Fuzzy arithmetic

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Almost forty years ago, Kaufmann and Gupta (1985) proposed a numerical calculus for uncertainty they dubbed fuzzy arithmetic based loosely on Lotfi Zadeh's influential ideas about fuzzy sets and an alternative to probability he called possibility theory (Zadeh 1978; Dubois and Prade 1988).  The basic objects in this scheme are fuzzy numbers, which are unimodal possibility distributions that reach the possibility level of one.  Within possibility theory, the max-min convolution had been proposed as a relaxation of the familiar sum-product convolution of probability (which is appropriate when distributions are stochastically independent).  Kaufmann and Gupta showed that, so long as the inputs are fuzzy numbers (i.e., unimodal and reach one), then possibilistic max-min convolution is the same as simple level-wise interval arithmetic on horizontal slices of the fuzzy numbers at each alpha level between zero and one.  Although often derided by probabilists, there have been many successful applications of fuzzy arithmetic within engineering.

It appears that Dominik and Ander (<<ref>>) have defined a new fuzzy arithmetic that is different from that described by Kaufmann and Gupta.  <<Hmm, was it Dominik and Ander, or was it Didier or others?>>  The new fuzzy arithmetic is a Fréchet calculus, which is to say, the results are guaranteed no matter what dependencies may exist between the operands.  A simple counterexample, discussed in our consideration (<<ref>>) of the Lazy Risk Analyst Conjecture by Lev Ginzburg, shows that Kaufmann and Gupta's level-wise approach is not a Fréchet calculus.  The new fuzzy arithmetic uses the level-wise interval arithmetic to combine its operands, but the membership function of the result is multiplicatively scaled by the number of fuzzy number inputs, with values truncated at one if they would be larger.  This scaling produces wider uncertainty in the results compared to level-wise interval arithmetic alone.

This new fuzzy arithmetic serves as the internal calculus for consonant structures used in the unification with p-boxes (<<ref>>).  It should perhaps also be adopted as the fuzzy arithmetic, replacing the Kaufmann and Gupta level-wise interval arithmetic, and perhaps calling into question max-min convolutions within possibility theory more broadly.

We note that the rescaling step loses information and would tend, over many calculations, to erase all structure in the membership functions so that the results quickly become simple intervals.  This  is similar to what Bob Williamson projected would occur with his calculus for p-boxes (<<ref>>).  However, as we saw with p-boxes, this blurring to intervals might be prevented by intersecting results at each binary operation with moment constraints if they are also available.

References

Dubois, Didier, and Henri Prade (1988). Possibility Theory: An Approach to Computerized Processing of Uncertainty.  Plenum Press, New York.

Kaufmann, Arnold, and Madan M. Gupta (1985). Introduction to Fuzzy Arithmetic: Theory and Applications. Van Nostrand Reinhold, New York.

Zadeh, Lotfi (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28.

Seresht, N.G., and A.R. Fayek (2018). Fuzzy arithmetic operations: theory and applications in construction engineering and management. , pp. 111-147 in Fuzzy Hybrid Computing in Construction Engineering and Management, edited by A.R. Fayek, Emerald Publishing Limited, Bingley. https://doi.org/10.1108/978-1-78743-868-220181003. ISBN: 978-1-78743-869-9, eISBN: 978-1-78743-868-2

Abstract

Fuzzy numbers are often used to represent non-probabilistic uncertainty in engineering, decision-making and control system applications. In these applications, fuzzy arithmetic operations are frequently used for solving mathematical equations that contain fuzzy numbers. There are two approaches proposed in the literature for implementing fuzzy arithmetic operations: the α-cut approach and the extension principle approach using different t-norms. Computational methods for the implementation of fuzzy arithmetic operations in different applications are also proposed in the literature; these methods are usually developed for specific types of fuzzy numbers. This chapter discusses existing methods for implementing fuzzy arithmetic on triangular fuzzy numbers using both the α-cut approach and the extension principle approach using the min and drastic product t-norms. This chapter also presents novel computational methods for the implementation of fuzzy arithmetic on triangular fuzzy numbers using algebraic product and bounded difference t-norms. The applicability of the α-cut approach is limited because it tends to overestimate uncertainty, and the extension principle approach using the drastic product t-norm produces fuzzy numbers that are highly sensitive to changes in the input fuzzy numbers. The novel computational methods proposed in this chapter for implementing fuzzy arithmetic using algebraic product and bounded difference t-norms contribute to a more effective use of fuzzy arithmetic in construction applications. This chapter also presents an example of the application of fuzzy arithmetic operations to a construction problem. In addition, it discusses the effects of using different approaches for implementing fuzzy arithmetic operations in solving practical construction problems.