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7. Decision Making


									       7. Decision Making

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  7.1. Fuzzy Inference System (FIS)
  Fuzzy inference is the process of formulating the mapping from a
  given input to an output using fuzzy logic. Fuzzy inference systems
  have been successfully applied in fields such as automatic control,
  data classification, decision analysis, expert systems, and
  computer vision.

       Input   Fuzzifier                     Defuzzifier   Output

                               Fuzzy           if then Rule base
                           Knowledge base

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  Step1: Fuzzification
  • Fuzzifier converts a crisp input into a fuzzy variable (linguistic
       variable) using the membership functions stored in the fuzzy
       knowledge base.
  • Definition of the membership functions must:
  – reflect the designer's knowledge
  – provide smooth transition between member and nonmembers of
    a fuzzy set
  – simple to calculate

  • Typical shapes of the membership function are Gaussian,
    trapezoidal and triangular.

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       For example assume we want to evaluate the health of a person
       based on his height and weight.
       The input variables are the crisp numbers of the person’s height
       and weight.
       Fuzzification is a process by which the numbers are changed into
       linguistic words

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  Fuzzification of Height:

  Fuzzification of Weight

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  Step2: Infgerence Engine (Rules)
  •    Rules   reflect experts decisions.
  •    Rules   are tabulated as fuzzy words
  •    Rules   can be grouped in subsets
  •    Rules   can be redundant
  •    Rules   can be adjusted to match desired results

  Rules function
  • Rules are tabulated as fuzzy words
  – Healthy (H)
  – Somewhat healthy (SH)
  – Less Healthy (LH)
  – Unhealthy (U)

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  Using If-Then type fuzzy rules converts the fuzzy input to the
  fuzzy output.

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  In our example:
  Rules function
  • Rules are tabulated as fuzzy words
  – Healthy (H)
  – Somewhat healthy (SH)
  – Less Healthy (LH)
  – Unhealthy (U)

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  Step3: Calculate
  • For a given person, compute the membership of his/her
    weight and height
  • Assume that a person height is 6’ 1”, and weight is 140 lb

       Membership of Height

                               Membership of Weight

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  Step4: Activate Rules

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  Step5: compute Decision Function

                 f = {U, LH, SH, H}
              f = {0.3, 0.7, 0.2, 0.2}

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          f = {U, LH, SH, H}
       f = {0.3, 0.7, 0.2, 0.2}

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  Step6: Compute Final Decision (defuzzify)
  Defuzzification converts the fuzzy output of the inference engine
  to a crisp output, among other methods two methods are often
    –Maximum Method (not often used)
    – Centroid of area

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  Maximum Method:
  Fuzzy set with the largest membership value is selected.

  In our example: Fuzzy decision
  • Final Decision (FD) = Less Healthy
  • If two decisions have same membership max, use the average
  of the two.

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   Centroid Method:

 By geometric decomposition
 The centroid of a plane figure X can be computed by dividing it into a finite
 number of simpler figures , computing the centroid Ci and area Ai of each part,
 and then computing

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  We examine a simple two-input one-output problem that includes
  three rules:

  Rule: 1
  IF      brakes are    good
  OR      driver is     sleepy
  THEN risk      is     low
  Rule: 2
  IF      brakes are    thin
  AND     driver is     alert
  THEN risk      is     normal
  Rule: 3
  IF      brakes are    bad
  THEN risk      is     high

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  Step1: Fuzzification
  The first step is to take the crisp inputs (state of the brakes and
  the state of the driver), and determine the degree to which these
  inputs belong to each of the appropriate fuzzy sets.

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  Step2:Rules’ Evaluation

  o The second step is to take the fuzzified inputs, and apply
  them to the antecedents of the fuzzy rules.

  o If a given fuzzy rule has multiple antecedents, the fuzzy
  operator (AND or OR) is used to obtain a single number that
  represents the result of the antecedent evaluation.

  o This number (the truth value) is then applied to the consequent
  membership function.

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  o Now the result of the antecedent evaluation can be applied to
  the membership function of the consequent.

  o The most common method of correlating the rule consequent
  with the truth value of the rule antecedent is to cut the
  consequent membership function at the level of the antecedent
  truth. This method is called clipping (alpha-cut).

  o Since the top of the membership function is sliced, the clipped
  fuzzy set loses some information.

  o However, clipping is still often preferred because it involves less
  complex and faster mathematics, and generates an aggregated
  output surface that is easier to defuzzify.

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  Step 3: Aggregation of the rule outputs
  oWe then take the membership functions of all rule consequents
  previously clipped and combine them into a single fuzzy set.
  o The input of the aggregation process is the list of clipped
  consequent membership functions, and the output is one fuzzy set
  for each output variable.

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  Step 4: Defuzzification
  o The last step in the fuzzy inference process is defuzzification.

  o Fuzziness helps us to evaluate the rules, but the final output of a
  fuzzy system has to be a crisp number.

  o The input for the defuzzification process is the aggregate output
  fuzzy set and the output is a single number.

  Maximum Method
  Risk is high

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  Centroid Method:
       Degree of

             0   10   20   30   40   50   60      70   80   90   100
                                               67.4               Z

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  7.2. MATLAB Fuzzy Logic Toolbox

   Introduction
   Graphical User Interface (GUI) Tools
   Example: Dinner for two

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  The Matlab fuzzy logic toolbox facilitates the development of
  fuzzy-logic systems using:
       • graphical user interface (GUI) tools
       • command line functionality
  The tool can be used for building
       • Fuzzy Expert Systems
       • Adaptive Neuro-Fuzzy Inference Systems (ANFIS)

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  Graphical User Interface (GUI) Tools
  There are five primary GUI tools for building, editing, and
  observing fuzzy inference systems in the Fuzzy Logic Toolbox:
  •    Fuzzy Inference System (FIS) Editor
  •    Membership Function Editor
  •    Rule Editor
  •    Rule Viewer
  •    Surface Viewer

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       Graphical User Interface (GUI) Tools

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         Graphical User Interface (GUI) Tools
 Fuzzy Inference System (FIS) Editor

Define number of
input and output

  Adjust fuzzy                              Name and edit
   inference                               names of input,
   functions                               output variables

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          Graphical User Interface (GUI) Tools
  Membership Function Editor

 Select & edit
 attributes of

 Display & edit                             Name & edit
values of current                           parameters of
    variable                                 membership

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         Graphical User Interface (GUI) Tools
Rule Editor

          Rules –

       Create and edit

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            Graphical User Interface (GUI) Tools
  Rule Viewer

Shows how input
variable is used in

                                              Shows how output
                                              variable is used in
                                                 rules; shows
                                               output of fuzzy

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           Graphical User Interface (GUI) Tools
  Surface Viewer

                                              Shows output
                                             surface for any
                                              system output
                                            versus any one (or
Specify input and                              two) inputs
 output variables

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  Now apply MATLAB Fuzzy Logic Toolbox for the same example
  Use the Fuzzy-instruction and adjust no. of inputs and outputs
  and method of defuzzification

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Step1: Fuzzification Breaks functions

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Driver functions

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Risk function (output)

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    Define input and find output



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  Surface Viewer:

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  Try the example:
  Example: Tip Calculator
  Golden rules for tipping:
  1. IF the service is poor OR the food is bad,      THEN tip is cheap (5%)*.

  2. IF the service is good,                         THEN tip is average (15%).

  3. IF the service is excellent OR the food is delicious,
                                                   THEN tip is generous (25%)

  Use Gaussian functions for service, and
  Use Trapezoidal functions for food, and
  Use Triangular functions for tip

  *Peak of Triangular function
  •    Universe of discourse for Service and Food=1 to 10
  •    Calculate Tip for Service=3 and food=6

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Areas in which Fuzzy Decision-making Systems can be used
They include the following:
   Manufacturing: Scheduling and planning materials flow, resource
allocation, routing, and machine and equipment design.
  Traffic systems: Routing and signal switching.
  Robotics: Path planning, task scheduling, navigation, and mission
  Computers: Memory allocation, task scheduling, and hardware
  Process industries: Monitoring, performance assessment, and
failure diagnosis.
 Science and medicine: Medical diagnostic systems, health
monitoring, and automated interpretation of experimental data.
  Business: Finance, credit evaluation, and stock market analysis.

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