dea.direct {Benchmarking} R Documentation

## Directional efficiency

### Description

Directional efficiency rescaled to an interpretation a la Farrell efficiency and the corresponding peer importance (lambda).

### Usage

```dea.direct(X, Y, DIRECT, RTS = "vrs", ORIENTATION = "in",
XREF = NULL, YREF = NULL, FRONT.IDX = NULL,
SLACK = FALSE, param=NULL, TRANSPOSE = 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)

`Y`

Outputs of firms to be evaluated, a K x n matrix of observations of K firms with n outputs (firm x input).

`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` (and `TRANSPOSE`), 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"`.

`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 (down-scaling, but not up-scaling), convexity, 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 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).

`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.

`SLACK`

See `dea` and `slack`.

`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`

see `dea`

### Details

When the argument `DIRECT=d` is used then component `objval` of the returned object for input orientation is the maximum value of e where for input orientation x-e d, and for output orientation y+e d are in the generated technology set. The returned component `eff` is for input 1-e d/X and for output 1+e d /Y to make the interpretation as for a Farrell efficiency. Note that when the direction is not proportional to `X` or `Y` the returned `eff` are different for different inputs or outputs and `eff` is a matrix and not just an array. The directional efficiency can be restricted to inputs (`ORIENTATION="in"`), restricted to outputs (`ORIENTATION="out"`), or both include inputs and output directions (`ORIENTATION="in-out"`). Directional efficiency is discussed on pages 31–35 and 121–127 in Bogetoft and Otto (2011).

The Farrell efficiency interpretation is the ratio by which a firm can proportionally reduce all inputs (or expand all outputs) without producing less outputs (using more inputs). The directional efficiecies have the same interpretation expect that the direction is not proportional to the inputs (or outputs) and therefore the different inputs may have different reduction ratios, the efficiency is an array and not just a number.

### Value

The results are returned in a Farrell object with the following components. The method `slack` only returns the three components in the list relevant for slacks.

 `eff` The Farrell efficiencies. Note that the the efficiencies are calculated to have the same interpretations as Farrell efficiencies. `eff` is a matrix if there are more than 1 good. `lambda` The lambdas, i.e. the weight of the peers, for each firm `objval` The objective value as returned from the LP program; the `objval` are excess values in DIRECT units of measurement. `RTS` The return to scale assumption as in the option `RTS` in the call `ORIENTATION` The efficiency orientation as in the call `TRANSPOSE` As in the call `slack` A vector with sums of the slacks for each firm. Only calculated in dea when option `SLACK=TRUE` `sx` A matrix for input slacks for each firm, only calculated if the option `SLACK` is `TRUE` or returned from the method `slack` `sy` A matrix for output slack, see `sx`

### Note

To handle fixed, non-discretionary inputs, one can let it appear as negative output in an input-based mode, and reversely for fixed, non-discretionary outputs. Fixed inputs (outputs) can also be handled by directional efficiency; set the direction, the argument `DIRECT`, equal to the variable, discretionary inputs (outputs) and 0 for the fixed inputs (outputs).

When the the argument `DIRECT=X` is used the then the returned efficiency is equal to 1 minus the Farrell efficiency for input orientation and equal to the Farrell efficiency minus 1 for output orientation.

### Author(s)

Peter Bogetoft and Lars Otto larsot23@gmail.com

### References

Directional efficiency is discussed on pages 31–35 and 121–127 in Bogetoft and Otto (2011).

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

`dea`

### Examples

```# Directional efficiency
x <- matrix(c(2,5 , 1,2 , 2,2 , 3,2 , 3,1 , 4,1), ncol=2,byrow=TRUE)
y <- matrix(1,nrow=dim(x))
dea.plot.isoquant(x[,1], x[,2],txt=1:dim(x))

E <- dea(x,y)
z <- c(1,1)
e <- dea.direct(x,y,DIRECT=z)
data.frame(Farrell=E\$eff, Perform=e\$eff, objval=e\$objval)
# The direction
arrows(x[,1], x[,2], (x-z)[,1], (x-z)[,2], lty="dashed")
# The efficiency (e\$objval) along the direction
segments(x[,1], x[,2], (x-e\$objval*z)[,1], (x-e\$objval*z)[,2], lwd=2)

# Different directions
x1 <- c(.5, 1, 2, 4, 3, 1)
x2 <- c(4,  2, 1,.5, 2, 4)
x <- cbind(x1,x2)
y <- matrix(1,nrow=dim(x))
dir1 <- c(1,.25)
dir2 <- c(.25, 4)
dir3 <- c(1,4)
e <- dea(x,y)
e1 <- dea.direct(x,y,DIRECT=dir1)
e2 <- dea.direct(x,y,DIRECT=dir2)
e3 <- dea.direct(x,y,DIRECT=dir3)
data.frame(e=eff(e),e1=e1\$eff,e2=e2\$eff,e3=e3\$eff)[6,]

# Technology and directions for all firms
dea.plot.isoquant(x[,1], x[,2],txt=1:dim(x))
arrows(x[,1], x[,2],  x[,1]-dir1, x[,2]-dir1,lty="dashed")
segments(x[,1], x[,2],
x[,1]-e1\$objval*dir1, x[,2]-e1\$objval*dir1,lwd=2)
# slack for direction 1
dsl1 <- slack(x,y,e1)
cbind(E=e\$eff,e1\$eff,dsl1\$sx,dsl1\$sy, sum=dsl1\$sum)

# Technology and directions for firm 6,
# Figure 2.6 page 32 in Bogetoft & Otto (2011)
dea.plot.isoquant(x1,x2,lwd=1.5, txt=TRUE)
arrows(x[6,1], x[6,2],  x[6,1]-dir1, x[6,2]-dir1,lty="dashed")
arrows(x[6,1], x[6,2],  x[6,1]-dir2, x[6,2]-dir2,lty="dashed")
arrows(x[6,1], x[6,2],  x[6,1]-dir3, x[6,2]-dir3,lty="dashed")
segments(x[6,1], x[6,2],
x[6,1]-e1\$objval*dir1, x[6,2]-e1\$objval*dir1,lwd=2)
segments(x[6,1], x[6,2],
x[6,1]-e2\$objval*dir2, x[6,2]-e2\$objval*dir2,lwd=2)
segments(x[6,1], x[6,2],
x[6,1]-e3\$objval*dir3, x[6,2]-e3\$objval*dir3,lwd=2)
```

[Package Benchmarking version 0.29 Index]