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

[Spencer Graves]

      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
>
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