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over which inefficient DMUs need to cross in order to become efficient.
Finally, we will compare the weight schemes with PCA-DEA (see Chapter
8), and view the results graphically.
Table 10-3. Variables over which Forestry Boards were analyzed
Input Output
Total Costs
1.
2.
Forest Management Planning
Woodlot-level forest management plans
Regional forest management planning
3.
4.
5.
Forest Ditching
Forest ditches planned
Forest ditching supervised
Forest ditches inspected and improved
6.
7.
8.
Forest Road Construction
Forest roads planned
Forest road building supervised
Forest roads inspected and approved
9.
10.
11.
Training and Extension
Forest owners attending group extension meetings
Forest owners offered face-to-face assistance
Training offered to Forestry Board personnel
12.
13.
Handling Administrative Matters of Forest Improvement
Forest improvement projects approved
Regeneration plans approved
14.
15.
Overseeing the Applications of Forestry Laws
Forest sites inspected for quality control
Forest sites inspected for tax relief
In Figure 10-3, the raw data is presented graphically using a plot with
θ=0.10 and the average of correlations equal to 0.83. We can
immediately see that DMU 1 is the smallest unit and that DMUs 11,
13, 17 and 19 are the largest and, by definition, will be BCC efficient
(Banker et al. (1984)) due to their individual specializations.
Furthermore, DMU 18 would appear to be a special unit in that it does
not belong to any cluster.
Co-Plot can also be used to analyze each variable individually and
we have chosen variable 9, the number of
forest owners attending group
extension meetings, randomly. Figure 10-4 presents a Co-Plot in which the
data was sorted automatically, with DMU 9 to the far right of the arrow
producing the smallest amount of this output and DMUs 10 and 12
producing the most. We can also see the large gap between the two most
prolific forestry boards and the next in line, namely DMU 2.
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Figure 10-3. Presentation of Raw Forestry Data using Co-Plot
Figure 10-4. Co-Plot Presentation of Output 9
Adler, Raveh & Yazhemsky, DEA Presented Graphically
180
Figure 10-5. Co-Plot of Finnish Forestry Boards over Ratios of Variables
In Figure 10-5, the Co-Plot display presents the ratios of data for 19
forestry boards over 15 ratios, with a coefficient of alienation θ=0.19 and an
average of correlations 0.647. The text output of this plot is presented in
Table 10-4. It would appear that ratios 8 and 9 have little value in this
analysis, hence their respective arrows are very short relatively and their
correlation values are 0.23 and 0.24 respectively. r8 and r9 were
consequently eliminated from in the subsequent analyses.
By removing ratios 8 and 9, we can improve the statistical significance of
the plot, as shown in Figure 10-6, where the coefficient of alienation is 0.16
and the average of correlations is 0.72. In this plot, the single BCC
inefficient DMU is lightly shaded, namely DMU3, and all the remaining
observations are BCC efficient (see Table 10-6 for full details). The DMUs
that become OW inefficient are colored white, the DMUs that become AR
inefficient are shaded and those that remain efficient in all models are
colored black. It becomes apparent that the OW weight restriction approach
removes DMUs on an inner circle that are neither specialists in any specific
area nor reasonably good at all criteria, namely those placed close to the
center-of-gravity in the plot. The AR weight restriction approach removes all
the inner DMUs from the envelope frontier, as demonstrated in Figure 10-6.
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This is true for all observations except DMU 18, which is efficient in all four
models apart from the assurance region model, where it is the closest unit to
the Pareto frontier with an efficiency score of 0.95.
Table 10-4. Correlation of Ratios
Coefficient of Alienation: 0.188
Center of Gravity: (55.20,40.70)
Average of Correlations: 0.647
Observation X
1i
X
2i
Variable Degree Correlation
1 44.31 0
1
66 0.62
2 68.43 7.35
2
-167 0.51
3 77.78 39.6
3
25 0.85
4 45.41 21.14
4
14 0.88
5 58.91 35.22
5
12 0.9
6 51.66 50.66
6
127 0.9
7 50.98 24.2
7
127 0.77
8 49.24 62.49
8
44 0.23
9 30.68 44.88
9
-54 0.24
10 50.18 451.18
10
113 0.71
11 31.67 89.48
11
-6 0.56
12 38.43 56.93
12
82 0.92
13 100 70.74
13
98 0.43
14 75.04 65.14
14
128 0.38
15 89.58 40.06
15
78 0.8
16 76.84 35.75
17 98.1 37.94
18 0 9.69
19 12.34 40.85
Figure 10-6. Co-plot graphic display for 19 forestry boards with 13 ratios
Adler, Raveh & Yazhemsky, DEA Presented Graphically
182
Co-plot could also be used to demonstrate the path over which an
inefficient DMU needs to travel in order to be considered efficient. For
example DMU 3 is considered inefficient in all models and in Figure 10-7
the inefficient DMU and its hypothetical, efficient counterpart can be viewed
and explained. Under the OW weight restricted model, the DMUs that
belong to the benchmark group appear dark and DMU 3 is lightly shaded.
The weight on DMU 11 is minimal, in other words the facet defined for
hypothetical DMU 3* includes DMUs 5 (17%), 13 (20%) and 14 (63%).
Figure 10-7. Co-Plot displaying the path to efficiency for DMU 3
An additional point that we would like to discuss in this section is the
potential use of PCA-DEA (described in detail in Chapter 8) to improve the
discriminatory power of DEA with such datasets. After running a principal
component analysis of the 15 outputs, we can see that three principal
components will explain more than 80% of the original data variance and
should be sufficient to analyze this dataset. The three components and their
powers of explanation are presented in Table 10-5 and a Co-Plot of the PCA-
DEA results is presented in Figure 10-8.
Table 10-5. Principal Component Analysis of Outputs
% information
PC1 56.6
PC2 15.5
PC3 12.8
Total Variance Explained 84.8
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Figure 10-8. PCA-DEA pictorially
Table 10-6. DEA scores for various models
DMU BCC CCR OW (VRS) AR (VRS) 3 PCA-DEA (VRS)
1 1 1 1 1 1
2 1 1 1 0.85 0.92
3 0.98 0.92 0.86 0.8 0.83
4 1 0.92 0.88 0.81 0.90
5 1 1 1 0.77 0.84
6 1 1 1 0.8 0.91
7 1 1 0.86 0.77 0.72
8 1 1 1 0.85 0.97
9 1 1 1 0.83 0.98
10 1 1 1 0.84 0.82
11 1 1 1 1 1
12 1 1 0.98 0.81 0.91
13 1 1 1 1 1
14 1 1 1 1 1
15 1 1 1 0.86 0.89
16 1 1 0.96 0.81 0.84
17 1 1 1 1 0.94
18 1 1 1 0.95 0.98
19 1 1 1 1 0.99
Adler, Raveh & Yazhemsky, DEA Presented Graphically
184
Figure 10-8 clearly presents the remaining 4 DMUs that are efficient
under the PCA-DEA model in the outer ring of data. In order to compare this
model with the results of the weight-constrained models, OW and AR, we
present the efficiency scores in the last column of Table 10-6 and a Co-Plot
of the different results in Figure 10-9.
Figure 10-9. Co-Plot displaying the results of various DEA models
Figure 10-9 presents a Co-Plot with value of alienation equal to 0.04 and
average of correlations equal to 0.91. PCA-DEA has provided the greatest
level of discrimination, with results very similar to the other models, but
removes an additional two DMUs from the efficient set of the AR model.
Indeed, one could argue that the most efficient set of DMUs, painted a dark
color in Figure 10-9, are likely to be truly relatively efficient, since all the
models are in agreement over this cluster. Other DMUs, such as DMU 3, 4
and 7 are likely to be either very small or strictly inefficient and DMUs 17
and 18 would appear to be very close to the efficient frontier, based on the
findings of all the models combined.
5. CONCLUSIONS
In this chapter we have discussed two methodologies that can be used to
analyze multiple variable data. Data envelopment analysis (DEA) is
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frequently used to analyze the relative efficiency of a set of homogeneous
units with multiple inputs and outputs. The results group the data into two
sets, those units that are considered efficient and define the Pareto frontier
and those that are not. Co-Plot is a graphical technique, which reduces each
unit to an observation in two dimensions, enabling a plot to be drawn of the
entire dataset. After applying Co-Plot over the set of variable ratios (each
output divided by each input), in order to align the technique to the idea of
efficiency as utilized in DEA, we can then use Co-Plot to graphically display
the DEA results.
This chapter has illustrated the approach using a problematic dataset
drawn from the literature, in which 19 Finnish Forestry Boards are compared
over one input and fifteen outputs. The Co-Plot presents the variable
direction in the form of arrows, DMUs in the form of circles and colors
separate the efficient from inefficient units. The further an observation
appears along a particular ray, the more efficient that DMU is with respect to
that ratio. The DEA efficient units can be clearly identified in the Co-Plot as
the (partial) outer-rings. Thus, each methodology can be used as an aid in
understanding the relative efficiency of the observations. Whilst DEA
provides a detailed analysis of how inefficient units can be improved to
attain relative efficiency, a series of Co-Plots can be used as graphical aids to
explain the results to management. Furthermore, Co-Plot is also useful in an
exploratory data analysis to identify extreme outliers, which at the least need
to be considered closely for potential data measurement errors, if not lack of
homogeneity amongst observations. It can also be used to identify
unnecessary variables that contribute little to the analysis, which may help
with the problem of excess dimensionality that occurs in DEA.
REFERENCES
1. Adler, N., A. Raveh. 2006. Presenting DEA graphically. forthcoming in
Omega.
2. Albriktsen, R., F. Forsund. 1990. A productive study of the Norwegian
building industry. Journal of Productivity Analysis 2 53-66.
3. Banker, R.D., A. Charnes, W.W. Cooper. 1984. Some models for
estimating technical and scale inefficiencies in data envelopment
analysis. Management Science 30/9 1078-1092.
4. Belton, V., S.P. Vickers. 1993. Demystifying DEA – A virtual
interactive approach based on multiple criteria analysis. Journal of the
Operational Research Society 44 883-896.
Adler, Raveh & Yazhemsky, DEA Presented Graphically
186
5. Charnes, A., W.W. Cooper, B. Golany, L. Seiford, J. Stutz. 1985.
Foundations of data envelopment analysis for Pareto-Koopmans efficient
empirical production functions. Journal of Econometrics 30 91-107.
6. Desai, A., L.C. Walters. 1991. Graphical presentations of data
envelopment analysis: Management implications from parallel axes
representations. Decision Sciences 22 335-353.
7. El-Mahgary, S., R. Lahdelma. 1995. Data envelopment analysis:
Visualizing the results. European Journal of Operational Research 85
700-710.
8. Gilady, R., Y. Spector, A. Raveh. 1996. Multidimensional scaling: An
analysis of computers 1980-2990. European Journal of Operational
Research 95 439-450.
9. Golany, B., J.E. Storbeck. 1999. A data envelopment analysis of the
operational efficiency of bank branches. Interfaces 29 14-26.
10. Guttman, L. 1968. A general non-metric technique for finding the
smallest space for a configuration of points. Psychometrica 33
469-506.
11. Hackman, S.T., U. Passy, L.K. Platzman. 1994. Explicit representation
of the two-dimensional section of a production possibility set. The
Journal of Productivity Analysis 5 161-170.
12. Joro, T., E-J Viitala. 2004. Weight-restricted DEA in action: From
expert opinions to mathematical tools. Journal of the Operational
Research Society 55 814-821.
13. Lipshitz, G., A. Raveh. 1994. Applications of the Co-Plot method in the
study of socioeconomic differences among cities: A basis for a
differential development policy. Urban Studies 31 123-135.
14. Maital, S., A. Vaninsky. 1999. Data envelopment analysis with a single
DMU: A graphic projected-gradient approach. European Journal of
Operational Research 115 518-528.
15. Raveh, A. 1986. On measures of monotone association. The American
Statistician 40 117-123.
16. Raveh, A. 2000a. The Greek banking system: Reanalysis of
performance. European Journal of Operational Research 120
525-534.
17. Raveh, A. 2000b. Co-Plot: A graphic display method for geometrical
representations of MCDM. European Journal of Operational Research
125 670-678.
18. Scheel, H. 2001. Efficiency Measurement System. http://www.wiso.uni-
dortmund.de/lsfg/or/scheel/ems.
19. Sinuany-Stern, Z., L. Friedman. 1998. DEA and the discriminant
analysis of ratios for ranking units. European Journal of Operational
Research 111 470-78.
Chapter 10
187
20. Talby, D. 2005. Visual Co-Plot 5.5. http://www.cs.huji.ac.il/
~davidt/vcoplot/
.
21. Talluri, S., M.M. Whiteside, S.J. Seipel. 2000. A nonparametric
stochastic procedure for FMS evaluation. European Journal of
Operational Research 124 529-538.
22. Viitala, H.J., H. Hänninen. 1998. Measuring the efficiency of public
forestry organizations. Forestry Science 44 298-307.
23. Weber, C.A., A. Desai. 1996. Determination pf paths to vendor market
efficiency using parallel coordinates representation: A negotiation tool
for buyers. European Journal of Operational Research 90 142-155.
24. Weber, Y., O. Shenkar, A. Raveh. 1996. National versus corporate
cultural fit in mergers and acquisitions: An exploratory study.
Management Science 42 1215-1227.
25. Zhu, J. 1998. Data envelopment analysis versus principal component
analysis: An illustrative study of economic performance of Chinese
cities. European Journal of Operational Research 111 50-61.
The authors would like to thank the Recanati Fund for partial funding of this
research.
Adler, Raveh & Yazhemsky, DEA Presented Graphically