Demystifying Fuzzy Inference Systems

Fuzzy logic aims to model situations where we need to capture the partial truthfulness of a given proposition. Unlike the two possible truthfulness values (1 or 0) in boolean logic, we may represent the truthfulness value between 0 and 1 (both included). For a basic understanding of fuzzy logic, you may explore the blog Demystifying Fuzzy Logic: A Beginner’s Guide on the Educative platform.

We may model decision support systems using fuzzy logic to represent the underlying inputs, outputs, and the relationships between inputs and outputs.

A fuzzy inference system is built to enable decision support in a particular domain where capturing the underlying logic is not concrete. In fuzzy inference systems, inputs and outputs are modeled as fuzzy variables. The value of each of the variables is modeled as a fuzzy membership function. The rules of the application domain are identified and enumerated. In each fuzzy rule, fuzzy variables and their corresponding fuzzy values are used, depicting the underlying principles of the application domain. The following steps are performed to model such systems:

These steps have been summarized in the following figure:

Let’s look at an example where we’ll build a fuzzy inference system to determine the percentage of the tuition fee to be waived for a university student who has disclosed both their monthly household income and the net worth of family assets in dollars.

There are two input variables involved in this process:

• Monthly household income (income)
• Net worth of family assets (assets)

There is one output variable in this system:

• Financial support to be determined as the percentage of the tuition fee to be waived (support)

For the sake of simplicity, we are categorizing each of the variables into one of three fuzzy values, listed below:

• income can be low, medium, or high.
• assets can be weak, average, or strong.
• support can be low, average, or high.

The fuzzy (linguistic) values of each of the variables have been modeled below:

Now, let’s define the fuzzy rules based on these fuzzy variables.

Based on the identified fuzzy variables and their corresponding values, a list of fuzzy rules is extracted from the application domain, which is supposed to depict the business logic. We will use the following three rules in our example:

• If income is low AND assets are weak, then support is high.
• If income is medium OR assets are average, then support is average.
• If income is high AND assets are strong, then support is low.

The following steps are performed to infer the output from the given inputs:

We’re going to explore how the fuzzy inference algorithm works by using it to calculate a university student’s tuition fee reduction. For this purpose, let’s suppose that the user enters the following inputs:

• income = 3,200 dollars
• assets = 200,000 dollars

The first step is to fuzzify the inputs. Membership values of the inputs are computed in each of the membership functions, as shown below:

The membership value $\mu$ of the input of income in each membership function of its variable has been given below:

• $\mu_{low}(3,200) = 0.80$
• $\mu_{medium}(3,200) = 0.46$
• $\mu_{high}(3,200) = 0$

Similarly, the membership value $\mu$ of the input of assets in each membership function of its variable has been given below:

• $\mu_{weak}(200,000) = 0.99$
• $\mu_{average}(200,000) = 0$
• $\mu_{strong}(200,000) = 0$

We have three fuzzy rules defined in this example. The fuzzy operators are applied on the antecedents of each rule using the membership values computed during the fuzzification process. The outcome of this step on each rule has been given below:

The antecedent of rule-1 is income is low AND assets are weak. It results in the following computation:

$\mu_{low}(income)$ AND $\mu_{weak}(assets) = min(\mu_{low}(income), \mu_{weak}(assets)) = min(0.80, 0.99) = 0.80$

This step results in 0.80 as the strength value of the antecedent of rule-1.

The antecedent of rule-2 is income is medium OR assets are average. It results in the following computation:

$\mu_{medium}(income)$ OR $\mu_{average}(assets) = max(\mu_{medium}(income), \mu_{average}(assets)) = max(0.46, 0) = 0.46$

This step results in 0.46 as the strength value of the antecedent of rule-2.

The antecedent of rule-3 is income is high AND assets are strong. It results in the following computation:

$\mu_{high}(income)$ AND $\mu_{strong}(assets) = min(\mu_{high}(income), \mu_{strong}(assets)) = min(0, 0) = 0$

This step results in 0 as the strength value of the antecedent of rule-3.

The strength values of all rules are applied to the membership functions of the respective consequents. It results in outputting the area of the respective membership function of the output variable. Following are the exact cut points of all of the membership functions of the output variable used in this example:

• Membership function high of support is cut at 0.80.
• Membership function average of support is cut at 0.46.
• Membership function low of support is cut at 0.

The areas under respective cuts of all membership functions are aggregated together, as shown below:

In this figure, the blue lines represent the cut points of the consequents, and the red-shaded area represents the aggregated output space.

Appropriate defuzzification methods need to be chosen and applied to the output space to decode the exact output of the question. There are different defuzzification methods available in the literature, including centroid, bisection, largest of maxima, smallest of maxima, and middle of maxima. Centroid is one of the more popular defuzzification methods. It computes the geometric center of the output space.

The green bubble in the following diagram shows the centroid. It is dropped to the horizontal axis to map it to the actual output shown as the green arrow. The output is 70%.

A fuzzy inference system is one of the most powerful choices to model decision support systems in different application domains. When the possible values of the variables are linguistic concepts, then the overlap between the linguistic values is inherent in the application domain. Fuzzy modeling is an attractive choice to capture such inherent ambiguity in the application domain. There is a wide variety of potential applications where fuzzy inference systems may be applied, including:

• Automatic speed control systems for vehicles
• Altitude controls of spacecraft
• Intelligent decision support systems in business intelligence
• Evaluation systems to grant loans in financial organizations
• Optimization in the manufacturing industry
• Medical diagnostic support systems
• Handwriting recognition systems

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