dea.add {Benchmarking}

Additive DEA model

Description

Calculates additive efficiency as sum of input and output slacks within different DEA models

Usage

dea.add(X, Y, RTS="vrs", XREF=NULL, YREF=NULL, 
        FRONT.IDX=NULL, param=NULL, TRANSPOSE=FALSE, LP=FALSE)

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 TRANSPOSE=TRUE the input matrix is transposed to input x firm.

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 TRANSPOSE=TRUE the output matrix is transposed to output x firm.

RTS

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	add	Additivity (scaling up and down, but only with integers), and free disposability

XREF

Inputs of the firms determining the technology, defaults to X

YREF

Outputs of the firms determining the technology, defaults to Y

FRONT.IDX

Index for firms determining the technology

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 param for other purposes.

TRANSPOSE

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.

LP

Only for debugging. If LP=TRUE then input and output for the LP program are written to standard output for each unit.

Details

The sum of the slacks is maximized in a LP formulation of the DEA technology. The sum of the slacks can be seen as distance to the frontier when you only move parallel to the axes of inputs and outputs, i.e. not a usual Euclidean distance, but what is also known as an L1 norm.

Since it is the sum of slacks that is calculated, there is no exogenous ORIENTATION in the problem. Rather, there is generally both an input and an output direction in the slacks. The model considers the input excess and output shortfall simultaneously and finds a point on the frontier that is most distant to the point being evaluated.

Value

sum

Sum of all slacks for each firm, sum=sum(sx)+sum(sy).

slack

A non-NULL vector of logical variables, TRUE if there is slack for the corresponding firm, and FALSE if the there is no slack, i.e. the sum of slacks is zero.

sx

A matrix of input slacks for each firm

sy

A matrix of output slack for each firm

lambda

The lambdas, i.e. the weights of the peers for each firm

Note

This is neither a Farrell nor a Shephard like efficiency.

The value of the slacks depends on the scaling of the different inputs and outputs. Therefore the values are not independent of how the input and output are measured.

Author(s)

Peter Bogetoft and Lars Otto [e-mail]

Source

Corresponds to Eqs. 4.34-4.38 in Cooper et al. (2007)

References

Bogetoft and Otto; Benchmarking with DEA, SFA, and R; Springer 2011

Cooper, Seiford, and Tone; Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software; Second edition, Springer 2007

Examples

x <- matrix(c(2,3,2,4,6,5,6,8),ncol=1)
y <- matrix(c(1,3,2,3,5,2,3,5),ncol=1)
dea.plot.frontier(x,y,txt=1:dim(x)[1])

sb <- dea.add(x,y,RTS="vrs")
data.frame("sx"=sb$sx,"sy"=sb$sy,"sum"=sb$sum,"slack"=sb$slack)