MATHEMATICAL PROGRAMMING PROBLEMS
For each of the following problem:
a) Express the problem in mathematical notation.
b) Write it GAMS notation and run the program to solve the problem.
c) Report the results on a separate paper. Specifically, report:
i) The optimum levels of the decision variables.
ii) The value of the objective.
iii) Interpret the marginal values of the constraints, including the meaning of a zero marginal value.
d) Open your *.LST file, edit out the page breaks, print, and attach after the report page.
1. San Bernardino Social Services agency is trying to design a minimum-cost meal that will be used in feeding the area's poor. The meal is to consist only of seaweed and tofu but must meet the following nutrient requirements: calories 300 units, protein 250 units, and vitamins 100 units. The nutrient content per ounce of each food is as follows:
|
|
Seaweed
|
Tofu
|
Minimum Nutrition Requirement
|
|
Calories
|
20
|
15
|
300
|
|
Protein
|
10
|
25
|
250
|
|
Vitamins
|
10
|
4
|
100
|
Seaweed costs $0.02 per ounce and Tofu costs $0.04 per ounce. Determine the amount of seaweed and tofu that should go into a meal to minimize cost.
2. The production of type A, B, and C transistors requires processing in each of five locations of a firm's manufacturing facility. The price per unit on these products are $1.07, $0.86, and $0.98, respectively and the costs per unit are $1, $0.80, and $0.90, respectively. The processing times required in each location (in minutes) and the total minutes available per production period are shown below. Determine the amount of each type of transistor this company should produce to maximize profits.
|
|
Time Required in Location (minutes)
|
|
Transistor
|
L1
|
L2
|
L3
|
L4
|
L5
|
|
A
|
3
|
1
|
2
|
5
|
2
|
|
B
|
1
|
1
|
4
|
3
|
5
|
|
C
|
2
|
2
|
1
|
4
|
4
|
|
Total time
|
|
|
|
|
|
|
Available in
|
3,500
|
2,000
|
3,000
|
5,000
|
3,000
|
|
Locations
|
|
|
|
|
|
3. Coyote Publishing prints its Mathematical Economics textbook in plants in Boston, Atlanta, Denver, Seattle, and San Bernardino. The books are then shipped to distribution centers in Newark, Chicago, Dallas, and Los Angeles. Table below gives the information on cost of shipping a book from the production plants to distribution centers, production capacity at each plant, and quantity demanded at each distribution center. Find the lowest cost way of supplying the textbook to each distribution center.
|
|
Distribution Center
|
|
Plant
|
Newark
|
Chicago
|
Dallas
|
Los Angeles
|
Production Capacity
|
|
Boston
|
$0.20
|
$0.30
|
$0.40
|
$0.60
|
40,000
|
|
Atlanta
|
0.35
|
0.40
|
0.45
|
0.50
|
10,000
|
|
Denver
|
0.30
|
0.20
|
0.30
|
0.40
|
15,000
|
|
Seattle
|
0.50
|
0.40
|
0.30
|
0.30
|
15,000
|
|
San- Bernardino
|
0.70
|
0.50
|
0.45
|
0.20
|
20,000
|
|
Demand
|
20,000
|
40,000
|
30,000
|
10,000
|
|
4. Marketing department must select a mix of advertising in order to reach a minimum of 1 million adult males, 2 million adult females, 0.5 million senior citizens, and 1.5 children. The cost per unit of advertising is: television $200, radio $15, magazines $90, and newspaper $30. Determine the number of units of each advertisement that will meet the marketing requirements at a minimum cost.
|
|
Number of People Reached per Unit
|
|
Target
|
Television
|
Radio
|
Magazine
|
Newspaper
|
|
Adult Males
|
100
|
5
|
50
|
30
|
|
Adult Females
|
300
|
20
|
160
|
5
|
|
Seniors
|
40
|
10
|
5
|
25
|
|
Children
|
100
|
40
|
10
|
5
|