Data Envelopment Analysis
7.1 Model Specifications
7.1.1 Analysis of individual years
Originally DEA models were not designed to handle time series data or to evaluate performance from a dynamic point of view. Therefore,the basic DEA model introduced here is based on individual runs for each of the 7 years under evaluation. Recall the input-oriented DEA model with VRS specification by Banker (1984):
minfo (7.1)
subject to:
λom im o io m
n
x ≤f x
∑
= 1i= 1, . . . ,r λom jm jo
m n
y ≥y
∑
= 1j= 1, . . . ,s λom
m n
=
∑
= 1 11 www.banxia.co.uk
2 www.emq.se/onfront1.htm
3 www.warwick.ac.uk/~bsrlu/dea/deas/deas1.htm
4 www.ideas2000.com
5 www.saitech-inc.com
6 www.parn.org.uk/
7 www.une.edu.au/econometrics/deap.htm
8 www.wiso.uni-dortmund.de/lsfg/or/scheel/ems/
In the present application to the AHRP database,there are four input variables i(r= 4),and three output variablesj(s= 3). For each of the 61 companies in the database a linear programming task had to be defined,which made 427 optimization runs in total.
The piecewise linear form of the non-parametric frontier in DEA can cause difficulties in efficiency measurement of some companies which define the frontier. Refer to Fig. 4.8 (p. 70) where the companies using input combina- tions P3, P4, P5and P6are the four efficient companies that define the frontier, and companies P1and P2are inefficient firms. However,it is questionable as to whether the point P6is an efficient point since one could reduce the amount of inputx2(until P6= P3) and still produce the same output. This phenomenon is known as input slack and can be found in output-oriented models as well (output slack). TheDEAPsoftware gives the user three choices regarding the treatment of slacks (Coelli, 1996: 14):
1. One-stage DEA,which solves Equation 4.28 (p. 68),but ignores the need for a second optimization step. Slacks are calculated residually;
2. Two-stage DEA,which introduces the non-Archimedean infinitesimal (see p. 71),a very small number in Equation 4.31 used by most DEA software packages; and
3. Multistage DEA,which conducts a sequence of radial linear programs to identify the efficient projected point (Coelli, 1997).
The advantage of the multistage specification for slack optimization is that it identifies projected efficient points which have input and output mixes that are as similar as possible to those of the efficient points,and it is also invariant to units of measurement (Coelli,1996: 15). Therefore,this specification was selected for the analysis.
A complete summary of the findings is presented in Appendix Table A.5, which contains the efficiency scores for all seven years and the number of times each company resides on the frontier,and thus was selected as a peer for the inefficient companies. The far right column sums up the peer counts for each hotel and gives an overall picture of an individual hotel’s performance. For example,hotels no. 2631 and no. 2788 are the two most preferred bench- marking partners with 90 and 88 times being a peer candidate,respectively.
Between 1991 and 1997,among all hotels,a hotel could be identified being inefficient 219 times,which is 51.3% of all observations in the database. The mean score of all inefficient hotels between 1991 and 1997 is 0.829 and varies between 0.467 (no. 396 in 1997) to 0.998 (no. 2287 in 1994).
Figure 7.1 shows the operating profit plotted against the DEA efficiency score for each of the 61 hotels in 1997. The hotels deemed efficient and given a score of 1 are plotted along the right border. Although the relationship between profitability and efficiency is obvious,the plot clearly shows that not all hotels that have been classified as efficient are necessarily also profitable.
There are some low-profitability hotels that are run efficiently,and some high-profitability hotels that are run inefficiently. The DEA solution plotted
in Fig. 7.1 illustrates that by the use of this methodology,unanticipated insights may be obtained and may thus redirect managerial action. In a broader sense,the DEA framework can create an approach for learning from outliers and for inducing new theories of best practice.
Next,the capabilities and limitations of this simple DEA approach will be demonstrated by a case example.
Target setting
Hotel no. 283 (the eighth hotel in the list of hotels in this study) is a property having 97 beds and was selected for more detailed analysis. The possibilities for using DEA findings for managerial target setting can be demonstrated by means of the 1996 data. In this year,hotel no. 283 achieved?2,081,000 in F&B revenue and?1,061,400 in room revenue, and spent?3,074,600 for operating expenses. The gross profit was ?67,700, or 2.2% of the total revenue. The average occupancy rate per number of opening days was 68.9%.
That compares favourably with the average set’s occupancy rate of 54.9%, F&B revenue of?755,610, and total room revenue of ?509,871 in 1996.
Hence hotel no. 283 exceeded the average output measures of all hotels in the data set on all three of the performance measures. Variable returns to scale DEA shows that four of the remaining 60 hotels constitute a peer group for company no. 283. However,as is typically the case,the characteristics of the peer-group members do not perfectly match those of the unit being evaluated.
Fig. 7.1. Profitability versus DEA efficiency for 61 Austrian hotels in 1997.
The linear program builds a weighted composite of the four efficient peer-group members identified so that they perfectly match the levels of outputs of the hotel being evaluated. Thus a composite,efficient benchmark company is derived that can be used to develop the targets for hotel no. 283.
Particularly,it shows the resource–expenditure targets for hotel no. 283 based on the management achieving at least the same room revenue,F&B revenue, and occupancy rate as benchmark. The benchmarking figures are summarized in Table 7.1.
The efficiency score for hotel no. 283 is 0.948. As it turns out,hotel no. 283 ranks only 40th out of the 61 hotels in efficiency,notwithstanding the fact that the hotel’s occupancy rate and revenue figures all exceed those of the average set. The peer group is matched to hotel no. 283 on the levels of all outputs. In relation to output factors the virtual benchmarking company either performs equal or even better than the hotel under evaluation. The benchmarked F&B revenue figures are the same as those for hotel no. 283,and the benchmark for the room revenue as well as the occupancy rate show higher levels compared to hotel no. 283.
The benchmarks that are most interesting are those for the eight resource expenditures. They constitute a set of guidelines for hotel no. 283 to work toward. If hotel no. 283 utilizes its resources more efficiently,it should be able to achieve the same output with lower expenditures. The DEA analysis indicates that hotel no. 283 has the potential to improve its gross profit by
t= 1996 m= 8
(no. 283) m= 4
(no. 42) m= 14
(no. 626) m= 49
(no. 2788) m= 43
(no. 588) BMb Total F&B revenue (?)
Total room revenue (?) Occupancya(%)
2,081,000.9 1,061,400.9 2,081,068.9
1,947,800.9 2,525,400.9 2,081,080.4
2,853,100.9 1,911,100.9 2,081,083.0
484,700.9 68,100.9 081,037.5
1,348,700.9 2,209,500.9 2,081,042.4
2,081,000.9 1,195,700.9 2,081,070.5 Number of beds
Number of seats Number of opening days
2,081,140.9 2,081,550.9 2,081,292.9
2,081,137.9 2,081,220.9 2,081,260.9
2,081,180.9 2,081,400.9 2,081,270.9
081,028.9 081,150.9 081,300.9
2,081,068.9 2,081,180.9 2,081,364.9
2,081,133.9 2,081,304.9 2,081,277.9 Total expenditures (?)
Payroll and related (?) Material-type exp. (?) Energycosts (?) Cleaning costs (?) Maintenance costs (?) Communication costs (?) Marketing costs (?) Administration costs (?)
3,074,600.9 1,580,100.9 2,552,800.9 2,199,600.9 2,089,200.9 2,210,800.9 2,050,100.9 2,072,200.9 2,319,800.9
2,097,600.9 2,877,300.9 2,580,600.9 2,108,800.9 2,091,300.9 2,160,700.9 2,018,400.9 2,138,900.9 2,121,700.9
4,351,100.9 1,933,500.9 2,892,400.9 2,167,700.9 2,079,100.9 2,630,300.9 2,065,300.9 2,252,600.9 2,330,200.9
310,500.9 171,200.9 52,600.9 21,400.9 11,000.9 17,800.9 6,400.9 7,300.9 22,800.9
1,330,800.9 2,666,000.9 2,378,600.9 2,083,700.9 2,019,600.9 2,071,700.9 2,019,900.9 2,031,000.9 2,060,200.9
2,915,300.9 1,295,800.9 2,620,200.9 2,119,600.9 2,063,000.9 2,391,100.9 2,042,100.9 2,168,900.9 2,214,700.9 Gross profit (?) 2,067,700.92,375,600.92,413,100.9242,300.92,227,400.92,361,300.9
aPer opening days;bbenchmark (composite group) based on LP solution withλ42*= 0.162, λ14*= 0.571,λ49*= 0.259 andλ43*= 0.008;e8*= 0.948.
Table 7.1. DEA benchmarks for hotel no. 283.
?293,600 (more than four times the observed gross profit), when it reduces it expenditures to ?2,915,300 (a decrease of 5.2%). The identification of the four peer properties can be made available to hotel no. 283 so that the management can ascertain from them (perhaps through site visits) the details of the processes and practices that enable them to perform better.
Finally it should be mentioned that the target-setting capabilities of DEA demonstrated by the example can be used to develop policy-making scenarios that would enable managers to identify the operating response to different managerial priorities. This kind of sensitivity analysis in conjunction with scenario planning for electricity generating plants was recently introduced by Athanassopouloset al. (1999).