**WHITE PAPER #4**

__Fuzzy Logic Technology__

**by Suzanne M. Rodriguez, Ph.D.**

**What is Fuzzy Logic?**

Fuzzy logic is an alternative to traditional Boolean (true/false) logic. Boolean logic requires all statements to be either completely true or completely false. Fuzzy logic allows statements to assume degrees of truth ranging from completely false to completely true and including all intermediate degrees. By providing a framework for reasoning with imprecise information, fuzzy logic technology enables computer systems to reach specific conclusions in accordance with general guidelines.

Many business decisions, such as determining assortment levels, setting prices or evaluating the success of a marketing program, are guided by broad strategies rather than precise rules. For example, a rule of thumb might dictate that "if a product is very popular then the assortment should be extensive." Without fuzzy logic, this rule of thumb is too imprecise to be utilized directly by a computerized system to make assortment recommendations. A fuzzy system, however, would be able to use the rule as stated to recommend stocking all five colors of a top-selling pump shoe.

**What's wrong with precision**?

Continuing with the assortment example, Boolean logic requires that every product be classified absolutely as either "very popular" or "not very popular" in order to determine whether to recommend an extensive assortment. For example, we might declare that a monthly unit sales value of exactly 900 units defines "very popular." Therefore, a product with monthly sales of 901 units is "very popular" while a product with monthly sales of 899 units is "not very popular." Because different decision-making rules will be applied based on product popularity, there may be an abrupt change in the recommended assortment even though there is only a 2 unit difference in monthly sales. In addition, moving the cutoff value by only 2 units in either direction has a significant impact on the resulting assortment decisions. At this point we shrug and move on, suspecting that our precise cutoff is a bit too precise.

We add a second rule of thumb: "if the product margin is low, the assortment should be limited." The problem is that we now have conflicting directives for products which are both popular and low margin. In the world of Boolean logic, an assortment can never be both extensive and limited at the same time. There is no compromise. To work around the problem, we make the existing rules more specific and add an explicit compromise such as "if the product margin is low and the product is very popular then the assortment should be moderate." Over time we create an unwieldy set of cutoff points and highly specialized rules. There is, however, an alternative -- fuzzy logic.

**How do Fuzzy Systems work**?

Fuzzy logic permits a more natural representation of concepts by acknowledging gradual contrasts. Instead of being represented by a single value in the form of a cutoff point, the concept of "very popular" is represented by a fuzzy set. We can construct a fuzzy set such as the following:

Products with monthly sales <= 700 units are absolutely not (0%) "very popular"

Products with monthly sales > 1000 units are absolutely (100%) "very popular"

All other products are "very popular" to the following extent:

((monthly unit sales - 700) / (1000 - 700)) x 100%

A product with monthly unit sales of 901 units is 67% "very popular." A product with monthly unit sales of 899 units is only slightly less "very popular" at 66%.

We still choose some cutoff points at the edges in order to define a fuzzy set, but we have solved the problem of abrupt transitions from absolutely not "very popular" to absolutely "very popular" based on small differences in the monthly sales figure. In addition, the choice of cutoff points is not as critical because relatively large differences have only minor effects on the popularity rating instead of causing a complete about face. For example, if we increase the requirement for absolutely "very popular" to 1100 units, a product with monthly unit sales of 901 units is still 50% "very popular."

By defining additional fuzzy sets we can also reason about "low" margins, "extensive" assortments and "limited" assortments. Any assortment over 3 colors may be considered "extensive," although 5 colors is obviously more "extensive" than 4 colors. The extent to which a product matches the popularity requirement determines just how "extensive" the selection should be. If a product is 100% "very popular," then the assortment should be 100% "extensive" (perhaps 6 colors). If a product is only 50% "very popular," then the assortment should be only 50% "extensive" (5 colors). Once the fuzzy sets are defined, the system can produce many different recommendations all based on the same general policy.

In a fuzzy system, all rules with some degree of truth contribute to the final solution. Through a process called defuzzification, the system produces a single answer that represents a compromise between all of the suggested conclusions. Rules that are 90% true influence the final solution more than rules that are only 50% true. In the case of a popular but low margin product, the popularity rule may suggest that 5 colors would be appropriate while the margin rule suggests that 1 color would be appropriate. Both of these suggested conclusions are combined to recommend, for example, that 3 colors should be stocked. Conflict resolution occurs automatically without an increase in the complexity of the knowledge base.

This very brief introduction illustrates some of the basic differences between fuzzy logic and Boolean logic. The bottom line (literally) is that fuzzy logic provides a more natural and compact problem representation in many situations characterized by inherent imprecision and conflicting points of view.

© 2015 Churchill Systems Inc.