Route-Based Performance Evaluation Using Data Envelopment Analysis Combined with Principal Component Analysis

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1 Route-Based Performance Evaluation Using Data Envelopment Analysis Combined with Principal Component Analysis Agnese Rapposelli Abstract Frontier analysis methods, such as Data Envelopment Analysis (DEA), seek to investigate the technical efficiency of productive systems which employ input factors to deliver outcomes. In economic literature one can find extreme opinions about the role of input/output systems in assessing performance. For instance, it has been argued that if all inputs and outputs are included in assessing the efficiency of units under analysis, then they will all be fully efficient. Discrimination can be increased, therefore, by being parsimonious in the number of factors. To deal with this drawback, we suggest to employ Principal Component Analysis (PCA) in order to aggregate input and output data. In this context, the aim of the present paper is to evaluate the performance of an Italian airline for 2004 by applying a model based upon PCA and DEA techniques. 1 Introduction Modern efficiency measurement begins with Farrell (1957), who introduced the seminal concept of technical efficiency. He drew upon the theory of Pareto optimality stating that a productive system is technically efficient if it either maximises output for a given amount of input or minimises input to achieve a given level of output. Moreover, he introduced the concept of the best practice frontier, also called efficiency frontier: according to him, the measure of technical efficiency is given by the relative distance between the observed production and the nearest benchmark production lying on the frontier. It was the seed for later exploitation, following its rediscovery by Charnes et al. (1978) and subsequent relabelling as CCR-efficiency under the broader heading of Data Envelopment Analysis (DEA) (Stone 2002). A. Rapposelli ( ) Dipartimento di Metodi Quantitativi e Teoria Economica, Università G.D Annunzio Chieti-Pescara, Viale Pindaro, 42 Pescara agnese.rapposelli@virgilio.it A. Di Ciaccio et al. (eds.), Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics, DOI / , Springer-Verlag Berlin Heidelberg

2 352 A. Rapposelli DEA method measures technical efficiency relative to a deterministic best practice frontier, which is built empirically from observed inputs and outputs using linear programming techniques. Its main advantage is that it allows several inputs and several outputs to be considered at the same time. The identification of the input and output variables to be used in an assessment of comparative performanceis the most important stage in carrying out the assessment: in order to examine relative efficiency of a set of units it is necessary to define a production function which captures the key points of the production process (Coli et al. 2010). Apart of the nature of the inputs and outputs used in assessing efficiency, it must be remembered that questions can also be raised concerning the appropriate number of inputs and outputs for describing an activity process. Introduction of too many, and especially redundant, variables tend to shift the units towards the efficiency frontier, resulting in a large number of units with high efficiency scores (Golany and Roll 1989; Lin 2008). Also in DEA context the first problem in the selection of inputs and outputs is to include factors indiscriminately. As DEA allows flexibility in the choice of inputs and outputs weights, the greater the number of factors included the less discriminatory the method appears to be. In order to individuate a significant number of inefficient organisations, the literature suggests in the case of static analysis that the number of units has to be greater than 3.mCs/, where m C s is the sum of the number of inputs and number of outputs (Friedman and Sinuany-Stern 1998). Another suggested rule (Dyson et al. 2001) is that, to achieve a reasonable level of discrimination, the number of units has to be at least 2ms. Thus the number of inputs and outputs included in a DEA assessment should be as small as possible in relation to the number of units being assessed. This can be achieved by using Principal Component Analysis (PCA), which is able to reduce the data to a few principal components whilst minimising the loss of information. This process provides therefore a more parsimonious description of a relatively large multivariate data set. Following on from the above discussion, our objective is to adapt the techniques of efficiency measurement, such as DEA, to airline industry. The production process of air transportation services is characterised by multiple outputs and a large number of categories of costs (inputs) (Banker and Johnston 1994). Hence, this study proposes a modified DEA model that includes PCA results and apply it to measure the technical efficiency of the Italian airline Air One, by comparing its domestic routes for The integration of both DEA and PCA techniques have already been proposed in literature. In the last decade, some researchers have made some contributions to combine PCA with DEA in the hope of improving the discriminatory power within DEA and achieving more discerning results. Adler and Golany (2001) tried to apply both DEA and PCA techniques to evaluate West European airline network configurations, Adler and Berechman (2001) used this methodology to determine the relative quality level of West European airports, Zhu (1998) and Premachandra (2001) applied the integrated approach to evaluate economic performance of Chinese cities, Liang et al. (2009) applied the PCA DEA formulation to evaluate the ecological efficiency of Chinese cities. However, it is the first time to see the

3 Route-Based Performance Evaluation Using Data Envelopment Analysis 353 application of PCA DEA formulation to route-based performance measurement: hence, this paper enhance the practicability of PCA DEA. This paper is organised as follows. Section 2 provides the technical framework for the empirical analysis, Sect. 3 describes inputs and outputs used and lists the results obtained. Finally, Sect. 4 presents conclusions of this study. 2 Methods This section describes both DEA and PCA techniques and presents the PCA DEA model to conducting performance measurement in the airline industry. 2.1 Data Envelopment Analysis (DEA) DEA is a linear-programming technique for measuring the relative efficiency of a set of organisational units, also termed Decision Making Units (DMUs). Each DMU represents an observed correspondence of input output levels. The basic DEA models measure the technical efficiency of one of the set of n decision making units, DMU j 0, temporarily denoted by the subscript 0, in terms of maximal radial contraction to its input levels (input orientation) or expansion to its output levels feasible under efficient operation (output orientation). Charnes et al. (1978) proposed the following basic linear model, known as CCR, which has an input orientation and assumes constant returns to scale of activities (CRS): e 0 D min 0 subject to 0 x ij0 j x ij 0; i D 1;:::;m (1) j y rj y rj0 ; r D 1; :::; s (2) j 0; 8 j (3) where y rj is the amount of the r-th output to unit j, x ij is the amount of the i-th input to unit j, j are the weights of unit j and 0 is the shrinkage factor for DMU j 0 under evaluation. The linear programming problem must be solved n times, once for each unit in the sample, for obtaining a value of for each DMU. The efficiency score is bounded between zero and one: a technical efficient DMU will have a score of unity. Subsequent papers have considered alternative sets of assumptions, such as Banker et al. (1984), who modified the above model to permit the assessment of the productive efficiency of DMUs where efficient production is characterised by variable returns to scale (VRS). The VRS model, known as BCC, differs from the

4 354 A. Rapposelli np basic CCR model only in that it includes the convexity constraint j D 1 in the id1 previous formulation. This constraint reduces the feasible region for DMUs, which results in an increase of efficient units; for the rest, CRS and VRS models work in thesameway. In this study we choose a VRS model, that is also in line with the findings of Pastor (1996) and we use the following output-oriented BCC formulation of DEA method: e 0 D max 0 subject to j x ij x ij0 ; i D 1;:::;m (4) 0 y rj0 j y rj 0 r D 1;:::;s (5) j D 1; (6) id1 j 0; 8 j (7) where 0 is the scalar expansion factor for DMU j 0.DMUj 0 is said to be efficient, according to Farrell s definition, if no other unit or combination of units can produce more than DMU j 0 on at least one output without producing less in some other output or requiring more of at least one input. 2.2 The PCA DEA Formulation As stated in Sect. 1, Principal Component Analysis (PCA) is a multivariate statistical method devised for dimensionality reduction of multivariate data with correlated variables. This technique accounts for the maximum amount of the variance of a data matrix by using a few linear combinations (termed principal components) of the original variables. The aim is to take p variables X 1, X 2,..., X p and find linear combinations of them to produce principal components X PC1, X PC2,..., X PCp that are uncorrelated. The principal components are also ordered in descending order of their variances so that X PC1 accounts for the largest amount of variance, X PC2 accounts for the second largest amount of variance, and so on: that is, var.x PC1 / var.x PC2 / ::: var.x PCp /. Often much of the total system variability can be accounted for by a small number k of the principal components, which can then replace the initial p variables without much loss of information (Johnson and Wichern 2002). We have already highlighted that an excessive number of inputs and outputs will result in an excessive number of efficient units in a basic DEA model: the

5 Route-Based Performance Evaluation Using Data Envelopment Analysis 355 greater the number of input and output variables, the higher the dimensionality of the linear programming solution space, and the less discerning the analysis. Dyson et al. (2001) argued that omitting even highly correlated variables could have a major influence on the computed efficiency scores. To deal with this drawback, PCA can be combined with DEA to aggregate and then to reduce inputs and outputs. We can use principal component scores instead of original inputs and outputs variables (Adler and Golany 2001): they can be used to replace either all the inputs and/or outputs simultaneously or alternatively groups of variables (Adler and Yazhemsky 2010). The general DEA formulation has to be modified to incorporate principal components directly into the linear programming problem: hence, the constraints have to be derived from the principal components of original data (Ueda and Hoshiai1997). In particular, constraint (4) is replaced with the following: j x PCij x PCij0 (8) j x Oij x Oij0 (9) and constraint (5) is replaced with the following: 0 y PCrj0 0 y Orj0 j y PCrj 0 (10) j y Orj 0 (11) where x Oij and y Orj denote original input variables and original output variables respectively. The combination of PCA and DEA techniques enable us to overcome the difficulties that classical DEA models encounter when there is an excessive number of inputs or outputs in relation to the number of DMUs, whilst ensuring very similar results to those achieved under the original DEA method. The advantage of this technique is also that it does not require additional expert opinion (Adler and Berechman 2001), unlike the earliest approach to reducing the number of variables. 3 Case Study As mentioned in the introduction, this study evaluates the comparative performance of Air One domestic routes for the year Set up in 1995, Air One was the leading privately owned domestic operator in Italy. It was a lower cost airline but not low cost (or no frills carrier) as it did not fit

6 356 A. Rapposelli the low fare model (Lawton 2002). Air One began operating with domestic flights: in addition to the increase in domestic operations (35% of market share and 20 airports served), it expanded its offer by opening international routes. Scheduled passenger air service was the company s core business and was generating approximately 80% of Air One s revenues. In addition to scheduled airline s service, Air One was also operating charter flights and executive flights for passengers and freight, including its postal service. It was also offering maintenance and handling services (Air One 2005). On 13th January 2009, Air One became part of Compagnia Aerea Italiana (CAI), which has taken over Alitalia and Air One as one whole company. 3.1 Data The sample analysed comprises 30 domestic routes. In order to respect homogeneity assumptions about the units under assessment, we have not included international routes, seasonal destinations and any routes which started during the year The domestic airline industry provides a particularly rich setting for this empirical study. In order to assess Air One domestic routes, the inputs and the outputs of the function they perform must be identified. However, there is no definitive study to guide the selection of inputs and outputs in airline applications of efficiency measurement (Nissi and Rapposelli 2008). In the production process under analysis we have identified seven inputs and four outputs to be included in the performance evaluation. The input selected are the number of seat available for sale, block time hours and several airline costs categories such as total variable direct operating costs (DOCs), total fixed direct operating costs (FOCs), commercial expenses, overhead costs and financial costs. We give a brief overview of inputs used. The number of seats available for sale reflects aircraft capacity. Block time hours is the time for each flight sector, measured from when the aircraft leaves the airport gate to when it arrives on the gate at the destination airport. With regard to the costs categories considered, variable or flying costs are costs which are directly escapable in the short run, such as fuel, handling, variable flight and cabin crew expenses, landing charges, passenger meals, variable maintenance costs. These costs are related to the amount of flying airline actually does, hence they could be avoided if a flight was cancelled (Doganis 2002). Fixed or standing costs are costs which are not escapable in the short or medium term, such as lease rentals, aircraft insurance, fixed flight and cabin crew salaries, engineering overheads. These costs are unrelated to amount of flying done, hence they do not vary with particular flights in the short run; they may be escapable but only after a year of two, depending on airlines (Doganis 2002). Both DOCs and FOCs are dependent on the type of aircraft beingflown(holloway 1997). Commercial expenses, such as reservations systems, commissions, passengers reprotection, lost and found, and overhead costs, such as certain general and administrative costs which do not vary with output (legal expenses, buildings, office equipment, advertising, etc.), are not directly dependent

7 Route-Based Performance Evaluation Using Data Envelopment Analysis 357 on aircraft operations (Holloway 1997). Finally, we have included financial costs, such as interests, depreciation and amortisation. With regard to the output side of the model, the output variables are the number of passengers carried, passenger scheduled revenue, cargo revenue and other revenues. Passenger scheduled revenue is the main output for a passenger focused airline, cargo revenue includes outputs that are not passenger-flight related such as freight and mail services and other revenues includes charter revenue and a wide variety of non-airline businesses (incidental services) such as ground handling, aircraft maintenance for other airlines and advertising and sponsor. Even if incidental services are not airline s core business, they are considered in the production process under evaluation because they utilise part of the inputs included in the analysis (Oum and Yu 1998). All data have been developed from Financial Statements as at 31st December 2004 and from various internal reports. 3.2 Empirical Results The airline production process defined in this empirical study is characterised by a higher number of inputs than those considered in a previous study (Nissi and Rapposelli 2008), where only two categories of costs have been included as inputs in the DEA model. Unlike the previous paper, in this study all the original input variables are very highly correlated. This is therefore good material for using PCA to produce a reduced number of inputs by removing redundant information. Table 1 gives the eigenanalysis of the correlation matrix of data set. The first principal component X PC1 explains 98.22% of the total variance of the data vector, so the input variables will be included in the DEA model via the first principal component. It should be noted that principal components used here are computed based on the correlation matrix rather than on covariance, as the variables are quantified in different units of measure. Generally inputs and outputs of DEA models need to be strictly positive, but the results of a PCA can have negative values (Adler and Berechman 2001). It has been argued (Pastor 1996) that the BCC output- Table 1 Eigenvalues and total variance explained Component Eigenvalue Proportion (%) Cumulative (%) : : : : : : : : :

8 358 A. Rapposelli Table 2 Efficiency ratings of Air One domestic routes DMU Score DMU Score DMU Score DMU Score DMU Score DMU Score TT 1 Z 1 EE F H JJ A 1 V 1 L U N OO CC 1 J 1 T W QQ SS VV 1 E FF Q G PP DD 1 S X AA B MM oriented model used in the current study is input translation invariant and vice versa. Hence the efficiency classification of DMUs is preserved if the values of principal component X PC1 are translated by adding a sufficiently large scalar ˇ (1 in this case) such that the resulting values are positive for each DMU j. We apply therefore DEA on the translated first component and not on the whole set of the original input variables. In order to incorporate X PC1 directly into the linear programming problem the general DEA formulation has to be modified (Sect. 2.2). With regard to the output variables, they are not included in terms of principal components. Hence, only constraint (8)and(11) are used in the DEA model applied. The DEA analysis has been performed by using DEA-Solver software (Cooper et al. 2000). The efficiency scores of Air One routes, in descending order of efficiency, are shown in Table 2. Two different remarks can be made. The average level of technical efficiency is Eight routes are fully efficient and three more routes are quite close to the best practice frontier. On the other hand, the remaining DMUs are sub-efficient but they do not show very low ratings. These results suggest that Air One routes are operating at a high level of efficiency, although there is room for improvement in several routes. 4 Conclusions and Future Research It is well known that the discriminatory power of DEA often fails when there is an excessive number of inputs and outputs in relation to the number of DMUs (Adler and Golany 2002). We have introduced a new model formulation within DEA framework that can be used in efficiency measurement when there is a large number of inputs and outputs variables that can be omitted with least loss of information. This approach has been illustrated with an application to an Italian airline. However, these results can be improved. This study suggest three main avenue for future research. First of all, further research could include the presence of undesirable outputs (Liang et al. 2009) in the PCA DEA model proposed, such as the number of delayed flights, which may reflect the service quality of the network. Besides, the usefulness of the method could be explored for large data sets: for example, in further application studies we could add international routes or other air carriers. Finally, we would like to explore the combination between canonical

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