Innovative Education
The Science of Decision-Making
A problem-based approach using Excel
By Eric V. Denardo
A sea change is underway in the teaching of ORMS, in its roles in college and professional education, and in its general usefulness. In decades past, our tools and methods had been coherent and potent, but they could be hard for beginners to grasp, and they could be difficult for non-experts to use effectively. Breathtaking developments in computer hardware and software have made our discipline increasingly easy to understand, increasingly easy to employ and a great deal more useful. Our perspective on problem-solving is becoming central to all sorts of decision-makers, be they engineers, economists, managers, bankers or medical personnel.
My perception of this sea change has propelled me to write a book that mirrors the title of this article. Like several other recent texts, mine is problem-based and spreadsheet-oriented. It differs, perhaps, by taking a somewhat broader view of our discipline and by probing it more deeply. My premise (and my personal experience) is that a problem-based approach to our methods, when coupled with spreadsheets, makes them accessible to all sorts of students, deepens the students' understanding of our methods, and helps students to perceive the relevance of our methods to their education and their careers. Here, I attempt to record my perception of this sea change and to suggest how students and their teachers can benefit from it.
Personal Experience
My approach to our discipline owes much to a problem-based course on quantitative methods that was taught until 1988 at Yale's School of Organization and Management (SOM). Several faculty members contributed to that course, Art Swersey being the de facto leader. SOM's students found the course to be challenging, interesting, integrative and relevant, a centerpiece of their educational experience.
More recently, this approach was adapted to the needs of Yale undergraduates, and Excel was introduced. Excel proved to be a godsend. It enabled students to learn more, to learn more rapidly, and to achieve deeper insight. Excel gave students access to intermediate-level material. It provided students with tools they could employ in other courses and later in their careers. Enrollment leapt, as did the students' enthusiasm for the material.
Problem-based Instruction
The advantages of problem-based instruction in ORMS are now widely understood. Well-chosen examples make the general ideas easy to grasp, show how to make effective use of our models, reveal the ways in which our models approximate reality, and avoid the thickets of mathematical notation that abstract presentations can require.
The trick, however, is to present students with intriguing examples that raise important issues — decision-making problems whose analyses are insightful, not just puzzles. Students will welcome such problems, and will attack them with vigor.
The spreadsheet as a calculator. An obvious benefit of a spreadsheet is that it automates calculation. A spreadsheet provides easy access to the standard mathematical functions and to the probability distributions. It makes easy work of the "what-if" calculations that explore the ways in which our models approximate reality. Solver (an Excel Add-In) provides seamless access to algorithms that solve linear programs, nonlinear programs and integer programs. By facilitating computation in these ways, a spreadsheet lets students grapple with larger and more realistic problems.
It's also true that students are eager to sharpen their spreadsheet skills. They anticipate using spreadsheets in other courses and throughout their professional careers.
The spreadsheet as a mathematical tool. A spreadsheet has more subtle benefits. It lets us rethink the content of an introductory course. Before spreadsheets, we were limited to the calculations that could be done with the elementary functions. Now, anything that can be computed on a spreadsheet is fair game.
Recursions illustrate this point. Before spreadsheets, recursions were a bit abstruse — hard to grasp and not much used in introductory courses. On a spreadsheet, we can "repeat a pattern" by "dragging" the mouse pointer. That's a recursion. Dragging executes all sorts of recursions that had been too difficult for beginners to grasp. On a spreadsheet, executing numerical integration with a drag is a breeze, for instance.
Recursions are one example, among many. Matrix operations are another. Before the spreadsheet era, techniques for manipulating simultaneous equations had been off-putting, as they can entail masses of subscripts. Spreadsheet functions make matrix operations easy to understand and very easy to execute.
A third example is the optimizer within Solver. Previously, the uses of optimization in introductory courses tended to be restricted to models that were linear and deterministic. In a spreadsheet environment, which includes the probability distributions, Solver's optimization feature analyzes models that are nonlinear and uncertain.
The spreadsheet as a laboratory. A spreadsheet is itself a laboratory, one that presents the stages of a calculation, showing lots of detail. If the calculation is incorrect, the spreadsheet is very likely to give a wacky answer, with clues as to what is wrong. This makes a spreadsheet a vehicle for interactive learning. You try something out. You get a nonsensical answer. You try again, and get more gibberish. And again until — aha, that's it!
Homework and Interactive Learning
The mathematical sciences, including ours, are learned by working problems. Interactive learning on a spreadsheet lets students attack — and solve — problems that had previously been well beyond their capabilities. With success comes positive reinforcement. With that comes self-confidence, enthusiasm for our methods, and the ability to apply them in novel ways. Again, the trick is to present students with problems that are challenging and intriguing, but doable.
A spreadsheet isn't a panacea. One important limitation of a spreadsheet may be its lack of "auditability;" on a spreadsheet, you can see either the values that are assigned to the functions or the functions themselves, but not both. A different limitation of a spreadsheet lies in its inability to change the value that is assigned to a cell during the course of computation. High-powered solution methods are by no means obsolete. But, for an introductory or intermediate-level account, spreadsheets are hard to beat.
In decades past, introductory courses in ORMS tended to focus either on deterministic models or on stochastic models. Economics was de-emphasized or omitted. Probability was taught elsewhere in the curriculum, as a prerequisite to the introductory course on stochastic models. This approach is dated.
Probability can be a bit mysterious, but it is easy to grasp when it is cast in a decision-making context with spreadsheets. Again, a well-chosen example indicates why the model was built, what it can accomplish, and how to use it. The models of probability are themselves an important facet of decision science. A firm understanding of probability is crucial to inventory theory, queueing, simulation and game theory. Probability is important to allied disciplines, including financial economics. For each of these reasons, integrating probability into our introductory courses helps the student learn our methods and utilize them.
Many of our techniques have been assimilated by economics. Indeed, economists have contributed to them. Yet the connections between the fields are often suppressed or omitted. Emphasizing these connections provides apt illustrations of our ideas. It links economic reasoning to concrete decision-making situations. Economics students will welcome this, and they are legion.
Opportunity cost provides an illustration. Opportunity cost is one of the cornerstones of economic reasoning. The ideal setting in which to learn about opportunity costs is a linear program. We should teach it there. Marginal analysis, mean-variance trade-off, efficiency, general equilibrium and arbitrage also illustrate this point; they make lovely applications of our methods.
Focus: The Science of Decision-making
In decades past, introductory accounts of our subject tended to emphasize the algorithms that we had developed. A spreadsheet helps us to refocus introductory courses on problem-solving — on formulating problems and on making sense of their solutions. The algorithms remain as important as ever. But, in an introductory account of our discipline, the algorithms must compete with problem-solving.
With a broadened scope, a focus on problem-solving, an emphasis on insight and a de-emphasis on algorithms, the phrases "operations research" and "management science" seemed to carry the wrong connotations. The "science of decision-making" seemed more apt. Hence, the title.
Spreadsheet-based instruction does not preclude the teaching of algorithms. To the contrary, a spreadsheet can make the algorithm easier to grasp. In a second-tier course, I use an Excel Add-In to "pivot" from one tableau to another. The student selects the coefficient on which the pivot occurs, and the computer does the number crunching. By experimenting with this Add-In, the student can learn how the simplex method works, and how to adapt it to various situations. In this way, the spreadsheet eases access to the simplex method itself and, through it, to computational complexity, duality, and its ramifications.
The Computer Revolution
The evolution of digital computation has made our methods manifestly more usable for two separate reasons. One of these is the many-thousand-fold improvement in computer hardware and software. Simulation illustrates this point. A quarter century ago, a reasonable-sized simulation might require an investment of a hundred thousand dollars, a special-purpose computer code and months of an expert's time. Today, a novice can design and run the same simulation in an afternoon on a personal computer with an Excel Add-In. Simulation is not atypical. Each of our methods has become vastly easier to learn and to employ.
The computer revolution has benefited us in a second way, one that is illustrated by production and distribution systems. We have long understood the fundamentals of managing these systems, but our techniques were not much used because they shared a serious flaw — the decision-makers were presumed to know where the material was, and that knowledge was precious. Until recently, counting was done manually; it was time-consuming, inaccurate and prohibitively expensive. Today, laser scanners count with seeming ease, and multi-access computer systems give all of the players in the "supply chain" instant access to information about the flow of material. The resulting increases in productivity and decreases in cost have been monumental. With Web-based distribution systems, parts of the supply chain are becoming virtual, which presents the opportunity for another leap in efficiency.
Making the Switch
Problem-based teaching with spreadsheets does require an investment, but it isn't onerous. Several books take this approach, with different emphases. Find a book that serves the needs of your students. Do not fear Excel — you will find it easy to learn. But if you are over a certain age, do try to team up with a young person who is facile with personal computers; he or she will prove to be a blessing in unforeseen ways.
There is a natural synergy between spreadsheets and decision science. A spreadsheet is a decision-making tool. Ours is a decision-making methodology. Each makes the other more potent, more useful. A spreadsheet fosters interactive learning; it helps students tackle challenging problems.
You will find that problem-based teaching with spreadsheets enables you to cover more material, to probe topics in greater depth, and to appeal to a broader range of students. Your courses will become markedly more effective and more popular.
Eric V. Denardo (eric.denardo@yale.edu) is a professor of operations research at Yale University.
OR/MS Today copyright © 2001 by the Institute for Operations Research and the Management Sciences. All rights reserved.
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