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Fuzzy Systems ToolBox, Mark Beale and Howard Demuth. Decision-Making Systems. Fuzzy Objectives and Crisp Constraints Fuzzy Objectives “My new shoes should be as pretty as possible.” “My new shoes should be as comfortable as possible” Crisp Constraint - PowerPoint PPT Presentation
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Decision-Making Systems
• Fuzzy Objectives and Crisp Constraints– Fuzzy Objectives
• “My new shoes should be as pretty as possible.”• “My new shoes should be as comfortable as
possible”
– Crisp Constraint• “My new shoes absolutely must not be smaller
than my feet.” (Crisp Statement.)• “My new shoes absolutely must not cost more
than I have with me.” (Crisp Statement.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Decision-Making Systems
– Objectives are typically fuzzy sets (different candidates meet objectives to varying degrees)
• Each shoe can be graded by observing and wearing them!
– Constraints can be represented by crisps sets (candidates are either adequate or not)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Decision-Making Systems: Example - MATLAB
– Automated Decision• Define possible options
– 5 shoes to choose» shoeS = 1:5;
• Describe the constraints– “My new shoes absolutely must not be smaller than
my feet.”» toosmallG = [0 1 0 0 0];
– “My new shoes absolutely must not cost more than I have with me.”
» toomuchG = [0 0 0 0 1];
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Decision-Making Systems: Example - MATLAB
– Automated Decision (Continued)• Obtain grades for each objective
– looking at the shoes and trying them on» looksG = [0.7 1.0 0.4 0.7 0.5];» ComfortG = [0.4 0.0 0.8 0.6 0.9];
• Grade each objective using the constraints and Objectives
» constraintG = not(or(toosmallG,toomuchG));» = [ 1 0 1 1 0 ]» the first, third, and fourth shoes meet all the
constraints.
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Decision-Making Systems: Example - MATLAB
– Automated Decision (Continued)• Find objective grade
– averaging their grades » for looks and comfort» objectiveG = mean([looksG; comfortG])» = [ 0.55 0.50 0.60 0.65 0.70 ]» Shoe five is the best, highest grade!
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Decision-Making Systems: Example - MATLAB
– Automated Decision (Continued)• Find the final grades
– Shoe Grades » Taking the intersection of the constraint and
objective grades» shoeG = and([constraintG; objectiveG]);» = 0.55 0.00 0.60 0.65 0.00» Shoe 2 and shoe 5 did not satisfy the constraints
• Finalize the decision– Pick the best shoe from the highest grade
» Shoe = highestgrade(shoeS,shoeG)» = 4 -----> shoe number 4 is picked!
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Objectives with Different Importance
– Not all objectives are equally important!•Hedging
– Making objectives more or less restrictive to differentiate among various importance.
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Objectives with Different Importance (Cont.)
– “The shoe should be somewhat attractive but very comfortable”
• objectiveG = mean([somewhat(lookG) ; very(comfortG)]) ; Or
• objectiveG = mean(hedge([looksG; comfortG], [0.5; 2]));
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Objectives with Different Importance (Cont.)
• Shoes = 1:5;• toosmallG = [ 0 1 0 0 0 ];• toomuchG = [ 0 0 0 0 1];• looksG = [ 0.7 1.0 0.4 0.7 0.5];• comfortG = [ 0.4 0.0 0.8 0.6 0.9];• constraintG = not (or (toosmallG, toomuchG));• objectiveG = mean(hedge([looksG; comfortG], [0.5; 2]));• shoeG = and([constraintG; objectiveG]);• Shoe = highestgrade(shoeS,shoeG) • = 3 ---> shoe 3, the more comfortable shoe, is
picked
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Paired Comparison weighting
•Use hedges to emphasize the importance of objectives - How?
–Estimate each hedge value, or–Estimate the importance of the
objectives and calculate hedge values from those estimates.
» i.e. taken in pairs, which of the two objectives is more important and to what extent?
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Paired Comparison weighting (Cont.)
•Example: a decision must be made using 3 criteria having importance 2.0, 1.0, and 1.7 from the interval [0:10]
Create an array containing relative importanceg = [2.0 1.0 1.7];
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Paired Comparison weighting (Cont.)
Create a pairwise comparison matrix by executing the following function:
function [p] = imp2pc(g)[gr,gc] = size(g);x = ones(gc,1)*g;p = x'-x+1;i = find(p < 1);pt = p';p(i) = 1 ./ pt(i);
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Paired Comparison weighting (Cont.)
%g = [2.0 1.0 1.7]
function [p] = imp2pc(g)
[gr,gc] = size(g); %Get size
x = ones(gc,1)*g; %Square matrix, x=% 2.0000 1.0000 1.7000
% 2.0000 1.0000 1.7000
% 2.0000 1.0000 1.7000
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Paired Comparison weighting (Cont.)
p = x'-x+1; % p = 1.0000 2.0000 1.3000
0 1.0000 0.3000 0.7000 1.7000 1.0000
i = find(p < 1); % i = [2,3,8]’pt = p'; % pt= 1.0000 0
0.70002.0000 1.0000 1.7000
1.3000 0.3000 1.0000
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Paired Comparison weighting (Cont.)
p(i) = 1 ./ pt(i); %p = 1.0000 2.0000 1.3000 0.5000 1.0000 0.5882
0.7692 1.7000 1.0000
% g = [2.0 1.0 1.7];
Pairwise comparison matrix:
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
1.0 2.0 1.30.5 1.0 0.5880.769 1.7 1.0
Objective #1Objective #2Objective #3
Objective #1 Objective#2 Objective #3
Pairwise Comparison to Hedge Algorithm
(O’Hagan)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Obtain eigenvectors and eigenvalues of pairwise matrix, p (nxn)– [v,d] = eig(p);
• Determine largest eigenvalue and its corresponding eigenvectors– e = diag(d);– i = find(e== max(e));
• Calculate normalize hedge value– h= (n*v(: ,i) / sum(v(: ,i)))’;
Pairwise Comparison to Hedge Algorithm
(O’Hagan)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• The MATLAB functionfunction [h,t] = pc2hed(p)[m,n] = size(p);[v,d] = eig(p);e = diag(d);i = find(e == max(e));i1 = i(1);h = (m * v(:,i1) ./ sum(v(:,i1)))';t = e(i);
Pairwise Comparison to Hedge Algorithm
(O’Hagan)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
function [h,t] = pc2hed(p)[m,n] = size(p); % m=3 n=3[v,d] = eig(p); % d – eigenvalues, v - eigenvectors
% v = -0.7320 0.7320 0.7320
% -0.3540 -0.1770 - 0.3066i -0.1770 + 0.3066i
% -0.5821 -0.2911 + 0.5041i -0.2911 - 0.5041i
% d = 3.0011 0 0
% 0 -0.0006 + 0.0577i 0 % 0 0 -0.0006 - 0.0577i
Pairwise Comparison to Hedge Algorithm
(O’Hagan)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
e = diag(d);% e = [ 3.0011 -0.0006 + 0.0577i -0.0006 -
0.0577i]’
i = find(e == max(e)); % i =1i1 = i(1);h = (m * v(:,i1) ./ sum(v(:,i1)))';% h = [1.3164 0.6367 1.0469]
t = e(i); % t = 3.0011
Importance to Hedge: Example
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• “comfort might be three times as important as looks”– i = [ 1.0 3.];
• calculate pairwise matrix, using imp2pc– p = imp2pc(i)
• Calculate hedge value, using pc2hed– h = pc2hed(p)
= 1 0.3333 3 1
= 0.5 1.5
Importance to Hedge: Example
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Shoes = 1:5;• toosmallG = [ 0 1 0 0 0 ];• toomuchG = [ 0 0 0 0 1];• looksG = [ 0.7 1.0 0.4 0.7 0.5];• comfortG = [ 0.4 0.0 0.8 0.6 0.9];• constraintG = not (or (toosmallG, toomuchG));
• i =[ 1.0 3.0]; %Emphasize comfort• h = imp2hed(i); %Calculate hedge value• objectiveG = mean(hedge([looksG; comfortG], h ));• shoeG = and([constraintG; objectiveG]);• Shoe = highestgrade(shoeS,shoeG) • = 3 ---> shoe 3, the more comfortable shoe, is picked
Perron Decision Making
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Treats importances and grades in the same way– Obtain Perron importances
• Calculate hedge value for importances
– Obtain Perron grades• hedge value from grades
– Combine grades and importances by multiplying the Perron importances by the Perron grades and normalizing the result.
Perron Decision Making: Example
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
– i = [ 1.0 3.0];– h = imp2hed(I)
• = 0.5 1.5
– h2 = imp2hed([looksG; comfortG])• = 1.0217 1.3036 0.7922 1.0217 0.8607• 0.8640 0.6468 1.1894 1.0128 1.2870
– ShoeG = normh(h*h2)• = 0.7653 0.6871 0.9235 0.8599
1.000
Zimmerman-Zysno Decision Making
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Defined “Compensatory AND”– Acts as a generalization of the algebraic
sum [ora] and product [anda] , i.e. • andc(c,g1,g2) = anda(g1,g2)^(1-c) *
ora(g1,g2)^c• WHERE c = 0 . . 1; and• c = 1 mean “high optimism”,• c = 0 mean “low optimism”
Zimmerman-Zysno Decision Making (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• An optimism level of 1.0 is like saying– “the criterion with the highest grade
is most important”
• An optimism level of 0.0 is like saying– “ we need to cover all bases.”
Zimmerman-Zysno Decision Making (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Example– Assume two criterion for three alternatives
• G = [0.3 0.9 0.4 ; 0.4 0.1 0.5]• = 0.3 0.9 0.4• 0.4 0.1 0.5
– Try optimism level of 0.0• X = andc(0.0, G) = 0.12 0.09 0.2• The third alternative is best because its
grades have high and fairly uniform values
Zimmerman-Zysno Decision Making (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Example– Assume two criterion for three alternatives
• G = [0.3 0.9 0.4 ; 0.4 0.1 0.5]• = 0.3 0.9 0.4• 0.4 0.1 0.5
– Try optimism level of 1.0• X = andc(1.0, G) = 0.58 0.91 0.7• The second alternative is preferred because its
list of objectives contains one high value that dominates.
Fuzzy Decision Making
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• All techniques discussed so far will do a good job on an arbitrary decision-making system.
• Picking the right technique, however, can maximize the correspondence between how a decision-making system operates and your intuitions about how it should operate!
Application: Pricing Decision
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Several legitimate and often contradictory objectives must be combined– President: “The price should be large”– Salesman: “The price should be small”– Marketing person : “The price absolutely must be
ending in 99.”– Manufacturing person: “The price should be greater
than the manufacturing cost”– Marketing person : The price should be less than our
competitor’s price.”
Application: Pricing Decision
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• priceS = minprice:maxprice;• obj1G = large(priceS);• obj2G = small(priceS);• obj3G = (rem(priceS,100) ==99)*0.5+0.5; %grade = 0.5 unless 99
=1• obj4G = greater(priceS,mancost);• obj5G = less(comprice);• objG = [obj1G; obj2G; obj3G; obj4G; obj5G;]• importances = [10 8 3 10 6];• Hedges = imp2hed(importantces);• hedgeG = hedge(objG, hedges);• priceG = mean(hedgeG);• Price = highestgrade(priceS, priceG);
Application: Pricing Decision
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Example Decision-Making system for price Setting
– Please answer the following questions about your product.
• What is a minimum price for the product? 150• What is a maximum price for the product?
1000• What is its manufacturing cost? 85• What is the competitor’s price? 700
Application: Pricing Decision
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• Please rate the importance of each objective.– Objective 1: “The price must be high for our shareholders.”
• How important is objective 1? [1-10] 9– Objective 2: “The price must be low to encourage sales.”
• How important is objective 2? [1-10] 6– Objective 3: “The price must end in 99.”
• How important is objective 3? [1-10] 7– Objective 4: “The price must exceed manufacturing cost.”
• How important is objective 4? [1-10] 10– Objective 5: “The price should beat our competitors.”
• How important is objective 5? [1-10] 6
• The final results: -----> The final price is : 699
Personal Decision: Choosing a Car
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT– Please answer the following questions:
• How many criteria? 4• What is criteria 1? High safety• What is criteria 2? Low maintenance• What is criteria 3? Low cost• What is criteria 4? High gas mileage
• Important rating for high safety? [0-10] 10• Important rating for Low maintenance? [0-10] 8• Important rating for Low cost? [0-10] 6• Important rating for High gas mileage? [0-10] 5
Personal Decision: Choosing a Car
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT (Cont.)• How many alternatives? 3• What is alternative 1? Car A• What is alternative 2? Car B• What is alternative 3? Car C
• high safety rating for car A? [0-10] 10• high safety rating for car B? [0-10] 7• high safety rating for car C? [0-10] 8
• Low maintenance rating for car A? [0-10] 8• Low maintenance rating for car B? [0-10] 9• Low maintenance rating for car C? [0-10] 6
Personal Decision: Choosing a Car
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT (Cont.)
• low cost rating for car A? [0-10] 7• low cost rating for car B? [0-10] 9• low cost rating for car C? [0-10] 7
• high gas mileage rating for car A? [0-10] 6• high gas mileage rating for car B? [0-10] 9• high gas mileage rating for car C? [0-10] 6
– Final Grades:• Car A 0.327578• Car B 0.472612 (Max)• Car C 0.215683 [Min]
Application: Selecting a Marketing Company
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• An industrious person has independently created software product and now has to choose between two marketing options:– #1 - A small marketing company that can be hired to
do the sales, product shipping, and billing. The author, however, would remain responsible for the cost of advertisements, technical support, enhancements, etc.
– #2 - A larger, well-established company that would handle all marketing, sell, shipping, billing, and first level-tech. Support. The author would responsible for second-level user questions and product enhancement.
Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• List of some of the concerns the author might have:– The author’s percentage of profit,– The marketing company’s responsibility for Technical
support,– The marketing company’s financial stability,– The marketing company’s experience in selling similar
products,– The marketing company’s long term commitment to the
author,– The marketing company’s responsiveness to author
concerns.
Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT– Please answer the following questions:
• How many criteria? 6• What is criteria 1? profit• What is criteria 2? responsibility• What is criteria 3? stability• What is criteria 4? experience• What is criteria 5? commitment• What is criteria 6? Responsiveness
Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT (Cont.)• Important rating for profit? [0-10] 7• Important rating for responsibility? [0-10] 6• Important rating for stability? [0-10] 8• Important rating for experience? [0-10] 3• Important rating for commitment? [0-10] 9• Important rating for responsiveness? [0-10] 9
• How many alternatives? 2• What is alternative 1? Small Company• What is alternative 2? Large Company
Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT (Cont.)• profit rating for Small Company? [0-10] 3• profit rating for Large Company? [0-10] 7• responsibility rating for Small Company? [0-10] 4• responsibility rating for Large Company? [0-10] 8• stability rating for Small Company? [0-10] 5• stability rating for Large Company? [0-10] 9• experience rating for Small Company? [0-10] 3• experience rating for Large Company? [0-10] 5
Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
• SAATY DECISION -MAKING SCRIPT (Cont.)• commitment rating for Small Company? [0-10] 8• commitment rating for Large Company? [0-10] 4• responsiveness rating for Small Company? [0-10] 8• responsiveness rating for Large Company? [0-10] 4
• Final Grades:– Small Company 0.01277 (Min)– Large Company 0.04801 (Max)
Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
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Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
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Application: Selecting a Marketing Company (Cont.)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
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