CN117273786A - Estimation method for direct operation cost of civil passenger plane market - Google Patents
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Abstract
The invention relates to a method for estimating direct operation cost of civil airliner market, which relates to the technical field of cost management, and establishes a three-interval digital airliner market direct operation cost mixed estimation regression model aiming at minimizing the upper and lower bounds of the direct operation cost of the civil airliner market and the square sum of expected value point errors by taking the fluctuation of the direct operation cost of the civil airliner market, the seat capacity and the airline flight time into consideration. Regression analysis proves that the linear relation between the direct operation cost of the civil aircraft market and the seat capacity and the flight time of the air route is established, and the regression fitting level is higher. The parameterized model is verified to be superior to the point data model as a whole through predictive verification, and has better adaptability. The method has the advantages that the fluctuation of the data is considered to be more beneficial to the estimation of the direct operation cost of the civil aircraft market, and more accurate guidance opinion can be provided for the optimization of the performance of the civil aircraft in the design stage.
Description
Technical Field
The invention relates to the technical field of cost management, in particular to a method for estimating direct operation cost of a civil airliner market.
Background
The estimation of the direct operation cost (Direct Operating Cost, DOC) of the civil aircraft market refers to a method for integrally predicting the cost directly related to the civil aircraft model by taking a certain technical and economic connotation as a principle according to the condition of the civil aircraft model and the economic condition of the intended operation market. In the aircraft design stage, the DOC value of the civil aircraft market is accurately obtained, and the DOC value has extremely important values for optimizing the design of the civil aircraft model number, improving the market competitiveness of the civil aircraft, improving the operation economy and the like. The DOC estimation method for the civil aircraft market which is widely applied earlier comprises the following steps: the ATA-67 method, boeing method, DOC+I method, harris method, etc. promulgated by the American air transportation Association (American Air Transport Association, ATA). However, the civil aircraft market DOC estimation model needs to obtain main performance parameters (such as maximum takeoff weight, maximum fuel-free weight, etc.) of the type of aircraft, and the parameters are just design parameters which need to be adjusted and optimized in the aircraft design stage and cannot be obtained, so that the model cannot be directly used for estimating the civil aircraft market DOC value in the aircraft design stage.
Further research finds that a more concise and optimized cost estimation model can be established by searching for parameter items with key influences on the civil aircraft market DOC and the cost estimation relation between the parameter items, so that the limitation of the civil aircraft market DOC estimation method is overcome. And a part of scholars study the cost estimation relation of the DOC in the civil aircraft market aiming at the civil aircraft base position capacity and the air range. Swan WM and AdlerN et al construct a conventional Cobb-Douglas cost equation to estimate the civil aircraft market DOC by performing logarithmic processing on the three variables of civil aircraft base position capacity, range and DOC. The demonstration shows that the cost increases with seat capacity and range and that the model is extremely effective for the design of an airline network. Wang Yu et al tried to quantitatively reflect the relationship between the civil aircraft base position capacity, design voyage and flight distance and the civil aircraft market DOC by using them, built DOC estimation regression model and further verified the rationality of the model based on 10 pieces of operation data of 4 sample models. LEE M et al quantitatively analyzed the impact of airline distance, fuel price, seating arrangement and passenger number on civil aircraft market DOC. The most cost effective wide aircraft is determined and evaluated by calculating DOC values for each flight scenario, thereby providing an effective decision basis for aircraft manufacturers and airlines. In addition, there are some studies on the impact of time of flight and fuel consumption on civil aircraft market costs. Mofoxing T and Mativenga P T et al verify significant correlation of civil aircraft repair costs to airline time of flight based on data reports from certain airlines 2014-2018. The giany et al study found that the fuel consumption during cruise phase was about 80% of the total flight life, and optimizing both could greatly help reduce the green direct operation costs of civilian aircraft (Green Direct Operating Cost, GDOC). The exploration provides an important reference for researching the relationship between the weight performance parameter of the airplane and the DOC of the civil aircraft market in the design stage of the airplane. However, the uncertainty faced by the designed civil aircraft in the future actual operation environment is hardly considered, such as factors of 'people (such as pilot operation deviation)' aircraft (such as aircraft performance attenuation) 'ring (such as operation environment fluctuation)' and 'pipe (such as management mode, means and mode difference)', and the fluctuation of the DOC value of the aircraft seat number, the flight time and the civil aircraft market caused by the factors of 'people (such as pilot operation deviation)', and further, deviation between an estimated result and an ideal value is often caused, and the objectivity and the accuracy of the DOC estimation of the civil aircraft market are affected.
Disclosure of Invention
Considering the influence of fluctuation caused by people, machines, rings and pipes in the actual running process on the estimation precision of the direct operation cost of the civil aircraft market, the main purpose of the invention is to provide an estimation method of the direct operation cost of the civil aircraft market.
In order to achieve the purpose, the invention adopts the following technical proposal, and the estimation method of the direct operation cost of the civil airliner market comprises the following steps:
collecting actual operation data of an airline company, and obtaining an interval value data set after processing; wherein the interval value dataset comprises a seat capacity observation interval and a route flight time observation interval;
introducing an upper and lower bound regression model and a regression model of expected value points, establishing a parameterized model, and increasing constraint conditions to obtain a mixed parameterized model with interval number regression and point data regression;
the objective function of the mixed parameterized model is the minimization of the sum of squares of error terms of upper and lower bounds of direct operation cost and the most probable value points of the machine market;
the constraint condition of the mixed parameterized model is that the regression upper bound is not smaller than the regression expected value point and not smaller than the regression lower bound, the upper bound of the regression interval is larger than or equal to the lower bound of the observation interval, and the upper bound of the observation interval is larger than or equal to the lower bound of the regression interval;
Respectively solving an objective function of the mixed parameterized model and a gradient function of the constraint condition, introducing a Lagrange multiplier to the constraint condition to obtain an optimal solution of the constraint condition, and further solving regression coefficients of the mixed parameterized model to obtain a direct operation cost estimation model of the civil passenger plane market based on the mixed parameterized interval;
and inputting the interval value data set into a direct operation cost estimation model of the civil passenger plane market to obtain a direct operation cost estimation result of the civil passenger plane market.
The upper and lower bound regression model includes:
an upper and lower boundary model of direct operation cost of civil aircraft market is introduced, as shown in formula (1):
where c denotes the seat capacity, t denotes the course time of flight, i denotes the number of samples,regression coefficient representing upper bound of seat capacity, +.>Regression coefficient representing the lower bound of seat capacity, +.>Representing a voyageRegression coefficient sum of line time-of-flight bound +.>Regression coefficients respectively representing the lower bounds of the flight time of the route,/->An error term representing the upper bound between the regression interval and the observation interval,error terms respectively representing the lower boundary between the regression interval and the observation interval, +.>Upper bound and +.f. of observation interval for expressing direct operation cost of civil machine market >Respectively representing the lower bound of the observation interval of the direct operation cost of the civil aircraft market;
respectively introducing geometric parameters of upper and lower bounds of seat capacity and flight time of the route And->I.e. the seat capacity and the points within the time of flight observation interval of the course can be used separatelyAndparameterized representation; wherein (1)>Reference point representing upper limit of seat capacity, +.>Reference point representing the lower limit of the seat capacity, +.>Reference point representing the upper boundary of the flight time of the route, < >>Respectively representing reference points of the lower bounds of the flight time of the route;
redefining new coefficients respectively And->Wherein (1)>Representing the estimated coefficient on seat capacity in the direct running cost lower bound estimation model, +.>Estimating coefficients representing the lower bound of seat capacity in the direct running cost lower bound estimation model, +.>Estimation coefficients and +.f representing the upper boundary of the time of flight of the route in the direct operation cost lower boundary estimation model>An estimation coefficient representing a lower bound of the flight time of the route in the direct operation cost lower bound estimation model; />An estimation coefficient representing the upper bound of seat capacity in the direct running cost upper bound estimation model, +.>An estimation coefficient representing the lower bound of seat capacity in the direct running cost upper bound estimation model, +.>Estimation coefficients and +.f representing the upper boundary of the time of flight of the route in the direct operational cost upper boundary estimation model >Representing estimation coefficients of a direct operational cost upper bound estimation model and a route flight time lower bound; then equation (1) can be converted to the one shown in equation (2):
i.e. geometric parametersAnd->Can be by->And->Respectively calculating to obtain the product.
Regression model of the expected value points:
the expected value points are modeled based on the midpoint radius method as shown in equation (3).
Wherein m represents a desired value point,and->Regression coefficients respectively representing expected value points of the seat capacity and the route flight time observation interval; />Error items representing expected value points of direct operation cost of civil aircraft markets; />Representing expected values of direct operation cost observation intervals of civil aircraft markets;
wherein the expected value point of the observation interval of the fixed seat capacity is representedThe generation process of (2) is as follows:
classifying according to the machine types, and dividing the seat capacity observation interval of each machine type into a plurality of equal parts;
calculating the number of times of occurrence of the seat capacity of each route in each equal partition under the same model, and further obtaining an expected value of the whole seat capacity observation interval, wherein the expected value is defined as the most possible value point of the seat capacity observation interval;
for expected value points representing time-of-flight observation intervalsThe generation process of (2) is as follows:
Classifying according to the model and the route, and dividing the flight time observation interval running under the same model and the same route into a plurality of equal parts;
calculating the number of times of the flight time of the route on the same model and route in each equal partition; and then the expected value of the whole flight time observation interval of the route is obtained.
The hybrid parameterized model: in order to ensure mathematical consistency between the upper and lower bounds of the regression interval and the expected value point, the following constraint conditions are added:
the regression upper bound is larger than or equal to the regression expected value point; the regression expected value point is larger than or equal to the regression lower limit; as shown in formula (4):
in order to maximize the regression accuracy of the hybrid parametric model, the objective function of the hybrid parametric model is as shown in equation (5):
when interval data fluctuation is large, the intersection of the regression interval and the observation interval is unstable, so that the regression accuracy of the mixed parameterized model is reduced; the following two constraints are added on the basis of the formula (4):
the upper bound of the regression interval is greater than or equal to the lower bound of the observation interval; the upper bound of the observation interval is greater than or equal to the lower bound of the regression interval, so that an intersection exists between the observation interval and the regression interval;
In summary, the established hybrid parameterized model with interval number regression and point data regression is shown in formulas (5) and (6).
Wherein:and->The estimation coefficients of the parameterized model are directly operated on the upper and lower bounds of the cost and the expected value point for the civil aircraft market.
The regression coefficient solving of the mixed parameterized model comprises the following steps:
converting the hybrid parameterized model into a matrix form representation as shown in equation (7):
wherein X is lu Upper and lower boundary value matrixes for the seat capacity and the flight time observation interval of the route; beta u ，β l And beta m Coefficient estimation value matrix of direct operation cost upper and lower bounds and expected value points of civil aircraft market respectively, y u ，y l And y m Respectively observing sample points epsilon for upper and lower bounds and expected value of direct operation cost of civil aircraft market u ，ε l And epsilon m Respectively observing error items of sample points and estimated values for the upper and lower bounds of the direct operation cost of the civil aircraft market and the expected value;
the hybrid parameterized model is then converted to the one shown in equation (5) and equation (8):
respectively solving an objective function of the mixed parameterized model and a gradient function of the constraint condition, and introducing a Lagrange multiplier to the constraint condition to obtain an optimal solution of the constraint condition;
wherein the constraint condition is g 1 ，g 2 ，g 3 And g 4 Lagrangian multipliers are introduced respectively:
and->Then the following equation (11) is obtained:
wherein,and->Representing the lagrangian multiplier introduced for each sample i in the four constraints, respectively.
When (when)When the formula (11) has an optimal solution, the optimal solution of the formula (11) can be written as:
compared with the prior art, the invention has the beneficial effects that: regression analysis proves that the linear relation between the direct operation cost of the civil aircraft market and the seat capacity and the flight time of the air route is established, and the regression fitting level is higher. The mixed parameterized model is verified to be superior to the point data model as a whole through predictive verification, and has better adaptability. The method has the advantages that the fluctuation of the data is considered to be more beneficial to the estimation of the direct operation cost of the civil aircraft market, so that more accurate guidance opinion is provided for the optimization of the performance of the civil aircraft in the design stage.
Drawings
FIG. 1 is a graph of the regression effect of the parameterized model of the present invention based on dataset I;
FIG. 2 is a graph of the regression effect of the parameterized model based on dataset II of the present invention;
FIG. 3 is a graph of the predictive effects of a parameterized model based on dataset III of the present invention;
FIG. 4 is a graph showing the effect of a parameterized model prediction part based on a data set III according to the present invention;
Fig. 5 is a graph of the predictive effect of the parameterized model of the present invention based on dataset vi.
FIG. 6 is a graph of the effect of the parameterized model prediction part based on the dataset VI of the present invention;
fig. 7 is a flow chart of the basic framework of the present invention.
Detailed Description
The invention will be further described with reference to the drawings and embodiments.
Example 1
The estimation method of the direct operation cost of the civil airliner market comprises the following steps:
collecting actual operation data of an airline company, and obtaining an interval value data set after processing; wherein the interval value dataset comprises a seat capacity observation interval and a route flight time observation interval;
introducing an upper and lower bound regression model and a regression model of expected value points, establishing a parameterized model, and increasing constraint conditions to obtain a mixed parameterized model with interval number regression and point data regression;
the objective function of the mixed parameterized model is the minimization of the sum of squares of error terms of upper and lower bounds of direct operation cost and the most probable value points of the machine market;
the constraint condition of the mixed parameterized model is that the upper bound of regression is not less than the lower bound of regression, the upper bound of the regression interval is greater than or equal to the lower bound of the observation interval, and the upper bound of the observation interval is greater than or equal to the lower bound of the regression interval;
Respectively solving an objective function of the mixed parameterized model and a gradient function of the constraint condition, introducing a Lagrange multiplier to the constraint condition to obtain an optimal solution of the constraint condition, and further solving regression coefficients of the mixed parameterized model to obtain a direct operation cost estimation model of the civil passenger plane market based on the mixed parameterized interval;
and inputting the interval value data set into a direct operation cost estimation model of the civil passenger plane market to obtain a direct operation cost estimation result of the civil passenger plane market.
The upper and lower bound regression models include:
an upper and lower boundary model of direct operation cost of civil aircraft market is introduced, as shown in formula (1):
where c denotes the seat capacity, t denotes the course time of flight, i denotes the number of samples,regression coefficient representing upper bound of seat capacity, +.>Regression coefficient representing the lower bound of seat capacity, +.>Regression coefficient and +.>Regression coefficients respectively representing the lower bounds of the flight time of the route,/->An error term representing the upper bound between the regression interval and the observation interval,error terms respectively representing the lower boundary between the regression interval and the observation interval, +.>Upper bound and y of observation interval for expressing direct operation cost of civil aircraft market i l Respectively representing the lower bound of the observation interval of the direct operation cost of the civil aircraft market;
respectively introducing geometric parameters of upper and lower bounds of seat capacity and flight time of the route And->I.e. the seat capacity and the points within the time of flight observation interval of the course can be used separatelyAndparameters (parameters)Representing the map by chemical means; wherein (1)>Reference point representing upper limit of seat capacity, +.>Reference point representing the lower limit of the seat capacity, +.>Reference point representing the upper boundary of the flight time of the route, < >>Respectively representing reference points of the lower bounds of the flight time of the route;
redefining new coefficients respectively And->Wherein (1)>Representing the estimated coefficient on seat capacity in the direct running cost lower bound estimation model, +.>Estimating coefficients representing the lower bound of seat capacity in the direct running cost lower bound estimation model, +.>Estimation coefficients and +.f representing the upper boundary of the time of flight of the route in the direct operation cost lower boundary estimation model>Representing direct operating costsEstimating coefficients of a lower boundary of the flight time of the route in the lower boundary estimating model; />An estimation coefficient representing the upper bound of seat capacity in the direct running cost upper bound estimation model, +.>An estimation coefficient representing the lower bound of seat capacity in the direct running cost upper bound estimation model, +.>Estimation coefficients and +.f representing the upper boundary of the time of flight of the route in the direct operational cost upper boundary estimation model >Representing estimation coefficients of a direct operational cost upper bound estimation model and a route flight time lower bound; then equation (1) can be converted to the one shown in equation (2):
i.e. geometric parametersAnd->Can be by-> Andrespectively calculating to obtain the product.
Regression model of the expected value points:
the expected value points are modeled based on the midpoint radius method as shown in equation (3).
Wherein m represents a desired value point,and->Regression coefficients respectively representing expected value points of the seat capacity and the route flight time observation interval; />Error items representing expected value points of direct operation cost of civil aircraft markets; />Representing expected values of direct operation cost observation intervals of civil aircraft markets;
wherein the expected value point of the observation interval of the fixed seat capacity is representedThe generation process of (2) is as follows:
classifying according to the machine types, and dividing the seat capacity observation interval of each machine type into a plurality of equal parts;
calculating the number of times of occurrence of the seat capacity of each route in each equal partition under the same model, and further obtaining an expected value of the whole seat capacity observation interval, wherein the expected value is defined as the most possible value point of the seat capacity observation interval;
for expected value points representing time-of-flight observation intervalsThe generation process of (2) is as follows:
Classifying according to the model and the route, and dividing the flight time observation interval running under the same model and the same route into a plurality of equal parts;
calculating the number of times of the flight time of the route on the same model and route in each equal partition; and then the expected value of the whole flight time observation interval of the route is obtained.
The hybrid parameterized model: in order to ensure mathematical consistency between the upper and lower bounds of the regression interval and the expected value point, the following constraint conditions are added:
the regression upper bound is larger than or equal to the regression expected value point; the regression expected value point is larger than or equal to the regression lower limit; as shown in formula (4):
in order to maximize the regression accuracy of the hybrid parametric model, the objective function of the hybrid parametric model is as shown in equation (5):
when interval data fluctuation is large, the intersection of the regression interval and the observation interval is unstable, so that the regression accuracy of the mixed parameterized model is reduced; the following two constraints are added on the basis of the formula (4):
the upper bound of the regression interval is greater than or equal to the lower bound of the observation interval; the upper bound of the observation interval is greater than or equal to the lower bound of the regression interval, so that an intersection exists between the observation interval and the regression interval;
In summary, the established hybrid parameterized model with interval number regression and point data regression is shown in formulas (5) and (6).
Wherein:and->The estimation coefficients of the parameterized model are directly operated on the upper and lower bounds of the cost and the expected value point for the civil aircraft market.
The regression coefficient solving of the mixed parameterized model comprises the following steps:
converting the hybrid parameterized model into a matrix form representation as shown in equation (7):
wherein X is lu Upper and lower boundary value matrixes for the seat capacity and the flight time observation interval of the route; beta u ，β l And beta m Coefficient estimation value matrix of direct operation cost upper and lower bounds and expected value points of civil aircraft market respectively, y u ，y l And y m Respectively observing sample points epsilon for upper and lower bounds and expected value of direct operation cost of civil aircraft market u ，ε l And epsilon m Respectively observing error items of sample points and estimated values for the upper and lower bounds of the direct operation cost of the civil aircraft market and the expected value;
the hybrid parameterized model is then converted to the one shown in equation (5) and equation (8):
respectively solving an objective function of the mixed parameterized model and a gradient function of the constraint condition, and introducing a Lagrange multiplier to the constraint condition to obtain an optimal solution of the constraint condition;
wherein the constraint condition is g 1 ，g 2 ，g 3 And g 4 Lagrangian multipliers are introduced respectively:
and->Then the following equation (11) is obtained:
wherein,and->Representing the lagrangian multiplier introduced for each sample i in the four constraints, respectively.
When (when)When the formula (11) has an optimal solution, the optimal solution of the formula (11) can be written as:
example 2:
based on example 1, another implementation and verification of its results. The method comprises the steps of attempting to use three interval numbers, namely using an upper boundary, a lower boundary and expected value points thereof to represent the direct operation cost value, the airplane seat capacity and the airplane line flight time of a civil aircraft market DOC, and providing a mixed parameterization method according to the direct operation cost value, the airplane seat capacity and the airplane line flight time, and establishing an interval regression model among the three interval numbers; then, proving the model as a convex planning problem, and further solving the regression coefficient of the model by utilizing a Kuhn-Tucker equation set; finally, carrying out demonstration research based on the observation data, and verifying whether the linear relation among the three variables is established and the accuracy and the effectiveness of the model.
The construction of the hybrid parameterized model comprises the following steps:
1.1 data acquisition and arrangement,
65530 pieces of airline operations data (including flight schedules, direct operation cost values for each flight using market DOC for model aircraft) are collected. The following processing is performed based on these data, thereby forming an interval value data set.
The direct operation cost items of the civil aircraft market DOC refer to cost items directly related to the model, including crew costs, fuel/oil costs, aircraft insurance costs, aircraft rental costs, aircraft maintenance costs, and aircraft depreciation/amortization costs.
And classifying and sorting the point data of the seat capacity, the flight time of the air route and the direct operation cost of the civil aircraft market DOC according to the same air route and model to form interval value data, and setting the maximum value and the minimum value of the point data as the upper bound and the lower bound of the interval value data respectively.
709 pieces of interval value data are formed according to the airline route operation data, wherein the interval value data comprise 322 pieces of narrow body machine data and 387 pieces of wide body machine data. The data set I is formed by 226 pieces of narrow-body machine interval data and is regarded as a regression data set; the remaining 96 pieces of narrow body machine interval data are considered as prediction data sets, called data set iii. The data set II is formed by 271 pieces of wide body machine interval data and is used for regression analysis of the model; the remaining 116 pieces of wide body machine interval data are used for predictive inspection of the model, referred to as dataset IV. The finally formed interval value data are shown in the table.
TABLE 1 partial interval value data
1.2 construction of Mixed parameterized models
Involving part of the parameter specification, let X c An interval argument representing the capacity of the seat,the i-th observation interval representing the seat capacity can be expressed as +.>Where i=1, 2, …, n. Given seat capacity observation intervalWherein-> And->Respectively representing the upper and lower bounds of the observation interval and the expected value point. Let X t Interval argument representing time of flight, +.>The i-th observation interval representing the time of flight, i.e. representing +.>Where i=1, 2, …, n. Given time of flight observation interval +.>Wherein-> And->Respectively representing the upper and lower bounds of the observation interval and the expected value point. To clearly compare the advantages of considering the volatility of data over direct operation cost estimation, based on literature [6]The invention also carries out the logarithmic processing on the data.
1.2.1 PM model for upper and lower bound regression
By means of parametersAnd->Points within the seat capacity and line time of flight observation interval, respectivelyAndand parameterized. />And->Regression coefficients respectively representing upper and lower bounds of seat capacity, +.>And->Regression coefficients respectively representing upper and lower bounds of flight time of the route,/->And->Error terms respectively representing upper and lower bounds between regression interval and observation interval, < ->And->Respectively represent the upper and lower bounds of the direct operation cost observation interval of the civil aircraft market DOC. Therefore, the direct operation cost upper and lower bound model of the civil aircraft market DOC in solving the three-interval digital parameterized regression problem is shown as the formula (1).
Defining new coefficients respectively And->Then formula (1) can be converted to formula (2).
I.e. parametersAnd->Can be by-> Andrespectively calculating to obtain the product.
1.2.2 regression model of desired value points
Based on Center-and-Range Method (CRM) [0 ]]Modeling the expected value points as shown in equation (3). Wherein the method comprises the steps ofAnd->Regression coefficients respectively representing expected value points of the seat capacity and the route flight time observation interval; />Error terms representing direct operational cost expectation points for civil aircraft markets DOC; />The most likely value of the direct operation cost observation interval of the civil aircraft market DOC is represented.
The generation process of (2) is as follows: classifying according to the machine types, dividing the seat capacity observation interval of each machine type into a plurality of equal parts, calculating the number of times of occurrence of the seat capacity of each air route in each equal part under the same machine type, and further obtaining the expected value of the whole seat capacity observation interval. The expected value is defined as the most likely point of view of the seating capacity. />The generation process of (2) is as follows: classifying according to the model and the route, equally dividing the flight time observation interval running under the same model and the same route into a plurality of equal parts, calculating the times of the flight time on the same model and the same route in each equal part, and further solving the expected value of the whole flight time observation interval. The expected value is defined as the most likely point of the observation interval for the time of flight of the route.
1.2.3 Mixed parameterized model
In order to ensure mathematical consistency between the upper and lower bounds of the regression interval and the expected value point, the following constraint is added: 1) The regression upper bound is larger than or equal to the regression expected value point; 2) The regression expectation value point is larger than or equal to the regression lower limit. As shown in formula (4).
In order to maximize the regression accuracy of the hybrid model, the objective function is defined as the minimization of the sum of squares of the error terms of the upper and lower bounds of the direct operating cost of the civil aircraft market DOC and the most likely value points, as shown in equation (5).
When interval data fluctuation is large, the model is difficult to ensure the intersection of the regression interval and the observation interval, and the regression accuracy of the model is reduced. Thus, the following two constraints are added on the basis of equation (4): 1) The upper bound of the regression interval is greater than or equal to the lower bound of the observation interval; 2) The upper boundary of the observation interval is larger than or equal to the lower boundary of the regression interval, so that the intersection of the observation interval and the regression interval is ensured;
in summary, the established hybrid parameterized model with interval number regression and point data regression is shown in formulas (5) and (6).
1.2.4 solution of regression coefficients
To facilitate the solution of the coefficient estimation values, the above model is converted into a matrix form representation, as shown in equation (7).
/>
Wherein,
the hybrid parameterized model may be converted to the equations (5) and (8).
As can be seen from the conversion of the constraint in the formula (8) to the expression (9), the constraint conditions of the model are all linear functions, so that the Hessian matrices are all real symmetric matrices and the primary and secondary formulas of each order are all 0, thereby explaining that the constraint conditions are all concave functions.
The objective function of the hybrid parameterized model may be expressed as:
from literature []It turns out that the objective function of the hybrid model is a convex function, i.e. the model is a convex planning model. In order to further solve the regression coefficients of the hybrid parameterized model, the objective function and the gradient function of the constraint condition are solved respectively. And is g 1 ，g 2 ，g 3 And g 4 Four constraint conditions are respectively introduced into Lagrange multipliersAnd->The K-T condition can be represented as formula (11).
Solving the model regression coefficient, and discussing the following possibilities;
if it isMake->And->In this case, in the formula (11)And->Establishment; that is, the existence of any observation interval i E {1,2, …, n } cannot guarantee the mathematical consistency of interval data, therefore +.>And->And cannot be 0 at the same time.
Based on (1), the following three cases are discussed further: specifically as shown in table 2;
table 2 discussion of the possibilities of solutions
To sum up, for any observation interval i ε {1,2, …, n }, the K-T equation is not solved as long as any of () - (11) in (1) and (2) exists; conversely, when the situation of 12) in (2) occurs, then there is an optimal solution for a particular observation interval i e {1,2, …, n }. Obviously, for either one ofThere is no solution as long as any one of cases 1) to 11) in (1) and (2) occurs. Then only when->All have->The solution of K-T equation (11) is only available when the optimal solution can be written as +.>
Example 2
Based on the embodiment 1, the direct operation cost estimation model of the civil airliner market based on the parameterized mixed interval is evaluated, and the existing regression model of the evaluation index is usually selected from the available coefficient (r 2 ) Root Mean Square Error (RMSE), average Accuracy (AR), average ratio (Average ratio ofobserved intervals containing regressed intervals, PCO) and number of observation interval and regression interval disjoint(N 0 ) As an evaluation index, the regression effect of the model is compared and analyzed, and the application is improved on the basis.
To verify whether the linear relation between the direct operation cost of civil aircraft market, the seat capacity and the line flight time is established and measure the overall fitting goodness of the civil aircraft market, an upper bound determinable coefficient is introduced Lower bound coefficient->And the expected value point determinable coefficient->The range of the determinable coefficient value is between 0 and 1, and the closer to 1, the higher the goodness of the regression fit is, the more specific calculation formulas are shown as (13) - (15).
In order to determine the influence of the three-interval hybrid regression model on the regression accuracy of each part of interval data, the application mainly uses the upper-bound Root Mean Square Error (RMSE) of the observation interval and the regression interval L ) Lower Root Mean Square Error (RMSE) U ) Root Mean Square Error (RMSE) m ) Three aspects are calculated. The smaller the RMSE value, the better the regression effect of the model. Specifically as shown in equations (6) - (8), whereinAnd->The range value, the upper limit value, the lower limit value and the most likely value of the direct operation cost regression interval of the civil aircraft market DOC are respectively.
/>
RMSE only measures root mean square error of regression data points relative to observation data points, and cannot reflect the area coincidence of the regression interval and the observation interval, so PCO is introduced. The index measures the ratio of the intersection of the regression interval and the observation interval relative to the observation interval, and the larger the PCO value is, the better the regression effect of the model is, as shown in an equation (9).
To determine if the regression interval range is excessively elongated due to the forced intersection constraint of the hybrid parameterized model, AR was introduced, which measures the ratio of the intersection of the regression interval and the observation interval relative to the union of the two intervals. For AR, a larger AR value indicates a better regression effect of the model, as shown in equation (20).
To reflect whether there is an extreme case of no intersection between the observation interval and the regression interval, N is introduced 0 The index of the index is that,representing the number of samples without intersections between the observation interval and the regression interval.
And carrying out model fitting and comparison analysis, carrying out regression fitting on the mixed parameterized model based on the data set I which is a narrow body machine and the data set II which is a wide body machine, so as to verify whether the linear relation between the direct operation cost of the civil aircraft market, the seat capacity and the route flight time is established or not, and evaluating the regression fitting effect based on the selected evaluation index.
For the narrow body machine, regression fitting is performed on the interval data based on the data set I, wherein the regression equation of the mixed parameterization model is shown as a formula (21). And carrying out comparative analysis on the regression effect of the mixed parameterized model based on the evaluation indexes selected above.
Wherein the method comprises the steps ofAnd->In order to maximize the regression accuracy of the hybrid parameterized model, the reference points selected by the direct operating cost regression lower bound of the civil aircraft market DOC at this time are the lower bound of seat capacity and the upper bound of course time of flight. As can be seen from the formula (21), the coefficients in the model are positive numbers, which indicates that the direct operation cost value of the civil aircraft market will increase with the increase of the seat capacity and the flight time of the route, and the direct operation cost value accords with the actual operation rule of the airline company. The lower bound of seat capacity (0.2657) has less impact on the direct operational cost return lower bound of the civil aircraft market than the airline time of flight has on it (0.4995,0.3789). For the direct operating cost return upper bound of the civil aircraft market, the impact of the upper bound of seat capacity (0.8387) is greater than the impact of the airline time of flight (0.2434,0.3705). In addition, the impact (0.1277) of the most likely point of seat capacity on the direct cost of operation regression expectations of the civil aircraft market is much less than the most likely point-to-point of airline time of flight Its influence (0.6545). The regression results (as shown in FIG. 1) were then subjected to comparative analysis based on the above-described evaluation index. />
Table 3 shows the regression effect of the hybrid parameterized model. The RMSE value is controlled below 0.2870, which means that the single-side average error of the direct operation cost regression result of the civil aircraft market after taking the logarithm is controlled below 824 yuan, which is far less than the direct operation cost of the civil aircraft market. The PCO value of the hybrid parameterized model was 0.8826, indicating that the average ratio of the intersection of each regression interval with the observation interval to the observation interval was 0.8826, indicating that the regression effect of the direct running cost upper and lower bound regression line of the civil aircraft market DOC in the hybrid parameterized model remained at a relatively high level. The AR value of the hybrid parameterized model is 0.4283, which indicates that the average ratio of the intersection of each regression interval and the observation interval to the union of the observation interval and the regression interval is 0.4283, which indicates that the intersection of the hybrid parameterized model and the observation interval is ensured in the regression fitting process, so that the regression interval is correspondingly elongated, as shown in fig. 1, that is, a part of the AR value is compensated to the PCO value. N of hybrid parameterized model 0 The values are all 0, meaning that the regression effect of the hybrid parameterized model is optimal and all observation intervals and regression intervals are intersected. Hybrid parameterized model And->The values of (2) are above 0.77, and +.>As can be seen in combination with RMSE and fig. 1, the regression upper bound error is significantly greater than the regression lower bound and the regression expectation point when the overall regression error is minimized under the forced intersection constraint. However three r 2 The overall average value is 0.7, the overall indicates that the linear relation between the direct operation cost of the civil aircraft market DOC, the seat capacity and the route flight time is established, and the regression fitting precision of the mixed parameterized model on the narrow body machine is higher.
TABLE 3 evaluation index results based on narrow body machine
For wide body machine
Regression fit analysis was performed based on dataset II generated by wide body machine operation using the hybrid parameterized model. As shown in fig. 2, although the fluctuations in the interval data of data set II are further exacerbated, the linear relationship between the direct operating cost of civil aircraft market DOC and the seat capacity and course time of flight still exists, and the regression equation of the corresponding hybrid parameterized model is shown in equation (22).
Wherein,meaning that the direct operating cost regression lower bound model of the civil aircraft market DOC at this point selected reference points are the lower bound of seat capacity and the upper and lower bounds of time of flight. The coefficients in this formula (22) are also positive, indicating that the direct operating cost value of the civil air market DOC increases with increasing seat capacity and course time of flight, as is the rule of a narrow machine. The extent of impact of seat capacity on the direct operating cost of the wide body machine market DOC is significantly increased compared to the narrow body machine (1.2465,0.4265,0.9743,1.6672).
The results of the nine evaluation indexes in table 4 have similar patterns to the data in table 3. The values of the mixed parameterized model r2 are all above 0.7, and in combination with RMSE and fig. 2, it can be seen that the linear relationship between the three variables is established at this time, and the goodness of fit between the regression upper bound and the expected value point is higher. Furthermore, the fluctuation of the data samples increases due to the presence of the wide body machine, resulting in a continuously deteriorated regression effect of RMSE. However, the RMSE value is controlled below 0.5344, which means that the direct operation cost of the log-taking civil aircraft market DOC is returnedThe unilateral average error control of the result is less than 2856 yuan, which is far less than the cost of the airplane. The AR value of the hybrid parameterized model is 0.2880, which is smaller than the AR value based on narrow-volume machine data regression fit, meaning that in the case of larger data fluctuation amplitude, in order to be able to ensure as much as possible that the observation interval intersects the regression interval, the regression interval is partially extended so that a part of the AR and RMSE compensates for PCO, as shown in fig. 2 in particular. N of hybrid parameterized model 0 The value still guaranteed to be 0 indicates that the three interval linear relationship is fully established, and the PCO value (0.9349) shows better regression effect even compared to the PCO value in table 3. In summary, the hybrid parameterized model can still maintain a high regression level in the case of large interval sample fluctuations.
Table 4 evaluation index results based on a wide body machine
Predictive test and comparative analysis in order to test whether the fluctuation of the considered data is more favorable for the estimation of the direct operation cost of the civil aircraft market DOC, the section carries out predictive test comparative analysis based on a mixed parameterized model and a point data model, and a specific predictive test flow is as follows:
predictive verification for interval data: based on the regression analysis of the hybrid parameterized model above, a fixed lambda value is obtained to achieve optimal regression accuracy. And substituting the interval data of the seat capacity in the data set III and the seat capacity in the data set VI, the flight time of the route and the direct operation cost of the civil aircraft market DOC into the obtained regression equation respectively to obtain a final prediction interval.
Predictive verification for the point data model: the point data model is built based on the linear relationship verified in the section above, using the basic principle of the model in the literature. Firstly, establishing a most likely value regression equation based on direct operation cost of civil aircraft markets DOC of the same model and route interval data in a data set I and a data set II respectively, and expected values of seat capacity and route flight time interval data. And substituting the most likely values of the seat capacity and the route flight time of each piece of interval data in the data set III and the data set VI into a formed point data regression equation respectively to obtain the direct operation cost point data of the predicted civil aircraft market DOC, namely, the point data simultaneously predicts an upper boundary, a lower boundary and an expected value point for the direct operation cost of the civil aircraft market DOC. And finally, comparing and analyzing the predicted direct operation cost of the civil aircraft market DOC with the data of the observation interval.
For a narrow body machine, based on the remaining operation data (i.e., 96 pieces of data in the data set iii) related to the narrow body machine type, a point data model and a mixed parameterization model are used for prediction analysis, and a specific prediction effect diagram is shown in fig. 3. Overall, the airline flight time has a linear relationship with the direct operating cost of the civil aircraft market DOC. Wherein the most likely value regression model based on the data set I point data model is shown in equation (20).
As can be seen from table 5, the point data prediction finally presents a plurality of line segments, compared with the prediction of the interval data, and the AR and PCO values of the point data model do not exist. PCO (0.8708) and AR (0.4541) of the mixed parameterized model are higher than the point data model, which indicates that the overlap ratio of the prediction interval and the observation interval is higher, and the prediction accuracy of the mixed parameterized model is better than that of the point data model. In addition, reference is made in particular to FIG. 4, i.e. there is a singular interval, N of the hybrid parameterized model 0 The values are also much smaller than the point data model. Although RMSE of mixed parameterized models m (0.1141) is slightly larger than the RMSE of the Point data model m (0.0981) but RMSE of hybrid parameterized model L And RMSEU values are smaller than the point data model, meaning that in order to ensure that the observed samples intersect the predicted samples as much as possible, a portion of the RMSEm value needs to be sacrificed to compensate for PCO. In summary, the hybrid parameterized model has significant advantages.
Table 5 evaluation index results based on narrow body machine
Note that: -indicating absence. Because the prediction upper and lower boundaries of the point data model coincide with the expected value points, i.e. the prediction interval of the point data does not exist, the calculation of the AR and PCO values cannot be performed for one line segment, i.e. the values of the two indexes do not exist.
For a wide-body machine, the prediction results are formed based on the remaining operational data (i.e., 116 pieces of data in data set IV) associated with the wide-body machine model as shown in FIG. 5. Although the fluctuations of the interval observation data may be largely fluctuating in the case of wide-body machine operation, a linear relationship similar to the data set iii is also exhibited as a whole. Wherein the most likely value regression model based on the data set II point data model is shown in equation (21).
As can be seen from table 6, the AR and PCO values of the same point data model are not present. Compared with the prediction test of a narrow body machine, PCO (0.9536) of the mixed parametric model is increased and AR (0.285) is reduced, which shows that constraint that the intersection of the observation interval and the prediction interval is forced to lead to the increase of the extension range of the prediction interval due to the increase of the fluctuation of interval data, so that a part of AR values are compensated for PCO. The same applies to the fact that the RMSE prediction effect of the mixed parameterized model is inferior to that of the point data model, namely the model fitting robustness is poor. Furthermore, even if the mixed parameterized model has two singular intervals (N 0 =2) (see fig. 6 for specific details), PCO and N in a mixed parameterized model 0 The values still maintain a good level of prediction. The RMSE values are higher and higher than the results of table 5, but still will take the single-sided average error limit of the regressive results of the direct operating costs of the log civil aircraft market DOCIs made at about 2463 yuan, which is far less than the cost of the airplane. In summary, the predictive effect of the hybrid parameterized model is better than that of the point data model.
Table 6 evaluation index results based on a wide body machine
Note that: -indicating absence. Because the prediction upper and lower boundaries of the point data model coincide with the expected value points, i.e. the prediction interval of the point data does not exist, the calculation of the AR and PCO values cannot be performed for one line segment, i.e. the values of the two indexes do not exist.
According to the method, factors such as 'people (pilot operating deviation), aircraft (aircraft performance difference), ring (operating environment fluctuation) and management (management mode)' in the whole life cycle are fully considered, so that the direct operating cost of the civil aircraft market DOC, the seat capacity and the fluctuation of the airline flight time are caused, and a direct operating cost mixed estimation regression model of the three-interval digital aircraft market DOC is established, wherein the direct operating cost upper and lower bounds of the aircraft market DOC and the most likely value error square sum are minimized as targets. 1) The regression analysis verifies that the direct operation cost of the civil aircraft market DOC is in a linear relation with the seat capacity and the route flight time, and the regression fitting level is higher. 2) The mixed parameterized model is verified to be superior to the point data model as a whole through predictive verification, and has better adaptability. The method has the advantages that the fluctuation of the considered data is more beneficial to the estimation of the direct operation cost of the DOC in the civil aircraft market, so that more accurate guidance opinion is provided for the optimization of the performance of the civil aircraft in the design stage. Notably, the forced constraints in the hybrid parameterized model lead to the fitting result of the partial regression interval being pulled large, which in turn leads to the partial AR and RMSE values being compensated onto PCO, thus how to flexibly balance the relationship between PCO and RMSE and AR is a problem to be further solved later.
It is noted that in the present invention, relational terms such as first and second, and the like are used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Moreover, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
The above embodiments are merely illustrative of the present invention and are not to be construed as limiting the scope of the present invention, and all designs which are the same or similar to the present invention are within the scope of the present invention.
Claims (5)
1. The method for estimating the direct operation cost of the civil airliner market is characterized by comprising the following steps:
collecting actual operation data of an airline company, and obtaining an interval value data set after processing; wherein the interval value dataset comprises a seat capacity observation interval and a route flight time observation interval;
Introducing an upper and lower bound regression model and a regression model of expected value points, establishing a parameterized model, and increasing constraint conditions to obtain a mixed parameterized model with interval number regression and point data regression;
the objective function of the mixed parameterized model is the minimization of the sum of squares of error terms of upper and lower bounds of direct operation cost and the most probable value points of the machine market;
the constraint condition of the mixed parameterized model is that the regression upper bound is not smaller than the regression expected value point and not smaller than the regression lower bound, the upper bound of the regression interval is larger than or equal to the lower bound of the observation interval, and the upper bound of the observation interval is larger than or equal to the lower bound of the regression interval;
respectively solving an objective function of the mixed parameterized model and a gradient function of the constraint condition, introducing a Lagrange multiplier to the constraint condition to obtain an optimal solution of the constraint condition, and further solving regression coefficients of the mixed parameterized model to obtain a direct operation cost estimation model of the civil passenger plane market based on the mixed parameterized interval;
and inputting the interval value data set into a direct operation cost estimation model of the civil passenger plane market to obtain a direct operation cost estimation result of the civil passenger plane market.
2. The method of estimating a direct operating cost for a commercial airliner as defined in claim wherein the upper and lower bound regression models include:
An upper and lower boundary model of direct operation cost of civil aircraft market is introduced, as shown in formula (1):
where c denotes the seat capacity, t denotes the course time of flight, i denotes the number of samples,regression coefficient representing upper bound of seat capacity, +.>Regression coefficient representing the lower bound of seat capacity, +.>Regression coefficient and +.>Regression coefficients respectively representing the lower bounds of the flight time of the route,/->Error term representing upper bound between regression interval and observation interval, +.>Error terms respectively representing the lower boundary between the regression interval and the observation interval, +.>Upper bound and +.f. of observation interval for expressing direct operation cost of civil machine market>Respectively representing the lower bound of the observation interval of the direct operation cost of the civil aircraft market;
respectively introducing geometric parameters of upper and lower bounds of seat capacity and flight time of the route Andi.e. the seat capacity and the points within the time of flight observation interval of the course can be used separatelyAndparameterized representation; wherein (1)>Reference point representing upper limit of seat capacity, +.>Reference point representing the lower limit of the seat capacity, +.>Reference point representing the upper boundary of the flight time of the route, < >>Respectively representing reference points of the lower bounds of the flight time of the route;
redefining new coefficients respectively And->Wherein (1)>Representing the estimated coefficient on seat capacity in the direct running cost lower bound estimation model, +.>Estimating coefficients representing the lower bound of seat capacity in the direct running cost lower bound estimation model, +.>Estimation coefficients and +.f representing the upper boundary of the time of flight of the route in the direct operation cost lower boundary estimation model>An estimation coefficient representing a lower bound of the flight time of the route in the direct operation cost lower bound estimation model; />An estimation coefficient representing the upper bound of seat capacity in the direct running cost upper bound estimation model, +.>An estimation coefficient representing the lower bound of seat capacity in the direct running cost upper bound estimation model, +.>Estimation coefficients and +.f representing the upper boundary of the time of flight of the route in the direct operational cost upper boundary estimation model>Representing estimation coefficients of a direct operational cost upper bound estimation model and a route flight time lower bound; then equation (1) can be converted to the one shown in equation (2):
i.e. geometric parametersAnd->Can be by->Andrespectively calculating to obtain the product.
3. A method of estimating the direct operation cost of a commercial airliner as defined in claim 2 wherein the regression model of the expected value points:
modeling the expected value point based on a midpoint radius method, as shown in a formula (3):
wherein m represents a desired value point, Regression coefficient and +.>Regression coefficients representing expected value points of the time-of-flight observation interval of the route; />Error items representing expected value points of direct operation cost of civil aircraft markets; />Representing the most possible value of the observation interval of the direct operation cost of the civil aircraft market;
wherein the expected value point of the observation interval of the fixed seat capacity is representedThe generation process of (2) is as follows:
classifying according to the machine types, and dividing the seat capacity observation interval of each machine type into a plurality of equal parts;
calculating the number of times of occurrence of the seat capacity of each route in each equal partition under the same model, and further obtaining an expected value of the whole seat capacity observation interval, wherein the expected value is defined as the most possible value point of the seat capacity observation interval;
wherein the expected value point representing the time of flight observation intervalThe generation process of (2) is as follows:
classifying according to the model and the route, and dividing the flight time observation interval running under the same model and the same route into a plurality of equal parts;
calculating the number of times of the flight time of the route on the same model and route in each equal partition; and then the expected value of the whole flight time observation interval of the route is obtained.
4. A method of estimating the direct operating cost of the commercial airliner market as defined in claim 3 wherein the hybrid parameterized model adds the following constraints to ensure mathematical consistency between the regression interval upper and lower bounds and the desired value points:
The regression upper bound is larger than or equal to the regression expected value point; the regression expected value point is larger than or equal to the regression lower limit; as shown in formula (4):
in order to maximize the regression accuracy of the hybrid parametric model, the objective function of the hybrid parametric model is as shown in equation (5):
when interval data fluctuation is large, the intersection of the regression interval and the observation interval is unstable, so that the regression accuracy of the mixed parameterized model is reduced; the following two constraints are added on the basis of the formula (4):
the upper bound of the regression interval is greater than or equal to the lower bound of the observation interval; the upper bound of the observation interval is greater than or equal to the lower bound of the regression interval, so that an intersection exists between the observation interval and the regression interval;
in summary, the established mixed parameterized model with interval number regression and point data regression is shown in formulas (5) and (6);
wherein the method comprises the steps of：And->The estimation coefficients of the parameterized model are directly operated on the upper and lower bounds of the cost and the expected value point for the civil aircraft market.
5. The method for estimating the direct operation cost of the commercial airliner as defined in claim 4 wherein the regression coefficient solution of the hybrid parameterized model comprises the steps of:
converting the hybrid parameterized model into a matrix form representation as shown in equation (7):
Wherein X is lu Upper and lower boundary value matrixes for the seat capacity and the flight time observation interval of the route; beta u ，β l And beta m Coefficient estimation value matrix of direct operation cost upper and lower bounds and expected value points of civil aircraft market respectively, y u ，y l And y m Respectively observing sample points epsilon for upper and lower bounds and expected value of direct operation cost of civil aircraft market u ，ε l And epsilon m Respectively observing error items of sample points and estimated values for the upper and lower bounds of the direct operation cost of the civil aircraft market and the expected value;
the hybrid parameterized model is then converted to the one shown in equation (5) and equation (8):
respectively solving an objective function of the mixed parameterized model and a gradient function of the constraint condition, and introducing a Lagrange multiplier to the constraint condition to obtain an optimal solution of the constraint condition;
wherein the constraint condition is g 1 ，g 2 ，g 3 And g 4 Lagrangian multipliers are introduced respectively:
and->Then the following equation (11) is obtained:
wherein,and->Respectively representing Lagrangian multipliers which are correspondingly introduced by each sample i in four constraint conditions;
when (when)When the formula (11) has an optimal solution, the optimal solution of the formula (11) can be written as:
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