

Finally, the aggregate value is obtained according to Equation (2) and it is used as the predicted value of the missing data of attribute s₁.

Similarly, the predicted values of attributes s₂, s₆, s₇ are 1.5, 20, and 1.4, respectively. Because the correlation attribute s₁ of the attribute s₄ has changed, the prediction of the missing value of s₄ goes through a second iteration.

The second iteration. The aggregate value of the attribute s₄ is 11.2 through the same steps above.

After the above processing, we get the complete radiator data in Table 3.

Table 3

Complete data of radiators after been processed by the proposed algorithm (experiment 1)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	6.8	1.5	72	14	32	102	0.3
x ₂	7.8	0.9	190	17	14	20	4.6
x ₃	5.8	1.3	120	13	21	47	1.2
x ₄	5.4	1.4	113	11.2	19	51	2.6
x ₅	8.4	0.4	213	20	4.3	5.6	5.9
x ₆	4.7	1.6	104	8	16	19	1.4

The evaluation matrix is constructed based on Table 3. $A = [\begin{matrix} 6.8 & 1.5 & 72 & 14 & 32 & 102 & 0.3 \\ 7.8 & 0.9 & 190 & 17 & 14 & 20 & 4.6 \\ 5.8 & 1.3 & 120 & 13 & 21 & 47 & 1.2 \\ 5.4 & 1.4 & 113 & 11.2 & 19 & 51 & 2.6 \\ 8.4 & 0.4 & 213 & 20 & 4.3 & 5.6 & 5.9 \\ 4.7 & 1.6 & 104 & 8 & 16 & 19 & 1.4 \end{matrix}]$ (22)

To eliminate the difference of attribute dimension and unit, based on the analysis of the attributes above, the benefit-type indexes and the cost-type indexes are processed according to Equation (5) and Equation (6) respectively. According to Ref. [29], platform height, platform distance, angle of attack and signal pulse width are cost-type indexes, namely, the greater the values of these indexes, the lower the threat degree of the radiator. Moreover, platform approach speed, signal radio frequency and signal pulse repetition frequency are benefit-type indexes, namely, the greater the values of these indexes, the higher the threat degree of the radiator. Then, we obtain the standardized evaluation matrix as follows. $B = [\begin{matrix} 0.4324 & 0.9167 & \dots & \dots & \dots & 1.0000 & 1.0000 \\ 0.1622 & 0.4167 & \dots & \dots & \dots & 0.1494 & 0.2321 \\ 0.7027 & 0.7500 & \dots & \dots & \dots & 0.4295 & 0.8393 \\ 0.8108 & 0.8333 & \dots & \dots & \dots & 0.4710 & 0.5893 \\ 0.0000 & 0.0000 & \dots & \dots & \dots & 0.0000 & 0.0000 \\ 1.0000 & 1.0000 & \dots & \dots & \dots & 0.1390 & 0.8036 \end{matrix}]$ (23)

(2) We assign the attribute weights based on Shapley value dominance relation rough set.

The missing data in the incomplete information system are completed and the standardized evaluation matrix is constructed above. We mainly assign attribute weights in this step.

At first, based on the evaluation matrix B, the identity matrix is generated as follows (in Table 4) according to definition 7.

Table 4

Identity matrix (experiment 1)

	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
x ₁	ø	ø	{s₁, s₄}	{s₁, s₄}	ø	{s₁, s₂, s₄}
x ₂	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$	ø	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$	ø	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{7}} \end{matrix}$
x ₃	{s₂, s₃, s₅, s₆, s₇}	ø	ø	{s₁, s₂, s₃, s₄, s₆}	ø	{s₁, s₂, s₃, s₄}
x ₄	{s₂, s₃, s₅, s₆, s₇}	ø	{s₅, s₇}	ø	ø	{s₁, s₂, s₃, s₄, s₇}
x ₅	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}} \end{matrix}$	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$	ø	$\begin{matrix} {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, \\ s_{6}, s_{7}} \end{matrix}$
x ₆	{s₃, s₅, s₆, s₇}	{s₆}	{s₅, s₆, s₇}	{s₅, s₆}	ø	ø

All non-empty elements contain, {s₆}, {s₁, s₄}, {s₅, s₆}, {s₅, s₇}, {s₁, s₂, s₄}, {s₅, s₆, s₇}, {s₁, s₂, s₃, s₄}, {s₃, s₅, s₆, s₇}, {s₁, s₂, s₃, s₄, s₆}, {s₁, s₂, s₃, s₄, s₇}, {s₂, s₃, s₅, s₆, s₇}, {s₁, s₂, s₃, s₄, s₅, s₆}, {s₁, s₂, s₃, s₄, s₅, s₇}, {s₁, s₂, s₃, s₄, s₅, s₆, s₇}. According to definition 8, the reduction set is V = {s₄, s₆, s₇}, and the evaluation matrix after reduction is as follows: $B = [\begin{matrix} 0.5000 & 1.0000 & 1.0000 \\ 0.2500 & 0.1494 & 0.2321 \\ 0.5833 & 0.4295 & 0.8393 \\ 0.7333 & 0.4710 & 0.5893 \\ 0.0000 & 0.0000 & 0.0000 \\ 1.0000 & 0.1390 & 0.8036 \end{matrix}]$ (24)

Then, the amount of information of s₄, s₆, s₇ can be calculated as Equation (14), and the Shapley values of s₄, s₆, s₇ can be calculated as Equation (18). ${\begin{matrix} δ (s_{4}) = \frac{41}{216} \\ δ (s_{6}) = \frac{47}{216} \\ δ (s_{7}) = \frac{50}{216} \end{matrix}$ (25) ${\begin{matrix} H (s_{4}) = \frac{15}{36} \\ H (s_{6}) = \frac{15}{36} \\ H (s_{7}) = \frac{15}{36} \end{matrix}$ (26)

Finally, according to Equation (19), the attribute weights are calculated as follows. ${\begin{matrix} ω_{s_{4}} = 0.3712 \\ ω_{s_{6}} = 0.3258 \\ ω_{s_{7}} = 0.3030 \end{matrix}$ (27)

(3) We evaluate the threat degrees of radiators based on TOPSIS algorithm. This part is the last step of the proposed algorithm, which is also the core step of radiator threat evaluation. First of all, combining the reduced evaluation matrix with attributes weights, the new evaluation matrix is obtained as follows. $P = [\begin{matrix} 0.1289 & 0.2867 & 0.3266 \\ 0.0000 & 0.1004 & 0.2175 \\ 0.1718 & 0.1729 & 0.1403 \\ 0.2149 & 0.1522 & 0.1538 \\ 0.0645 & 0.0000 & 0.0000 \\ 0.3867 & 0.1211 & 0.0454 \end{matrix}]$ (28)

Following this, the decision environment is built. According to Equation (8) and Equation (9), positive and negative ideal solutions are generated. $C^{+} = {0.3867, 0.2867, 0.3266}$ (29) $C^{-} = {0, 0, 0}$ (30)

After that, the relative “distance” between the radiator and positive and negative ideal solutions can be calculated as Equation (10) and Equation (11) respectively. $D_{1}^{+} = 0.2578, D_{1}^{-} = 0.4533$ (31) $D_{2}^{+} = 0.4429, D_{2}^{-} = 0.2396$ (32) $D_{3}^{+} = 0.3063, D_{3}^{-} = 0.2812$ (33) $D_{4}^{+} = 0.2783, D_{4}^{-} = 0.3050$ (34) $D_{5}^{+} = 0.5410, D_{5}^{-} = 0.0645$ (35) $D_{6}^{+} = 0.3263, D_{6}^{-} = 0.4078$ (36)

Finally, we obtain the nearness degrees of radiators as follows, according to Equation (12). ${\begin{matrix} D_{1} = 0.7220 \\ D_{2} = 0.2166 \\ D_{3} = 0.5948 \\ D_{4} = 0.6055 \\ D_{5} = 0.0000 \\ D_{6} = 0.6088 \end{matrix}$ (37)

According to the nearness degrees, the threat degree ranking result of the five radiators is x₁ > x₆ > x₄ > x₃ > x₂ > x₅, which is consistent with the result obtained in Ref. [29] based on complete data.

(2) Experiment 2

To further verify the effectiveness and reliability of the proposed algorithm, the incomplete radiator data in Table 5 are processed based on the proposed algorithm.

Table 5

Incomplete data of radiators (experiment 2)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	Null	1.8	72	14	32	102	0.3
x ₂	7.8	Null	190	17	14	32	4.6
x ₃	6.2	1.3	120	Null	21	Null	1.2
x ₄	5.4	1.4	113	12	Null	51	2.6
x ₅	8.4	0.4	213	20	4.3	5.6	5.9
x ₆	4.7	1.6	104	8	16	19	1.7

According to the step1-step3 in experiment 1, the ranking result of the radiator threat degree is x₁ > x₆ > x₄ > x₃ > x₂ > x₅, which is consistent with the result obtained in Ref. [29] based on complete data.

(3) Experiment 3

In experiment 3, the incomplete radiator data in Table 6 are processed based on the proposed algorithm similarly.

Table 6

Incomplete data of radiators (experiment 3)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	6.8	1.8	72	Null	32	102	0.3
x ₂	7.8	0.9	Null	17	14	32	4.6
x ₃	6.2	1.3	120	13	21	47	Null
x ₄	Null	1.4	113	12	19	Null	2.6
x ₅	8.4	Null	213	20	4.3	5.6	5.9
x ₆	4.7	1.6	104	8	16	19	1.7

According to the processing steps in experiment 1, the ranking result of the radiator threat degree is x₁ > x₆ > x₄ > x₃ > x₂ > x₅, which is consistent with the result obtained in Ref. [29] based on complete data.

(4) Experiment 4

In experiment 4, the incomplete radiator data in Table 7 are processed based on the proposed algorithm similarly.

Table 7

Incomplete data of radiators (experiment 4)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	Null	1.8	72	14	32	102	0.3
x ₂	Null	0.9	190	17	14	32	Null
x ₃	6.2	Null	120	13	21	47	1.2
x ₄	5.4	1.4	113	12	Null	51	2.6
x ₅	8.4	0.4	213	Null	4.3	5.6	Null
x ₆	4.7	1.6	Null	8	16	19	1.7

According to the processing steps in experiment 1, the ranking result of the radiator threat degree is x₁ > x₆ > x₃ > x₄ > x₂ > x₅, which is inconsistent with the result obtained in Ref. [29] based on the complete data. In the light of the ranking result, the threat degree of radiator x₃ is higher than that of radiator x₄, which is reverse with the actual situation.

(5) Experiment 5

In experiment 5, the incomplete radiator data in Table 8 are processed based on the proposed algorithm similarly.

Table 8

Incomplete data of radiators (experiment 5)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	6.8	1.8	72	14	32	102	Null
x ₂	Null	0.9	190	17	14	Null	4.6
x ₃	6.2	Null	120	13	Null	47	1.2
x ₄	Null	1.4	Null	12	19	51	2.6
x ₅	8.4	0.4	213	Null	4.3	5.6	Null
x ₆	4.7	1.6	104	8	16	Null	1.7

According to the processing steps in experiment 1, the ranking result of the radiator threat degree is x₆ > x₃ > x₁ > x₄ > x₂ > x₅, which is inconsistent with the result obtained in Ref. [29] based on complete data. In the light of the ranking result, the threat degrees of radiator x₃ and x₆ are higher than that of radiator x₁. But actually, the threat degree of radiator x₁ is the highest one.

(6) Experiment 6

In experiment 6, the incomplete radiator data in Table 9 are processed based on the proposed algorithm similarly.

Table 9

Incomplete data of radiators (experiment 6)

Object	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆	s ₇
x ₁	6.8	1.8	Null	14	Null	102	Null
x ₂	7.8	Null	190	17	14	32	4.6
x ₃	6.2	1.3	Null	13	21	47	1.2
x ₄	5.4	1.4	113	12	Null	51	2.6
x ₅	Null	0.4	Null	Null	4.3	5.6	5.9
x ₆	4.7	Null	104	Null	16	Null	1.7

According to the processing steps in experiment 1, the ranking result of the radiator threat degree is x₄ > x₁ > x₆ > x₅ > x₃ > x₂, which is inconsistent with the result obtained in Ref. [29] based on the complete data. In the light of the ranking result, the threat degree of radiator x₄ is higher than that of radiator x₁, and the threat degree of radiator x₅ is higher than that of radiator x₃ and x₂. But actually, the threat degrees of radiator x₁ and x₅ are the highest one and the lowest one respectively.

7 Conclusion and future work

It can be demonstrated that the proposed algorithm eliminates incompleteness and achieves threat degree evaluation. In this work, the algorithm effectively realizes the threat evaluation of radiators with missing data, and the ranking results of threat degree in the first three groups (experiment 1-experiment 3) are consistent with the actual situation. However, the ranking results of threat degree in the last three groups (experiment 4-experiment 6) are inconsistent with the actual situation. As can be seen from the experiments, the proposed algorithm works effectively when the number of missing values is 5 or 6, and the performance of it degrades when the number of missing values is 8, 10 or 12. The reason for the above phenomenon is that the amount of information for the rest radiator data is scant and then the missing data cannot be predicted accurately when the number of missing values is 8, 10 or 12, but it can be predicted with some accuracy when the number of missing values is 5 or 6. In other words, when the number of missing values is too large, the reliability and availability of the algorithm will descend gradually with the increasing of the number of missing values.

In considering the actual demands of radiator threat evaluation under incomplete conditions, an evaluation algorithm based on improved TOPSIS is proposed. Firstly, we introduce the null estimation algorithm based on IOWA to comprehensively analyze the relationship between correlation attributes and to obtain the predicted values of the missing data. Then, the attribute set is reduced, and the weights are given based on the algorithm of Shapley Value Dominance Relation Rough Set, which can objectively measure the influence degrees of each attribute to the radiator threat evaluation process. Finally, the ranking result is obtained through the TOPSIS algorithm. Compared with the ICW-RCM evaluation algorithm in Ref. [29], the algorithm in this work is highly automated and does not rely on prior knowledge and expert systems. Furthermore, the model in this work makes up the limitations of radiator threat evaluation in the absence of data and can be applied to military practices.

In future work, weapon configuration information of the radiator platform should be considered in the research of radiator threat evaluation to improve the rationality of the model, and how to select a reasonable evaluation index is significant with the developing trend of multi-aircraft cooperative operation. In addition, due to changes in the combat situation and the tasks of the combatants involved, the threat degrees of the radiators are constantly changing, so it is necessary to consider the threat evaluation under dynamic variable parameters. Furthermore, the applicability of threat evaluation model needs to be improved to tackle the huge amount of missing values.

Footnotes

Acknowledgment

Here, Yuheng Xu expressed his heartfelt thanks to Siyi Cheng and all friends who contributed to this work.

References

Jiang

, Hu

W.-L.

and Sun

, Fuzzy multi-attribute method of emitter threating grade evaluation, Acta Armamentarii 25(1) (2004), 56–59.

Kumar

and Tripathi

B.K.

, Modelling of threat evaluation for dynamic targets using bayesian network approach, Procedia Technology 24 (2016), 1268–1275.

Meng

G.-L.

and Gong

G.-H.

, Threat assessment of aerial targets based on hybrid bayesian network, Systems Engineering and Electronics 32(11) (2010), 1102–1109.

Benavoli

, Ristic

, Farina

, Oxenham

and Chisci

, Anapproach to threat assessment based on evidential networks, 2007.

Feng

L.-Y.

, Xue

and Liu

M.-X.

, Threat evaluation model of targets based on information entropy and fuzzy optimization theory, IEEE International Conference on Idustrial Engineering and Engineering Management 2011; 1789–1793.

Liang

Q.-L.

and Cheng

X.-Z.

, KUPS: Knowledge-based ubiquitous and persistent sensor networks for threat assessment, IEEE Transactions on Aerospace and Electronic Systems 44(3) (2008), 1060–1069.

Chen

, Yu

G.-H.

and Gao

X.-G.

, Cooperative threat assessment of multi-aircrafts based on synthetic fuzzy cognitive map, Journal of Shanghai Jiaotong University 17(2) (2012), 228–232.

Y.-J.

, Wang

Y.-C.

and Miu

X.-D.

, Multi-attribute decision making method for air target threat evaluation based on intuitionistic fuzzy sets, Journal of Systems Engineering and Electronics 23(6) (2012), 891–897.

Wang

, Liu

S.-Y.

, Niu

and Liu

, Threat assessment method based on intuitionistic fuzzy similarity measurement reasoning with orientation, Communications China 11(6) (2014), 119–128.

10.

Zhang

, Kong

W.-R.

, Liu

P.-P.

, Shi

, Lei

and Zou

, Assessment and sequencing of air target threat based on intuitionistic fuzzy entropy and dynamic VIKOR, Journal of Systems Engineering and Electronics 29(2) (2018), 305–310.

11.

Shanian

and Savadogo

, TOPSIS multiple-criteria decision support analysis for material selection of metallic bipolar plates for polymer electrolyte fuel cell, Journal of Power Sources 159(2) (2006), 1095–1104.

12.

Wang

Y.-J.

and Lee

H.-S.

, Generalizing TOPSIS for fuzzy multiple-criteria group decision-making, Computers & Mathematics with Applications 53(11) (2007), 1762–1772.

13.

Guo

, Xu

H.-J.

and Liu

, Threat assessment for air combat target based on interval TOPSIS, Systems Engineering and Electronics 31(12) (2009), 2914–2917.

14.

Yang

Y.-Z.

, Wang

H.-W.

, Suo

Z.-Y.

, Chen

and Fan

X.-Y.

, A emitter threat evaluation method based on rough set and TOPSIS, Acta Armamentarii 37(5) (2016), 945–952.

15.

Zhang

H.-W.

, Xie

J.-W.

, Ge

J.-A.

, Zhang

Z.-J.

and Zong

B.-F.

, Intuitionistic fuzzy set threat assessment based on improved TOPSIS and multiple times fusion, Systems Engineering and Electronics 40(10) (2018), 112–118.

16.

Liu

J.-C.

and Li

D.-F.

, A note on “TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets”, IEEE Transactions on Fuzzy Systems 18(2) (2018), 299–311.

17.

Seker

and Aydin

, Hydrogen production facility location selection for Black Sea using entropy based TOPSIS under IVPF environment, International Journal of Hydrogen Energy 2019; https://doi.org/10.1016/j.ijhydene.

18.

Hajek

and Froelich

, Integrating TOPSIS with interval-valued Intuitionistic fuzzy fognitive maps for effective group decision making, Information Sciences 485 (2019), 394–412.

19.

Abootalebi

, Hadi-Vencheh

and Jamshidi

, Ranking the alternatives with a modified TOPSIS method in multiple attribute decision making problems, IEEE Transactions on Engineering Management 99 (2019), 1–6.

20.

Y.-H.

, Cheng

S.-Y.

, Zhou

Y.-P.

, Suo

Z.-Y.

and Peng

S.-M.

, Radiator Threat Evaluating Based on Shapley-TOPSIS, Journal of Air Force Engineering University (Natural Science Edition) 21(2) (2020), 91–96.

21.

Zhang

, Wang

H.-W.

and Chen

, Emitter threat grade evaluation of incomplete information systems, Telecommunication Engineering 58(4) (2018), 443–449.

22.

Sun

H.-W.

, Xie

X.-F.

, Sun

and Zhang

L.-J.

, Threat assessment method of warships formation air defense based on DBN under the condition of small sample data missing, Systems Engineering and Electronics 41(6) (2019), 1300–1308.

23.

Z.-H.

, Chen

Z.-C.

, Pei

, Ma

X.-Z.

and Liu

, An improved ranking strategy for fuzzy multiple attribute group decision making, International Journal of Computational Intelligence Systems 6(1) (2013), 38–46.

24.

Z.-H.

, Null values estimation method based on IOWA operators and application for incomplete information systems, Xihua University, 2013; 6–9+20–22.

25.

Alhichri

, Bazi

, Alajlan

, Melgani

, Malek

and Yager

R. A novel fusion approach based on induced ordered weighted averaging operators for chemometric data analysis, Journal of Chemometrics 27(12) (2013), 447–456.

26.

Xian

S.-D.

and Xue

W.-T.

, Intuitionistic fuzzy linguistic induced ordered weighted averaging operator for group decision making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23(4) (2015), 627–648.

27.

Sun

, Feng

, Gao

X.-G.

and Liu

, Dynamic threats assessment based on intuitionistic fuzzy set under missing data condition, Fire Control & Command Control 43(8) (2018), 93–97.

28.

Zheng

X.-L.

and Deng

, Dependence assessment in human reliability analysis based on evidence credibility decay model and IOWA operator, Annals of Nuclear Energy 112 (2018), 673–684.

29.

Zhang

, Wang

H.-W.

and Chen

, Combined emitter threat assessment based on ICW-RCM, Systems Engineering and Electronics 40(3) (2018), 557–562.

Radiator threat evaluation with missing data based on improved technique for order preference by similarity to ideal solution

Abstract

Keywords

1 Introduction

2 Null estimation algorithm based on IOWA

6 Experiment

Table 1 Data of radiators Object s 1 s 2 s 3 s 4 s 5 s 6 s 7 x 1 6.8 1.8 72 14 32 102 0.3 x 2 7.8 0.9 190 17 14 32 4.6 x 3 6.2 1.3 120 13 21 47 1.2 x 4 5.4 1.4 113 12 19 51 2.6 x 5 8.4 0.4 213 20 4.3 5.6 5.9 x 6 4.7 1.6 104 8 16 19 1.7

Footnotes

Acknowledgment

References

Table 1
Data of radiators

Object s ₁ s ₂ s ₃ s ₄ s ₅ s ₆ s ₇

x ₁ 6.8 1.8 72 14 32 102 0.3

x ₂ 7.8 0.9 190 17 14 32 4.6

x ₃ 6.2 1.3 120 13 21 47 1.2

x ₄ 5.4 1.4 113 12 19 51 2.6

x ₅ 8.4 0.4 213 20 4.3 5.6 5.9

x ₆ 4.7 1.6 104 8 16 19 1.7