Trident OPM300 module 4 case SLP 2015

LINEAR PROGRAMMING

Case Assignment

1. Make a sketch of the feasibility region defined by the following constraints. Label the edges of the region with numbers; label the extrema with letters. Find and present the coordinates of the extrema. Assume that x and y are both equal to or greater than zero.

3y<=4x, 2x+3y<=8, x>=3

2. The constraints on a particular manufacturing process are shown on the right. The extrema of the feasibility region have been calculated and plotted. Using the profit function given below, calculate the profit (value of P) at each extrema. P=3x-2y At which extremum is the profit the maximum? The minimum? (A negative profit is a loss. The minimum profit is either the smallest positive profit, or the largest loss.)

3. Eye-Full Optics assembles astronomical telescopes (x), premium binoculars (y) and student-grade microscopes (z) from imported parts. Each telescope takes one hour to assemble, each pair of binoculars two hours, and each microscope four hours; the availability of skilled labor limits assembly work to 1000 hours per day. Eye-Full has a contract with FedEx, and must ship no less than 400 items per day. A contract with a major retailer requires them to deliver a minimum of 100 telescopes, 250 binocs, and 50 microscopes per day. But there are supply limitations. The telescopes and binocs are shipped with the same eyepieces; each scope has one, and each pair of binocs has two. The subcontractor who supplies the eyepieces can only furnish 800 per day. Similarly, both the binocs and the microscopes use the same prisms; each pair of binocs needs two, and each microscope needs four. The prism supplier can only ship Eye-Full 1600 per day.

If Eye-Full makes a profit on $100 on each scope, $200 on each pair of binocs, and $350 on each microscope, how many of each should the company manufacture each day? What is its daily profit?

(Since the feasibility region is a volume in three-dimensional space, a sketch is not required.)

Assignment Expectations

• Graphics must be neat, clear and complete. A graphics app can be used, but a freehand sketch is also acceptable.
• All calculations should be shown.
• All answers must be clearly stated.
• Relevant theory should be cited as necessary to explain which procedures were used to arrive at the answers, and why.

LINEAR PROGRAMMING

Complete the wrapup of a three-round Delphi decision-making exercise, following the detailed example cited in the Home Page discussion. As before, you may copy and / or adapt verbiage from the example without citing it.

SLP Assignment Expectations

The SLP writeup should consist of:

• The Letters to the Participants, which include
• Thanks for their participation
• A summary of their third-round responses
• A short narrative discussing the evolution of the decision-making process, how opinions shifted, what relevant factors the group identified, and what consensus (if any) the group arrived at.

SLP General Expectations

• Follow the instructions in the BSBA Writing Style Guide (July 2014 edition), available online at
  https://mytlc.trident.edu/files/Writing-Guide_Trident_2014.pdf.
• There are no guidelines concerning length. Write what you need to write – neither more, nor less.
• In the SLP ONLY, references and citations are NOT required. However: If you state a fact, express an opinion, or use a turn of phrase that isn’t your own, then you should credit the source, just like you would in everyday conversation. (Example: “In the words of Monty Python, ‘And now for something completely different.’ “)

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TEST YOUR UNDERSTANDING: Answer

The linear equations corresponding to the constraints are:

1. 2y = 3x

2. 2x + 3y = 15

3. 3y = x

Here’s the plot, with the lines, extrema, and the region labeled. It was created with Relplot, and the labels were added using the Snagit graphics editor. Using Relplot, it’s possible to create the sketch without knowing the coordinates of the extrema. That’s because the app takes the line equations as input.

Here’s another version. It’s less elaborate, but still perfectly acceptable. If you want to upload a hand sketch, however, you’ll have to do the calculations first, so you’ll know where to put the extrema.

Here’s how to find the coordinates of the extrema:

A: The only values of x and y that satisfy the equation 1 (that is, 2y=3x) is (0,0) . Ditto for equation 3. So the coordinates for A are

A(0,0)

B: This point is the simultaneous solution of equations 1 and 2; that is, of

2y=3x

2x + 3y = 15.

We’ll use the Webmath solver (Discovery, 2014) to find the values of x and y that satisfy both equations. There are many such apps on the Web; look for them using Google, or your favorite search engine.

Here’s what the setup looks like:

Proceed in the same way to find the coordinates of point C, which is simultaneous solution of equations 2 and 3; that is,

2x + 3y = 15

3y=x

The answer is C(5, 1.67).

Summary answer: Extrema are

A(0,0)

B(2,31, 3.46)

C(5, 1.67)