8-7 (Media selection problem) The advertising director for Diversey Paint and Supply, a chain of four retail stores on Chicago's North Side, is considering two media possibilities. One plan is for a series of half-page ads in the Sunday Chicago Tribune newspaper, and the other is for advertising time on Chicago TV.
The stores are expanding their lines of do-it-yourself tools, and the advertising director is interested in an exposure level of at least 40% within the city's neighborhoods and 60% in northwest suburban areas.
The TV viewing time under consideration has an exposure rating per spot of 5% in city homes and 3% in the northwest suburbs. The Sunday newspaper has corresponding exposure rates of 4% and 3% per ad. The cost of a half-page Tribune advertisement is $925; a television spot costs $2,000.
Diversey Paint would like to select the least costly advertising strategy that would meet desired exposure levels.
a. Formulate using LP.
b. Solve the problem.
8-10 (High school busing problem) The Arden County, Maryland, superintendent of education is responsible for assigning students to the three high schools in his county. He recognizes the need to bus a certain number of students, for several sectors of the county are beyond walking distance to a school.
The superintendent partitions the county into five geographic sectors as he attempts to establish a plan that will minimize the total number of student miles traveled by bus.
He also recognizes that if a student happens to live in a certain sector and is assigned to the high school in that sector, there is no need to bus that student because he or she can walk to school. The three schools are located in sectors B, C, and E.
The following table reflects the number of high-school-age students living in each sector and the busing distance in miles from each sector to each school:
|
DISTANCE TO SCHOOL
|
|
|
SECTOR
|
SCHOOL IN SECTOR B
|
SCHOOL IN SECTOR C
|
SCHOOL IN SECTOR E
|
NUMBER OF STUDENTS
|
|
A
|
5
|
8
|
6
|
700
|
|
B
|
0
|
4
|
12
|
500
|
|
C
|
4
|
0
|
7
|
100
|
|
D
|
7
|
2
|
5
|
800
|
|
E
|
12
|
7
|
0
|
400
|
|
|
|
|
2,500
|
Each high school has a capacity of 900 students. Set up the objective function and constraints of this problem using LP so that the total number of student miles traveled by bus is minimized. Then solve the problem.
8-15 (Agricultural production planning problem) Margaret Black's family owns five parcels of farmland, referred to as the southeast sector, north sector, northwest sector, west sector, and southwest sector.
Margaret is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production plan for next year.
The Pennsylvania Water Authority has just announced its yearly water allotment, with the Black farm receiving 7,400 acre-feet. Each parcel can tolerate only a specified amount of irrigation per growing season, as specified in the following table:
|
PARCEL
|
AREA (ACRES)
|
WATER IRRIGATION LIMIT (ACRE-FEET)
|
|
Southeast
|
2,000
|
3,200
|
|
North
|
2,300
|
3,400
|
|
Northwest
|
600
|
800
|
|
West
|
1,100
|
500
|
|
Southwest
|
500
|
600
|
Each of Margaret's crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follow:
|
CROP
|
MAXIMUM SALES
|
WATER NEEDED PER ACRE (ACRE-FEET)
|
|
Wheat
|
110,000 bushels
|
1.6
|
|
Alfalfa
|
1,800 tons
|
2.9
|
|
Barley
|
2,200 tons
|
3.5
|
Margaret's best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre.
- Formulate Margaret's production plan.
- What should the crop plan be, and what profit will it yield?
- The Water Authority informs Margaret that for a special fee of $6,000 this year, her farm will qualify for an additional allotment of 600 acre-feet of water. How should she respond?