A multiple criteria methodology for prioritizing and selecting portfolios of urban projects

A multiple criteria methodology for prioritizing and selecting portfolios of urban projects

Abstract

This paper presents an integrated methodology supporting decisions in urban planning. In particular, it deals with the prioritization and the selection of a portfolio of projects related to buildings of some values for the cultural heritage in cities. More precisely, our methodology has been validated to the historical center of Naples, Italy. Each project is assessed on the basis of a set of both quantitative and qualitative criteria with the purpose to determine their level of priority for further selection. This step was performed through the application of the Electre Tri-nC method which is a multiple criteria outranking based method for ordinal classification (or sorting) problems and allows to assign a priority level to each project as an analytical “recommendation” tool. To identify the efficient portfolios and to support the selection of the most adequate set of projects to activate, a set of resources (namely budgetary constraints) as well as some logical constraints related to urban policy requirements have to be taken into consideration together with the priority of projects in a portfolio analysis model. The process has been conducted by means of the interaction between analysts, municipality representative and experts. The proposed methodology is generic enough to be applied to other territorial or urban planning problems. We strongly believe that, given the increasing interest of historical cities to restore their cultural heritage, the integrated multiple criteria decision aiding analytical tool proposed in this paper has significant potential to be used in the future.

keywords:
Multiple criteria analysis, Decision support systems, Electre Tri-nC, Urban planning, Portfolio selection.
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Additional_Material

1 Introduction

Today, cities have to face big challenges due to the contemporary environmental, socio-economic, and institutional crises (García-Hernández et al., 2017), that add to an already problematic context (Rees and Wackernagel, 1996). Indeed, in the last century, urban sprawl has grown impetuously, leading to some difficulties for urban sustainability. Often, the natural resources have been so much eroded that it is not possible to divide the city from rural or natural areas. The urban space has been expanded without providing adequate services (e.g. public transportation or waste collection) and infrastructures (e.g. social housing or leisure spaces) (McGreevy, 2017). Consequently, the objectives and the constraints of urban policies, such as the approaches to define them, have to be revised.

With respect to the objectives, the general paradigm for urban policies is changing, taking into account the need to control further geographic expansion of the cities and use a more systemic and holistic approach in planning (Hansen et al., 2015). Attention has shifted, consequently, to urban sustainability, focusing on:

  • Management and optimization of resources (Agudelo-Vera et al., 2012).

  • Waste and pollution reduction, and the improvement of liveability (Newman, 1999).

  • Enhancement of the natural and cultural heritage (Gražulevičiūtė, 2006).

With respect to the constraints, a definition of urban policy also has to consider some significant restrictions, that are becoming tighter and more selective. Among these, the current reduction of public expenditures for implementing the planned policies has a specific relevance, which calls for the definition of a proper prioritization of the actions to be funded.

With respect to the approaches, moreover, the increasingly complex urban context requires that the decision aiding methodologies for defining the urban policy must be:

  • More accurate, specific, and selective with respect to the characterization of the potential actions (Büyüközkan et al., 2018).

  • More flexible, adaptive, and robust with respect to their implementation (Ahern, 2011).

  • More pluralist and participatory, to take into account the plurality of stakeholders, experts, and policy makers (Thabrew et al., 2009).

  • More user friendly, taking into consideration the behavioural aspects of decision making, with the aim of defining an effective procedure of interaction with the involved actors (Abastante et al., 2018).

In this paper we propose a general methodology for defining urban policies that takes into account the above remarks and, therefore, is characterized by:

  1. A formulation in terms of a multiple criteria problem allowing the consideration of a family of heterogeneous points of views.

  2. Consideration of integrated families of actions, technically called portfolios (Salo et al., 2011), as feasible solutions of the problem at hand.

  3. An interactive procedure for the optimization problem as well as for the sorting problem, in order to involve stakeholders, experts and policy makers in the whole decision process.

  4. Inclusion in the optimization problem of a sorting procedure to assign a priority level to each potential action.

Despite its general applicability, this methodology has been applied to a real world resource optimization problem for the regeneration of a large and complex historical centre in the city of Naples (Italy). Therefore, we will present the real and specific arguments that helped us to identify the four features of the methodology listed above. In Naples, a large part of the historical centre has been inscribed as a World Heritage Site, a program with the objective of safeguarding the identity of communities worldwide. At the same time, other objectives, related to the preservation of the cultural heritage, have to be pursued. These objectives can be both of:

  • Economic nature, such as the promotion of tourism (e.g., McKercher et al., 2005), and local entrepreneurship and business (e.g., Tuan and Navrud, 2008); and,

  • Non-economic nature (Blake, 2000), such as social inclusion (Vasile et al., 2015), community engagement (Waterton, 2015), and improvement of the environment and the urban landscape (Veldpaus et al., 2013).

This suggests adopting a multiple criteria optimization procedure to choose from among the potential actions, which is point in the list of characteristics of our methodology.

Moreover, for the sake of having an integrated vision, the potential actions should not be considered in isolation, but as parts of a comprehensive program to be evaluated in its totality. The output of the decision will be the definition of a portfolio of actions to implement. This suggests approaching the problem at hand in terms of portfolio decision analysis, which is point in the list of characteristics of our methodology.

UNESCO requires a shared understanding of cultural heritage, which involves not only experts and policy makers, but also stakeholders and, in general, citizens, should be involved in the process of its enhancement and protection. This suggests adopting an interactive procedure for the decision problem, which is point in the list of characteristics of our methodology.

For all sites inscribed in the World Heritage List, UNESCO requires a management plan with the identification and allocation of the necessary resources. However, optimizing and collecting the resources necessary for the management plan of sites that include a complex and big area, such as a historic centre, is often quite problematic. In the case of Naples, the problem is exacerbated because the local authorities (often the principal owners) do not have enough resources for preserving a huge cultural heritage. This requires an accurate, specific, and selective prioritization of actions in order to distinguish those that are deferrable from those ones requiring a prompt intervention. This suggests defining these priorities with a well established sorting procedure that has to be integrated in the selection of the optimal portfolio of action, which is point in the list of characteristics for our methodology. In fact, this represents the most innovative contribution in the methodology we are proposing.

In this sense, there emerges the possibility of using Multiple Criteria Decision Aiding (MCDA) methods (see e.g., Greco et al., 2016; Ishizaka and Nemery, 2013) that can guide the Decision Makers (DMs) throughout the process. In the literature, several methods have been introduced to deal with real cases of the preservation of cultural heritage. The main scope is the identification and the selection of the possible actions to be made. Usually, several criteria are considered, as in Wang and Zeng (2010), where the cultural aspect was taken into account together with economic, architectural, environmental, social and continuity aspects.

The Analytic Hierarchic Process (AHP) was adopted by Wang and Zeng (2010), as well as others, for the selection of the reuse of historic buildings in Tapei City. In Hong and Chen (2017) as well, AHP helps to rank the historic buildings to be reused with a different scope in the Grand Canal Area in China. Also, Kutut et al. (2014) used AHP combined with the additive ratio assessment method for the definition of preservation actions for the historic buildings in Vilnius or in Lolli et al. (2017), where it is combined with the -means algorithm for selecting the energy requalification interventions.

Often, the weighted-sum of the criteria is considered, as in Dutta and Husain (2009), for the selection of historic buildings to preserve in the historical city of Calcutta or in Giuliani et al. (2018), to classify the possible reuse of historic grain silos in Italy. In the same perspective, Ferretti et al. (2014) used multi-attribute value theory for ranking the different reuses of mills located in the metropolitan area of Turin.

Furthermore, Giove et al. (2011) tested some MCDA methods, such as the use of the Choquet integral, to evaluate the sustainability of projects for the reuse of the Old Arsenal in Venice.

Some authors have also proposed to integrate MCDA methods within a GIS environment, as in Tarragüel et al. (2012) or Fusco Girard and De Toro (2007), or with an integrated spatial assessment, as in Cerreta and Toro (2010). Hamadouche et al. (2014) integrated them to select the sites that need urgent conservation in an archaeological site in Algeria. Similarly, Oppio et al. (2015) integrated spatial MCDA analysis with GIS and SWOT analysis for studying the reuse of castles in the Valle D’Aosta region. Another aspect to consider is the inclusion in the decision process of several stakeholders, as in Yung and Chan (2013) for the preservation of historic buildings in Hong Kong or in Cerreta and Panaro (2017) for the design of resilient landscapes.

While all these works refer to single case studies with only some insights provided on how to extend the methods to different applications, Ferretti and Comino (2015) have proposed an integrated framework involving MCDA methods for the evaluation of heritage systems. They stress the importance of interacting with experts and the DMs, in a transparent process, and how MCDA must support the public authorities in the definition of their strategic planning.

In this sense, Nesticò et al. (2018) models the investment selection of historic buildings as a knapsack problem. The adoption of such an approach allows defining a plan that does not consider only one project but the whole portfolio of projects. This strand of methods, called portfolio decision analysis (Salo et al., 2011), helps DMs to make more informed decisions. It has been implemented in several contexts, from the location of wind farms (Cranmer et al., 2018) to the selection of research and development projects (e.g., Çağlar and Gürel, 2017), and it can be integrated with many other multiple criteria methods, as in Barbati et al. (2018).

We aim to integrate a portfolio decision problem with an MCDA sorting method and in particular with the Electre Tri-nC (Almeida-Dias et al., 2012), embedded in a transparent and interactive process, to allow the assessment and the prioritization of the investment in cultural heritage. We aim to show how the methodology can help DMs to make rational decisions. We apply it to a case study related to one of the culturally outstanding cities in Europe, Naples (Italy), and in particular to its historical centre, where a huge number of interventions have been planned but without assigning priorities to these interventions. We also show how our approach works through the interaction with different actors.

This paper is organized as follows. In Section 2, we illustrate the context of the historical centre of Naples. In Section 3, we illustrate our methodology. In Section 4, we list the stakeholders, the actors involved, the criteria, and the projects of the case study. In Section 5, we explain how the interaction with the different actors was conducted. In Section 6, we formulate the portfolio selection problem. Section 7 presents the results of our experiments. Lastly, in Section 8, we discuss some insights coming from this practical experience and we conclude the paper. An Appendix provides the basics of Electre Tri-nC.

2 Context

The historic city centre of Naples was inscribed as a World Heritage Site in 1995. Naples is a major city in the south of Italy and is among the most ancient cities in Europe. Its structure and its culture have been created throughout history with influences from several civilizations, from its Greek foundation in 470 B.C., through the Roman period, to the Aragonese period, until the 19th century. A multitude of governmental and ecclesiastic buildings are still well preserved. The ancient city centre is well recognizable with its Greek rectangular grid layout and its Aragonese walls. Moreover, the underground is made of several cavities used over the course of the years for different aims. Furthermore, we find peculiar attractions and many businesses activities.

In addition to this part of the cultural heritage, there is a long tradition of an intangible heritage. In fact, Naples has been for long periods a leading cultural city, attracting artists and scholars in philosophy, art, literature, music, and theatre. Its influence is still well-known worldwide and events such as exhibits and shows are continually being planned. Recently, the image of the city has been relaunched in Italy and worldwide, attracting an increasing number of tourists.

Being a World UNESCO site, the government of the city of Naples approved a management plan for its historic centre which defines the vision, the general objectives and the approach to preserve the universal values associated with its cultural heritage (see Piano-Napoli, 2011). This plan highlights the stakeholders involved as well as the strategic challenges, and lists a series of possible improvements of the area concerning both the tangible and intangible cultural heritage. Although the plan indicates a set of projects to be implemented, and their associated benefits and disadvantages, it fails to indicate which projects should be prioritized and which are the most beneficial for addressing the strategic challenges previously mentioned. Anyway, from the plan there does not emerge a picture of the whole that would help to decide which investments are the most fundamental ones. Unfortunately, in a period in which there is an increasing need for the rationalization of expenses and resources, it is not realistic to think that all the projects will be funded or will be funded in different periods. Furthermore, during the long period of time that had passed after the development of the plan and obtaining the funds, the number of projects grew, thus the plan had to be revised. For example, in the last few years, some of the already planned projects have been developed in more detail, and others have been added. Based on the management plan, funds are being sought by the local authorities. A systematic methodology could help the DMs to systematize their choices, when the funds become available.

In particular, this seems to be a case in which we need to select a portfolio of projects in a complex process in which more than a single criterion needs to be considered for their prioritization and subsequently their selection. A systematic procedure that could support the DMs in this process should be employed to make informed decisions, integrating both economic and non-economic aspects. This is even truer in the case of cultural heritage, where the economic aspect of the problem needs to be integrated with criteria related to very different aspects, often intangible and difficult to quantify. A multiple criteria methodology helps the DMs throughout the whole process, supporting their decision even in the presence of qualitative evaluations.

3 An integrated methodology

Our methodology is based on the integration of a portfolio decision problem with a sorting procedure. In particular, the methodology includes the following steps:

  1. Identification of the urban planning problem. From the analysis of the urban context the following elements should be identified:

    • The stakeholders involved in the decision process and their specific aims.

    • The actions to implement in the urban context, i.e. the projects that could be implemented.

    • The criteria, i.e. the models used to assess the performance of the projects.

    • The performances of each single project in terms of each criterion.

    • The constraints related to the execution of the projects (e.g. budget constraints).

  2. Prioritization. Using a sorting method, the projects will be prioritized according to the criteria defined. In particular, we suggest using a multiple criteria sorting method in a constructive perspective (Almeida-Dias et al., 2010), in which the stakeholder representatives are involved in the decision model building process. In this way, we assign to each project a priority, which is then used in the subsequent portfolio optimization problem.

  3. Selection of a portfolio of projects. We build an optimization problem in which we maximize the number of projects with the highest priorities, taking into account some constraints related to the specific characteristics of the decision problem at hand (e.g. budget constraints). In this way, we define “the best portfolio”, i.e. a set of projects that should be executed taking into account the different points of view of the stakeholders and the criteria used to assess the projects themselves. This best portfolio should be presented and recommended to the stakeholders as their best option in the current context.

  4. Robustness Analysis. In order to verify that the portfolio obtained by our methodology is sufficiently robust with respect to variations of the parameters of the model, an appropriate robustness analysis should be carried out (Roy, 1998, 2010a). Indeed, we aim to verify that even if we change some of the parameters used in the model, e.g. the weights representing the importance of the criteria or the formulation of some of the constraints, the stakeholders will still be comfortable with the proposed solution.

We would like to stress that in each of these steps, several actors can be involved at different levels, making our methodology strongly participatory. Therefore, the portfolio defined should be accepted by all the stakeholders. Moreover, the use of an optimization model allows us to be very flexible and adaptable, with the ability to include in our model different aspects as constraints.

In the following we present, for our case study, a description of each step including how the interaction with the actors involved (experts, analysts, a representative of the municipality, and a specific expert who played the role of the DM) has been considered.

  1. Identification of the urban planning problem. The first preliminary activity started by the identification of the stakeholders. We clarify what they expect to obtain from the implementation of the regeneration works for the tangible cultural heritage projects. This step was implemented by means of a careful analysis of the management plan published by the municipality. Then, we carried out an initial determination of the criteria for our analysis from the management plan and, later, we particularized these criteria by means of interaction with the expert. The projects and their evaluations were developed in detail in relation to the available information.

  2. Prioritization. By means of a socio-technical approach based on the interaction with different actors and the implementation of the Electre Tri-nC method (Almeida-Dias et al., 2012), we defined for each project its own category, prioritizing the most important ones according to the set of determined criteria. This step, therefore, is composed of several stages:

    • Definition of the weights for the criteria through the interaction with the experts.

    • Construction of the reference actions and the categories to which to assign the projects by means of the interaction with the experts and the analysts.

    • Modelling the imperfect knowledge of the data and arbitrariness through the construction of the discriminating (indifference and preference) thresholds by means of the interaction with the experts and the analysts.

    • Definition of the veto thresholds through the interaction with experts and analysts.

    • Application of the Electre Tri-nC ordinal classification method.

  3. Modelling the selection of projects. We have defined a binary linear programming model that includes a specific objective function that optimizes the number of projects with highest priority and a set of specific constraints derived from the interaction with the municipality representative and the DM. The application of this model determines the portfolio of projects to implement. The results were discussed again with the DM.

  4. Performing a robustness analysis. We tried several scenarios, changing the weights of the criteria and the veto thresholds, which we discussed with the DM.

In the following we describe the case study.

4 A case study involving the city of Naples

In this section, we present the concrete elements of our case study: the stakeholders, the criteria, and the set of projects related to the urban tangible cultural heritage to be evaluated.

4.1 Stakeholders listed in the management plan of the historic city centre: Their aims and expectations

Several stakeholders are involved in the decision aiding process and they have different interests generated by the re-qualification of the historic city centre of Naples. The various stakeholders can be placed into three main categories: institutional bodies, social and cultural organizations, and local businesses. A brief description of each of them is presented below.

  1. Institutional bodies (which include the Municipality of Naples, the Province of Naples, the Campania Region, the Ministry of Cultural Heritage, the Port authority, and the local healthcare authority): Their aim is the conservation and enhancement of the cultural heritage (tangible and intangible), the improvement of the local economy, and the wellbeing of the citizens. The governmental bodies often own the buildings whose use they want to improve. Therefore, they are promoters of the interventions and have an active interest in managing the buildings. Moreover, they expect that the whole local and international community will benefit from the general improvement of the area.

  2. Social and cultural organizations (which include, for example, universities, ecclesiastical communities, and nonprofit organizations): Their aim is to reduce the number of degraded/abandoned buildings as well as to promote the local culture and traditional knowledge. They want to increase the cultural offerings and the spaces for activities such as exhibitions, shows and local community events. They also expect that the reuse of the buildings (of which they are sometimes the owners) will increase the number of services for the citizens, students and visitors, contributing to the liveability and social cohesion of the area.

  3. Local businesses (which include, for example, building companies, tourism and hospitality companies, and craft associations): Their aim is to promote the restoration of the buildings and to provide new services to satisfy the increasing flow of tourists. They want to improve the general aspect of the historical city centre, making it a better tourist attraction, safeguarding the traditional artisanship and local products. They are also interested in the general improvement of the city. They expect that the increasing number of visitors will bring additional income by means of the opening of new activities or the improvement of traditional ones, such as shops of artisans.

4.2 Criteria identified in our study

Criteria were built to make operational the different points of view that were identified from the values and concerns of the stakeholders. These were adapted from the management plan for the historical centre of the city of Naples, in Italy. The management plan for the historical centre of Naples had identified four significant points of view (or strategic challenges) for the conservation and the enhancement of the UNESCO site (see Piano-Napoli, 2011). In our study, we have taken into account different points of view, to make operational the adopted criteria, always according to the values and concerns of the stakeholders in our specific context.

There are four major points of view in our study: (1) conservation of the tangible cultural heritage; (2) promotion of the traditional craftsmanship, tourism, and local businesses; (3) improvement of the quality of the urban environment; and (4) social benefits for the community. These points of view along with their respective criteria (and possible subcriteria) are presented next.

  1. Conservation of the tangible cultural heritage point of view.
    In this point of view, the stakeholders are concerned with the care and protection of the tangible cultural heritage, in particular with the possibility of restoring old buildings to enrich and improve their current use or to adapt them to an alternative use. These aspects can be rendered operational by taking into account the following two criteria.

    1. Compatibility of the project with the affected tangible cultural heritage (notation: ; label: CON-COMP; unit: qualitative levels; preference direction: to maximize). This criterion consists of an evaluation of the compatibility of the building with the project defined. For each project, we evaluate the compatibility with the proposed use for the building. This criterion can be modelled through a 4-level qualitative scale with the following verbal statements and respective labels: very high (VH), high (H), medium (M), low (L).

    2. Increasing the usability of the tangible heritage (notation: ; label: CON-USAB; unit: percentage; preference direction: maximize). This criterion consists of considering the increased usability of the building. We adopt a percentage scale in order to take into account the fact that beyond a certain level a project cannot improve any more the usability of the tangible heritage.

  2. Promoting traditional craftsmanship-Tourism-Business point of view.
    In this point of view, the stakeholders are concerned with the actions that should be taken for promoting the activities of traditional craftsmanship and the expansion of the production of local products, as well as tourism and the establishment of new local businesses. This is important for improving the local economy as well as preserving the intangible heritage and the local traditions.

    1. Promotion of traditional craftsmanship and traditional knowledge (notation: ; label: PRO-CRAF; unit: qualitative levels; preference direction: to maximize). This criterion consists of the enhancement and safeguarding in the historical city centre of the traditional craftsmanship and knowledge, and also the enrichment of the offerings of local products. It comprises two subcriteria:

      1. Creation of new jobs and expansion of local products. It uses a 4-level qualitative scale as in criterion .

      2. Dissemination of the intangible cultural heritage. Availability of activities to promote traditional knowledge. The same type of scale as in the previous subcriterion is used.

      The two subcriteria can be combined to generate the scale of the criterion defining a 16-level qualitative scale of profiles for the criterion: VH-VH(16), VH-H(15),…, H-VH(12),…, M-VH(8),…, L-VH(4),…, L-L(1).

      In a first moment, the numbers from 16 to 1 were only used to code the profiles, so that, e.g. 16 is not a value, but a code for VH-VH. Indeed, taking into account all profiles resulting from the 16 combinations of ordinal evaluations using the two criteria, we can define only a partial ordering, for which combination (,) dominates combination (,) if is not worse than and is not worse than . In Figure 1 we represent the possible profiles with respect to the two considered subcriteria by a Hasse diagram, where it is possible to see all the possible preference relations between the different possible profiles. More precisely, the profile is preferred to the profile if there is an arrow directed from to (as is the case, e.g. for 16 and 12) or if there is a set of consecutive arrows starting from and arriving at (as is the case, e.g. for and , which are linked through the arrows going from to , from to , and from to ). Observe also that if it is not possible to reach starting from by one or more consecutive arrows, then and are not comparable (as is the case, e.g. for and ). In a second moment, after discussing with the DM, the assumption that the first subcriterion is more important than the second one was adopted. Consequently, the qualitative levels were ordered lexicographically, so that one level represents a better evaluation of another level if

      • the first criterion has a better evaluation, or

      • on the first criterion the two evaluations are the same but on the second evaluation the first level has a better evaluation.

      For example, VH-M() is now better than L-VH() because has a better evaluation on the first criterion. The levels are therefore now all comparable among them and ordered according to their codes.

      Figure 1: Hasse diagram for the two subcriteria of criterion
    2. Promotion of local entrepreneurship and businesses (notation: ; label: PRO-BUSI; unit: qualitative levels; preference direction: maximize). New local companies created, use of a 4-level qualitative scale as in criterion .

    3. Promotion of tourism (notation: ; label: PRO-TOUR; unit: qualitative levels; preference direction: to maximize). Flow of visits in the city. Use of a 4-level qualitative scale as in criterion .

  3. Quality of the urban environment point of view.
    In this point of view, stakeholders are concerned with urban aesthetics as an important factor for improving the perception of the historical urban landscape and its conservation. In this concrete case, it includes the assessment of such actions as refurbished facades, the surrounding urban spaces, the urban layout, and green spaces. There is only one criterion to render operational the quality of the urban environment point of view.

    1. Maintenance of urban spaces (notation: ; label: ENV-MAIN; unit: ; preference direction: to maximize). This criterion consists of the urban area that has been restyled.

  4. Social benefits for the community point of view.
    In this point of view, the stakeholders are concerned with improving the identity of the local communities and their knowledge of their cultural heritage. This is a point of view with the utmost importance and is rendered operational using two different criteria, as stated below.

    1. Enrichment of the cultural offerings (notation: ; label: SOC-CULT; unit: qualitative levels; preference direction: to maximize). This criterion consists of the initiatives to enrich the cultural offerings, aiming to increase public awareness and cultural identity of the city.

      1. New cultural proposals. Use of a 4-level qualitative scale as in criterion

      2. Increasing attendance at museums and other cultural sites. Use of a 4-level qualitative scale as in criterion .

      The overall scale of this criterion is built as in .

    2. Social cohesion (notation: ; label: SOC-COHE; unit: quantitative; preference direction: maximize). Number of non-profit organizations participating in the management of the tangible cultural heritage and their planned cultural activities. Use of a 4-level qualitative scale as in criterion

4.3 Actors involved in our study: The representative of the municipality and experts

In this subsection we present the actors involved in our study (experts, analysts, and a representative of the municipality), the interaction with them, and the contribution of this interaction. As in Bottero et al. (2015) and Bottero et al. (2018), a very interesting situation for dealing with this type of problems is when we can interact with all the stakeholders’ views and work within a focus group scheme. In the current situation, having a representative from each stakeholder was not possible. For this reason, we interacted with different actors, as follows:

  1. First, we interviewed an expert for each of the points of view presented in the previous subsection. We asked them to rank the criteria for each of them, as we will explain in Section 5.5.

  2. Second, we asked these four experts to interact in a focus group scheme. In this way, we have gathered the different points of view that it is essential to consider when dealing with such a difficult problem. The experts, who suggested in the first phase different rankings for the criteria, have reached an agreement and they have accepted the method.

  3. Third, we dialogued with a representative of the local municipality, who has a thorough knowledge of the ongoing projects in the historic city centre. He has provided important information about the details and the peculiarities of the projects. He has also highlighted the difficulties and restrictions linked to the different projects.

  4. Lastly, we interacted with a different expert with more technical knowledge about the problem, the projects, and the data. This expert contributed throughout the entire decision process, with a continuous interaction with the analysts. For this reason, he has played the role of the DM in this study, while the authors played the role of the analysts.

Therefore, the interaction with the actors had two significant steps:

  • interaction between the analysts, the experts and the representative of the municipality, which has permitted clarifying the elements of the real problem and collecting information about the preferences and constraints;

  • interaction between the analysts and the expert playing the role of the DM, which has permitted modelling the problem and selecting a possible portfolio of projects.

4.4 The tangible urban cultural heritage: Projects

In this subsection we present a list of the projects related to the tangible urban cultural heritage included in our study. These are thus the objects of the decision, i.e. the actions of our problem. More details about these projects can be seen in the plan for the city of Naples (Piano-Napoli, 2011). This plan considers also some of the intangible cultural heritage, which is not included in our study. In addition, we have integrated the plan with some additional projects added over time, and reported on the website of the municipality. In some cases, a work started but was never completed. In Table 1 we report the selected projects and their associated notation and label. In Appendix B, for the selected projects, we have listed the typology of the tangible cultural heritage, e.g. a building and area and so on. Moreover, we think it is appropriate to highlight what is the state of the building, e.g. usable, degraded, and what type of intervention has been planned, e.g. improving the use, making it usable, or establishing new functions.

Project Notation Label
Murazione Aragonese di Porta Capuana ( Mura-Capua)
Castel Capuano ( Cast-Capua)
Complesso ex-ospedale di Santa Maria della Pace ( Comp-Maria)
Complesso S. Lorenzo Maggiore ( Comp-Loren)
Complesso S. Gregorio Armeno ed ex Asilo Filangieri ( Comp-Grego)
Insula del Duomo – Area archeologica ( Area-Duomo)
Complesso di S. Lorenzo maggiore (Area archeologica) ( Area-Loren)
Teatro antico di Neapolis ( Teat-Neapo)
Chiesa SS. Cosma e Damiano ( Chie-Cosma)
Castel dell’Ovo ( Cast-D’Ovo)
Complesso dei Girolamini ( Comp-Gerol)
San Gioacchino a Pontenuovo ( SanG-Corvo)
Sant’Aniello a Caponapoli ( SanA-Capon)
Complesso Trinità’ delle Monache ( Comp-Monac)
Mercatino S. Anna Di Palazzo ( Merc-Palaz)
Chiesa San Giovanni Battista delle Monache ( Chie-Monac)
Complesso Santa Maria della Fede ( Comp-Mfede)
Carminiello al Mercato ( Carm-Merca)
Complesso di S. Paolo Maggiore ( Comp-Paolo)
Villa Ebe alle rampe di Lamont Young ( Vill-EbeRa)
Table 1: List of Projects

In Figure 2 we can see the locations of all the sites on a city map of Naples.

Figure 2: Projects and their locations

Moreover, for each of the projects, one or more functions, intended as the planned activities, have been identified. These are classified as:

  • Tourist facilities (label );

  • Museum and archaeological sites (label );

  • Accomodations for students and the elderly (label );

  • Leisure Activities (label );

  • Record Office Archives (label );

  • Public Services (label ).

The functions have been listed for each project in Table 13 in Appendix B.

4.5 The performance table

To evaluate the performance of the projects, we have examined them in detail. Some information was not directly available, therefore, we have estimated it. For example, the quantitative criterion related to the maintenance of the urban spaces has been judged considering the area of the interventions and the surrounding areas that will benefit from it. See Appendix B for Table 11 with all the performances for each project and each criterion.

5 Interaction with actors involved: A behavioural and socio-technical approach

In this section, we present a method to assign priority levels to the tangible urban heritage projects. These projects have different levels of priority of implementation, the assignment of which is not obvious, since they are based on several conflicting criteria.

5.1 Why an Electre Method ?

Before choosing an MCDA method, we checked for some basic requirements. This eventually led to the choice of an Electre method (for more details see Figueira et al., 2013). The requirements are mainly the following:

  • The number and the nature of the criteria. In our problem we have eight criteria, which fits perfectly within the adequate number for using an MCDA method (in general, in between five and twelve). In addition, the problem uses both quantitative and qualitative criteria. The scales are rather heterogeneous, with different units: , K€, number, and verbal levels.

  • The choice of the scales for some criteria was rather arbitrary, as for example for the criterion PRO-CRAF . In addition, it not easy to define the concrete meaning of all the qualitative levels of some scales. Thus, the data are only imperfectly known, which is related to this arbitrariness when building the criteria, since there is some imprecision and/or uncertainty in this process.

  • Through the use of some dummy projects, i.e. fictitious projects that do not exist and are provided with ad hoc characteristics in terms of implementation and restoration of the cultural heritage, we could observe that the DM could prefer one project over a second one, the second one over a third, but the third could be preferred to the first.This means that intransitivities are accepted.

  • Some examples with dummy buildings, e.g. buildings that do not exist but are provided with evaluations or characteristics defined only for the purpose of the analysis, were presented to the DM to test his sensitivity to compensatory phenomena. We concluded there was a non relevance of compensatory effects.

  • When comparing two different buildings, the DM pointed out the advantages of one building (reasons for) over the other, as well as the disadvantages (reasons against) of the first compared with the second.

With these aspects of our problem, as summarized in Figueira et al. (2016), the choice of an Electre method is the most appropriate one from among the variuos MCDA methods.

Since the problem is by its very nature a sorting problem, the next question was, which one of the Electre methods for sorting problems was the most adequate. We started by asking one of the experts if it was possible to delimit the categories by (lower and upper) limiting profiles as in Electre Tri-B and nB (see Fernández et al., 2017). This was rather difficult for the expert. Then, we tried to see if a more or less “central” reference action for each category could be easier to identify, as in Electre Tri-B (see Almeida-Dias et al., 2010). Again, the expert did not feel comfortable and was not able to define a “central” action. When we gave him the possibility of choosing any representative reference action per category, he felt much more comfortable providing such information, which led us to adopt the Electre Tri-nC (Almeida-Dias et al., 2010) and pursue our study with this method. A brief description of the method is provided in Appendix A.

5.2 Determining the discriminating (indifference and preference) thresholds

For the determination of the indifference and preference thresholds, also called discriminating thresholds, we follow the method proposed in Roy et al. (2014). These thresholds are used for modelling the imperfect knowledge of data and should result from an interaction between the analysts and the expert.

If there are important reasons leading to rejecting the hypothesis of constant thresholds, the expert can consider the use of affine functions to model the discriminating thresholds. These functions can be direct (when varying with the worst performance) or inverse (when varying with the best performance). The two are equivalent, as can be seen in Roy et al. (2014). Let us consider two actions and , where a criterion has to be maximized, and where the performances of are better than the performances of , i.e. . Consider also the direct affine threshold function (computed with respect to the worst performance): . The inverse thresholds are easy to compute, as can be seen in Roy et al. (2014).

In what follows we present the thresholds for all the criteria. We start with criterion .

Criterion (Env-Main)

The area of the buildings to be restyled is not precisely defined, leading to imperfect knowledge. The analyst felt there were strong reasons favouring the need for building variable functions. The area in is a criterion to be maximized. The following procedure was employed.

  • Consider a project with reference area of in the bottom of the scale. A project with a difference greater than or equal to was considered as significantly better in terms of restyled area than a project with a reference area of . Since we are making the difference with respect to the worst performance, we are computing the direct thresholds (the inverse thresholds are easy to compute). We have thus

    It should be remarked that a project with a value of was considered as the first value allowing the establishment of the threshold. Other values have been considered, for example, was considered too high, while was considered slightly low, which led us to increase the value a little bit to .

  • Consider a reference area of in the upper level of the scale. A project with an area greater than or equal to was considered as significantly better than a project with an area of . This led to putting

  • From these two equations we can derive the values of the two and ,

    with the solutions and .

  • This leads to the following direct preference threshold function:

  • This function was tested and validated for values in the middle of the scale and it was taken to model the imperfect knowledge of the data in our model.

  • A similar process can be used to derive the values of and :

    with the solutions and .

  • This leads to the following direct indifference threshold function:

Criterion (Con-Usab)

The process was similar to the previous one, leading to the following direct discriminating thresholds. However, the expert, this time, felt more comfortable starting with the indifference thresholds.

  • A project with reference percentage of in the bottom of the scale was considered, while in the top of the scale we considered a reference project with a reference percentage of . A project with a difference greater than or equal to in the bottom of the scale and a project with a percentage of in the top of the scale were considered by the expert indifferent to the respective reference projects. These pieces of information lead to the following system:

    with the solutions and .

  • A similar process was followed for the preference thresholds, giving rise to the system

    with the solutions and .

Criteria (Pro-Craf) and (Soc-Cult)

These two criteria possess a 16-level qualitative scale. The same process as in criterion was used, but now for a discrete scale. This led to consider only constant thresholds, and the same for both criteria: and .

Criteria (Con-Comp), (Pro-Busi), (Pro-Tour) and (Soc-Cohe)

For these three criteria the scale is the same, with only four qualitative levels. There were no strong reasons to consider indifference and preference thresholds.

5.3 Constructing the reference actions

The reference actions were defined in a co-constructive way with the participation of the expert and the analysts. A graphical representation helped in making an adequate choice of the representative alternatives of each category. The categories of priority with respect to conservation, promotion, environmental and social points of view, are the following.

  • Category : This category contains the projects with low priority.

  • Category : This category contains the projects with medium priority.

  • Category : This category contains the projects with high priority.

  • Category : This category contains the projects with very high priority.

First, the analysts started by asking the expert if he could identify in the performance table representative projects of each category, or at least for some of the categories. This was not possible. He found the task difficult given the set of projects provided in the performance table.

Then, along with the expert, the analysts made a graphical representation of all criteria scales, one after the other, on the same sheet of paper and asked if he could draw a representative project of category . After some reflection they drew an action with the following profile: .

The process was repeated for the other categories. Table 2 presents the set of representative actions that were co-constructed during the interaction process between the analysts, from one side, and the expert, from the other side.

Representative
Actions CON-COMP CON-USAB PRO-CRAF PRO-BUSI PRO-TOUR ENV-MAIN SOC-CULT SOC-COHE
1 20 5 1 1 1000 5 1
\hdashline 2 40 7 2 1 3000 7 2
2 30 6 2 1 3500 6 3
\hdashline 3 70 11 3 2 10000 11 4
3 50 7 2 2 5000 11 3
\hdashline 4 80 12 3 3 30000 15 4
Table 2: Reference actions

5.4 Determining the veto thresholds

The process for assessing the veto is similar to the one used for the discriminating thresholds. The concepts were clearly explained to the expert playing the role of the DM using an illustrative example, and were accepted. The veto thresholds were easily assessed. Their values are presented in Table 12 in Appendix B. It was, of course, more laborious than the task for assessing the variable thresholds. Let us remark that for some criteria, no discriminating threshold has been identified. A veto threshold, along with the weights of the criteria and preference-based parameters, are related with the role each criterion plays in the construction of the outranking relation.

5.5 Determining the weights of the criteria

Criteria differe in their importance. The procedure applied for determining the relative importance of the criteria is the Simos–Roy–Figuiera (SRF) method, proposed in Figueira and Roy (2002). The interaction was conducted first with each expert and subsequently in the focus group.

  1. First, we used a deck of cards with the name of each criterion on a card. We also added a brief description of each criterion and provided each expert with this set of cards.

  2. Second, we asked that the cards be ranked, from the least important to the most important of the related criteria. If some criteria are equally important, the corresponding cards should be clipped together in the same pack. This yields a ranking of equally important subsets or packs of criteria.

  3. Third, we explained to each expert that the difference between two successive pairs of subsets of criteria can be more or less close. When determining the weights, we should take into account such smaller or bigger differences of importance (or intensity). We provided each expert with a deck of blank cards and asked them to insert these blank cards in the intervals between successive pairs of subsets of criteria in the ranking. The meaning of the blank cards is as follows: having no blank cards means that the difference between the weights of the subsets of criteria is minimal, say one unit; one blank card means twice the unit, two blank cards means three times the unit, and so on.

  4. Finally, we asked each expert to tell us by how many times is the most important criterion more important than the least important one. This was the most difficult question for the experts, even with the possibility of providing us with three different values, or even a range. This major drawback was mitigated when we told the experts that they can reason in terms of votes. Assume the least important criterion has been given one mark, how many marks would you assign to the most important one?

In this way, we have the opportunity to perform a series of analyses considering even more than one set of weights, defining different versions of the problem for which the solution will be acceptable, and therefore robust, even having different sets of values for the weights.

In our case, in the focus group, the experts provided the following information for each step:

  1. The provided ranking is the following, where the symbol is used to denote “strictly less important than”, and to denote “equally important than”:

  2. After a rich discussion and many exchanges, a decision was taken on the following distribution of blank cards (the number of blank cards is given in brackets):

  3. Finally, we asked the experts to let us know by how many times the most important criterion, , is more important than the least important one, . This was a rather difficult question. We rephrased it in the following way. If we assign one vote to criterion , how many votes do you assign to criterion ? The experts, after some reflection, said 10. This answered our first question.

This classification leads to finding the weights, denoted by for , and presented in Table 3.

Anyway, after some comments and a quite long discussion, the expert provided also an additional ranking and classification. In particular, they felt that they could change the position in the ranking of some criteria, as follows:

They also said that in this case they would not add any blank cards, and the distance between the first and the last level should be equal to 10. This different classification of criteria leads to a different set of weights, denoted by , and presented in Table 3.

In Table 3 we have also included the weights originating from the individual interaction with every expert, in particular:

  • The expert in the conservation of the tangible cultural heritage point of view (label: EXP-CONV), weights ;

  • The expert in the promotion of the traditional craftsmanship and local products point of view (label: EXP-PROM), weights ;

  • The expert in the quality of the urban environment point of view (label: EXP-URBE), weights ;

  • The expert in the social benefits for the community point of view (label: EXP-COMM), weights ;

The ranking of the criteria that was generated by those weights has been presented in Tables 1417 in Appendix B.

Criteria
weights CON-COMP CON-USAB PRO-CRAF PRO-BUSI PRO-TOUR ENV-MAIN SOC-CULT SOC-COHE
20.0 8.0 14.0 8.0 2.0 17.0 14.0 17.0
22.7 6.4 14.5 10.5 2.3 18.6 6.4 18.6
13.4 18.3 6.1 18.3 6.1 18.3 13.4 6.1
11.3 11.3 16.1 16.1 1.6 11.3 16.1 16.1
4.3 14.7 11.2 11.2 11.2 21.5 7.8 18.1
20.8 6.3 16.6 4.2 18.7 12.5 16.7 6.1
Table 3: Sets of weights of the criteria

6 Selecting a portfolio of reusable physical urban cultural heritage artifacts

In this section we describe how the selection of the highest priority projects is modelled. First, we introduced an objective function that maximizes the number of projects to introduce in our portfolio with the highest priority. Second, we discussed with the representative of the municipality and the DM the potential constraints that should be considered for decisions about the urban planning.

6.1 Objective function

After assigning a priority level to each project, we must construct the portfolio of projects to be proposed for funding. This decision problem was handled by defining a knapsack problem with additional logical constraints related to budget limitations and urban planning requirements. Since we cannot select all the projects at the same time due to the multiple constraints, we should start by selecting as many as possible of the projects with the highest priority, then those with the second highest priority, and so on. More precisely, we can associate a decision variable, , with each , such that if is selected, and , otherwise. Then, the number of projects in the maximal priority category is maximised, solving the following optimization problem:

subject to all the constraints of the problem. Assume that the optimal value of this problem is . Then, for the maximization of the number of artifacts in , the second highest priority category, we can add the constraint to the initial set of constraints and proceed with the next optimization:

The process is repeated until the lowest category is explored.

This sequential process can, however, be replaced by the solution of an equivalent single optimization problem. Instead of several objective functions, we define a single objective function as follows.

where

We associate a coefficient with each category . We define also the total weight of a category by multiplying the coefficient of the category by the number of its elements, i.e. .

The idea is that the coefficient of category should be strictly larger than the sum of all the values associated with all the categories having a lower priority.

With the proposed procedure, projects with the highest priority are selected first, unless this is not possible due to the constraints. In that case, the optimization process goes to the next priority level, until it reaches the lowest one.

6.2 Constraints

Constraints for the problem were discussed with the representative of the municipality and the expert acting as DM. This allows having a real perspective on the problem. While the representative of the municipality helped in the understanding of the broad problems of the area, the DM was able to provide a more detailed description of them, allowing the analysts to formulate the constraints. We have several constraints related to different aspects.

The most important constraint is the one related to the budget available for the implementation of the projects. Each project is associated with a cost (see Table 11 in Appendix B for the costs of each project). Given that at the moment the budget allocated to these projects was still not uniquely defined, the representative of the municipality and the DM suggested implementing several scenarios with different available budgets. Denoting the available budget by , this constraint can be formulated as

(1)

In addition, the DM and the representative of the municipality have remarked that we can have constraints related to the type of functions that each building or area will have after its restoration. Furthermore, the location of the project is a strategic element in the decision. On this basis, several logical constraints can be formulated.

First, in the ancient city centre there are three main roads, called “decumano”, highlighted in Figure 3. These roads are the main attractions for tourists and citizens and there is an urgent need to restore. For this reason, the DM suggested having a minimum number of regeneration projects to be implemented in this part of the city. From the map, we can see that the set of projects , are located on one of the “decumano”. The DM suggested that among those projects, a minimum number of projects, let us say , should be carried out. We formulate this as

(2)

The DM will be asked to supply a desired value for .

Figure 3: “Decumano”, “Insula” and Quadrants in the historic city centre

Second, the ancient city centre can be divided into smaller areas, called “insulae”, derived from the intersection of the “decumano” with the smaller roads called “cardines”. Some projects are located In 6 of these areas, as represented in Figure 3. In our case, the DM has suggested that if two projects are located on the same “insula”, they can generate a greater benefit than the sum of each of their own separate benefits, creating a synergy, but he could not specify by how much was the improvement. For this reason, he has indicated that he is willing to consider portfolios where at least a given number, called , of those synergies will be implemented. To model that, we define the set of “insula” , such that , which contains all the projects in the th “insula”. and the decision variables with and such that if both and are activated and otherwise. The following set of constraints, then, should be imposed:

(3)
(4)
(5)
(6)

Constraints 3 to 6 specify that if two projects are located in the same “insula”, then the variable associated with that two projects should be equal to 1. We want that at least of the variables will be equal to 1, meaning we want at least synergies.

Third, another important aspect is the set of functions that each building will have in relation to the area in which it is located. For each project, we have identified all the possible functions (see Table 13 in Appendix B). In addition, the DM has also requested considering bigger areas of the “insula”, called “quadrants”, which represent four important areas of the ancient city centre. Note that a project can have one or more functions. In particular, if indexes the set of the functions, and the set of the four quadrants, we can define the binary coefficients as equal to 1 if the project is in quadrant and delivers function , but otherwise. The DM has indicated two types of constraints concerning these two aspects. For a specific function, a given number of projects should be implemented. This constraint can be modelled as

(7)

He has also suggested that it would be ideal to distribute equally some of the functions in some quadrants. For this purpose, we define the binary variables . The constraints are of the following type:

(8)

with being a large number, and

(9)

with being a number between and .

Those two constraints imply that at least one project with a given function should be open in at least quadrants. For example, if is equal to 2, two quadrants should have at least one project implemented for one of the functions defined.

The monobjective binary programming model can be solved by any linear optimization solver. We have used the CPLEX 12.1 software.

7 Case study

The results have been obtained with the interaction with the DM throughout the whole of the decision process. First, we carried out several possible scenarios for assigning priorities, and after that, we selected the projects to carry out.

7.1 Assigning priority levels to the projects

According to the data reported in Appendix B, we have used the MCDA-ULAVAL software 2, which implements the majority of the Electre methods, including the Electre Tri-nC method, allowing the insertion of all the data and parameters elicited.

First, the analysts suggested to apply the method for each set of weights from Table 3 using the other necessary data of Appendix B. We have called the configuration to specify that we used weights , for weights and so on. The results are shown in Table 4.

Weights
Configuration
( Mura-Capua)
( Cast-Capua)
( Comp-Maria)
( Comp-Loren)
( Comp-Grego)
( Area-Duomo)
( Area-Loren)
( Teat-Neapo)
( Chie-Cosma)
( Cast-D’Ovo)
( Comp-Gerol)
( SanG-Corvo)
( SanA-Capon)
( Comp-Monac)
( Merc-Palaz)
( Chie-Monac)
( Comp-Mfede)
( Carm-Merca)
( Comp-Paolo)
( Vill-EbeRa)
Table 4: Categorization for each defined set of weights

The projects belong to similar categories even after changing the weights. In particular, we can see that projects can be allocated to a single category or they can belong to two consecutive categories. Only for some projects does the use of a different set of weights lead to their being assigned to different categories ( and ) or falling between two categories (i.e., , , and ).

At this point, the analysts showed these categorizations to the DM. He found that the most interesting classification, i.e. the one most in accord with his perspectives, is provided by configuration . It is the one closest to his opinions in terms of projects that should be prioritized.

The analysts wanted to conduct some further analysis to validate the performance of this categorization and wished to perform a robustness analysis, as follows:

  1. First, he tried to increase the number of cards introduced in the ranking that generated the weights of the preferred classification. In particular, he tried to increase the distance between the first, the second and the third levels, adding five cards to every level. This scenario was analysed because, during the focus group, there were many exchanges about the number of cards to insert between each level. Some of the experts in the focus group pointed out how the criteria in the first and second levels should be the most important, but they did not provide enough arguments to convince the others to insert some additional cards. Anyway, even if this change results in different weights, it did not change the categorization of the projects. Moreover, he attempted to put in 20 cards, distributing them between the last two criteria. With this, the categorization was the same as the one obtained with weights , i.e. configuration .

  2. Second, the analysts felt that it was worth testing if the ratio between the last and the first level had any influence on the solution. He increased this value to 20. Again, the same categorization as the preferred one was obtained.

  3. Third, the analysts felt that in order to test the robustness of the categorization, the preference information and the veto thresholds should be changed. The analyst performed this analysis by inserting different preference information, veto thresholds and the reference actions are given in Appendix D. They used the weights derived from the focus group (Configuration ) and those expressed by expert EXP-URBE, which were those that differed the most (Configuration ).

Weights
Configuration
( Mura-Capua)
( Cast-Capua)
( Comp-Maria)
( Comp-Loren)
( Comp-Grego)
( Area-Duomo)
( Area-Loren)
( Teat-Neapo)
( Chie-Cosma)
( Cast-D’Ovo)
( Comp-Gerol)
( SanG-Corvo)
( SanA-Capon)
( Comp-Monac)
( Merc-Palaz)
( Chie-Monac)
( Comp-Mfede)
( Carm-Mercato)
( Comp-Paolo)
( Vill-EbeRa)
Table 5: Categories obtained with a second set of veto thresholds, reference actions, and preference information

These two new obtained categorizations were discussed with the DM. Both configurations were considered inadequate. This is due to the presence of many projects that can be assigned to two different categories and in some cases to even more (e.g. and ). The DM commented that this classification does not represent what he values as being a good prioritization. In particular, concerns were raised for the presence of project in the last category. Even if it will restore the largest building and surrounding area (the highest value for criterion ), project does not represent the DM’s opinion of being a project to prioritize, and so it should be in a different category than that of project . Also, project has been classified in category , while the expert felt that this project should be at least in the upper category, since it is a strategic attraction for the city. At this point the analyst, felt that the best option was to move to the selection of the projects adopting the priorities obtained with the first set of reference of veto thresholds, reference actions and preference information.

7.2 Selection

The analysts started with the preferred classification, generated by configuration , as this was indicated by the DM as the most preferred one.

First, the computation of the objective function was discussed. As the calculation of the coefficients depends on the number of categories, the analysts asked the DM if he wanted to assign the projects lying in between two categories to one of the two categories. The DM did not feel that this operation was useful: he preferred to consider these intermediate categories each as a single different category. In light of this, the calculation of the was carried out for every possible configuration (these are given in Table 18 in Appendix C) .

After this, the DM and the analysts considered which constraints should be included among the ones defined in Section 6.2, fixing the values of every parameter included in the constraints.

This is because, given the economic situation, he felt that one possibility was to verify which are the projects that will be selected considering those different scenarios. This allows the DM to learn, depending on the different amounts of budget available, what are the projects that will be selected on the basis of the assigned priorities. The analysts suggested that the scenarios with the following budgets could be tested. We denote by the possible budgets available, and the configuration generated together with weights (see Table 6).

Budget B(7)
52240 45710 39180 32650 26120 19590 13060
Table 6: Budgets available in K€

Next, he specified that the ideal portfolio will have at least 4 projects in the “decumano”, therefore in constraint (2) the was set equal to 4. For the third set of constraints (3)–(6) the DM specified that at least one synergy among the ones presented in every “insula” should be realized. For this reason, was set to 1 for constraint (6).

The DM stressed that in what concerns the functions of the projects, he would like to have at least three projects for function , i.e. accomodation for students and the elderly, in constraint (7)).

For the set of constraints (8)–(9), the DM indicated that he would consider a good portfolio to be one with at least one project that provides facilities for the citizens and one project that provides facilities for the tourists in at least three of the four quadrants. This way, in constraint (9) should be set equal to 1. The formulation of all these constraints is given in Appendix C. The solution of the binary programming problem for the different available budgets has led to the portfolios in Table 7 (note that the symbol ✓means that a project should be carried out, while the symbol denotes that the project should not be implemented).

We would like to stress that for budgets and there is no possible solution to the model that satisfies all the constraints at the same time. Therefore, we asked the DM to remove or weaken some of the constraints. He advised us to lower the value of the projects expected to be in the “decumano” to 1 and the number of student and elderly housing facilities to 1. The portfolios given in Table 7 have been obtained considering these two relaxed constraints. As expected, the number or projects to be implemented decreases with a decrease of the available budget. The DM was invited to reflect on the portfolios identified. He has provided the following comments:

Configuration
( Mura-Capua)
( Cast-Capua)
( Comp-Maria)
( Comp-Loren)
( Comp-Grego)
( Area-Duomo)
( Area-Loren)
( Teat-Neapo)
( Chie-Cosma)
( Cast-D’Ovo)
( Comp-Gerol)
( SanG-Corvo)
( SanA-Capon)
( Comp-Monac)
( Merc-Palaz)
( Chie-Monac)
( Comp-Mfede)
( Carm-Mercato)
( Comp-Paolo)
( Vill-EbeRa)
Table 7: Projects selected for different available budgets, weights
  • The portfolio of configuration was welcomed by the DM. Indeed, the DM said that in a scenario with such a budget, he would prefer not to carry out project but still implement all the others, in order to improve most of the areas of the historic city centre. From his point of view, since project is the one with the highest cost, it is not an extremely attractive project and he would happily compromise in doing all the other projects.

  • The portfolio of configuration could represent a good compromise because it suggests implementing all the most important projects with a reasonable budget that is more likely to be obtained.

  • The portfolio of configuration has some areas of concerns due to the presence of projects , and belonging to category and not including projects from the higher category, . Then, the analysts solved the model again, excluding those projects from the optimization procedure. Once the new portfolio produced by this procedure was shown to the DM, he advised us that the original portfolio was more representative of his preferred portfolio for that amount of budget available.

  • The DM considered as a good portfolio even the one for configuration . In fact, even with a very limited budget, all the listed constraints will still be satisfied and the best projects of the last two categories will still be in the portfolio.

  • For the portfolios of configurations , and , the DM has disclosed some concerns about the quality of these portfolios with very few projects. He has also raised concerns about the fact that projects in the lower categories were introduced. The analysts explained that this was due to the constraint regarding the distribution of functions (8)–(9), and that he could try to do some other simulations relaxing also these constraints. Eventually, the DM really thought that this is a major and fundamental constraint that he will not happily compromise on, therefore he accepted the portfolio obtained.

The analysts asked the DM if he was satisfied with the analysis concerning the different budget scenarios and he replied that he had enough information to allow him to better understand the process and to have a better grasp of how much funding will be necessary to have a portfolio that will be the closest to the preferred one.

7.3 Robustness Analysis

A robustness analysis is important to verify that the results obtained can be replicated even in slightly changed conditions (Roy, 1998, 2010b). We focussed on taking into account the different configurations derived from the different opinions of the experts involved.

Configuration
( Mura-Capua)
( Cast-Capua)
( Comp-Maria)
( Comp-Loren)
( Comp-Grego)
( Area-Duomo)
( Area-Loren)
( Teat-Neapo)
( Chie-Cosma)
( Cast-D’Ovo)
( Comp-Gerol)
( SanG-Corvo)
( SanA-Capon)
( Comp-Monac)
( Merc-Palaz)
( Chie-Monac)
( Comp-Mfede)
( Carm-Mercato)
( Comp-Paolo)
( Vill-EbeRa)
Table 8: Projects selected for different weights, budget
Configuration
( Mura-Capua)
( Cast-Capua)
( Comp-Maria)
( Comp-Loren)
( Comp-Grego)
( Area-Duomo)
( Area-Loren)
( Teat-Neapo)
( Chie-Cosma)
( Cast-D’Ovo)
( Comp-Gerol)
( SanG-Corvo)
( SanA-Capon)
( Comp-Monac)
( Merc-Palaz)
( Chie-Monac)
( Comp-Mfede)
( Carm-Mercato)
( Comp-Paolo)
( Vill-EbeRa)
Table 9: Projects selected for different weights, budget
Configuration
( Mura-Capua)
( Cast-Capua)
( Comp-Maria)
( Comp-Loren)
( Comp-Grego)
( Area-Duomo)
( Area-Loren)
( Teat-Neapo)
( Chie-Cosma)
( Cast-D’Ovo)
( Comp-Gerol)
( SanG-Corvo)
( SanA-Capon)
( Comp-Monac)
( Merc-Palaz)
( Chie-Monac)
( Comp-Mfede)
( Carm-Mercato) </