[R] optimize simultaneously two binomials inequalities using nlm( ) or optim( )

[Emir Toktar]

Dear Marc,

I'd been used the acceptance sampling but it is not possible to minimize
given that problem. It fits very well to calculate, but it's necessary have
an acceptance plan defined to assess the probabilities (p1,p2) and consumer
and producer risks...

Thank you for the tips.

EToktar
 

> -----Original Message-----
> From: Marc Schwartz [mailto:marc_schwartz at comcast.net] 
> Sent: Tuesday, August 05, 2008 1:12 PM
> To: Spencer Graves
> Cc: emir.toktar at gmail.com; r-help at r-project.org
> Subject: Re: [R] optimize simultaneously two binomials 
> inequalities using nlm( ) or optim( )
> 
> Just a quick follow up to Spencer's post, you might want to 
> look at the AcceptanceSampling package on CRAN:
> 
> http://cran.r-project.org/web/packages/AcceptanceSampling/index.html
> 
> HTH,
> 
> Marc Schwartz
> 
> 
> on 08/05/2008 09:00 AM Spencer Graves wrote:
> >      I saw your post on 7/29, and I have not seen a reply, 
> so I will 
> > attempt a response to the question at the start of your email:
> > obtain the smallest value of 'n' (sample size) satisfying both 
> > inequalities:
> >             (1-alpha) <= pbinom(c, n, p1) && pbinom(c, n, 
> p2) <= beta, 
> > where alpha, p1, p2, and beta are given, and I assume that 
> 'c' is also 
> > given, though that's not perfectly clear to me.
> >       Since 'n' is an integer, standard optimizers like 
> optim and nlm 
> > are not really appropriate.  This sounds like integer programming.
> > RSiteSearch('integer programming', 'fun') just produced 138 
> hits for 
> > me.  You might find something useful there.
> >       However, before I tried that, I'd try simpler things first.  
> > Consider for example the following: 
> > c. <- 5
> > # I used 'c.' not 'c', because 'c' is the name of a function in R.  
> > alpha <- .05
> > beta <- .8
> > p1 <- .2
> > p2 <- .9
> > n <- c.:20
> > p.1 <- pbinom(c., n, p1)
> > p.2 <- pbinom(c., n, p2)
> > 
> > good <- (((1-alpha) <= p.1) & (p.2 <= beta))
> > min(n[good])
> > 
> > op <- par(mfrow=c(2, 1)) plot(n, p.1)
> > abline(h=1-alpha)
> > plot(n, p.2)
> > abline(h=beta)
> > par(op)
> > 
> >       In this case, n = 6 satisfies both inequalities, 
> though n = 15 
> > does not.  If min(n[good]) = Inf either no solution exists 
> or you need 
> > to increase the upper bound of your search range for 'n'.
> >       If you'd like more help, PLEASE do read the posting guide 
> > 'http://www.R-project.org/posting-guide.html' and provide 
> commented, 
> > minimal, self-contained, reproducible code.
> > 
> >       Hope this helps.        Spencer Graves
> > 
> > emir.toktar at gmail.com wrote:
> >> Dear R users,
> >>
> >> I´m trying to optimize simultaneously two binomials inequalities 
> >> (used in acceptance sampling) which are nonlinear 
> solution, so there 
> >> is no simple direct solution. Please, let me explain 
> shortly the the 
> >> problem and the question as following.
> >>
> >> The objective is to obtain the smallest value of 'n' (sample size) 
> >> satisfying both inequalities:
> >> (1-alpha) <= pbinom(c, n, p1) && pbinom(c, n, p2) <= beta
> >>
> >> Where p1 (AQL) and p2 (LTPD) are probabilities parameters 
> (Consumer 
> >> and Producer), the alpha and beta are conumer and producer 
> risks, and 
> >> finally, the 'n' represents the sample size and the 'c' an 
> acceptance 
> >> number or maximum number of defects (nonconforming) in sample size.
> >>
> >> Considering that the 'n' and 'c' values are integer 
> variables, it is 
> >> commonly not possible to derive an OC curve including the both
> >> (p1,1-alpha)
> >> and (p2,beta) points. Some adjacency compromise is 
> commonly required, 
> >> achieved by searching a more precise OC curve with respect 
> to one of 
> >> the points.
> >> I´m using Mathematica 6 but it is a Trial, so I would like use R 
> >> intead (or better, I need it)!
> >>
> >> To exemplify,  In Mathematica I call the function using NMinimize 
> >> passing the restriction and parameters:
> >>
> >> /* function name "findOpt" and parameters... */
> >>
> >> restriction = (1 - alpha) <= 
> CDF[BinomialDistribution[sample_n, p1], 
> >> c]         &&  betha >= CDF[BinomialDistribution[sample_n, p2], c] 
> >>         &&   0 < alpha < alphamax && 0 < betha < bethamax  
>          
> >> &&   1 < sample_n <= lot_Size   &&   0 <= c < lot_size
> >>         &&  p1 < p2 < p2max ;
> >>
> >> fcost =     sample_n/lot_Size;
> >> result = NMinimize[{fcost, restriction}, {sample_n, c, 
> alpha, betha, 
> >> p2max}, Method -> "NelderMead", AccuracyGoal -> 10];
> >>
> >> /* Calling the function findOpt */
> >> findOpt[p1=0.005, lot_size=1000, alphamax=0.05, bethamax 
> =0.05, p2max 
> >> = 0.04]
> >>
> >> /* and I got the return of values of; minimal "n", "c", "alpha", 
> >> "betha" and the "p2" or (LTPD) were computed */ {0.514573, 
> {alpha$74 
> >> -> 0.0218683,
> >> sample_n$74 -> 155.231, betha$74 -> 0.05,
> >> c$74 -> 2, p2$74 -> 0.04}}
> >>
> >> Now, using R, I would define the "pbinom(c, n, prob)" 
> given only the 
> >> minimum and maximum values to "n" and "c" and limits to p1 and p2 
> >> probabilities (Consumer and Producer), similar on the 
> example above.
> >> I found some examples to optimize equations in R and some 
> tips, but I 
> >> not be able to define the sintaxe to use with that 
> functions. Among 
> >> the functions that could be used to resolve the problem 
> presented, I 
> >> found the function
> >> optim() that it is used for unconstrained optimization and 
> the nlm() 
> >> which is used for solving nonlinear unconstrained minimization 
> >> problems. May I wrong, but the nlm() function would be 
> appropriate to 
> >> solve this problem, is it right?
> >>
> >> Can I get a pointer to solve this problem using the nlm() 
> function or 
> >> where could I get some tips/example to help me, please?
> >>
> >> // (1-alpha) <= pbinom(c, n, p1) && pbinom(c, n, p2) <= 
> beta It was 
> >> used "betha" parameter name to avoid the 'beta' function used in 
> >> Mathematica...
> >>
> >>
> >> findS <- function(p1='numeric', lot_size='numeric', 
> >> alphamax='numeric', bethamax ='numeric', p2max ='numeric') {
> >>     (1 - alpha) <= pbinom(c, sample_n, p1) &&  betha >= pbinom(c, 
> >> sample_n, p2)
> >>     &&   0 < alpha < alphamax && 0 < betha < bethamax      
>  &&   1 < 
> >> sample_n <= lot_Size   &&   0 <= c < lot_size
> >>     &&  p1 < p2 < p2max ;
> >> }
> >>
> >> Parameters:
> >>     p1=0.005,     lot_size=1000,     alphamax=0.05,     bethamax 
> >> =0.05,     p2max = 0.04
> >>
> >>
> >> Minimize results should return/printing the following values:
> >>     sample_n,     (minimal sample size)
> >>     c ,         (critical level of defectives)
> >>     alpha ,     (producer's risk)
> >>     betha ,     (consumer's risk)
> >>     p2max    (consumer's probability p2)
> >>
> >>
> >> Could one help me understand how can desing the optimize nonlinear 
> >> function using R for two binomials or point me some tips?
> >>
> >>
> >> Thanks in advance.
> >>
> >> EToktar
> >>