(Adapted from a paper printed in the Society of Industrial Engineers Envoy Vol. 1 No. 3, January/February 1992.)
Two Fractal Models for Operations Research
Timothy D. Greer
Fractals are mathematical objects frequently seen nowadays in situations intended to flaunt computer graphics display capabilities. Their appearance ranges from the bizarre to very natural-looking objects such as clouds and ferns [1]. Because fractals offer a means to define, using relatively few parameters, mathematical objects that resemble natural objects, they have been suggested as models for such natural objects [2]. In certain contexts, such "models" have already proven their value; the obvious example is in creating artificial scenery [3]. But engineers have been trained to think not just of natural objects that can be seen, but also to recognize such things as processes and relationships. Operations Research, usually taught as a part of Industrial Engineering, seeks to characterize such "operations" and thereby show how they can be advantageously manipulated. However, apparently no fractal models have been offered for processes interesting to IEs or operations researchers. What follows are two models intended to help fill this gap. They are necessarily toy models such as one might find in a textbook. Their purpose is to demonstrate the potential viability of fractals as a modeling tool in Industrial Engineering, and to give the reader a couple of examples to aid in conceptualizing other models.
Organizational Structure
Consider the traditional organizational structure. It is usually presented as a tree with a single important dimension -- how high up it are you? Also, although mathematically a tree structure is a partially ordered set, usually an organizational structure is regarded as a fully ordered set: associates rank below senior associates, who rank below staff, and so on. But while the ordering is only one dimensional, it is generally acknowledged that multi-dimensional factors are involved. For example, technically talented people will not advance up the management ladder if they lack political skills. Here is a fractal model of organizational structure intended to illustrate how multiple dimensions can be taken into account. An advantage of this approach is that facets of the interplay of these multiple dimensions may appear, which would not otherwise be noticed.
Figure 1 -- Standard organizational structure
Figure 1 is a fractal made of five contractions mappings. The reader should readily recognize it as a traditional organizational hierarchy. A key feature is that the organizational structure looks the same below each level as below the first level. The vice presidents can view themselves as Chief Executive Officer (CEO) of their own company (except that they have someone to report to), and so on for their lieutenants. Of course, because the figure is of a fractal, the organization has an infinite number of levels.
Suppose now that both vertical and horizontal directions represent attributes of the people in the organization. Let us say, for example, that the vertical axis represents political skill (the CEO is then the most politically astute person in the organization), and the horizontal axis represents technical skill. Then, as one descends through the organization, one encounters subordinates of consistently less political skill, but both greater and lesser technical skill. One might then ask, "Who is the most technically skilled person in the organization?" The answer is the person farthest out on the horizontal axis. With an infinite organization, as shown, there is no "farthest out" person. In real life, the organization terminates at some level; still, even within the fractal model, it is apparent that technical skill approaches some limit.
In addition to changing the meaning of the axes, it is possible to change this model by rotating the fractal. Suppose, for example, that the fractal is tilted 30 degrees or so clockwise. The model is now that the CEO is rather more technically skilled than average and still the most politically skilled. At the second level, one vice president is more technically inclined while the other is more politically inclined, and so on for other levels. By adjusting the rotation, it can be arranged, for example, that one of the third-line "technical side" managers is exactly as technical as the "political side" second-line manager. It would also be appropriate to adjust the lengths of the line segments and the angles at which the organizational lines come together, in order to better fit a desired model.
Figure 2 -- Organizational structure with 5-degree tilt
The fractal can also be changed to be asymmetric, to add more subordinates at each level, or to adjust subordinates' relationships technically and politically to their superiors. Consider Figure 2. Here a 5-degree tilt is introduced, with the interesting consequences that the lower-level organizations begin to overlap, and "technical skill" reaches a maximum only a few levels down the management chain. (Note that the tilt is at every level. The tilt angle relative to the preceding level decreases with increasing depth because as this particular fractal is defined, for each mapping the horizontal scale contracts more than the vertical scale.) But an organization with only a 5-degree tilt still resembles the "standard" organization of Figure 1. Suppose the tilt is changed to 45 degrees, as in Figure 3. Here, a very strange-looking mix of skills appears within the organization, perhaps a consequence of management at each level being out of touch with their underlings. The tilts introduced with Figures 2 and 3 could have been in the other direction, of course, and combined with a rotation of the entire figure, so that modeling of many variations of management/subordinate skill relationships would be straightforward. Additional dimensions may also be introduced in order to model interactions of more than two skill types.
Figure 3 -- Organizational structure with 45-degree tilt
Market Share
Computer hardware maker XYZ makes both Personal Computers (PCs) and intelligent terminals. From time to time, XYZ comes out with new models of each product, affecting its market share of both markets. The markets for PCs and intelligent terminals are interrelated; PCs are frequently used as terminal emulators, and intelligent terminals often conveniently embody functions that are not conveniently available on the PC used as an emulator. So sometimes a new PC product will lure customers who would have otherwise bought intelligent terminals, and sometimes customers would buy a new model of intelligent terminal rather than an older model PC.
The market share of XYZ in these two markets at a given time can be represented as a point in 2-dimensional space; let the X coordinate represent XYZ's market share of PCs, and the Y coordinate represent XYZ's market share of intelligent terminals. When a new product of either type is introduced, XYZ's market share should change. The introduction of a new product will then be a function mapping 2-D space to itself -- introducing a new product changes the XYZ's point in the coordinate system. It is reasonable to expect that the new market shares will depend partly on current market shares and partly on the new product.
Introduction of any new product should be modeled as a bounded mapping, because market share cannot grow without bound. It is also reasonable to require it to be a contraction mapping, meaning that if XYZ continually introduced nothing but one type of product, its market share would tend to go to some fixed value. This is an approximation that needs more justification and refinement. However, to keep the mathematics straightforward (i.e. linear), the contraction mapping assumption is convenient. In this model a new product introduction's affect on market share describes that product introduction.
Affine transformations (matrix multiplication followed by addition of a vector constant) can be used for our contraction mappings. In two dimensions, this looks like
(Xnew) [ a b ](Xold) ( e ) ( ) = [ ]( ) + ( ) (Ynew) [ c d ](Yold) ( f )
If XYZ only comes out with products of type x, the market share for products of type y should go to zero, and vice versa. To satisfy this, the matrix entries c and f must be zero for mappings representing new releases of product x, and the entries b and e must be zero for new releases of product y. Other constraints are required to insure that the mappings are contractive and that market share for each product is always between zero and one. Figure 4 shows a fractal with five mappings satisfying these constraints, plus one additional mapping representing an advertising campaign. The plotted points represent feasible market shares under these constraints. Each point is associated with a particular sequence of product introductions and advertising campaigns.
Figure 4 -- Market Share of PCs vs. Smart Terminals
One way a planner might use a real model such as this is to look at current market shares and select the best available product introduction to move toward desired market shares. "Today we have a 50% share of the terminal market but only a 10% share of the PC market. We want to double our PC market share, but not drop below 25% share of the terminal market. Can we do it? How?" Sequential product introductions might also be considered. Additional dimensions would probably be added to a real model to allow for other types of products, simultaneous introductions, etc.
References
- Barnsley, Michael F. Fractals Everywhere. Academic Press, Boston, 1988, pp. 103, 379.
- Mandelbrot, Benoit B. The Fractal Geometry of Nature. W. H. Freeman and Co., New York, 1983, pp. 1-5.
- Sørensen, Peter, and Pfitzer, Gary. "Algorithmic Advancements", Computer Graphics World, June 1991 (Vol. 14, No. 6), pp. 42-48.
Tables
The following tables give the iterated function system coefficients used to generate the figure of the corresponding number. The coefficients a-f for each mapping are defined as discussed in the "Market Share" section.
Mapping Description | a | b | c | d | e | f |
---|---|---|---|---|---|---|
Left Vice-President | 0.00000 | 0.25000 | 0.00000 | 0.05000 | 0.2500 | 0.8500 |
Right Vice-president | 0.00000 | -0.25000 | 0.00000 | 0.05000 | 0.7500 | 0.8500 |
Left side of Organization | 0.50000 | 0.00000 | 0.00000 | 0.85000 | 0.0000 | 0.0000 |
Right side of Organization | 0.50000 | 0.00000 | 0.00000 | 0.85000 | 0.5000 | 0.0000 |
CEO | 0.00000 | 0.00000 | 0.00000 | 0.10000 | 0.5000 | 0.9000 |
The next table, Table 2, has been scaled to fit in a 320x240 window centered in a 640x480 screen. The scaling method is described in another article.
Mapping Description | a | b | c | d | e | f |
---|---|---|---|---|---|---|
CEO | 0.000000 | 0.008749 | 0.000000 | 0.100000 | 329.9827 | 107.9999 |
Right Vice-president | 0.00000 | -0.25000 | 0.00000 | 0.05000 | 363.002 | 136.508 |
Left Vice-president | 0.00000 | 0.25000 | 0.00000 | 0.05000 | 303.002 | 136.508 |
Left side of Organization | 0.499339 | -.043687 | 0.025698 | 0.848877 | 116.6777 | 43.38929 |
Right side of Organization | 0.499339 | -.043687 | 0.025698 | 0.848877 | 229.2157 | 43.38929 |
Mapping Description | a | b | c | d | e | f |
---|---|---|---|---|---|---|
Left side of Organization | 0.430967 | 0.430967 | -0.253510 | 0.732644 | -0.353354 | 0.218760 |
Right side of Organization | 0.430967 | 0.430967 | -0.253510 | 0.732644 | 0.146646 | 0.218760 |
Left Vice-President | 0.0 | 0.345938 | 0.00000 | 0.069188 | 0.1625 | 0.8325 |
Right Vice-president | 0.00000 | -0.345938 | 0.00000 | 0.069188 | 0.8375 | 0.8325 |
CEO | 0 | -0.135 | 0 | .135 | .535 | .865 |
Mapping Description | a | b | c | d | e | f |
---|---|---|---|---|---|---|
Convert product y users to product x | 0.4 | 0.5 | 0 | 0.5 | 0.1 | 0 |
Get more product x users | 0.7 | -0.1 | 0 | 0.8 | 0.25 | 0 |
New x product bombs | 0.5 | 0.1 | 0 | 0.6 | 0.1 | 0 |
Convert product x users to product y | 0.6 | 0 | 0.6 | 0.3 | 0 | 0.1 |
Get more product y users | 0.8 | 0 | -0.1 | 0.7 | 0 | 0.25 |
Advertising campaign | 0.76 | 0.095 | 0.095 | 0.76 | 0.1 | 0.1 |
Although not referred to in the text above, I have included the following table as yet another demonstration.
Mapping Description | a | b | c | d | e | f |
---|---|---|---|---|---|---|
Left side of Organization | 0.497332 | 0.087693 | -0.051584 | 0.845464 | -0.077590 | 0.029418 |
Right side of Organization | 0.497332 | 0.087693 | -0.051584 | 0.845464 | 0.422410 | 0.029418 |
Left Vice-President | 0.00000 | 0.234375 | 0.00000 | 0.046875 | 0.265625 | 0.853125 |
Right Vice-president | 0.00000 | -0.234375 | 0.00000 | 0.046875 | 0.734375 | 0.853125 |
CEO | 0 | -0.016531 | 0 | .09375 | .5 | .90625 |
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