Sports of All Sorts produces, distributes, and sells high-quality skateboards. Its supply chain consists of three factories (located in Detroit, Los Angeles, and Austin) that produce skateboards. The Detroit and Los Angeles facilities can produce 350 skateboards per week, but the Austin plant is larger and can produce up to 700 skateboards per week. Skateboards must be shipped from the factories to one of four distribution centers, or DCs (located in Iowa, Maryland, Idaho, and Arkansas). Each distribution center can process (repackage, mark for sale, and ship) at most 500 skateboards per week.

Factory/DCs	Shipping Costs ($ per skateboard)
	Iowa 4	Maryland 5	Idaho 6	Arkansas 7
Detroit 1	25.00	25.00	35.00	40.00
Los Angeles 2	35.00	45.00	35.00	42.50
Austin 3	40.00	40.00	42.50	32.50

Skateboards are then shipped from the distribution centers to retailers. Sports of All Sorts supplies three major U.S. retailers: Just Sports, Sports 'N Stuff, and The Sports Dude. The weekly demands are 200 skateboards at Just Sports, 500 skateboards at Sports 'N Stuff, and 650 skateboards at The Sports Dude. The following tables display the per-unit costs for shipping skateboards between the factories and DCs and for shipping between the DCs and the retailers.

Retailers/DCs	Shipping Costs ($ per skateboard)
	Iowa 4	Maryland 5	Idaho 6	Arkansas 7
Just Sports 8	30.00	20.00	35.00	27.50
Sports 'N Stuff 9	27.50	32.50	40.00	25.00
The Sports Dude 10	30.00	40.00	32.50	42.50

(a)

Draw the network representation for this problem. (Submit a file with a maximum size of 1 MB.)

(b)

Build a model to minimize the transportation cost of a logistics system that will deliver skateboards from the factories to the distribution centers and from the distribution centers to the retailers. (Let Detroit be node 1, Los Angeles be node 2, Austin be node 3, Iowa be node 4, Maryland be node 5, Idaho be node 6, Arkansas be node 7, Just Sports be node 8, Sports Stuff be node 9, and Sports Dude be node 10. Express your answers in the form x_ij where x_ij represents the number of units shipped from node i to node j.)

Min

s.t.Detroit Production

Los Angeles Production

Austin Production

Iowa Shipments

Maryland Shipments

Idaho Shipments

Arkansas Shipments

Iowa Processing

Maryland Processing

Idaho Processing

Arkansas Processing

Just Sports Demand

Sports Stuff Demand

Sports Dude Demand

x_ij 0 for all i and j.

What is the optimal production strategy and shipping pattern for Sports of All Sorts? Enter the number of units shipped where x_ij represents the number of units shipped from node i to node j.

x14	x15	x16	x17
x24	x25	x26	x27
x34	x35	x36	x37
x48	x58	x68	x78
x49	x59	x69	x79
x410	x510	x610	x710

=

What is the minimum attainable transportation cost (in dollars)?

$

(c)

Sports of All Sorts is considering expansion of the Iowa DC capacity to 800 units per week. The annual amortized cost of expansion is $40,000. Should the company expand the Iowa DC capacity so that it can process 800 skateboards per week? (Assume 50 operating weeks per year.)

The cost of the optimal solution associated with this increased capacity at the Iowa distribution center is $ per week. Therefore, the potential savings over 50 weeks is 50

$ $

= $ per year. The $40,000 cost of expansion ---Select--- exceeds does not exceed the potential savings, so Sports of All Sorts should not expand.