# Math 204--Calculus III: Project 2, Fall 1998

## Costs of Production...

### by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, November 1998

permission granted to use and distribute free in an academic setting.

PostScript version of project

### Chemproc, Inc.

20000 Ryan-ears Blvd.
Lonlinc, SK 08685

4 November 1998

Sophisticated and Independent Mathematical Contractors (SIMaC), Inc.
Lonlinc, SK 04685

Dear SIMaC:

Having received your report on the subject of our ``ChipLess'' microchip coating process, it is with considerable faith that we write you again to request your assistance in our work on the marketing of our latest product line expansion. This is to include two new products, which we for the sake of proprietary secrecy shall refer to as ``X'' and ``Y.'' As these are related they share some production costs and resources, resulting in some difficulty in ascertaining the optimum production quantities x and y of each, and it is to determine this best marketing strategy that we are contacting you.

We have found that, appropriately scaled, the production costs of the manufacturing processes for these products is reasonably given by

C(x,y) = awx + bzy + xy,
where a and b are positive constants and w and z are variables that are dictated by costs of materials which are out of our control. In the scaled variables, w and z vary between 0 and 1, and are hypothesized to be either random values between these two extremes or be most likely found near 0.75, with vanishing probabilies that they will be equal to 0 or 1. As we are unable to vary our production in response to the values of w and z, we need your analysis to treat their possible values in the aggragate, and would like you to consider both of the indicated possible models for their variation.

We also find that the production quantities x and y are constrained by the fact that they share common resources and are influenced by interrelated market forces which, while they are again beyond our control, are more predictable than the material costs noted above. In a series of test marketing efforts we were able to vary the sustainable production of x and y with time, finding that they at maximum production behaved as shown in figure 1 to the right. We believe this to indicate that x and y have some sort of direct relationship that will constrain the production when we commence with our long-term manufacturing and marketing of X and Y.

Insofar as possible we would like your analysis to be applicable for general values of a and b, though we do expect that a will be approximately 1 and b approximately 2.

We look forward to receiving your final report on or before the 30th of November. To assure your success, we have again arranged for the most estimable of mathematicians, Dr. P. Gavin LaRose, to answer any questions you may have in the course of your investigation. Please note, however, that he will unfortunately be unavailable to assist with this project over the weekend of the 27th--29th of November. You should plan on meeting with him before the week of the 16th of November to verify your initial progress. We have also made available, through him, a number of sample reports that may prove useful as you develop your report.

Sincerely
E. Idu Pont
President, Chemproc, Inc.

eip:glr

Gavin's Calc III Project 2, Fall 1998