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Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 6

Development of Membership Functions - all with Video Answers

Educators

Chapter Questions

Problem 1

Using your own intuition, develop fuzzy membership functions on the real line for the fuzzy number 3 , using the following function shapes:
(a) Symmetric triangle
(b) Trapezoid
(c) Gaussian function

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Problem 2

Using your own intuition, develop fuzzy membership functions on the real line for the fuzzy number "approximately 2 or approximately $8^{\prime \prime}$ using the following function shapes:
(a) Symmetric triangles
(b) Trapezoids
(c) Gaussian functions

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Problem 3

Using your own intuition, develop fuzzy membership functions on the real line for the fuzzy number " approximately 6 to approximately $8^{\prime \prime}$ using the following function shapes:
(a) Symmetric triangles
(b) Trapezoids
(c) Gaussian functions

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02:24

Problem 4

Using your own intuition and your own definitions of the universe of discourse, plot fuzzy membership functions for the following variables:
(a) Age of people
(i) Very young
(ii) Young
(iii) Middle-aged
(iv) Old
(v) Very old
(b) Education of people
(i) Fairly educated
(ii) Educated
(iii) Highly educated
(iv) Not highly educated
(v) More or less educated

Christopher Stanley
Christopher Stanley
Numerade Educator
01:39

Problem 5

Using the inference approach outlined in this chapter, find the membership values for each of the triangular shapes (I, R, IR, E, T) for each of the following triangles:
(a) $80^{\circ}, 75^{\circ}, 25^{\circ}$
(b) $55^{\circ}, 65^{\circ}, 60^{\circ}$
(c) $45^{\circ}, 75^{\circ}, 60^{\circ}$

Steven Clarke
Steven Clarke
Numerade Educator

Problem 6

Develop a membership function for rectangles that is similar to the algorithm on triangles in this chapter. This function should have two independent variables; hence, it can be plotted.

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Problem 7

The following raw data were determined in a pairwise comparison of new premium car preferences in a poll of 100 people. When it was compared with a Porsche ( P ), 79 of those polled preferred a BMW (B), 85 preferred a Mercedes (M), 59 preferred a Lexus (L), and 67 preferred an Infinity (I). When a BMW was compared, the preferences were $21-\mathrm{P}, 23-\mathrm{M}$, $37-\mathrm{L}$, and $45-\mathrm{I}$. When a Mercedes was compared, the preferences were $15-\mathrm{P}, 77-\mathrm{B}, 35-\mathrm{L}$, and $48-1$. When a Lexus was compared, the preferences were $41-\mathrm{P}, 63-\mathrm{B}, 65-\mathrm{M}$, and $51-\mathrm{I}$. Finally, when an Infinity was compared, the preferences were $33-\mathrm{P}, 55-\mathrm{B}, 52-\mathrm{M}$, and $49-\mathrm{L}$. Using rank ordering, plot the membership function for "most preferred car."

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Problem 8

For the data shown in the accompanying table, show the first iteration in trying to compute the membership values for the input variables $x_1, x_2$, and $x_3$ in the output regions $\mathrm{R}^1$ and $\mathrm{R}^2$. Assume a random set of weights for your neural network.
(a) Use a $3 \times 3 \times 1$ neural network.
(b) Use a $3 \times 3 \times 2$ neural network.
(c) Explain the difference in results when using (a) and (b).

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Problem 9

For the data shown in the following table, show the first iteration in trying to compute the membership values for the input variables $x_1, x_2, x_3$, and $x_4$ in the regions $\mathrm{R}^1, \mathrm{R}^2$, and $\mathrm{R}^3$.
Use a $4 \times 3 \times 3$ neural network with a random set of weights.

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03:01

Problem 10

For the data shown in the accompanying Table A , show the first two iterations using a genetic algorithm in trying to find the optimum membership functions (use right-triangle functions) for the input variable $x$ and output variable $y$ in the rule table, Table B.
For the rule table, the symbols SM, MD, and LG mean small, medium, and large, respectively.

Harshita Goel
Harshita Goel
Numerade Educator
03:01

Problem 11

For the data shown in the following Table A , show the first two iterations using a genetic algorithm in trying to find the optimum membership functions (use right-triangle functions) for the input variable $x$ and output variable $y$ in the rule table, Table B. For the rule table, Table B, the symbols ZE, S, and LG mean zero, small, and large, respectively.

Harshita Goel
Harshita Goel
Numerade Educator
05:34

Problem 12

The results of a price survey for 30 automobiles is presented here:
Consider the automobile prices as a variable and the classes as economy, midsize, and luxury automobiles. Develop three membership function envelopes for car prices using the method of inductive reasoning.

Sneha Ravi
Sneha Ravi
Numerade Educator