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

In Benchmarking: Benchmark and Frontier Analysis Using DEA and SFA

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)

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.

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

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.

Note

The index for peer units can be returned by the method peers and the weights are returned in lambda. Note that the peers now are the firms for the optimal input and/or output allocation, not just the technical efficient firms.

Author(s)

Peter Bogetoft and Lars Otto [e-mail]

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")

Benchmarking documentation built on Nov. 10, 2022, 5:56 p.m.