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Churchill Systems Inc WP#4.pdf Suite 106, 3155 W. Big Beaver Rd.
[email protected], MI 48084(248) 649-1800 Copyright
Churchill Software Inc. 1998-2010. All Rights Reserved.
White Paper #4
Fuzzy Logic Technology by
Suzanne M. Rodriguez, PhD
What is Fuzzy Logic?
Fuzzy logic is an alternative to traditional Boolean
(true/false) logic. Boolean logic requires all statementsto 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.
Byproviding 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 successof 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." Withoutfuzzy logic, this
rule of thumb is too imprecise to be utilized directly by a
computerized system to makeassortment recommendations. A fuzzy
system, however, would be able to use the rule as stated
torecommend 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 classifiedabsolutely as either "very popular"
or "not very popular" in order to determine whether to recommend
anextensive assortment. For example, we might declare that a
monthly unit sales value of exactly 900 unitsdefines "very
popular." Therefore, a product with monthly sales of 901 units is
"very popular" while aproduct with monthly sales of 899 units is
"not very popular." Because different decision-making rules willbe
applied based on product popularity, there may be an abrupt change
in the recommended assortmenteven though there is only a 2 unit
difference in monthly sales. In addition, moving the cutoff value
by only2 units in either direction has a significant impact on the
resulting assortment decisions. At this point weshrug 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." Theproblem is that we now have
conflicting directives for products which are both popular and low
margin. Inthe 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 addan explicit compromise such as "if the product
margin is low and the product is very popular then theassortment
should be moderate." Over time we create an unwieldy set of cutoff
points and highlyspecialized rules. There is, however, an
alternative -- fuzzy logic.
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Churchill Systems Inc WP#4.pdf Suite 106, 3155 W. Big Beaver Rd.
[email protected], MI 48084(248) 649-1800 Copyright
Churchill Software Inc. 1998-2010. All Rights Reserved.
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 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 theproblem of abrupt
transitions from absolutely not "very popular" to absolutely "very
popular" based onsmall differences in the monthly sales figure. In
addition, the choice of cutoff points is not as criticalbecause
relatively large differences have only minor effects on the
popularity rating instead of causing acomplete about face. For
example, if we increase the requirement for absolutely "very
popular" to 1100units, 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 isobviously more "extensive" than 4 colors. The extent to
which a product matches the popularityrequirement determines just
how "extensive" the selection should be. If a product is 100%
"verypopular," 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 aredefined, 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 processcalled
de-fuzzification, 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 areonly 50% true. In the case of a popular but low
margin product, the popularity rule may suggest that 5colors 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 Booleanlogic. The bottom line
(literally) is that fuzzy logic provides a more natural and compact
problemrepresentation in many situations characterized by inherent
imprecision and conflicting points of view.
