Dr Yun Ye

Dr Marija Jankovic
email: marija.jankovic@ecp.fr

Dr Gül E Kremer
email: gkremer@psu.edu

Understanding the Impact of SubjectiveUncertaintyon Architecture and Supplier Identificationin Early Complex Systems Design

Keywords: expert estimation uncertainty, subjective uncertainty, early conceptual design stage, early complex systems design, fuzzy set theory, 2-tuple fuzzy linguistic representation

TheArchitecture & Supplier Identification Tool(ASIT) is a design support tool, which enables identification ofthe most suitable architectures and suppliers in early stages of complex systemsdesign, with consideration of overall requirements satisfaction and uncertainty.During uncertainty estimation,several types of uncertainties that are essential in early design (i.e., uncertainty of modules due to new technology integration, compatibility between modules, and supplier performance uncertainty) have been considered in ASIT. However, it remains unclear whetherexpert estimation uncertainties should be taken into account. From one perspective, expert estimation uncertainties may significantly influence the overall uncertaintysince early complex systems designgreatly depends on expert estimation; while an opposing perspective argues that expert estimation uncertaintiesshould be neglected because they are relatively much smaller in scale. In order to understand how expert estimation uncertainties(especially subjective uncertainty) influence the architecture and supplier identification results achieved through ASIT, a comprehensive study of possible modelling approaches has been discussed; and type-1 fuzzy sets and 2tuple fuzzy linguistic representation are selected to integrate subjective uncertainties into ASIT. A powertrain design case is used to compare results between considering subjective uncertainties and not considering them.

Finally,the consideration of subjective uncertainty in early conceptual design as well as other design stages is discussed.

INTRODUCTION

In early designphases of complex systems, the OEMs (Original Equipment Manufacturers) strive to explore possibilities, delay decisions, and at the same time control both the performance and the uncertainty of the future system [START_REF] Ye | Managing uncertainty in potential supplier identification[END_REF]. Nowadays,more and more OEMs tend to involve their suppliers early in design in the context of an extended enterprise [START_REF] Nguyen | System engineering for collaborative data management systems: Application to design/simulation loops[END_REF], especially in airplane and automotive industries. For example, Airbus has several thousand suppliers from more than 100 countries (Airbus Group, 2014).Among these suppliers, some arecalled first-tier suppliers, such as Snecma and Roll Royce for engines.These first-tier suppliersusually have the full responsibility of anentire module, and become involved starting in early conceptual design phase of the system in most cases. Therefore, it is important to consider the performance and uncertainty of these suppliers when estimating performance and uncertainty of the future system. However, very few methods consider both system architecture design as well as supplier information [START_REF] Ye | Managing uncertainty in potential supplier identification[END_REF]. In order to fill this gap,we proposed an Architecture & Supplier Identification Tool(ASIT) [START_REF] Ye | Managing uncertainty in potential supplier identification[END_REF] to support design teams when considering these issues. Given the lack of data, fuzziness and different uncertainties inherent to early design, ASIT takes into account uncertainties related to interfaces, suppliers' capability, and the capability of one subsystem/module to reach a certain performance. These data are estimated by experts using predefined linguisticterms (as shown in Tab.1 and Tab.2), and these linguistic terms are then translated intonumerical scales to facilitate calculation.

We have observed that when using information coming from expert estimation, researchers use methods such as fuzzy sets(e.g.,J. (Ray) [START_REF] Wang | Ranking engineering design concepts using a fuzzy outranking preference model[END_REF] and rough sets(e.g., [START_REF] Zhai | Design concept evaluation in product development using rough sets and grey relation analysis[END_REF]to model the information in order to represent the uncertainty in expert estimation. Because of new technology integration, new modules and new suppliers, ASIT also greatly relies on expert estimation. However, different from most of the existing design supporting methods, ASIT is developed for early conceptual design stage, where overall uncertainty level is much greater than in later design stages. Therefore, since the expert uncertainty is relatively small comparing to overall uncertainty, it is unclear whether using mathematical methods to model expert estimation will influence the overall results.

In this paper, we propose to analyzethe influence of considering expert estimation uncertainties on ASIT resultsby answering the following two research questions:

1. Which uncertainty modeling method is most suitable for representing the expert estimationuncertainties within the context of ASIT? 2. Does the consideration of expert estimation uncertainties influence ASIT results?

In the following sections, we start with an overview of the ASIT, and the uncertainties caused by expert estimation in ASIT. In section three, we give a review of different uncertainty representation methods that can be used to represent subjective uncertainty.

The suitability of the uncertainty representation methods is analysed based on the characteristics of the early design stage and the ASIT, making up the set of comparison criteria. In section four, the operations of fuzzy sets, the selection of suitable membership functions, and the utilization of defuzzification methods are discussed. In section five, the selected representation methods are integrated into ASIT by using a powertrain case study. In section six, the results are compared and discussed before conclusions are presented in section seven.

THE EXPERT ESTIMATION UNCERTAINTIES IN ASIT

The ASIT is an early design support tool that aims at generating possible system architectures given a list of suppliers that are co-designing the system. First, possible architectures are generated by integrating new technologies in order to better satisfy new requirements. Then, the generated architectures are filtered by uncertainty and requirement satisfaction thresholds in order to identify the architectures with relatively high performance and low uncertainty. Three major types of uncertainties are taken into account: (1) uncertainties related to capability of one subsystem/module to reach defined system performances, (2) interface related uncertainties, and(3) uncertainty related to supplier capabilities.

The ASIT contains four phases, as shown in Fig. 1. In phase one, the satisfaction of new requirements by OEM's existing products is calculated with data from existing products stored in a database. Subsequently, the un-satisfied requirements, un-satisfied functions, and responsible modules for the un-satisfaction are identifiedthrough the mapping of -requirementfunction -module type‖.In phase two, new solutions for the responsible module types are foundby experts in existing or new supplier companies. Then, all possible architectures are generated by taking one module from each module type. In phase three, uncertainty of generated architectures is calculated based on experts' estimation of three types of uncertainties; requirements satisfaction level of architectures is calculated based on experts' estimation of -function satisfaction by modules‖. Finally, inphase four, by using the uncertainty and requirements satisfaction thresholds (which are defined by the OEM based on the tolerance of requirements satisfaction, and the risk that they are willing to take), the generated architectures are filtered to identify potential architectures and corresponding suppliers. During ASIT process, experts are consulted for two types of information: satisfaction levels, and uncertainties.Although expert evaluations are often expressed in an informal way, there are many methodstomake them more precise; one such mostly used method is pre-defined linguistic terms or ordinal scales. In ASIT, the satisfaction levels [START_REF] Fiod-Neto | Assessment of product conception: A critical review[END_REF]and uncertainties are predefined as shown in Tab.1 and Tab.2. Here, the two forms (i.e., linguistic and numerical) are provided together since the linguistic terms provide explanatory notes while the numerical levels facilitate aggregation. (Meyer and Booker,2001). As human beings, experts generate two categories of uncertainty: aleatory (random) and epistemic (subjective) [START_REF] Medsker | Knowledge acquisition from multiple experts: Problems and issues[END_REF]. The aleatory uncertainty is also referred to as inherent uncertainty, irreducible uncertainty, and variability. It describes uncertainty due to random variation or inherent variation [START_REF] Booker | The role of expert knowledge in uncertainty quantification (are we adding more uncertainty or more understanding?[END_REF]. This type of uncertainty is considered to be rooted in the way the brain processes information [START_REF] Dror | Why experts make errors[END_REF]. The epistemic uncertainty is also referred to as subjective uncertainty and reducible uncertainty. The fundamental cause for this type of uncertainty is incomplete information or incomplete knowledge onsystem characteristics or the environment [START_REF] Oberkampf | Mathematical representation of uncertainty[END_REF]. Aleatory and epistemic uncertainties both exist in expert estimation. However, in early conceptual design phase of complex systems, very limited amount of information exists;this causes high level of epistemic uncertainty. In comparison to epistemic uncertainty, thealeatory uncertainty, in this phase, is much smaller in scale. Therefore when integrating the two types of uncertainties, the aleatory uncertainty can be covered by the epistemic uncertainty. That is why we propose to consider only the epistemic uncertainty in the context of this study.

One of the strategies to reduce epistemic uncertainty in complex system design is to use expert group evaluations [START_REF] Medsker | Knowledge acquisition from multiple experts: Problems and issues[END_REF].In this study, we assume that a group of experts is used for each estimation in order to prevent from the situation that one expert does not have sufficient knowledge in a specific domain, or multiple domains.

As per the discussion above, the most important uncertainty caused by expert estimation in ASIT is the subjective (epistemic) uncertainty, which is caused by the limited information in early design. In order to test the sensitivity of the ASIT tothis type of uncertainty, we need to first choose a suitable uncertainty representation method to model this type of uncertainty. In the next section, several approaches that can model the subjective uncertainty are reviewed and their appropriateness for use within ASIT is discussed.

REPRESENTING SUBJECTIVE UNCERTAINTY IN ASIT

In order to identify the most suitable method for representing subjective uncertainty in ASIT, we have compiled uncertainty representation methods that have been the most commonly used for representing this type of uncertainty [START_REF] Ng | Uncertainty management in expert systems[END_REF][START_REF] Oberkampf | Mathematical representation of uncertainty[END_REF][START_REF] Booker | The role of expert knowledge in uncertainty quantification (are we adding more uncertainty or more understanding?[END_REF]: Epistemic uncertainty has been traditionally modeled as probability distributions. The probability theory has four perspectives including classical, empirical, subjective and axiomatic probabilities [START_REF] Asadoorian | Essentials Of Inferential Statistics[END_REF]. The expert estimation can be modeled as subjective probability, which represents an individual's measure of belief that an event will occur. The information gathered for the distribution could be a mixture of limited experimental data and a person's experience, or the elicitation of multiple expert opinion [START_REF] Oberkampf | Mathematical representation of uncertainty[END_REF]. The main concern of this method is that the -fuzziness‖ of information is usually lost since in probability theory, an event either occurs or not [START_REF] Ng | Uncertainty management in expert systems[END_REF].


The imprecise probability [START_REF] Walley | Statistical reasoning with imprecise probabilities[END_REF] is a generalization of probability theory; it is used when a unique probability distribution is hard to identify. In imprecise probability, a lower probability and an upper probability are used instead of one single probability.

For an uncertain event A, instead of assigning a single probability P(A), the imprecise probability assigns an interval ( ), ( ) P A P A   , with 0 ( ) ( ) 1 P A P A  , where () PAis the lower probability for A, () PAis the upper probability for A, and ( ) ( )

A P A P A   
is the imprecision for event A [START_REF] Coolen | On the Use of Imprecise Probabilities in Reliability[END_REF]. Similar to the probability theory, the fuzziness of information is thought to be lost when using imprecise probability.

The evidence (Dempster-Shafer) theory [START_REF] Dempster | Upper and Lower Probabilities Induced by a Multivalued Mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF])is a generalization of the Bayesian theory of subjective probability. It allows considering the confidence one has in the probabilities assigned to the outcomes. The evidence theory uses an interval to represent the probabilities with a lower bound called -believe‖ and an upper bound called -plausibility‖. The -believe‖ is the sum of the evidence that supports the hypothesis, while the -plausibility‖ is 1 minus the sum of the evidence that opposes the hypothesis. According to [START_REF] Ng | Uncertainty management in expert systems[END_REF], one obvious problem of the evidence theory is its implementation complexity since experts must provide all the beliefs for all subsets of possible hypotheses.

In [START_REF] Zadeh | Fuzzy sets[END_REF][START_REF] Zadeh | Fuzzy sets[END_REF] started a revolution in uncertainty thinking by introducing the fuzzy set theory. This theory uses a membership function to represent the degree of membership of an element to a set of objects. The degree to which an element belongs to a set is defined by a value between 0 and 1; the higher the value is the greater its belongingness, and an element can partly belong to a fuzzy set. The fuzzy set is widely used in describing linguistic information since it can effectively represent the gradual changes of people's perception of a concept in a certain context [START_REF] Dalalah | Diagnosis and management of strep throat: remote multicriteria fuzzy reasoning approach[END_REF]. Moreover, the fuzzy set theory also allows mathematical operations that help to provide quantitative methods to deal with qualitative data.

The possibility theory [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF][START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF][START_REF] Cooman | Foundations and Applications of Possibility Theory[END_REF] is an extension of the theory of fuzzy sets. It can be used to express the vague terms used by human experts with precision and accuracy [START_REF] Ng | Uncertainty management in expert systems[END_REF].

Interval analysis [START_REF] Moore | Methods and Applications of Interval Analysis[END_REF][START_REF] Kearfott | Applications of Interval Computations[END_REF] is an approach that treats an interval as a new kind of number [START_REF] Moore | Methods and Applications of Interval Analysis[END_REF] and follows the following elementary properties [START_REF] Moore | Interval analysis and fuzzy set theory[END_REF]:

[ , ] [ , ] [ , ] a b c d a c b d     (1) [ , ] [ , ] [ , ] a b c d a d b c     (2) [ , ][ , ] [min( , , , ), max( , , , )] a b c d ac ad bc bd ac ad bc bd  (3) [ , ] [ , ] [ , ][1/ ,1/ ] a b c d a b d c  where, 0 [ , ] cd  (4)
The rough sets theory [START_REF] Pawlak | Rough sets[END_REF] uses a pair of sets to give the lower and upper approximation of the original set. This theory is used when objects are characterized by the same information and thus are indistinguishable [START_REF] Pawlak | Rough set approach to knowledge-based decision support[END_REF]. Each rough set has boundary-line elements, which cannot be properly classified using the available knowledge.

These methods have all been used in previous works to represent expert's subjective uncertainty. However, it is not clear how to choose a method according to a specific context (e.g., for ASIT). In order to identify approaches that can be seen as most suitable in the context of ASIT, we propose to consider the following four criteria:

1. The method is able to represent numerical levels:In ASIT, experts use predefined linguistic terms and related numerical levels to represent their estimation. Therefore, the chosen method should be able to represent these types of dis-continuous numerical levels.

The method requires reasonable amount of information:

In ASIT, the estimation is provided by experts. Due to the limitationsin time, budget and human capacity, the amount of information required by the mathematical representation has to be reasonable.

The method should capture the "fuzziness" of expert estimation:

Human knowledge is imprecise. The chosen method should be able to capture this kind of imprecision.

Support multi-criteria group decision making:

The aim of the potential supplier identification is to provide candidates for the supplier selection, which is usually a multi-criteria group decision making problem. In order to be able to use the supplier identification results directlyduring the supplier selection stage, the selected uncertainty representation method should be able to combine with other methods to support the multi-criteria group decision making.

With regard to the first criterion (representing the numerical levels), the rough set theory is not suitable since it assumes that some of the elements are characterized by the same information hence indistinguishable. However, the elements defined by the numerical levels are clearly distinguishable (e.g., numbers 1 -10 used in Tab.1).

In view of the second criterion (requiring reasonable amount of information), the evidence theory is inappropriate. According to the principles of evidence theory, experts need to provide 2 x beliefs foreach estimation, where x represents the number of elements. For example, when using satisfaction levels provided in Tab.1, x is equal to 10. Therefore, when estimating -how well module A satisfies function B‖, experts need to provide 2 10 beliefs. With the increase of module and function counts, the number of estimation needed also increases.

The third criterion (capturing the -fuzziness‖ of expert estimation) makes the subjective probability theory, the imprecise probability theory, the possibility theory and the interval analysis theory inappropriate since they are not designed to capture the -fuzziness‖.

With regards to the fourth criterion (support multi-criteria group decision making), many fuzzy set based multi-criteria group decision making methods exist, including:

 Fuzzy set theory + TOPSIS [START_REF] Chen | A fuzzy approach for supplier evaluation and selection in supply chain management[END_REF],  Intuitionistic fuzzy sets + TOPSIS [START_REF] Boran | A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method[END_REF],

 Fuzzy AHP [START_REF] Kahraman | Multi-criteria supplier selection using fuzzy AHP[END_REF][START_REF] Haq | Fuzzy analytical hierarchy process for evaluating and selecting a vendor in a supply chain model[END_REF][START_REF] Chan | Global supplier selection: a fuzzy-AHP approach[END_REF],

 Fuzzy AHP + cluster analysis [START_REF] Bottani | An adapted multi-criteria approach to suppliers and products selection-An application oriented to lead-time reduction[END_REF],

 Fuzzy ANP [START_REF] Vinodh | Application of fuzzy analytic network process for supplier selection in a manufacturing organisation[END_REF],

 Fuzzy ANP + TOPSIS [START_REF] Önüt | Long term supplier selection using a combined fuzzy MCDM approach: A case study for a telecommunication company[END_REF],

 Fuzzy multi-objective programming [START_REF] Amid | Fuzzy multiobjective linear model for supplier selection in a supply chain[END_REF],  Fuzzy arithmetic operation [START_REF] Bayrak | A fuzzy approach method for supplier selection[END_REF],  Fuzzy SMART [START_REF] Chou | A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach[END_REF],  Fuzzy QFD [START_REF] Bevilacqua | A fuzzy-QFD approach to supplier selection[END_REF],  Fuzzy DEMATEL + TOPSIS [START_REF] Dalalah | A fuzzy multi-criteria decision making model for supplier selection[END_REF].

With regard to previously discussed criteria, the fuzzy set theory appears to be the most suitable mathematical representation to express subjectivity in expert estimations. In recent years, several branches of fuzzy set theory were developed; the most popular ones among these with applications in supplier identification and selection are:

 Type-1 fuzzy sets (e.g., Önüt et al., 2009),  Interval type-2 fuzzy sets (e.g., Chen & Lee, 2010),  Intuitionistic fuzzy sets (e.g., [START_REF] Boran | A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method[END_REF],  2-tuple fuzzy linguistic representation (e.g., [START_REF] Wang | A fuzzy linguistic computing approach to supplier evaluation[END_REF].

The type-1 fuzzy setsis the same as the ordinary fuzzy set theory. The re-name is for the purpose of distinguishing from the type-2 fuzzy sets. This theory uses a membership function to represent the degree of membership of an element to a set. The degree to which an element belongs to a set is defined by a value between 0 and 1.Higher the value is the greater its belongingness, and an element can partly belong to a fuzzy set.

The fuzzy set is widely used in describing linguistic information as iteffectively represents the gradual changes of people's perception of a concept in a certain context [START_REF] Dalalah | Diagnosis and management of strep throat: remote multicriteria fuzzy reasoning approach[END_REF].It is also widely used for representing human uncertainty.

The interval type-2 fuzzy sets is a simplified form of type-2 fuzzy sets, which is defined by [START_REF] Mendel | Type-2 fuzzy sets made simple[END_REF] based on Zadeh's Extension Principle [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning-I[END_REF]. The The intuitionistic fuzzy sets was proposed by [START_REF] Atanassov | Intuitionistic fuzzy sets[END_REF] twenty years after Zadeh's fuzzy sets. The intuitionistic fuzzy sets use dual membership degrees in each of the sets by giving both a degree of membership and a degree of non-membership.

Similar to the interval type-2 fuzzy sets, it might be difficult for experts to define two membership functions foreach estimation; thus, the intuitionistic fuzzy setsdo not seem convenient for representing subjective uncertainty in expert estimation either.

The 2-tuple fuzzy linguistic representation is developed by [START_REF] Herrera | A 2-tuple fuzzy linguistic representation model for computing with words[END_REF] based on the fuzzy set theory and the symbolic method [START_REF] Delgado | On aggregation operations of linguistic labels[END_REF]. The linguistic values (e.g., expert estimation) are usually modelled as fuzzy sets.

When aggregating the linguistic values (fuzzy sets), the result may not exactly match any of the initial linguistic terms, and thus an approximate linguistic term must be found.

However, the imprecision of this approximation is lost. In the 2-tuple fuzzy linguistic representation, the linguistic information is expressed by a 2-tuple ( , ) s  , where s represents the approximate linguistic term, and  represents the imprecision of this approximation. This representation can efficiently prevent the loss of information and thus help the ranking of alternatives.

Given the discussion above, we think that the type-1 fuzzy sets and the 2-tuple fuzzy linguistic representation are both suitable for representing subjectivity in expert estimations during early design. Therefore, we propose to use these two approaches to represent the subjective uncertainty and compare them to the initial results where subjectivity is not taken into account. Before integrating the two fuzzy methods into ASIT, the fuzzy techniques to be used should be defined, e.g., the operations of fuzzy sets,the selection of suitable membership functions, and the utilization of defuzzification methods. Theseare discussed in the next section.

FUZZY TECHNIQUES FOR REPRESENTING SUBJECTIVE UNCERTAINTY IN ASIT

Many different fuzzy membership functionsexist. It is not possible or appropriate to test all of them in this work. Therefore, it is necessary to identify the most appropriate membership function within our research context. In addition, the simplification for fuzzy number operations should also be reviewed, since operations such as the multiplication of several fuzzy members can be very tedious.

A fuzzy number is a special fuzzy set  

( , ( )),

N N x x x R   
, where x is a real value, 1 : R X     and ()

N x
 is a continuous mapping from 1 R to (0,1) [START_REF] Haq | Fuzzy analytical hierarchy process for evaluating and selecting a vendor in a supply chain model[END_REF].Operations of fuzzy numbers can be defined based on the extension principle proposed by [START_REF] Zadeh | The concept of a linguistic variable and its application to approximate reasoning-I[END_REF]. If M  and N  are fuzzy numbers, membership of (*)

M N   is
defined as follow [START_REF] Gao | Multiplication Operation on Fuzzy Numbers[END_REF]:

(*) * ( ) sup min{ ( ), ( )} M N M N z x y z x y          (5)
Where * stands for any of the four algebraic operations including addition, subtraction, multiplication and division.

Fuzzification is the process of making a crisp quantity fuzzy (Ross, 2009, p. 93), which is normally the first step in using fuzzy set theory. The main objective of fuzzification is to define a membership function for each fuzzy quantity. The commonly used membership functions for linguistic terms are triangular, trapezoidal, left shoulder, right shoulder, Gaussian and Sigmoid [START_REF] Garibaldi | Choosing membership functions of linguistic terms[END_REF]. Among the various shapes of membership functions, the triangular membership function has been frequently used in many fuzzy set applications [START_REF] Pedrycz | Why triangular membership functions? Fuzzy Sets and Systems[END_REF]. [START_REF] Pedrycz | Why triangular membership functions? Fuzzy Sets and Systems[END_REF] The basic features of triangular fuzzy numbers can be found in the work of [START_REF] Dubois | Operations on fuzzy numbers[END_REF], and the basic operations can be found in the work of Chou andChang(2008). A triangular fuzzy number can be denoted as

( , , ) N l m u  , its membership function ( ) : [0,1] N xR  
is represented as [START_REF] Chang | Applications of the extent analysis method on fuzzy AHP[END_REF]:

1 1 , [ , ], 1 ( ) , [ , ], 0, otherwise. N x x l m m l m l u x x x m u m u m u                    (6)
Where l m u  , l , m and u are the lower bound of the support, the core, and the upper bound of the support of N, respectively.

Given two triangular fuzzy numbers Addition of two fuzzy numbers  :

1 1 2 2 3 3 ( , , ) A B a b a b a b        (7)
Addition of a real number k and a fuzzy number  :

1 2 3 ( , , ) k B k b k b k b       (8)
Multiplication of a real number k and a fuzzy number  :

1 2 3 ( , , ) k B kb kb kb   (9)
Although the multiplication of a real number and a fuzzy triangular number is easy to calculate, a higher level operation (i.e., multiplication of two or several fuzzy numbers) is cumbersome with insurmountable computational effort [START_REF] Triantaphyllou | Fuzzy Multi-Criteria Decision Making[END_REF]. Gaoet al. ( 2009) demonstrated that the result from multiplication of two triangular fuzzy numbers is not a triangular fuzzy number, and the result can be obtained by using nonlinear programming method, analytical method, computer drawing method and computer simulation method. [START_REF] Gao | Multiplication Operation on Fuzzy Numbers[END_REF] demonstrated that by using an analytical method, for two triangular fuzzy numbers such as

1 2 3 1 2 3 ( , , ), ( , , ) A a a a B b b b     , the membership function of () N A B     is: 2 1 2 1 2 1 1 1 2 1 2 2 3 2 1 1 1 2 2 2 1 2 1 2 3 2 3 2 3 3 3 2 3 2 2 3 2 3 2 2 3 3 2 3 2 3 ( 2 ) ( ) 4( )( ) , 2( )( ) ( 2 ) ( ) 4( )( ) ( ) , 2( )( ) 0,
                           otherwise            (10)
Onenotices that the multiplication of several fuzzy numbers requires significant computational effort to obtain precise results. Therefore, in order to facilitate its application in engineering problems, a simplified formula is usually used [START_REF] Chiou | Evaluating sustainable fishing development strategies using fuzzy MCDM approach[END_REF][START_REF] Tzeng | Multiple Attribute Decision Making: Methods and Applications[END_REF]:

1 1 2 2 3 3 ( , , ) A B a b a b a b    (11)
In decision making problems, a single scalar is preferred as output of a fuzzy process in order to facilitate ranking or selection. To transform fuzzy results into a scalar, defuzzification is performed. Defuzzification is defined as a mapping of fuzzy sets to elements of the universe considered significant with respect to this fuzzy set [START_REF] Runkler | Selection of appropriate defuzzification methods using application specific properties[END_REF]. Widely used defuzzification methods are maximum (max) membership principle, centroid method, weighted average method, mean max membership, center of sums, and center of largest area [START_REF] Ross | Fuzzy Logic with Engineering Applications[END_REF].Within the context of this study, the triangular fuzzy membership functions are used, which are peaked output functions with their maximum equal to the-significant element‖. Since in peaked output functions, the max membership principle is commonly used [START_REF] Ross | Fuzzy Logic with Engineering Applications[END_REF], we propose to use this defuzzification method in the 2-tuple linguistic representations.The max membership principle is given by the expression [START_REF] Ross | Fuzzy Logic with Engineering Applications[END_REF]):

( *) ( ), for all

A A z z z Z       (12)
Where z* is the defuzzified value.

In the 2-tuple linguistic representation, the single number obtainedfrom defuzzification istransformed again to the initial expression domain (i.e., the predefined linguistic terms). The 2-tuple linguistic representation uses a 2-tuple ( , ) s  to represent the results, where s refers to the closest linguistic term, α is a numerical value expressing the value of the translation from the original result to the closest linguistic term [START_REF] Herrera | A 2-tuple fuzzy linguistic representation model for computing with words[END_REF].

Definition: Let 0 { ,..., } g S s s 
be a linguistic term set and

[0, ] g  
a value representing the result of a symbolic aggregation operation.Then, the 2-tuple that expresses the equivalent information to  , which is obtained with the following function (adopted from [START_REF] Herrera | A 2-tuple fuzzy linguistic representation model for computing with words[END_REF]:

:[0, ] [ 0.5 , 0.5 ) g S i i      (13)
, ( ) ( ) ( , ),with , [ 0.5 ,0.5 )

i i s i round s i i i                  (14)
Where round (·) is the usual round operation, s i has the closest term to -β‖, and -α‖ is the value of the symbolic translation. δi is the gap between i and i-1 for {1, 2,... } i g  .

Because of the lack of additional information in early conceptual design stage, the isosceles triangular membership function isused in this work. We assume that the support of the triangular fuzzy number is twice the predefined scale, which indicates the assumption that the group of experts are able to choose the correct linguistic term. In case of greater or smaller subjective uncertainty, the support of the triangular fuzzy number can also be changed, but the principle and the reasoning of this workremains the same, and the results obtained in this work will not be greatly influenced.

In the next section, the two fuzzy methods are integrated into ASIT using a powertrain design case as the context.

THE POWER TRAIN DESIGN CASE

This powertrain design case is used to show the initial ASIT results, and theintegration ofsubjective uncertaintiesthrough the two selected fuzzy methods.

A powertrain is a system of mechanical parts in a vehicle that first provides energy, then converts it in order to propel the vehicle. Due to the increasing demand of lower emissions and higher fuel efficiency, the OEM plans to design a new powertrain for their motor vehicle to better satisfy market needs. Although the powertrain is normally an in-house subsystem, with only few modules outsourced (the battery and the engine for example), we assume in this case study that each module in the powertrain is planned to be outsourced to one supplier, for the purpose of illustration. The powertrain design case used in this work is very similar to the one used in [START_REF] Ye | Managing uncertainty in potential supplier identification[END_REF]; the data used is the same except the added uncertainty information shown in Figure 6. See [START_REF] Ye | Managing uncertainty in potential supplier identification[END_REF] for more information about ASIT and the powertrain design case.

ASIT without Considering Subjective Uncertainty

In ASIT phases 1 and 2, the satisfaction of new customer needs by existing products is estimated, the modules that should be improved are identified, and new modules are found. In phase 3,experts start providing estimation on satisfaction levels and uncertainties, after the generation of all possible architectures in which new modules are integrated. Since the subjective uncertainty studied in this work is caused by expert estimation, we focus on phases 3, 4, and 5 in this case study because of space limitation.

Please see [START_REF] Ye | Managing uncertainty in potential supplier identification[END_REF] for further information about the other phases of ASIT.

We assume that modules that can sufficiently satisfy new requirements are found. All possible architectures with integration of these new modules are generated, as shown in Fig. 2. The number -1‖ represents that the module belongs to the architecture, while the -0‖ represents the module does not belong to the architecture. In phase 3, to calculate the overall uncertainty and satisfaction level ofeach architecture, estimations are provided by experts using scales defined in Tab.1 and Tab.2. Since the structure of the product is complex and the requirements vary from project to project, it is difficult for experts to estimate how well a product satisfies a requirement directly. In comparison, it is much easier to estimate how well a module satisfies a function, as shown in Fig. 3. The numbers in Fig. 3 represent satisfaction levels defined in Tab.1. For example, the -engine 1‖ is a -weak solution‖ for satisfying the function -respect environment‖. Therefore, using information from Tab.1, the satisfaction level 2 is assigned to this estimation. Since the significant figures include all the precise digits and the first estimated digit [START_REF] Serway | Physics for Scientists and Engineers with Modern Physics[END_REF], we keep two significant figures for the estimation. Then, using the composition of architectures (Fig. 2), the -satisfaction of functions by modules‖ is propagated to the -satisfaction of functions by architectures‖. For simplicity, we assume that how an architecture satisfies a function depends on how the modules in the architecture satisfy the function. The function satisfaction level by an architectureis defined as the average of its modules' satisfaction levelsfor this function.

The -satisfaction of functions by architectures‖ is then propagated to the -satisfaction of requirements by architectures‖ by using the relations between requirements and functions (Fig. 4).The requirementfunction relations in Fig. 4 represent the percentage that a function satisfies a requirement. For example, the requirement -CAFE standard‖ is satisfied 50% by the function -Economize fuel‖, 50% by -provide power‖. For propagating the function satisfactions to requirements satisfactions, weused:

1 req arch fun arch M M M   (15)
Assuming equal importance of the requirements (this assumption can be changed if needed), an overall requirements satisfaction score is obtained for each of the possible architectures by calculating the average (Fig. 5).This score represents how well the architecture satisfies the entire requirements set (with a 1-10 scale). We round the final result to two significant figures. The overall uncertainty of an architecture is calculated based on uncertainty of modules (M 5 in Fig. 6), compatibility between modules (M 4 ), and uncertainty of suppliers' capabilities (M 6 ). The matrix M 3 shows the supplier of each module. Since the uncertainties and compatibilities can all be considered as probabilities, -the overall uncertainty of an architecture‖ is defined as the product of all its modules' uncertainties, its suppliers' uncertainties, and the compatibilities between its modules (independence of probabilities is assumed).Experts provide their estimations in linguistic terms defined in Tab.2.These linguistic terms are then converted intonumerical scales.

The overall uncertainties of possible architectures are shown in Fig. 7, which represents the percentage that an architecture can be developed without any problem. The confidence that one company has on the capabilities of a given supplier is also integrated in this overall uncertainty.

Taking into Account Subjective Uncertaintyin ASIT

Using Type-1 Fuzzy Sets

Using this method, each expert's estimation of satisfaction levels (using Tab.1)is converted to a 1 10   fuzzy number scale. The isosceles triangular membership functions are used as shown in Fig. 8. These fuzzy numbers are then aggregated by using fuzzy operations defined in section 4. Let us take the satisfaction of requirement -CAFE standard‖forarchitecture 6 as an example. In order to calculate how the architecture 6 satisfies the requirement -CAFE standard‖, the functions -Economize fuel‖ and -Provide power‖ need to be considered since they each satisfy 50% of this requirement (Fig. 4).The possible architecture 6 is composed of modules -engine 2‖, -battery 2‖, -transmission 1‖, -electric motor 1‖, -driveshaft 1‖, and -final drive 1‖. In order to calculate how the architecture satisfies the function -Economize fuel‖, we need to calculate the average of how the engine 2 and the battery 2 satisfy this function. Therefore: 2 (0.5 4) (0.5 (9,10,11)) 2 (4.5,5,5.5) (6.5,7,7.5)

FSL       
Here, the satisfaction level -4‖ is not converted to a fuzzy number, since the engine 2 is an existing module. Therefore, the information of engine 2 comes from previous projects, instead of expert estimation, so that the subjective uncertainty is not considered. Using the same principle, satisfaction of function -Provide power‖ can be calculated. Then, the satisfaction of requirement -CAFE standard‖ can be calculated by: 1 (0.5 (6.5, 7, 7.5)) (0.5 (5,5.5, 6))

(3.25,3.5,3.75) (2.5, 2.75,3) (5. 75, 6.25, 6.75)

RSL       
Assuming equal importance of the requirements (which can be changed if necessary), an overall requirements satisfaction score is obtained by calculating the average for each possible architectures (shown in Fig. 9). In this case, it is obvious that architectures 1, 2, 3, and 4 are below the satisfaction threshold (since the entire fuzzy number is below the threshold), while architectures 9, 10, 11, and 12 are above the threshold. The situation for architectures 5, 6, 7, and 8 is more complicated, because they are partly above and partly below the threshold. We discuss this kind of situations later in this chapter.

When estimating uncertainty of modules (M 5 ), compatibility between modules (M 4 ), and uncertainty of suppliers' capabilities (M 6 ), probability levels are needed from Tab.2.

In order to integrate subjective uncertainty, we convert each numerical level in Tab.2 into a triangular fuzzy number.The membership function of the fuzzy number set is shown in Fig. 11. Since -0‖ represents -impossible‖ and -1‖ represents -certain‖, we have low fuzziness level, thus we keep them as crisp values during fuzzification.

The overall uncertainty of an architecture is defined as the product of all its modules' uncertainties, its suppliers' uncertainties and the compatibilities between the modules.

Therefore, the multiplication of several triangular fuzzy numbers is needed. Simplified as described in section 4, the result achieved is shown in Fig. 12. The illustration of overall uncertainties is shown in Fig. 13, with the threshold set at 0.1. In this case, it is quite obvious that architectures 1, 5, and 9 are below the threshold, while the architectures 2, 6, 3, 4, and 7 are above it. However, it is more difficult to define the situation of architectures 10, 11, and 12. We focus on the fuzzy results that are partly above and partly below the threshold for both satisfaction levels and uncertainties, as shown in Fig. 14. The utilization of fuzzy methods is to the purpose of considering fuzziness in human estimation, which is also represented in the results shown in Fig. 14. Therefore, when considering whether these results pass the threshold, the fuzziness should also be considered, which means that the belongingness of these results to the set that passes the threshold is also fuzzy. Therefore, we think that each of these results should have a degree of -passing the threshold‖, and whether the architecture or supplier belongs to the identified architecture or supplier depends on decision maker's tolerance about the result.

There are many ways to represent decision makers' tolerance level. In this work, we use the α-cut to represent the tolerance, and define that the result passes the threshold if the maximum value of the fuzzy number after the α-cut passes the threshold. With this definition, the decision totally depends on the value of αwhich represents the tolerance level of the decision maker (bigger α is, smaller the tolerance is).For example, if we set α at 0.7 for the overall uncertainty as shown in Fig. 15, the architectures 10 and 12 will pass the threshold and the architecture 11 does not pass the threshold. However, the purpose of using the α-cut is only to show that these kinds of fuzzy results should be considered fuzzy regarding their belonging to the identified candidates. They can be below or above the threshold regarding decision makers' tolerance. Therefore, in architecture and supplier identification results, we represent this kind of candidates as -possible candidates depend on tolerance level‖.

Using 2-Tuple Fuzzy Linguistic Representation

The 2-tuple fuzzy linguistic representation is exactly the same as the type-1 fuzzy sets from fuzzification until obtaining fuzzy results (shown inFig.9 and Fig. 12). After obtaining fuzzy results, the 2-tuple fuzzy linguistic representationdefuzzifies the results, and converts the defuzzification result back to the closest initial linguistic terms, and the distance between the original results to the closest linguistic term.

According to the reasoning in section 4, the max membership principle is used for defuzzification, and the results are shown in Fig. 16 and Fig.17 represents the closest linguistic term, and α represents the distance. For example, the overall requirements satisfaction level of architecture 1 is equal to 4.7 (see Fig. 16), its closest linguistic terms is represented by the level 5, which is -a satisfactory solution‖.

The distance between 4.7 and 5 is equal to -0.3. That is why in 2-tuple fuzzy linguistic representation, the overall requirements satisfaction level of architecture 1 is (N 5 , -0.3).

The resultsof requirement satisfaction and uncertainty when using the 2-tuple fuzzy linguistic representationare shown in Fig. 18 and Fig. 19, respectively. comparison of the results, we see that without considering subjective uncertainty, ASIT identifies three architectures (6, 7 and 8), which are also identified with consideration of subjective uncertainty using the two fuzzy methods.

When using type-1 fuzzy sets, three -possible architectures depend on tolerance level‖ are also identified (numbers 10, 11 and 12). Whether these three architectures belong to the identified architectures depends on the decision makers' tolerance level.

When using the 2-tuple fuzzy linguistic representation, aside from architectures 6, 7 and 8, two additional architectures (10, 12)are identified. This expansion of thescopein results is because that the 2-tuple fuzzy linguistic representation converts the defuzzification result to the initial linguistic terms, which rounds up the results, thus increases the tolerance level.

From the comparison one can observe that (1) the architectures and suppliers identified without considering subjective uncertainty are also identified with consideration of subjective uncertainty by using the two fuzzy methods; (2) considering subjective uncertainty enlarges the result's scope mainly because the consideration of subjective uncertainty increases the level of tolerance (i.e. more tolerant) when filtering candidates.

However, the tolerance level can also be changed without considering subjective uncertainty, and by simply changing the thresholds. As shown in Fig. 21, by changing the uncertainty threshold to 0.05, the samescope for results can be obtained as integrating subjective uncertainty using 2-tuple linguistic representation. By changing the uncertainty threshold to 0.04, same results can be obtained as using type-1 fuzzy sets to model subjective uncertainty. According to the analysis above, we found that (1) the result of ASIT without considering subjective uncertainty is reliable because the identified architectures are also within the result's scope with consideration of subjective uncertainty;

(2) same results as considering subjective uncertainty can be obtained without considering subjective uncertainty by simply changing the value of thresholds.Therefore, we conclude that the consideration of subjective uncertainty does not considerably influence ASIT results, so that it is not necessary to consider the subjective uncertainty in ASIT in early conceptual design stage.However, it is very important to note that this conclusion is obtained under the situation of using triangular membership functions and which is due to the lack of information in early conceptual design stage. In other design stages, this conclusion may not be valid since when more information is available, other types of fuzzy membership functions can be used, which may lead to different results.

This result also points out that under certain situation, it is not useful to consider expert estimation uncertainty. Considering expert estimation uncertainty without analyzing the situation may result in wasted effort.

There are several limitations in this work, however. Firstly, although the max membership principleis demonstrated as a proper choice for defuzzification, it is still interesting to test other defuzzification methods, which may lead to different outcomes.

Secondly, we assumed that using a group of experts is able to correct the cognitive bias of each expert. However, the cognitive bias is potentially a fruitful issue that is worth investigating further under the topic of expert uncertainty. In addition,this work is carried out specifically in the context of ASIT. It may benefit from a generalization to a broader context.

CONCLUSIONS

Due to innovation integration in early design phases, aside from previous project data, expert estimations are often used. In this work, we investigated how subjective uncertainty resulting from expert estimations influences the result of ASIT; ASIT is an early design support tool proposed in our previous work. It is important to understand to what extent this approach can be used, and can yield robust results in an industrialcontext, and whether it is necessary to consider subjective uncertainty in ASIT.

After analysing different uncertainty representation methods for subjective uncertainty, both Type-1 and 2-tuples fuzzy sets were found suitable for representing this type of uncertainty in ASIT. A powertrain design case was used to compare the results of the original ASIT and the results with integration of subjective uncertainties. The comparison shows that considering subjective uncertainty enlargesthe result set (i.e., more architectures and suppliers are identified) because the consideration of subjective uncertainty increases the level of tolerance (i.e., more tolerant) when filtering candidates. However, the initial set without considering subjective uncertainty was found to match the results in part; note that with the uncertainty integration the set of results expanded. Therefore, considering subjective uncertainty in ASIT will not have a considerable impact on the overall ASIT results, so that it is unnecessary to consider subjective uncertainty in ASIT in early conceptual design stage.

The result of this work is valid only in the context of early conceptual design, where information is lacking, so that only triangular membership function can be used. The results also pointed out that it is not always necessary to consider expert estimation uncertainty. Before considering this type of uncertainty, the context should be analysed to prevent the waste of effort.

REFERENCE

Airbus Group. (2014). Airbus -Global Website -Our Suppliers. Retrieved April 14, 2014, from http://www.airbus-group.com/airbusgroup/int/en/our-company/Oursuppliers.html

Figure

  Figure 1 OVERVIEW OF ASIT

  Subjective probability theory,  Imprecise probability theory,  Evidence (Dempster-Shafer) theory,  Fuzzy sets,  Possibility theory,  Interval analysis theory, and  Rough sets.

  type-2 fuzzy setsis able to model one additional degree of uncertainty than the type-1 fuzzy sets. In type-1 fuzzy sets the membership functions are crisp, but in type-2 fuzzy sets the membership functions are themselves fuzzy. Therefore, a type-2 membership function is a three-dimensional membership function, which is sometimes difficult to understand and define. The interval type-2 fuzzy sets simplified the fuzziness of the primary membership function of type-2 fuzzy sets by assuming that the fuzziness of the primary membership function is equal to one. Therefore, the interval type-2 fuzzy sets can be seen as composed of an upper membership function and a lower membership function, which are both of type-1 membership functions. The main problem with this representation for representing expert estimation is that it might be already difficult for experts to define one membership function; defining two is even harder. Moreover, in the context of representing expert estimation, the definition of the second membership function does not provide further understanding of the problem. Consequently, the interval type-2 fuzzy setsdoes not seem to beeffectiveto represent subjective uncertainty in expert estimation.

  explained the reason of the popularity of the triangular membership function: -Undoubtedly, if the semantics of a certain linguistic term has to be specified, then the simplest form of the membership function one could think of would be to provide a modal (typical) value of the considered term along with the lower and upper bounds. The distribution of the grades of membership between these boundaries is then linearan acceptance of any other form of relationship to bear some legitimacy may definitely call for some auxiliary information about the membership values to be furnished at the selected intermediate points distributed within these bounds.‖ In this paper, we propose to use the triangular fuzzy membership function mainly because of the lack of information in early conceptual design stage. As presented previously, experts use predefined linguistic terms and numerical levels to give their estimation. Therefore, we have two types of information for defining a fuzzy membership function: (1) the estimation that is given by a group of experts; (2) the predefined numerical levels, and the distance between the levels. When using the triangular membership function, we assign the expert estimation as the mode of the triangular fuzzy number, and twice the distance between two adjacentlevels as the support. To the best of our knowledge, other types of membership functions all require more information than this.Therefore, we propose to use the triangular membership function to model experts' subjective uncertainty in ASIT in early conceptual design stage.

Figure 2 .

 2 Figure 2. GENERATION OF ALL POSSIBLE ARCHITECTURES

Figure 3 .

 3 Figure 3. FUNCTION SATISFACTION LEVEL BY MODULES

Figure 4 .

 4 Figure 4. REQUIREMENT-FUNCTION RELATIONS

Figure 5 .

 5 Figure 5. REQUIREMENTS SATISFACTION (WITHOUT CONSIDERING EXPERT UNCERTAINTY)

Figure 6

 6 Figure 6. UNCERTAINTY INFORMATION

Figure 8 .

 8 Figure 8. FUZZY NUMBERS FOR SATISFACTION LEVELS

Figure 9 .

 9 Figure 9. REQUIREMENTS SATISFACTION BY USING TYPE-1 FUZZY SETS (VALUES)The requirements satisfactions obtained in Fig.9can be illustrated using triangular fuzzy

Figure 10 .

 10 Figure 10. REQUIREMENTS SATISFACTION BY USING TYPE-1 FUZZY SETS (MEMBERSHIP FUNCTIONS)

Figure 11 .

 11 Figure 11. FUZZY MEMBERSHIP FUNCTION FOR POSSIBILITIES

Figure 13 .

 13 Figure 13. UNCERTAINTY USING TYPE-1 FUZZY SETS (MEMBERSHIP FUNCTIONS)

Figure

  Figure 14FUZZY RESULTS THAT ARE PARTLY ABOVE THE THRESHOLD

Figure

  Figure 15USING α -CUT TO REPRESENT TOLERANCE LEVEL

  .

Figure

  Figure 16REQUIREMENTS SATISFACTION AFTER DEFUZZIFICATION

Figure 18 .

 18 Figure 18. REQUIREMENTS SATISFACTION USING 2-TUPLE LINGUISTIC REPRESENTATION

Figure 20 .

 20 Figure 20. COMPARISON OF SUPPLIER IDENTIFICATION RESULTS

Figure 21

 21 Figure 21 Changing thresholds without considering subjective uncertainty

Table 1 .

 1 SATISFACTION LEVELS

Table 2 PROPABILITIES

 2

Expert knowledge is-what is known by qualified individuals, responding to complex, difficult (technical) questions, obtained through formal expert elicitation‖


Evolutionary game design, Springer briefs in computer science series

How can computers create interesting games? What makes a game interesting and how can a machine decide if a game is of interest to humans? These are among the questions I asked myself when I was asked to review Cameron Brown's book Evolutionary game design. Fortunately, after reading it I had some answers.

It begins by the introduction of "Yavalath", the first commercially released board game that was entirely designed by a machine. After reading this short introduction, I was glued to the book by the desire to understand the central thesis: how can humans design machines that design games for humans? It lead me through the different design steps, as if I was reading a story; the story of the genesis of Yavalath.

Following this introduction, chapter 2 describes the class of combinatorial games and their elements. Combinatorial games include those that are discrete and deterministic with no hidden information, such as chess. Chapter 2 focuses on the elements of such games that can be modified to create new games variants. In chapter 3, Cameron Brown describes the "Ludi" system, the software architecture that designs games. In a nutshell, this architecture invents new games from a set source games using genetic programming. The components of this architecture are discussed individually in later chapters. Chapter 3 introduces the Game Description Language a high level lisp style language that fully describes the game rules, the board and the winning conditions. It goes on to describe the General Game Player, the second component of the Ludi system, which plays games by simulating two players using a set of predefined strategies and using alpha-beta adversarial search. At this stage the reader has a general idea on how the machine creates games. The other question, "how does it decide if a game is interesting?" is answered in chapter 4.

The fourth chapter is the most important as it describes the criteria used to measure the quality of a game. It starts by introducing the attributes that measure, as Cameron Browne elegantly puts it, the "intellectual appeal; the elegance of the rules ... and the quality of the competition they produce". These quality measures are described in detail in an appendix, chapter 4 only gives a few details on some of them. Enough that the reader has the general idea. For exam-

ple "drama" measures the chance a player has to recover from bad positions. Or "uncertainty" which measures the uncertainty in the game outcome. The most interesting part in chapter 4 is the way these subjective measures are quantified.

At this stage, we have all the ingredients: a mean of representing games and a mean of evaluating their quality, all that is left is a way to evolve them and produce new games. This is the theme of chapter 5 which describes the elements of the genetic programming engine used to evolve the game description language. It describes how games are combined to produce new candidates; how they are modified and how their viability is verified. Chapter 5 ends with an interesting experiment that compares the quality of the evolved games as given by the quality measures with measures given by human test subjects.

Chapters 6 and 7 describe different games produced using Ludi. These games are presented in order of human preference along with their rules as produced by the evolutionary process. In chapter 7, the focus is on Yavalath. This game was ranked second by the human subjects and fourth by the machine. Incidentally Pentalath, the game that was ranked first by the machine was also ranked first by the human subjects. It however is more complex and deeper than Yavalath, which is simpler and more engaging. Yavalath is described and the interesting game situations are detailed. Chapter 7 also discusses the features of the game that lead to its commercialization.

Cameron Browne finishes with a discussion of the Ludi system, evolutionary game design and computational creativity. He leaves the reader with an interesting fact that raises further questions, he explains that there is nothing in the rules of the games designed by Ludi, that was not introduced in the initial rules given to the system. He also argues that the next step should be one where the "creative process should be attributed more to the machine than to the programmer". This part is rather short and I was left hanging expecting a more in-depth discussion.

Finally, the interested reader can find in the appendices many details about the game description language and the quality measures that evaluates games. The references section is also a good source of information, it includes references from many fields ranging from psychology to evolutionary computation passing by game theory and artificial intelligence.

The electronic version of the book is available on the "springerlink" web site as PDF where the chapters are packaged individually along with the appendices, the references and the index. One advantage over the printed version is that the figures are in color which makes them more legible than on the printed version where they are in black and white. Furthermore, all the citations to the figures, tables and the references are in the form of hyperlinks, which is useful.

Evolutionary Game Design is a valuable contribution to evolutionary computation and more generally to artificial intelligence. It is engaging to read, easy to follow and lives up to its promises. Furthermore, it delivers insights that should be helpful to anyone interested in AI and games. In the beginning, Cameron Browne cites Leibniz who once said "Human being are never more ingenious than in the invention of games". In my opinion, Evolutionary Game Design says how to transfer parts of this ingenuity to the machine.


La fin de l'hypothèque islamique ? Analyse des principales tendances de l'élection présidentielle algérienne de novembre 1995 Saïd Belguidoum1 L'élection présidentielle algérienne du 16 novembre 1995 intervient dans un contexte de crise aiguë, la plus importante depuis l'indépendance du pays. Le processus de recomposition sociale de ces trente dernières années a provoqué une exacerbation des tensions qui se sont cristallisées dans les émeutes d'octobre 1988, puis dans les importants succès électoraux du Front Islamique du Salut en juin 1990 (municipales) et en décembre 1991 (premier tour des législatives). L'interruption du processus électoral, la démission de Chadli ont donné à la crise algérienne une nouvelle dimension : le pays entre alors dans un cycle de violence proche de la guerre civile. Les islamistes du FIS après avoir échoué dans leur tentative de soulèvement insurrectionnel, organisent des maquis armés et se lancent dans un terrorisme systématique qui plonge le pays dans un profond chaos. La tenue d'une élection présidentielle constitue une initiative forte mais périlleuse. Après quatre années de violence, comment allait se comporter l'électorat ? L'enjeu consiste à savoir où se trouve la légitimité politique. Liamine Zeroual, soutenu par l'Armée, à la tête de l'État depuis deux ans, allait-il être investi d'une légitimité politique, consolidant ses succès militaires. Au contraire, les islamistes, à travers la candidature de Mahfoud Nahnah, le leader de Hamas, organisation islamique légale, conserveraient-ils leur influence dans la société algérienne ? Derrière cet enjeu central une question essentielle se pose : celle de la participation électorale. Sera-t-elle suffisamment significative pour légitimer l'élection ? En appelant au boycott, les trois "fronts", le FIS, le FLN et le FFS, qui avaient totalisé 80% des suffrages exprimés lors des législatives de 1991, allaient-ils être suivis ? Cette élection devait donc répondre à deux grandes inconnues : le rapport de forces électoral et le taux de participation. Avec cette dernière, allait se mesurer la légitimité du futur élu mais aussi la représentativité des partis du boycott. En clair, l'enjeu est celui du contrôle de la société algérienne.

Un rapport de forces inversé

Un seul tour aura suffi pour élire Liamine Zeroual. Avec un peu plus de 7 millions de voix et 61,34% des suffrages exprimés (soit 44% des inscrits), Zeroual relègue ses adversaires loin derrière : Nahnah, le candidat islamiste recueillant un quart des suffrages exprimés (25,38%), Sadi le candidat démocrate, 9,30% et Boukrouh ( candidat se prévalant d'un islam moderniste) 3,78%. 2 Cette victoire est d'autant plus large que le second pari de cette élection est également gagné, la participation a été massive, près de 75%, nettement supérieure à celle des législatives où le taux avait alors été de 59%.

Du vote anomique au vote refuge

Cette élection présidentielle inverse les tendances des élections précédentes. Le FIS en rassemblant 54% des suffrages lors des municipales et 47% aux législatives (auxquels il faut adjoindre les 7% réalisés par les deux autres partis islamistes) avait su capter le mécontentent de la population. Le vote islamiste était un vote de contestation, un vote révélateur du caractère anomique de la crise algérienne : crise à la fois économique (crise de la croissance) mais surtout sociale (crise du modèle de la promotion sociale, accroissement des inégalités provoquant frustrations et désarrois), culturelle (brouillage des référents symboliques) et donc politique (Belguidoum, 1994 ;Cherrad, 1991). Les islamistes, par la radicalité de leur discours, mais aussi en profitant d'un champ idéologique que le FLN leur avait balisé, ont récupéré ce profond malaise de la société. Ce sont dans les villes, là où les effets des bouleversements sociétaux ont été les plus fulgurants, que le vote islamiste a été le plus puissant. Aujourd'hui, la réduction de l'influence islamique à un quart de l'électorat, montre qu'une partie importante de la population s'est retournée : au vote anomique fait place un vote sécuritaire, un vote refuge.

Un enjeu dans l'enjeu : la participation

La participation est sans doute une des caractéristiques les plus importantes de ce scrutin. Avec un taux de 74,9%, les participations enregistrées lors des scrutins précédents (65,1% en 1990 et 59% en 1991) sont largement dépassées. Comparaison importante, puisque en 1991, hormis les communistes, toutes les forces politiques participaient et notamment les trois fronts. Forts des 80% de suffrages exprimés qu'ils totalisaient (soit plus de 40% du corps électoral) ils appelaient la population à s'abstenir. Si l'on considère que les consignes islamistes étaient relayées par des menaces physiques et que l'implantation militante du FFS en Kabylie lui permet de jouer un véritable rôle de contrôle social de la population, alors il était raisonnable de penser que les partis du boycott tireraient forcément profit d'une participation moyenne. Dans un tel contexte, la participation de 74, 9%, soit une progression de 16 % par rapport à 1991, est plus qu'un échec pour les partis du boycott, il est un désaveu complet. La progression est générale. Toutes les wilayas sans exception ont un taux de votants supérieur à celui de 1991 et si seules 4 wilayas ont une progression inférieure à 5%, 28 dépassent la moyenne nationale. Cette progression est cependant inégale. Les wilayas urbaines se sont relativement moins mobilisées. Les wilayas de l'Est, rurales comme urbaines, enregistrent les progressions les plus fortes et les taux de participation les plus élevés. Sur les cinq wilayas qui dépassent 90% de votants, quatre sont de l'Est (dont les trois wilayas des Aurès). Dans l'Ouest, la participation est bonne et conforme à la moyenne nationale. Seules les wilayas du Centre et de Kabylie n'ont pas eu la même intensité : 9 des 10 wilayas de la région ont une progression inférieure à 16%. C'est-à-dire là où l'appel au boycott avait certaines chances d'être relayé. Il s'agit de la Kabylie, d'Alger, de la Mitidja et des zones où les maquis islamistes sont bien implantés. L'impact a donc été réel mais limité. Encore ne faut-il pas exagérer son importance : ces wilayas ne mobilisaient pas leur électorat lors des scrutins précédents et la participation a été supérieure à celle de 1991. Sur les 25% d'abstentionnistes, au plus peut-on concéder aux partis du boycott la moitié des abstentionnistes. Dans l'ensemble du pays c'est toujours entre 60 et 97% de l'électorat qui en votant a défié le diktat des groupes armés. Cette participation massive profite essentiellement à Zeroual.Le taux de participation et le vote Zeroual ont les mêmes caractéristiques d'ensemble, les mêmes tendances se retrouvant à travers leur distribution spatiale (villes et campagnes) et géographique. Illustration de cette forte corrélation : parmi les 10 wilayas qui votent le plus, on dénombre les 8 meilleurs résultats de Zeroual.

Le vote

Un électorat urbain mobilisé

Principaux centres de la contestation du pouvoir et lieux des succès les plus éclatants du FIS, les villes, comme le reste du pays, votent Zeroual. Dans les wilayas à fort réseau urbain son succès est indéniable.Toutes les wilayas urbaines, hormis Bejaïa, lui donnent la majorité de leurs suffrages. 9 sur 15 lui accordent plus de 61%. Certes parmi celles-ci (6 sont de l'Est, 3 de l'Ouest, aucune du Centre). Certes les trois grandes métropoles (Alger, Oran, Constantine) ne l'élisent qu'avec 50 à 57% des suffrages, mais il est évident que le vote urbain s'est considérablement modifié. Les villes, lieu de la radicalité de la contestation, ne se sont pas identifiées à l'islamisme.

La réactivation des appartenances régionalistes

Une autre tendance notable de cette élection, c'est la réactivation des réflexes régionalistes. Zeroual, originaire de Batna dans les Aurès, comme le sont nombre des cadres supérieurs de l'Armée et de l'appareil d'État, réalise un véritable raz-de-marée dans l'Est algérien. 13 des 16 wilayas de l'Est l'élisent avec des pourcentages supérieurs à sa moyenne nationale. L'engouement est moindre dans l'Ouest (8 sur 14, mais aucune des wilayas de l'Oranie) et dans le Sud (5 sur 8) et nettement plus faible dans le Centre (1 sur 8). Sur les 10 wilayas qui ont le plus massivement voté pour Zeroual, 7 sont de l'Est dont celles des Aurès où il obtient 94% à Khenchela, 92% à Batna et 87% à Oum el Bouaghi. L'importance des scores réalisés montre à l'évidence que dans l'Est Zeroual a bénéficiéd'une prime "au candidat régional". La proportion supérieure de 15 régionalistes qu'un appareil politique local a su travailler. A contrario, dans l'Oranie et l'Algérois, la réactivation du régionalisme aura joué au détriment de Zeroual et profité au candidat islamiste, originaire de Blida.

Qui vote Zeroual ?

L'aire d'attraction électorale de Zeroual est large, son électorat est hétérogène, et ses soutiens politiques divers.Mobilisant les villes et les campagnes, il a été le principal bénéficiaire du vote des nouveaux électeurs (3 millions de plus, soit un accroissement de 20% depuis 1991), c'est-à-dire les nouvelles classes d'âge et les abstentionnistes de 1991. Il a aussi bénéficié du soutien de la puissante UGTA (syndicat des travailleurs), du ralliement de l'électorat démocrate qui a vu en lui le seul rempart crédible contre l'islamisme, mais aussi d'une large fraction de l'électorat islamiste de 1991, et de tous ceux qui ont refusé de cautionner les groupes armés.

Le FLN phagocyté ?

Mais, politiquement ce qui semble se dessiner c'est l'absorption du FLN. Malgré ses défaites le FLN était demeuré une force politique nationale disposant d'un réservoir électoral de 23%. Or on constate que c'est souvent là où le FLN avait fait de bons scores en 1991, que Zeroual réalise ses meilleurs résultats. (Sur les 15 wilayas où le FLN faisait ses meilleurs scores, 10 ont massivement voté Zeroual). En occupant le même champ idéologique, celui du nationalisme et de l'héritage de la guerre de libération, Zeroual phagocyte le FLN dont un grand nombre d'appareils locaux lui sont déjà acquis. Désavoué par sa base électorale, le FLN n'a pas d'autres choix, pour continuer d'exister, de rallier Zeroual.

Le vote islamiste

La fin d'une hégémonie électorale En rassemblant sur son nom 25,38% des suffrages exprimés, 2.907.857 voix, la défaite du candidat islamiste est sans appel. En données relatives, Nahnah divise par deux le pourcentage de l'électorat islamiste de 1991 (25% contre 47% au FIS et 54% à l'ensemble de la mouvance islamiste). Il ne retrouve que les 3/4 du potentiel de voix islamiste, soit une perte en données absolues de 871.155 voix, alors que le corps électoral progresse de 20% et le taux de participation croit de 16% (preuve que la perte d'audience n'est pas liée à l'abstention prônée par le FIS clandestin). Du fait de la bipolarisation de l'électorat dans la plupart des wilayas, les scores de Nahnah sont souvent inversement proportionnels à ceux de Zeroual. Dans aucune des 48 wilayas il n'arrive premier et il ne maintient qu'une seule fois (Ghardaïa) le pourcentage islamiste de 1991. Si comme en 1991, le vote islamiste est complètement marginalisé en Kabylie (3%), le fait nouveau est qu'il l'est également dans des wilayas où le vote FIS avait été quasi hégémonique (4 à 6% dans les Aurès). Parmi les 16 wilayas où le recul est supérieur à la moyenne nationale (plus de 25%), 6 sont urbaines, dont Alger (avec une perte de 160 000 voix) et Constantine.

Une implantation toujours réelle

Mais l'islamisme n'est pas mort pour autant. Il reste une force importante même s'il a perdu son caractère majoritaire et hégémonique. La distribution du vote islamiste montre qu'il dispose de solides points d'ancrage. 28 wilayas lui donnent un score supérieur à sa moyenne nationale et dans 37 d'entre elles il dépasse la barre des 20%. La comparaison en données absolues entre 1991 et 1995 fait ressortir que si dans 28 wilayas le nombre de voix recueillies par Nahnah est inférieur à celui des islamistes en 1991, dans 20 il est plutôt égal ou légèrement supérieur. C'est surtout notable dans 6 wilayas où le gain se situe entre 10 000 et 20 000 voix (Ouargla, Mostaganem, Mascara, Laghouat, M'sila et Djelfa). Malgré l'effondrement du vote islamiste dans un certain nombre de régions : l'Est, mais aussi celles les plus touchées par le terrorisme (Alger, Constantine, Jijel...), on constate que les wilayas où il résiste le mieux sont urbaines, plutôt du centre et de l'Ouest. Ainsi sur les 28 wilayas où son score est supérieur à 25%, 10 sont urbaines. 7 des 8 wilayas du Centre et 10 des 14 wilayas de l'Ouest font partie de ces wilayas où le vote islamiste est encore influent. S'il ne fait aucun doute que Nahnah a bénéficié du ralliement de la grande majorité de la sensibilité islamiste, il est par contre intéressant de constater que dans le classement de ces meilleurs résultats des tendances particulières apparaissent. Oran, Tlemcen et AïnTemouchent peu touchées par le terrorisme lui accordent une proportion importante de leurs suffrages. Est-ce une réaction à une candidature Zeroual symbolisant la domination de l'appareil d'État par l'Est au détriment de l'Oranie ? Ou encore la volonté d'une population peu touchée par le terrorisme de choisir un candidat dont la campagne a été basée sur la modération du ton, reflétant le compromis national ? On notera aussi la présence d'un certain nombre de wilayas où l'influence des réseaux confrériques est importante. C'est le cas de M'sila, Tlemcen, El Oued, Ouargla, Laghouat et Médea. A M'sila par exemple, wilaya où il réalise son meilleur score (44,5%), les réseaux confrériques actifs dans cette wilaya l'ont soutenu. A El Oued, Tlemcen, mais aussi Ghardaïa, il semble que les bourgeoisies locales liées par ailleurs aux réseaux confrériques aient également été séduites par la candidature "rassurante" de cet islamiste "modéré".

Le vote Sadi ou le vote identitaire kabyle

Saïd Sadi a tenté lors de sa campagne de rassembler les démocrates. A travers sa candidature, c'est donc le poids de la mouvance démocrate et son aptitude à représenter un pôle alternatif possible qui allait pouvoir être mesurés.Mais Sadi, militant de la première heure du combat pour l'identité berbère est aussi le président du Rassemblement pour la Culture et la Démocratie (RCD), essentiellement implanté parmi la population kabyle. Les résultats qu'il obtient montrent que c'est essentiellement cette dimension qui sera retenue. Sadi réalise un raz-de-marée en Kabylie et échoue dans sa volonté de rassembler le vote démocrate. En recueillant 9,29% des suffrages, soit 1.064.532 voix, il réalise certes un score convenable. Mais en y regardant de plus près, force est de constater le caractère essentiellement kabyle de ce vote.

A Tizi Ouzou et à Bejaïa, les deux wilayas kabyles, il arrive largement en tête avec respectivement 88,42% et 84,62%. Dans ces deux seules wilayas il recueille 478.734 voix soit 45% de son score national. Si on rajoute Alger (125.778 voix), Bouira, Boumerdès, et Tipaza, c'est-à-dire les régions abritant de fortes populations kabyles, c'est 70% de son score qui est réalisé dans ces six wilayas. Dans 38 wilayas, il fait des scores inférieurs à 5%. Seules Oran et Chlef lui donnent un pourcentage de voix (4 à 5%) sans doute supérieur au taux de la population Kabyle résidant dans ces départements. Dans 15 wilayas il arrive néanmoins en troisième position, mais dans 30 autres il est dernier, battu même par NourdineBoukrouh, le candidat prônant un islam moderne. Certes les wilayas urbaines font ressortir de meilleurs résultats que les wilayas rurales, mais dans des proportions trop infimes pour être réellement significatives. 9 wilayas urbaines sur 16 lui donnent plus de 2,5% contre 2 rurales sur 7. L'électorat démocrate potentiel a été aspiré par Zeroual, marginalisant complètement Sadi dans les trois quart du pays. En 1991, le vote FFS -RCD rassemblait 10,3% des suffrages et 710.928 voix, dont plus de la moitié dans les deux wilayas kabyles, c'est incontestablement ce potentiel qui a été retrouvé par Sadi dont le seul véritable succès aura été de s'imposer face au FFS. En 1989, la Kabylie s'était massivement abstenue suivant l'appel au boycott du FFS (25% de participation). Ce qui avait même permis au FLN de remporter les élections à Bejaïa. En 1995, non seulement elle vote plus qu'en 1991 (le FFS participait) mais elle "plébiscite" Sadi. En défiant à la fois le FFS et Zeroual, les Kabyles expriment aussi un vote de résistance. La réconciliation nationale qui aura été un des thèmes de la campagne, ne pourra pas se faire au détriment des revendications identitaires de la Kabylie.

Que fera Zeroual de sa nouvelle légitimité ? L'islamisme qui depuis 1988 s'est imposé comme l'alternative à un pouvoir usé par 30 ans d'exercice, a subi une double défaite : son influence électorale est réduite de moitié et l'interdit des groupes armés a été défié. Ce "retournement" de l'électorat est la preuve que le vote islamiste n'était pas un vote adhésion mais qu'il reposait sur une dynamique de contestation du pouvoir. La mobilisation populaire pour l'islamisme ne s'est réellement exprimée que lors des scrutins électoraux. La grève insurrectionnelle de juin 1991 comme l'appel au soulèvement populaire généralisé de janvier 1992 sont des échecs auxquels le recours au terrorisme comme stratégie de blocage de la société algérienne tente de palier. Il était donc important que la dernière légitimité qui restait aux islamistes, qui lui avait été donnée par les élections, lui soit retirée par d'autres élections. En perdant plus de la moitié de son influence électorale après quatre années d'une lutte armée déclenchée par le FIS, l'islamisme a montré ses limites. Avec cette élection présidentielle, l'hypothèque qu'il faisait peser sur l'Algérie semble levée, provisoirement tout au moins car les raisons du vote islamiste existent encore. La crise économique et sociale, véritable terreau au radicalisme islamique, mais aussi la crise politique trouveront-elles un début de réponse dans les initiatives que le pouvoir devra prendre ? C'est en permettant à une véritable citoyenneté d'émerger dans le cadre d'un véritable Etat de droit, c'est en répondant au besoin de moralité de la vie publique réclamée par la population, c'est en laissant la société civile se construire, c'est aussi en répondant au besoin de justice sociale d'une population que les plans du F.M.I. risquent de replonger dans la désespérance sociale, que l'issue à la crise sera trouvée et que ce vote refuge, loin d'être un plébiscite, pourra prendre son véritable sens. L'hypothèque islamiste levée, le champ redevient libre pour que les réponses politiques à la crise sociale et économique puissent être apportées, à condition que le vote ne soit pas détourné et que sa signification sécuritaire ne soit pas comprise comme un plébiscite, un blanc-seing à la mise en oeuvre d'une politique continuant celle des années 1980. Les véritables lignes de clivage opérant au sein de la société, longtemps brouillées par l'idéologie religieuse, le nationalisme ou la réactivation des solidarités régionales et communautaires, pourront alors avoir une nouvelle visibilité.

Bibliographie

Belguidoum Saïd, 1994, De l'anomie à l'islamisme : exacerbation des tensions sociales en Algérie, Avis de Recherche, Institut de Recherches Marxistes Sud Cherrad Salah-Eddine, 1992, "Élections municipales et législatives en Algérie : les scrutins du 12 juin 1990 et du 26 décembre 1991", Espace rural, n°29, Laboratoire de géographie rurale de l'Université́ Paul-Valery et de l'U.R.A.

Zeroual, ses principales caractéristiques Un électorat national... moins la Kabylie La

  victoire de Zeroual ne prête aucune contestation. Premier dans 46 wilayas sur 48, il dépasse les 50% des suffrages exprimés dans 45 d'entre elles. Seule la Kabylie le prive d'un succès total.

Sociologue, Aix Marseille Université, CNRS, IREMAM UMR 7310, 13094 Aix-en-Provence, France

Nous utilisons dans cet article les résultats par wilaya tels que publiés dans la presse au lendemain des élections. La non disponibilité de données plus fines (par commune) limitera l'analyse, notamment dans l'observation de l'évolution des électorats selon le type d'agglomération. La classification que nous avons faite entre les wilayas urbaines, semi-urbaines, rurales et du Sud permet néanmoins de rendre compte des logiques électorales selon les différents cadres socio-spatiaux.


Wassila Ouerdane
email: wassila.ouerdane@lamsade.dauphine.fr

Alexis Tsoukiàs

Nicolas Maudet

Multiple Criteria Decision Aiding: a dialectical perspective

Keywords: Decision aiding process, Multiple criteria evaluation, Argumentation theory

The thesis is

Introduction

Deciding is not an easy task: we are often facing problem situations where we are not sure what to decide (it is the case particularly when several points of view have to be considered in order to assess the quality of the different alternatives). Decision analysis is concerned with the process of providing decision support in such situations. We call such an activity "decision aiding". Decision aiding is the result of an interaction between an "analyst" (or expert) and a "client'" (or decision maker), where the aim of the analyst is to support the decision maker to find a solution to his problem and to be convinced that this solution is a reasonable one. For that, the analyst and the decision maker are committed in an interactive process, where the main objective is eventually to reach a consensus upon the solution to a decision problem. Systems which aim at assisting people in decision making help the user to shape a problem situation, formulate a problem and possibly try to establish a viable solution to it. Decision theory and Multiple Criteria Decision Analysis (MCDA) have established the theoretical foundation upon which many decision support systems have blossomed. These approaches (and the formal tools coming along with them) focused for a long time on how a "solution" should be established. But it is clear that the process involves many other aspects that are handled more or less formally by the analyst. For instance, (i) the problem of accountability of decisions is almost as important as the decision itself. The decision maker should then be convinced by a proper explanation that the proposed solution is indeed the best (see [START_REF] Bouyssou | Evaluation and decision models: a critical perspective[END_REF]), and (ii) it should be possible, for the client, to refine, or even contradict, a given recommendation. Indeed, the decision-support process is often constructive, in the sense that the client refines its formulation of the problem when confronted to potential solutions.

Nowadays, decision-aiding situations are pervasive: they can occur in situations where the role of the analyst is taken by a non-expert, even in some extreme cases by an automatic tool. This means that several aspects usually delegated to the human analyst should (in these situations) ideally be handled by the decision-support system. The task is ambitious: in a "human-to-human" interaction -even though the dialogue is possibly supported by standard protocols (as in the case of constructing a value or an utility function or assessing importance parameters) which fix some explicit formal rules on how such a process can be conducted-the dialogue is handled through typical human interaction. A tool will be able to structure the dialogue on a formal basis in order to be able to control and assess what the device concludes as far as the user preference models are concerned and what type of recommendations (if any) is going to reach. In short, we need on the one hand some formal theory about preferences (and this is basically provided by decision analysis), on the other hand some formal language enabling to represent the dialogue, to explain it, to communicate its results, to convince the decision maker that what is happening is both theoretically sound and operationally reasonable.

Although there was until recently very little attention in the decision analysis literature to the use of decision theories and decision aiding methodology when the interaction occurs between a human and an automatic device (see [START_REF] Klein | Decision analytic intelligent systems: automated explanation and knowledge acquisition[END_REF] for a noticeable exception), the recent surge of automatic decision aiding tools on the Internet (recommender systems) have motivated a great deal of research, studying for instance the impact on the efficiency of recommendations when greater interaction with the user is allowed, or when explanations are provided [START_REF] Pu | Trust Inspring explaining interfaces for recommender systems[END_REF]. Due to the context however, only very simple interactions and models of preferences are envisaged (a typical consumer is not prepared to enter in a long preference elicitation process, or to discuss endlessly the benefits of a given options as opposed to another one). Our perspective in this work is different, as it is not to fully automatize the process. Instead, we rely on existing decisionaiding tools as developed in the MCDA community, and seek to enhance them with additional features as discussed above. Under such a perspective we consider that argumentation theory (Amgoud ( 2009)) provides a useful framework within which develop such a dialogue and a tool.

How to use argumentation in a decision aiding process?

Our aim through this work is to build a system that allows: (i) to present a recommendation that can be explicitly justified; (ii) to revise any piece of reasoning involved in this process, and be informed of the consequences of such moves; and (iii) to stimulate the user by representing naturally (sometimes visually) the exchanged arguments, possibly generating counterarguments. These different aspects trigger different problems and involve different models and mechanisms. As a consequence, our contributions in this work are of different nature, from the modeling exercise to the implementation of a prototype exhibiting the different features discussed in this thesis. More specifically, the following points have been investigated:

A hierarchical structure of arguments. We propose a first approach allowing to specify in argumentative terms the steps involved in an evaluation stage of a decision aiding process (see [START_REF] Ouerdane | Argument schemes and critical questions for decision aiding process[END_REF]). To do that, we make use of the popular notion of argument schemes. Argument schemes are forms of arguments that capture stereotypical patterns of humans reasoning, especially defeasible ones. Different approaches have investigated the use of argument schemes to decision-making [START_REF] Ouerdane | Argumentation Theory and Decision Aiding[END_REF]). They greatly extend our understanding of the construction of argument schemes for action. Our claim in this work is that by presenting the reasoning steps of an evaluation process under the form of arguments schemes it makes justification possible and offers in turn the possibility to identify how these steps can be dialectically defeated. Thus, we propose a hierarchical structure where we identify three levels of argument schemes that are embedded. At the highest level the multi-criteria level, which is based on the aggregation level, which is in turn based on the Unicriteria level.

Choosing and justifying an aggregation procedure. The aim at the aggregation step is to construct a set of supporting reasons (or evidences) that allow to justify a given claim (or decision). What conclusion to provide depends entirely on the procedure that we should use to aggregate all the arguments that are both in favour and against that claim. Different procedures necessarily provide different results. In argumentation, such procedures are called proof standards [START_REF] Gordon | The Carneades Model of Argument and Burden of Proof[END_REF]. Usually, such proofs that are used are fixed a priori by the context. However (multicriteria) Decision-Aiding (DA) is a context where it may be modified dynamically during the process, depending on the responses of the decision-maker. In fact, a delicate step in DA is to identify what kind of aggregation mechanism is appropriate in the decision context considered. Thus, the system needs to adapt and refine its choice of an appropriate method of aggregating arguments, so that it fits the preference model inferred from the interaction. In this work we address how this aspect can be handled in an argumentationbased decision-aiding framework. The first contribution is conceptual: the notion of a concept lattice based on simple properties and allowing to navigate among the different proof-standards is put forward. We then show how this can be integrated within the Carneades model [START_REF] Gordon | The Carneades Model of Argument and Burden of Proof[END_REF], in order to discuss the acceptability of a procedure during the interaction.

Generating minimal explanations. At each step of the evaluation process, the decision maker can ask for explanations to clarify the reasoning steps assumed by the system, as well as the constructed recommendations and solutions. The idea is that the system generates automatically explanations by instantiating argument schemes constructed in the hierarchy. Moreover, the system seeks to provide the most "simple", "natural", explanation for a given recommendation. More precisely, the system provides the minimal evidence which allows to support a given conclusion. Thus, in this work we propose to examine different definitions of "minimality", depending on the language used to construct the explanation.

Other aspects were discussed in this thesis, such as handling revision and changes during the process, however, there are still many open questions and problems which deserve further attention. For instance, we believe that an experimental study aiming at analyzing the behavior of the decision maker in a real situation of decision support should allow to refine the types of reactions that the system should be prepared to deal with.