9. Economic Order Quantity. The economic order quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for that model are that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed.

Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products.

The following model describes this situation:

Let Dj 5 annual demand for item j Cj 5 unit cost of item j Sj 5 cost per order placed for item j wj 5 space required for item j W 5 the maximum amount of space available for all goods i 5 inventory carrying charge as a percentage of the cost per unit The decision variables are Qj, the amount of item j to order. The model is Minimize 2

s.t 0 1, 2, . . ., 1 1







 



 

=

=

C D S D Q

iC Q

w Q W Q j N In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (Dj /Qj is the number of orders), and the last term is the annual inventory holding cost (Qi /2 is the average amount of inventory).
Construct and solve a nonlinear optimization model for the following data:
Item 1 Item 2 Item 3 Annual Demand 2,000 2,000 1,000 Item Cost ($) 100 50 80 Order Cost ($) 150 135 125 Space Required (sq. feet)
5000 0.20 W i 5 5 50 25 40