- Linear programming can be used to solve real-world optimization problems by setting up constraints and an objective function in terms of variables. - Several examples are provided to demonstrate how to set up and solve linear programming problems to determine the optimal quantities of different options to maximize a given objective like number of passengers transported or number of people fed. - The optimal solutions are found by testing the objective function values at the vertices of the feasible region defined by the constraints.

1. Linear Programming
– set up and solve linear programming
problems to find real life minimums and
maximums.

2. Linear Programming Hints
 To write the constraints and objective
function:
– Identify the variables.
– Write the objective function
– Identify each constraint. Then write an
inequality for each.
• Start with x > 0 and y > 0 (usually).

3. Example 1
 The West Hartford Senior Center is trying to establish a
transportation system of small and large vans. It can
spend no more than $100,000 for both sizes of vehicles
and no more than $500 per month for maintenance. The
WHSC can purchase a small van, which carries up to 7
passengers, for $10,000 and maintain it for $100 per
month. The large vans, which carry up to 15
passengers, cost $20,000 each and can be maintained for
$75 per month. How many of each type of van should
they purchase if they want to maximize the number of
passengers?
s 0
s # of small vans
l # of large vans l 0 Initial purchase cost
10,000s 20,000l 100, 000
# of passengers 100 s 75l 500 Monthly maintenance
P = 7s + 15l

4. s 0 s # of small vans
l 0 Example 1 l # of large vans
10, 000s 20, 000l 100, 000 P 7 s 15l
100s 75l 500
The vertices that should
10
be tested are
(0,5), (2,4), and (5,0).
8
# of large vans, l
P(0,5) 7(0) 15(5) 75
6 P(2,4) 7(2) 15(4) 74
P(5,0) 7(5) 15(0) 35
4 The WHSC should buy 5
large vans which can
2 hold 75 passengers.
00 2 4 6 8 10
# of small vans, s

5. Example 2
 Pancakes Waffles
3 cups Bisquick 2 cups Bisquick
1 cup Milk 2 cups Milk
2 Eggs 2 Eggs
Serves 6 Serves 5
 You have 24 cups of Bisquick, 18 cups of milk, and 20
eggs. If you want to feed as many people as
possible, how many batches of each should you make?
p # of batches of pancakes p 0
w # of batches of waffles w 0
Servings Bisquick 3 p 2w 24
S 6 p 5w Milk p 2 w 18
Eggs 2 p 2w 20

6. 3p 2w 24
p 2w 18 Example 2
2p 2w 20 p # of batches of pancakes
p 0 w # of batches of waffles
w 0 S 6 p 5w
10 The vertices that should
be tested are
# of batches of waffles, w
8 (0,9), (2,8), (4,6), and
(8,0). 6(0) 5(9) 45
S(0,9)
6 S(2,8) 6(2) 5(8) 52
S(4,6) 6(4) 5(6) 54
4
S(8,0) 6(8) 5(0) 48
2 Make 4 batches of
pancakes and 6
00 batches of waffles
2 4 6 8 10 feed 54 people.
to
# of batches of pancakes, p

7. Example 3
 Kayla works no more than 20 hours per
week during the school year. She is
paid $10 an hour for tutoring Geometry
students and $7 an hour for babysitting.
She wants to spend at least 3 hours but
no more than 8 hours a week tutoring.
Find Kayla’s maximum weekly earnings.
t # of hrs spent tutoring t 3
b # of hrs spent babysitting t 8
Earnings b 0
E 10t 7b Total hours worked t b 20

8. t 3
t 8 Example 3
b 0 t # of hrs spent tutoring
t b 20 b # of hrs spent babysitting
E 10t 7b
20
# of hrs spent babysitting, b
The vertices that should
be tested are (3,17) and
16
(8,12).
E(3,17) 10(3) 7(17) 149
12
E(8,12) 10(8) 7(12) 164
8 Kayla should tutor for 8
hours and babysit for 12
4 hours.
00 2 4 6 8 10
# of hrs spent tutoring, t

9. Example 4
 As part of your weight training regimen, you want to
consume lean sources of protein. You want to
consume at least 300 Calories a day from at least 48
grams of protein. One ounce of chicken provides 35
Calories and 8.5 g of protein. One ounce of tofu
provides 20 Calories and 2.5 g of protein. Your local
supermarket charges $5 a pound for chicken and
$2.50 a pound for tofu. How much of each food
should you eat each day if you want to meet your
requirements with the lowest cost? What is this daily
cost?
c # of lbs of chicken c 0
t # of lbs of tofu t 0
Calories 560c 320t 300
Price Paid
P 5c 2.50t Protein 136c 40t 48

10. 560c 320t 300 c # of lbs of chicken
136c 40t 48 Example 4 t # of lbs of tofu
c 0 P 5c 2.50t
t 0 The vertices that should
be tested are approx.
c 0.536
(0, 1.2), (0.16, 0.65), and
t 0.9375 1.0
c 0.35 (.536,0).
t 1.2 P(0,1.2) 5(0) 2.5(1.2) 3
0.8
# of lbs of tofu, t
P(0.16,0.65) 5(0.16) 2.5(0.65) 2.425
0.6 P(0.536,0) 5(0.536) 2.5(0) 2.68
You should eat about 0.16
0.4 pounds of chicken and 0.65
pounds of tofu, which will
0.2 cost about $2.43 per day.
00 0.2 0.4 0.6 0.8 1.0
# of lbs of chicken, c