MAT 230, Optimization Modeling Project, Spring 2017 Name ___________________________

A. Itarsi Silicon Materials Company (ISMC) is looking to break into the market with a new product. For security reasons, their company materials merely refer to the product as Brand Q. To gauge how much they can charge for their new product, they survey 1000 people about how likely they are to purchase Brand Q when it comes onto the market at a given price. The table below shows the results of their study with proportions translated to an estimate of products sold per month based on market share of currently available products.

Price ($) $199 $176 $166 $160 $151 $136 $119.30 $111.40 $102.40

Quantity Sold (Expected) In Thousands 0 6 8 10 12 15 19 22 25

B. On the graph below, plot the x-values along the horizontal axis, and the Price along the vertical

axis, to create a scatterplot. What do you notice about the graph you created? Be sure to label your axes appropriately.

Check with your instructor.

C. Our next step is to construct a regression equation in our calculator to model the price of Brand Q by sales. Follow the steps below to enter data into a TI-84.

Press the key and then select Edit. You will now see the data screen with the six lists, L1 through L6 (use your right arrow to see L4 through L6).

If there is already data in one of the lists, we need to clear the list. Do this by placing your

cursor at the top of the list you wish to clear and then press the button. DO NOT press the DEL button - this will delete the list from your calculator.

Enter your x data into L1 and your volume data into L2. You can navigate from list to list with the cursor arrow buttons.

D. Sketch the graph in your calculator. To draw the graph in the TI, select to

enter StatPlot. Turn on Plot1, press , and move the cursor to ON and press again. On the Type line, the first graph type is the scatterplot. Put the cursor over it and press

to select. We used the default XList (L1) and the default YList (L2), so we don’t need to adjust those settings.

After setting up the graphing screen, press ,

followed by for 9: ZoomStat. This will adjust your Window settings to match the data sorted in your lists.

E. Compare the results to the scatterplot you drew above. Does it match? (If the answer is ‘no’, find the error before proceeding!)

F. Next we need to set up the calculator to find the regression equation for us. Complete the following steps.

1. Turn on Diagnostics:

a. On newer TI-84s, select and scroll down toward the bottom on the list of commands (onto the second page). You will see a line for the Diagnostics. Select ON to be highlighted.

b. On older machines (TI-83s, and some older 84s), press for CATALOG. Scroll down through the alphabetical list to the line that reads Diagnostics

On. Select when you get to that line and then press again to execute the command.

(You should only have to do this once unless someone clears the memory.)

2. Select and then scroll over the CALC menu. Scroll down to 4: LinReg (ax+b) . This

is the linear regression function. Select or . On newer TI-84s, your screen will look like this:

Accept the defaults for the List locations unless you used a different list. Leave the frequency line blank. On the Store RegEQ line type:

then select Y-VARS, then FUNCTION, then

Y1. Then scroll down to Calculate and press .On older machines your screen will look like this:

Before hitting anything else, follow the steps

above to select Y1 regression equation location. Then press to execute the command.

Your screen will look something like what is shown here, though, with different numbers.

The equation the coefficients belong to appears first: y=ax+b, and the values of the coefficients are shown below in order. The

final value show up only when the Diagnostic setting is ON. The R2 measures how good the fit to the data is. Closer to 1 is a better fit.

G. What equation did your calculator give you? Write the result in function notation.

Check with your instructor.

H. Graph the equation against your plotted data. Visually inspect how good the fit is. Does the equation do a good job of modeling the data you have? If it does not, consult with your instructor about obtaining a different equation type. Add your final equation to the scatterplot in F.

I. Now that we’ve created a price model, ISCM wants to use this information to predict their production to obtain a maximum profit. They estimate the start-up costs for production of the first unit is $50,000. Once the factory is up and running, additional units of Brand Q will cost $0.0037 per unit. Use this information to create a cost model for Brand Q. [Recall that units in the price function were in thousands of units. Your cost model should be in the same units.] Write your equation in function notation.

J. Using the information about price and sales obtained from the survey data, find the Revenue function for Brand Q. Write your equation in function notation.

K. Use the results of I and J to find the model Profit function for the first month. Write your equation in function notation.

L. What is the domain of your profit function? [Hint: Check the price model. Where are units sold positive? Where is price positive?]

M. If the start-up costs are only in the first month, what is the cost model for each future month?

N. Assume that the start-up costs are applied at the start of the first year. What is the profit function for the entire first year? [Hint: Use the original cost function for the first month, and the continuing cost model for the remaining 11 months of the year. Don’t forget that the Revenue function is also monthly—what is it for the first year?] Write your equation in function notation.

O. What would your cost, revenue and profit functions be in year two?

Check with your instructor.

P. Sketch the graph of your profit function for the first month on the first graph, and then for the first year on the second graph. Label the axes appropriately.

Q. Estimate from the graph the approximate value of maximum profit and where it occurs for the first month graph.

R. Use calculus to find the exact solution: Where is profit maximized in the first month? (how many units sold)? And what is the maximum profit in the first month? [Recall that your equations are in thousands of units of Brand Q.]

S. Does your work confirm your estimate? Why or why not?

T. Estimate from the graph the approximate value of maximum profit and where it occurs for the first year graph.

U. Use calculus to find the exact solution: Where is profit maximized in the first year? (how many units sold)? And what is the maximum profit in the first year? [Recall that your equations are in thousands of units of Brand Q.]

V. Does your work confirm your estimate? Why or why not? Did your estimate change from the first month compared with the whole year?

W. Did you make money in the first year? If not, use your equations for the first and second years and add them to find the profit over the first two years together. Is that enough time to make a profit?

X. Suppose you were the production manager for Brand Q. Describe how you would use this information to plan the number of items to be produced in plants per month? How would you account for the uncertainty in your original price model? How would you plan around production delays for broken equipment, sick workers or supply chain problems to ensure there was enough product to sell? Write a paragraph to explain your general strategy.