7. Decision Making

Transcription

1 7. Decision Making 1

2 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 Inference Engine Fuzzy Knowledge base Defuzzifier Output if then Rule base 2

3 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. 3

4 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 4

5 Fuzzification of Height: Fuzzification of Weight 5

6 Step2: Infgerence Engine (Rules) Rules Rules Rules Rules Rules reflect experts decisions. are tabulated as fuzzy words can be grouped in subsets can be redundant 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) 6

7 Step2:Rules Using If-Then type fuzzy rules converts the fuzzy input to the fuzzy output. 7

8 In our example: Rules function Rules are tabulated as fuzzy words Healthy (H) Somewhat healthy (SH) Less Healthy (LH) Unhealthy (U) 8

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10 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 10

11 Step4: Activate Rules 11

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13 Step5: compute Decision Function f = {U, LH, SH, H} f = {0.3, 0.7, 0.2, 0.2} 13

14 f = {U, LH, SH, H} f = {0.3, 0.7, 0.2, 0.2} 14

15 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 used: Maximum Method (not often used) Centroid of area 15

16 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. 16

17 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 17

18 but 18

19 example: We examine a simple two-input one-output problem that includes three rules: Rule: 1 IF brakes are OR driver is THEN risk is good sleepy low Rule: 2 IF brakes are AND driver is THEN risk is thin alert normal Rule: 3 IF brakes are THEN risk is bad high 19

20 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. 20

21 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. 21

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23 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. 23

24 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. 24

25 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 25

26 Centroid Method: Degree of Membership Z 26

27 7.2. MATLAB Fuzzy Logic Toolbox Introduction Graphical User Interface (GUI) Tools Example: Dinner for two 27

28 Introduction 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) 28

29 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 29

30 Graphical User Interface (GUI) Tools 30

31 Graphical User Interface (GUI) Tools Fuzzy Inference System (FIS) Editor Define number of input and output variables Adjust fuzzy inference functions Name and edit names of input, output variables 31

32 Graphical User Interface (GUI) Tools Membership Function Editor Select & edit attributes of membership function Display & edit values of current variable Name & edit parameters of membership function 32

33 Graphical User Interface (GUI) Tools Rule Editor Rules automatically updated Create and edit rules 33

34 Graphical User Interface (GUI) Tools Rule Viewer Shows how input variable is used in rules Shows how output variable is used in rules; shows output of fuzzy system 34

35 Graphical User Interface (GUI) Tools Surface Viewer Specify input and output variables Shows output surface for any system output versus any one (or two) inputs 35

36 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 Hany Selim 36

37 Step1: Fuzzification Breaks functions Hany Selim 37

38 Driver functions 38

39 Risk function (output) 39

40 Rules Hany Selim 40

41 Define input and find output output input Hany Selim 41

42 Surface Viewer: 42

43 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 43

44 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 planning. Computers: Memory allocation, task scheduling, and hardware design. 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. 44