Modificación de la Ley 1/1970, de 4 de abril, de Caza.

Fuzzy logic enables the user to employ qualitative variables in their planning through the incorporation of fuzzy sets. The idea of fuzzy sets was first introduced by Zadeh (1965) in order to express the ambiguity, uncertainty and so-called “fuzziness” in an elements membership to a particular set. Unlike the ordinary sets, element membership is more like binary logic (i.e. either belong to a set which is 1 or doesn’t belong to a set which is 0). The concept of fuzzy sets introduced by Zadeh (1965) allows the membership values of an element to vary between 0 and 1, thus giving a more accurate representation of an elements membership—especially when the boundaries of the set can’t be determined in terms of crisp values. Support of a membership function are usually the range of values over which the membership function of a fuzzy set is defined (Ross, 2010). This support of membership function is divided into regions which are the core with a membership value equal to 1 and the boundaries where the membership function varies between 0 and 1 (Ross, 2010).

To incorporate the knowledge of the user and to establish the link between the input variables and outputs, fuzzy inference systems are used by employing an “if-then” framework. Fuzzy rules combine the antecedents and consequents to determine the output. In case of multiple variables “AND” or “OR” operators are used to combine the antecedents and consequent (Ross, 2010). Using “AND” operator results in the intersection of the rule outputs, whereas using “OR” operator results in the union of the rule outputs. Fuzzy rules are processed by using either Mamdani inference method (Mamdani and Assilian, 1975) or Sugeno (Takagi and Sugeno, 1985) method. The former method is the most widely used inference method where the consequent is modeled as

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a fuzzy function (Ross, 2010) while in the latter; the consequent is modeled as a crisp function of the input variables (Ross, 2010). It has been reported that Sugeno inference method performs better with different optimization techniques and can be integrated with genetic algorithm or any other optimization technique which is considered an advantage when compared to Mamdani FIS (Kaur and Kaur, 2012).

Fuzzy rules normally take the form “𝑖𝑓 𝑥 𝑖𝑠 𝑋𝑖 𝑡ℎ𝑒𝑛 𝑦 = 𝑓𝑖(𝑥)” in which “𝑥 𝑖𝑠 𝑋𝑖" is the antecedent and “𝑦 = 𝑓𝑖(𝑥)" is the consequent (Ross, 2010). In case of multiple inputs “AND” and “OR” operators are used to connect the antecedents. In Sugeno fuzzy inference method, each fuzzy rule is used to determine the area under the corresponding consequent fuzzy set. The resultant output for each fuzzy rule is combined to determine a crisp output by using weighted average method as shown in Equation 2.4.

𝑦(𝑥) =

∑𝑛_𝑖=1µ_𝑋𝑖(𝑥) (2.4)

Where 𝑦(𝑥): Consequent,

µ𝑋𝑖(𝑥): Membership function value for variable (x),

𝑓𝑖(𝑥): Corresponding value of the membership function (singleton) and n: is the number of fuzzy rules.

As previously discussed, risk is the integration of probability of failure and the consequences of that failure. The failure represents the failure to meet the designed level of service which can be calculated using system models or analytical techniques. As for the consequences of failure it represents the socioeconomic losses and environmental impact due to that failure. When representing the likelihood and consequences of failure on ordinal scale and risk of failure consequently, there is no definitive or one way to select their levels. Based on psychologists’ experiments, the pieces of information handled should be in order of 7 plus or minus 2. As such,

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and as recommended by Karwowski and Mital (1986), the number of categories shouldn’t be more than seven.

The IPCC (2012) uses a seven level to assess the likelihood as shown in Table 2.5. Also, selecting a particular shape of a membership function depends mainly on the information that is required to be represented and how certain the user is. Several methods are used to select the appropriate shape for membership functions; however, in engineering applications the shape of membership functions has little influence on the results (Klir and Yuan, 1995). Membership functions could be triangular, trapezoidal, Gaussian and others; Triangular Fuzzy Numbers (TFNs) are usually used because of their suitability and simplicity in engineering applications (Lee, 1996). Table 2.5: Risk Level as per IPCC (2012)

Risk Level Description Probability 1 Virtually certain 99-100% 2 Very likely 90-100% 3 Likely 66-100% 4 About as likely 33-66% 5 Unlikely 10-33% 6 Very unlikely 0-10% 7 Exceptionally unlikely 0–1%

Figure 2.5 shows the triangular fuzzy membership functions for the likelihood, consequences and risk of failure and the corresponding membership value (α).

0 1 α

Likelihood/Consequence/Risk Values

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Figure 2.5: Fuzzy Membership Functions for Likelihood, Consequences and Risk of Failure Since the membership function shape is triangular, the likelihood 𝑙̃(𝑥)and consequences of failure 𝐶̃(𝑥)for any value x can be denoted by:

𝑥−𝑎1 𝑏1−𝑎1 , 𝑎1 < 𝑥 < 𝑏1 𝑙̃(𝑥) = 𝑐1−𝑥 𝑏1−𝑎1, 𝑏1 < 𝑥 < 𝑐1 (2.5) 0, otherwise 𝑥−𝑎2 𝑏2−𝑎2 , 𝑎2 < 𝑥 < 𝑏2 𝐶̃(𝑥) = 𝑐2−𝑥 𝑏2−𝑎2, 𝑏2 < 𝑥 < 𝑐2 (2.6) 0, otherwise 𝛼 = 𝜇𝐿(𝑥) = 𝐿1 𝛼_−𝑎 1 𝑏1−𝑎1 𝑎𝑛𝑑 𝑐1−𝐿𝛼2 𝑏1−𝑎1 (2.7) 𝛼 = 𝜇𝐶(𝑥) =𝐶1 𝛼_−𝑎2 𝑏2−𝑎2 𝑎𝑛𝑑 𝑐2−𝐶2𝛼 𝑏2−𝑎2 (2.8) Therefore 𝐿𝛼1 = (𝑏1− 𝑎1)𝛼 + 𝑎1 (2.9) 𝐿𝛼2 = 𝑐1− (𝑐1− 𝑏1)𝛼 (2.10) 𝐶1𝛼= (𝑏2− 𝑎2)𝛼 + 𝑎2 (2.11) 𝐶2𝛼= 𝑐2− (𝑐2− 𝑏2)𝛼 (2.12)

And since risk is the product of likelihood and consequences, then:

𝑅1𝛼 = 𝐿𝛼1 ∗ 𝐶1𝛼= (𝑏1− 𝑎1)(𝑏2− 𝑎2)𝛼2+ [(𝑏1− 𝑎1)𝑎2+ (𝑏2 − 𝑎2)𝑎1]𝛼 + 𝑎1𝑎2 (2.13) 𝑅2𝛼 = 𝐿𝛼2∗ 𝐶2𝛼= (𝑐1− 𝑏1)(𝑐2− 𝑏2)𝛼2+ [(𝑐1− 𝑏1)𝑐2+ (𝑐2− 𝑏2)𝑐1]𝛼 + 𝑐1𝑐2 (2.14) To determine the value of membership function corresponding to (x), Equations 2.13 and 2.14 are solved as a quadratic equation with the roots the value of (α) as per Equation 2.15.

35 𝛼 = 𝜇𝑅(𝑥) =−[(𝑏1− 𝑎1)𝑎2+ (𝑏2− 𝑎2)𝑎1] ± √[(𝑏_2(𝑏1− 𝑎1)𝑎2+ (𝑏2− 𝑎2)𝑎1]2− 4[(𝑏1− 𝑎1)(𝑏2− 𝑎2)][𝑎1𝑎2] 1− 𝑎1)(𝑏2 − 𝑎2) , 𝑎1𝑎2 < 𝑥 < 𝑏1𝑏2 = [(𝑐1− 𝑏1)𝑐2+ (𝑐2− 𝑏2)𝑐1] ± √[(𝑐1_2(𝑐− 𝑏1)𝑐2+ (𝑐2− 𝑏2)𝑐1]2− 4[(𝑐1− 𝑏1)(𝑐2− 𝑏2)][𝑐1𝑐2] 1− 𝑏1)(𝑐2− 𝑏2) , 𝑏1𝑏2 < 𝑥 < 𝑐1𝑐2 = 0, otherwise (2.15) Salman and Salem (2012) developed a risk assessment tool to determine criticality of sewer pipelines using fuzzy inference systems. To determine the consequences of failure, a weighted scoring method was used after identifying important factors, their relative importance and summarizing the overall performance of sewer pipes in terms of these factors based on collected historical data. Three different logistic regression techniques were used to develop a deterioration model for sewer pipes that identifies the likelihood of failure. Binary logistic regression was found to be the most suitable technique. Risk values resulting from combining the consequences and likelihood of failure values using simple multiplication, risk matrices, and fuzzy inference systems were compared, and a risk map was created to help in identifying sewer pipe sections that require immediate action.

Mamdani Fuzzy inference system was used to determine the risk of failure values based on probability and consequences of failure enabling users to use “what-if” scenarios. This research provided a useful comparison between different available techniques to determine the risk of failure; however, the risk assessment performed was only based on the structural condition of pipelines without considering the operational conditions. Multinomial logistic regression was used to develop a model for forecasting the need for rehabilitation of sewage networks over a specific

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planning horizon of years in terms of monetary amounts (El-Assaly et al. 2006). The cost in this model was estimated as the product of the predicted defected pipe and the cost of the repair method for the same pipe. Different pipes were arranged in an ascending order to determine the ones with the highest cost, and which would need to be rehabilitated.

Fuzzy inference systems were used by Kleiner et al. (2005) in order to model the failure risk of water mains. In this research, deterioration of water main pipes was modeled using rule based fuzzy Markov procedures, which assumed that the consequences of failure value was known by the model developer. In this particular study, the deterioration of a certain pipe was modeled on results of only two inspection points and a rule based database. The probability values obtained from this procedure were combined with the consequences value—which was assumed to be known by using fuzzy rule based method. The same procedure was applied to sewerage infrastructure as well (Kleiner et al. 2007). While the use of fuzzy rule based Markov deterioration modeling is applicable to water mains due to scarce data on water main failures, data related to sewer inspections is more readily available due to the ease associated with CCTV inspections compared to inspection of pressurized water mains.

Fares and Zayed (2010) used a Hierarchical Fuzzy Expert System in which likelihood of failure was modeled under three categories, namely, “environmental”, “physical”, and “operational” and consequences of failure was modeled under the category of “post failure.” A hierarchical structure was selected in order to reduce the number of rules to be evaluated by the system. Also, Kleiner et al. (2004) applied fuzzy sets (Zadeh, 1965) to combine the consequences and possibility of failures to determine the risk of failure by using a fuzzy-rule based system. By using fuzzy rules, the expert opinion regarding important factors that affect the likelihood and consequences of failure of water mains was incorporated into a fuzzy risk assessment procedure. The main advantage of using

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Fuzzy inference system in determining the risk of failure in pipelines is its ability to handle problems with a combination of numeric and linguistic variables, resulting in availability of system knowledge in the problem. In addition, fuzzy inference system provides an easily understood and robust algorithm when dealing with vague or little data.