THE SPECIALIST WITH A UNIVERSAL MINDANDREW VAZSONYI, Feature Editor, McLaren School of Business, University of San Francisco
EXPERIMENTAL MATHEMATICS AND FUNCTIONSby Andrew Vazsonyi, McLaren School of Business, University of San Francisco In a previous column I discussed the task of how to teach math for students of decision sciences and focussed on functions. Now I extend the discussion to piece-wise linear functions. These functions are defined by a set of pairs of points, the anchors. Each function consists of the straight line segments connecting the neighboring points. Immediate applications are prices with discounts and tax calculations. But these functions are important as approximations, and substitutes to nonlinear functions such as the cumulative probability distribution function, CDS, and nonlinear utility functions. Exhibit 1 shows an empirical cumulative distribution function of sales which was obtained by statistical data and/or managerial judgment. Five pair of points, the first anchors to approximate the curve, are also shown. The exhibit is a floating (embedded) graph and the anchors are to be ``dragged'' around by the mouse to give a good approximation (you can do this in EXCEL). After a few trials Exhibit 2 is generated, a piece-wise linear function, giving a close approximation to the CDS. To do a Monte Carlo simulation requires the inverse of the CDS. Exhibit 3 shows the inverse function which illustrates that the values of the random variables are a function of random numbers generated between 0 and 1. The chart is obtained either by the rotate style of the graph or by using an XY plot, x being probability and y sales. The spreadsheet automatically connects the anchors in the diagram; but to do the actual Monte Carlo analysis requires the calculation of the values. This is done by calculating the piece-wise linear function by the @VLOOKUP or @HLOOKUP functions. I tried all sorts of popular functionþnormal, binomial, exponential, beta, and was surprising that five points give excellent approximations to all. Exhibit 4 shows the normal distribution. This is a symmetrical distribution, so I used only three points. Spreadsheet packages like Microsoft Excel and @RISK provide the inverse functions for a number of CPDs. You may want to use these, if you have them, and the floating method of adjusting parameters makes it easy to find the distribution that satisfies managerial judgment. Conclusions Traditionally a variety of nonlinear functions are used, and curve fitting becomes a problem. For example, a popular suggestion is to use such functions as the exponential, logarithmic, or logistic curves as approximations. All these approaches pose the problem of curve fitting. The new world of spreadsheets allows visual curve fitting, leading to simplification and avoidance of complicated mathematics. Piece-wise linear functions are prime candidates for approximating and replacing traditional nonlinear functions. Using piece-wise linear functions is a natural process and requires no sophisticated mathematics. For example, teaching Monte Carlo simulation, and multiattribute utility theory, is greatly simplified. Write, call, or fax to my home: |