Answered: (a) Draw the network representation for…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter5: Network Models

Section: Chapter Questions

Problem 68P

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Question

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.

(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.

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