# harrington1: One-sided Harrington type desirability function In desire: Desirability functions

## Description

Returns a one-sided desirability function of the Harrington type. Density, distribution function, quantile function and random number generation for the distribution of the one-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 14 15 16 17``` ```harrington1(y1, d1, y2, d2) ## S3 method for class 'harrington1' ddesire(x, f, mean, sd) ## S3 method for class 'harrington1' pdesire(q, f, mean, sd) ## S3 method for class 'harrington1' qdesire(p, f, mean, sd) ## S3 method for class 'harrington1' edesire(f, mean, sd) ## S3 method for class 'harrington1' vdesire(f, mean, sd) dharrington1(x, y1, d1, y2, d2, mean, sd) pharrington1(q, y1, d1, y2, d2, mean, sd) qharrington1(p, y1, d1, y2, d2, mean, sd) rharrington1(n, y1, d1, y2, d2, mean, sd) eharrington1(y1, d1, y2, d2, mean, sd) vharrington1(y1, d1, y2, d2, mean, sd) ```

## Arguments

 `x,q` vector of quantiles. `p` vector of probabilies. `n` number of observations. `f` one-sided Harrington type desirability function. `y1,d1,y2,d2` Two values `y1` and `y2` of variable Y with respective desirability values `d1` and `d2` determine the shape of the desirability function. `mean` vector of expected values of normal distributions. `sd` vector of standard deviations of normal distributions.

## Details

`harrington1(y1, d1, y2, d2)` is the one-sided desirability function of Harrington type (Harrington (1965)). It aims at the specification of desired values of a variable Y which has to be minimized or maximized. Y is transformed onto a unitless scale to the interval [0,1].

Harrington's one-sided desirability function `d` given a normally distributed variable Y with E(Y)= `mean` and sd(Y)=`sd` has the Double Lognormal Distribution (Holland and Ahsanullah (1989)).

## Value

`harrington1(y1, d1, y2, d2)` returns a function object of the one-sided desirability function of the Harrington type (see example below). Values b_0 and b_1 of the desirability function formula are determined.

`ddesire` /`dharrington1` give the density, `pdesire` / `pharrington1` give the distribution function, `qdesire` / `qharrington1` give the quantile function, and `rdesire` / `rharrington1` generate random deviates. `edesire` / `eharrington1` and `vdesire` / `vharrington1` 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 [email protected], Detlef Steuer [email protected] and Olaf Mersmann [email protected]

## References

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

B. Holland, M. Ahsanullah (1989): Further Results on the Distribution of Meinhold and Singpurwalla. The American Statistician 43 (4): 216-219.

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

`harrington2` for two sided Harrington type desirabilities
 ``` 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 28 29 30 31 32 33``` ```##Assigning the function object to h: h <- harrington1(-2, .1, 2, .9) ## Plot of desirability function: plot(h) ## Desirability function of a vector: h(seq(-2,2,0.1)) ## d/p/q/r/e/v examples: ddesire(.8, h, 0, 1) dharrington1(.8, -2, .1, 2, .9, 0, 1) ddesire(.8, h, c(0,0.5), c(1,1.5)) pdesire(.8, h, 0, 1) pharrington1(.8, -2, .1, 2, .9, 0, 1) qdesire(.8, h, 0, 1) qharrington1(.8, -2, .1, 2, .9, 0, 1) rdesire(1e6, h, 0, 1) rharrington1(1e6, -2, .1, 2, .9, 0, 1) edesire(h,3,0.5) eharrington1(-2, .1, 2, .9,3,0.5) vdesire(h,3,0.5) vharrington1(-2, .1, 2, .9,3,0.5) ## b_0 and b_1 values: environment(h)\$b0 environment(h)\$b1 ```