**Antonio Nesticò \* and Piera Somma**

Department of Civil Engineering, University of Salerno, 84084 Fisciano (SA), Italy **\*** Correspondence: anestico@unisa.it; Tel.: +39-089-964318

Received: 26 July 2019; Accepted: 18 August 2019; Published: 21 August 2019

**Abstract:** 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 REalité (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.

**Keywords:** economic evaluation of projects; Multi Criteria Decision Making (MCDM); historical building; economic enhancement

### **1. 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 integration with the context, the respect of the building, and investment opportunities. In the process of recovery and enhancement of the real estate property, the identification of the best use maximizes the effects of the project, not only in purely financial terms, but also in social, cultural, and environmental terms [9–11].

On the one hand, the functional destination must satisfy the existing technical-urban planning constraints, which usually put significant limits on the possible alternatives; on the other hand, it must look at the financial sustainability of the investment project, and at maximizing the benefits for the community [12,13].

But what approaches are possible to establish the optimal destination with regard to the multiple needs of a community? The economic evaluation of the projects provides multi-criteria methodologies for analysis. A peculiarity of these methodologies is precisely the ability to consider the multi-dimensional character of the evaluation problem, allowing a comparative analysis of the alternatives according to multiple criteria, both quantitative and qualitative [14,15].

The recovery, protection, and valorization of historic buildings represent a demanding, yet unavoidable bet, where careful actions on the vast existing heritage can certainly contribute to the relaunch of the building sector and to the achievement of all the benefits of urban regeneration [16–18].

### **2. 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.

### **3. 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].

The literature offers a wide range of techniques, such as: the Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), ELimination Et Choix Traduisant la REalité (ELECTRE), Preference Ranking Organization METHods for Enrichment Evaluations (PROMETHEE), Tecnique for Order

Preference by Similarità to Ideal Solution (TOPSIS), Compromise Rankimg Method (VIKOR), and EVAluation of MIXed criteria (EVAMIX). Use of the AHP, ELECTRE, TOPSIS, and VIKOR methods are certainly widespread, and these are the subject of this research [24–35].

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, *aij*, are obtained by comparing the criteria and the alternatives with reference to each criterion.

The *A* matrices of pairwise comparisons are of the type:

$$A = \begin{bmatrix} a\_{11} & a\_{12} & \dots & a\_{1m} \\ a\_{21} & a\_{22} & \dots & a\_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ a\_{n1} & a\_{n2} & \dots & a\_{nm} \end{bmatrix} \tag{1}$$

These matrices are:


As suggested by Saaty, comparisons can be made according to the semantic scale with scores 1, 3, 5, 7, and 9 [33].

Commonly, in real cases, the relationship *wi*/*wj* is not known, so it is necessary to look for *aij*, such that *aij* ≈ *wi*/*wj*. If *W* is the matrix of the weights and *w* the column vector of the sought variables, then:

$$\mathcal{W} \cdot w = n \cdot w \tag{2}$$

In extended form:

$$\mathcal{W} \cdot w = \begin{bmatrix} \frac{w\_1}{w\_1} & \frac{w\_1}{w\_2} & \dots & \frac{w\_1}{w\_n} \\ \frac{w\_2}{w\_1} & \frac{w\_2}{w\_2} & \dots & \frac{w\_2}{w\_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{w\_n}{w\_1} & \frac{w\_n}{w\_2} & \dots & \frac{w\_n}{w\_n} \end{bmatrix} \cdot \begin{bmatrix} w\_1 \\ w\_2 \\ \vdots \\ w\_n \end{bmatrix} = \begin{bmatrix} n \, w\_1 \\ n \, w\_2 \\ \vdots \\ n \, w\_n \end{bmatrix} = n \begin{bmatrix} w\_1 \\ w\_2 \\ \vdots \\ w\_n \end{bmatrix} = n \cdot w \tag{3}$$

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, ..., *wn*, 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 *wi*/*wj* (*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 *wi*/*wj* 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*:

$$
\mathcal{N} \cdot \mathfrak{n} = \ \lambda\_{\text{max}} \cdot w \qquad \text{con} \qquad \lambda\_{\text{max}} \ge \mathfrak{n} \tag{4}
$$

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 *wi*/*wj* 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:

$$CI = \frac{\lambda\_{\text{max}} - n}{n - 1} \tag{5}$$

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*):

$$CR = \frac{CI}{RCI} \tag{6}$$

Binary comparisons are sufficiently coherent with one another if:


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 *fkp* are:

$$f\_{kp} = 1 \text{ if } c\_{kp} \ge \underline{\mathfrak{c}} \quad f\_{kp} = 0 \text{ if } c\_{kp} < \underline{\mathfrak{c}}.$$

and the dominance matrix of discordance *G,* whose *gkp* elements are:

$$\mathcal{g}\_{k\mathcal{p}} = 1 \text{ if } d\_{k\mathcal{p}} \ge \underline{d} \quad \mathcal{g}\_{k\mathcal{p}} = 0 \text{ if } d\_{k\mathcal{p}} < \underline{d}.$$

where *ckp* represents the set of all the criteria for which the alternative *Ak* is preferable to the alternative *Ap*, and *dkp* is the complement of *ckp*. The product of the homologous elements of the matrices *F* and *G* returns the aggregate dominance matrix, *E*:

$$E = \begin{bmatrix} \mathfrak{e}\_{11} & \mathfrak{e}\_{12} & \dots & \mathfrak{e}\_{1p} \\ \mathfrak{e}\_{21} & \mathfrak{e}\_{22} & \dots & \mathfrak{e}\_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ \mathfrak{e}\_{k1} & \mathfrak{e}\_{k2} & \dots & \mathfrak{e}\_{kp} \end{bmatrix} \tag{7}$$

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 *Si \** and *Si* −. It is possible to determine the relative distance *Ci\** of the generic alternative *Ai*, with respect to the ideal solution:

$$\mathbf{C}\_{i}^{\*} = \frac{\mathbf{S}\_{i}^{-}}{\mathbf{S}\_{i}^{-} + \mathbf{S}\_{i}^{\*}} \tag{8}$$

For the final classification, the best solution is the one that presents the minimum distance from *A\** and the maximum distance from *A*<sup>−</sup> at the same time—that is, the one with the highest value of *Ci \** .

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 *Qi* scalar corresponds to this alternative:

$$Q\_i = \nu \frac{S\_i - S^\*}{S^- - S^\*} + (1 - \nu) \frac{R\_i - R^\*}{R^- - R^\*} \tag{9}$$

In (9), the variable ν is between 0 and 1, and allows to give different weight to the single addends. Moreover: 

$$S\_i = \sum\_{j=1}^{m} \frac{w\_j (a\_j^\* - a\_{ij})}{a\_j^\* - a\_{ij}} \; ; \; R\_i = \max\_j \left[ \frac{w\_j (a\_j^\* - a\_{ij})}{a\_j^\* - a\_{ij}} \right]$$

$$S^\* = \min\_i S\_i \; ; \; S^- = \max\_i S\_i \; ; \; R^\* = \min\_i R\_i \; ; \; R^- = \max\_i R\_i$$

According to VIKOR, the optimal solution must comply with two criteria of acceptability:


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<sup>N</sup>* is the last solution, taken in the order obtained according to *Qi*, for which the inequality is still valid:

$$Q\left(A^N\right) \cdot Q\left(A'\right) < DQ.\,.$$

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.


