Amidou Kpoumié
email: amidou.kpoumie@lamsade.dauphine.fr

Sébastien Damart

Alexis Tsoukiàs

Integrating Cognitive Mapping Analysis into Multi-Criteria Decision Aiding

Keywords: Hydrogen technologies, Social acceptability, Problem structuring, Cognitive mapping, Value trees of objectives, Multi-criteria decision aiding

Multi-criteria decision aiding (MCDA) is a process implying two distinctive actors (the client and the analyst) which aims at providing transparent and coherent support for complex decision situations, taking into account values of decision makers involved in a specic decision context. The theoretical framework of MCDA traditionally addresses problems involving a single decision maker. However, MCDA ought to investigate the case where the decision maker is made up of groups of individuals with conicting interests.

In contrast, cognitive mapping (CM) is frequently used in order to capture the values in a group of individuals and to reduce the antagonism between such values. Its ability to capture multiple values and reduce their conicting aspects provides a rationale for decision problem analysis with multiple stakeholders. Nevertheless, capturing values by CM is not always intended for a subsequent multi-criteria analysis. This paper explores the integration of both techniques combining their respective strengths as well as their application in assessment of hydrogen technologies scenarios in terms of their perception and social acceptability by the general public.

Introduction

The work report on this paper is conducted within the context of the AIDHY project, in which distributed expertise on hydrogen technologies is brought together to address the issue of the social acceptability of hydrogen technologies scenarios. Power planning marked by the predicted decline of fossils fuels and the need for consideration of environmental concerns and energy independence, lead governments to think in terms of energy mix. The term energy mix refers to the distribution, within a given geographical area, of energy originating from various energy sources (crude oil, natural gas, coal, nuclear energy, and renewable energy). It depends on (i) the availability of usable ressources (possibility of local or import ressources), (ii) the extent and nature of energy needs to be meet, (iii) the social, economic, environmental and geopolitical context and (iv) the political choice resulting from the previous points. As a result the choice of energy mix is a complex decision with important consequences in society. Dierent energy mix will require dierent types of energy carriers for eective transformation, storage and consumption. This resulted in developing new technologies about energy carriers such as the hydrogen. To ensure that energy using such new technologies is not rejected, a study of social acceptability must be conducted.

The decision makers face a complex situation, since assessing hydrogen technologies involves the evaluation of many conicting objectives, expression of various multiple stakeholders. This decision context is even more dicult because of its social dimension. This diculty is particularly important when the social group is extended to the general public, which by denition consists of heterogeneous opinions. Since the sum of individual rationalities does not necessary lead to a collective rationality it is unlike that consensus self-emerges. Hence the necessity to study the problem of the legitimacy of the decision and its acceptability by the stakeholders.

In order to face the particular complexity of decision problems in such contexts, Munda [START_REF] Munda | Social Multi-Criteria Evaluation: Methodological Foundations and Operational Consequences[END_REF] suggests a methodological framework called Social Multi-criteria Evaluation . This methodology emphasises uncertainty and signicant conicts of values, an issue specic to public decision processes.

In addition to a technical dimension of uncertainty, which is quantitative and relative to the inaccuracy of the parameters and can be apprehended by tools such as sensitivity analysis, robustness and Monte Carlo methods, it oers three other dimensions: (i) a methodological dimension which is related to the reliability of the methods used , (ii) an epistemological dimension which is linked to the lack of knowledge w.r.t the problem studied and (iii) a social dimension due to the social mess [START_REF] Horn | Knowledge Mapping for Complex Social Messes. A presentation to the[END_REF]. In the latter case of social uncertainty, decisions are not completely determined by scientic facts (see also [START_REF] Marchi | From Evidence -Based Policy -Making to Policy Analytics[END_REF]). Assuming that a good decision involves a socio-technical process, scientic arguments can be debated by the arguments based on the values of the actors. The actors being taken into the sense of socio-economic public and private stakeholders. This is for instance the case of the hydrogen scenarios assessment when it comes to evaluate technologies scenarios, based on scientic expertise and taking into account the values of the general public.

Multi-criteria decision aiding (MCDA) is often chosen as the basis for decision support systems in prospect of energy issues (see [START_REF] Montignac | An MCDA approach for evaluating hydrogen storage systems for futur vehicles[END_REF], [START_REF] Montignac | An MCDA approach for evaluating hydrogen storage systems for future vehicles[END_REF], [START_REF] Thery | Energy planning: a multi-level and multicriteria decision making structure proposal[END_REF]), since MCDA aims at providing transparent and coherent support for the comprehension of complex decision situations with possibly conicting objectives. Typically, depending on the approach or a combination of approaches adopted (Normative, Descriptive, Prescriptive, or Constructive) [START_REF] Tsoukiàs | On the concept of decision aiding process: an operational perspective[END_REF], a decision aiding process consists in producing four cognitive artifacts: (1) a representation of problem situation, (2) a problem formulation, (3) an evaluation model, and (4) a nal recommandation [START_REF] Tsoukiàs | On the concept of decision aiding process: an operational perspective[END_REF]. Many MCDA evaluation models are based on deterministic evaluations of the consequences of each alternative on each attribute in relation to the views of a single and specic decision maker. Traditional evaluation methods have diculties solving problems involving several possible decision makers with potentially conicting objectives. Hence, mechanisms that guarantee for the consistency of the problem situation and its development should be included. Another problem is that there are no features inherent in classical MCDA allowing to capture values for more than one decision maker or considering social uncertainty in public decisions. Under such a perspective there are substantial benets to be expected from a framework that integrates Cognitive mapping (CM) into MCDA going beyond from social choice inspired methods or from methods eliciting sound trade-os (see [START_REF] Bouyssou | Evaluation and Decision Models: A Critical Perspective[END_REF], [START_REF] Bouyssou | Evaluation and Decision Models with Multiple Criteria: Stepping stones for the analyst[END_REF]).

Cognitive mapping has been applied predominantly in psychology and behaviourial sciences [START_REF] Kelly | The psychology of personal constructs[END_REF], management (see [START_REF] Eden | SODA-journey making and mapping in practice[END_REF], [START_REF] Cossette | Cartes cognitives et organisation[END_REF], [START_REF] Fiol | Maps for managers: where are we? where do we go from here[END_REF], [START_REF] Langeld | Exploring the need for a shared cognitive map[END_REF], [START_REF] Lee | A system for organizational learning using cognitive maps[END_REF], [START_REF] Sahin | Cognitive mapping expert views for policy analysis in the public sector[END_REF], [START_REF] Weick | Organizations as cognitive maps: charting ways to success and failure[END_REF]), politics (see [START_REF] Axelrod | Structure of Decision: The Cognitive Maps of Political Elites[END_REF], [START_REF] Eden | Cognitive mapping expert views for policy analysis in the public sector[END_REF]), economics (see [START_REF] Coban | Prediction of socio-economical consequences of privatization at the rm level with fuzzy cognitive mapping[END_REF], [START_REF] Kwahk | Supporting business process redesign using cognitive maps[END_REF]) and other areas (see [START_REF] Montibeller | Reasoning Maps for Decision Aiding: an integrated approach for problem structuring and multi-criteria evaluation[END_REF], [START_REF] Montibeller | Supporting Factoring Transactions in Brazil using Reasoning Maps: a language-based DSS for evaluating accounts receivable[END_REF]). Although CM have been initially t for individual decision making representations, they are nowadays mainly used to support group decision contexts where one should consider judgments of experts and group participation in an environment (focus groups) that fosters creativity. A prime aim of cognitive maps is to graphically represent the ideas of a group of individ-uals through a network of interrelated concepts. Cognitive mapping allows to build a shared vision of the decision problem and facilitate the identication of values and their conicting elements that may have an impact on the consequence of decision [START_REF] Eden | Cognitive mapping: a review[END_REF]. The way cognitive mapping allows to deal with values diers from conventional methods. These usually present one single decision maker objectives, including his values and interests in terms of criteria and preferences. Instead, cognitive mapping address complexity by presenting several stakeholders objectives that encompass all their relevant values, so as to reach a cluster of consensual values through negotiation of ideas [START_REF] Cossette | Cartes cognitives et organisation[END_REF] between individuals. In addition, the design of cognitive maps through the interactive setting of focus groups is likely to be attractive to stakeholders, for it provides additional means of decision legitimacy by ensuring transparency and participation. Cognitive mapping indeed provides support for mapping the participation of multiples stakeholders as shown in the methodology proposed by Damart [START_REF] Damart | Cognitive Mapping Approach to Organizing the Participation of Multiple Actors in Problem Structuring Process[END_REF].

The paper is structured as follows: Section 2 starts with a brief outline of MCDA illustrating how MCDA methods can be applied. Then the analysis of cognitive maps and the structuration of a decision problem on the basis of cognitive mapping ndings are explained. Subsequently, the problem of cognitive maps items conversion into value tree for objectives by clustering them under High-level and lower-level hierarchically is considered. We demonstrate how a judicious choice of graphical models can facilitate this conversion. Then, an example for the AIDHY project is introduced to highlight the key points of our approach. The last section gathers conclusions.

Value trees and problem structuring

According to Simon [START_REF] Simon | Rational choice and the structure of the environment[END_REF], decision making is a process consisting of three main stages: (1) Intelligence, (2) design and (3) choice (see Fig. 1). In the intelligence phase, we try to determine if the problem to face requires a decision.

Simon considers the design step as the true structuring phase of the problem since it allows the identication of alternatives, criteria and attributes.

However, following the authors of the so called soft operational research (for a discussion, see the opposition between Soft Operations Research (OR) and Hard Operations Research in [START_REF] Checkland | Systems Thinking, Systems Practice[END_REF], [START_REF] Rosenhead | Rational Analysis for a Problematic World Revisited:Problem Structuring Methods for Complexity, Uncertainty and Conict[END_REF]), we consider that the intelligence stage is an integral and most important part of problem structuring because it prevents type III errors: dening the wrong problem, leads to the wrong solution (see Raia [START_REF] Raia | Decision Analysis: Introductory Lectures on Choice under Uncertainty[END_REF]). Many other authors also focus on this crucial phase of decision analysis as a starting point for problem structuring (see [START_REF] Bouyssou | Evaluation and Decision Models: A Critical Perspective[END_REF], [START_REF] Dewey | How we think[END_REF], [START_REF] Rosenhead | Rational Analysis for a Problematic World Revisited:Problem Structuring Methods for Complexity, Uncertainty and Conict[END_REF], [START_REF] Tsoukiàs | On the concept of decision aiding process: an operational perspective[END_REF]).

Stages in problem solving (Dewey, 1910) Phases of the decision-making process (Simon, 1960 Which alternative is best ?

Decision analysis

Figure 1: General framework of decision analysis. Sources: Galves [START_REF] Galves | structuring decision situations: a brief overview[END_REF] From Figure 1, we can see that the results of structuring is an input to a multi-criteria evaluation model. This necessary link between the structuring and evaluation model has been the subject of numerous studies (see [START_REF] Belton | Multiple Criteria Decision Analysis: An integrated approach[END_REF], [START_REF] Montibeller | Reasoning Maps for Decision Aiding: an integrated approach for problem structuring and multi-criteria evaluation[END_REF]). Since for most MCDA evaluation models the criteria are deduced from the objectives, the later have to be elaborated and made clear. Using the principles of value-focused thinking proposed by Keeney [START_REF] Keeney | Value Focused Thinking: A Path to Creative Decisionmaking[END_REF] seems adequate in order to address this issue. These principles allow to specify objectives in terms of decision-making context, purpose and preferential direction. Objectives are statements of something that one desires to achieve. According to Keeney [START_REF] Keeney | Value Focused Thinking: A Path to Creative Decisionmaking[END_REF], objectives are characterised by three features:

• decision context • object • direction of preferences
For example, two objectives for power planning decisions could be to minimise costs and to maximise security. For the former objective, the decision context could be the choice of a good power plan, the object is costs for a chosen plan, and less costs are preferred to more costs. For the last objective, the decision context remains the same, the object is systems' security for a chosen plan, and more security is preferred to less security.

For decisions using multiple attributes, Keeney and Raia [START_REF] Keeney | Decision with Multiple Objectives:Preferences and Value Tradeos[END_REF] propose to structure the decision maker's objectives, beginning with dening their area of concern, which must provide a formal specication of these objectives, so that multiple points of view are comprehensively considered. Keeney [START_REF] Keeney | Value Focused Thinking: A Path to Creative Decisionmaking[END_REF] distinguishes two types of objectives: the fundamental and the means objectives. He states the dierence as follows: on the one side, The fundamental objective characterises an essential reason for interest in decision situation;

on the other side, A means objective is of interest in the decision context because of its implications for the degree to which another (more fundamental)

objective can be achieved [...]

For example, higher control system may appear to be an important objective, but it may be seen important only because it would allow a plan to increase its security standards. Thus, higher control system could be seen as a means objective and increasing security standards as a fundamental objective.

In traditional MCDA methods, structuring objectives (assuming the perspective of an evaluation) results in a value tree hierarchy of objectives referring to the fundamental objectives hierarchy and criteria associate with it (cfr. an illustrative example in Figure 2). This is a three level value tree 3-level value tree of fundamental objectives. The construction of such fundamental objectives is based on a top-down approach. In this approach the overall fundamental objective is identied, then it is detailed into more specic objectives. The decomposition of objectives is carried out iteratively until a suciently low level, that can be associated with an attribute or a measurable criterion, is reached. This type of representation of decisionmaking structure has been used by many authors and applied eectively in many studies (see [START_REF] Adelman | An Experimental Investigation of the Relative Eectiveness of two Techniques for Structuring Multiattribute Hierarchies[END_REF], [START_REF] Borcheding | The Eect of Varying Value Trees on Multiattribute Evaluations[END_REF], [START_REF] Mustajoki | Web-HIPRE: Global Decision Support by Value Tree and AHP Analysis[END_REF], [START_REF] Pöyhönen | Behavioral and Procedural Consequences of Structural Variation in Value Trees[END_REF], [START_REF] Saaty | The Analytic Hierarchy Process[END_REF], [START_REF] Salo | On the Measurement of Preferences in the Analytic Hierarchy Process[END_REF]) particularly in the eld of energy issues (see [START_REF] Jones | A Multi-Attribute Value Model for the Study of UK Energy Policy[END_REF], [START_REF] Phdungslip | Decision Support for the Selection of Electric Power Plants Generated from Renewable Sources[END_REF]). In order to help the structuring of objectives, Belton et al. [START_REF] Belton | Integrated support from problem structuring through to alternative evaluation using COPE and VISA[END_REF] propose the use of cognitive mapping [START_REF] Eden | SODA-journey making and mapping in practice[END_REF] which we develop in section 3 .

Cognitive mapping (CM)

In traditional MCDA setting, two individuals, the decision maker (DM)

and the analyst interact with respect to a problem situation. This interaction is intended to help a decision-maker to structure his ideas for handling the problem that he faces. An informal dialogue between the decision maker and the analyst may be sucient in the case of a single decision maker. In the case of multiple decision-makers or group of stakeholders, this task becomes much more dicult. On the basis of former works about animal psychology [START_REF] Tolman | Cognitive maps in rats and men[END_REF], human psychology [START_REF] Kelly | The psychology of personal constructs[END_REF], or strategic choice approach [START_REF] Friend | Planning under Pressure: the Strategic Choice Approach[END_REF], Bougon [START_REF] Bougon | Uncovering Cognitive Maps: The Self-Q Technique[END_REF],

Eden [START_REF] Eden | SODA-journey making and mapping in practice[END_REF], Ackermann [START_REF] Eden | SODA-journey making and mapping in practice[END_REF] and Komocar [START_REF] Komocar | Participant cause maps of a work setting: an approach to cognition and behavior in organizations[END_REF] proposed a more formal tool for this kind of interaction: cognitive mapping (CM). The general idea of cognitive mapping is to graphically represent the ideas of a group of actors through a network of concepts and possible causal links. A cognitive map is co-constructed by the participants and the facilitator in a format that is viewable by all participants in the focus group (cf. section 1). These groups aim to promote open discussion among participants and stimulate their imagination to make them produce the most ideas in the shortest possible time (brain-storming). The facilitator is the person responsible to conduct and supervise the discussion in a group of approximately fteen individuals. The CM activity can be conducted through a focus group conversational mode.

Focus groups are a special type of group used to gather information from members of a clearly dened target audience. Such audience is composed of six to twelve people who are similar in one or more ways, are guided through a facilitated discussion, on a clearly dened topic to gather information about the perceptions, opinions, beliefs, etc. of the group members.

CM is generally used in a process of decision support for dening a problem through a network of explanations and consequences associated with a unique situation [START_REF] Cossette | Cartes cognitives et organisation[END_REF]. Its visualisation helps to think, explore and transform or conrm more or less shared ideas . In this sense it can resolve conicting objectives through negotiating ideas between individuals. Beyond the interest to tell what the problem is, we will focus on analysis and exploitation of its contents for purposes of structuring a multi-criteria analysis. In our study, this implies to consider cognitive maps not as a goal but as a mean.

For this purpose we will retain the following denition: cognitive map is a graphical representation of the mental representation that the researcher [facilitator] gets from a set of discursive representations expressed by a subject from its own cognitive representations, about a particular object [START_REF] Cossette | Cartes cognitives et organisation[END_REF].

Several graphic forms that adopt dierent conventions have been proposed by Bougon and al [START_REF] Bougon | Cognition in organizations: an analysis of the Utrecht Jazz Orchestra[END_REF], Axelrod [START_REF] Axelrod | Structure of Decision: The Cognitive Maps of Political Elites[END_REF] and Eden [START_REF] Eden | Cognitive mapping: a review[END_REF] in order to represent cognitive maps (see Fig. 3, 4 and 5).

Figure 3: Graphical form used by Bourgon. Sources: [START_REF] Eden | Cognitive mapping: a review[END_REF] Figure 4: Graphical form used by Axelrod.

Sources: [START_REF] Eden | Cognitive mapping: a review[END_REF] Figure 5: Graphical form used by Eden.

Sources: [START_REF] Eden | Cognitive mapping: a review[END_REF] Example 3.1. Figure 6 represents a partial cognitive map using Eden [START_REF] Eden | Cognitive mapping: a review[END_REF] convention describing the acceptance of H 2 powered cars by a group of individuals. More details and explanation about this map will be given in the next subsection. At this level, we will only present an overview of this map.

Indeed, it is part of a cognitive map constructed by interaction with a group of individuals who expressed their views on the issue of hydrogen (see [START_REF] Kpoumié | Model for structuring and evaluating hydrogen technologies scenarios in terms of social acceptability[END_REF]).

In the next section, we will return to this example to demonstrate the potential of such a representation regarding its possible transformation into a value tree, particularly that of stakeholders objectives. Our goal is to provide a consistent methodology for moving from one representation to another one in a process of multi-criteria decision aid involving groups of individuals rather than a single decision maker.

Conversion of cognitive maps in value trees

The conversion of cognitive maps in value trees can be based both on the physical structure of these graphical representations and their semantic aspects. Here, we rst present briey the theoretical framework underlying this conversion, then we propose handling practices to reect the characteristics of these graphs in terms of a decision-aiding context.

Theoretical approach

Formally, the two objects that are the subject of this section ie, cognitive maps and value trees, are graphs; the former being a simple graph and the last being a particular type of graph. Many textbooks provide a broad development of this objects in graph theory (see [START_REF] Deo | Graph theory with applications to engineering and computer science[END_REF], [START_REF] Gondran | Graphes et algorithmes[END_REF], [START_REF] Wilson | Introduction to Graph Theory[END_REF] and [START_REF] Diestel | Graph theory[END_REF]). To apply the graph formalism in our model, we propose the following denitions: Lemma 4.1.

C o M contains an arborescence V T o if an only if each vertex in C o M is reachable from r.
Where r is a vertex called root and there is a path from r to every other vertex v∈ O.

To prove this lemma, we recall the following denition (see [START_REF] Wilson | Introduction to Graph Theory[END_REF]):

Denition 4.5. A spanning tree T of a graph G is a subgraph of G contain- ing all the vertices of G such that V (T ) = V (G). Where V (T ) (respectively V (G)) is the set of vertices of T (respectively of G). Proof 4.1 (Lemma 4.1). proof of "1 ⇐⇒ 2 ′′ : 1 ≡ C o M contains an arborescence V T o ; 2 ≡ each vertex in C o M is reachable from r.
(a) proof of "1 ⇒ 2 ′′ : assume that the direction of arcs is ignored in C o M (without loss of generality) and that C o M contains an arborescence V T o , hence V T o is a spanning tree (cfr.(i) in denition 4.4) and V (V T o ) = V (C o M ) (cfr. denition 4.5), from theses consequences, the conclusion is straightforward: each vertex in C o M is reachable from any vertex in the spanning tree V T o , since there is no cycle (cf. denition 4.4). In particular each vertex in C o M is reachable from the the root r of the spanning tree.

(b) proof of "1 ⇐ 2 ′′ : assume that each vertex in C o M is reachable from r, hence there is (i) a path from r to any other vertex u ′ ∈V T o , a subgraph of C o M ; and (ii) V T o has no cycle.

(i) and (ii)

⇒ V T o is a spanning tree , hence V T o is an arborescence contained in C o M .
The consequence of this lemma is that, by construction, a cognitive map still contains an arborescence since the main concept is the root from which all other concepts are directly (strongly connected) or indirectly (weakly connected) related.

Within this formalism, the transfer of CM into a value tree follows some rules. Since graphs can have closed circuits, this transfer is achieved by meaningfully (R 1 )opening the circuits by either duplication of one of its vertices (which may contain more distinctive information) or (R 2 )merging two or more vertices (which may contain redundant information). This transfer patterns, using graph theory, and the previous rule will be illustrated by a real-world case study in section 5.

Practical approach

To consider a conversion from cognitive maps to a value tree of objectives, a set of features should be identied between the two graphs. These may be dierences or similarities such that by a minimum of simple manipulations, we can move from one graph to another and possibly vice versa. In general, the concepts are represented by circles connected by arrows indicating the presence and direction of the relationship of inuence between them. In some cases, the direction of the link when it exists, is represented by the signs (+) or (-) indicating a positive or negative correlation. It appears that the graphical formalism to draw a cognitive map is not always the same.

Depending on the chosen formalism, a graphical representation can be more or less adapted to a given problem. If the modelling of the decision problem is oriented to the construction of an evaluation model, therefore based on the principle of value tree of objectives, the formalisms of Axelrod [START_REF] Axelrod | Structure of Decision: The Cognitive Maps of Political Elites[END_REF] and Eden [START_REF] Eden | SODA-journey making and mapping in practice[END_REF] are more suitable. According to these graphical formalism and the value tree features (see section 2), we propose to articulate the transition from cognitive maps to value trees through the following matrix connecting them (see Table . The connection between the two graphs is a table of equivalences, which allows the transition from cognitive maps to values trees of objectives. Fol-lowing such equivalences, the conversion between the two graphs becomes possible. The two graphs are dierential graphical representations of similar underlying mental operations, assuming that all the concepts are the objectives, such operations being conducted to gather informations. With some manipulations, a cognitive map of the type Eden [START_REF] Eden | SODA-journey making and mapping in practice[END_REF] or Axelrod [START_REF] Axelrod | Structure of Decision: The Cognitive Maps of Political Elites[END_REF] suggested, can be drawn as a value tree. Depending on the problem on hand, Axelrod cognitive maps can also be used to directly evaluate options by modelling them as fuzzy cognitive maps (FCM). The discussion of this approach is beyond the scope of this paper, but it can be found e.g in ( [START_REF] Kosko | Fuzzy Cognitive Maps[END_REF], [START_REF] Kosko | Neural Networks and Fuzzy Systems: A dynamic system approach to machine intelligence[END_REF], [START_REF] Kosko | Fuzzy Engineering[END_REF]).

Example 4.1. Let consider again the acceptance of H 2 powered cars given in example 3.1 and its derived cognitive map (see Figure 6).

According to denition 4.1, the cognitive map in Figure 6 The polarities are attached to the following meanings:

(R 3 ): c 5 has a positive inuence on c 2 i.e more c 5 implies more c 2 , hence the following proportional relation for any two concepts i, j :

(iP + j) =⇒ (c i ↗) ∧ (c j ↗) (1)
(R 4 ): c 2 has a negative inuence on c 1 i.e more c 2 implies less c 1 , hence the following conversely proportional relation for any two concepts i, j:

(iP -j) =⇒ (c i ↗) ∧ (c j ↘) (2)
For instance, (1) more mature will be the H 2 technologies (c 7 ), more the cars will be present in the market (c 6 ); (2) more will be the safety in H 2 cars (c 4 ), less will be the risk of H 2 explosion (c 5 ).

The nal objectives hierarchy created by applying the previous theoretical and practical rules (R 1 , R 2 , R 3 , R 4 ) on example 6 map, w.r.t the matrix connecting items, consist of a 3-levels value tree of objectives: The rst level starts with the main goal of improving the acceptance of H 2 systems. To reach this main objective, the second level objectives are to maximise safety and minimise costs. The third level is achieved by three objectives: rst the objective of maximising safety is assumed to be reached by minimising risk of explosion and maximising control of systems, second the objective of minimising costs is reached by maximising the number of H 2 powered cars present in the market, etc. Practically this consists in transforming Figure 6 to Figure 7. [START_REF] Kpoumié | Model for structuring and evaluating hydrogen technologies scenarios in terms of social acceptability[END_REF]). The depletion of fossil fuels, the environmental concerns and the rise of renewable energy, provide an overview of the current energy environment. The analysis of such information allows the formulation of concrete decision problems. Hydrogen is an energy carrier, i.e is a form of energy transposable, to be used in a place dierent from where it is produced. It's a way to store energy for later use.

An energy carrier does not exist in nature but is produced using dierent primary energy sources. For dierent uses, hydrogen needs to be produced, stored, and converted into useful energy in technical systems as shown in Figure 8 representing the hydrogen chain. There are several technologies for each of the activities in this chain, each with advantages and disadvantages. In addition, the introduction of these new technologies in the circuit of mass consumption could meet the opposition or even rejection by the general public. Thus, in such condition of multiple alternatives with dierent consequences, decisions must be taken in order to establish which technologies or group of technologies should be promoted w.r.t to social acceptability. This constitutes an assessment problem, an issue that arises in energy planning.

This particular assessment problem is characterised by a high level complexity, regarding both the multiple stakeholders and the social dimensions to be considered. The complexity of the problem suggests the need to adopt an integrated methodology to assist the hydrogen social acceptability process, providing a better understanding of it without leaving important features unattended. For this purpose, a problem structuring approach was adopted.

Keeping in mind that at this stage we are interested in understanding how dierent types of stakeholders could react with respect to dierent scenarios of H 2 technologies deployment, we identied three classes of stakeholders: political decision makers, hydrogen industry actors, and the general public (citizens). In this paper, we focus only on the structure of the objectives of the public. Initially, cognitive maps relating to groups of individuals who are representative of dierent sensitivities of the public in relation to energy issues, were co-constructed. Then we implemented the approach described in section 4 to convert these cognitive maps to a value tree of the objectives of the public. To this end, through several rounds of discussions with participants including hydrogen experts, in addition to a literature review, we constructed a graphic encompassing its key points (see Figure 8). This rst study structured the knowledge about hydrogen, and then submit it to the validation of the expert group in order to focus our work on a shared vision of the problem of hydrogen. This framework is a result of our problem structuring, combining group interactions with feedback from other pilot projects in the same eld. At this stage of the process, only technical considerations were taken into account. The integration of the social acceptability in the process really began with the construction of the cognitive maps [START_REF] Damart | Focus groups de l'acceptabilité de l'hydrogène: rapport de synthèse phase qualitative[END_REF].

Three focus groups were conducted by the second author in order to gather informations on the perception of hydrogen by dierent interest groups.

The rst author participated as an observer in order to ensure that the need to bring out useful information for an implementation in a valuation model was taken into account within the discussions. Ahead of focus groups, we have identied specic needs for a multi-criteria analysis perspective such as (i) setting goals and establishing priorities and trade-os between the competitive ones, and (ii) setting criteria and alternatives. In the implementation of the focus groups, three citizen panels representing the general public were selected on the basis of their anity with the problem of energy (for more details about these specic focus groups see [START_REF] Damart | Focus groups de l'acceptabilité de l'hydrogène: rapport de synthèse phase qualitative[END_REF]):

1. Frequent users of public transport 2. Frequent users of personal car

Users of green technologies of power generation

The activity of cognitive mapping that follows a particular protocol, allowed the facilitator to build the following cognitive maps of the previous categories (Figures [START_REF] Bouyssou | Evaluation and Decision Models with Multiple Criteria: Stepping stones for the analyst[END_REF][START_REF] Checkland | Systems Thinking, Systems Practice[END_REF][START_REF] Coban | Prediction of socio-economical consequences of privatization at the rm level with fuzzy cognitive mapping[END_REF].

Value tree of objectives

The value tree representing the objectives of the public resulting from the application of the graphical conversion described in section 4 is displayed in Fig. 12. A principal characteristic of these value trees of objectives is that they branch with increasing specicity from top to bottom. This characteristic is illustrated by the fact that the lowest level (third level) contains the greatest detail. The level selected to be used as evaluation criteria in a decision aiding process needs to be suciently detailed in order to allow quantication and measurement, but not that detailed to confuse analysis by drowning decision makers in a plethora of information, deviating them from the main goal of the process. The process of shaping the value tree into an operable form is an important aspect in developing a multi-criteria based decision-aiding process, where an appropriate balance between being too general and too detailed needs to be found. Therefore, some of the detailed objectives in cognitive maps shown in Fig. 9, 10, 11 were eliminated and categorised in a dierent way, so as to have more dening objectives in the value tree, inclusive of details that were removed.

Using the theoretical foundations and the practical tips described in section 4, and following the steps in Table 2, we obtained the Meta-value tree in Fig. 12 where the concerns about the acceptability is distributed following three generic categories of actors of the public.

Step N • Description of the step The objectives of these categories of actors are detailed in a meta-tree which is a tree constructed from the trees of each category of stakeholders groups by merging dierent trees. Only the resulting meta-tree is given here (Figure 12). The rst level of the meta-tree is a separator which divide dierent stakeholders into the three generic categories of actors above. The following levels represent objectives, sub-objectives, etc.

The overall hierarchy of the value tree we obtain consists of four levels, starting with the main goal: capturing the social acceptability of H 3 and4. Then the criteria are derived from the lowest level objectives as shown in Table 5. 12 so that they can be used in a valuation model, but not only. Indeed, all subobjectives of the hierarchy of objectives are not directly measurable. Table 4 therefore allows to solve this problem by associating each low-level objective to an attribute or criterion. This is justied by the presence of Table 5 which is logically accompanied by Tables 3 and4 to ensure a consistent presentation of the hierarchical decomposition.

1

Concluding remarks

In this paper, we discussed how evidence from cognitive mapping analysis can be translated into multiple criteria decision analysis by the mean of value trees of stakeholders objectives. Our claim is that this tools integration can be done with some theoretical and practical manipulations based on some rules, exploiting and taking advantage of appropriate graphical representation of the issue w.r.t the problem formulation.

More specically, the work performed aimed at developing a methodological framework to inform the integration of CM into MCDA in the context of assessing hydrogen technology scenarios w.r.t their social acceptability. As this decision situation consists of a broad range of stakeholders with possibly conicting and unstructured views, it appears dicult to make a good or rational decision in such a social mess. In such ill-dened decision context, it was crucial that the related decision problem is structured in order to build consensus among stakeholders' objectives. However, structuring this problem needs to take specically into account how to construct such a consensus and this is the reason for which CM comes into play. A small example combining CM and value tree of objectives (VTO) has been used to illustrate our approach, paying special attention to theoretical and practical standards we propose to operate the transfer from one map to another.

Then this approach has been applied in a real world case dealing with the problem of the social acceptability of hydrogen technologies scenarios. The obtained results of this project showed that, in spite of some limitations, the framework has been able to structure the decision problems, leading to an operational and consensual evaluation model [START_REF] Kpoumié | Model for structuring and evaluating hydrogen technologies scenarios in terms of social acceptability[END_REF]. The developed methodology is quite dierent from other approaches documented in the literature where one can nd either direct assessment of options with fuzzy cognitive maps (FCM) or the generation of VTO by a wish list, but not the combined use of both techniques. It encompass both paradigms in a framework that is able to accommodate a decision context with multiple stakeholders and multiples possibly conicting objectives.

The suggestion for further developments concerns designing further experiments to test the impact of our two-stage methodology (cognitive mapping and value tree) on the consistency and eectiveness of the family of criteria obtained in the sense of Bouyssou et al. (see [START_REF] Bouyssou | Evaluation and Decision Models: A Critical Perspective[END_REF], [START_REF] Bouyssou | Evaluation and Decision Models with Multiple Criteria: Stepping stones for the analyst[END_REF]) w.r.t criteria axioms [START_REF] Kpoumié | Model for structuring and evaluating hydrogen technologies scenarios in terms of social acceptability[END_REF]. Whereupon, framing and formalising an algorithmic procedure of our integrated methodology is to be investigated.

Figure 2 :

 2 Figure2: Illustrative 3-level (partial) value tree of objectives. Sources[START_REF] Kpoumié | Model for structuring and evaluating hydrogen technologies scenarios in terms of social acceptability[END_REF]

Figure 6 :

 6 Figure6: Illustrative (partial) cognitive map on H 2 powered cars. Sources:[START_REF] Kpoumié | Model for structuring and evaluating hydrogen technologies scenarios in terms of social acceptability[END_REF]

Denition 4 . 1 .

 41 A directed graph (also called digraph) is an ordered pair of sets G = (V, A) where V = V(G) is a set of vertices and A = A(G)⊆ V(G)×V(G) a set of arcs consisting of ordered pairs of vertices of V. Denition 4.2. A digraph is said to be connected if there is path (an alternating sequence of vertices and arcs, beginning and ending with vertices, with no repeated vertices), between any two vertices. Denition 4.3. A cognitive map (CM) is a connected digraph CM = (C, A) with concepts like concerns, objectives, events, key issues, ideas, or/and opinions as vertices (C) and relationships between concepts as arcs (A). Here a CM digraph is loopless, i.e ∀u ∈ C, (u, u) / ∈A. Concepts represent ideas, opinions and key issues an individual or group of individuals associate with the investigated issue. For example, in our case, in cognitive maps capturing the perception and social acceptability of hydrogen technologies, concepts represent key issues and main options a specic group of individuals associated with the idea of hydrogen technologies and their consequences. As we state in denition 4.3, concepts on cognitives maps may be heterogeneous items (e.g concerns, opinions, ideas, etc.). In addition, cognitive maps capture in a hierarchical format (although inaccurate, imprecise and biased) how an individual explains its perspective, and why situations (strategic issues) might matter for the strategic future of an organisation (eliciting goals, objectives, values). Thus we can derive from the CM digraph, a connected subdigraph denoted by C o M = (O, A o ) where O = O(C o M ) represents a set of stakeholders objectives derived from the initial set of concepts C (O ⊆ C) and A o the corresponding subset of A consisting of ordered pairs of objectives so that A o =A∩(O×O).

Denition 4 . 4 .

 44 An arborescence with a vertex r called the root is a subdigraph T = (V ′ , A ′ ) of digraph G which does not contain a paire of opposite arcs (no cycle) and such that the following conditions hold: (i) if the directions of arcs are ignored, then T is a spanning tree; (ii) there is a path from r to every other u ′ ∈V ′ 10 Thus a value tree of objectives (V T o ) is an arborescence whose root is the overall objective.Given these denitions, the question is under which conditions we can move from a graph representation (C o M ) to a tree representation (V T o ) in accordance with the objective of our study. The following lemma gives us a tool to eectively address this issue.

  is a signed digraph CM = (C, A) with circuit (c 5 c 4 c 3 ) where C = {c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 , c 8 } is a set of concepts, A= {(c 2 , c 1 ), (c 3 , c 1 ), (c 8 , c 1 ), (c 5 , c 2 ), (c 7 , c 2 ), (c 3 , c 5 ), (c 4 , c 3 ), (c 5 , c 2 ), (c 5 , c 4 ),(c 5 , c 6 ), (c 6 , c 8 ), (c 7 , c 8 ), (c 8 , c 6 )} are relation of inuence between pairs of concepts and P + (respectively P -) the positive (respectively negative) polarities of the edges.

Figure 7 :

 7 Figure 7: Partial value tree on H 2 powered cars.

Figure 8 :

 8 Figure 8: Integrated hydrogen chain

Figure 9 :Figure 10 :

 910 Figure 9: Collective cognitive map of frequent users of public transport.

Figure 11 :

 11 Figure 11: Collective cognitive map of users of green technology of power generation.

Table 1 :

 1 Matrix connecting items between value tree and cognitive map

	1):
	Key transfer points Objet	Cognitive Map Concepts	Value Tree Objectives
	Starting point	Central concept	Overall objective
	Structure	Relational	Hierarchical
	Type of relation	Inuence	Top-down
	Nature of relation	Correlation	Preferential
	Direction	Positive/Negative Maximize/Minimize
	Stakeholders	Multiple DMs	Single DM

Table 2 :

 2 Stages of interactive elicitation of value tree of objectives

	The three generic categories of actors of the public we mentioned above
	are:
	1. Users of H 2 technical systems
	2. Neighbours of H 2 technical systems

  sales service and maintenance service. Maximising condence is achieved by maximising information sources, maximising condence in National and European norms, and maximising condence in manufacturers of H 2 systems. This description is obtained from Tables

	2 tech-
	nologies. The next (second) level is about to achieve this main objective, by
	minimising economics impacts, maximising safety, minimising environmental
	impacts, maximising services, and maximising condence. In the third level,
	the objective of minimising economics impacts is supposed to be reached by
	minimising purchase cost, minimising utilisation cost and minimising main-
	tenance cost. Maximising safety is achieved by maximising security and
	maximising reliability. Minimising environmental impacts is obtained by
	minimising nuisance, minimising climatic change, minimising batteries for
	recycling and minimising nuclear waste. Maximising services is reached by
	maximising usage autonomy, maximising the number of service stations, and
	maximising after-

Table 3 :

 3 rst level objectives 2 nd level objectives objectives hierarchyTables 3,[START_REF] Belton | Multiple Criteria Decision Analysis: An integrated approach[END_REF] and 5 are a way of presenting the information in Figure

	Economic aspects
	Safety aspects Social acceptability Environmental impacts
	Supply security
	Services
	Condence