# An Evaluation Method for Sortie Generation Capacity of Carrier Aircrafts with Principal Component Reduction and Catastrophe Progression Method

Authors: Guoqing Xia; Tiantian Luan; Mingxiao Sun
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

This study proposes a new comprehensive evaluation method with principal component reduction and catastrophe progression method, considering the complexity, hierarchy, contradiction, and relevance of the factors in sortie generation of carrier aircrafts. First of all, the index system of sortie generation capacity is ascertained, which contains correlative indexes. The principal component reduction method is applied to transform the correlative indexes into independent indexes. This method eliminates the effect of correlativity among indexes. The principal components are determined with their contributions. Then the reduced principal components are evaluated in catastrophe progression method. This method is easy to realize without weights, which is more objective. In fact, catastrophe progression method is a multidimensional fuzzy membership function, which is suitable for the incompatible multiobjective evaluation. Thus, a two-level evaluation method for sortie generation capacity of carrier aircrafts is realized with principal component reduction and catastrophe progression method. Finally, the Surge operation of aircraft carrier “Nimitz” is taken as an example to evaluate the sortie generation capacity. The results of the proposed method are compared with those of Analytic Hierarchy Process, which verify the usefulness and reliability of the proposed method.


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## Body

## 1. Introduction

Aircraft carrier is the important part in the modern naval warfare. The research on the warfare capacity of aircraft carrier has become a hot issue with the increasing attention of the security in the territorial sea [1–3]. The comparison of sortie generation capacity of aircraft carrier in different operational schemes is helpful to determine the final plan [4–7]. Therefore, the evaluation for sortie generation capacity of aircraft carrier has important theoretical significance and application value.The sortie generation capacity of aircraft carrier is mostly evaluated by Analytic Hierarchical Process (AHP) at present. The evaluation results are obtained by subjective scores of experts. Reference [8] studied the application of AHP in the measurement process. Reference [9] evaluated the original purchase process with AHP. Reference [10] developed an evaluation tool for the information sharing of supply chain using AHP. Reference [11] discussed the application of AHP in the process of risk assessment. An improved AHP in [12] was used in the priority scheduling problems. Reference [13] researched the application of AHP in business management. Reference [14] proposed the combination of fuzzy theory and AHP and discussed the consistency problem of the evaluation method. Reference [15] pointed out the shortcomings and improvements of AHP. Reference [16] studied the evaluation process of comprehensive method of fuzzy AHP. Reference [17] solved mining selection problem based on AHP and fuzzy mathematics. However, these evaluation methods are one-sided and subjective, which ignore the correlation and contradiction of indexes. AHP is difficult to evaluate the multivariate evaluation objects objectively [18–26].A new evaluation method of principal component reduction (PCR) and catastrophe progression method (CPM) is proposed to evaluate the sortie generation capacity of carrier aircrafts in this study. The proposed method can avoid the subjectivity and complexity in the traditional evaluation method. The main contents are as follows. Firstly, the hierarchy structure of index system for sortie generation capacity is determined. Secondly, the related indexes are transformed to independent principal components by PCR. Then, independent principal components are evaluated by CPM. Finally, the usefulness and reliability of the new method are verified by comparing with the traditional evaluation method.


## 2. Index System for Sortie Generation Capacity of Carrier Aircrafts

The index system for sortie generation capacity of carrier aircrafts is established with related research results. A three-level index system with complexity, hierarchy, contradiction, and relevance is established by recursive hierarchy method. The index system for sortie generation capacity of carrier aircrafts is shown in Figure1.Figure 1
Index system for sortie generation of carrier aircrafts.These indexes are defined as follows.(1) Emergency sortie generation rate (ESGR): it is the maximum number of ready aircrafts taking off in a few minutes.(2) Surge sortie generation rate (SSGR): it is the average number of aircrafts per day in the Surge operation (4 days).(3) Last sortie generation rate (LSGR): it is the average number of aircrafts per day in the continuous operation (30 days).(4) Performing tasks proportion (PTP): it is the time proportion that the aircrafts can carry out one task at least under a certain flight plan and logistics condition.(5) Missing tasks proportion waiting for parts (MTPWP): it is the proportion of aircrafts missing the tasks due to waiting for parts.(6) Missing tasks proportion waiting for repair (MTPWR): it is the proportion of aircrafts missing the tasks due to waiting for repair.(7) Scheduled completion proportion (SCP): it is the proportion of completed number in the planned number of aircrafts.(8) Pilot utilization rate (PUR): it is the average utilization rate of the pilots per day.(9) Plan implementation probability per aircraft (PIPA): it is the plan implementation probability per aircraft under the certain constraints in a given period of time.(10) Sortie generation rate per aircraft (SGRA): it is the sortie generation rate per aircraft under the certain constraints.(11) Preparation time for next sortie (PTNS): it is the preparation time for next sortie under the condition of a certain resource allocation.(12) Ejection interval (EI): it is the average time for ejecting a single aircraft per catapult.(13) Take-off outage proportion (TOOP): it is the proportion of canceled number in the ready number of aircrafts.(14) Recovery interval (RI): it is the average time for recovering a single aircraft.(15) Overshoot proportion (OP): it is the proportion of number of aircrafts failed to recover in the number of aircrafts ready to recover.


## 3. Principal Component Reduction Method

### 3.1. Principal Component Reduction Principle

There are correlations between various indexes for sortie generation capacity of carrier aircrafts, which will bring repetitive information. The independent indexes can be obtained from related indexes using principal component reduction method. This method can minimize the information loss after reduction.Principal component reduction uses dimension reduction techniques to obtain less comprehensive variables instead of the original variables. These comprehensive variables cover the most information of the original variables. Then the objective phenomenon is evaluated by calculating the score of comprehensive principal component.


### 3.2. Steps of Principal Component Reduction

Steps of principal component reduction are as shown in Figure2.Figure 2
Steps of principal component reduction.Specific steps are as follows.Step 1 (parameters standardization).
Each index is nondimensionalized due to the different dimensions of indexes. The numerical transformation can eliminate the dimensional effect of indexes.Z-Score method is applied to transform the original matrix X=[xij]n×m to standardized matrix Z=[zij]n×m, where n is the number of scenarios and m is the number of indexes:(1)zij=xij-xj¯sj,where xj¯ is the mean of jth index and sj is the standard deviation of jth index.Step 2 (correlation coefficient matrixR).
(2)
r
j
k
=
1
n
-
1
∑
i
=
1
n
z
i
j
z
i
k
,where R=[rjk]m×m, rii=1, and rjk=rkj.Step 3 (characteristic roots ofR).
The characteristic roots ofR can be calculated by(3)λgIm-R=0,where λgg=1,2,…,m is the characteristic root, which is the variance of principal component. It denotes the effect of each principal component on the evaluated object.Step 4 (feature vectors ofR).
The feature vectors ofR can be obtained from(4)λgIm-RLg=0,where L is a real-valued vector of m dimensions. Lg is the feature vector of λg and the coefficient of zj in the new coordinate system.Step 5 (contribution ofR).
α
g is the information amount of each component in the total information amount, which is the contribution(5)αg=λg∑g=1mλg.Step 6 (number of principal componentsK).
If the number of original variables is more, the firstK principal components are analyzed instead of the original variables and the other variables are ignored. The proportion of the K principal components in the original variable information is αK:(6)αK=∑g=1Kλg∑g=1mλg-1.
Thus, the number of principal components is determined with a balance betweenK and αK. On the one hand, the smaller K is better. On the other hand, the larger αK is better. It will keep enough information with few components in this way. In this study, αK≥95%.Finally,m related indexes can be transformed to K independent principal components fi1-fiK:(7)fig=∑j=1mLijzij.The index system after reduction is shown in Figure3.Figure 3
Index system after reduction.


## 3.1. Principal Component Reduction Principle

There are correlations between various indexes for sortie generation capacity of carrier aircrafts, which will bring repetitive information. The independent indexes can be obtained from related indexes using principal component reduction method. This method can minimize the information loss after reduction.Principal component reduction uses dimension reduction techniques to obtain less comprehensive variables instead of the original variables. These comprehensive variables cover the most information of the original variables. Then the objective phenomenon is evaluated by calculating the score of comprehensive principal component.


## 3.2. Steps of Principal Component Reduction

Steps of principal component reduction are as shown in Figure2.Figure 2
Steps of principal component reduction.Specific steps are as follows.Step 1 (parameters standardization).
Each index is nondimensionalized due to the different dimensions of indexes. The numerical transformation can eliminate the dimensional effect of indexes.Z-Score method is applied to transform the original matrix X=[xij]n×m to standardized matrix Z=[zij]n×m, where n is the number of scenarios and m is the number of indexes:(1)zij=xij-xj¯sj,where xj¯ is the mean of jth index and sj is the standard deviation of jth index.Step 2 (correlation coefficient matrixR).
(2)
r
j
k
=
1
n
-
1
∑
i
=
1
n
z
i
j
z
i
k
,where R=[rjk]m×m, rii=1, and rjk=rkj.Step 3 (characteristic roots ofR).
The characteristic roots ofR can be calculated by(3)λgIm-R=0,where λgg=1,2,…,m is the characteristic root, which is the variance of principal component. It denotes the effect of each principal component on the evaluated object.Step 4 (feature vectors ofR).
The feature vectors ofR can be obtained from(4)λgIm-RLg=0,where L is a real-valued vector of m dimensions. Lg is the feature vector of λg and the coefficient of zj in the new coordinate system.Step 5 (contribution ofR).
α
g is the information amount of each component in the total information amount, which is the contribution(5)αg=λg∑g=1mλg.Step 6 (number of principal componentsK).
If the number of original variables is more, the firstK principal components are analyzed instead of the original variables and the other variables are ignored. The proportion of the K principal components in the original variable information is αK:(6)αK=∑g=1Kλg∑g=1mλg-1.
Thus, the number of principal components is determined with a balance betweenK and αK. On the one hand, the smaller K is better. On the other hand, the larger αK is better. It will keep enough information with few components in this way. In this study, αK≥95%.Finally,m related indexes can be transformed to K independent principal components fi1-fiK:(7)fig=∑j=1mLijzij.The index system after reduction is shown in Figure3.Figure 3
Index system after reduction.


## 4. Catastrophe Progression Evaluation Method

### 4.1. Description of Catastrophe Progression Method

The index system for sortie generation capacity of carrier aircrafts after reduction is applied to evaluate. The contradiction decomposition of evaluated objects is the first step of catastrophe progression method. Then catastrophe fuzzy membership function is the combination of catastrophe theory and fuzzy mathematics. This method considers the relative importance of evaluation indexes instead of index weight, which reduces the subjectivity and simplifies calculation.In the process of formulating combat scenario, a variety of scenarios are designed due to the influence of various factors. The scenarios are evaluated comprehensively in the process of selecting the optimal scenario. The evaluation process is conducted from the indexes in lower levels to the indexes in upper levels according to catastrophe progression method. Finally, a catastrophe progression between 0 and 1 can be obtained. If the catastrophe progression is bigger, the scenario is better.


### 4.2. Steps of Catastrophe Progression Method

The steps of catastrophe progression method are as shown in Figure4.Figure 4
Steps of catastrophe progression method.Step 1 (type of catastrophe system).
The type of catastrophe system is determined by the number of subindexes, which is shown in Table1.Table 1
Type of catastrophe system.
Type
Number of subindexes
System model
Sharp point type
2
f
x
=
x
4
+
a
x
2
+
b
x
Dovetail type
3
f
x
=
1
5
x
5
+
1
3
a
x
3
+
1
2
b
x
2
+
c
x
Butterfly type
4
f
x
=
1
6
x
6
+
1
4
a
x
4
+
1
3
b
x
3
+
1
2
c
x
2
+
d
xIn Table1, fx is the potential function of x. a, b, c, and d are subindexes, which are sorted from high importance to low importance.Step 2 (unitary formula).
The critical points of potential functionfx gather to a balance surface based on catastrophe theory, which can be obtained from the first-order derivative of fx:(8)f′x=0.The singular points of potential functionfx can be obtained by the second-order derivative:(9)f′′x=0.The unitary formula can be derived from bifurcation set, which will transform different states of subindex to the same state.(1) Bifurcation set equations of sharp point system are(10)a=-6x2,b=8x3.Then the normalization formula can be derived from(11)xa=a1/2,xb=b1/3,where xa is the value of x corresponding a. xb is the value of x corresponding b.(2) Bifurcation set equations of dovetail system are(12)a=-6x2,b=8x3,c=-3x4.Then the normalization formula can be derived from(13)xa=a1/2,xb=b1/3,xc=c1/4.(3) Bifurcation set equations of butterfly system are(14)a=-10x2,b=20x3,c=-15x4,d=4x5.Then the normalization formula can be derived from(15)xa=a1/2,xb=b1/3,xc=c1/4,xd=d1/5.Normalization formula is a multidimensional fuzzy membership function.Step 3 (comprehensive evaluation with normalization formula).
The ideal strategy is obtained from (16), when the fuzzy targets are A1,A2,…,Am in the same scenario(16)C=A1∩A2∩⋯∩Am.The membership function is(17)μx=μA1x∧μA2x∧⋯∧μAmx,where μAix is the membership function of Ai. If the indexes are complementary, the membership function is the average value of μAix.


## 4.1. Description of Catastrophe Progression Method

The index system for sortie generation capacity of carrier aircrafts after reduction is applied to evaluate. The contradiction decomposition of evaluated objects is the first step of catastrophe progression method. Then catastrophe fuzzy membership function is the combination of catastrophe theory and fuzzy mathematics. This method considers the relative importance of evaluation indexes instead of index weight, which reduces the subjectivity and simplifies calculation.In the process of formulating combat scenario, a variety of scenarios are designed due to the influence of various factors. The scenarios are evaluated comprehensively in the process of selecting the optimal scenario. The evaluation process is conducted from the indexes in lower levels to the indexes in upper levels according to catastrophe progression method. Finally, a catastrophe progression between 0 and 1 can be obtained. If the catastrophe progression is bigger, the scenario is better.


## 4.2. Steps of Catastrophe Progression Method

The steps of catastrophe progression method are as shown in Figure4.Figure 4
Steps of catastrophe progression method.Step 1 (type of catastrophe system).
The type of catastrophe system is determined by the number of subindexes, which is shown in Table1.Table 1
Type of catastrophe system.
Type
Number of subindexes
System model
Sharp point type
2
f
x
=
x
4
+
a
x
2
+
b
x
Dovetail type
3
f
x
=
1
5
x
5
+
1
3
a
x
3
+
1
2
b
x
2
+
c
x
Butterfly type
4
f
x
=
1
6
x
6
+
1
4
a
x
4
+
1
3
b
x
3
+
1
2
c
x
2
+
d
xIn Table1, fx is the potential function of x. a, b, c, and d are subindexes, which are sorted from high importance to low importance.Step 2 (unitary formula).
The critical points of potential functionfx gather to a balance surface based on catastrophe theory, which can be obtained from the first-order derivative of fx:(8)f′x=0.The singular points of potential functionfx can be obtained by the second-order derivative:(9)f′′x=0.The unitary formula can be derived from bifurcation set, which will transform different states of subindex to the same state.(1) Bifurcation set equations of sharp point system are(10)a=-6x2,b=8x3.Then the normalization formula can be derived from(11)xa=a1/2,xb=b1/3,where xa is the value of x corresponding a. xb is the value of x corresponding b.(2) Bifurcation set equations of dovetail system are(12)a=-6x2,b=8x3,c=-3x4.Then the normalization formula can be derived from(13)xa=a1/2,xb=b1/3,xc=c1/4.(3) Bifurcation set equations of butterfly system are(14)a=-10x2,b=20x3,c=-15x4,d=4x5.Then the normalization formula can be derived from(15)xa=a1/2,xb=b1/3,xc=c1/4,xd=d1/5.Normalization formula is a multidimensional fuzzy membership function.Step 3 (comprehensive evaluation with normalization formula).
The ideal strategy is obtained from (16), when the fuzzy targets are A1,A2,…,Am in the same scenario(16)C=A1∩A2∩⋯∩Am.The membership function is(17)μx=μA1x∧μA2x∧⋯∧μAmx,where μAix is the membership function of Ai. If the indexes are complementary, the membership function is the average value of μAix.


## 5. Evaluation for Sortie Generation Capacity of Carrier Aircrafts

### 5.1. Evaluation Samples

The object of evaluation is the Surge operation of “Nimitz” carrier in 1997. Ten scenarios are selected randomly in order to ensure the scientific nature, which are shown in Tables2–5. In Tables 2–5, X11 is emergency sortie generation rate, X12 is surge sortie generation rate, X13 is last sortie generation rate, X21 is performing tasks proportion, X22 is missing tasks proportion waiting for parts, X23 is missing tasks proportion waiting for repair, X31 is scheduled completion proportion, X32 is pilot utilization rate, X33 is plan implementation probability per aircraft, X34 is sortie generation rate per aircraft, X41 is preparation time for next sortie, X42 is ejection interval, X43 is take-off outage proportion, X44 is recovery interval, and X45 is overshoot proportion. The data in Tables 2–5 are the original data.Table 2
IndexX1 of sortie generation rate capacity.
Scenario
X
11 (sortie)
X
12 (sortie/day)
X
13 (sortie/day)
1
30
250
200
2
31
240
180
3
29
235
210
4
33
260
220
5
32
210
170
6
29
245
194
7
27
267
230
8
32
211
183
9
25
261
201
10
32
232
196Table 3
IndexX2 of aircraft availability capacity.
Scenario
X
21 (%)
X
22 (%)
X
23 (%)
1
80
11
9
2
85
20
5
3
90
4
6
4
75
11
14
5
82
10
18
6
91
3
5
7
78
23
9
8
84
9
7
9
85
1
4
10
72
2
16Table 4
IndexX3 of tasks completion capacity.
Scenario
X
31 (%)
X
32 (sortie/day)
X
33 (%)
X
34 (sortie/day)
1
85
2.5
90
6
2
74
2.2
80
7
3
81
2.0
84
5
4
90
1.8
75
6
5
61
1.5
68
5
6
86
1.9
86
8
7
78
2.1
88
5
8
65
2.3
94
6
9
79
2.4
81
7
10
83
1.7
82
5Table 5
IndexX4 of support, ejection, and recovery capability.
Scenario
X
41 (minute)
X
42 (minute)
X
43 (%)
X
44 (minute)
X
45 (%)
1
30
1
1
1.5
3.3
2
32
2
0.5
1.8
5
3
28
1.4
1.2
1.4
1
4
25
1.6
1.6
1.9
7
5
33
2.5
2
2.2
6
6
45
1.1
3
1.1
2
7
27
0.6
0.6
1.2
10
8
26
0.7
0.8
1.6
3
9
36
0.5
1.6
1.7
4.5
10
29
1.2
0.4
2.1
6


### 5.2. Indexes Reduction

Take the reduction process of indexX4 as an example.Step 1 (standardization).
TheZ-Score method is used to standardize indexes, and the results are as shown in Table 6. In Table 6, Z41, Z42, Z43, and Z44 are standardize indexes.Table 6
Standardization of indexX4.
Scenario
Z
41
Z
42
Z
43
Z
44
Z
45
1
-
0.1854
-
0.4086
-
0.3352
-
0.4134
-
0.5609
2
0.1517
1.1630
-
0.9558
0.4134
0.0834
3
-
0.5224
0.2200
-
0.0869
-
0.6890
-
1.4326
4
-
1.0280
0.5343
0.4096
0.6890
0.8414
5
0.3202
1.9487
0.9062
1.5157
0.4624
6
2.3425
-
0.2515
2.1475
-
1.5157
-
1.0536
7
-
0.6909
-
1.0372
-
0.8317
-
1.2402
1.9784
8
-
0.8595
-
0.8801
-
0.5834
-
0.1378
-
0.6746
9
0.8258
-
1.1944
0.4096
0.1378
-
0.1061
10
-
0.3539
-
0.0943
-
1.0799
1.2402
0.4624Step 2 (correlation coefficient matrix).
One has(18)R=10.01590.7398-0.2916-0.37310.015910.16000.59190.02790.73980.16001-0.2186-0.3401-0.29160.5919-0.218610.2588-0.37310.0279-0.34010.25881.Step 3 (characteristic roots).
One has(19)λ1λ2λ3λ4λ5=2.17811.52060.73100.23810.3322.Step 4 (feature vectors).
One has(20)L1L2L3L4L5=-0.5778-0.2117-0.2907-0.6141-0.39960.1322-0.73420.0912-0.29760.5888-0.5450-0.3264-0.31130.69720.11610.4056-0.55420.09270.2136-0.68860.43260.0500-0.8954-0.05120.0780.Step 5 (contribution).
One has(21)α1α2α3α4α5=0.43560.30410.14620.04760.0664.Step 6 (number of principal components).
LetαK≥95%; then sort α from big to small:(22)α1α2α3α5α4=0.43560.30410.14620.06640.0476.
WhenK=4, αK=95.24%≥95%.Step 7 (principal components).
The principal components of indexX4 are f41, f42, f43, and f44, which are determined by characteristic roots, feature vectors, and the number of principal components:(23)f41=-0.3915X41+0.1072X42-0.6375X43+0.8312X44+0.7505X45,f42=-0.1434X41-0.5954X42-0.3818X43-1.1359X44+0.0868X45,f43=-0.1970X41+0.0740X42-0.3641X43+0.1900X44-1.5535X45,f44=-0.2708X41+0.4774X42+0.1357X43-1.4112X44+0.1354X45.
Similarly, the principal components of indexX1, X2, and X3 can be derived by repeating Steps 1–7:(24)f11=-1.2109X11+0.7943X12+0.3977X13,f12=2.0214X11+0.2061X12+0.3569X13,f13=0.5796X11+0.9406X12-0.4140X13,f21=0.5329X21-0.4290X22-0.6711X23,f22=-0.0095X21-1.8995X22+0.2937X23,f23=-0.5352X21-0.3935X22-0.6795X23,f31=0.5573X31+0.7273X32+0.5758X33+0.3004X34,f32=1.4342X31-0.3351X32-0.3749X33+0.3459X34,f33=1.1895X31-0.0913X32+0.3033X33-0.5228X34,f34=0.2282X31+0.8156X32-0.6306X33-0.1828X34.The comprehensive scoring model can be obtained from (23) and (24) and the contributions:(25)Y1=0.4180X11+0.6767X12+0.3430X13,Y2=0.2630X21-0.9130X22-0.3558X23,Y3=0.8846X31+0.2985X32+0.1979X33+0.1235X34,Y4=-0.2740X41-0.0964X42-0.4599X43-0.0518X44-0.1420X45.The weights of indexes are shown in Figure5 according to (25). In Figure 5, the horizontal axis is the evaluated index and the vertical axis is the weight of index.Figure 5
Weights of sortie generation capacity indexes.Figure5 shows that the most important subindexes are the pilot utilization rate and scheduled completion proportion, the weights of which are greater than weights of other indexes.


### 5.3. Catastrophe Progression Evaluation

The index system after reduction is shown in Figure6.Figure 6
Index system after reduction.The steps of catastrophe progression evaluation are as follows.Step 1 (normalization).
Take the principal componentsf11, f12, and f13 of index X1 as the example. The results are shown in Table 7. In Table 7, f11, f12, and f13 are normalization of principal components.Table 7
Normalization of principal components.
Scenario
f
11
f
12
f
13
1
0.5301
0.5716
0.6225
2
0.3134
0.6165
0.7314
3
0.5451
0.4590
0
4
0.4320
1
1
5
0
0.6593
0.2127
6
0.5541
0.4282
0.4463
7
0.9863
0.3299
0.3273
8
0.0532
0.6986
0.0815
9
1
0
0.2813
10
0.2390
0.7688
0.4574Step 2 (calculate evaluation value).
The evaluation values of catastrophe progression for indexesX1, X2, X3, and X4 are calculated. The number of subindexes for indexes X1 and X2 is three; then the type of catastrophe is dovetail type. The number of subindexes for indexes X3 and X4 is four; then the type of catastrophe is butterfly type. Thus the evaluation values are shown in Table 8. In Table 8, X1 is sortie generation rate capacity, X2 is availability capacity, X3 is tasks completion capacity, and X4 is support, ejection, and recovery capacity. The data in Table 8 are the evaluation values of the above four indexes.Table 8
Evaluation values of indexesX1, X2, X3, and X4.
Scenario
X
1
X
2
X
3
X
4
1
0.8154
0.7687
0.9483
0.8721
2
0.7786
0.6630
0.8140
0.8525
3
0.5032
0.9201
0.8629
0.8776
4
0.8858
0.5842
0.9191
0.8212
5
0.5165
0.3618
0.3380
0.6127
6
0.7718
0.9456
0.6934
0.6853
7
0.8135
0.3654
0.8388
0.7291
8
0.5508
0.8386
0.5526
0.8818
9
0.5761
0.9757
0.8906
0.8003
10
0.7425
0.6427
0.8199
0.6519Step 3 (calculate evaluation value ofX).
The number of subindexes for indexX is four; then the type of catastrophe is butterfly type. The evaluation results of 10 scenarios are shown in Table 9. X is sortie generation capacity. The data in Table 9 are the evaluation values of sortie generation capacity.Table 9
Evaluation values of indexX.
Scenario
1
2
3
4
5
6
7
8
9
10
X
0.9663
0.9554
0.7237
0.9719
0.6233
0.9268
0.9239
0.8832
0.9453
0.9380


### 5.4. Analysis of Evaluation Results

The evaluation results of the proposed method are compared with that of AHP, in order to verify the usefulness of the proposed method. The comparison is shown in Figure7 and the deviations of evaluation results are shown in Figure 8. In Figure 7, the horizontal axis is the evaluated scenario and the vertical axis is the evaluation value. In Figure 8, the horizontal axis is the evaluated scenario and the vertical axis is the deviation of proposed method and AHP.Figure 7
Comparison of evaluation results.Figure 8
Deviations of evaluation results.Figure7 shows that that evaluation results of two methods are similar. Figure 8 shows the deviations of evaluation results, and the maximum absolute value of deviation is less than 0.05, which verifies usefulness and reliability of the principal component reduction and catastrophe progression evaluation method. The proposed method can evaluate scenarios more objectively.Therefore, the selected 10 scenarios can be sorted according to the comprehensive evaluation results, which is shown in Figure9. In Figure 9, the horizontal axis is the evaluated scenario and the vertical axis is the evaluation value.Figure 9
Order of scenarios index.The Figure9 shows that the best scenario is scenario 4. Its comprehensive evaluation value is 0.9719. And the worst scenario is scenario 5. Its comprehensive evaluation value is 0.6233. The best scenario can be elected from the selected scenarios. Thus, the evaluation method will help decision-maker to draw up a plan.Therefore, the principal component reduction and catastrophe progression evaluation method can analyze the importance of indexes for sortie generation capacity and sort the selected scenarios objectively and reliably.


## 5.1. Evaluation Samples

The object of evaluation is the Surge operation of “Nimitz” carrier in 1997. Ten scenarios are selected randomly in order to ensure the scientific nature, which are shown in Tables2–5. In Tables 2–5, X11 is emergency sortie generation rate, X12 is surge sortie generation rate, X13 is last sortie generation rate, X21 is performing tasks proportion, X22 is missing tasks proportion waiting for parts, X23 is missing tasks proportion waiting for repair, X31 is scheduled completion proportion, X32 is pilot utilization rate, X33 is plan implementation probability per aircraft, X34 is sortie generation rate per aircraft, X41 is preparation time for next sortie, X42 is ejection interval, X43 is take-off outage proportion, X44 is recovery interval, and X45 is overshoot proportion. The data in Tables 2–5 are the original data.Table 2
IndexX1 of sortie generation rate capacity.
Scenario
X
11 (sortie)
X
12 (sortie/day)
X
13 (sortie/day)
1
30
250
200
2
31
240
180
3
29
235
210
4
33
260
220
5
32
210
170
6
29
245
194
7
27
267
230
8
32
211
183
9
25
261
201
10
32
232
196Table 3
IndexX2 of aircraft availability capacity.
Scenario
X
21 (%)
X
22 (%)
X
23 (%)
1
80
11
9
2
85
20
5
3
90
4
6
4
75
11
14
5
82
10
18
6
91
3
5
7
78
23
9
8
84
9
7
9
85
1
4
10
72
2
16Table 4
IndexX3 of tasks completion capacity.
Scenario
X
31 (%)
X
32 (sortie/day)
X
33 (%)
X
34 (sortie/day)
1
85
2.5
90
6
2
74
2.2
80
7
3
81
2.0
84
5
4
90
1.8
75
6
5
61
1.5
68
5
6
86
1.9
86
8
7
78
2.1
88
5
8
65
2.3
94
6
9
79
2.4
81
7
10
83
1.7
82
5Table 5
IndexX4 of support, ejection, and recovery capability.
Scenario
X
41 (minute)
X
42 (minute)
X
43 (%)
X
44 (minute)
X
45 (%)
1
30
1
1
1.5
3.3
2
32
2
0.5
1.8
5
3
28
1.4
1.2
1.4
1
4
25
1.6
1.6
1.9
7
5
33
2.5
2
2.2
6
6
45
1.1
3
1.1
2
7
27
0.6
0.6
1.2
10
8
26
0.7
0.8
1.6
3
9
36
0.5
1.6
1.7
4.5
10
29
1.2
0.4
2.1
6


## 5.2. Indexes Reduction

Take the reduction process of indexX4 as an example.Step 1 (standardization).
TheZ-Score method is used to standardize indexes, and the results are as shown in Table 6. In Table 6, Z41, Z42, Z43, and Z44 are standardize indexes.Table 6
Standardization of indexX4.
Scenario
Z
41
Z
42
Z
43
Z
44
Z
45
1
-
0.1854
-
0.4086
-
0.3352
-
0.4134
-
0.5609
2
0.1517
1.1630
-
0.9558
0.4134
0.0834
3
-
0.5224
0.2200
-
0.0869
-
0.6890
-
1.4326
4
-
1.0280
0.5343
0.4096
0.6890
0.8414
5
0.3202
1.9487
0.9062
1.5157
0.4624
6
2.3425
-
0.2515
2.1475
-
1.5157
-
1.0536
7
-
0.6909
-
1.0372
-
0.8317
-
1.2402
1.9784
8
-
0.8595
-
0.8801
-
0.5834
-
0.1378
-
0.6746
9
0.8258
-
1.1944
0.4096
0.1378
-
0.1061
10
-
0.3539
-
0.0943
-
1.0799
1.2402
0.4624Step 2 (correlation coefficient matrix).
One has(18)R=10.01590.7398-0.2916-0.37310.015910.16000.59190.02790.73980.16001-0.2186-0.3401-0.29160.5919-0.218610.2588-0.37310.0279-0.34010.25881.Step 3 (characteristic roots).
One has(19)λ1λ2λ3λ4λ5=2.17811.52060.73100.23810.3322.Step 4 (feature vectors).
One has(20)L1L2L3L4L5=-0.5778-0.2117-0.2907-0.6141-0.39960.1322-0.73420.0912-0.29760.5888-0.5450-0.3264-0.31130.69720.11610.4056-0.55420.09270.2136-0.68860.43260.0500-0.8954-0.05120.0780.Step 5 (contribution).
One has(21)α1α2α3α4α5=0.43560.30410.14620.04760.0664.Step 6 (number of principal components).
LetαK≥95%; then sort α from big to small:(22)α1α2α3α5α4=0.43560.30410.14620.06640.0476.
WhenK=4, αK=95.24%≥95%.Step 7 (principal components).
The principal components of indexX4 are f41, f42, f43, and f44, which are determined by characteristic roots, feature vectors, and the number of principal components:(23)f41=-0.3915X41+0.1072X42-0.6375X43+0.8312X44+0.7505X45,f42=-0.1434X41-0.5954X42-0.3818X43-1.1359X44+0.0868X45,f43=-0.1970X41+0.0740X42-0.3641X43+0.1900X44-1.5535X45,f44=-0.2708X41+0.4774X42+0.1357X43-1.4112X44+0.1354X45.
Similarly, the principal components of indexX1, X2, and X3 can be derived by repeating Steps 1–7:(24)f11=-1.2109X11+0.7943X12+0.3977X13,f12=2.0214X11+0.2061X12+0.3569X13,f13=0.5796X11+0.9406X12-0.4140X13,f21=0.5329X21-0.4290X22-0.6711X23,f22=-0.0095X21-1.8995X22+0.2937X23,f23=-0.5352X21-0.3935X22-0.6795X23,f31=0.5573X31+0.7273X32+0.5758X33+0.3004X34,f32=1.4342X31-0.3351X32-0.3749X33+0.3459X34,f33=1.1895X31-0.0913X32+0.3033X33-0.5228X34,f34=0.2282X31+0.8156X32-0.6306X33-0.1828X34.The comprehensive scoring model can be obtained from (23) and (24) and the contributions:(25)Y1=0.4180X11+0.6767X12+0.3430X13,Y2=0.2630X21-0.9130X22-0.3558X23,Y3=0.8846X31+0.2985X32+0.1979X33+0.1235X34,Y4=-0.2740X41-0.0964X42-0.4599X43-0.0518X44-0.1420X45.The weights of indexes are shown in Figure5 according to (25). In Figure 5, the horizontal axis is the evaluated index and the vertical axis is the weight of index.Figure 5
Weights of sortie generation capacity indexes.Figure5 shows that the most important subindexes are the pilot utilization rate and scheduled completion proportion, the weights of which are greater than weights of other indexes.


## 5.3. Catastrophe Progression Evaluation

The index system after reduction is shown in Figure6.Figure 6
Index system after reduction.The steps of catastrophe progression evaluation are as follows.Step 1 (normalization).
Take the principal componentsf11, f12, and f13 of index X1 as the example. The results are shown in Table 7. In Table 7, f11, f12, and f13 are normalization of principal components.Table 7
Normalization of principal components.
Scenario
f
11
f
12
f
13
1
0.5301
0.5716
0.6225
2
0.3134
0.6165
0.7314
3
0.5451
0.4590
0
4
0.4320
1
1
5
0
0.6593
0.2127
6
0.5541
0.4282
0.4463
7
0.9863
0.3299
0.3273
8
0.0532
0.6986
0.0815
9
1
0
0.2813
10
0.2390
0.7688
0.4574Step 2 (calculate evaluation value).
The evaluation values of catastrophe progression for indexesX1, X2, X3, and X4 are calculated. The number of subindexes for indexes X1 and X2 is three; then the type of catastrophe is dovetail type. The number of subindexes for indexes X3 and X4 is four; then the type of catastrophe is butterfly type. Thus the evaluation values are shown in Table 8. In Table 8, X1 is sortie generation rate capacity, X2 is availability capacity, X3 is tasks completion capacity, and X4 is support, ejection, and recovery capacity. The data in Table 8 are the evaluation values of the above four indexes.Table 8
Evaluation values of indexesX1, X2, X3, and X4.
Scenario
X
1
X
2
X
3
X
4
1
0.8154
0.7687
0.9483
0.8721
2
0.7786
0.6630
0.8140
0.8525
3
0.5032
0.9201
0.8629
0.8776
4
0.8858
0.5842
0.9191
0.8212
5
0.5165
0.3618
0.3380
0.6127
6
0.7718
0.9456
0.6934
0.6853
7
0.8135
0.3654
0.8388
0.7291
8
0.5508
0.8386
0.5526
0.8818
9
0.5761
0.9757
0.8906
0.8003
10
0.7425
0.6427
0.8199
0.6519Step 3 (calculate evaluation value ofX).
The number of subindexes for indexX is four; then the type of catastrophe is butterfly type. The evaluation results of 10 scenarios are shown in Table 9. X is sortie generation capacity. The data in Table 9 are the evaluation values of sortie generation capacity.Table 9
Evaluation values of indexX.
Scenario
1
2
3
4
5
6
7
8
9
10
X
0.9663
0.9554
0.7237
0.9719
0.6233
0.9268
0.9239
0.8832
0.9453
0.9380


## 5.4. Analysis of Evaluation Results

The evaluation results of the proposed method are compared with that of AHP, in order to verify the usefulness of the proposed method. The comparison is shown in Figure7 and the deviations of evaluation results are shown in Figure 8. In Figure 7, the horizontal axis is the evaluated scenario and the vertical axis is the evaluation value. In Figure 8, the horizontal axis is the evaluated scenario and the vertical axis is the deviation of proposed method and AHP.Figure 7
Comparison of evaluation results.Figure 8
Deviations of evaluation results.Figure7 shows that that evaluation results of two methods are similar. Figure 8 shows the deviations of evaluation results, and the maximum absolute value of deviation is less than 0.05, which verifies usefulness and reliability of the principal component reduction and catastrophe progression evaluation method. The proposed method can evaluate scenarios more objectively.Therefore, the selected 10 scenarios can be sorted according to the comprehensive evaluation results, which is shown in Figure9. In Figure 9, the horizontal axis is the evaluated scenario and the vertical axis is the evaluation value.Figure 9
Order of scenarios index.The Figure9 shows that the best scenario is scenario 4. Its comprehensive evaluation value is 0.9719. And the worst scenario is scenario 5. Its comprehensive evaluation value is 0.6233. The best scenario can be elected from the selected scenarios. Thus, the evaluation method will help decision-maker to draw up a plan.Therefore, the principal component reduction and catastrophe progression evaluation method can analyze the importance of indexes for sortie generation capacity and sort the selected scenarios objectively and reliably.


## 6. Conclusions

This study proposes a new comprehensive evaluation method based on principal component reduction and catastrophe progression method. First of all, the index system of sortie generation capacity is ascertained in Figure1 and Tables 2–5, which contains correlative indexes. The principal component reduction method is applied to transform the correlative indexes into independent indexes in Figures 2, 5, and 6 and Table 6. This method eliminates the effect of correlativity among indexes. The principal components are determined with their contributions. Then the reduced principal components are evaluated in catastrophe progression method in Figures 3 and 4 and Table 1. This method is easy to realize without weights, which is more objective. In fact, catastrophe progression method is a multidimensional fuzzy membership function, which is suitable for the incompatible multiobjective evaluation. Thus, a two-level evaluation method for sortie generation capacity of carrier aircrafts is realized with principal component reduction and catastrophe progression method. The principal component reduction and catastrophe progression evaluation method can analyze the importance of indexes for sortie generation capacity and sort the selected scenarios objectively and reliably in Figures 7–9 and Tables 7–9. At the same time, the proposed method is suitable for other evaluated objects.



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*Source: 2678216-2017-05-28.xml*