Fuzzy Logic with Engineering Applications

Timothy Ross

Chapter 10 Decision Making with Fuzzy Information - all with Video Answers

Educators

Chapter Questions

Problem 1

For Example 10.2 change the first fuzzy set $I_1$ to $\left\{\frac{1}{3}+\frac{0.7}{5}+\frac{0.4}{9}\right\}$ and recalculate the same quantities as those in Example 10.2.

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18:10

Company Z makes chemical additives that are ultimately used for engine oil lubricants. Components such as surfactants, detergents, rust inhibitors, etc., go into the finished engine oil before it is sold to the public. Suppose that company Z makes a product D739.2 that is a detergent additive. You are asked to determine if a particular batch of D739.2 is good enough to be sold to an oil company, which will then make the final product. The detergent is evaluated on the following parameters: actual color of the material, consistency, base number (BN, measure of detergent capacity), and flash point (FP, ignition temperature of the material). After making several hundred batches of the detergent additive D739.2, the following relation matrix is obtained:

$$
\mathrm{R}=\begin{aligned}
& \text { color } \\
& \text { consistency } \\
& \mathrm{BN} \\
& \mathrm{FP}
\end{aligned}\left[\begin{array}{ccc}
\text { excellent } & \text { very } & \text { food } \\
0.3 & 0.4 & 0.3 \\
0.1 & 0.5 & 0.4 \\
0.5 & 0.4 & 0.1 \\
0.4 & 0.3 & 0.3
\end{array}\right]
$$

The weight factor for the detergent is $\mathrm{a}=\{0.1,0.25,0.4,0.25\}$. Evaluate the quality of the detergent.

Caleb Huber

Numerade Educator

01:07

Problem 3

In making a decision to purchase an aircraft, airline management will consider the qualities of the plane's performance with respect to the competition. The Boeing 737 is the best-selling plane in aviation history and continues to outsell its more modern competitor, the A320, manufactured by the Airbus consortium. The four factors to be considered are these: range, payload, operating costs, and reliability. The criteria will be a comparison of the 737 with respect to the A320: superior (sup.), equivalent (eq.), and deficient (def.).

Given a typical airline's weighting factor of the four factors as $\mathrm{a}=\{0.15,0.15,0.3,0.4\}$, evaluate the performance of the 737 with respect to the A320.

Jodi Folley

Numerade Educator

Problem 4

A power supply needs to be chosen to go along with an embedded system. Four categories of evaluation criteria are important. The first is the physical size of the power supply. The second is the efficiency of the power supply. The third is the "ripple" voltage of the output of the power supply. This is a measure of how clean the power provided is. The fourth criterion is the peak current provided by the power supply. The following matrix defines the type of power supply required for the embedded system application:

From this matrix, one can see that for the embedded system in mind, the power supply's physical size is very important as well as its ripple voltage. Of lesser importance is its efficiency, and lesser yet, is its peak current. So a small power supply with clean output voltage is needed. It needs to be somewhat efficient and is not required to provide very much "inrush current" or peak current. Evaluate a power supply with the following characteristics:

$$
\begin{aligned}
& \text { Power } \\
& \text { supply }
\end{aligned}=\left[\begin{array}{ll}
0.5 \\
0.1 \\
0.2 \\
0.2
\end{array}\right] \begin{aligned}
& \text { physical size } \\
& \text { efficiency } \\
& \text { ripple voltage } \\
& \text { peak current }
\end{aligned}
$$

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Problem 5

An aircraft control system is a totally nonlinear system when the final approach and landing of an aircraft are considered. It involves maneuvering flight in an appropriate course to the airport and then along the optimum glide path trajectory to the runway. We know that this path is usually provided by an instrument landing system, which transmits two radio signals to the aircraft as a navigational aid. These orthogonal radio beams are known as the localizer and the glide slope and are transmitted from the ends of the runway in order to provide the approaching aircraft with the correct trajectory for landing. The pilot executing such a landing must monitor cockpit instruments that display the position of the aircraft relative to the desired flight path and make appropriate corrections to the controls. Presume that four positions are available to the pilot and that four corrections $P_1, P_2, P_3$, and $P_4$ from the actual position $P$ are required to put the aircraft on the correct course. The pairwise comparisons for the four positions are as follows:

$$
\begin{array}{llll}
f_{P_1}\left(P_1\right)=1 & f_{P_1}\left(P_2\right)=0.5 & f_{P_1}\left(P_3\right)=0.6 & f_{P_1}\left(P_4\right)=0.8 \\
f_{P_2}\left(P_1\right)=0.3 & f_{P_2}\left(P_2\right)=1 & f_{P_2}\left(P_3\right)=0.4 & f_{P_2}\left(P_4\right)=0.3 \\
f_{P_3}\left(P_1\right)=0.6 & f_{P_1}\left(P_2\right)=0.4 & f_{P_1}\left(P_3\right)=1 & f_{P_3}\left(P_4\right)=0.6 \\
f_{P_4}\left(P_1\right)=0 & f_{P_4}\left(P_2\right)=0.3 & f_{P_4}\left(P_3\right)=0.6 & f_{P_4}\left(P_4\right)=1
\end{array}
$$

Now, from these values, compute the comparison matrix, and determine the overall ranking.

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Problem 6

When designing a radar system for imaging purposes, we frequently need to set priorities in accomplishing certain features. Some features that need to be traded off against each other are these:
1. The ability to penetrate foliage and even the ground to some depth.
2. The resolution of the resulting radar image.
3. The size of the antenna required for the radar system.
4. The amount of power required to operate at a given frequency.

It is useful to determine the order of importance of these features in selecting an operating frequency for the radar. Let $x_1$ represent penetration; $x_2$, resolution; $x_3$, antenna size; and $x_4$, power. A crisp ordering will have trouble resolving the importance of penetration compared to resolution, resolution compared to antenna size, and antenna size compared to penetration. These are entities that can only be compared in a very subjective manner, ideal for fuzzy techniques and difficult for crisp techniques.

Let $f_{x_i}\left(x_j\right)$ be the relative importance of feature $x_j$ with respect to $x_i$. The comparisons $f_{x_i}\left(x_j\right)$ are subjectively assigned as follows:
Develop a comparison matrix and determine the overall ranking of the importance of each feature.

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Problem 7

In tracking soil particles, a tracked particle can be occluded by other objects. To find out which one is the tracked particle, one can choose or pick a particle that is a certain distance from the tracked particle. Suppose there are four particles in the region of interest where the tracked particle is. Furthermore, let $x_1, x_2, x_3$, and $x_4$ resemble the tracked particle with fuzzy measurement $0.3,0.4,0.6,0.7$, respectively, when they alone are considered. Note $f_{x_j}\left(x_i\right)$ means how close $x_i$ is to the tracked particle with respect to $x_j$.

$$
\begin{array}{llll}
f_{x_1}\left(x_1\right)=1 & f_{x_1}\left(x_2\right)=0.6 & f_{x_1}\left(x_3\right)=0.4 & f_{x_1}\left(x_4\right)=0.3 \\
f_{x_2}\left(x_1\right)=0.7 & f_{x_2}\left(x_2\right)=1 & f_{x_2}\left(x_3\right)=0.1 & f_{x_2}\left(x_4\right)=0.4 \\
f_{x_3}\left(x_1\right)=0.2 & f_{x_3}\left(x_2\right)=0.4 & f_{x_1}\left(x_3\right)=1 & f_{x_3}\left(x_4\right)=0.3 \\
f_{x_4}\left(x_1\right)=0.5 & f_{x_4}\left(x_2\right)=0.3 & f_{x_4}\left(x_3\right)=0.4 & f_{x_4}\left(x_4\right)=1
\end{array}
$$

Develop a comparison matrix, and determine which particle is closest to the tracked particle.

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Problem 8

Suppose a wine manufacturer was interested in introducing a new wine to the market. A very good but somewhat expensive Chenin Blane was already available to consumers and was very profitable to the company. To enter the lower-priced market of wine consumers, the wine manufacturer decided to make a less expensive wine that tasted similar to the very profitable Chenin Blanc already sold. After much market research and production knowledge, the manufacturer settled on four possible wines to introduce into the market. The fuzzy criteria of evaluation is taste, and we would like to know which wine tastes the most like the expensive Chenin Blanc. Define the following subjective estimations: universe $\mathrm{X}=\left\{x_1, x_2, x_3, x_4\right\}$. A panel of wine tasters tasted each of the wines $x_1, x_2, x_3$, and $x_4$ and made the following estimations:

$$
\begin{array}{llll}
f_{x_1}\left(x_1\right)=1 & f_{x_1}\left(x_2\right)=0.4 & f_{x_1}\left(x_3\right)=0.8 & f_{x_1}\left(x_4\right)=0.5 \\
f_{x_2}\left(x_1\right)=0.2 & f_{x_2}\left(x_2\right)=1 & f_{x_2}\left(x_3\right)=0.7 & f_{x_2}\left(x_4\right)=0.4 \\
f_{x_1}\left(x_1\right)=0.3 & f_{x_3}\left(x_2\right)=0.2 & f_{x_3}\left(x_3\right)=1 & f_{x_1}\left(x_4\right)=0.5 \\
f_{x_1}\left(x_1\right)=0.7 & f_{x_1}\left(x_2\right)=0.5 & f_{x_4}\left(x_3\right)=0.8 & f_{x_4}\left(x_4\right)=1
\end{array}
$$

Develop a comparison matrix, and determine which of the four wines tastes most like the expensive Chenin Blanc.

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Problem 9

. The Environmental Protection Agency (EPA) is faced with the challenge of cleaning up contaminated groundwaters at many sites around the country. In order to ensure an efficient cleanup process, it is crucial to select a firm that offers the best remediation technology at a reasonable cost. The EPA is deciding among four environmental firms. The professional engineers at the EPA compared the four firms and created a consensus matrix, shown here:

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

Compute the distance to Type fuzzy consensus.

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Problem 10

A manufacturing company is planning to purchase a lathe and is assessing the proposals from four lathe manufacturers. The company has developed a reciprocal relation for the four manufacturers based on the speed of delivery of the lathes and the cost. The relation is

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

Calculate the degree of preference measures, and the distance to Type I, Type II, and Type fuzzy consensus. Explain the differences between the distances to the three consensuses.

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Problem 11

Four methods for determining control are being considered for a navigation project. These methods include: Global Positioning System (GPS), Inertial Navigational System (INS), Surveying (S), and Astronomical Observation (ASTR). For the development of a network of control points, which will improve the accuracies of various applications that can use these points as a control point for their own networks, experts were asked to determine a reciprocal relation for these four methods. Generally, GPS is preferred to INS because of its superior long-term stability of measurements and it is not necessary to travel from one network point to another. GPS is somewhat less preferred than S because it cannot match the accuracy of S , which is important for a control network. GPS is much preferred over ASTR because it requires much less time and the skill requirement for ASTR is significant. INS is generally about the same as S; INS is preferred over ASTR for the same reasons as GPS. While S is somewhat less preferred than ASTR because both are slow and labor intensive, ASTR can generally offer better results and the equipment is not as expensive.

Using the following reciprocal relation determine the average fuzziness, average certainty, and the distance to consensus for Type I, II, and fuzzy consensus.

$$
\mathrm{R}=\begin{gathered}
\\
\text { GPS } \\
\text { INS } \\
\mathrm{S} \\
\text { AST }
\end{gathered}\left[\begin{array}{cccc}
\text { GPS } & \text { INS } & \mathrm{S} & \text { AST } \\
0 & 0.6 & 0.4 & 0.8 \\
0.4 & 0 & 0.5 & 0.7 \\
0.6 & 0.5 & 0 & 0.4 \\
0.2 & 0.3 & 0.6 & 0
\end{array}\right]
$$

Victor Salazar

Numerade Educator

Problem 12

A chemical plant reactor has yields lower than expected because the reactor is getting old. There are four feasible alternatives to solve the problem:

$$
\begin{aligned}
& A_1=\text { Buy a new reactor and replace the old one. } \\
& A_2=\text { Buy a used reactor and replace the old one. } \\
& A_3=\text { Add a new smaller unit at the end of the reactor to complete the reaction } \\
& \quad \text { to the expected yields. } \\
& A_4=\text { Do major repair to the old reactor. }
\end{aligned}
$$

Each alternative has its own advantages and disadvantages according to cost, maintainability, and physical space available in the plant. The engineers involved in selecting one of the options have created a relation to show their consensus:

$$
\mathrm{R}=\begin{gathered}
\\
\mathrm{A}_1 \\
\mathrm{~A}_2 \\
\mathrm{~A}_3 \\
\mathrm{~A}_4
\end{gathered}\left[\begin{array}{cccc}
\mathrm{A}_1 & \mathrm{~A}_2 & \mathrm{~A}_3 & \mathrm{~A}_4 \\
0 & 0.8 & 0.5 & 0.3 \\
0.2 & 0 & 0.7 & 0.4 \\
0.5 & 0.3 & 0 & 0.1 \\
0.7 & 0.6 & 0.9 & 0
\end{array}\right]
$$
Find the average fuzziness, average certainty, distance to consensus, and distances to consensus for a Type I, Type II, and Type fuzzy.

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Problem 13

An automotive manufacturing company is buying vision sensors for its assembly verification to its client. The engineering team in the company has done some pairwise comparisons among four types of sensors ( S ); the results of these comparisons are given by the consensus relation matrix below:

$$
\mathrm{R}=\begin{gathered}
\mathrm{S}_1 \\
\mathrm{~S}_2 \\
\mathrm{~S}_3 \\
\mathrm{~S}_4
\end{gathered}\left[\begin{array}{cccc}
\mathrm{S}_1 & \mathrm{~S}_2 & \mathrm{~S}_3 & \mathrm{~S}_4 \\
0 & 0.6 & 0.4 & 0.8 \\
0.4 & 0 & 0.5 & 0.7 \\
0.6 & 0.5 & 0 & 0.4 \\
0.2 & 0.3 & 0.6 & 0
\end{array}\right]
$$

Calculate the degree of preference measures, and the distance to Type I, Type II, and Type fuzzy consensus.

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Problem 14

A carcinogen, trichloroethylene (TCE), has been detected in soil and groundwater at levels higher than the EPA maximum contaminant levels (MCLs). There is an immediate need to remediate soil and groundwater. Three remediation alternatives - (1) pump and treat with air stripping (PTA), (2) pump and treat with photooxidation (PTP), and (3) bioremediation of soil with pump and treat and air stripping (BPTA) - are investigated.

The objectives are these: cost $\left(\mathrm{O}_1\right)$, effectiveness $\left(\mathrm{O}_2\right.$, capacity to reduce the contaminant concentration), duration $\left(\mathrm{O}_3\right)$, and speed of implementation $\left(\mathrm{O}_4\right)$. The ranking of the alternatives on each objective are given as follows:

$$
\begin{aligned}
& \mathrm{O}_1=\left\{\frac{0.7}{\text { PTA }}+\frac{0.9}{\text { PTP }}+\frac{0.3}{\text { BPTA }}\right\} \\
& \mathrm{O}_2=\left\{\frac{0.4}{\text { PTA }}+\frac{0.6}{\text { PTP }}+\frac{0.8}{\text { BPTA }}\right\} \\
& \mathrm{O}_3=\left\{\frac{0.7}{\text { PTA }}+\frac{0.3}{\text { PTP }}+\frac{0.6}{\text { BPTA }}\right\} \\
& \mathrm{O}_4=\left\{\frac{0.8}{\text { PTA }}+\frac{0.5}{\text { PTP }}+\frac{0.5}{\text { BPTA }}\right\}
\end{aligned}
$$

The preferences for each objective are $\mathrm{P}=\{0.6,0.8,0.7,0.5\}$. Determine the optimum choice of a remediation alternative.

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Problem 15

Evaluate three different approaches to controlling conditions of an aluminum smelting cell (with respect to voltage across the cell and alumina concentration in the bath). The control approaches are

$$
\begin{aligned}
& a_1=\text { AGT: aggressive control tuning (very reactive) } \\
& a_2=\text { MOD: moderate control tuning (mildly reactive) } \\
& a_3=\text { MAN: essentially manual operation (very little computer control) }
\end{aligned}
$$

There are several objectives to consider:
$\mathrm{O}_1$ : minimum power consumption (power/lb of aluminum produced)
$\mathrm{O}_2$ : overall operating stability
$\mathrm{O}_3$ : minimum environmental impact
The control approaches are rated as follows:

$$
\begin{aligned}
& \mathrm{O}_1=\left\{\frac{0.7}{\mathrm{AGT}}+\frac{0.6}{\mathrm{MOD}}+\frac{0.3}{\mathrm{MAN}}\right\} \\
& \mathrm{O}_2=\left\{\frac{0.45}{\mathrm{AGT}}+\frac{0.8}{\mathrm{MOD}}+\frac{0.6}{\mathrm{MAN}}\right\} \\
& \mathrm{O}_3=\left\{\frac{0.5}{\mathrm{AGT}}+\frac{0.62}{\mathrm{MOD}}+\frac{0.4}{\mathrm{MAN}}\right\}
\end{aligned}
$$

The preferences are given by $b_1=0.8, b_2=0.5$, and $b_3=0.6$. What is the best choice of control?

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Problem 16

In the tertiary treatment process for wastewater, the disinfection process is an important procedure that focuses on the destruction of disease-causing organisms. There are a lot of disinfection technologies available; of these three popular methods for disinfecting are to use: chlorine ( Cl ), ozone $(\mathrm{Oz})$, or UV radiation (UV). A new wastewater treatment plant is to be built and the designers are having difficulty selecting a disinfecting method and thus elect to use a multiobjective decision approach. It is concluded that the selection of a disinfection method should be based on: efficiency and performance (EP), availability of large quantities of the disinfectants and reasonable prices (Av), maintenance and operation (MO), and environmental impact (Ev). The sets of alternatives (A), objectives (O), and preferences (P) are shown below. Using the ratings given for each objective and the preference specified by the facility owner, make a decision on which disinfection technology to use.

$$
\begin{aligned}
& \mathrm{A}=\{\mathrm{C} 1, \mathrm{Oz}, \mathrm{UV}\}=\left\{a_1, a_2, a_3\right\} \\
& \mathrm{O}=\{\mathrm{EP}, \mathrm{Av}, \mathrm{MO}, \mathrm{Ev}\}=\left\{\mathrm{O}_1, \mathrm{O}_2, \mathrm{O}_3, \mathrm{O}_4\right\} \\
& \mathrm{P}=\left\{b_1, b_2, b_3, b_4\right\}=\{0.8,0.9,0.6,0.5\}
\end{aligned}
$$

Objectives:

$$
\begin{aligned}
& \mathrm{O}_1=\left\{\frac{0.8}{a_1}, \frac{0.9}{a_2}, \frac{0.7}{a_3}\right\}, \quad \mathrm{O}_2=\left\{\frac{0.9}{a_1}, \frac{0.4}{a_2}, \frac{0.5}{a_3}\right\}, \quad \mathrm{O}_3=\left\{\frac{0.8}{a_1}, \frac{0.7}{a_2}, \frac{0.7}{a_3}\right\} \\
& \mathrm{O}_4=\left\{\frac{0.5}{a_1}, \frac{0.8}{a_2}, \frac{0.9}{a_3}\right\}
\end{aligned}
$$

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Problem 17

For environmental modeling, remote sensing data play an important role in the data acquisition. Researchers must decide which type of sensor data best meet their preferences. Among the many alternative sensors available, the list of candidates has been reduced to three: LANTSAT 7 (LS7), GOES (GS), and TERRA (TA). The researchers have defined four objectives that impact their decision: (1) cost of the data (COST), (2) time to deliver data (TIME), (3) resolution of the data collected (RES), and (4) time for the sensor to return to the same spot cycle (CT). There was some disagreement as to how to define the importance of each objective in the preference set so the researchers decided to define two sets of preferences, $P_1$ and $P_2$.

Alternatives: $\mathrm{A}=\{\mathrm{LS} 7, \mathrm{GS}, \mathrm{TA}\}$
Objectives: $O=\{$ COST, TIME, RES, CT $\}$

$$
\begin{array}{ll}
\text { Preferences : } & P_1=\left\{b_1, b_2, b_3, b_4\right\}=\{0.8,0.4,0.8,0.7\} \\
& P_2=\left\{b_1, b_2, b_3, b_4\right\}=\{0.4,0.6,0.4,0.5\}
\end{array}
$$
The degree of membership of each alternative in the objectives is as follows:

$$
\begin{aligned}
& \mathrm{O}_1=\left\{\frac{0.2}{\mathrm{LS} 7}, \frac{0.8}{\mathrm{GS}}, \frac{0.4}{\mathrm{TA}}\right\}, \quad \mathrm{O}_2=\left\{\frac{0.6}{\mathrm{LS} 7}, \frac{1}{\mathrm{GS}}, \frac{0.2}{\mathrm{TA}}\right\}, \quad \mathrm{O}_3=\left\{\frac{1}{\mathrm{LS} 7}, \frac{0.4}{\mathrm{GS}}, \frac{0.8}{\mathrm{TA}}\right\} \\
& \mathrm{O}_4=\left\{\frac{0.8}{\mathrm{LS} 7}, \frac{0.7}{\mathrm{GS}}, \frac{0.2}{\mathrm{TA}}\right\}
\end{aligned}
$$

Find the decision for each preference.

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Problem 18

In the city of Calgary, Alberta, subdivisions constructed before 1970 were not required to retain overland storm-water flow on a site during major storm events to the level that has been accepted under current design criteria. In order to properly mitigate flooding and property damage in older subdivisions prone to flooding they are being upgraded based on technical feasibility and public acceptance of the works. Presently a subdivision is being considered for an upgrade of its storm-water sewer. It has been determined that there are two different methods to achieve the mitigation, either larger storm sewers have to be installed through the affected neighborhoods (pipe network) or storm-water retention facilities (pond) have to be built close enough to the neighborhood to reduce the flood threat. The mitigation alternatives (A) and the considered impacts or objectives $(\mathrm{O})$ are described below.

Alternatives: $\mathrm{A}=\{$ pipe, pond $\}$
Objectives: Additional land required $\left(\mathrm{O}_1\right)$, Cost $\left(\mathrm{O}_2\right)$, Flood damage $\left(\mathrm{O}_3\right)$, Public acceptance $\left(\mathrm{O}_4\right)$, and Environmental constraints $\left(\mathrm{O}_5\right)$ :

$$
\mathrm{O}=\left\{\mathrm{O}_1, \mathrm{O}_2, \mathrm{O}_3, \mathrm{O}_4, \mathrm{O}_5\right\}
$$

Based on previous experience with other subdivisions the city design engineer has determined the following ratings for this subdivision:

$$
\begin{aligned}
& \mathrm{O}_1=\left\{\frac{0.8}{\text { pipe }}, \frac{0.6}{\text { pond }}\right\}, \quad \mathrm{O}_2=\left\{\frac{0.8}{\text { pipe }}, \frac{0.4}{\text { pond }}\right\}, \quad \mathrm{O}_3=\left\{\frac{0.6}{\text { pipe }}, \frac{0.8}{\text { pond }}\right\} \\
& \mathrm{O}_4=\left\{\frac{0.4}{\text { pipe }}, \frac{0.9}{\text { pond }}\right\}, \quad \mathrm{O}_5=\left\{\frac{0.8}{\text { pipe }}, \frac{0.5}{\text { pond }}\right\}
\end{aligned}
$$

The city council has given the administration the following preference values for each objective. Using the above objectives and preferences determine which system to use for this subdivision:

$$
P=\left\{\mathrm{b}_1, \mathrm{~b}_2, \mathrm{~b}_3, \mathrm{~b}_4, \mathrm{~b}_5\right\}=\{0.6,0.4,0.6,0.7,0.6\}
$$

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01:28

Problem 19

A company produces printed circuit boards as a subcomponent for a system that is integrated (with other subcomponents) by another company. The system integration company cannot give precise information on how many PC boards it needs other than "approximately 10,000 ." It may require more or less than this number. The PC board manufacturer has three courses of action from which to choose: (1) build somewhat less than $10,000 \mathrm{PC}$ boards, $\mathrm{A}_1$; (2) build approximately $10,000 \mathrm{PC}$ boards, $\mathrm{A}_2$; and (3) build somewhat more than $10,000 \mathrm{PC}$ boards, $\mathrm{A}_3$.
The systems integration company will need the PC boards to meet the demand for its final product. The following are the three fuzzy states of nature:
1. Low demand, $\mathrm{D}_1$
2. Medium demand, $\mathrm{D}_2$
3. High demand, $D_3$

The utility function is given in this table:
There are six discrete states of nature, $s_1-s_6$, on which the fuzzy states are defined. The membership functions for the fuzzy states and the prior probabilities $p\left(s_i\right)$ of the discrete states are shown in the following table:
The demand for the system integrator's product is related to the growth of refineries, as the final product is used in refineries. The new samples of refinery growth information are $x$; and $\mathrm{M}_i$ are the fuzzy sets on this information, defined as
1. Low growth, $\mathrm{M}_1$
2. Medium growth, $\mathrm{M}_2$
3. High growth, $\mathrm{M}_3$
The likelihood values for the probabilistic uncertain information for the data samples are shown here:
The likelihood values for the probabilistic perfect information for the data samples are shown next:
For the information just presented, compare the following for perfect and imperfect information:
(a) Posterior probabilities of fuzzy state $2\left(\mathrm{D}_2\right)$ given the fuzzy information $3\left(\mathrm{M}_3\right)$.
(b) Conditional expected utility for action $1\left(\mathrm{~A}_1\right)$ and fuzzy information $2\left(\mathrm{M}_2\right)$.

Dominador Tan

Numerade Educator

Problem 20

In a particular region a water authority must decide whether to build dikes to prevent flooding in case of excess rainfall. Three fuzzy courses of action may be considered:
1. Build a permanent dike $\left(A_1\right)$.
2. Build a temporary dike $\left(\mathrm{A}_2\right)$.
3. Do not build a dike ( $\mathrm{A}_3$ ).

The sets $A_1, A_2$, and $A_3$ are fuzzy sets depending on the type and size of the dike to be built. The utility from each of these investments depends on the rainfall in the region. The crisp states of nature, $\mathrm{S}=\left\{s_1, s_2, s_3, s_4, s_5\right\}$, are the amount of total rainfall in millimeters in the region. The utility for each of the alternatives has been developed for three levels of rainfall, (1) low ( $F_1$ ), (2) medium ( $F_2$ ), and (3) heavy ( $F_3$ ), which are defined by fuzzy sets on $S$. The utility matrix may be given as follows:
The membership functions of $E_1, F_2, F_3$, and the prior probabilities are given here:
Let $\mathrm{X}=\left\{x_1, x_2, x_3, x_4\right\}$ be the set of amount of rainfall in the next year. This represents the new information. The conditional probabilities $p\left(x_j \mid s_i\right)$ for probabilistic uncertain information are as given below:
Consider a fuzzy information system,

$$
\underset{\sim}{\mathrm{M}}=\left\{\mathrm{M}_1, \mathrm{M}_2, \mathrm{M}_3\right]
$$

where $\quad \mathrm{M}_1=$ rainfall is less than approximately 35 mm
$\mathrm{M}_2$ = rainfall is equal to approximately 35 mm
$\mathrm{M}_3=$ rainfall is greater than approximately 35 mm
The membership functions for the new fuzzy information that satisfy the orthogonality condition are given here:
Determine the following:
(a) Posterior probabilities for fuzzy state $\mathrm{F}_2$ and fuzzy information $\mathrm{M}_1$, and for fuzzy state $\mathrm{F}_3$ and fuzzy information $\mathrm{M}_3$.
(b) Conditional expected utility of building a permanent dike $\left(\mathrm{A}_1\right)$ when fuzzy information $\mathrm{M}_3$ is given.

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Problem 21

Your design team needs to determine what level of technology to incorporate in a new product. As is usually the case, current technology is least expensive whereas the most advanced or leading-edge technology is the most expensive. A given technology usually comes down in price with time. The decision cycle of your project is several years. The team must decide what level of technology to incorporate in the product based on the future expected cost. If the technology is still expensive by the time the product goes to the market, the product will not sell. If you do not incorporate the latest affordable technology, your product may not be so advanced as that of the competition and therefore sales may be poor. Consider the following:

Actual discrete states of nature:
$s_1$ : Cost is low
$s_2$ : Cost is moderate
$s_3$ : Cost is high
Fuzzy actions:
$\mathrm{A}_1$ : Use current/well-established technology
$\widetilde{A}_2$ : Use newer/leading-edge/advanced technology
Fuzzy states on fuzzy information system, $\mu$ :
$\mathrm{M}_1$ : Cost is approximately the cost of implementing with current technology
$\widetilde{\mathbf{M}}_2$ : Cost is approximately 2 times the cost of the current technology
$\mathrm{M}_3$ : Cost is approximately 10 times the cost of current technology
Let $\mathrm{X}=\left\{x_1, x_2, x_3, x_4, x_5\right\}$ be the set of rates of increase in usage of advanced technology in the next term. Then we have the following:

Fuzzy states of nature:
$\mathrm{F}_1$ : Low cost
$\mathrm{F}_2$ : Medium cost
$\mathrm{F}_3$ : High cost
Prior probabilities:

$$
p\left(s_i\right)=\left[\begin{array}{c}
0.25 \\
0.5 \\
0.25
\end{array}\right] \begin{aligned}
& s_1 \\
& s_2 \\
& s_3
\end{aligned}
$$

Utility matrix:

Membership values for each orthogonal fuzzy state on the actual state system:

$$
\mu_{\mathrm{F}}=\left[\begin{array}{ccc}
s_1 & s_2 & s_3 \\
0.8 & 0.1 & 0 \\
0.2 & 0.8 & 0.2 \\
0 & 0.1 & 0.8
\end{array}\right]{\underset{\sim}{\mathrm{F}}}_3
$$

Membership values for each orthogonal fuzzy set on the fuzzy information system:

$$
\mu_{\mathrm{M}}=\left[\begin{array}{ccccc}
x_1 & x_2 & x_3 & x_4 & x_5 \\
1 & 0.5 & 0 & 0 & 0 \\
0 & 0.5 & 1 & 0.5 & 0 \\
0 & 0 & 0 & 0.5 & 1
\end{array}\right] \begin{aligned}
& \mathrm{M}_1 \\
& \mathrm{M}_2 \\
& \mathrm{M}_3
\end{aligned}
$$

Utility matrix for fuzzy information:

Likelihood values for probabilistic (uncertain) information for the data samples:

$$
p\left(x_i \mid s_k\right)=\left[\begin{array}{ccccc}
x_1 & x_2 & x_3 & x_4 & x_5 \\
0.1 & 0.25 & 0.15 & 0.35 & 0.15 \\
0.3 & 0.05 & 0.1 & 0.1 & 0.45 \\
0.2 & 0.4 & 0.35 & 0 & 0.05
\end{array}\right] \begin{aligned}
& \\
& s_1 \\
& s_2 \\
& s_3
\end{aligned}
$$

Likelihood values for probabilistic perfect information for the data samples:

$$
p\left(x_i \mid s_k\right)=\left[\begin{array}{ccccc}
x_1 & x_2 & x_3 & x_4 & x_5 \\
0 & 0 & 0 & 1 & 0 \\
0.4 & 0 & 0 & 0 & 0.6 \\
0 & 0.55 & 0.45 & 0 & 0
\end{array}\right] \begin{gathered}
\\
s_1 \\
s_2 \\
s_3
\end{gathered}
$$

(a) Determine the value of information for the fuzzy states and fuzzy actions for uncertain probabilistic information.
(b) Determine the value of information for the fuzzy states and fuzzy actions for perfect probabilistic information.

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