EM 505 - Decision Models, Fall 2012 Homework 3 - December 10-11, 2012 1. The diagram below depicts a system of aqueducts that originate at three rivers (nodes R1, R2 and R3) and terminate at a major city (node T) where the other nodes are junction points in the system. Using units of thousands of acre feet, the tables below show the maximum amount of water that can be pumped through each aqueduct per day and the following diagram shows the network of the system. From/To A R1 75 R2 40 R3 B 65 50 80 C 60 70 From/To D A 60 B 70 C E 45 55 70 F 45 90 From/To T D 120 E 190 F 130
The city water manager wants to determine a flow plan that will maximize the flow of water of the city. Formulate this problem as a max flow problem by identifying a source, a sink and transshipment nodes, and then drawing the complete network that shows the capacity of each arc. 2. The Quick Chip gravel company has received a contract to supply two new construction projects in the towns of Brock and Wurst. A total of 60 truckloads are needed at Brock in the next month and 90 at Wurst. Quick Chip has idle gravel pits in the towns of Nova, Scova, and Tova, each with a monthly production capacity of 50 truckloads. Travel distances from each pit to each project site are shown in the following table: Pit Nova Scova Tova To Brock 23 8 53 To Wurst 77 94 41
The truck company wants to fulfill its contract at least total truck travel distance. Formulate an LP to choose an optimal shipping plan. 3. The Makonsel Company is a fully integrated company that both produces goods and sells them at its retail outlets. After production, the goods are stored in the company’s two warehouses until needed by the retail outlets. Trucks are used to transport the goods from two plants to the warehouses, and then from the warehouses to the three retail outlets. Using the units of full truckloads, the following table shows each plant’s monthly output, its shipping cost per truckload sent to each warehouses, and the maximum amount that it can ship per month to each warehouse.
From/To p1 p2
unit shipping cost Wh1 Wh2 $425 $560 $510 $600
From/To p1 p2
shipping capacities Wh1 Wh2 125 150 175 200
output 200 300
For each retail outlet (RO), the next table shows its monthly demand, its shipping cost per truckload from each warehouse, and the maximum amount that can be shipped per month from each warehouse. unit shipping cost From/To Wh1 Wh2 demand RO1 470 390 150 RO2 505 410 200 RO3 490 440 150 shipping capacities RO1 100 125 RO2 150 150 RO3 100 75
Management now wants to determine a distribution plan (# of truckloads shipped per month from each plant to each warehouse and from each warehouse to each retail outlet) that will minimize the total shipping cost. (a) Draw a network that depicts the company’s distribution network. (b) Formulate this problem as a minimum cost flow problem. 4. RentCar is developing a replacement plan for its car fleet for a 5-year (2008 to 2012) planning horizon. At the start of each year, a decision is made as to whether a car should be kept in operation or replaced. A car must be in service at least 1 year but must be replaced after 3 years. The following table provides the replacement cost as a function of the year a car is acquired and the number of years in operations. year acquired 2008 2009 2010 2011 1 4000 4300 4800 4900 2 5400 6200 7100 3 9800 8700 -
Formulate the problem that will minimize the total cost. 5. The Chicago Board of Education is taking bids on the city’s four school bus routes. Four companies have made the bids in table below: Route 1 Company 1 Company 2 Company 3 Company 4 $2500 $3000 Route 2 $3000 $4000 Route 3 $1500 $2000 $4000 Route 4 $4000 $5000
(a) Suppose each bidder can be assigned only one route. How can Chicago minimize its cost of running the four bus routes? (b) Suppose that each company can be assigned two routes. How can Chicago minimize its cost of running the four bus routes?
6. The R&D...
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