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LOGIC

ANDFUZZY SYSTEM

LESSON 6:FUZZY NUMBERS

In this Chapter we will take a good long look at fuzzy logic,and a brief look at fuzzy sets and fuzzy numbers. (Laterchapters will look more deeply into fuzzy sets and fuzzynumbers.)1.3 Fuzzy Numbers.

The final major member of our fuzzy systems arsenal is thefuzzy number . A fuzzy number is simply an ordinary numberwhose precise value is somewhat uncertain. Here, for example,is a fuzzy two.

A fuzzy subset of the universe of a numerical number. Forinstance fuzzy real number is a fuzzy subset of the domain of real numbers. A fuzzy integer is a subset of domain of integers. An example of fuzzy integer, about-20 is as shown inthe figure.

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Fig : A fuzzy number

Here our confidence that numbers 1.5 or less belong to thefuzzy 2 is zero, as is our confidence that numbers 2.5 or greaterbelong to the fuzzy 2. Our confidence that 1.6 belongs to thefuzzy 2 is 200; for 1.7, 400; and so on to confidence 1000 for 2.For numbers greater than 2, the confidence declines as the

number increases, being 800 for 2.1, 600 for 2.2, and so on tozero confidence for 2.5.A very convenient way to describe fuzzy numbers is to usemodifying words. For example, the fuzzy two shown in Figure1 could be completely specified by roughly 2. Othermodifying words available are nearly, about and crudely,with progressively larger uncertainties. These words are calledhedges in fuzzy math circles.With fuzzy numbers, we can make approximate comparisons.It is quite possible, for example, to ask if an input persons age

is approximately equal to about 30. This is often very usefulwhen our data or imprecise, or when we dont want the rigidityof accepting a person 30 years old but rejecting one thirty yearsplus one day old.A very convenient way to describe fuzzy numbers is to usemodifying words. For example, the fuzzy two shown in Figure2 could be completely specified by roughly 2. Othermodifying words available are nearly, about and crudely,with progressively larger uncertainties. These words are calledhedges in fuzzy math circles.With fuzzy numbers, we can make approximate comparisons.It is quite possible, for example, to ask if an input persons ageis approximately equal to about 30. This is often very usefulwhen our data or imprecise, or when we dont want the rigidityof accepting a person 30 years old but rejecting one thirty years

plus one day old.

Operations on Fuzzy Numbers: Addition andSubtraction

Illustration of Arithmetic Operations on Fuzzy numbers

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Operations on Fuzzy Numbers: Multiplication andDivision

2.6.1 Introduction This lesson deals with discrete fuzzy arithmetic operations.Well be using Fuzzy L ogic , along with standard Mathemati ca functions.

This loads the package.In[1]:=

2.6.2 Creating Fuzzy NumbersWhen working with fuzzy numbers and performing fuzzyarithmetic, we should use a large universal space because theintervals over which fuzzy numbers are defined widen asarithmetic operations are performed. Also, like traditionalnumbers, fuzzy numbers can be negative or positive, so theuniversal space should be symmetric around zero. Taking thisinto consideration, we investigate fuzzy arithmetic with auniversal space from -100 to 100. We change the defaultuniversal space to this range with the following command.

In[2]:= The following function is used to create fuzzy numbers. Thefunction accepts an integer as its argument and returns atriangular fuzzy set centered around the integer. This is acommon representation of a fuzzy number.In[3]:= We can create a set of fuzzy numbers by using the function we

just defined and Mathemati ca s Table function. Here we createfuzzy numbers ranging from NegSIX (Negative Six) to SIX.In[4]:=

2.6.3 Fuzzy Arithmetic

Shortening Arithmetic Names The Fuzzy L ogic package comes with a number of convenientfunctions for performing discrete arithmetic, but the names of these functions are quite long. To save some typing, we renamethe functions here to shorten their names.

In[5]:=

Using Infix Form

When performing arithmetic operations, it is traditional to writeout the operation from left to right, with the sign for theoperation to be performed located between the two numberson which it is to be performed. To simulate the traditionalform, we use Mathemati ca s infix form in the followingexamples. Instead of surrounding the operands with squarebrackets and the function name (e.g., Func[ A , B ]), the infix formlocates the function name between the operands and surroundsthe operand with tildes (e.g., A ~Func~ B ). Lets look at thefollowing example to see how this works. In this example, weperform fuzzy addition on the fuzzy numbers NegFOUR and

THREE.In[9]:=

Out[9]=

After defuzzifying the result with theMeanOfMax method, wesee that the result is a fuzzy number centered around -1. This isthe answer we would expect to get when performing theoperation -4 + 3.In[10]:=

Example 1 The following example demonstrates the fuzzy equivalent of the operation: (5 - (-1)) * (-3)In[11]:=By using the MeanOfMax defuzzification, we receive -18, whichwould be the solution to the equivalent nonvisual operation.In[12]:=

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