Mixture Problems For All Practical Chapter 4: Linear Programming - - PowerPoint PPT Presentation
Special Topics Chapter 4.1 Mixture Problems For All Practical Chapter 4: Linear Programming Purposes Lesson Plan Mixture Problems Combining Resources to Maximize Profit Mathematical Literacy in Todays World, 8th ed. Finding
Chapter 4: Linear Programming Lesson Plan
Mixture Problems
Combining Resources to Maximize Profit
Finding the Optimal Production Policy Why the Corner Point Principle Works
Decreasing-Time-List Algorithm
Linear Programming
Life Is Complicated
A Transportation Problem
Delivering Perishables
Improving on the Current Solution
Mathematical Literacy in Today’s World, 8th ed.
For All Practical Purposes
Chapter 4: Linear Programming Mixture Problems
Mixture Problem
Limited resources are combined into products in such a way that the profit from selling those products is a maximum.
Linear Programming
A management science technique that helps a business allocate the resources it has on hand to make a particular mix of products that will maximize profit. One of the most frequently used management science techniques.
Chapter 4: Linear Programming Mixture Problems
Production Policy A solution to a linear-programming mixture problem is a production policy that tells us how many units of each product to make. Optimal Production Policy Has Two Properties First, it is possible; that is, it does not violate any of the limitations under which the manufacturer
Second, the optimal production policy gives the maximum profit.
Chapter 4: Linear Programming Mixture Problems
Common Features of Mixture Problems
for time period.
resources.
needed.
make to maximize profit without exceeding any of the resource limitations.
Chapter 4: Linear Programming Mixture Problems
Mixture Problem: Making Skateboards and Dolls Skateboards require five units of plastic and are sold for $1 profit. Dolls require two units of plastic and are sold for $0.55 profit. If 60 units of plastic are available, what numbers of skateboards and/or dolls should be manufactured to maximize the profits?
Chapter 4: Linear Programming Mixture Problems
chart that includes the unknown variables (―x‖ units
Chapter 4: Linear Programming Mixture Problems
Example 2 Make a mixture chart to display this situation: A clothing manufacturer has 60 yards of cloth available to make shirts and decorated vests. Each shirt requires 3 yards of cloth and provides a profit of $5. Each vest requires 2 yards of cloth and provides a profit of $3. Let x = Let y =
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
What must be true about the sign of the numbers we can use for ―x‖ and ―y‖ in the skateboard/doll problem? Both ―x‖ and ―y‖ cannot be negative numbers. How can we write this information using an inequality sign, like >, or ≥, or <, or ≤? x ≥ 0 y ≥ 0
Chapter 4: Linear Programming Mixture Problems
These inequalities are called minimum constraints. Which means that one cannot manufacture negative numbers of objects.
Chapter 4: Linear Programming Mixture Problems
The next problem is that we only have so much plastic available to make skateboards and dolls. How can we represent this information? Since we need five units of plastic for each skateboard, we can write that information mathematically as needing 5x units of plastic for each skateboard. Since we need 2 units of plastic for each doll, we can write that information mathematically as needing 2y units of plastic. Hence we will need 5x + 2y units of plastic for the mixture
Chapter 4: Linear Programming Mixture Problems
Reading from the table, we have only a limited number of units of plastic available. How can we represent this information mathematically? 5x + 2y ≤ 60 This is called the resource constraint. Notice that all of the numbers in this inequality can be
reasons we construct a mixture chart is that it helps us speed up the conversion into inequalities of the information about the problem we wish to solve.
Chapter 4: Linear Programming Mixture Problems
Use the mixture chart to write an equation for the amount of profit that will be produced when we manufacture different mixture of skateboards and dolls. 1x + 0.55y = P, where P = profit Our goal is to find which values of ―x‖ and ―y‖ (skateboards and dolls) make this profit as large as possible.
Chapter 4: Linear Programming Mixture Problems
Example 3 Write the minimum constraints inequalities, the resource constraint inequality, and the profit equation for example 2. x ≥ 0 y ≥ 0 3x + 2y ≤ 60 P = 5x + 3y
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
Feasibility Set or Feasibility Region Our goal is to find the best mixture of ―x‖ and ―y‖ (skateboards and/or dolls) to produce the largest profit — two phases:
subject to limited resources. Graph line below 5x + 2y 60 (plastic)
largest profit.
Chapter 4: Linear Programming Mixture Problems
To draw the graph of an inequality, let’s first review how to draw the graph of the equation of a straight line. Remember that two points can be used to uniquely determine a straight line. Let’s use the equation associated with the resource constraint in example 1. The resource constraint is 5x + 2y ≤ 60. The equation associated with this inequality is 5x + 2y = 60
Chapter 4: Linear Programming Mixture Problems
There are two points that are easy to find on this line. When x = 0, this gives rise to one point on the line, and when y = 0, we can find another point. Find these two points.
Chapter 4: Linear Programming Mixture Problems
Let x = 0 so, 5(0) + 2y = 60 0 + 2y = 60 2y = 60 y = 30 The point (0, 30) is on the line.
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
Let y = 0 so, 5x+ 2(0) = 60 5x + 0 = 60 5x = 60 x = 12 The point (12, 0) is on the line.
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
Now that we know the graph of the equation 5x + 2y = 60 looks , we can think through where points (x, y) that satisfy 5x + 2y < 60 are located. The points that are either on the line 5x + 2y = 60 or satisfy 5x + 2y < 60 will satisfy 5x + 2y ≤ 60. Any line, for example, 5x + 2y = 60, divides the xy-plane into three parts: those points on the line, and the points in
have the points for which 5x + 2y < 60 and in the other we have the points for which 5x + 2y > 60. How can we tell which of the two half-planes is above the line 5x + 2y = 60 and which is below?
Chapter 4: Linear Programming Mixture Problems
The key is the use of a test point (x, y) that is not on the line and whose half-planes we wish to distinguish. We saw that (3, 10) is not on the line 5x + 2y = 60 and is below the line. This enables us to see that the half-plane for which 5x + 2y < 60 consists of the points below the line 5x + 2y = 60.
are true: 5x + 2y 60, and where x ≥ 0 , y ≥ 0
Chapter 4: Linear Programming Mixture Problems
Chapter 4: Linear Programming Mixture Problems
Example 4 In the earlier clothing manufacturer example, we developed a resource constraint of 3x + 2y ≤ 60. Draw the feasible region corresponding to that resource constraint, using the reality minimums of x ≥ 0 and y ≥ 0.
Let x = 0 0 + 2y = 60 2y = 60 y = 30 Graph (0, 30).
Chapter 4: Linear Programming Mixture Problems
Let y = 0 3x + 0 = 60 3x = 60 x = 20 So graph (20, 0). Which way are you going to shade? Below the line 3x + 2y = 60.
Chapter 4: Linear Programming Mixture Problems
Textbook Pages 139-140 #2,4,6,12,16