YAO Tianle,WANG Weili,MIAO Run,DONG Jun,and YAN Xuefei

1.Ordnance Engineering College,Naval University of Engineering,Wuhan 430000,China;2.Xi’an Modern Chemistry Research Institute,Xi’an 710065;3.Institute of Systems Engineering,Academy of Military Science,Beijing 100082

Abstract: The research on the damage effectiveness assessment of anti-ship missiles involves system science and weapon science,and has essential strategic research significance.With comprehensive analysis of the specific process of the damage assessment process of anti-missile against ships,a synthetic damage effectiveness assessment process is proposed based on the double hierarchy linguistic term set and the evidence theory.In order to improve the accuracy of the expert’s assessment information,double hierarchy linguistic terms are used to describe the assessment opinions of experts.In order to avoid the loss of experts’ original information caused by information fusion rules,the evidence theory is used to fuse the assessment information of various experts on each case.Good stability of the assessment process can be reflected through sensitivity analysis,and the fluctuation of a certain parameter does not have an excessive influence on the assessment results.The assessment process is accurate enough to be reflected through comparative analysis and it has a good advantage in damage effectiveness assessment.

Keywords:anti-ship missile,damage effect assessment,linguistic term set,evidence reasoning.

1.Introduction

The damage effect assessment of anti-ship missiles against ship targets is an important long-term continuous research in various countries around the world,and is an important basis for assessing real-time battle conditions in actual combat environment.Therefore,there is an urgent need for an efficient and low-cost assessment method that can effectively assess the damage effect of antiship missiles.In actual situations,the damage effect is the result of the interaction between the damage source and the damaged object,so it induces strong causality [1,2].This is a comprehensive research involving weapon science and system science,and many factors need to be considered comprehensively.In the field of damage assessment and effectiveness assessment,scholars have conducted research and established related models for analyzing the relationship between criteria and the probability of damage.For example,according to the characteristics of infrared imaging and the Global Positioning System (GPS),Zhao et al.[3]analyzed the characteristics of terminal ballistics to determine the relationship between damage effect,fuze explosion height,and guidance accuracy.Wang et al.[4]established a calculation model of warhead damage probability based on the Monte Carlo method,and obtained the damage probability of different warhead’s modes of action to two targets.Xiong et al.[5]analyzed the damage mechanism of torpedoes to ship targets,established a damage assessment model based on damage trees,and simulated the effects of different factors on the damage effect of ships.Guo et al.[6]designed four levels of combat effectiveness assessment criteria for air-to-surface missile weapon systems and provided mathematical expressions of effectiveness criteria,forming a complete air-to-surface missile weapon system combat effectiveness assessment method.In terms of the combination of qualitative judgment and quantitative calculation,Li [7]used the fishbone diagram method to construct an assessment model that supported causal analysis,and provided a basis for causal analysis of the effectiveness of weapon equipment system assessment results.In order to improve the shortcomings of the analytic hierarchy process (AHP) combat effectiveness assessment,Peng et al.[8]applied the gray cloud model to the whitening weight function,and established a combat effectiveness assessment model that integrated the advantages of the gray cloud model and the improved AHP.Aiming at the situation of the anti-radiation weapon range test,Liu et al.[9]proposed an anti-radiation weapon combat effectiveness assessment model based on nonlinear criteria aggregation,and gave the corresponding criteria system model and criteria aggregation method.Li et al.[10]constructed a fuzzy Bayesian network model to quickly assess the damage effect to target under various attack conditions.Analyzing the above works,it can be seen that there are two key steps to conduct damage effectiveness assessment.The first is to construct an appropriate assessment criteria system based on the relationship between the damage source and the damage object,and to form a reasonable assessment analytic structure.The second is to select an appropriate assessment method based on the correlation between the criteria,and form a suitable criteria data fusion process.Much research has been done in both aspects.

Although work has been done on the research of damage effect assessment,there are still two problems that have not been fully resolved in the actual assessment process,which need to be further considered.The first problem is that it is difficult to control the accuracy of the data assessed by the experts when the assessment process is conducted.The more accurate the data assessed by the experts,the more difficult it is for the experts to distinguish the performance of adjacent data.Therefore,how to enable the experts to better distinguish and judge the granularity of data without reducing the accuracy of the assessment is also an urgent problem in the process of assessing the damage effect of the anti-ship missile.The second is that the mapping between the hierarchical structure of the assessment criteria system is non-linear.How to effectively fuse the information assessed by the experts and how to prevent the assessment information from being lost in the assessment process are problems that need to be solved.Therefore,it is necessary to construct a novel assessment process to analyze and handle the complex damage effect assessment of anti-ship missiles.

Generally,in complex assessment situations,the score often cannot directly reflect the preference and personal judgment of the assessment expert.At this time,because linguistic term can describe the assessment information vividly,the greater assessment advantages than numbers can be fully reflected [11,12].Double hierarchy linguistic term set divides complex linguistic information into two simple linguistic hierarchies in which the first hierarchy linguistic term set is the main linguistic hierarchy and the second hierarchy linguistic term set is the linguistic feature or detailed supplementary of each linguistic term in the first hierarchy linguistic term set [13].Scholars such as Zhang et al.[14]and Xue et al.[15]developed extensions of the double hierarchy linguistic term set and successfully applied it in haze management and strategic capability assessment.In order to prevent the loss of information due to information fusion rules,the evidence theory proposed by Dempster et al.[16]can preserve the integrity of the information to the greatest extent,and it can fuse uncertain information from different experts by using evidence inference rules [17]and reduce the loss of information in the assessment process [18].Moreover,the evidence theory has also been widely used in weight calculation.Fei et al.[19]introduced the concept of evidence contradiction coefficient and proposed a method for calculating the weight of evidence that can effectively use conflict information,overcoming the shortcomings of the classic Dempster-Shafer (D-S)evidence theory.It also has been widely used in the field of information fusion,and better fusion results can be obtained in an effective reasoning form without prior probability [20].Based on the above analysis,damage effect assessment of anti-ship missiles against ship targets is taken as the background in this paper,and a damage effect assessment process is proposed based on the double hierarchy linguistic term set and the evidence theory.While improving the accuracy of expert assessment during the assessment process,this process can minimize the information loss and effectively integrate multiple opinions from different experts,thus improving the accuracy of damage effect assessment.

2.Construction of the hierarchical structure of assessment criteria system for anti-ship missile

As shown in Fig.1,the hierarchical structure of the assessment criteria system is divided into three layers,whereF1,F2,andF3represent the three layers,andCrepresents the damage effect criteria.There are many actual operational factors that need to be considered in the actual combat environment,including the command status of the commander,the interference of the electromagnetic environment,and so on.Therefore,it is difficult to accurately assess the damage effectiveness of the damage source.Fig.1 only shows the result of the interaction between the damage source and the damage object.This result can only represent the physical result produced by the causal interaction between the damage source and the damaged object,which is the damage effect of the damage source against the damaged object.Yet it can lay a useful foundation for further assessing the damage effectiveness of the damage source under actual combat conditions.Therefore,it is meaningful to conduct damage effect assessment.Because damage effect is mainly the result of the interaction between the warhead damage capability of the anti-ship missile and the damage resistance of ship,the criteria in the second layer are the warhead damage capabilityC1and the damage resistance of shipC2.The criteria in the third layer are mainly used to describe the performance of the criteria in layer.The criteria in the lower layer of the warhead damage capabilityC1include missile hit positionC11,hole sizeC12,number of penetrating cabinsC13,range of shock waveC14,shock wave overpressureC15,positive pressure durationC16,number of fragmentsC17,and initial velocity of fragmentsC19.The criteria in the lower layer of the damage resistance of shipC20include geometry of the shipC21,hull bulkhead structureC22,hull deck structureC23,special structureC24,main material of the shipC25,ship stiffenerC26,and welding material for shipC27.The symbols of criteria in the second layer and the third layer are shown in Fig.1.In Fig.1,there are three criteria in the first layer and the second layer,so there are a total of three times of information fusion.

Fig.1 Hierarchical structure of damage effect assessment criteria system

3.Information processing and fusion model

3.1 Information processing model based on double hierarchy linguistic term set

Before assessing the damage effect,the relevant sets and variables should be defined.The lower layer criteria ofC2are used as examples.LetCs={Csi|i=1,2,···,7} be a set of criteria;w={wi|i=1,2,···,7} be a set of weights,where。LetA={Aj|j=1,2,3} be a set of cases of damage.In this paper,a total of four experts are invited to participate in the assessment,so the expert set isE={eq|q=1,2,3,4}.is a set of weights of the experts,whereIn the assessment criteria system shown in Fig.1,Cw1?Cw9are the criteria of the benefit type andCs1?Cs7are the criteria of the cost type.The cost type is converted into the benefit type according to

As shown in the following equation,the first linguistic term set is denoted as

The second linguistic term set is denoted as

Among them,according to the different values oft,the meaning ofOti s slightly different when it is integrated into the first hierarchy linguistic term set,as shown in Fig.2.

Fig.2 Information expression form of the second hierarchy linguistic term set

After experts express their opinions through the first hierarchy linguistic term set,the second hierarchy linguistic term set is used to adjust the assessment opinions.The elements in the second hierarchy linguistic term set shown in Fig.2 are adverbs or adphrases and can be used to modify the adjectives in the first hierarchy linguistic term set.There are five elements in Fig.2:“just right”,“a little”,“far from”,“much”,and “entirely”.These elements are used to adjust the degree of the elements in the first hierarchy linguistic term set.

Therefore,the double hierarchy linguistic term set can be expressed as

Therefore,the decision matrix formed by an expert can be obtained as

Two scoring functions are used to quantify the opinions of each layer of linguistic term set assessed by experts.The first hierarchy linguistic term set has obtained the main judgment of the expert,and the second hierarchy linguistic term set is a subtle adjustment to the expert’s judgment on the basis of the first hierarchy linguistic term set,which makes the expert’s judgment more accurate.Therefore,the final scoring function is the linear superposition of the two scoring functions.The score is represented by fraction before the final result is obtained.

The first score function is

The judgment of experts makesktake ?2,?1,0,1,2 respectively.

The second-level score function is

Therefore,the final score function is shown in the following equation and the possible scores of all criteria are shown in Fig.3:

Fig.3 Information expression form of the double hierarchy linguistic term set

3.2 Information transformation of criteria

In the fusion process,in order to fully fuse the information of opinions expressed by experts and not cause information loss,the opinions are formed into evidence,so that the information of opinions can be fully reflected in the assessment results.In order to facilitate analysis and information fusion,transforming all information into a belief structure is considered [21].Belief structure is actually a method of using belief degree to represent the distribution of assessment degree of a criterion.The specific expression is

whererepresents the assessment degree，and βirepresents the belief degree ofinCi.

In order to calculate βi,the information shown in (5)should be handled first.Using Hamming distance to measure the difference in opinions between experts,the difference in opinions between any two experts can be expressed as

Then the similarity degree between the linguistic elements assessed by any two experts and the support degree for one of the expert’ assessment result can be expressed as

Finally，through calculating the support degree of all expert’ assessment results,the belief degree of any expert’assessment result can be expressed as

Based on the above analysis,the evidence matrix that integrate all experts’ assessment information and their belief degree can be obtained:

3.3 Information fusion based on evidence reasoning

Through information transformation,all qualitative information is expressed by belief structure.This section uses evidence reasoning to aggregate the belief structure of each criterion into up-layer criteria.The seven lowerlayer criteria ofC2are still used as examples.First,combine the relative weights and belief degrees of the criteria and the belief degrees are transformed into the basic probability masses by

After transforming the lower-layer criteria ofCsifrom the belief structure to the basic probability mass,the following equations [22]are used to aggregate the basic probability mass:

wheren=0,1,···,24，i=1,2,···,9，N=4 . βnand βHrepresent the belief degrees of the aggregated assessment,which respectively correspond to degreeand setS.Yang [23]proved thatand the above evidence reasoning is reasonable and effective.The normalization process is provided by (17).

For the final assessment results,linear aggregation should be performed according to the weight of expert.It is more objective to obtain the weight of expert based on the experts’ assessment results.The richness of the expert’s subjective experience is often reflected in the consistency of the assessment results,which has been studied by Yu et al.[24].Euclidean distance which is used to measure the consistency of experts’ assessment results is an effective method to optimize expert’s weights.

The weighted Euclidean distance betweenandas the assessment results between any two experts is

The optimization model that minimizes the sum of squared distances between the assessment results of all experts is constructed as follows:

The Lagrangian algorithm is used to solve (19),the final weight vector of expert can be obtained as

Let us see what is in the other casket, before we get into a bad humor, saidthe Emperor. So the nightingale came forth5 and sang so delightfully7 that atfirst no one could say anything ill-humored of her.

whereI=(1,1,···,1)Tand Ψ is

4.Case study

4.1 Collection of criteria information

In order to obtain the evidence matrix shown in (14),experts should first be asked to assess the criteria shown in Fig.1,and then the results of the linguistic assessment result should be transformed into the double hierarchy linguistic term set according to Fig.2 and Fig.3.This paper invites four experts from scientific research institutes and industrial departments to assess the three damages in the form of a survey report.The assessment results are shown in Table 1.

Table 1 Assessment linguistic results assessed by four experts

Substituting the assessment linguistic information in Table 1 into (8) can get the scores of all the criteria.Substituting the score corresponding to each criterion into (9)?(13),the Hamming distance,the similarity degree,and the support degree related to criteria can be obtained.And the belief degrees of the linguistic information of all criteria are finally obtained as shown in Table 2.After obtaining the data in Table 2,all the evidence elements of the decision matrix in (14) are obtained.

Table 2 Belief degree of the linguistic information of all criteria

4.2 Fusion of criteria information

In order to aggregate the data information in Table 1 and Table 2,the weight of each expert should be solved first.Substituting the score corresponding to the linguistic information in Table 1 into the optimization model (19),the matrix Ψ can be obtained as

Substitute the matrix Ψ into (20),and the weight vector of experts can be obtained as

After the four experts have assessed the weights of the eighteen criteria in the second and third layers in Fig.1,the weight of each expert and the weights of criteria are linearly summed.The final weight shown in the brackets of each criterion can be obtained as shown in Table 3.

Substitute the scores corresponding to the linguistic information in Table 1,the belief degree of the scores of criteria in Table 2,and the weights of criteria in Table 3 into (16) and (17).The distributions of the assessment results of three damages assessed by four experts are shown in Fig.4?Fig.7.In Fig.4?Fig.7,the horizontal axis represents the numerator of the score corresponding to the assessment degree,and the denominators are all twenty-four.The vertical axis represents the belief degree of corresponding to each assessment degree.

Table 3 Weight of all criteria in the second and the third layers

Table 4 Expected values of the assessment results of the four experts for the three damages

Fig.4 Distribution of belief degree of the assessment information expressed by Expert 1

Fig.5 Distribution of belief degree of the assessment information expressed by Expert 2

Fig.6 Distribution of belief degree of the assessment information expressed by Expert 3

Fig.7 Distribution of belief degree of the assessment information expressed by Expert 4

According to the results in Table 4,the expected values of the assessment results assessed by four experts are ranked as

The expected values in Table 4 and the weights of experts are linearly summed,and the final damage effect assessment result is shown in Table 5.

Table 5 Final assessment result

Therefore,the expected values of the assessment results are ranked as

It can be seen from the above rank that the final assessment result is consistent with the assessment results of Expert 1 and Expert 2,and partly consistent with the assessment results of Expert 3 and Expert 4.Among the four experts,Expert 3 has the highest weight,while Expert 1 and Expert 2 have relatively low weights.This is closely related to the consistency of the original assessment information of several experts.However,the difference of consistency does not make the gap of weight between experts too large.Therefore,the final assessment result is still the assessment result that combines the common preferences of the four experts.

4.3 Sensitivity analysis of weight

The weights of criteria may have a greater influence on the final assessment result,so it is necessary to analyze its influence on the result according to the fluctuation of the weights.The weight in the paper is divided into two parts:the weights of criteria and the weights of experts.Since the weights of experts used in this paper are objectively solved based on a large number of assessment data of experts,it has good stability.The weights of criteria come entirely from the assessment of experts,which is too subjective,so different assessment results of the same expert can be different.From Fig.4 to Fig.8,it can be seen that the assessment results of different experts will inevitably produce different results.Based on above analysis,the influence of the fluctuation of the weight on the final assessment result should be analyzed.First,we should analyze the sensitivity of the weight’s fluctuations to the results.As shown in Table 3,the number of criteria at the bottom layer is large,so the weights of criteria are generally small,and the fluctuation of a single weight has a little influence on other weights.However,there are only two criteria in the second layer of the assessment criteria system hierarchy structure,and fluctuations in the weight of each criterion may have more obvious influence on the final result.Therefore,assessment ofA1handled by Expert 1 is taken as an example to analyze the weight sensitivity.The weight of the criteria in the second layer will fluctuate as

The fluctuation range is ?30%?30%,and the fluctuation step is 10 %.θis the fluctuation range of the weight value ofw2.As shown in Table 6,the fluctuation range is from?30% to 30%.

Table 6 Changed criteria weight values

It can be seen from Fig.8 that whenθf(wàn)luctuates from?30% to 30%,the belief degree of the assessment result in the low score area becomes higher,while the belief degree in the high score area becomes lower.When the assessment degree is less than 10/24,the belief degree increases with the increase ofθ;when the assessment degree is greater than 15/24,the belief degree decreases with the increase ofθ.

Fig.8 Distribution of assessment results of A1 under different weights

It can be seen from Table 1 that after the normalization,the scores of warhead damage capability are mainly distributed in the high score area,while the scores of damage resistance of the ship are mainly distributed in the low score area.From (21) and Table 6,we can see thatw1decreases monotonously asθincreases.Sincew1is the weight of the warhead damage capability andw2is the weight of damage resistance of the ship,it shows that the influence of the change of weight is transmitted to the result of information fusion along with evidence reasoning,and mainly influence the results related to criteria.It can be seen from Fig.8 that the change in the distribution of the assessment result is very obvious as the weights are constantly changing.

In order to further analyze the influence of the fluctuation of weight on the final assessment result,the expected value of the assessment result in Fig.8 under the fluctuation of weight is solved.Moreover,the trend graph of distribution of the expected value shown in Fig.9 is obtained.It can be seen from Fig.9 that the change of the expected value of the assessment degree presents a curve of the convex function.Whenθf(wàn)luctuates from ?30% to 30%,the expected value of the assessment result is monotonously reduced,and the variation does not exceed 0.05.In the actual assessment process,the fluctuation ofθgenerally does not reach 60%.Whenθf(wàn)luctuates from?10% to 10%,the change in the expected value of the assessment result does not exceed 0.02.At this time,the fluctuation ofθalso reaches 20%,which is still a small probability event.Therefore,although the fluctuation of the weight of the criteria has an influence on the assessment result,it can be considered that the influence is very limited.If the weight value given by the expert does not have a major error,the assessment result can be considered accurate.

Fig.9 Change trend of the expected value of the assessment result of A1 under different weights

5.Comparative analysis

5.1 Comparison with simulation result and test result

As shown in Table 7,the data reflecting the specific attribute values of the warhead damage capability in each case are provided,and the data can be used as a basis to verify the accuracy of the assessment process.It can be seen from Table 5 that the anti-ship missile has the highest damage effect in caseA3.By verifying the accuracy of the case with the highest damage effect,the assessment process of several cases can be considered reasonable.Therefore,the relevant conditions ofA3are used to verify the damage effect through dynamic simulation and actual explosion test.In order to simulate the damage of the missile against the cabin,the simulation is carried out according to the relevant conditions of caseA3.As shown in Fig.10,the simulation of the damage of the missile to the proportionally reduced single cabin is carried out in LS-DYNA software.The hole on the right side of the cabin in Fig.10 describes the part hit by the missile.It can be seen from Fig.10(a) that the stress is the highest at the welds and corners in the cabin,so the disintegration of the cabin starts from the edge.It can be seen from Fig.10(b) that the disintegration effect of the cabin is better,and the bulkhead stress begins to decrease after cabin disintegration,which shows that the damage effect of the missile is very good.

Fig.10 Von-Mises (V-M) equivalent stress cloud diagram of the bulkhead

Table 7 Normalized values of each criterion of warhead damage capability

According to the relevant conditions of caseA3,the actual explosion test of the proportionally reduced single cabin is carried out,and the test results is shown in Fig.11.It can be seen from Fig.11 that the cabin has been completely dismantled,and the bulkhead has been seriously damaged.It proves once again that the damage effect of the missile in caseA3is high,and it also proves the accuracy of the assessment process proposed in this paper.

Fig.11 A part of the cabin after being damaged

5.2 Comparison with other assessment processes

In order to verify the accuracy of the synthetic assessment process proposed in this paper,this sub-section presents a comparison with two existing assessment process to determine the effectiveness of the proposed synthetic assessment process.

Literature [10?12]mainly used the hesitant fuzzy linguistic term sets theory to reflect information,which is helpful to solve the problem of disagreement of experts or hesitation of experts.Therefore,this section combines the hesitant fuzzy linguistic term sets (HFLTS) theory with the evidence theory for joint assessment.In addition,as a classic decision-making theory,the technique for order preference by similarity to an ideal solution (TOPSIS)plays an essential role in many assessment occasions.Therefore,TOPSIS is often used as a way for comprehensive assessment.According to the analysis in Subsection 4.2,the damage effect of the missile in caseA3is the highest.Therefore,taking the caseA3as an example,the above two assessment processes are used for assessment.As shown in Fig.12,the three curves respectively represent the assessment results of caseA3in different assessment processes under the fluctuation of weights.

In Fig.12,the black curve represents the assessment results of the assessment process proposed in this paper;the red curve represents the assessment results of the assessment process using HFLTS and evidence theory;the blue curve represents the assessment results of the assessment process using TOPSIS.The results of the TOPSIS method is relatively close,so the final assessment result has a larger value.In order to show the changing trend of the curve more clearly,the blue curve in Fig.12 represents the result of a downward shift of 0.4-unit length.It can be seen from Fig.12 that the assessment results of the assessment process using hesitant fuzzy linguistic term sets theory and the evidence theory are very close to the assessment results in this paper.Actually,these two processes are very close in principle.Hesitant fuzzy linguistic term sets are more suitable for fuzzy environments.When experts are hesitant to many assessment values,hesitant fuzzy sets provide a powerful tool to solve problems.In this paper,a distance-based optimization model is adopted to optimize the weights of experts.Therefore,the final assessment result excludes the subjective factors of the experts and considers all the information of the experts.Therefore,changes in the final assessment results should be gentler.If the TOPSIS method is used for assessment,it can be seen that the change in the assessment results using the TOPSIS method are the gentlest.As the weights change,the assessment results change very little.The TOPSIS method cannot obtain information as finely as the linguistic term set,and the TOPSIS method uses Euclidean distance and close degree to characterize the assessment results,which cannot provide a good explanation for the fusion process with nonlinear characteristics.Therefore,the TOPSIS method as a quick assessment method is very convenient,but the assessment results are relatively rough.

Fig.12 Fluctuation of the assessment result of A3 with different assessment processes

6.Conclusions and future work

Based on the damage effect assessment of the anti-ship missile against the ship target,this paper proposes a damage effect assessment process based on double hierarchy linguistic term set and evidence theory.The following two main conclusions can be drawn:

(i) Aiming at solving the problem of low accuracy of experts’ assessment information in the assessment process,double hierarchy linguistic term set is used to describe experts’ opinions,and an appropriate scoring function is used to convert experts’ opinions into quantifiable scores.The overall accuracy of the missile damage effect assessment process is improved.

(ii) This evidence reasoning is used to fuse experts’ assessment information in this paper,which can reduce information loss caused by information fusion rules.Sensitivity analysis shows that the assessment process proposed in this paper has good stability,and the assessment results will not change greatly due to the fluctuation of a certain parameter.Comparative analysis shows that the assessment process proposed in this paper has good accuracy.

The hierarchical structure of the assessment criteria system established in this paper only considers the performances of damage source and the damage object,but many actual combat factors have great influence on the capabilities of both the source and the object.Therefore,in future work,if it is needed to accurately assess the damage effectiveness of the anti-ship missile against the ship in the actual combat environment,it is necessary to further investigate the influencing factors in the actual combat environment based on the assessment criteria system established in this paper,and the key influencing factors should be included as criteria into the assessment criteria system.In this way,the comprehensive process proposed in this paper can be used to further quantitatively assess the damage effectiveness.Moreover,the interaction between criteria and the effect of damage on the time scale can be considered.On the basis of the work in this paper,the accuracy of damage effectiveness assessment can be further improved.