Percepción de la calidad de los servicios deportivos públicos

Fuzzy set theory was introduced by Zadeh (1965) as a method of representing the concept of partial set membership as opposed to the classical binary (two-valued) logic where set membership is represented as either completely true or completely false. It was based on the observation that human reasoning can utilize concepts and knowledge that do not have well-defined, sharp boundaries, as an alternative approach to overcome difficulties in developing and analyzing complex systems encountered by conventional mathematical tools (Yen & Langari 1999). Fuzzy logic has two different meanings in literature; according to Klir et al. (1997) it is viewed as either a system of concepts, principles and methods for dealing with various modes of reasoning which are approximate rather than exact in nature, or as a generalization of the various proposed multi-valued logics. Zadeh (1973), referring to the inability of conventional quantitative techniques of system analysis to deal with complex systems such as humanistic systems (human-centred), introduced the principle of incompatibility based on the following reasoning: “...as the complexity of a system

74 increases, our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics.”(Zadeh 1973, p28). This statement implies that the behaviour of complex systems (e.g. human or environmental systems) cannot be quantified with meaningful accuracy using conventional quantitative approaches.

Despite its innovative problem-solving capabilities, it was not until a decade later that fuzzy set theory was successfully applied as control system by Mamdani (1974) and became popular primarily in industrial applications in Europe and Japan. Since then the concept of fuzzy logic has found a wide range of applications in various areas such as, control systems, decision-making, artificial intelligence and spatial analysis.

Fuzzy set theory can be considered as an extension of classical set theory (Ross 1995). In the classical set theory, an element has a clearly defined relationship with a set, which means that the element either belongs (1) or does not belong (0) to the set, therefore its membership degree value can be either 0 or 1.

( ) {

(3.1)

where the ( ) is the indicator or characteristic function of element representing the membership of element in the set .

In the fuzzy set theory, elements have varying degrees of membership in [0, 1] interval.

( ) [ ] (3.2)

where ( ) is the degree of membership of the element in the fuzzy set ; 1 represents full membership and 0 non-membership (Fig. 3.1).

75 Figure 3.1 Membership functions for “cold” (green), “hot” (orange) and “warm” (blue) temperatures.

Boolean logic (left) assumes that temperature is either “cold” below a specific threshold or “hot” above that threshold (never both). Fuzzy logic allows a gradual transition between “cold” and “hot” introducing intermediate values represented by the “warm” membership function.

3.2.1.1 Fuzzy logic in spatial analysis

In spatial analysis concepts like close to, far from, steep or suitable often do not have clearly defined boundaries, demonstrating a degree of vagueness or uncertainty. To illustrate the application of fuzzy set theory in spatial analysis we consider a hypothetical site suitability analysis problem. The problem involves identifying a suitable location to build a house on the basis of environmental criteria. The criteria to classify a site as suitable could be the following: low slope, favourable slope aspect, close to a State Highway and not close to faults. Although the above statements are vague or imprecise compared to precise values (e.g. slope angle ≤ 10^o, 135^o < slope aspect < 225^o, ≤ 1 km distance from State Highway, ≥ 1 km distance from faults), they correspond to the way a human perceives the factors controlling the suitability of a location. Using conventional approaches the above factors would be converted into classes with crisp boundaries. Therefore, if a location falls within the range of the assumed threshold values we would consider it, otherwise, even if it were very close to the class boundaries, it would be excluded from the analysis (e.g. if the distance from faults is 1 km, then we consider the location, but if the distance is 999 m we reject it). As a result neighbouring areas with a slight difference of criteria values may have different suitability. However, by introducing degrees of membership to the classification, locations close to class boundaries will also be included in the analysis by getting an appropriate membership value (e.g. locations with distance 999 m from faults will get a low membership value but they will still be considered), resulting in a more realistic representation of the site suitability (Fig. 3.2). In this sense (Ross 1995), points out that fuzzy systems are useful in two general contexts: in situations involving highly complex systems whose behaviours are not well understood, and in situations where an approximate, but fast solution is required.

76 Figure 3.2 Example of site suitability based on distance from faults (≥ 1km) using (A) crisp boundaries and (B) a fuzzy linear membership function. Discrete boundaries are rarely applicable to spatial properties and natural phenomena.

3.2.1.2 Degree of membership vs. probability

Degrees of membership and probabilities may be easily confused as being the same, as they both have values ranging between 0 and 1. However they have a small, yet important difference. The following example illustrates this difference. Assume that we have been asked to identify an area with “very low landslide hazard” in order to build critical infrastructure. According to the available information there are two areas that match this criterion, the area X where there is 0.9 probability of no landslides within the next 100 years and the area Y which has 0.9 membership in the set “very low landslide hazard”. Interpreting the above information the X area has 90% chance of having not a single slope failure in the next 100 years and 10% chance of having a landslide, even a catastrophic rock avalanche! On the other hand, the 0.9 membership means that the Y area has “very low landslide hazard” and even if a slope failure occurs it is highly unlikely to be a large magnitude mass movement. Thus, the prior probability 0.9 becomes a posterior probability of 1 or 0 after the 100 years period at the X area, whereas the 0.9 membership value remains the same regardless how many years have passed, as it indicates the relationship between the landscape and the fuzzy concept “very low landslide hazard”.