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Fuzzy Reasoning in Decision Making and Optimization

Christer Carlsson, Robert Fullér, Fuzzy Reasoning in Decision Making and Optimization. Studies in Fuzziness and Soft Computing Series, Springer-Verlag, 2002.

Abstract:

Many decision-making tasks are too complex to be understood
quantitatively, however, humans succeed by using knowledge that is
imprecise rather than precise.

{\em Fuzzy logic} resembles human reasoning in its use of imprecise
information to generate decisions. Unlike classical logic which requires a
deep understanding of a system, exact equations, and precise numeric
values, fuzzy logic incorporates an {\em alternative way of thinking}, which
allows modeling complex systems using a higher level of abstraction
originating from our knowledge and experience.

{\em Fuzzy logic} allows expressing this knowledge with subjective
concepts such as {\em very big} and a {\em long time} which are mapped
into exact numeric ranges. Since knowledge can be expressed in a more
natural by using fuzzy sets, many decision (and engineering) problems can
be greatly simplified.

{\em Fuzzy logic} provides an inference morphology that enables
approximate human reasoning capabilities to be applied to
knowledge-based systems. The theory of fuzzy logic provides a
mathematical strength to capture the uncertainties associated with human
cognitive processes, such as thinking and reasoning. The conventional
approaches to knowledge representation lack the means for representating
the meaning of fuzzy concepts. As a consequence, the approaches based
on first order logic do not provide an appropriate conceptual framework for
dealing with the representation of commonsense knowledge, since such
knowledge is by its nature both lexically imprecise and noncategorical.
The developement of fuzzy logic was motivated in large measure by the
need for a conceptual framework which can address the issue of lexical
imprecision. Some of the essential characteristics of fuzzy logic relate to
the following \cite{Zad92}: (i) In fuzzy logic, exact reasoning is viewed as a
limiting case of approximate reasoning; (ii) In fuzzy logic, everything is a
matter of degree; (iii) In fuzzy logic, knowledge is interpreted a collection of
elastic or, equivalently, fuzzy constraint on a collection of variables; (iv)
Inference is viewed as a process of propagation of elastic constraints; and
(v) Any logical system can be fuzzified.

There are two main characteristics of fuzzy systems that give them better
performance for specific applications: (i) Fuzzy systems are suitable for
uncertain or approximate reasoning, especially for systems with
mathematical models that are difficult to derive; and (ii) Fuzzy logic allows
decision making with estimated values under incomplete or uncertain
information.

This monograph summarizes the authors' works in the nineties on fuzzy
optimization and fuzzy reasoning.

The book is organized as follows. It begins, in Chapter 1 'Fuzzy Sets and
Fuzzy Logic', with a short historical survey of development of fuzzy thinking
and progresses through an analysis of the extension principle, in which we
derive exact formulas for t-norm-based operations on fuzy numbers of
$LR$-type, show a generalization of Nguyen's theorem \cite{Ngu78} on
$\alpha$-level sets of sup-min-extended functions to sup-t-norm-extended
ones and provide a fuzzy analogue of Chebyshev's theorem \cite{Che67}.

Fuzzy set theory provides a host of attractive aggregation connectives for
integrating membership values representing uncertain information. These
connectives can be categorized into the following three classes {\em union,
intersection} and {\em compensation} connectives. Union produces a high
output whenever any one of the input values representing degrees of
satisfaction of different features or criteria is high. Intersection connectives
produce a high output only when all of the inputs have high values.
Compensative connectives have the property that a higher degree of
satisfaction of one of the criteria can compensate for a lower degree of
satisfaction of another criteria to a certain extent. In the sense, union
connectives provide full compensation and intersection connectives provide
no compensation. In a decision process the idea of {\em trade-offs}
corresponds to viewing the global evaluation of an action as lying between
the {\em worst} and the {\em best} local ratings. This occurs in the presence
of conflicting goals, when a compensation between the corresponding
compabilities is allowed. Averaging operators realize trade-offs
\index{trade-offs } between objectives, by allowing a positive compensation
between ratings. \\ In Chapter 2 'Fuzzy Multicriteria Decision Making', we
illustrate the applicability of Ordered Weighted Averaging \cite{Yag88}
operators to a doctoral student selection problem. In many applications of
fuzzy sets such as multi-criteria decision making, pattern recognition,
diagnosis and fuzzy logic control one faces the problem of weighted
aggregation. In 1994 Yager \cite{Yag94a} discussed the issue of weighted
$\min$ and $\max$ aggregations and provided for a formalization of the
process of importance weighted transformation. We introduce fuzzy
implication operators for importance weighted transformation containing as
a subset those ones introduced by Yager. \\ Then we discuss the issue of
weighted aggregations and provide a possibilistic approach to the process
of importance weighted transformation when both the importances
(interpreted as {\em benchmarks}) and the ratings are given by symmetric
triangular fuzzy numbers. Furthermore, we show that using the possibilistic
approach (i) small changes in the membership function of the importances
can cause only small variations in the weighted aggregate; (ii) the weighted
aggregate of fuzzy ratings remains stable under small changes in the
{\em nonfuzzy} importances; (iii) the weighted aggregate of crisp ratings still
remains stable under small changes in the crisp importances whenever
we use a continuous implication operator for the importance weighted
transformation.

In 1973 Zadeh \cite{Zad73} introduced the compositional rule of inference
and six years later \cite{Zad79} the theory of approximate reasoning. This
theory provides a powerful framework for reasoning in the face of imprecise
and uncertain information. Central to this theory is the representation of
propositions as statements assigning fuzzy sets as values to variables. In
Chapter 3 'Fuzzy Reasoning', we show two very important features of the
compositional rule of inference under triangular norms. Namely, we prove
that (i) if the t-norm defining the composition and the membership function
of the observation are continuous, then the conclusion depends
continuously on the observation; (ii) if the t-norm and the membership
function of the relation are continuous, then the observation has a
continuous membership function. The stability property of the conclusion
under small changes of the membership function of the observation and
rules guarantees that small rounding errors of digital computation and small
errors of measurement of the input data can cause only a small deviation in
the conclusion, i.e. every successive approximation method can be applied
to the computation of the linguistic approximation of the exact conclusion.

Possibilisitic linear equality systems (PLES) are linear equality systems
with fuzzy coefficients, defined by the Zadeh's extension principle.
Kov\'acs \cite{Kov88} showed that the fuzzy solution to PLES with
symmetric triangular fuzzy numbers is stable with respect to small changes
of centres of fuzzy parameters. First, in Chapter 4 'Fuzzy Optimization',
we generalize Kov\'acs's results to PLES with (Lipschitzian) fuzzy numbers
and flexible linear programs, and illustrate the sensitivity of the fuzzy
solution by several one- and two-dimensional PLES. Then we consider
linear (and quadratic) possibilistic programs and show that the possibility
distribution of their objective function remains stable under small changes
in the membership functions of the fuzzy number coefficients. Furthermore,
we present similar results for multiobjective possibilistic linear programs
with noninteractive and weakly-noninteractive fuzzy numbers.

In Chapter 5 'Fuzzy Reasoning for Fuzzy Optimization', we interpret fuzzy
linear programming (FLP) problems with fuzzy coefficients and fuzzy
inequality relations as multiple fuzzy reasoning schemes (MFR), where the
antecedents of the scheme correspond to the constraints of the FLP
problem and the fact of the scheme is the objective of the FLP problem.
Then the solution process consists of two steps: first, for every decision
variable, we compute the (fuzzy) value of the objective function, via sup-min
convolution of the antecedents/constraints and the fact/objective, then an
(optimal) solution to FLP problem is any point which produces a maximal
element of the set of fuzzy values of the objective function (in the sense of
the given inequality relation). We show that this solution process for a
classical (crisp) LP problem results in a solution in the classical sense,
and (under well-chosen inequality relations and objective function) coincides
with those suggested by Buckley \cite{Buc89}, Delgado et al.
\cite{Del87,Del88}, Negoita \cite{Neg81}, Ramik and Rimanek \cite{Ram85},
Verdegay \cite{Ver82,Ver84} and Zimmermann \cite{Zim75}. \\ Typically, in
complex, real-life problems, there are some unidentified factors which
effect the values of the objective functions. We do not know them or can not
control them; i.e. they have an impact we can not control. The only thing we
can observe is the values of the objective functions at certain points. And
from this information and from our knowledge about the problem we may be
able to formulate the impacts of unknown factors (through the observed
values of the objectives). First we state the multiobjective decision problem
with independent objectives and then adjust our model to reality by
introducing interdependences among the objectives. Interdependences
among the objectives exist whenever the computed value of an objective
function is not equal to its observed value. We claim that the real values of
an objective function can be identified by the help of feed-backs from the
values of other objective functions, and show the effect of various kinds
(linear, nonlinear and compound) of additve feed-backs on the compromise
solution.

Even if the objective functions of a multiobjective decision problem are
exactly known, we can still measure the {\em complexity} of the problem,
which is derived from the {\em grades of conflict} between the objectives.
Then we introduce concave utility functions for those objectives that support
the majority of the objectives, and convex utility functions for those ones
that are in conflict with the majority of the objectives. Finally, to find a good
compromise solution we employ the following heuristic: increase the value
of those objectives that support the majority of the objectives, because the
gains on their (concave) utility functions surpass the losses on the (convex)
utility functions of those objectives that are in conflict with the majority of
the objectives.

In Chapter 6 'Applications in Management' we present four management
applications. In the first case, {\em Nordic Paper Inc.}, we outline an
algorithm for strategic decisions for the planning period 1996-2000 based on
the interdependencies between the criteria.

{\em Strategic Management} is defined as a system of action programs
which form sustainable competitive advantages for a corporation, its
divisions and its business units in a strategic planning period. A research
team of the IAMSR institute has developed a support system for strategic
management, called the {\em Woodstrat}, in two major Finnish forest
industry corporations in 1992-96. The system is modular and is built around
the actual business logic of strategic management in the two corporations,
i.e. the main modules cover the {\em market position}, the {\em competitive
position}, the {\em productivity position} and the {\em profitability} and {\em
financing positions}. The innovation in {\em Woodstrat} is that these
modules are linked together in a hyperknowledge fashion, i.e. when a strong
market position is built in some market segment it will have an immediate
impact on profitability through links running from key assumptions on
expected developments to the projected income statement. There are
similar links making the competitive position interact with the market
position, and the productivity position interact with both the market and the
competitive positions, and with the profitability and financing positions.
In the second case, {\em The Woodstrat projec} we briefly decsribe a
support system for strategy formation and show that the effectiveness and
usefulness of hyperknowledge support systems for strategy formation can
be further advanced using adaptive fuzzy cognitive maps.

{\em Real options} in option thinking are based on the same principals as
financial options. In real options, the options involve {\em real} assets as
opposed to financial ones. To have a {\em real option} means to have the
possibility for a certain period to either choose for or against something,
without binding oneself up front. Real options are valued (as financial
options), which is quite different with from discounted cashflow investment
approaches. The {\em real option rule} is that one should invest today only
if the net present value is high enough to compensate for giving up the value
of the option to wait. Because the option to invest loses its value when the
investment is irreversibly made, this loss is an opportunity cost of investing.

However, the pure (probabilistic) {\em real option rule} characterizes the
present value of expected cash flows and the expected costs by a single
number, which is not realistic in many cases. In the third case, {\em A
fuzzy approach to real option valuation}, we consider the {\em real option
rule} in a more realistic setting, namely, when the present values of
expected cash flows and expected costs are estimated by trapezoidal
fuzzy numbers.

In the fourth case, {\em Soft computing methods for reducing the bullwhip
effect}, we consider a series of companies in a supply chain, each of which
orders from its immediate upstream collaborators. Usually, the retailer's
order do not coincide with the actual retail sales. The {\em bullwhip effect}
refers to the phenomenon where orders to the supplier tend to have larger
variance than sales to the buyer (i.e. demand distortion), and the distortion
propagates upstream in an amplified form (i.e. variance amplification). We
show that if the members of the supply chain share information with
intelligent support technology, and agree on better and better fuzzy
estimates (as time advances) on future sales for the upcoming period, then
the bullwhip effect can be significantly reduced.

In the quest to develop faster and more advanced, intelligent support
systems the introduction of software agents a few years ago has opened
new possibilities to build and implement useful support systems. The
reason for wanting more advanced and intelligent systems is simple: we
want to be able to cope with complex and fast changing business contexts.

In Chapter 7 'Future Trends in Fuzzy Reasoning and Decision Making'
we will introduce some software agents we have built and implemented and
then show how fuzzy reasoning schemes can be included in the agent
constructs in order to enhance their functionality.

BibTeX entry:

@BOOK{bCaFu02a,
  title = {Fuzzy Reasoning in Decision Making and Optimization},
  author = {Carlsson, Christer and Fullér, Robert},
  volume = {82},
  series = {Studies in Fuzziness and Soft Computing Series},
  publisher = {Springer-Verlag},
  year = {2002},
  ISBN = {3-7908-1428-8},
}

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