A Declarative Modeling Framework for Intuitive Multiple Criteria Decision Analysis in a Visual Semantic Urban Planning Environment

Abstract

The magnitude and multiformity of recorded and readily available urban data from independent sensors and mobile user devices have paved new possibilities for city environment planning and evolution. In previous works, a visual semantic decision support system for urban planning was presented, and the capability of machine learning approaches, ranging from random forests to graph-based convolutional neural networks, to infer preferable directions for future development was explored, extrapolating upon previous opted locations and selected alternatives for publicly commercial services. In this work, the anterior decision-making process leading to establishment choices is addressed with the same sets of criteria and samples within the same environment. A Declarative Modeling shell is proposed, and Multiple Criteria Decision Analysis processes are adopted to encompass the DM’s/DMs’ rationale, solely relying on methodologies close to human intuition. To this end, outranking representatives from the PROMETHEE family as well as weighted sum approaches are employed, fueled by the interpretation of the declarative description of decision parameters on behalf of the DM(s), exploring the ability to achieve classifications in a straight synthetic manner, i.e., in the absence of previous knowledge, thus exhibiting the potential of decision analysis methodologies, enhanced by Declarative Modeling, to be used as powerful intuitive tools in similar paradigm contexts.

1. Introduction

Declarative Modeling has attempted to alleviate the burden of imperative procedural design, repositioning the modeling process closer to human intuition [1], at the cost of increased computational complexity [2]. The core concept was the ability of the designer to express ideas and requirements in a declarative manner, in less concrete terms than the ones required by a strict geometric model or a detailed imperative description. The abstract elements of the declarative description allowed for more than one interpretation, which introduced increased processing for the generation and evaluation of the potential outcomes. On the other hand, this multiplicity would lead to alternative interpretations of the declarative description, which might be of interest yet not originally conceived by the designer. Initially applied for scene modeling in the area of computer graphics [3,4], it was subsequently adopted in a wide range of areas with the attempt to bring its benefits to relevant stages of business process modeling [5], application deployment [6], systems simulations [7], and curriculum modeling [8], to name a few. In the programming realm, declarative languages, e.g., SQL, have provided the description of the information required (“what”) without requiring the description of the algorithmic procedural manner to be obtained from the underlying data tables (“how”) [9].

In the current work, in the context of Decision Analysis [10], the role of the designer is assumed by the Decision Maker (DM) and the focus is on capturing the DM’s preferences in the context under examination. Multiple Criteria Decision Analysis (MCDA), also termed Multiple Criteria Decision Making (MCDM) or Multiple Attribute Decision Making (MADM) [11,12], offers a wide variety of methodologies for decision making under diverse criteria. These methodologies range from pure weighted sum approaches, relying on utility functions and requiring or leading to the definition of a set of weights on behalf of the DM for the relevant computations, to convoluted outranking methods often demanding the definitions of thresholds and other parameters expressing the DM’s outlook on the alternatives, their characteristics, and relevant comparison criteria [13]. Prominent representatives of the former end of the spectrum of MCDA methodologies, representing the American school of thought [12], include the Multi-Attribute Utility Theory (MAUT) [13,14], relying on weight assignment to attributes and a per-attribute evaluation of alternatives, aggregated as a linear function yielding an overall performance index per alternative. In the same category, the Analytic Hierarchy Process (AHP) [13,15] extends this notion with an attribute hierarchy, allowing for weight assignment to attributes per hierarchy level, as well as to alternatives against the lowest level attributes, for their per-attribute evaluation. The attributes’ weights and per-attribute evaluation of alternatives are elicited exclusively from pair-wise comparisons performed by the DM at each level of the hierarchy. On the other end of the MCDA spectrum reside the outranking methods, including the ELECTRE (ÉLimination Et Choix Traduisant la REalité) [13,16] and PROMETHEE (PReference Ranking Organization METHod for Enriched Evaluation) [13,17] families as prominent representatives of the European school of thought [12]. The majority of the methods in these families allow for non-comparability between alternatives, thus relaxing the total or partial order implied by the weighted sum and the subsequent mandated preference transitivity. This choice potentially enhances the realism of the preference model at the cost of the requirement for additional DM’s input.

Despite this diversity, all MCDA approaches build upon a core set of fundamental assumptions: (a) a set of alternatives to be evaluated by the DM, (b) a set of common dimensional attributes representing each alternative, and (c) a preference model for the criteria of the DM’s evaluation of the alternatives against these given attributes representing a valid preference structure [18]. In the current work, Declarative Modeling is employed at this stage of the decision-making process. The aim is to provide an intuitive manner for the DM to express and subsequently apply the aspects of their personal interpretation of the variables involved in the process and required by the method at hand. To this end, the taxonomy proposed by Watrobski et al. [12] is adopted, isolating the characteristics of employed MCDA methodologies to be expressed in a declarative manner in the context of the current work.

Relevant works include a Declarative Decision Support Framework, which has been proposed for scheduling groups of orders [19], making it possible to formulate the scheduling problem through a series of general, specific, logical, etc., questions, transforming the input into a set of constraints to be satisfied. A similar approach relies on a set of questions for a Declarative Decision Support Platform allowing the implementation of relevant decision-making models and implied constraints focused on sustainable supply chain problems [20]. A Declarative Recommender System for cloud service selection has also been proposed, aiming to map a user’s specified application requirements to available cloud service configurations [21]. The Decision EXpert—DEX [22]—represents a multiple criteria decision modeling method combining hierarchical structured criteria with symbolic attribute values, requiring the definition of a set of decision rules instead of weights, on behalf of the DM, to carry out performance aggregation of alternatives for sorting or classification among predefined categories.

In the area of decision analysis for urban planning, an approach has explored and assessed the integration of DEX for Architecture and Urban Design competitions in regard to its impact on the transparency, objectivity, and efficiency of the decision-making process [23]. In the same context, an effort towards resilient open urban spaces has commenced from citizens’ needs, collected through questionnaires, and they were used as the basis for the construction of alternative urban regeneration scenarios [24]. The latter have been subsequently evaluated through simulations in ENVI-met, a 3D modeling software for urban cooling and climate adaptive planning [25]. An alternative hybrid method, aiming to achieve the generation and evaluation of sustainable future scenarios for an underdeveloped area in Northern Italy, comprises a sequence of phases starting with a Stakeholder Analysis, followed by STEEP and SWOT analyses as well as Scenario Building, and it is eventually evaluated by a weighted-sum MCDA mechanism [26]. A review study investigated the efforts connecting MCDA mechanisms with Building Information Modeling (BIM), highlighting gaps in the established integration process as well as suggesting strategies to improve this synergy [27]. A critical review on the application of MCDA methods for green spaces and nature-based solutions highlights the benefits of adopting MCDA methods in the specific context but also identifies the problems of the imbalanced distribution of the criteria used in their adoption in the literature, the locality of the relevant case studies, and the lack of adaptability to alternative contexts [28].

In the current work, the Declarative Modeling methodology is employed for the generation of alternative instances of well-established MCDA mechanisms to be subsequently employed in the decision-making process. The intuitive input allowed by the Declarative Modeling methodology, in combination with the wide scope of the MCDA mechanisms employed, ensures the pertinence of the concept independent of the domain of application. The adaptation and application of the proposed framework to any field of interest requires no additional overhead compared to the adaptation and application of the same MCDA method included in the framework for the same field. Based on the discussion above, the adoption of the Declarative Modeling methodology for the intuitive definition of alternative instances of established MCDA mechanisms to be applied to a decision problem, as proposed in the current work, is a novel point in the relevant literature.

The focus is set on a popular representative MCDA methodology requiring varied inputs on behalf of the DM. Outranking methods PROMETHEE I and II, with increased DM participation in the form of decision variables and criterion functions [13], serve as the main testbed for the Declarative Modeling of encompassed preferences. A weighted sum approach, relying on a subset of PROMETHEE’s required input, representing the utility function methodologies, is also employed. Illustrative examples are presented to exhibit the details of the proposed methodology and highlight its benefits.

The dataset and corresponding characteristics upon which the combined proposed methodology will be applied originate from the domain of urban planning and have already served as the basis for machine learning approaches for decision support [29,30]. This allows for the positioning of the proposed method in a wider context of an existing urban planning environment, relying on the same samples and dimensional attributes. From this perspective, within the current context, although previous knowledge regarding the DM and their corresponding preferences is not granted, positive representatives of the dataset serve as the baseline DM’s outcome and offer a comparison basis. In addition, negative representatives approved by the method stand as eligible candidates for the decision problem addressed, further underlining the benefit from the connection of the proposed method with an already existing urban planning environment.

Section 2 discusses and highlights the key points of the contribution of the current work. Section 3 presents the methodology of Declarative Modeling as well as the employed PROMETHEE I/II and weighted sum mechanisms, representing the MCDA approaches of outranking methods and utility functions, respectively. Section 4 describes in detail the proposed methodology in the context of PROMETHEE I/II, including weight elicitation, also exploited by the weighted sum mechanism. Section 5 presents the experimental organization and execution. In Section 6, the experimental results are discussed, revealing the benefits of the proposed method and identifying its limitations. Section 7 presents the conclusions of the current work.

2. Paper Contributions

Multiple Criteria Decision Analysis methodologies require the definition of a number of parameters on behalf of the DM in order to be employed in any given context. These parameters are concrete values representing weights of importance, thresholds for veto activation, for strong preference or for indifference relations, etc., depending on the approach adopted. In the current work, Declarative Modeling is employed with the aim of providing the DM with a declarative shell, allowing them to express their view of the parameters’ values in an intuitive manner. The rationale is to alleviate the DM from the exact value definition process, allowing their indirect discovery through abstract terms and expressions. Nevertheless, the intricacies and particularities of the domain of application are taken into consideration towards the limitation of the otherwise vast range of alternatives.

In order to achieve this, the focus is set on representative methodologies of the two main directions of MCDA bearing the characteristic features of each school of thought [12]: the European school, represented by outranking methodologies, where preference is not necessarily transitive and a wider range of relations between alternatives is covered, including incomparability, and the American school, represented by utility function methodologies, where alternatives are mapped to representative weighted sum indices, thus offering a global partial order of preference for the entire alternatives’ population, eliminating incomparability and implying transitivity. Incomparability as well as lack of preference transitivity often arise in realistic decision scenarios [13]; thus, their support represents a justifiable trade-off. The latter lies between increased fidelity to the DM’s input, at the cost of extra required input to address them if necessary, against increased efficiency in the comparison of large numbers of alternatives at the cost of reduced realism.

For each method’s required concrete input, on behalf of the DM, a set of declarative objects, relations, and properties are proposed, closer to human intuition and allowing alternative interpretations. The latter is expected to lead to alternative instances of each methodology applied, thus possibly yielding differentiated results regarding the considered alternatives’ ordering or grouping. This is anticipated to be the main benefit of the current approach since it may reveal implications of the DM’s declarative input that might escape their consideration in the traditional procedure of concrete value input.

The full scope of the contributions of the current work can be summarized as follows:

• A Declarative Modeling shell is proposed, allowing the DM to parametrize the MCDA mechanism of choice in an intuitive manner through a Declarative Description;
• Alternative instances of the employed MCDA mechanism are generated, all compliant with the DM’s intuitive input;
• For each instance, the best-performing samples are selected as the set of top-class representatives;
• The potential of discovery, among the top-class representatives of alternative instances, of eligible samples that might have escaped the DM’s attention through traditional concrete-valued input to the employed MCDA mechanism is one of the main benefits of Declarative Modeling;
• Based on top-class representatives of various instances, the DM has the ability to gain insights and refine the Declarative Description to be closer to their preferences, restarting the Declarative Modeling cycle;
• The intuitive input allowed by the Declarative Modeling methodology and the wide scope of the MCDA mechanisms parametrized through it ensure the applicability and adaptability of the proposed framework to an extensive range of domains with zero overhead relative to the application of the same MCDA mechanisms sans the declarative shell.
• The entire scope of the proposed framework is applied as a use case in an existing visual semantic urban planning environment for PROMETHEE I/II and weighted sum methodologies, representing outranking and utility function approaches, respectively.

3. Materials and Methods

3.1. Declarative Modeling Overview

The Declarative Modeling methodology comprises three distinct interconnected stages. Each of these stages represents a fundamental step in the transition from the stakeholder’s intuition to the desired outcome. Relying on the latter, the stakeholder may conclude the process or reconsider their initial approach, revising the initial input based on acquired insight and feedback, thus restarting the cycle. An overview of the Declarative Modeling cycle is shown in Figure 1.

The first stage is the declarative description synthesis. Depending on the context, the stakeholder amalgamates ideas, demands, preferences, and requirements, either personal or originating from the environment they represent. The declarative description is typically synthesized upon a repertoire of declarative objects, relations, and properties relevant to the domain of application. These domain-specific objects, relations, and properties serve as the balance between the ability to express an intuitive description in a declarative manner while conforming with the intricacies of the domain of application. The outcome of this stage is the declarative description, serving as the input to the subsequent stage of solution generation.

The stage of solution generation commences with the translation of the declarative description constituents into a series of alternative interpretations. This may be carried out with a set of constraints input to a solution generation mechanism responsible for producing interpretations of the description in the form of entities of the domain of application, which are compliant with the implied stakeholder’s restrictions and requirements. This set of solutions is then offered as input to the final stage of evaluation and understanding of the implications of the declarative description.

At the final stage of the Declarative Modeling methodology, the stakeholder is able to inspect and evaluate the produced output. This stage compares and contrasts the initial abstract description with the produced output, offering insights into the former and its implications. The initial description may have allowed a wider than desired range of solutions to be obtained. On the other hand, a traditional advantage of Declarative Modeling is the opportunity to encounter alternative, innovative solutions, which might have escaped the stakeholder’s consideration despite being compliant with the input description, conveying their preferences and requirements in a novel manner.

3.2. Representative MCDA Approach—PROMETHEE I/II

Multicriteria Decision Analysis (MCDA) represents a domain of methodologies employed in a wide range of enterprises and organizations for decision making on behalf of one or more stakeholders acting as Decision Makers (DMs) [12]. The corresponding field can be considered as part of the wider area of Decision Analysis that was established during the 1960s as the formal procedure for the analysis of decision problems [13]. Recent surveys reveal the lasting relevance and wide penetration of the domain’s methodologies [11,12]. In the current work, members from both main directions of the area are represented, i.e., the outranking methodologies and the utility function approaches.

One of the prominent representatives of the outranking methodologies is the PROMETHEE family proposed in the early 1980s that is still adopted today in numerous domains of application [13]. Its variations PROMETHEE I and II are popular due to the former’s support for incomparability and the absence of implied transitivity, which is often the case in real-world preference situations, whereas the latter enhances the former’s calculations to yield a partial order of the entire alternatives’ population if this is desired. The methods rely on pair-wise comparisons of the alternatives upon the domain criteria, taking into consideration the DM’s interpretation of the difference in the alternatives’ performance in the form of generalized criteria functions. The latter are responsible for mapping the difference in performance between alternatives for each specific criterion to a normalized value between 0 and 1 according to the DM’s interpretation. The methodology combines these values, based on weights, also assigned by the DM, for each criterion into aggregated preference indices for each ordered pair of alternatives which, in turn, contribute to the overall positive and negative outranking flows for each alternative.

Formally, let A be the set of alternatives, C the set of criteria, and $g_c$ the function defined in Equation (1), mapping each alternative to a value with respect to criterion c:

$$g_c:A⟶\mathbb{R},c\in C$$

(1)

The difference in performance for any two alternatives with respect to a specific criterion c is expressed by d_c, as defined in Equations (2) and (3):

$$d_c:A\times A\to \mathbb{R}, c\in C$$

(2)

$$d_c\left(a,b\right)=g_c\left(a\right)-g_c\left(b\right), a\in A,b\in A, c\in C$$

(3)

The generalized criterion function P, defined in Equation (4), maps this difference to the normalized range between 0 and 1 according to the DM’s interpretation:

$$P:\mathbb{R}\to \left[0,1\right]$$

(4)

Usually, for clarity, the composite function represented by the two aforementioned steps is symbolized as a single function F_c, defined in Equations (5) and (6), mapping an ordered pair of alternatives directly to the normalized value:

$$F_c:A\times A\to \left[0,1\right], c\in C$$

(5)

$$F_c\left(a,b\right)=P\left(d_c\left(a,b\right)\right), a\in A,b\in A, c\in C$$

(6)

The generalized criterion functions, proposed by the method, to be used for P appear in

Table 1. PROMETHEE generalized criterion functions (adapted from [13]).

Function Graph	Function Definition	DM-Defined Parameters
	$P\left(d\right)=\left\{\begin{array}{c}0, d\le 0 \\ 1, d>0 \end{array}\right.$	-
	$P\left(d\right)=\left\{\begin{array}{c}0, d\le p \\ 1, d>p \end{array}\right.$	p
	$P\left(d\right)=\left\{\begin{array}{c}0, d\le 0 \\ {\displaystyle \frac{d}{p}}, 0<d\le p\\ 1, d>p \end{array}\right.$	p
	$P\left(d\right)=\left\{\begin{array}{c}0, d\le q \\ {\displaystyle \frac{1}{2}}, q<d\le p\\ 1, d>p \end{array}\right.$	q, p
	$P\left(d\right)=\left\{\begin{array}{c}0, d\le q \\ {\displaystyle \frac{d-q}{p-q}}, q<d\le p\\ 1, d>p\end{array}\right.$	q, p
	$P\left(d\right)=\left\{\begin{array}{c}0, d\le 0 \\ 1-e^{-{\displaystyle \frac{d^2}{{2s}^2}}}, d>0 \end{array}\right.$	s

, adapted from [13].

One of the DM’s responsibilities, in view of applying PROMETHEE I or II, is selecting the desired generalized criterion function and the corresponding parameters q, p, and/or s to control its behavior. Parameter q represents the threshold from indifference to weak preference, while p stands for the limit between weak and strong preference, respectively. In the case of the gaussian criterion, regarding the inflection point s, it is recommended for the DM to initially define desired q and p values and then assign the s value between these two limits, in which case an s value close to q makes a stronger preference for small deviations, while an s value closer to p softens the effect [13]. The latter is a typical example where the DM’s required input will benefit from the declarative enhancement, as will become evident in Section 4.4, covering the declarative function form and relevant parameters’ abstraction.

For the next stage of the method, the DM has to define the criteria weights to be used for the aggregation of the individual results for each criterion. Formally, given the set of normalized criteria weights provided by the DM,

$$W=\left\{w_1,w_2,\cdots ,w_{\left|C\right|}\right\}$$

(7)

For each pair of alternatives, a and b, the aggregated preference indices, are defined in Equations (8) and (9):

$$\mathsf{\pi }\left(a,b\right)=\sum _{i=1}^{\left|C\right|}{w_{i}F}_i\left(a,b\right),w\in W,a\in A,b\in A$$

(8)

$$\mathsf{\pi }\left(b,a\right)=\sum _{i=1}^{\left|C\right|}{w_{i}F}_i\left(b,a\right),b\in A,a\in A$$

(9)

These indices are often depicted using a complete weighted directed graph, as shown in Figure 2, for an example decision problem with four alternatives {a,b,c,d}.

Given these indices, the positive and negative outranking flows for each alternative a may be calculated against all other alternatives in A according to Equations (10) and (11), respectively:

$${\phi }^{+}\left(\alpha \right)={\displaystyle \frac{1}{n-1}}\sum _{x\in A}\pi \left(a,x\right), a\in A, n=\left|A\right|$$

(10)

$${\phi }^{-}\left(\alpha \right)={\displaystyle \frac{1}{n-1}}\sum _{x\in A}\pi \left(x,a\right), a\in A, n=\left|A\right|$$

(11)

The final stage of PROMETHEE I involves the combination of the positive and negative outranking flows to compose a partial ranking supporting preference P, indifference I, or incomparability R between any two alternatives (α and b), as defined in Equations (12)–(14):

$$a P b\iff \left\{\begin{array}{c}{\phi }^{+}\left(a\right)>{\phi }^{+}\left(b\right)\wedge {\phi }^{-}\left(a\right)<{\phi }^{-}\left(b\right)\\ \begin{array}{c}\vee \\ {\phi }^{+}\left(a\right)={\phi }^{+}\left(b\right)\wedge {\phi }^{-}\left(a\right)<{\phi }^{-}\left(b\right)\end{array}\\ \begin{array}{c}\vee \\ {\phi }^{+}\left(a\right)>{\phi }^{+}\left(b\right)\wedge {\phi }^{-}\left(a\right)={\phi }^{-}\left(b\right)\end{array}\end{array},a\in A,b\in A\right.$$

(12)

$$a I b⟺{\phi }^{+}\left(a\right)={\phi }^{+}\left(b\right)\wedge {\phi }^{-}\left(a\right)={\phi }^{-}\left(b\right),a\in A,b\in A$$

(13)

$$a R b\iff \left\{\begin{array}{c}{\phi }^{+}\left(a\right)>{\phi }^{+}\left(b\right)\wedge {\phi }^{-}\left(a\right)>{\phi }^{-}\left(b\right)\\ \begin{array}{c}\vee \\ {\phi }^{+}\left(a\right)<{\phi }^{+}\left(b\right)\wedge {\phi }^{-}\left(a\right)<{\phi }^{-}\left(b\right)\end{array}\end{array},\right.a\in A,b\in A$$

(14)

PROMETHEE I halts at this stage, leaving the final outcome at the discretion of the DM. It may be the case that more than one alternative appears as preferred over the rest of the population. Moreover, some may not participate in any preference relation presenting non-comparability with the rest, and, possibly, indifference among them. A top-class T can be defined as the set of alternatives not being out-preferred by any other and being preferred over at least one. Formally, T, with respect to set A of all alternatives, is defined in Equation (15) as follows:

$$T=\left\{t\in A|\forall t\in T,\nexists a\in A:\alpha P t\wedge \exists b\in A:t P b\right\}$$

(15)

PROMETHEE II proceeds to aggregate the two flows for each alternative, reaching an overall outranking net flow, thus implying a total order, i.e., among all available alternatives:

$$\phi \left(\alpha \right)={\phi }^{+}\left(\alpha \right)-{\phi }^{-}\left(\alpha \right), a\in A$$

(16)

Due to its definition, the net flow exhibits the properties shown in Equations (17) and (18):

$$\forall a\in A:-1\le \phi \left(\alpha \right)\le 1$$

(17)

$$\sum _{a\in A}\phi \left(a\right)=0$$

(18)

A total order is directly inferred by the representation of each alternative by a single number, thus eliminating incomparability and implying transitivity. The preference relations are formally defined in Equations (19) and (20):

$$a P b\iff \phi \left(a\right)>\phi \left(b\right),a\in A,b\in A$$

(19)

$$a I b\iff b I a \iff \phi \left(a\right)=\phi \left(b\right),a\in A,b\in A$$

(20)

Due to the law of trichotomy for numbers, all pairs of alternatives are now related either by indifference or by preference in one of the two directions, as formally shown in Equation (21):

$$\begin{gathered} \forall a,b\in A: \\ \left[\left(a P b\right)\wedge \neg \left(b P a\right)\wedge \neg \left(a I b\right)\right]\vee \left[\left(b P a\right)\wedge \neg \left(a P b\right)\wedge \neg \left(a I b\right)\right]\vee \left[\left(a I b\right)\wedge \neg \left(a P b\right)\wedge \neg \left(b P a\right)\right] \end{gathered}$$

(21)

3.3. Representative MCDA Approach—Utility Function

Utility function methodologies incorporate the DM’s preferences as a set of weights representing their assessment of the importance of each one of the common criteria for the evaluation of alternatives. These weights are subsequently applied to the performance index of each alternative per criterion, aggregating the alternative’s overall performance index as a weighted sum. The set of weights may be directly suggested by the DM or may be extracted using an appropriate mechanism [13,31,32,33], whereas the individual alternative’s performance per attribute originates from quantities measured in the real world or from the DM’s interpretation of the alternative’s performance against each criterion.

Formally, similar to Section 3.2, let A be the set of alternatives; C the set of criteria; $g_c$ the function mapping each alternative to a value with respect to criterion c, as defined in Equation (1); and W the set of normalized criteria weights, as defined in Equation (7). In order to apply the normalized weights to each alternative’s criteria performance metrics, the latter are mapped to the normalized range between 0 and 1, as defined in Equation (22) for any criterion c:

$$k_c:\mathbb{R}\to \left[\mathrm{0,1}\right],c\in C$$

(22)

For clarity, the composite function mapping each alternative from set A to its normalized performance is symbolized as a single function, as shown in Equations (23) and (24):

$$v_c:A\to \left[\mathrm{0,1}\right],c\in C$$

(23)

$$v_c\left(a\right)=k_c\left(g_c\left(a\right)\right),a\in A,c\in C$$

(24)

The aggregated performance u(a) for each alternative is calculated upon these normalized values, taking advantage of the corresponding normalized weights, as shown in Equation (25):

$$u\left(a\right)=\sum _{i=1}^{\left|C\right|}{w_{i}v}_i\left(a\right),w\in W,a\in A$$

(25)

4. Declarative PROMETHEE I/II

4.1. Decision Parameters

In the following, it is assumed that, for the domain of application, the alternatives are well defined in terms of a set of dimensional attributes representing the criteria. The performance of each alternative, concerning each attribute, is expressed as a numerical or categorical value. In the former case, it is assumed that the DM has defined the direction of preference and the extrema to be adopted for their normalization, whereas in the latter case, they have provided the necessary interpretation for their numerical mapping and subsequent normalization. Without the loss of generality, it is assumed that normalization has already been performed and, within the context of any specific attribute, a higher normalized value represents a higher performance of the alternative with respect to the attribute.

The DM’s intervention parameters to the PROMETHEE I/II methodology, as presented in detail in Section 3.2, are summarized in

Table 2. PROMETHEE I/II and the DM’s intervention parameters (function graphs are detailed in Table 1).

Decision Parameter	Decision Methodology Entity	DM’s Action
Criteria Weights	$W=\{w_1,w_2,\cdots ,w_{\left|C\right|}\}$	Definition of values
Generalized Criterion Function		Choice of function
Indifference/Weak Preference Threshold (dependent on function choice)	q	Definition of value (dependent on function choice)
Weak/Strong Preference Threshold (dependent on function choice)	pws	Definition of value (dependent on function choice)
Indifference/Preference Threshold (dependent on function choice)	pip	Definition of value (dependent on function choice)
Inflection Point of Gaussian Function (dependent on function choice)	s	Definition of value (dependent on function choice)

. Due to the duality of the role of parameter p, depending on the generalized criterion function opted by the DM, it was chosen to be represented as two separate entities to conform with the declarative description mapping that will follow. Nevertheless, the role of p in PROMETHEE will be assumed by the appropriate entity depending on the function selected, thus leaving the functionality of the methodology intact. The current section presents the core of the declarative description of these parameters and the relevant declarative entities to be used in the construction of the declarative description in the form of declarative objects, relations, and properties.

4.2. Declarative Mapping

The goal is to aid the DM in defining the aforementioned parameters in an intuitive manner. To that end, the construction of the declarative description may be considered as the response to a series of questions, focused on the requirements and desired outcome for the methodology adopted, in the form of relevant decision variables. In order to achieve this, each parameter is mapped to a set of declarative entities aiming to capture the notions the DM is trying to convey. Because the set of parameters is practically the communication between the DM and the MCDA methodology adopted, the declarative description serves as a shell aiming to provide a friendlier interface for this communication. The proposed declarative entities offer one or more alternative options for the expression of the DM’s ideas regarding each parameter. Declarative objects represent the fundamental representation of the corresponding parameters, and their declarative properties typically provide a desired range instead of a single value. This fact offers the aforementioned possibility of discovering new interpretations of interest at the cost of increased complexity. Declarative relations allow the DM to interconnect parameters, thus offering the ability to intuitively restrict the interpreted range of declaratively defined values, hence limiting the computational cost while expressing their understanding of the problem parameters. Finally, declarative modifiers may further narrow areas of interest in the suggested ranges and interconnections, providing additional control of the implied computational cost.

Table 3. Declarative representation options for methodology parameters.

Methodology Parameter	Declarative Object	Declarative Relation(s)	Declarative Property/Ies
criterion weight	criterion	muchMoreImportant moreImportant lessImportant muchLessImportant equallyImportant	crucial major average minor negligible (hierarchical)
generalized criterion function	comparativePerformanceInterpretation		stepped linear non-linear weak&strong single strict
q (indifference to weak preference threshold)	comparativeIWThreshold		early late wide narrow
pws (weak to strong preference threshold)	comparativeWSThreshold		early late wide narrow
pip (indifference to preference threshold)	comparativeIPThreshold		early late wide narrow
s (inflection point of Gaussian function)			strict

summarizes the proposed declarative entities with respect to the methodology parameters.

4.3. Declarative Weight Abstraction

Weight assignment has been extensively studied in the literature in the context of decision analysis [13,31,32,33]. Regarding the DM’s cognitive intervention to the process, methods range from minimal input in the form of a simple ranking of the criteria [32,33] to weighted pair-wise comparisons for every pair, required by AHP [13,15]. In the current context, the DM is offered a series of declarative options that have to be interpreted into one or more sets of weight assignments. In the following, the alternative scenarios and subsequent interpretation of this part of the declarative description are explored.

For each declarative object criterion, the DM may assign a declarative property or use it in a binary declarative relation to abstractly compare it with another criterion. A lack of preference transitivity, despite appearing as a logical paradox, often arises in realistic decision scenarios [13], even based on simple rules for pair-wise choice. A minimal example is the selection among three alternatives, a(1,2,3), b(2,3,1), and c(3,1,2), where the numbers in parentheses represent each alternative’s attribute values, under the simple rule that between any two alternatives, the one with the larger number of higher attribute values is preferred. According to this rule c P b, i.e. c is preferred over b, due to the first and third attribute; b P a due to the first and second attribute; a P c due to the second and third attribute, thus closing the paradoxical preference cycle.

The aim is to offer a declarative model with the ability to allow cyclic preferences, thus maintaining the capacity to more accurately capture the DM’s preferences. Nevertheless, it is also aimed to support a straightforward operation implying non-cyclic preferences if desired by the DM. In order to serve this twofold aim, it was selected to construct the declarative description of the criteria weights in two stages. During the first stage, the DM is offered the option to characterize a criterion object through a declarative property. Declarative properties obey an implied importance hierarchy and, by assigning such a property to a criterion, it is implied to be at least as important as every other criterion of the same or lower hierarchical property in the population. This interpretation allows for the intuitive straightforward characterization of criteria however it restricts at a fast rate (without immediately eliminating) the potential to form cycles. This is because the assignment of such a property to a criterion implicitly relates this criterion to all other characterized criteria. The latter fact rapidly reduces the non-related pairs of criteria that allow cycles. At the end of this first stage, all declarative properties are horizontally transformed into the declarative relations implied by the hierarchy. For example, a major criterion is represented as a series of moreImportant relations against average criteria, notablyMoreImportant relations against minor criteria, and considerablyMoreImportant relations against negligible criteria, as well as a series of equallyImportant relations against major criteria and lessImportant relations against crucial criteria. The complete mapping of every declarative property is shown in

Table 4. Declarative properties and their translation into declarative relations.

DM-Assigned Declarative Property	Translation into Declarative Relations (Row Criterion $\stackrel{\mathit{r}\mathit{e}\mathit{l}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}{\to }$ Column Criterion)
	Crucial	Major	Average	Minor	Negligible
crucial	equally Important	more Important	notablyMore Important	considerably MoreImportant	vastlyMore Important
major	lessImportant	equally Important	moreImportant	notablyMore Important	considerably MoreImportant
average	notablyLess Important	lessImportant	equally Important	moreImportant	notablyMore Important
minor	considerably LessImportant	notablyLess Important	lessImportant	equally Important	moreImportant
negligible	vastlyLess Important	considerablyLess Important	notablyLess Important	less Important	equally Important

.

In the second stage, two distinct criterionImportance declarative objects may be associated through a declarative relation by the DM, implying a relation between their weights. Nevertheless, declarative relations between pairs of criteria respect already existing declarative relations from the previous stage. Hence, it is not possible to deem a crucial criterion lessImportant than a minor one since there already exists the implied relation between the two as a result of the previous stage. It is worth noting that, in the absence of declarative properties from the previous stage for a group of criteria, the DM is still able to express cyclic preferences at this stage, if desired. At the end of this second stage, pairs or criteria not related by the DM directly, during the second stage, or indirectly, through declarative properties, are related as equallyImportant.

Once the DM has concluded this second stage, the part of the declarative description pertaining to the criteria weights is final. The respective declarative entities then undergo a series of computational steps towards their interpretation as a set of weights conforming to the description. In particular, the declarative relations are interpreted as ratios of importance according to pre-defined ranges of values, and an approach similar to that employed by AHP for weight calculation is used at any hierarchy level [13,15]. In particular, a $\left|C\right|\times \left|C\right|$ matrix is completed with the pair-wise ratios implied by the declarative relations. The processing of alternative versions of the matrix, ensuing from the alternative ratios within the pre-defined ranges implied by the corresponding relations, leads to alternative sets of weights to be used by PROMETHEE. Τhe corresponding ranges for the superiority declarative relations are summarized in

Table 5. Declarative relations and implied alternative ratios.

Declarative Relation Modifier	Implied Alternative Ratios
vastlyMoreImportant	Scenario A: 9/1, Scenario B: 7/1
considerablyMoreImportant	Scenario A: 7/1, Scenario B: 5/1
notablyMoreImportant	Scenario A: 5/1, Scenario B: 3/1
moreImportant	Scenario A: 3/1, Scenario B: 2/1
equallyImportant	Scenario A and B: 1/1

—the inverse ratios are used for the respective inferiority relations. In the current work a minimal computational footprint was opted, but the scope and granularity of the parameters’ interpretation is subject to the desired computational requirements and may be widened and deepened, respectively.

4.4. Declarative Function Form and Parameter Abstraction

The generalized criteria functions employed by PROMETHEE exhibit specific characteristics pertaining to their behavior. The latter is controlled by their respective parameters. At this stage of the declarative description synthesis, the aim is to aid the DM in opting for these characteristics and specifying the corresponding parameters through a set of declarative entities closer to their intuition. Concentrating on a subset of these functions’ characteristics, three distinct groups may be isolated. The first group contains Types 1, 2, and 4, which are stepped functions, offering a few distinct output values in the normalized output range, each assigned to a continuous part of the input range. Among them, Type 4 is the most flexible, allowing for the intermediate value of 0.5 to be denoted as weak preference, requiring the definition of two thresholds. Type 0 is the strictest, assigning any non-zero performance difference to 1. For the second group, Types 3 and 5 feature a linear mapping part at some range between 0 and 1. Between the two, Type 5 is more flexible, also allowing for weak preference, again with the corresponding thresholds. It is worth noting that Type 4 is the general case for Type 1 and Type 2 since the latter is practically Type 4, with p = q, and Type 1 is practically Type 4, with p = q = 0. Similarly, Type 5 is a general case for Type 3 since the latter is Type 5, with q = 0. The third group comprises only Type 6, which carries out non-linear mapping with one control parameter for the value in the input range that corresponds to the change in curvature—intuitively, the point where the ascending rate starts to drop.

Function choice is abstracted by the comparativePerformanceInterpretation declarative object and relevant properties. The stepped, linear, and non-linear properties represent the aforementioned groups, but their choice is not mandatory. The weak&strong, single, and strict properties represent an alternative grouping based on the availability of weak and strong preference thresholds, with just one threshold or none. The combination of these groups of options offers the DM the ability to narrow their choice down to a single function or allow alternatives. The Type 6 function has been included in the weak&strong grouping due to the suggested policy to define q and p and subsequently define s between these two values, varying it close to q or p to express strictness or softness in the preference mapping the difference in performance [13]. This point is covered in the last step of the declarative description synthesis, concerning the declarative abstraction of thresholds.

The last step of the declarative description synthesis is the DM’s understanding of the selected function(s) parameters. Depending on the functions implied by the declarative properties opted for in the previous step, this last step offers the ability to define the corresponding thresholds. Positioning a value in an interval starting from 0 may be intuitively captured regarding how early, i.e., close to 0, or late, i.e., closer to a maximum difference of 1, it is positioned. In the case weak&strong has been selected, the DM expects to also intuitively determine the span between the two thresholds. In this case, the DM may opt for narrow or wide regarding the distance between q and p. Combining the options allows varied alternatives, e.g., for Type 5, early with narrow implies a steep ascend relatively close to 0. The strict property, when combined with non-linear, implies that the positioning of s is closer to the q value. The declarative properties for each type of generalized criterion and respective thresholds are summarized in

Table 6. Generalized criteria functions and relevant declarative properties (function graphs detailed in Table 1): symbol ✓ denotes relevance of declarative property (column) to function (row).

	Declarative Properties
Generalized Criterion Function	Step	Linear	Non-Linear	Weak & Strong	Single	Strict	Early	Late	Narrow	Wide
Type 1.	✓					✓
Type 2.	✓				✓		✓	✓
Type 3.		✓			✓		✓	✓
Type 4.	✓			✓			✓	✓	✓	✓
Type 5.		✓		✓			✓	✓	✓	✓
Type 6.			✓	✓		✓	✓	✓	✓	✓

.

The declarative properties step, linear, non-linear, weak&strong, single, and strict imply the morphology and, as a result, the generalized criterion function(s) to be used. The declarative property strict may imply Type 1, if combined with step, or the location of the inflection point s when used for Type 6. The declarative properties early, late, narrow, and wide refer to the thresholds pertaining to the generalized criteria functions, except Type 1, and are interpreted as alternative values. The properties and their connection with the two main categories of generalized functions, i.e., those with a single threshold and those with two thresholds, are summarized in the following table. The category distinction is suggested by the DM by the weak&strong property. Similarly to the interpretation of the declarative relations in Section 4.3, a minimum implied computational overhead is maintained in the definition of the scope and granularity of the numerical rendering of the declarative properties, appearing in

Table 7. Declarative relations and implied alternative thresholds and combinations.

Declarative Property	Alternative Values
early	pip = 0.15/pip = 0.3 (single threshold)
late	pip = 0.7/pip = 0.85 (single threshold)
neither early nor late	pip = 0.4/pip = 0.6 (single threshold)
narrow (positioning implied by early, late, or none)	pws − q = 0.1 early: q = 0.1, pws = 0.2/q = 0.2, pws = 0.3 late: q = 0.7, pws = 0.8/q = 0.8, pws = 0.9 none: q = 0.45, pws = 0.55
wide (positioning implied by early, late, or none)	pws − q = 0.7 early: q = 0, pws = 0.7/q = 0.1, pws = 0.8 late: q = 0.2, pws = 0.9/q = 0.3, pws = 1 none: q = 0.15, pws = 0.85
neither narrow nor wide	pws − q = 0.4 early: q = 0, pws = 0.4/q = 0.1, pws = 0.5 late: q = 0.5, pws = 0.9/q = 0.6 pws = 1 none: q = 0.3, pws = 0.7
weak&strong non-linear	$\left(\mathit{strict}\right) s=q+0.1·(p_{ws}-q)$$/s=q+0.2·(p_{ws}-q)$ $\left(\mathrm{none}\right) s=q+0.5·(p_{ws}-q)$$/s=q+0.85·(p_{ws}-q)$

.

5. Experimental Setup

5.1. Experiments Organization

The experimental procedure initiates with the interpretation of the declarative description submitted by the DM into one or more alternative instances of the PROMETHEE I/II parameter set. The latter comprises (a) a set of weight ranges, originating from the DM’s choices of declarative relations and properties at the respective stage, (b) one or more generalized criterion function(s), and (c) the respective morphological tuning parameters (thresholds’ distance and positioning, inflection point). These alternative instances of the PROMETHEE I/II parameter set are subsequently employed to form alternative working versions of the PROMETHEE I/II methodology to be applied to the urban data samples population. In the following, their characteristics will be presented in detail before demonstrating the application of the proposed approach on these real-world data.

The availability of criteria weights at this stage, as a result of the declarative description interpretation, in combination with the availability of the normalized performance of each alternative against each criterion, inherent in the samples representation, allows for the parallel application of a utility function as a comparative methodology, representing the American school of thought of Multiple Criteria Decision Analysis. A direct weighted sum, taking advantage of the aforementioned weights and normalized performance scores per criterion, yields a global evaluation and subsequent partial ordering of the alternatives. The latter is directly comparable to the outcome of PROMETHEE II, which also yields global scores and a respective partial order of all alternatives but through a considerably different sequence of steps.

5.2. Data Sample Population

The data sample population used for the experiments is a subset of the dataset used in related previous works for the experimental evaluation of machine learning approaches in the same context [29,30] extracted from the raw data available for the city of Lyon [34]. It comprises 100 positive and 100 negative parking establishment representatives, with the former being randomly selected from the positive representatives, whereas the latter are randomly selected from the entire (non-parking) building’s population. Each sample is represented by a vector comprising the following features:

• F1—the number of other parking places in the area (1000 m);
• F2—the distance to the next nearest parking establishment;
• F3—the number of ATMs at a distance of 1000 m;
• F4—the distance from the nearest ATM;
• F5—the number of spots of tourist interest (1000 m);
• F6—the distance to the nearest spot of tourist interest;
• F7—the building area (in m²).

5.3. Experiment 1

5.3.1. Experiment 1—Declarative Description

The experimental setup initiates with the declarative description submitted by the DM. The DM’s rationale was to opt for locations close to tourist spots, with enough area to support a parking establishment. In the first experiment, the DM submitted the description comprising the elements presented below. In particular,

Table 8. Declarative properties of features—Experiment 1.

Feature	Declarative Property
F5—Number of spots of tourist interest (1000 m)	crucial
F6—Distance to nearest spot of tourist interest	average
F7—Building area (in m2)	average
F4—Distance from nearest ATM	negligible

and

Table 9. Declarative relations between features A (declarativeRelation) and B—Experiment 1.

Feature A	Declarative Relation	Feature B
F7—Building area (in m2)	vastlyMoreImportant	F3—Number of ATMs at distance of 1000 m
F7—Building area (in m2)	notablyMoreImportant	F2—Distance to nearest next parking
F7—Building area (in m2)	notablyMoreImportant	F1—Number of other parking places in area (1000 m)

summarize the declarative properties requested for selected features and the declarative relations requested for certain pairs of features, respectively. Cells containing values controlled by the declarative properties are colored blue, whereas cells containing values controlled by the declarative relations are colored green. Gray cells indicate automatically calculated inverse ratios, whereas white cells represent equally important features either due to not being paired explicitly or implicitly by the DM or due to being identical.

Moreover, the DM has requested the following properties for the generalized criterion function(s): linear, single, and early.

5.3.2. Experiment 1—Declarative Description Interpretation

The aforementioned description has to be interpreted before enabling the invocation of the corresponding instances of the PROMETHEE I/II method. Practically, a set of alternative tables are created due to the alternative ratios implied by the description, leading, in turn, to alternative sets of weights.

Table 10. Pair-wise ratios implied by declarative relations and properties: blue cells indicate values controlled by the declarative properties; green cells indicate values controlled by the declarative relations.; gray cells indicate automatically calculated inverse ratios of blue and green cells; white cells represent pairs of features equally important, either due to not being paired by the DM or due to being identical.

	F1	F2	F3	F4	F5	F6	F7
F1	1	1	1	1	1	1	1/5 or 1/3
F2	1	1	1	1	1	1	1/5 or 1/3
F3	1	1	1	1	1	1	1/9 or 1/7
F4	1	1	1	1	1/5 or 1/3	1/5 or 1/3	1/5 or 1/3
F5	1	1	1	5 or 3	1	3 or 2	3 or 2
F6	1	1	1	5 or 3	1/5 or 1/3	1	1
F7	5 or 3	5 or 3	9 or 7	5 or 3	1/5 or 1/3	1	1

summarizes the alternative interpretations of the part of the declarative description pertaining to the features, presenting all alternative ratios in certain cells.

The generalized criterion function implied by the declarative description, according to Table 6, is of Type 3. The early threshold property is interpreted as 0.15 or 0.3 for the corresponding function parameter. The above table and these alternative values for the threshold imply that there are 4 different scenarios for the specific experiment. The outcome for each scenario focuses on the highest-ranking alternatives according to PROMETHEE I. These are the samples not being out-preferred by any other while being preferred over at least one other sample. For these samples, the corresponding total flow by PROMETHEE II is presented as well as their score according to the utility function implied by the feature weights and normalized feature values. These results also imply a total ordering of the samples among the entire samples’ population and, therefore, a corresponding rank for these samples, which is also presented. The actual classification of the selected samples, i.e., being a positive or negative representative of a parking facility, is also included.

5.3.3. Experiment 1—Experimental Results

The above table and the alternative values for the threshold imply that there are 4 different scenarios for the specific experiment, as summarized in

Table 11. Alternative values for declarative parameters of declarative description—Experiment 1.

Declarative Parameter	Scenario 1 Value	Scenario 2 Value	Scenario 3 Value	Scenario 4 Value
vastlyMoreImportant	7	7	9	9
notablyMoreImportant	3	3	5	5
early	pip = 0.15	pip = 0.3	pip = 0.15	pip = 0.3
linear, single	Type 3	Type 3	Type 3	Type 3

. In the following table, the results for these scenarios are presented.

Both PROMETHEE I and II are applied to the input samples for all scenarios in Table 11. For PROMETHEE I, aggregated preference indices are calculated for each pair of alternatives and, upon them, outranking positive flow ${\phi }^{+}\left(t\right)$ and outranking negative flow ${\phi }^{-}\left(t\right)$ for each alternative, as detailed in Section 3.2. Each scenario ultimately yields the partial ranking implied by the P, I, and R relationships, as defined in Equations (12)–(14). The set of samples representing the top-class T is the main outcome of PROMETHEE I. It contains alternatives for which no other alternative is preferred over them in the partial ranking adopted by the method, while they are preferred over at least one other alternative, as formally defined in Equation (15).

Regarding PROMETHEE II, relying on the outranking positive and negative flows, the net outranking flow $\phi \left(t\right)$ is calculated yielding a complete ranking of all alternatives. For comparison, an alternative complete ranking is obtained using the feature weights suggested by the scenario in combination with the normalized performance of each alternative per feature, thus calculating the corresponding weighted sum in lieu of the utility function $u\left(t\right)$.

In the following, the members of class T suggested by PROMETHEE I are presented, with their outranking positive and negative flows, their net flows suggested by PROMETHEE II, as well as from the aforementioned utility function. The real-world classification of the samples as positive or negative parking establishments is also included in the results. Finally, in order to facilitate the interpretation of the results, the rankings of each sample ($r_{II}\left(t\right)$ and $r_u\left(t\right)$) implied by their respective PROMETHEE II and utility function evaluations are included. The results for scenario 1 of Table 11 are presented in

Table 12. Scenario 1: 7, 3, and 0.15.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
79	positive	0.744	0.153	0.592	1	0.652	2
101	positive	0.699	0.135	0.564	5	0.391	9

.

Table 13. Scenario 2: 7, 3, and 0.3.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
79	positive	0.670	0.125	0.545	2	0.652	2
80	positive	0.381	0.095	0.286	11	0.358	12
85	positive	0.703	0.146	0.557	1	0.659	1
101	positive	0.497	0.101	0.396	9	0.391	9

presents the results for scenario 2 of Table 11.

By studying the detailed results to justify the absence of samples 80 and 85 in the previous scenario, it is observed that 101 was preferred over 80, while 79 was preferred over 85 in scenario 1, thus preventing them from being included in T. Since the weight parameters remain unchanged, in comparison with the previous scenario, the utility function values do not fluctuate. It is also interesting to observe that the rankings between PROMETHEE II and the utility function vary only slightly.

The results for scenario 3 of Table 11 are presented in

Table 14. Scenario 3: 9, 5, and 0.15.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
79	positive	0.781	0.131	0.650	1	0.662	2
101	positive	0.720	0.120	0.600	8	0.360	10

below.

Samples 80 and 85 do not appear in scenario 3 due to being out-preferred by 101 and 79, respectively, similarly with scenario 1. Sample 85 is also out-preferred by sample 44 in scenario 7.

Finally,

Table 15. Scenario 4: 9, 5, and 0.3.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
44	positive	0.710	0.109	0.600	3	0.501	5
79	positive	0.709	0.107	0.602	2	0.662	2
80	positive	0.386	0.087	0.299	12	0.326	12
85	positive	0.746	0.125	0.620	1	0.670	1
101	positive	0.504	0.090	0.414	10	0.360	10

contains the results for scenario 4 of Table 11.

Sample 44 is included in T for the first time in this series of scenarios for the experiment. When re-examining the detailed results, it is observed that 44 is consistently out-preferred by 79 in all previous scenarios, thus preventing it from being included in T.

It is worth noting that, in regard to PROMETHEE II, the top 15 samples for all scenarios are almost the same, with the only exception being sample 31 ranked as 18th and 16th in scenarios 3 and 4, respectively. It may also be observed that the samples classified as top options by PROMETHEE I are always among the top according to PROMETHEE II in the current experiment. However, their rankings according to the total order of PROMETHEE II fluctuate. The rankings according to the utility function, despite differing, point to an overall very similar set of samples.

Regarding the construction of PROMETHEE I, its top class does not necessarily correspond to the highest scoring samples of PROMETHEE II. In order to illustrate this,

Table 16. Net flow and rank of top 15 PROMETHEE II representatives for all 4 scenarios. Shaded cells indicate net flow and rank per scenario (column pair) of samples (rows) which are members of top-class T for PROMETHEE I. Empty cells indicate samples (rows) not belonging to the top-class T for PROMETHEE I or the top 15 PROMETHEE II representatives for the specific scenario (column pair).

ID	Positive/Negative	Sc.1 Net Flow	Sc. 1 Rank	Sc.2 Net Flow	Sc.2 Rank	Sc.3 Net Flow	Sc.3 Rank	Sc.4 Net Flow	Sc.4 Rank
8	positive	0.299	12	0.270	12	0.342	12	0.304	11
19	positive					0.279	15
28	positive	0.527	9	0.429	8	0.582	9	0.478	8
29	positive	0.559	7	0.497	5	0.616	6	0.551	5
30	positive	0.568	3	0.486	6	0.624	4	0.538	6
31	positive	0.249	15	0.179	14
44	positive	0.589	2	0.544	3	0.643	2	0.600	3
60	positive	0.258	14	0.168	15	0.290	14	0.184	14
64	positive							0.176	15
79	positive	0.592	1	0.545	2	0.650	1	0.602	2
80	positive	0.479	11	0.286	11	0.507	11	0.299	12
85	positive	0.560	6	0.557	1	0.623	5	0.620	1
86	positive	0.566	4	0.518	4	0.625	3	0.575	4
96	positive	0.524	10	0.382	10	0.575	10	0.420	9
99	positive	0.556	8	0.443	7	0.614	7	0.494	7
101	positive	0.564	5	0.396	9	0.600	8	0.414	10
329	negative	0.262	13	0.240	13	0.303	13	0.274	13

summarizes the performance of the top 15 PROMETHEE II samples in all scenarios—shaded cells represent the top-class samples for PROMETHEE I in each scenario. Empty cells represent samples not included in the top 15 PROMETHEE II samples for the specific scenario.

Figure 3 summarizes the normalized performance of these samples per feature; higher normalized values represent improved performance.

The selected sample 101 appears in Figure 4 in the supporting visual urban planning environment in relation to other parking samples and feature elements.

5.4. Experiment 2

5.4.1. Experiment 2—Declarative Description

For the second experiment, the DM submitted a declarative description characterizing all features by declarative properties, as summarized in the following table. The rationale here is to opt for locations far from other parking establishments yet close to at least one spot of tourist interest. Regarding the generalized criterion function(s), the request is for stepped, weak&strong, and early. 

Table 17. Declarative properties of features—Experiment 2.

Feature	Declarative Property
F2—Distance to nearest next parking	crucial
F6—Distance to nearest spot of tourist interest	crucial
F1—Number of other parking places in area (1000 m)	major
F7—Building area (in m2)	major
F3—Number of ATMs at distance of 1000 m	negligible
F4—Distance from nearest ATM	negligible
F5—Number of spots of tourist interest	negligible

summarizes the declarative properties requested by the DM for all available features.

5.4.2. Experiment 2—Declarative Description Interpretation

The aforementioned declarative properties imply relations between participating features. Since all features have been declaratively characterized, they are all related according to the ratios indicated by the declarative relations associating their respective declarative properties, as detailed in Table 5 and Table 9.

Table 18. Pair-wise ratios implied by declarative relations and properties: blue cells indicate values controlled by the declarative properties; gray cells indicate automatically calculated inverse ratios of blue cells; white cells represent pairs of features equally important, either due to not being paired by the DM or due to being identical.

	F1	F2	F3	F4	F5	F6	F7
F1	1	1/3 or 1/2	7 or 5	7 or 5	7 or 5	1/3 or 1/2	1
F2	3 or 2	1	9 or 7	9 or 7	9 or 7	1	3 or 2
F3	1/7 or 1/5	1/9 or 1/7	1	1	1	1/9 or 1/7	1/7 or 1/5
F4	1/7 or 1/5	1/9 or 1/7	1	1	1	1/9 or 1/7	1/7 or 1/5
F5	1/7 or 1/5	1/9 or 1/7	1	1	1	1/9 or 1/7	1/7 or 1/5
F6	3 or 2	1	9 or 7	9 or 7	9 or 7	1	3 or 2
F7	1	1/3 or 1/2	7 or 5	7 or 5	7 or 5	1/3 or 1/2	1

summarizes the alternative interpretations of the part of the declarative description pertaining to the features, presenting all alternative ratios in certain cells.

In the current experiment, the implied generalized criterion function is Type 4 and, according to the interpretation of its declarative properties, the p and q thresholds will take the values 0 and 0.4, respectively, for one set of scenarios, as well as 0.1 and 0.5 for an alternative set of scenarios. In combination with the features’ declarative properties and implied ratios, a total of 4 alternative scenarios will be explored, as summarized in the

Table 19. Alternative values for declarative parameters of declarative description—Experiment 2.

Declarative Parameter	Scenario 1 Values	Scenario 2 Values	Scenario 3 Values	Scenario 4 Values
vastlyMoreImportant	7	7	9	9
considerablyMoreImportant	5	5	7	7
notablyMoreImportant	3	3	5	5
moreImportant	2	2	3	3
weak&strong, early	q = 0 pws = 0.4	q = 0.1 pws = 0.5	q = 0 pws = 0.4	q = 0.1 pws = 0.5
stepped, weak&strong	Type 4	Type 4	Type 4	Type 4

.

In the following table, the members of class T suggested by PROMETHEE I are presented again, with their outranking positive and negative flows, their net flows suggested by PROMETHEE II, as well as those from the aforementioned utility function; the real-world classification of the building is also included. The results for Scenario 1 of Table 19 are presented in

Table 20. Scenario 1: 7, 5, 3, 2, 0, and 0.4.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
27	positive	0.333	0.189	0.144	5	0.482	19
44	positive	0.413	0.224	0.189	1	0.533	3
54	positive	0.341	0.209	0.131	7	0.466	42
80	positive	0.343	0.213	0.131	9	0.473	30
87	positive	0.332	0.186	0.147	4	0.500	10

.

This first experiment exhibits the largest discrepancy between the PROMETHEE II and utility function rankings to date. Apart from sample 44, the rest of the members of the group are high-ranking both for PROMETHEE I and II but not for the utility function.

Table 21. Scenario 2: 7, 5, 3, 2, 0.1, and 0.5.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
44	positive	0.324	0.157	0.168	2	0.533	3
47	positive	0.191	0.090	0.101	9	0.495	14
85	positive	0.330	0.199	0.131	5	0.576	1
787	negative	0.225	0.111	0.114	6	0.520	4
443	negative	0.288	0.115	0.172	1	0.544	2

presents the results for Scenario 2 of Table 19.

It is interesting to observe that the small perturbation for the values of the generalized criterion function thresholds yields a largely (4 out of 5) different set of samples in set T for PROMETHEE I in this series of scenarios. When examining the detailed results for the samples not appearing in this scenario, samples 27 and 54 can be discerned, being out-preferred by sample 47, whereas samples 80 and 87 are both out-preferred by sample 443. Sample 54 is also out-preferred by samples not included in T. Moreover, the utility function is in more unison with PROMETHEE II in this scenario than in the previous one.

The results for Scenario 3 are presented in

Table 22. Scenario 3: 9, 7, 5, 3, 0, and 0.4.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
14	positive	0.411	0.260	0.151	3	0.511	10
27	positive	0.329	0.195	0.133	4	0.488	20
35	positive	0.411	0.301	0.110	11	0.514	8
44	positive	0.399	0.233	0.166	2	0.531	3
54	positive	0.336	0.218	0.117	8	0.473	45
75	positive	0.412	0.369	0.043	57	0.434	140
87	positive	0.325	0.195	0.130	6	0.505	15
758	negative	0.331	0.200	0.131	5	0.510	12
443	negative	0.395	0.227	0.167	1	0.556	2

.

The change in the interpretation of the values for the declarative relations to the set of higher ratios has altered considerably the set of samples comprising T. In particular, samples 14, 35, 75, and 758 are included for the first time in this series, whereas samples 27, 75, and 87 re-emerge in this scenario after being absent from the results of scenario 2. Focusing on samples absent from the current scenario in comparison to scenario 1, only sample 80 is identified, which is now out-preferred by 54, thus preventing it from being included in T. In comparison with scenario 2, samples 47, 85, and 787 are absent. Sample 47 is out-preferred by samples 18, 27, 87, and 758, all members of the top 10 ranked samples according to PROMETHEE II for the current scenario. Sample 85 is eliminated due to samples 14, 34, 44, and 443, which are also all members of the top 10 ranked samples of PROMETHEE II. For sample 787, a larger group of preferable samples can be identified, namely 27, 44, 54, 68, 80, 101, 129, 758, and 443, of which 27, 44, 54, 129, 758, and 443 belong to the top 10 samples of PROMETHEE II and 80, 68, and 101 are also highly ranked as 13th, 14th and 15th, respectively. Last but not least, the discrepancy between PROMETHEE II and the utility function is large again, mainly attributed to the thresholds’ values, which are similar to those of scenario 1.

Finally,

Table 23. Scenario 4: 9, 7, 5, 3, 0.1, and 0.5.

t: Sample ID	Class	${\mathit{\phi }}^{+}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{\phi }}^{-}\mathbf{\left(}\mathit{t}\mathbf{\right)}$	$\mathit{\phi }\mathbf{\left(}\mathit{t}\mathbf{\right)}$	${\mathit{r}}_{\mathit{I}\mathit{I}}\left(\mathit{t}\right)$	$\mathit{u}\left(\mathit{t}\right)$	${\mathit{r}}_{\mathit{u}}\left(\mathit{t}\right)$
44	positive	0.306	0.162	0.144	2	0.531	3
47	positive	0.198	0.094	0.104	7	0.508	13
81	positive	0.313	0.327	−0.015	127	0.447	112
85	positive	0.309	0.203	0.106	6	0.565	1
787	negative	0.229	0.118	0.111	5	0.530	4
443	negative	0.298	0.124	0.174	1	0.556	2
601	negative	0.214	0.116	0.098	8	0.519	7

presents the results for Scenario 4 of Table 19.

Similarly to the transition from scenario 1 to scenario 2, the modest change in p and q thresholds results in a significantly different group of samples in T. Samples 81 and 601 appear for the first time in the series of scenarios. It is interesting to observe that sample 81 is not out-preferred by any other sample despite demonstrating a significantly low PROMETHEE II ranking, thus highlighting the difference and potential value of employing both methods. Examining the absent samples which appeared in previous scenarios reveals the reasons. In comparison with scenario 1, samples 27, 54, 80, and 87 are absent. Sample 27 is out-preferred by samples 47, 638, and 601, all ranked in the top 15 of PROMETHEE II. Sample 54 is eliminated from T due to samples 32, 47, 53, 100, 129, 638, and 601. Among the latter samples, 47, 638, and 601 belong to the top 15 of PROMETHEE II, whereas samples 32, 53, 100, and 129 are lower ranked as 40th, 19th, 28th, and 27th, respectively. Sample 80 is out-preferred by samples 787 and 443, both high-ranked PROMETHEE II samples, and 443 actually demonstrates the top net flow, whereas 787 is ranked 5th. Finally, sample 87 is also absent due to being out-preferred by sample 443.

Regarding scenario 2, set T of the current scenario is practically a superset of the respective set of scenario 2, containing all of the samples of the latter.

Finally, in comparison with scenario 3, samples 14, 27, 35, 54, 75, 87, and 758 are absent from the current scenario. Sample 14 is out-preferred by samples 44, 101, and 443, which are all among the highest-ranking samples for PROMETHEE II (2nd, 4th, and 1st, respectively). Sample 27 is eliminated due to samples 47, 638, and 601 being preferred over it, which are all among the top 15 ranked samples of PROMETHEE II. Regarding sample 35, it is out-preferred by samples 44 and 85, both members of the top set T of PROMETHEE I. Sample 54 is out-preferred by the same set of samples as in the previous scenario, namely 32, 47, 53, 100, 129, 638, and 601, bearing the same ranks in the PROMETHEE II ordering as in the analysis of the previous scenario.

Sample 75 represents an interesting case, being out-preferred only by sample 81 which, despite being included among the top-ranking samples of PROMETHEE I, is only 127th according to the net flow ordering of PROMETHEE II. Delving into the detailed results of the experiments, it is observed that sample 81 is deemed mostly incomparable (i.e., related as R against 193 out of the 200 total sample population), indifferent against 1, and only preferable over 6. However, it is never out-preferred by any sample, thus being included in the top-class T. The specific sample exhibits a relatively high positive flow and analogously high negative flow, a combination that often leads to incomparability in PROMETHEE I yet makes it preferable in certain matchings. However, when net flow is considered, the negative flow being greater than the positive flow leads to a negative sum and a low overall ranking for PROMETHEE II. A similar situation actually arises with sample 75 itself: it is included in the top-class T of the previous scenario, exhibiting high positive and negative flows and leading to incomparability with the majority of the population, yielding a rather low ranking in PROMETHEE II due to antagonizing positive and negative flows leading to a low net flow.

Finally, sample 87 is eliminated from the current scenario in comparison to scenario 3 due to being out-preferred by the top sample 443, while sample 758 is out-preferred by samples 787 and 601, both among the top 10 samples of PROMETHEE II. The utility function’s rankings are again similar to PROMETHEE II in comparison to scenarios 1 and 3.

In contrast to the previous experiment, the samples occupying the highest ranks for PROMETHEE II according to net flow are not almost identical in each scenario. Given this variance, the top 10 ranking samples were selected for each scenario, and a union of the sets of the top 10 PROMETHEE II samples from all scenarios is presented in the following table. Samples included in T for certain scenarios despite exhibiting a ranking lower than top 10 in terms of PROMETHEE II have also been appended. The samples comprising the top-ranking set T of PROMETHEE I for the corresponding scenarios are shaded. The aforementioned results are presented in

Table 24. The net flow and rank of the union of the top 10 PROMETHEE II representatives for all 4 scenarios. Shaded cells indicate net flow and rank per scenario (column pair) of samples (rows) which are members of top-class T for PROMETHEE I. Empty cells indicate samples (rows) not belonging to the top-class T for PROMETHEE I or the top 10 PROMETHEE II representatives for the specific scenario (column pair).

ID	Positive/Negative	Sc.1 Net Flow	Sc. 1 Rank	Sc.2 Net Flow	Sc.2 Rank	Sc.3 Net Flow	Sc.3 Rank	Sc.4 Net Flow	Sc.4 Rank
14	positive	0.159	3			0.151	3
18	positive					0.117	9
27	positive	0.144	5			0.133	4
34	positive	0.133	6			0.128	7
35	positive					0.11	11
44	positive	0.189	1	0.168	2	0.166	2	0.144	2
47	positive			0.101	9			0.104	7
54	positive	0.131	7			0.117	8
75	positive					0.043	57
79	positive			0.105	8
80	positive	0.131	9	0.107	7			0.088	10
81	positive							−0.015	127
85	positive			0.131	5			0.106	6
87	positive	0.147	4	0.132	4	0.13	6
101	positive			0.138	3			0.118	4
129	negative	0.127	10			0.117	10	0.119	3
443	negative	0.163	2	0.172	1	0.167	1	0.174	1
601	negative			0.097	10			0.098	8
758	negative	0.131	8			0.131	5	0.088	9
787	negative			0.114	6			0.111	5

.

The above samples are also presented in Figure 5 in terms of their normalized performance per feature.

6. Discussion

In the current work, a Declarative Modeling framework for intuitive Multiple Criteria Decision Analysis, in the context of a visual semantic urban planning environment, has been proposed and implemented. The operating paradigm is the application of the PROMETHEE I/II methodologies for the selection of the most prominent representatives among a population of candidate buildings for a parking establishment.

The proposed framework allows the DM to express the parameters of the PROMETHEE I/II methodologies in a declarative manner, using intuitive terms instead of opting for concrete numerical values and mathematical functions. The DM’s declarative description is subsequently interpreted as a set of alternative instances of the PROMETHEE I/II methodologies, employed upon the sample buildings’ population. The latter includes positive and negative examples of parking establishments, offering additional insights into the DM’s understanding of the decision parameters in view of the results. In addition, the feature weights extracted as part of the PROMETHEE procedure are incorporated into a utility function, offering an additional alternative evaluation of the selected samples. The application of each PROMETHEE or utility function instance per se is not affected by the proposed framework. This is due to the fact that the proposed declarative shell intervenes between the DM and the parameter definition required by the MCDA mechanism of choice.

In the experiments conducted, two alternative declarative descriptions were submitted as input. Each description yielded four alternative scenarios representing alternative instances for the PROMETHEE I/II parameters and corresponding mechanism. For the first declarative description, the samples selected by PROMETHEE I method as its top-class T, i.e., samples not out-preferred by any other sample but preferred over at least one, vary among scenarios but form a relatively consistent set of five samples. Moreover, the global order proposed by PROMETHEE II includes these samples in its top 15 performing samples for all scenarios. Nevertheless, for three out of four scenarios, the isolated invocation of the corresponding PROMETHEE instance would have yielded only a subset of T. This implies that in a non-declarative context, valid alternative samples would have been eliminated due to the isolated application of the method. Moreover, the set of top-n performing samples of PROMETHEE II, where $n=\left|T\right|$ for each scenario, do not coincide with the members of T from PROMETHEE I, i.e., the two versions do not agree upon their top performing representatives. These differences highlight one of the main strengths of the Declarative Modeling methodology, which is its potential to reveal desirable solutions not originally conceived by the stakeholders involved in the process. The utility function results, despite the small variance in the exact ranking, tend to agree with those of PROMETHEE II.

For the second experiment, the declarative description again yields four alternative scenarios representing four instances of PROMETHEE. In this case, the variance of T members among scenarios is greater, leading to an overall union of 15 samples. Certain samples included in the top class of PROMETHEE I exhibit a significantly low ranking according to PROMETHEE II for the same scenario, highlighting the difference between the two members of the PROMETHEE family and the value of exploration of parameter alternatives offered by Declarative Modeling. In this series, it is also evident that small fluctuations in parameters may lead to significant differences. For example, changing the weak and strong preference thresholds from q = 0 and p = 0.4 to q = 0.1 and p = 0.5 between scenarios 1 and 2, respectively, led to the replacement of four out of the total five samples included in T. In this experiment, the utility function demonstrates increased divergence from the results of PROMETHEE II but maintains semblance.

Despite the different rationale followed by the DMs in the synthesis of the declarative description in each experiment, there is a degree of overlap. The majority of the selected samples in both experiments are positive representatives. The latter fact reveals a degree of conformity of both DMs’ requirements, implied by their respective declarative descriptions, to real-world actual parking establishments and the underlying rationale for their choice. This fact, apart from an informal validation of the DMs’ intuition expressed through the declarative description, offers a set of eligible prospective candidates, due to their presence in the group, in the form of the negative representatives included in the results.

The most significant computational overhead of the proposed methodology originates from the generation of different instances of the PROMETHEE parameters and their subsequent application to the alternatives’ population. The main trade-off is between the desire for a wider scope and finer granularity of the explored PROMETHEE parameters and the desire for computational efficiency. In addition, the evaluation of the generated alternative PROMETHEE instances is based on the DM’s inspection and evaluation of the subset of alternative samples eventually selected by each instance, thus introducing additional time overhead for the DM.

7. Conclusions

The proposed approach employs Declarative Modeling in the early phase of the decision-making process in order to allow the DM to formulate the parameters of the MCDA method of choice in an intuitive manner. PROMETHEE I and II were selected as example methodologies for the application of the concept due to their diversity on required input components on behalf of the DM. This input is implicitly interpreted on the basis of a declarative description, synthesized by the DM upon the intuitive interpretation of their understanding of the decision parameters. This declarative description is translated into a range of choices for the components of PROMETHEE, leading to more than one interpretation of the DM’s preferences, which are then applied as distinct instances of PROMETHEE to the alternatives’ population. At the final stage, the DM is able to inspect the selected alternatives for each interpretation, thus gaining insights into the declarative input and potentially reinitiating the Declarative Modeling cycle.

The application of the approach offers an intuitive shell and alternative interpretations of the DM’s intuition at the cost of an increased computational overhead. However, the incurred computational cost, when compared to the application of a single instance of the PROMETHEE I and II methods, guided by parameters concretely defined by the DM, is comparable and far from prohibitive, thus representing an interesting trade-off towards the potential of the discovery of a closer interpretation of the DM’s understanding of the decision parameters. In the current work, a minimal computational footprint was maintained, but the scope and granularity of the proposed method’s interpretation of the invoked MCDA method’s parameters, in this case, PROMETHEE’s, are subject to the desired computational requirements and may be adjusted accordingly.

The experiments utilized a subset of the urban data sample population from the city of Lyon in regard to a decision problem regarding the applicability of alternative buildings as parking establishments; positive representatives of existing parking establishments were equally included in the population. Most experimentally selected alternatives were positive parking establishment representatives, which revealed not only a common baseline between the DM’s understanding and the actual real-world situation but also the ability of the proposed method to capture and manifest this understanding. Buildings selected by the method originating from the non-parking class represent valid candidates, compliant with the DM’s requirements, with their set being elicited and expanded through the alternative instances of the declarative decision parameters.

The experimental results of these alternative interpretations of the submitted declarative descriptions, in the form of the employed MCDA mechanisms’ instances, exhibit controlled variance among the selected samples. All selected samples conform to the DM’s intuitive input, thus validating the potential of discovering novel desired solutions through these alternative instances of the MCDA mechanism, achieved through the declarative description. It also stresses the added value that Declarative Modeling provides to the inherent power of the MCDA methodology invoked. The intuitive interface offered by the Declarative Modeling paradigm to the MCDA methodologies and the wide penetration of the latter allow for the applicability and adaptability of the proposed framework in an extensive range of domains.

The latter stage of evaluation may be accelerated by offering the DM the option to apply aspects of the machine learning approaches employed in related previous works, requiring only the choice of positive representatives as examples. These representatives, although not replacing the outcome of the PROMETHEE instances, may be optionally used to automate the DM’s a posteriori inspection and evaluation, if desired.