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Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 1

Introduction - all with Video Answers

Educators

Chapter Questions

03:13

Problem 1

Develop a reasonable membership function for the following fuzzy sets based on height measured in centimeters:
(a) "Tall"
(b) "Short"
(c) "Not short"

Supratim Pal
Supratim Pal
Numerade Educator

Problem 2

Develop a membership function for laminar and turbulent flow for a typical flat plate with a sharp leading edge in a typical air stream. Transition usually takes place between Reynolds numbers ( Re ) of $2 \times 10^5$ and $3 \times 10^6$. An Re of $5 \times 10^5$ is usually considered the point of turbulent flow for this situation.

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01:21

Problem 3

Develop a reasonable membership function for a square, based on the geometric properties of a rectangle. For this problem use $L$ as the length of the longer side and $l$ as the length of the smaller side.

Yujie Wang
Yujie Wang
College of San Mateo
02:41

Problem 4

For the cylindrical shapes shown in Fig. 1.2, develop a membership function for each of the following shapes using the ratio $d / h$, and discuss the reason for any overlapping among the three membership functions:
(a) Rod
(b) Cylinder
(c) Disk

Gaurav Kalra
Gaurav Kalra
Numerade Educator
04:10

Problem 5

The question of whether a glass of water is half-full or half-empty is an age-old philosophical issue. Such descriptions of the volume of liquid in a glass depend on the state of mind of the person asked the question. Develop membership functions for the fuzzy sets "half-full," "full," "empty," and "half-empty" using percent volume as the element of information. Assume the maximum volume of water in the glass is $V_0$. Discuss whether the terms "half-full" and "half-empty" should have identical membership functions. Does your answer solve this ageless riddle?

Anas Venkitta
Anas Venkitta
Numerade Educator
04:23

Problem 6

Landfills are a primary source of methane, a greenhouse gas. Landfill caps, called biocaps, are designed to minimize methane emission by maximizing methane oxidation; these caps are classified as "best" if they are capable of oxidizing $80 \%$ of the methane that originates in the landfill's interior. Complete oxidation was found to be difficult to establish. Develop a reasonable membership function of the percent methane oxidation to show the performance of the biocap and emissions of methane.

Lottie Adams
Lottie Adams
Numerade Educator
01:30

Problem 7

Industry A discharges wastewater into a nearby river. Wastewater contains high biological oxygen demand (BOD) and other inorganic contaminants. The discharge rate of rivers and wastewater is constant through the year. From research, it has been found that BOD values not exceeding $250 \mathrm{mg} / \mathrm{L}$ do not cause any harmful effect to aquatic ecosystems. However, BOD values higher than $250 \mathrm{mg} / \mathrm{L}$ have significant impact. Draw both a crisp and fuzzy membership function to show the effects of the BOD value on aquatic ecosystems.

Anand Jangid
Anand Jangid
Numerade Educator
03:13

Problem 8

A fuzzy set for a major storm event in Calgary, Alberta, could be described as a rainstorm in a subdivision that raised the level of the storm-water pond to within $70 \%$ of its design capacity. The membership function for a major storm set could be described as having full membership when $70 \%$ of the pond volume has been reached but varying from zero membership to full membership at $40 \%$ capacity and $70 \%$ capacity, respectively. Draw a typical membership function as it is described.

Supratim Pal
Supratim Pal
Numerade Educator

Problem 9

In Alberta a waste is orally toxic if it has an oral toxicity ( $\mathrm{LD}_{50}$ ) of less than $5000 \mathrm{mg} / \mathrm{kg}$. Develop and draw a crisp and a fuzzy membership function for the oral toxicity.

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02:37

Problem 10

Using the ratios of internal angles or sides of a hexagon, draw the membership diagrams for "regular" and "irregular" hexagons.

Jay Patel
Jay Patel
Numerade Educator

Problem 11

Develop algorithms for the following membership function shapes:
(a) Triangular
(b) Gamma function
(c) Quadratic S-function
(d) Trapezoid
(e) Gaussian
( $f$ ) Exponential-wire function

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04:40

Problem 12

In soil mechanics soils are classified based on the size of their particles as clay, silt, or sand (clays having the smallest particles and sands having the largest particles). Though silts have larger particles than clays, it is often difficult to distinguish between these two soil types; silts and sands present the same problem. Develop membership functions for these three soil types, in terms of their grain size, $S$.

Noraney Ocampo
Noraney Ocampo
Numerade Educator
02:44

Problem 13

A sour natural gas stream is contacted with a lean amine solution in an absorber; this allows the amine to remove the sour component in the natural gas producing a rich amine solution and a "sales gas" which is the natural gas with a much lower sour gas concentration than the feed gas as shown in Fig. P1.13. Concentrations above $\mathrm{C}_2$, which is the pipeline specification for sour gas concentration, are considered to have full membership in the set of "high concentrations." A concentration below $\mathrm{C}_1$, which is the lower limit of sour gas concentration that can be detected by analysis instrumentation, is considered to have full membership in the set of "low concentrations." Sketch a membership function for the absorber "sales gas" sour gas concentration as a function of concentration, C ; show the points $\mathrm{C}_1$ and $\mathrm{C}_2$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator

Problem 14

A circular column loaded axially is assumed to be eccentric when the load is acting at $5 \%$ of the axis, depending on the diameter of the column, $d$ as shown in Fig. P1.14. We have the following conditions: $e / d=0.05$ eccentric; $e / d<0.05$ not-very-eccentric; $e / d>0.05$, very eccentric. Develop a membership function for "eccentricity" on the scale of e/d ratios.

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13:45

Problem 15

A rectangular sheet of perimeter $2 L+2 h$ is to be rolled into a cylinder with height, $h$. Classify the cylinder as a function of the rectangular sheet as being a rod, cylinder, or disk by developing membership functions for these shapes.

Carlos Pinilla
Carlos Pinilla
Numerade Educator
04:11

Problem 16

Enumerate the nonfuzzy subsets of the power set for a universe with $n=4$ elements, i.e., $\mathrm{X}=\left\{x_1, x_2, x_3, x_4\right\}$, and indicate their coordinates as the vertices on a 4 -cube.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator

Problem 17

Probability distributions can be shown to exist on certain planes that intersect the regions shown in Fig. 1.4. Draw the points, lines, and planes on which probability distributions exist for the one-, two-, and three-parameter cases shown in Fig. 1.4.

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