THE SPECIALIST WITH A UNIVERSAL MIND

ANDREW VAZSONYI, Feature Editor, McLaren School of Business, University of San Francisco

Are Mathematical Models Obsolete for the Business School Curriculum?

by Andrew Vazsonyi, McLaren School of Business, University of San Francisco

Why do we teach mathematical models? Sounds like a real stupid question. Of course we do. Obvious. What else?

I tried to find an answer and the only reference I could find is in Hillier and Lieberman. Apparently there are two reasons:

They are better than verbal descriptions.

They provide a bridge to the computer.

I realize now, after 40 years of hard trying, that there is a fatal flaw in the argument, so far as teaching is concerned. Better for whom? Sure enough, it is better for Hillier, Lieberman, Vazsonyi and the entire OR/MS gang. But what about the unfortunate student? The story of the saber tooth fish is relevant.

The Saber Tooth Fish Curriculum

Once upon a time there was a tribe living along a brook that was full of saber tooth fish. It was a clear mountain brook and it provided the main food for the tribe. However, the brook was swift, and it was quite a trick to catch the fish. So there was an elaborate school curriculum to teach catching the fish.

One day there was an earthquake, the ground moved and the brook became muddy. It was impossible to catch the fish and an alternate method of fishing with nets was developed. However, the curriculum continued to teach catching fish by hand. There was no practical benefit, but it was claimed that the curriculum helped to develop a logical, clear mind, and it was þgood for the soul.þ

Far fetched?

To get more realistic, consider the teaching of Latin. During the middle ages, Latin flourished as the language of universities, and writers. It was impossible to be a scholar without Latin. For instance, the official language of Hungary, until the middle of the 19th century, was Latin. But long after Latin became useless, the teaching of Latin continued. It was "good for the soul."

Calculus is an enormously useful subject in the physical sciences, but seems to have not much use in the social sciences. Still we are teaching it to a large number of innocent victims. It is "good for the soul."

Mathematics or Numeracy?

There is no doubt that verbal models have serious flaws, and in the mid-50's there was no choice but words or math. Fundamentally there are two reasons for the success of math: it provides an insight, clarity to problems beyond words, and enables us to do calculations. One of the reasons that algebra was discovered was to make calculations easier. Much of applied math is for calculations.

John Paulus' book Innumeracy was a best seller in the late 80's, advocating that people work with numbers. He stressed the need to calculate and the insight and clarity given by numbers. His point was well taken: in today's culture calculations are considered a nuisance, most people are uncomfortable with numbers, and the calculating person is despised. I see no evidence that Paulus delivered an impact and changed people's attitude. But the computer revolution did cause a change in business.

Today there are millions of people using spreadsheets and making calculations: this is the emergence of the numerate manager. Can spreadsheet models serve as models of business problems? I am making a review of the management science textbooks and finding some encouraging answers. Let me illustrate.

The Transportation Problem

The traditional mathematical model is a real bear. The use of doubly-subscripted notation with the Greek sigma is not salable to the majority of students or managers. But the spreadsheet model is straightforward, insightful and leads directly to minimum cost solutions.

We have two rectangular ranges of matching size: one has the cells to change, the second the transportation costs. The SUMPRODUCT function with one swoop replaces the subscript and doubly-subscripted notation. Calculations of the slacks are direct. To solve the problem requires the trivial task of filling in a form on the screen.

But I go further. One serious objection to linear programming is that it provides solutions widely different from the initial, base-case solution. Managers would like to start with a base solution and improve it. But the simplex method does nothing of the sort. It starts with the North-West Rule. So, I introduce the base-case, adjust it, and use it as an artificial constraint to the problem. The manager can step-by-step relax the artificial constraints and find better and better solutions.

Conclusions

I am convinced that there is no need to teach the algebraic formulation of the transportation problem, and I am finding that this holds for much of the material in the standard texts. Looks to me that the primary model is the spreadsheet, and math serves only as a last resource. Unfortunately, I have no empirical support for my theory.

There are some textbooks on the market that use spreadsheets as supporting material. Also Wayne Winston of Indiana University reports on a highly successful spreadsheet-based management science course for MBA's. I am in the process of developing a great deal of material along these lines, and will be happy to collaborate with you in further developing the concept.