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 twolevel 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.
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 [
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 [
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.
The index system for sortie generation capacity of carrier aircrafts is established with related research results. A threelevel 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 Figure
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) Takeoff 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.
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.
Steps of principal component reduction are as shown in Figure
Steps of principal component reduction.
Specific steps are as follows.
Each index is nondimensionalized due to the different dimensions of indexes. The numerical transformation can eliminate the dimensional effect of indexes.
The characteristic roots of
The feature vectors of
If the number of original variables is more, the first
Thus, the number of principal components is determined with a balance between
Finally,
The index system after reduction is shown in Figure
Index system after reduction.
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.
The steps of catastrophe progression method are as shown in Figure
Steps of catastrophe progression method.
The type of catastrophe system is determined by the number of subindexes, which is shown in Table
Type of catastrophe system.
Type  Number of subindexes  System model 

Sharp point type  2 

Dovetail type  3 

Butterfly type  4 

In Table
The critical points of potential function
The singular points of potential function
The unitary formula can be derived from bifurcation set, which will transform different states of subindex to the same state.
(
Then the normalization formula can be derived from
(
Then the normalization formula can be derived from
(
Then the normalization formula can be derived from
Normalization formula is a multidimensional fuzzy membership function.
The ideal strategy is obtained from (
The membership function is
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 Tables
Index
Scenario 




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  196 
Index
Scenario 




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  16 
Index
Scenario 





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  5 
Index
Scenario 






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 
Take the reduction process of index
The
Standardization of index
Scenario 






1 





2 





3 





4 





5 





6 





7 





8 





9 





10 





One has
One has
One has
One has
Let
When
The principal components of index
Similarly, the principal components of index
The comprehensive scoring model can be obtained from (
The weights of indexes are shown in Figure
Weights of sortie generation capacity indexes.
Figure
The index system after reduction is shown in Figure
Index system after reduction.
The steps of catastrophe progression evaluation are as follows.
Take the principal components
Normalization of principal components.
Scenario 




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  0.4574 
The evaluation values of catastrophe progression for indexes
Evaluation values of indexes
Scenario 





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  0.6519 
The number of subindexes for index
Evaluation values of index
Scenario  1  2  3  4  5  6  7  8  9  10 


0.9663  0.9554  0.7237  0.9719  0.6233  0.9268  0.9239  0.8832  0.9453  0.9380 
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 Figure
Comparison of evaluation results.
Deviations of evaluation results.
Figure
Therefore, the selected 10 scenarios can be sorted according to the comprehensive evaluation results, which is shown in Figure
Order of scenarios index.
The Figure
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.
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 Figure
Sortie generation rate capacity
Availability capacity
Tasks completion capacity
Support, ejection, and recovery capacity
Emergency sortie generation rate
Surge sortie generation rate
Last sortie generation rate
Performing tasks proportion
Missing tasks proportion waiting for parts
Missing tasks proportion waiting for repair
Scheduled completion proportion
Pilot utilization rate
Plan implementation probability per aircraft
Sortie generation rate per aircraft
Preparation time for next sortie
Ejection interval
Takeoff outage proportion
Recovery interval
Overshoot proportion
Original input matrix
Standardized matrix
Number of scenarios
Number of indexes
Mean of
Standard deviation of
Correlation coefficient matrix
Characteristic root
Realvalued vector
Feature vector
Information amount of each component in the total information amount
Number of principal components
Proportion of the
Independent principal components
Potential function
Subindex
Fuzzy targets
Membership function.
The authors declare that there are no conflicts of interest regarding the publication of this study.