The Efficiency Evaluation of DEA Model Incorporating Improved Possibility Theory

Abstract

The data envelopment analysis (DEA) models have been widely recognized and applied in various fields. However, these models have limitations, such as their inability to globally rank DMUs, the efficiency values are definite numerical values, they are unable to reflect potential efficiency changes, and they fail to adequately reflect the degree of the decision maker’s preference. In order to address these shortcomings, this paper combines possibility theory with self-interest and non-self-interest principles to improve the DEA model to provide a more detailed reflection of the differences between DMUs. First, the self-interest and non-self-interest principles are employed to establish the DEA evaluation model, and the determined numerical efficiency is transformed into efficiency intervals. Second, an attitude function is added to the common possible-degree formula to reflect the decision maker’s preference, and a more reasonable method for solving the attitude function is presented. Finally, the improved possible-degree formula proposed in this paper is used to rank and compare the interval efficiencies. This improved method not only provides more comprehensive ranking information but also better captures the decision maker’s preferences. This model takes preference issues into account and has improved stability and accuracy compared with existing models. The application of the improved model in airlines shows that the model proposed in this paper effectively achieved a full ranking. From a developmental perspective, the efficiency levels of Chinese airlines were generally comparable. Joyair and One Two Three performed poorly, exhibiting significant gaps compared with other airlines.

1. Introduction

The DEA models, as methods for evaluating the effectiveness of DMUs, have been applied in various fields [1,2,3], such as education, healthcare [4], and energy conservation, receiving widespread recognition and attention. Through continuous improvement and adaptation, the DEA model has not only developed various branch models, such as network DEA [5], window DEA [6], cross-efficiency evaluation [7], and directional distance functions [8], but has also been extensively integrated with other domains, such as cluster analysis [9] and machine learning [10], demonstrating significant research potential and application versatility. These expansions and integrations make the DEA model more flexible and applicable to complex problems in various fields, providing powerful tools and methods for interdisciplinary research.

When assessing the efficiency of DMUs, the traditional DEA model excessively selects a set of input–output weight values to maximize its own efficiency. Additionally, it can only dichotomize DMUs into effective or ineffective categories, without the ability to classify or rank them hierarchically. The efficiency scores of DMUs are between zero and one inclusive. It is very often the case that more than one DMU are evaluated as DEA-efficient DMUs or are related near the maximum efficiency score, which makes the ranking of the DMUs impossible to carry out. A lack of discrimination power is a drawback of DEA that has aroused considerable research interest in the DEA literature. There is no lack of survey studies in the ranking DMUs in DEA. Liu et al. [11] summarized the literature related to ranking, Izadikhah and Farzipoor [12] proposed an algorithm to rank all DMUs, An et al. [13] ranked DMUs using AHP and DEA models, Kritikos [14] performed a comprehensive ranking based on the TOPSIS method, Jahanshahloo et al. [15] gave a methodology for efficient DMUs to be ranked, and Ekiz and Tuncer Sakar [16] discussed the drawbacks of the existing ranking methodologies and proposed a new methodology for efficiency ranking. Moreover, traditional DEA models represent efficiency values as fixed numerical values, which hinder the detection of potential efficiency changes. Arana-Jiménez et al. [17] used slack-based interval DEA to calculate efficiency scores, Wei et al. [18] proposed two forms of intervals for performance assessment, Lei et al. [19] calculated DEA sorting intervals and ordering, Aliasghar et al. [20] investigated the sorting problem of integer interval DEA, and Toloo et al. [21] proposed an interval efficiency approach that considers dual role variables.

To address these shortcomings, many researchers have endeavored to enhance the DEA model. Lei et al. [19] proposed a ranking method for the network DEA model, defining the superiority relationship and computational model between DMUs by considering the impact of stage efficiency on the ranking results and presenting a DEA full ranking method that considering the stage efficiency. The cross-efficiency evaluation model, initially introduced by Sexton, Silkman, and Hogan [22], aims to enhance the discriminative ability of DEA and achieve a full ranking of DMUs. However, the non-uniqueness of the optimal input–output weights in DEA means that there are multiple optimal solutions for optimal weights in DEA, which diminishes the effectiveness of the cross-efficiency method. Although some measures were proposed, such as implementing secondary objectives to enhance the variability of cross-efficiency scores, the issue of non-uniqueness in solutions remains unresolved. Dominikos et al. [23] measured DMUs’ self-efficiency scores using optimistic and pessimistic coefficients, combined with the theory of peer evaluation, and used cross-efficiency to rank DMUs. Ruiz [24] combined cross-efficiency evaluation with a directional distance function to achieve a full ranking of DMUs. Chen et al. [25] introduced cross-efficiency into the meta frontier analysis framework to ensure the stability of the optimal solution. Wu et al. [26] improved the cross-efficiency model by incorporating self-evaluation, peer evaluation, and common weight evaluation, leading cross-efficiency scores that produce Pareto-optimal solutions. Lin and Tu [27] combined a cross-efficiency evaluation and directional distance function model in the efficiency decomposition and ranking of network DEA. Cross-efficiency is a model developed for effective ordering, which solves the full-ordering problem and has been widely used and improved, where many researchers made a series of improvements to address the phenomenon where the optimal solution of cross-efficiency is not unique, and the typical existing solutions are optimistic and pessimistic models; however, these two models limit the examination of the performance of the DMU, which leads to a certain degree of irrationality in the final results.

Andersen and Petersen [28] later proposed a method for sorting DMUs called the super-efficiency method. The super-efficiency method achieves the full ranking of DMUs by removing the evaluated DMU from the reference set. However, the frequent changes in the frontier increase the instability of the model, and when the model is applied to VRS technology, it may encounter situations where no feasible solution exists. Xie et al. [29] compared the performances of the super-efficiency model and the cooperative game model in various aspects. Qu et al. [30] noticed the infeasibility of super-efficiency under VRS technology and provided necessary and sufficient conditions for the infeasibility of the super-efficiency model, as well as improvements to the super-efficiency model. Yan et al. [31] applied the super-efficiency SBM to measure the energy efficiency of the Silk Road Economic Belt. Guan et al. [32] used a three-level super-efficiency SBM to measure the comprehensive technical efficiency, pure technical efficiency, and scale efficiency of the ship industry chain. The super-efficient model is a typical model of DMU full ordering that makes up for the inherent defects of the traditional model, and it can be seen that the super-efficient model has been greatly applied through the above studies. However, the frequently changing front surface of the super-efficient model leads to the model not being sufficiently stable; there is the problem of infeasibility under the conditions of the VRS; and even though the sufficient and necessary conditions for the occurrence of infeasible solutions were given, there is still no effective solution to the infeasibility problem.

Although the abovementioned methods for cross-efficiency and super-efficiency sorting achieve a full ranking of DMUs, they have certain drawbacks, such as the non-uniqueness of optimal solutions in cross-efficiency and the instability and infeasibility issues in the super-efficiency model. Additionally, the limitation of deterministic efficiency values has not been addressed. In subsequent studies, many scholars attempted to address these limitations by employing interval efficiency methods, preserving data information more comprehensively, and combining interval efficiency with the interval comparison method of possibility theory.

Nakahara et al. [33] proposed an interval-number-sorting method based on the concept of definite integrals in 1992. Facchinetti et al. [34] proposed a possibility formula that considers optimistic and pessimistic scenarios based on triangular fuzzy numbers. Lan et al. [35] applied the interval DEA evaluation model with a common weight, addressing the decision maker’s preference issue in possibility theory. Song and Liu [36] proposed a variance coefficient method based on Shannon entropy to determine the weights of DEA cross-efficiency aggregation. Chen and Wang [37] introduced prospect theory and distance entropy into the evaluation of the DEA model, considering the subjective preferences of different types of decision makers and preserving decision information to the maximum extent. Zhang et al. [38] drew on the principles of probability theory, presented an analysis method for interval numbers with uncertainty, and introduced the concepts of interval advantage and possibility. Ke et al. [39] addressed the differences in interval number sorting methods and the failure of possibility functions by constructing a possibility function based on binary contact numbers, thereby establishing an interval number sorting method based on binary contact numbers that comprehensively reflects the deterministic and uncertain information of interval numbers. Sun et al. [40] applied interval DEA to evaluate the performance of underwater vehicles and used the possibility formula to rank efficiency intervals. Ta et al. [41] aimed to reduce the information loss caused by converting interval models into deterministic models, proposed a particle swarm-based interval optimization algorithm to handle uncertainty interval planning problems, and ranked the problems using the possibility formula. Liu et al. [42] summarized existing possibility formulas and proposed a generalized possibility formula that considers decision maker attitudes.

By analyzing and summarizing the above literature, it was found that many researchers paid attention to the defects of the ranking and deterministic efficiency representations of the DEA model and improved them. However, the existing improvements are still not perfect, with problems such as model instability and non-unique optimal solutions, and only for one of the defects in the full ranking or deterministic efficiency. Based on the above considerations, this paper proposes a novel efficiency calculation method based on the DEA model. First, the self-interest and non-self-interest principles are introduced to construct two independent DEA models. Second, the efficiency values evaluated by these two models are taken as the endpoints of the efficiency interval to capture the potential fluctuation range of the efficiency. Subsequently, the possibility theory is improved, and a ranking method based on the possibility theory is proposed and applied to the interval efficiency for ranking. In addition, considering the issue of decision maker attitude preference, an attitude function is introduced to more accurately reflect the actual decision environment. The theoretical framework of this method aims to deepen the understanding of efficiency assessment and provide decision makers with more comprehensive and actionable information to formulate and implement optimization strategies more effectively.

The rest of this paper is structured as follows. Section 2 introduces the basic knowledge of the DEA model and possibility theory. Section 3 introduces the innovations of this paper and the principles of model improvement, and clarifies the properties and steps of the improved model. Section 4 compares the improved model with existing models and applies it to the efficiency evaluation of airlines. Section 5 summarizes the theoretical models and case studies of this paper.

2. Materials and Methods

2.1. DEA Model

Efficiency is the degree to which resources are applied efficiently given conditions such as the inputs and technology. In the efficiency evaluation of a DEA model, each DMU has multiple input and output indicators, and the goal of the DEA is to find an optimal combination of weights that allows each DMU to maximize its input and output indicators to achieve the highest efficiency.

The DEA model, proposed by Charnes et al. [43], was designed to assess the efficiencies of similar DMUs. Assuming there are n DMUs, each DMU uses m inputs to produce s outputs, where x_ij represents the i-th input of the j-th DMU (j = 1, 2, …, n; i = 1, 2, …, m), u_ij denotes the weight of the corresponding input variable, y_rj represents the r-th output of the j-th DMU (r = 1, 2, …, s), and v_rj denotes the weight of the output variable. For the evaluated DMU_d, the foundational evaluation model of the DEA can be expressed as

(1)$$\begin{gathered} max\mathrm{}{\displaystyle \frac{\sum _{r=1}^s{v}_{rd}{y}_{rd}}{\sum _{i=1}^m{u}_{id}{x}_{id}}} \\ {\displaystyle \frac{\sum _{r=1}^s{v}_{rk}{y}_{rk}}{\sum _{i=1}^m{u}_{ik}{x}_{ik}}}\le 1,\mathrm{}\mathrm{k}=1,\mathrm{}2,\mathrm{}\dots ,\mathrm{}n \\ {v}_{rk},u_{ik}\ge 0 \end{gathered}$$

Linearizing the traditional model, the input-oriented DEA model can be expressed as

(2)$$\begin{gathered} max\sum _{r=1}^s{v}_{rd}{y}_{rd} \\ {\sum }_{i=1}^m{u}_{ik}{x}_{ik}=1 \\ \sum _{r=1}^s{v}_{rk}{y}_{rk}-\sum _{i=1}^m{u}_{ik}{x}_{ik}\le 0,k=\mathrm{1,2},\dots ,n \\ {v}_{rk},u_{ik}\ge 0 \end{gathered}$$

The efficiency values of models (1) and (2) both lie within the interval [0,1]. DMUs with efficiency values less than 1 are defined as inefficient DMUs, while DMUs with efficiency values equal to 1 are defined as efficient DMUs. However, the traditional model cannot distinguish the difference between efficient DMUs, which is a typical limitation of the traditional model.

2.2. Full-Ranking Models

The super-efficiency DEA model is a typical model used to achieve a full ranking in DEA. The specific model is as follows:

(3)$$\begin{gathered} max{\displaystyle \frac{\sum _{r=1}^s{v}_{rd}{y}_{rd}}{\sum _{i=1}^m{u}_{id}{x}_{id}}} \\ {\displaystyle \frac{\sum _{r=1}^s{v}_{rk}{y}_{rk}}{\sum _{i=1}^m{u}_{ik}{x}_{ik}}}\le 1,\mathrm{}k=1,\mathrm{}2,\mathrm{}\dots ,\mathrm{}n,\mathrm{}k\ne d \\ {v}_{rk},\mathrm{}{\mathrm{u}}_{ik}\ge 0 \end{gathered}$$

The super-efficiency model achieves the full ranking of DMUs by removing the evaluated DMU from the reference set. However, the frequently changing frontier causes the model to be unstable.

From the perspective of the frontier, the reason that efficient DMUs cannot be ranked is that they are located on the frontier, which results in an efficiency score of 1 for all efficient DMUs. One approach to address this issue is to shift the frontier so that all DMUs are positioned within it, thus achieving a full ranking of the DMUs.

From Figure 1, it is evident that after a translation, the efficiency of all decision units is less than 1. The efficiency evaluation model of the virtual frontier can be represented as

(4)$$\begin{gathered} \mathit{Max}{\sum }_{r=1}^s{v}_{rd}{y}_{rd} \\ {\sum }_{i=1}^m{u}_{ik}{x}_{ik}=1 \\ \sum _{r=1}^s{v}_r{y}_{rk}-\sum _{i=1}^m{u}_i{x}_{ik}\le 0,k=1,2,\dots ,n,k\in \Omega \\ d\in \mathrm{Ф} \\ {v}_r,u_i\ge 0 \end{gathered}$$

Ф represents the set of evaluated decision units, and Ω represents the set of virtual reference units. According to model (4), it is known that the evaluated unit set and the reference unit set are two completely different sets.

Ghasemi [44] proposed a new method to address ranking problems, considering them from a relaxation perspective. Decision units are ranked based on relaxation criteria. The model is shown as follows:

(5)$$\begin{gathered} \mathit{min}{d}_o \\ \sum _{i=1}^m u_i{x}_{io}=1, \\ \sum _{r=1}^s u_r{y}_{rj}-\sum _{i=1}^m u_i{x}_{ij}+d_j-c_o=0,j=1,\dots ,n, \\ {u}_r\ge 0,r=1,\dots ,s, \\ {v}_i\ge 0,i=1,\dots ,m, \\ {d}_j\ge 0,j=1,\dots ,n \\ {c}_{o}freeinsign \end{gathered}$$

From model (5), the optimal solution for the deviation variables of each decision unit can be obtained and expressed as ${(d_1,d_2,\dots ,d_n)}_{n_1},$…$,{(d_1,d_2,\dots ,d_n)}_{n_k}$. Using the value ${\displaystyle \frac{d_1+d_2+\dots {+d}_n}{n}}$, the decision units are fully ranked based on their performances.

The above model is a common method in existing research for ranking efficient decision units. Different methods have their own characteristics and limitations. This study conducted a thorough comparative analysis between the above model and the method proposed in this study through case studies.

2.3. Possibility Theory

Interval analysis has been studied for several decades [45], where real numbers can be regarded as a special case with equal upper and lower bounds in interval numbers. In interval analysis, how to rank two or more interval numbers is a crucial problem, and various methods were proposed. These methods mainly fall into two categories: probability-based possibility formulas and area-based possibility formulas. Assuming there are two uncertain intervals A = [a_l, a_r] and B = [b_l, b_r], the comparison between intervals A and B can be expressed in the following scenarios.

Nakahara et al. [33], based on probability and integral theory, proposed possibility formulas:

(6)$P\left(A\le B\right)={\displaystyle \frac{\omega -a_l}{a_r-a_l}}+{\displaystyle \frac{v-\omega }{{(a}_r-a_l)\cdot {(b}_r-b_l)}}\left(b_r-{\displaystyle \frac{v+\omega }{2}}\right)$

where $\nu =max\left\{min\left\{a_r,b_r\right\},a_l\right\}$ and $\omega =min\left\{max\left\{a_l,b_l\right\},a_r\right\}$. Similarly, there are multiple expressions for possibility formulas:

(7)$P\left(A\ge B\right)=min\left\{max\left({\displaystyle \frac{a_r-b_l}{w\left(A\right)+w\left(B\right)}},0\right),1\right\}$

(8)$P\left(A\ge B\right)={\displaystyle \frac{max\left\{0,w\left(A\right)+w\left(B\right)-max\left(b_r-a_l,0\left)\right\}\right.\right.}{w\left(A\right)+w\left(B\right)}}$

(9)$P\left(A\ge B\right)={\displaystyle \frac{min\left\{w\left(A\right)+w\left(B\right),max\left(a_r-b_l,0\right)\right\}}{w\left(A\right)+w\left(B\right)}}$

(10)$P\left(A\ge B\right)={\displaystyle \frac{max\left(0,a_r-b_l\right)-max\left(0,a_l-b_r\right)}{w\left(A\right)+w\left(B\right)}}$

(11)$P\left(A\ge B\right)=min\left\{max\left({\displaystyle \frac{m\left(A\right)-m\left(B\right)}{w\left(A\right)+w\left(B\right)}}+{\displaystyle \frac{1}{2}},0\right),1\right\}$

The above formulas were proposed by Facchinetti et al. [34], Da and Liu [46], Xu and Da [47], Wang et al. [48], and Sun and Yao [49]. These formulas are based on probability. Additionally, Fang Liu et al. [50] proposed a more concise possibility formula from a graphical perspective (Figure 2):

(12)$P\left(A\ge B\right)={\displaystyle \frac{S^{\prime }}{S^{\prime }+S^{″}}}$

Model (12) is a general expression for P(A ≥ B). The straight line y=x divides the rectangle into two parts, and it is easy to see that P(A ≥ B) can be expressed in terms of S′, and the probability of S′ occurring is expressed in terms of area as the ratio of the area of S′ to the total area. Similarly, P(B ≥ AA) can be expressed as S″/(S′ + S″) or 1 − S′/(S′ + S″).

The abovementioned possibility formulas all provide comparisons of interval numbers from different perspectives and are commonly used in interval number comparisons. However, they do not take into account the preferences of decision makers. Liu et al. [42], based on the aforementioned possibility formula, proposed a generalized possibility function that incorporates subjective preferences. The possibility formula can be expressed as $P(A\ge B)={\displaystyle \frac{{\int }_{b_l}^{a_r} f(x)dx}{{\int }_{b_l}^{a_r} f(x)dx+{\int }_{a_l}^{b_r} f(x)dx}}$, where f(x) is the attitude function, which is used to reflect the decision maker’s attitude and thoughts toward the solutions. However, the selection of the attitude function is typically limited to 1/x, 1/$\sqrt{x}$, c, $\sqrt{x}$, or x, which is overly restrictive and does not consider the impact of probability on the ranking. Therefore, based on the analysis of existing possibility formulas, this paper proposes an improved possibility formula for ranking interval efficiency in Section 3 to address the abovementioned shortcomings.

3. Improved Model

3.1. Interval DEA Improvement Model

Based on the principles of the interval DEA model, we can consider the lower bound of interval efficiency as the worst performance of the DMUs, and the upper bound of interval efficiency as the best performance of the DMUs. After obtaining the efficiency of the best- and worst-performing DMUs, we can use the possibility evaluation model for the final efficiency ranking.

The evaluation principle of traditional models is to optimize oneself, selecting the weights that yield the best performance without considering the efficiencies of other DMUs. This is a self-centered model evaluation approach. Therefore, the efficiency calculated by traditional models represents the best performance of DMUs. The model can be expressed as

(13)$$\begin{gathered} max\sum _{r=1}^s{v}_{rd}{y}_{rd} \\ {\sum }_{i=1}^m{u}_{ik}{x}_{ik}=1 \\ \sum _{r=1}^s{v}_{rk}{y}_{rk}-\sum _{i=1}^m{u}_{ik}{x}_{ik}\le 0,k=\mathrm{1,2},\dots ,n \\ {v}_{rk},u_{ik}\ge 0 \end{gathered}$$

The efficiency of the worst-performing DMU can be considered from the perspective of the opposite of self-interest, namely, the non-self-interest principle [51]. When constructing a non-self-interest model, considering the literal meaning of “non-self-interest”, the goal of DMU evaluation is not to optimize its own efficiency but to optimize the efficiencies of the other DMUs. This is the approach to constructing a non-self-interest DEA model. The model can be expressed as

(14)$$\begin{gathered} max\sum _{r=1}^s{v}_{rj}{y}_{rj},j\ne d \\ {\sum }_{i=1}^m{\mathrm{u}}_{ik}{x}_{ik}=1 \\ \sum _{r=1}^s{v}_{rk}{y}_{rk}-\sum _{i=1}^m{\mathrm{u}}_{ik}{x}_{ik}\le 0,k=\mathrm{1,2},\dots ,n \\ {v}_{rk},{\mathrm{u}}_{ik}\ge 0 \end{gathered}$$

Model (14) is a non-self-interest DEA model, where the objective of the evaluation is to optimize all DMUs except for the evaluated DMU_d. Considering that DMUs evaluated by the DEA model are homogeneous and of the same type, it is understood that all DMUs are equally significant during evaluation. Therefore, the expression of n − 1 objective functions can be rewritten as $max\sum _{r=1}^s{v}_{r1}{y}_{r1}+max\sum _{r=1}^s{v}_{r2}{y}_{r2}+\dots +max\sum _{r=1}^s{v}_{rd-1}{y}_{rd-1}++max\sum _{r=1}^s{v}_{rd+1}{y}_{rd+1}+\dots +max\sum _{r=1}^s{v}_{rn}{y}_{rn}$. By merging identical terms, the objective function can be obtained as $max\sum _{k=1,k\ne d}^n\sum _{r=1}^s{v}_{rk}{y}_{rk}$. The rewritten non-self-interest model can be represented as

(15)$$\begin{gathered} max\sum _{k=1,k\ne d}^n\sum _{r=1}^s{v}_{rk}{y}_{rk} \\ {\sum }_{i=1}^m{\mathrm{u}}_{ik}{x}_{ik}=1 \\ {\sum }_{r=1}^s{v}_{rk}{y}_{rk}-{\sum }_{i=1}^m{\mathrm{u}}_{ik}{x}_{ik}\le 0, k=\mathrm{1,2},\dots ,n \\ {v}_{rk},{\mathrm{u}}_{ik}\ge 0 \end{gathered}$$

The results obtained from model (15) represent the efficiencies of all DMUs, except for the evaluated DMU. On this basis, a system of linear equations can be established to calculate the efficiency values of DMUs evaluated using the non-self-interest principle.

Let the efficiency intervals obtained from models (13) and (15) be denoted as ${[{\theta }_l,{\theta }_u]}_{{DMU}_1}$, ${[{\theta }_l,{\theta }_u]}_{{DMU}_2}$, …, ${[{\theta }_l,{\theta }_u]}_{{DMU}_n}$. With the known efficiency intervals, the comparison and ranking of interval efficiencies can be conducted using the possibility model.

3.2. Improving DEA Model with Possibility Theory

The research in the field of interval ranking is extensive and can be divided into two categories: order-based interval number ranking methods and degree-based interval number ranking methods. Order-based interval ranking typically selects a representative number to signify the interval, achieving interval ranking by ordering these representative numbers. In other words, this method converts intervals into deterministic values, which may result in significant data loss and overlook the impact of the interval length, thus providing a partial view. Degree-based ranking methods determine the ranking of interval numbers by measuring the extent to which one interval number is greater than another using models such as possibility, credibility, and acceptability. Among these, the possibility measure has been the most extensively studied and widely applied method and has yielded the most abundant results. However, existing possibility ranking models use fixed values or functions to represent preferences when addressing preference issues, which does not align with real-world situations. In light of this, our paper adopts the possibility method for ranking efficiency intervals and proposes a new method for constructing attitude functions to better reflect actual conditions.

In this section, we use DMUs A and B as examples to improve the possibility formula. The interval efficiencies of DMUs A and B are represented as [${\theta }_l^a,{\theta }_u^a$] and [${\theta }_l^b,{\theta }_u^b$], respectively. Their efficiency relationships can be classified into the following six scenarios, as shown in Figure 3.

In Nakahara et al., 1992 [33], the possibility formula is obtained through integration as $P\left(A\le B\right)={\displaystyle \frac{\omega -a_l}{a_r-a_l}}+{\displaystyle \frac{v-\omega }{{(a}_r-a_l)\cdot {(b}_r-b_l)}}\left(b_r-{\displaystyle \frac{v+\omega }{2}}\right)$. In the solving process, it is assumed in advance that the probability of each number in the interval being chosen is equal; hence, it is solved as an example of a uniform distribution, which represents a neutral possibility formula. However, in real-life situations, there may inevitably be certain decision preferences, which can be represented by an attitude function f(x). Therefore, the possibility formula with decision preferences can be expressed as follows:

1. When the interval relationship is as shown in Figure 3a:

(16)$p\left(B\ge A\right)=1$

2. When the interval relationship is as shown in Figure 3b:

(17)$p\left(B\ge A\right)={\displaystyle \frac{{\int }_{{\theta }_l^a}^{{\theta }_l^b}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}$

3. When the interval relationship is as shown in Figure 3c:

(18)$p\left(B\ge A\right)={\displaystyle \frac{{\int }_{{\theta }_u^a}^{{\theta }_u^b}f\left(x\right)dx}{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}}+{\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^a}f\left(x\right)dx}{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}}\ast {\displaystyle \frac{{\int }_{{\theta }_l^a}^{{\theta }_l^b}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^a}f\left(x\right)dx}{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}}\ast {\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^a}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}$

where f(x) is an attitude function defined on (0, +∞) that reflects the decision maker’s attitude toward alternative options. When the interval relationships are as shown in Figure 3d–f, the above formula can be referenced, and individual explanations are no longer provided.

To further illustrate the attitude function graphically, Figure 4 corresponds to the six different situations mentioned above.

1. When the interval relationship is as shown in Figure 4a:

(19)$p\left(B\ge A\right)=1$

2. When the interval relationship is as shown in Figure 4b:

(20)$p\left(B\ge A\right)={\displaystyle \frac{S_1}{S_1+S_2+S_3}}+{\displaystyle \frac{S_2}{2\left(S_1+S_2+S_3\right)}}$

3. When the interval relationship is as shown in Figure 4c:

(21)$p\left(B\ge A\right)={\displaystyle \frac{S_3}{S_2+S_3}}+{\displaystyle \frac{S_2}{S_2+S_3}}\ast {\displaystyle \frac{S_2}{S_1+S_2}}+{\displaystyle \frac{S_2}{2\left(S_2+S_3\right)}}\ast {\displaystyle \frac{S_2}{\left(S_1+S_2\right)}}$

When the interval relationships are as shown in Figure 4d–f, the formulas mentioned earlier can be referenced accordingly. When p(A) = p(B), since $S_1=S_2$, then $p\left(B\ge A\right)=0.5$.

The new possibility formulas corresponding to the six interval relationships have been provided. However, the attitude function f(x) has not been determined yet. From the definition of the attitude function, it is known that when f(x) increases, the decision maker is optimistic, while when f(x) decreases, the decision maker is pessimistic. When f(x) is a constant, the decision maker is neutral, without any decision preference. We can determine the decision maker’s preferences based on the variation in f(x), and the choice of function has a significant impact on the interval ordering. Liu et al. [42] directly specified the attitude function as 1/x, 1/$\sqrt{x}$, c, $\sqrt{x}$, or x, which represent five common increasing, decreasing, and constant functions but may not reflect subtle differences between preferences. To avoid this, in this study, f(x) was determined through data-based fitting.

The newly proposed possibility formulas are for sorting the interval efficiencies of DEA. Therefore, we can predict the next stage of data from the perspective of existing data. The predicted data represent the preference data for this stage.

Taking the efficiencies of DMUs a and b, namely, [${\theta }_l^a,{\theta }_u^a$] and [${\theta }_l^b,{\theta }_u^b$], as an example, assume that the above efficiencies represent the interval efficiencies of DMUs for a certain year. Then, the decision maker certainly hopes for an improvement in efficiency in the next year, that is, a reduction in inputs and an increase in outputs. Assuming the inputs decrease by a% and the outputs increase by b% in the next stage, the reduced inputs and increased outputs are defined as predicted inputs and predicted outputs, which represent the efficiencies the decision maker hopes to achieve in the next stage. In the current stage, a series of measures will be taken to make the efficiency infinitely close to the predicted inputs and outputs, which represent the decision preference for the current stage. By fitting the predicted data, the fitted function serves as the attitude function f(x) for this period, which avoids the robustness and subjectivity brought about by directly specifying the attitude function.

3.3. Model Steps and Properties

The model presented in this paper combines the strengths of DEA and possibility theory. Both the DEA model and the possibility theory model are individually enhanced and integrated into a unified framework. The theoretical model framework can be illustrated as shown in Figure 5.

The specific evaluation steps are as follows:

Step 1: Calculate the self-interest efficiency and non-self-interest efficiency using models 13 and 15 and represent them in the form of interval efficiencies.

Step 2: Calculate the predicted efficiencies based on the existing interval efficiencies.

Step 3: Fit the predicted efficiencies to obtain the attitude function f(x).

Step 4: Utilize the newly proposed possibility model in this paper, combined with the corresponding interval relations, to derive the possibility matrix.

Step 5: Sort the possibility matrix to obtain the final efficiency full ranking. When ranking the possibility matrix, the comprehensive possibility method or the ordinal priority number method can be used to determine the ranking results. Specifically, it can be expressed as follows:

(22)$P_j^{*}={\sum }_{k=1}^n{P}_{jk}^{*}$

(23)${\lambda }_{jk}=\left\{\begin{array}{cc}0,& P_{jk}^{*}\le 0.5\\ 1,& P_{jk}^{*}>0.5\end{array}\right.$

(24)${\lambda }_j={\sum }_{k=1}^n{\lambda }_{jk}$

The possibility model proposed in this paper is an improvement based on existing possibility theory and possesses the following properties:

Property 1.

0 < $p\left(B\ge A\right)$< 1.

Property 2.

When ${\theta }_u^a\le {\theta }_l^b$, $p\left(B\ge A\right)=1$.

Property 3.

When${\theta }_u^b\le {\theta }_l^a$, $p\left(B\ge A\right)=0$.

Property 4.

Transitivity: if A > B and B > C, then A > C.

Proof of Property 4.

Since A > B and B > C, we have $p\left(A\ge B\right)\ge 0.5$ and $p\left(B\ge C\right)\ge 0.5$, implying ${\displaystyle \frac{A_l+A_u}{2}}\ge {\displaystyle \frac{B_l+B_u}{2}}$ and ${\displaystyle \frac{B_l+B_u}{2}}\ge {\displaystyle \frac{C_l+C_u}{2}}$. Therefore, ${\displaystyle \frac{A_l+A_u}{2}}\ge {\displaystyle \frac{C_l+C_u}{2}}$, which leads to A > C. □

Property 5.

P(A ≥ B) + P(B ≥ A) = 1.

Proof of Property 5.

Taking Figure 4c as an example, $p\left(B\ge A\right)={\displaystyle \frac{{\int }_{{\theta }_u^a}^{{\theta }_u^b}f\left(x\right)dx}{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}}+{\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^a}f\left(x\right)dx}{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}}\ast {\displaystyle \frac{{\int }_{{\theta }_l^a}^{{\theta }_l^b}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^a}f\left(x\right)dx}{{\int }_{{\theta }_l^b}^{{\theta }_u^b}f\left(x\right)dx}}\ast {\displaystyle \frac{{\int }_{{\theta }_l^b}^{{\theta }_u^a}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}$. This is equivalent to $p\left(B\ge A\right)={\displaystyle \frac{S_3}{S_2+S_3}}+{\displaystyle \frac{S_2}{S_2+S_3}}\ast {\displaystyle \frac{S_2}{S_1+S_2}}+{\displaystyle \frac{S_2}{2(S_2+S_3)}}\ast {\displaystyle \frac{S_2}{(S_1+S_2)}}$. Similarly, $p\left(A\ge B\right)$ can be represented as $p\left(A\ge B\right)={\displaystyle \frac{S_2}{2(S_2+S_3)}}\ast {\displaystyle \frac{S_2}{(S_1+S_2)}}$. Thus, it can be deduced that $p\left(A\ge B\right)+p\left(B\ge A\right)=1$. The same applies for the other five cases. □

Property 6.

Scalar multiplication preserves order: for any k > 0, if A > B, then Ka > kB.

Proof of Property 6.

Since A > B, then $p\left(A\ge B\right)\ge 0.5$, which implies ${\displaystyle \frac{A_l+A_u}{2}}\ge {\displaystyle \frac{B_l+B_u}{2}}$. Because k > 0, it follows that ${\displaystyle \frac{{k(\mathrm{A}}_l+{\mathrm{A}}_u)}{2}}\ge {\displaystyle \frac{k(B_l+B_u)}{2}}$. Therefore, kA > kB. □

Property 7.

Additive preservation of order: if A > B, then A + C > B + C.

Proof of Property 7.

A + C = [$A_l+C_l$, $A_u+C_u$] and B + C = [$B_l+C_l$,$B_u+C_u$]. Since A > B, then $p\left(A\ge B\right)\ge 0.5$, implying ${\displaystyle \frac{A_l+A_u}{2}}\ge {\displaystyle \frac{B_l+B_u}{2}}$ and ${\displaystyle \frac{A_l+A_u+C_l+C_u}{2}}\ge {\displaystyle \frac{B_l+B_u+C_l+C_u}{2}}$. Therefore, A + C > B + C. □

When ${\theta }^a$ and ${\theta }^b$degenerate into a real number, such as when ${\theta }^b$ is a real number, the relationship between ${\theta }^a$ and ${\theta }^b$ can be represented as shown in Figure 6.

The definition of the possibility degree is as follows:

1. When the interval relationship is as shown in Figure 6a:

(25)$p\left(B\ge A\right)=1$

2. When the interval relationship is as shown in Figure 6b:

(26)$p\left(B\ge A\right)={\displaystyle \frac{{\int }_{{\theta }_l^a}^{{\theta }^b}f\left(x\right)dx}{{\int }_{{\theta }_l^a}^{{\theta }_u^a}f\left(x\right)dx}}$

3. When the interval relationship is as shown in Figure 6c:

(27)$p\left(B\ge A\right)=0$

When both ${\theta }^a$ and ${\theta }^b$ degenerate into real numbers, such as when both ${\theta }^a$ and ${\theta }^b$ are real numbers, the comparison between DMU_a and DMU_b is a comparison of ordinary real numbers.

4. Illustrations

In this chapter, numerical examples and specific instances are used to verify the feasibility and rationality of the method proposed in this paper.

4.1. Numerical Examples

As discussed in this section, numerical examples were used to illustrate the effectiveness of the improved model proposed in this paper and compare it with existing methods. This comparison served to test whether the proposed method achieved comparable or superior performance to the comparison models. Data from Wong and Beasley [52] were used for the subsequent comparative analysis.

Based on the data in Table 1 and the models, the interval efficiencies of each DMU were calculated first using the selfish and non-selfish models, as shown in Table 2.

Table 2 presents the results of the selfish and non-selfish models. The efficiencies shown in the table are relative efficiencies. Due to the lack of a 0–1 interval constraint in the non-selfish model, there was a phenomenon where the efficiency values were greater than those of the selfish model. The interval efficiencies normalized for the non-selfish model are shown in Table 3.

Table 3 presents the interval efficiency values. From the table, it can be observed that the interval upper and lower bounds for DMU₅ were both 1, indicating a case where the interval degenerated into a real number. A rough assessment could be made based on Table 3. For instance, the efficiency of DMU₅ should be greater than that of all other DMUs. Additionally, as the lower bound of the interval efficiency for DMU₇ was 0.9896, which was greater than the upper bound of the interval efficiency for DMU₄ (0.82), the efficiency of DMU₇ should be greater than that of DMU₄. However, it was not possible to make a comparison between DMU₄ and DMU₆, as their interval efficiencies overlapped. The sorting of interval efficiencies was determined using the improved possibility degree model proposed in this paper.

Figure 7 depicts the fitting results of the predicted output relative to the predicted input. The obtained fitting functions served as the attitude functions for the current stage. The prediction performance and the fitting functions for each scenario are shown in Figure 7.

In the evaluation of the DEA model in this paper, all inputs and outputs were considered equally important. This implies that the statuses of the nine fitting functions mentioned above were the same. To determine the final attitude function, this study took the sum of each fitting function’s parameters multiplied by one-ninth. Thus, the final attitude function was given by f(x) = −0.0204x² + 2.8545x − 8.1161. It was increasing within the input range, indicating relative optimism on the part of the decision maker given the current input–output situation.

The possibility degree matrix obtained using the improved Formulas (13)–(15) and the results of row-wise summation are presented in Table 4.

In Table 4, the diagonal elements are all 0.5, which indicates that in comparison with themselves, the DMUs had equal degrees of superiority or inferiority. The possibility values corresponding to DMU₅ were all 0 in the columns and all 1 in the rows, indicating that DMU₅ performed the most outstandingly in terms of efficiency. By summing the rows of the possibility matrix, it can be observed that DMU₅ was in the most favorable position overall, further confirming that DMU₅ had interval efficiencies with upper and lower bounds both equal to 1, making it the most efficient. Conversely, DMU₁ had the lowest ranking, which was attributed to its excellent selfish efficiency but lowest non-selfish efficiency; this corresponded to the phenomenon in real life where individuals pursue personal development while neglecting cooperation and engaging in malicious competition between DMUs.

Comparisons of the ranking results between our approach and existing literature methods are presented in Table 5.

It can be observed from Table 5 and Figure 8 that the ranking trends of the four methods were generally consistent, although there were some differences in the rankings of individual DMUs across different methods. When evaluating with the CCR model, except for DMU₄, all other DMUs were deemed efficient and could not be ranked. From this perspective, DMU₄ should be the least-efficient DMU. However, in the method of shifting the frontier, as well as in the approach combining the possibility ranking, DMU₄ was not ranked last in terms of efficiency. This could be attributed to the way the shifting frontier method separated the evaluation set and the reference set, ensuring that the efficiency values of the evaluation set were all lower than those of the reference set, thus resulting in all evaluated DMUs having efficiency values less than 1. When constructing the reference set, the method used x_io = min{x_ij} and yio = max{x_rj}, and the input–output pairs in the reference set were randomly generated within the intervals [0.95x_io, x_io] and [y_io,1.05y_io]. This led to a slight deviation of the virtual frontier compared with the original frontier, resulting in an improvement in the efficiency of DMU₄. In the approach that combines the possibility theory, the consideration of the non-self-interest principle on top of the original CCR model suggests that although DMU₄ had the lowest efficiency in the selfish CCR model, it performed well in the non-self-interest model, leading to an increase in its final ranking.

Although the super-efficiency method removed the evaluated DMUs from the reference set to obtain a complete ranking of the DMUs, it also introduced the drawback of frequent changes in the frontier, leading to unstable results. It was evident that both the virtual frontier model and the super-efficiency model had significant shortcomings. In contrast, the proposed model improved the model stability while considering the non-self-interest principle and integrating the performances of the other DMUs. Furthermore, it incorporated the concept of development and took into account the preferences of decision makers at the current stage, which resulted in a more comprehensive and thoughtful approach.

4.2. Airline Efficiency Evaluations

In order to validate the effectiveness of the proposed improved model in practical applications, we took 53 airlines in 2020 as an example (excluding the entire freight company) and applied the DEA improved model that combines possibility theory to evaluate the efficiency of airlines. The data source was “From the statistical view of Civil Aviation”.

Indicators for scientific evaluation are the prerequisite for efficiency evaluation, different indicators will have an important impact on the results, and there are several principles for constructing indicators: (1) Principle of representativeness, where the constructed indicator system can reflect the operation of the DMUs. (2) Suitability with the evaluation method. With different evaluation methods, the constructed indicators should be adjusted. If the carbon emissions reduction efficiency of airlines is measured, carbon emissions should be taken as one of the output indicators. Indicator selection and evaluation methods are compatible with each other in order to maximize the role of evaluation methods. The selection of airline indicators in the existing literature is shown in Table 6.

For airlines, their main task is the efficient and timely transport of passengers and goods. For customers choosing civil aviation transport, one of the reasons is the speed and efficiency of air travel. To reflect the time advantage of civil aviation transport, flight hours are considered as an input, and another input is the number of takeoffs (aircraft). In the existing literature on efficiency evaluation in civil aviation, the number of aircraft is mostly used as an input. However, the number of takeoffs, compared with the total number of aircraft, excludes idle aircraft. The utilization rate of aircraft by airlines is not 100%, and for idle aircraft, only maintenance and care costs are incurred. Therefore, this study considered the number of takeoff aircraft in operation as an input. In terms of the outputs, considering the main tasks of airlines, the passenger volume and cargo volume were chosen as output indicators. In civil aviation transport, different passengers have different destinations, and when transporting goods, the routes and starting points also vary. Therefore, two additional indicators, passenger turnover and cargo turnover, were added to the cargo volume and passenger volume indicators. The six indicators were selected for the efficiency evaluation, which were split into two input indicators: takeoff frequency (unit: aircraft) and flight hours (unit: hours), and four output indicators: passenger transport volume (unit: individuals), passenger turnover (unit: 10,000 passenger-kilometers), cargo and mail transport volume (unit: tons), and cargo and mail turnover (unit: 10,000 ton-kilometers). Table 7 provides explanations of the indicator meanings.

Since the non-selfish principle has no 0–1 restriction, the non-selfish efficiency obtained by the model is greater than the selfish efficiency. The model values obtained in this study were relative values and not absolute. When calculating the interval efficiency, the lower bound of the efficiency should be normalized to satisfy the requirements of interval upper and lower bounds. The interval efficiency before and after the normalization is shown in Table 8.

In Table 8, the meanings of the efficiency values before and after normalization were the same, while only the scales differed. The upper limit of the interval was obtained through the selfish CCR model. From the table, it can be seen that Urumqi Air, 9 Air, Suparna Airlines, West Air, Beijing Capital Airlines, Spring Airlines, and China Southern Airlines each had an interval upper efficiency value of 1, indicating highly efficient airlines. The CCR model lacked the ability to distinguish between efficient airlines, as from the perspective of the selfish CCR model, there was no distinction between these highly efficient airlines. Joyair had an efficiency value of 0.421488, making it the least efficient airline. Further research revealed that Joyair has consistently had low operational efficiency since its establishment. This phenomenon can be considered in terms of the context of Joyair’s founding purpose. Joyair was approved for establishment in 2008, with the aim of increasing the proportion of regional flights and serving as a signal to promote the civil aircraft manufacturing industry and the development of regional aviation. However, the development of regional aviation and domestically manufactured aircraft in China has faced significant challenges, which has directly impacted the efficiency of Joyair’s development.

The lower limit of the interval was obtained through the non-selfish model (12). China Southern Airlines had the highest efficiency value among the airlines, indicating its ability to utilize resources more effectively to achieve a higher efficiency. This, coupled with a strong sense of cooperation, further aided in attracting customers. In the non-selfish model, Joyair had a relatively higher efficiency, further explaining its goal of increasing the proportion of regional aviation and promoting the development of domestically manufactured aircraft.

Based on the known efficiency intervals, it was necessary to rank the different efficiency intervals to better reflect the differences in potential efficiencies between the DMUs. Using the improved possibility algorithm proposed in this paper to rank the interval efficiencies, the fitting of different outputs relative to inputs is illustrated in Figure 9.

In this example, all the outputs and weights were equally important. The eight scenarios depicted in Figure 9 were assigned equal weights of one-eighth each. The final attitude function was determined to be f(x) = 25.0133x + 12712.867. It was an increasing function within the range of inputs, indicating that the decision makers held an optimistic attitude toward future development given the current input–output conditions. Based on the known attitude function, the partial airline possibility matrix obtained using the proposed improved possibility ranking model in this paper is shown in Table 9.

The final ranking could be obtained by cumulatively adding the possibility matrix row by row. The final ranking is shown in Table 10.

Table 10 presents the final ranking results by incorporating both the selfish and non-selfish principles. From the table, it can be observed that China Southern Airlines was ranked first, indicating that it maintained good efficiency, both in terms of its own development and the overall development, making it the benchmark enterprise in the aviation industry. Joyair ranked at the bottom. Through the specific analysis, it was evident that Joyair had a low efficiency, and even though its efficiency improved when considering the selfish principle, overall, its efficiency remained low.

5. Conclusions

This paper considers the phenomenon of bias in weights caused by the traditional DEA model’s focus on optimizing itself during evaluation without considering the impact on other DMUs. By integrating the non-selfish principle and considering the overall development of all DMUs, efficiency is represented in the form of efficiency intervals, preserving more potential efficiency variations. When sorting the efficiency intervals, the knowledge of possibility theory is combined, taking into account the preferences of decision makers in reality. In contrast to other papers that directly define attitude functions, this paper adopts the concept of fitting the original data, focusing on the present, and fitting the input–output functions for the next stage as the attitude function of this stage, thereby adding a developmental perspective and improving possibility theory. The improved possibility theory is then used to rank DMUs, forming the theoretical research framework of this paper.

In the empirical analysis section, numerical examples from other studies are first cited and compared with existing methods, revealing that the proposed method achieved similar results to existing methods while providing more comprehensive consideration. This method was then applied to the efficiency evaluation of civil aviation in 2020, achieving a complete ranking of the airlines. Based on the model results, the reasons for the low efficiencies of airlines were analyzed, and improvement suggestions are provided.

Although this study combined DEA models and possibility theory, it still had some limitations: (1) The DEA model in this paper does not consider the internal structure of DMUs. (2) In the empirical analysis section, only airlines from 2020 were analyzed, without summarizing the development of airlines in recent years. Future research could combine the theoretical framework of this paper with various DEA branch models, such as network DEA, window DEA, and integrating the time factor to systematically analyze the development of airlines in recent years to better predict future trends in airline development.

Author Contributions

Conceptualization, S.Y. and G.Z.; methodology, S.Y.; software, S.Y. and F.L.; validation, S.Y.; formal analysis, S.Y. and G.Z.; investigation, S.Y. and F.L.; resources, S.Y.; data curation, S.Y., G.Z. and F.L.; writing—original draft preparation, S.Y. and F.L.; writing—review and editing, S.Y.; visualization, S.Y. and F.L.; supervision, S.Y. and G.Z.; funding acquisition, S.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this paper are openly available from the official websites of Chinese airlines and the Civil Aviation Administration of China (CAAC) database.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The principle diagram of the virtual frontier.

Figure 2. The principle diagram of possibility based on area.

Figure 3. Diagrams of six interval relationships. (a–f) represent six cases of interval relations for interval numbers A, B.

Figure 4. Diagrams of interval number relationships under six given f(x) scenarios. (a–f) represent the distributions of interval numbers A, B corresponding to different interval relationships.

Figure 5. Theoretical framework diagram.

Figure 6. Situation where one number degenerates into a real number. (a–c) represent three relationships between interval numbers and real numbers.

Figure 7. Fitting of outputs to inputs.

Figure 8. Four methods ranking comparison graph.

Figure 9. Fitting graphs of airline output relative to input.

Table 1

Data from Wong and Beasley.

DMU	x1	x2	x3	y1	y2	y3
1	12	400	20	60	35	17
2	19	750	70	139	41	40
3	42	1500	70	225	68	75
4	15	600	100	90	12	17
5	45	2000	250	253	145	130
6	19	730	50	132	45	45
7	41	2350	600	305	159	97

Table 2

Efficiency table under two scenarios.

DMU	1	2	3	4	5	6	7
Selfish model (13)	1	1	1	0.82	1	1	1
Non-selfish model (15)	−10.1524	1.336643	5.268163	−1.57278	7.280709	0.993284	7.099546

Table 3

The normalized interval efficiency table.

DMU	1	2	3	4	5	6	7
Interval efficiency	[0,1]	[0.6590,1]	[0.8846,1]	[0.4291,0.82]	[1,1]	[0.6393,1]	[0.9896,1]

Table 4

The table of possibility degrees.

DMU	1	2	3	4	5	6	7	Sum
1	0.5	0.146722	0.046936	0.301884	0	0.155947	0.00411	1.1556
2	0.853278	0.5	0.159949	0.883573	0	0.529575	0.014007	2.940381
3	0.953064	0.840051	0.5	1	0	0.849512	0.043786	4.186413
4	0.698116	0.116427	0	0.5	0	0.139267	0	1.453811
5	1	1	1	1	0.5	1	1	6.5
6	0.844053	0.470425	0.150488	0.860733	0	0.5	0.013179	2.838878
7	0.99589	0.985993	0.956214	1	0	0.986821	0.5	5.424918

Table 5

Comparison of rankings between four methods.

DMU	CCR	Shifted Frontier		Super-Efficiency		Ghasemi		The Proposed Method
	Eff. Values	Eff. Values	Rank	Eff. Values	Rank	Eff. Values	Rank	Rank
DMU1	1	0.20683	7	1.829615	1	0.218	7	7
DMU2	1	0.270049	2	1.048895	7	0.145	5	4
DMU3	1	0.197749	6	1.198308	5	0.805	6	3
DMU4	0.82	0.221479	5	0.819737	3	-	2	6
DMU5	1	0.250576	4	1.219992	6	0.049	1	1
DMU6	1	0.25645	1	1.190642	2	0.126	3	5
DMU7	1	0.277708	3	1.266094	4	0.045	4	2

Table 6

Indicator collation from the existing literature.

Representative Study	Input Indicators	Output Indicators
Ngo and Tsui [53]	Available seat-kilometers (ASK)Available tonne-kilometers (ATK)Operating expenses (EXPENSES)	Revenue passenger-kilometers (RPK)Revenue tonne-kilometers (RTK)Operating revenues (REVENUES)
Lee and Worthington [54]	The average number of employeesTotal assets in USDKilometers flown	Available tonne-kilometers (ATK)
Min and Joo [55]	UnderutilizationOperating expenses	PassengersRPKsOperating revenueService rating
Barbot et al. [56]	LaborCapital(the airline’s fleet)FuelOther operating inputs	Passenger serviceCargo serviceAncillary output

Table 7

Indicator explanations.

	Indicators	Explanation
Input indicators	Number of take-offs (unit: aircraft)	The aggregate number of aircraft sorties within a designated period of time.
	Flight hours (unit: hours)	The duration of the aircraft flying time.
Output indicators	Passenger volume (unit: individuals)	The actual number of passengers transported within a specific period.
	Passenger turnover (unit: 10,000 passenger-kilometers)	The passenger turnover reflects the total volume of passenger transportation work during a specific period.
	Cargo volume (unit: tons)	The freight transport volume represents the actual quantity of goods transported during a specific period.
	Cargo turnover (unit: 10,000 ton-kilometers)	The freight turnover reflects the total volume of goods transportation work during a specific period.

Table 8

Airlines efficiency interval table.

Airlines	Before Normalization		After Normalization
	Efficiency Upper Bound	Efficiency Lower Bound	Efficiency Upper Bound	Efficiency Lower Bound
China Southern Airlines	1	7605.413	0.421488	1
Air China	0.977608	7602.522	0.421484	0.977608
China Eastern Airlines	0.858185	7603.034	0.421485	0.858185
Sichuan Airlines	0.905655	7593.871	0.421475	0.905655
Xiamen Air	0.857561	7594.547	0.421476	0.857561
Shenzhen Airlines	0.816271	7595.956	0.421477	0.816271
Hainan Airlines	0.989872	7589.414	0.42147	0.989872
Spring Airlines	1	7588.01	0.421468	1
Shandong Airlines	0.892908	7589.532	0.42147	0.892908
Juneyao Airlines	0.899937	7577.694	0.421457	0.899937
Beijing Capital Airlines	1	7567.888	0.421446	1
Zhejiang Loong Airlines	0.864505	7564.184	0.421442	0.864505
China Eastern Airlines Yunnan Limited	0.738032	7573.442	0.421452	0.738032
Shanghai Airlines	0.774332	7568.362	0.421447	0.774332
China Eastern Airlines Jiangsu Limited	0.780659	7566.109	0.421444	0.780659
Lucky Air	0.957777	7549.363	0.421426	0.957777
Tianjin Airlines	0.825296	7558.067	0.421436	0.825296
West Air	1	7539.486	0.421415	1
Chengdu Airlines	0.825793	7549.955	0.421427	0.825793
Suparna Airlines	1	7454.055	0.421321	1
China Express	0.592403	7565.231	0.421443	0.592403
China Xinhua Airlines Group	0.8613	7526.507	0.421401	0.8613
China United	0.684278	7546.518	0.421423	0.684278
Hebei Airlines	0.836645	7516.756	0.42139	0.836645
China Southern Airlines Henan Limited	0.84781	7518.408	0.421392	0.84781
Tibet Airlines	0.785782	7520.074	0.421394	0.785782
Shenzhen Donghai Airlines	0.910002	7500.322	0.421372	0.910002
Kunming Airlines	0.829705	7508.146	0.421381	0.829705
9 Air	1	7484.573	0.421355	1
Qingdao Airlines	0.839825	7499.946	0.421372	0.839825
Okay Airways	0.944351	7477.43	0.421347	0.944351
Ruili Airlines	0.921448	7479.333	0.421349	0.921448
Chongqing Airlines	0.733775	7504.117	0.421376	0.733775
Air Guizhou	0.865709	7474.284	0.421344	0.865709
China Eastern Airlines Wuhan Limited	0.919896	7469.817	0.421339	0.919896
Guangxi Airlines	0.764911	7463.234	0.421331	0.764911
Urumqi Air	1	7330.962	0.421186	1
Zhuhai Airlines	0.792984	7418.66	0.421282	0.792984
Fuzhou Airlines	0.857159	7402.27	0.421264	0.857159
Shantou Airlines	0.821189	7420.001	0.421284	0.821189
Air Changan	0.951132	7358.437	0.421216	0.951132
Dalian Airlines	0.773927	7370.418	0.42123	0.773927
Air Travel	0.91094	7290.883	0.421142	0.91094
Jiangxi Airlines	0.855918	7303.761	0.421156	0.855918
Air China Inner Mongolia	0.820033	7350.895	0.421208	0.820033
Air Guilin	0.855987	7283.549	0.421134	0.855987
Colorful Guizhou Airlines	0.618158	7380.347	0.42124	0.618158
Grand China Air	0.924141	6267.371	0.420019	0.924141
Longjiang Airlines	0.804375	6227.493	0.419975	0.804375
Beijing Airlines	0.950829	6511.255	0.420286	0.950829
Joyair	0.421488	7020.024	0.420845	0.421488
Genghis Khan Airlines	0.541277	5500.45	0.419177	0.541277
One Two Three	0.50545	-376311	0	0.50545

Table 9

Airlines possibility matrix (selected airlines).

	China Southern Airlines	Air China	China Eastern Airlines	Sichuan Airlines	Xiamen Air	Shenzhen Airlines	Hainan Airlines
China Southern Airlines	0.5	0.519367	0.622623	0.58159	0.623171	0.658865	0.508774
Air China	0.480633	0.5	0.607416	0.56473	0.607986	0.645117	0.489219
China Eastern Airlines	0.377377	0.392584	0.5	0.450968	0.500726	0.548017	0.384124
Sichuan Airlines	0.41841	0.43527	0.549032	0.5	0.549687	0.592338	0.425888
Xiamen Air	0.376829	0.392014	0.499274	0.450313	0.5	0.547357	0.383566
Shenzhen Airlines	0.341135	0.354883	0.451983	0.407662	0.452643	0.5	0.347238
Hainan Airlines	0.491226	0.510781	0.615876	0.574112	0.616434	0.652762	0.5

Table 10

Final ranking situation.

Airlines	Possibility Accumulation	Ranking
China Southern Airlines	33.7404	1
Air China	32.96998	9
China Eastern Airlines	27.95349	25
Sichuan Airlines	30.1364	19
Xiamen Air	27.92244	26
Shenzhen Airlines	25.82609	38
Hainan Airlines	33.39712	8
Spring Airlines	33.73934	2
Shandong Airlines	29.57464	21
Juneyao Airlines	29.88564	20
Beijing Capital Airlines	33.73813	3
Zhejiang Loong Airlines	28.25687	23
China Eastern Airlines Yunnan Limited	21.53633	46
Shanghai Airlines	23.56419	43
China Eastern Airlines Jiangsu Limited	23.91198	42
Lucky Air	32.24218	10
Tianjin Airlines	26.29467	35
West Air	33.73641	4
Chengdu Airlines	26.31985	34
Suparna Airlines	33.73121	6
China Express	12.92083	50
China Xinhua Airlines Group	28.09983	24
China United	18.4483	48
Hebei Airlines	26.87465	32
China Southern Airlines Henan Limited	27.43652	30
Tibet Airlines	24.18898	41
Shenzhen Donghai Airlines	30.31785	18
Kunming Airlines	26.51913	33
9 Air	33.73307	5
Qingdao Airlines	27.03476	31
Okay Airways	31.72312	13
Ruili Airlines	30.80009	15
Chongqing Airlines	21.2903	47
Air Guizhou	28.30871	22
China Eastern Airlines Wuhan Limited	30.73475	16
Guangxi Airlines	23.03551	45
Urumqi Air	33.72369	7
Zhuhai Airlines	24.57313	40
Fuzhou Airlines	27.89012	27
Shantou Airlines	26.07175	36
Air Changan	31.97793	11
Dalian Airlines	23.52835	44
Air Travel	30.34448	17
Jiangxi Airlines	27.82282	29
Air China Inner Mongolia	26.00671	37
Air Guilin	27.82484	28
Colorful Guizhou Airlines	14.49835	49
Grand China Air	30.83279	14
Longjiang Airlines	25.10228	39
Beijing Airlines	31.91201	12
Joyair	1.358619	53
Genghis Khan Airlines	9.517831	51
One Two Three	1.570545	52

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