 # probability mass function of Possion distribution NAT1301

Hello all!

I have to code a program that gives an answer to the question: Which quantity of a given item do I have to take on stock, if I want to cover the incoming orders with a probability of 95%? The average of items ordered per period should be a parameter.

Input-Parameter #1: Percentage
Input-Parameter #2: Average
Output-Parameter: Needed stock

This mathematical problem can be solved with the so-called “probability mass function of Possion distribution”. See http://en.wikibooks.org/wiki/Statistics:Distributions/Poisson
To keep it simple I took the following restaurant-example out of wikibooks.org.
The question here is: How many dishes do I have to prepare, if I want to cover 95% of the orders?
My first draft of a program is:

``````
define data local
1 #average         (f8)   init <4>
1 #percentage      (N1.5) init <0.95> /* has to be < 1
*
1 #-average        (f8)
1 #needed_dishes   (I4)
1 #sum_probability (f8)   (EM=9.9(10))
1 #factorial       (F8)
end-define
*
#-average := #average * -1
#needed_dishes := -1
*
repeat
if #needed_dishes = 0    /* 0! = 1
#factorial := 1
else
#factorial := #factorial * #needed_dishes
end-if
*
#sum_probability := #sum_probability + #average ** #needed_dishes /
#factorial * EXP(#-average)
*
display #needed_dishes #sum_probability  #factorial
until #sum_probability >= #percentage
end-repeat
*
write 'Needed dishes:' #needed_dishes
*
end``````

Now the problem is:
The program works good for a very small average number of dishes (an average of 4 dishes and 95% --> 8 dishes). But let’s say I have a very very big restaurant and an average of 500 dishes per day. Then the factorial in the formula becomes very big and an NAT1301 occurs.

Any ideas how to solve the problem?

Thanks,

matthias

Hi Matthias;

In an earlier life I was pursuing a PhD in Operations Research. Will thumb through old (okay, really old) notes to see about alternative algorithm.

I am sure you played with this a bit. 125 works for #average, 130 does not.

steve

Hello Steve and thanks for answering!

Right! But it also depends on #percentage. I put the limit to -100 <= #average <= +100 with #average <> 0