Estimates a DEA frontier and calculates efficiency measures a la Farrell.
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 Farrell
print(x, digits=4, ...)
# S3 method for Farrell
summary(object, digits=4, ...)
Inputs of firms to be evaluated, a K x m matrix
of observations of K firms with m inputs (firm x input). In case
TRANSPOSE=TRUE the input matrix is transposed to input x
firm.
Outputs of firms to be evaluated, a K x n matrix
of observations of K firms with n outputs (firm x input). In case
TRANSPOSE=TRUE the output matrix is transposed to output x
firm.
Text string or a number defining the underlying DEA technology / returns to scale assumption.
0 |
fdh | Free disposability hull, no convexity assumption |
1 |
vrs | Variable returns to scale, convexity and free disposability |
2 |
drs | Decreasing returns to scale, convexity, down-scaling and free disposability |
3 |
crs | Constant returns to scale, convexity and free disposability |
4 |
irs | Increasing returns to scale, (up-scaling, but not down-scaling), convexity and free disposability |
5 |
irs2 | Increasing returns to scale (up-scaling, but not down-scaling), additivity, and free disposability |
6 |
add | Additivity (scaling up and down, but only with integers), and free disposability; also known af replicability and free disposability, the free disposability and replicability hull (frh) -- no convexity assumption |
7 |
fdh+ | A combination of free disposability and restricted or local constant return to scale |
Input efficiency "in" (1), output efficiency "out"
(2), and graph efficiency "graph" (3). For use with DIRECT,
an additional option is "in-out" (0).
Inputs of the firms determining the technology, defaults
to X
Outputs of the firms determining the technology, defaults
to Y
Index for firms determining the technology
Calculate slack in a phase II calculation by an intern
call of the function slack. Note that the precision
for calculating slacks for orientation graph is low.
Calculate dual variables, i.e. shadow prices; not calculated for orientation graph as that is not an LP problem.
Directional efficiency, DIRECT is either a
scalar, an array, or a matrix with non-negative elements.
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 ORIENTATION.
If the argument is a matrix then different directions are used for
each firm. The dimensions depends on the ORIENTATION (and
TRANSPOSE), the number of firms must correspond to the
number of firms in X and Y.
DIRECT must not be used in connection with
DIRECTION="graph".
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 param for other purposes.
Input and output matrices are treated as firms
times goods matrices for the default value TRANSPOSE=FALSE
corresponding to the standard in R for statistical models. When
TRUE data matrices are transposed to good times firms
matrices as is normally used in LP formulation of the problem.
Only for debugging. If LP=TRUE then input and
output for the LP program are written to standard output for each
unit.
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 slack and peers.
Possible controls to lpSolveAPI, see the documentation for that package.
Optional parameters for the print and summary methods.
An object of class Farrell (returned by the
function dea) -- R code uses ‘object’ and
‘x’ alternating for generic methods.
digits in printed output, handled by format in print.
when LPK=k then a mps file is written for firm
k; it can be used as input to an alternative LP solver
to check the results.
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.
The efficiencies. Note when DIRECT is used then the efficencies are not Farrell efficiencies but rather exces values in DIRECT units of measurement
The lambdas, i.e. the weight of the peers, for each firm
The objective value as returned from the LP program;
normally the same as eff, but for slack it is the the sum
of the slacks.
The return to scale assumption as in the option RTS
in the call
The efficiency orientation as in the call
As in the call
A logical vector where the component for a firm is
TRUE if the sums of slacks for the corresponding firm is
positive. Only calculated in dea when option SLACK=TRUE
A vector with sums of the slacks for each firm. Only
calculated in dea when option SLACK=TRUE
A matrix for input slacks for each firm, only calculated if
the option SLACK is TRUE or returned from the
method slack
A matrix for output slack, see sx
Dual variable for input, only calculated if DUAL is
TRUE.
Dual variable for output, only calculated if DUAL is
TRUE.
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 dimentional 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"). Dirctional efficiency is
discussed on pages 31--35 and 121--127 in Bogetoft and Otto (2011).
Peter Bogetoft and Lars Otto; Benchmarking with DEA, SFA, and R; Springer 2011
Paul W. Wilson (2008), “FEAR 1.0: A Software Package for Frontier Efficiency Analysis with R,” Socio-Economic Planning Sciences 42, 247--254
# NOT RUN {
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))
# }
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