Chapter 12, Fuzzy Arithmetic and the Extension Principle Video Solutions, Fuzzy Logic with Engineering Applications

Problem 1

Perform the following operations on intervals:
(a) $[2,3]+[3,4]$
(b) $[1,2] \times[1,3]$
(c) $[4,6] \div[1,2]$
(d) $[3,5]-[4,5]$

Derrick Hanson

Numerade Educator

Problem 2

Given the following fuzzy numbers and using Zadeh's extension principle, calculate $K=1$ - J and explain (or show) why 6 is nonconvex:

$$
\begin{aligned}
& I=3=\frac{0.2}{2}+\frac{1}{3}+\frac{0.1}{4} \\
& J=2=\frac{0.1}{1}+\frac{1}{2}+\frac{0.3}{3}
\end{aligned}
$$

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This problem makes use of Zadeh's extension principle. You are given the fuzzy sets $A$ and $\underset{\sim}{B}$ on the real line as follows:
If $x$ and $y$ are real numbers defined by sets $\underset{\sim}{\mathrm{A}}$ and $\underset{\sim}{\mathrm{B}}$, respectively, calculate the fuzzy set C representing the real numbers $z$ given by
(a) $z=3 x-2$
(b) $z=4 x^2+3$
(c) $z=x^2+y^2$
(d) $z=x-x$
(e) $z=\min (x, y)$

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Problem 4

For the function $y=x_1^2+x_2^2-4 x_1+4$ and the membership functions for fuzzy variables $x_1$ and $x_2$ shown in Fig. P12.4, find and plot the membership function for the fuzzy output variable, $y$, using
(a) A discretized form of the extension principle
(b) The vertex method
(c) The DSW algorithm

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02:54

Problem 5

The voltage drop across an element in a series circuit is equal to the series current multiplied by the element's impedance. The current, 1 , impedance, R , and voltage, V , are presumed to be fuzzy variables. Membership functions for the current and impedance are as follows:

$$
\begin{aligned}
\underline{I} & =\left\{\frac{0}{0}+\frac{0.8}{0.5}+\frac{1}{1}+\frac{0.8}{1.5}+\frac{0}{2}\right\} \\
\underline{R} & =\left\{\frac{0.5}{500}+\frac{0.9}{750}+\frac{1}{1000}+\frac{0.9}{1250}+\frac{0.5}{1500}\right\}
\end{aligned}
$$

Find the arithmetic product for $\mathrm{V}=\mathrm{I} \cdot \mathrm{R}$ using the extension principle.

Dharmendra Jain

Numerade Educator

05:09

Problem 6

Determine equivalent resistance of the circuit shown in Fig. P12.6, where $R_1$ and $R_2$ are fuzzy sets describing the resistance of resistors $R_{\mathrm{I}}$ and $R_2$, respectively, expressed in ohms. Since the resistors are in series they can be added arithmetically. Using the extension principle, find the equivalent resistance,

$$
\mathrm{R}_{\mathrm{eq}}=\mathrm{R}_1+\mathrm{R}_2
$$

The membership functions for the two resistors are

$$
\mathrm{R}_1=\left\{\frac{0.5}{3}+\frac{0.8}{4}+\frac{0.6}{5}\right\} \quad \text { and } \quad \mathrm{R}_2=\left\{\frac{0.3}{8}+\frac{1.0}{9}+\frac{0.4}{10}\right\}
$$

Thomas Thompson

Numerade Educator

03:55

Problem 7

In Newtonian mechanics the equivalent force on a body in motion can be found by taking the product of its mass and acceleration; this is commonly referred to as Newton's second law. For an object in a particular state, suppose the acceleration under the present force is given by the fuzzy set

$$
A=\left\{\frac{0}{0}+\frac{0.2}{1}+\frac{0.7}{2}+\frac{1}{3}+\frac{0}{4}\right\}
$$

and the mass is given by the fuzzy set

$$
M=\left\{\frac{0}{1}+\frac{0.5}{2}+\frac{1}{3}+\frac{0.5}{4}+\frac{0}{5}\right\}
$$
Assume both sets are in nondimensionalized units.
(a) Find the fuzzy set representing the force on the object using the extension principle.
(b) Develop analogous continuous membership functions, and plot them, for the fuzzy acceleration and mass and solve for the fuzzy force using (i) the vertex method and (ii) the restricted DSW algorithm.

Alexander Allen

Numerade Educator

04:33

Problem 8

For fluids, the product of the pressure $(P)$ and the volume $(V)$ of the fluid is a constant for a given temperature, i.e.,

$$
P V=\text { constant }
$$

Assume that at a given temperature a fluid of fuzzy volume

$$
\mathrm{V}_1=\left\{\frac{0.0}{0.5}+\frac{0.5}{0.75}+\frac{1.0}{1.0}+\frac{0.5}{1.25}+\frac{0.0}{1.5}\right\}
$$

is under a fuzzy pressure

$$
\mathrm{P}_1=\left\{\frac{0.0}{0.5}+\frac{0.5}{1.75}+\frac{1.0}{2.0}+\frac{0.5}{2.25}+\frac{0.0}{2.5}\right\}
$$

(a) Using the extension principle, determine the pressure $\mathrm{P}_2$ if the volume is reduced to

$$
\mathrm{V}_2=\left\{\frac{0.0}{0.4}+\frac{0.5}{0.45}+\frac{1.0}{0.5}+\frac{0.5}{0.55}+\frac{0.0}{0.6}\right\}
$$

(b) Develop analogous continuous membership functions for the fuzzy pressure $\mathrm{P}_1$ and volume $\mathrm{V}_1$ and solve for the pressure $\mathrm{P}_2$ using (i) the vertex method and (ii) the DSW algorithm. Plot the resulting membership function.
(c) Explain why $\mathrm{P}_2 \cdot \mathrm{V}_2$ would not be the same as ${\underset{\sim}{P}}_1 \cdot \mathrm{V}_1$ -

Regina Hays

Numerade Educator

Problem 9

A circle is govemed by the equation $x^2+y^2=8^2$. Its fuzzy $x$ coordinate is defined by the fuzzy set

$$
x=\left\{\frac{0}{0}+\frac{0.6}{2}+\frac{0.65}{3}+\frac{0.7}{4}+\frac{0.75}{5}+\frac{0.8}{6}\right\}
$$

Find the fuzzy $y$ coordinate, and plot its membership function for the equation of a circle.
(a) Use the DSW algorithm.
(b) Perform the same calculation using the restricted DSW algorithm.
(c) Comment on the nature of the results using a fuzzy $x$ that is non-normal.

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Problem 10

For the function $y=x_1^2 \cdot x_2-3 x_2$, where the membership functions of $x_1$ and $x_2$ are given in Fig. P12.10, find and plot the fuzzy membership function for $y$ using
(a) The vertex method
(i) Ignoring any extreme points
(ii) Including any extreme points
(b) The restricted DSW algorithm

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04:07

Problem 11

Define a fuzzy set $X$ with the membership function

$$
x=\left\{\frac{0.1}{1}+\frac{1}{2}+\frac{0.4}{3}\right\}
$$

Using the extension principle, determine the membership function for $z$ written in two different forms, i.e., for
(a) $z=x^2$
(b) $z=\underline{x} \cdot \underline{x}$

For parts (a) and (b) use the direct extension principle, the vertex method, and the DSW method, and compare the three results.
(c) $\mathrm{z}=\mathrm{x}^2$ and $\mathrm{z}=\underline{\mathrm{x}} \cdot \mathrm{x}$ using the vertex method
(d) $z=x^2$ and $z=x \cdot x$ using the DSW algorithm
(e) Discuss your answers from the different forms and methods.

James Kiss

Numerade Educator

Problem 12

Now suppose $x$ has membership function

$$
x=\left\{\frac{0.1}{-3}+\frac{0.3}{-2}+\frac{0.7}{-1}+\frac{1}{0}+\frac{0.7}{1}+\frac{0.3}{2}+\frac{0.1}{3}\right\}
$$

Repeat steps (a), (b), (c), and (d) of Problem 12.11 and (e) comment on any differences or similarities.

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03:55

Problem 13

When taking hydrostatic measurements, hydrostatic pressure is given by

$$
P=\rho g h
$$

where $\rho$ is density, $g$ is acceleration due to gravity, and $h$ is the height of the column of fluid. In a well-drilling environment, the density of the drilling mud has some uncertainty due to the inconsistent nature of the fluid. The well depth can also possess significant uncertainty due to stretching in the drill pipe used to measure the well depth. A membership function for density and depth is given in Fig. P12.13. Using the DSW algorithm determine the membership function for hydrostatic pressure, $P$.

Supratim Pal

Numerade Educator

01:33

Problem 14

A dissipated power, $P$, in a resistor can be described by $P=R \cdot I^2$, where $R$ is the resistance, in ohms, and $I$ is the current, measured in amps, passing through the resistor. Let a fuzzy set R be defined on the universe $x_1=\{10,20,30,40,50\}$ ohms and a fuzzy set $I$ be defined on the universe $x_2=\{0,1,2,3\} \mathrm{amps}$. We wish to map elements of these fuzzy sets to the dissipated power universe, y , under the relation $P=R \cdot I^2$. We have a medium resistance given by

$$
\underset{R}{R}=\left\{\frac{0.3}{10}+\frac{0.9}{20}+\frac{1}{30}+\frac{0.8}{40}+\frac{0.4}{50}\right\}=\text { "medium resistance" }
$$

and a low current given by

$$
\mathrm{I}=\left\{\frac{0.5}{0}+\frac{1}{1}+\frac{0.4}{2}+\frac{0.1}{3}\right\}=\text { "low current" }
$$

Using the discretized form of the extension principle, determine the membership values for $P$.

Stanley Enemuo

Numerade Educator

03:30

Problem 15

An airport passenger terminal has two activities with specific time intervals: processing times ( $\left.t_1\right)$ and waiting times $\left(t_2\right)$. The universe of time is $\mathrm{X}=\{10,20,30\}$ in minutes. For each of these two activities there is a membership function relating the level of service to the total time the passengers spend waiting in line: (1) tolerable service, or (2) good service. For this exercise, suppose each of the times is "good," as given below:

$$
\begin{aligned}
& t_1=\left\{\frac{1.0}{10}+\frac{0.8}{20}+\frac{0.5}{30}\right\} \\
& t_2=\left\{\frac{1.0}{20}+\frac{0.6}{30}+\frac{0.3}{40}\right\}
\end{aligned}
$$

Using a discretized form of the extension principle, find the membership function for the total time (processing time + waiting time), i.e., for the total time defined as $t=t_1+t_2$.

Willis James

Numerade Educator

05:50

Problem 16

Flue gas is used to heat a process stream using a counter-current heat exchanger. The process stream is intended to meet a required temperature of $190^{\circ} \mathrm{C}$ with an average heat capacity rate $W C p_{p s}$. The flue gas entering the heat exchanger has $W C p=0.3 \mathrm{~kW} /{ }^{\circ} \mathrm{C}$ and $T=1000^{\circ} \mathrm{C}$ (Fig. P12.16).
The outlet temperature of the gas $\left(T_{\mathrm{fg}}\right)$ is considered a discrete fuzzy set of values related to optimum operation conditions $\left(\mu\left(T_{\text {fg }}\right)\right)$, and the process stream inlet temperature depends on both $T_{\text {fe }}$ and the approach temperature ( $\Delta T_{\text {app }}$ ) that is also considered a discrete fuzzy set of values $\left(\mu\left(\Delta T_{\text {app }}\right)\right)$ :

$$
\begin{aligned}
\mu_{T_{\text {k }}} & =\left\{\frac{0}{160^{\circ} \mathrm{C}}+\frac{0.5}{170^{\circ} \mathrm{C}}+\frac{0.75}{180^{\circ} \mathrm{C}}+\frac{0.9}{190^{\circ} \mathrm{C}}+\frac{1}{200^{\circ} \mathrm{C}}\right\} \\
\mu_{\Delta T_{40 p}} & =\left\{\frac{0}{20^{\circ} \mathrm{C}}+\frac{0.55}{30^{\circ} \mathrm{C}}+\frac{0.85}{40^{\circ} \mathrm{C}}+\frac{1}{50^{\circ} \mathrm{C}}+\frac{0}{60^{\circ} \mathrm{C}}\right\}
\end{aligned}
$$

The following equation from the energy balance is needed:

$$
W C_{p_{\mathrm{ps}}}=\frac{\left(0.3 \mathrm{~kW} /{ }^{\circ} \mathrm{C}\right)\left(1000^{\circ} \mathrm{C}-T_{\mathrm{fg}}\right)}{190^{\circ} \mathrm{C}-T} \quad \text { with } T=T_{\mathrm{fg}}-\Delta T_{\mathrm{app}}
$$

Implementing the extension principle, find the fuzzy values of the heat capacity rate of the process stream for optimum operation.

Lottie Adams

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16