Problem 10-13 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. 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. Shipping Costs ($ per skateboard) Factory/DCs Iowa Maryland Idaho Arkansas Detroit $25.00 $25.00 $35.00 $40.00 Los Angeles $35.00 $45.00 $35.00 $42.50 Austin $40.00 $40.00 $42.50 $32.50 Retailers/DCs Iowa Maryland Idaho Arkansas Just Sports $30.00 $20.00 $35.00 $27.50 Sports 'N Stuff $27.50 $32.50 $40.00 $25.00 The Sports Dude $30.00 $40.00 $32.50 $42.50 a. Choose the correct network representation of this problem. (i) Supply Demand DC Capacity Demand 500 8 25 1 30 Just 30 8 350 Detroit 200 35 27.5 Sports Iowa Just 27.5 200 Sports 40 30 500 30 20 25 20 5 32.5 2 350 45 32.5 9 Maryland LA Sports 500 9 40 Stuff Sports 500 40 500 Stuff 40 35 40 6 35 27.5 25 Idaho 32.5 35 10 3 500 25 27.5 42.5 700 Austin 42.5 Sports Dude 650 10 Sports 650 7 42.5 Dude Ark Supply DC Capacity 500 (iv) Supply DC Capacity 500 Demand 1 25 4 25 4 30 1 Iowa Iowa Detroit Detroit 350 25 350 25 27.5 Just Sports 200 35 500 35 30 500 40 40 20 5 5 32.5 35 45 Maryland 35 45 Maryland 350 ~ 2 2 9 350 40 LA LA Sports 500 35 500 35 500 Stuff 35 42.5 42.5 40 6 6 40 Idaho 40 Idaho 32.5 40 40 500 500 25 42.5 42.5 27.5 3 3 700 Austin 700 Austin 10 Sports 650 7 7 Dude 32.5 Ark 32.5 42.5 Ark Model (iv) 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. What is the optimal production strategy and shipping pattern for Sports of All Sorts? What is the minimum attainable transportation cost? If required, round your answers to two decimal places. Let x = units shipped from node i to node j. Min 24 X X1,4+ 25 X1,5 + 35 X1,6 + 40 X1,7 + 35 X2.4 + 45 X2,5 + 35 X2,6 + 42.5 42.5 X3,6 + 32.5 X3,7 + 30 X4,8 + 27.5 X4,9 + 30 X4,10 + 20 X2,7 + X5,8 + 40 X3,4 + 40 x3 32.5 X5,9 + 40 X5 35 X6.8 + 40 X6.9 + 32.5 X6.10 + 27.5 X7,8 + 25 X7,9 + 42.5 X7,10 subject to 150 X X1,4 + 200 X X1,5 + 0 X X1,6 + 0 X X1,7 s 350 200 X X2,4 + 0 X X2,5 + 150 X X2,6 + 0 X X2,7 350 150 X X3,4 + 0 X X3,5 + 0 X X3,6 + 500 X X3,7 s 750 150 X X1,4 + 200 X X2,4 + 150 X X3,4 = 0 X X4,8 + 0 X X4,9 + 500 X X4,10 200 X X1,5 + 0 X X2,5 + 0 X X3,5 200 = X X5,8 + 0 X X5,9 + 0 X X5,10 0 X X1,6 + 150 X X2,6 + 0 X X3,6 = 0 0 X X6,8 + X X 6,9 + 150 XX6,10 0 X X1,7 + 0 X X2,7 + 500 X X3,7 = 0 X X7,8 + 500 X X7,9 + 0 X X7,10 150 X X1,4 + 200 X X2,4 + 150 X X3,4 500 0 200 X X1,5 + X X2,5 + 0 X X3,5 500 0 X X1,6 + 150 X X2,6 + 0 X X3,6 < 500 0 X X1,7 + 0 X X2,7 + 500 X X3,7 500 0 X X4,8 + 200 X X5,8 + 0 X X 6,8 + 0 X X7,8 200 0 X X4,9 + 0 X X5,9 + 0 X X6.9 + 500 X X7,9 500 500 X X4,10 + 0 X X5,10 + 150 X X 6,10 + 0 X X7,10 650 for all i and j. Solving the formulation above, the optimal cost is $ 18,750 X per week. c. Sports of All Sorts is considering expansion of the lowa DC capacity to 800 units per week. The annual amortized cost of expansion is $40,000. Should the company expand the lowa DC capacity so that it can process 800 skateboards per week? (Assume 50 operating weeks per year.) Yes X Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor. The annual savings is less than the annual amortized cost of expansion. Therefore, the company should not expand the Iowa DC Capacity.