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Comparison Analysis of Airline Energy Efficiency Under Weak Disposability and Strong Disposability Using a Virtual Frontier Slack-Based Measure Model.

Abstract

In this article, the energy efficiency of airlines has been studied with number of employees, capital stock, and tons of aviation kerosene as the inputs and revenue tonne kilometers, revenue passenger kilometers, total business income, and C[O.sub.2] emission as the outputs. Two new models, Virtual Frontier SBM (slack-based measure) with strong disposability and Virtual Frontier SBM with weak disposability, are proposed to calculate the energy efficiencies of 22 airlines from 2008 to 2012. We prove two important properties of Virtual Frontier SBM, which can show its advantages over traditional model. The main findings are (1) Scandinavian Airlines has the largest average energy efficiency in the period both in strong disposability and weak disposability; and (2) C[O.sub.2] emission has significant impacts on energy efficiency change in strong disposability, while it has little impact in weak disposability.

Keywords

Virtual Frontier slack-based measure, airline energy efficiency, strong disposability, weak disposability, undesirable output

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In recent years, with the rapid development of the world economy and the improvement of the household consumption level, the gap between the demand and supply of energy has widened. According to the statistical data of the International Air Transport Association (IATA), in 2012, the total energy cost of all of the airlines in the world was more than $160 billion, and the carbon dioxide emission volume was more than 0.676 billion tons. The airline industry is one of the few sectors where energy consumption has increased at a rate of more than 6 percent over the past 10 years. However, energy production has lagged behind, increasing at less than 6 percent over the same period. The gap between the energy supply and demand is becoming more and more pronounced. Meanwhile, according to the Commercial Aircraft Corporation of China (http://www.comac.cc/xwzx/gsxw/201411/11/ t20i4iin_2075388.shtml) forecast for the coming 20 years, the revenue passenger kilometers (RPK) of the total aviation industry will increase by 4.8 percent a year, and the total passenger transport demand will be 2.6 times the current level. This huge demand for air transport will stimulate a much higher level of energy consumption. Furthermore, in 2014, the aviation industry produced approximately 2 percent of the total global C[O.sub.2] emission. Thus, the energy utilization problem of the airline industry has drawn great public attention. Energy efficiency is defined to reflect whether energy has been used efficiently (Clinch, Healy, and King 2001; Blomberg, Henriksson, and Lundmark 2012).

Moreover, the carbon dioxide emissions of the airline industry have attracted much attention. According to the statistical data of the IATA, in 2014 air transport was responsible for about 2 percent of manmade carbon emissions annually. Although this proportion is relatively small, the industry recognizes that it must work even harder on behalf of the environment to achieve long-term sustainability. Furthermore, the International Civil Aviation Organization predicts that in the absence of mitigation measures, driven by a sevenfold increase in air traffic, total greenhouse gas (GHG) emissions associated with aviation will be 400-600 percent higher in 2050 than in 2010. In this context, some policies have been proposed to control aircraft emissions to achieve the sustainable development of the airline industry. The European Union (EU) enacted the 2008/101/EC decree in November 2008, in which international airline business was brought into the European Union Emission Trading System (EU ETS). From January 1, 2012, each international flight taking off and landing in the European Union will be given an emission permit (Cui, Wei, and Li 2016).

This policy caused great controversy all over the world. In the face of the great diplomatic pressure, the European Union suspended the emission taxes of non-EU airlines and continued to apply it on the EU airlines. For another, the 39th Session of the ICAO in 2016 implemented the "carbon neutral growth from 2020" strategy to realize the air transport carbon-neutral growth. The cores of this strategy are integrating the market-based measures (MBM) into the overall strategy and determining the road for the airlines to share the abatement costs. Therefore, for airlines, the improvement of energy efficiency is not just the need of the production and business operation, but the requirement of environmental protection. The evaluation of energy efficiency for airlines should consider the carbon dioxide emissions, which can make the results more reasonable.

For the energy efficiency of airlines, Babikian, Lukachko, and Waitz (2002) analyzed the fuel efficiency of different aircraft types, and the results showed fuel-efficiency differences could be explained largely by differences in aircraft operations. Morrell (2009) analyzed the potential for greater fuel efficiency through using larger aircraft and different operational patterns. Miyoshi and Merkert (2010) evaluated carbon and fuel efficiency of 14 European airlines during the period from 1986 to 2007 to understand the relationship between fuel efficiency and fuel price, distance flown and load factors. Zou et al. (2014) employed ratio-based, deterministic, and stochastic frontier approaches to investigate fuel efficiency of 15 large jet operators in the United States. The results showed that potential cost savings of mainline airlines could reach approximately $1 billion in 2010. Cui and Li (2015a) proposed a Virtual Frontier Benevolent DEA Cross Efficiency model to evaluate the energy efficiency of 11 airlines during 2008-2012. Li, Wang, and Cui (2016a) proposed a Virtual Frontier Dynamic Range Adjusted Measure to evaluate the energy efficiency of 22 international airlines during 2008-2012.

Other efficiencies of airline is another popular topic (Truitt and Haynes 1994; Fu, Oum, and Zhang 2009; Wen and Yeh 2010). In recent years, Wu and Liao (2014) applied standard DEA and balanced scorecard (BSC) to evaluate the efficiency of 38 international airlines in 2010 and concluded that leading and lagging factors of BSC were adapted to the evaluation of operational performance of airlines along with DEA. Chang et al. (2014) used a slack-based measure (SBM) DEA to determine the efficiency of 27 international airlines in 2010 and found that fuel consumption and revenue structure were the major causes of inefficient airlines. Lee and Worthington (2014) used bootstrapped DEA and bootstrapped truncated regression to evaluate the efficiency of 42 US and European airlines and discovered that DEA scores were estimated simultaneously with a bootstrapped truncated regression model to explain efficiency drivers. Lozano and Gutierrez (2014) applied a two-stage SBM to analyze the efficiency of 16 European airlines in 2007 and concluded that Network DEA approach had more discriminative power than the single-process DEA. Mallikarjun (2015) applied three-stage unoriented network DEA to evaluate the efficiency of 27 US airlines in 2012 and discovered that reducing operating expenses and increasing fare revenue were the main methods to improve efficiency. Li, Wang, and Cui (2016b) employed Network SBM with weak disposability and strong disposability to evaluate the efficiency of 22 international airlines during 2008-2012 and found strong disposability was a more reasonable way in treating undesirable outputs. Cui et al. (2016) built a Virtual Frontier Dynamic SBM to calculate the energy efficiencies of 21 airlines from 2008 to 2012 and concluded that per capita gross, domestic product, the average service age of fleet size and average haul distance had significant impacts on the efficiency score.

However, the evaluation of fuel efficiency or energy efficiency for airlines in the above papers has not considered the undesirable output. In the existing energy efficiency papers, the main undesirable outputs are C[O.sub.2] emission (e.g., Wei, Liao, and Fan 2007; Mandal 2010; and Tao, Li, and Xia 2012). This article chooses C[O.sub.2] emission as the undesirable output.

The remainder of this article is organized as follows: the next section proposes the method, followed by a case study. The final section summarizes the conclusions.

Methodology

Data envelopment analysis (Charnes, Cooper, and Rhodes 1978) is a data-planning method to evaluate the relative efficiency of decision-making units (DMUs) with multi-inputs and multi-outputs. It has been applied to theory innovation, model development and practical application.

Suppose the data set is (Y,X), Ystands for the K x N matrix of the outputs and X denotes K x M matrix of the inputs, [mathematical expression not reproducible] K,N,M stand for the number of decision-making units, the outputs and the inputs respectively.

The DEA model attempts to measure the ratio of outputs to inputs, such as u [y.sub.i]/v [x.sub.i], where u, v are the weight vectors of outputs and inputs. For each DMU, the following linear programming problem is formulated:

maxu [y.sub.i]

s.t. v' [x.sub.i] = 1 u'[y.sub.k] - v'[x.sub.k] [less than or equal to] o, k = i,2,***,K u [greater than or equal to] o, v [greater than or equal to] o (1)

Any of the DMUs may or may not be on the frontier when the ratio is measured. The distance from the actual allocation of a particular DMU to the frontier is believed to represent the inefficiency of the DMU, which may be caused by various factors, specific to the DMU. If the efficiency of DMU i is 1, DMU i is technically efficient DMU; if its efficiency is less than 1, it is technically inefficient. The above problem assumes constant returns to scale (CRS).

In order to observe both the efficiency and the slacks, many SBM DEA models are proposed--SBM in Tone (2001) and some derived models, such as SBM in Network DEA (Tone and Tsutsui 2009) and SBM in Dynamic DEA (Tone and Tsutsui 2010).

The basic model in Tone (2001) is:

[mathematical expression not reproducible] (2)

X,Y stand for the inputs and outputs, [s.sup.+],[s.sup.-] stand for the input excess and output shortfall. [lambda] is the weight.

However, the original SBM model has not considered the undesirable outputs. Aiming at this problem, many SBM models with undesirable outputs are proposed (Avkiran and Rowlands 2008; Fukuyama and Weber 2010; Liu et al. 2010; Barros, Managi, and Matousek 2012). The basic SBM model with undesirable outputs is:

T = f(x,y,u)

[mathematical expression not reproducible] (3)

X,Y,U stand for the inputs, the desirable outputs and the undesirable outputs. [s.sup.+], [s.sup.-] and [ss.sup.-] stand for the input slacks, the desirable output slacks and the undesirable output slacks.

However, in the traditional SBM model, each decision-making unit compares its production ability with the production ability of an optimal real frontier (Xue and Harker 2002). When its result is 1, the DMU is technically efficient; otherwise, the DMU is technically inefficient. However, it cannot distinguish the differences between efficient DMUs.

To overcome this disadvantage, we propose a Virtual Frontier SBM model with weak disposability and strong disposability based on the principle of Virtual Frontier. Bian and Xu (2013) first proposed the idea of virtual frontier and applied it on the traditional DEA model to build the Virtual Frontier DEA model. Cui and Li (2015a) applied the Virtual Frontier DEA model to evaluate the transportation carbon efficiencies of 15 countries. Cui and Li (2014) proposed a three-stage Virtual Frontier DEA model and the model can eliminate the impacts of some environmental factors on efficiency. Cui and Li (2015b) combined the Virtual Frontier DEA model and the benevolent cross-efficiency model to propose a Virtual Frontier Benevolent DEA Cross Efficiency model. These models have not considered the slacks, which are effective to put forward efficiency improvements for DMUs.

Li, Wang, and Cui (2015) applied the idea of virtual frontier to the network model and proposed a Virtual Frontier Network SBM model. The model considered the internal or linking activities of efficiency, but it had not taken the effects of carryover activities into account. Li, Wang, and Cui (2016a) proposed a Virtual Frontier Dynamic RAM to evaluate airline energy efficiency, but it had not analyzed the influence of external factors on the efficiency. Cui et al. (2016) proposed a Virtual Frontier Dynamic SBM to calculate the energy efficiencies of 21 airlines from 2008 to 2012.

However, these models have not considered the disposability of undesirable outputs. If undesirable outputs are considered, they can be processed in two ways: weak disposability and strong disposability. The weak disposability thinks the desirable outputs entail the increase of undesirable outputs and the undesirable outputs entail the decrease of desirable outputs. The strong disposability believes the processing capacity of the environment on undesirable outputs and thinks the environment can handle as many undesirable outputs as possible. These two ways have been critically debated in academic journals (Hailu and Veeman 2001; Fare and Grosskopf 2003; Hailu 2003), indicating the importance of the method used to dispose of undesirable outputs. Li, Wang, and Cui (2016b) has compared the results of Network SBM model with weak disposability and the model with strong disposability in measuring airline energy efficiency.

If [zeta] denotes the evaluating DMU set and [psi] is the reference DMU set (the virtual frontier), the Virtual Frontier SBM model with undesirable outputs is

[mathematical expression not reproducible] (4)

[xx.sub.in], [yy.sub.rn], [uu.sub.jn] stand for the inputs, the outputs, and the undesirable outputs in the virtual reference set. M,N,Q,K are the number of the inputs, the desirable outputs, the undesirable outputs and the DMUs.

In this model, the reference DMU set and the evaluating DMU set are two different sets; this offers the possibility of distinguishing between the efficient DMUs in the traditional SBM model. And in the evaluating process, the reference DMU set remains unchanged so that its results may be more reasonable than existing models.

Next, this article will introduce the selection of a reference DMU set. Set [mathematical expression not reproducible] represent the DMUs, [x.sub.in] denotes the ith input of DMU n, [y.sub.rn] denotes the rth output of DMU n, [u.sub.jn] is the jth undesirable output of DMU n. For the reference set, the inputs are set as [xx.sub.in] = 0.95[x.sub.i*], its output are set as [yy.sub.rn] = 1.05[y.sub.r*], its undesirable output [uu.sub.jn] = 0.95[u.sub.j*]..

We can get two properties for the Virtual Frontier SBM model.

Property 1. The efficiency of the Virtual Frontier SBM model is lower than that of the corresponding SBM model.

The proof process can be found in appendix A.

Property 2. There will not be two DMUs with the same efficiencies in Virtual Frontier SBM model, unless the inputs, the desirable outputs, and the undesirable outputs of these two DMUs are completely the same.

The proof process can be found in appendix B.

The first property can guarantee all of the DMUs' efficiencies are less than 1 and the second property can ensure none of the DMUs has the same efficiencies. These two properties can distinguish the DMUs completely and embody the new model's merits.

Empirical Study

The Selection of Inputs and Outputs

In this article, based on the former literature review and the reality of the airline industry, the inputs and outputs of airlines' energy efficiency are selected. Three measurable variables are selected as inputs: labor (number of employees, NE), capital (capital stock, CS) and energy (tons of aviation kerosene, AK). Because more than 95 percent of the energy consumption is aviation kerosene, this article chooses it as the index of energy input. Four measurable variables are chosen as outputs: revenue tonne kilometers (RTK), revenue passenger kilometers (RPK), total business income (TBI), and C[O.sub.2] emission (C[O.sub.2]).

Considering the impact of undesirable output on energy efficiency, this article employs C[O.sub.2] emission as the undesirable output. It should be noted that different types of jet fuel have different C[O.sub.2] emissions. However, because the C[O.sub.2] emission in this article is from the sustainability, environment, and corporate social responsibility reports, it has high reliability to take it as an output.

The Data

An empirical study in this article will be performed with the data of a five-year period, from 2008 to 2012. Since 2008 the financial crisis in the United States has deeply affected the global airline and the energy markets. To minimize the effect of the financial crisis, many airlines anchor their hope on improving overall efficiency; on account of the increasing energy price, energy efficiency has become an important consideration. It is meaningful to study the energy efficiencies of some of the major airlines during this period.

The empirical data are obtained from 22 airlines: China Eastern Airlines, China Southern Airlines, Korean Air, Qantas Airways, Air France-KLM, Lufthansa Airlines, Scandinavian Airlines, Delta Air Lines, Alaska Airlines, Air China, Hainan Airlines, Emirates, Ethiopian Airlines, Air Greenland, Air Canada, Cathay Pacific Airways, Kenya Airways, Malaysia Airlines, Asiana Airlines, Southwest Airlines, Singapore Airlines, and All Nippon Airways. According to IATA World Air Transport Statistics, out of these 22 airlines, the number of revenue passengers of six ranked in the top 10 worldwide in 2012 (Delta Air Lines, Southwest Airlines, China Southern Airlines, China Eastern Airlines, Lufthansa Airlines, and Air France-KLM). Ten ranked in the top 20 worldwide in 2012 (the other four are Air China, Qantas Airways, All Nippon Airways and Emirates). These 22 airlines come from Asia, America, Europe, Africa, and Oceania, so they are representative of global airlines. The data on number of employees, capital stock, total business income, revenue tonne kilometers, and revenue passenger kilometers are collected from the annual reports. The data on tons of aviation kerosene and C[O.sub.2] emission volume are taken from the sustainability, environment, and corporate social responsibility reports of the 22 companies. Because the topic of this article is to discuss the energy efficiency of airlines, we have not considered the difference of low-cost carriers and full service carriers.

Descriptive statistics of the inputs and outputs are provided in table 1. From this table it can be concluded that C[O.sub.2] emission has no significant linear relationship with the amount of aviation kerosene.

Table 2 shows the Pearson correlation coefficients (Pearson et al. 2002; Cui et al. 2013) between the inputs and the outputs. As shown in the table, all the coefficients are positive and relatively high, which provides assurance that the inputs and outputs are closely related. Furthermore, the correlation coefficient between AK and C[O.sub.2] is 0.687, which verifies that C[O.sub.2] has no significant linear relationship with the amount of aviation kerosene.

Each airline is defined as a decision-making unit (DMU) and each DMU has three inputs, three outputs and one undesirable output.

Results of the Traditional SBM Model with Undesirable Outputs

To verify the reasonability of the new model, first, this article uses the traditional SBM model with undesirable outputs in model (3) to calculate the energy efficiencies. The results are shown in table 3. The efficiencies of many DMUs are 1, and traditional SBM cannot distinguish them.

Results of the Virtual Frontier SBM Model with Undesirable Outputs

In this article, the Virtual Frontier SBM model with undesirable outputs model in model (4) is conducted through MATLAB programming; its results are shown in table 4. It can be concluded that the Virtual Frontier SBM model can distinguish the DMUs, as all DMUs' efficiencies are not the same. Because the reference DMUs have larger outputs and lower levels of inputs, all DMUs' efficiencies are less than those from the traditional SBM model. When the reference DMUs are efficient, all evaluated DMUs' efficiencies are less than those from the traditional SBM model.

In the Virtual Frontier SBM model, all of the airlines are inefficient; so the difference in the efficiencies can be shown, which improves the limitation of the traditional SBM model. If the undesirable outputs are weakly disposed, we refer to Fare et al. (1989) and Li, Wang, and Cui (2015) to build the Virtual Frontier SBM model with weak disposability. If [zeta] denotes the evaluating DMU set and [psi] is the reference DMU set (the virtual frontier), the Virtual Frontier SBM model with weak disposability is

[mathematical expression not reproducible] (5)

[xx.sub.in], [yy.sub.rn], [uu.sub.jn] stand for the inputs, the outputs and the undesirable outputs in virtual reference set. M,N,Q,K are the number of the inputs, the desirable outputs, the undesirable outputs, and the DMUs. For the reference set, the inputs are set as [xx.sub.in] =0.95[x.sub.i*], its output are set as [yy.sub.rn] = 1.05[y.sub.r*], its undesirable output [uu.sub.jn] = [u.sub.jn]. Because the objective function has no slacks of undesirable output, the undesirable output has not direct impact on the efficiency score. Therefore, the undesirable output in reference set remains unchanged. The results are shown in table 5.

As shown in tables 4 and 5, the average energy efficiency of Scandinavian Airlines over the period from 2008 to 2012 is the highest, that is, the benchmarking airlines are the same whether the undesirable outputs are strongly disposed or weakly disposed. This result verifies the robustness of the Virtual Frontier models.

In table 4, Scandinavian Airlines' average energy efficiency from 2008 to 2012 is 0.022 and that in table 5 is 0.017. The main reason lies in its high income production efficiency and its capacity in handling carbon dioxide. Its average total business income of unit aviation kerosene ranks first among the 22 airlines and is approximately 4.550, whereas that of the least one Air Canada is approximately 0.007. Scandinavian Airlines' average total business income of unit employee ranks first and is approximately 0.035, while that of the least one, Malaysia Airlines, is approximately 0.001. Its average total business income of unit capital stock ranks second and is approximately 4.809 (Kenya Airways ranks first), while that of the least one, Hainan Airlines, is approximately 0.298. Its average carbon dioxide of unit capital stock ranks twenty-second and is approximately 3.366, and that of the largest one, Asiana Airlines, is approximately 426.50. Thus, the high income production efficiency and the mature capacity in handling carbon dioxide have significant impacts on Scandinavian Airlines' energy efficiency

If the undesirable outputs are not considered, the Virtual Frontier SBM model changes into:

[mathematical expression not reproducible] (6)

The inputs remain as labor, capital and energy. The outputs change as: revenue tonne kilometers, revenue passenger kilometers and total business income. For the reference set, the inputs are set as [xx.sub.in] = 0.95[x.sub.i*], its output are set as [yy.sub.rn] = 1.05[y.sub.r*]. And the energy efficiency of airlines is calculated by Virtual Frontier SBM model, the results are shown in Table 6.

In order to compare tables 4-6 more clearly, this article defines two influence indices of undesirable output: the strong influence index and the weak influence index. The former one is defined as the quotient of the efficiency in table 4 and the corresponding efficiency in table 6, and the latter one is defined as the quotient of the efficiency in table 5 and the corresponding efficiency in table 6. For example, in table 4, the energy efficiency of China Eastern Airlines in 2008 is 0.010 and the corresponding efficiency in table 6 also is 0.008, so the influence index of undesirable output for China Eastern Airlines in 2008 is 1.269. The results of influence index are shown in table 7.

From table 7, we can conclude that under strong disposability, C[O.sub.2] emission has conspicuous impacts on the energy efficiency change of airlines. Among the 22 airlines, Air Canada's energy efficiency suffers most from C[O.sub.2] emission; it should pay important attentions on the C[O.sub.2] emission as it has important impacts on energy efficiency. For the 22 airlines, the largest influence of C[O.sub.2] emission occurs in 2012. This result has close relationships with the disposability way of C[O.sub.2] emission. In strong disposability, the C[O.sub.2] emission is considered as an input and most airlines' C[O.sub.2] emission increased during the period of 2008-2012. Therefore, the largest influence appears in 2012.

For another, from table 7, we can know that the efficiency scores of weak disposability are the same as those of the situation when undesirable output is not considered. This result indicates that although model (5) has a weak disposability constraint (C3) compared with model (6), the constraint (C3) has little impact on the weight [lambda], and then the results have little difference with those of model (6).

Policy Implication

From the principle of the resolution of the 39th ICAO conference in October 2016, we can find that each airline has great responsibility to control the carbon emission. And the carbon emission offsetting is an important driving force for airlines to improve energy efficiency. From the results in this article, we can get the following policies:

1. In this article, the average energy efficiency of Scandinavian Airlines is the highest among the 22 airlines and its some measures can be a reference for other airlines. Scandinavian Airlines have taken many measures to improve energy efficiency and protect environment. Since 2007 the airline has reported a climate index, which refers to weighted climate impact excluding noise, that is, emissions of carbon dioxide (C[O.sub.2]) and nitrogen oxides (NOx). The index measures the airline's overall climate impact relative to traffic measured in RPK. Furthermore, since 1996, the airline has been measuring eco-efficiency using an environmental index in which environmental impact is measured relative to production. These indices are a tool for managing and following up the airline's environmental performance. Starting in February 2009, the airline has offered a simplified payment solution for carbon offsets. Carbon offsets have been integrated into the emissions trading scheme as of 2012. Carbon offset revenues go entirely to the airline's partner, the Carbon Neutral Company, which is responsible for funding energy projects based on renewables and verified/certified projects. All business travel is offset and corresponds to emissions of 4,000-5,000 tonnes per year.

Other airlines should build their own climate index or environmental index to track the effects of their efforts on controling emissions. More important, other airlines should seek the third-party partners to participate in the carbon offsets, such as the CarbonNeutral Company of Scandinavian Airlines. The carbon offset partners can help airlines to achieve carbon reduction targets.

2. The airlines should attach great importance to the undesirable outputs. According to the results of the virtual frontier models in this paper, C[O.sub.2] emission has conspicuous impacts on the energy efficiency change of airlines in strong disposability, while it has little impact on the airline energy efficiency change in weak disposability. This indicates that strong disposability can reflect the effects of undesirable outputs more clearly than weak disposability. This conclusion is the same as Li, Wang, and Cui (2016b), in which strong disposability is more reasonable than weak disposability in treating undesirable outputs. According the principle of strong disposability, this result shows that the current emissions have not exceeded the carrying capacity of the environment. However, the principle of weak disposability is more suitable for the long-term sustainable development of the aviation industry and the airlines must take some measures to reduce undesirable outputs.

Conclusions

The topic of airlines' energy efficiency is studied in this article. The number of employees, capital stock, and tons of aviation kerosene are chosen as the inputs. Revenue tonne kilometers, revenue passenger kilometers, and total business income are selected as the outputs, C[O.sub.2] emission is the undesirable output. A new model, the Virtual Frontier SBM model, is proposed and applied to evaluate the energy efficiencies of 22 airlines from 2008 to 2012. The results verify the rationality of the new model.

Overall, the contribution of this article to the literature is embodied in two aspects. First, based on the existing paper on airlines' energy efficiency, this article considers the undesirable output. Except for sustainable management, the rapidly growing carbon emission is another important factor to prompt airlines to improve energy efficiency. The idea in this article enriches the theory and method of energy research and supplies a new view on evaluating the development of the airlines. Second, a new model, the Virtual Frontier SBM model, is proposed. It can resolve the limitation of traditional SBM model in distinguishing efficient DMUs. The results verify the rationality of the new model.

Future research could focus on exploring important influencing factors for airlines' energy efficiency.

Appendix A

The Proof Process of Property l

Proof. We define the efficiency, the inputs, the weights, the input slacks, the outputs, the desirable output slacks and the undesirable output slacks of Virtual Frontier SBM model as [theta][theta], [lambda][lambda], xx, s[s.sup.-.sub.i], yy, s[s.sup.+.sub.r] and s[s.sup.-.sub.j]. Those of corresponding SBM model can be labeled as [theta], [lambda], x, [s.sup.-.sub.i], y, [s.sup.+.sub.r] and [s.sup.-.sub.j].

Because [K.summation over (n=1)][lambda][[lambda].sub.n] = [K.summation over (n=1)] [[lambda].sub.n] = 1,

then we have

[mathematical expression not reproducible]

The proof process of s[s.sup.-.sub.j] > [s.sup.-.sub.j] is same. Similarly,

[mathematical expression not reproducible]

And other parameters are the same, so [theta][theta] < [theta], and the efficiency of the Virtual Frontier SBM model is lower than that of the corresponding SBM model.

Appendix B

The Proof Process of Property 2

Proof. We discuss three situations: when the inputs are different but the desirable outputs and the undesirable outputs are the same; when the desirable outputs are different but the inputs and the undesirable outputs are the same; when the undesirable outputs are different but the inputs and the desirable outputs are the same.

For the first situation, we label the two DMUs as A and B, the inputs of A and B are [x.sub.A] and

[mathematical expression not reproducible]

this reference set is the same for A and B. Because [x.sub.A] [not equal to] [x.sub.B], then s[s.sup.-] is different for A and B.

Therefore, the efficiency is different for A and B. The proof process of the second situation is the same. When only the desirable outputs are different for A and B, then

[K.summation over (n=1)][[lambda].sub.n][yy.sub.rn] = 1.05*max([y.sub.rn])* [K.summation over (n=1)][[lambda].sub.n] = 1.05* max([y.sub.rn]).

This reference set is the same for A and B. Because [y.sub.A] [not equal to] [y.sub.B], then [SS.sup.+] is different for A and B, the efficiency is different for A and B.

Qiang Cui

Corresponding Author

Southeast University

cuiqiang1011@16B.com

Ye Li

Nanjing University of Finance and Economics

Yi-Ming Wei

Beijing Institute of Technology

Note

This research is funded by National Nature Science Foundation of China: Nos. 71403034 and 71521002), Nature Science Foundation of Liaoning Province (No. 201601841), the Fundamental Research Funds for the Central Universities (No. 20110116204), China Postdoctoral Science Foundation (No. 2016M590050) and National Key R&D Program (2016YFA0602603).

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Table 1/Descriptive Statistics of the Inputs and Outputs

Variable                            Mean       Std.dev.

The Inputs

Number of employees              35,593.68    29,448.21

Capital stock                       144.48       115.93
([10.sup.8] dollars)

Aviation kerosene                 1,053.51     2,611.41
([10.sup.4] tons)

Desirable Outputs

Revenue tonne kilometers          6,240.11     6,632.17
([10.sup.6] ton-kilometers)

Revenue passenger kilometers     85,882.58    72,566.22
([10.sup.6] person-kilometers)

Total business income               144.48       144.09
([10.sup.8] dollars)

Undesirable Outputs

C[O.sub.2] emission               1,415.73     1,252.29
([10.sup.4] tons)

Variable                           Min         Max

The Inputs

Number of employees              626.00    119,084.00

Capital stock                      0.01        580.33
([10.sup.8] dollars)

Aviation kerosene                 33.80     12,838.75
([10.sup.4] tons)

Desirable Outputs

Revenue tonne kilometers          47.96     23,672.00
([10.sup.6] ton-kilometers)

Revenue passenger kilometers     442.45    310,875.37
([10.sup.6] person-kilometers)

Total business income              0.04        735.42
([10.sup.8] dollars)

Undesirable Outputs

C[O.sub.2] emission               82.43      6,027.14
([10.sup.4] tons)

Note: The sales costs and total business income
are expressed in purchasing power parity dollars.

Table 2/Input-Output Correlations

            RTK                  RPK

NE   0.501 (Sig. 0.001)   0.649 (Sig. 0.002)
AK   0.621 (Sig. 0.001)   0.830 (Sig. 0.004)
CS   0.531 (Sig. 0.005)   0.723 (Sig. 0.003)

         C[O.sub.2]              TBI

NE   0.535 (Sig. 0.002)   0.811 (Sig. 0.008)
AK   0.687 (Sig. 0.001)   0.617 (Sig. 0.002)
CS   0.623 (Sig. 0.005)   0.711 (Sig. 0.003)

Note: Sig. denotes the significance level.

Table 3/Results of the Traditional SBM Model with Undesirable Outputs

Airlines                   2008    2009    2010    2011    2012

China Eastern Airlines     1.000   1.000   1.000   1.000   1.000
China Southern Airlines    1.000   1.000   1.000   1.000   1.000
Korean Air                 0.787   1.000   1.000   0.642   0.624
Oantas Airways             1.000   1.000   0.937   1.000   1.000
Air France-KLM             1.000   1.000   1.000   1.000   1.000
Lufthansa Airlines         1.000   1.000   1.000   1.000   1.000
Scandinavian Airlines      1.000   1.000   1.000   1.000   1.000
Delta Air Lines            1.000   1.000   1.000   1.000   1.000
Alaska Airlines            0.215   1.000   1.000   1.000   1.000
Air China                  1.000   0.553   0.868   1.000   1.000
Hainan Airlines            1.000   1.000   1.000   1.000   1.000
Emirates                   1.000   1.000   1.000   1.000   1.000
Ethiopian Airlines         1.000   1.000   1.000   1.000   1.000
Air Greenland              1.000   1.000   1.000   1.000   1.000
Air Canada                 1.000   1.000   1.000   1.000   1.000
Cathay Pacific Airways     1.000   1.000   1.000   1.000   1.000
Kenya Airways              1.000   1.000   1.000   1.000   1.000
Malaysia Airlines          1.000   1.000   1.000   1.000   1.000
Asiana Airlines            1.000   1.000   1.000   1.000   1.000
Southwest Airlines         1.000   1.000   1.000   1.000   1.000
Singapore Airlines         0.550   0.679   0.513   1.000   1.000
All Nippon Airways         0.422   1.000   1.000   1.000   1.000

Table 4/Energy Efficiency When Undesirable
Output Is Strongly Disposed

Airlines                     2008       2009      2010

China Eastern Airlines      0.010      0.011      0.016
China Southern Airlines     0.011      0.011      0.015
Korean Air                  0.011      0.011      0.014
Oantas Airways              0.008      0.008      0.010
Air France-KLM              0.011      0.010      0.012
Lufthansa Airlines          0.009      0.009      0.013
Scandinavian Airlines       0.015      0.020      0.023
Delta Air Lines             0.007      0.007      0.010
Alaska Airlines             0.002      0.002      0.002
Air China                   0.009      0.009      0.012
Hainan Airlines             0.001      0.001      0.001
Emirates                    0.011      0.011      0.014
Ethiopian Airlines          0.002      0.003      0.004
Air Greenland               0.002      0.001      0.002
Air Canada                  0.003      0.002      0.003
Cathay Pacific Airways      0.012      0.010      0.014
Kenya Airways               0.003      0.003      0.004
Malaysia Airlines           0.003      0.003      0.005
Asiana Airlines            5.22E-05   5.79E-05    0.015
Southwest Airlines          0.005      0.006      0.006
Singapore Airlines          0.011      0.011      0.011
All Nippon Airways          0.005      0.007      0.008

Airlines                     2011       2012     Average

China Eastern Airlines      0.015      0.025      0.015
China Southern Airlines     0.018      0.020      0.015
Korean Air                  0.015      0.018      0.014
Oantas Airways              0.011      0.013      0.010
Air France-KLM              0.012      0.015      0.012
Lufthansa Airlines          0.015      0.015      0.012
Scandinavian Airlines       0.024      0.027      0.022
Delta Air Lines             0.011      0.014      0.010
Alaska Airlines             0.002      0.003      0.002
Air China                   0.013      0.013      0.011
Hainan Airlines             0.001      0.001      0.001
Emirates                    0.016      0.020      0.015
Ethiopian Airlines          0.003      0.005      0.004
Air Greenland               0.001      0.002      0.002
Air Canada                  0.003      0.003      0.003
Cathay Pacific Airways      0.015      0.019      0.014
Kenya Airways               0.002      0.004      0.003
Malaysia Airlines           0.005      0.006      0.004
Asiana Airlines            9.32E-05    0.020      0.007
Southwest Airlines          0.007      0.008      0.006
Singapore Airlines          0.014      0.016      0.013
All Nippon Airways          0.009      0.009      0.007

Table 5/Energy Efficiency When Undesirable Output Is Weakly Disposed

Airlines                     2008       2009      2010

China Eastern Airlines      0.008      0.008      0.012
China Southern Airlines     0.008      0.009      0.012
Korean Air                  0.009      0.009      0.013
Oantas Airways              0.006      0.007      0.009
Air France-KLM              0.007      0.008      0.009
Lufthansa Airlines          0.008      0.007      0.011
Scandinavian Airlines       0.012      0.016      0.019
Delta Air Lines             0.005      0.004      0.006
Alaska Airlines             0.001      0.001      0.001
Air China                   0.008      0.007      0.009
Hainan Airlines             0.001      0.001      0.001
Emirates                    0.009      0.009      0.012
Ethiopian Airlines          0.002      0.003      0.004
Air Greenland               0.002      0.001      0.002
Air Canada                  0.001      0.001      0.001
Cathay Pacific Airways      0.010      0.009      0.013
Kenya Airways               0.002      0.002      0.004
Malaysia Airlines           0.003      0.004      0.006
Asiana Airlines            6.16E-05   6.77E-05    0.014
Southwest Airlines          0.005      0.005      0.006
Singapore Airlines          0.009      0.009      0.009
All Nippon Airways          0.004      0.005      0.006

Airlines                     2011       2012     Average

China Eastern Airlines      0.010      0.015      0.011
China Southern Airlines     0.014      0.015      0.012
Korean Air                  0.012      0.014      0.011
Oantas Airways              0.009      0.010      0.008
Air France-KLM              0.009      0.011      0.009
Lufthansa Airlines          0.012      0.011      0.010
Scandinavian Airlines       0.019      0.020      0.017
Delta Air Lines             0.005      0.006      0.005
Alaska Airlines             0.001      0.001      0.001
Air China                   0.010      0.009      0.009
Hainan Airlines             0.001      0.001      0.001
Emirates                    0.013      0.016      0.012
Ethiopian Airlines          0.003      0.005      0.003
Air Greenland               0.001      0.002      0.002
Air Canada                  0.001      0.001      0.001
Cathay Pacific Airways      0.012      0.015      0.012
Kenya Airways               0.002      0.004      0.003
Malaysia Airlines           0.006      0.007      0.005
Asiana Airlines            1.07E-04    0.018      0.006
Southwest Airlines          0.006      0.006      0.005
Singapore Airlines          0.012      0.013      0.010
All Nippon Airways          0.007      0.006      0.006

Table 6/Energy Efficiency When Undesirable Output Is Not Considered

Airlines                     2008       2009     2010

China Eastern Airlines      0.008      0.008     0.012
China Southern Airlines     0.008      0.009     0.012
Korean Air                  0.009      0.009     0.013
Oantas Airways              0.006      0.007     0.009
Air France-KLM              0.007      0.008     0.009
Lufthansa Airlines          0.008      0.007     0.011
Scandinavian Airlines       0.012      0.016     0.019
Delta Air Lines             0.005      0.004     0.006
Alaska Airlines             0.001      0.001     0.001
Air China                   0.008      0.007     0.009
Hainan Airlines             0.001      0.001     0.001
Emirates                    0.009      0.009     0.012
Ethiopian Airlines          0.002      0.003     0.004
Air Greenland               0.002      0.001     0.002
Air Canada                  0.001      0.001     0.001
Cathay Pacific Airways      0.010      0.009     0.013
Kenya Airways               0.002      0.002     0.004
Malaysia Airlines           0.003      0.004     0.006
Asiana Airlines            6.16E-05   6.77E-05   0.014
Southwest Airlines          0.005      0.005     0.006
Singapore Airlines          0.009      0.009     0.009
All Nippon Airways          0.004      0.005     0.006

Airlines                     2011       2012

China Eastern Airlines      0.010      0.015
China Southern Airlines     0.014      0.015
Korean Air                  0.012      0.014
Oantas Airways              0.009      0.010
Air France-KLM              0.009      0.011
Lufthansa Airlines          0.012      0.011
Scandinavian Airlines       0.019      0.020
Delta Air Lines             0.005      0.006
Alaska Airlines             0.001      0.001
Air China                   0.010      0.009
Hainan Airlines             0.001      0.001
Emirates                    0.013      0.016
Ethiopian Airlines          0.003      0.005
Air Greenland               0.001      0.002
Air Canada                  0.001      0.001
Cathay Pacific Airways      0.012      0.015
Kenya Airways               0.002      0.004
Malaysia Airlines           0.006      0.007
Asiana Airlines            1.07E-04    0.018
Southwest Airlines          0.006      0.006
Singapore Airlines          0.012      0.013
All Nippon Airways          0.007      0.006

Table 7/Influence Indices of Undesirable Output

Airlines                    2008             2009

                           Strong   Weak    Strong   Weak

China Eastern Airlines     1.269    1.000   1.317    1.000
China Southern Airlines    1.263    1.000   1.264    1.000
Korean Air                 1.195    1.000   1.155    1.000
Oantas Airways             1.213    1.000   1.186    1.000
Air France-KLM             1.598    1.000   1.214    1.000
Lufthansa Airlines         1.217    1.000   1.231    1.000
Scandinavian Airlines      1.261    1.000   1.265    1.000
Delta Air Lines            1.617    1.000   1.696    1.000
Alaska Airlines            1.403    1.000   1.536    1.000
Air China                  1.205    1.000   1.247    1.000
Hainan Airlines            1.127    1.000   1.236    1.000
Emirates                   1.277    1.000   1.266    1.000
Ethiopian Airlines         1.005    1.000   1.013    1.000
Air Greenland              1.123    1.000   1.123    1.000
Air Canada                 3.228    1.000   3.173    1.000
Cathay Pacific Airways     1.153    1.000   1.192    1.000
Kenya Airways              1.336    1.000   1.376    1.000
Malaysia Airlines          0.800    1.000   0.795    1.000
Asiana Airlines            0.849    1.000   0.856    1.000
Southwest Airlines         1.159    1.000   1.151    1.000
Singapore Airlines         1.231    1.000   1.229    1.000
All Nippon Airways         1.268    1.000   1.272    1.000
Average                    1.309    1.000   1.309    1.000

Airlines                    2010             2011

                           Strong   Weak    Strong   Weak

China Eastern Airlines     1.345    1.000   1.481    1.000
China Southern Airlines    1.246    1.000   1.281    1.000
Korean Air                 1.152    1.000   1.251    1.000
Oantas Airways             1.173    1.000   1.251    1.000
Air France-KLM             1.252    1.000   1.298    1.000
Lufthansa Airlines         1.230    1.000   1.328    1.000
Scandinavian Airlines      1.245    1.000   1.280    1.000
Delta Air Lines            1.640    1.000   2.053    1.000
Alaska Airlines            1.555    1.000   1.855    1.000
Air China                  1.240    1.000   1.281    1.000
Hainan Airlines            1.217    1.000   1.259    1.000
Emirates                   1.188    1.000   1.217    1.000
Ethiopian Airlines         0.936    1.000   1.091    1.000
Air Greenland              1.000    1.000   1.114    1.000
Air Canada                 2.863    1.000   4.180    1.000
Cathay Pacific Airways     1.148    1.000   1.222    1.000
Kenya Airways              1.059    1.000   1.370    1.000
Malaysia Airlines          0.793    1.000   0.800    1.000
Asiana Airlines            1.084    1.000   0.868    1.000
Southwest Airlines         1.096    1.000   1.179    1.000
Singapore Airlines         1.212    1.000   1.176    1.000
All Nippon Airways         1.254    1.000   1.298    1.000
Average                    1.269    1.000   1.415    1.000

Airlines                    2012

                           Strong   Weak

China Eastern Airlines     1.636    1.000
China Southern Airlines    1.344    1.000
Korean Air                 1.293    1.000
Oantas Airways             1.298    1.000
Air France-KLM             1.317    1.000
Lufthansa Airlines         1.372    1.000
Scandinavian Airlines      1.351    1.000
Delta Air Lines            2.138    1.000
Alaska Airlines            1.942    1.000
Air China                  1.455    1.000
Hainan Airlines            1.540    1.000
Emirates                   1.247    1.000
Ethiopian Airlines         1.031    1.000
Air Greenland              1.006    1.000
Air Canada                 3.234    1.000
Cathay Pacific Airways     1.229    1.000
Kenya Airways              1.079    1.000
Malaysia Airlines          0.803    1.000
Asiana Airlines            1.121    1.000
Southwest Airlines         1.273    1.000
Singapore Airlines         1.213    1.000
All Nippon Airways         1.431    1.000
Average                    1.425    1.000
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Author:Cui, Qiang; Li, Ye; Wei, Yi-Ming
Publication:Transportation Journal
Date:Jan 1, 2018
Words:8536
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