It All Adds Up
Mathematical models can predict system performance

Managers must utilize various methods to predict future results, given a course of actions. Management scientist have focused on this need over the years by developing various mathematical models the help practitioners predict, with a high degree of reliability, outcomes of various alternative decision variables. Again, the exact outcome is sometime in the future, but the manager must make present predictions about that future state. These models help predict, with a high degree of accuracy, what the future outcome is likely to be. Among the useful models often utilized are Linear Programming, Network Analysis, Probability Theory and Queuing Theory

Linear programming
Linear Programming (LP) was developed during the 1940s and has been applied to a wide variety of industrial and nonprofit activities since about 1950. Given all the other mathematical models, LP has perhaps been the most valuable to management. This leads one to ask, “What is the nature of this remarkable tool and how may we summarize its characteristics?”

Simply stated, this tool helps managers make decisions about the allocations of limited resources among competing activities.

Let us suppose, for example, that ABC Company is a producer of high quality windows and glass doors for the residential construction market. ABC has three plants. Aluminum frames and hardware are made in Plant 1, wood frames are made in Plant 2, and Plant 3 is utilized to manufacture glass and assemble the final products.

ABC currently is experiencing a decline in sales of several unprofitable products. This decline has freed up excess capacity in each of its three plants. Plant 1 has five percent excess capacity available, Plant 2 has 15 percent capacity available, and Plant 3 has 20 percent capacity available. ABC’s management team has the option of utilizing this excess capacity to manufacture two new products recently developed by the research and development department.

However, because both products would be competing for the excess capacity in Plant 3, it is not exactly clear to management which product or combination of products should be manufactured in order to maximize profits. The marketing department has stated that it could sell as many of either of the products that it could produce. It is estimated that Product X will contribute $5 per unit to profit and Product Y would contribute $7 per unit to profits.

The management team decides that this is a problem that should be given to their Operations Research Department. The O. R. Department recognizes this problem as one of linear programming. The Department decides the key to the choices is to first express the problem as an objective function:

Z = 5X + 7Y with Z = Profits.

This means that there would be some combination of the two products that would maximize profits. It is beyond the scope of this article to further develop this problem. However, it does illustrate the usefulness of LP mathematical models in helping managers allocate limited resources among competing activities.

Network analysis
A network is equivalent to that of a city street system, a state highway system, or a computer logic chip circuit system. The common characteristic of all these various entities is a flow from a start point to a destination point. A common problem in network analysis is to find the shortest route through a network to maximize some flow.

A manufacturing machine shop may have its grinding machines in one department, its lathes in another department, and its broaching machines in another department. Manufacturing engineers develop routings that specify what paths various products will take through the machine shop. This, in effect, is an attempt to develop the optimum route or flow of materials through the machine shop. Knowledge of network analysis can help manufacturing engineers determine the shortest routes.

Probability theory
When we make decisions under degrees of uncertainty, one has to rely on the theory of probability. Uncertainty is caused by variation or inconsistency of natural phenomena. These variations have the possibility of being managed, depending of course on what one wants to learn about the population, by utilizing probability theory.

Suppose, for example, the demand for a product is projected, say over a six months period, to be 10,000 units. One could take samples from the buying population to predict the validity of this projection. A manufacturing example might be the reliability of the assembly process that is utilized to manufacture these 10,000 units.

Again, it would be appropriate to take a series of samples to measure the accuracy of the various manufacturing operations as a means of predicting the overall manufacturing process reliability.

Queuing theory
Every Christmas season we run into a number of waiting lines. These range from traffic flow patterns around a shopping mall to customers waiting to receive service in a department store.

Queuing theory involves the mathematical characteristics of queues, or what we generally know to be waiting lines. Queues happen because the demand for a service is exceeded by the capacity to provide the service. Decisions must be made regarding the amount of service one wants to provide.

We find a queue of inventory at almost every operation in a manufacturing process. This inventory serves as a buffer stock to prevent running out or shutting down the assembly process.

A game plan would be to develop a method, utilizing queuing theory, to assure with a high degree of reliability that we do not run out of stock. Yet we would like the ideal state where only one item of stock is actually waiting in a queue.

Back to Industry Index

Back to February Issue

Copyright 1996-2001, by Kentucky Business Online, LLC.  All rights reserved.

Editorial content is copyright 2001, Lane Communications Group
All editorial materials is fully protected and must not be reproduced in any manner without prior permission. 

Buzzword and the Buzzword balloon are registered trademarks of Buzzword, Inc.  The Lane Report is a trademark of Lane Communications Group.  All other trademarks are the property of their respective owners.