A Fuzzy Multicriteria Categorization of Water Scarcity in Complex Water Resources Systems

Abstract

Although water scarcity is a well-understood concept, it is difficult to define it precisely and furthermore, to address the water scarcity which requires a complicated set of water actions. In this work, with the aim to face the complexity of water scarcity, we propose the establishment of several categories in order to characterize the water scarcity. These categories are guided by the corresponding roadmap so as to address the water scarcity. Consequently, the considered categories are used to characterize the scarcity are non-ordered. The methodology is based on a developed fuzzy multi-criteria filtering approach. The Water Demand Satisfaction, Water Reliability, Management Effectiveness and Water Sustainability, are selected as criteria. Each category is characterized by reference to a fictitious alternative, the prototype. Τhe water scarcity categorization process is based on the fuzzy binary comparison between the water scarcity evaluation of the criteria in the watersheds and the prototype of each category. This binary comparison over all criteria consists of the concordance and non-discordance principle. The use of fuzzy analysis enables us to express the grey region of the monocriterion comparison between the basins and the prototypes and furthermore, to achieve a suitable aggregation of the achieved monocriterion comparison between the basins and the prototypes. Eventually, the use of the veto thresholds and discordance indices of the fuzzy aggregation allowing compensation, are to be avoided when criteria are strongly conflicting, which is a very useful property in classification problems. The described methodology was applied to the Ebro river basin, identifying the existing problems.

Appendices

Appendix I

Τhe S-weak preference (or outranking with the same meaning) relation can be defined axiomatically as follows:

Definition: For any criterion j whose scale is X_j the valued binary relation S_j is said to be a one – dimensional weak preference if there exists a function θ, defined from X_j × X_j into [0, 1] verifying S _j (α, b) = θ _j (α _j , b _j ) for all α, b in A and such that (Perny and Roy, 1992):

∀ y ₀ ∈ X _j , θ _j (x, y) is a non – decreasing function of x,

∀ x ₀ ∈ X _j , θ _j (x, y) is a non – increasing function of y,

$$ \forall z\in {X}_j,\;{\theta}_j\left(z,\;z\right)=1 $$

(Α.1)

Example: In Electre III the partial weak preference indices which were used satisfy the axiomatic properties of the weak preference relation presented above in Eq. A.1 (Perny and Roy, 1992):

$$ {S}_j\left(\alpha, b\right)=\frac{p_j- \min \left\{{b}_j-{\alpha}_j,\;{p}_j\right\}}{p_j- \min \left\{{b}_j-{\alpha}_j,\;{q}_j\right\}} $$

(Α.2)

It is quite easy to confirm that the above relation verifies the definition of the weak preference and therefore this index is a weak preference relation.

It should be clarified that where α _j , b _j are the score of the alternatives α,b with respect to the criterion j. Ιn this article, as alternative we use the alternative α which represents the examined basin and y ^r which represents the prototype r.

As mentioned before the indifference relation can be defined with the use of the weak preference relation (Eq. 1).

Appendix II: Discordance Principle: Respect of the Minority

The main concept of the discordance principle is to take into consideration strongly conflicting minorities (Perny 1998). In a preference oriented problem, this amounts to checking whether some criterion wants to make use of its right of veto against an overall statement αΙy _j ^r, in favour of a (Perny 1998 & Tsakiris and Spiliotis 2011). The construction of the overall discordance relation is based firstly on the monocriterion discordance relation and secondly on the overall discordance relation. The first concept which is used in this section is the veto threshold which can be defined axiomatically as follows (Perny 1998):

Definition: For any criterion j, the veto threshold with respect to S (weak preference) is in general a real – valued function υ _j defined on X_j such that, for any pair (α, y^r), υ _j , is the maximal of a score difference of type y _j ^r − α _j that may be compatible with the proposition αSy _j ^r:

$$ \forall \left(\alpha, {y_j}^r\right)\in A\times F,\;\left(\exists j\in \left\{1,\dots,\;n\right\}{y_j}^r-{\alpha}_j>{\upsilon}_j\right)\Rightarrow \neg \left(\alpha {S}_j{y}^r\right) $$

(Α.3)

(where ¬ denotes the complement logical operation).

Here, the veto threshold is considered as a constant value different for each category.

The veto threshold has a vital contribution in the monocriterion discordance formulation as it can been seen from the next definition.

Definition: For any criterion j whose scale is X_j, the fuzzy binary relation D_j on A² is said to be a monocriterion discordance relation of the weak preference if there is a threshold υ _j and a real – valued function d_j, defined on ℜ², verifying that D _j (α, y ^r) = d _j (α, y ^r) for all a, b in A such that:

∀ y ₀ ∈ X _j , d _j (α, y ^r) is a non – increasing function of α,

∀ y ₀ ∈ X _j , d _j (α, y ^r) is a non – decreasing function of y ^r,

$$ \forall x,y\in {X}_j,\kern0.36em {\theta}_j\left(\alpha,\;{y}^r\right)>0\iff {d}_j\left(\alpha,\;{y}^r\right)=0 $$

$$ \forall x,y\in {X}_j,\kern0.24em {y_j}^r-{\alpha}_j>{\upsilon}_j\iff {d}_j\left(\alpha,\;{y}^r\right)=1 $$

where X _j  ∈ ℜ (Α.4)

The discordance monocriterion relation can be defined also for the strict preference relations by changing the definition of the veto threshold.

Example: In Electre III, the monocriterion discordance indices of the weak preference which were used, satisfy also the axiomatic conditions of the discordance relation (with respect to the weak preference) presented above in Eq. A.4 (Perny and Roy, 1992):

$$ {D}_{Sj}\left(\;\alpha,\;{y}^r\right)= \min \left\{1,\; \max \left\{0,\kern0.24em \frac{{y_j}^r-{a}_j-{p}_j}{\upsilon_j-{p}_j}\right\}\right\} $$

(Α.5)

The presented monocriterion evaluations of the weak preference and the discordance with respect to the weak preference, were used in Electre III while they are used also in this work. Perny 1998 proposed the following monocriterion discordance for indifference relations which is based on the monocriterion discordance relation of the weak preference relations:

$$ {D}_{Ij}\left(a,\kern0.24em {y_j}^r\right)= \max \left({D}_{Sj}\left(a,\;{y_j}^r\right),\kern0.24em {D}_{Sj}\left({y_j}^r,\;a\right)\right) $$

(Α.6)

The monocriterion discordance relation with respect to indifference (Eq. A.6) are depicted in Fig. 2. Thus, the discordance monocriterion relation can be defined with the use of the weak preference discordance relation. The comprehensive discordance relation (over all criteria) is defined for every indifference relation.

Actually, the overall discordance index measures the degree to which exists at least one discordant criterion, whereas the concordance index measures the overall importance of the concordant coalition. For this reason, a disjunctive aggregative operator is most often chosen for the overall discordance index (Perny 1998).

In this framework, a proper aggregator of the monocriterion discordance relations is the max-union (which is the union of the crisp logic, in fuzzy terms) since it verifies the general axiom of the overall discordance relation (Tsakiris and Spiliotis 2011 and Perny 1998). Thereafter, the comprehensive discordance relation for n criteria can be analyzed as follows:

$$ {D}_I\left(\alpha, {y}^r\right)= \max \left\{{D}_{I_1}\left({a}_1,\kern0.24em {y_1}^r\right),\dots, {D}_{I_2}\left({a}_j,\kern0.24em {y_j}^r\right),\dots, {D}_{In}\left({a}_n,\;{y_n}^r\right)\right\} $$

(Α.7)