sdea {Benchmarking} R Documentation

## Super efficiency

### Description

The method `sdea` calculates super-effciency and returns the same class of object as `dea`.

### Usage

```sdea(X, Y, RTS = "vrs", ORIENTATION = "in", DIRECT = NULL, param = NULL,
TRANSPOSE = FALSE, LP = FALSE, CONTROL = NULL)
```

### 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; the same values as for `dea`.

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

Input efficiency "in" (1), output efficiency "out" (2), and graph efficiency "graph" (3). For use with `DIRECT`, an additional option is "in-out" (0).

`DIRECT`

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 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`, 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"`.

`param`

Argument is at present only used when `RTS="fdh+"`, see `dea` for a description.

`TRANSPOSE`

See the description in `dea`.

`LP`

Only for debugging, see the description in `dea`.

`CONTROL`

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

### Details

Super-efficiency measures are constructed by avoiding that the evaluated firm can help span the technology; ie. if the firm in qestuen is a firm on the frontier in a normal dea approach then this firm in super efficiency might be outside the technology set.

### Value

The object returned is a Farrell object with the component described in `dea`. The relevant components are

 `eff` The efficiencies. Note when DIRECT is used then the efficencies are not Farrell efficiencies but rather exces values in DIRECT units of measurement `lambda` The lambdas, i.e. the weight of the peers, for each firm `objval` The objective value as returned from the LP program; normally the same as eff. `RTS` The return to scale assumption as in the option `RTS` in the call `ORIENTATION` The efficiency orientation as in the call

### Note

Calculation of slacks for super efficiency should be done by using the option `SLACK=TRUE` in the call of the method `sdea`. If the two phases are done in two steps as first a call to `sdea` and then a call to `slacks` the user must make sure to set the reference technology to the one corresponding to super-efficiency in the call to `slack` and this requires a loop with calls to `slack`.

### Author(s)

Peter Bogetoft and Lars Otto larsot23@gmail.com

### References

Peter Bogetoft and Lars Otto; Benchmarking with DEA, SFA, and R; Springer 2011. Sect. 5.2 page 115

P Andersen and NC Petersen; “A procedure for ranking efficient units in data envelopment analysis”; Management Science 1993 39(10):1261–1264

`dea`

### Examples

```x <- matrix(c(100,200,300,500,100,200,600),ncol=1)
y <- matrix(c(75,100,300,400,25,50,400),ncol=1)
se <- sdea(x,y)
se

# Leave out firm 3 as a determining firm of the technology set
n <- 3
dea.plot.frontier(x[-n], y[-n], txt=(1:dim(x)[1])[-n])
# Plot and label firm 3
points(x[n],y[n],cex=1.25,pch=16)