[R] Cost-benefit/value for money analysis

[David Winsemius]

On Jan 4, 2011, at 9:03 PM, Ben Bolker wrote:

> Graham Smith <myotistwo <at> gmail.com> writes:
>
>>
>> I assume this has a "proper" name, but I don't know what it is and  
>> wondered
>> if anyone knew of a package that might do the following, or something
>> similar.
>>
>> As an example, assume I have borrowed and read 10 books on R , and   
>> I have
>> subjectively given each of them a "value" score in terms of how  
>> useful I
>> think they are. I also know how much each costs in terms of money.
>>
>> What I would like to do is to calculate the costs of every possible
>> combination of the 10 books, and plot the total monetary value for  
>> each of
>> these possible combination  with their associated subjective value  
>> totals,
>> to help decide which combination of books represents the best value  
>> for
>> money.
>>
>> I know that some specialist decision analysis software does this  
>> sort of
>> thing, but was hoping R might have an appropriate package.
>
>  Perhaps you can specify your question more precisely, or differently.
> The way I interpret it, if there are no interactions in price
> (e.g. you get a discount for buying more than one book at a time)
> or in value (e.g. you learn more from one book having read another),
> then you get the best value/price ratio by taking only the book with
> the highest value/price.  (If you take no books at all, your value/ 
> price
> ratio is undefined.)  The algebra below shows that combining a lower
> value/price book with a higher one always lowers your overall value/ 
> price
> ratio.

I think a similar argument "at the margins" would show that even if  
the task were specified as maximal value with a budget, simply  
ordering by the value/price and buying until the cumsum of the price  
was greater than budget would solve the alternate statement of the  
problem. I suppose there might be situations where there were marginal  
choices of buying two books whose value/price was less than marginally  
maximal because two other marginally maximal choices would break the  
budget. This sounds like a homework problem and I don't see any  
student effort yet. Search terms include: "decision analysis" , "cost- 
benefit analysis", or "utility theory".

-- 
David.
>
>  If you redefine your problem, you might find the combn() or
> expand.grid() functions, along with various versions of apply(), to
> be useful.  If you have too large a search space you might take a look
> at the simulated annealing (SANN) option of optim().
>
> ===================
> if a1/b1 > a2/b2   (1)
>
> and a1, b1, a2, b2 > 0
>
> show
>
> a1/b1 > (a1+a2)/(b1+b2)  (2)
>
> i.e.
>
> a1/b1 - (a1+a2)/(b1+b2) > 0
>
> or
> (a1(b1+b2)-(a1+a2)b1)/(b1+b2) =
>  (a1*b2-a2*b1)/(b1+b2) > 0
>
> the numerator is (a1*b2-a2*b1):
>
>  (1) implies that a1*b2>a2*b1
> so the numerator is positive
>
> qed
>
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David Winsemius, MD
West Hartford, CT