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Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 15

Monotone Measures: Belief, Plausibility, Probability, and Possibility - all with Video Answers

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Chapter Questions

Problem 1

In structural dynamics a particular structure that has been subjected to a shock environment may be in either of the fuzzy sets "damaged" or "undamaged," with a certain degree of membership over the magnitude of the shock input. If there are two crisp sets, functional ( F ) and nonfunctional (NF), then a monotone measure would be the evidence that a particular system that has been subjected to shock loading is a member of functional systems or nonfunctional systems. Given the evidence from two experts shown here for a particular structure, find the beliefs and plausibilities for the focal elements.

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Problem 2

Suppose you have found an old radio (vacuum tube type) in your grandparents' attic and you are interested in determining its age. The make and model of the radio are unknown to you; without this information you cannot find in a collector's guide the year in which the radio was produced. Here, the year of manufacture is assumed to be within a particular decade. You have asked two antique radio collectors for their opinion on the age. The evidence provided by the collectors is fuzzy. Assume the following questions:
1. Was the radio produced in the 1920 s?
2. Was the radio produced in the 1930 s?
3. Was the radio produced in the 1940 s?

Let R, D, and W denote subsets of our universe set $P$ - the set of radio-producing years called the 1920s (Roaring 20s), the set of radio-producing years called the 1930s (Depression years), and the set of radio-producing years called the 1940s (War years), respectively. The radio collectors provide beas as given in the accompanying table.
(a) Calculate the missing belief values for the two collectors.
(b) Calculate the missing plausibility values for the two collectors.
(c) Calculate the missing combined evidence values.
(d) Calculate the missing combined belief and plausibility values.

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Problem 3

The quality control for welded seams in the hulls of ships is a major problem. Ultrasonic defectoscopy is frequently used to monitor welds, as is x-ray photography. Ultrasonic defectoscopy is faster but less reliable than x-ray photography. Perfect identification of flaws in welds is dependent on the experience of the person reading the signals. An abnormal signal occurs for three possible types of situations. Two of these are flaws in welds: a cavity (C) and a cinder inclusion (I); the former is the more dangerous. Another situation is due to a loose contact of the sensor probe (L), which is not a defect in the welding seams but an error in measuring. Suppose we have two experts, each using a different weld monitoring method, who are asked to identify the defects in an important welded seam. Their responses in terms of beas are given in the table. Calculate the missing portions of the table.

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Problem 4

You are an aerospace engineer who wishes to design a bang-bang control system for a particular spacecraft using thruster jets. You know that it is difficult to get a good feel for the amount of thrust that these jets will yield in space. Gains of the control system depend on the amount of the force the thrusters yield. Thus, you pose a region of three crisp sets that are defined with respect to specific gains. Each set will correspond to a different gain of the control system.

You can use an initial estimate of the force you get from the thrusters, but you can refine it in real time utilizing different gains for the control system. You can get a foree estimate and a belief measure for that estimate for a specific set. Suppose you define the following regions for the thrust values, where thrust is in pounds:
$\mathrm{A}_1$ applies to a region $0.8 \leq$ thrust value $\leq 0.9$.
$\mathrm{A}_2$ applies to a region $0.9 \leq$ thrust value $\leq 1.0$.
$\mathrm{A}_3$ applies to a region $1.0 \leq$ thrust value $\leq 1.1$.
Two expert aerospace engineers have been asked to provide evidence measures reflecting their degree of belief for the various force estimates. These beas along with calculated belief measures are given here. Calculate the combined belief measure for each focal element in the table.

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04:50

Problem 5

Consider the DC series generator shown in Fig. P15.5. Let $R_a=$ armature resistance, $R_s=$ field resistance, and $R=$ load resistance. The voltage generated across the terminal is given by $V_I=E_g-\left(I_x R_x+I_u R_\alpha\right)$. Note: If there is no assignment for $R$, i.e., if the value of $R$ is infinity, then the generator will not build up because of an open circuit. Also $R_a$ can have a range of values from a low value to a high value. To generate different load voltages required, we can assign values for $R, R_a$, and $R_s$ in different ways to get the voltage. They are very much interrelated and the generated voltage need not have a unique combination of $R, R_\alpha$, and $R_s$. Hence, nesting of focal elements for these resistances does have some physical significance.
Let the basic evidence assignment for the elements of universe $\mathrm{X}\left\{R, R_a, R_x\right\}$ be as shown in the accompanying table:
(a) Does $m_1$ or $m_2$ represent a possibility measure?
(b) If either of the evidence measures (or both) is nested, find the possibility distributions.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator

Problem 6

A general problem in biophysics is to segment volumetric MRI (Magnetic Resonance Imaging) data of the head given a new set of MRI data. We use a "model head" that has already had the structures in the head (mainly brain and brain substructures) labeled. We can use the model head to help in segmenting the data from the new head by assigning beas to each voxel (a voxel is a three-dimensional pixel) in the new MRI data set, based on what structures contain, or are near, the corresponding voxel in the model head. In this example a nested subset corresponds to the physical containment of a head structure within another structure. We will select a voxel for which the beas form a consonant body of evidence.

$$
\begin{aligned}
& \mathrm{X}=\{\text { structures in the MRI data }\} \\
& \mathrm{H}=\text { head } \\
& \mathrm{B}=\text { brain } \\
& \mathrm{N}=\text { neocortex } \\
& \mathrm{L}=\text { occipital lobe } \\
& \mathrm{C}=\text { calcarine fissure }
\end{aligned}
$$

Thus, we have $\mathrm{C} \subset \mathrm{L} \subset \mathrm{N} \subset \mathrm{B} \subset \mathrm{H}$. For voxel V we have a basic distribution of

$$
m=\left(\mu_{\mathrm{C}}, \mu_1, \mu_{\mathrm{N}}, \mu_{\mathrm{B}}, \mu_{\mathrm{H}}\right)=(0.1,0.1,0.2,0.3,0.3)
$$

Find the corresponding possibility distribution and draw the nesting diagram.

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Problem 7

A test and diagnostics capability is being developed for a motion control subsystem that consists of the following hardware: a motion control IC (Integrated Circuit), an interconnect between motion control IC, an H-switch current driver, an interconnect between H-switch current driver, a motor, and an optical encoder.
The elements of the motion control subsystem are as follows:

$$
\begin{aligned}
& x_1=\text { motion control IC } \\
& x_2=\text { interconnect } 1 \\
& x_3=\text { H-switch current driver } \\
& x_4=\text { interconnect } 2 \\
& x_5=\text { motor } \\
& x_6=\text { optical encoder }
\end{aligned}
$$

If a motion control subsystem failure exists, a self-test could describe the failure in the following bea: $m=(0.2,0,0.3,0,0,0.5)$. This nested structure is based on the level of hardware isolation of the diagnostic software. This isolation is hierarchical in nature. You first identify a motion control subsystem failure $m\left(\mathrm{~A}_6\right)$ that includes a possibility of any component failure $\left(x_1, x_2, x_3, x_4, x_5, x_6\right)$. The test then continues and, due to isolation limitations, a determination can be made of the failure possibility consisting of $m\left(A_3\right)$, subset $\left(x_1, x_2, x_3\right)$, followed by the ability to isolate to an $x_1$ failure if $x_1$ is at fault. Basic evidence assignments are constructed from empirical data and experience.

Find the associated possibility distribution and draw the nesting diagram.

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08:34

Problem 8

Design of a geometric traffic route can be described by four roadway features: a comer, a curve, a U-turn, and a circle. The traffic engineer can use four different evaluation criteria (expert guidance) to use in the design process:
$m_1=$ criteria: fairly fast, short distance, arterial road, low slope points
$m_2=$ criteria: slow, short distance, local road, low slope points
$m_3=$ criteria: fast, long distance, ramp-type road, medium slope points
$m_4=$ criteria: very fast, medium distance, highway, medium slope points
Using the $15\left(2^4-1\right)$ focal elements shown in the accompanying table, determine which, if any, of the four evidence measures $\left(m_1-m_4\right)$ results in an ordered possibility distribution.

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator

Problem 9

Given a communication link with a sender, receiver, and interconnecting link, an error in a message could occur at the sender, receiver, or on the interconnecting link. Combinations such as an error on the link that is not corrected by the receiver are also possible. Let $\mathrm{S}, \mathrm{R}$, and L represent sources of error in the sender, receiver, and link, respectively. If E is the universe of error sources, then

$$
P(E)=(\emptyset,\{S\},\{R\},\{L\},\{S, R\},\{S, L\},\{R, L\},\{S, R, L\})
$$

Now assume each source has its own expert and each of these provides their basic assignment of the actual source of an error as follows:
Indicate which experts, if any, have evidence that is consonant. For each of these, do the following:
(a) Determine the possibility distribution.
(b) Draw the nesting diagram.
(c) Give the physical significance of the nesting.

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Problem 10

There are a number of hazardous waste sites across the country that pose significant health risk to humans. However, due to high costs involved in exposure analysis only a limited amount of information can be collected from each site to determine the extent of contamination.
Suppose it is determined that one of the sites is contaminated by a new carcinogenic chemical identified as Tox. The table shows the results from the chemical analysis of the groundwater samples collected from one of the sites. Given these sparse data, determine the possibility distribution of exposure concentrations for the chemical Tox.

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03:04

Problem 11

Due to their excellent self-healing properties, rock salt caverns are used to store nuclear waste from various nuclear plants. One of the properties useful in determining the suitability of a cavern for nuclear waste storage is the creep rate of salt; salt creeps very slowly with time. This creep rate determines the strength of the cavern and the duration that the cavern can be accessible to human operations. The table shows the strain rate results from creep tests conducted on rock salt cores from four locations of the waste repository. Given these data, determine the strain rate interval that is $80 \%$ possible (possibilistic weight $=0.8$ ). Also, find the degree of confirmation.

Bin Chen
Bin Chen
Numerade Educator

Problem 12

Predicting interest rates is critical for financial portfolio management and other investment decisions. Based on historical variations and other factors, the following interest rates are predicted for the next two months.
(a) What is the possibility that the interest rates will be higher than $2 \%$ ?
(b) Give the reason for your choice of consonant intervals.
(c) Find the degree of confirmation.

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