Linear Programming Practice Problems Sheet No. 5
Formulate the mathematical model for the following LP problems:
Problem No. 1:
The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year’s program. Advertising alternatives include television, radio, and newspaper. Audience estimates, costs, and maximum media usage limitations are as shown.
Constraint / Television / Radio / NewspaperAudience per advertisement / 100,000 / 18,000 / 40,000
Cost per advertisement / $2,000 / $300 / $600
Maximum media usage / 10 / 20 / 10
To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.
If the promotional budget is limited to $18,200, write the mathematical model to determine how many commercial messages should be run on each medium to maximize the total audience contact.
Problem No. 2:
The management of Hartman Company is trying to determine the amount of each of the two products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability.
Product / (hours/unit)Department / 1 / 2 / Labor-Hours Available
A / 1.00 / 0.35 / 100
B / 0.30 / 0.20 / 36
C / 0.20 / 0.50 / 50
Profit contribution/unit / $30.00 / $15.00
Problem No. 3:
The employee credit union at StateUniversity is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue-producing investments together with annual rates of return are as follows:
Type of Loan/Investment / Annual Rate of Return (%)Automobile loans / 8
Furniture loans / 10
Other secured loans / 11
Signature loans / 12
Risk-free securities / 9
The credit union will have $2,000,000 available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments.
Risk-free securities may not exceed 30% of the total funds available for investment.
Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, other secured, and signature loans).
Furniture loans plus other secured loans may not exceed the automobile loans
Other secured loans plus signature loans may not exceed the funds invested in risk-free securities.
Write the mathematical model to allocate the $2,000,000 to each of the loan/investment alternatives to maximize total annual return.
Problem No. 4:
Ajax Fuels, Inc., is developing a new additive for airplane fuels. The additive is a mixture of three ingredients: A, B, and C. For proper performance, the total amount of additive (amount of A + amount of B + amount of C) must be at least 10 ounces per gallon of fuel. However, because of safety reasons, the amount of additive must not exceed 15 ounces per gallon of fuel. The mix or blend of the three ingredients is critical. At least 1 ounce of ingredient A must be used for every ounce of ingredient B. The amount of ingredient C must be greater than one-half the amount of ingredient A. If the costs per ounce for ingredients A, B, and C are $0.10, $0.03, and $0.09, respectively, write the mathematical model to find the minimum cost mixture of A, B, and C for each gallon of airplane fuel.
Problem No. 5:
The Clark County Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 A.M., noon, 4:00 P.M., 8:00 P.M., midnight, and 4:00 A.M. An officer beginning a shift at one of these times works for the next 8 hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty:
Time of Day / Minimum Officers on Duty8:00 A.M. – noon / 5
Noon- 4:00 P.M. / 6
4: 00 P.M. - 8:00 P.M. / 10
8:00 P.M. – Midnight / 7
Midnight – 4:00 A.M. / 4
4:00 A.M. – 8:00 A.M. / 6
Determine the number of police officers that should be scheduled to begin the 8-hour shifts at each of the six times (8:00 A.M., noon, 4:00 P.M., 8:00 P.M. midnight, and 4:00 A.M.) to minimize the total number of officers required.
Problem No. 6:
Seastrand Oil Company produces two grades of gasoline: regular and high octane. Both gasolines are produced by blending two types of crude oil. Although both types of crude oil contain the two important ingredients required to produce both gasolines, the percentage of important ingredients in each type of crude oil differs, as does the cost per gallon. The percentage of ingredients A and B in each type of crude oil and the cost per gallon are shown.
Crude Oil / Cost / Ingredient A / Ingredient B1 / $0.10 / 20% / 60%
2 / $0.15 / 50% / 30%
Each gallon of regular gasoline must contain at least 40% of ingredient A, whereas each gallon of high octane can contain at most 50% of ingredient B. Daily demand for regular and high-octane gasoline is 800,000 and 500,000 gallons, respectively. Formulate the mathematical model to find how many gallons of each type of crude oil should be used in the two gasolines to satisfy daily demand at a minimum cost.
Problem No. 7:
The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum.
The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The accompanying table shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes.
Feed MixDiet Requirement (Ingredient) / Oat Product (units) / Enriched Grain (units) / Mineral product (units) / Minimum Daily Requirement
(units)
A / 2 / 3 / 1 / 6
B / ½ / 1 / ½ / 2
C / 3 / 5 / 6 / 9
D / 1 / 1½ / 2 / 8
E / ½ / ½ / 1½ / 5
Cost/unit / $0.09 / $0.14 / $0.17
In addition, the stable owner determines that 6 pounds of feed per day are the most that any horse needs to function properly.
Problem No. 8:
A company imports goods at two ports: Philadelphia and New Orleans. Shipments of one product are made to customers in Atlanta, Dallas, Columbus, and Boston. For the next planning period, the suppliers at each port, customer demands, and shipping costs per case from each port to each customer are as follows:
CustomersPort / Atlanta / Dallas / Columbus / Boston / Port Supply
Philadelphia / 2 / 6 / 6 / 2 / 5000
New Orleans / 1 / 2 / 5 / 7 / 3000
Demand / 1400 / 3200 / 2000 / 1400
Write the mathematical model for the above problem.
Problem No. 9:
Tri-County Utilities, Inc. supplies natural gas to customers in three-country area. The company purchases natural gas from two companies: Southern Gas and Northwest Gas. Demand forecasts for the coming winter season are HamiltonCounty, 400 units; ButlerCounty, 200 units; and ClermontCounty, 300 units. Contracts to provide the following quantities 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:
ToFrom / Hamilton / Butler / Clermont
Southern Gas / 10 / 20 / 15
Northwest Gas / 12 / 15 / 18
Problem No. 11:
Fowle Marketing Research has three project leaders available for assignment to three clients. Find the assignment of project leaders to clients that will minimize the total time to complete all projects. The estimated project completion times in days are as follows:
ClientsProject Leader / 1 / 2 / 3
Terry / 10 / 15 / 9
Carle / 9 / 18 / 5
McClymonds / 6 / 14 / 3
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