**I. Production Management**

Metal Containers, Inc. is reviewing the way it submits bids on U.S. Army contracts. The army often requests open-top boxes, with square bases and of specified volumes. The army also specifies the materials for the boxes, and the base is usually made of a different material than the sides. The box is assembled by riveting a bracket at each of the eight corners. For Metal Containers, the total cost of producing a box is the sum of the cost of the materials for the box and the labor costs associated with affixing each bracket.

Instead of estimating each job separately, the company wants to develop an overall approach that will allow it to cost out proposals more easily. To accomplish this, company managers need you to devise a formula for the total cost of producing each box and determine the dimensions that allow a box of specified volume to be produced at minimum cost. Use the following notation to help you solve this problem.

Cost of the material for the base
=
A
per square unit

Cost of the material for the sides
=
B
per square unit

Cost of each bracket
=
C

Cost to affix each bracket
=
D

Length of the sides of the base
=
x

Height of the box
=
h

Volume specified by the army
=
V

Metal Containers asks you to determine how best to order the brackets it uses on its boxes. You are able to obtain the following information: The company uses approximately 100,000 brackets a year, and the purchase price of each is $5. It buys the same number of brackets (say,
n
) each time it places an order with the supplier, and it costs $60 to process each order. Metal Containers also has additional costs associated with storing, insuring, and financing its inventory of brackets. These carrying costs amount to 15% of the average value of inventory annually. The brackets are used steadily and deliveries are made just as inventory reaches zero, so that inventory fluctuates between zero and *n* brackets.

If the total annual cost associated with the bracket supply is the sum of the annual purchasing cost and the annual carrying costs, what order size *n* would minimize the total cost?