### Write down the coordinates of the extrema

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1. Discuss the problems you encountered while completing the exercise. Would you expect to encounter the same problems if you were using LP for a real-world allocation decision? Why or why not? Explain in the context of an actual allocation problem with which you are familiar.

2. List some characteristics of an allocation problem that may make it difficult to optimize using LP. Here are two:

• Constraints may not be totally predictable. For example, the price one can charge for a particular product may vary unpredictably over time.

• All of the constraints may not be known. For example, a new pollution regulation may limit effluents, thereby limiting a particular production process.

To summarize: When given a problem with constraints,

1. Determine the function to be maximized or minimized.

2. Plot the constraints to determine the feasibility region.

3. Determine the (x,y) values of the extrema (the "corners")

4. Test the coordinates of each extremum (corner) by substituting it into the function. Tabulate the results.

5. The maximum and minimum value of the function will be determined by the (x,y) values of two of the extrema.

To solve an LP problem,

1. Determine the function to be maximized or minimized. Typically, this is profit. It can, however, be anything.

2. Plot the constraints to determine the feasibility region. Use an application such as Relplot, or a pencil and a piece of graph paper.

3. Determine the (x,y) values of the extrema (the "corners"). In simple cases, this can be done by inspection. Otherwise, the simultaneous solution of the equations determining the constraints needs to be found, either manually or by using an app.

4. Test the coordinates of each extremum (corner) by substituting it into the function to be maximized or minimized. Tabulate the results.

5. Examine the tabulated results. The maximum and minimum values of the function will be determined by the (x,y) values of two of the extrema.

Given the following two-dimensional constraints:
2y<= 3x
2x + 3y <=15
3y >= x

1. Plot the feasibility region in the X-Y plane. You may use a plotting function such as Relplot (Myers, 2012), or make a neat sketch, take a picture of it, and paste the picture into your assignment.

2. Label the lines that are the boundaries of the feasibility region, using numbers. (1, 2, ... etc.) Remember that x and y are always greater than or equal to zero. You can't make a negative quantity of anything!

3. Label the extrema using letters; A, B, C, etc.

4. Write down the coordinates of the extrema. You may calculate the coordinates by hand, or use an online app to find the simultaneous solution of the two linear equations that determine each extremum.

5. Clearly indicate the feasibility region by placing an "R" in the middle of it.

Case:

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.

Version B: 2y<=2x, 2x+3y<=15, 3y>=x, x>=12. 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=x-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 three 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 \$150 on each scope, \$220 on each pair of binocs, and \$300 on each microscope, how many of each should the company manufacture each day? What is its daily profit?

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