%DQ[LD)URQWLHU$QDO\VW
3OXV3DFN
8VHU·V*XLGH
4XLFN6WDUW"
4XLFN6WDUW"
Cross efficiency analysis is a complex facility, and requires a high
level of understanding. Please read the whole of this manual before
contacting Banxia for assistance.
•
Purchased with main software:
If you purchased Plus Pack 2 at the same time as the main Frontier
Analyst software, then Plus Pack 2 is already enabled and working.
•
Purchased at a later date:
If you purchased Plus Pack 2 at a later time than the main software
purchase, you will be supplied with a new install code. To activate Plus
Pack 2, the following steps should be followed.
1.
Run the Frontier Analyst Professional software (it does not matter
whether you open a project or not).
2.
Select the File menu Change install code item.
3.
The licence information
required dialog appears.
4.
Enter the details supplied
on the updated registration
information supplied when
your purchased the
upgrade (if you cannot
find it, contact Banxia
Software). Note that the
details must be entered exactly – if the name or organisation have
changed, you must enter them too.
5.
When the install code is correct, click the OK button. The Plus
Pack is immediately available.
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
LLL
4XLFN6WDUW"
Published in the UK, April 2001
LY
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
&RS\ULJKWDQG'LVFODLPHU
&RS\ULJKWDQG'LVFODLPHU
The software, manual and other materials relating to the software are
protected by United Kingdom copyright law and international treaty
provisions. They are provided to you under licence, a copy of which is
included in the package.
All parts of the Frontier Analyst Professional program and packaging,
documentation and other related items are © Copyright 1995-2001
Banxia Holdings Limited and Andrew Tait unless specifically noted.
Frontier Analyst is marketed and distributed by Banxia Software Ltd.
Information in this document is subject to change without notice and
does not represent a commitment on the part of Banxia Holdings
Limited, Banxia Software Limited or any reseller. This document does
not represent a fixed specification or warranty. It is intended only to
give an idea of what Frontier Analyst Professional and Plus Pack 2 may
be like. We reserve the right to change any part of Frontier Analyst
Professional without prior notice.
Banxia Holdings Limited and Banxia Software Limited specifically
disclaim all warranties either expressed or implied, including but not
limited to, implied warranties of merchantability and fitness for a
particular purpose with respect to defects in the software, distribution
media and documentation. Any documentation supplied with the
package does not represent a specification for the software. Similarity
to results obtained by other software shall not be taken as indication that
the results will be similar in any other situation. In no event shall
Banxia Holdings Limited or Banxia Software Limited be liable for any
loss of profit or any other damage, including but not limited to special,
incidental, consequential or other damages caused by any product or
service provided.
Banxia and Frontier Analyst are registered trademarks of Banxia
Holdings Limited.
Microsoft and MS-DOS are registered trademarks, and Microsoft Windows is a
trademark of Microsoft Corporation. Other trademarks are registered to their respective
owners. Banxia Software Limited is registered in England #2966533.
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
Y
2EWDLQLQJ)URQWLHU$QDO\VW3URIHVVLRQDO,QIRUPDWLRQ
2EWDLQLQJ)URQWLHU$QDO\VW3URIHVVLRQDO
,QIRUPDWLRQ
We intend to continuously develop Frontier Analyst Professional, so
the content and operation of the software may change. This User’s
Guide should represent a basic source of data - but you must not rely
upon it absolutely. It is designed primarily as a reference manual and
describes the commands and actions, and their main purposes.
The most up-to-date information, describing changes to Frontier
Analyst Professional is documented in the README.HTML file on the
CD-ROM or floppy disks supplied with any software updates you may
receive. Print this file before proceeding with any new version of the
software. Also, the on-line help will have current information on new
commands and options. This is available at any time in the program by
pressing the O key.
For further information about Frontier Analyst Professional, contact:
Banxia Software Ltd
141 St James Road
Glasgow, G4 0LT, UK.
Telephone:
Fax:
Email:
WWW:
+44 (0) 870 787 2994
+44 (0) 870 787 2995
[email protected]
http://www.banxia.com
7UDLQLQJ&RQVXOWDQF\
Training and consultancy in the use of Frontier Analyst is available
from a variety of sources. Please contact Banxia Software or visit our
web pages for current details.
Please note that although we may put you in touch with independent
trainers and consultants, we cannot warrant their performance.
YL
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
&RQWHQWV
&RQWHQWV
Quick Start?.................................................................. iii
Copyright and Disclaimer ..............................................v
Obtaining Frontier Analyst Professional Information...vi
Training & Consultancy........................................vi
Contents ...................................................................... vii
Welcome! .......................................................................1
Why Frontier Analyst Professional Plus Pack 2? ..1
Installing.........................................................................3
Running Frontier Analyst.......................................4
Accessing the Cross Efficiency Analysis ...............4
Cross Efficiency Window...............................................5
Calculation Basis ........................................5
Why Might Cross Efficiency Scores Be Useful? ...8
Cross Efficiency and Variable Returns to
Scale. ..........................................................9
Cross Efficiency in Operation...................10
Performance Tip .......................................10
Troubleshooting ...........................................................12
Feedback Welcome ......................................................13
Index ............................................................................14
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
YLL
&RQWHQWV
YLLL
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
:HOFRPH
:HOFRPH
Welcome to the Frontier Analyst Professional edition Plus Pack 2
User’s Guide. This guide will tell you how to install and operate the
Plus Pack 2 software and a little about how it may be used. It is written
to complement the main Frontier Analyst Professional User’s Guide.
:K\)URQWLHU$QDO\VW3URIHVVLRQDO
3OXV3DFN"
This “Plus Pack” is designed to enhance the analysis capabilities of
Frontier Analyst. Plus Pack 2 contains the following items:
•
Cross Efficiency module
The cross efficiency module allows you to investigate further the
characteristics of your data and hence the operating characteristics of
your organisational units.
Plus Pack 2 gives you an advanced analysis option. It is therefore
important that you understand the mathematics and theory of data
envelopment analysis before making extensive use of the cross
efficiency analysis option. This manual tells you how to install and
operate the software but does not, and cannot, give you an
understanding of the mathematical theory associated with the more
complex results which cross efficiencies can yield. Please see the
bibliography on our web site at www.banxia.com for an extensive
reading list on the theory of data envelopment analysis.
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
:HOFRPH
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
,QVWDOOLQJ
,QVWDOOLQJ
Note: If you have trouble with the installation, please contact Banxia
Software for assistance. Email [email protected] with details of the
problem.
•
Purchased with main software:
If you purchased Plus Pack 2 at the same time as the main Frontier
Analyst software, then Plus Pack 2 is already enabled and working.
•
Purchased at a later date:
If you purchased Plus Pack 2 at a later time than the main software
purchase, you will be supplied with a new install code. To activate Plus
Pack 2, the following steps should be followed.
1.
Run the Frontier Analyst Professional software (it does not matter
whether you open a project or not).
2.
Select the File menu Change install code item.
3.
The licence information
required dialog appears.
4.
Enter the details supplied
on the updated registration
information supplied when
your purchased the
upgrade (if you cannot
find it, contact Banxia
Software). Note that the
details must be entered exactly – if the name or organisation have
changed, you must enter them too.
5.
When the install code is correct, click the OK button. The Plus
Pack is immediately available.
If the Plus Pack is not enabled, and you cannot access its features,
please contact Banxia Software. The install code information will
specify which Plus Packs are enabled (if any).
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
,QVWDOOLQJ
Cross efficiency analysis is a complex facility, and requires a high
level of understanding. Please read the whole of this manual before
contacting Banxia for assistance.
5XQQLQJ)URQWLHU$QDO\VW
Frontier Analyst is in the Programs menu, Banxia Frontier Analyst submenu, and you need only to select it to start the application.
The operation of most aspects of the software is covered in the main
Frontier Analyst Professional documentation.
$FFHVVLQJWKH&URVV(IILFLHQF\$QDO\VLV
To view the cross efficiency analysis information, select the View menu
Cross efficiency item. Alternatively, select the Analysis section of the
option bar and select the Cross efficiency icon. (If these items are
disabled, Plus Pack 2 is not installed. Please see the installation
information on page 3.)
The Cross efficiency window will appear, and automatically start to
perform the analysis. Because the calculation is very large, a progress
display is shown. For large projects the analysis can take a long time
(many calculations have to be performed – variables * units * units).
You can interrupt the analysis at any time by closing the window, or by
performing any other operation. The calculation will be done in the
background, and will automatically restart if the main DEA analysis is
recalculated. Once complete the window will be updated and the values
displayed. You can easily copy the values to the clipboard for further
analysis.
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
&URVV(IILFLHQF\:LQGRZ
&URVV(IILFLHQF\:LQGRZ
The Cross Efficiency window is used to display a post-analysis
calculation which compares each unit with every other. The basis of the
comparison is applying the weights of the other units to the unit being
considered, to see what score it would have been given under another
unit’s weighting scheme.
The analysis results are available in a tabular form (which can be copied
to the clipboard or saved to disk), or as a graph. The graph displays the
values across and/or down the grid for a particular unit. Click on the
unit in the list to display its graphs.
&DOFXODWLRQ%DVLV
The average cross efficiency of a DMU is calculated using the unit’s
variable values and the weights derived (from analysis) for the other
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
&URVV(IILFLHQF\:LQGRZ
units in the data set. The variable weights used in the cross efficiency
calculation are those derived from a “simple” (non-weight restricted)
DEA solution. So a “simple” DEA calculation (based either on the BCC
or CCR model) must be carried out for the data set prior to the
calculation of the cross efficiencies. A matrix is then constructed to
calculate each unit’s cross efficiency score.
In the cross efficiency matrix, reading down the columns, the
relationship is - “my unit’s efficiency scores, when viewed using my
variables with other peoples’ (units’) weights”. Reading along the rows
the relationship is - “my unit’s weights applied to other peoples’ (units’)
variables”.
The averages obtained by averaging down each column are known as an
“averaged appraisal by peers” - the average cross efficiency obtained
for the unit when each of the other unit’s weights have been applied to
its variables. This gives a peer assessment in the sense of the unit’s
performance being examined when the unit is rated as if it is operating
under other units’ trade-offs (as illustrated through their weighting
schemes). Under an analysis based on the CCR model with input
minimisation and no uncontrolled variables, units with a high averaged
appraisal by peers are representative of good all round performers,
since even when portrayed under other units’ trade-offs (weights), they
have a high efficiency score.
The averages obtained by averaging along each row is known as an
“averaged appraisal of peers”, an average of all the units efficiencies
using one units set of weights. Again, under an analysis based on the
CCR model with input minimisation and no uncontrolled variables a
unit might be said to be “truly” efficient, if it is the only unit in the data
set (in the row) which is 100% efficient under its own weighting
scheme.
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
&URVV(IILFLHQF\:LQGRZ
A cross efficiency matrix for four units would look like this:
Rated DMU
2
Averaged
Rating
DMU
1
3
4
appraisal
1
E11
E12
E13
E14
A1
2
E21
E22
E23
E24
A2
3
E31
E32
E33
E34
A3
4
E41
E42
E44
E44
A4
e1
e2
e3
e4
of peers
Averaged appraisal by peers
For each cell in the matrix, the cross efficiency of unit j is calculated
with the optimal weights derived (from a “simple” DEA calculation) for
unit i. Looking at the second column (unit 2), the cross efficiency E12 is
the cross efficiency for unit 2, using unit 2’s variables and unit 1’s
weights. The cross efficiency E22 is the cross efficiency for unit 2, using
unit 2’s variables and unit 2’s weights (in other words this is the
efficiency score from the initial, non-weight restricted analysis). The
cross efficiency E32 is the cross efficiency for unit 2, using unit 2’s
variables and unit 3’s weights. The cross efficiency E42 is the cross
efficiency of unit 2, using unit 2’s variables and unit 4’s weights. The
results reading down the columns tell us how consistently (or otherwise)
a unit is performing. The result reading along the rows give a “peer selfcomparison” (explained in more detail in item 4 of the cross efficiency
uses section, below). The matrix shown here illustrates the general
principles for the calculation of cross efficiencies. The interpretation of
the results is relatively straightforward for the CCR, input minimisation
situation and the comments given in this and the next section apply to
that situation.
After each set of weights has been applied to each unit, the average
cross efficiency for each unit is calculated by simple summation and
division (the sum of the column values divided by the number of units).
This average can be inclusive or exclusive of the leading diagonal in the
matrix. The values in the leading diagonal represent each unit’s own
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
&URVV(IILFLHQF\:LQGRZ
efficiency score from the “simple”/ initial DEA calculation. In a large
data sent the inclusion of the leading diagonal is likely to make very
little difference to the average efficiency scores. In a smaller data set
you may want to exclude the leading diagonal, so that the unit is not
contributing to its own averaged appraisal by peers.
:K\0LJKW&URVV(IILFLHQF\6FRUHV%H8VHIXO"
Cross-efficiencies are useful because they help to:
1.
Overcome the problem of “maverick” DMU’s .
A “maverick DMU” is a unit whose efficiency score is based only on
one or a small number of all of the available inputs/ outputs. (This
might be typified by the attitude that: “we’re great at one thing, so never
mind the rest”). The existence of such units may suggest the need to
introduce weight restrictions into the model. Cross efficiencies can be
used to identify maverick DMU’s – the greater the difference between
the “simple” efficiency score of a unit and the average cross efficiency
for the unit, the more of a “maverick” that unit is. And cross efficiencies
allow you to see how these “maverick units” would perform if they
were operating under other units’ trade-offs (so you may choose not to
introduce weight restrictions into the model, but to use the cross
efficiency results as the basis for identifying performance
improvements).
2.
Assess the similarity of the appraisal by peers
Cross efficiencies make it possible for the DMU’s included in the
analysis to be classified into separate groupings. By calculating the
correlation coefficient between a pair of columns, it is possible to
identify how similarly the two units have been appraised by their peers.
By doing this for each of the columns, it is possible to put DMU’s into
clusters grouping similar units.
3.
Sub-classify 100% efficient peers
Through using cross efficiencies you can distinguish between 100%
efficient units, providing a more meaningful ranking amongst efficient
peers. The “simple”/ initial DEA calculation results in the identification
of a number of “peer” or reference units. The cross efficiency scores in
the matrix and the average cross efficiencies give you insight into
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
&URVV(IILFLHQF\:LQGRZ
whether or not these 100% efficient peers are consistently good
performers (in other words whether or not they have a high average
cross efficiency score). The units with the highest average cross
efficiency by peer need not necessarily be 100% efficient based on a
“simple” efficiency calculation, (however in most cases they will be).
4.
Differentiate “truly” efficient peers
A “truly efficient” peer is one which is not only 100% efficient based
on the “simple” DEA calculation, but also one whose own efficiency
score (Ejj) is greater than all other DMU’s cross efficiency scores (Eij
for all i) using j’s weights (i.e. looking at the rows in the cross
efficiency matrix, the 100% efficient unit has the highest value in the
row). The question is: “if I am 100% efficient, is anyone else 100%
efficient when using my weights?”. If not, I am truly differentiated from
the rest of the units”. The unit appraises itself better, than its trades-offs
appraise the other units in the data set.
5.
Identify good all round performers – units with a high average
cross efficiency appraisal by peers are representative of “good all
round performers”
&URVV(IILFLHQF\DQG9DULDEOH5HWXUQVWR
6FDOH
Under the constant returns to scale (CCR) model cross efficiency scores
are in the range 0 to 100%, as would be expected from a “simple” DEA
calculation. Under the variable returns to scale (BCC) model cross
efficiency results are somewhat different. The non-linear relationship
between variables and scale effects in the model mean that it is possible
to have negative cross efficiency scores. This makes the interpretation
of the cross efficiency scores rather problematic. Similarly if there are
uncontrolled variables in the analysis. As with any analysis package, if
you are going to use cross efficiency analysis with variable returns to
scale and uncontrolled variables then it is important that you research
and understand the underlying mathematics.
In the calculation of efficiency scores under the BCC model an
“intercept value” is calculated for each unit. So, rather than the
efficiency score being based on the weighted sum of the outputs divided
by the weighted sum of the inputs, the calculation includes an intercept
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
&URVV(IILFLHQF\:LQGRZ
value (either added to the numerator or subtracted from the
denominator depending on the input/ output orientation of the model).
The intercept value relates to the scale size of each unit. In Frontier
Analyst, when calculating cross efficiency scores, the intercept values
used for each unit are those which are calculated from the “simple”,
unweighted DEA analysis. The cross efficiency calculations in the
columns of the matrix are based on “my unit’s variables and my unit’s
intercept value, with other units’ weights” and, reading along the rows,
the calculation is - “my unit’s weights applied to other peoples’ (units’)
variables with their associated intercept values”. The results in this
mode require a good understanding of the mathematics to be any use.
&URVV(IILFLHQF\LQ2SHUDWLRQ
Opening or closing the cross efficiency window determines whether or
not the cross efficiency calculations will be carried out. Cross
efficiencies are calculated automatically for every data set. If you are
working with a very large data set (thousands of units), you will find
that unless you have a very powerful computer the cross efficiency
calculations will take quite some time. It is for this reason that the
calculations are not done automatically.
If you do want to calculate cross efficiencies then you must first carry
out a “simple” DEA analysis. Enter your data in the data viewer, choose
your DEA options and analyse the data set. Using either the View
menu, Cross efficiency item, or the cross efficiency icon from the
Analysis Option Bar, open the cross efficiency window. When you do
this Frontier Analyst will automatically start the calculations needed to
populate the cross efficiency matrix.
If you decide that you do not want to continue with these calculations,
perhaps because you want to change the original analysis, then simply
close the cross efficiency window to stop the calculations.
3HUIRUPDQFH7LS
For best performance of the software, keep the cross efficiency window
closed when not in use. Although it recalculates in the background, and
does not stop you redoing the analysis at any time, calculating cross
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
&URVV(IILFLHQF\:LQGRZ
efficiencies is time consuming and will slow down your work
unnecessarily.
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
7URXEOHVKRRWLQJ
7URXEOHVKRRWLQJ
The cross efficiency window is designed to allow you to work
on other aspects of the results while it is generating the table.
For very large data sets, the analysis can take a long time due to
the number of calculations that are required (variables * units *
units).
If the calculation is causing trouble in other operation, closing
the cross efficiency window will stop any delays.
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
)HHGEDFN:HOFRPH
)HHGEDFN:HOFRPH
We always welcome feedback on how we can improve the product to
better suit your needs. Please write to or email us with your suggestions.
All ideas are added to a list, and you may find them appear in later
versions.
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
,QGH[
,QGH[
A
Accessing the Cross Efficiency
Analysis, 4
Averaged appraisal by peers, 6
Averaged appraisal of peers, 6
Averages, 6
I
Install code, 3
Installing, 3
Interrupting the analysis, 4
Introduction, 1
L
B
Licence information dialog, 3
Banxia Software Ltd, vi
Bibliography, 1
C
Calculation basis, 5
Cancelling the analysis, 4
Contact information, vi
Contents, vii
Copyright and disclaimer, v
Cross Efficiency and Variable Returns
to Scale, 9
Cross Efficiency in Operation, 10
Cross Efficiency window, 5
M
Matrix, 6
Maverick DMU, 8
P
Performance, 10
Plus Pack 1, 1
R
README.HTML, vi
D
S
Differentiate “truly” efficient peers, 9
Sub-classify 100% efficient peers, 8
F
T
Feedback welcome, 13
Trademark, v
Troubleshooting, 12
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
,QGH[
U
Uncontrolled variables, 9
V
W
Why Might Cross Efficiency Scores Be
Useful?, 8
WWWeb page, vi
Variable Returns to Scale, 9
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
,QGH[
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG
3OXV3DFN
,QGH[
3OXV3DFN
&RS\ULJKW%DQ[LD+ROGLQJV/LPLWHG