Interval efficiency evaluation method in combination with possibility degree measurement

Abstract

The invention relates to the technical field of interval efficiency evaluation, in particular to an interval efficiency evaluation method in combination with possibility degree measurement. The invention discloses an interval efficiency evaluation method in combination with possibility degree measurement, and particularly relates to improvement of efficiency evaluation and data envelope analysis evaluation methods in the field of civil aviation flight training, comprising evaluation of potential efficiency change by an interval efficiency model, reflection of training preference by an attitude function, and training efficiency comprehensive sorting based on the possibility degree measurement. According to the method, a profit model and a non-profit model are constructed, the profit model pays attention to the maximum efficiency of trainees, and the non-profit model considers the efficiency distribution of the whole flight training system to obtain a flight training efficiency interval; the attitude function is combined with the possibility degree measurement, so that the evaluation technology based on the possibility degree measurement is improved, the full sorting of interval efficiency is realized, and the identification degree and the difference of efficient trainee training results are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of interval efficiency evaluation, and specifically relates to an interval efficiency evaluation method combined with possibility measurement. Background Art

[0002] Civil aviation transportation is one of the main transportation modes in China. With its characteristics of fast speed, high efficiency and safety, it has become an important choice for passenger travel and cargo circulation. The safety and efficiency of flight training are the key links to ensure civil aviation safety and have received extensive attention. In this context, the Civil Aviation Administration has put forward an important strategic deployment to deepen the reform of flight training, requiring both to carry out flight training evaluation based on competency and to improve flight training efficiency. The flight training efficiency of civil aviation is not only directly related to the operation of general aviation airspace, but also affects the use of airspace resources in other scenarios such as unmanned aircraft operations and low-altitude flights. Improving flight training efficiency and optimizing resource allocation have become the key driving forces for the high-quality development of the future general aviation industry and low-altitude economy.

[0003] The Data Envelopment Analysis (DEA) model, as a method for evaluating the efficiency of decision-making units, has been widely used in many fields such as education, medical care, energy conservation and emission reduction, and has received significant attention and recognition. With the continuous development and improvement of the DEA model, in addition to the basic model, many branches such as network DEA, window DEA, cross-efficiency evaluation, and directional distance function have been developed, and combined with research in fields such as cluster analysis and machine learning, showing great potential and application space. These expansions and integrations make the DEA model more flexible, capable of solving complex problems in different fields, and providing powerful tools and methods for interdisciplinary research. Although the DEA model has broad application potential in flight training efficiency evaluation, the existing evaluation methods still have certain limitations:

[0004] (1) Lack of full ranking ability and unable to comprehensively distinguish the efficiency levels of different decision-making units;

[0005] (2) Ignore preference factors and fail to fully consider the educational preferences of flight instructors and the specific needs of students at different training stages.

[0006] These defects make it difficult for the traditional DEA model to accurately reflect the training differences among students in the application of flight training efficiency evaluation and also unable to effectively support the teaching decisions of flight instructors at different stages. Therefore, it is urgent to introduce new evaluation methods that comprehensively consider the subjective preferences of instructors and the personalized needs of students to improve the comprehensiveness and accuracy of evaluation results. The purpose of the present invention is to provide an interval efficiency evaluation method that can combine the preferences existing in flight training, perform possibility measurement, and achieve the evaluation of local efficiency and overall efficiency optimization, so as to provide more accurate evaluation results for optimizing flight training management. Summary of the Invention

[0007] In view of the above-mentioned disadvantages existing in the prior art, the present invention provides an interval efficiency evaluation method combined with possibility measure, which can effectively solve the problem that the flight training efficiency cannot be comprehensively and accurately evaluated in the prior art.

[0008] To achieve the above object, the present invention is realized through the following technical solutions:

[0009] The present invention provides an interval efficiency evaluation method combined with possibility measure, including the following steps:

[0010] S1. Construct a self-interested model and a non-self-interested model, and calculate the interval of the training efficiency of the trainees;

[0011] S2. Calculate the predicted input and the predicted output according to the input-output data;

[0012] S3. Fit the attitude function through the predicted input and output data;

[0013] S4. Calculate the possibility measure based on decision preference based on the attitude function;

[0014] S5. Calculate the comprehensive ranking of the interval efficiency of the trainees' flight training based on the possibility measure relation matrix and the solution of the weights of the decision-making units.

[0015] In S1, based on the data envelopment analysis model to maximize the efficiency of the evaluated decision-making unit, the self-interested model is represented by establishing the following expression:

[0016]

[0017] s.t.

[0018]

[0019] Among them, represents the upper bound of the efficiency of the d-th evaluated decision-making unit, represents maximizing the upper bound of the efficiency of the d-th evaluated decision-making unit, y rd represents the r-th output of the d-th evaluated decision-making unit, v rd represents the weight of the r-th output of the d-th evaluated decision-making unit, u id represents the weight of the i-th input of the d-th evaluated decision-making unit, x id represents the i-th input of the d-th evaluated decision-making unit, and the notation s.t. indicates that the following formula is the constraint condition of the optimization model, y rk represents the r-th output of the k-th evaluated decision-making unit, v rk represents the weight of the r-th output of the k-th decision-making unit, u ikrepresents the weight of the i-th input of the k-th decision-making unit, x ik represents the i-th input of the k-th decision-making unit to be evaluated, n represents the number of decision-making units, that is, the total number of efficiency evaluation objects. Therefore, 1 ≤ d ≤ n; m represents the total number of input resources, and s represents the total number of outputs; by solving the above self-interested model, what is obtained is the upper bound of the evaluation efficiency of the d-th decision-making unit to be evaluated. There are a total of n decision-making units, so n self-interested models are solved to obtain the upper bounds of the evaluation efficiencies of all decision-making units;

[0020] Construct a non-self-interested model to obtain the lower bound of the efficiency of the decision-making unit, then there is:

[0021]

[0022] s.t.

[0023]

[0024] Among them, y rk represents the r-th output of the k-th decision-making unit to be evaluated, v rk represents the weight of the r-th output of the k-th decision-making unit, u ik represents the weight of the i-th input of the k-th decision-making unit, x ik represents the i-th input of the k-th decision-making unit to be evaluated; n represents the number of decision-making units, m represents the total number of input resources, and s represents the total number of outputs; It means that the objective function is to maximize the efficiency of all other decision-making units to be evaluated except the d-th decision-making unit to be evaluated.

[0025] The non-self-interested model has multiple optimization objectives. The optimization objective is to optimize the other n - 1 decision-making units except the evaluated decision-making unit d. For the solution of the non-self-interested model, the objective function is rewritten. Since the decision-making units evaluated in the model are homogeneous and of the same type, and the importance of all decision-making units is the same, the sum of the n - 1 maximization objective functions is changed into one objective function, that is:

[0026]

[0027] Among them, v rk y rk represents the r-th weighted output of the k-th decision-making unit to be evaluated, represents the sum of the weighted outputs of the k-th decision-making unit to be evaluated, k = 1, 2…, n and k ≠ d, that is, y rn represents the r-th output of the n-th decision-making unit to be evaluated, v rn represents the weight of the r-th output of the n-th decision-making unit to be evaluated, v rn y rnDenote the $r$-th weighted output of the $n$-th decision-making unit to be evaluated; since the optimization objective is to maximize the efficiency of all decision-making units to be evaluated except decision-making unit $d$, the efficiency of the $d$-th decision-making unit to be evaluated is not included in the summation; the objective function is transformed into maximizing the cumulative sum of the efficiencies of $n - 1$ decision-making units:

[0028]

[0029] The processed non-selfish model is expressed as:

[0030]

[0031] s.t.

[0032]

[0033] where $y$ rk denotes the $r$-th output of the $k$-th decision-making unit to be evaluated, $v$ rk denotes the weight of the $r$-th output of the $k$-th decision-making unit, $u$ ik denotes the weight of the $i$-th input of the $k$-th decision-making unit, $x$ ik denotes the $i$-th input of the $k$-th decision-making unit to be evaluated; $n$ represents the number of decision-making units, $m$ represents the total number of input resources, and $s$ represents the total number of outputs; the optimization objective of the non-selfish model is the cumulative sum of the efficiencies of other decision-making units except the decision-making unit $d$ to be evaluated, and its optimal value is denoted as $\varPhi$ d ;

[0034] For the optimal value $\varPhi$ in the processed non-selfish model d , by establishing a system of multiple linear equations, the lower bound of the efficiency value of each decision-making unit evaluated based on the non-selfish principle is obtained. The system of multiple linear equations is:

[0035]

[0036] where denotes the lower bound of the evaluation efficiency of the $k$-th decision-making unit to be evaluated obtained from the non-selfish model. The superscript indicates that the obtained lower bound of the efficiency has not been normalized, $k = 1, \ldots, n$; $\varPhi$ k denotes the optimal objective value in the non-selfish model of the $k$-th decision-making unit to be evaluated, that is, the optimal value of the cumulative sum of the efficiencies of other decision-making units except the $k$-th decision-making unit to be evaluated, $k = 1, \ldots, n$; the efficiencies obtained from the selfish model and the non-selfish model represent the upper and lower bounds of the efficiency interval respectively.

[0037] The obtained interval efficiency lower bound is normalized using the principle of interval mapping, and the unnormalized interval efficiency lower bound is mapped to the interval $[0, 1]$. The calculation formula for normalizing the efficiency lower bound is as follows:

[0038]

[0039] Among them, represents the lower bound of the interval efficiency of n unnormalized decision-making units where d = 1, …, n, take the maximum value, denoted as represents the lower bound of the interval efficiency of n unnormalized decision-making units where d = 1, …, n, take the minimum value among them, denoted as θ l ; represents the lower bound of the interval efficiency of the normalized decision-making unit d.

[0040] In S2, it is recorded that the input is reduced by a% and the output is increased by b%. The reduced input and increased output are defined as the predicted input and predicted output. Thus, the calculation formulas for the predicted input and predicted output are as follows: Among them, x ij represents the i-th input of the j-th evaluated decision-making unit, and y rj represents the r-th output of the j-th evaluated decision-making unit. respectively represent the i-th predicted input and the r-th predicted output of the j-th evaluated decision-making unit.

[0041] In S3, by fitting the predicted data, the obtained fitting function will be used as the attitude function f(x) of this period; among them, a polynomial function is used to fit the predicted data in S2 to obtain the polynomial fitting functions of the predicted input and output. To determine the order of the polynomial fitting function, by comparing the mean squared error between the estimated values corresponding to the fitting functions under different orders and the true data values to be fitted, the one with the smallest mean squared error is the optimal fitting function. Select the fitting function corresponding to the optimal fitting order, and average the coefficients of the same order of multiple fitting functions to obtain the finally estimated attitude function f(x).

[0042] In S4, measure the possibility in the uncertain decision-making, and calculate the decision preference in combination with the attitude function obtained in S3 to obtain the binary order relationship of the interval efficiency; conduct the possibility measurement based on the attitude function, specifically as follows:

[0043] For any two decision-making unit interval efficiencies a and b obtained in S1, they are respectively denoted as Among them represents the lower bound of the interval efficiency A of decision-making unit a, represents the upper bound of the interval efficiency A of decision-making unit a; represents the lower bound of the interval efficiency B of decision-making unit b, Denote the upper bound of the interval efficiency \(B\) of decision-making unit \(b\) as \(P(B\geq A)\) which represents the possibility that interval efficiency \(B\) is superior to interval efficiency \(A\), \(f(x)\) represents the attitude function, and \(dx\) represents the integral with respect to \(x\). The possibility measure based on decision-making preference is determined according to different interval relationships, including:

[0044] If the interval is located on the left side of the interval , then

[0045] \(P(B\geq A)=1\),

[0046] The above formula indicates that the possibility that interval efficiency \(B\) is superior to interval efficiency \(A\) is 1;

[0047] If the interval completely contains the interval , then

[0048]

[0049] The above formula shows that to make interval efficiency \(B\) superior to interval efficiency \(A\), there are two cases: one is that the efficiency of decision-making unit \(a\) falls within the interval , and the corresponding possibility can be expressed as the ratio of the integral of the attitude function on to the integral of the attitude function over the entire efficiency interval of decision-making unit \(a\), that is The other is that the efficiencies of both decision-making unit \(a\) and decision-making unit \(b\) fall within the interval , and the corresponding possibility can be expressed as the ratio of the integral of the attitude function on to the integral of the attitude function over the entire efficiency interval of decision-making unit \(a\) multiplied by , that is Therefore, the sum of the two possibilities represents the possibility that interval efficiency \(B\) is superior to interval efficiency \(A\) in this case;

[0050] If the interval overlaps with the interval , and the interval is relatively small, then

[0051]

[0052] The above formula shows that to make interval efficiency \(B\) superior to interval efficiency \(A\), there are three cases: one is that the efficiency of decision-making unit \(b\) falls within the interval , and the corresponding possibility can be expressed as the ratio of the integral of the attitude function on to the integral of the attitude function over the entire efficiency interval of decision-making unit \(b\), that is The second is that the efficiency of decision-making unit \(b\) falls within the interval above, and the efficiency of decision-making unit a falls within the interval above, and the corresponding possibility can be expressed as the product of the proportion of the integral of the attitude function in the corresponding interval Thirdly, the efficiencies of both decision-making unit b and decision-making unit a fall within the interval above, and the corresponding possibility can be expressed as the product of the proportion of the integral on That is Therefore, the sum of the three possibilities represents the possibility that the interval efficiency B is better than the interval efficiency A in this case;

[0053] If the interval is located on the right side of the interval then

[0054] P(B≥A) = 0,

[0055] The above formula shows that the possibility that the interval efficiency B is better than the interval efficiency A is 0;

[0056] If the interval completely contains the interval then

[0057]

[0058] The above formula shows that the possibility that the interval efficiency B is better than the interval efficiency A is equivalent to 1 minus the possibility that the interval efficiency A is better than the interval efficiency B. To make the interval efficiency A better than the interval efficiency B, there are two cases: The efficiency of decision-making unit b falls within the interval above, and the corresponding possibility can be expressed as the proportion of the integral based on the attitude function Secondly, the efficiencies of both decision-making unit a and decision-making unit b fall within the interval above, and the corresponding possibility can be expressed as the proportion of the integral based on the attitude function multiplied by That is If the interval overlaps with the interval and the interval is relatively small, then

[0059]

[0060] The above formula shows that the possibility that the interval efficiency B is better than the interval efficiency A is equivalent to 1 minus the possibility that the interval efficiency A is better than the interval efficiency B. To make the interval efficiency A better than the interval efficiency B, there are three cases: Firstly, the efficiency of decision-making unit a falls within the interval above, and the corresponding possibility can be expressed as the proportion of the integral based on the attitude function Secondly, the efficiency of decision-making unit a falls within the interval above, and the efficiency of decision-making unit b falls within the interval Above, the corresponding possibility can be expressed as the product of the proportion of the integral of the attitude function in the corresponding interval Thirdly, the efficiencies of decision-making unit b and decision-making unit a both fall within the interval Above, the corresponding possibility can be expressed as the product of the proportion of the integral of the attitude function multiplied by That is Through pairwise comparison of the interval efficiencies obtained in S1 by the possibility measure, a relationship matrix P of the interval efficiencies based on the possibility is obtained, and its matrix form is as follows:

[0061]

[0062] Among them, p ij represents the possibility value obtained by comparing decision-making unit i and decision-making unit j, i = 1, …, n; j = 1, …, n. When 0 < p ij < 0.5, it means that the interval efficiency of decision-making unit i is inferior to that of decision-making unit j; when 0.5 < p ij < 1, it means that the interval efficiency of decision-making unit i is superior to that of decision-making unit j; the magnitude of the value represents the degree of superiority or inferiority of the interval efficiency.

[0063] In S5, the evaluation weights of the decision-making units are calculated by constructing a weight solution model that comprehensively reflects the relative efficiency contributions of the decision-making units. The weight solution model is:

[0064]

[0065] s.t.

[0066]

[0067] Among them, is the upper bound of the efficiency value of the d-th decision-making unit, that is, the upper bound of the efficiency obtained in S1. x ij is the i-th input of the j-th decision-making unit, y rj represents the r-th output of the j-th decision-making unit. n represents the number of decision-making units, m represents the total number of input resources, s represents the total number of outputs, λ dj represents the weight assigned by the evaluated decision-making unit d to different decision-making units j. δ takes a positive number close to 0, and 1 / (λ dj + δ) is the penalty factor;

[0068] The weight matrix obtained by the weight solution model is denoted as:

[0069]

[0070] Among them, λ ijrepresents the weight assigned by the evaluated decision-making unit \(i\) to different decision-making units \(j\), where \(i = 1,\ldots,n\); \(j = 1,\ldots,n\); the \(d\)-th row of the weight matrix represents the weights assigned to all other decision-making units when evaluating the \(d\)-th decision-making unit; the weights of decision-making unit \(i\) are obtained by accumulating and normalizing the weight matrix column-wise, denoted as \(w\) i , that is, the weight of the \(i\)-th decision-making unit

[0071] Combined with the possibility relation matrix \(P\) calculated in S4, the ranking score value of decision-making unit \(j\) is \(P\) j :[[]]END]]

[0072] The technical solution provided by the present invention has the following beneficial effects compared with the known prior art:

[0073] The present invention evaluates efficiency by constructing two DEA models of local optimum (egoistic model) and overall optimum (non-egoistic model) to form the upper and lower bounds of the flight training efficiency interval. This evaluation method can effectively distinguish the differences between the training efficiency of individual students and the efficiency of the entire system, and provide a clear identification of possible efficiency bottlenecks in the training process.

[0074] The present invention constructs an attitude function using a data-driven method, further introduces the subjective preferences of flight instructors into the evaluation process, and ensures that the training efficiency evaluation better meets the actual needs by dynamically setting development goals. On this basis, by improving the calculation of possibility measurement, the problem that the traditional efficiency evaluation based on the DEA model fails to fully consider training preferences is solved, and the accuracy and practicality of the evaluation results are improved.

[0075] By combining the attitude function with the improved possibility measurement method and solving the weights of decision-making units, the present invention realizes the full ranking of interval efficiency. This not only solves the problem that the traditional DEA model cannot achieve full ranking in the efficiency evaluation process, but also significantly improves the ability to identify the efficiency differences between high-efficiency and low-efficiency students. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention, and those of ordinary skill in the art can also obtain other drawings without creative efforts based on these drawings.

[0077] Figure 1 is the flowchart of the steps described in the present invention;

[0078] Figure 2(a) is the interval is located in the interval The relationship diagram on the left;

[0079] Figure 2(b) is the relationship diagram of the interval completely containing the interval ;

[0080] Figure 2(c) is the relationship diagram of the interval overlapping with the interval and the interval being relatively small;

[0081] Figure 2(d) is the relationship diagram of the interval located on the right side of the interval ;

[0082] Figure 2(e) is the relationship diagram of the interval completely containing the interval ;

[0083] Figure 2(f) is the relationship diagram of the interval overlapping with the interval and the interval being relatively small;

[0084] Figure 3(a) is the optimal fitting function diagram obtained from the predicted data of input x1 and output y;

[0085] Figure 3(b) is the optimal fitting function diagram obtained from the predicted data of input x2 and output y;

[0086] Figure 3(c) is the optimal fitting function diagram obtained from the predicted data of input x3 and output y;

[0087] Figure 3(d) is the optimal fitting function diagram obtained from the predicted data of input x4 and output y;

[0088] Figure 3(e) is the optimal fitting function diagram obtained from the predicted data of input x5 and output y;

[0089] Figure 3(f) is the optimal fitting function diagram obtained from the predicted data of input x6 and output y;

[0090] Figure 4 is the possibility relationship matrix diagram described in the present invention. Detailed implementation manners

[0091] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0092] The present invention will be further described below in conjunction with embodiments.

[0093] In the implementation of the present invention, the specific steps are as follows:

[0094] S1. Construct an egoistic model and a non-egoistic model, and calculate the training efficiency intervals of n = 20 trainees. The steps include:

[0095] H1: Establish an evaluation system for training efficiency indicators, clarify the meanings of the indicators, and select key input and output indicators through scientific methods;

[0096] (1) Input indicators: The cumulative number of takeoffs and landings in 13 hours (x1), the cumulative number of takeoffs and landings for the first solo flight (x2), the cumulative flight experience time in the practical exam (x3), the cumulative number of takeoffs and landings in the practical exam (x4), the number of practical exams (x5), the training time span (x6);

[0097] (2) Output indicator: Competence intensity (y);

[0098] In this implementation, first construct and clarify the input and output indicator systems for efficiency evaluation. This indicator system is shown in Table 1:

[0099] Table 1 Input and Output Indicator System Table

[0100]

[0101] The egoistic model is expressed as:

[0102]

[0103] s.t.

[0104]

[0105] Wherein, represents the upper bound of the efficiency of the d-th trainee to be evaluated, represents maximizing the upper bound of the efficiency of the d-th trainee to be evaluated, y rd represents the r-th output of the d-th trainee to be evaluated, v rd represents the weight of the r-th output of the d-th trainee to be evaluated, u iddenotes the weight of the \(i\)-th input of the \(d\)-th trainee to be evaluated, \(x\) id represents the \(i\)-th input of the \(d\)-th trainee to be evaluated; the notation s.t. indicates that the following formula is a constraint of the optimization model; \(y\) rk represents the \(r\)-th output of the \(k\)-th trainee to be evaluated, \(v\) rk denotes the weight of the \(r\)-th output of the \(k\)-th trainee, \(u\) ik denotes the weight of the \(i\)-th input of the \(k\)-th trainee, \(x\) ik represents the \(i\)-th input of the \(k\)-th trainee to be evaluated; the number of decision-making units \(n = 20\), that is, the total number of flight trainees as the efficiency evaluation objects is 20, so \(1\leq d\leq20\); \(m\) represents the total number of input resources, that is, \(m = 6\); \(s\) represents the total number of outputs, that is, \(s = 1\); by solving the above self-interested model, the upper bound of the evaluation efficiency of the \(d\)-th trainee to be evaluated is obtained. There are a total of \(n = 20\) trainees, so 20 self-interested models are solved to obtain the upper bounds of the evaluation efficiencies of all trainees.

[0106] The non-self-interested model is expressed as:

[0107]

[0108] s.t.

[0109]

[0110] where, \(y\) rk represents the \(r\)-th output of the \(k\)-th trainee to be evaluated, \(v\) rk denotes the weight of the \(r\)-th output of the \(k\)-th trainee, \(u\) ik denotes the weight of the \(i\)-th input of the \(k\)-th trainee, \(x\) ik represents the \(i\)-th input of the \(k\)-th trainee to be evaluated; indicates that the objective function is to maximize the efficiency of all other trainees to be evaluated except the \(d\)-th trainee to be evaluated.

[0111] Furthermore, the lower bound of the efficiency value of each decision-making unit evaluated based on the non-self-interested principle can be obtained by establishing a system of multiple linear equations.

[0112]

[0113] where, denotes the lower bound of the evaluation efficiency of the \(k\)-th trainee to be evaluated obtained by the non-self-interested model. The superscript indicates that the obtained lower bound of the efficiency has not been normalized, \(k = 1,\cdots,20\); \(\varPhi\) k represents the optimal value in the non-self-interested model of the \(k\)-th trainee to be evaluated, that is, the optimal value of the sum of the efficiencies of other trainees except trainee \(k\), \(k = 1,\cdots,20\); The efficiencies obtained from the self-interested model and the non-self-interested model represent the upper and lower bounds of the training efficiency interval respectively.

[0114] The egoistic model and the non-egoistic model are used to evaluate the efficiency of trainees, calculate the local optimal efficiency and the overall optimal efficiency of trainees, and the evaluation results of the obtained efficiency intervals are shown in Table 2 below:

[0115] Table 2 Results of efficiency intervals after solving with two optimization models

[0116]

[0117]

[0118] To ensure the upper and lower bound relationship of the interval efficiency, the present invention standardizes the evaluation results of the non-egoistic model and maps the lower bound of the efficiency to the range of [0, 1]. This processing step aims to eliminate the scale differences in the original data, ensure the comparability of the efficiency evaluation results of different decision-making units, and at the same time maintain their effectiveness within the preset range. The standardized results are shown in Table 3:

[0119] Table 3 Results of standardized efficiency intervals

[0120]

[0121]

[0122] S2. Generate predicted inputs and predicted outputs according to the input-output data.

[0123] The predicted inputs and predicted outputs are the ideal inputs and outputs in the next stage from the perspective of this stage. Generally, decision-makers hope that a certain improvement can reduce the input and increase the output. Therefore, considering a 2% reduction in input and a 3% increase in output, the input-output calculation of the trainees in the next stage of training is as follows:

[0124]

[0125] S3. Fit the attitude function through the predicted input and output data.

[0126] Based on each predicted input Denoted as xi (i = 1,..., 6); the predicted output Denoted as y, the optimal fitting function obtained by data fitting is shown in Figures 3(a)-(f).

[0127] S4. Calculate the possibility measure of decision preference based on the attitude function

[0128] Adopt the improved possibility calculation formula proposed by the present invention:

[0129] Including:

[0130] When the interval relationship is as shown in Figure 2(a),

[0131] P(B≥A) = 1

[0132] When the interval relationship is as shown in Fig. 2(b),

[0133]

[0134] When the interval relationship is as shown in Fig. 2(c),

[0135]

[0136] When the interval relationship is as shown in Fig. 2(d),

[0137] P(B≥A) = 0

[0138] When the interval relationship is as shown in Fig. 2(e),

[0139]

[0140] When the interval relationship is as shown in Fig. 2(f),

[0141]

[0142] Compare the interval efficiencies to obtain the possibility matrix P, as shown in Figure 4 (The numerical size is represented by the shade of color).

[0143] S5. Based on the possibility measurement relationship matrix and the solution of the weights of the decision-making units, calculate the comprehensive ranking of the interval efficiencies of the students' flight training.

[0144] After completing the construction and ranking of the possibility matrix, conduct a comprehensive weighted evaluation of all students according to the improved possibility ranking model, and finally obtain the ranking results of the students. The present invention proposes a ranking method based on the weight solution model when ranking the possibility matrix. The weight solution model can be expressed as:

[0145]

[0146] s.t.

[0147]

[0148] Wherein, is the upper bound of the efficiency value of the d-th student, that is, the upper bound of the student's efficiency, λ djIt represents the weight assigned by the evaluated trainee d to different trainees j. δ takes a positive value close to 0, and here δ is taken as 0.00001. Combining with the upper bound of the efficiency interval obtained in S1, the weights of each decision-making unit are solved; after normalization, they are multiplied by the possibility matrix P and accumulated to calculate the final interval efficiency ranking score value; this ranking comprehensively considers the interval efficiency of each trainee, the preference factors of the decision-maker, and their relative contributions in the evaluation process, so as to ensure the rationality and accuracy of the ranking result. The ranking result is shown in Table 4 as follows:

[0149] Table 4 Final Efficiency Interval Evaluation Results of Each Flight Trainee

[0150]

[0151]

[0152] The working principle of the present invention is as follows: Based on the principle of data envelopment analysis (DEA), an egoistic model and a non-egoistic model are constructed to solve the local optimal efficiency and the overall optimal efficiency of the trainees, so as to obtain the efficiency interval. The attitude function is introduced into the possibility theory, and through the full ranking of the efficiency interval, the comprehensive evaluation of the trainees' performance is realized. The introduced attitude function can reflect the preference of the decision-maker, making the ranking result more accurate and fully reflecting the difference between the actual decision-making needs and the trainees' performance.

[0153] The above is only the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. The structures, devices, and operation methods not specifically described and explained in the present invention, unless otherwise specified and limited, are implemented according to the conventional means in the art.

[0154] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements will not make the essence of the corresponding technical solutions deviate from the protection scope of the technical solutions of the embodiments of the present invention.

Claims

1. An interval efficiency evaluation method combined with possibility measure, characterized in that, It includes the following steps: S1. Construct an egoistic model and a non-egoistic model, and calculate the training efficiency interval of trainees; S2. Calculate the predicted input and predicted output according to the input-output data; S3. Fit the attitude function through the predicted input and output data; S4. Calculate the possibility measure based on decision preference based on the attitude function; S5. Calculate the comprehensive ranking of the efficiency of the trainees' flight training interval based on the possibility measure relationship matrix and the solution of the weights of the decision-making units.

2. The interval efficiency evaluation method combined with possibility measure according to claim 1, characterized in that, In S1, based on the data envelopment analysis model, maximize the efficiency of the evaluated decision-making unit. The egoistic model is represented by establishing the following expression: s.t. Among them, represents the upper bound of the efficiency of the d-th decision-making unit to be evaluated, max represents maximizing the upper bound of the efficiency of the d-th decision-making unit to be evaluated, y rd represents the r-th output of the d-th decision-making unit to be evaluated, v rd represents the weight of the r-th output of the d-th decision-making unit to be evaluated, u id represents the weight of the i-th input of the d-th decision-making unit to be evaluated, x id represents the i-th input of the d-th decision-making unit to be evaluated; the notation s.t. indicates that the following formula is the constraint condition of the optimization model; y rk represents the r-th output of the k-th decision-making unit to be evaluated, v rk represents the weight of the r-th output of the k-th decision-making unit, u ik represents the weight of the i-th input of the k-th decision-making unit, x ik represents the i-th input of the k-th decision-making unit to be evaluated; n represents the number of decision-making units, that is, the total number of efficiency evaluation objects, so 1 ≤ d ≤ n; m represents the total number of input resources, and s represents the total number of outputs; by solving the above self-interested model, what is obtained is the upper bound of the evaluation efficiency of the d-th decision-making unit to be evaluated. There are a total of n decision-making units, so n self-interested models are solved to obtain the upper bounds of the evaluation efficiencies of all decision-making units; Construct a non-egoistic model to obtain the lower limit of the efficiency of the decision-making unit, then there is: s.t. Among them, y rk represents the r-th output of the k-th decision-making unit to be evaluated, v rk represents the weight of the r-th output of the k-th decision-making unit, u ik represents the weight of the i-th input of the k-th decision-making unit, x ik represents the i-th input of the k-th decision-making unit to be evaluated; n represents the number of decision-making units, m represents the total number of input resources, and s represents the total number of outputs; indicates that the objective function is to maximize the efficiency of all decision-making units to be evaluated except the d-th decision-making unit to be evaluated.

3. The interval efficiency evaluation method combining possibility degree measurement according to claim 2, characterized in that The non-egoistic model has multiple optimization objectives. The optimization objective is to optimize the other n - 1 decision-making units except the evaluated decision-making unit d. For the solution of the non-egoistic model, rewrite the objective function. Since the decision-making units evaluated in the model are homogeneous and of the same type, and the importance of all decision-making units is the same, sum the n - 1 maximization objective functions into one objective function, that is: Among them, v rk y rk represents the r-th weighted output of the k-th decision-making unit to be evaluated, represents the sum of the weighted outputs of the k-th decision-making unit to be evaluated, k = 1, 2,..., n and k ≠ d, that is, y rn represents the r-th output of the n-th decision-making unit to be evaluated, v rn represents the weight of the r-th output of the n-th decision-making unit to be evaluated, v rn y rn represents the r-th weighted output of the n-th decision-making unit to be evaluated; since the optimization objective is to maximize the efficiency of all other decision-making units to be evaluated except decision-making unit d, the efficiency of the d-th decision-making unit is not included in the summation; the objective function is transformed into maximizing the cumulative sum of the efficiencies of n - 1 decision-making units:

4. The interval efficiency evaluation method combining possibility degree measurement according to claim 3, characterized in that The processed non-egoistic model is: Among them, y rk represents the r-th output of the k-th decision-making unit to be evaluated, v rk represents the weight of the r-th output of the k-th decision-making unit, u ik represents the weight of the i-th input of the k-th decision-making unit, x ik represents the i-th input of the k-th decision-making unit to be evaluated; n represents the number of decision-making units, m represents the total number of input resources, and s represents the total number of outputs; the optimization objective of the non-selfish model is the sum of the efficiencies of other decision-making units except the decision-making unit d to be evaluated, and its optimal value is denoted as Ф d ; The optimal value Ф in the processed non-selfish model d , the lower bound of the efficiency value of each decision-making unit evaluated based on the non-selfish principle is obtained by establishing a system of multiple linear equations, and the system of multiple linear equations is as follows: Among them, represents the lower bound of the evaluation efficiency of the k-th decision-making unit to be evaluated obtained by the non-selfish model. The superscript indicates that the obtained lower bound of the efficiency has not been normalized, where k = 1, …, n; Ф k represents the optimal objective value in the non-selfish model of the k-th decision-making unit to be evaluated, that is, the optimal value of the sum of the efficiencies of other decision-making units except the k-th decision-making unit to be evaluated, where k = 1, …, n; The efficiencies obtained by the selfish model and the non-selfish model respectively represent the upper and lower bounds of the efficiency interval.

5. The interval efficiency evaluation method combining possibility degree measurement according to claim 4, characterized in that, Adopt the principle of interval mapping to normalize the lower bound of efficiency, and map the unnormalized interval efficiency lower bound to the interval [0, 1]. The calculation formula for normalizing the lower bound of efficiency is as follows: Among them, represents the lower bound of the interval efficiency of a total of n unnormalized decision-making units for d = 1, …, n take the maximum value, denoted as represents the lower bound of the interval efficiency of a total of n unnormalized decision-making units for d = 1, …, n take the minimum value from it, denoted as θ l ; represents the lower bound of the interval efficiency of the normalized decision-making unit d.

6. The interval efficiency evaluation method combining the possibility measure according to claim 5, characterized in that, In S2, it is recorded that the input decreases by a%, and the output increases by b%. The reduced input and the increased output are defined as the predicted input and the predicted output. Therefore, the calculation formulas for the predicted input and the predicted output are as follows: where x ij represents the i-th input of the j-th decision-making unit to be evaluated, and y rj represents the r-th output of the j-th decision-making unit to be evaluated, respectively representing the i-th predicted input and the r-th predicted output of the j-th decision-making unit to be evaluated.

7. An interval efficiency evaluation method combining possibility degree measurement according to claim 6, characterized in that, In S3, through fitting the prediction data, the obtained fitting function will be used as the attitude function f(x) of this period; among them, use a polynomial function to fit the prediction data in S2 to obtain the polynomial fitting functions of the predicted input and output; determine the order of the polynomial fitting function by comparing the mean squared error between the estimated values corresponding to the fitting functions under different orders and the values of the data to be fitted. The one with the smallest mean squared error is the optimal fitting function. Select the fitting function corresponding to the optimal fitting order, and average the coefficients of the same order of multiple fitting functions to obtain the finally estimated attitude function f(x).

8. An interval efficiency evaluation method combining possibility degree measurement according to claim 7, characterized in that In S4, measure the possibility in uncertain decision-making, and calculate the decision preference in combination with the attitude function obtained in S3 to obtain the binary order relationship of the interval efficiency; conduct the possibility measure based on the attitude function, specifically as follows: The interval efficiencies of any two decision-making units a and b obtained in S1 are respectively denoted as where represents the lower bound of the interval efficiency A of decision-making unit a, represents the upper bound of the interval efficiency A of decision-making unit a; represents the lower bound of the interval efficiency B of decision-making unit b, represents the upper bound of the interval efficiency B of decision-making unit b. Denote P(B≥A) as the possibility that the interval efficiency B is better than the interval efficiency A, f(x) as the attitude function, and dx as the integral of x; The possibility measure based on decision preference is determined according to different interval relationships, including: If the interval is located to the left of the interval , then P(B≥A) = 1, The above formula indicates that the possibility that the interval efficiency B is better than the interval efficiency A is 1; If the interval completely contains the interval then The above formula shows that there are two cases for the interval efficiency B to be better than the interval efficiency A: one is that the efficiency of decision-making unit a falls within the interval and the corresponding possibility can be expressed as the ratio of the integral of the attitude function over to the integral of the attitude function over the entire efficiency interval of decision-making unit a ; The other is that the efficiencies of decision-making unit a and decision-making unit b both fall within the interval and the corresponding possibility can be expressed as the ratio of the integral of the attitude function over to the integral of the attitude function over the entire efficiency interval of decision-making unit a multiplied by i.e., Therefore, the sum of the two possibilities represents the possibility that the interval efficiency B is better than the interval efficiency A in this case; If the interval overlaps with the interval and the interval is small, then The above formula shows that there are three situations in which interval efficiency B is better than interval efficiency A: one is that the efficiency of decision unit b falls within the interval The corresponding possibility can be expressed as the attitude function in The integral on the efficiency interval of the attitude function over the entire decision-making unit b is The proportion of the above points The second is that the efficiency of decision-making unit b falls within the interval On, and the efficiency of decision-making unit a falls within the interval The corresponding possibility can be expressed as the product of the integral proportion of the attitude function in the corresponding interval, that is, Third, the efficiency of decision-making unit b and decision-making unit a both fall within the interval The corresponding possibility can be expressed as the attitude function based on The product of the percentage of points on Right now Therefore, the sum of the three possibilities indicates the possibility that interval efficiency B is better than interval efficiency A in this case; If the interval is located to the right of the interval , then P(B≥A) = 0, The above formula indicates that the possibility that the interval efficiency B is better than the interval efficiency A is 0; If the interval completely contains the interval then The above formula shows that the possibility that the interval efficiency B is better than the interval efficiency A is equivalent to 1 minus the possibility that the interval efficiency A is better than the interval efficiency B. To make the interval efficiency A better than the interval efficiency B, there are two cases: the efficiency of decision-making unit b falls within the interval above, and the corresponding possibility can be expressed as the integral proportion based on the attitude function Second, the efficiency of decision-making unit a and decision-making unit b falls within the interval above, and the corresponding possibility can be expressed as the integral proportion based on the attitude function multiplied by That is If the interval overlaps with the interval and the interval is relatively small, then The above formula shows that the possibility that the interval efficiency B is better than the interval efficiency A is equivalent to 1 minus the possibility that the interval efficiency A is better than the interval efficiency B. To make the interval efficiency A better than the interval efficiency B, there are three cases: The first is that the efficiency of decision-making unit a falls within the interval above, and the corresponding possibility can be expressed as the integral proportion based on the attitude function The second is that the efficiency of decision-making unit a falls within the interval above, and the efficiency of decision-making unit b falls within the interval above. The corresponding possibility can be expressed as the product of the integral proportions based on the attitude function in the corresponding intervals The third is that the efficiencies of both decision-making unit b and decision-making unit a fall within the interval above, and the corresponding possibility can be expressed as the product of the integral proportions based on the attitude function multiplied by That is Through pairwise comparison of the training efficiency intervals obtained in S1 by the possibility measure, obtain the relationship matrix P of the interval efficiency based on the possibility, and its matrix form is as follows: Among them, p ij represents the possibility value obtained by comparing the interval efficiency between decision-making unit i and decision-making unit j. i = 1, …, n; j = 1, …, n; when 0 < p ij < 0.5, it indicates that the interval efficiency of decision-making unit i is inferior to that of decision-making unit j; when 0.5 < p ij < 1, it indicates that the interval efficiency of decision-making unit i is superior to that of decision-making unit j; the magnitude of the value represents the degree of superiority or inferiority of the interval efficiency.

9. An interval efficiency evaluation method combining possibility degree measurement according to claim 8, characterized in that, In S5, calculate the evaluation weights of the decision-making units by constructing a weight solution model that comprehensively reflects the relative efficiency contribution of the decision-making units. The weight solution model is: s.t. Among them, is the upper bound of the efficiency value of the d-th decision-making unit, that is, the upper bound of the efficiency obtained in S1, x ij is the i-th input of the j-th decision-making unit, y rj represents the r-th output of the j-th decision-making unit, n represents the number of decision-making units, m represents the total number of input resources, s represents the total number of outputs, λ dj represents the weight assigned by the evaluated decision-making unit d to different decision-making units j. δ takes a positive number close to 0, and 1 / (λ dj +δ) is the penalty factor; The weight matrix obtained by the weight solution model is denoted as: where λ ij represents the weight assigned by the decision-making unit \(i\) to be evaluated to different decision-making units \(j\), \(i = 1,\ldots,n\); \(j = 1,\ldots,n\); the \(d\)-th row of the weight matrix represents the weights assigned to all other decision-making units when evaluating the \(d\)-th decision-making unit; the columns of the weight matrix are accumulated and normalized, and the weight result of the decision-making unit \(i\) is expressed as \(w\) i That is 10. A method for evaluating interval efficiency by combining possibility degree metrics according to claim 9, characterized in that Calculate the ranking score value of decision-making unit j as P according to the possibility relation matrix P calculated in S4 j :

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