Introduo Inteligncia Artificial e Engenharia de Conhecimento

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AIA 2002 Malaga, Spain
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Curso de Pós-Graduação em Inteligência
Computacional
Paper 362-309
Adding Compensatory Terms in the Fuzzy Integral
for Synthetic Evaluation of Multi-attribute
Alternatives
Paulo Sergio S. Borges
Universidade Federal de Santa Catarina Florianópol
is, SC - BRASIL
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Outline of the presentation

• Synthetic evaluations of multi-attribute
  alternatives
• The Fuzzy Integral (FI)
• Adding compensatory terms to the Fuzzy Integral
• Conclusion

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• Synthetic evaluations of multi-attribute
  alternatives
• Why care about it?

The problem of ranking multi-attribute
alternatives according to the order of their
overall preferences can be frequently found in
real life situations.
For example, when choosing a spouse, one may may
rank the prospective candidates by subjectively
assigning scores to the attributes, or factors,
involved in the decision, such as Handsome? Intel
ligent? Professionally successful? Sincere? etc.
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Though infrequently, the ranking problem becomes
very simple if the concept of dominance can be
applied to all the alternatives, so that they can
be ordered, in terms of preference, from the best
to the worst.
An object x(i) dominates another object x(j) if
x(i) is not worse than x(j) in any of the factors
(attributes) involved and if it is better than
x(j) in at least one factor.
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It is easy to see that x(3) ? x(2) ? x(1) and
that x(4) ? x(2) ? x(1).
But how about x(3) and x(4) or x(4) and x(5) ?
Common solution Weighted mean
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It would be advantageous and convenient if to
each option an index could be ascribed,
consisting of a real number indicating its
overall degree of preference, a synthetic
evaluation.
Though often employed, the weighted mean may
render unsuitable as a general method for
obtaining reliable synthetic evaluations, mainly
because of its assumption of independence of the
factors.
An attractive option to overcome those
difficulties is the method of the fuzzy integral,
also called Sugenos Integral.
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2. The Fuzzy Integral Suppose V v1, v2, ...,
vn is a finite set of factors regarding all the
aspects, or qualities, of an object under
appraisal. A global evaluation is supposed to be
reached taking into consideration the assessment
of each aspect of the factor space V, which very
often do not have the same importance.
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A real number g(E) in the interval 0, 1 is
associated to each subset E ? V, indicating the
importance of the subset E in the global
evaluation and considering only the factors vi
that are present in it. The function g(.) obeys
the following conditions

• g(?)0
• g(V)1
• If E ? F ? V, then g(E) ? g(F), and it is a
  fuzzy measure on (V, 2V).

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In the objects judgment process, for each of the
factors vi ? V, a score h(vi) ? 0,1 is settled
by the examiner. Then, g(vi) and h(vi) are
compounded by means of a fuzzy integral, denoted
by , corresponding to a synthetic evaluation of
the item.
max min (min(h(v)), g(E)) E ? V v ?
E
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The function g(E) is a measure of how the subset
of factors E fulfills the importance concept,
characterized by g. g(E) should be estimated
independently and ahead of the examination of the
specific object under appraisal, and therefore
should consist in a widely accredited
postulation. h(v) regards the degree of
satisfaction of a particular subset of qualities
pertaining the object. Thus, h(v) corresponds to
the score to be attributed to that subset.
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The operation min h(v) over all the elements v ?
E supplies the extent to which those elements
satisfy h. The portion min (min(h(v)), g(E))
of the fuzzy integral matches the degree to which
all v ? E carry out the demands brought into the
process by the functions g(.) and h(.)
simultaneously. Finally, the max of all minima
concerning the subsets E is taken, yielding what
is called the objects synthetic evaluation.
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APPLYING THE FUZZY INTEGRAL A NUMERICAL EXAMPLE
Evaluation of a model of a car Features
considered performance (P), comfort (C) and
price (M). These are the qualities (attributes)
or factors vi, i1, 2, 3, of the object, the car,
so V P, C, M. The set function g(.)
appoints an importance measure to the subsets
that result from the possible combinations that
can be formed employing the individual factors P,
C and M.
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For the example, the values of g(.) have been
arbitrated as
The values of g(.) have neither the additive
nor the independency properties that are assumed
to be present in the traditional weighted mean
method.
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The attributes, or factors, performance and
price, when evaluated alone, have an importance
of g(P) 0.20 and g(M) 0.60, respectively,
but g(P, M) 0.9 ? g(P) g(M). The
assignment of zero to g(C) means that the
factor comfort, taken isolatedly, does not have
any importance in the items appraisal. Another
peculiar aspect of the method is that the factors
can interact, or exhibit a mutual reinforcement
effect when considered conjoinedly, as shown by
the pairs P, C, P, M and C, M.
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To each factor, a score h(vi) is adjudicated by
the evaluator, being assumed here as h(P)
0.4 h(C) 0.6 h(M) 0.7 min h(vi) h(P) 0.4,
corresponds to the factor performance min h(vi)
works as the parameter ? of a ?-level subset E?
of V, such that E? v h(vi) ? ? In other
words, E? is the collection of importance factors
whose scores are greater than or equal to min
h(vi), which correspond to the whole set of
importance factors, V. Therefore, g(E?0.4)
g(V) 1.
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The other values of g(E?) for ? ? min h(vi) are
determined in a similar way, so E?0.6 V
P C, M, g(E?0.6) g(C, M) 0.8
E?0.7 V P C M, g(E?0.7)
g(M) 0.6.
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maxminh(P), g(V), minh(C), g(C,M),
minh(M), g(M)
Substituting the numerical values in the fuzzy
integral depicted above, comes
maxmin0.4, 1, min0.6, 0.8, min0.7,
0.6
0.6, which stands for the overall rate, or
the synthetic evaluation of that particular
car.
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A brief discussion of the result
Regarding the example, it can be noted that the
resulting value of the synthetic evaluation is
not affected by some particular changes in the
scores. The synthetic evaluation that resulted
is 0.6, which may have been originated from
either h(C) ? g(C, M) min0.6, 0.8, or
h(M) ? g(M) min0.7, 0.6.
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The values of g(.), which have been previously
defined, are fixed for the evaluator and are
independent of the assignment of the scores h(.)
to the object under appraisal. The variable that
is actually governing the synthetic evaluation is
h(C) 0.6, because neither h(P) nor h(M) are
having any influence whatsoever on
until the benchmark established by h(C) ? g(C,
M) 0.6 is surpassed.
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For instance, suppose that in the example
presented, the value of h(P) had assumed a value
different from 0.4. In this case, h(P) could lie
anywhere between zero and h(C) 0.6, without
having any impact on the previous result, that
is, 0.6.
An equivalent scenario occurs when the score
appointed to the attribute Price, h(M), varies
between 0.6 and 1.0, thus not interfering in the
final value of 0.6, because minh(M),
g(M) is restrained by g(M) 0.6.
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This means that if two items A and B are compared
in regard to the importance factors of the
example, being assigned the scores, say,
h(PA) 0.4, h(CA) 0.6 and h(MA) 0.7
h(PB) 0.5, h(CB) 0.6 h(MB) 0.9
both would yield the same synthetic evaluation,
namely 0.6. Yet, according to the concept of
dominance, item B should be preferred to A.
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Adding compensatory terms to the Fuzzy Integral
In order to remedy that intuitively undue feature
of the original FI, a compensation is proposed,
consisting of two terms, namely an increment (?)
and a decrement (??). The reasoning concerning
this modification is as follows
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Because the fuzzy integral is defined as a
maximum operation over two or more minima, which
on their turn are always represented by
intersections of two fuzzy measures (score and
importance), the final value yielded by must be
necessarily equal to one of those minima, which
will be henceforth designated as the governing
value of . The governing value of
shall be indicated by the superscript ?.
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The variations of the scores concerning the
non-governing measures, which originally do not
affect if the results from the operations
min(h(.), g(.) are less than the governing
value, now will be taken into account, and will
be called relevant variations.
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The ranges of the relevant variations are always
restrained by (a) the governing value h?(.)
or g?(.), (b) the other scores h(.), and (c)
the importance factor g(.) that is associated
with the score of the non-governing value under
consideration inside the fuzzy integral.
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A relevant variation may imply in either an
increment or decrement over , depending
on whether it has originated from a value that is
greater or lesser than the governing value,
respectively, and taken as the difference between
it and the nearest of (a), (b) or (c) as
described above. In other words, the absolute
value of every relevant variation is the minimum
of the absolute values of the differences between
each of the attributes on its turn to be
considered and each one of (a), (b) or (c).
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The increment (?) and the decrement (??) to be
added to results from weighting each relevant
variation to a function of the importance factor
isolatedly associated to it. The final synthetic
evaluation of the item shall then be given by
? ??. To better clarify the application of the
compensatory terms as described above, it will be
worked through the example.
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Implementation of the Compensatory Terms The
governing value of is given by h(C) ? g(C,
M) min0.6, 0.7 h?(C) 0.6. To calculate
the relevant variations concerning the factors of
the non-governing values, it follows that (a)
h? (C) 0.6 (the governing value) (b) h(P)
0.4 h(M) 0.7 (c) g(V) 1 (because h(P)lt
h(C)lt h(M)), g(C, M) 0.8 and g(M) 0.6.
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For the example, the values of g(.) themselves
have been used as the weights for the relevant
variations that compound the increments and
decrements, but they could be any function of
g(.) . The attributes (factors) Performance and
Price respectively yield the decrement ?? (P)
because h(P) 0.4 lt h? (C) and the increment
? (M) because h(M) 0.7 gt h? (C). For the
decrements, the nearest value to the score for
which it is being accounted is the minimum of
(a), (b) and (c) for increments it is the
maximum.
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Relevant variation
Weight
??(P) g(P) ? (h(P) ? min(h? (C), h(M), g(V))
0.2 ? (0.4 ? 0.6) 0.04 ?(M)
g(M) ? (h(M) ? max(h? (C), h(P), g(M))
0.6 ? (0.7 ? 0.6) 0.06 Adding the
compensatory terms ??(P) and ?(M) to the
original computation of the fuzzy integral, the
final synthetic evaluation of the car of the
example is ??(P) ?(M) 0.6 0.04 0.06
0.62.
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For the other car, with the scores h(PB) 0.5,
h(CB) 0.6 and h(MA) 0.9 the final
synthetic evaluation (with the compensatory
terms) would result in 0.76 (instead of 0.6),
which seems to be consistent with the common
sense.
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CONCLUSION The original fuzzy integral is
susceptible to show regions of indifference
regarding some variations in the scores of the
attributes. With the proposed method, a hybrid
approach is performed, with the convenience of
making intermediary results possible. This system
of performing a synthetic evaluation of an item
with fuzzy measures can be advantageously
employed in applications of Expert Systems and
other Decision Support Systems involving
multi-attribute alternatives, as well as in NNs
and GAs.
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Thank you! Questions and comments welcome.
Universidade Federal de Santa Catarina Florianópol
is, SC - BRASIL