Data Envelopment Analysis
7.1 Model Specifications
7.1.4 Procedure for ranking efficient companies
real applications are constrained by the need to limit the number of companies labelled ‘best practice’. Consequently,for the majority of companies in a data set nothing can be said about performance other than the implicit presumption that,in attaining best-practice,their performance is satisfactory and ‘equiva- lent’. Since best-practice is not necessarily adequate in any absolute sense,it would be useful for the decision-maker to have more guidance on the quality of best-practice performance.
Andersen and Petersen (1993) suggested a modified version of DEA, which allows the ranking of efficient companies. Their basic idea was to compare the company under evaluation with a linear combination of all other companies in the sample,however,the company itself is excluded. Under this approach it is possible for an efficient company to increase its input vector proportionally while preserving efficiency. The company obtains,in that case, an efficiency score above one. The score reflects the radial distance from the company under evaluation to the production frontier estimated with that company excluded from the sample. The approach provides an efficiency rating of efficient companies similar to the rating of inefficient companies.
An illustration of Andersen and Petersen’s (1993) idea is given by Fig. 7.6.
Consider the evaluation of the efficient company P3in Fig. 7.6. According to the definition of the DEA efficiency measure,the reference point in the evalua- tion of P3is the observation itself,and P3is assigned the index one. Elimination of P3in the spanning of the reference set implies that P3is compared to that (inefficient) point in the input possibility set spanned by the remaining set of observations with the minimal distance of P3. The reference point thus becomes P3′. In analogy to the inefficiency index,the efficiency index is calculated by 0P3′/0P3and has the same interpretation as the Farrell measure:
P3may increase its input vector proportionally up to the efficiency index and still remain efficient,but it will be dominated by a combination of P2and P4if the proportional increase in the input vector exceeds the efficiency score.
Although this technique by Andersen and Petersen is obviously an impor- tant extension and can enhance existing DEA models,it is rarely applied in
Fig. 7.6. Ranking efficient companies (Andersen and Petersen, 1993).
performance research applications. One of the few exceptions is Jammernegg et al. (1997).
In the present study,the Andersen and Petersen procedure for ranking efficient companies is applied to the previously achieved DEA results of discretionary and non-discretionary input factors. In this model,out of 427 evaluations,208 (48.7%) hotels were identified as being efficient. Hence for almost 50% of the hotels in the database nothing else could be said as they performed equally efficiently.
Repeating the analysis with the Andersen and Petersen modifications for efficient companies resulted in a high number of infeasible solutions (59.1%).
This is in line with results by others who have found that estimates for a considerable number of companies are undefined because of the infeasibility of the set of constraints of the modified DEA model (Pastoret al.,1999). BoljunCic (1999) gives an explanation for this high number of infeasible solutions.
He identifies that the main problems are caused either by zero values in the variable set or by cases where some companies show extremely high efficiency values (BoljunCic,1999: 243). In general,infeasible solutions occur in attempts to solve large,complicated linear programming problems,where the constraints specified cannot be simultaneously satisfied (Schrage,1997: 7).
There is nothing much to do with infeasible linear programming solutions, hence the cases impacted must be excluded from further analysis.
The complete results are summarized in Appendix Table A.10. For an individual efficient company the analysis can provide two meaningful insights.
First,a company can monitor its efficiency development even for years when the firm has been classified as a ‘best-practice’ company in the sense of Farrell.
Figure 7.7 gives an example for hotel no. 14 which was a best-practice company in the years 1993,1994,1996 and 1997 (white bars) and an
Fig. 7.7. Inefficiency (n) and efficiency (o) scores for hotel no. 14.
inefficiently operated company in the years 1991, 1992 and 1995 (black bars). The figure clearly indicates the very regular development, gradually increasing between 1991 and 1993, and declining slowly from 1993 to 1995.
Note that hotel no. 14 crossed the ‘efficiency line’ twice, first from an inefficient to an efficient company and then back to an inefficient group of companies.
1996 was an extraordinary year for hotel no. 14, which certainly has to be considered in an overall evaluation.
There are several new procedures suggested in the literature for the analy- sis of efficient companies using DEA results. However, all of these extensions have in common that they introduce a complex form of two-stage approach in model formulation and computation (e.g. the ‘slack adjusted DEA model’
suggested by Sueyoshiet al. (1999); the DR/DEA model by Sinuany-Stern and Friedman (1998); or the ‘single price system extension’ by Ballestero (1999), which classifies efficient but not inefficient companies).
The model selected here is an input-oriented model, which seeks to identify technical inefficiency as a proportional reduction in input usage. As discussed in Chapter 4, it is also possible to measure technical efficiency as a proportional increase in output production, however, the former is the more adequate model for the manager. This is because hotel managers usually have objectives to fulfil, either set by corporate management goals or by self-defined business plans, and hence the input quantities appear to be the primary decision variables. In other applications it will also be possible that managers may be given a fixed quantity of resources and asked to produce as much output as possible. In this case an output orientation would be more appropriate.