1. A food distribution company ships fresh spinach from its four packing plants to large East-Coast cities. The shipping costs per crate and the supply and demand are shown in the table below.
a) Formulate a model that will permit the company to meet its demand at the lowest possible cost.
b) The company wishes to spread out the source for each of its markets to the maximum extent possible. To accomplish this, it will accept a 5% increase in its total transportation cost from part (a). What is the new transportation plan, and what is the new cost?
|
MARKETS
|
|
Packing Plants
|
Atlanta
|
Boston
|
Charlestown
|
Dover
|
Supply
|
|
Eaglestown
|
$6.00
|
$7.00
|
$7.50
|
$7.50
|
8,000
|
|
Farrier
|
$5.50
|
$5.50
|
$4.00
|
$7.00
|
10,000
|
|
Guyton
|
$6.00
|
$5.00
|
$6.50
|
$7.00
|
5,000
|
|
Hayesville
|
$7.00
|
$7.50
|
$8.50
|
$6.50
|
9,000
|
|
Demand
|
8,000
|
9,000
|
10,000
|
5,000
|
|
2. A security firm needs to connect alarm systems to the firm's main control site from five potential trouble locations. Since the systems must be fail-safe, the cables must be run in special pipes. These pipes are very expensive but large enough to simultaneously handle five cables (the maximum that might be needed). Use the minimal-spanning tree model to find the minimum total length of pipes needed to connect the locations. Node 6 represents the main control site.
3. Theo Harris earns $55,000 a year and has $9,000 to invest in a portfolio. His investment alternatives and their expected returns are shown in the table below.
|
Investment
|
Description
|
Expected Return
|
|
A
|
IRA (retirement)
|
3.5%
|
|
B
|
Employer's retirement plan
|
4.5%
|
|
C
|
Deferred income (retirement)
|
8.0%
|
|
D
|
Unity mutual fund
|
7.0%
|
|
E
|
Liberty mutual fund
|
7.5%
|
|
F
|
Money market
|
5.5%
|
Theo's investment goals are as follows and can be ranked according to the weights shown in parenthesis. Which investments should be included in Theo's portfolio, and how much should he invest in each?
Goal 1: (25) Invest all funds available.
Goal 2: (20) Maximize the total annual return in dollars, with a target of $1,000.
Goal 3: (15) Invest at least 3% of salary in employer's retirement plan.
Goal 4: (15) Invest at least 10% of the total investment in the money market.
Goal 5: (10) Invest at most 25% of the total investment in retirement plans.
Goal 6: (10) Invest at least 50% of the total investment in non-retirement plans.
Goal 7: (5) Invest at most 50% of the total investment in mutual funds.
Which investments should be included in Theo's
4. Jason Scott (see Problem 8-42) has decided to incorporate utility theory into his decision with his mortgage application. The following table describes Jason's utility function:
|
Monetary Value
|
Utility
|
|
-$4800
|
0.00
|
|
-$2900
|
0.10
|
|
-$2400
|
0.12
|
|
-$1000
|
0.15
|
|
-$500
|
0.19
|
|
$0
|
0.21
|
|
$1900
|
0.26
|
|
$2400
|
0.30
|
|
$4800
|
1.00
|
(a) How can you best describe Jason's attitude toward risk? Justify your answer.
(b) Will the use of utilities affect Jason's original decision in Problem 8-42?
5. A plant engineering group needs to set up an assembly line to produce a new product. The following table describes the relationships between the activities that need to be completed for this product to be manufactured:
|
Activity
|
Days
|
Immediate Predecessors
|
|
a
|
m
|
b
|
|
A
|
3
|
6
|
8
|
|
|
B
|
5
|
8
|
10
|
A
|
|
C
|
5
|
6
|
8
|
A
|
|
D
|
1
|
2
|
4
|
B, C
|
|
E
|
7
|
11
|
17
|
0
|
|
F
|
7
|
9
|
12
|
D
|
|
G
|
6
|
8
|
9
|
D
|
|
H
|
3
|
4
|
7
|
F, G
|
|
I
|
3
|
5
|
7
|
E, F, H
|
If using Crystal Ball, assume that the duration of each activity follows a BetaPert distribution, with the three time estimates shown for that activity. Otherwise, assume that each activity time is normally distributed with expected time and standard deviation computed as shown in equations 7-6 and 7-8, respectively, on page 283. Round off all activity times to two decimal places.
(a) Use simulation to determine the probability that the project will finish in 37 days or less.
(b) Use simulation to determine the probability that the project will take more than 32 days.