1. You are provided the following linear program:
Maxz = 3x1 +4x2,
s.t.
x1+ x2 ≤ 34
2x1 + 5x2 ≤ 120
2x1 + x2 ≤ 60
x2 ≤ 20
x1, x2 ≥ 0
a. On the graph paper provided on the following page, use the graphical solution method to determine the optimal solution to the given linear program. Identify the feasible region on your graph.
b. Provide the optimal solution and optimal objective function value. Report your answers to two decimal places.
Optimal solution: x = _______ ; y = ________
Optimal objective function value: __________
2. Ethan Steel, Inc. has two factories that manufacture steel components for four different rail projects located at four different sites. The demand for the steel components for the four projects, Project A, Project B, Project C, and Project D, are 3220, 3675, 4125, and 2975, respectively. The shipping details are as below:
Production Details:
|
Factor
|
Maximum Capacity
|
|
1
|
6500
|
|
2
|
8500
|
Shipping Details (with per0unit shipping cost):
|
Factory
|
Project
|
|
A
|
B
|
C
|
D
|
|
1
|
$7
|
$7
|
$8
|
$4
|
|
2
|
$6
|
$5
|
$7
|
$3
|
Develop a linear programming model to minimize the cost for this transportation problem?
3. A construction company must decide on the size of the shopping mall, i.e. Large, Medium or Small, that has to be constructed in their acquired plot in the sub-urban area of Seattle. Due to the market conditions, the number of visitors to the mall will be High, Moderate, or Low. The profit payoff table for management (in millions of dollars) after 5 years is provided below.
|
|
Number of visitors
|
|
Size of the mall
|
High
|
Moderate
|
Low
|
|
Large
|
25
|
15
|
-20
|
|
Medium
|
20
|
12
|
-10
|
|
Small
|
15
|
13
|
5
|
The probabilities are P(High) = 0.35, P(Moderate) = 0.40, and P(Low) = 0.25.
a. Construct a decision tree for this problem.
b. Use the expected value approach and recommend the best decision.
4. Suppose the Durr family is considering purchasing a new home. Three mortgage options are available: a I-year adjusted-rate mortgage (ARM) at a low interest rate, a 3-year ARM at a slightly higher rate, and a 30-year fixed mortgage at the highest rate. However, both ARMs are sensitive to interest rate changes and the rates may change resulting in either higher or lower interest charges; thus the potential changes in interest rates are the uncertain outcomes. Because the family anticipates staying in the home at least 5 years and want to minimize the amount of interest they pay on their mortgage during those 5 years, they want to know the total interest costs they might incur; these represent the payoffs associated with their choice and the future change in interest rates. The payoff table is as follows:
|
|
States of Nature
|
|
Decision
|
Rates Rise
|
Rates Stable
|
Rates Fall
|
|
1-year ARM
|
$49,392
|
$35,494
|
$27,192
|
|
3-year ARM
|
$39,283
|
$33,214
|
$32,234
|
|
30-year fixed
|
$34,567
|
$34,567
|
$34,567
|
a. Which mortgage type should the Durr family choose if they make their decision using the conservative approach?
b. Which mortgage type should the Durr family choose if they make their decision using the minimax regret approach?
c. Which mortgage type should the Durr family choose if they make their decision using optimistic approach?
5. A Cake & pastry shop makes 3 types of cakes which require three significant ingredients, given the combination of other ingredients vary. The data for the amount of these ingredients needed to make the cakes are provided in the table below:
|
Cake
|
Small
|
Medium
|
Large
|
Available
|
|
Plain flour (Ounce)
|
8
|
16
|
21
|
400
|
|
Caster sugar (Ounce)
|
18
|
22
|
25
|
500
|
|
Cocoa powder (Ounce)
|
3
|
5
|
11
|
150
|
|
Profit/Unit
|
$18
|
$25
|
$32
|
|
Assuming that the A Cake & Pastry shop is interested in maximizing the total profit, answer the following questions based on given the linear programming model and sensitivity reports below:
a. Optimal Profit:
b. # of small cakes made:
c. # of medium cakes made:
d. # of large cakes made:
e. If 100 more ounces of caster sugar is available, what is the change in profit?
f. What is the change in profit if 5 more ounces of plain flour is available?
g. If 23 more ounces of cocoa powder is available, what is the change in profit?
Let S = Numbers of small cakes made
M = Number of medium cakes made
L = Number of large cakes made
Max 18S + 25M + 32L
s.t. 8S + 16M + 21L ≤ 400
18S + 22M + 25L ≤ 550
3S + 5M + 11L ≤ 150
S, M, L ≥ 0
The Sensitivity Report:
|
Variable Cells
|
|
Cell
|
Name
|
Final Value
|
Reduced Cost
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
$C$3
|
S
|
3
|
0
|
18
|
2
|
4
|
|
$D$3
|
M
|
16.5
|
0
|
25
|
2
|
2
|
|
$E$3
|
L
|
5
|
0
|
32
|
8
|
2
|
|
Constraints
|
|
Cell
|
Name
|
Final Value
|
Shadow Price
|
Constraint R.H. Side
|
Allowable Increased
|
Allowable Decrease
|
|
$F$9
|
Plain flour
|
400
|
2
|
400
|
14
|
72
|
|
$F$10
|
Caster sugar
|
550
|
0.75
|
550
|
350
|
22
|
|
$F$11
|
Cocoa powder
|
150
|
0.47
|
150
|
49
|
25
|