Question

A&C Distributors is a company that represents many outdoor products companies and schedules deliveries to discount stores, garden centers, and hardware stores. Currently, scheduling needs to be done for two lawn sprinklers: the Water Wave and Spring Shower models. Requirements for shipment to a warehouse for a national chain of garden centers are shown below.

Month	Shipping Capacity	Product	Minimum Requirement	Unit Cost to Ship	Per Unit Inventory Cost
March	8000	Water Wave	3000	.30	.06
		Spring Shower	1800	.25	.05
April	7000	Water Wave	4000	.40	.09
		Spring Shower	4000	.30	.06
May	6000	Water Wave	5000	.50	.12
		Spring Shower	2000	.35	.07

Let S_ij be the number of units of sprinkler i shipped in month j, where i = 1 or 2, and j = 1, 2, or 3. Let W_ij be the number of sprinklers that are at the warehouse at the end of a month, in excess of the minimum requirement.

a.	Write the portion of the objective function that minimizes shipping costs. (5 marks)
b.	An inventory cost is assessed against this ending inventory. Give the portion of the objective function that represents inventory cost. (5 marks)

Answer 1

As asked,

Let S_ij be the number of units of sprinkler i shipped in month j, where i = 1 or 2, and j = 1, 2, or 3. Let W_ij be the number of sprinklers that are at the warehouse at the end of a month, in excess of the minimum requirement.

a)

Cost function of Shipping - Min  0.30S₁₁ + 0.25S₁₂ + 0.40S₂₁ + 0.30S₂₂ + 0.50S₃₁ + 0.35S₃₂

b)

Cost function of Inventory - Min 0.06W₁₁ + 0.05W₁₂ + 0.09W₂₁ + 0.06W₂₂ + 0.12W₃₁ + 0.07W₃₂