Warhead power assessment based on double hierarchy hesitant fuzzy linguistic term sets theory and gained and lost dominance score method

Tinle YAO, Weili WANG,*, Run MIAO, Qiwei HU, Jun DONG,Xuefei YAN

a Ordnance Engineering College, Naval University of Engineering, Wuhan 430000, China

b Department of Equipment Command and Management, Army Engineering University, Shijiazhuang 050003, China

c Xi’an Modern Chemistry Research Institute, Xi’an 710065, China

d Institute of Systems Engineering, Academy of Military Science, Beijing 100082, China

KEYWORDS Evidence theory;Fuzzy hesitant linguistic term set;Gained and lost dominance score method;Optimal alternative of warhead design;Warhead power assessment

Abstract Warhead power assessment of the anti-ship missile plays a vital role in determining the optimal design of missile, thus having important strategic research significance. However, in the assessment process, expert’s judgement will directly affect the assessment accuracy. In addition,there are many criteria involved in the missile design alternatives. Some criteria with poor performance may be compensated by other criteria with excellent performance, and then it is impossible to find the truly optimal alternative. Aimed at solving these problems, this paper proposes a synthetical assessment process based on fuzzy hesitant linguistic term set and the Gained and Lost Dominance Score (GLDS) method. In order to improve the assessment accuracy of experts and solve the problem that experts generate different opinions, combined with the advantages of fuzzy hesitant sets and linguistic term sets,the double hierarchy hesitant fuzzy linguistic term sets are used in this paper to improve the accuracy of expert’s judgement.In order to effectively combine expert’s experience with the data of criteria, the evidence theory and entropy weight method are used to transfer the expert’s judgement to the weight. In order to avoid selecting defective alternative of missile design, the GLDS is used to fuse expert information and criteria information. Sensitivity analysis shows that the assessment process has sensitivity to some extent. However, when the fluctuation of expert’s assessment makes the fluctuation of θ in the range of-5%to 5%,the impact on the results is not quite conspicuous. The analysis of calculation result and comparative analysis show that the assessment process proposed in this paper is accurate enough, has great advantage in selecting the current and potential optimal alternative of missile design, and avoids the alternatives with low criteria performance that cannot be compensated by other criteria being selected.

1. Introduction

As the main combat weapon against ships, anti-ship missiles have continuously improved requirements for their power and combat effectiveness. The assessment of anti-ship missile warhead power has become an important subject for longterm continuous research in various countries around the world.The traditional anti-ship missile warhead power assessment is generally carried out in the way of full-size projectile shooting, and the damage to the target is used to assess the anti-ship missile warhead power. However, with the acceleration of the development of new missile warhead, the use of a large number of full-size projectiles to conduct assessments takes long time and requires huge investment in manpower,material and financial resources. Moreover, the assessment of warhead power is an indispensable part of optimizing missile warhead design. Therefore, an efficient and low-cost assessment method that can effectively assess the anti-ship missile warhead power is urgently needed.

Generally speaking, the assessment process of the anti-ship missile warhead power can be regarded as a process of multiattribute group decision-making. In the actual physical process, the assessment of the anti-ship missile warhead power has a strong correlation with the mechanical calculations,involving various damage factors such as shock wave and fragment.After determining the assessment criteria set,it is necessary to obtain basic data through simulation calculation, and the assessment value with good results should be tested for verification.At the same time,qualitative data should be given by experts for the criteria weights to determine the importance of each criterion in the assessment process.Furthermore,it is necessary to fully fuse the information, and solve the problems of incommensurability,ambiguity and even contradiction of data information in the process of fusion, and obtain reasonable assessment results. This is a synthetical research involving weapon science and system science, and many factors need to be considered comprehensively.

In the past 10 years, some scholars developed some methods to analyze the effectiveness of weapon systems in various environments,and gave relevant analysis cases.These methods are very inspiring for the assessment of the anti-ship missile warhead power.For example,aimed at the damage mechanism of torpedoes to ship targets,Xiong et al.established a damage assessment model based on damage trees, and simulated the effects of different factors on the damage effect of ships. In order to improve the shortcomings of the analytic hierarchy process in assessing combat effectiveness, Peng et al.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 analytic hierarchy process. Aimed at the current situation of the anti-radiation weapon range test,Liu et al.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. According to actual fault situation, Sun et al.proposed a multi-criteria fusion feature selection algorithm to identify an optimal feature subset from the highdimensional original feature space for fault diagnosis,and validated the effectiveness of algorithm through six fault recognition datasets from a real helicopter transmission platform.Based on the structure and working principle of the system,Chen et al.proposed a new Evidential Reasoning(ER) based approach with uncertain parameters to construct a nonlinear optimization model to evaluate the system performance, and analyzed the sensitivity of the evaluation result to propose the maintenance strategy on the subsystems that have a great impact on the performance of the whole system.Li et al.combined fuzzy theory and Bayesian network to establish a fuzzy Bayesian network model for target damage assessment, which can quickly assess the target damage effect under various attack conditions. Li et al.analyzed the characteristics and requirements of system target damage assessment, and used the exponent method to solve the quantitative relationship between the sub-target physical damage degree and the effectiveness attenuation degree. Wang et al.established a rod/fragment dual-mode warhead damage probability calculation model based on the Monte Carlo method, and obtained the damage probability of the warhead’s different action modes to two targets and the law of the change with the miss amount.

Although some work has been done on this type of assessment problem,there are still three problems that have not been fully resolved in the actual assessment process, which requires further thinking and new solutions.

The first problem is that experts in the assessment process will hesitate and information will be lost when fusing the information of lower-level criteria.In the process of using the expert group to assess the weight of each criterion,for a certain criterion, the assessment results of the experts may produce inconsistent opinions,and the experts cannot agree with each other.Or a certain expert will still be hesitant to assess a certain indicator,so there will be several assessment values that cannot be finalized. Therefore, this complex decision-making problem with uncertainty still needs to be resolved. At the same time,effective information fusion is still needed for multiple indecisive results produced by experts. The hesitant fuzzy set theory proposed by Torra in 2010is a powerful tool for dealing with uncertain and fuzzy information. When the assessment expert or expert group hesitates among several possible criteria values,the hesitant fuzzy set containing these possible criteria values can accurately represent this situation. A large number of studies have been published in the use of hesitant fuzzy set theory to solve many complex and uncertain decision-making and assessment problems.

In the process of information fusion, a large number of existing assessment rules may cause some original information to be lost, which may have an important impact on the final assessment result. In order to prevent the loss of information due to the assessment rules,the D-S evidence theory proposed by Dempsterand Shafercan preserve the integrity of the information to the greatest extent, and it can fuse uncertainty information of different experts by using evidence reasoning rules and reduce the loss of information in the assessment process.Thanks to these achievements,many complex assessment and decision-making problems can be solved.

Second,it is difficult to control the accuracy of the opinions expressed by the experts during the assessment.When processing assessment information,information processors often hope to obtain more accurate original assessment information,which is more conducive to obtaining more accurate assessment results. However, when the granularity of the opinions expressed by experts becomes smaller and smaller,it will cause the experts to find it difficult to distinguish adjacent attributes.For example, in the process of expert scoring, when an expert wants to express an excellent judgment on a certain criterion,the distinction among 8 points, 9 points and 10 points is difficult to grasp, which may eventually lead the information processor to think ‘‘8 points=9 points=10 points”. But this result is very inaccurate, and even ridiculous. Therefore, how to enable assessment experts to better distinguish the granularity of opinions without reducing the accuracy of the assessment result is an urgent problem in the process of assessing the anti-ship missile warhead power.

Generally,in more complex assessment situations,the score cannot often directly reflect the preference and personal judgment of the assessment expert. At this time, because linguistic term can describe the assessment information vividly, their assessment advantages greater than numbers can be fully reflected. Rodriguez et al.combined the advantages of hesitant fuzzy set and linguistic term in 2012 and developed hesitant fuzzy linguistic term set. Studies have shown that hesitant fuzzy linguistic term setcan express uncertain information better than hesitant fuzzy set. Moreover,when solving the problem of too small granularity of the opinions expressed by experts, hesitant fuzzy linguistic term set can provide more solutions. The concept of double hierarchy hesitant fuzzy linguistic term setprovides more linguistic features and further improves the hesitant fuzzy linguistic term set. Double hierarchy hesitant fuzzy linguistic term set can describe hesitation more accurately and reasonably by adding a second hierarchy hesitant fuzzy linguistic term set,and the problem of too small assessment granularity using single hierarchy hesitant fuzzy linguistic term set can be solved.Due to the advantages of double hierarchy hesitant fuzzy linguistic term set in assessment,they have been widely used to solve practical decisionmaking problems.

The third problem is that when comprehensively assessing the anti-ship missile warhead power, criteria with lower effectiveness may be compensated by other criteria with higher effectiveness. In the process of comprehensively assessing the anti-ship missile warhead power, due to missile design and other reasons, some criteria may have poor performance, but other criteria have very good performance, which makes the final assessment result still good. However, the bad performance of a certain criterion may cause extreme effect in the actual use of the missile. Therefore, it is necessary to consider the comprehensive performance of all criteria and the performance of a specific criterion at the same time in the assessment process to avoid the good assessment result containing bad criteria.The Gained and Lost Dominance Score(GLDS)method proposed by Wu and Liaoprovides an inspiration for solving this problem. GLDS is developed from the most well-known TODIM in modern behavior decision theory. Its basic idea is to calculate the dominance degree of each alternative to other alternative through the utility function derived from prospect theory,which fully takes into account the bounded rationality behavior character of the assessment experts.It considers both the gained dominance score and the lost dominance score of the alternatives under uncertain circumstances, and describes the weighted maximum lost dominance score as an‘‘individual regret” value, so that the poor performance of some criteria cannot be compensated by the good performance of other criteria.Therefore,applying the GLDS method to the assessment of the anti-ship missile warhead power can avoid selecting very poor assessment results for certain key criteria. Motivated by original GLDS method, many useful works have been done.

Based on the analysis of the above problems and solutions,combined with evidence theory, this paper establishes a set of warhead power assessment process based on the double hierarchy hesitant fuzzy linguistic term set and the GLDS:

(1) Collect criteria data and expert’s assessment information to form the decision matrixes of double hierarchy hesitant fuzzy linguistic term set of different assessment experts.

(2) Aggregate the decision matrixes of double hierarchy hesitant fuzzy linguistic term set of different experts into the final evidence matrix by using evidence theory.

(3) Calculate the information entropy of each criterion by using the characteristics of the evidence matrixes.

(4) Calculate the weight of each criterion by using information entropy.

(5) Combine the weight of each criterion with the data of each criterion itself, and rank the power of each missile warhead according to the GLDS.

(6) Compare the ranking results with the experimental data,and analyze the calculation errors of the assessment model.

According to the above assessment process,the three problems raised can be effectively solved, and the information of assessment expert and testing and simulation data can be fully combined to obtain more scientific and accurate assessment result.

2. Assessment criteria and basic data collection

The design of the warhead involves a wide range of technologies, including knowledge in many fields such as mechanics and chemistry. When designing an anti-ship missile warhead,because of the cognitive differences of different design experts,the final damage results produced by different alternatives of warhead design will be quite different. Therefore, it is necessary to collect the data representing the warhead power in the damage process,carry out effective data fusion,and assess the alternative of warhead design,which is helpful in discovering the alternative that has the best comprehensive performance. The warhead power of anti-ship missile is mainly reflected in the terminal effect of the damage process. Therefore, from the perspective of the terminal effects, the damage of the anti-ship missile warhead to the ship comes from the penetration of the projectile, the damage from fragments and the shock wave generated by the explosion in the cabin, so the selected criteria should reflect the damage in these three aspects.

Combined with the related projectile shooting tests in recent years,three experts from the industrial sector and scientific research units were invited to assess the warhead power,so the seven criteria which are key criteria to measure the warhead power of anti-ship missile can be obtained.The seven criteria include the hole size produced by the penetration of the projectile,the impact velocity corresponding to the penetration of the projectile,the shock wave overpressure corresponding to the shock wave effect in the cabin,the range of shock wave,the number of fragments corresponding to the fragment damage,the damage radius and initial velocity of fragments.Let P represent the warhead power of each alternative, and c(i=1,2,···,7) represent the criteria of the warhead power. The finally constructed assessment criteria system hierarchy structure is shown in Fig. 1.

In order to obtain the data of each criterion in Fig. 1 formed by three different alternatives of warhead design, on the basis of historical test data, ANSYS-LS-DYNA is used for simulation calculation.

Fig. 2 is the Von Mises equivalent stress cloud diagram of the bulkhead when the warhead damages the ship’s command cabin, and the process of dynamic simulation can show the changing cabin state during the explosion in cabin. As shown in Fig. 2, there is a connecting space between the upper and lower floors of the ship’s command cabin. The upper cabin is a simulated command cabin with cabinets and so on. The simulation calculation of the explosion effect in the connected upper and lower cabin structure is carried out, and the charge of warhead is TNT.It can be seen from Fig.2 that the material is broken when t=50 ms.The aluminum cabinet in the upper cabin is severely deformed under the action of detonation waves and detonation products,but at the same time,the presence of the aluminum cabinet affects the movement of detonation waves and detonation products in the upper cabin,resulting in the difference of the stress in the left and right wall of cabin.

Through the dynamic simulation,it is possible to obtain the force situation when the cabin is destroyed and disintegrated in

Fig. 1 Assessment criteria system of warhead.

Fig. 2 Von Mises equivalent stress cloud diagram of bulkhead.

the condition of considering the reinforcement state of the stiffener and the real welding state, so as to obtain a more accurate final state of the cabin.The dynamic simulation result can be compared with the test result, and the consistency between the dynamic simulation result and the test result is used to ensure the accuracy of the basic data. According to the changes in the stress of the bulkhead, the vital basic data such as shock wave overpressure used to assess the warhead power can be solved. According to the alternative of missile design, other data are determined by empirical calculation equations and historical measured data. After collecting enough data, the representative data of each alternative are selected for subsequent assessment.

3. Information processing

3.1. Assessment information processing model based on double hierarchy fuzzy hesitant linguistic term set and evidence method

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

Fig. 4 Information expression form of the second hierarchy linguistic term set.

Combining Eqs. (1)-(4), the decision matrix formed by an expert can be obtained as

For the double hierarchy hesitant fuzzy linguistic term set, two scoring functions are used to quantify it. Fig. 3 shows the first hierarchy linguistic term set, which corresponds to Eq. (1). It can be seen intuitively from Fig. 3 that the first hierarchy linguistic term set has obtained the main judgment of the experts. In order to facilitate data processing, the highest assessment degree ‘‘perfect” is defined as 1,and the lowest assessment degree ‘‘terrible” is defined as 0.The first score function adopts a linear function, and the assessment degrees respectively correspond to the value of k in Eq. (1). Moreover, 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. There are also five elements in the second hierarchy linguistic term set. The midpoint of the scores of two adjacent elements in the first hierarchy linguistic term set is used as the assessment boundary. The midpoint between the assessment degree and the higher assessment degree indicates that the assessment degree is fully agreed, and the midpoint between the assessment degree and the lower assessment degree indicates that the assessment degree is completely disapproved. The second score function also adopts a linear function, and the assessment degrees respectively correspond to the value of t in Eq.(2). Since both the first score function and the second score function are linear functions, the final scoring function adopts the linear superposition of the two score functions.For the convenience of calculation and presentation, the score is represented by fraction before the final result is obtained.

The first-level score function is

When the judgment of expert makes k take -2, -1, 0, 1, 2 respectively, the score of the criterion is 0, 1/4, 1/2, 3/4, 1 respectively.

In order to adjust the score of the first hierarchy linguistic term set, the second-level score function is

Finally, the possible scores of all criteria are shown in Fig. 5.

Evidence theory can retain original information in the process of uncertain information fusion, which has great advantages in the field of assessment. In order to fuse the information of all experts without causing loss of information during the fusion process, the opinions of all experts can be transformed into evidence,so that the opinions of experts can be fully reflected in the final weight of criteria.

Using Hamming distance to measure the difference in opinions between experts, the difference in opinions between any two experts can be expressed as

Fig.5 Information expression form of double hierarchy linguistic term set.

3.2. Information collection by experts and processing results

In order to obtain the evidence matrix shown in Eq. (13),experts should be asked to assess the criteria shown in Fig. 1 using linguistic terms, and then the assessment results should be transformed into the double hierarchy hesitant fuzzy linguistic term set according to Fig.3 and Fig.4.The double hierarchy hesitant fuzzy linguistic term set formed by assessment of criteria by three experts is shown in Tables 1-3.

According to the data in Tables 1-3, combined with Eqs.(9)-(12), the belief degree of each expert’s assessment result can be obtained as

Up to Eq. (16), all the information assessed by experts on the warhead power has been collected and processed.

4. Ranking of warhead power based on the gained and lost dominance score method

The above process makes full use of the group wisdom of the expert group, and forms an evidence matrix that reflects the richness of expert knowledge. In the process of solving the weight of criteria, the comprehensive use of the information of Eq. (13) and the objective modeling method will make full use of the expert’s assessment information, making the final assessment result more accurate.

Table 1 Transformed curriculum-assessment values from the first expert.

Table 2 Transformed curriculum-assessment values from the second expert.

Table 3 Transformed curriculum-assessment values from the third expert.

4.1. Solving criteria weight based on entropy weight method

Entropy is a measure of information uncertainty, and it is an important concept obtained through probability theory.The difference of the assessment data and the uncertainty of the assessment criteria show a positive correlation. In the calculation, the entropy values of the criteria are positively correlated with the similarity of the assessment data of the criteria, and negatively correlated with the weights of the criteria. In the assessment process, using the entropy weight method to determine the weight can effectively prevent the influence of multiple subjectivity and human factors. Therefore, the weight solved by the entropy weight method using Eq. (13) will be more objective than the weights directly assessed by the experts. The specific process is as follows.

As shown in Eq. (17), the deviation of all missile design alternatives under the same criteria should be calculated first.

Combining the data in Tables 1-3 and Eqs.(14)-(16),using Eqs.(17)-(19),the weight of each criterion finally obtained are shown in Table 4.

4.2. Criteria information fusion based on the gained and lost dominance score method

In order to simultaneously consider the comprehensive influence of all criteria and individual criterion on the final assessment results, the gained and lost dominate score methodis used to rank the power of warhead.

According to Eq. (20), dominance flow of alternative Ato alternative Awith respect to criterion cshould be calculated first.

It can be seen from the above process that the global score simultaneously takes into account the values of both ‘‘group utility” and ‘‘individual regret”, which is helpful in avoiding selecting an alternative that performs too badly on some criteria.Therefore,the warhead power can be ranked effectively to obtain the optimal alternative of missile design.

4.3. Ranking of warhead power

From the simulation data in Section 2, combined with the empirical data of historical tests, the basic assessment data shown in Table 5 can be obtained. Moreover, the normalized values of each criterion can be obtained as shown in Table 6.Substituting the data in Table 4 and Table 6 into Eqs.(20)-(22), the gained dominance scores of A, Aand Acan be obtained as -0.2992, 0.1862 and 0.1131 respectively, which represent the ‘‘group utility” value of A, Aand A. The lost dominance scores of A, Aand Aare 0.1056, 0.1671 and 0.0835 respectively, which represent ‘‘individual regret” value of A, Aand A. By substituting the gained and lost dominance scores into Eq. (23), the final global scores of A, Aand Aare -0.0851, 0.0095 and 0.0238 respectively. Based on the global scores of A,Aand A,according to the assessment results of the warhead power, the final ranking of the three alternatives of warhead design can be obtained as

From the above ranking results,it can be seen that Ais the best alternative of missile warhead.

4.4. Analysis of calculation results of warhead power

Table 4 Weight of each criterion.

It can be seen from the above calculation results.Alternative Ahas the highest global score,so Ais the best alternative of missile warhead. However, when analyzing the gained dominance scores and the lost dominance scores of A, Aand A, it can be concluded that the‘‘group utility”of Ais the highest.Moreover,the‘‘group utility”value of Ais significantly higher than that of A,which shows that the alternative Ais a relatively successful alternative of missile design. However, the ‘‘individual regret”value of Ais also much higher than that of A,which leads to the global score of Abeing lower than that of A.It also shows that there are some defects in the design of alternative A,which makes the value of some certain criteria of Abe relatively low to the overall alternative.It can be seen that both the gained dominance score and the final global score of Aare less than zero by further analyzing the mathematical characteristics of the calculation results. This is because the alternative Adoes not have any advantage in overall performance compared with Aand A,which makes the gained dominance score of Aless than zero, thus reflecting A’s disadvantages. Moreover, after considering the impact of criteria with poor performance on the overall performance of the alternative A, the final global score of Ashould be less than zero.

The data in Table 5 are analyzed and the values of criteria c-cin alternative Aand alternative Aare compared. It can be seen that the values of some criteria of Aare higher than those of A, and the values of another part of criteria of Aare lower than those of A.The values of all criteria of alternative Aand alternative Aare relatively close overall.However,in the criteria values of Alower than those of A, it can be seen that the difference of cis relatively large.cis the number of fragments, which is determined by the number of prefabricated fragments when the missile warhead is designed. Therefore,this is a design criterion that is easier to modify.It can be seen from Table 5 that the number of fragments in alternative Ais as high as 4000. Therefore, the number of fragments in alternative Acan be increased by optimizing the design.

Assuming that the number of fragments in Ais increased from 2500 to 3500 through optimized design, the final global scores of A,Aand Aare-0.0814,0.0289 and 0.0029 respectively.In this case,when the defect of criterion cis eliminated,Ahas the highest global score,and the‘‘utility group”value is more obvious.When the number of fragments of Ais further increased to 4000, the final global scores of A, Aand Aare-0.0890,0.0753 and-0.0123 respectively.In this case,the global score of Ais much higher than the other two alternatives,and the alternative Acan be regarded as the optimal alternative of warhead design.

By comprehensively assessing the‘‘group utility”value and the ‘‘individual regret” value of the alternative of warhead design, the current optimal alternative can be obtained, and the potential optimal scheme can be obtained by improving the criteria with defect. If the criterion with defects is difficult to make up,even if the‘‘group utility”of the alternative is the highest, it can be avoided being chosen. This is the biggest advantage of the warhead power assessment process proposed in this paper.

Table 5 Original values of each criterion.

Table 6 Normalized values of each criterion.

5. Sensitivity analysis

The data in this paper come from two parts:the weights of criteria and the values of the criteria. When the alternatives of warhead design are determined, the values of the criteria itself are generally difficult to change. The weight comes from the subjective judgment of experts,so the value of the weight fluctuates during each assessment process, which may affect the assessment result. Whether the process proposed in this paper has good stability, the impact of the fluctuation of weights on the final result is an essential criterion. Therefore, it is necessary to test the extent of the impact of the fluctuation of weights on the final result and determine the reasonable range of the fluctuation of weights. The sensitivity of the change in weights to the assessment result should be analyzed. The change of weight of each criterion is shown in Table 7, and the step size of change is 5%.

As shown in Table 4, the weight value of criterion cis the largest,so the weight value of criterion cis taken as the main variable value, and the weight values of other criteria change simultaneously. In the existing research, it is very intuitive and effective to use the method of multiplying the parameter and the variable to be analyzed to control the change of the variable.Therefore, the parameter θ is introduced to control the change of weights. Criterion cand other criteria fluctuate according to

where θ is the fluctuation range of the weight value of each criterion except criterion c.As shown in Table 7,the fluctuation range is from 15% to -15%.

By substituting the weights of the criteria in Table 4 and the criteria values in Table 6 into Eqs.(20)-(23),the global score of A,Aand Acan be obtained as shown in Table 8 in the case of fluctuation of θ.

As shown in Fig.6,under the condition of fluctuation of θ,the trend of the global scores of A,Aand Acan be obtained.As θ fluctuates in the range of 15%to-15%,Ashows a fluctuating trend. However, the global score of Ahas been lower than the global score of Aand A. Therefore, the fluctuation of θ has little effect on alternative A. As criteria of Ahave little advantage in their own data, it will not be selected as the optimal alternative of warhead design.

The global scores of Aand Ashow opposite trends with the fluctuation of θ. As θ fluctuates from 15% to -15%, the global score of Ashows an upward trend, while the global score of Ashows a downward trend. When θ is greater than-5%, the global score of Ais higher than that of A. When θ is less than -5%, the global score of Ais lower than that of A. It can be seen from Eq. (25) that the fluctuation of θ has the greatest effect on the criterion c. It can be seen from Table 5 that the value of criterion cin alternative Ais higher than that in A.By combining Eq.(25)with Fig.6 to analyze,it can be seen that when θ is greater than-5%,the advantage of this single criterion is not enough to make the comprehensive advantage of Asurpass A.However,when θ is less than-5%, the weight of criterion cis already much higher than that of other criteria. Therefore, the advantage of criterion cin alternatives Aand Acan largely affect the global scores of Aand A,thereby changing the ranking of the global scores of Aand A.

Through the above analysis, it can be concluded that the warhead power assessment process proposed in this paper is sensitive to the weights of criteria. Since the initial weight of the criterion cis relatively large, the fluctuation of θ has a greater effect on the weight of c.For a criterion with a smaller initial weight, it is less affected by the fluctuation of θ. However, since the basic data for solving weights are derived from the subjective experience of experts,the experts still need to be cautious when making judgments.When the fluctuation of θ is in the range of 5%to-5%,it can be considered that it has no influence on the assessment result.

6. Comparative analysis

6.1. Comparison with other assessment processes

In this sub-section, the proposed assessment process is compared with two existing assessment processes to show the good performance. The first assessment process is TODIM, which inspires the proposal of GLDS. The second is TOPSIS, which is the general method to conduct assessment in many literatures. When conducting the two assessment processes in this sub-section, θ in Section 5 still fluctuates in the range of 15%to-15%,which is convenient for comparing and analyzing the assessment results of the three processes.

As shown in Figs. 6-8, the assessment result derived from the synthetical assessment process proposed in this paper is A＞A＞Awhen θ=0, and the results calculated using TODIM and TOPSIS are both A＞A＞A.As θ fluctuates between 15% and -15%, all the three assessment processes show that Ais the poorest alternative, which shows that Adoes not have any overall advantage over Aand A.However,with the fluctuation of θ,the changes of the assessment results of Aand Aare different in the three assessment procedures.When the assessment process with GLDS is used,it can be seen from the analysis of the results in Section 4.4 that the gained dominance score of Ais significantly higher than that of A.However, the lost dominance score of Ais also obviously higher than that of A, which makes the final global score of Alower than that of A.When θ is less than-5%,the impact of the criteria with poor performance on the final assessmentresult of the alternative is less than that of the overall advantages of the alternative, so there exists A＞A.

Table 7 Fluctuation value of the weight of each criterion.

Table 8 Fluctuation of the global value of each alternative.

Fig. 6 Fluctuation of global value of each alternative.

As shown in Fig.7,when the assessment process with TOPSIS is used,as θ fluctuates between 15%and-15%,the assessment result of Ais always higher than that of A. When θ fluctuates to 15%, the assessment result of Aand Aare the closest, but there is already a large error between the weight and the optimal weight at this time. This is because TOPSIS is a fully compensatory model. In the assessment process, the poor performance of the criteria will be fully compensated by the good performance of other criteria.Therefore,the TOPSIS method cannot be used to find the defects of the alternatives. When the alternative has a large number of criteria or the alternative contains many vital criteria, it is not appropriate to adopt this kind of fully compensatory method, which may cause serious consequences such as assessment failure or alternative cost maximization.

As shown in Fig.8,if the assessment process with TODIM is used, the final ranking of the alternatives is A＞A＞Awhen θ<10%is established and the final ranking of the alternatives is A＞A＞Awhen θ ＞10% is established. By combining Table 6 and Fig. 7, it can be seen that cis the criterion of the alternative Awith the best performance. With the increase of θ, the weight of cgradually decreases, and the overall advantage of alternative Agradually decreases.Therefore, there exists A＞A＞A. When -15%≤θ<10% is established, the overall advantage of Ais the highest among all alternatives, so there exists A＞A＞A.This is because TODIM has a function incomplete compensation. with the use of TODIM, the dominance score of each alternative to other alternatives under each criterion can be obtained, and then the overall dominance score of all alternatives can be obtained through summation. In this assessment process,the poor performance of criteria cannot be completely compensated by the good performance of other criteria. If the alternatives are not sensitive to poor performance,an accurate ranking can be obtained by using this method for assessment.Because the poor performance of the criteria will not cause any serious consequence,it is enough to compare the overall dominance score of the alternatives to get accurate results. However, this method is inapplicable to the assessment process of warhead power of anti-ship missile, which has a high cost of assessment failure. Moreover,TODIM does not have the normalization process of criteria in the calculation process,so different units and different orders of magnitude will affect the assessment results and reduce the stability of the assessment results.Based on the above analysis,the synthetical assessment process proposed in this paper has advantages in the assessment of warhead power of anti-ship missile.

Fig. 7 Fluctuation of result of each alternative with TOPSIS.

Fig. 8 Fluctuation of comprehensive value of each alternative with TODIM.

6.2. Comparison with test results

In order to test the advantage and accuracy of the assessment process proposed in this paper, experiments are carried out to test all the criteria in Fig.1.The alternative Aand alternative Aof warhead design are used as test alternatives. According to the test results, the assessment results using test data are compared with the assessment results in Section 4 to observe the accuracy of the assessment results. Experts are invited to directly assess the warhead power based on the results of the warhead damaging cabin, and the assessment results are compared with the assessment results in Section 4 to observe the advantage of the process proposed in this paper.

Combined with the historical data of actual projectile shooting, many experiments have been carried out in this paper and the required criteria have been tested. Tests including ‘‘hole effect” and other related content have been carried out. The content of each test and the corresponding carrier are shown in Table 9.

Fig.9 shows the exterior structure of the test cabin,Fig.10 shows the interior structure of the test cabin, and the hole in Fig. 10 is preset armor-piercing hole. The test conditions are:the ambient temperature is 26.8°C, the humidity is 58.7%RH, the air pressure is 975.9 hPa, the dew point is 18.00 Dp,and the wind speed is 0 m/s.

The test system is mainly composed of pressure sensor,lownoise cable and Digital Pressure Recorder (DPR) digital pressure recorder.The wall sensor is a pressure sensor with a range of 0.01-50 MPa. DPR is selected for data collection. In order to achieve system synchronization triggering,each DPR is connected in turn via the network synchronization line to form a closed-loop test network.

As shown in Fig. 11, in order to more accurately analyze the shock wave overpressure inside the cabin, several test points have been set.The lower cabin is described as an example. Since the lower cabin is symmetrical, 4 test points are set on the right bulkhead (D1) surface, 2 test points are set on the top bulkhead (D3) surface, and 1 test point is set on the rear bulkhead(D5).Fig.12 shows the overpressure-time curve of each test point in the lower cabin,which is the superposition effect of the incident shock wave overpressure and the multiple reflected shock wave overpressure on the bulkhead. Each test point reaches the peak point at different time according to the distance from the initiation point. Among them, point D3-1 reaches the peak point first, which is the point near the hole between the upper and lower cabin.The maximum shock wave overpressure peak in the cabin appears at points D1-4 at the corner where the three bulkheads D1, D3 and D5 meet.Fig. 13 shows a part of the cabin after being damaged.

As shown in Fig. 14, the three scores of the assessment results of this paper are compared with the three scores of the test control group. It can be seen from Fig. 14 that the assessment conclusion of the test control group is completely consistent with the assessment conclusion in Section 4.In Section 4,the‘‘group utility”value of both Aand Aare greater than the ‘‘individual regret” value, which makes the global scores of both Aand Apositive. And because the defects of Ain its criteria make ‘‘individual regret” value of Avery large, although the ‘‘group utility” value of Ais greater than that of A,this results in the fact that the global score of Ais still smaller than that of A.

The above conclusion is completely consistent in the test control group, and it is more obvious. In the test control group, the defects of A’s criteria are more obvious, which makes the‘‘individual regret”value of Aonly slightly smaller than the ‘‘group utility” value of A, resulting in the global score of Aclose to zero. Compared with alternative A, Adoes not have any criteria defect,which makes the global score of Ain the test control group larger than that of Ain Section 4, leading to the more obvious advantage of A.

Therefore, the assessment process proposed in this paper is accurate. Under the situation that the weights of criteria have not changed, small fluctuations in the criteria data will not change the assessment result. Moreover, when the assessment data are closer to the actual data,the advantages of the assess-ment process are more obvious,and the assessment results are more objective.

Table 9 Content of each test.

Fig. 9 Exterior structure of test cabin.

Fig. 10 Interior structure of test cabin.

Fig. 11 Layout of test points in the cabin.

Fig. 12 Overpressure-time curve of each test point in lower cabin.

It can be seen from Fig. 15 that the conclusion of the experts’ scoring is completely opposite to that of the assessment process proposed in this paper. To further analyze the reasons, several values in Fig. 15 are compared and analyzed.It can be seen that the‘‘group utility”value of Ain Fig.15 is higher than that of A,which is consistent with the conclusion of the expert’s scoring. In order to further analyze the reasons for the difference between the assessment results of the two methods, the basic data for assessment in Table 6 are normalized, and the scores of Aand Aare obtained by linear summation of the values of criteria and weights. The result is that the score of Ais 0.6231 and the score of Ais 0.5952.

From the above analysis, it can be concluded that the experts mainly considered the ‘‘group utility” of the alternatives when assessing.Although experts may consider the influence of the weight of a single criterion when making the assessment, this consideration is very limited. It is difficult for experts to pay too much attention to a specific criterion when assessing an alternative with a large number of criteria.Therefore,for criteria containing defects,the defects of the criteria are likely to be compensated by the excellent performance of other criteria during the assessment process. However, this is a flaw in most assessment processes. When the alternative involves more criteria,the compensation will be more obvious,and the defects of certain criteria will be hidden and more difficult to be discovered. In these cases, the advantages of the assessment process proposed in this paper will be proven again.

Fig. 13 A part of cabin after being damaged.

Fig. 14 Conclusion in this paper compared with conclusion of control group.

Fig. 15 Comparison of results of A2 and A3 under three methods.

7. Conclusions and future research work

The main contribution of this paper is to design a set of highprecision assessment process to assess the warhead power of the anti-ship missile, and then to judge the performance of the alternatives of warhead design. The main work is reflected in the following two aspects:

(1) The process of assessing the warhead power of the missile relies heavily on the experience of experts.Therefore,the use of the double hierarchy fuzzy hesitant linguistic term set to collect expert’s opinions can help improve the accuracy of expert’s opinions in assessment process.The use of evidence theory and entropy weight method can transmit the opinions of experts in the weight of criteria to the greatest extent, so that the assessment process makes full use of the subjective experience of experts. The comparative analysis shows that the conclusion of the assessment process of this paper is reliable.

(2) The analysis of calculation result and comparative analysis in this paper show that the process proposed in this paper has great advantage in selecting the optimal alternative of missile design. Because when designing the warhead of missile, not only the synthetical effects of the alternative must be considered, but also the poor influence of some specific criteria should be noticed.The gained and lost dominance score method can prevent the bad criteria in the alternatives from being compensated by the good performance of other high-quality criteria, which is helpful in selecting the potential optimal alternative of missile design and avoids the alternatives with low criteria performance that cannot be compensated by other criteria being selected.

Based on the above analysis, the synthetic assessment process proposed in this paper can be further studied in two aspects.The first is to consider the interdependence of criteria,which helps to quantitatively describe the influence of interdependence between criteria on weights,thereby further improving the accuracy of the assessment results. The second is to consider using a unified model to deal with the psychological effects of experts in assessment process, thus improving the accuracy of experts’ assessment opinions. On the basis of the process proposed in this paper,the accuracy of the assessment results can be further improved by studying these two aspects in future work.

The criteria of the research object in this paper are static criteria, which means that the value of the criteria will not change in the time domain. Therefore, this factor should be paid attention to when making the assessment. On this basis,by collecting reasonable assessment criteria and establishing correct assessment criteria system, the synthetic assessment process proposed in this paper can also be applied to other assessment problem such as the reliability assessment and safety assessment of missile systems.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.