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 importance
g = [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
0.7000
2.0000 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:
Objective #1
Objective #1 1.0
Objective #2 0.5
Objective #3 0.769
Objective#2
2.0
1.0
1.7
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Objective #3
1.3
0.588
1.0
Pairwise Comparison to Hedge
Algorithm (O’Hagan)
• 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)))’;
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Pairwise Comparison to Hedge
Algorithm (O’Hagan)
• The MATLAB function
function [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);
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Pairwise Comparison to Hedge
Algorithm (O’Hagan)
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
%
%
-0.1770 - 0.3066i
-0.2911 + 0.5041i
-0.1770 + 0.3066i
-0.2911 - 0.5041i
% d = 3.0011
0
0
%
%
-0.0006 + 0.0577i
0
0
-0.0006 - 0.0577i
-0.3540
-0.5821
0
0
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Pairwise Comparison to Hedge
Algorithm (O’Hagan)
e = diag(d);
% e = [ 3.0011 -0.0006 + 0.0577i -0.0006 - 0.0577i]’
i = find(e == max(e));
% i =1
i1 = i(1);
h = (m * v(:,i1) ./ sum(v(:,i1)))';
% h = [1.3164 0.6367 1.0469]
t = e(i); % t = 3.0011
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Importance to Hedge: Example
• “comfort might be three times as important as looks”
– i = [ 1.0 3.];
• calculate pairwise matrix, using imp2pc
– p = imp2pc(i)
= 1
3
0.3333
1
• Calculate hedge value, using pc2hed
– h = pc2hed(p)
= 0.5 1.5
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Importance to Hedge: Example
•
•
•
•
•
•
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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Perron Decision Making
• 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.
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Perron Decision Making: Example
– 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Zimmerman-Zysno Decision
Making
• 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”
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Zimmerman-Zysno Decision
Making (Cont.)
• 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.”
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Zimmerman-Zysno Decision
Making (Cont.)
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Zimmerman-Zysno Decision
Making (Cont.)
• 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 Systems ToolBox, Mark Beale and Howard Demuth
Fuzzy Decision Making
• 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!
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Pricing Decision
• 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.”
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Pricing Decision
•
•
•
•
•
•
•
•
•
•
•
•
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);
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Pricing Decision
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Pricing Decision
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Personal Decision: Choosing a Car
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Personal Decision: Choosing a Car
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Personal Decision: Choosing a Car
• 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]
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company
• 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.
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company (Cont.)
• 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.
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company (Cont.)
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company (Cont.)
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company (Cont.)
• 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
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company (Cont.)
• 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)
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
Application: Selecting a Marketing
Company (Cont.)
Grades For Each Criteria
1
0.9
0.8
0.7
Grade
0.6
0.5
0.4
0.3
0.2
0.1
0
1
1.2
1.4
Alternative
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
1.6
1.8
2
Application: Selecting a Marketing
Company (Cont.)
Hedged Grades For Each Criteria
1
0.9
0.8
0.7
Grade
0.6
0.5
0.4
0.3
0.2
0.1
0
1
1.2
1.4
1.6
Alternatives
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
1.8
2
Application: Selecting a Marketing
Company (Cont.)
Final Grades
1
0.9
0.8
0.7
Grade
0.6
0.5
0.4
0.3
0.2
0.1
0
1
1.2
1.4
1.6
Alternatives
Fuzzy Systems ToolBox, Mark Beale and Howard Demuth
1.8
2