• Humans learn to read and write.
• Humans invent machines (culminating in the industrial age).
• Machines learn to read and write (dawn of the information age).

I don't know much about how we learned to read and write, but the industrial age was characterized by harnessing the power of physics. This required chains of technologies, starting at the top with either mathematics or physics (depending on whether you ask a mathematician or physicist) and then moving on through engineering to the often ignored field of industrial design. To allow us to grasp the power of physics with our own hands, an industrial designer first had to develop an appropriate handle. Some famous examples include the steering wheel, the electric light switch and the typewriter keyboard.

Now at the dawn of the information age, the power we are harnessing is not physical and you can't grasp it with your hand. You grasp it with your mind. Thus the role of the informational designer is not to develop a handle but a mindle. Operations research and management science are the quintessential "information age" fields, and are in dire need of better mindles as a necessary step toward WMI.

Connecting the Seat of the Intellect to the Seat of the Pants


A homeless man once had a baseball cap that read: "I'm not completely worthless. At least I can serve as a bad example." In this spirit, I present an example of a bad mindle for something you already understand:

These are the equations of motion of a bicycle, and you solve them every time you ride; not, however, through the seat of your intellect, but through the seat of your pants.

Experiential learning in an interactive environment is a powerful form of WMI. Can this be applied to OR/MS? Absolutely.

The Fall of the Algebraic Curtain


When I started teaching management science at the University of Chicago Graduate School of Business in 1974, I was disappointed to find that only 10 percent of my students understood the material, and of those who did, only 10 percent would go on to apply it. And no wonder. The steps of the simplex algorithm probably looked to my students like the bicycle equations. An algebraic curtain separated management from management science. Convinced that the field was moribund, I abandoned management science for 10 years, until 1985, when Linus Schrage and I worked together on the development of What'sBest! to bring linear programming to the spreadsheet.

At the time I described our work in terms of an automotive analogy. If linear programming was an internal combustion engine and the electronic spreadsheet was a wagon, then we were developing an automobile. The user interface, shown in Figure 1, was the dashboard, and contained the minimum mindles required to grasp optimization. Note that the Extension command was used to create vector products in the days before the SUMPRODUCT formula.

What's Best Commands Optimize F1 F2   Adjustable Cell F3 F4 Fixed Cell Maximize F5 F6 Minimize Dual Value F7 F8 Integer Extension F9 F10 Less or Equal < > Greater or Equal Figure 1: The original What'sBest! dashboard.

What'sBest! began to bring down the algebraic curtain, and I was reborn as a management scientist. Instead of teaching algebra, I now presented students with small interactive spreadsheet models containing what I called the developmental necessities of the application or DNA for short [1]. The DNA, usually no more than a handful of cells in a spreadsheet, provides a mindle for an entire class of optimization application. It is a living, breathing solution to a problem that can be replicated, scaled up and recombined with other DNA to create integrated models.

This at last was a weapon of mass instruction. Students and practitioners loved it. But on several occasions I found professors of management science painstakingly building tableaus in the spreadsheet to use with What'sBest! This bypassed the powerful modeling capability of the spreadsheet, and was analogous to loading the engine onto the back of the wagon and then pulling the whole thing along with a horse.

In the words of George Dantzig, the father of linear programming, "The final test for a theory is its capacity to solve the problems which originated it." [2] Internal combustion engines did not become truly successful until drivers didn't need to know they were using one. As a powerful testament to Dantzig's own theory, thousands of people are now performing optimization with no knowledge of the simplex algorithm. In fact, today, between What'sBest! and the Excel Solver, one can solve linear, non-linear and combinatorial problems, and build logical statements into optimization models.

Understanding Uncertainty


In the late 1980s I turned my attention from optimization to other areas of management science. @Risk and Crystal Ball had by that time brought powerful Monte Carlo simulation to the spreadsheet, and I believed this technique had an even wider audience than optimization.

I began to use simulation to teach stochastic modeling to students who had already taken introductory statistics. I tested their intuition on the outcomes of the single spin of a game board spinner, and on the averages of two spins. At first I thought that much of the class didn't understand the central limit theorem. Then I made the shocking discovery that many students didn't even have a mindle on the concept of probability distributions.

Testing intuition with a spinner.

In a given class, between a third to half of my students drew a bell shaped distribution for a single spin. If probability were a bicycle, these students wouldn't know the front from the back, yet they had all passed the equivalent of bicycles 101. In the words of Mark Twain, their schooling had interfered with their education. Presumably scores would be higher among Eskimos on an ice floe, whose only contact with probability theory came from playing Spin the Walrus Tusk.

This work was described in my 1994 article in OR/MS Today [3] and also in "Statistical Analysis for the Masses" [4]. More formal testing in this area has been performed by Armann Ingolfson and David Zalkind, as described in 1999 articles in Interfaces [5, 6].

Using spinners to connect the seat of the intellect to the seat of the pants was a central theme in the seminal 1959 book on portfolio theory by Harry Markowitz [7]. When I first saw this classic a few years ago, I marveled that much of it could be understood by a bright high school student.

Merton Miller, who shared the Nobel Prize in Economics with Markowitz and Sharpe in 1990, believed that Harry had never received the respect he deserved in the academic community because he had written a book that could be understood by the masses. As I will discuss further on, academia is probably not the place to look for WMI.

Interactive Histograms


Where Markowitz encouraged people to think of spinners to improve their intuition into uncertainty, today we can run simulations in Excel.

A particularly gratifying case involved an investment banker client of mine in the entertainment industry. Rick Medress is president of Cineval LLC, a media-consulting firm in Los Angeles that provides valuations of film investments. When he asked my opinion of simulating the outcomes of portfolios of movies, I responded that it would be dereliction of duty if he didn't. So, based on the actual box office results of 28 "live action" films, I helped him create an interactive histogram [8], allowing him to scroll through the distributions of outcomes from a single film, a portfolio of two films, three films and so on. This is depicted in Figure 2 (click on the image to view a larger version in a seperate browser window) and is viewable as an animation online [9].



Figure 2: Distributions of outcomes for films (click here to view a larger, animated version in a seperate browser window).
This figure was produced using the Blitzogram^TM interactive histogram feature of the author's XLSim. Courtesy of Cineval LLC.

Selected frames are shown in Figure 2, with the one in the upper left displaying the historical distribution of revenues of the 28 live action films. Assuming that future movies of this genre are drawn independently from the same distribution, this would represent the distribution of outcomes of investing in a single film. Thanks to the central limit theorem, as one diversifies across additional films, the distribution approaches the normal. As a result, the chance of losing your shirt drops from above 20 percent with a single film to about 1 percent with four films. When I pointed this out to Rick, he said: "It takes some people decades to learn that."

A few weeks after we had developed this simulation, Rick phoned me up. "It works with cartoons!" he reported excitedly. "What works?" I inquired. "When you simulate portfolios based on historical cartoon revenues instead of live action films, the outcomes become bell shaped again when you diversify," he explained.

I grabbed my probability textbook off the shelf and looked it up. Sure enough, Rick was right; the central limit theorem does apply to cartoon revenues as well as live action films! Unlike most of my graduate students, Rick Medress now has an almost carnal knowledge of one the most profound results in mathematics.

The Flaw of Averages


Perhaps even more important than the central limit theorem in today's business world is Jensen's Inequality. With a name like that, no wonder it never received the attention it deserves. So with apologies to Johan Ludwig William Valdemar Jensen, I call it the "Flaw of Averages" [9, 10].

Briefly, it states that a non-linear function of a random variable, evaluated at the average of the random variable, is not the average of the function, or

F(E(X)) ≠ E(F(X)) for short.

"Functions of random variables," you say. "That's an exotic topic in advanced stochastic modeling." No. A spreadsheet model is just a function. And if you are uncertain of the inputs then it is function of random variables. Do the math. There are 40 million spreadsheet users of whom roughly 100 percent are uncertain about their inputs.

In this light, the Flaw of Averages may be stated as follows:

When you plug average values into a spreadsheet, you don't get average outputs unless the model is linear (and most people don't know if their models are linear or not).

Less than 5 percent of my students seem to be aware of this crucial fact.

Some examples,

• The profit associated with average demand is probably not the average profit.
• The rate of loan defaults associated with average economic conditions is probably not the average number of defaults.
• The value of a call option given the average price of the underlying stock is definitely not the average value of the call option.
  

The entire area of developing intuition into probabilistic models has just received a shot in the arm with Daniel Kahneman's Nobel Prize for showing how people (quoting from the academy) "take shortcuts that systematically depart from basic principles of probability." Could this be due to the way we teach the subject? Don't get me wrong, the classical theory of probability and statistics is powerful and elegant. But so is the steam locomotive, and they were developed around the same time. The difference is that although we no longer teach steam locomotion, we continue to teach steam-era statistics. Stanford's Bradley Efron, a pioneer in computational statistics and inventor of the statistical bootstrap, puts it this way: "As far as what we are teaching new students, statistics stopped dead in 1950." [11].

No wonder people don't know what a distribution is.

A Conflict of Interest


One stumbling block on the road to WMI involves educators themselves. As an undergraduate in mathematics I had an insecure professor in an important class. The main thing he had over his students was that he understood the material and they didn't, and by golly he wasn't going to give up his last advantage. In desperation I called my father (also a mathematician) for help, and he was able to explain the salient concepts of the course in a 15-minute phone conversation.

This is an extreme example of a conflict of interest inherent in the teaching profession: by illuminating your subject, you increase the stature of your students; by obfuscating it, you increase your own apparent stature. If educators are worried that WMI could undermine their job security, can you blame them? Yes. They are holding our field back.

Targeting WMI


At the recent INFORMS conference in Phoenix, Gerald Brown of the Naval Postgraduate School questioned whether information technology had made operations research obsolete. He pointed out that the powerful new job title of chief information officer is in a position to spearhead analytical initiatives, but unfortunately these managers are not well enough trained in the OR/MS disciplines to tell good science from bad. A pessimist listening to Brown might wring his hands and look for another profession. An optimist would thank Brown for identifying, in the CIOs of the world, large strategic targets to be bombarded with WMI. But please don't poison the well by teaching them about internal combustion engines. Let's teach them how to drive instead.

References

1. 1. Savage, S. L., "Weighing the Pros and Cons of Decision Technology in Spreadsheets," OR/MS Today, February 1997, Vol. 24, No. 1, pp. 42-45.
2. 2. Dantzig, George B., "Linear Programming and Extensions," Princeton University Press, 1963.
3. 3. Savage, S. L., "Consumer Stochastics," OR/MS Today, December 1994, Voll. 21, No. 6, pp. 39-43.
4. 4. "Statistical Analysis For The Masses," in "Statistics in Public Policy," Bruce Spencer, ed., Oxford University Press, 1998.
5. 5. Ingolfson, Armann, "Obvious Abstractions: The Spinner Experiment," Interfaces, Vol. 29, No. 6, November-December 1999, pp. 112-122.
6. 6. Zalkind, David, "Another Take on the Spinner Experiment," Interfaces, Vol. 29, No. 6, November-December 1999, pp. 122-126.
7. 7. Markowitz, H. M., "Portfolio Selection, Efficient Diversification of Investments," John Wiley, 1959.
8. 8. Savage, S.L. Blitzograms, "Interactive Histograms," Informs Transactions on Education, January 2001, http://ite.pubs.informs.org/Vol1No2/Savage/Savage.php.
9. 9. http://www.stanford.edu/dept/MSandE/faculty/savage/FOA%20Index.htm.
10. 10. Savage, S.L., "The Flaw of Averages," Harvard Business Review, November 2002, pp. 20-21.
11. 11. Efron, Bradley, Journal of the American Statistical Association, December 2000, Vol. 95, No. 452, Vignettes.

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