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5.4 to 5.6 Optimization.notebook
1
September 24, 2012
Sep 1712:28 PM
Optimization ProblemsOptimization problems involve finding the best possible solution in order to maximize or minimize a quantity following a set of guidelines or conditions.Vocabulary to know: (p.244) constraintobjective function feasible region
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1712:48 PM
constraint a limiting condition of the optimization problem being modelled, represented by a linear inequality.
objective function the equation that represents the relationship between the two variables in the system of linear inequalities and the quantity to be optimized.
feasible region the solution region for a system of linear inequalities that is modelling an optimization problem
One solution in the feasible region will represent a maximum, and another a minimum.
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 178:37
Steps to Solving an Optimization Problem
1. Identify the quantity to be optimized (maximized or minimized). Look for key words, such as maximize or minimize, largest or smallest, and greatest or least.
2. Define the variables that affect the quantity to be optimized. Identify any restrictions on these variables.
3. Write a system of linear inequalities to describe all constraints of the problem. Graph the system.
4. Write an objective function to represent the relationship between the variables and the quantity to be optimized.
5. Determine the feasible region (possible solutions region).
6. The optimal solutions will be represented by the points of intersection of the boundaries of the feasible region, known as the vertices. Evaluate the objective function by substituting the values of the coordinates of each vertex.
7. Verify that the solution(s) satisfies the constraints of the problem situation.
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 178:56 PM
Example 1:
Three teams are travelling to a basketball tournament in cars and minivans. • Each team has no more than 2 coaches and 14 athletes.• Each car can take 4 team members, and each minivan can take 6 team members.• No more than 4 minivans and 12 cars are available.The school wants to know the combination of cars and minivans that will require the minimum and maximum number of vehicles.
5.4 to 5.6 Optimization.notebook
5
September 24, 2012
Sep 178:37
Steps to Solving an Optimization Problem
1. Identify the quantity to be optimized (maximized or minimized). Look for key words, such as maximize or minimize, largest or smallest, and greatest or least.
number of vehicles
2. Define the variables that affect the quantity to be optimized. Identify any restrictions on these variables.
let x = number of cars; must be whole numberslet y = number of minivans; must be whole numbers
3. Write a system of linear inequalities to describe all constraints of the problem. Graph the system. (on next page)
number of cars available: x≤ 12number of minivans available: y ≤ 4maximum number of team members = 16x3 = 48so travel arrangements for team members: 4x + 6y ≤48
4. Write an objective function to represent the relationship between the variables and the quantity to be optimized.
V = total number of vehicles, soV = x + y
5. Determine the feasible region (possible solutions region).
6. The optimal solutions will be represented by the points of intersection of the boundaries of the feasible region, known as the vertices. Evaluate the objective function by substituting the values of the coordinates of each vertex.
7. Verify that the solution(s) satisfies the constraints of the problem situation.
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1710:17 PM
number of cars
number of m
inivans
2 8 1410
4
12
2
64 16 18
6
8
10
12 x ≤12
x = 12 is a vertical line
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1710:17 PM
number of cars
number of m
inivans
2 8 1410
4
12
2
64 16 18
6
8
10
12
y ≤ 4y = 4 is a horizontal line
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1710:17 PM
number of cars
number of m
inivans
2 8 1410
4
12
2
64 16 18
6
8
10
12
4x + 6y < 48xint, let y=04x = 48x = 12(12, 0) yint, let x=0
6y = 48y = 8(0,8)
Where is the region of overlap? This becomes the feasible region.
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 179:32 PM
number of cars
number of m
inivans
2 4
6
4
2
8
10
12
6 8 10 12 14 16 18 20
feasible region(6,4)(0,4)
(0,0) (12,0)
Substitute coordinates of each vertex into the objective function:V = x + y V = x + y V = x + y V = x + y(0,0) 0 + 0 (0,4) 0 + 4 (6,4) 6 + 4 (12,0) 12+0V = 0 V = 4 V = 10 V = 12
# team members: 4x+6y 4(0)+6(0) 4(0)+6(4) 4(6)+6(4) 4(12)+6(0)0 + 0 0 + 24 24 + 24 48 + 0 0 24 48 48
So,• 0 vehicles could transport 0 team members.• 4 vehicles can transport 24 team members.• 10 vehicles can transport 48 team members.• 12 vehicles can transport 48 team members.
Which solution is the best one? Why?
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1711:22 PM
Example 2:
L&G Construction is competing for a contract to build a fence.• The fence will be no longer than 50 yd and will consist of narrow boards that are 6 in. wide and wide boards that are 8 in. wide.• There must be no fewer than 100 wide boards and no more than 80 narrow boards.• The narrow boards cost $3.56 each, and the wide boards cost $4.36 each.Determine the maximum and the minimum costs for the lumber to build the fence.
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1711:27 PM
Ed found spiders and crickets in his storage room.• There were 20 or fewer spiders 20 or more crickets.• There were 45 of fewer crickets and spiders, in total. Spiders have 8 legs, and crickets have 6 legs.a) What combination of spiders and crickets would have the greatest number of legs?b) What combination would have the least number of legs?
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1711:31 PM
Sophie has two summer jobs.
• She works no more than a total of 32 hours a week. Both jobs allow her to have flexible hours, but in whole hours only.• At one job, Sophie works no less than 12 hours and earns $8.75/h.• At the other job, Sophie works no more than 24 hours and earns $9.00/h.
What combination of numbers of hours will allow her to maximize her earnings? What can she expect to earn?
5.4 to 5.6 Optimization.notebook
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September 24, 2012
Sep 1711:38 PM
A Saskatchewan farmer is planting wheat and barley.
• He wants to plant no more than 1000 ha altogether.• The farmer wants at least three times as many hectares of wheat as barley.• The yield per hectare of wheat averages 50 bushels, and the yield per hectare of barley averages 38 bushels.• Wheat pays the farmer $5.25 per bushel, and barley pays $3.61 per bushel.
Help the farmer determine the combination of wheat and barley he should plant that will maximize revenue.
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