Rule bases

The rules used in each inference step express linguistically the dependence of a composite indicator on other, more elementary indicators. This section describes a compact representation of the rule bases, which avoids storing all rules in the computer memory. This is done in three steps outlined below.

  1. The fuzzy sets of Fig. 6 are assigned integer values 0, 1, 2, …, where 0 corresponds to the fuzzy sets with the lowest sustainability. The fuzzy set Weak in Fig. 6a is assigned the value 0, Medium is assigned the value 1, and Strong is assigned the value 2. The corresponding weights for the composite indicators of Fig. 6b are Very Bad→0, Bad→1, Average→2, Good→3, and Very Good→4, and for OSUS (Fig. 6c) Extremely Low→0, Very Low→1, …, Extremely High→8. Moreover, each indicator used as input to an inference engine is also assigned a positive weight, which measures its relative importance against the other inputs. Currently, all inputs of the SAFE inference engines are assigned the weight 1.
  2. For each rule, a weighted sum of inputs is computed and assigned to the output variable.
  3. if
    	‘Threatened Mammals’ is Medium
    	‘Threatened Birds’ is Strong
    	and ‘Threatened Plants’ is Medium
    	and ‘Threatened Fishes’ is Weak
    	and ‘Threatened Reptiles’ is Strong
    	and ‘Threatened	Amphibians’ is Strong
    then
      	PR(BIOD) is Bad.
    
    	
    The weighted sum of its inputs is:
    	weight of PR(BIOD) = 1 + 2 + 1 + 0 + 2 + 2 = 8.
    
  1. The resulting weight is assigned to some fuzzy set. The larger the weight the larger or better the fuzzy set of the output. For example, the rule base for the composite indicator PR(BIOD) comprises 729 rules (36 six-tuples of the fuzzy sets W, M, and S). It is represented compactly as follows
\[{\rm{fuzzy}}\,{\rm{set}}\,{\rm{of}}\,\;{\rm{PR}}\left( {{\rm{BIOD}}} \right){\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if }}\;{\rm{weight}}}& \le &{\rm{7}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{8}}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{9}}\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&{{\rm{10}}}\\{{\rm{VG}}}&{{\rm{weight}}}&{\rm{ = }}&{{\rm{11}}{\rm{,12}}}\end{array}} \right.\]

The same rule base is used for PR(LAND), which has also six inputs as shown in Table 1. The rule bases used to assess other composite indicators are given below.

Tertiary components with only one input have the same fuzzy sets and membership grades as their inputs. RE(WATER) depends solely on the basic indicator “Public wastewater treatment plants (percent of population connected)” and RE(AIR) depends on “Renewable energy production (percent of total primary energy supply).” Contrary to the other basic indicators which are mapped on three fuzzy sets, these two indicators are fuzzified using the five fuzzy sets VB, B, A, G, and VG and the resulting membership grades are passed on to RE(WATER) and RE(AIR).

Tertiary components with two inputs:

\[\left\{ {\begin{array}{*{20}{l}}{{\rm{ST(LAND)}}}\\{{\rm{PR(BIOD)}}}\\{{\rm{ST(BIOD)}}}\\{{\rm{PR(AIR)}}}\\{{\rm{RE(POLICY)}}}\\{{\rm{RE(WEALTH)}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}&{\rm{ = }}&{\rm{0}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{1}}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{2}}\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{3}}\\{{\rm{VG}}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{4}}\end{array}} \right.\]

Tertiary components with three inputs:

\[{\rm{PR}}\left( {{\rm{WATER}}} \right) = \left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}&{\rm{ = }}&{{\rm{0}}{\rm{,1}}{\rm{,2}}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&3\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&4\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&5\\{{\rm{VG}}}&{{\rm{weight}}}&{\rm{ = }}&6\end{array}} \right.\] \[\left\{ {\begin{array}{*{20}{l}}{{\rm{ST(WATER)}}}\\{{\rm{PR(POLICY)}}}\\{{\rm{ST(WEALTH)}}}\\{{\rm{PR(KNOW)}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}&{\rm{ = }}&{{\rm{0}}\;{\rm{or}}\;{\rm{1}}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{2}}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{3}}\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{4}}\\{{\rm{VG}}}&{{\rm{weight}}}&{\rm{ = }}&{{\rm{5}}\;{\rm{or}}\;{\rm{6}}}\end{array}} \right.\]

Freshwater availability and quality have become an increasingly crucial concern for many countries. The rule base of PR(WATER) is more pessimistic than those of the other tertiary components. Indeed, out of the seven possible weights (0–6) of PR(WATER), the four smallest ones or 60% correspond to the fuzzy sets VB and B. This is in agreement with widely accepted practices for the assessment of environmental pressures. For example OECD (2004) considers water stress to be high when the annual water withdrawals are at least 40% of the total renewable water resources. Equivalently, 60% of values are VB or B. The same reasoning is followed in the rule bases of pressure indicators PR(LAND) and PR(BIOD) which have six inputs.

Tertiary components with four inputs:

\[\left\{ {\begin{array}{*{20}{l}}{{\rm{RE(LAND)}}}\\{{\rm{ST(AIR)}}}\\{{\rm{PR(HEALTH)}}}\\{{\rm{ST(HEALTH)}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}&{\rm{ = }}&{{\rm{0}}{\rm{,1}}{\rm{,2}}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{3}}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&4\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&5\\{{\rm{VG}}}&{{\rm{weight}}}&{\rm{ = }}&{6,7,8}\end{array}} \right.\] \[{\rm{ST(POLICY) = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}& \le &{\rm{3}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{{\rm{4}}\;{\rm{or}}\;{\rm{5}}}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&6\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&7\\{{\rm{VG}}}&{{\rm{weight}}}&{\rm{ = }}&8\end{array}} \right.\]

ST(POLICY) gives the state of human rights and is assessed using more strict criteria than the other components.

Tertiary components with five inputs:

\[\left\{ {\begin{array}{*{20}{l}}{{\rm{RE(HEALTH)}}}\\{{\rm{RE(KNOW)}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}& \le &{\rm{3}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{\rm{4}}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&5\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&6\\{{\rm{VG}}}&{{\rm{weight}}}& > &7\end{array}} \right.\]

Tertiary components with six inputs:

\[\left\{ {\begin{array}{*{20}{l}}{{\rm{PR(LAND)}}}\\{{\rm{PR(BIOD)}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}& \le &7\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&8\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&9\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&{10}\\{{\rm{VG}}}&{{\rm{weight}}}& > &{{\rm{11}}\;{\rm{or}}\;{\rm{12}}}\end{array}} \right.\] \[\left\{ {\begin{array}{*{20}{l}}{{\rm{PR(HEALTH)}}}\\{{\rm{ST(KNOW)}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}& \le &4\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&5\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&6\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&7\\{{\rm{VG}}}&{{\rm{weight}}}& \ge &8\end{array}} \right.\]

Environmental pressures are judged using stricter rules, as discussed previously.

Secondary components with three inputs (PR, ST, RE):

\[\left\{ {\begin{array}{*{20}{l}}{{\rm{LAND}}{\rm{, WATER}}}\\{{\rm{BIOD}}{\rm{,AIR}}}\\{{\rm{POLICY}}{\rm{,WEALTH}}}\\{{\rm{HEALTH}}{\rm{, KNOW}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}& = &{{\rm{0}}\;{\rm{or}}\;{\rm{1}}}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{2,3,4}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&{5,6,7}\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&{8,9,10}\\{{\rm{VG}}}&{{\rm{weight}}}& = &{{\rm{11}}\;{\rm{or}}\;{\rm{12}}}\end{array}} \right.\]

Finally, the rule bases of the primary components of sustainability and the overall sustainability index are:

\[\left\{ {\begin{array}{*{20}{l}}{{\rm{ECOS}}}\\{{\rm{HUMS}}}\end{array}} \right\}{\rm{ = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{VB}}}&{{\rm{if}}\;{\rm{weight}}}& = &{0,1,2}\\{\rm{B}}&{{\rm{weight}}}&{\rm{ = }}&{3,4,5,6}\\{\rm{A}}&{{\rm{weight}}}&{\rm{ = }}&{7,8,9,10}\\{\rm{G}}&{{\rm{weight}}}&{\rm{ = }}&{11,12,13}\\{{\rm{VG}}}&{{\rm{weight}}}& = &{{\rm{15}}\;{\rm{or}}\;{\rm{1}}6}\end{array}} \right.\] \[{\rm{OSUS = }}\left\{ {\begin{array}{*{20}{r}}{{\rm{EL}}}&{{\rm{if}}\;{\rm{weight}}}& = &{\rm{0}}\\{{\rm{VL}}}&{{\rm{weight}}}&{\rm{ = }}&1\\L&{{\rm{weight}}}&{\rm{ = }}&2\\{\rm{I}}&{{\rm{weight}}}&{\rm{ = }}&4\\{{\rm{FH}}}&{{\rm{weight}}}& = &{\rm{5}}\\{\rm{H}}&{{\rm{weight}}}& = &6\\{{\rm{VH}}}&{{\rm{weight}}}& = &7\\{{\rm{EH}}}&{{\rm{weight}}}& = &8\end{array}} \right.\]