3. Tri-County Utilities, Inc. supplies natural gas to customers in three-county area. The company purchases purchases natiural gas from two companies: Southern Gas and Northwest Gas. Deman forecasts for the coming winter season are as follows: Hamilton County, 400 units; Butler County, 200 units; and Clermont County, 300 units. Contracts to provide the following quanities have been written: Southern Gas, 500 units; and Northwest Gas, 400 units. Distribution costs for the counties vary, depending upon the location of the suppliers. The distribution costs per unit (in thousands of dollars) are as follows:
From Hamilton Butler Clermont
Southern Gas 10 20 15
Northwest Gas 12 15 18

a. Develop a network representation of this problem.
b. Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs.
c. Describe the distribution plan and show the total distribution cost.
d. Recent residential and industrial growth in Butler County has the potential for increasing demand by as much as 100 units. Which supplier should Tri-County contract with to supply the additional capacity?

5. Premier Consulting's two consultants. Avery and Baker, can be scheduled to work for clients up to a maximum of 160 hours each over the next four weeks. A third consultant, Campbell, has some administrative assignments already planned and is available for clients with projects in process. The estimated hourly requirements for each of the clients over the four-week period as follows:

Client Hours
A 180
B 75
C 100
D 85

Hourly rates vary for the consultant-client combination and are based on several factors, including project type and the consultant's experience. The rates (dollars per hour) for each consultant-client combination are as follows:

Consultant A B C D
Avery 100 125 115 100
Baker 120 135 115 120
Campbell 155 150 140 130

a. Develop a network representation of the problem.
b. Formulate the problem as a linear program, with the optimal solution providing the hours each consultant should be scheduled for each client to maximize the consulting firm's billings. What is the schedule and what is the total billing?
c. New information shows that Avery doesn't have the experience to be scheduled for client B. If this consulting assignment is not permitted, what impact does it have on total billings? What is the revised schedule?

7. Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in Los Angeles, Tulsa, and Seattle. The following table shows Aggie Power Generation's major residential markets, the annual demand in each market (in megawatts or MWs) and the cost to supply electricity to each market from each power generation plant (prices are in $/MW).

Distribution Costs
City Los Angeles Tulsa Seattle Demand (MWs)
Seattle $356.25 $593.75 $59.38 950.00
Portland $356.25 $593.75 $178.13 831.25
San Francisco $178.13 $475.00 $296.88 2375.00
Boise $356.25 $475.00 $296.88 593.75 
Reno $237.50 $475.00 $356.25 950.00
Bozeman $415.63 $415.63 $296.88 593.75
Laramie $356.25 $415.63 $356.25 1187.50
Park City $356.25 $356.25 $475.00 712.50
Flagstaff $178.13 $475.00 $593.75 1187.50
Durango $356.25 $296.88 $593.75 1543.75

7b. If at most 4000 MWs of power can be supplied by any one of the power plants, what is the optimal solution? What is the annual increase in power distribution cost that results from adding these constraints to the original formulation?

9. The Ace Manufacturing Company has orders for three similar products:
Product (units)
A 2000
B 500
C 1200

Three machines are available for the manufacturing operations. All three machines can be produce all the products at the same products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are as follows:
Machine (units) Machine A B C
1 1500 1 $1.00 $1.20 $0.90
2 1500 2 $1.30 $1.40 $1.20
3 1000 3 $1.10 $1.00 $1.20

Use the transportation model to develop the minimum cost production schedule for the products and machines. Show the linear programming formulation.

19. A local television station plans to drop four Friday evening programs at the end of the season. Steve Botuchis, the station manager, developed a list of six potential replacement programs. Estimates of the advertising revenue ($) that can be expected for each of the new programs in the four vacated time slots are as follows. Mr. Botuchis asked you to find the assignment of programs to time slots that will maximize total advertising revenue.

5:00- 5:30- 7:00- 8:00-
5:30 PM. 6:00 PM. 7:30 PM. 8:30 PM.
Home Improvement 5000 3000 6000 4000
World News 7500 8000 7000 5500
NASCAR Live 8500 5000 6500 8000
Wall Street Today 7000 6000 6500 5000
Holiday Briefings 7000 8000 3000 6000
Ramundo & Son 6000 4000 4500 7000

a. Develop a network representation of the problem.
b. Formulate and solve a linear programming model to determine the minimum cost assignment.