Fuzzy Logic with Engineering Applications, 4th Edition

4Properties of Membership Functions, Fuzzification, and Defuzzification

“Let’s consider your age, to begin with—how old are you?” “I’m seven and a half, exactly.” “You needn’t say ‘exactually,’” the Queen remarked; “I can believe it without that. Now I’ll give you something to believe. I’m just one hundred and one, five months, and a day.” “I can’t believe that!” said Alice.

“Can’t you?” the Queen said in a pitying tone. “Try again; draw a long breath, and shut your eyes.” Alice laughed. “There’s no use trying,” she said; “one can’t believe impossible things.”

Lewis Carroll, Through the Looking Glass, 1871

It is one thing to compute, to reason, and to model with fuzzy information; it is another to apply the fuzzy results to the world around us. Despite the fact that the bulk of the information we assimilate every day is fuzzy, such as the age of people in the Lewis Carroll quotation, most of the actions or decisions implemented by humans or machines are crisp or binary. The decisions we make that require an action are binary, the hardware we use is binary, and certainly the computers we use are based on binary digital instructions. For example, in making a decision about developing a new engineering product, the eventual decision is to go forward with development or not; the fuzzy choice to “maybe go forward” might be acceptable in planning stages, but eventually funds are released for development or they are not. In giving instructions to an aircraft autopilot, it is not ...