COMPOSITIONS
OF FUZZY PREFERENCES
ANNIBAL
PARRACHO SANT’ANNA[1]
Abstract
In this paper a probabilistic methodology to evaluate finite sets of
alternatives under multiple criteria is developed. This methodology is based on
randomising initial classifications and determining the probability of choice
of each alternative as the best according to each criterion. Pessimistic and
optimistic points of view are determined. The possibilities of criteria with
different importance and statistical correlation between criteria are
considered.
Keywords: fuzzy preferences –
probability of choice – decision aid
1. Introduction
Petrovic and Petrovic (2002)
developed an approach for multicriteria decision-making that takes into account
the optimistic or pessimistic mood that the evaluator desires to assume. Other
important feature of their methodology is that not only the evaluation of each
alternative according to each criterion but also the weight previously
attributed to each criterion may have a linguistic form and a fuzzy
representation. The evaluation of the probability of each alternative being
chosen according to each criterion plays also an important role in it.
Sant’Anna and Sant’Anna (2001) have developed a
different approach that has the same basis on the transformation of initial
evaluations into probabilities of choice. Here we extend this approach to
provide optimistic and pessimistic extreme evaluations and propose a
probabilistic form of combining them. The advantage of this approach is that,
these final evaluations being set in strictly probabilistic terms, the
composition of the optimist and pessimist value is less susceptible to the
influence of scales of measurement.
The optimistic evaluation is here set in terms of
maximizing the probability of choice according to at least one criterion while
the pessimistic is set in terms of maximizing the probability of avoiding the
worst performance with respect to any criterion. The assumptions underneath
this optimistic evaluation are that any strategy to excellence is equally
satisfactory and an option is excellent whenever it presents the best results
under the criterion that favours it most. On the other hand, the pessimistic
idea is that failure to satisfy any criterion will disqualify the effort toward
excellence. Besides, while the optimistic evaluation is taken in comparison to
the observed options, to the pessimistic evaluation a threshold of worst
behaviour with respect to each criterion must also be set. These thresholds may
be derived from theoretical limits or from the observed performances.
In the following section, I describe the procedure
adopted to transform the initial evaluations into probabilities of choice.
Different approaches to randomisation are there discussed. To provide an
example, the modelling approach taken by Sant’Anna and Sant’Anna (2001), based
on independent, continuous and uniform membership functions with the same
range, derived from the observed ranges, is compared to other approaches to the
randomisation of a five options qualitative ranking.
In Section 3, the optimistic and pessimistic
composition approaches proposed are applied to Petrovic and Petrovic (2002)
example of three alternative options compared under seven criteria. Final
evaluations are calculated under the hypothesis of equally important criteria
and under a hypothesis of dominance. The results are similar to those obtained
by those authors. The possibility of dependence between the criteria is also
studied.
Finally, in Section 4, Petrovic and Petrovic (2002) proposal
of associating final measures to predetermined intermediate levels of pessimism
or optimism is further developed.
2. Fuzzy Preferences and Probabilities of
Choice
I believe that the precise form of quantifying
preference for a particular option in a set of possible options is through the
probability of its choice. But the probability of choice is difficult to access
directly. For each preference criterion there is a more natural form of
expliciting the relative position of the options: in terms of ranks, in terms
of values of numerical variables, such as cost or speed, or in terms of the
common language, such as low, moderate or high preference.
Preferences presented in linguistic terms suggest representation by fuzzy sets
determined through membership functions such as those employed by Petrovic and
Petrovic (2002). Randomisation of ranks is also easy to perform (Sant’Anna and
Sant’Anna, 2001). Even criteria naturally presented through numerical measures
always involve some imprecision. For instance, the preference represented by a
precise measurement such as the length of a section road to be built is in fact
less deterministic than that measurement. Even if there is no possibility of
future changes affecting the length, the inconvenience for the traveller
associated to the length, which is what in fact matters and the criterion
intends to measure, may be affected by unforeseeable traffic or scenic factors.
We may access the probabilities of choice through a
three stages procedure. First obtain a vector of preferences for the available
options in the easiest terms. Then transform these punctual measurements into
random variables. Finally derive from the distribution of these random
variables the probabilities of choice.
As in the first,
different alternatives are also available in the second stage, of modelling the
distribution of probabilities of the options evaluations. The initial
measurements provide reference points, from which immediately derive the mean
of the random distribution. Equalization assumptions may speed the process of
modelling the other parameters and simplify the future interpretation of
results. Sant’Anna and Sant’Anna (2001) develop assumptions of independence
between the distributions representing different options, symmetry around the
means and identical dispersion parameters. A uniform continuous distribution is
there also proposed as a basic option to be assumed whenever no contrary hints
on the distribution form are available. Once
a uniform distribution is assumed, determining the range will complete
modelling.
Range estimates
may also be derived from the initial measurements. The range should be
large enough to allow for change of position between the admissible options, so
that, if any two production units belong to the set under comparison, there
must be a nonnull probability of inversion of their positions. And, since this
probability should be small when the considered units are those with the
largest and smallest values, we can simplify matters by adding a small parcel
to the observed range to generate a common estimate for the range of the
distribution of each measure.
In this
article we will adopt the above assumptions and determine the range by adding
to the observed range a fraction of 1/(n-1) of it, where n is the number of
options. An algorithm to perform the computation of the probabilities of choice
under these assumptions is provided in the Appendix.
In Sant’Anna (2002), a comparison between uniform and
normal assumptions is presented. The distributional assumptions do not
substantially affect the results there obtained. Petrovic and Petrovic (2002)
employ discrete, asymmetric and different distributions to represent a
classification stated in terms of verbal evaluations. The probabilities of
choice may then be derived in the same fashion. We shall consider here the case
of 5 options with preferences verbally set as very low, low, moderate,
high and very high.
Petrovic and Petrovic (2002) represent these 5
classifications, in a fuzzy set with 5 values, through membership functions of
different shapes. In their representation, the graph of the membership function
to the random set corresponding to the linguistic classification very high
with minimum at 0 and maximum at 1, would be a parable passing through (0,0), (1,1)
and (1/2,1/4). Its algebraic expression is given by fvh(x) = x2.
To the term very low, symmetrically, the membership function is given by
fvl(x) = (1-x)2. To low and high correspond
linear membership functions with the same extremes of those corresponding to very
low and very high, respectively. Finally the graph of the membership
function corresponding to moderate has a triangular shape with vertices
at (0,0), (1/2,1) and (1,0). Employing a discrete support set with 5 points,
this corresponds to the probability distributions of Table 2.1.
|
|
lowest |
second
lowest |
median |
second
highest |
highest |
|
very
low |
8/15 |
3/10 |
2/15 |
1/30 |
0 |
|
low |
2/5 |
3/10 |
1/5 |
1/10 |
0 |
|
moderate |
0 |
1/4 |
1/2 |
¼ |
0 |
|
high |
0 |
1/10 |
1/5 |
3/10 |
2/5 |
|
very
high |
0 |
1/30 |
2/15 |
3/10 |
8/15 |
Petrovic and Petrovic (2002)
derive probabilities of choice of each of the five options by the minimum of
the probabilities of outranking other options. Thus, the probability of the
option verbally classified as very high to receive the
highest numerical classification is estimated by its probability of obtaining a
numerical classification highest than that given to the option with a
linguistic classification of high. And the
probabilities of choice of the other linguistic classifications are estimated
by their probabilities of obtaining a numerical classification highest than
that given to the option with the linguistic classification of very high.
Table
2.2 presents, in the first line, the probabilities of choice derived from
Petrovic and Petrovic (2002) approach. The second line presents the
probabilities of choice obtained by computing precisely the probabilities of
attributing to each fuzzy option the highest numerical values when applying the
membership functions of Table 2.1. By looking at the
linguistic classification to untie the numerical ties, each of these
probabilities may be calculated adding the products of the probabilities of
P[r(A) = c], for c varying along the support set, by the product of four
summands P[r(B) < c]+P[r(B) = c]1[A > B], obtained
by making B assume the four classifications different from A. And in the third line, we present the same
probabilities calculated from the randomisation of five equally spaced ranks
assuming the uniform distribution as above described and applying the algorithm
developed in the Appendix.
Among the three lines of
Table 2.2, the last one presents the closest resemblance to the geometric
distribution, that has been seen to adequately fit many real practice situations
(Lootsma, 1993). This is a possible advantage associated with the uniform
model. But the main point in favour of this distribution has
to do with its capability of representing ignorance when nothing is known about
how possible distortions affect the preferences elicited. A continuous
distribution reflects more accurately uncertainty about a given classification
than a discrete one and placing a classification as the centre of a uniform
distribution is the simplest way to utter a fuzzy preference for this
classification.
|
|
very low |
Low |
Moderate |
high
|
very high |
||||||
probability of passing the highest
|
4,8% |
9,0% |
20,8% |
57,7% |
75,7% |
|
||||||
|
probability of being the highest |
0,1% |
0,4% |
1,4% |
24,1% |
74,0% |
|
||||||
uniformly being the highest
|
0,3% |
2,5% |
10,3% |
27,7% |
59,1% |
|
||||||
3. Probabilistic Composition of Preferences
From the ordering according to different criteria in
terms of probabilities of choice we must pass to the global classification of
the options. This will be done here by a systematic procedure of composition of
the particular probabilities of choice into a global probability. This
aggregation procedure must consider at least three different aspects: the
relative importance of the criteria, the optimistic or pessimistic global approach
to the consequences of the choice and the correlation between the criteria.
3.1. The importance of the
Criteria
Determining weights to be attributed to the criteria
is a much harder task than applying them to classify the concrete options.
Simpler than trying to identify, besides values for the options according to
the criteria, values for the criteria, as tools to evaluate the same options,
is dividing the set of criteria into only two parts: criteria of primary and of
secondary importance. I propose here a simple hypercriterion to perform this
separation task. The subset of criteria of primary importance will be formed by
those criteria that cannot be replaced when computing the probabilities of the
options being chosen as the best. The second subset will be formed by those
criteria replaceable when considering probabilities of being the best and
individually important only when probabilities of avoiding bad performances are
computed.
3.2. Optimistic and Pessimist points of View
I propose to characterize an optimistic and a
pessimistic approach in terms of determining, respectively, optimal or not very
bad options. The approach is optimistic when the goal is optimising some
performance and pessimistic when what matters is avoiding the proximity of any
inferior threshold. The optimistic approach will result in preferring options
that present the best indicator according to some criterion. The pessimistic
approach will prefer options that do not present any bad evaluation.
Consider, for instance, a criterion given verbally in
terms of low, high, very low, very high and moderate satisfaction. If all the
alternatives come to be classified in the highly or very highly satisfactory
points, from a pessimistic point of view the differences should be given less
importance than if the options are classified in the low and very low terms. On
the other side, from the optimistic point of view, the difference between high
and very high satisfaction should be given much more importance than that given
to the difference between low and very low.
The optimistic or pessimistic approach must affect
only secondary criteria. Only after the options are classified according to its
probability of choice by the primary criteria, we will try to find, in the
secondary criteria, some large probability of choice or no large probability of
rejection according as an optimistic or pessimistic approach is taken.
Supposing, to simplify terms, all criteria presented in an increasing scale
from the less desirable to the most desirable situation, the optimistic
evaluation will then be put in terms of maximizing the probability of choice
according to all the primary criteria and at least one secondary criterion,
while the pessimistic evaluation will be put in terms of, besides maximizing all
primary criteria, not minimizing any secondary criterion.
If there is no distinction between the criteria, we
must attach primary importance to none. Then, in the optimistic approach, we
will classify the options by the probability of being chosen according to at
least one criterion. Analogously, in the pessimistic approach, we will classify
according to the probability of not minimizing any particular probability of
choice only.
A pessimist bound will also be added, through a
fictitious unit with the lowest admissible position according to each
criterion, one step below very low if the criterion is applied in verbal
terms.
3.3. Correlation
Independence between the criteria should be generally
desired, to make easier to evaluate according to them and consequently more
reliable the information gathered. But one advantage of the probabilistic
approach is that estimates of the correlation can be employed to improve the
calculation of the joint probabilities.
A strategy to estimate the probability of joint choices
can be based on the knowledge of the correlation coefficient between the
choices according to the criteria and the separate probabilities of choice
according to each criterion. The correlation coefficient between two Bernoulli
random variables is obtained, by definition, dividing the difference between
the probability of joint choice and the product of the probabilities of choice
according to each criterion by the square root of the product of four terms,
these probabilities of choice and their complements. From this we derive an
estimate for the joint dependent probability as the sum of the product of the
probabilities of choice according to each criterion with this square root
multiplied by an estimate of the correlation coefficient.
We combine below probabilistic preferences on
roads sections alternatives, assuming independence and, alternatively, assuming
a positive correlation between the choices according to length and maintenance
cost criteria. Two hypotheses are also considered with respect to the importance
attributed to the criteria. First we assume all criteria equally important.
Alternatively we apply the methodology to the case of one criterion, pollution,
being more important than all the others, as assumed by Petrovic and Petrovic
(2002). Computations are also performed under the optimistic and pessimistic
points of view above described.
3.4. Results
The data set analysed is reproduced in Table
3.1. Length, costs and pollution criteria are to be considered in reverse
terms. A last line is added to the original data, presenting the inferior
thresholds. Since there are three options, these thresholds were calculated,
for the numerical values, by increasing the observed range in one half. For the
evaluations in terms from very low to very high, made to
correspond to ranks 1 to 5, they were set as zero.
Table 3.2 presents the probabilities of choice
according to each criterion and Table 3.3 the probabilities of reaching the
lower threshold, under the uniform assumptions of Section 2.
Assuming independence and equal importance to all
criteria, option A2 is the best in the pessimistic point of view but option A3
is slightly better if we take the optimistic point of view. If the pollution
criterion is considered more important than the others, then option A2 is
clearly the best under any approach.
Conceivable correlation levels do not change the
classifications. Only if the correlation between length and maintenance cost
reaches 0.95, the advantage of option A3 under the optimistic point view with
equal importance disappears and option A2 becomes the best under any
composition of other assumptions. Table 3.4 presents the final probabilistic
classifications under the four different approaches considered, under
independence and under correlation of 0.95 between length and maintenance cost.
|
|
length
|
construction
cost |
maintenance
|
speed |
security |
impact |
pollution
|
|
option A1 |
16,2 |
156 |
16 |
12 |
high |
Moderate |
very low |
|
option A2 |
12,5 |
151 |
15 |
12 |
high |
High |
very low |
|
option A3 |
21,2 |
128 |
14 |
11 |
very high |
High |
low |
|
threshold |
23,7 |
170 |
17 |
10,5 |
0 |
0 |
0 |
|
|
length
|
construction
cost |
maintenance
|
speed |
Security |
impact |
pollution
|
|
option A1 |
25% |
4% |
3% |
49% |
23% |
14% |
41% |
|
option A2 |
72% |
10% |
22% |
49% |
23% |
43% |
41% |
|
option A3 |
3% |
86% |
75% |
1% |
53% |
43% |
27% |
|
|
length
|
construction
cost |
maintenance
|
speed |
Security |
impact |
pollution
|
|
option A1 |
2% |
19% |
22% |
0% |
2% |
3% |
0% |
|
option A2 |
0% |
10% |
3% |
0% |
2% |
0% |
0% |
|
option A3 |
30% |
0% |
0% |
22% |
0% |
0% |
2% |
|
|
INDEPENDENT
|
LENGHT AND MAINTENANCE HIGHLY
CORRELATED |
||||||
|
|
Equal importance |
pollution primary |
equal importance |
pollution primary |
||||
|
|
OPTIM. |
PESSIM. |
OPTIM. |
PESSIM. |
OPTIM. |
PESSIM. |
OPTIM. |
PESSIM. |
|
option A1 |
86% |
59% |
32% |
24% |
88% |
54% |
33% |
22% |
|
option A2 |
97% |
85% |
40% |
35% |
99% |
85% |
41% |
35% |
|
option A3 |
99% |
54% |
17% |
9% |
99% |
54% |
17% |
9% |
4.
Weighting Pessimism and Optimism
Petrovic
and Petrovic (2002) suggest combining the optimist and pessimist evaluations
and derive a pessimistic-optimistic index a from pessimistic-optimistic positions in a scale of five verbal
descriptions. The value of a
corresponding to each linguistic classification would be the probability of the
random number representing this classification appearing as the best in the
case of independent assignation of numerical values to the linguistic terms
according to predefined membership functions.
The n-th value of a in the scale from the pessimistic to the optimistic extreme would then
be given by the probability of the truly n-th best option appearing as the
best. The first value, representing the next to maximum optimistic approach,
with the identical uniform continuous membership function, would then be 0,74.
The value 1-a = 0,26 complementarily corresponds to weight the probability of not
being the worst in any criterion by the probability of a classification as very
low not generating the lowest numerical rank. A simplest approach would be to
consider 5 equally spaced values for a: 0,
0.25, 0.5, 0.75 and 1.
These a
values do not take into account the scale in which the final preferences
reflecting the two opposite points of view are set. Since our pessimistic
approach asks for satisfaction of a series of simultaneous conditions and the
optimistic point of view asks for satisfaction of alternative conditions, the
first values tend to be smaller than the later. One way to avoid the effect of
this difference is to calculate the final probabilities conditionally on the
probability that some option satisfy the respective conditions. In the
optimistic evaluation, this total probability is always equal to 1, while, in
the present case, for the independent equal importance assumption, the pessimistic
total is about 97%. Table 4.1, for this hypothesis of independent and equally
important criteria, presents the final conditional probabilities under
different prior preference levels for the optimistic and pessimistic
approaches.
It may also be interesting to determine the value of
alpha for which the choice changes, if such a change may occur. In our
application, under the hypotheses of independent and equally important
criteria, only when the preference for the optimistic point view is given by a
value of 95% or more the choice of option A3 would prevail.
|
|
0 |
0,05 |
0,25 |
0,5 |
0,75 |
0,95 |
1 |
|
option A1 |
60% |
62% |
67% |
73% |
80% |
85% |
86% |
|
option A2 |
88% |
88% |
90% |
93% |
95% |
97% |
97% |
|
option A3 |
55% |
58% |
66% |
77% |
88% |
97% |
99% |
5.
Final Comments
A general approach to take into
account uncertainty on evaluations and to compose with different levels of
pessimism and optimism was here developed on pure probabilistic terms. Other
forms of composition could be explored and other probability distributions
might be used to mirror more precise information on criteria importance or on
uncertainty. Particularly, the use of discrete random sets associated with the
empirical determination of the statistical model for the preferences can be
implemented without any changes in the general formulation proposed.
References
F. A. Lootsma (1993), Scale Sensitivity in the Multiplicative AHP and SMART, Journal of Multicriteria Decision Analysis,
2, 87-110.
S. Petrovic and R. Petrovic
(2002), A New Fuzzy Multi-criteria Methodology for Rankin of Alternatives, International
Transactions in Operational Research, 9, 73-84.
A. P. Sant’Anna and L. F. Sant’Anna,
Randomisation as a Stage in Criteria Combining. Proceedings of the VII ICIEOM, 248-256, Salvador, BR (2001).
A. P. Sant’Anna (2002), Data Envelopment
Analysis of Randomized Ranks, Pesquisa Operacional, to appear.
APPENDIX
Computation
of the probabilities of choice assuming independent uniform distributions with
the same range.
Let x(1),…x(n) denote the observed values in decreasing order and x(n+1)
the fictitious smallest value added to the sample. The probabilities will not
change if we subtract x(n+1) of every value and divide by the range x(1) –
x(n+1). So we can assume without loss of generality x(n+1) = 0, x(1) = 1 and
the range equal to 1.
Let M(i) denote the maximum x(i) +1/2 of the random distribution of the
(i)-th option evaluated and m(i) its minimum, equal to x(i)-1/2.
By the uniform assumption, the probability that the random value X(i)
representing the (i)-th option is the maximum can be computed by integrating
from m(i) to M(i) the conditional probability of that happening given X(i).
Since the conditional probability of X(i) being the maximum given X(i) = x is the
product of the probabilities of X(j) < x, for j ¹1, this integral will be the sum of n integrals, each one corresponding
to a subinterval of integration determined by the M(j). In each such
subinterval, there is a different number of options with nonnull probability of
surpassing X(i).
Consider first i = 1. The first subinterval goes from M(2) to M(1) and
along this subinterval the conditional probability is 1. Thus the first summand
is M(1)-M(2).
The second summand gives the integral from M(3) to M(2) of the
probability of X(2) assuming a value smaller than the integrand x(1). This
probability is given by x(1) – m(2) and its integral from M(3) to M(2) is given
by M(2)2/2 -M(3)2/2 +(M(2) –M(3))*m(2).
Analogously the third summand is the difference between three sums. The
parcels in the minuend of this difference are M(3)3/3, M(3)2/2
multiplied by m(2)+m(3) and M(3) multiplied by m2*m3. The parcels in
the subtrahend have the same form with M(3) replaced by M(4).
The same procedure will follow with increasing number of powers, until
reaching the lowest subinterval.
For i ¹ 1, the same conditioning approach and the subsequent partitioning
works. For i =2, n-1 integrals will have to be computed, since X(2) does not
take values in the subinterval (M(2),M(1)). For i = 3, n-2 integrals, and so
on.
To
speed computation we may also use the fact that the sum of all the products of
a number m among the k first m(j) is equal to the sum of all the products of
m-1 factors among the first k-1 multiplied by m(k) plus the sum of all products
of m factors in the same set of the first k-1.