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 Today’s World, 8th ed.  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

Chapter 4: Linear Programming Mixture Problems  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.  Mixture Problem  Limited resources are combined into products in such a way that the profit from selling those products is a maximum.

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 operates, such as availability of resources.  Second, the optimal production policy gives the maximum profit.

Chapter 4: Linear Programming Mixture Problems Common Features of Mixture Problems  Resources – Available in limited, known quantities for time period.  Products – Made by combining, or mixing, the resources.  Recipes – How many units of each resource are needed.  Profits – Each product earns a known profit per unit.  Objectives – To find how much of each product to 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  Step 1  Mixture Chart – display the verbal information into a chart that includes the unknown variables (―x‖ units of Skateboards, and ―y‖ units of dolls).

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 Translating Mixture Charts into Mathematical Form

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 of skateboards and dolls we make.

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 obtained from a column of the mixture chart. One of the 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 Feasibility set (feasibility region) - A collection of all physically possible solutions, or choices, that can be made.

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: 1. Find the feasible set for the mixture problem subject to limited resources. Graph line below 5 x + 2 y 60 (plastic) 2. Determine the mixture that gives rise to the 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 one of the two half-planes. In one of the half-planes we 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.

Chapter 4: Linear Programming Mixture Problems 1. Graph of 5 x + 2 y = 60 2. Shade in the feasible region is where all equations 60, and where x ≥ 0 , y ≥ 0 are true: 5x + 2y

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. 1. Graph 3x + 2y = 60. 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

Assignment  Textbook Pages 139-140 #2,4,6,12,16