MATH121 Linear
Programming Word Problems
1) Candy Kane Cosmetics (CKC)
produces Leslie Perfume, which requires chemicals and labor. Two production processes are available:
Process 1 transforms 1 unit of labor and 2 units of chemicals into 3 oz of
perfume. Process 2 transforms 2 units of
labor and 3 units of chemicals into 5 oz of perfume. It costs CKC $3 to purchase a unit of labor
and $2 to purchase a unit of chemicals.
Each year, up to 20,000 units of labor and 35,000 units of chemicals can
be purchased. In the absence of
advertising, CKC believes it can sell 1000 oz of perfume. To stimulate demand for Leslie, CKC can hire
the lovely model Jenny Nelson. Jenny is
paid $100/hour. Each hour Jenny works
for the company is estimated to increase the demand for Leslie Perfume by 200
oz. Each ounce of Leslie Perfume sells
for $5. Use LP to determine how CKC can
maximize profits.
2) Televco
produces TV picture tubes at three plants. Plant 1 can produce up to 150 tubes
per week; plant 2 can produce up to 100 tubes per week; plant 3 can produce up
to 80 tubes per week. Tubes are shipped to five customers. The profit earned
per tube depends on the site where the tube was produced and on the customer
who purchases the tube (see table below). Customer 1 is willing to purchase up
to 70 tubes per week; customer 2, up to 50 tubes per week; customer 3, up to 65
tubes per week; customer 4, up to 90 tubes per week; and customer 5, up to 85 tubes per
week. Televco wants to find a shipping and production
plan that will maximize profits.
|
|
CUSTOMER
1 |
CUSTOMER
2 |
CUSTOMER
3 |
CUSTOMER
4 |
CUSTOMER
5 |
|
PLANT1 |
$75 |
$60 |
$69 |
$55 |
$80 |
|
PLANT
2 |
$79 |
$73 |
$68 |
$60 |
$85 |
|
PLANT
3 |
$85 |
$76 |
$70 |
$50 |
$65 |
3) Turkeyco produces two types of
turkey cutlets for sale to fast food restaurants. Each type of cutlet consists of white meat
and dark meat. Cutlet 1 sells for $4/lb
and must consist of at least 70% white meat.
Cutlet 2 sells for $3/lb and must consist of at least 60% white
meat. At most 50 lbs of cutlet 1 and 30
lbs of cutlet 2 can be sold. The two
types of turkey used to manufacture the cutlets are purchased from the GobbleGobble Turkey Farm.
Each type 1 turkey costs $10 and yields 5 lbs of white meat and 2 lbs of
dark meat. Each type 2 turkey costs $8
and yields 3 lbs of white meat and 3 lbs of dark meat. Use LP to maximize Turkeyco's
profit.
4) City 1 produces 500 tons of waste
per day, and city 2 produces 400 tons of waste per day. Waste must be incinerated at incinerator 1 or
2 , and each incinerator can process up to 500 tons of waste per day. The cost to incinerate waste is $40/ton at
incinerator 1 and $30/ton at incinerator 2.
Incineration reduces each ton of waste to 0.2 tons of debris, which must
be dumped at one of two landfills. Each
landfill can receive at most 200 tons of debris per day. It costs $3 per mile to transport a ton of
material (either debris or waste).
Distances (in miles) between locations are shown in the table
below. Use LP to to
minimize the total cost of disposing of the waste of both cities.
|
|
Incinerator 1 |
Incinerator 2 |
|
City 1 |
30 |
5 |
|
City 2 |
36 |
42 |
|
|
|
|
|
|
Landfill 1 |
Landfill 2 |
|
Incinerator 1 |
5 |
8 |
|
Incinerator 2 |
9 |
6 |