DEA 8 Working Paper

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Using Excel for Basic Data Envelopment Analysis

BOOK · JANUARY 2000

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The University of Manchester

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Using Excel for Basic Data Envelopment Analysis Nathan Proudlove

1

Proudlove, N.C. (2000), “Using Excel for Data Envelopment Analysis”, Manchester School of

Management Working Paper No. 2007, ISBN 1 86615076 8

Using Excel for Basic Data Envelopment Analysis

Nathan Proudlove

Manchester School of Management, UMIST

Abstract

This paper describes the use of Excel to implement the basic Data Envelopment Analysis (DEA)

algorithms. Excel’s Solver can perform the optimisation required, whether nonlinear or linear

programming formulations are used. DEA requires repeated optimisation. Since the macro language for

Excel (Visual Basic for Applications) can control the Solver, a macro can be written to automate the

process of calculating the efficiency of each unit. It is argued that making use of the developing power

of spreadsheets to implement OR techniques such as DEA can help make OR more accessible to

practitioners and students.

Introduction

Since Charnes et al ’s (1978) description of the use of ratios of weighted inputs and outputs to measure

efficiency, Data Envelopment Analysis (DEA) has developed into a successful OR tool. Interest in

applying the technique has been growing rapidly, and many papers have been published on public and


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private sector applications, often leading to technical developments. DEA is starting to appear as a topic

in recent editions of general OR teaching texts (e.g. Andersen et al , 2000 and Winston, 1994) as well as

in those concentrating on mathematical programming (e.g. Williams, 1999). Applying DEA to

problems of realistic size requires a very large amount of computation, and so specialist software

packages have been developed (e.g. Thanassoulis et al , 1996 and Banxia Software , 1998).

At the same time, the capabilities of spreadsheet packages such as Microsoft Excel have been growing

rapidly. By making use of features such as the Solver and the Data Analysis tools, many of the

statistical analysis, optimisation and simulation tasks involved in OR education and practice can now

be tackled using a spreadsheet. Some teaching texts are taking this approach to statistics (e.g. Levine at

al , 1999) and OR techniques, for example linear programming (LP) (including Andersen et al , 2000

and Pidd, 1996). Excel’s macro programming language (Visual Basic for Applications, VBA) can be

used to automate tasks such as replication of simulations and repeated optimisation. The Macro

Recorder is particularly useful for novice users, recording user actions as code to generate pieces of

prototype VBA code.

Several authors (including Powell, 1997 and Winston, 1996) stress the value of using spreadsheets

(alongside or instead of specialist packages) to encourage students and managers to become more

active develo pers and users of OR modelling, if of only prototype or ‘quick and dirty’ models, rather

than just ‘consumers’. Both practising managers and students, particularly at MBA level, have

spreadsheets readily available and generally already make use of them for other purposes, whereas

access to specialist OR software is often restricted by cost, licensing or lack of familiarity. An

important by-product of OR education or consultancy can be promoting awareness of the power of

spreadsheet modelling and developing skills with a tool that most students and managers have at their

fingertips every day.


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Using a familiar, trusted tool can help integrate OR modelling with other analyses and reduce the

distrust in solutions produced by what some users may feel to be ‘black- box’ software packages. For

example, spreadsheets make it easy for users to experiment with solutions to get a feel for how the

model behaves, before using the Solver for optimisation. For example, the decision variables of a linear

programme can easily be adjusted and the effects on the objective and constraints observed.

DEA models

Following the notation used in Norman and Stoker (1991), the basic constant returns to scale output

maximisation (output-oriented) DEA model with n decision making units (DMUs), s output variables

and r input variables is:

MIN

s

1 j

0 j j

r

1i

0ii'

0

yw

xv

e (1)

subject to:

s

1 j

jm j

r

1i

imi

yw

xv

1 for m = 1 to n (2)

w j 0 for j = 1 to s (3)

vi 0 for i = 1 to r (4)

Note that this is the reciprocal measure of the input minimisation model, which considers the ratio of

weighted outputs over weighted inputs. In the above formulation, e' 0 is therefore a reciprocal of the


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usual efficiency score. The variables w j and vi are weights on the input variables, often known as virtual

multipliers (El-Mahgary, 1995). The unit under evaluation is known as unit 0.

This is usually converted to linear form:

MIN

r

1i

0ii

'

0 xve (5)

subject to:

s

1 j

jm j

r

1i

imi ywxv 0 for m = 1 to n (6)

s

1 j

0 j jyw = 1 (7)

w j 0 for j = 1 to s (8)

vi 0 for i = 1 to r (9)

The above formulation (Equations 5 to 9) constitutes a basic version of the LP generally labelled the

primal, following Charnes et al (1978). It is also known as the multiplier form, as it uses weights on the

input and output variables which form the model’s decision variables. However, the following dual

form (derived from the ‘primal’ using the standard primal-dual relationships of linear programming) is

regarded as often easier to interpret.

MAX '0f (10)

subject to:

0i

n

1m

im

'

m0 xxL

for i = 1 to r (11)

0 j

'

0

n

1m

jm

'

m0 yf yL

for j = 1 to s (12)


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'

m0L 0 for m = 1 to n (13)

'

0f unrestricted

The variable f' 0, the intensity factor, is the factor by which unit 0’s output must be increased to achieve

efficiency. An optimal value of one indicates that unit 0 is efficient relative to the set of DMUs under

consideration (the ‘field’), whereas a result greater than one indicates inefficiency. The ratio 1/f’0 is a

measure of efficiency (the efficiency score). The variables L' 0m are sometimes known as the dual

multipliers and represent the set of weights which, if applied to the DMUs, would produce an efficient

composite unit consuming no more inputs than unit 0, but producing at least as much output. They can

be used to set (output) targets for inefficient units.

At the optima, of course, the primal and dual formulations give the same objective value and the dual

prices of each constraint of the primal give the values of the decision variables of the dual, and vice

versa. One advantage of using the output maximisation form is that it is unnecessary to make a

distinction between controllable and uncontrollable inputs in the model (Norman and Stoker, 1991).

Note: the above formulation is often put in standard form, using slacks (or deviational) variables to

transform the constraints to equalities. The sum of these deviations, weighted by a very small constant,

can then be added to the objective. This removes a mathematical ambiguity which can give rise to dual

minima (Norman and Stoker, 1991). For simplicity of presentation and understanding, this refinement

will not be pursued here.

Using the Solver for DEA analysis of a single DMU


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A recent technical note in this journal demonstrated the use of Excel together with its Solver and VBA

to generate an efficient frontier in portfolio analysis (Jackson and Staunton, 1999). Here the use of

these tools for DEA is illustrated using the data from an example given in Williams’ (1999)

mathematical programming text. This example contains 28 DMUs (garage franchises) with 6 input

variables and 3 output variables. Figure 1 shows an extract. Cells A8 to L35 are defined as an array

called data.

Figure 1 Extract of Data worksheet (from Williams (1999, p. 254))

The basic formulation of DEA (Equations 1 to 4) is non-linear. Since the Solver can perform non-linear

optimisation, the form can be built and solved in Excel. However, as noted above, linear forms are

more commonly used. The primal LP form (Equations 5 to 9) is useful for illustrative experimentation

with different values of decision variables (the weights on the inputs and outputs). The dual weights

( L' 0m), from which the efficient composite unit can be constructed, can be found from the sensitivity

analysis sheet which the Solver can be instructed to produce.

In the spreadsheet described here the dual form (Equations 10 to 13) is used, since the composite unit

information is produced more directly, and recording of this will, ultimately, be automated. This is also

the form used in Williams’ (1999) text.


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Figure 2 shows the Excel worksheet called DEA - outmax Dual LP , which contains the dual LP model.

The user enters the identification number of the unit to be evaluated in cell B13. This is then the value

of the variable unit_0. The worksheet then uses the data array to insert in row 13 the unit’s name and

the values of the unit’s input variables, and the output variables multiplied by the intensity factor (i.e.

the right-hand sides of Equations 11 and 12). The intensity factor (cell M13) and the dual multipliers of

each unit (D21:D48) are the decision variables, which can be set by the user or the Solver. The

composite unit’s inputs and outputs consist of the sum of the multiplier or weight on each unit

multiplied by its inputs and outputs (i.e. the left-hand sides of Equations 11 and 12). For example, the

formula in cell D11 is =SUMPRODUCT(dual_multipliers,INDEX(data,,COLUMN(D11))). The

composite unit must consume no more inputs than unit 0 and produce at least as much output. A

feasible solution can always be found to this problem, as the composite unit could be unit 0 itself. In

this case unit 0 is efficient. For inefficient units it is possible to find a set of dual multipliers to produce

a composite unit which, whilst consuming no more inputs than unit 0, produces more on each of the

output variables. The intensity factor is the amount by which unit 0’s outputs could be scaled up whilst

maintaining the constraint that the composite must produce at least as much. The final efficiency score

of unit 0 (N13) is the reciprocal of the intensity factor.


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Figure 2 DEA - outmax Dual LP worksheet with Solver window superimposed

In Figure 2 the Solver window containing the model attached to this sheet is superimposed on the view

of the worksheet. It shows the objective (‘Target Cell’), decision variables (‘Changing Cells’) and

constraints. The Solver’s options are set to assume a linear model, in which case it uses a variant of the

simplex method for solution, and, to avoid adding further constraints, to assume non-negativity of the

‘Changing Cells’ (the decision variables). Although in theory f' 0 is unrestricted, as noted above it will

always have a value greater than or equal to one. Clicking ‘Solve’ runs the Solver and puts the values

of the dual multipliers and intensity factor into the worksheet, as shown in Figure 2 for the evaluation

of the first DMU (Winchester). The user can then choose to keep or discard these values, and obtain a

sensitivity analysis including the virtual multipliers mentioned earlier.


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Using the Solver for DEA analysis of all DMUs

DEA requires the reformulation and solution of the model to calculate the efficiency of each unit.

Therefore, using the implementation presented in this paper, an analysis of all the units requires the

user to enter the identifier number of each unit as the value of unit 0 in cell B13 of the DEA - outmax

Dual LP worksheet, the Solver rerun and record the results. Since VBA can be used to control the

Solver, a macro can be written to automate this process. Appendix A contains an extract of VBA code

that does this for the example presented. Here the results are recorded on a separate worksheet called

All Results, see Figure 3.

Figure 3 All Results worksheet: all DEA results produced by macro


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Conclusions

The great advantages of using spreadsheets for DEA are that they are available and familiar to a wide

audience. Using specialised DEA applications brings advantages in basic data handling and

computational efficiency (Ali, 1993). For example, they are often set up to handle the very small

constants sometimes used as lower limits on the virtual multipliers and as objective weights on the

slacks incorporated in some formulations. Further, solution times of large problems can be reduced by

automatically removing a unit found to be inefficient from calculations for subsequent units as it cannot

be an efficient peer. However, the Solver can cope with formulations involving very small constants,

and the speed of PCs means solution time is rarely a practical consideration.

In addition to accessibility, other advantages arise from the flexibility of spreadsheets. Additional

constraints can easily be incorporated to reflect special features of the situation modelled in order, for

example, to restrict unevenness in setting values on the virtual multipliers (e.g. Wong and Beasley,

1990; Beasley, 1990; Beasley, 1995) or to impose the convexity constraint of the variable returns to

scale (VRS) DEA model. Further analysis or displays can also be conveniently added. For example the

cross efficiency matrix which records the results of applying each unit’s own optimal weights to all the

units to produce a further comparison of relative efficiencies (Doyle and Green, 1994) or the

‘efficiency versus profitability’ segmentation of DMUs proposed by Dyson et al (1990) and used, for

example, by Camanho and Dyson (1999).


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Appendix A

Macro to apply Solver to all units

The following VBA code is an extract of a macro that performs the following actions for each unit:

enter the identifier number of the unit as unit 0 in the DEA - outmax Dual LP worksheet (the

spreadsheet then does an automatic recalculation which updates the values on each side of all the

constraints)

run the Solver set up for the DEA - outmax Dual LP worksheet

copy the efficiency score and dual multipliers (units weights) from DEA - outmax Dual LP (see

Figure 2) and paste them to another worksheet, here called All Results (see Figure 3).

Note: in order for Solver commands (such as SolverSolve() ) to be available to VBA, Solver.xla must

be added to the Available References in the Visual Basic Editor by locating it using Browse. The

developers of the Solver provide details of this and of the Solver VBA commands (Frontline Systems,

forthcoming).

' Switch off screen updating to speed up performance

Application.ScreenUpdating = False

' Run the required number of iterations

For m = 1 To 28

Application.StatusBar = "Calculating Efficiency for unit " & Str(m)

' Paste unit 0's number to model worksheet

Sheets("DEA - outmax Dual LP").Select

Application.Goto Reference:="unit_0"

Selection.Value = m

' Run the Solver (with the dialog box turned off)


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SolverSolve (True)

' Paste unit 0's number and name to All Results sheet

Application.Goto Reference:="unit_0_name"

Selection.Copy

Sheets("All Results").Select

Range("A12").Offset(m - 1, 0).Select

Selection.Value = m


Selection.PasteSpecial Paste:=xlValues

' Paste unit 0's efficiency to All Results sheet


Application.Goto Reference:="efficiency_score"

Selection.Copy



Selection.PasteSpecial Paste:=xlValues

' Paste unit 0's composite unit's weights to All Results sheet


Application.Goto Reference:="dual_multipliers"

Selection.Copy



Selection.PasteSpecial Paste:=xlValues, Transpose:=True

Next m

' Switch screen updating back on

Application.ScreenUpdating = True


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References

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Andersen DR, Sweeney DJ and Williams TA (2000). An introduction to management science 9th

edition. South-Western: Cincinnati.

Banxia Software (1998). Frontier Analyst software (Version 1.1). Banxia Software: 141 St James

Road, Glasgow, G4 OLT. www.banxia.com.

Beasley JE (1990). Comparing university departments. Omega 18: 171-183.

Beasley JE (1995). Determining teaching and research efficiencies. J Opl Res Soc 46: 441-452.

Camanho AS and Dyson RG (1999). Efficiency, size, benchmarks and targets for bank branches: an

application of data envelopment analysis. J Opl Res Soc 50: 903-915.

Charnes A, Cooper W and Rhodes E (1978). Measuring the efficiency of decision making units. Eur J

Opl Res 17: 35-44.

Doyle J and Green R (1994). Efficiency and cross-effectiveness in DEA: derivations, meanings and

uses. J Opl Res Soc 45: 567-578.

Dyson RG, Thanassoulis E and Boussofiane A (1990). Data envelopment analysis. In: Henry LC and

Eglese R (eds). Operational Research Tutorial 1990. Operational Research Society: Birmingham, pp

13-28.

El-Mahgary S (1995). Data envelopment analysis – a basic glossary. OR Insight 8(4): 15-22.

Frontline Systems, Inc., P.O. Box 4288, Incline Village, NV 89450-4288, USA. www.frontsys.com see

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Interfaces.

Jackson M and Staunton MD (1999). Quadratic programming applications in finance using Excel. J

Opl Res Soc 50: 1256-1266.


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Levine DM, Berenson ML and Stephan D (1999). Statistics for managers using Microsoft Excel 2nd

edition. Prentice-Hall: Upper Saddle River, NJ.

Norman M and Stoker B (1991). Data envelopment analysis: the assessment of performance. Wiley:

Chichester.

Pidd M (1996). Tools for thinking: modelling in management science. Wiley: Chichester.

Powell SG (1997). The teachers' forum: from intelligent consumer to active modeler, two MBA

success stories. Interfaces 27(3): 88-98.

Thanassoulis E, Athanassopoulos AD and Dyson RG (1996). Warwick DEA software (Windows

version). Warwick Busiess School, University of Warwick: Coventry.

www.warwick.ac.uk/~bsrlu/dea/deas/deas1.htm

Williams HP (1999). Model building in mathematical programming 4th edition. Wiley: Chichester.

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Res Soc 41: 829-835.

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