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Friday: - Linear Programming - Exercises - Quiz

The document discusses linear programming techniques for determining optimal solutions subject to constraints, including defining an objective function, forming inequalities from constraints, plotting the feasible region to find vertices, and substituting vertices into the objective function to maximize or minimize value. Examples of investment applications and a corn/soybean farming problem are provided to illustrate the linear programming process. Students are assigned to study systems of linear equations and inequalities as well as linear programming for an upcoming summative test that prohibits calculators.

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66 views11 pages

Friday: - Linear Programming - Exercises - Quiz

The document discusses linear programming techniques for determining optimal solutions subject to constraints, including defining an objective function, forming inequalities from constraints, plotting the feasible region to find vertices, and substituting vertices into the objective function to maximize or minimize value. Examples of investment applications and a corn/soybean farming problem are provided to illustrate the linear programming process. Students are assigned to study systems of linear equations and inequalities as well as linear programming for an upcoming summative test that prohibits calculators.

Uploaded by

김나연
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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FRIDAY

• Linear Programming
• Exercises
• Quiz
Linear Programming
• Objective: Maximize or minimize the value, given a set of
constraints
• Objective Function – the quantity to be optimized
• The optimal solution(s) must occur at one of the corner
points of the solution region or feasible region
*Investment Application: what are the optimal solutions?
how do we determine optimal or maximized values?
• Arrange given variables and values in a table
• determine your objective function.
• Form the inequalities given the constraints.
• Plot them and solve for the vertices of the overlapping region.
• Substitute the vertices to the objective function and determine
which vertex gives the highest value.
Example 4, Page 332
Investment Application 𝐵 + 𝐶 ≤ 30
𝐶 ≥ 10
• Invest up to $30,000
𝐶 ≤ 30
• Bonds or B at 9%
𝐵 ≤ 15
• MMC or C at 5%
𝐵≥0
• Invest no more than $15,000 in B
𝐶≥0
• Invest at least $10,000 in C

Objective Function: what will give the highest interest?


I = .09B+.05C
I = 9B+5C
VERTICES?
𝐵 + 𝐶 ≤ 30 C
𝐶 ≥ 10 I = .09B+.05C
𝐶 ≤ 30 I = 9B+5C
𝐵 ≤ 15
𝐵≥0
𝐶≥0
To graph, we let
B=x and C=y

B
Let us look at Example 7, page 335
What is the Objective – or that which we need to Maximize or
minimize?
PROFIT
What is the Objective function?
P(r,d) = $3 r + $4 d
We are able to form 2 inequalities given the constraints.
We plot them, then determine the vertices.
To get optimized profit, we substitute the vertices to the
objective function.
So again how do we determine optimal or maximized
values?
• Arrange given variables and values in a table and determine your
objective function.
• Form the inequalities given the constraints.
• Plot them and solve for the vertices of the overlapping region.
• Substitute the vertices to the objective function and determine
which vertex gives the highest value.
Game Question:
#55, page 340
Corn (x) Soybean (y) Constraints
profit 900/acre 800/acre -
area x y 500 acres
Hours to 3hrs/acre 2hrs/acre 1300 hours
plant

Objective function: profit = 900x + 800y


Other inequalities: 3x + 2y ≤ 1300 and x + y ≤ 500
Objective function: profit = 900x + 800y
Assignment:
Study for the Summative Test on Monday:
2-variable systems of linear equations
3-variable systems of linear equations
systems on linear inequalities
linear programming

Btw, no calculators are allowed during the test.

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