Linear Programming Assignment

Due: Monday, November 28th(Blue Day) or Tuesday, November 29th (White Day)

(the class period after we return from Thanksgiving Break!)

REQUIRED: On a separate sheet of paper, complete ALL of the following problems (#s 1 - 5). Make sure you show all your work to receive credit. This is worth a 50 point quiz grade!!

Find the minimum AND maximum values of the given objective function for the given feasible region. (Evaluate the vertices in the given objective function.)

1.)  C = 5x + 7y 2.) C = 6x – 4y

Complete the following application problems for Linear Programming:

3.)  Craft Fair: Piñatas are made to sell at a craft fair. It takes 2 hours to make a mini piñata and 3 hours to make a regular-sized piñata. The owner of the craft booth will make a profit of $14 for each mini piñata sold and $22 for each regular-sized piñata sold. If the craft booth owner has no more than 40 hours available to make piñatas and wants to have at least 16 piñatas to sell, how many of each size piñata should be made to maximize profit?

4.)  Manufacturing: A company manufactures inkjet printers and laser printers. The company can make a total of 60 printers per day, and it has 120 labor-hours per day available. It takes 1 labor-hour to make an inkjet printer and 3 labor-hours to make a laser printer. The profit is $45 per inkjet printer and $65 per laser printer. How many of each type of printer should the company make to maximize its daily profit?

5.)  Manufacturing: A bicycle company produces two models of bicycles. The table below shows the number of hours it takes for assembling, painting, and packaging each model. The total times available are 4000 hours for assembling, 4800 hours for painting, and 1500 hours for packaging. The profits are $55 for model A and $60 for model B. How many of each model of bicycle should the company make to maximize profit? What is the maximum profit?

Assembling / Painting / Packaging
Hours, Model A / 2 / 4 / 1
Hours, Model B / 2.5 / 1 / 0.75

OPTIONAL: If you would like an opportunity for extra credit, then read the following. This is worth a 100 point test grade!!

Choose ONE of the application problems above (#s 3 - 5), and make a project. You may choose one of the following formats for your project:

·  Poster

·  Powerpoint

·  Video

Your project must include all the steps for linear programming and explain the problem correctly. You MUST present your project to the class the day the project is due (the class period after returning from Thanksgiving Break).

Vocabulary of Linear Programming

Constraints - the inequalities limiting the problem at hand.

Feasible Region - the shaded polygonal area created by the intersection of the graphs of inequalities. It is the location where every constraint is met or satisfied.

Unbounded - a function where no maximum value exists. The shaded regions of the constraints do not form a closed figure.

Objective Function - a function in two variables f(x,y) that is the objective to maximize or minimize, such as profit, etc.

Property–Vertex Principle of Linear Programming--If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices of the feasible region.
The Procedure
1.)  Identify your variables, x and y. (You will only have 2 variables!)
2.)  Identify your objective function pertaining to the situation.
3.)  Identify your constraints/restrictions (inequalities)
4.)  Identify the non-negative constraints (Ask yourself: can the situation involve negative numbers?)
5.)  Graph each inequality on a coordinate plane. (Make sure you shade correctly!)
6.)  Identify the feasible region. (Look for where all the shaded regions overlap)
7.)  Determine the vertices of the feasible region (polygon) that are formed.
8.)  Evaluate each vertex in the objective function from Step 2 above.
9.)  Determine the maximum(highest)/minimum(lowest) value from evaluating your vertices.