Chapter 3
FInd corner points using z = 4x + 2y
(0,8) | 16 (3,4) | 20 (13/2,2) | 30 (12,0) | 48
Graph feasible region x + 3y ≤ 6 2x + 4y ≥ 7
7/4 , 7/2
Corner Point Theorem
If an optimum value of the objective function exists, it will occur at one of the corner points of the feasible region
Linear Inequality
This is a linear function that consists of the greater than less than or equal to symbols.
Constraints
a set of restrictions written as equasions
Feasible Region
area where all the inequalities work
Find max and min z = 3x + 2y z = x + 4y
max 29 @ (7,4) min 10 @ (0,5)
using corner points (0,8) (3,4) (13/2,2) (12,0) and z = 2x + 3y find the max and min
max = none min = 18 @ (3,4)
maximize z = 2x + 4y subject to 3x + 2y ≤ 12 5x + y ≥ 5 x ≥ 0 y ≥ 0
max of 24 @ (0,6)
Objective Function
maximum or minimum value of a function
using corner points (6,0) and (3,4) with z= 4x +7y find the max and min
min 24 @ (6,0) max 40 @ (3,4)
using corner points (0,0) (8,0) (7,3) (4,8) (0,12) and z = 1.5x + .25y find the max and min
min = 0 @ (0,0) max = 12 @ (8,0)
Minimize z = 4x + 7y subject to: x - y ≥ 1 3x +2y ≥ 18 x ≥ 0 y ≥ 0
minimum of 24 when x = 6 and y = 0
Half - Plane
region on one side of the line
graph linear inequality x + y ≤ 2
rise over run = -2/2
Boundaries
separates the points in the solution from the points that are not a solution
Corner Point
this is where the boundary lines of two constraints cross in the feasible region
System of Inequalities
two or more inequalities
Unbounded
when the feasible region has a side(s) that is not closed off.
Bounded
when the feasible region is closed off by boundaries on all sides
what do variables stand for Y = mx + b
y= y variable m= slope b= y intercept x= x variable