We’ve come a long way from the days when Henry Ford famously said “Any customer can have a car painted any colour that he wants - so long as it is black.” These days, consumers expect to get a car in the colour they want, when they want it. But how can auto manufacturers set up their supply chain to ensure they are providing the right mix of car colours to the dealerships at the right time, rather than all yellow cars one month and all red the next?

For that matter, how can any industry that delivers customized products deliver the right products in proportion to the demand?

A recently published book by Dr. Wieslaw Kubiak, Faculty of Business professor and university research professor, Proportional Optimization and Fairness, hopes to provide an answer to those questions by exploring a new model of proportional optimization and fairness that can be applied across a variety of industries and issues.

Proportional optimization is a concern in many sectors and industry other than business, and in many instances the fairness of the solution must be considered. For example, in the United States electoral system, seats in the House of Representatives are allocated in proportion to the population of a given state. Dr. Kubiak cites the Alabama paradox as one case of when the mathematical rules for proportional optimization led to a surprising, and unfair, allocation.

In the 19^th century United States, the House of Representatives had 400 seats that were allocated proportionally according to certain mathematical rules. The House decided to increase the number of seats to 401, with the expectation that no state loses a seat but one state gets one more. Unfortunately, the formula they were using at the time created an anomaly and allocated one less seat to the state of Alabama. 

“People realized that certain rules that lead to certain ‘optimal solutions’ cannot be used in practice because they create the kind of anomaly that people would not agree to,” explained Dr. Kubiak. “Therefore, rules need to be set up in such a way to avoid these paradoxes. This means you have to look at the problem from two different angles, what is the optimal solution and what is fair.”

Dr. Kubiak’s work builds on previous research in proportional optimization and fairness, especially in social sciences, and offers a fresh approach to the mathematical foundation. “I created a model that allows a subject to be looked at from the same theoretical angle no matter what area they come from,” he said. 

He hopes this new approach to proportional optimization and fairness will create new avenues of study for researchers. “There are a number of issues and problems left open in the book and I hope it will generate more interest in the topic that will lead to more research and study into proportional optimization and fairness in supply chain management.”

Proportional Optimization and Fairness was published by Springer in January 2009.