Show all work by describing variables, stating assumptions, illustrating model and showing output solution to the problem.
1. Market Research Real Estate
A market research firm has been retained to conduct focus group interviews for a real estate company. Plans are made to classify participants to age: over 45 or 45 and under and as to whether or not they have recently bought and/or sold a house (within the last two years). Specific guidelines have been developed:
- At least 100 people must be selected
- At least 75% of the participants must have bought or sold a house recently
- Of those who have not bought or sold recently, at least 40% must be over 45 years of age
- No more than 40% of participants should be 45 or under
There are preliminary screenings with associated costs. The cost for each participant begins at $10. Additional contact with those over 45 raises the cost by $12. Additional time with the recent buyers and sellers adds $7 to the cost of including those participants. Formulate a linear integer programming model to determine the amount of and type of participants that minimizes total cost. Use integer programming to establish whole numbers for the solution variables
2. Coffee Beans
Mrs. Olsen, a coffee processor, markets three blends of coffee. They are; Brand X , Minim and Taster's Reject. Olsen uses two types of coffee beans to make the three different brands; Columbian and Mexican Beans. The following chart list the composition of the blends:
|
Brand
|
Percentage of Columbian Beans
|
Percentage of Mexican Beans
|
|
Brand X
|
80%
|
20%
|
|
Minim
|
50%
|
50%
|
|
Tasters Reject
|
30%
|
70%
|
Ms. Olsen has already purchased 20,000 pounds of Columbian beans at 90 cents per pound, and she has purchased 30,000 pounds of Mexican beans at 50 cents per pound. At such, these resources are available for use. Unused Columbian beans can be sold at cost to another processor, but unused Mexican beans can be sold only for 35 cents per pound. Due to warehouse space limitations, Ms. Olsen must dispose of all unused beans.
Brand X sells for $2.60 per pound, Minim sells for $2.50 per pound, and Taster's Reject brings $2.34 per pound. All three products have the same production and packaging costs of $1.20 per pound.
Ms. Olsen is interested in finding the production schedule that will maximize profit. Formulate a relevant linear programming model or this problem to maximize profit and determine the solution.
3. McDonald and Thomas Advertising Agency
McDonald and Thomas Advertising Agency has been hired to put together an advertising plan for the Healthy Heart Charity Banquet. The advertising media under consideration are listed in the table below:
|
Medium
|
Cost Per Use
|
Effective Audience Reached
|
Maximum Number of times Available
|
|
Outdoor
|
$500
|
2000
|
3
|
|
Radio
|
$100
|
800
|
10
|
|
Newspaper
|
$200
|
1500
|
2
|
|
TV Commercial
|
$1000
|
3000
|
4
|
|
PSA
|
0
|
500
|
*
|
* The TV station will award up to three public service announcements (PSA) for each commercial purchased.
The budget for the campaign is $6000. Every medium must be used at least once. The total number of appearances on TV (including PSA) should be no more than twice the number of all the other advertisements. Develop a linear programming model that list objective function and constraints to maximize the effective audience reached. Use integer programming to make the variables in whole numbers
4. Susan Wong's Personal Budget
After Susan Wong graduated from State University with a degree in Operations Research, she went to work for a computer systems development firm in the Washington, D.C., area. As a student at State, Susan paid her normal monthly living expenses for apartment rent, food, and entertainment out of a bank account set up by her parents. Each month they would deposit a specific amount of cash into Susan's account. Her parents also paid her gas, telephone, and bank credit card bills, which were sent directly to them. Susan never had to worry about things like health, car, homeowners', and life insurance; utilities; driver's and car licenses; magazine subscriptions; and so on. Thus, while she was used to spending within a specific monthly budget in college, she was unprepared for the irregular monthly liabilities she encountered once she got a job and was on her own.
In some months Susan's bills would be modest and she would spend accordingly, only to be confronted the next month with a large insurance premium, or a bill for property taxes on her condominium, or a large credit card bill, or a bill for a magazine subscription, and so on the next month. Such unexpected expenditures would result in months when she could not balance her checking account; she would have to pay her bills with her bank credit card and then payoff her accumulated debt in installments while incurring high interest charges. By the end of her first year out of school she had hoped to have some money saved to begin an investment program, but instead she found herself in debt.
Frustrated by her predicament, Susan decided to get her financial situation in order. First, she sold the condominium that her parents had helped her purchase and moved into a cheaper apartment. This gave her enough cash to clear her outstanding debts with $3,800 left over to start the new year with. Susan then decided to use some of the operations research she had learned in college to help her develop a budget. Specifically, she decided to develop a linear programming model to help her decide how much she should put aside each month in short-term investments to meet the demands of irregular monthly liabilities and save some money.
First, Susan went through all of her financial records for the year and computed her expected monthly liabilities for the coming year, as shown in the following table:
|
MONTH
|
BILLS ($)
|
MONTH
|
BILLS ($)
|
|
January
|
$2,750
|
July
|
$3,050
|
|
February
|
$2,860
|
August
|
$2,300
|
|
March
|
$2,335
|
September
|
$1,975
|
|
April
|
$2,120
|
October
|
$1,670
|
|
May
|
$1,205
|
November
|
$2,710
|
|
Jun
|
$1,600
|
December
|
$2,980
|
Susan's after-taxes-and-benefits salary is $29,400 per year, which she receives in 12 equal monthly paychecks that are deposited directly into her bank account.
Susan has decided that she will invest any money she doesn't use to meet her liabilities each month in either a one-month, three-month, or seven-month short-term investment vehicles, rather than just leaving the money in an interest-bearing checking account. The yield on one-month investments is 6% per year nominal; on three-month investments the yield is 8% per year nominal; and on a seven-month investment the yield is 12% per year nominal. As part of her investment strategy, any time one of the short term investments comes due she uses the principal as part of her budget, but she transfers any interest earned to another long-term investment (which she doesn't consider in her budgeting process). For example, if she has $100 left over in January that she invests for three months, in April when the investment matures she uses the $100 she originally invested in her budget, but any interest on the $100 is invested elsewhere. (Thus, the interest is not compounded over the course of the year.)
Susan wants to develop a linear programming model that will maximize her investment return during the year so she can take that money and reinvest it at the end of the year in a longer-term investment program. However, she doesn't have to confine herself to short-term investments that will all mature by the end of the year; she can continue to put money toward the end of the year in investments that won't mature until the following year. Her budgeting process will continue to the next year, where she can take out any surplus left over after December and reinvest it in a long term program if she wants to.
Develop a linear programming model that to help Susan determine her best investment options to maximum her return. (Hint: You need to create variables that represent monthly investment options. Your objective is maximize return taking into account that money invested each month is left over after expenses have been paid)
5. The Hoggs Corn and Wheat Farm
The Hogg company are a group of Oklahoma agricultural producers. The Hoggs want to derive a production process that will satisfy their goals. These farmers can produce corn, wheat, and/or hogs. The acreage they can cultivate for crops is 1100 acres, of which 600 acres is for corn production and 500 acres for wheat production. The Hogs have a maximum of 2000 hr of labor available for crop operations and 600 hr of labor for their hog operation. Corn production requires 1.6 hr of labor per acre, wheat requires 2.0 hr of labor per acre, and the hog operation requires 20 hr of labor per hog unit. This farm can produce 120 bushel of corn per acre and 32 bushel of wheat per acre. Corn and wheat can be sold on the market at $2.50 per bushel and $2.35 per bushel, respectively, and/or can be used as feed in the hog operation. Also, corn and wheat may be purchased from the city feed store at $2.70 per bushel and $2.55 pr bushel, respectively, for feeding the hogs. The hog unit requires 169.1 bushel of feed during a single production period. The corn and wheat used as hog feed can be transformed into corn equivalent units; that is, 1.1 bushel of wheat equals 1 bushel of corn.
The cost per acre to produce corn and wheat are $146.40 per acre and $48.60 per acre. The returns to the hog operation are $919.20 per hog unit. A hog unit is a sow and her litter, which are sold on the market. Returns to the hog operation are net values, for example, after maintenance and replacement stock cost.
The Hogg company is considering the following strategy:
Strategy 1
As the goal with the highest priority, P1 they would like to see returns to the agriculture operation of at least $100,000. As their second priority, P2, at least three fourths of the land available for corn production and at least one-half of the land available for wheat production must be planted. The producers also desire that the under-planting of corn be weighed three times greater than that for wheat. The third and last priority goal P3, is that the hog operation be no greater than 40 hog units.
Formulate the problem in a goal programming format and solve for strategies 1 and 2. Discuss the results of whether the goals were achieved in each strategy and the corn, wheat and hog production output
6. Environmental Protection
Refer to the separate document "environ_lp_problem