In short, this work deals with composite indexes from an MCDA point of view. More precisely, some concepts and tools developed in the context of MCDA are applied to the ranking problem underlying a composite index. Two main topics are addressed: the “rank reversal” effect, and the application of visual support tools. It must be remarked that these topics may be of concern in a broad context, virtually for any index or benchmarking technique, not limited to sustainability issues. However, no efforts towards generality are made in this work, on the contrary, the above topics are addressed only in the context of the SDEWES Index. This implies that some observations and results are motivated by, and related to, the features and aims of the SDEWES Index, however, it must be kept in mind that here the SDEWES Index is adopted essentially as a paradigmatic example. The motivations for this choice should be clear in light of the above discussion, and include the availability of data, the wide scientific literature on the subject and, last but not least, the promising potential of the SDEWES Index as a decision supporting tool. In particular, the analyses conducted in the present work were made possible by the availability of the underlying data both at the dimension level [8] and (almost completely) at the indicator level [9]-[14]. For other indexes, data are not always made available, e.g., the Sustainable Cities Index [3] only provides the final scores and a graphical representation of normalized data.

In MCDA terminology, a rank reversal occurs if the order of preference between two alternatives changes when an alternative is added to or removed from the decision problem. This means that the relative ranking of a pair of alternatives depends on the whole set of alternatives, and not only on the pair itself, in fact, an MCDA method exposed to rank reversal lacks of the so called “independence” property. Rank reversal, or equivalently “non-independence”, is known to affect many MCDA methods, and has been the subject of a long-lasting debate, see for example [23], [24] and the references therein. In the context of MCDA methods for sustainability assessment rank reversal is specifically addressed in Cinelli et al. [22]. In practice, the occurrence of rank reversals is likely to be rather limited, and can be considered a minor problem in light of the aim and scope of the SDEWES Index. On the other hand, since the index is designed to address an inherently evolving reality, non-independence implies that the results may not be computed consistently throughout a wide time interval. This can be limiting if the goal is to adopt the index as a tool for evaluating and assessing policies, to this aim, it would be useful to devise a “stable” (i.e., “independent”, or “rank-reversal free”) version of the SDEWES Index. Here it will be shown that the MCDA theory offers a viable approach to obtain such version. It turns out that devising a stable index, although demanding in terms of technological expertise, is relatively easy from a mathematical point of view.

A plethora of data visualization techniques are available today, and they are widely used in support of sustainability assessment analyses. As for the SDEWES Index, just to mention a couple of examples: radar (or spider-web) charts are used in the SDEWES Index Benchmarking Tool [11], while geographical maps and stacked charts are combined in the “SDEWES City Index Atlas” [8], [15]. In addition, special techniques have been developed, e.g. the three-dimensional visualization proposed for the City Sustainability Index [25]. Visualization tools have been widely studied also in the context of MCDA, to address the “description problematic” [16], see Miettinen [26] for an overview. A well-known example is the Graphical Analysis for Interactive Aid (GAIA) methodology, that has been developed as a visual companion to PROMETHEE methods [16]. GAIA includes several graphic and interactive tools, in particular the “GAIA plane”, based on the projective approach originally proposed in Mareschal and Brans [27]. The GAIA plane allows to reveal interesting relations between criteria and/or alternatives of an MCDA problem. Other tools similar to the GAIA plane have been proposed, e.g., the “Co-plot” method: see Raveh [28] for a description and a comparison to the GAIA plane. It is worth noting that Co-plot exploits more sophisticated statistical data analysis techniques compared to GAIA, on the other hand, Co-plot is not linked to a specific MCDA method, as GAIA is. In the present work, the GAIA plane is adapted to work with dimensions and indicators of the SDEWES Index. The benefits of this approach are shown, and some drawbacks are pointed out. Consequently, a new and simple visual tool is proposed. This tool shows explicitly and exactly some information that is somehow hidden or approximated in the GAIA plane. Some particular features of the new tool are exploited, first to provide a graphical representation of the scenario analyses discussed in Kilkiş [5] and Kilkiş [13], and then to point out some aspects related to the decoupling of energy use from CO₂ emissions.

The layout of this paper is as follows. In Section 2 rank reversal is discussed, together with a possible approach for obtaining a stable index. Section 3 provides the definitions of the visual tools and points out their main features, while Section 4 shows the application of the tools to the SDEWES Index, in particular to data aggregated at the dimension level. The last section contains a few conclusions and suggestions for further work.

The weighted sum method is known to be exposed to rank reversal, due to its normalization phase. An expository example is shown in Wang and Luo [23], note the normalization technique in that example is the same as in eq. (1) and eq. (2), except that the factor 10 is missing. It follows that also the SDEWES Index is potentially exposed to rank reversal, indeed, it is easy to see that the normalization process violates the independence property, since the results of eq. (1) and eq. (2) depend on m_x,y and M_x,y, and thus on the whole city sample. Clearly, the bounds for an indicator can change with time, as long as new cities are added to the sample, or due to changes in the city evaluations. It is interesting to point out what happens to the normalized values I_x,y(C_j) when either m_x,y or M_x,y changes. It can be shown (technical details are rather straightforward and omitted here) that improving the best evaluation (respectively, worsening the worst evaluation) makes the I_x,y(C_j) decrease (respectively, increase).

This can be considered as a reasonable and fair reaction to the setting of a higher or a lower standard, due to the arrival of a new very good or very poor city. Note that the use of winsorization may prevent these changes to take place, or at least filter off the ones with the most consistent impact. On the other hand, a straightforward application of the winsorization process to wider and wider samples may lead to unexpected effects, as shown by the following example.

Observe that the combination of winsorization and normalization implicitly define for each criterion a (normalized and piecewise linear) utility function F_x,y(v) mapping each evaluation E_x,y(C_j) onto the interval [0, 10]. For a maximization criterion, this function assigns full utility to evaluations above the threshold M_x,y, and null utility to evaluations below the threshold m_x,y, evaluations in [m_x,y, M_x,y] are linearly mapped onto [0, 10]. For a minimization criterion, the role of the thresholds is symmetric, as shown in Figure 1. Note that utility functions are defined on the whole domain of possible criterion evaluations, that (at least in principle) may be unbounded, even if in most cases (e.g., for a percentage) upper and lower bounds are readily available. Therefore, the flat zones on the left and on the right appear whenever the interval [m_x,y, M_x,y] does not cover the domain, clearly, if winsorization detects outliers, they end up falling below these flat zones. It can be observed that the utility function (for maximization) resembles the “Linear” preference function of PROMETHEE [16], [31]. In particular, m_x,y and M_x,y play the role of the “indifference” and “preference” thresholds Q and P, respectively. Although in different contexts, these function share the common approach of mapping high and low values onto the extremes of the normalization interval.

Based on the above utility functions, the SDEWES Index may be interpreted in terms of Multiple Attribute Utility Theory (MAUT: see e.g. [16], [29]) as an additive utility model:

$SI\left(C_j\right)={\displaystyle \sum }_{x=1}^7{\displaystyle \sum }_{y=1}^5{f}_{x,y}{F}_{x,y}\left[E_{x,y}\left(C_j\right)\right]$ (5)

Figure 1.

Utility function for maximization and minimization criteria

where each utility function F_x,y is given the weight f_x,y = a_x. Such additive model, once defined, can be later applied to any sample of cities, and applied again to the same city whenever some of its evaluations have changed. Remark that the score computed by eq. (5) does no longer depend on the city sample, since each city is evaluated independently, thus the additive model satisfies the independence property, i.e. is not exposed to rank reversal. As observed explicitly in Cinelli et al. [22], MAUT (not necessarily restricted to additive models) is the only MCDA approach that provides completely rank reversal free solutions. Clearly, in order to define the additive model in eq. (5) the utility functions F_x,y(v) must be defined once and for all. This amounts to say that a stable version of the SDEWES Index is obtained if the current bounds m_x,y and M_x,y are fixed as definitive, possibly after some suitable adjustments. Unfortunately, as discussed earlier, this operation is in general not safe: for some indicators, current bounds may not be representative of future trends. In these cases, reasonable bounds should be found, and this may be a challenging task for those indicators (in particular, maximization ones) for which evaluations are expected to improve substantially in the future. Note that the task can be simplified in light of a few preliminary observations, including (but not limited to) the following:

• Some indicators (e.g. those for dimension D₂, but also I_5,5, I_6,2, I_7,1, I_7,2) are measured on essentially qualitative scales, that are specifically defined to aggregate the results of sub-indicators, and for which it should be easy to derive reasonable bounds;

• For some indicators measured in percentage (e.g. I_3,4, I_3,5, I_4,2, I_7,5), m_x,y and/or M_x,y are equal or very close to 0 and 100, respectively;

• for some indicators (e.g. I_1,1, I_1,2, I_1,4, I_4,5, I_5,1, I_5,2, I_7,4), M_x,y is two or three orders of magnitude larger than m_x,y: in these cases, it should be safe to shift m_x,y to zero,

• For most maximization (minimization) indicators it seems suitable to set the current m_x,y(M_x,y) as a minimum performance thresholds under (over) which a null utility must be assigned.

Moreover, an additive model is not restricted to use the utility functions F_x,y(v) described above. These functions are not monotonically increasing or decreasing, since they show flat zones where evaluations are not distinguished from each other. Thus it may be appealing to consider smoothed version of these functions. A smooth utility function does not require to set the bounds m_x,y and M_x,y, and may provide a more sophisticated model including e.g. saturation effects. A suggestion for a smooth function is again offered by PROMETHEE, in particular by the “Gaussian” preference function, a version rescaled within [0, 10] will be considered here:

$G\left(v\right)=10-10e^{-v^2/2d^2}$ (6)

In addition, a “cubic” version of G(v) will be used:

$H\left(v\right)=10-10e^{-v^3/2d^3}$ (7)

Both functions G(v) and H(v) are monotonically increasing, with shape changing from convex to concave, and such that G(d) = H(d), as shown in Figure 2. The following example gives a hint of how these smooth functions may be used for one particular indicator.

Figure 2.

Piecewise linear and smooth utility functions

In the GAIA plane, alternatives, criteria and weights are jointly represented by points on a plane. More precisely, each criterion is graphically represented by a “vector”, i.e. a segment from the origin to the corresponding point, the vector representing the weights is usually referred to as the “stick”. The primary plane is identified by the axes U (horizontal) and V (vertical), a third axis W allows to define the secondary planes (U, W) and (V, W). This representation is obviously approximated, since it only shows the projections on a plane of points in a space of dimension p, i.e. the number of criteria. The method also provides a measure of the quality of the representation, which can be seen as the percentage of information retained after projection. The GAIA plane allows to visualize several aspects of an MCDA problem, such as conflicting criteria or sensitivity to changes in the weights, see Mareschal and Brans [27] and Greco et al. [16] for a detailed discussion. The interesting features for the present work can be summarized as follows:

• Alternatives with similar characteristics appear close to each other in the plane;

• Criteria expressing similar (respectively: opposite, uncorrelated) preferences are represented by vectors oriented in approximatively similar (respectively: opposite, orthogonal) directions;

• Points corresponding to better alternatives for a criterion are likely to be found moving in the direction of the corresponding criterion vector, similarly, the stick shows the direction where globally better alternatives can be found;

• The length of a vector denotes the reliability of the visual information it conveys: a short length suggests a high loss of information due to projection.

In order to find the axes U, V and W, GAIA applies Principal Component Analysis (PCA) to the matrix of profiles computed by the PROMETHEE method. The profile of an alternative is a (row) vector of scores, one for each criterion. Profiles are particularly suitable for PCA since they are normalized in [-1, 1] and centered, i.e. the sum of scores over all alternatives is zero for each criterion. Obviously, in the context of the SDEWES Index profiles do not exist, but PCA can be applied to available data that are normalized, even if not centered. There are two possibilities here, namely, apply PCA at the top level or at the bottom level of the criteria hierarchy. In the former case, a city C_j is represented by the sub-indexes A_x(C_j), for x = 1, 2, …, 7, recall that the sub-indexes are normalized in [0, 50]. In the latter case, PCA is applied separately for each dimension D_x, and a city C_j is represented by its scores I_x,y(C_j), y = 1, 2, …, 5, normalized in [0, 10]. From now on, the generic term “criterion” is used to denote either a dimension (at top level) or an indicator (at bottom level). In both cases, let M denote the n × p “score matrix”, where n is the number of cities and p is either 7 or 5, each city C_j is represented by row j of M. The PCA method for finding the axes U, V, W and the qualities of the projection planes can be summarized as follows:

Given the axes, the coordinates of relevant points on the plane (U, V) can be computed as shown below, the computation for the secondary planes (U, W) and (V, W) is similar:

• City C_j has coordinates (N_j.U, N_j.V), where N_j. is row j of N;

• Criterion k has coordinates (U_k, V_k);

• The stick has coordinates (w^TU, w^TV).

Note that the weights are represented by a unit-length vector w, where w = α/||α||₂ at top level, while at bottom level w = u/||u||₂, where u = [1, 1, 1, 1]^T. Recall that the weight vector is not considered in the PCA algorithm. Consequently, the aggregated score (the SDEWES Index at top level, a sub-index at bottom level) is not represented exactly on the GAIA plane. The stick shows a direction of expected growth, but this information may be quite approximated, in particular if the stick length is short. This is one of the motivations that lead to the proposal of a different visual representation.

As discussed earlier, the weighted sum method is totally compensatory, i.e. it is not sensitive to dispersion of criteria values. Accordingly, the SDEWES Index does not take dispersion into consideration. On the other hand, distribution patterns of sub-indexes are relevant for city pairing [5], [12], [15], thus a measure of dispersion may be helpful in that context. The GAIA plane reveals some information about dispersion, since the direction orthogonal to the stick is somehow related to dispersion, however, this information is not explicit, and in some cases can be quite approximated. The tool proposed here aims at visualizing an aggregated score (horizontal coordinate) and the corresponding dispersion (vertical coordinate) explicitly and exactly. Also in this case, the method can be applied at two levels: at top level, the aggregated score is the SDEWES Index, while at bottom level it is the sub-index A_x(C_j) for a given dimension x, in both cases, the input data are contained in the n × p score matrix M, as defined for the GAIA plane.

Clearly, many different measures of dispersion can be adopted: a straightforward geometrical approach is followed here. Consider first the top level, where each city C_j is represented by the sub-indexes A_x(C_j), contained in row j of the score matrix M. Thus city C_j is a point $M_{j.}^T$ in the space of dimension p = 7, and weights are represented by the unit length vector w = α/||α||₂. For each city C_j let $M_{j.}^T={\pi }_{j}w+d_j$, where π_j = M_j.w. Note that π_j is the length of the projection of $M_{j.}^T$ onto the axis defined by w, while the vector d_j is orthogonal to this axis, thus ||d_j||₂ is the Euclidean distance of $M_{j.}^T$ from the axis. It can be easily checked that:

$SI\left(C_j\right)={‖\alpha ‖}_2{\pi }_j$ (8)

Therefore, to obtain homogeneous scales, the normalized distance:

${\delta }_j={‖\alpha ‖}_2{‖d‖}_j{}_2$ (9)

is chosen as a measure of the dispersion of the sub-indexes representing city C_j. The extension to the bottom level is immediate: in this case, for dimension x, city C_j is represented by indicators $I_{x,y}\left(C_j\right)$, $M_{j.}^T$ is a point in the space of dimension p = 5, and the vector u = [1, 1, 1, 1, 1]^T replaces the vector α, i.e. w = u/||u||₂. The value π_j and the vector d_j are defined as for the top level, and it turns out that:

$A_x\left(C_j\right)={‖u‖}_2{\pi }_j$ (10)

Therefore, similar to eq. (9), dispersion is measured by ${\delta }_j=u_2{d}_j{}_2$.

Based on the representation of cities, a two-dimensional visualization of the criteria can be defined. The idea is that criterion k is represented by the point of coordinates $\left({\rho }_k^{\left(1\right)},{\rho }_k^{\left(2\right)}\right)$, where ${\rho }_k^{\left(1\right)}$ and ${\rho }_k^{\left(2\right)}$ are a measure of the correlation of the criterion with the aggregated score and the dispersion, respectively. In particular, the Pearson correlation coefficient will be used, recall that this coefficient is normalized in [-1, 1]. Both at top and bottom level, criterion k corresponds to column k of the score matrix M, while aggregated score and dispersion can be represented by the n-dimensional vectors π and δ defined above. Thus the horizontal coordinate of criterion k is defined as:

${\rho }_k^{\left(1\right)}=\frac{{{\displaystyle \sum }}_{j=1}^n\left(M_{jk}-{\overline{M}}^k\right)\left({\pi }_j-\overline{\pi }\right)}{\sqrt{{{\displaystyle \sum }}_{j=1}^n{\left(M_{jk}-{\overline{M}}^k\right)}^2}\sqrt{{{\displaystyle \sum }}_{j=1}^n{\left({\pi }_j-\overline{\pi }\right)}^2}}$ (11)

where ${\overline{M}}^k$ is the average of column k of M, while π ̅ is the average value of vector π. The vertical coordinate ${\rho }_k^{\left(2\right)}$ is defined as in eq. (11), replacing π by δ. Graphically, criterion k is represented by a vector from the origin to the point $\left({\rho }_k^{\left(1\right)},{\rho }_k^{\left(2\right)}\right)$. Remark that (differently from GAIA) each of the two coordinates conveys sound information on the underlying MCDA problem. The horizontal coordinate shows whether, and up to what extent, the aggregated preferences agree with the ones expressed by the criterion. The vertical coordinate shows whether a good performance on the criterion comes at the expenses of a larger dispersion among criteria. An interesting feature of the Index/Dispersion approach is that criteria coordinates can be computed considering only a subset of the n cities. This allows, in particular, to partition the overall sample into sub-samples (e.g., quartiles) and obtain distinct representations of criteria separately for each sub-sample. These representations can be compared to each other, in order to spot those cases where the relations between criteria differ depending on the sub-sample. Note that a similar process is not possible with the GAIA plane, where the representation of criteria is determined univocally by the axis U and V, and cannot be related to sub-samples. It is worth mentioning that data analysis separated by quartiles has been exploited in [5], [12], [13], [15], see e.g. figure 4 in Kilkiş [5], where graphical representations are given both at the index and at the sub-index level.

The first pair of figures illustrates the main difference between the two approaches, namely, the capability of representing exactly the value of the SDEWES Index. To this aim, the four quartiles individuated by the index values are represented. Figure 3 shows the GAIA plane with the stick showing the approximate direction of increase for the SDEWES Index. Observe that the length of the stick is 0.912, that is sufficiently close to one, this means that the unit length vector w = α/||α||₂ is quite close to its projection on the (U, V) plane. However, some information is lost in the projection process, as can be expected from the quality value. Indeed, quartiles appear in the right order along the stick direction, but with overlapping convex hulls. Obviously, no overlapping appears in the Index/Dispersion representation of Figure 4, that allows to point out two interesting details:

• For cities in the bottom quartile (91-120) both the SDEWES Index and the dispersion are spread in a much wider interval compared to the middle quartiles, up to a minor extent, the same holds true for the top quartile too;

• Within the top quartile, the best index values are found in the top-right corner, i.e. show a relatively high dispersion, in other words, and excellent overall performance can be obtained even with relatively wide differences between single dimension performances.

Figure 3.

GAIA: city quartiles and stick

Figure 4.

Index/Dispersion: city quartiles and simulation trajectories

Figure 4 also shows two trajectories individuated by simulating the evolution of a city along time. The simulation for Rio de Janeiro is based on the results for the normative scenario addressed in Kılkış [13]. The simulation for the “Average City” is taken from a scenario addressed in [5], [15], where a fictitious city evolves, in each dimension, from the average to the maximum of the corresponding sub-index values in the current 120-city sample. In both cases, the simulation assumes a transition from the current situation (year 2019) to the situation foreseen for the target year 2050. Here, for simplicity, the transition is assumed to be linear in time, markers show the expected situation for each year in the time horizon. Even if the transition is linear, the trajectory is not necessarily linear, since the measure of dispersion is not linear. It can be observed that the proposed scenarios lead to a substantial increase not only for the SDEWES Index (as expected) but also for dispersion.

The goal of the next figures is to show that the two tools have the same capability of representing the relations among criteria and the relations between criteria and cities. Moreover, the information they convey have similar reliability. To this aim, conjoint plots of criteria and cities are provided, and further visual information is added as follows. In each plot, a specific dimension D_x is selected, and the best ten and the worst ten cities for D_x are highlighted. This gives a visual intuition of the quality of the visual ranking of the cities. Moreover, the measure of reliability defined above, here denoted by “Correlation”, is provided for each plot. In Figure 5 and Figure 6, the selected dimension is D₅. For both tools, the correlation value is very close to one, and indeed, the visual ranking of the cities seems quite close to the actual one, with best and worst cities appearing on opposite sides of the city cloud. Note that for both tools the length of the vector representing D₅ is high.

Figure 5.

GAIA: conjoint plot, best/worst cities for D₅

Figure 6.

Index/Dispersion: conjoint plot, best/worst cities for D₅

In Figure 7 and Figure 8, the selected dimension is D₃. In this case, the correlation value is rather low, and indeed the visual ranking appears much less reliable, with some of the best and worst cities mixing together around the origin. Note that both the length of the vector representing D₃ and the correlation value are higher for Index/Dispersion than for GAIA. The relatively poor visual performance of dimension D₃ can be related to its low correlation to the index, and this behaviour seems to suggest that some cities do not yet fully exploit their renewable energy potential in their energy supply systems, in this area, a remarkable example of best practice is given by the city of Reykjavík (the diamond close to the point defining D₃ in Figure 8) that decarbonized its power sector [5], [14].

Figure 7.

GAIA: conjoint plot, best/worst cities for D₃

Figure 8.

Index/Dispersion: conjoint plot, best/worst cities for D₃

Besides the above observations on the visual ranking of cities, Figures 5-8 show that the two tools provide remarkably similar representations of the criteria. The only evident difference is the length of the vectors representing dimensions D₃ and D₄. Keeping in mind the meaning of the stick for GAIA, it is possible to draw some conclusions supported by both tools:

• Up to different extents, the first six dimensions bring some similarity to the overall index, with D₅ showing the strongest correlation ${\rho }_5^{\left(1\right)}=0.8056$;

• On the contrary, dimension D₇ is almost uncorrelated to the index, this seems to suggest that an innovative city is not necessarily a sustainable city;

• There is a partial conflict (or better a lack of correlation) between two sets of dimensions, namely, D₁ and D₅ opposed to D₂, D₆ and D₇, in particular, this is revealed by dispersion, or equivalently by the direction orthogonal to the stick on the GAIA plane.

As for the last observation, it is not easy to find a simple interpretation. A possible explanation may be as follows. On one side, dimensions D₂, D₆ and D₇ are somehow related to the degree of social-cultural development of a city (in terms of sustainability awareness, welfare, education, innovation, etc.) and thus can be expected to be related to each other. On the other side, D₁ and D₅ measure the level of evolution of energy systems (in particular in terms of energy/emission decoupling) and thus are likely to be correlated, more details on this aspect are given at the end of this section. Yet, these two general aspects (social-cultural development and evolution of energy systems) appear essentially independent of each other, despite their potential for mutual enhancement.

Figure 9 and Figure 10 show conjoint plots for the (overall 58) cities addressed in [9]-[11]: the three samples are distinguished, and referred to as MED, SEE and WC1, respectively. Note that the city of Istanbul, addressed both in Kılkış [9] and in Kılkış [10], is showed separately: this choice allows to point out more clearly the differences between the MED and SEE samples. To begin with, note that the representation of dimensions for both tools is very similar to the one obtained for the 120-city sample, as for the GAIA plane, also in this case the length of the stick is high (0.928) while the quality (65.88%) is lower than the one for the 120-city sample. As for the comparison of city samples, the following observations can be drawn:

• MED and SEE samples (both located in specific geographical areas) are concentrated within relatively small areas of the plane, while the world cities in WC1 are spread in a larger area;

• Overall, SEE cities have a slightly better index w.r.t. MED cities, while WC1 cities have a larger dispersion;

• WC1 cities show a better performance in terms of social-cultural development (dimensions D₂, D₆ and D₇) while MED (and up to some extent, SEE) cities are better in terms of energy systems and emissions (D₁ and D₅).

Figure 9.

58 cities, GAIA: conjoint plot, distinguished samples

Figure 10.

58 cities, Index/Dispersion: conjoint plot, distinguished samples

Figure 11 shows the criteria representations computed by the Index/Dispersion method separately for each quartile. The four plots show rather evident differences, following an apparent trend: roughly speaking, the pattern of criteria vectors seems to rotate clockwise from top to bottom quartiles. In particular, it is interesting to point out the behaviour of dimensions D₁ and D₅, that are closely related to a main focus of the SDEWES Index, namely the energy/emission decoupling. For the top quartile D₁ and D₅ have the strongest correlations to the index, meaning that they have the most relevant impact on the city ranking. This is consistent with the results in Kilkiş [5] (mentioned earlier in the present work) showing that the top ten cities propose best practices in energy saving and/or reducing emission. Moving towards lower quartiles, the role of D₁ and D₅ becomes less and less relevant, in favour of other dimensions, in particular D₆ and D₇. In the bottom quartile, D₁ and D₅ have the weakest correlation to the index, and are rather weakly correlated to each other. Overall, this behaviour seems to suggest that the leading cities are those that adopted integrated measures to reduce energy consumption and emissions at the same time, while for the less sustainable cities these two aspects seem to be rather disregarded, or at least, far from being addressed in a systematic way.

Figure 11.

Index/Dispersion: criteria representation by quartile