Research ArticleAircraft Combat Survivability Calculation Based onCombination Weighting and Multiattribute Intelligent GreyTarget Decision Model
Lintong Jia Zhongxiang Tong Chaozhe Wang and Shenbo Li
Aeronautics and Astronautics Engineering Institute Air Force Engineering University Xirsquoan 710037 China
Correspondence should be addressed to Lintong Jia jialintong406163com
Received 20 September 2015 Revised 22 December 2015 Accepted 13 January 2016
Academic Editor Danielle Morais
Copyright copy 2016 Lintong Jia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Aircraft combat survivability is defined as the capability of an aircraft to avoid or withstand aman-made hostile environment whichhas been increasingly important In order to give a rational calculation of aircraft combat survivability an integrated method basedon combination weighting andmultiattribute intelligent grey target decision model is proposed Firstly an evaluation index systemcontaining susceptibility index and vulnerability index as well as their subindexes is established Then a multiattribute intelligentgrey target decision model is introduced A combination weighting method is brought up based on a modified AHP (analytichierarchy process) method and the entropymethod offering a rational weight for various indexes Finally utilize the multiattributeintelligent grey target decisionmodel to assess the aircraft combat survivability of aircraft verified by a practical case of five aircraftThe results show that the proposed method is effective and has a great value in engineering application which will provide usefulreferences for other projectsrsquo evaluation
1 Introduction
Survivability is defined as the capability of a system includingits crew to avoid or withstand a hostile environment withoutsuffering an abortive impairment of its ability to accomplishits designated mission [1] For the aircraft combat survivabil-ity Robert E Ball defined it as the capability of an aircraft toavoid or withstand a man-made hostile environment Withthe development of precision guided weapons particularlyradar-guided missiles and infrared-guided missiles aircraftcombat survivability is becoming increasingly importantTheAmerican Department of Defense has taken aircraft combatsurvivability as a basic design criterion For example theNavy MIL-HDBK-2069-1997 aircraft survivability stipulatesthat the survivability criterion should be carried out through-out the cycle lifeThenewest combat aircraft FA-18EF SuperHornets FA-22 Raptor and F-35 Lighting II to name a fewhas adopted survivability strengthening measures from theinitial research phases
Usually survivability can be subdivided into susceptibilityand vulnerability referring to the inability of an aircraft to
avoid and withstand the man-made hostile environmentrespectively Aircraft combat survivability can be also definedas the probabilistic values that the aircraft would survive inman-made hostile environment with the antithesis killabilityThemore susceptible and vulnerable the aircraft in the hostileenvironment the more killable and lower survivable theaircraft
Wang et al construct an analytic model for aircraftsurvivability assessment based on the theory of stochasticduel considering the encounter process [2] Konokman et alcarry out the aircraft survivability analysis considering vul-nerability against fragmenting warhead threat [3] Li et alpropose a vulnerability modeling and computation methodbased on product structure and CATIA and assess the effectsof redundant technology [4] Erlandsson andNiklasson arguea five-state survivability model including undetected statedetected state tracked state engaged state and hit state [5]Shi et al build an aircraft antagonistic model and a warfaremodel based on the agent theory [6]These simulationmeth-ods bring a great deal of calculation and complex procedure
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8934749 9 pageshttpdxdoiorg10115520168934749
2 Mathematical Problems in Engineering
Aircraft combatsurvivability
Susceptibility
Infrared signal character
Visual signal character
Maneuverability character
ECM character
Radar signal character
Vulnerability
Ratio of fatal components
Redundant ratio of fatal components
Average safety factor
Shelter ratio of fatal components
Average killing probability
Objective Main index Subindex
First evaluation
Second evaluation
Figure 1 Aircraft combat survivability evaluation index system
which impose restrictions on the application especially in theinitial research without accurate data
Multicriteria decision method is a useful replacementthrough ranking and selecting a finite number of alter-native plans [7] Grey target theory is the application ofnonuniqueness principle in decision theory which has beenused in many good fields A new multiattribute intelligentgrey target decision model is introduced into the aircraftcombat survivability analysis In this paper firstly the aircraftcombat survivability evaluation index system is establishedcontaining susceptibility index and vulnerability index Sec-ondly a multiattribute intelligent grey target decision modelis established based on the index systemThen a combinationweighting method is brought up based on a modified AHP(analytic hierarchy process)method and the entropymethodoffering the combination weight to the multiattribute intel-ligent grey target decision model In the end a numericalexample containing five aircraft is given proving that thismethod is practicable and effective
2 Aircraft Combat SurvivabilityEvaluation Index System
Survivability can be subdivided into susceptibility and vul-nerability So the index system contains susceptibility indexand vulnerability indexThe index is established in hierarchyThe main index is susceptibility index and vulnerabilityindex The subindexes of susceptibility contain radar signal
character infrared signal character visual signal charactermaneuverability character and ECM (electronic counter-measure) character while the subindexes of vulnerabilitycontain ratio of fatal components redundant ratio of fatalcomponents shelter ratio of fatal components average killingprobability and average safety factor as is shown in Figure 1The details are as follows
21 Aircraft Susceptibility Subindexes Considering the char-acter of the aircraft radar signal character infrared signalcharacter visual signal character maneuverability characterand ECM character are brought out
Radar Signal Character Radar signal is the most importantsignal character of the aircraft for locating identifying andtracking Radar is one of the most lethal threats for aircraftFor example earlywarning radars can provide airborne targetinformation out of hundreds of miles while ground controlinterceptor can even provide an accurate target locationFrom a simplified form of radar range equation in (1) wecan see that RCS (radar cross section) is the one and onlycontrollable parameter for the aircraft
1198774
max
=
(1198751199031198622
1199051205822
(4120587)2
sdot1
119878min119871) (120590) 119865
2
11198652
2
Radar Capability RCS Atmospheric propagation
(1)
Mathematical Problems in Engineering 3
where 119877max is the maximum range at which an aircraft canbe detected 119862
119905is the characteristic parameter of radar which
varies for different type and 1198651and 119865
2are the propagation
loss of signal emission and returnSo we choose radar cross section as the evaluation index
denoted by the sign 11987811with the unit square meter m2
Infrared Signal Character Infrared signal is another impor-tant signal character of the aircraft With the developmentof stealth and antistealth techniques of radars the aircraftrsquosRCS has decreased significantly However compared withthe environment infrared signal is still significant even forthe stealth aircraft In the war area for 20 years about 90of airplanes were damaged by the infrared-guided missile[8] Nowadays IRST (infrared search and track) systemFLIR (forward looking infrared) system and infrared-guidedmissiles have been significant threats by accurately locatingaircraft For point source infrared detector the infrared rangeequation under uniform background can be written as 119877 prop
radic119868 where 119877 is the range at which an aircraft can be detectedand 119868 is the aircraftrsquos infrared radiation intensity For mostaircraft the engines are the largest sources of thermal energySo we choose turbine inlet temperature as the evaluationindex denoted by the sign 119878
12with the unit Kelvin K
Visual Signal Character Visual signal is another importantfactor in determining overall aircraft detectability [9] Aircombat in visual range is still essential and visual acquisitionbefore launch may be required Contrails engine exhaustglow cockpit lighting and luminescence may provide visualcues Here we choose size factor which is closely related tothe probability of visual detection as the evaluation indexdenoted by the sign 119878
13with the unit meter m Size factor is
defined as
11987813
=3radic3 lowast 119871 lowast 119867 lowast119882
4 lowast 120587 (2)
where 119871 is the length of aircraft 119867 is the height of aircraftand119882 is the wingspan of aircraft
Maneuverability Character Maneuverability is an effectivemeans of defense for the aircraft against detection andattack Supersonic maneuver is now the standard of newgeneration aircraft Supersonic maneuver gives an aircraft alower susceptibility and a higher survivability Maneuverabil-ity character can be defined with the maximum allowableoverload 119899
119910max the maximum steady turn overload 119899cir andspecific excess power SEP as is shown in the following
11987814
= 119899119910max + 119899cir + SEP times
9
300 (3)
Electronic Countermeasure Character Electronic counter-measure plays an important role in modern military affairsincluding active and passive jamming which is an effectivemean to decrease aircraft susceptibility and enhance aircraftsurvivability Electronic countermeasure equipment con-tains omnidirectional radar warning equipment radar chaffdispensing device infrared jammer and infrared-guided
Table 1 Electronic countermeasure subindexes of the aircraft
Number Airborne electroniccountermeasure equipment 119878
15
1 Omnidirectional radar warningequipment 105
2 Ibid + passive jamming dispensingdevice 110
3 Ibid + active infrared andelectromagnetic jammer 115
4 Ibid + missile approach warningdevice 120
Table 2 Aircraft vulnerability subindexes and definition
Subindexes Sign DefinitionRatio of fatal components
11987821
11987821=119873119865
119873
Redundant ratio of fatal components11987822
11987822=119873119877
119873119865
Shelter ratio of fatal components11987823
11987823=
119873119865
sum
119894=1
119873119878119894
119873119865
Average killing probability11987824
11987824=
119873119865
sum
119894=1
119875119870119894
119873119865
Average safety factor11987825
11987825=
119873119865
sum
119894=1
(119899119863119894
119899119863119894max
)
missiles However for electronic countermeasure capabilitywe can only give a fuzzy value Electronic countermeasurecharacter can be denoted by the sign 119878
15with dimensionless
unit A typical value can be read in Table 1
22 Aircraft Vulnerability Subindexes Aircraft vulnerabilityrefers to the inability of an aircraft to withstand the man-made hostile environment which lies on ratio of fatal com-ponents redundant ratio of fatal components shelter ratioof fatal components average killing probability and averagesafety factor These factors can be defined as is shown inTable 2 where 119873
119865is the numbers of fatal components119873 is
the numbers of whole components and119873119878is the numbers of
redundant fatal components 119873119878119894is the redundant degree of
the 119894th fatal component 119875119870119894
is the killing probability of the119894th component while the aircraft is hit 119899
119863119894and 119899
119863119894max arethe designed overload andmaximum overload of the 119894th fatalcomponent respectively
Although there are clear equations for the calculation ofaircraft vulnerability subindexes the value of every parameteris comparatively subjective for different definition such asldquofatalrdquo and different granularity analysis
3 A Multiattribute Intelligent GreyTarget Decision Model
Here we introduce a multiattribute intelligent grey targetdecision model proposed by Liu et al [10] This model takes
4 Mathematical Problems in Engineering
the situation of the shoot and miss of the bullrsquos eye of theobjectiversquos effect value and vector based on four kinds ofuniform effect measures
31 Problem Description Assume that 119860 = 1198861 1198862 119886
119894119895in objective 119896 referring to the similar
level or the removed level between the sample and the criticalsample
Assume that 119889(119896)1
and 119889(119896)2
are the upper and lower criticalvalue 119904
119894119895in objective 119896 Then 119878
1= 119903 | 119889
(119896)
1le 119903 le 119889
(119896)
2 is
the one-dimension grey target and 120583(119896)
119894119895
isin [119889(119896)
1 119889(119896)
2] is the
pleased effect in objective 119896Multidimensional grey target canbe discussed in the same way Details are shown in [10]
32 Uniform Effect Measures Considering the decisionobjectives with different meaning different dimensionandor different nature the effect value of the objectiveshould be transferred to the uniform effect measures
Assume that 11990611989611989401198950
is the critical value of objective 119896 andthen the grey objective decision of 119896 is designed as fol-lows
119906119896
119894119895isin
[119906119896
11989401198950max119894
max119895
119906119896
119894119895] 119896 isin BTO
[min119894
min119895
119906119896
119894119895 119906119896
11989401198950] 119896 isin CTO
[119860 minus 119906119896
11989401198950 119860 + 119906
119896
11989401198950 ] 119896 isin MTO
(4)
where 119896 isin BTOmeans that objective 119896 belongs to benefit typeobjective 119896 isin CTO means that objective 119896 belongs to costtype objective and 119896 isin MTO means that objective 119896 belongsto moderate type objective (same as what follows) And 119860 isthe moderate value of the moderate type objective
For the decision objective of benefit type and cost typethe effect measures 119903119896
119894119895can be shown as
119903119896
119894119895=
119906119896
119894119895minus 119906119896
11989401198950
max119894max119895119906119896
119894119895 minus 119906119896
11989401198950
119896 isin BTO
119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950minusmin
119894min119895119906119896
119894119895
119896 isin CTO
(5)
For the decision objective of moderate type the effectmeasure can be divided as the upper effect measure and thelower effect measure according to the scale of effect measure119903119896
119894119895 which can be shown as follows
119903119896
119894119895=
119906119896
119894119895minus 119860 + 119906
119896
11989401198950
119906119896
11989401198950
119906119896
119894119895isin [119860 minus 119906
119896
11989401198950 119860] lower effect measure
119860 + 119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950
119906119896
119894119895isin [119860 119860 + 119906
119896
11989401198950] upper effect measure
(6)
The effectmeasure of the decision objective of benefit typereflects the similar level between the sample and the biggestsample as well as the removed level between the sampleand the critical sample The effect measure of the decisionobjective of cost type reflects the similar level between thesample and the smallest sample as well as the removed levelbetween the sample and the critical sample The lower effectmeasure of the decision objective of moderate type reflectsthe level between the sample less than moderate value 119860 andthe lower critical sample The upper effect measure of thedecision objective of moderate type reflects the level betweenthe sample greater than moderate value 119860 and the uppercritical sample
The decision objective of benefit type is a type of objectivewith an expectance the bigger the better or the more thebetterThe decision objective of cost type is a type of objectivewith an expectance the smaller the better or the less the betterThe decision objective of moderate type is a type of objectivewith an expectance neither too big nor too small or neithertoo many nor too few
The miss of the bullrsquos eye under different conditions canbe shown as
119906119896
119894119895lt 119906119896
11989401198950119896 isin BTO
119906119896
119894119895gt 119906119896
11989401198950119896 isin CTO
119906119896
119894119895lt 119860 minus 119906
119896
11989401198950
or 119906119896119894119895gt 119860 + 119906
119896
11989401198950
119896 isin MTO
(7)
The effect measure 119903119896
119894119895is satisfied with the following
requirements(1) 119903119896
119894119895is dimensionless (2) 119903
119896
119894119895is a uniform variable
namely 119903119896119894119895isin [minus1 1] (3) The greater the 119903119896
119894119895 the more ideal
the effect
Mathematical Problems in Engineering 5
Thus in order to satisfy the standardization the selectionof critical value 119906119896
11989401198950usually satisfies the following
119906119896
119894119895ge minusmax119894
max119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin BTO
119906119896
119894119895le minusmin119894
min119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin CTO
119906119896
119894119895ge 119860 minus 2 lowast 119906
119896
11989401198950
or 119906119896119894119895le 119860 + 2 lowast 119906
119896
11989401198950
119896 isin MTO
(8)
Assume that 120596119896is the decision weight of objective 119896 and
sum119904
119896=1120596119896= 1 Thus 119903
119894119895= sum119904
119896=1120596119896lowast 119903119896
119894119895will be the synthetic
effect measures of decision approach 119904119894119895and 119877 = 119903
119894119895 will be
the matrix of synthetic effect measures of decision set 119878The synthetic effect measure 119903
119894119895is satisfied with the
following requirements(1) 119903119894119895is dimensionless (2) 119903
119894119895is a uniform variable
namely 119903119894119895isin [minus1 1] (3) The greater the 119903
119894119895 the more ideal
the effectWhile 119903
119894119895isin [minus1 0] means the miss of the bullrsquos eye
and 119903119894119895isin [0 1] means the hit of the bullrsquos eye through the
comparison of the value of 119903119894119895 we can judge the performance
of 1198861198940 1198871198950 and 119904
11989401198950according to the definition shown as
follows
Definition 1 1198871198950
is the best decision of event 119886119894
ifmax1le119895le119898
119903119894119895 = 119903
1198941198950 1198861198940
is the best event correspondingwith the decision 119887
1198950if max
1le119894le119899119903119894119895 = 119903
1198940119895 11990411989401198950
is the bestdecision approach if max
1le119894le119899max1le119895le119898
119903119894119895 = 11990311989401198950
33 Algorithm Steps Algorithm steps of the multiattributeintelligent grey target decision model are as follows
Step 1 Form the decision set 119878 = 119904119894119895= (119886119894 119887119895) | 119886119894isin 119860 119887
Step 3 Confirm the decision weight of objective 119896 120596119896(119896 =
1 2 119904)
Step 4 Form the matrix 119880119896
= 119906119896
119894119895 of effect measures of
decision set 119878
Step 5 Set the critical value of the objective
Step 6 Obtain the uniform matrix 119877119896
= 119903119896
119894119895 of effect
measures
Step 7 Obtain the uniformmatrix119877 = 119903119894119895 of synthetic effect
measures
Step 8 Confirm the best decision 1198871198950
or the best decisionapproach 119904
11989401198950according to the definitions
Table 3 Exponential scale the golden section
Exponential scale One factor compared to another1000 Equally important1618 Slightly more important2618 More important4236 Greatly more important6854 Absolutely more importantReciprocal Reversed scale
4 Combination Weighting Method
In Step 3 the decision weight of objective 119896 120596119896is needed
In this paper according to requirements of the evaluation ofaircraft combatant survivability the evaluation system con-sists of various criteria with unequal importance which has abiggish influence on the evaluation So a rational weighting isnecessary In order to elicit weights a combination weightingmethod is brought up based on a modified AHP (analytichierarchy process) method and the entropy method offeringthe subjective weight and the objective weight respectively
41 Analytic Hierarchy Process Based on Exponential ScaleThe AHP is a proven decision-making tool integratingthe quantitative analysis and qualitative analysis togetherconsidering the quantitative weight and qualitative weightinto the decision-making process The AHP has been usedwidely in a far-ranging field especially in solving complexdecision-making problems with numerous criteriaThe AHPestablishes a hierarchical structure showing the relationshipof the target criteria and index using the decision matrixTheAHP can be broken into five steps as building a hierarchymaking comparisons calculating weights checking consis-tency and producing the result The detailed steps are shownin [11]
Dr Saaty developed a 9-point scale in the pairwisecomparisons which states whether one factor compared toanother is important or not Assign value of 1 to equallyimportant and values of 3 5 7 and 9 to slightly moreimportant more important greatly more important andabsolutely more important Values of 2 4 6 and 8 arereserved for intermediate values The 9-point scale gives thedifference of importance but the ratio of importance in thepairwise comparisons is what we need according to the AHPSo the AHP based on exponential scale is built Introduce theratio of importance 120572 into the comparison as the exponentialscale Replace the values of 1 3 5 7 and 9 with the valuesof 1 120572 1205722 1205723 and 120572
4 Assume 120572meets the rule of ladder byleaps namely 119886119896 = 119886
119896minus1+ 119886119896minus2 where 119896 isin 2 3 4 5 It turns
out to be that 119886 = 1618 which satisfied the golden section ofimportanceThe reciprocal scale is achievedwhen comparingthe factors in the opposite direction that is if 119860 is moreimportant than 119861 (2618) 119861 could be said to be less importantthan 119860 (12618) The exponential scale is shown in Table 3
For the given matrix 119872 through pairwise comparisonsthe vector of weights 120596
lowast is the normalized eigenvector ofthe matrix associated with the largest eigenvalue 120582max using
6 Mathematical Problems in Engineering
Table 4 Susceptibility subindexes of different aircraft
= 120582max120596lowast Thus after calculating and
checking of consistency in the matrix the result is produced
42 Entropy Weight Method Entropy according to Shan-nonrsquos information theory reflects the degree of disorder ofinformation [12] In the information system the smaller theentropy value the greater the degree of order and the greaterthe amount of informationThe entropy weight method is aneffective method calculating the objective weight just cor-relating with the data distribution of the evaluation matrixwithout relying on the subjective preference of decision-maker
Suppose that there is a matrix 119872 = [119898119894119895]119899times119898
isthe normalized matrix of119872 So the entropy of the 119895th indexis defined as
119864119895= minus
1
log 119899
119899
sum
119894=1
119901119894119895log119901119894119895 (9)
Then the entropy weight of 119895th index is defined as
1205961015840
119895=
(1 minus 119864119895)
(119898 minus sum119898
119895=1119864119895)
(10)
43 Algorithm Steps The AHP based on exponential scaleand entropy weight method are both available in processingthe weight however they have their own advantages anddisadvantagesThe combinationweightingmethod can evadethe disadvantages The effective combination of subjectiveweight and objective weight reconciles the expertrsquos preferencefor the index and the decrease of fuzzy random color thusproducing an advantage of both weighting methods So the
combination weighting method produces a scientific andrational weight
The combination weight can be defined as
120596119895= 120582120596lowast
119895+ (1 minus 120582) 120596
1015840
119895 (11)
where 120596119895is the combination weight of the 119895th index 120596
lowast
119895is
the subjective weight given by the AHP based on exponentialscale 1205961015840
119895is the objective weight given by the entropy weight
method120582 is the share of subjectiveweight in the combinationweight
5 Evaluation of Aircraft Survivability
51 Data Collection and Pretreatment Through literaturereview data of five aircraft is acquired in susceptibilitysubindexes which is shown in Table 4
But for the vulnerability subindexes as can be seen everwhich are comparatively subjective we cannot give a precisevalue The vague set is an effective way to solve this problem[13] For the restriction of the length of this paper the detailswill not be discussed here We know that linguistic variablesare available in certainty Seven-grade linguistic variables119878 = VGG FGM FPPVP presented in vague values areshown in Table 5
However with vague value and precise value in oneevaluation it is hard to handleHerewe can transfer the usefulinformation of vague value into a precise value through ascore functionWe know that a vague value such as [04 07]can be interpreted as ldquothe vote for a resolution is 4 in favor3 against and 3 abstentions [13]rdquo Score function can give acertain value of the vague set correlated with a certain valueof the fuzzy set through the undefined and unascertainedinformation of the vague set
A proved score function contrived by Guo et al [14] isgiven as
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
Figure 1 Aircraft combat survivability evaluation index system
which impose restrictions on the application especially in theinitial research without accurate data
Multicriteria decision method is a useful replacementthrough ranking and selecting a finite number of alter-native plans [7] Grey target theory is the application ofnonuniqueness principle in decision theory which has beenused in many good fields A new multiattribute intelligentgrey target decision model is introduced into the aircraftcombat survivability analysis In this paper firstly the aircraftcombat survivability evaluation index system is establishedcontaining susceptibility index and vulnerability index Sec-ondly a multiattribute intelligent grey target decision modelis established based on the index systemThen a combinationweighting method is brought up based on a modified AHP(analytic hierarchy process)method and the entropymethodoffering the combination weight to the multiattribute intel-ligent grey target decision model In the end a numericalexample containing five aircraft is given proving that thismethod is practicable and effective
2 Aircraft Combat SurvivabilityEvaluation Index System
Survivability can be subdivided into susceptibility and vul-nerability So the index system contains susceptibility indexand vulnerability indexThe index is established in hierarchyThe main index is susceptibility index and vulnerabilityindex The subindexes of susceptibility contain radar signal
character infrared signal character visual signal charactermaneuverability character and ECM (electronic counter-measure) character while the subindexes of vulnerabilitycontain ratio of fatal components redundant ratio of fatalcomponents shelter ratio of fatal components average killingprobability and average safety factor as is shown in Figure 1The details are as follows
21 Aircraft Susceptibility Subindexes Considering the char-acter of the aircraft radar signal character infrared signalcharacter visual signal character maneuverability characterand ECM character are brought out
Radar Signal Character Radar signal is the most importantsignal character of the aircraft for locating identifying andtracking Radar is one of the most lethal threats for aircraftFor example earlywarning radars can provide airborne targetinformation out of hundreds of miles while ground controlinterceptor can even provide an accurate target locationFrom a simplified form of radar range equation in (1) wecan see that RCS (radar cross section) is the one and onlycontrollable parameter for the aircraft
1198774
max
=
(1198751199031198622
1199051205822
(4120587)2
sdot1
119878min119871) (120590) 119865
2
11198652
2
Radar Capability RCS Atmospheric propagation
(1)
Mathematical Problems in Engineering 3
where 119877max is the maximum range at which an aircraft canbe detected 119862
119905is the characteristic parameter of radar which
varies for different type and 1198651and 119865
2are the propagation
loss of signal emission and returnSo we choose radar cross section as the evaluation index
denoted by the sign 11987811with the unit square meter m2
Infrared Signal Character Infrared signal is another impor-tant signal character of the aircraft With the developmentof stealth and antistealth techniques of radars the aircraftrsquosRCS has decreased significantly However compared withthe environment infrared signal is still significant even forthe stealth aircraft In the war area for 20 years about 90of airplanes were damaged by the infrared-guided missile[8] Nowadays IRST (infrared search and track) systemFLIR (forward looking infrared) system and infrared-guidedmissiles have been significant threats by accurately locatingaircraft For point source infrared detector the infrared rangeequation under uniform background can be written as 119877 prop
radic119868 where 119877 is the range at which an aircraft can be detectedand 119868 is the aircraftrsquos infrared radiation intensity For mostaircraft the engines are the largest sources of thermal energySo we choose turbine inlet temperature as the evaluationindex denoted by the sign 119878
12with the unit Kelvin K
Visual Signal Character Visual signal is another importantfactor in determining overall aircraft detectability [9] Aircombat in visual range is still essential and visual acquisitionbefore launch may be required Contrails engine exhaustglow cockpit lighting and luminescence may provide visualcues Here we choose size factor which is closely related tothe probability of visual detection as the evaluation indexdenoted by the sign 119878
13with the unit meter m Size factor is
defined as
11987813
=3radic3 lowast 119871 lowast 119867 lowast119882
4 lowast 120587 (2)
where 119871 is the length of aircraft 119867 is the height of aircraftand119882 is the wingspan of aircraft
Maneuverability Character Maneuverability is an effectivemeans of defense for the aircraft against detection andattack Supersonic maneuver is now the standard of newgeneration aircraft Supersonic maneuver gives an aircraft alower susceptibility and a higher survivability Maneuverabil-ity character can be defined with the maximum allowableoverload 119899
119910max the maximum steady turn overload 119899cir andspecific excess power SEP as is shown in the following
11987814
= 119899119910max + 119899cir + SEP times
9
300 (3)
Electronic Countermeasure Character Electronic counter-measure plays an important role in modern military affairsincluding active and passive jamming which is an effectivemean to decrease aircraft susceptibility and enhance aircraftsurvivability Electronic countermeasure equipment con-tains omnidirectional radar warning equipment radar chaffdispensing device infrared jammer and infrared-guided
Table 1 Electronic countermeasure subindexes of the aircraft
Number Airborne electroniccountermeasure equipment 119878
15
1 Omnidirectional radar warningequipment 105
2 Ibid + passive jamming dispensingdevice 110
3 Ibid + active infrared andelectromagnetic jammer 115
4 Ibid + missile approach warningdevice 120
Table 2 Aircraft vulnerability subindexes and definition
Subindexes Sign DefinitionRatio of fatal components
11987821
11987821=119873119865
119873
Redundant ratio of fatal components11987822
11987822=119873119877
119873119865
Shelter ratio of fatal components11987823
11987823=
119873119865
sum
119894=1
119873119878119894
119873119865
Average killing probability11987824
11987824=
119873119865
sum
119894=1
119875119870119894
119873119865
Average safety factor11987825
11987825=
119873119865
sum
119894=1
(119899119863119894
119899119863119894max
)
missiles However for electronic countermeasure capabilitywe can only give a fuzzy value Electronic countermeasurecharacter can be denoted by the sign 119878
15with dimensionless
unit A typical value can be read in Table 1
22 Aircraft Vulnerability Subindexes Aircraft vulnerabilityrefers to the inability of an aircraft to withstand the man-made hostile environment which lies on ratio of fatal com-ponents redundant ratio of fatal components shelter ratioof fatal components average killing probability and averagesafety factor These factors can be defined as is shown inTable 2 where 119873
119865is the numbers of fatal components119873 is
the numbers of whole components and119873119878is the numbers of
redundant fatal components 119873119878119894is the redundant degree of
the 119894th fatal component 119875119870119894
is the killing probability of the119894th component while the aircraft is hit 119899
119863119894and 119899
119863119894max arethe designed overload andmaximum overload of the 119894th fatalcomponent respectively
Although there are clear equations for the calculation ofaircraft vulnerability subindexes the value of every parameteris comparatively subjective for different definition such asldquofatalrdquo and different granularity analysis
3 A Multiattribute Intelligent GreyTarget Decision Model
Here we introduce a multiattribute intelligent grey targetdecision model proposed by Liu et al [10] This model takes
4 Mathematical Problems in Engineering
the situation of the shoot and miss of the bullrsquos eye of theobjectiversquos effect value and vector based on four kinds ofuniform effect measures
31 Problem Description Assume that 119860 = 1198861 1198862 119886
119894119895in objective 119896 referring to the similar
level or the removed level between the sample and the criticalsample
Assume that 119889(119896)1
and 119889(119896)2
are the upper and lower criticalvalue 119904
119894119895in objective 119896 Then 119878
1= 119903 | 119889
(119896)
1le 119903 le 119889
(119896)
2 is
the one-dimension grey target and 120583(119896)
119894119895
isin [119889(119896)
1 119889(119896)
2] is the
pleased effect in objective 119896Multidimensional grey target canbe discussed in the same way Details are shown in [10]
32 Uniform Effect Measures Considering the decisionobjectives with different meaning different dimensionandor different nature the effect value of the objectiveshould be transferred to the uniform effect measures
Assume that 11990611989611989401198950
is the critical value of objective 119896 andthen the grey objective decision of 119896 is designed as fol-lows
119906119896
119894119895isin
[119906119896
11989401198950max119894
max119895
119906119896
119894119895] 119896 isin BTO
[min119894
min119895
119906119896
119894119895 119906119896
11989401198950] 119896 isin CTO
[119860 minus 119906119896
11989401198950 119860 + 119906
119896
11989401198950 ] 119896 isin MTO
(4)
where 119896 isin BTOmeans that objective 119896 belongs to benefit typeobjective 119896 isin CTO means that objective 119896 belongs to costtype objective and 119896 isin MTO means that objective 119896 belongsto moderate type objective (same as what follows) And 119860 isthe moderate value of the moderate type objective
For the decision objective of benefit type and cost typethe effect measures 119903119896
119894119895can be shown as
119903119896
119894119895=
119906119896
119894119895minus 119906119896
11989401198950
max119894max119895119906119896
119894119895 minus 119906119896
11989401198950
119896 isin BTO
119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950minusmin
119894min119895119906119896
119894119895
119896 isin CTO
(5)
For the decision objective of moderate type the effectmeasure can be divided as the upper effect measure and thelower effect measure according to the scale of effect measure119903119896
119894119895 which can be shown as follows
119903119896
119894119895=
119906119896
119894119895minus 119860 + 119906
119896
11989401198950
119906119896
11989401198950
119906119896
119894119895isin [119860 minus 119906
119896
11989401198950 119860] lower effect measure
119860 + 119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950
119906119896
119894119895isin [119860 119860 + 119906
119896
11989401198950] upper effect measure
(6)
The effectmeasure of the decision objective of benefit typereflects the similar level between the sample and the biggestsample as well as the removed level between the sampleand the critical sample The effect measure of the decisionobjective of cost type reflects the similar level between thesample and the smallest sample as well as the removed levelbetween the sample and the critical sample The lower effectmeasure of the decision objective of moderate type reflectsthe level between the sample less than moderate value 119860 andthe lower critical sample The upper effect measure of thedecision objective of moderate type reflects the level betweenthe sample greater than moderate value 119860 and the uppercritical sample
The decision objective of benefit type is a type of objectivewith an expectance the bigger the better or the more thebetterThe decision objective of cost type is a type of objectivewith an expectance the smaller the better or the less the betterThe decision objective of moderate type is a type of objectivewith an expectance neither too big nor too small or neithertoo many nor too few
The miss of the bullrsquos eye under different conditions canbe shown as
119906119896
119894119895lt 119906119896
11989401198950119896 isin BTO
119906119896
119894119895gt 119906119896
11989401198950119896 isin CTO
119906119896
119894119895lt 119860 minus 119906
119896
11989401198950
or 119906119896119894119895gt 119860 + 119906
119896
11989401198950
119896 isin MTO
(7)
The effect measure 119903119896
119894119895is satisfied with the following
requirements(1) 119903119896
119894119895is dimensionless (2) 119903
119896
119894119895is a uniform variable
namely 119903119896119894119895isin [minus1 1] (3) The greater the 119903119896
119894119895 the more ideal
the effect
Mathematical Problems in Engineering 5
Thus in order to satisfy the standardization the selectionof critical value 119906119896
11989401198950usually satisfies the following
119906119896
119894119895ge minusmax119894
max119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin BTO
119906119896
119894119895le minusmin119894
min119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin CTO
119906119896
119894119895ge 119860 minus 2 lowast 119906
119896
11989401198950
or 119906119896119894119895le 119860 + 2 lowast 119906
119896
11989401198950
119896 isin MTO
(8)
Assume that 120596119896is the decision weight of objective 119896 and
sum119904
119896=1120596119896= 1 Thus 119903
119894119895= sum119904
119896=1120596119896lowast 119903119896
119894119895will be the synthetic
effect measures of decision approach 119904119894119895and 119877 = 119903
119894119895 will be
the matrix of synthetic effect measures of decision set 119878The synthetic effect measure 119903
119894119895is satisfied with the
following requirements(1) 119903119894119895is dimensionless (2) 119903
119894119895is a uniform variable
namely 119903119894119895isin [minus1 1] (3) The greater the 119903
119894119895 the more ideal
the effectWhile 119903
119894119895isin [minus1 0] means the miss of the bullrsquos eye
and 119903119894119895isin [0 1] means the hit of the bullrsquos eye through the
comparison of the value of 119903119894119895 we can judge the performance
of 1198861198940 1198871198950 and 119904
11989401198950according to the definition shown as
follows
Definition 1 1198871198950
is the best decision of event 119886119894
ifmax1le119895le119898
119903119894119895 = 119903
1198941198950 1198861198940
is the best event correspondingwith the decision 119887
1198950if max
1le119894le119899119903119894119895 = 119903
1198940119895 11990411989401198950
is the bestdecision approach if max
1le119894le119899max1le119895le119898
119903119894119895 = 11990311989401198950
33 Algorithm Steps Algorithm steps of the multiattributeintelligent grey target decision model are as follows
Step 1 Form the decision set 119878 = 119904119894119895= (119886119894 119887119895) | 119886119894isin 119860 119887
Step 3 Confirm the decision weight of objective 119896 120596119896(119896 =
1 2 119904)
Step 4 Form the matrix 119880119896
= 119906119896
119894119895 of effect measures of
decision set 119878
Step 5 Set the critical value of the objective
Step 6 Obtain the uniform matrix 119877119896
= 119903119896
119894119895 of effect
measures
Step 7 Obtain the uniformmatrix119877 = 119903119894119895 of synthetic effect
measures
Step 8 Confirm the best decision 1198871198950
or the best decisionapproach 119904
11989401198950according to the definitions
Table 3 Exponential scale the golden section
Exponential scale One factor compared to another1000 Equally important1618 Slightly more important2618 More important4236 Greatly more important6854 Absolutely more importantReciprocal Reversed scale
4 Combination Weighting Method
In Step 3 the decision weight of objective 119896 120596119896is needed
In this paper according to requirements of the evaluation ofaircraft combatant survivability the evaluation system con-sists of various criteria with unequal importance which has abiggish influence on the evaluation So a rational weighting isnecessary In order to elicit weights a combination weightingmethod is brought up based on a modified AHP (analytichierarchy process) method and the entropy method offeringthe subjective weight and the objective weight respectively
41 Analytic Hierarchy Process Based on Exponential ScaleThe AHP is a proven decision-making tool integratingthe quantitative analysis and qualitative analysis togetherconsidering the quantitative weight and qualitative weightinto the decision-making process The AHP has been usedwidely in a far-ranging field especially in solving complexdecision-making problems with numerous criteriaThe AHPestablishes a hierarchical structure showing the relationshipof the target criteria and index using the decision matrixTheAHP can be broken into five steps as building a hierarchymaking comparisons calculating weights checking consis-tency and producing the result The detailed steps are shownin [11]
Dr Saaty developed a 9-point scale in the pairwisecomparisons which states whether one factor compared toanother is important or not Assign value of 1 to equallyimportant and values of 3 5 7 and 9 to slightly moreimportant more important greatly more important andabsolutely more important Values of 2 4 6 and 8 arereserved for intermediate values The 9-point scale gives thedifference of importance but the ratio of importance in thepairwise comparisons is what we need according to the AHPSo the AHP based on exponential scale is built Introduce theratio of importance 120572 into the comparison as the exponentialscale Replace the values of 1 3 5 7 and 9 with the valuesof 1 120572 1205722 1205723 and 120572
4 Assume 120572meets the rule of ladder byleaps namely 119886119896 = 119886
119896minus1+ 119886119896minus2 where 119896 isin 2 3 4 5 It turns
out to be that 119886 = 1618 which satisfied the golden section ofimportanceThe reciprocal scale is achievedwhen comparingthe factors in the opposite direction that is if 119860 is moreimportant than 119861 (2618) 119861 could be said to be less importantthan 119860 (12618) The exponential scale is shown in Table 3
For the given matrix 119872 through pairwise comparisonsthe vector of weights 120596
lowast is the normalized eigenvector ofthe matrix associated with the largest eigenvalue 120582max using
6 Mathematical Problems in Engineering
Table 4 Susceptibility subindexes of different aircraft
= 120582max120596lowast Thus after calculating and
checking of consistency in the matrix the result is produced
42 Entropy Weight Method Entropy according to Shan-nonrsquos information theory reflects the degree of disorder ofinformation [12] In the information system the smaller theentropy value the greater the degree of order and the greaterthe amount of informationThe entropy weight method is aneffective method calculating the objective weight just cor-relating with the data distribution of the evaluation matrixwithout relying on the subjective preference of decision-maker
Suppose that there is a matrix 119872 = [119898119894119895]119899times119898
isthe normalized matrix of119872 So the entropy of the 119895th indexis defined as
119864119895= minus
1
log 119899
119899
sum
119894=1
119901119894119895log119901119894119895 (9)
Then the entropy weight of 119895th index is defined as
1205961015840
119895=
(1 minus 119864119895)
(119898 minus sum119898
119895=1119864119895)
(10)
43 Algorithm Steps The AHP based on exponential scaleand entropy weight method are both available in processingthe weight however they have their own advantages anddisadvantagesThe combinationweightingmethod can evadethe disadvantages The effective combination of subjectiveweight and objective weight reconciles the expertrsquos preferencefor the index and the decrease of fuzzy random color thusproducing an advantage of both weighting methods So the
combination weighting method produces a scientific andrational weight
The combination weight can be defined as
120596119895= 120582120596lowast
119895+ (1 minus 120582) 120596
1015840
119895 (11)
where 120596119895is the combination weight of the 119895th index 120596
lowast
119895is
the subjective weight given by the AHP based on exponentialscale 1205961015840
119895is the objective weight given by the entropy weight
method120582 is the share of subjectiveweight in the combinationweight
5 Evaluation of Aircraft Survivability
51 Data Collection and Pretreatment Through literaturereview data of five aircraft is acquired in susceptibilitysubindexes which is shown in Table 4
But for the vulnerability subindexes as can be seen everwhich are comparatively subjective we cannot give a precisevalue The vague set is an effective way to solve this problem[13] For the restriction of the length of this paper the detailswill not be discussed here We know that linguistic variablesare available in certainty Seven-grade linguistic variables119878 = VGG FGM FPPVP presented in vague values areshown in Table 5
However with vague value and precise value in oneevaluation it is hard to handleHerewe can transfer the usefulinformation of vague value into a precise value through ascore functionWe know that a vague value such as [04 07]can be interpreted as ldquothe vote for a resolution is 4 in favor3 against and 3 abstentions [13]rdquo Score function can give acertain value of the vague set correlated with a certain valueof the fuzzy set through the undefined and unascertainedinformation of the vague set
A proved score function contrived by Guo et al [14] isgiven as
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
where 119877max is the maximum range at which an aircraft canbe detected 119862
119905is the characteristic parameter of radar which
varies for different type and 1198651and 119865
2are the propagation
loss of signal emission and returnSo we choose radar cross section as the evaluation index
denoted by the sign 11987811with the unit square meter m2
Infrared Signal Character Infrared signal is another impor-tant signal character of the aircraft With the developmentof stealth and antistealth techniques of radars the aircraftrsquosRCS has decreased significantly However compared withthe environment infrared signal is still significant even forthe stealth aircraft In the war area for 20 years about 90of airplanes were damaged by the infrared-guided missile[8] Nowadays IRST (infrared search and track) systemFLIR (forward looking infrared) system and infrared-guidedmissiles have been significant threats by accurately locatingaircraft For point source infrared detector the infrared rangeequation under uniform background can be written as 119877 prop
radic119868 where 119877 is the range at which an aircraft can be detectedand 119868 is the aircraftrsquos infrared radiation intensity For mostaircraft the engines are the largest sources of thermal energySo we choose turbine inlet temperature as the evaluationindex denoted by the sign 119878
12with the unit Kelvin K
Visual Signal Character Visual signal is another importantfactor in determining overall aircraft detectability [9] Aircombat in visual range is still essential and visual acquisitionbefore launch may be required Contrails engine exhaustglow cockpit lighting and luminescence may provide visualcues Here we choose size factor which is closely related tothe probability of visual detection as the evaluation indexdenoted by the sign 119878
13with the unit meter m Size factor is
defined as
11987813
=3radic3 lowast 119871 lowast 119867 lowast119882
4 lowast 120587 (2)
where 119871 is the length of aircraft 119867 is the height of aircraftand119882 is the wingspan of aircraft
Maneuverability Character Maneuverability is an effectivemeans of defense for the aircraft against detection andattack Supersonic maneuver is now the standard of newgeneration aircraft Supersonic maneuver gives an aircraft alower susceptibility and a higher survivability Maneuverabil-ity character can be defined with the maximum allowableoverload 119899
119910max the maximum steady turn overload 119899cir andspecific excess power SEP as is shown in the following
11987814
= 119899119910max + 119899cir + SEP times
9
300 (3)
Electronic Countermeasure Character Electronic counter-measure plays an important role in modern military affairsincluding active and passive jamming which is an effectivemean to decrease aircraft susceptibility and enhance aircraftsurvivability Electronic countermeasure equipment con-tains omnidirectional radar warning equipment radar chaffdispensing device infrared jammer and infrared-guided
Table 1 Electronic countermeasure subindexes of the aircraft
Number Airborne electroniccountermeasure equipment 119878
15
1 Omnidirectional radar warningequipment 105
2 Ibid + passive jamming dispensingdevice 110
3 Ibid + active infrared andelectromagnetic jammer 115
4 Ibid + missile approach warningdevice 120
Table 2 Aircraft vulnerability subindexes and definition
Subindexes Sign DefinitionRatio of fatal components
11987821
11987821=119873119865
119873
Redundant ratio of fatal components11987822
11987822=119873119877
119873119865
Shelter ratio of fatal components11987823
11987823=
119873119865
sum
119894=1
119873119878119894
119873119865
Average killing probability11987824
11987824=
119873119865
sum
119894=1
119875119870119894
119873119865
Average safety factor11987825
11987825=
119873119865
sum
119894=1
(119899119863119894
119899119863119894max
)
missiles However for electronic countermeasure capabilitywe can only give a fuzzy value Electronic countermeasurecharacter can be denoted by the sign 119878
15with dimensionless
unit A typical value can be read in Table 1
22 Aircraft Vulnerability Subindexes Aircraft vulnerabilityrefers to the inability of an aircraft to withstand the man-made hostile environment which lies on ratio of fatal com-ponents redundant ratio of fatal components shelter ratioof fatal components average killing probability and averagesafety factor These factors can be defined as is shown inTable 2 where 119873
119865is the numbers of fatal components119873 is
the numbers of whole components and119873119878is the numbers of
redundant fatal components 119873119878119894is the redundant degree of
the 119894th fatal component 119875119870119894
is the killing probability of the119894th component while the aircraft is hit 119899
119863119894and 119899
119863119894max arethe designed overload andmaximum overload of the 119894th fatalcomponent respectively
Although there are clear equations for the calculation ofaircraft vulnerability subindexes the value of every parameteris comparatively subjective for different definition such asldquofatalrdquo and different granularity analysis
3 A Multiattribute Intelligent GreyTarget Decision Model
Here we introduce a multiattribute intelligent grey targetdecision model proposed by Liu et al [10] This model takes
4 Mathematical Problems in Engineering
the situation of the shoot and miss of the bullrsquos eye of theobjectiversquos effect value and vector based on four kinds ofuniform effect measures
31 Problem Description Assume that 119860 = 1198861 1198862 119886
119894119895in objective 119896 referring to the similar
level or the removed level between the sample and the criticalsample
Assume that 119889(119896)1
and 119889(119896)2
are the upper and lower criticalvalue 119904
119894119895in objective 119896 Then 119878
1= 119903 | 119889
(119896)
1le 119903 le 119889
(119896)
2 is
the one-dimension grey target and 120583(119896)
119894119895
isin [119889(119896)
1 119889(119896)
2] is the
pleased effect in objective 119896Multidimensional grey target canbe discussed in the same way Details are shown in [10]
32 Uniform Effect Measures Considering the decisionobjectives with different meaning different dimensionandor different nature the effect value of the objectiveshould be transferred to the uniform effect measures
Assume that 11990611989611989401198950
is the critical value of objective 119896 andthen the grey objective decision of 119896 is designed as fol-lows
119906119896
119894119895isin
[119906119896
11989401198950max119894
max119895
119906119896
119894119895] 119896 isin BTO
[min119894
min119895
119906119896
119894119895 119906119896
11989401198950] 119896 isin CTO
[119860 minus 119906119896
11989401198950 119860 + 119906
119896
11989401198950 ] 119896 isin MTO
(4)
where 119896 isin BTOmeans that objective 119896 belongs to benefit typeobjective 119896 isin CTO means that objective 119896 belongs to costtype objective and 119896 isin MTO means that objective 119896 belongsto moderate type objective (same as what follows) And 119860 isthe moderate value of the moderate type objective
For the decision objective of benefit type and cost typethe effect measures 119903119896
119894119895can be shown as
119903119896
119894119895=
119906119896
119894119895minus 119906119896
11989401198950
max119894max119895119906119896
119894119895 minus 119906119896
11989401198950
119896 isin BTO
119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950minusmin
119894min119895119906119896
119894119895
119896 isin CTO
(5)
For the decision objective of moderate type the effectmeasure can be divided as the upper effect measure and thelower effect measure according to the scale of effect measure119903119896
119894119895 which can be shown as follows
119903119896
119894119895=
119906119896
119894119895minus 119860 + 119906
119896
11989401198950
119906119896
11989401198950
119906119896
119894119895isin [119860 minus 119906
119896
11989401198950 119860] lower effect measure
119860 + 119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950
119906119896
119894119895isin [119860 119860 + 119906
119896
11989401198950] upper effect measure
(6)
The effectmeasure of the decision objective of benefit typereflects the similar level between the sample and the biggestsample as well as the removed level between the sampleand the critical sample The effect measure of the decisionobjective of cost type reflects the similar level between thesample and the smallest sample as well as the removed levelbetween the sample and the critical sample The lower effectmeasure of the decision objective of moderate type reflectsthe level between the sample less than moderate value 119860 andthe lower critical sample The upper effect measure of thedecision objective of moderate type reflects the level betweenthe sample greater than moderate value 119860 and the uppercritical sample
The decision objective of benefit type is a type of objectivewith an expectance the bigger the better or the more thebetterThe decision objective of cost type is a type of objectivewith an expectance the smaller the better or the less the betterThe decision objective of moderate type is a type of objectivewith an expectance neither too big nor too small or neithertoo many nor too few
The miss of the bullrsquos eye under different conditions canbe shown as
119906119896
119894119895lt 119906119896
11989401198950119896 isin BTO
119906119896
119894119895gt 119906119896
11989401198950119896 isin CTO
119906119896
119894119895lt 119860 minus 119906
119896
11989401198950
or 119906119896119894119895gt 119860 + 119906
119896
11989401198950
119896 isin MTO
(7)
The effect measure 119903119896
119894119895is satisfied with the following
requirements(1) 119903119896
119894119895is dimensionless (2) 119903
119896
119894119895is a uniform variable
namely 119903119896119894119895isin [minus1 1] (3) The greater the 119903119896
119894119895 the more ideal
the effect
Mathematical Problems in Engineering 5
Thus in order to satisfy the standardization the selectionof critical value 119906119896
11989401198950usually satisfies the following
119906119896
119894119895ge minusmax119894
max119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin BTO
119906119896
119894119895le minusmin119894
min119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin CTO
119906119896
119894119895ge 119860 minus 2 lowast 119906
119896
11989401198950
or 119906119896119894119895le 119860 + 2 lowast 119906
119896
11989401198950
119896 isin MTO
(8)
Assume that 120596119896is the decision weight of objective 119896 and
sum119904
119896=1120596119896= 1 Thus 119903
119894119895= sum119904
119896=1120596119896lowast 119903119896
119894119895will be the synthetic
effect measures of decision approach 119904119894119895and 119877 = 119903
119894119895 will be
the matrix of synthetic effect measures of decision set 119878The synthetic effect measure 119903
119894119895is satisfied with the
following requirements(1) 119903119894119895is dimensionless (2) 119903
119894119895is a uniform variable
namely 119903119894119895isin [minus1 1] (3) The greater the 119903
119894119895 the more ideal
the effectWhile 119903
119894119895isin [minus1 0] means the miss of the bullrsquos eye
and 119903119894119895isin [0 1] means the hit of the bullrsquos eye through the
comparison of the value of 119903119894119895 we can judge the performance
of 1198861198940 1198871198950 and 119904
11989401198950according to the definition shown as
follows
Definition 1 1198871198950
is the best decision of event 119886119894
ifmax1le119895le119898
119903119894119895 = 119903
1198941198950 1198861198940
is the best event correspondingwith the decision 119887
1198950if max
1le119894le119899119903119894119895 = 119903
1198940119895 11990411989401198950
is the bestdecision approach if max
1le119894le119899max1le119895le119898
119903119894119895 = 11990311989401198950
33 Algorithm Steps Algorithm steps of the multiattributeintelligent grey target decision model are as follows
Step 1 Form the decision set 119878 = 119904119894119895= (119886119894 119887119895) | 119886119894isin 119860 119887
Step 3 Confirm the decision weight of objective 119896 120596119896(119896 =
1 2 119904)
Step 4 Form the matrix 119880119896
= 119906119896
119894119895 of effect measures of
decision set 119878
Step 5 Set the critical value of the objective
Step 6 Obtain the uniform matrix 119877119896
= 119903119896
119894119895 of effect
measures
Step 7 Obtain the uniformmatrix119877 = 119903119894119895 of synthetic effect
measures
Step 8 Confirm the best decision 1198871198950
or the best decisionapproach 119904
11989401198950according to the definitions
Table 3 Exponential scale the golden section
Exponential scale One factor compared to another1000 Equally important1618 Slightly more important2618 More important4236 Greatly more important6854 Absolutely more importantReciprocal Reversed scale
4 Combination Weighting Method
In Step 3 the decision weight of objective 119896 120596119896is needed
In this paper according to requirements of the evaluation ofaircraft combatant survivability the evaluation system con-sists of various criteria with unequal importance which has abiggish influence on the evaluation So a rational weighting isnecessary In order to elicit weights a combination weightingmethod is brought up based on a modified AHP (analytichierarchy process) method and the entropy method offeringthe subjective weight and the objective weight respectively
41 Analytic Hierarchy Process Based on Exponential ScaleThe AHP is a proven decision-making tool integratingthe quantitative analysis and qualitative analysis togetherconsidering the quantitative weight and qualitative weightinto the decision-making process The AHP has been usedwidely in a far-ranging field especially in solving complexdecision-making problems with numerous criteriaThe AHPestablishes a hierarchical structure showing the relationshipof the target criteria and index using the decision matrixTheAHP can be broken into five steps as building a hierarchymaking comparisons calculating weights checking consis-tency and producing the result The detailed steps are shownin [11]
Dr Saaty developed a 9-point scale in the pairwisecomparisons which states whether one factor compared toanother is important or not Assign value of 1 to equallyimportant and values of 3 5 7 and 9 to slightly moreimportant more important greatly more important andabsolutely more important Values of 2 4 6 and 8 arereserved for intermediate values The 9-point scale gives thedifference of importance but the ratio of importance in thepairwise comparisons is what we need according to the AHPSo the AHP based on exponential scale is built Introduce theratio of importance 120572 into the comparison as the exponentialscale Replace the values of 1 3 5 7 and 9 with the valuesof 1 120572 1205722 1205723 and 120572
4 Assume 120572meets the rule of ladder byleaps namely 119886119896 = 119886
119896minus1+ 119886119896minus2 where 119896 isin 2 3 4 5 It turns
out to be that 119886 = 1618 which satisfied the golden section ofimportanceThe reciprocal scale is achievedwhen comparingthe factors in the opposite direction that is if 119860 is moreimportant than 119861 (2618) 119861 could be said to be less importantthan 119860 (12618) The exponential scale is shown in Table 3
For the given matrix 119872 through pairwise comparisonsthe vector of weights 120596
lowast is the normalized eigenvector ofthe matrix associated with the largest eigenvalue 120582max using
6 Mathematical Problems in Engineering
Table 4 Susceptibility subindexes of different aircraft
= 120582max120596lowast Thus after calculating and
checking of consistency in the matrix the result is produced
42 Entropy Weight Method Entropy according to Shan-nonrsquos information theory reflects the degree of disorder ofinformation [12] In the information system the smaller theentropy value the greater the degree of order and the greaterthe amount of informationThe entropy weight method is aneffective method calculating the objective weight just cor-relating with the data distribution of the evaluation matrixwithout relying on the subjective preference of decision-maker
Suppose that there is a matrix 119872 = [119898119894119895]119899times119898
isthe normalized matrix of119872 So the entropy of the 119895th indexis defined as
119864119895= minus
1
log 119899
119899
sum
119894=1
119901119894119895log119901119894119895 (9)
Then the entropy weight of 119895th index is defined as
1205961015840
119895=
(1 minus 119864119895)
(119898 minus sum119898
119895=1119864119895)
(10)
43 Algorithm Steps The AHP based on exponential scaleand entropy weight method are both available in processingthe weight however they have their own advantages anddisadvantagesThe combinationweightingmethod can evadethe disadvantages The effective combination of subjectiveweight and objective weight reconciles the expertrsquos preferencefor the index and the decrease of fuzzy random color thusproducing an advantage of both weighting methods So the
combination weighting method produces a scientific andrational weight
The combination weight can be defined as
120596119895= 120582120596lowast
119895+ (1 minus 120582) 120596
1015840
119895 (11)
where 120596119895is the combination weight of the 119895th index 120596
lowast
119895is
the subjective weight given by the AHP based on exponentialscale 1205961015840
119895is the objective weight given by the entropy weight
method120582 is the share of subjectiveweight in the combinationweight
5 Evaluation of Aircraft Survivability
51 Data Collection and Pretreatment Through literaturereview data of five aircraft is acquired in susceptibilitysubindexes which is shown in Table 4
But for the vulnerability subindexes as can be seen everwhich are comparatively subjective we cannot give a precisevalue The vague set is an effective way to solve this problem[13] For the restriction of the length of this paper the detailswill not be discussed here We know that linguistic variablesare available in certainty Seven-grade linguistic variables119878 = VGG FGM FPPVP presented in vague values areshown in Table 5
However with vague value and precise value in oneevaluation it is hard to handleHerewe can transfer the usefulinformation of vague value into a precise value through ascore functionWe know that a vague value such as [04 07]can be interpreted as ldquothe vote for a resolution is 4 in favor3 against and 3 abstentions [13]rdquo Score function can give acertain value of the vague set correlated with a certain valueof the fuzzy set through the undefined and unascertainedinformation of the vague set
A proved score function contrived by Guo et al [14] isgiven as
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
119894119895in objective 119896 referring to the similar
level or the removed level between the sample and the criticalsample
Assume that 119889(119896)1
and 119889(119896)2
are the upper and lower criticalvalue 119904
119894119895in objective 119896 Then 119878
1= 119903 | 119889
(119896)
1le 119903 le 119889
(119896)
2 is
the one-dimension grey target and 120583(119896)
119894119895
isin [119889(119896)
1 119889(119896)
2] is the
pleased effect in objective 119896Multidimensional grey target canbe discussed in the same way Details are shown in [10]
32 Uniform Effect Measures Considering the decisionobjectives with different meaning different dimensionandor different nature the effect value of the objectiveshould be transferred to the uniform effect measures
Assume that 11990611989611989401198950
is the critical value of objective 119896 andthen the grey objective decision of 119896 is designed as fol-lows
119906119896
119894119895isin
[119906119896
11989401198950max119894
max119895
119906119896
119894119895] 119896 isin BTO
[min119894
min119895
119906119896
119894119895 119906119896
11989401198950] 119896 isin CTO
[119860 minus 119906119896
11989401198950 119860 + 119906
119896
11989401198950 ] 119896 isin MTO
(4)
where 119896 isin BTOmeans that objective 119896 belongs to benefit typeobjective 119896 isin CTO means that objective 119896 belongs to costtype objective and 119896 isin MTO means that objective 119896 belongsto moderate type objective (same as what follows) And 119860 isthe moderate value of the moderate type objective
For the decision objective of benefit type and cost typethe effect measures 119903119896
119894119895can be shown as
119903119896
119894119895=
119906119896
119894119895minus 119906119896
11989401198950
max119894max119895119906119896
119894119895 minus 119906119896
11989401198950
119896 isin BTO
119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950minusmin
119894min119895119906119896
119894119895
119896 isin CTO
(5)
For the decision objective of moderate type the effectmeasure can be divided as the upper effect measure and thelower effect measure according to the scale of effect measure119903119896
119894119895 which can be shown as follows
119903119896
119894119895=
119906119896
119894119895minus 119860 + 119906
119896
11989401198950
119906119896
11989401198950
119906119896
119894119895isin [119860 minus 119906
119896
11989401198950 119860] lower effect measure
119860 + 119906119896
11989401198950minus 119906119896
119894119895
119906119896
11989401198950
119906119896
119894119895isin [119860 119860 + 119906
119896
11989401198950] upper effect measure
(6)
The effectmeasure of the decision objective of benefit typereflects the similar level between the sample and the biggestsample as well as the removed level between the sampleand the critical sample The effect measure of the decisionobjective of cost type reflects the similar level between thesample and the smallest sample as well as the removed levelbetween the sample and the critical sample The lower effectmeasure of the decision objective of moderate type reflectsthe level between the sample less than moderate value 119860 andthe lower critical sample The upper effect measure of thedecision objective of moderate type reflects the level betweenthe sample greater than moderate value 119860 and the uppercritical sample
The decision objective of benefit type is a type of objectivewith an expectance the bigger the better or the more thebetterThe decision objective of cost type is a type of objectivewith an expectance the smaller the better or the less the betterThe decision objective of moderate type is a type of objectivewith an expectance neither too big nor too small or neithertoo many nor too few
The miss of the bullrsquos eye under different conditions canbe shown as
119906119896
119894119895lt 119906119896
11989401198950119896 isin BTO
119906119896
119894119895gt 119906119896
11989401198950119896 isin CTO
119906119896
119894119895lt 119860 minus 119906
119896
11989401198950
or 119906119896119894119895gt 119860 + 119906
119896
11989401198950
119896 isin MTO
(7)
The effect measure 119903119896
119894119895is satisfied with the following
requirements(1) 119903119896
119894119895is dimensionless (2) 119903
119896
119894119895is a uniform variable
namely 119903119896119894119895isin [minus1 1] (3) The greater the 119903119896
119894119895 the more ideal
the effect
Mathematical Problems in Engineering 5
Thus in order to satisfy the standardization the selectionof critical value 119906119896
11989401198950usually satisfies the following
119906119896
119894119895ge minusmax119894
max119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin BTO
119906119896
119894119895le minusmin119894
min119895
119906119896
119894119895 + 2 lowast 119906
119896
11989401198950 119896 isin CTO
119906119896
119894119895ge 119860 minus 2 lowast 119906
119896
11989401198950
or 119906119896119894119895le 119860 + 2 lowast 119906
119896
11989401198950
119896 isin MTO
(8)
Assume that 120596119896is the decision weight of objective 119896 and
sum119904
119896=1120596119896= 1 Thus 119903
119894119895= sum119904
119896=1120596119896lowast 119903119896
119894119895will be the synthetic
effect measures of decision approach 119904119894119895and 119877 = 119903
119894119895 will be
the matrix of synthetic effect measures of decision set 119878The synthetic effect measure 119903
119894119895is satisfied with the
following requirements(1) 119903119894119895is dimensionless (2) 119903
119894119895is a uniform variable
namely 119903119894119895isin [minus1 1] (3) The greater the 119903
119894119895 the more ideal
the effectWhile 119903
119894119895isin [minus1 0] means the miss of the bullrsquos eye
and 119903119894119895isin [0 1] means the hit of the bullrsquos eye through the
comparison of the value of 119903119894119895 we can judge the performance
of 1198861198940 1198871198950 and 119904
11989401198950according to the definition shown as
follows
Definition 1 1198871198950
is the best decision of event 119886119894
ifmax1le119895le119898
119903119894119895 = 119903
1198941198950 1198861198940
is the best event correspondingwith the decision 119887
1198950if max
1le119894le119899119903119894119895 = 119903
1198940119895 11990411989401198950
is the bestdecision approach if max
1le119894le119899max1le119895le119898
119903119894119895 = 11990311989401198950
33 Algorithm Steps Algorithm steps of the multiattributeintelligent grey target decision model are as follows
Step 1 Form the decision set 119878 = 119904119894119895= (119886119894 119887119895) | 119886119894isin 119860 119887
Step 3 Confirm the decision weight of objective 119896 120596119896(119896 =
1 2 119904)
Step 4 Form the matrix 119880119896
= 119906119896
119894119895 of effect measures of
decision set 119878
Step 5 Set the critical value of the objective
Step 6 Obtain the uniform matrix 119877119896
= 119903119896
119894119895 of effect
measures
Step 7 Obtain the uniformmatrix119877 = 119903119894119895 of synthetic effect
measures
Step 8 Confirm the best decision 1198871198950
or the best decisionapproach 119904
11989401198950according to the definitions
Table 3 Exponential scale the golden section
Exponential scale One factor compared to another1000 Equally important1618 Slightly more important2618 More important4236 Greatly more important6854 Absolutely more importantReciprocal Reversed scale
4 Combination Weighting Method
In Step 3 the decision weight of objective 119896 120596119896is needed
In this paper according to requirements of the evaluation ofaircraft combatant survivability the evaluation system con-sists of various criteria with unequal importance which has abiggish influence on the evaluation So a rational weighting isnecessary In order to elicit weights a combination weightingmethod is brought up based on a modified AHP (analytichierarchy process) method and the entropy method offeringthe subjective weight and the objective weight respectively
41 Analytic Hierarchy Process Based on Exponential ScaleThe AHP is a proven decision-making tool integratingthe quantitative analysis and qualitative analysis togetherconsidering the quantitative weight and qualitative weightinto the decision-making process The AHP has been usedwidely in a far-ranging field especially in solving complexdecision-making problems with numerous criteriaThe AHPestablishes a hierarchical structure showing the relationshipof the target criteria and index using the decision matrixTheAHP can be broken into five steps as building a hierarchymaking comparisons calculating weights checking consis-tency and producing the result The detailed steps are shownin [11]
Dr Saaty developed a 9-point scale in the pairwisecomparisons which states whether one factor compared toanother is important or not Assign value of 1 to equallyimportant and values of 3 5 7 and 9 to slightly moreimportant more important greatly more important andabsolutely more important Values of 2 4 6 and 8 arereserved for intermediate values The 9-point scale gives thedifference of importance but the ratio of importance in thepairwise comparisons is what we need according to the AHPSo the AHP based on exponential scale is built Introduce theratio of importance 120572 into the comparison as the exponentialscale Replace the values of 1 3 5 7 and 9 with the valuesof 1 120572 1205722 1205723 and 120572
4 Assume 120572meets the rule of ladder byleaps namely 119886119896 = 119886
119896minus1+ 119886119896minus2 where 119896 isin 2 3 4 5 It turns
out to be that 119886 = 1618 which satisfied the golden section ofimportanceThe reciprocal scale is achievedwhen comparingthe factors in the opposite direction that is if 119860 is moreimportant than 119861 (2618) 119861 could be said to be less importantthan 119860 (12618) The exponential scale is shown in Table 3
For the given matrix 119872 through pairwise comparisonsthe vector of weights 120596
lowast is the normalized eigenvector ofthe matrix associated with the largest eigenvalue 120582max using
6 Mathematical Problems in Engineering
Table 4 Susceptibility subindexes of different aircraft
= 120582max120596lowast Thus after calculating and
checking of consistency in the matrix the result is produced
42 Entropy Weight Method Entropy according to Shan-nonrsquos information theory reflects the degree of disorder ofinformation [12] In the information system the smaller theentropy value the greater the degree of order and the greaterthe amount of informationThe entropy weight method is aneffective method calculating the objective weight just cor-relating with the data distribution of the evaluation matrixwithout relying on the subjective preference of decision-maker
Suppose that there is a matrix 119872 = [119898119894119895]119899times119898
isthe normalized matrix of119872 So the entropy of the 119895th indexis defined as
119864119895= minus
1
log 119899
119899
sum
119894=1
119901119894119895log119901119894119895 (9)
Then the entropy weight of 119895th index is defined as
1205961015840
119895=
(1 minus 119864119895)
(119898 minus sum119898
119895=1119864119895)
(10)
43 Algorithm Steps The AHP based on exponential scaleand entropy weight method are both available in processingthe weight however they have their own advantages anddisadvantagesThe combinationweightingmethod can evadethe disadvantages The effective combination of subjectiveweight and objective weight reconciles the expertrsquos preferencefor the index and the decrease of fuzzy random color thusproducing an advantage of both weighting methods So the
combination weighting method produces a scientific andrational weight
The combination weight can be defined as
120596119895= 120582120596lowast
119895+ (1 minus 120582) 120596
1015840
119895 (11)
where 120596119895is the combination weight of the 119895th index 120596
lowast
119895is
the subjective weight given by the AHP based on exponentialscale 1205961015840
119895is the objective weight given by the entropy weight
method120582 is the share of subjectiveweight in the combinationweight
5 Evaluation of Aircraft Survivability
51 Data Collection and Pretreatment Through literaturereview data of five aircraft is acquired in susceptibilitysubindexes which is shown in Table 4
But for the vulnerability subindexes as can be seen everwhich are comparatively subjective we cannot give a precisevalue The vague set is an effective way to solve this problem[13] For the restriction of the length of this paper the detailswill not be discussed here We know that linguistic variablesare available in certainty Seven-grade linguistic variables119878 = VGG FGM FPPVP presented in vague values areshown in Table 5
However with vague value and precise value in oneevaluation it is hard to handleHerewe can transfer the usefulinformation of vague value into a precise value through ascore functionWe know that a vague value such as [04 07]can be interpreted as ldquothe vote for a resolution is 4 in favor3 against and 3 abstentions [13]rdquo Score function can give acertain value of the vague set correlated with a certain valueof the fuzzy set through the undefined and unascertainedinformation of the vague set
A proved score function contrived by Guo et al [14] isgiven as
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
Step 3 Confirm the decision weight of objective 119896 120596119896(119896 =
1 2 119904)
Step 4 Form the matrix 119880119896
= 119906119896
119894119895 of effect measures of
decision set 119878
Step 5 Set the critical value of the objective
Step 6 Obtain the uniform matrix 119877119896
= 119903119896
119894119895 of effect
measures
Step 7 Obtain the uniformmatrix119877 = 119903119894119895 of synthetic effect
measures
Step 8 Confirm the best decision 1198871198950
or the best decisionapproach 119904
11989401198950according to the definitions
Table 3 Exponential scale the golden section
Exponential scale One factor compared to another1000 Equally important1618 Slightly more important2618 More important4236 Greatly more important6854 Absolutely more importantReciprocal Reversed scale
4 Combination Weighting Method
In Step 3 the decision weight of objective 119896 120596119896is needed
In this paper according to requirements of the evaluation ofaircraft combatant survivability the evaluation system con-sists of various criteria with unequal importance which has abiggish influence on the evaluation So a rational weighting isnecessary In order to elicit weights a combination weightingmethod is brought up based on a modified AHP (analytichierarchy process) method and the entropy method offeringthe subjective weight and the objective weight respectively
41 Analytic Hierarchy Process Based on Exponential ScaleThe AHP is a proven decision-making tool integratingthe quantitative analysis and qualitative analysis togetherconsidering the quantitative weight and qualitative weightinto the decision-making process The AHP has been usedwidely in a far-ranging field especially in solving complexdecision-making problems with numerous criteriaThe AHPestablishes a hierarchical structure showing the relationshipof the target criteria and index using the decision matrixTheAHP can be broken into five steps as building a hierarchymaking comparisons calculating weights checking consis-tency and producing the result The detailed steps are shownin [11]
Dr Saaty developed a 9-point scale in the pairwisecomparisons which states whether one factor compared toanother is important or not Assign value of 1 to equallyimportant and values of 3 5 7 and 9 to slightly moreimportant more important greatly more important andabsolutely more important Values of 2 4 6 and 8 arereserved for intermediate values The 9-point scale gives thedifference of importance but the ratio of importance in thepairwise comparisons is what we need according to the AHPSo the AHP based on exponential scale is built Introduce theratio of importance 120572 into the comparison as the exponentialscale Replace the values of 1 3 5 7 and 9 with the valuesof 1 120572 1205722 1205723 and 120572
4 Assume 120572meets the rule of ladder byleaps namely 119886119896 = 119886
119896minus1+ 119886119896minus2 where 119896 isin 2 3 4 5 It turns
out to be that 119886 = 1618 which satisfied the golden section ofimportanceThe reciprocal scale is achievedwhen comparingthe factors in the opposite direction that is if 119860 is moreimportant than 119861 (2618) 119861 could be said to be less importantthan 119860 (12618) The exponential scale is shown in Table 3
For the given matrix 119872 through pairwise comparisonsthe vector of weights 120596
lowast is the normalized eigenvector ofthe matrix associated with the largest eigenvalue 120582max using
6 Mathematical Problems in Engineering
Table 4 Susceptibility subindexes of different aircraft
= 120582max120596lowast Thus after calculating and
checking of consistency in the matrix the result is produced
42 Entropy Weight Method Entropy according to Shan-nonrsquos information theory reflects the degree of disorder ofinformation [12] In the information system the smaller theentropy value the greater the degree of order and the greaterthe amount of informationThe entropy weight method is aneffective method calculating the objective weight just cor-relating with the data distribution of the evaluation matrixwithout relying on the subjective preference of decision-maker
Suppose that there is a matrix 119872 = [119898119894119895]119899times119898
isthe normalized matrix of119872 So the entropy of the 119895th indexis defined as
119864119895= minus
1
log 119899
119899
sum
119894=1
119901119894119895log119901119894119895 (9)
Then the entropy weight of 119895th index is defined as
1205961015840
119895=
(1 minus 119864119895)
(119898 minus sum119898
119895=1119864119895)
(10)
43 Algorithm Steps The AHP based on exponential scaleand entropy weight method are both available in processingthe weight however they have their own advantages anddisadvantagesThe combinationweightingmethod can evadethe disadvantages The effective combination of subjectiveweight and objective weight reconciles the expertrsquos preferencefor the index and the decrease of fuzzy random color thusproducing an advantage of both weighting methods So the
combination weighting method produces a scientific andrational weight
The combination weight can be defined as
120596119895= 120582120596lowast
119895+ (1 minus 120582) 120596
1015840
119895 (11)
where 120596119895is the combination weight of the 119895th index 120596
lowast
119895is
the subjective weight given by the AHP based on exponentialscale 1205961015840
119895is the objective weight given by the entropy weight
method120582 is the share of subjectiveweight in the combinationweight
5 Evaluation of Aircraft Survivability
51 Data Collection and Pretreatment Through literaturereview data of five aircraft is acquired in susceptibilitysubindexes which is shown in Table 4
But for the vulnerability subindexes as can be seen everwhich are comparatively subjective we cannot give a precisevalue The vague set is an effective way to solve this problem[13] For the restriction of the length of this paper the detailswill not be discussed here We know that linguistic variablesare available in certainty Seven-grade linguistic variables119878 = VGG FGM FPPVP presented in vague values areshown in Table 5
However with vague value and precise value in oneevaluation it is hard to handleHerewe can transfer the usefulinformation of vague value into a precise value through ascore functionWe know that a vague value such as [04 07]can be interpreted as ldquothe vote for a resolution is 4 in favor3 against and 3 abstentions [13]rdquo Score function can give acertain value of the vague set correlated with a certain valueof the fuzzy set through the undefined and unascertainedinformation of the vague set
A proved score function contrived by Guo et al [14] isgiven as
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
= 120582max120596lowast Thus after calculating and
checking of consistency in the matrix the result is produced
42 Entropy Weight Method Entropy according to Shan-nonrsquos information theory reflects the degree of disorder ofinformation [12] In the information system the smaller theentropy value the greater the degree of order and the greaterthe amount of informationThe entropy weight method is aneffective method calculating the objective weight just cor-relating with the data distribution of the evaluation matrixwithout relying on the subjective preference of decision-maker
Suppose that there is a matrix 119872 = [119898119894119895]119899times119898
isthe normalized matrix of119872 So the entropy of the 119895th indexis defined as
119864119895= minus
1
log 119899
119899
sum
119894=1
119901119894119895log119901119894119895 (9)
Then the entropy weight of 119895th index is defined as
1205961015840
119895=
(1 minus 119864119895)
(119898 minus sum119898
119895=1119864119895)
(10)
43 Algorithm Steps The AHP based on exponential scaleand entropy weight method are both available in processingthe weight however they have their own advantages anddisadvantagesThe combinationweightingmethod can evadethe disadvantages The effective combination of subjectiveweight and objective weight reconciles the expertrsquos preferencefor the index and the decrease of fuzzy random color thusproducing an advantage of both weighting methods So the
combination weighting method produces a scientific andrational weight
The combination weight can be defined as
120596119895= 120582120596lowast
119895+ (1 minus 120582) 120596
1015840
119895 (11)
where 120596119895is the combination weight of the 119895th index 120596
lowast
119895is
the subjective weight given by the AHP based on exponentialscale 1205961015840
119895is the objective weight given by the entropy weight
method120582 is the share of subjectiveweight in the combinationweight
5 Evaluation of Aircraft Survivability
51 Data Collection and Pretreatment Through literaturereview data of five aircraft is acquired in susceptibilitysubindexes which is shown in Table 4
But for the vulnerability subindexes as can be seen everwhich are comparatively subjective we cannot give a precisevalue The vague set is an effective way to solve this problem[13] For the restriction of the length of this paper the detailswill not be discussed here We know that linguistic variablesare available in certainty Seven-grade linguistic variables119878 = VGG FGM FPPVP presented in vague values areshown in Table 5
However with vague value and precise value in oneevaluation it is hard to handleHerewe can transfer the usefulinformation of vague value into a precise value through ascore functionWe know that a vague value such as [04 07]can be interpreted as ldquothe vote for a resolution is 4 in favor3 against and 3 abstentions [13]rdquo Score function can give acertain value of the vague set correlated with a certain valueof the fuzzy set through the undefined and unascertainedinformation of the vague set
A proved score function contrived by Guo et al [14] isgiven as
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
Through the seven-grade linguistic variables of vaguevalue we give vulnerability subindexes of different aircraft thevalue shown in Table 6
Thus the subindexes can be normalized as is shown inTable 7 For vulnerability subindexes a certain value shouldbe given by score function and then the vector normalizationmethod is used The type of index is also given Here for thesubindexes there are only too types where ldquo+rdquo representsbenefit type and ldquominusrdquo represents cost type
52 Calculation ofWeight According toAHPbased on expo-nential scale the subjective weight can be calculated throughthe introduced five steps Pairwise comparisons are quiteimportant Each of the subfactors will need to be comparedto each other There are two approaches generally availableThe first is comparing the factors of susceptibility andvulnerability followed by comparing each subfactor undereach factor separately The other approach is comparing eachsubfactor to every other one directly which involves morecomparisonsThis approach would be difficult and lead to anuncertain result For instance trying to compare radar signalcharacter to ratio of fatal components would be difficult asthey are so dissimilar So the first approach will be used
Three matrixes of pairwise comparisons given by expertsare established Matrix 119875 is the comparison of susceptibilityindex and vulnerability index Define susceptibility indexas more important than vulnerability index thus producingthe two by two matrix 119875
1and 119875
2are the comparison of
subindexes of susceptibility and vulnerability respectively
119875 = [
[
1000 2618
1
26181000
]
]
(13)
1198751=
[[[[[[[[[[[[[[
[
1000 1618 4236 2618 1618
1
16181000 2618 1618 1618
1
4236
1
26181000
1
1618
1
1618
1
2618
1
16181618 1000 1000
1
1618
1
16181618 1000 1000
]]]]]]]]]]]]]]
]
1198752=
[[[[[[[[[[[[[[
[
1000 1618 1618 2618 4236
1
16181000 1000 1618 2618
1
16181000 1000 1618 1618
1
2618
1
1618
1
16181000 2618
1
4236
1
2618
1
1618
1
26181000
]]]]]]]]]]]]]]
]
(14)
The next steps are calculating weights and checkingconsistency The calculation of 119875 is 120596lowast
119868= (07236 02764)
and the calculation of 1198751and 119875
2is as follows
120596lowast
1= (03553 02399 00916 01483 01649)
120596lowast
2= (03496 02160 01993 01502 00848)
(15)
Reference [11] gives the method of checking consistencyusing the equation CR = CIRI where CI = (120582max minus 119899)(119899 minus
1) RI is the random consistency index 120582max is maximumeigenvalue of the matrix and 119899 is the number of factorsThe consistency of matrixes 119875
1and 119875
2is 00063 and 00168
respectively which is less than 10 percent that is the matrixis consistent enough and can be used for calculating results
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
53 Multiattribute Intelligent Grey Target Decision Here wegive an evaluation of aircraft survivability The evaluation ofaircraft survivability is the event 119886
1 so 119860 = 119886
1 The aircraft
119860 119861 119862 119863 and 119864 are the countermeasures 1198871 1198872 1198873 1198874 1198875 so
119861 = 1198871 1198872 1198873 1198874 and 119887
As is shown above the susceptibility subindexes andvulnerability subindexes of different aircraft are chosen asthe decision objective 119896 = 1 2 10 For the decisionobjective of benefit type the critical value is 119906119896
11989401198950= 1922 119896 =
1 2 3 6 9 For the decision objective of cost type the criticalvalue is 119906119896
11989401198950= 05 119896 = 4 5 7 8 10 The decision weight of
objective 119896 has been given above The matrix 119880119896= 119906119896
119894119895 of
effect measures of decision set 119878 is formed according to thecollected dataThus the uniform effect measures matrix 119877119896 =119903119896
119894119895 is obtained with the defined effect measure for benefit
type objective and effect measure for cost type objectiveshown as follows
119877119896=
[[[[[[[[[[[[[[[[[[[[[[
[
00000 05476 06190 06111 10000
05967 10000 05967 03962 00000
00000 10000 08577 02321 00380
05313 05568 06386 00000 10000
06667 00000 06667 10000 10000
06234 00000 06234 10000 10000
05577 00000 00000 10000 05577
10000 00000 00000 00000 10000
06234 00000 06234 10000 10000
06234 00000 00000 10000 06234
]]]]]]]]]]]]]]]]]]]]]]
]
(19)
Through the equation 119903119894119895
= sum119904
119896=1120596119896lowast 119903119896
119894119895 the uniform
matrix 119877 = 119903119894119895 of synthetic effect measures can be
obtained as 119877 = [04533 03795 05237 06138 07383]Then according to the definitions the best countermeasure
is 1198875 that is the aircraft 119864 is the best decision and has the best
aircraft survivability Through the ranking of 119903119894119895 we can give
a ranking of aircraft 119864 gt 119863 gt 119862 gt 119860 gt 119861A rational weight is essential An opposite example with
equal weight for every subindex is given The weight is 01For the restriction of the length of this paper we just list theresult of the uniformmatrix of synthetic effect measures 119877 =
[08782 07764 07527 08742 08393] Here the aircraft119860 isthe best decision and has the best aircraft survivability Thisresult is quite different And the difference of every aircraft islittle which is hard to give a rational evaluation
With the calculated weight according to AHP basedon exponential scale and the entropy weight method wecan just calculate the susceptibility of the aircraft using thesame method According to the algorithm steps the uniformmatrix of synthetic effect measures can be obtained as119877119878= [08244 08136 08304 08106 07580]Then according
to the definitions the best countermeasure is 1198873 that is
the aircraft 119862 is the best decision and has the best aircraftsusceptibility Through the ranking of 119903
119894119895 we can give a
ranking of aircraft 119862 gt 119860 gt 119861 gt 119863 gt 119864 However this resultis so different with the aircraft survivability Some reasonscan be listed Firstly aircraft susceptibility and aircraft surviv-ability are different in definitions and evaluation subindexesThen although turbine inlet temperature is a good evaluationindex for the infrared signal character different from RCSturbine inlet temperature is not enough especially for thenew generation aircraft Turbine inlet temperature is not thesole influence factor of infrared signal character Infraredsuppressing functions such as stealth materials and thermalisolation have no influence on turbine inlet temperature buthave great influence on infrared signal character This is thefuture work to establish a more rational index system
Then we calculate the vulnerability of the aircraft Thecombinationweight has been given in the former sectionTheuniform matrix of synthetic effect measures is obtained as119877119881= [07300 02279 05395 09248 09183] Then the best
countermeasure is 1198874 that is the aircraft119863 is the best decision
Mathematical Problems in Engineering 9
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
and has the best aircraft vulnerability The ranking of aircraftvulnerability is119863 gt 119864 gt 119860 gt 119862 gt 119861
6 Conclusions
With the development of precision guided weapons air-craft combat survivability has been increasingly importantMulticriteria decision method is a useful method throughranking and selecting a finite number of alternative plans Inthis paper in order to give a rational evaluation of aircraftcombat survivability a new multiattribute intelligent greytarget decision model is introduced Conclusions can beshown as follows
(1) The aircraft combat survivability evaluation indexsystem is established in hierarchy containing sus-ceptibility index and vulnerability index and theirsubindexes which is essential to aircraft survivability
(2) The multiattribute intelligent grey target decisionmodel is introduced which not only can be used inthe evaluation of aircraft combat survivability butalso can be used in any similar evaluation questions
(3) Then a combination weighting method is broughtup based on a modified AHP (analytic hierarchyprocess)method and the entropymethod offering thecombination weight to the multiattribute intelligentgrey target decision model which can evade thedisadvantages This method produces a scientific andrational weight improving the accuracy and objec-tivity of the evaluation which can be used in anyevaluation requiring a rational weight
(4) In the end a numerical example containing fiveaircraft is given proving that this method is prac-ticable and effective The result of more numericalcalculation can provide more details to find the mostimportant influencing factors which can be used inthe program design of new aircraft and modifieddesign of the old aircraft
However we did not consider the real states in thecombat such as detected tracked or hit as well as theinfluence of aircraft combat capability So future work willbe carried out on in directions (1) establish a more rationalevaluation system and findmore rational subindexes such asthe representation of infrared signal character (2) considerthe real states in the combat the threats and aircraft combatcapability to give capability of an aircraft to avoid orwithstanda man-made hostile environment
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Military Handbook Aircraft Survivability MIL-HDBK-2069 US Department of Defense 2069
[2] X Wang B-F Song and Y-F Hu ldquoAnalytic model for aircraftsurvivability assessment of a one-on-one engagementrdquo Journalof Aircraft vol 46 no 1 pp 223ndash229 2009
[3] H E Konokman A Kayran and M Kaya Analysis of AircraftSurvivability Against Fragmenting Warhead Threat AmericanInstitute of Aeronautics andAstronautics NationalHarborMdUSA 2014
[4] J Li W Yang Y Zhang Y Pei Y Ren and W Wang ldquoAircraftvulnerability modeling and computation methods based onproduct structure and CATIArdquo Chinese Journal of Aeronauticsvol 26 no 2 pp 334ndash342 2013
[5] T Erlandsson and L Niklasson ldquoA five states survivabilitymodel formissionswith ground-to-air threatsrdquo inModeling andSimulation for Defense Systems and Applications VIII vol 8752of Proceedings of SPIE pp 1ndash7 May 2013
[6] S Shi B-F Song Y Pei and T Cheng ldquoAssessment method ofaircraft susceptibility based on agent theoryrdquo Acta Aeronauticaet Astronautica Sinica vol 35 no 2 pp 444ndash453 2014 (Chi-nese)
[7] S F Liu and N M Xie Grey Systems Theory and ApplicationScience Press Beijing China 4th edition 2008 (Chinese)
[8] S-H Ahn Y-C Kim T-W Bae B-I Kim and K-H KimldquoDIRCM jamming effect analysis of spin-scan reticle seekerrdquo inProceedings of the IEEE 9th Malaysia International Conferenceon Communications with a Special Workshop on Digital TVContents (MICC rsquo09) pp 183ndash186 Kuala Lumpur MalaysiaDecember 2009
[9] J Paterson ldquoOverview of low observable technology and itseffects on combat aircraft survivabilityrdquo Journal of Aircraft vol36 no 2 pp 380ndash388 1999
[10] S-F Liu W-F Yuan and K-Q Sheng ldquoMulti-attribute intelli-gent grey target decision modelrdquo Control and Decision vol 25no 8 pp 1159ndash1163 2010
[11] M J Tabar Analysis of Decisions Made Using the AnalyticHierarchy Process Naval Postgraduate School Monterey CalifUSA 2013
[12] Z Yi andW Zhuo-Fu ldquoBid evaluation research of constructionproject based on two-stage entropy weightrdquo in Proceedingsof the International Conference on Information ManagementInnovation Management and Industrial Engineering (ICIII rsquo09)vol 2 pp 133ndash135 IEEE Xirsquoan China December 2009
[13] W-L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993
[14] R Guo J Guo Y-B Su and Y-D Zhang ldquoRanking limitationand improvement strategy of vague sets based on score func-tionrdquo Systems Engineering and Electronics vol 36 no 1 pp 105ndash110 2014 (Chinese)
