1. Chapter 4: Linear Programming Lesson Plan
For All Practical Purposes
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. 1
© 2009, W.H. Freeman and Company

2. Chapter 4: Linear Programming Mixture Problems
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. 2

3. 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.
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. 3

4. 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?
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).
Step 2
Translate the mixture chart into mathematical form from the chart.
5x + 2y ≤ (plastic)
P = 1x y (profit) 4

5. 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:
Find the feasible set for the mixture problem subject to limited resources. Graph line below 5x + 2y  60 (plastic)
Determine the mixture that gives rise to the largest profit (next slide).
Feasibility set (feasibility region) - A collection of all physically possible solutions, or choices, that can be made.
Shade in the feasible region is where all equations are true:
5x + 2y  60, and where x ≥ 0 , y ≥ 0
Graph of x + 2y = 60 5

6. Chapter 4: Linear Programming Finding the Optimal Production Policy
Corner point principle – States that in a linear programming problem, the maximum value for the profit formula always corresponds to a corner point of the feasible region.
Feasibility Set or Feasibility Region
Next step is to find the optimal production policy, a point within that region that gives a maximum profit.
Find the corner points of the feasible region.
Evaluate the profit at each corner point.
Choose the corner point with the highest profit as the production policy.
Optimal production policy – Corresponds to a corner point of the feasible region where the profit formula has a maximum value.
Calculation of the Profit Formula for Skateboards and Dolls
Corner Point Value of the Profit Formula: $1.00x + $0.55y
(0,0)
$1.00(0)
+ $0.55(0)
= $0.00
+ $0.00
= $0.00
(0,30)
+$0.55(30)
+ $16.50
= $16.50
(12,0)
$1.00(12)
= $12.00
= $12.00
Optimal production policy would be the point (0,30), which gives the maximum profit of $16.50. 6

7. Chapter 4: Linear Programming Finding the Optimal Production Policy
Example: Mixture of Two Fruit Juices
Using the data from the mixture chart (two products, two resources):
Determine the profit and constraint equations.
Graph the equations and find the feasibility region.
Feasible region
Maximize profit formula: 3x + 4y
Constraints: Cranberry: 3x + 2y  200
Apple: 1x + 2y  100
Minimums: x ≥ 0 and y ≥ 0 7

8. Chapter 4: Linear Programming Why the Corner Point Principle Works
Example Continued: Mixture of Two Fruit Juices
Using the corner point principle, the highest profit value on a polygonal feasible region is always at a corner point.
Evaluate the profit formula at these corner points:
The profit line for 360 lies outside the feasible region
Finding the Optimal Production Policy for Beverages
Corner Point Value of the Profit Formula: 3x + 4y cents
(0, 0)
3(0) +
4(0)
= 0 cents
(0, 50)
4(50)
= cents
(50, 25)
3(50) +
4(25)
= cents
(66.7, 0)
3(66.7) +
= cents (rounded)
Optimal production policy is (50, 25) with max profit = 250 cents.
Profit Line: If a profit line corresponding to a certain profit does not touch the feasible region, that profit is not possible. Example 3x + 4y = 360 does not work. However, a profit = 160 is feasible, but we can do better. The line shifts from 160 to 360 until it reaches a corner point (50,25) for maximum profit. 8