Comparative Analysis of Multi-Criteria Methods for the Enhancement of Historical Buildings

: The protection of cultural heritage is essential to preserve the memory of the territory and its communities, but its enhancement is also important. In this perspective, the theme of choosing the best use for historic buildings, which often make up a substantial and widespread part of real estate and which can become a driving force for the sustainable development of cities, is important. These decision-making processes find effective support tools in Multi-Criteria Decision Making (MCDM) methods, able to consider the multiple financial, social, cultural, and environmental effects that the enhancement project generates. In order to identify the most appropriate evaluation approach to select the best use of the building, this paper proposes a comparison between some of the best-known MCDM methods: Analitic Hierarchy Process (AHP), ELimination Et Choix Traduisant la REalit é (ELECTRE), Tecnique for Order Preference by Similarity to Ideal Solution (TOPSIS), and the Compromise Ranking Method (VIKOR). The comparative analysis gives rise to the validity of the AHP, which is useful for reducing the problem into its essential components, so as to make a rational comparison among the design alternatives based on different criteria. The novelty of the research is the characterization of the hierarchical structure of the model, as well as the selection of criteria and indicators of economic evaluation. The application of the model to a real case of recovery and enhancement of a former convent in the province of Salerno (Italy) verifies the effectiveness of the tool and its adaptability to the specificities of the case study.


Introduction
In recent decades, growing interest in historic buildings has grown in the European context. In fact, these possess a cultural, social, and economic value capable of generating virtuous processes for the development of the surrounding territory [1,2]. In preserving this heritage, to interrupt the frequent degradation processes and promote recovery activities, it is necessary to give legal force to the actions necessary for its protection, as the historical environment is vulnerable [3].
However, it is important to highlight that just protection of the historic building alone does not guarantee the economic development of the surrounding communities, and does not always have a positive impact on their quality of life. Indeed, citizens often do not have the opportunity to access or use the historic building. It is therefore necessary to guarantee public space and accessibility, and more generally, the inclusion of architectural heritage in management and development programs aimed at meeting the needs of residents [4,5].
The recovery and enhancement of the historic building can be important for the sustainable development of the city [6,7]. From this point of view, it is sufficient to think about the saving of soil that is generated where derelict buildings are recovered and reused [8]. The indication of the right function provides the decision-maker with indispensable data to evaluate the correctness of the intervention, the

Aim of the Paper
The research mainly pursued three goals. Starting from a comparative evaluation of different methods of multicriterial analysis, the objective was to identify an effective approach to establish the best use of a historic building, despite the multiple technical-urbanistic and financial constraints that characterize any option of investment.
The second objective was to select multiple social, cultural, and financial criteria that would be useful for logically structuring the economic analysis model. This is fundamental to reconcile the multiple evaluative aspects involved and to rationally organize the available information.
The third objective was the characterization of a model based on the Analytic Hierarchy Process (AHP) algorithms able to compare functional alternatives based on different selected criteria, so as to choose the optimal use for the historic building.
The resolution of a real multicriterial evaluation problem concerning the recovery and enhancement of a former convent in the province of Salerno (Italy) allows for the verification of both the usefulness of the proposed model and its easy adaptability to the specificities of the case study.

MCDM Methods Comparative Analysis
With reference to urban and territorial planning interventions and the building heritage, the multi-criteria analysis methods allow for comparisons of different design solutions on the basis of multiple criteria (financial, social, cultural, and environmental), which can be expressed through quantitative or qualitative indicators [19]. Thus, unlike the cost-benefit analysis that expresses the judgment of economic convenience only on the monetary criterion, the multicriteria analysis rationalizes the selection process through the optimization of a multi-criteria vector, weighted according to the decision-maker s priorities. In this way, it is possible to include both monetizable and extra-economic criteria, measurable only in physical or qualitative terms, in a single evaluation process [20,21].
Through logical-mathematical algorithms structured on parameters specific to the technical, economic, managerial, social, environmental, and psychological disciplines, the methods of multicriterial analysis allow for the ordering of the possible solutions to the problem, even when there is no alternative that clearly prevails over the others-in other words, one that is capable of simultaneously maximizing all the evaluation criteria [22]. These methods make it possible to identify the optimal solution that can better combine the set of objectives [23].
In general, the choice of the most appropriate multi-criteria method to solve the evaluation problem depends on the characteristics of the case study, in terms of the objectives of the analysis, reference territory, and specificity of the intervention, nature, and quality of the information available to the analyst [36,37].
The AHP method is more effective in the presence of the criteria and sub-criteria of evaluation, since through a hierarchical structure, at several levels, it allows the complex problem to be broken down into simpler sub-problems that can be analyzed in greater detail. The other three methods only consider criteria and do not include the presence of sub-criteria.
The hierarchical structure of the AHP has: the main goal of the decision-maker at the top; the sub-goals at the underlying level; and the criteria that lead to the achievement of sub-goals at an even lower level [30,33]. The criteria can be expressed by sub-criteria up to the most appropriate breakdown level for full understanding of the problem. The last level is that of alternatives [29,32,33,38].
The hierarchical analysis requires the estimation of the weights w to be associated with each criterion and the various alternatives. This estimate is carried out using evaluation matrices whose individual elements, a ij , are obtained by comparing the criteria and the alternatives with reference to each criterion.
The A matrices of pairwise comparisons are of the type: These matrices are: -Positive-that is, all the principal minors are positive, where by 'principal minor', we mean the determinant of the square sub-matrix formed by the first n rows and m columns (with 1 ≤ m ≤ n); -Reciprocal, being a ij = 1/a ji and therefore, the elements on the main diagonal are all unitary (a ii = 1). This relationship of reciprocity arises from the need to guarantee the symmetry of judgments of importance; -Constituted by finite elements, since for each criterion, C, considered, we have a ij ∞.
Commonly, in real cases, the relationship w i /w j is not known, so it is necessary to look for a ij , such that a ij ≈ w i /w j . If W is the matrix of the weights and w the column vector of the sought variables, then: In extended form: Therefore, the matrix W has, as its sole eigenvalue, its order n, and as a corresponding eigenvector, the vector of the sought variables. This means that it is possible to obtain the values of the variables w 1 , w 2 , ..., w n , starting from the relationships between them, taken two at a time. This allows us to affirm that, if we do not have the exact value of the relations w i /w j (i, j = 1, 2, ..., n), but their estimate, the method of the eigenvalue can still be usefully used for the approximate evaluation of the variables. The values thus obtained are closer to the exact ones if the estimates of the w i /w j ratios are consistent between them [39]. In this case, all the n eigenvalues are almost zero, and if the ratios are coherently estimated, the maximum eigenvalue λ max is not far from the value n, which can therefore be assumed as an approximate estimate of the vector w: Alternatively, the vector w can be determined by normalizing the matrix W through the ratio of each of its elements to the sum of the elements placed in the same column. The arithmetic mean of each of its lines is then calculated [40].
The consistency measure of the values assigned to the w i /w j ratios derives from the difference between λ max and n. This difference is null for perfectly consistent estimates. The relationship consistency index CI is defined as: The CI index must be compared to the RCI random consistency index, whose value is a function of the number n of variables according to Table 1. The relationship between CI and RCI is defined as the Consistency Ratio (CR): Binary comparisons are sufficiently coherent with one another if: -CR < 5% for n = 3; -CR < 9% for n = 4; -CR < 10% for n > 4.
The final step is to calculate the overall weights (or priorities) of the actions. For this, it applies the principle of hierarchical composition, by virtue of which the local weights of each element are multiplied by those of the corresponding superordinate elements. Finally, the products obtained are added together [30].
Compared to AHP, the ELECTRE method is not preferred when there is a large number of alternatives and a large number of criteria. Depending on the number of alternatives, the values of the concordance threshold c and the discordance threshold d are calculated. On the basis of these values, we constructed the dominance matrix of concordance F, whose elements f kp are: and the dominance matrix of discordance G, whose g kp elements are: where c kp represents the set of all the criteria for which the alternative A k is preferable to the alternative A p , and d kp is the complement of c kp . The product of the homologous elements of the matrices F and G returns the aggregate dominance matrix, E: which allows the construction of a partial ordering of the alternatives, since the columns in which the value 1 appears indicate the alternatives to be discarded because they are dominated by the others. Therefore, ELECTRE, unlike the other models in question, does not provide a final ranking of scores, but only an order of preference. Precisely, the peculiarities of the calculation algorithms can make ELECTRE easier to use if the number of evaluation criteria is particularly limited.
Despite being very versatile, as it can be applied even in the presence of a large number of criteria and alternatives, the TOPSIS method tends to reject those alternatives that have low values in most of the attributes. Specifically, for each criterion, TOPSIS searches for the ideal solution, A * and the ideal negative solution, A − (respectively better and worse performance offered by the alternatives considered), with respect to which it calculates the distances S i * and S i − . It is possible to determine the relative distance C i * of the generic alternative A i , with respect to the ideal solution: For the final classification, the best solution is the one that presents the minimum distance from A * and the maximum distance from A − at the same time-that is, the one with the highest value of C i * .
The VIKOR method substantially has the same characteristics of TOPSIS, but unlike the latter, does not provide for the presence of an ideal negative solution. In fact, VIKOR considers the optimal alternative to be the one closest to the ideal solution. The lowest value of the Q i scalar corresponds to this alternative: In (9), the variable ν is between 0 and 1, and allows to give different weight to the single addends. Moreover: According to VIKOR, the optimal solution must comply with two criteria of acceptability: , where n is the number of alternatives; -Acceptable stability of the decision, that is, the alternative A must also be the best, or in the ranking must present the minimum value in terms of S i and/or R i .
If one of the two conditions is not verified, it is impossible to directly determine the best solution, which is obviously a limitation of the method. In this case, a set of compromised solutions is established in the set of alternatives under consideration. This set is given by: A N is the last solution, taken in the order obtained according to Q i , for which the inequality is still valid: The alternatives of the set thus determined are characterized by a sensitive reciprocal "closeness". The essential examination of the four methods, AHP, ELECTRE, TOPSIS, and VIKOR aimed at a useful comparison to highlight the peculiarities of each of them, is summarized in Table 2.

Selection of the Best Use of a Historic Building. Characterization of a Hierarchical Analysis Model
The Analytic Hierarchy Process is defined as "analytical" because it breaks down the complex decision problem into fundamental constituent elements [30,31,41,42]; "hierarchical" because the breakdown of the problem itself occurs at successive levels characterized by ever-increasing detail [27,28,43,44]. At the top of the hierarchy, there is the goal to be reached, which, in the case under consideration, is given by the best use for the historic building.
The choice of indicators is fundamental for the correct application of the model. Here, they have been selected from literature datasets [45][46][47][48].
At the first level, we consider the social (C1), cultural (C2), and financial (C3) criteria, which are described in turn in the sub-criteria. In particular, the social criterion includes the sub-criteria: - Community involvement (C 1 ), intended as the average number of daily users. It is measured as the average daily number of people who attend the structure; -New workers (C 2 ), measured as the number of new workers in the structure. - The cultural criterion includes the sub-criteria: - Cultural effects (C 3 ), understood as the attraction of the structure with respect to cultural events; -Compatibility of the function with the historical-architectural characteristics of the property (C 4 ). This criterion depends on the three sub-criteria: -Representativeness of the use function (C 41 ), evaluated as the aptitude to express the cultural peculiarities of the reference territory, in respect of the material and spiritual reality of the architectural artefact; - Residential impact (C 42 ), understood as the average number of daily users, in this case with a negative meaning, in the sense that higher this number is, the lower the score attributed to the alternative is. In fact, with the aim of protecting the cultural asset, it is advisable to prefer functional activities with a moderate residential impact, in order to avoid excessive loads on the structure that may require interventions of static adaptation such as to compromise the authenticity of the historical matter; - Respect of the criterion of minimum intervention (C 43 ), that is, safeguarding the characteristics of the building. It is necessary to avoid intrusive interventions, such as new openings, tracks for installations, partitions, kitchens, and toilets, so as not to reduce the artistic-monumental quality of the property. This criterion represents one of the cardinal principles of architectural restoration and conservation of historic buildings. It is therefore a qualitative, rather than a quantitative criterion [49,50].
The financial criterion includes the sub-criterion: -Return On Investment (C 5 ). This is measured by ROI (Return On Investment), which expresses the rate of return on the total investments of a company for the assumed activity.
According to the existing technical-urban planning constraints and in relation to the socio-demographic and economic characteristics of the territory in which the recovery and enhancement project of the historic building is inserted, the analysts define the possible investment alternatives A 1 , A 2 , ..., A n . Among these, it is necessary to make the choice that guarantees the optimal use, according to the multiple criteria C1, C2, C3 and sub-criteria C 1 , C 2 , C 3 , C 4 , C 5 , as well as C 41 , C 42 , and C 43 defined above. The logical-operative scheme outlined finds graphic representation in Figure 1. enhancement project of the historic building is inserted, the analysts define the possible investment alternatives A1, A2, ..., An . Among these, it is necessary to make the choice that guarantees the optimal use, according to the multiple criteria C1, C2, C3 and sub-criteria C1, C2, C3, C4, C5, as well as C41, C42, and C43 defined above. The logical-operative scheme outlined finds graphic representation in Figure 1.

Case Study and Results
The aim of the paper was to define the best use for the recovery and enhancement of a former convent in the province of Salerno (Italy). For this purpose, in accordance with the provisions of paragraph 3, the Analogue Hierarchy Process was used. This is a multi-criteria approach able to take into account both the use value of cultural heritage as a source of economic growth in the short term, and its intrinsic value as an identity element of a community, to be preserved in the long term [51].
The monastic complex under study is of particular historical-architectural interest, but is currently in a very poor state of preservation. Thus, the public administration acquired the property with the aim of recovering it [52][53][54][55]. Figure 2 shows the plants, sections, and main elevations of the building in the actual state.

Case Study and Results
The aim of the paper was to define the best use for the recovery and enhancement of a former convent in the province of Salerno (Italy). For this purpose, in accordance with the provisions of paragraph 3, the Analogue Hierarchy Process was used. This is a multi-criteria approach able to take into account both the use value of cultural heritage as a source of economic growth in the short term, and its intrinsic value as an identity element of a community, to be preserved in the long term [51].
The monastic complex under study is of particular historical-architectural interest, but is currently in a very poor state of preservation. Thus, the public administration acquired the property with the aim of recovering it [52][53][54][55]. Figure 2 shows the plants, sections, and main elevations of the building in the actual state.   The Figures A1-A12 in the Appendix A report the functional distribution of the internal spaces at the three levels of the building for each of the four solutions of use.
The AHP was developed according to the first-level evaluation criteria: social (C 1 ), cultural (C 2 ), and financial (C 3 ). For the case in question, the sub-criteria are estimated as follows: - Community involvement (C 1 ). As there is no data on the average number of users, a filling index was envisaged as a rate of the maximum capacity of the structure; -New workers (C 2 ). This data was obtained through surveys carried out in similar structures in the territory; - Cultural effects (C 3 ). This is expressed as the ratio between the surface used for cultural activities (S C ) and the total area of the building (S T ); -Compatibility of the function with the historical-architectural characteristics of the property (C 4 ). This criterion depends on the three sub-criteria: -Representativeness of the use function (C 41 ), according to a judgment scale from 1-7; - Residential impact (C 42 ), assessed as for criterion C 1 ; -Respect of the criterion of minimum intervention (C 43 ). The judgment was assigned by the decision-maker according to a qualitative scale from 1-7, depending on the level of protection that is guaranteed to the original structure; -Return On Investment, ROI (C 5 ). For each of the four alternatives, the ROI value was derived from information on the profitability of the reference economic sector in the area of investigation.
Once the alternatives, criteria, and sub-criteria for the evaluation was defined, the hierarchical decision model was outlined, as shown in Figure 3.  Local cuisine restaurant (A3), which proposes traditional dishes, cooking workshops, as well as workshops for the valorization of the typical craftsmanship of the places, with an adjoining museum of rural civilization;  Public and private offices (A4). The Figures A1-A12 in the Appendix report the functional distribution of the internal spaces at the three levels of the building for each of the four solutions of use.
The AHP was developed according to the first-level evaluation criteria: social (C1), cultural (C2), and financial (C3). For the case in question, the sub-criteria are estimated as follows:


Community involvement (C1). As there is no data on the average number of users, a filling index was envisaged as a rate of the maximum capacity of the structure;  New workers (C2). This data was obtained through surveys carried out in similar structures in the territory;  Cultural effects (C3). This is expressed as the ratio between the surface used for cultural activities (SC) and the total area of the building (ST);  Compatibility of the function with the historical-architectural characteristics of the property (C4). This criterion depends on the three sub-criteria:  Representativeness of the use function (C41), according to a judgment scale from 1-7;  Residential impact (C42), assessed as for criterion C1;  Respect of the criterion of minimum intervention (C43). The judgment was assigned by the decision-maker according to a qualitative scale from 1-7, depending on the level of protection that is guaranteed to the original structure;  Return On Investment, ROI (C5). For each of the four alternatives, the ROI value was derived from information on the profitability of the reference economic sector in the area of investigation.
Once the alternatives, criteria, and sub-criteria for the evaluation was defined, the hierarchical decision model was outlined, as shown in Figure 3.   Table 3 shows the scalarized decision matrix, useful for the implementation of the AHP. In this matrix the generic element a ij (attribute) expresses the performance of the generic alternative Ai (i = 1, 2, ..., n) with respect to the generic criterion C j (j = 1, 2, ..., m) [56,57]. The dominance coefficients were assigned according to Saaty's semantic scale in Table 4. They are representative of the relative importance between the compared elements [33]. In paired comparisons of the alternatives with respect to the criteria, graduated scales were used, each divided into equal bands corresponding to the intensity of the Saaty scale, where the maximum value of the indicator in question corresponds to the upper end. On the scale thus defined, the numerical values of the indicators have been positioned for each alternative (Figure 4). If there is only one band between the indicators, then the strongest value was given coefficient 3, and to the other, the reciprocal; if there are two bands, the strongest value was assigned coefficient 5, and to the other, the reciprocal, and so on. In intermediate situations, there were intermediate intensities.  Table 3 shows the scalarized decision matrix, useful for the implementation of the AHP. In this matrix the generic element aij (attribute) expresses the performance of the generic alternative Ai (i = 1, 2, ..., n) with respect to the generic criterion Cj (j = 1, 2, ..., m) [56,57].
The dominance coefficients were assigned according to Saaty's semantic scale in Table 4. They are representative of the relative importance between the compared elements [33].
In paired comparisons of the alternatives with respect to the criteria, graduated scales were used, each divided into equal bands corresponding to the intensity of the Saaty scale, where the maximum value of the indicator in question corresponds to the upper end. On the scale thus defined, the numerical values of the indicators have been positioned for each alternative (Figure 4). If there is only one band between the indicators, then the strongest value was given coefficient 3, and to the other, the reciprocal; if there are two bands, the strongest value was assigned coefficient 5, and to the other, the reciprocal, and so on. In intermediate situations, there were intermediate intensities.     Table 5 contains the matrices of pairwise comparisons and the matrices of normalized pair comparisons. From these matrices, processed by implementing Microsoft Excel software, it was possible to obtain the weights of each alternative with respect to each sub-criterion. In all cases, the matrices are consistent. In fact, being of rank 4, they have a value of consistency ratio lower than 9. Table 5. Comparison matrices in pairs between the alternatives with respect to each criterion.    In the preparation of the matrices of pairwise comparisons, both between criteria and between sub-criteria, we assigned equal weight to each criterion, and with reference to the single criterion, we assigned the same weight to each sub-criterion. All matrices were consistent.

C 43 -Criterion of minimum interventation C 43 -Criterion of minimum interventation
The total priorities are shown in Table 6. The AHP therefore identifies the local cuisine restaurant as the best destination for the recovery and enhancement of the former convent.

Discussion
The selection of possible alternatives of use for the historical building under study was derived from a careful analysis of the needs of citizenship, and from the economic and socio-demographic characteristics of the reference territory. In particular, the choice of the B&B was born from the need to improve the reception services connected to the use of the existing historical, architectural, and archaeological heritage. The multi-purpose hall option was aimed at the cultural growth of the city, which was also poorly equipped with facilities for conferences, seminars, and professional training. The local cuisine restaurant destination with its museum of rural civilization intends to recover the building as the driving force of an economy capable of enhancing the typical local products, as in the past, the convent-thanks to the presence of Benedictine monks-was a point of reference for the agricultural development of a marginal area. The office function can allow the urban center to play a central economic role. Table 5 of the priorities shows that, due to the multiple evaluation criteria considered, the AHP method considers A 3 as the best alternative. Thus, the optimal use is that of the local cuisine restaurant, with a score of 0.474. In the order of the alternatives: A 2 (multi-purpose rooms) with a score of 0.220, that is less than half of the valid score for the first solution in the ranking; A 4 (offices) with a score of 0.175; and A 1 (bed & breakfast) with a score of 0.130.

Conclusions
Historical environments are an exhaustible resource, so it requires great attention in terms of protection, but also of enhancement. This can happen through the functional recovery of the building with the identification of its best use, intended as a function able to simultaneously maximize the social, cultural, and financial effects that the intervention on the historic building generates on the reference territory. This approach respects the principles of sustainable urban development.
The aim of the research is to outline a multi-criteria evaluation model able to select the optimal function for a historic building in a state of neglect. From this point of view, a comparative analysis was carried out on some of the most well-known Multi-Criteria Decision Making methods: the Analitic Hierarchy Process (AHP), ELimination Et Choix Traduisant la REalité (ELECTRE), Tecnique for Order Preference by Similarity to Ideal Solution (TOPSIS), and the Compromise Ranking Method (VIKOR). The theoretical examination highlights that, due to its hierarchical structure, the AHP model allows the problem to be broken down to the level of detail necessary for the analysis, thus being more effective in solving the problem in question, where, very often, there are multiple criteria and sub-criteria of evaluation.
For the correct implementation of the hierarchical analysis algorithms, the rigorous selection of the evaluation criteria and sub-criteria, as well as the corresponding indicators, is fundamental. This selection represents an element of novelty of the research. The study led us to recognize: the social criteria, which can be expressed through the community involvement and the new workers; cultural criteria, translated in terms of cultural effects and compatibility of the function with the historical-architectural characteristics of the property; and financial criteria, to be expressed quantitatively by the Return On Investment. It should be noted that a careful study on the compatibility of the function with the historical-architectural characteristics of the property leads to a breaking down of this criterion into the three interventions.
The operational coherence of the evaluation model is verified through a case study, concerning the selection of the optimal destination for a former convent in the province of Salerno (Italy).
The multi-criteria analysis model proved to be an effective decision support tool, guaranteeing to both public administrations and private individuals an optimal allocation of available resources, with obvious and important implications of economic policy. criteria, translated in terms of cultural effects and compatibility of the function with the historical-architectural characteristics of the property; and financial criteria, to be expressed quantitatively by the Return On Investment. It should be noted that a careful study on the compatibility of the function with the historical-architectural characteristics of the property leads to a breaking down of this criterion into the three interventions.
The operational coherence of the evaluation model is verified through a case study, concerning the selection of the optimal destination for a former convent in the province of Salerno (Italy).
The multi-criteria analysis model proved to be an effective decision support tool, guaranteeing to both public administrations and private individuals an optimal allocation of available resources, with obvious and important implications of economic policy.