cost.opt {Benchmarking} R Documentation

## 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

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

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

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 larsot23@gmail.com

### References

Bogetoft and 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

### 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)

# 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")
```

[Package Benchmarking version 0.29 Index]