Fuzzy Preference Ordering of Interval Numbers in Decision Problems
In conventional mathematical programming, coefficients of problems are usually determined by the experts as crisp values in terms of classical mathematical reasoning. But in reality, in an imprecise and uncertain environment, it will be utmost unrealistic
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Introduction
“There must be an ideal world, a sort of mathematician’s paradise where everything happens as it does in textbooks.” -- Bertrand Russell (1872-1970)
In our quest for such an ideal world with utmost precision and certainty we often try to replicate the pervasiveness of the real world into a formal model that aims to be a kind of abstraction of domain and the forces and dynamics of the environment. But in reality, formal (mathematical) models happen to be based on precise information, certain assumptions, crisp hypotheses and well-defined theories, which are logically sound but very little, can mimic reality. For example, formal (mathematical) models used in decision aid are usually characterized by multiple parameters. A situation with precise information occurs when the Decision Makers (DMs) are able to indicate a precise value for each parameter. However, there are always many difficulties for obtaining precise values for all the parameters: •
the performance of each action on each criterion may be unknown at the time of the analysis – it may result from arbitrariness in constructing parts of the model or from the aggregation of several aspects having impact on different criteria, and it may result from a measuring instrument or from a statistical tool (which usually involves certain or uncertain extent of error);
A. Sengupta and T.K. Pal: Fuzzy Preference Ordering, STUDFUZZ 238, pp. 1–23. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
Chapter 1 Introduction
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not all but many parameters, which define relations between the effects and consequences while framing parts of the model, generally have seldom objective existence – they are mostly reflected by the DMs’ subjective perception and opinion, which the DM may find difficult to express in so-called ‘mathematical precision’ and an objective measure of which may change from man-to-man in time-to-time; in either case, the result of a quest for numerical precision and certainty is neither real nor accurate. A decision situation related to human aspect, in fact, has only a little to do with the absolute attributes – certainty and precision – which are not present in our cognition, perception, reasoning and thinking. There are so many things which can only be defined by vague and ambiguous predicates and as a result it has been increasingly clear that formal modelling of a real decision situation does not reflect the pervasiveness of human perception, cognition and mutual interaction with the outside world (Gupta (1988)).
I would like to quote from the works of Sir Karl Popper (1974) on the philosophy of science in this context: Both precision and certainty are false ideals. They are impossible to attain, and therefore dangerously misleading if they are uncritically accepted as guides. The quest for precision is analogous to the quest for certainty, and both should be abandoned... one should never try to be more precise than the problem situation demands… Quite a similar note can be found in the words of one of the greatest scien
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