symbols: less than, fewer than greater than, more than...
TRANSCRIPT
Optimisation.notebook
1
December 20, 2013
Optimisation
Symbols: less than, fewer than
greater than, more than, exceeds
less than or equal to, at most, maximum of, no more than
greater than or equal to, at least, minimum of, no less than
<
>
≥
≤
Optimisation.notebook
2
December 20, 2013
Examples with two variables:
1) At a school dance, students paid $3.00 and guests paid $5.00. The proceeds were more than $600.00.
2) At a high school, at least twice as many girls as boys take enriched science.
Optimisation.notebook
3
December 20, 2013
3) Mario recycles empty bottles. Small bottles are worth 10 cents and large ones, 40 cents. He never collects more than $40.00.
4) John and Sheila are going to New York and Boston. They want to spend at least twice as much time in New York than Boston.
Optimisation.notebook
4
December 20, 2013
The inequalities presented in a situation are known as constraints - conditions that must be met.
Most situations have two extra constraints that are not mentioned. These are called the non-negative constraints and they exist when it is not possible for the variables to take on negative values.
Optimisation.notebook
5
December 20, 2013
Example: The maximum number of seats in a plane is 100. There must be at least 4 times as many seats in economy class as in business class.
The constraints are:
Optimisation.notebook
6
December 20, 2013
Showing the Solution Set of a Linear Inequality
Example: Graph the solution set of .
1. Graph the line *.
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
* Because the inequality includes the equal sign, the solution includes the line. Draw a solid line.
X Y
Optimisation.notebook
7
December 20, 2013
2) Choose a test point (the point cannot be on the line). Test the point to see if it satisfies the inequality.
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
Optimisation.notebook
8
December 20, 2013
3) Shade on one side of the line - the side containing points that satisfy the inequality. This half-plane shows all the possible points that satisfy the inequality.
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
The solution set of a linear inequality is called a half plane.
Optimisation.notebook
9
December 20, 2013
Example: Graph the solution set of the inequality<
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
Because this is a "strict inequality", the line is not part of the solution. Draw a broken line.
X Y
Optimisation.notebook
10
December 20, 2013
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
Optimisation.notebook
11
December 20, 2013
Systems of Linear Inequalities
Solving a system of linear inequalities means finding all the points that satisfy all the inequalities.
Example: Determine the solution set of the following system.
Optimisation.notebook
12
December 20, 2013
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
1) 2)
X YX Y
Optimisation.notebook
13
December 20, 2013
Example: An orchestra has no more than 30 members. There are at least twice as many musicians who play string instruments as musicians who play wind instruments. Graph the solution set.
Optimisation.notebook
14
December 20, 2013
0 5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
x
y
The figure created by the solution set of all the inequalities is called the polygon of constraints.
Optimisation.notebook
15
December 20, 2013
Objective Function
A system of linear inequalities has many solutions. Depending on the situation, some of these solutions (usually one solution) is better than the others. What determines the optimal solution is the objective function.
Optimisation.notebook
16
December 20, 2013
Example: A glass company manufactures both summer and winter windows. They have to manufacture at least 10 summer windows and 30 winter windows per week. They can make no more than 100 windows per week.The profit made on each summer window is $70 and on each winter window is $85.
• Graph the polygon of constraints.
Optimisation.notebook
17
December 20, 2013
0 10 20 30 40 50 60 70 80 90 100 110 120
10
20
30
40
50
60
70
80
90
100
110
120
x
y
This company's objective is to make a profit; in fact it is to make the maximum profit possible given the constraints. The objective rule for this company is
Optimisation.notebook
18
December 20, 2013
0 10 20 30 40 50 60 70 80 90 100 110 120
10
20
30
40
50
60
70
80
90
100
110
120
x
y
Points Profit
Notice that it is one of the vertices of the polygon of constraints that provides the maximum profit.
Optimisation.notebook
19
December 20, 2013
Example: Joan wants to give at least 12 chocolates to her friends for Easter. She intends to buy at least twice as many dark chocolates as milk chocolates, but no more than 20 dark chocolates. One milk chocolate costs $2.00 and one dark chocolate costs $4.00. How many of each type of chocolate should Joan buy in order to minimise her costs?
Optimisation.notebook
20
December 20, 2013
0 2 4 6 8 10 12 14 16 18 20 22 24
12345678910111213141516171819202122232425
x
y
Vertices Cost C = 4x + 2y
Optimisation.notebook
21
December 20, 2013
Solving Optimisation Problems
1. Define the variables.2. List the constraints.3. Determine the objective function.4. Graph the polygon of constraints.5. Determine the vertices of the polygon.6. Calculate which vertex (vertices) creates a
maximum or minimum value.
Optimisation.notebook
22
December 20, 2013
Example: A store specializes in selling paper for photocopiers and printers.
The constraints with respect to the purchase of x boxes of photocopy paper and y boxes of printer paper are represented by the polygon to the right.
The net revenue R from the sale of these two kinds of paper is obtained from the following relation: R = 3x + 4y
How many ordered pairs maximize the net revenue for this store?
Optimisation.notebook
23
December 20, 2013
If the optimal solution (maximum or minimum) is achieved at two consecutive vertices (say M and N), then each point along the edge MN of the polygon of constraints is also an optimal solution.
How many solutions are there to this problem?
Optimisation.notebook
24
December 20, 2013
Example: A school wants to minimize the transportation costs involved in taking students on a field trip. The following polygon of constraints represents the solutions for this situation. The values shown in the table below were calculated in order to determine the minimum transportation cost.
In this situation, how many solutions minimize the transportation cost involved in taking students on this field trip?
Coordinates of the vertices of the polygon of constraints
Transportation cost
P (1, 9) $440
Q (11, 7) $1 160
R (11,3) $1 000
S (5, 1) $440
Optimisation.notebook
25
December 20, 2013
Example: Vincent works for a company that makes storage racks for DVDs. Each week, he divides his time between
assembly work and finishing work. The polygon of constraints represents the different constraints that Vincent faces.
x : number of hours spent on assembly work each week
y: number of hours spent on finishing work each week
Vincent is told that from now on he faces the following additional constraint: the number of hours spent on finishing work must be less than or equal to the number of hours spent on assembly work. He makes $10 an hour for assembly work and $8 an hour for finishing work. By how many dollars does this constraint decrease Vincent's maximum possible weekly income?
Vertices
P(5, 35)
Q(20, 5)
R(15, 5)
S(5, 15)