## WPI Builds on Industrial Ties to Create REU Program

**October 1, 1999**

Ellen Phifer (left) and Ane Coughlin, members of the student team that worked on the optimal control formulation of an HVAC system that is designed to reduce utility bills, incorporate metropolitan weather data, and maintain air quality in buildings.

**Paul Davis**

"The new mathematics learned includes: optimization, especially linear and quadratic programming; dynamic programming; Fourier analysis, especially the discrete Fourier transform; basic statistics; simulation and probability distributions; modeling with differential equations; numerical solution of differential equations; calculus of variations and optimal control and discretization for optimization."

Is that the summary of an ambitious master's program in applied mathematics? Hardly. It's a list of the new areas of mathematics studied by the ten undergraduates who spent the summer of 1998 working in the innovative Industrial Research Experience for Undergraduates at Worcester Polytechnic Institute. The two-month program was supported by the National Science Foundation, by WPI's provost, and by the corporations whose problems the students solved.

The industrial REU program was organized by WPI mathematics faculty members Bogdan Vernescu and Arthur Heinricher, with the assistance of their colleague Ann Wiedie. These faculty members guided four student teams, each of which worked full time to solve one of the four problems posed by United Technologies Research Center, The Premier Insurance Company, John Hancock Insurance, and Morgan Construction Company.

**Finding Industrial Problems**The three WPI mathematicians had a head start in organizing the industrial REU-all WPI undergraduates are required to complete two projects, one in their major field and one in which they solve a problem at the interface between society and technology. Many of the projects have off-campus sponsors; indeed, half of the interface projects are completed abroad.

Three of the four REU projects arose from technical projects that had been completed by WPI mathematics majors working through the department's Center for Industrial Mathematics and Statistics. Contact with United Technologies Research, the one newcomer to industrial mathematics at WPI, was established with the assistance of WPI's Corporate Relations Office.

Before the summer REU, John Hancock had sponsored two WPI project teams working on similar problems. Based on that work and on the success of the 1998 REU project, it subsequently sponsored two more, completely different projects for the 1999 REU program.

In a similar pattern, Premier Insurance has sponsored a series of WPI project teams, each building on the work of its predecessors. The most recent in the sequence was the summer REU.

Morgan Construction Company is an old Worcester firm with long-standing ties to WPI, primarily through its engineering departments. Several years ago, mathematics faculty took advantage of those connections to introduce themselves to Morgan's technical staff. The result was a series of problems that have been tackled by WPI students at all levels---BS, MS, PhD, and, most recently, those in the REU program.

**Useful Solutions, New Insights**The problems posed by the corporate sponsors involved operating and product decisions that ranged in value from $50,000 to several million dollars. The students' solutions were useful, to the point, and sometimes even surprising.

But the program was even more valuable to the students. One student's comment captured a common insight: "Applied math is harder than it looks!" Another gave clear evidence of broadened horizons: "Mathematics is not only solving problems with pencil and paper, it is also modeling, deciphering, computer programming, and all sorts of other things."

Despite their experience in advising undergraduate projects in industry, the faculty advisers were no less affected. "It was more work than I expected but also much more fun," said Bogdan Vernescu. Art Heinricher agreed, adding, "No textbook can begin to kindle the drive and energy that comes when students get to see their work put to real use. The summer experience, especially the students' excitement about the projects, reminded me that WPI's own project program is incredibly special."

Ann Wiedie learned "that I can let students take more of the lead and that as a result they will learn more and actually get more done. I also think that the students learned a lot about working with people they had never met before and about dealing with the personalities of people they might not like to work with if they had the choice."

**Being Right Isn't Enough**The student teams met daily with their faculty advisers and periodically but less frequently with their industrial sponsors at corporate headquarters. The daily meetings on campus were often driven by the need to prepare for presentations to the sponsors. For a presentation to technical and marketing staff, Wiedie pointed out, "It isn't enough to be right. You have to be clear and convincing too."

The teams also attended weekly "staff" meetings and presentations by other industrial mathematicians, including Keith Hartt of Fidelity Investments, Stanislav Oks of Microsoft, and Bill Browning of Applied Mathematics, Inc. Browning had called on his corporate contacts to arrange a tour of an active nuclear submarine, and he treated all the REU students and advisers to a lobster dinner, a particularly tasteful example of corporate support for mathematics!

Personal development was as important as intellectual. In Heinricher's words: "What happens when you bring 10 very bright and very motivated mathematics students together for eight weeks in the summer? There is nothing to distract them. They may not be working on the same project but they are facing the same problems. The program worked because the students built their own support structure." Guidance in building that structure came from WPI alum and graduate student Jef Spaleta, who lived with the REU students and shared his own project experiences with them.

**Problem: Simulating an Office Building's HVAC System**The heating, ventilation, and air conditioning (HVAC) system in an office building can be responsible for as much as half of its electricity bill. United Technologies Research, prompted by its corporate sibling Carrier Corporation, was seeking an optimal control formulation of a building's HVAC system that would reduce the utility bill while incorporating actual metropolitan weather data and maintaining minimum air-quality levels.

The United Technologies team consisted of Ane Coughlin of Bates College, Brian Rieksts of Kutztown University, George Ellington of Wake Forest University, and Ellen Phifer of Trinity University. The liaison at United Technologies was James Fuller, senior principal engineer.

Beginning with a heat and mass transfer model that incorporated the major elements of the ventilation system, the students sought to minimize the cost of electricity plus a penalty to be imposed for falling outside a comfort zone defined by a temperature-humidity relation. Instead of a straightforward gradient-based minimization with commercial software (projected to require 40 years!), the team developed a dynamic programming formulation that demonstrated the feasibility of their approach and identified the critical steps in the next phase of the research.

**Problem: Optimal Auto Insurance Strategies**The Premier Insurance Company is required by the Massachusetts State Insurance Commission to issue auto insurance to anyone who applies, regardless of driving record, at rates set by the state. The company does have the option of "ceding" risky policies to a state agency, but with different costs and risks. Richard Welch and Martin Couture, vice president and chief actuary, respectively, were looking for an algorithm that would help them decide which policies to keep and which to cede, based on available driver data, such as age, driving experience, and traffic violation record.

Kristin Blenk of the University of Dayton and Jesse Field of the University of North Texas responded by formulating a bucketing algorithm to summarize a range of qualitative factors that characterize potential customers. They then tested various strategies, many of them from prior projects completed at Premier by teams of WPI mathematics majors. Their bucketing algorithm added a new element-an assessment of risk similar to that included in analyses of stock portfolios.

In their final presentation, the students also offered insight into the insurance industry. To Kristen's observation that perfect predictions of results require cession data "too old for use in the real world," Jesse added, "An insurance company is like a car with marketing on the gas, the comptroller on the brake, and the CEO at the wheel. The actuaries are navigating by looking out the back window."

**Problem: Variable-rate Life Insurance**Variable-rate life insurance combines a guaranteed death benefit with an investment vehicle. The variability in the return on the investment introduces uncertainty into the time required for full funding of the policy's face amount.

Rejecting projections based on fixed rates of return from the investment, Bruce Kearnan and Stephen J. O'Brien, senior and associate actuaries, respectively, at John Hancock Insurance, were seeking the probability distribution that governs the number of premiums necessary to fully fund a policy, given realistic assumptions about rates of return and account costs.

Jonathan Van Haste of Calvin College and Heather Stultz of Xavier University began by refining a program written the previous year by a WPI project team. They then developed a Monte Carlo simulation that led them to the startling discovery that, in certain market situations, there is only a 40% chance of fully funding the insurance policy!

**Problem: Rolling Mills**Morgan Construction Company is the nation's largest manufacturer of steel mill rolling equipment. The machines built by Morgan use a sequence of rollers to squeeze a red-hot slab of steel into a thin rod of hot steel that leaves the last set of rollers traveling at nearly 250 mph. A corkscrew-shaped piece of hollow tube called a bending pipe must stop that rod by coiling it into a spiral in less than five feet!

Bruce Kiefer, manager of mill process modeling and analysis at Morgan, wanted to find a shape for the bending pipe that would make it both long-lived and easy to fabricate. As a step toward that goal, he challenged the REU team to develop an accurate description of the wear distribution in a pipe of given shape and to recommend an optimal shape.

Peter Trautman of Baylor University and Jon Kennedy of WPI developed a one-dimensional continuum model whose predictions of wear profiles along the pipe were in excellent agreement with Morgan's experimental data. On the basis of that work and the students' discretized optimization formulation, Morgan is now able to analyze and develop new pipe shapes for a variety of production environments.

*The REU teams met daily, often to prepare for presentations to their corporate sponsors. Shown here are (left to right) Duane Ellington, Ellen Phifer, Ane Coughlin, WPI alum and graduate student Jef Spaleta, Brian Rieksts, and faculty member Ann Wiedie. *

**Good News and Bad**The good news at the end of two months of hard work was a well-attended round of thoroughly professional presentations by each student team. Hard questions were answered well, sponsors were obviously pleased by the results they received, and the students had come to grips with the inevitable gaps between aspirations and practicality when solving real problems in limited time.

After the last presentation, Art Heinricher, with a wilted smile appropriate to the hot July day, delivered the bad news: "Most of us are exhausted after seven and half weeks of hard work and a hot and humid day of intense presentations. But United Technologies is all excited. They want to talk to the students one more time before they leave for home. So we'll be down there bright and early on our very last morning!"

And they were. And this past summer, another group of WPI Industrial REU students found themselves working just as hard.

Postscript: Jesse Field, who was in an advanced high school program at North Texas State, has matriculated at Harvard. At least three of the four seniors in the group are beginning graduate school this fall---Ane Coughlin at Boston College, Ellen Phifer at the University of Delaware, and Kristin Blenk at Ohio State. Most of those who haven't yet graduated participated in another REU this summer.

Information about WPI's Industrial REU program is available at http://www.wpi.edu/Academics/Depts/Math/CIMS/REU/.

*Paul Davis is a professor of mathematical sciences at Worcester Polytechnic Institute. *

**What the Students Learned**

*"The hardest part is what to do, not how to do it. [We were] constantly reformulating all summer." "I already had encountered optimization, but I had never had to work so hard to actually get the objective function and constraints before!" *

*"It seems that mathematics is used everywhere; there is not a 'math' department at a company." *

*The most valuable thing learned was "how to understand and decipher an industrial problem and make it into a mathematical problem." *

*"Though math is a strictly a quantitative language, it is always used qualitatively in industry. Approximations and assumptions always limit the mathematical models." *