Chapter 4, Properties of Membership Functions, Fuzzification, and Defuzzification Video Solutions, Fuzzy Logic with Engineering Applications

Problem 1

Two fuzzy sets A∼ and B∼, both defined on X, are as follows:
Express the following $\lambda$-cut sets using Zadeh's notation:
(a) $(\overline{\mathrm{A}})_{0.7}$
(e) $(\mathrm{A} \cup \overline{\mathrm{A}})_{0.7}$
(b) $(\mathrm{B})_{0.4}$
(f) $(\underline{B} \cap \overline{\mathrm{B}})_{0.5}$
(c) $(\mathrm{A} \cup \mathrm{B})_{0.7}$
(g) $(\overline{\mathrm{A}} \cap \mathrm{B})_{0.7}$
(h) $(\overline{\mathrm{A}} \cup \overline{\mathrm{B}})_{0.7}$

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Problem 2

[Klir and Folger, 1988] Show that all $\lambda$-cuts of any fuzzy set A defined in $\mathrm{R}^n$ space ( $n \geq 1$ ) are convex if and only if

$$
\mu_{\mathrm{A}}[\lambda r+(1-\lambda) s] \geq \min \left[\mu_{\mathrm{A}}(r), \mu_{\mathrm{A}}(s)\right]
$$

for all $r, s \in R^n$, and all $\lambda \in[0,1]$.

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Problem 3

The fuzzy sets $\underset{\sim}{A}, B$, and $\underset{\sim}{C}$ are all defined on the universe $X=[0,5]$ with the following membership functions:

$$
\mu_{\mathrm{A}}(x)=\frac{1}{1+5(x-5)^2} \quad \mu_{\mathrm{B}}(x)=2^{-x} \quad \mu_{\underline{C}}(x)=\frac{2 x}{x+5}
$$

(a) Sketch the membership functions.
(b) Define the intervals along the $x$ axis corresponding to the $\lambda$-cut sets for each of the fuzzy sets $\underset{\sim}{\mathrm{A}}, \underset{\sim}{\mathrm{B}}$, and $\underset{\sim}{\mathrm{C}}$ for the following values of $\lambda$ :
(i) $\lambda=0.2$
(ii) $\lambda=0.4$
(iii) $\lambda=0.7$
(iv) $\lambda=0.9$
(v) $\lambda=1.0$

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Problem 4

Determine the crisp $\lambda$-cut relations for $\lambda=0.1 j$, for $j=0,1, \ldots, 10$, for the following fuzzy relation matrix R :

$$
R=\left[\begin{array}{cccc}
0.2 & 0.7 & 0.8 & 1 \\
1 & 0.9 & 0.5 & 0.1 \\
0 & 0.8 & 1 & 0.6 \\
0.2 & 0.4 & 1 & 0.3
\end{array}\right]
$$

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Problem 5

For the fuzzy relation $\mathrm{R}_4$ in Example 3.11 find the $\lambda$-cut relations for the following values of $\lambda$ :
(a) $\lambda=0^{+}$
(b) $\lambda=0.1$
(c) $\lambda=0.4$
(d) $\lambda=0.7$

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Problem 6

For the fuzzy relation $R$ in Problem 3.9 (a) sketch (in 3D) the $\lambda$-cut relations for the following values of $\lambda$ :
(a) $\lambda=0^{+}$
(b) $\lambda=0.3$
(c) $\lambda=0.5$
(d) $\lambda=0.9$
(e) $\lambda=1$

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Problem 7

Show that any $\lambda$-cut relation (for $\lambda>0$ ) of a fuzzy tolerance relation results in a crisp tolerance relation.

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Problem 8

Show that any $\lambda$-cut relation (for $\lambda>0$ ) of a fuzzy equivalence relation results in a crisp equivalence relation.

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Problem 9

In metallurgy materials are made with mixtures of various metals and other elements to achieve certain desirable properties. In a particular preparation of steel, three elements, namely iron, manganese, and carbon, are mixed in two different proportions. The samples obtained from these two different proportions are placed on a normalized scale, as shown in Fig. P4.9 and are represented as fuzzy sets $A_1$ and $A_2$. You are interested in finding some sort of "average" steel proportion. For the logical union of the membership functions shown we want to find the defuzzified quantity. For each of the seven methods presented in this chapter assess (a) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

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Problem 10

Two companies bid for a contract. A committee has to review the estimates of those companies and give reports to its chairperson. The reviewed reports are evaluated on a nondimensional scale and assigned a weighted score that is represented by a fuzzy membership function, as illustrated by the two fuzzy sets, $\mathrm{B}_1$ and $\mathrm{B}_2$, in Fig. P4.10. The chairperson is interested in the lowest bid, as well as a metric to measure the combined "best" score. For the logical union of the membership functions shown we want to find the defuzzified quantity. For each of the seven methods presented in this chapter assess (a) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

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Problem 11

A landfill is the cheapest method of solid waste treatment and disposal. Once disposed into a landfill, solid waste can be a major source of energy due to its potential to produce methane. However, all the solid waste disposed cannot generate methane at the same rate and in the same quantities. Based on its biodegradability, solid waste is classified into three distinct groups, namely: rapidly biodegradable, moderately biodegradable, and slowly biodegradable. Design of a landfill gas extraction system is based on gas production through the first two groups; both have different gas production patterns. The data collected from experiments and experiences are sets rapidly biodegradable and slowly biodegradable, respectively, in units of years. In order to properly design the gas extraction system we need a single representative gas production value. For the logical union of the membership functions shown we want to find the defuzzified quantity. For each of the seven methods presented in this chapter assess (a) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

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01:09

Problem 12

Uniaxial compressive strength is easily performed on cylindrical or prismatic ice samples and can vary with strain rate, temperature, porosity, grain orientation, and grain size ratio. While strain rate and temperature can be controlled easily, the other variables cannot. This lack of control yields an uncertainty in the uniaxial test results.

A test was conducted on each type of sample at a constant strain rate of $10^{-4} \mathrm{~s}^{-1}$, and a temperature of $-5^{\circ} \mathrm{C}$. Upon inspection of the results the exact yield point could not be determined; however, there was enough information to form fuzzy sets for the failure of the cylindrical and prismatic samples A and B, respectively, as shown in Fig. P4.12. Once the union of A and B has been identified (the universe of compressive strengths, megapascals $\mathrm{N} / \mathrm{m}^2 \times 10^6$ ) we can obtain a defuzzified value for the yield strength of this ice under a compressive axial load. For each of the seven methods presented in this chapter assess (a) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

Manik Pulyani

Numerade Educator

Problem 13

In the field of heat exchanger network (HEN) synthesis, a chemical plant can have two different sets of independent HENs. The term "optimum cost" is considered fuzzy, because for the design and construction of the HENs we have to consider other parameters in addition to the capital cost. The membership function of fuzzy sets HEN1 and HEN2 is shown in Figs.P4.13(a) and P 4.13 (b), respectively.

We wish to determine the optimum capital cost of a project to optimize the plant using both independent networks (HEN1 and HEN2); hence, the logical union of their membership functions, as shown in Fig. P4.13(c). For each of the seven methods presented in this chapter assess ( $a$ ) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

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Problem 14

In reactor design, it is often best to simplify a reactor by assuming ideal conditions. For a continuous stirred tank reactor (CSTR), the concentration inside the reactor is the same as the concentration of the effluent stream. In a plug flow reactor (PFR), the concentrations of the inlet and outlet streams are different as the concentrations of the reactants change along the length of the tube. For a fluidized bed in which catalyst is removed from the top of the reactor, there exists both characteristics of a CSTR and PFR. The difference between inside reactor concentration $\left(C_i\right)$ and effluent concentration $\left(C_e\right)$ gives the membership of either CSTR or PFR, as seen in Fig. P4.14.

Find the difference in concentration that represents the optimum design, i.e., find the most representative value for the union of PFR and CSTR. For each of the seven methods presented in this chapter assess (a) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

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Problem 15

Often in chemical processing plants there will be more than one type of instrumentation measuring the same variable at the same instance during the process. Due to the nature of measurements they are almost never exact, and hence can be represented as a fuzzy set. Due to the differences in instrumentation the measurements will usually not be the same. Take for example two types of temperature sensors, namely a thermocouple (TC) and an RTD (Resistance Temperature Detector) measuring the same stream temperature. The membership function of the two types of temperature sensors may look as in Fig. P4.15.

When an operator who prefers one measuring device ends his or her shift, and then is replaced by another operator with a different preference in measuring device, there may be a problem in determining the actual value of a variable. To avoid this problem it was decided to plot the membership functions of the two types of sensors, take their union, and employ defuzzification to select one temperature for the operator to use. To find this temperature, for each of the seven methods presented in this chapter, assess (a) whether each is applicable and, if so, (b) calculate the defuzzified value, $z^*$.

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