| dea {Benchmarking} | R Documentation |
DEA efficiency
Description
Estimates a DEA frontier and calculates efficiency measures a la Farrell.
Usage
dea(X, Y, RTS="vrs", ORIENTATION="in", XREF=NULL, YREF=NULL,
FRONT.IDX=NULL, SLACK=FALSE, DUAL=FALSE, DIRECT=NULL, param=NULL,
TRANSPOSE=FALSE, FAST=FALSE, LP=FALSE, CONTROL=NULL, LPK=NULL)
## S3 method for class 'Farrell'
print(x, digits=4, ...)
## S3 method for class 'Farrell'
summary(object, digits=4, ...)
Arguments
X |
Inputs of firms to be evaluated, a K x m matrix
of observations of K firms with m inputs (firm x input). In case
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Y |
Outputs of firms to be evaluated, a K x n matrix
of observations of K firms with n outputs (firm x input). In case
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RTS |
Text string or a number defining the underlying DEA technology / returns to scale assumption.
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ORIENTATION |
Input efficiency "in" (1), output efficiency "out"
(2), and graph efficiency "graph" (3). For use with | |||||||||||||||||||||||||||
XREF |
Inputs of the firms determining the technology, defaults
to | |||||||||||||||||||||||||||
YREF |
Outputs of the firms determining the technology, defaults
to | |||||||||||||||||||||||||||
FRONT.IDX |
Index for firms determining the technology | |||||||||||||||||||||||||||
SLACK |
Calculate slack in a phase II calculation by an intern
call of the function | |||||||||||||||||||||||||||
DUAL |
Calculate dual variables, i.e. shadow prices; not calculated for orientation graph as that is not an LP problem. | |||||||||||||||||||||||||||
DIRECT |
Directional efficiency, If the argument is a scalar, the direction is (1,1,...,1) times the scalar; the value of the efficiency depends on the scalar as well as on the unit of measurements. If the argument is an array, this is used for the direction for every
firm; the length of the array must correspond to the number of
inputs and/or outputs depending on the If the argument is a matrix then different directions are used for
each firm. The dimensions depends on the
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param |
Possible parameters. At the moment only used for
RTS="fdh+" to set low and high values for restrictions on lambda;
see the section details and examples for its use. Future versions
might also use | |||||||||||||||||||||||||||
TRANSPOSE |
Input and output matrices are treated as firms
times goods matrices for the default value | |||||||||||||||||||||||||||
LP |
Only for debugging. If | |||||||||||||||||||||||||||
FAST |
Only calculate efficiencies and just return them as a
vector, i.e. no lambda or other output. The return when using
FAST cannot be used as input for | |||||||||||||||||||||||||||
CONTROL |
Possible controls to lpSolveAPI, see the
documentation for that package; use | |||||||||||||||||||||||||||
... |
Optional parameters for the print and summary methods. | |||||||||||||||||||||||||||
object, x |
An object of class Farrell (returned by the
function | |||||||||||||||||||||||||||
digits |
digits in printed output, handled by format in print. | |||||||||||||||||||||||||||
LPK |
when |
Details
The return from dea and sdea is an object of class
Farrell. The efficiency in dea is calculated by the LP method
in the package lpSolveAPI. Slacks can be calculated either in
the call of dea using the option SLACK=TRUE or in a
following call to the function slack.
The directional efficiency when the argument DIRECT is used,
depends on the unit of measurement and is not restricted to be less
than 1 (or greater than 1 for output efficiency) and is therefore
completely different from the Farrell efficiency.
The crs factor in RTS="fdh+" that sets the lower and upper bound can
be changed by the argument param that will set the lower and
upper bound to 1-param and 1+param; the default value is
param=.15. The value must be greater than or equal to 0 and strictly
less than 1. A value of 0 corresponds to RTS="fdh". To get an
asymmetric interval set param to a 2 dimensional array with values for
the low and high end for interval, for instance
param=c(.8,1.15). The FDH+ technology set is described in
Bogetoft and Otto (2011) pages 73–74.
The technology RTS="vrs+" uses the parameter param to set
restrictions on lambda, the convexity parameters. The elements of param
are param=(low, high, sum_low, sum_high) where "low" and "high"
are restrictions on the individual lambda and "sum_low" and "sum_high" are
restrictions on the sum of lambdas. The individual lambda must be in the
interval from low to high or be zero. With one parameter the restrictions
set are (param, 1+1-(param),1,1), with two parameters
(param[1], param[2],1,1), and with four parameters
(param[1], param[2],param[3], param[4]). The resulting technology set is
not necessarily convex.
The graph orientated efficiency is calculated by bisection between feasible and infeasible values of G. The precision in the result is less than for the other orientations.
When the argument DIRECT=d is used then the returned value
e for input orientation is the exces input measured in d
units of measurements, i.e. x-e d, and for output orientation
y+e d. The directional efficency can be restricted to inputs
(ORIENTAION="in"), restricted to outputs
(ORIENTAION="out"), or both include inputs and output
directions (ORIENTAION="in-out"). Directional efficiency is
discussed on pages 31–35 and 121–127 in Bogetoft and Otto (2011).
Value
The results are returned in a Farrell object with the
following components. The last three components in the list are
only part of the object when SLACK=TRUE.
eff |
The efficiencies. Note when DIRECT is used then the efficencies are not Farrell efficiencies but rather exces values in DIRECT units of measurement |
lambda |
The lambdas, i.e. the weight of the peers, for each firm |
objval |
The objective value as returned from the LP program;
normally the same as eff, but for |
RTS |
The return to scale assumption as in the option |
ORIENTATION |
The efficiency orientation as in the call |
TRANSPOSE |
As in the call |
slack |
A logical vector where the component for a firm is
|
sum |
A vector with sums of the slacks for each firm. Only
calculated in dea when option |
sx |
A matrix for input slacks for each firm, only calculated if
the option |
sy |
A matrix for output slack, see |
ux |
Dual variable for input, only calculated if |
vy |
Dual variable for output, only calculated if |
Note
The arguments X, Y, XREF, and YREF are
supposed to be matrices or numerical data frames that in the function
will be converted to matrices. When subsetting a matrix or data frame
to just one column then the class of the resulting object/variable is
no longer a matrix or a data frame, but just a numeric (array,
vector). Therefore, in this case a numeric input that is not a matrix
nor a data frame is transformed to a 1 column matrix, and here the use
of the argument TRANSPOSE=TRUE gives an error.
The dual values are not unique for extreme points (firms on the boundary with an efficiency of 1) and therefore the calculated dual values for these firms can depend on the order of firms in the reference technology. The same lack of uniqueness also makes the peers for some firms depend on the order of firms in the reference technology.
To calucalte slack use the argument SLACK=TRUE or use the
function slack directly.
When there is slack, and slack is not taken into consideration, then the peers for a firm with slack might depend on the order of firms in the data set; this is a property of the LP algorithm used to solve the problem.
To handle fixed, non-discretionary inputs, one can let it appear as
negative output in an input-based mode, and reversely for fixed,
non-discretionary outputs. Fixed inputs (outputs) can also be handled
by directional efficiency; set the direction, the argument
DIRECT, equal to the variable, discretionary inputs (outputs)
and 0 for the fixed inputs (outputs).
When the the argument DIRECT=X is used the then the returned
effiency is equal to 1 minus the Farrell efficiency for input
orientation and to the Farrell effiency minus 1 for output
orientation.
To use matrices X and Y prepared for the methods in the
package FEAR (Wilson 2008) set the options
TRANSPOSE=TRUE; for consistency with FEAR the options
RTS and ORIENTATION also accepts numbers as in
FEAR.
The tolerance that lambda is zero or one is 1e-7, the default value of
'epsint' in the package lpSolveAPI, i.e. values closer than
1e-7 from zero or one are set to respective integer value. The 'epsint'
is the tolerance that is used to determine whether a floating-point
number is in fact an in teger. The same tolerance is used for
efficiency value near one.
Some scaling is done in the function, but this does not always work
satisfactory, i.e. sometime, a solution cannot always be found – the
program prints a warning and the efficiency for the firm is set to NA.
Often this is due to a bad scaling of the data. Either the user can try
a different scaling of data when calling the function or one can use the
option CONTROL to try a different scaling by the program. For
instance one can insert CONTROL=list(scaling=c("geometric", "equilibrate")
or CONTROL=list(scaling=c("curtisreid", "equilibrate", "dynupdate")
in the option list for the function call. The full list of possible
scaling options can be found found from ?lp.control.options under
"scaling".
If a numerical problem occurs, status=5, the best solution is probably to
scale the input X and output Y yourself or use a different
scaling option as desribed above. The best results are obtained when the
variables are close to 1. If some variable are in the millions, then let
the unit of measure be a million.
Author(s)
Peter Bogetoft and Lars Otto larsot23@gmail.com
References
Peter Bogetoft and Lars Otto; Benchmarking with DEA, SFA, and R; Springer 2011
See Also
Paul W. Wilson (2008), “FEAR 1.0: A Software Package for Frontier Efficiency Analysis with R,” Socio-Economic Planning Sciences 42, 247–254
Examples
x <- matrix(c(100,200,300,500,100,200,600),ncol=1)
y <- matrix(c(75,100,300,400,25,50,400),ncol=1)
dea.plot.frontier(x,y,txt=TRUE)
e <- dea(x,y)
eff(e)
print(e)
summary(e)
lambda(e)
# Input savings potential for each firm
(1-eff(e)) * x
(1-e$eff) * x
# calculate slacks
el <- dea(x,y,SLACK=TRUE)
data.frame(e$eff,el$eff,el$slack,el$sx,el$sy)
# Fully efficient units, eff==1 and no slack
which(eff(e) == 1 & !el$slack)
# fdh+ with limits in the interval [.7, 1.2]
dea(x,y,RTS="fdh+", param=c(.7,1.2))