Data Envelopment Analysis
7.1 Model Specifications
7.1.2 Longitudinal analysis in DEA
?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).
application of the window technique to DEA was a study of brand efficiency among various segments in the US carbonated beverage industry by Charnes et al. (1994a).
DEA window analysis considers the same companies in different time periods as separate observations. Out of a total ofTyears,DEA is performed on all companies defined in the earliestt<Tcontiguous years,that is,years 1–t. It is then performed on years 2–t+ 1, 3–t+ 2,and so on. Each company is thus analysed several times with slightly different comparison sets. In general, Charneset al. (1994a),who applied the windows technique on a quarterly data set,found thatt= 3 or 4 tended to yield the best balance of quality of information and stability of the efficiency scores.
The windows analyses using the AHRP database were performed for 3- and 4-year windows. The results were essentially identical,which indicates that the analysis of data is robust and not sensitive to the choice of the number of years in the moving window. The results presented here focus on the three-years solution,for which there are five window runs.9Table 7.2 shows the results for five hotels taken from the much larger window DEA. The four columns added on the right of Table 7.2 provide diagnostics for the stability of each brand’s efficiency ratings. The first two of these columns contain the mean efficiency rating and its variance over the 15 evaluations of each hotel.
The third column shows the largest difference in efficiency scores recorded for a single period,and the fourth column gives the difference between the maximum and minimum scores over all evaluations. These results show that the efficiency scores are fairly stable. This was usual for all window analysis runs. A summary of the complete analysis for all hotels is provided in Appendix Table A.6.
ANALYSIS OF PEER APPEARANCE When a DEA is performed for a single period,one might suspect that some of the peer hotels appear there only by chance and that they will not necessarily be a peer on other time periods. With a window analysis,each hotel is evaluatedm×ktimes,wheremis the number of windows andkis the number of periods in a window. Thus there arem×k lists of peer hotels (with their corresponding optimalλvalues) for each hotel in the database.
Charneset al. (1994a) proposed to analyse these results either by counting the number of times efficient companies appear in the reference set of all com- panies (including its own) or by summing the optimalλvalues corresponding to each efficient company over all peers. In order to derive the efficient reference companies that are most important in determining a particular company’s efficiency,they established a ‘facet participation table’ (Charnes et al.,1994a: 157),which allows one to rank order the efficient brands by their overall influence on the reference set. The table they introduced also
9 In general, the number of window runs can be calculated by the number of available periods – the number of periods in the moving window + 1.
gives insights into the geometric properties of the empirical production function. Companies with high counts tend to be located near the centre of the production frontier whereas those with low counts are located near the edges.
The ordering of some hotels starting from the highest peer appearance is displayed in Fig. 7.2. Hotel no. 2788,for instance,is the most frequent refer- ence hotel. This is based on 285 peer appearances. Note that the maximum number of times a hotel may be counted as a peer member is 915 (= 15×61).
Such a graph gives one a perspective on the most robustly efficient companies,i.e. companies that appear as reference firms most often. Based on the pattern seen one can establish a cut-off point in terms of the number of peer appearances and then select those companies that meet or exceed this standard as ‘exemplary cases’.
A second possible way to gain insight from the peer statistics,which was suggested by Charneset al. (1994a) but not examined in detail,was to consider the weights of the peers in the analysis. From the perspective of what makes a company inefficient,for instance,greater attention might be paid to those
j Hotel
no. 1991 1992 1993 1994 1995 1996 1997 Mean Var Column range Total
range 1
2
3
4
5 14
25
38
42
81 0.835
1.000
0.932
1.000
0.880 0.929 0.940
1.000 1.000
1.000 1.000
1.000 1.000
0.906 0.910
0.961 1.000 0.984
1.000 1.000 1.000
1.000 1.000 1.000
0.977 0.978 0.999
0.857 0.858 0.855
0.987 0.970 0.983
1.000 1.000 1.000
1.000 1.000 1.000
1.000 1.000 1.000
0.811 0.813 0.876
0.995 0.994 0.991
0.951 0.951 0.983
0.884 0.865 0.861
0.903 0.900 0.924
0.803 0.861 0.876
1.000 1.000
1.000 1.000
0.832 0.831
1.000 1.000
0.890 0.914
0.968
0.997
0.832
1.000
0.942 0.969
0.992
0.936
0.979
0.870 0.002
0.000
0.006
0.001
0.002 0.126
0.049
0.168
0.100
0.099 0.165
0.049
0.169
0.100
0.139 Table 7.2. Summaryof window DEA results for five hotels (t= 3).
peers with the highest λ values in a particular evaluation. In multiple evaluation situations, such as in window analysis, peer members with the largest sums ofλvalues across the total of their evaluations may be considered for comparative purposes.
For hotel no. 472 this is illustrated in Table 7.3. Between 1991 and 1997, based on the windows analysis, hotel no. 472 was identified ten times as inefficient and five times as efficient (column N in Appendix Table A.7). During the inefficient occurrences the hotels listed in Table 7.3 presented them- selves as comparison partners of hotel no. 472. Note that hotel no. 472 itself appeared six times in its own peer group, since, by definition, window analysis allows a company to be compared with its past and future performance. Also note that the mean weights (second column) clearly deviate from the number of counts (third column), thus leading to a different ranking for the optimal selection of comparison partners. Recommendations concerning the fine tuning of partner choice therefore differ depending on the strategy selected.
In summary, efficient companies with the largest sums ofλvalues across all observations most strongly evaluate all other companies. Because of this they may be considered as the benchmarks industrywide. To allow insights Fig. 7.2. Hotels with the most peer appearances.
DEA solutions to window width) is currently determined by trial and error.
Furthermore,representing each company as if it were a different company for each period in the window must be replaced by an approach that recognizes the continuity of firms over time. From an overall point of view,the high dependence on intuition involved in using the window analysis is motivation to think about alternative ways of time series analysis with DEA such as the Malmquist DEA.
The Malmquist DEA approach
Färeet al. (1992) combined Farrell’s ideas of efficiency with some work of Caveset al. (1982) on the measurement of productivity to a Malmquist index of productivity change. Caveset al. defined their Malmquist productivity index as the ratio of two input distance functions,while assuming no technical inefficiency in the sense of Farrell (1957). Färeet al. extended this approach by dropping the assumption of no technical inefficiency and developed a Malmquist index of productivity that can be decomposed into indices describing changes in technology and efficiency. Finally,Färe and Grosskopf (1996) summarized their findings on intertemporal production frontiers in a comprehensive textbook.
Recently,there were two extensions to the original Färe et al. (1992) approach. First,Simar and Wilson (1999) give a statistical interpretation to the Malmquist productivity index and its components,and present a bootstrap algorithm which may be used to estimate confidence intervals for the indices.
Second,Löthgren and Tambour (1999) modify and apply the DEA concept to model both production and consumption activities in Swedish pharmacies.
Färeet al. (1992) specify an input-based ‘Malmquist productivity change index’ as:
Peer no. Mean n SD Min Max
1061472 1206626 20051134 921226 277642 26312914 2771
0.583 0.472 0.302 0.196 0.184 0.093 0.084 0.079 0.072 0.037 0.022 0.007 0.005
62 105 28 23 71 11 1
0.388 0.031 0.194 0.140 0.210 0.064 0.021 0.053 0.058
0.035 0.450 0.089 0.016 0.035 0.008 0.069 0.019 0.007 0.037 0.022 0.007 0.005
0.920 0.494 0.486 0.362 0.332 0.184 0.098 0.119 0.169 0.037 0.022 0.007 0.005 Table 7.3. Peers for hotel no. 472 in 1991–1997.
( ) ( )
( )
M M x y x d x y
d x y
d x y
i t t t t i
t
t t
i
t t t
i
t t t
+ +
+ + + +
=
1 1
1
1 1 1
, , , ,
,
(
,)
(
+) ( ( ) )
+ + + × +
1 1
1 1 1
1 2
d x y
d x y
d x y
i t
t t
i
t t t
i t
t t
,
, ,
/
(7.1) The notation dit+1
(
y xt, t)
represents the distance from the observation in periodtto periodt+ 1. Hence,the input-oriented productivity measure com- pares the input requirements for producing output levelyt,produced in period t,with the input that would have been required if the production technology was the same as in periodt+ 1 (Grosskopf,1993: 183). This means the input oriented index essentially comparesxtwith what would have been required in periodt+ 1. The subscriptiin Equation 7.1 indicates the input-orientation of the measures.10ValuesMi(t1,t2) < 1 indicate improvements in productivity betweent1and t2,whereas valuesMi(t1,t2) > 1 indicate decreases in productivity from timet1
tot2;Mi(t1,t2) = 1 would indicate no change in productivity. The ratio outside the square brackets of Equation 7.1 measures the change in Farrell’s input technical efficiency between periodst1 and t2,and defines the input-based index ‘efficiency change’:
( ) ( )
( )
e y x y x d x y
d x y
i t t t t
i t
t t
i
t t t
+ +
+ + +
=
1 1
1
1 1
, , , ,
, (7.2)
Values ofei(t1,t2) less than 1 indicate improvements in efficiency betweent1
andt2,and vice versa. Similarly,the remaining part of the right-hand side of Equation 7.1 defines an input-based measure of ‘technical change’:
( ) ( )
( )
T y x y x d x y
d x y
d x
i t t t t ti
t t
i t
t t
i
t t
+ + + +
+ + +
= ×
1 1
1 1
1
1 1
, , , ,
,
( ) ( )
, ,
/
y
d x y
t i t
t t +
1
1 2
(7.3) As withMi(t1,t2) andei(t1,t2),values ofTi(t1,t2) less than 1 indicate technical growth between timest1andt2, and vice versa.
DEA can be used to measure the distance functions which make up the Malmquist index (Färeet al.,1994). In a two-period case,four distance func- tions must be calculated for each company in the database,thus requiring one to solve four linear programming problems. Recall the basic input-oriented DEA model for constant returns to scale in Equation 4.29 (p. 69). Including the time subscript gives the first linear program:
( )
[
dtio xio,minyto]
−1= fo (7.4)subject to:
λom itm o ito m
n
x −f x + =s−
∑
= 11
0 i= 1, . . . ,r
10 Note that the output-oriented Malmquist index is defined similarly to the input-oriented measures presented here (see Coelliet al., 1998).
λom jtm j jto m
n
y − =s+ y
∑
= 1j= 1, . . . ,s
The other three linear programs are logical derivations from Equation 7.4:
( )
[
dtio+1 xt+1o,minyt+1o]
−1= fo (7.5)subject to:
λom it m o it o m
n
x + f x + s−
=
− + =
∑
1 1 11
0 i= 1, . . . ,r λom jt m j jt o
m n
y + s+ y +
=
∑
1 − = 11
j= 1, . . . ,s
( )
[
dtio xt+1o,minyt+1o]
−1= fo (7.6)subject to:
λom itm o it o m
n
x −f x + + =s−
∑
= 1 1 10 i= 1, . . . ,r λom jtm j jt o
m n
y − =s+ y +
∑
= 11
j= 1, . . . ,s
( )
[
dtio+1 xto,minyto]
−1= fo (7.7)subject to:
λom it m o ito m
n
x + f x s−
=
− + =
∑
1 11
0 i= 1, . . . ,r λom jt m j jto
m n
y + s+ y
=
∑
1 − =1
j= 1, . . . ,s
For an output-oriented formulation of the linear programs see Coelli (1996:
28). The approach can also be extended to determine a DEA solution based on variable returns to scale. This requires the solution of two additional linear programming problems with convexity restrictions added to each. Compared to the constant returns to scale Malmquist approach,this increases the num- ber of linear programs fromN×(3×t−2) toN×(4×t−2). However,Grifell- Tatjé and Lovell (1995) demonstrated that the Malmquist index does not in general correctly measure changes when variable returns to scale technology is assumed. Several authors have recommended the use of constant returns to scale specifications to avoid these problems (e.g. Coelliet al., 1998: 228).
To apply the Malmquist DEA to the data set of this study,a minimum of 1159 linear programs had to be solved for a single constant returns to scale Malmquist DEA run. The software packageDEAP was used. Table 7.4
summarizes the results obtained from these optimization runs with the 1991 to 1997 data.
It should be noted that the first year of observation is used to initialize the Malmquist indices and is therefore set to 1. The cumulative indices of technical efficiency change,technical change and productivity change are shown in Fig. 7.3.
The decline in total factor productivity of Austrian accommodation pro- viders between 1991 and 1996 can be seen in Fig. 7.3 (dotted lines). Although
Year Efficiencychangeb Technical changeb Productivitychangeb 1991a
19921993 19941995 19961997
1.000 1.006 0.967 0.991 1.013 1.012 0.989
1.000 1.018 1.004 0.993 0.993 0.959 1.034
1.000 1.025 0.971 0.984 1.006 0.971 1.023
a1991 set to unity;ball Malmquist index averages are geometric means.
Table 7.4. Malmquist index summaryof annual means.
Fig. 7.3. Comparing the performance of hotel no. 688 with industry performance.
these results are based on a relatively small experimental data set,the negative developments in the Austrian hotel industry during 1991 and 1996 described earlier (see pp. 81–82) are observed. In the Malmquist analysis the total factor productivity index is decomposed to efficiency and technical change indices.
The favourable course of the efficiency change index between 1993 and 1996 indicates the efforts of the industry to balance the drop in demand with improved efficiency (e.g. profit improvement programmes).
Malmquist indices for individual hotels can be used to assist hotel manag- ers in benchmarking their performance against the decomposed industry performance indicators. A complete listing of the Malmquist indices for all companies in the database is given in Appendix Table A.8. For an example consider Fig. 7.3 in which the solid lines represent the performance of com- pany no. 688. Overall,the manager of hotel no. 688 experienced unfavourable conditions between 1991 and 1997. In spite of 1994,the productivity change index is always clearly below the cumulative index,which indicates a poor productivity compared to the performance of the industry.
The decomposition of the productivity index allows multiple insights. One can assess whether the bad overall performance refers solely to inefficiencies in operations or involves poor anticipation of technological changes in the industry by the management of the company. Comparing the development of the technical change index with the cumulative function reveals that hotel no. 688 anticipated technological changes fairly well,thus leaving the primary explanation of the poor performance to the relative inefficiencies, which especially occurred between the periods 1991–1993 and 1996–1997.
The Malmquist extension to DEA has several merits compared to the window analysis approach demonstrated in the previous section. The major benefit is that it permits total factor productivity to be decomposed into techni- cal change and technical efficiency change. It also has methodological merits;
it neither requires the parameterization of a window width nor does it have idiosyncratic characteristics like the multiple representation of one and the same company in each window.
The disadvantage of the Malmquist extension is that there are no peer groups which derive automatically from the analysis as is possible with traditional DEAs. The dynamic version of the efficiency score,the efficiency index,is defined by two vectors ofλs,one for the efficiency score evaluation in periodt,and one for the efficiency score evaluation int+ 1. It is still unclear from the literature how these two vectors can be combined to form one common reference vector for all periods of observation.