Homework Set 1
1. Enrollment Management. An engineering department chair wants to maximize the use of
scholarship dollars to shape the enrollment of the incoming freshmen. All applicants take an
engineering aptitude test and receive a scholarship based on their score. Here is the scholarship
award amount:
Test Score
Scholarship Amount
80
$2,290
85
$2,590
90
$3,250
95
$3,880
100
$4,150
The department chair wants to
 award scholarships to 100 students
 with at least 5 scholarships given to students in each test score
 and a total of $350,000 spent on the 100 scholarships.
What is the maximum average test score the freshmen class can have?
2. Make or Buy. A sudden increase in the demand for smoke detectors has left Acme Alarms with
insufficient capacity to meet demand. The company has seen monthly demand from its retailers for
its AC and battery-operated detectors rise to 20,000 and 10,000 respectively. Acme’s production
process involves three departments: fabrication, assembly, and shipping. The relevant quantitative
data on production and prices are summarized as follows:
Monthly Hours Hours/Unit Hours/Unit
Available
(AC)
(Battery)
2,800
0.1
0.14
Fabrication
Assembly
Shipping
3,000
0.11
0.15
2,300
0.1
0.13
In House
Variable cost/unit
Retail Price
AC
$19.90
Battery
$17.50
Subcontracted
AC
Battery
$21.30
$21.30
$31.90
$28.10
$31.90
$28.10
The company also has the option to obtain additional units from a subcontractor, who has offered
to supply up to 20,000 units per month in any combination of outlet and battery-operated models at
the charge shown in the chart. For this price, the subcontractor will test and ship its models directly
to the retailers without using Acme’s production process. What is the maximum profit? How many
of each type should they fabricate themselves, and how many should they buy from the
subcontractor? (fractional decisions are acceptable)?
Homework Set 2
3. Transporting Coal. The Calcio Coal Company produces coal at four mines (Mine1-Mine 4)and ships
it to four power plants (P1-P4). The cost per ton of producing coal and the production capacity (in
tons) for each mine are known. The number of tons of each coal demanded by each customer is
also known. The cost (in dollars) of shipping a ton of coal from a mine to each plant is available as
well. The following table provides the data:
P1
P2
P3
P4
Capacity
Cost
Mine 1
13
8
11
9
109
59
Mine 2
12
6
10
9
136
64
Mine 3
9
6
11
10
130
51
Mine 4
11
7
13
13
139
62
Demands
109
149
108
118
a. Calcio wishes to minimize the cost of producing and transporting coal from its mines to its
plants. What is the minimum cost?
b. What is the transportation schedule that achieves the minimum cost in (a)? (In other words,
how many tons should each mine ship to each plant?)
4. Assigning Engineers. A small engineering firm has 4 senior engineers (E1 – E 4) available to work on
the firm’s 4 current projects (P1-P4) over the next 2 weeks. The firm’s manager has developed the
following table of quality scores, which show each engineer’s quality on each type of project, on a
scale of 100. Also shown is the estimate of the time (in hours) required for each project.
P1
P2
P3
P4
E1
55
50
67
65
E2
73
76
70
42
E3
59
27
79
87
E4
70
67
36
85
Time
61
50
85
35
a. Assume that one engineer is assigned to each project. What assignment of engineers to
projects maximizes the sum of the quality scores assigned?
b. Suppose that each engineer has 80 hours available over the next two weeks. Assuming that
more than one engineer can work on a project, what assignment schedule maximizes the sum of
quality scores assigned?
Hint: In Excel, create different tabs for part a. and b. Solve independently of each other.
Homework 3
5. The blood bank wants to determine the least expensive way to transport available blood donations
from Pittsburg and Staunton to hospitals in Charleston, Roanoke, Richmond, Norfolk, and Suffolk.
The supply and demand for donated blood is shown along with the unit cost of shipping along each
possible arc. The numbers inside the nodes indicate the demand for units of blood.
a. What is the optimal solution?
b. Suppose that no more than 800 units of blood can be transported over any one arc. What is
the optimal solution to this revised problem?
6.
A furniture manufacturer has warehouses in cities represented by nodes 1, 2, and 3. The values on
the arcs indicate the per unit shipping costs required to transport living room suites between the
various cities. The supply of living room suites at each warehouse is indicated by the negative
number in the nodes 1, 2, and 3. The demand for living room suites is indicated by the positive
number in the remaining nodes.
a. Determine the least costly shipping plan for this problem.
b. Indicate how many suites should be sent from each node to each node.
Homework 4
7. Selecting Projects. The Texas Electronics Company (TEC) is contemplating a research and
development program encompassing eight major projects. The company is constrained from
embarking on all projects by the budget available for projects ($300,000). Following are the resource
requirements and the estimated profit for each project:
Expense ($000)
Profit ($000)
Engineers
P1
50
41
5
P2
79
39
8
P3
98
83
8
P4
70
82
7
P5
113
42
7
P6
99
33
6
P7
93
43
7
P8
114
78
7
a. What is the maximum profit, and which projects should be selected?
b. Suppose that management decides that projects 2 and 3 are mutually exclusive. (In other words,
TEC will not undertake both). What is the revised project portfolio and the revised maximum
profit?
c. Suppose that management also decides to undertake at least two of projects 5-8. As a result,
what are the revised project portfolio and maximum profit? (Assume b. is also true)
Hint: In Excel, create different tabs for part a., b, and c.. Solve independently of each other.
8. Establishing a Product Line. National Metals Company (NMC) manufactures titanium shafts. Its
equipment is capable of producing shafts in 10 lengths (in cm) as shown in the chart below,
reflecting settings on its machinery. Setting up the machinery to produce one of these results costs
$250. As a result, NMC has decided to make only a selected number of lengths. When a customer
requests a given length, NMC may supply it from stock, if it happens to match one of the lengths in
the production schedule. Otherwise, NMC trims a longer length to meet the order. The variable
cost for producing the shafts is $20 per cm, and NMC receives revenue of $40 per cm. Trim waste
can be sold to a recycler for $15 per cm.
The demand requirements for the coming week are tabulated as follows; all demand must be
satisfied.
Length
Demand
32
12
34
4
36
7
38
8
40
16
42
7
44
12
a. What is the optimal assortment of lengths for NMC to manufacture?
b. What is the optimal profit in the coming week?
46
5
48
8
50
3
Homework 5
9. Construction Bids. Gamma Construction Company has been asked to bid on the construction of 20
lighted tennis courts for State University. Each court will cost $19,000 in construction costs, and, in
addition, there will be a fixed expense of $8,000 to cover the preparation and submittal of the bid.
Gamma is considering five different bid levels. Each level involves a different profit margin,
calculated as a percentage above total construction cost (TCC). Fixed expenses are excluded from
this calculation, but they are relevant for profitability that it will win the bid at each level being
considered. The bids and the probabilities are summarized in the following table.
Bid 1
Bid 2
Bid 3
Bid 4
Bid 5
Amount of Bid
TCC + 5%
TCC + 10%
TCC + 15%
TCC + 20%
TCC + 25%
Probability of Winning
0.620
0.525
0.430
0.335
0.240
a. What is the optimal bid for Gamma to make?
b. What is the expected profit associated with the optimal bid?
10. System Options. JR Davidson recently started a practice in Landscape Design and is considering the
purchase of an automated drafting system. JR can purchase a system with three possible drafting
capacities. The payoffs for having any of these systems depend on the demand for drafting services
over the next few years. The costs for each system are shown as follows along with JR’s assessment
of the probabilities that demand will match the capacity of each one.
Small system
Medium system
Large system
Total Cost
$11,000
$13,000
$19,000
Probability
0.23
0.34
0.43
Working at capacity, each system would generate net cash flow at a yearly rate of 50% of its total
cost. If a system is chosen that is smaller than demand, it would work at capacity. If a system is
chosen that is larger than demand, revenue from the system would be limited by demand. For
convenience, JR has initially decided to count cash flow for three years, without discounting. For
example, if JR chooses the Medium system and demand is Small, then the profit is calculated as
follows:
Profit = 3*(0.5 x 11,000) – 13,000 = $3,500
a. What is the best decision under the maximax criterion?
b. What is the best decision under the maximin criterion?
c. What is the best decision under the minimax regret criterion?
d. What is the best decision under the expected payoff criterion?
Homework 6
11. Profit Analysis. A consumer electronics firm produces a line of battery rechargers for cell phones.
The following distributions apply:
Unit price
 triangular with a minimum of $18.95, most likely value of $24.95, and a maximum of $26.95.
Unit cost
 uniform with a minimum of $12.00 and a maximum of $15.00.
Quantity sold
 10,000 – 250 x unit price, plus a random term given by a normal distribution with a mean of 0
and a standard deviation of 10.
Fixed costs
 Normal with a mean of $30,000 and a standard deviation of $5,000.
a. What is the expected profit?
b. What is the probability of a loss?
c. What is the maximum loss?
12. LED Current. An LED with a current limiting resistor is attached to an output pin of a
microcontroller. To be visible, the LEC needs at least 7 mA of current going through it. The
microcontroller can provide a maximum of 10 mA. Here is the data:
 Resistor (RD): a normal distribution with a nominal value of 360 Ohms and standard deviation
of 10% of the nominal value.
 LED (VD): a normal distribution with a nominal value of 2 V and standard deviation of 10% of the
nominal value.
 Output Pin (Vout): a uniform distribution from 4.9 to 5.1 V.
The current provided to the LED is ID = (Vout-VD)/RD
a. Running a Monte Carlo simulation, what is the expected value of the LED current?
b. What percent of the time will the current be too small to light the LED?
c. What percent of the time will the current exceed to maximum current of the microcontroller?