Teaching Trends
Frame, Formulate and Solve
It's the one skill that OR professionals must have, so why aren't we teaching more of it?
By Richard C. Larson
Operations research. It's a wonderful subject to teach, to do, to research. It contains all the elements of disciplined, thoughtful approaches to complex problems. Once a learner is exposed to OR and masters its fundamental elements, he or she is able to tackle a wide range of problems, even if he or she is not visibly "doing" OR in the process. OR is a structured approach to the framing, formulation and solving of complex problems. As teachers/mentors who bring OR to our learners, in the classroom, over the Web, in professional meetings, are we in fact developing in them the skills needed to do these things: frame, formulate and solve complex problems?
Consider this problem: a yardstick is marked at two random points. "Random" here means uniform distribution over the length of the yardstick and that the two points are independently selected. At each of the two marks a saw is used to cut through the yardstick, resulting in three pieces whose respective length's sum is one yard. Question: a priori, what is the probability that such an experiment will yield three pieces that can be used to create a triangle?
This is a classic problem in applied probability that requires one to know manipulation in sample spaces, especially identification of relevant events. I have not met anyone who can do this in their head unless they have seen the problem before.
In teaching this problem, I usually start by describing the experiment and then asking students in the class their personal estimates of the probability. I write their estimates next to their names on the blackboard. If there are 10 estimates, I usually get at least one that states the probability is 1.0. The median typically is 0.7. I then produce a wooden yardstick (37 cents at Ace Hardware!) and an old rusty saw. We then carry out the experiment.
The random numbers are obtained from the second hands of students' watches (from those who have not set them in over a month). Very quickly any student who has guessed a probability of 1.0 learns that the ability to create a triangle from the three pieces is far from certain. We usually solve this together at the blackboard, as I feign amnesia at certain key points. The students must come up to the board to move things along. What is your personal estimate of the probability? Can you work out the answer? My belief is that a growing fraction of our students are unable to formulate and solve this relatively simple applied probability problem. One might say that applied probability is not part of OR, but if that were the case, how would a student tackle more complex problems involving queueing, supply chain management, production line planning, disease progression — just to name a few.
No, applied probability is a core curriculum underpinning of OR. But often it is taught in a way that encourages either rote learning, say by memorization of certain popular probability distributions, or "pattern recognition." Pattern recognition, while more sophisticated than rote memorization, is nonetheless worrisome. It is the process of reading a problem statement and then trying to match it to one of N new problems types learned since the last exam, and then pulling out the algorithm for that type of problem and, as we say, "turning the crank."
An OR professional, faced with a new and difficult problem, will not have the luxury of being able to choose from N problem types learned since the last exam. He must go back to basics and derive everything from basic principles, just as the broken stick problem requires. A true OR professional must be able to derive almost everything he or she does from basic principles, or else one is doing rote work or pattern recognition — both fraught with the risk of applying the wrong tool to the current problem.
If you would like to see the broken stick problem solved in an animated fashion, go to http://web.mit.edu/urban_or_book/www/animated-eg/stick/f1.0.html. There you will see the stick broken by laser guns aimed from flying saucers. How does your estimate of the probability compare to the correct answer?
On our Web site that is dedicated to these types of problems, we have some other classics as well. Students line up in a straight line 10 meters away from a bottle that is lying on the ground. Another student spins the bottle and finds that it points to one of the students in the line. One might call this a "straight version" of that child's game, "Spin the Bottle." Question: What is the probability that the bottle points to any given student standing in line? Or, more generally, what is the probability distribution of the point x on the line to which to bottle points?
Working this problem from basic principles yields the Cauchy probability distribution, that troublesome probability law that OR doctoral students are often convinced is saved for their oral exams. The Cauchy distribution has an ill-defined mean and infinite variance, and yet it can occur in a very simple probability experiment. See http://web.mit.edu/urban_or_book/www/animated-eg/bottle/. You may have heard of Buffon's Needle experiment. You drop a spinning needle on the red and white alternating stripes of an American Flag. Question: What is the probability that the needle after landing touches both colors, red and white? For the problem formulation and solution, see http://web.mit.edu/urban_or_book/www/animated-eg/needle/. (For the creative visual animations of these educational Web sites, we thank Elaine Chew, formally a doctoral student at the MIT Operations Research Center and now assistant professor at USC's Daniel J. Epstein Department of Industrial and Systems Engineering and USC's Integrated Media Systems Center, www-rcf.usc.edu/~echew/.)
Each of the above applied probability problems requires "problem framing, formulation and solution in-the-small." By this we mean that that the situation being analyzed is quite constricted and contained. Problem framing, formulation and solution in-the-small is one skill that OR professionals must have. We should, we must, teach it in our introductory applied probability courses. And in our optimization courses as well! We should be on the lookout for students who try to "get by" using rote or pattern recognition techniques and design our courses so these flawed learning approaches do not succeed.
Now consider an even more daunting task: "problem framing, formulation and solution in-the-large." This is the OR professional's assignment when he or she is brought into a complex real-world situation riddled with problems and difficulties and crying out for an intelligent and inspired OR analysis. Things are difficult, messy, fuzzy, nonlinear and hard to measure and quantify. Customers are screaming for better service. Perhaps stockholders are demanding decent profits. Here is where OR earns its "mettle." Let me go out on a limb and suggest the following: The majority of the value-added of an OR analysis is in problem framing and formulation. Once the problem is wrestled to the ground through intelligent framing and formulation, the task of "solution" represents less than half of the value added. Why? Because "solution" can be farmed out to a technician who is accustomed to working on well-formulated mathematical problems.
In the broken stick problem, framing and formulation would result in defining variables, creating an appropriate sample space and identifying the area of the sample space corresponding to the possibility of a triangle being formed. "Solution" may require multivariate integration. Yes, our OR analyst should know how to do all of that in the simple applied probability problems above. But he or she does not need to know all of the intricate details of more advanced methods of "solution," such as the latest heuristic optimization algorithm to "solve" NP-hard problem X. That can be farmed out to a specialist. But mastery of the art and science of problem framing and formulation in-the-large is the most important skill that an OR professional can have. Determining that the problem to be solved can be reduced to X rather than Y or Z is a huge contribution. Or, in many cases — recognizing the uniqueness of many real world problems — he or she identifies a new problem type, W, that has not been seen before!
Are we teaching our students how to approach problem framing, formulation and solution in-the-large? Back in the beginnings of OR, in the 1950s and 60s, we did, if for no other reason than there were few textbooks that nicely laid out the more mechanical methods of OR. Now, after five or six generations of doctoral students, we have scores of textbooks, many focusing on only small subsets of the technical solution methods of OR.
As teachers we often feel stressed to get the latest solution techniques crammed into our curricula. What subject matter is often excised to fit in the latest solution algorithm? In my experience, it is often the fuzzier area of problem framing, formulation and solution. Too often we teach as if the problem formulation is not problematic. Our textbooks often start with all the equations nicely and tidily in place and show the student how to optimize. Not to single out mathematical optimization, the same criticism can be directed at probabilistic modeling, say in which the student is given a Markov chain that is said to represent the dynamics of a baseball game. The student is never asked whether the Markov ("no memory") assumption is an appropriate model for a baseball game.
What happens if we do not address the issue of problem framing, formulation and solution in our teaching? We run the risk of graduating students who have OR solution tools looking for problems to apply them to, but with little or no skill at assessing which set of tools is appropriate for which real-world problems. To make matters worse, if such students graduate as rote or pattern recognition learners, they may be stranded when faced with a complex real-world problem, having mastered neither tools in-the-small or problem framing, formulation and solution in-the-large. I worry that the trend in teaching OR to MBA students via Excel spreadsheets may be exacerbating these problems.
What can be done? At MIT's Operations Research Center, years ago we added a requirement for each doctoral student to undertake an "OR Practicum." This effort, either on campus under the guidance of a faculty mentor or off campus at a company site, requires the student to undertake problem framing, formulation and solution in-the-large for a problem encountered by a real company or other organization. Often a letter testifying to the benefit of the analysis is obtained from an executive at the company. The requirement is there for all OR doctoral students, even those who are what you might call "pure math" OR researchers, i.e., those who will focus their careers solely on inventing and rigorously analyzing new algorithms and solution methods.
Amedeo Odoni, Arnold Barnett and I used to operate our subject, "Urban Operations Research," with student teams that went out to urban and public organizations to undertake problem framing, formulation and solution in-the-large in these organizations. For the "final exam" for this part of the course, the students made live presentations of their completed work, spoken before the class and senior decision-makers from the organizations. The results of some of these projects can be found online at http://web.mit.edu/urban_or_book/www/book/chapter8/contents8.html. Unfortunately we had to stop this practice, because we found it too labor intensive for the faculty members, who among other things had to find the set of projects each year and mentor the student teams as they progressed.
A challenge is to see if there may be less labor-intensive ways of immersing students into the difficult but important domain of problem framing, formulation and solution in-the-large. In this regard INFORMS is sitting on a huge asset: 30 years of Edelman Competition finalists. That's more than 150 successful case studies using OR and the management sciences. Most of these have on-site videotapes incorporated into their Edelman presentations. These videotape segments provide a type of "site visit" to the organization being studied and demonstrate the problems faced in a compelling way. In addition, each Edelman finalist has a nicely written summary article in Interfaces, highlighting the process and outcome of the OR/MS analysis.
Problem framing, formulation and solution play a big role in all of these materials, written and videotapes. These materials could be reassembled and manipulated in a way to create an instructional Web site for all OR students. The site would allow them to engage in problem framing, formulation and solution in-the-large, without excessive demands on faculty time and without the need to identify and work with real clients. Such a Web site would not be a substitute for the real thing, but it could come close. And it scales at no additional cost — it could be available to us all. I'd welcome your thoughts on this idea. Our students stand to become the primary beneficiaries.
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Richard Larson (rclarson@MIT.EDU) is professor of Electrical Engineering at MIT and director of MIT's Center for Advanced Educational Services. Over two separate periods totaling 14 years, he served as co-director of MIT's Operations Research Center. He has served in various capacities in our professional organizations, including president of ORSA (1994). He is a member of the National Academy of Engineering. His primary areas of research are OR in the services industries and technology-enabled education.
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