Mathematical Programming
(1)
(2)
(3)
(4)
write the algebraic formulation (be sure to explicitly define your decision variables),
implement and solve the model in Excel, and
get some insight on the optimal solution and try do find a solution pattern.
write a paragraph on modeling simplifications/assumptions that you think are most troubling
from a practical perspective.
1. Production Allocation: Howie’s Hot Tub Problem (Classic Elementary LP Problem):
Blue Ridge Hot Tubs manufactures and sells two models of hot tubs: the Acqua-Spa and the
Hydro-Lux. Howie Jones, the owner and manager of the company, needs to decide how
many of each type of hot tub to produce during his next production cycle. Howie buys
prefabricated fiberglass hot tub shells from a local supplier and adds the pump and tubing to
the shells to create his hot tubs. (This supplier has the capacity to deliver as many hot tub
shells as Howie needs.) Howie installs the same type of pump into both hot tubs. He will
have only 200 pumps available during his next production cycle. From a manufacturing
standpoint, the main difference between the two models of hot tubs is the amount of tubing
and labor required. Each Acqua-Spa requires 9 hours of labor and 12 feet of tubing. Each
Hydro-Lux requires 6 hours of labor and 16 feet of tubing. Howie expects to have 1,566
production labor hours and 2,880 feet of tubing available during the next production cycle.
Howie earns a profit of $350 on each Aqua-Spa he sells and $300 on each Hydro-Luc he
sells. He is confident that he can sell all the hot tubs he produces. The question is, how
many Acqua-Spas and Hydro-Luxes should Howie produce if he wants to maximize his
profits during the next production cycle? Taken from Ragsdale’s Book
2. Deans Course Mix: The Dean of an esteemed professional school is trying to improve her
planning of what courses to offer. Specifically, she wants to serve her customer (the
students) better by offering a large number of courses, particularly smaller seminar classes as
opposed to larger lecture hall courses. However, there are limits on faculty time, on class
room space, on the types of courses which can be offered in each classroom (large lecture
classes cannot be offered in small classrooms and seminars cannot be offered in the big
lecture hall). Also, she knows she cannot offer only seminars because they are of limited size
and she must ensure that there are enough course-slots for all of the students to take their full
load of courses. The Dean’s assistant reports that there are enough classrooms to offer up to
10 lectures and 20 seminars. There are enough faculty to teach 34 courses, but those who
teach large lectures are customarily given credit for teaching two courses. Hence, there
would be enough faculty to teach 34 seminars or 17 lectures or some mixture of the two.
Finally, enrollment in seminars is limited to 25 students, in lectures to 100 students. There
are 160 students who need to enroll in at least five courses each, and the Dean judges that
students value options to take seminars 50% more highly than they do options to take lecture
courses. That is, students would be indifferent between having three additional lectures
offered or two additional seminars. The question facing the Dean is: How many seminars and
how many lectures should the school offer in order to make the course offering as appealing
as possible to the students? Since this is a quantitative professional school, she quickly
realizes that this problem can readily be formulated and solved as a linear program.
3. Wealth Maximization: You currently have $10,000. Your goal is to maximize total wealth
after 10 years. At the beginning of each year, you have five investment opportunities.
Investment #1 returns a payoff of 10% after 2 years (i.e., if you invested $4,000 in #1 at the
beginning of year 5, it would return $4,400 at the beginning of year 7). Investment #2
returns a payoff of 17% after three years. Investment #3 returns a payoff of 35% after 5
years. Investment #4 returns a payoff of 52% after 7 years. Investment #5 returns a payoff
of 70% after 9 years. Since the goal is to maximize wealth after exactly 10 years,
investments should not be made if they won’t payoff until after the 10-year period. For
example, investing in #4 at the beginning of year 5 would not payoff until the beginning of
year 12 (or end of year 11), which is too late. Thus, the only opportunities for investment #4
are at the beginning of the first four years. Investments #2 and #4 are limited to $5,000 per
year, and investment #3 is limited to $2,500 per year. Develop a linear programming
formulation that will determine the amount of money placed in each investment at the
beginning of each year so as to maximize total wealth at the end of 10 years.
4. Evaluation: Five car salespeople had the following sales for the past two months:
Salesperson
Fred
Mary
John
Jane
Chris
SUV’s
Luxury Cars
3
7
1
2
5
6
4
4
3
5
Mid-sized
12
15
18
24
16
The general manager believes that total dollar sales doesn’t adequately capture performance
and would like to use a weighted average of luxury car, SUV, and mid-sized sales instead.
The manager asks each salesperson to come up with a (positive) weight for each car
category, but stipulates that weights cannot allow anyone’s total weighted score to exceed
100. For example, defining w1 = luxury weight, w2 = SUV weight, and w3 = mid-sized
weight, Fred’s weighted score would be: 3 w1 + 6 w2 + 12 w3. Develop and solve an LP
model that will find a set of weights that will make John’s weighted score as large as
possible.
5. Shift planning: A computer lab operates 10 hours per day, Monday through Friday. Two
five-hour work shifts (9am-2pm and 2pm-7pm) are used for scheduling purposes and two
attendants must be present during each shift. The lab currently uses 7 employees. The shifts
for which each employee is unavailable are shown in the table below. Each employee must
get at least two shifts. Use LP to find a feasible schedule assignment.
Tanya
Todd
Terry
Tom
Thelma
Theo
Theresa
Mon & Wed (9-2)
Tue & Thu (2-7)
Tue & Thu (9-2)
Mon (9-2 and 2-7)
Mon, Wed, & Fri (2-7)
Fri (9-2, 2-7)
Tue, Thu, & Fri (9-2)
6. Transportation Problem: You run a business that supplies natural gas to customers in
Tampa, Orlando, Jacksonville, and Pensacola. You have three distribution centers (DC) in
Gainesville, Lake City, and Tallahassee. Distribution costs are primarily a function of travel
distance and include fuel and equipment costs, as well as labor costs. Each DC has a
capacity restriction that limits the number of units of gas it can supply each month. Each
customer area has a demand that must be satisfied each month. Your goal is to arrive at a
distribution plan that minimizes total distribution costs. The per unit cost of shipping a one
unit of natural gas from each DC to each customer area is shown in the table below, along
with the demand and capacity levels (in thousands of units). Develop and solve an LP model
to find the distribution plan.
Gainesville
Lake City
Tallahassee
Demand
Tampa
$1.40
$1.55
$1.90
Orlando
$0.95
$1.40
$1.75
Jacksonville
$1.55
$1.20
$1.80
Pensacola
$3.25
$2.80
$2.15
900
600
700
600
Capacity
1250
1250
500
7. Coffee Blending: Bolger’s House Coffee manufactures two coffees (regular and mountain)
by blending three types of coffee beans. The cost per pound of beans 1, 2, and 3 are $0.65,
$0.75, and $0.45, respectively. The available pounds of beans 1, 2, and 3 during the coming
planning period are 800, 600, and 600, respectively. Consumer coffee tests were used to
provide ratings on a scale of 0-100, with higher ratings indicating higher quality. Product
quality standards for regular coffee require a consumer rating for aroma to be at least 75 and
a consumer rating for taste to be at least 80, whereas standards for mountain coffee are an
aroma rating of 70 and a taste rating of 85. The individual ratings of aroma for coffee made
from 100% of beans 1, 2, and 3 are 75, 85, and 60, respectively. The individual ratings of
taste for coffee made from 100% of beans 1, 2, and 3 are 86, 88, and 75, respectively.
Assume that the aroma and taste attributes of the regular and mountain coffee blends will be
a weighted average of the attributes of the beans used in the blend. Develop a linear
programming formulation for a minimum-cost blends that will meet the quality standards and
provide 900 pounds regular coffee and 600 pounds of mountain coffee products.
8. Production Allocation: The Acme Axle Company produces both car and track axles for
national and international markets. Each axle must complete two manufacturing processes:
molding and finishing. Each car axle requires 16 units of molding and 10 units of finishing,
whereas a truck axle requires 24 units of molding and 20 units of finishing. Weekly, 480
units of molding and 360 units of finishing are available. The demand for Acme‘s axles is
such that the firm may sell all it produces. Acme achieves a profit of $50 per car axle and $60
per truck axle. Acme also has an agreement with the Spitz Motor Company to supply 12 car
axles and 8 truck axles weekly. Given the above constraints and requirements, Acme desires
to know what amounts of car and truck axles to produce weekly in order to maximize profit.
Solve for the optimum production mix, using the graphic method. Determine the values of
the dual variables, using the complementary slackness conditions.
9. Kilgore's Deli Product Portfolio Mix: Kilgore’s Deli does a large walk-in carry-out lunch
business. It offers two chili specials, the “Wimpy” and the “Dial 911”. Each morning,
Kilgore needs to decide how much of each special to make (he always sells out whatever he
makes). The profit on one serving of Wimpy is $0.45, on one serving of Dial 911, $0.58.
Each serving of Wimpy requires 0.25 pounds of beef, 0.25 cups of onions, and 5 ounces of
special sauce. Each serving of Dial 911 requires 0.25 pounds of beef, 0.4 cups of onions, 2
ounces of special sauce, and 5 ounces of hot sauce. Today Kilgore has 20 pounds of beef, 15
cups of onions, 88 ounces of special sauce, and 60 ounces of hot sauce on hand.
10. G&S Insurance: carries an investment portfolio of bonds, stocks, and other investments. At
the beginning of next year, $500,000 will be available to invest in four instruments. The
expected annual rates of return on these four instruments are 0.06, 0.09, 0.07, and 0.11,
respectively. The risk factor per dollar invested is a measure of the inherent uncertainty of
the expected return, so a low risk factor is desirable. The risk factors per dollar invested are
subjective estimates of the possible loss in the worse case (0 means no risk, 1 means the
investor could lose everything). They are 0.02, 0.05, 0.04, and 0.075, respectively. For
purposes of diversification, no more than $200,000 can be invested in any one instrument.
Management needs a return of at least 0.08, but would like to minimize the risk taken to
achieve that return.
11. Calhoun Fabrics - Production Allocation: The Calhoun Textile Mill is deciding on a
production schedule. It wants to know how to weave various fabrics over the next quarter.
The sales department has confirmed orders for 15 fabrics, as indicated in the table. Calhoun
has 15 dobbie looms and 90 regular looms for weaving, which have different efficiencies for
different fabrics (and certain fabrics can only be made on the dobbie looms). Ignore set up
and change over times for this problem. The looms run 24 hours a day, 7 days a week, for all
13 weeks in the quarter. Demand that cannot be met in house is outsourced at the indicated
cost. Management would like to know how to allocate looms to fabrics and which fabrics to
outsource.
Fabric
Demand
(meters)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16500
52000
45000
22000
76500
110000
122000
62000
7500
69000
70000
82000
10000
380000
62000
Dobbie
Regular
Product Cost
(meters/hour) (meters/hour)
($/meter)
5
4
5
4.5
5
4
4.2
6
5
7
4
4.3
4.6
5.2
4.2
NA
NA
NA
NA
5
4
4.2
6
5
7
4
4.3
4.6
5.2
4.2
0.65
0.55
0.65
0.55
0.61
0.62
0.65
0.49
0.5
0.44
0.64
0.57
0.49
0.31
0.5
Outsource
Cost ($/hour)
0.8
0.7
0.85
0.7
0.75
0.75
0.8
0.6
0.7
0.6
0.8
0.75
0.65
0.45
0.7