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# Multi-attribute decision analysis & the fun of fuzzy

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Between the fuzzy sets of Lotfi Asker Zadeh and the fuzzy math of George W. Bush, what precisely are the implications for multi-attribute decision analysis (MADA)? First of all, let’s define the ‘fuzzy’ we’re talking about here. It’s not the chop logic that Bush accused Gore of using in the 2000 US Presidential campaign. ‘Fuzzy’ as in multi-attribute decision analysis refers to ‘the degree of possessing a certain quality’ (Zadeh’s definition from the 1960s). Something that absolutely has that quality scores 1, something that has none of that quality scores 0, and something that has ‘a bit of that quality’ scores… something in between. Does this sound like probability? And how does it apply to MADA?

## MADA and MODA; probabilistic, stochastic and fuzzy methods

Multi-attribute decision analysis can be considered to be analysis using deterministic data and methods. Decision trees and AHP (analytic hierarchical process) are examples of MADA. There are a finite number of outcomes of each parameter involved, or a direct comparison between parameters is possible, or both. In short, we’re in a probabilistic mode. When outcomes start to be expressed in terms of continuous probability curves so that outcomes are effectively infinite, then we move to multiple objective decision analysis (MODA). In this case (and even in complicated MADA cases), stochastic methods or Monte Carlo methods become more suitable for evaluating outcomes. However, ‘fuzziness’ is not probability. It is a notion in itself. For example, you don’t have to examine a population to ascribe a degree of fuzziness to something or someone – unlike probability, for which you have to consider a ‘universe’ in order to assign a probability to an individual entity or group.

The idea is to express uncertainty and vagueness, but in different way compared to stochastic approaches. For example, a factor that influences a decision and that has a fuzziness of 0.8 suggests the decision-maker considers the factor to be an important one, but not a critical one (1.0 would be critical, and 0 would be irrelevant). Using the fuzzy logic developed by Zadeh and others as applied to combinations of sets of factors, fuzziness can then be applied to multi-attribute decision theory as well.

## Who likes fuzzy?

Different industry sectors have used fuzzy analysis of outcomes, some in comparisons with stochastic methods. They include fuzziness for geotechnics (soil properties), structural engineering and the pricing of stock options. Authors have come up with different ways of expressing the difference between fuzzy and stochastic approaches: for example, Buckley’s comment that ‘stochastic MCDM (multi criteria decision making) counts all ways to accomplish a task but fuzzy MCDM looks for a best way to do the job. Others rate fuzzy models higher and easier to work with in some respects than stochastic models.

## So what’s the future of fuzzy?

Nobody would claim that fuzzy methods have taken over in multi-attribute decision analysis, even if they are being used in different instances today. Some reasons that explain the tempering of enthusiasm include lingering doubts that the theoretical fuzziness matches real-life vagueness, whether prescriptive modeling of imprecision is realistic compared to descriptive modeling, and whether fuzziness values can be ascribed in a logical, systematic way. For the moment they provide a means of comparison with stochastic methods and simulations, rather than a replacement.

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