Optimum number

Question

A restaurant has an annual demand for 887 bottles of wine. It costs $4 to store 1 bottle for 1 year and it costs $6 to place a reorder. Find the optimum number of bottles per order.

Answer 1

Jeff this using calculus to determine the minimum cost for storage and per order.

 

Let x = the number of orders per year

This implies 887/x = the number of wine bottles per order Note: this is the number of wine bottles put in storage per year.

 

C(x) = The total cost for storage and the number of orders.

 

Thus,

 

C(x)=4(887/x)+6x = 3548/x+6x

 

Take the derivative of the function with respect to x and set it to equal zero:

 

C'(x)=-3548/x^2+6=0

 

This implies 6=3548/x^2.

 

With some algebraic manipulation, the equation becomes:

 

x^2=3548/6

 

x≈24 orders which implies 887/24≈37 bottles of wine per order

 

Thus C(24)=4*37+6*24=$292 per year 

Answer 2

Okay, you can set this up either by time, by number of orders per year, or by order size. 

 

So, suppose that you order N times a year. What is the order size? 

 

O = 887/n.

 

What is the average number of bottles on hand?  half that number, or 887/2n

 

What is the average annual cost of holding?  4 * 887/2n

 

What is the cost of reordering?  6N  

 

So, your annual cost of wine is

 

C = 6N + 4*887/2N  = 6N +1774/N

 

Minimize that number C, then calculate O.  Presumably, this is a calculus class, so you'd be taking a derivative etc.    

 

If not, then you'd do some quick extrapolating, or like me plotting in an excel spreadsheet, and you'd find a number around 17 times a year, and in the 50-52 bottles per order range.