Symbols: less than, fewer than greater than, more than...

Optimisation.notebook

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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

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≥

≤

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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.

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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.

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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.

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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:

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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

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* Because the inequality includes the equal sign, the solution includes the line. Draw a solid line.

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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

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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.

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The solution set of a linear inequality is called a half plane.

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Example: Graph the solution set of the inequality<

­10 ­8 ­6 ­4 ­2 0 2 4 6 8 10

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Because this is a "strict inequality", the line is not part of the solution. Draw a broken line.

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­10 ­8 ­6 ­4 ­2 0 2 4 6 8 10

­10­9­8­7­6­5­4­3­2­1

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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.

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­10 ­8 ­6 ­4 ­2 0 2 4 6 8 10

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1) 2)

X YX Y

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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.

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The figure created by the solution set of all the inequalities is called the polygon of constraints.

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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.

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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.

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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

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Points Profit

Notice that it is one of the vertices of the polygon of constraints that provides the maximum profit.

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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?

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Vertices Cost C = 4x + 2y

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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.

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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?

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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?

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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

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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)