Download Optimization Exercise

Group members:
Table:
It’s all about Optimization!
1. Let’s see how well you can invest. You have $12, 000 to invest, and three different types of financial
instruments from which to choose.
(a) The municipal bond fund has a 7% return,
(b) The local bank’s Certificate of Deposits (CDs) (or Fixed Deposits) have an 8% return, and
(c) The high-risk account has an expected (hoped-for) 12% return.
To minimize risk, you decide not to invest any more than $2, 000 in the high-risk account. For tax reasons,
you need to invest at least three times as much in the municipal bonds as in the bank CDs. Assuming the
year-end yields are as expected, what are the optimal investment amounts?
Figure 1: Savings are very beneficial and in general can help improve the quality of one’s life
Mathematical Model
Feasible Region
References: Stapel, Elizabeth. ”Linear Programming: Word Problems.” www.purplemath.com
May 15, 2010, St. Petersburg
1
2010 IEEE Technical English Program
Group members:
Table:
It’s all about Optimization!
2. Let’s see if you can help this electrical wiring supply company: Being electrical engineers, we know
how important electrical wiring is to the design of proper electrical layout of a building. A electrical wiring
supply company has two warehouse locations in a city. The office receives orders from two customers, each
requiring electrical wire rolls. Customer A needs fifty (50) rolls and Customer B needs seventy (70) rolls. The
warehouse on the east side of the city has eighty (80) rolls in stock; the west-side warehouse has forty-five
(45) rolls in stock. Delivery costs per roll are as follows:
Table 1: Delivery Costs from different locations to the customers
East Side Warehouse
West Side Warehouse
Customer A
$0.50
$0.40
Customer B
$0.60
$0.55
Find the shipping arrangement which minimizes costs.
Mathematical Model
Feasible Region
References: Stapel, Elizabeth. ”Linear Programming: Word Problems.” www.purplemath.com
May 15, 2010, St. Petersburg
2
2010 IEEE Technical English Program
Group members:
Table:
It’s all about Optimization!
3. Let’s see if you can maximize revenue of this petrochemical refinery: At a certain refinery, the
refining process requires the production of at least two (2) gallons of gasoline for each gallon of fuel oil. To
meet the anticipated demands of winter, at least three (3) million gallons of fuel oil a day will need to be
produced. The demand for gasoline, on the other hand, is not more than 6.4 million gallons a day. If gasoline
is selling for $1.90 per gallon and fuel oil sells for $1.50 per gallon, how much of each should be produced in
order to maximize revenue?
Figure 1: Petrochemical refineries are a crucial part of a country’s economy. They bring in revenue and support
a substantial number of jobs.
Mathematical Model
Feasible Region
References: Stapel, Elizabeth. ”Linear Programming: Word Problems.” www.purplemath.com
May 15, 2010, St. Petersburg
3
2010 IEEE Technical English Program
Group members:
Table:
It’s all about Optimization!
4. Let’s see if you can help maximize profit for this calculator company: A calculator company
produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected
demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production
capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping
contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results
in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily
to maximize net profits?
Figure 1: Scientific and Graphing calculators have become ubiquitous in engineering education
Mathematical Model
Feasible Region
References: Stapel, Elizabeth. ”Linear Programming: Word Problems.” www.purplemath.com
May 15, 2010, St. Petersburg
4
2010 IEEE Technical English Program
Group members:
Table:
It’s all about Optimization!
5. Let’s see if you can find the optimal dietary balance for this lab technician: In order to ensure
optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet
containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protien. But the rabbits
should be fed no more than five ounces of food a day. Rather than order rabbit food that is custom-blended,
it is cheaper to order Food X and Food Y, and blend them for an optimal mix. What is the optimal blend?
Table 1: Nutrition values of the two different types of food
Ingredient
Fat (in grams per ounce)
Carbohydrates (in grams per ounce)
Protein (in grams per ounce)
Cost per ounce
Food X
8
12
2
$0.20
Food Y
12
12
1
$0.30
Mathematical Model
Feasible Region
References: Stapel, Elizabeth. ”Linear Programming: Word Problems.” www.purplemath.com
May 15, 2010, St. Petersburg
5
2010 IEEE Technical English Program
Group members:
Table:
It’s all about Optimization!
6. You need to buy some filing cabinets. You know that Cabinet X costs 10 per unit, requires six square feet
of floor space, and holds eight cubic feet of files. Cabinet Y costs 20 per unit, requires eight square feet of
floor space, and holds twelve cubic feet of files. You have been given 140 for this purchase, though you don’t
have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of
which model should you buy, in order to maximize shorage volume?
Figure 1: Next time your boss asks you to select cabinets, you know what to do!
Mathematical Model
Feasible Region
References: Stapel, Elizabeth. ”Linear Programming: Word Problems.” www.purplemath.com
May 15, 2010, St. Petersburg
6
2010 IEEE Technical English Program