Chapter 4: Linear ProgrammingLesson 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

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Mathematical Literacy in Today’s World, 8th

ed.

For All Practical Purposes

© 2009, W.H. Freeman and Company

Chapter 4: Linear ProgrammingMixture Problems

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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 ProgrammingMixture Problems

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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.

Common Features of Mixture ProblemsResources – 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.

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Chapter 4: Linear ProgrammingMixture 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 ≤ 60 (plastic)

P = 1x + 0.55y (profit)

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Chapter 4: Linear ProgrammingMixture 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 5x + 2y 60 (plastic)

2. 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 5x + 2y = 60

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Chapter 4: Linear ProgrammingFinding the Optimal Production Policy Feasibility Set or Feasibility Region

Next step is to find the optimal production policy, a point within that region that gives a maximum profit.

1. Find the corner points of the feasible region.

2. Evaluate the profit at each corner point.

3. Choose the corner point with the highest profit as the 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.

Optimal production policy – Corresponds to a corner point of the feasible region where the profit formula has a maximum value.

Optimal production policy would be the point (0,30), which gives the maximum profit of $16.50.

(0,0) $1.00(0) + $0.55(0) = $0.00 + $0.00 = $0.00

(0,30) $1.00(0) +$0.55(30) = $0.00 + $16.50 = $16.50

(12,0) $1.00(12) + $0.55(0) = $12.00

+ $0.00 = $12.00

Calculation of the Profit Formula for Skateboards and Dolls

Corner Point Value of the Profit Formula: $1.00x + $0.55y

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Chapter 4: Linear ProgrammingFinding 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.

Maximize profit formula: 3x + 4y

Constraints: Cranberry: 3x + 2y 200

Apple: 1x + 2y 100

Minimums: x ≥ 0 and y ≥ 0

Feasible region

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Chapter 4: Linear ProgrammingWhy 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:

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.

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) 3(0) + 4(50) = 200 cents

(50, 25) 3(50) + 4(25) = 250 cents

(66.7, 0) 3(66.7) + 4(0) = 200 cents (rounded)

Optimal production policy is (50, 25) with max profit = 250 cents.