# Two-sided Harrington type desirability function

### Description

Returns a two sided desirability function of the Harrington type.
Density, distribution function, quantile function and random number
generation for the distribution of the two-sided Harrington
desirability function are computed given a normally distributed
variable *Y* with expected value equal to `mean`

and standard
deviation equal to `sd`

.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
harrington2(LSL, USL, n)
## S3 method for class 'harrington2'
ddesire(x, f, mean, sd)
## S3 method for class 'harrington2'
pdesire(q, f, mean, sd)
## S3 method for class 'harrington2'
qdesire(p, f, mean, sd)
dharrington2(x, LSL, USL, n, mean, sd)
pharrington2(q, LSL, USL, n, mean, sd)
qharrington2(p, LSL, USL, n, mean, sd)
rharrington2(ns, LSL, USL, n, mean, sd)
eharrington2(LSL, USL, n, mean, sd)
vharrington2(LSL, USL, n, mean, sd)
``` |

### Arguments

`x,q` |
vector of quantiles. |

`p` |
vector of probabilies. |

`ns` |
number of observations. |

`f` |
two-sided Harrington type desirability function. |

`LSL` |
Lower Specification Limit of |

`USL` |
Upper Specification Limit of |

`n` |
Kurtosis parameter of desirability function. Values > 1
result in smoother shapes around the target value |

`mean` |
vector of means. |

`sd` |
vector of standard deviations. |

### Details

`harrington2(LSL, USL, n)`

is the two-sided desirability function
of Harrington type (Harrington (1965)). It aims at the specification
of desired values of a variable *Y* which has to be optimized
regarding a target value *T*. *Y* is transformed onto a
unitless scale to the interval [0,1]. *LSL* and *USL* are
associated with a desirability of *1/e.,
approx. 0.37*. *LSL* and *USL* have to be chosen
symmetrically around the target value *T*.

The density and distribution functions of Harrington's two-sided
desirability function `d`

given a normally distributed variable
*Y* with *E(Y)=* `mean`

and *sd(Y)=*`sd`

can be
determined analytically, see Trautmann and Weihs (2006).

### Value

`harrington2(LSL, USL, n)`

returns a function object of the
two-sided desirability function of the Harrington type (see example
below).

`ddesire`

/ `dharrington2`

give the density, `pdesire`

/ `pharrington2`

give the distribution function, `qdesire`

/
`qharrington2`

give the quantile function, and `rdesire`

/
`rharrington2`

generate random deviates. `edesire`

/
`eharrington2`

and `vdesire`

/ `vharrington2`

compute
the expected value and the variance of the desirability function for a
normally distributed random variable *Y* with
*E(Y)=*`mean`

and *sd(Y)=*`sd`

.

### Author(s)

Heike Trautmann trautmann@statistik.tu-dortmund.de, Detlef Steuer steuer@hsu-hamburg.de and Olaf Mersmann olafm@statistik.tu-dortmund.de

### References

J. Harrington (1965): The desirability function.
*Industrial Quality Control*,
**21**:494-498.

H. Trautmann, C. Weihs (2006):
On the Distribution of the Desirability Index using Harrington's Desirability Function.
*Metrika* **63**(2): 207-213.

### See Also

`harrington1`

for one sided Harrington type desirabilities

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | ```
##Assigning the function object to h:
h <- harrington2(3,7,1)
## Plot of desirability function:
plot(h)
## Desirability function of a vector:
h(seq(2,8,0.1))
## d/p/q/r/e/v examples:
ddesire(4, h, 0, 1)
dharrington2(4, 3, 7, 1, 0, 1)
ddesire(4, h, c(0,0.5),c(1,1.5))
pdesire(4, h, 0, 1)
pharrington2(4, 3, 7, 1, 0, 1)
qdesire(0.8, h, 0, 1)
qharrington2(0.8, 3, 7, 1, 0, 1)
rdesire(1e6, h, 0, 1)
rharrington2(1e6, 3, 7, 1, 0, 1)
edesire(h,3,0.5)
vdesire(h,3,0.5)
``` |