Sustainable tourism: a group parsimonious analytic hierarchy approach Open Access

Tourism is one of the main drivers of countries' economies (Mihalic, 2002). It produces effects on employment (Nguyen et al., 2025), on infrastructure investments (Yanase, 2015), and on the value of land (Romão, 2018). With respect to this last aspect, in particular, tourism has a crucial role in the strategic decision-making processes (Bousset et al., 2007) in local development. With regard to the differences between internal and peripheral areas (Jewell et al., 2004), the latter are those in which the most significant and critical issues, such as demographic decline, the reduction of essential services (Hospers and Reverda, 2015) and, worse still, the devaluation of the landscape and cultural heritage, occur. In this context, even more than in others, strategic choices are needed that encourage tourism as a driver of socio-economic development to enhance the whole of the territorial heritage and local resources (Boniface, 2000). These considerations give a central role to political decision-makers (DMs), who must adopt strategic choices to develop tourism in internal and peripheral areas. In order to support DMs in making strategic choices regarding inland and peripheral areas and to address decision-making problems in the tourism sector (Elbelehy and Crispim, 2026), multi-criteria decision-making (MCDM) models can be adopted, and these have proved to be particularly useful for modelling complex problems (Doumpos and Zopounidis, 2002). MCDM models are known for their versatility in analysing different types of decision-making problems. They consider a plurality of alternatives and criteria (which are often in conflict with each other) (Ishizaka and Nemery, 2013), and they are useful decision support tools for a DM in different applications and contexts (see, for example, Fattoruso et al., 2024; Fattoruso and Marcarelli, 2022; Cavallo et al., 2014). With reference to the tourism sector, a growing number of scholars have been working in recent years to provide new MCDM methodological approaches to support DMs in defining tourism development policies, especially when these involve networks of actors and territories with heterogeneous needs. In such situations, DMs face a complex decision-making process that can be characterized as multidimensional problem. The decision-making process becomes more complex when decisions are made in groups and in a public context. Starting from these critical issues, this study aims to support group decision-making in the public context. The increased complexity of the decision-making process in group and public contexts is due to several factors. For example, group size influences decision-making timing and dynamics; smaller groups often take longer to make decisions; and leaders tend to prevail within groups (Wu, 2024). The social and organizational context can influence the way in which groups reach decisions, adding further layers of complexity without necessarily improving the quality of decisions (Mullick and Sen, 2025). At the same time, social ties can reduce costs but complicate decision-making (Selivanovskikh et al., 2025). Overall, the interaction between social influence, group structure and environmental complexity makes group and public decision-making processes intrinsically more complex than individual decisions. Starting from these observations, this study aims to support group decision-making in the public context. To achieve this goal, the article, through the analysis of a real case study, aims to provide an appropriate MCDM methodology to support the development of tourism strategies that integrate sustainability-oriented decision criteria and promote sustainable tourism in internal and peripheral areas, that foster the enhancement of cultural and territorial heritage and that strengthen inter-municipal cooperation. In particular, the work involves DMs from six municipal administrations located in the Campania hinterland that have participated in national programmes to encourage and recognize tourism as a strategic sector to increase the territorial attractiveness. The problem at hand is, therefore, configured as a group decision problem. In this sense, our work proposes a methodological extension of the Parsimonious AHP method to support group strategic choices in complex and heterogeneous contexts. To this end, we propose, for the first time, the use of the aggregation of individual priorities (AIP) in a group Parsimonious AHP method.

The main originality of the methodology proposed in this paper is the integration of an algebraic approach into the Parsimonious AHP method. Traditionally, the Parsimonious AHP method uses the Saaty scale for the elicitation of preferences; this is based on verbal judgments translated into a discrete numerical scale $S={19,18,17,16,15,14,13,12,1,2,3,4,5,6,7,8,9}$ (Saaty, 1977, 2001). It is well known that the assumption of the Saaty scale leads to a boundary problem (Cavallo and D'Apuzzo, 2009) that is caused by the consistency rule (e.g. if the DM prefers alternative A 8 times more strongly than alternative B and alternative B 8 times more strongly than alternative C, he/she should prefer alternative A 64 times more strongly than alternative C, but the upper limit of the scale is 9). Several other scales have also been proposed in the literature (Cavallo and Ishizaka, 2023), in addition to the Saaty scale, in order to represent the data in a more realistic way. However, the assumptions of any scale give rise to the same boundary problem. The problem has been addressed and solved by Cavallo and D'Apuzzo (2009), who provide a rigorous algebraic approach to Pairwise Comparison Matrices (PCMs); they propose PCMs defined over real and continuous Abelian linearly ordered (Alo)-groups.

The main methodological innovation of our paper is the integration of this algebraic approach into the Parsimonious AHP; in this way, we remove some of the drawbacks related to the classical Saaty approach, such as the shown boundary problem described above and the independence of the scale-inversion condition of the weighting vector (Cavallo and D'Apuzzo, 2012). The details of this algebraic approach are provided in Section 3.2.

The use of Alo-groups offers an elegant and rigorous way of modelling, generalizing and solving problems; moreover, by means of proper isomorphisms, PCMs defined over two different Alo-groups are formally equivalent and therefore we can naturally extend concepts and properties from one representation to another.

This algebraic structure has been widely studied and accepted in the literature for generalizing PCMs, and several authors follow this Alo-group approach. For example, Hou (2016) applies a model based on a multiplicative Alo-group to a layout planning problem for an aircraft maintenance base; Koczkodaj et al. (2020) stress that an algebraic structure for generalizing PCMs has to be a torsion free group (i.e. an Alo-group as proposed by Cavallo and D'Apuzzo (2009), Kulakowski et al. (2019) focus on the conditions for order preservation for PCMs over Alo-groups; Ramík (2015a, b) focuses on PCMs with fuzzy elements on Alo-groups; and Xia and Chen (2015) deal with consistency and consensus in PCMs over Alo-groups. Compared with the existing evolutions of the Parsimonious AHP, our approach retains all the advantages of the method while replacing the classical Saaty approach with a rigorous algebraic approach to PCMs defined over Alo-groups. This allows for better control over the consistency and reliability of judgments without increasing the cognitive load for the DMs. Furthermore, although approaches based on Alo-groups have been analysed in the literature, they have not been integrated with the Parsimonious AHP or applied to real decision problems. Our work fills this gap by providing the first real case study (since 2009) that applies this integrated method and demonstrates its operational feasibility.

The proposed extension of the Parsimonious AHP emerges as a methodologically advanced solution to addressing group decision-making problems in real-world, highly complex contexts. We point out that the choice of the Parsimonious AHP is linked to the characteristics of the decision problem, and we discuss the advantages that the method presents. We recall that the Parsimonious AHP is an evolution of the classical AHP. Unlike the AHP, the method allows the analysis of problems with a high number of alternatives and criteria by reducing the number of pairwise comparisons. Furthermore, the Parsimonious AHP also solves the rank inversion problem (Abastante et al., 2018; Fattoruso and Marcarelli, 2022).

The paper is structured as follows: in Section 2, we analyse the main literature related to the use of MCDM methods in the tourism sector; in Section 3, we provide the basic notions on the Parsimonious AHP and the PCMs over Alo-groups; in Section 4, we describe our methodology; in Section 5, we describe the case study and report the main results; and in Sections 6 and 7, we discuss and conclude the paper.

To analyse the use of MCDM methods in the tourism sector, we searched the Scopus database for academic papers using the following search string: (TITLE-ABS-KEY(“multi-criteria decision making” OR “multi criteria decision making” OR “MCDM” OR “multi-criteria analysis” OR ”multicriteria analysis” OR “AHP” OR “analytic hierarchy process” OR “decision support system”)) AND (TITLE-ABS-KEY(“tourism” OR “eco-tourism” OR “sustainable tourism” OR “tourism development”)), limiting the search to scientific articles from 2015 onwards. The articles collected were analysed with VOSviewer software (Aria and Cuccurullo, 2017; Layeghi et al., 2026; Vesperi and Coppolino, 2023; Wong, 2018), which allowed us to examine the relationships between the keywords of the articles and to define the relevant scientific landscape, so that we could generate analyses regarding consistency, density, the temporal evolution of the scientific landscape and the main areas of analysis. In line with the consolidated literature on management (Selivanovskikh et al., 2025; Vesperi et al., 2025), we report only the results relating to the temporal evolution of the scientific landscape (Figure 1), which allows us to confirm that there is a substantial literature on the use of MCDM in the tourism sector. As can be seen from Figure 1, scholars' interest in different areas of analysis of the scientific landscape regarding the use of MCDM methods in the tourism sector has seen an increase in the last decade.

In our literature review, we refer to the yellow and green clusters provided by VOSviewer, which record papers reporting the use of MCDM tools in the tourism sector in the most recent period.

The scientific landscape (Figure 1) shows a growing use of MCDM to support decision-making processes in the managerial and decision science literature (e.g. Kumaresan et al., 2026). Among the recent studies focusing on the use of MCDM in the tourism sector, we can note that of Elbelehy and Crispim (2026), who propose a group multi-criteria decision-making method using interval-valued Pythagorean fuzzy variables based on the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) in order to evaluate social sustainability in the hospitality and tourism supply chain. A similar approach is used by Abdelsalam and Crispim (2026) to assess tourism supply chain resilience. By contrast, Aria et al. (2026) use a stochastic multi-criteria acceptability analysis (SMAA) method to improve the measurement of tourism sustainability in different regions and analyse an Italian case study. Ciano and Ferrara (2025) integrate explainable artificial intelligence with MCDM to evaluate the performance of tourism accommodation facilities. Ayvaz-Çavdaroğlu et al. (2024) use the best-worst method (BWM) and clustering methods to evaluate the quality of intelligent services in the hospitality and tourism industry. Cavallo et al. (2015) propose the use of the AHP method to study how the sustainable urban development of the port area of Naples can promote the development of tourism and the improvement of commercial activities. In applying the method, the authors pay particular attention to the economic, environmental and social impacts. Liu et al. (2013) propose a hybrid MCDM model that combines a decision-making trial and evaluation laboratory (DEMATEL), a DEMATEL-based analytic network process (DANP), and VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) to improve a metro-airport connection service to develop tourism. The methods are mainly used to analyse service quality, satisfaction and behavioural intentions. Talebi et al. (2019) carried out a study that evaluated and modified an existing road network for tourism purposes in the Arasbaran protected area. In particular, they follow a Geographic Information Systems (GIS) AHP methodology using a fuzzy logic approach. The road network was modified, taking into account the results obtained by the authors, thus encouraging tourism development in the area. By contrast, Racioppi et al. (2015) analyse a problem concerning the energy and environmental requalification of existing buildings which were intended to be used as innovative tourist structures. In particular, the authors use the AHP method to identify and propose different intervention scenarios to encourage tourism development. Stević et al. (2019) use the Simple Additive Weighting (SAW) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) methods to evaluate the attractiveness of six heritage sites in Porto, Portugal. Their paper sets out how to understand the attractiveness levels of cultural sites and discusses possible strategies to increase tourism. Referring to the development of policies to encourage tourism in internal and peripheral areas, Jena and Dwivedi (2023) propose the integrated use of Interpretive Structural Modelling (ISM) and DEMATEL methods to identify the major barriers to rural tourism in India. Their findings produce useful suggestions for increasing rural tourism. Pazhuhan and Shiri (2020) use the fuzzy accreditation tool and a TOPSIS model to determine the main regional tourism axes by analysing the spatial distribution of tourist attractions in a GIS environment. They highlight the importance of regional tourism planning for urban and rural socioeconomic development. Sadeghi et al. (2024) use the VIKOR method in a GIS environment to assess the tourism capacity of peri-urban areas that require specific tourism services.They also use the AHP method for further assessment of services. Moslem (2025b) addresses a strategic planning problem for the sustainable development of urban areas by proposing the integration of the Parsimonious AHP with the Multi-Objective Optimization by Ratio Analysis (MOORA) method. The paper presents a real-world optimization study of the public bus system in Budapest, demonstrating the effectiveness of the model. Also in the field of urban transport, the work of Moslem (2025a) proposes the Z-number extension of the Parsimonious Best Worst Method. As can be seen from this analysis of the main studies, MCDM methods are useful for supporting DMs in developing tourism strategies. As evidenced by the literature, several studies have evaluated performance and service quality for the purpose of incentivizing tourism or tourism sustainability, using MCDM approaches. Our work does not aim to analyse individual aspects (performance, service quality and so on) in the tourism sector but aims to provide an integrated approach to support public DMs in defining strategic actions to analyse and resolve decision-making problems aimed at incentivizing the development and growth of territories in internal and peripheral areas, focusing on tourism strategies. Similar approaches do not emerge from the analysis of the main literature. This work aims to bridge this gap and contribute to the academic debate on the effectiveness of MCDM approaches for group choices by describing a study that analyses group choices in the tourism sector.

In this section, we discuss the notions of the Parsimonious AHP method and PCMs defined over an Alo-group that are relied on in what follows.

The Parsimonious AHP method, introduced to the literature by Abastante et al. (2019), arose from the need to provide a methodology capable of analysing decision problems with a large number of decision elements, such as alternatives and criteria. The method is used, in particular, for the analysis of problems that allow the evaluation of any possible alternative under consideration with respect to competing criteria (consider, for example, repetitive standard evaluation procedures such as credit score analysis; see, e.g. Mareschal and Brans (1991) and Zopounidis (1987)).

The approach essentially involves different phases.

1. The direct evaluation of the decision elements (e.g. alternatives, criteria) under consideration;

2. The definition of reference points in agreement with the analyst;

3. The input of the pairwise comparisons of the criteria, and the calculation of the weights using the AHP method;

4. The input of the pairwise comparisons of the reference points using the AHP method;

5. The interpolation of the values obtained by the AHP in the previous phase, to give the local priorities of all alternatives; and

6. The computation of the weighted sum of the local priorities to calculate the global priorities.

In the following, we illustrate the main innovations compared to the classical AHP method. As mentioned previously, the first innovation is defined in the first phase of the method, which involves the DM's direct assessment of the elements characterizing the decision problem. The assessments can be expressed using a finite and discrete scale of values (e.g. Abastante et al., 2019) use a scale from 1 to 100). This allows a decision matrix to be constructed that collects the assessments expressed by the DM for each alternative with respect to each criterion. According to Abastante et al. (2019), using direct ratings to construct the decision matrix simplifies the DM's preference elicitation process, as it only requires them to make pairwise comparisons only between representative points. We note that the Parsimonious AHP represents an extension of the approach proposed by Corrente et al. (2016) to reduce the cognitive effort required of the DM and increase the evaluation efficiency in complex contexts by obtaining final ratings through the use of interpolation. After defining the decision matrix, the method includes a second important innovation regarding the definition of some reference points in agreement with the analyst. The concept of reference points derives from Prospect Theory (Kahneman and Tversky, 1979); they represent neutral values, the starting level or status quo perceived by the DM, against which gains and losses are evaluated. DMs do not evaluate the absolute value of an outcome, but rather how far it deviates from the reference point. In the Parsimonious AHP, the concept of the reference point is reinterpreted in a technical way, but its role is similar to that in Prospect Theory. The reference point can be considered as a reference (or hypothetical) alternative that allows the other alternatives of the decision problem to be evaluated via interpolation. Abastante et al. (2019) suggest that the number of reference points used to build the method should be determined according to the size of the decision problem; for example, they recommend using six reference points for a problem with at least 19 alternatives (see Abastante et al., 2019, for more details). It is also worth mentioning that Miller (1956) observed that it is difficult for DMs to evaluate more than seven elements.

We recall that the reference points are defined on the range of ratings assigned to the alternatives for each decision criterion, as provided by the DM. These points are constructed within the rating interval [min, max ], where min = 0 and max correspond to the highest rating assigned by the DM. The reference points are defined by the DM (Abastante et al., 2019). Alternatively, they are obtained by dividing the interval equidistantly (Ishizaka et al., 2020; Fattoruso et al., 2023).

The use of reference points allows the number of comparisons between alternatives to be reduced. In fact, reference points are used and treated as alternatives in the construction of the PCMs. This represents a further innovation of the Parsimonious AHP: the DM, rather than expressing a preference from the (excessively large) number of alternatives characterizing the decision problem, compares the reference points in pairs, just as if they were alternatives. Note that in the Parsimonious AHP, the alternatives are characterized by a numerical value assigned in the first evaluation phase; therefore, the DM will have no difficulty expressing her/his preferences regarding reference points that effectively simulate a reference (or hypothetical) alternative to the decision problem. The final innovation is that the priorities of all other alternatives are obtained by interpolation, based on the priority values obtained for the reference points. We highlight that the main advantage of the Parsimonious AHP over the classical AHP (besides the possibility of analysing decision problems with a large number of elements) is that it avoids rank reversal problems and the bias resulting from comparing more relevant objects with less relevant ones (Abastante et al., 2019).

Let X = {x₁, x₂, …, x_n} be a set of decision elements, such as criteria or alternatives; then, in the classical approach used by Saaty, the DM expresses the preference ratio m_ij of x_i over x_j by means of the PCM $M=(mij)n×n$⁠, with m_ij belonging to the Saaty scale. Strict preference of x_i over x_j, indifference and reciprocity are expressed as follows:

The DM is fully coherent if the following consistency condition is satisfied:

As introduced in Section 1, the assumption of the Saaty scale restricts the possibility for the DM to be consistent; for example, if X = {x₁, x₂, x₃} and the DM expresses the following preference ratios m₁₂ = 8 and m₂₃ = 8 then he/she will not be consistent because he/she should express m₁₃ = 8 ⋅ 8 = 64, but the preference ratio 64 exceeds the maximum value of 9. Thus, the maximum preference ratio that the DM can express, to be as consistent as possible, is m₁₃ = 9, but the following PCM:

Is not reliable; indeed, by using the Consistency Ratio (CR) proposed by Saaty, CR = 46.3% denotes high inconsistency and the PCM in (3) has to be adjusted to improve the consistency.

In order to avoid this boundary problem and to generalize the various approaches to PCMs introduced in the literature, Cavallo and D'Apuzzo (2009) propose that the PCM is defined over an Alo-group (G, ʘ, ≤), where G is a proper interval of the real line, ʘ a continuous group operation and ≤ the simple order on G inherited from the usual order on the real line. Within this algebraic approach, the entries m_ij of the PCM belong to G, and the strict preference of x_i over x_j, the indifference and the reciprocity in (1) are generalized as follows:

where e is the identity element and $mij(−1)$ the inverse of m_ij with respect to the group operation ʘ. The consistency in (2) is generalized as follows:

Figure 2 shows the Alo-groups and the related isomorphisms proposed in the literature, and Table 1 shows the corresponding group operations and identity elements; for further details, the reader can refer to Cavallo and D'Apuzzo (2009) for (]$−∞,+∞,+,≤$⁠, (]$0+∞,⋅,≤$ and (]$01,⊗,≤$⁠, to Ramík (2015a) for (]$−∞,+∞,+f,≤$ and to Cavallo (2025) for (]$1α,α,ʘα,≤$ and (]$−β,β,⊕β,≤$⁠.

By applying the well-known representation of the preferences in (1), if the DM expresses the preferences m₁₂ = 8 and m₂₃ = 8, used as examples in Section 1, then he/she can be consistent in both the Alo-group (]$0+∞,⋅,≤$ (i.e. by choosing m₁₃ = 64) and in the Alo-group (]$1α,α,ʘα,≤$ (e.g. by choosing m₁₃ = 8.97 for α = 9). In this paper, we choose to use PCMs defined over the α-multiplicative Alo-group (]$1α,α,ʘα,≤$⁠, with α = 9, because the DMs are often confused if asked to use unlimited intervals for expressing preferences and these PCMs have entries belonging to the limited interval ]$19,9$[ that are close to the values used in the Saaty scale.

We provide only the notions about α-multiplicative PCMs (i.e. PCMs defined over the Alo-group (]$1α,α,ʘα,≤$⁠) that are necessary for understanding STEPS 3–5 in the methodology described in the next section. For further details, the reader can refer to Cavallo and D'Apuzzo (2009) and Cavallo (2025).

The isomorphism between the α-multiplicative Alo-group (] $1α,α,ʘα,≤$ and the well known multiplicative Alo-group (]$0,+∞,⋅,≤$ is as follows (see Figure 2):

And the inverse isomorphism is as follows:

Thus, by applying the isomorphism in (7) to each entry of an α-multiplicative PCM $Mα=(mijα)n×n$⁠, we compute the isomorphic multiplicative PCM

With

And, by applying the isomorphism in (6), we compute the inconsistency index of $Mα=(mijα)n×n$⁠:

where I(M) is the inconsistency index of the multiplicative PCM $M=(mij)n×n$ in (8):

Note that.

1. I_α(M_α) belongs to the interval [1, α[;

2. I_α(M_α) = 1 if and only if M_α is consistent;

3. $Mα=(mijα)n×n$ is consistent if and only if $M=(mij)n×n$ is consistent.

The weighting vector of $Mα=(mijα)n×n$ is computed as follows:

That is, by applying the isomorphism in (6) to each component of the following vector:

where $m(mi)=mi1⋅mi2⋯minn$ is the geometric mean of the elements of the ith row of $M=(mij)n×n$ in (8).

Let {a₁, …, a_h, …, a_H} be a set of alternatives that represent all the tourism strategies defined to encourage tourism. Let {g₁, …, g_j, …g_J} be a set of criteria and {d₁, …, d_l, …, d_L} the set of DMs. Given these elements characterizing the decision problem, we perform the following steps.

• STEP 1. Identification of the decision matrix. A matrix $Z=(zhj)HxJ$ collects the evaluations expressed by all the DMs together for each alternative a_h with respect to each criterion g_j.

• STEP 2. Identification of the reference points. For each criterion g_j, the following P reference points are fixed (Miccoli and Ishizaka, 2017): r_j = (r_j1, …, r_jp, …r_jP).

• STEP 3. Construction of the PCMs. Fixed a value α > 1, PCMs over the Alo-group $Mα=1α,α,ʘα,≤$ are built from the answers obtained by submitting a questionnaire. In particular, for each DM d^l, with l ∈ {1, …, L}, the PCM $Al=(aikl)JxJ$ pairwise compares the criteria, and J PCMs $Bjl=(bikjl)PxP$⁠, with j ∈ {1, …, J}, compare the P reference points with respect to each criterion g_j. Then, by applying (8), the following multiplicative PCMs are computed:

• STEP 4. Measurement of the inconsistency of the PCMs. For each DM d^l, with l ∈ {1, …, L}, the inconsistency index of the PCMs $Al=(aikl)JxJ$ and $Bjl=(bikjl)PxP$ are computed by applying (9); that is:

In the case of high inconsistency, the DMs can revise their evaluations to obtain more tolerable inconsistent pairwise comparisons. The inconsistency index assumes a value in the interval [1, α[, and it is equal to 1 only in the case of consistency (see Section 3.2.1). In order to evaluate the goodness of the inconsistency indices, as done by Saaty, we required that our inconsistency indices are less than a threshold equal to 10% of the random inconsistency indices.

• STEP 5. Computation of the priorities. For each DM d^l, with l ∈ {1, …, L}, by applying (11), the following weighting vectors are computed:

These are the priority vectors of the criteria {g₁, …, g_j, …g_J} and the priority vectors of the reference points r_j = (r_j1, …, r_jp, …r_jP).

• STEP 6. Normalization and checking the monotonicity. We normalize the priority vectors w^l and $ujl$ so that the sum of their components is equal to 1, in accordance with standard practices in AHP-based models. The normalization is performed by dividing each component by the sum of all components in the respective vector, thus obtaining the normalized vectors $w¯l=(w¯1l,…,w¯jl…w¯Jl)$ and $u¯jl=(u¯j1l,…,u¯jpl,…,u¯jPl)$⁠. After that, as suggested by Abastante et al. (2019), we proceed to check the priorities w_α(B^jl) and the corresponding reference points r_j; that is, we check that there are no situations for which $rjp1>rjp2$ while $u¯jp1l≤u¯jp2l$ (Abastante et al., 2018; Corrente et al., 2016). If monotonicity is not met, the DMs can modify their assessments.

• STEP 7. Computation of local priorities. For each DM d_l, the local priorities of the alternatives with respect to each criterion are computed with the following linear interpolation formula (Abastante et al., 2019):

• STEP 8. Computation of global priorities for each DM. For each DM d_l, the global priority of the alternative a_h is computed in the following way:

• STEP 9. Computation of group global priorities. To compute the global priority of the group, we chose to use the AIP with the weighted geometric mean method (Ossadnik et al., 2016; Ramanathan and Ganesh, 1994; Forman and Peniwati, 1998):

where η_l represents the weight associated to the DM d_l; with η_l ≥ 0 and $∑l=1Lηl=1$⁠.

The framework of the model is shown in Figure 3. For STEPS 3–5, representing the integration of the Alo-group in the Parsimonious AHP, we implemented Algorithm 1; this runs for each α-multiplicative PCM given as the input.

• Require: n ≥ 2

• Require: α > 1

• Require: α-multiplicative PCM $A=(aij)nxn$

•   M = matrix(1, nrow = n, ncol = n); ⊳ Multiplicative PCM isomorphic to the α-multiplicative PCM A

•   for(i in 1:n) for (j in 1:n) M[i, j] = (1 + log(A[i, j], base = alpha))/(1 − log(A[i, j], base = alpha))

•   IM = 1; ⊳ Inconsistency index of M

•   for(i in 1:(n-2))for(j in (i+1):(n-1))for(k in (j+1):(n))IM = IM∗(max(M[i, j]∗M[j, k]/M[i, k], M[i, k]/(M[i, j]∗M[j, k])))∧(6/(n∗(n − 1)∗(n − 2)))

•   IA = alpha ∧ ((IM − 1)/(IM + 1)); ⊳ Inconsistency index of A

•   w < − matrix(nrow = 1, ncol = n); ⊳ Geometric mean vector of M

•   for(i in 1:n)w[1, i] = (prod(M[i, ])) ∧ (1/n)

•   wA = alpha ∧ ((w − 1)/(w + 1)); ⊳ Weighting vector of A

We emphasize that among the innovations presented in this work, this is the first time the AIP is used for a group Parsimonious AHP. The AIP procedure, as highlighted by Ossadnik et al. (2016), is more suitable than any other (such as the aggregation of individual judgments (AIJ)) for decisions of small and large groups in decision contexts with common goals; furthermore, the procedure respects the axioms of rationality. Note that AIP is not used for the rating aggregation in the PCMs.

We highlight that our proposal intentionally avoids consensus-building mechanisms because the proposed framework does not rely on group negotiation or iterative convergence. In decision-making contexts involving public administrations, imposing consensus can artificially homogenize preferences and increase the cognitive load for participants. Furthermore, it can introduce political or hierarchical biases, allowing individuals with greater influence to influence group decisions. This would contradict the goal of ensuring fair and impartial aggregation, especially in contexts with multiple DMs or where sensitive public decisions must be made, as envisaged in our real case study. In line with the Parsimonious AHP, our goal is to minimize the cognitive load on DMs and preserve the individuality of their judgments. For this reason, aggregation is performed ex post via AIP on the global priority of each DM; this avoids the need for iterative consensus rounds. Indeed, in our proposal, each DM provides independent and unconstrained judgments. The AIP is applied only to the global priorities obtained for each DM, which are combined using a weighted geometric mean as defined in formula (15). This ensures a rational and axiomatically sound combination of individual assessments. This is highlighted in the literature; indeed, Forman and Peniwati (1998) and Ossadnik et al. (2016) show that AIP aggregation can effectively replace consensus rounds when the goal is to combine the knowledge of DMs without forcing convergence. Similarly, Ishizaka and Labib (2011) emphasize that bypassing consensus mechanisms helps maintain decision-making neutrality and prevents the prevalence of bias in group contexts in which one member's influence could affect the collective outcomes.

The decision-making problem analysed in this work concerns a real case involving six municipal administrations in southern Italy, particularly in the hinterland of Campania. The municipal administrations are the Municipality of Paupisi (project leader) and the Municipalities of Ponte, San Lupo, San Lorenzo Maggiore, Torrecuso and Vitulano. These six territorial entities constitute a network of municipalities that occupies the geographical heart of the province of Benevento (Figure 4). In this region, which has a total surface area of approximately 121.5 km², there are just over 12.200 inhabitants, with a population density of 100.7 inhabitants/km². The administrations have focused on inter-municipal cooperation to increase the visibility and attractiveness of the territory as a whole and to overcome the limitations of the individual local entities. These municipal administrations, together, participated in a project in a national programme (FSC, 2021/2027) that encourages and recognizes tourism as a strategic sector. The local project was the Programme of cultural, naturalistic and food and wine tourist itineraries for the tourism promotion of Campania, period June 2025 – December 2025, as part of the Festival of flavours and street artists project which has among its objectives, the valorization of local resources, the promotion of sustainable tourism and the creation of new economic and employment opportunities.

For each municipal administration, we identified a DM who had a strategic role in its choices. The set {d₁, d₂, d₃, d₄, d₅, d₆} represents our set of DMs, who correspond, respectively, to the municipalities of Paupisi (d₁), Ponte (d₂), San Lupo (d₃), San Lorenzo Maggiore (d₄), Torrecuso (d₅) and Vitulano (d₆).

A description of the DMs (role, training and expertises) is provided in Table 2.

The DMs identified a set {a₁, …, a₆₈} of alternatives to be evaluated with respect to the set {g₁, g₂, g₃, g₄, g₅, g₆} of criteria. The descriptions of the criteria are shown in Table 3 and the descriptions of the alternatives in Table 4.

In this subsection, we show the application of the methodology proposed in Section 4 to our case study, with α = 9, L = 6, H = 68, J = 6 and P = 6. Thus, the following nine steps are performed.

• STEP 1. All the DMs together provided the decision matrix $Z=(zhj)68x6$ reported in Table 5.

• STEP 2. The DMs, by agreement, decided that the reference points, which are reported in Table 6, should be obtained by equidistantly dividing the reference intervals [min(z_j), max(z_j)], with z_j being the jth column of the decision matrix $Z=(zhj)68x6$ provided from STEP 1.

• STEP 3. Each DM d_l, with l ∈ {1, …, 6}, provided the PCM $Al=(aikl)6x6$ and the PCMs $Bjl=(bikjl)6x6$⁠, with j ∈ {1, …, 6}.

The DMs’ preferences were elicited by giving them a questionnaire with explanations and examples for how to fill it in.

The entries of each PCM, belonging to the interval ] $19,9$[, were provided by the DMs in agreement with the representation of the preferences in (1), the DMs were asked to provide values greater than or equal to 1 (i.e. strict preference and indifference), and then the reciprocal elements were computed by applying the reciprocity in (1). An extract of the criteria questionnaire is shown in Figure A.1 in the Appendix. Thus, the DMs provided the following PCMs for the criteria:

And the following PCMs for the reference points:

All the PCMs are reported in the Appendix; as an example, we report here the PCM $A2=(aik2)6x6$ for the criteria:

Then, for all j, l ∈ {1, …, 6}, the following multiplicative PCMs were computed:

As an example, we report the multiplicative PCM $ψ9−1(A2)$⁠: