The cost of keeping a unit of inventory is a.
The cost of being short a unit is b.
The expected demand is normally distributed with mean m and std. dev. c.
I'm trying to find a formula for the optimum amount of inventory.
I'm currently plugging in values for q in the following formula:
aq+b*Integral of(normal distribution*(x-q)) from q to infinite
q is the inventory in excess of the expected demand.
Is this correct? If so, what would the derivative of this formula be so I can find the maximum.
So (if I have done the derivative correctly):
and so the stationary point (maximum presumably) is:
where is the cumulative normal distribution function of a normal RV .
Then in terms of the standard normal cumulative distribution function:
(Check that to make sure it is correct. for close to zero it will still give a negative , but that is because with a normal distribution there is always a finite probability of negative sales)
The above may still have errors you will have to check it, but the key idea is the use of the fundamental theorem of calculus:
If a>.5b then the value inside of is less than .5 and the value of q is less than 0. But if the cost of inventory is less than the cost of shortage (a<b), we know that optimal inventory I=m+q must be at least the expected demand m. Am I misunderstanding something?
Thank you for the time you've spent so far btw.