To Weight or Not to Weight, That Is the Question: The Design of a Composite Indicator of Landscape Fragmentation

: Composite indicators (CIs), i.e., combinations of many indicators in a unique synthetizing measure, are useful for disentangling multisector phenomena. Prominent questions concern indicators’ weighting, which implies time-consuming activities and should be properly justiﬁed. Landscape fragmentation (LF), the subdivision of habitats in smaller and more isolated patches, has been studied through the composite index of landscape fragmentation (CILF). It was originally proposed by us as an unweighted combination of three LF indicators for the study of the phenomenon in Sardinia, Italy. In this paper, we aim at presenting a weighted release of the CILF and at developing the Hamletian question of whether weighting is worthwhile or not. We focus on the sensitivity of the composite to different algorithms combining three weighting patterns (equalization, extraction by principal component analysis, and expert judgment) and three indicators aggregation rules (weighted average mean, weighted geometric mean, and weighted generalized geometric mean). The exercise provides the reader with meaningful results. Higher sensitivity values signal that the effort of weighting leads to more informative composites. Otherwise, high robustness does not mean that weighting was not worthwhile. Weighting per se can be beneﬁcial for more acceptable and viable decisional processes.


Introduction
The study of complex phenomena requires the adoption of frameworks enabling multisector assessment of a variety of aspects. A typical solution is provided by the use of composite indicators (CIs), i.e., combinations of many indicators often structured in nested hierarchical ensembles, in diverse fields of knowledge, including environmental science and sustainability [1], vulnerability and climate changes [2], socioeconomics [3], and engineering [4]. The design of CIs is a critical process, which presents scientific and technical issues, even though it cannot be reduced to a unique pass par tout procedure [5]. In the public domain, the criticalities are even more evident since CIs influence or enter decision-making support systems concerning the use of funding for addressing specific policies [6]. In this respect, CIs support the ranking of a set of entities (countries, counties, municipalities, homogeneous zones, etc.), termed decision-making units (DMUs), that can be positively or negatively affected by the consequential distribution of resources. Final rankings depend on the variety of choices adopted to frame the indicators, though. The selection of indicators is crucial-different pools of indicators lead to different CIs. In addition, the same indicators may be combined, according to different algorithms leading to different expression of the CI. Algorithms usually include two elements-(i) the rule of indicator aggregation and (ii) the weighting of the indicators (see the review by Greco et al. [7]). A CI can be adopted in decision making when it is reliable, i.e., it is stable. In the other way around, a lower robustness is a sign that the modified CI conveys new information and prompts further assessment of the reasons why some inputs influence the CI. Thus, the sensitivity of the CI versus the algorithms used for its design and construction should be accurately assessed [8]. Sensitivity analysis is a powerful instrument and can be used to address properly the Hamletian question concerning the opportunity to introduce the weights or not.
In the domain of landscape analysis and planning, several metrics have been designed and applied to the assessment of landscape fragmentation (LF), the phenomenon of subdivision of habitats in smaller and more isolated patches (see the review by Wang et al. [9]). LF jeopardizes the normal evolution of animal species since it reduces the home range, i.e., the possibility to move freely in a favorite habitat. LF can be triggered by natural and human drivers, such as the construction of infrastructures and settlements. The indicators of LF provide a useful indication to effectively addressing green-based urban and regional management and planning. In a scientific panorama, in which contributions on composite indicators of LF are still rare, De Montis et al. [10] propose the composite index of landscape fragmentation (CILF). CILF combines three LF indicators bearing the same importance (i.e., weight)-the infrastructural fragmentation index (IFI), the urban fragmentation index (UFI), and the mesh size density (S eff ). This composite is applied to the assessment of LF processes through 51 landscape units of the island of Sardinia, Italy, and provides a basis for addressing defragmentation policies.
In this paper, we aim at presenting a weighted release of the CILF applied to the assessment of LF in Sardinia in 2003. We focus on the sensitivity of the composite to different algorithms combining three weighting patterns (equal weights, principal component analysis-based weights, and expert judgment-based weights) and three indicators aggregation rules (weighted average mean, weighted geometric mean, and weighted generalized geometric mean). We assess the sensitivity of the weighted CILF by measuring the volatility of the position in ranking assumed by the DMUs (i.e., the landscape units). Since the weighting implies a significant amount of extra work, we are interested in ascertaining whether the weighting-i.e., the time-consuming elaborations on the importance of the indicators-is a worthwhile exercise.

State-of-the-Art Summary
Many scientists have built and applied composite indicators to the assessment of complex phenomena in several domains. In addition, they often propose frameworks, in which indicators are aggregated and weighted, according to a variety of patterns. Additionally, they are confronted with the sensitivity of the composite upon the different ways they are obtained. With respect to application field, indicator algorithm structuring, and sensitivity analysis, we comment on the contribution provided by a selection of essays, which are scrutinized in Table A1 (in Appendix A), according to focus, composite and component indicators, method, data, and details about the weights.
A very prominent stream of research studies deals with sustainability. Babcicky et al. [1] illustrate the re-engineering of the environmental sustainability index (ESI). The equivalized ESI is obtained through a weighted summation and supplies a rank with higher robustness. Dulvy et al. [11] develop a marine biodiversity threat indicator, which was calculated from the weighted average of the threat scores of individual fishes species in each year. Foster et al. [12] construct a measure able to gauge the robustness of country rankings associated with some well-known composite indicators-the human development index (HDI), the index of economic freedom (IEF), and the environmental performance index (EPI). Zhou et al. [13] recalculate the HDI using a multiplicative optimization approach to reduce subjectivity in defining the weights for the sub-indicators. Hassan [14] focuses on the sustainability of a region by building the total index of sustainability (TIS). The composite is constructed through multi-attribute utility theory as a weighted summation, where the weights represent the marginal variation of preference corresponding to the change of an attribute value. Huang et al. [15] review nine urban sustainability indicators (USIs), with respect to the three pillars of sustainability-weak and strong sustainability, aggregation and weighting, and spatialization. Kurtener et al. [16] assess agricultural land suitability through a composite indicator obtained by applying fuzzy logic. Munda and Saisana [8] develop a sensitivity analysis of a composite indicator of regional sustainability with respect to different indicator aggregation patterns. The authors consider weighted linear formula, nonlinear and non-compensatory multicriteria analysis, and data envelopment analysis (DEA), which is used to extract the weights. Pert et al. [17] study vegetal conditions in Australia by building a 25-point scale composite threat index combining five indicators normalized with a five-point scale. The aggregation rule is the unweighted summation because the indicators are considered of equal importance.
Given the relevance of the topic for this paper, some studies focus on landscape. Paracchini et al. [18] assess rural landscape through the societal landscape awareness indicator. The composite is based on three dimensions and six indicators. The indicators are normalized according to the min-max formula and aggregated according to the unweighted summation. Schüpbach et al. [19] focus on the impact of individual farms on visual landscape quality in Switzerland, according to the agricultural life cycle assessment (ALCA). They build the composite landscape indicator (CLI) by combining two sub-indicators. The aggregation rule consists of the unweighted summation: sub-indicators are assigned the same importance.
Another important set of articles tackle the questions connected to climate change. Ahsan et al. [2] construct the socioeconomic vulnerability index (SeVI) to assess climate change vulnerability in southwestern coastal Bangladesh. SeVI is obtained by aggregating the component indicators into a unique measure through weighted summation. Bastin et al. [20] develop the directional leakiness index (DLI), a composite index based on remotely sensed imagery. A component indicator concerning the patch size is obtained through weighted summation considering the importance of each patch size class. Christensen et al. [21] focus on methods useful for weighting regional climate models (RCMs). The authors considered six model performance indices and examined three ways for combining them into one weight per regional climate model. Garriga and Pérez Foguet [22] propose a revision of the water poverty index (WPI). The new index is studied with a combination of different aggregation rules and weighting, and a sensitivity analysis is applied. Weights are obtained through expert judgment and multivariate (principal component) analysis. The best aggregation function implies a principal component analysis (PCA)-based selection of indicators and a weighted geometric mean of subindices. Machado and Ratick [23] consider the sensitivity of a composite index of vulnerability to flooding with respect to four aggregation rules-weighted linear combination (WLC), ordered weighted average (OWA), DEA, and compromise programming (CP). Salvati and Zitti [24] focus on land degradation by applying the land vulnerability index (LVI). LVI combines three dimensions and nine indicators, according to the weighted summation rule. Szlafsztein and Sterr [25] build the coastal vulnerability index (CVI) to assess the vulnerability of coastal zones. CVI combines two dimensions through an unweighted summation. Each dimension is obtained as weighted summation aggregation of 16 indicators.
The next cluster gathers essays concerning socioeconomic issues. Busu and Busu [3] construct the Shannon entropy composite index (SECI). The SECI is obtained as a weighted summation of simple indicators processed through an algorithm based on Shannon entropy. Lee and Lim [26] propose a composite for forecasting the daily change of the Korea composite stock price index (KOSPI). Manthalu et al. [27] focus on a composite indicator obtained through a weighted summation of 27 indicators. Weights are calculated through PCA. Rahman [28] analyses the sensitivity of a quality of life index (QOLI) to two aggregation and weighting patterns-the first, a PCA-based weighted summation, and the second, the unweighted summation of Borda scores. The composite combines eight indicators and is calculated for 43 developing countries. Ferrant et al. [29] evaluate gender inequality through the multidimensional gender inequality index (MGII). The MGII is obtained as a weighted summation of the squares of subcomponent indices measuring eight dimensions of inequality. The last group of essays attains engineering. Gerpott and Ahmadi [4] propose a revision of the Information and telecommunication technologies development index (IDI), a composite showing 3 sub-indices and 11 indicators. This study aims at tuning the set of weights to maximize the reliability of IDI, with respect to its ability to gauge the achievement of socioeconomic targets. Gitelman et al. [30] study a composite index that combines the main dimensions characterizing the road safety pyramid. The authors combine the indicators applying principal component analysis and factor analysis to obtain weights in five trials. Rocco et al. [31] develop a composite indicator by considering the ordered weighted averaging aggregation approach. Sharifuddin [32] focuses on the energy security index (ESI) combining five dimensions (availability, stability, affordability, consumption efficiency, and environmental impact), 13 elements, and 35 indicators. Indicators, elements, and dimension scores are normalized between 0 and 1 and aggregated according to unweighted summation for every level but the indicator tier. Shen et al. [33] construct a composite road safety performance indicator, combining seven indicators through normalization in the interval 0-1 and weighted summation aggregation rule.
The analysis of the literature provides the reader with some leitmotiv that is worth being recalled. Oftentimes, scientists have attempted to revise and re-engineer already existing composite indicators with more sophisticated algorithms, including the introduction of weights meant as the importance of the indicators. This has implied usually a great methodological and applicative effort, which has not necessarily led to a better or finer release of the composite. A variety of weighting frameworks has been adopted to extract figures representing the importance of each indicator: some methods focus on the percentage share of information embedded in each indicator, while others on the elaboration of the judgment of selected experts. The weighting is a relevant part of the indicator aggregation rule that can be compensatory, i.e., allowing the balancing of opposite performances in two or more indicators, or not. A cornerstone phase of the composite indicator assessment consists of the sensitivity analysis, which scrutinizes the changes of the output, i.e., the final ranking, with respect to the variation of the input, i.e., the weighting and aggregation patterns. If the changes are negligible, the ranking is stable, and the composite is robust. As a corollary, if the comparison of the resulting ranking of a composite with its weighted release reveals low volatility, the weighted composite does not add much to the picture already provided by the unweighted composite indicators. Otherwise, in presence of high sensitivity, the analyst deals with a weighted composite that describes a richer outline of the phenomenon at hand. This is clearly a sign that the weighting effort has been worthwhile.

Methods
The experimented method is a complement of the framework presented by De Montis et al. [10], who build, apply, and evaluate the robustness of the composite index of landscape fragmentation (CILF) with respect to the volatility of the ranking of 51 landscape units (LUs) established by the Regional Landscape Plan of Sardinia (RLP) [34]. De Montis et al. [10] (i) considered a complete spatial data set, which consisted of human settlements and transport and mobility infrastructures (roads and railways), (ii) applied the generalized geometric mean (GGM) as aggregation algorithm, and (iii) performed a sensitivity analysis of the findings to assess the robustness of CILF.
CILF is designed as a three-indicator unweighted composite, according to the scheme proposed by the Organisation for Economic Co-operation and Development (OECD) [35] and Nardo and Saisana [5]. The scheme includes the following steps: theoretical framework, variables, normalization, aggregation, and robustness and sensitivity [10]. Since we now focus on a weighted release of the CILF, we add a new step concerning the assessment of values representing the importance of the indicators. The simulated theoretical framework consists of aiding a decisional process, in which stakeholders are interested in managing landscape fragmentation starting from a unique simple measure (namely, the CILF) melting three major determinants, i.e., transport and mobility infrastructures, human settlements, and subdivision per se. The indicators selected for measuring those determinants are the infrastructural fragmentation index (IFI), the urban fragmentation index (UFI), and the effective mesh density (S eff ). In Table 1, we resume their main characteristics and rationales. It gauges the effects of urban settlements (soil consumption and negative effects on depletion of habitat, flora, and fauna).
[ [38][39][40] S eff A i : surface area of the i-th patch; M eff : effective mesh size; C: connectivity coefficient measuring the probability that two random points are directly connectable.
It measures the patch density (number of patches included in a 1 sqkm area). [41,42] Variables have been calculated by applying advanced geographical analysis and processing spatial data sets in shapefile in the GIS environment provided by QGIS (https://www.qgis.org/it/site/) and standard spreadsheet programs (Microsoft Excel). Information concerning the geography of Sardinia and-typically-polygonal elements was obtained by processing the land use map available for free at the institutional website of the regional administration. Sardinia Geoportal is the interface of the regional geographic information system and the related spatial data infrastructure.
Indicators are expressed in different measurement units and need to be normalized. In this case, we chose the min-max transformation rescaling the original values into figures calculating the distance from the minimum normalized with respect to the range of the original indicator. We selected this model because it was chosen by De Montis et al. [10] for designing the baseline release of the CILF. The next step concerns the indicator weighting, where components are attributed a value measuring the relevance of the issue represented by the indicator. As often presented in the literature, weights are obtained according to two main procedures-quantitative statistical calculations and expert judgment. The first procedure implies the scrutiny of the matrix of indicator values through multivariate statistical analysis, for instance, the widely adopted in the literature principal component analysis (PCA) and factor analysis (FA), and the association of weights values to each indicator depending on the percentage share of variance explained. In this respect, weights are referred to the variable loading calculated by means of factor analysis for the prevalent (i.e., explaining most of the variance) component. This procedure is based on the analysis of the results of an online questionnaire proposed to a panel of selected individuals. The survey aims at understanding the characteristics of the sample (position, age, education, familiarity with LF) and the importance attributed to each indicator. Interviewees are requested to express the level of importance, according to a five-step Likert-type scale (very low, low, medium, high, and very high). This method was introduced in psychology by Likert and is currently widely used [43][44][45]. Following El Gibari et al. [46], the weights for the ith indicator are extracted by (i) calculating the arithmetic mean of the judgments mw i ; (ii) computing the adjusted weights defined as aw i = 1.5 (mw i −1) to be applied in a multiplicative environment, where the ratio between two consecutive elements of the Likert-type scale is constant and equal to 1.5; and (iii) rescaling the adjusted weights so that they sum up to 1.
The next step consists of the selection of the patterns of indicators' aggregation. We used the weighted version of three algorithms selected by De Montis et al. [10] for the CILF, as detailed in Table 2. Table 2. Synopsis of the indicators combined in the CILF (after De Montis et al. [10]).

Aggregation Rule Level of Compensation Formula Variables
Weighted arithmetic mean (WAM) Full Weighted geometric mean (WGM) None The three algorithms are designed to yield composite indicators with full (weighted arithmetic mean (WAM), no (weighted arithmetic mean (WGM), and partial (weighted generalized geometric mean (WGGM) compensation between the indicators. WGGMbased aggregation rules merge compensatory (i.e., an increase of one indicator can be balanced by the decrease of another indicator) and non-compensatory frameworks [47]. The parameter β is set greater than zero and attaining a maximum of 1 (corresponding to the arithmetic mean allowing full compensation among the indicators). Following De Montis et al. [10], we chose an intermediate framework by setting β = 0.50.
The final step regards sensitivity analysis to ascertain the level of robustness of the composite with respect to different weighting and aggregation patterns. We performed sensitivity analysis by verifying the variability of the position occupied by each decisionmaking unit (DMU), according to the various rankings resulting from the different composites obtained. Since we are interested in the variation of the position in the ranking and not in its increase or decrease, we slightly modify what was proposed in other studies [10,48]. Thus, we set as a general measure of divergence GD between two rankings the absolute value of the average shift in the ranking (GD = |ASR|), where |ASR| obeys the following formula: where m is the number of DMUs, and Rank i 1 and Rank i 2 stand for the position occupied by the i-th DMU, according to composite 1 and, respectively, composite 2. The higher the GD is, the higher the sensitivity of the CI to changes in the way indicators are weighted and aggregated; otherwise, the lower the GD, the higher the robustness. Final outcomes do not change much when selecting different frameworks. However, statistics that suggest, on average, a given outcome are many times partially or totally contradicted by the analysis of average shifts in the ranking (|SR|) reported by individual DMUs. Thus, we add the study of measures of specific divergence (SD), namely, the maximum absolute shift in the ranking (|SR| max ), the share of higher than five positions absolute shifts in the ranking (|SSR| >5 ), and the share of higher than 10 positions absolute shifts in the ranking (|SSR| >10 ). These metrics provide the analysts with detection of the DMUs mostly affected by selecting a given CILF expression versus another one. The higher the SD matrices are, the higher the volatility of the CILF selected with respect to other composites.

Application and Results: A Weighted Composite Indicator of Landscape Fragmentation
In this section, we apply the methodology explained above to the assessment of a weighted release of the CILF. We simulate a decisional environment, where relevant stakeholders-including the Autonomous Regional Administration-are interested in managing LF in the 51 (27 coastal and 24 internal) LUs established by the RLP [34], the main strategic regional plan of Sardinia, Italy ( Figure 1A). Sardinia shows a surface area of about 24,000 km 2 ; thus, is the second largest Italian island. The island is scarcely populated, however, since it has about 1.6 million inhabitants. Sardinia is an autonomous region with special power in landscape management and planning. Decision makers can be assisted by the interpretation of the values CILF of the LUs, which are selected as decision-making units (DMUs) of this exercise; the spatial pattern of the LUs is illustrated in Figure 1B. While De Montis et al. [10] monitored the variation from 2003 to 2008, we refer just to the initial year 2003 for comparing the unweighted CILF to two weighted releases. Indicators have been obtained by processing spatial information described in Table 3.

tation
In this section, we apply the methodology explained above to the assessment of a weighted release of the CILF. We simulate a decisional environment, where relevant stakeholders-including the Autonomous Regional Administration-are interested in managing LF in the 51 (27 coastal and 24 internal) LUs established by the RLP [34], the main strategic regional plan of Sardinia, Italy ( Figure 1A). Sardinia shows a surface area of about 24,000 km 2 ; thus, is the second largest Italian island. The island is scarcely populated, however, since it has about 1.6 million inhabitants. Sardinia is an autonomous region with special power in landscape management and planning. Decision makers can be assisted by the interpretation of the values CILF of the LUs, which are selected as decisionmaking units (DMUs) of this exercise; the spatial pattern of the LUs is illustrated in Figure  1B. While De Montis et al. [10] monitored the variation from 2003 to 2008, we refer just to the initial year 2003 for comparing the unweighted CILF to two weighted releases. Indicators have been obtained by processing spatial information described in Table 3. Table 3. Spatial data set used for calculating the indicators of landscape fragmentation (LF).    Indicator values have been normalized, according to the min-max transformation, and projected in the range of 0-1.

Key Elements
Weights have been obtained, according to the two methods, and are reported in Table 4. The second set of weights has been computed by applying PCA to the min-max normalized values of the indicators. A preliminary check of the determinant (equal to 0.48) indicates that the structure of the dataset justifies the application of PCA. The Keiser-Meyer-Olkin (KMO) test (suitability measure equal to 0.34) reveals a limited utility of PCA; this is expected for a dataset with a reduced number of variables. The Bartlett test (significance equal to 0) indicates that the variables are poorly correlated. We have extracted three components, as reported in Table 5. Component 1 explains the largest share of variance (nearly 54%) and has been selected for the extraction of the variable loading values for the indicators, as indicated in Table 6. Variable loading values have been normalized in the triplet reported in Table 4 so that their sum is equal to 1.
The third set of weights has been obtained by processing the results of an online questionnaire submitted to selected stakeholders in January 2021. The link to the questionnaire was sent to scientists of the two universities of Sardinia, public officials belonging to Sardinian municipalities, and associates to the professional boards of the engineers of the province of Cagliari and agronomists of the provinces of Sassari and Oristano. The questionnaire was completed by 165 individuals. Most of them were academics (45%), freelance professionals (31%), and public officials (15%). The sample includes persons 36-50 (40%) and 51-65 (44%) years of age. The level of education was generally high: 53% of the respondents held a masters' degree and 45% a post lauream degree. Most of the respondents (50%) declared to have a medium level of acquaintance with LF, 25% a high level, 16% a low level, while 8% admitted having never heard of the concept. Weights vary depending on the many segmentation of the audience interviewed because academics may have a different perception of the phenomenon, compared to freelance professionals. In this study, we do not inspect these variations.
In Table A2 in Appendix B, we gather the values of the CILF obtained for 2003, with min-max normalization of the indicators, and different combinations of weighting and aggregation rules. We now apply sensitivity analysis to understand the volatility of the various rankings focusing on two issues, namely, (i) the influence of weighting patterns, being equal the aggregation rule and (ii) the influence of aggregation rules, being equal the weighting pattern. The resulting figures are shown in Table A3 in Appendix B, where the detailed absolute values of the shift in rankings are reported for each DMU. In Tables 7 and 8 and Figures 2 and 3, we report the GD and SD metrics.  aggregation rules. We now apply sensitivity analysis to understand the volatility of the various rankings focusing on two issues, namely, (i) the influence of weighting patterns, being equal the aggregation rule and (ii) the influence of aggregation rules, being equal the weighting pattern. The resulting figures are shown in Table A3 in Appendix B, where the detailed absolute values of the shift in rankings are reported for each DMU. In Tables  7 and 8 and Figures 2 and 3, we report the GD and SD metrics.  The analysis of the |SR| indicates that the less volatile aggregation rule is the WGM-in this domain, the most robust comparison concerns the interplay between composites obtained with weights extracted through PCA and expert judgment elaboration. In this case, |ASR|, |SR|max, |SR|>5, and |SR|>10 are equal to 0. Slightly higher figures are   As for the assessment of the impact of the change of aggregation rule keeping the weighting patterns fixed, much higher figures are obtained-a clear sign of a higher sensitivity of the resulting rankings to the different ways the indicators are weighted. The most robust is the WAM vs. WGGM comparison with EW and EJ weighting patterns, even though |SR|>5 = 7.84% is a sign of local perturbations. The most volatile comparison is WAM vs. WGM with the EJ weighting pattern.

Discussion
In this section, we discuss the outcomes of this paper with respect to the premises presented in the introduction.
First, in the framework of the assessment of LF throughout Sardinian LUs, we are interested to check whether the weighted versions of the CILF perform better-i.e., add more information-than its unweighted releases (all weights equal). We would like to stress that the development of both weight assessments implies demanding and time-con- The analysis of the |SR| indicates that the less volatile aggregation rule is the WGMin this domain, the most robust comparison concerns the interplay between composites obtained with weights extracted through PCA and expert judgment elaboration. In this case, |ASR|, |SR| max , |SR| >5 , and |SR| >10 are equal to 0. Slightly higher figures are associated with the other two comparisons within the WGM aggregation rule and the equal weight (EW) vs. expert judgment (EJ) comparisons within the WAM and the WGGM aggregation rules. The most volatile interplays are EW vs. PCA comparisons with WGGM and WAM aggregation rules.
As for the assessment of the impact of the change of aggregation rule keeping the weighting patterns fixed, much higher figures are obtained-a clear sign of a higher sensitivity of the resulting rankings to the different ways the indicators are weighted. The most robust is the WAM vs. WGGM comparison with EW and EJ weighting patterns, even though |SR| >5 = 7.84% is a sign of local perturbations. The most volatile comparison is WAM vs. WGM with the EJ weighting pattern.

Discussion
In this section, we discuss the outcomes of this paper with respect to the premises presented in the introduction.
First, in the framework of the assessment of LF throughout Sardinian LUs, we are interested to check whether the weighted versions of the CILF perform better-i.e., add more information-than its unweighted releases (all weights equal). We would like to stress that the development of both weight assessments implies demanding and time-consuming activities, such as the mobilization of scientific knowledge, the application of sophisticated methods and algorithms, the design, test, and submission of ad hoc questionnaires, the selection of the target audience, etc. Weighting is never for free because it implies costs, which are also difficult to evaluate precisely in advance. Thus, it is not surprising that scientists question the worthiness of the weighting effort. In our exercise, we obtained the weights following two streams of research. According to the first set of contributions (for instance, Huang et al. [15]), we opted for an "objective" quantitative assessment of the importance of the indicators, which are calculated as the percentage of variance (information) explained. This is an informational weighting method that yields values by processing characteristics embedded in the indicators' dataset. The PCA-based triplet of weights reported in Table 4 signs a clear prevalence of IFI and UFI with respect to the S eff . This per se diverges from the scenario represented by the equalization of the weights. According to a stream of contributions (see [43,46]), we have calculated another weight triplet applying a "subjective" assessment that considers the judgment of a set of selected experts. This triplet is representative of the average judgment of the experts and may vary when considering a different set of participants. The endorsers of subjective assessment frameworks question that quantitative PCA-based methods may be too neutral and abstract for policy and decision makers. In addition, they point out that a clear presentation of subjective assessment methods may strengthen the acceptability and viability of weighted CIs. In our case, the expert judgment-based weight triplet (see Table 4) converges to the scenario with all weights equal. Therefore, an argument purely focused on the values of the weight triplets would suggest that the first process of weighting was more worthwhile than the second because the first one potentially leads to a different picture with respect to the situation described by the unweighted CILF (all weights equal). However, we need to widen our perspective to complete the assessment of the influence of the algorithm on the new release of the CILF. Thus, we have jointly analyzed the combined impacts of both the weighting pattern and the aggregation rule.
Therefore, as for the argument concerning the evidence of the complete sensitivity analysis, it does not show us a monochrome picture and needs to be carefully interpreted. If we consider the outcomes reported in Tables 7 and 8, the figures do not address a monolithic message. In the first case, figures suggest that sensitivity is overall contained (|ASR| < 2.98 and many other SR statistics equal to zero). The high robustness emerging is a sign that-keeping the aggregation rule constant-CILF is not sensibly affected by changes in the weighting pattern. Thus, from this point of view, the weighting effort was not worthwhile. Similar evidence has been discussed by some of the selected contributions (by, for instance [12,21]). A different picture arises when we consider the outcomes reported in Table 8. In this case, sensitivity measures show a much higher range (|ASR| < 5.39, |SR| max < 36, |SR| >5 < 39.22%, and |SR| >10 < 13.73%). This means that keeping the weighting patterns constant and allowing changes in aggregation rules produce sensibly different rankings. Therefore, under these lenses, weighting was a worthwhile process, as documented by another set of selected contributions ( [1,4,22]). In line with what Foster et al. [12] found for the weighted releases of the HDI, IEF, and the EPI, the exercise developed in this paper documents a controversial outcome: some CIs are robust and some others are not. This means that weighting is sometimes (i.e., for some releases of the weighted CILF) justified, while at other times, is not.

Conclusions
In this paper, we have developed the Hamletian question revolving on whether to introduce weights or not in the design of composite indicators. We examined the impact of using two weighting extraction modes on the CILF, an unweighted composite indicator proposed by De Montis et al. [10] for measuring landscape fragmentation. We have focused on the sensitivity of the CILF to different indicator aggregation rules and weighting patterns. The analysis provides us with twofold outcomes. Even though high sensitivity is associated with lower robustness, in our case, it signals that the new composite conveys richer information, i.e., that the weighting has been worthwhile. The analysis of the SR statistics reveals higher values when rankings are compared pairwise changing aggregation rules and fixing the weighting patterns (Table 8). In this case, the weighted versions of the CILF embed new information and lead to different rankings-a circumstance that justifies the adoption of the composites and the effort for introducing the weights. The cases reported in Table 7 describe a panorama, in which low sensitivity values would support that the new weighted composites do not add much to the information provided by the original CILF and that the weighting has not been worthwhile.
However, this can be questionable since weighting per se adds advantages, as intangible as they may be. There are processual benefits connected to a procedure based on composites with weights extracted through both PCA and EJ. PCA provides important indications on the structure of the information contained in the indicators and may support the choice of "objective" weights. EJ-driven weights convey per se a powerful symbolic meaning, which can increase political awareness and endorsement for the widest possible viability of the composite. This holds also in our case, where the weights triplet obtained with EJ does not diverge much from the EW triplet. Still, it represents the combination of the opinions of many individuals representing sometimes the sentiment of relevant political parties or stakeholders. EJ-driven weighted composites often are accepted with higher appreciation in deliberative contexts, where PCA-based weights may be perceived as abstract constructs of meaningless technicality.
As a general message, this paper confirms that the design of a composite indicator is context specific and never a neutral activity and weighting processes are time and resources consuming. The main practical implication is that the cornerstones of the procedure must be clarified step by step so that the audience (institutional partners, decision-makers, stakeholders, etc.) can adhere to and use the composite indicator with full awareness of its impacts. In this respect, the development and correct communication of sensitivity analyses are fundamental.
In this study, we have developed the analyses under precise assumptions; thus, the outcomes have some limitations we would like to clarify for departing with further research studies. Firstly, we have tested the method on the CILF, a composite combining three indicators. The method-with the same steps and procedures-can be exported to other composites, but the results may be affected by the different structure of the indicator nesting and pattern and some adjustments may be required. Additionally, PCA statistics vary, in front of a higher number of indicators; and the extraction of EJ-driven weights may lead to diverse patterns when judging the importance of a larger number of indicators. Secondly, we have considered the CILF proposed by De Montis et al. [10] as a possible combination of three out of all the available measures of LF. However, we are aware that different formulations of the CILF may be obtained by including a larger number of indicators with a multi-layer nested structure. Thirdly, we have addressed the weighted CILF by choosing the min-max rescaling of the original indicators. Different courses of action would have followed a selection of alternative normalization patterns (Borda distance, standardization, ratio to the maximum, etc.). Similarly, we have focused on the GGM aggregation rule calibrated on a fixed compensation parameter β. We are aware that sensitivity analysis can be directed also to the assessment of the interplay between the composite and a changing compensation of the indicators. We will delve into this in future studies. Fourthly, EJ weights are extracted through the method proposed by El Gibari [46] and the Likert scale. Other methods can be selected for processing expert judgment (see, among others, the regime method by Hinloopen and Nijkamp [49], the analytical hierarchy process by Saaty [50], and goal programming and DEA methods [51,52]). This choice has its impacts on the entire study, which would have implied different data harvesting procedures and outcomes with the selection of a different EJ processing framework. Fifthly, regardless of the EJ processing pattern, we have extracted a unique weight triplet from the consideration of the entire dataset of answers provided by the experts. However, experts can be profiled by profession, age, education, and familiarity with LF issues in different clusters, which may be associated to many corresponding weight triplets. Finer sensitivity analyses may be applied to understand the influence of experts profiling on the weights and, overall, on the composite indicator. In addition, the sample was designed for including academics and professionals, i.e., individuals with a presumably medium-high familiarity with technical aspects connected to LF management and planning. However, this composition could not correspond to the characterization of a typical deliberative body operating in the public domain, where the share of decision makers with no exposure to LF issues might be greater. Further research will be directed to seek the most suitable profile of candidate stakeholders.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.    Hassan [14] Sustainability of a region.

Appendix A
Total Index of Sustainability (TIS) is based on three indicators (economy, society, and environment), which include several criteria.
Multi-attribute utility theory is applied to construct the TIS as a weighted summation.
Data on GDP, solid waste, income disparity, and crime rate. An example for three countries is made.
The weights represent the change in the strength of preferences as an attribute varies from the worst to the best level. Composite index of vulnerability to flooding based on the combination of three dimensions (exposure, sensitivity, and adaptive capacity) and six constituent indicators, i.e., exposed population, exposed area, forecasted frequency of floods, landscape disturbance, protection from financial loss, and financial resources.  Sensitivity of indicators of regional sustainability.