Benchmarking (version 0.33)

cost.opt: DEA optimal cost, revenue, and profit

Description

Estimates the input and/or output vector(s) that minimize cost, maximize revenue or maximize profit in the context of a DEA technology

Usage

cost.opt(XREF, YREF, W, YOBS=NULL, RTS="vrs", param=NULL,
         TRANSPOSE=FALSE, LP=FALSE, CONTROL=NULL, LPK = NULL)  
revenue.opt(XREF, YREF, P, XOBS=NULL, RTS="vrs",  param=NULL,
            TRANSPOSE = FALSE, LP = FALSE, CONTROL=NULL, LPK = NULL)
profit.opt(XREF, YREF, W, P, RTS = "vrs",  param=NULL,
           TRANSPOSE = FALSE, LP = FALSE, CONTROL=NULL, LPK = NULL)

Value

The values returned are the optimal input, and/or optimal output. When saved in an object the following components are available:

xopt

The optimal input, returned as a matrix by cost.opt and profit.cost.

yopt

The optimal output, returned as a matrix by revenue.opt and profit.cost.

cost

The optimal/minimal cost.

revenue

The optimal/maximal revenue

profit

The optimal/maximal profit

lambda

The peer weights that determines the technology, a matrix. Each row is the lambdas for the firm corresponding to that row; for the vrs technology the rows sum to 1. A column shows for a given firm how other firms are compared to this firm, i.e. peers are firms with a positive element in their columns.

Arguments

XREF

Input of the firms defining the technology, 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 as input x firm.

YREF

output of the firms defining the technology, 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 as output x firm.

W

Input prices as a matrix. Either same prices for all firms or individual prices for all firms, i.e. either a 1 x m or a K x m matrix for K firms and m inputs

P

Output prices as a matrix. Either same prices for all firms or individual prices for all firms, i.e. either a 1 x n or K x n matrix for K firms and n outputs

XOBS

The input for which an optimal, revenue maximizing, output vector is to be calculated. Defaults is XREF. Same form as XREF

YOBS

The output for which an optimal, cost minimizing input vector is to be calculated. Defaults is YREF. Same form as YREF

RTS

A 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, downscaling 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
6	fdh+	A combination of free disposability and restricted or local constant return to scale

param

Possible parameters. Now only used for RTS="fdh+" to set low and high values for restrictions on lambda; see the section details and examples in dea for its use. Future versions might also use param for other purposes.

TRANSPOSE

Input and output matrices are treated as firms times goods for the default value TRANSPOSE=FALSE corresponding to the standard in R for statistical models. When TRUE data matrices, quantities and prices, are transposed to goods times firms matrices.

LP

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

CONTROL

Possible controls to lpSolveAPI, see the documentation for that package. For examples of use see the function dea.

LPK

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.

Author

Peter Bogetoft and Lars Otto

Details

Input and output matrices are in the same form as for the method dea.

The LP optimization problem is formulated in Bogetoft and Otto (2011, pp 35 and 102) and is solved by the LP method in the package lpSolveAPI.

The methods print and summary are working for cost.opt, revenue.opt, and profit.opt

References

Bogetoft and 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(2,12, 2,8, 5,5, 10,4, 10,6, 3,13), ncol=2, byrow=TRUE)
y <- matrix(1,nrow=dim(x)[1],ncol=1)
w <- matrix(c(1.5, 1),ncol=2)

txt <- LETTERS[1:dim(x)[1]]
dea.plot(x[,1],x[,2], ORIENTATION="in",  cex=1.25)
text(x[,1],x[,2],txt,adj=c(-.7,-.2),cex=1.25)

# technical efficiency
te <- dea(x,y,RTS="vrs")
xopt <- cost.opt(x,y,w,RTS=1)
cobs <- x %*% t(w)
copt <- xopt$x %*% t(w)
# cost efficiency
ce <- copt/cobs
# allocaltive efficiency
ae <- ce/te$eff
data.frame("ce"=ce,"te"=te$eff,"ae"=ae)
print(cbind("ce"=c(ce),"te"=te$eff,"ae"=c(ae)),digits=2)

# isocost line in the technology plot
abline(a=copt[1]/w[2], b=-w[1]/w[2], lty="dashed")