Linear Programming

1. Republic of the Philippines
Mindanao State University – General Santos City
College of Natural Sciences and Mathematics
Reporters: Carcero, John Carl P. Subject: MathEd103
Delos Reyes, Abbygale Jade P. Date: March 15, 2018
Topic: Linear Programming
References:
 Buist, L. (2013). Linear programming History (background). Retrieved from
https://prezi.com/k5z8xuwjuupq/linear-programming-history-background/
 Davis, D. (2013). Part 1: Linear Programming. Retrieved from
https://www.youtube.com/watch?v=kpzIxQbLhME
Linear Programming - is one of the main applications of mathematics used in business
and the social sciences. The process known as linear programming is used to find
minimum cost, maximum profit, the maximum amount of learning that can take place
under given conditions, and so on. The procedures for solving linear programming
problems were developed in 1947 by George Dantzig, while working on the problem of
allocating supplies for Air Force troops during World War II, in a way that minimized
total cost.
Steps in Solving Linear Programming Problems:
1. Read the problem. Then READ
again!
2. Define variables.
3. Find the objective quantity (this is
what you are trying to minimize or
maximize).
4. Find constraints (these are
inequalities).
5. Graph the inequalities.
6. Find all corner points of your
feasible region.
7. Test all corner points in your
objective quantity (to find the
max/min).
8. Answer the question in a
sentence.
Example:
1. You are taking a test in which items of type A are worth 10 points and itmes of
type B are worth 15 points. It takes 3 minutes for each item of type A and 6
minutes for each item of type B. The toltal time allowed is 60 minutes, and you
may not answer more than 16 questions. Assuming all your answers are correct,
how many items of each type should you answer in order to get the best score?
Solution: Let x – be the number of type A questions.
y – be the number of type B questions.
Objective Quantity: (to maximize the score)
10x + 15y

2. Constraints: 3x + 6y ≤ 60 (allotted time)
x + y ≤ 16 (number of items to answer)
For 3x + 6y = 60, when x = 0, y = 10.
y = 0, x = 20
For x + y = 16, when x = 0, y = 16.
y = 0, x = 16.
For the intersection of the two lines,
3x + 6y = 60 Substituting y=4 to any of the equation, we have
-3(x + y)=(16)-3 x + y = 16
3y= 12 x + 4 = 16
y = 4 x = 12
Points to Test the Maximum Score using the Objective Quantity: 10x + 15y
A(0, 10)
10x + 15y = 10 (0) + 15 (10) = 150 points
B(12, 4) – point intersection of the two constraints
10x + 15y = 10 (12) + 15 (4) = 180 points
C(16, 0)
10x + 15y = 10 (16) + 15(0) = 160 points
Therefore, in order for you to get the best score, you must answer 12 type
A items and 4 type B items.
Activity:
Mechanics:
1. The class will be divided into 6 groups. Each member shall be assigned with
numbers 2-8 that correspond to the steps they are going to answer.
2. The question will be shown using the powerpoint presentation.
3. Each group shall do the peer-teaching activity within 10 minutes and must
discuss their answers silently and cooperatively.
4. Assigned members will individually answer on the manila paper provided for
them. Members assigned for Step 5 must have their graphs ready to be posted.
5. Each step corresponds to a given time limit depending on its level of difficulty.
6. The first group to answer the question correctly and accurately will be declared
as winner. A corresponding prize shall be given.
Question: A snack bar cooks and sells hamburgers and hotdogs during football
games. To stay in business, it must sell at least 10 hamburgers but cannot cook
more than 40. It must also sell at least 30 hotdogs but cannot cook more than 70.
It cannot cook more than 90 sandwiches all together. The profit on a hamburger
is $0.33 and $0.21 on a hotdog. How many of each kind of sandwich should the
stand sell to make a maximum profit?