Minimizing inventory cost

Question

A business sells 4000 units of a product a year. Each unit costs $1.25. The ordering fee is $50 per shipment. The storage cost is $2.50 per unit per year. How many orders will minimize inventory cost?

Answer 1

This is an inventory control management question usually asked in a graduate MBA course on Quantitative Concepts in Management.

 

Cost of each unit = C = 1.25

Cost of holding unit inventory = H = 2.50

Number of items demanded or sold each year = D = 4000

Cost for each order = S = 50

Number of items ordered each time = Q to be optimized

Thus annual ordering cost = D/Q*S

and annual holding cost = H*Q/2 since if Q is ordered Q/2 is the average number held in inventory as items are assumed to be sold continuously

 

Total cost = H*Q/2 + S*D/Q + C*D

 

to find the optimal Q to minimize cost of inventory we find the derivative of the total cost function wrt Q and set it equal to zero.

 

H/2 - S*D/Q² = 0 

 

Q* = √(2DS/H)

 

substituting the value given in this problem, we get

 

Q* = √(2*4000*50/2.5) = 400 or 4000/400 = 10 orders each year

 

----

annual ordering cost = $500

annual storage cost = 400/2*2.50 = $500

in general minimum inventory costs are incurred when annual ordering cost = annual storage cost