Evaluating Environmental Sustainability: An Integration of Multiple-Criteria Decision-Making and Fuzzy Logic

Abstract

While pursuing economic development, countries around the world have become aware of the importance of environmental sustainability; therefore, the evaluation of environmental sustainability has become a significant issue. Traditionally, multiple-criteria decision-making (MCDM) was widely used as a way of evaluating environmental sustainability, Recently, several researchers have attempted to implement this evaluation with fuzzy logic since they recognized the assessment of environmental sustainability as a subjective judgment Intuition. This paper outlines a new evaluation-framework of environmental sustainability, which integrates fuzzy logic into MCDM. This evaluation-framework consists of 36 structured and 5 unstructured decision-points, wherein MCDM is used to handle the former and fuzzy logic serves for the latter, With the integrated evaluation-framework, the evaluations of environmental sustainability in 146 countries are calculated, ranked and clustered, and the evaluation results are very helpful to these countries, as they identify their obstacles towards environmental sustainability.

Appendix: Fuzzy logic

An example of fuzzy reasoning, in which a new fuzzy value is derived on the basis of a fuzzy rule (i.e., the Mi rule in a fuzzy-rule base) with three antecedents and three fuzzy facts, is represented as follows:

$$ {IF\,X_{1}\,is\,{\tilde F}_{i1}\,AND\,X_{2}\,is\,{\tilde F}_{i2}\,AND\,X_{3}\,is\,{\tilde F}_{i3}\,THEN\,Y\,is\,{\tilde G}_{i}\,X_{1}\,is\,{\tilde F}_{1}^{\prime}\,AND\,\,X_{2}\,is\,{\tilde F}_{2}^{\prime}\,X_{3}\,is\,{\tilde F}_{3}^{\prime}\over Y\,is\,{\tilde G}_{i}^{\prime}} $$

(1)

where X _j and Y are linguistic variables; ${\tilde F}_{ij}$ and ${\tilde F}_{j}^{\prime}$ are fuzzy sets of $U_{j}$; ${\tilde G}_{i}$ and ${\tilde G}_{i}^{\prime}$ are fuzzy sets of V. In the framework of the compositional rule of inference (Zadeh 1975), ${\tilde G}_{i}^{\prime}$ is computed by

$$ {\tilde G}_i= ({\tilde F}_{1}^{\prime}\,\wedge\,{\tilde F}_{2}^{\prime}\,\wedge{\tilde F}_{3}^{\prime})\circ(({\tilde F}_{i1}\,\wedge\,{\tilde F}_{i2}\,\wedge\,{\tilde F}_{i3})\rightarrow {\tilde G}_i) $$

(2)

where Λ denotes a t-norm operator, ο is a composition operator and → indicates an implication operator.

Selection of operators is an important issue for calculating ${\tilde G}_{i}^{\prime}$. If “sup-min” is chosen as the composition operator (Zadeh 1973), the membership function of ${\tilde G}_{i}^{\prime}$ is computed by:

$$ \mu_{{\tilde G}_{i}^{\prime}}\left(\upsilon\right) = \mathop{\max\limits_{u_1,u_2,u_3}}\,\min\left[\mu_{{\tilde F}_{1}^{\prime}\,\wedge\,{\tilde F}_{2}^{\prime}\,\wedge\,{\tilde F}_{3}^{\prime}}\,({u_{1},u_{2},u_{3}}),\mu_{{\tilde F}_{i1}\,\wedge\,{\tilde F}_{i2}\,\wedge\,{\tilde F}_{i3}}\,\to\,{\tilde G}_{i}^{(u_{1},u_{2},u_{3},{\upsilon})}\right] $$

(3)

Furthermore, if “min” is treated as the t-norm operator (i.e., a Λ b = min(a, b)) and Mamdani’s implication operators are used (i.e.,a→b = mm(a,b)), Equation (3) becomes the well-known “Mamdani’s fuzzy reasoning” (Mamdani 1977), which can be expressed as

$$ \mu_{{\tilde G}_{i}^{\prime}}\left(\upsilon\right) = \mathop{\max\limits_{u_1,u_2,u_3}}\,\min\left[\mu_{{{\tilde F}_{1}}^{\prime}}(u_1),\mu_{{{\tilde F}_{2}}^{\prime}}(u_2),\mu_{{\tilde F}_{3}^{\prime}}(u_3),\mu_{{\tilde F}_{i1}^{\prime}}(u_{1}),\mu_{{\tilde F}_{i2}^{\prime}}(u_{2}),\mu_{{\tilde F}_{i3}^{\prime}}(u_{3}),\mu_{{\tilde G}_{i}}(\upsilon)\right] $$

(4)

Equation (4) can be further depicted in another form:

$$ \mu_{{\tilde G}_{i}^{\prime}}\left(\upsilon\right) = \min\left[\mathop{\max\limits_{u_1}}\mu_{{\tilde F}_{1}^{\prime}\wedge{\tilde F}_{i1}}(u_{1}),\,\mathop{\max\limits_{u_2}}\mu_{{\tilde F}_{2}^{\prime}\wedge{\tilde F}_{i2}}(u_2),\mathop{\max\limits_{u_{3}}\mu_{{\tilde F}_{3}^{\prime}\wedge{\tilde F}_{i3}}(u_{3})},\mu_{{\tilde G}_{i}(\upsilon)}\right] $$

(5)

where $({\tilde F}_{j}^{\prime}\wedge{\tilde F}_{ij})$ denotes the intersection of fuzzy sets. ${\tilde F}_{j}^{\prime}$ and ${\tilde F}_{ij}$; $\max_{\mu j}\mu_{{\tilde F}_{j}^{\prime}\wedge{\tilde F}_{ij}}(u_{j})$ is the highest degree of membership of the intersection and can be interpreted as the compatibility C _ij between ${\tilde F}_{j}^{\prime}$ and ${\tilde F}_{ij}$ $\min\left\{\max_{u_{1}}\mu_{{\tilde F}_{1}^{\prime}\wedge{\tilde F}_{i1}}(u_{1}),\max_{u_{2}}\mu_{{\tilde F}_{2}^{\prime}\wedge{\tilde F}_{i1}}(u_{2}),\max_{u_{3}}\mu_{{\tilde F}_{3}^{\prime}\wedge{\tilde F}_{i1}}(u_{3})\right\}$ can be viewed as the overall compatibility C _i between the facts and the rule; and C _i is used to truncate ${\tilde G}_{i}$ to obtain ${\tilde G}_{i}^{\prime}$. Moreover, if ${\tilde F}_{j}^{\prime}$ is a precise value (i.e., say ${\bar u}_{j}$), Equation (5) becomes:

$$ \mu_{{\tilde G}_{i}^{\prime}}(\upsilon)\,=\,\min\left\{\mu_{{\tilde F}_{i1}}({\bar u}_{1}),\mu_{{\tilde F}_{i2}}({\bar u}_{2}),\mu_{{\tilde F}_{i3}}({\bar u}_{3}),\mu_{{\tilde G}_{i}}(\upsilon)\right\} $$

(6)

where $\left(\min\left\{\mu{\tilde F}_{i1}({\bar u}_{1}),\mu{\tilde F}_{i2}({\bar u}_{2}),\mu{\tilde F}_{i3}({\bar u}_{3})\right\}\right)$ can be viewed as the overall compatibility C _i between the facts and the rule; C _i is used to truncate ${\tilde G}_{i}$ to obtain ${\tilde G}_{i}^{\prime}$.

If “product” substitutes for “min” as the t-norm operator (i.e., a Λ b = a · b), Equation (5) is modified as

$$ \mu_{{\tilde G}_{i}^{\prime}}(\upsilon) = \mathop{\max\limits_{u_1,u_2,u_3}}\min\left[\mu_{{\tilde F}_{1}}(u_1)\cdot \mu_{{\tilde F}_{2}}(u_2)\cdot \mu_{{\tilde F}_{3}}(u_3)\cdot \mu_{{\tilde F}_{i1}}(u_1)\cdot \mu_{{\tilde F}_{i2}}(u_2)\cdot \mu_{{\tilde F}_{i3}}(u_3),\mu_{{\tilde G}_{i}}(\upsilon)\right] $$

(7)

Likewise, if ${\tilde F}_{j}^{\prime}$ is a precise value (i.e.,${\bar u}_{j}$), Equation (7) evolves into:

$$ \mu_{{\tilde G}_{i}^{\prime}}(\upsilon) = \min\left\{\mu_{{\tilde F}_{i1}}({\bar u}_{1})\cdot \mu_{{\tilde F}_{i2}}({\bar u}_{2})\cdot \mu_{{\tilde F}_{i3}}({\bar u}_{3}),\mu_{{\tilde G}_{i}}(\upsilon)\right\} $$

(8)

where $\left(\mu_{{\tilde F}_{i1}}({\bar u}_{1})\cdot \mu_{{\tilde F}_{i2}}({\bar u}_{2})\cdot \mu_{{\tilde F}_{i3}}({\bar u}_{3})\right)$ can be viewed as the overall compatibility C _i between the facts and the rule; C _i is used to truncate ${\tilde G}_{i}$ to obtain ${\tilde G}_{i}^{\prime}$.

In this paper, “product,” “sup-min,” and “min” are selected as the t-norm, composition, and implication operators, respectively. It should be noted that “product” is chosen as the t-norm operator instead of another more widely used t-norm operator, “min,” because the t-norm operator “product” makes the result of ${\tilde G}_{i}^{\prime}$ sensitive to every input; whereas, only one input will control ${\tilde G}_{i}^{\prime}$ in the case of the t-norm operator “min.”