Lecture Topic: Linear Programming

Instructor: Tim Huber

Institution: UTPA

Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Set up and solve linear programming problems.
• Explain why the techniques used to solve linear programming problems work.
• Appreciate the variety of contexts in which linear programming problems arise.
• Sub Objectives- The objectives will require that students be able to:
• Understand the terminology used in linear programming problems.
• Graph systems of inequalities and plot the fundamental region for a set of constraints.
• Find corner points for feasible regions by solving appropriate systems of equations.
• Difficulties- Students may have difficulty:
• Graphing lines and corresponding inequalities.
• Solving systems of linear equations
• Avoiding arithmetic mistakes.
• Seeing the big picture – Understanding both the “how” and “why” of linear optimization methods.
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• In manufacturing processes, where only a limited supply of parts or labor exist, linear optimization can be used to determine the parameters that optimize revenue and profit while minimizing resources used.

Model of Knowledge

• Concept Map
• Writing a set of restrictions as a system of inequalities.
• Understand how to plot feasible regions.
• Understand how to find corner points of feasible regions.
• Determine optimum values of the parameters by testing the objective function at the corner points.
• Understanding the significance of the points found in the last step.
• Content Priorities
• Enduring Understanding
• Set up and solve linear programming problems.
• Explain why the techniques used to solve linear programming problems work.
• Appreciate the variety of contexts in which linear programming problems arise.
• Important to Do and Know
• Writing a set of requirements as a system of inequalities.
• Understand how to plot feasible regions.
• Understand how to find corner points of feasible regions.
• Determine optimum values of the parameters by testing the objective function at the corner points.
• Understanding the significance of the points found in the last step.
• Worth Being Familiar with
• Appreciate the variety of contexts in which linear programming problems arise.

Assessment of Learning

• Formative Assessment
• In Class (groups)
• Form groups to exchange ideas for optimizing revenue functions with constraints.
• Homework (individual)
• Students will solve linear programming problems on their own.
• Summative Assessment
• Exchange solutions to linear programming problems to determine if the points found optimize the objective function while satisfying the required inequalities.

Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Set up and solve linear programming problems.
• Explain why the techniques used to solve linear programming problems work.
• Appreciate the variety of contexts in which linear programming problems arise.
• Understand the terminology used in linear programming problems.
• Graph systems of inequalities and plot the fundamental region for a set of constraints.
• Find corner points for feasible regions by solving appropriate systems of equations.

The objectives will require that students be able to:

• Graph lines and corresponding inequalities.
• Solve systems of linear equations
• Solve linear optimization problems.
• Be able to clearly organize calculations to avoid arithmetic mistakes.

THE CHALLENGE

Your company makes an equal number of two types of skis: downhill and cross-country. Your task is to determine if a different production scheme can increase the profit for the company. To complicate matters, your company operates under the following restrictions:

 Manufacturing Time Finishing Time Cross-Country Skis 3 hours 1 hour Downhill Skis 3 hours 2 hours

There are only 15 hours available each week for the manufacturing process and 8 hours for the finishing process.

A profit of \$80 is made on each downhill ski, and a profit of \$83 is made on each cross-country ski.

GENERATE IDEAS

If the company insists on making an equal number of skies of each type, what is the maximum profit that can made? Can you change the production scheme while still satisfying the restrictions in the manufacturing and finishing processes? Describe other schemes that satisfy the necessary requirements?

MULTIPLE PERSPECTIVES

Can you graphically describe the set of all possible production schemes that satisfy the requirement? Draw some pictures in xy-plane.

RESEARCH & REVISE

What is the profit function? What kind of equation results from a particular (fixed) profit? Vary the fixed profit value so that the curve describing the profit function intersects with the region that describes the feasible production schemes?

Find an optimum production scheme with the constraints given in the challenge.

GO PUBLIC

Share your work with the class.

Pre-Lesson Quiz

1. Let C be the number of cross-country skis produced and let D denote the number of downhill skis produced. Without doing significant calculation, decide what the best approach the business would be if there were only 3.5 labor hours available. Assume the manufacturing criteria stated in the challenge.
2. Suppose the downhill skis require 2 hours for manufacturing and 3 hours for finishing. Suppose the cross country skis require 1 hour for manufacturing and 1 hour for finishing. Write down the corresponding system of inequalities.
3. Can you identify the points in the graph from the last problem that you would describe as "corners".
4. If cross country skis sell for 80 dollars each and downhill skis sell for 50 dollars each, write down the function that gives the revenue from total ski sales.
5. Find the value of the revenue function at each corner point.