Diadie Sow

Abdelhak Imoussaten

Pierre Couturier

Jacky Montmain

A Possibilistic Approach to Set Achievable and Feasible Goals while Designing Complex Systems

Keywords: design of complex system, imprecise assessment, multiple-criteria decision analysis, possibility theory. 

How to make, early in the development cycle of complex products, the most promising design choices as regards the customer's requirements and being at the limit of what is technically feasible by the manufacturer? To contribute to solve such a difficult problematic, we propose an original approach based on possibility theory that aims finding the best alternative according to the preferences of the stakeholders and being feasible by the designer team. Customer's and manufacturer's preferences are captured in a multi attribute utility theory (MAUT) framework that is extended to uncertain and imprecise evaluation of the alternatives' characteristics since available knowledge about the future system is mostly qualitative in preliminary design stages.

INTRODUCTION

The survival of a company is heavily dependent on its capacity to identify new customer needs and develop new products [START_REF] Shen | An integrated approach to Innovative Product Development using Kano's model and QFD[END_REF]. Industries must always search for sustainable advantages, improving their performance. However, designing new products or improving existing ones in today's highly competitive market presents significant risks. Many system requirements must be taken into account when designing or improving a product [START_REF] Baykasoglu | Prediction and multi-objective optimization of high strength concrete parameters via soft computing approaches[END_REF]; [START_REF] Ng | A critical analysis of current engineering design methodologies from a decision making perspective[END_REF]. Decisional strategies are required to define, compare and select potential design alternatives with respect to the relationships existing between performance expressions [START_REF] Bititci | Modelling of performance measurement systems in manufacturing enterprises[END_REF]. These relationships may be of a different nature, e.g. operational, physical or preferential. Operational relationships between two variables refer to the existence of improvement actions that allow or prevent the conjoint improvement of a subset of criteria (e.g., there does not exist improvement actions that both allow working better and more quickly). Physical relationships express influences, constraints or balances between variables related to performance expressions (e.g., it is difficult to reduce the friction force while increasing the speed of a vehicle because the friction force varies as the speed or the square of speed). Finally, preferential relationships refer to subjective interactions among performance expressions (e.g., "I would like my new car to be both roomy and fuel-friendly" refers to a conjunctive interaction whereas "I would like my new car to be either comfortable or sportive" is a disjunctive interaction). Whereas the customers' preferences regarding the system to be designed are considered to be expressed in their needs, the manufacturers' preferences are rather related to the effort the system achievement will necessitate: the more complex the system, the more uncertain the achievement and the more time the project risks to consume and finally the less worthwhile the cost/benefit ratio for the manufacturer. So, design decisions require large analysis and forecasting capacities especially during the preliminary design stage when system requirements, product models and performances' interactions are merely based on unprecise information. Therefore, identifying new solutions to satisfy customers in such a context appears as a complicated task [START_REF] Moulianitis | A model for concept evaluation in design--an application to mechatronics design of robot grippers[END_REF], [START_REF] Couturier | Tracking the consequences of design decisions in mechatronic systems engineering[END_REF]. The industrial manufacturers must design new products/systems from past experience according to customers' needs at the limit of what is technically feasible as they are aware of their available enterprise-level skills. Defining achievable targets is a matter of situation awareness to relevantly manage the balance between strategic ambition and manufacturing realism [START_REF] Sow | A qualitative approach to set achievable goals during the design phase of complex system[END_REF], [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF]. Thus, design alternatives to be retained as a priority are those which allow both significant positive impacts on product/system performance but also correspond to actions that are derived from the expertise of the manufacturer. This will help the designer to avoid focusing on the implementation of alternatives that would be too far from the genuine ability and know-how of the manufacturer.

Despite their interest in design decision making, few works address the modeling of technical feasibility. In [START_REF] Bause | Feasibility Stuties in the Product Development process[END_REF], the authors clarify the concept of « technical feasibility study» often used in the context of product development process and explain how such concept concerns usually the activities: "idea detection", "modeling of principle and embodiment", "detection of alternative solutions" and "analysis of consequences". However the authors do not deepen methods and tools required to evaluate. [START_REF] Clivillé | Feasible Optimization[END_REF] are interested with this duality between expected and feasible performances. They do not limit the decision making process to the satisfaction of alternatives but introduce a feasibility function such as "a configuration a is more feasible than a configuration b" if one can pass from a to b. For each configuration, they then seek to maximize the satisfaction that can be expected starting from this configuration under feasibility constraints. However the work does not consider relationships existing between performance expressions and all information about feasibility is supposed to be given by experts. In [START_REF] Chinkatham | Early feasibility evaluation of solution concepts in an inventive design method framework: approach and support tool[END_REF] is proposed the 'Inventive Design Method' to prevent the surrender of good solution concepts and to reject unfeasible ones as early as possible when designing a product. The approach is based on finding first doubts or uncertain conditions of any solution after reaction of designers or experts. The estimated feasibility is then found by considering one or more behavior model(s), but also design objectives and constraints. However no preference model is discussed. Also, this paper proposes selecting the most relevant design alternative for the product/system to meet the customer's requirements subject to the enterprise-level skills in taking into account the uncertain environment and the operational, physical or preferential relationships between performance expressions. At this aim, a fuzzy model of the expected performances and of the ability to achieve is defined. The paper will be organized as follows. Section 2 introduces the problematic and the necessary notations to the problem formalization in the context of design/improvement of complex system. Section 3 presents the possibilistic model of preference required to select a design alternative. Section 4 considers an experimented application. Further prospects for this work are considered in the conclusion.

PROBLEMATIC AND CHARACTERIZATION

Problematic: Evaluation in System Engineering

Designing a system generally imposes to solve an ill-posed problem admitting multiple solutions and whose definition becomes more and more accurate as the choices for developing a satisfying solution are made. The most critical stage in the design process is the preliminary stage where most of the system development costs are committed [START_REF] Phillips | Committed Costs vs. Uncertainty in New Product Development[END_REF]. Therefore it is crucial to evaluate concepts and design alternatives against technical and economic criteria very early in the preliminary design stage (i.e. the conceptual and embodiment stages) even if, at this design stages, the available knowledge and descriptions of the system are incomplete, imprecise and subject to change. To respond to this problematic, we assume that some experts can provide advices, as regards design choice performances and feasibility, which can be formulated in form of possibility distributions and we propose an approach that aims at identifying among the possible design solution alternatives, the ones that better satisfy the customer's criteria and that are achievable by the designer team. This is done by estimating an overall satisfaction of design alternatives on several criteria with respect to the preference of the stakeholders' respectively to its feasibility on these criteria. Some definitions and results will be recalled about multi attribute utility theory that manages multiple criteria context and possibility theory before formalizing the proposed approach.

Characterization and Notations

In order to design a complex system, we characterize it by a set of parameters 12 ( , ,..., ) n    whose values have to be fixed by the designers. Let  be the set of all possible values of the vector 12 ( , ,..., ) n    . A system is then defined by a design solution or configuration  . Improving a system is to make it evolve from a configuration   to a configuration '   which gives better satisfaction regarding the objectives that have been fixed for the system by the customer's and taking into account cost constraints of the manufacturer (effort, money, risk, time etc.). The satisfaction of the objectives will be evaluated in a multi-criteria decision analysis (MCDA) framework using multi attribute utility theory.

Multi Attribute Utility Theory (MAUT)

Let us denote by {1, 2,..., } N n  a set of attributes where the i th attribute takes its values in a set denoted i X . The MAUT allows establishing an analytical model of the decision maker's preference relationships over i i X X   . Let be a preferential relation over X . The MAUT proposes to model it through a utility function :

[0,1] UX  such that: 2 ( , ') , ' ( ) ( ') x x X x x U x U x     (1)
The function U of the equation ( 1) can take several forms; the most often used is the additive model:

1 , ( ) ( ) N i i i i x X U x w u x      where 1 1 N i i w    , , 0 i i N w    ,
and each :

i i uX  is an elementary utility function that synthetizes the preference of the decision maker regarding the i th attribute (it translates the value i x into a utility value ()

i i
ux, here a performance with regard to i th criterion).

The additive aggregation intrinsically tolerates compensation between criteria and required independence between them [START_REF] Keeney | Decision analysis with multiple conflicting objectives[END_REF]. This additive form is the most widespread because of its simplicity and its intuitive interpretation. It is generally, a very simplifying assumption because in reality attributes interact between them. To solve this problem, a more general model of the overall utility U in equation ( 1) has been proposed in [START_REF] Krantz | Foundations of measurement (additive and polynomia representations)[END_REF] where U is written under certain conditions of separability and independence:

1 1 2 2
, ( ) ( ( ), ( ),..., ( ))

nn x X U x F u x u x u x    (2)
where : n F  is an aggregation operator. To represent F in the quantitative case, the Choquet integral is an interesting candidate. Indeed, it is an aggregation operator that generalizes several classical operators, the weighted sum of the additive model, the ordered weighted sum (OWA), the min, the max etc. [START_REF] Grabisch | Agrégation multicritère. Utilisations de la logique floue, Hermes Grabisch[END_REF]. Moreover, the Choquet integral allows modeling the interaction between attributes through a fuzzy measure Marichal (1999). In the following, we propose a brief reminder about the concepts of fuzzy measure and the Choquet integral.

The Choquet Integral

Definition 1: A fuzzy measure  on N is a set function: : 2 [0,1] N  
that satisfies the following conditions:

( ) 0; ( ) 1; N     
 is monotonic non decreasing for inclusion, i.e., for any , , ( )

( ) A B N A B A B       (3)
In the context of multi-criteria decision analysis, () I  represents the importance coefficient of the combination of criteria IN  . Definition 2: In the context of multi-criteria decision analysis, the evaluation of the Choquet integral of an alternative 12 ( , ,..., )

n x x x x  with respect to  denoted by () Cx  is defined by: ( ) ( 1) ( ) 1 ( ) ( ) ( ) n i i i i C x x x A           (4) where 0 (1) (2) ( ) ( ) 0 ... ; { ( ), ( 1),..., ( )} n i x x x x A i i n              
with  a permutation on N (to simplify the notations, () i x  denotes here a utility value, it should be formally noted

( ) ( ) ( ) i i u x  
).

Furthermore, it is important to note that our approach aims first to select design solutions in a context of imprecision and uncertainty where values of attributes cannot be precisely known in the design stage. Uncertain scores are then modeled as possibility distributions in our framework.

Possibility Theory

Possibility Theory is an appropriate framework to represent imprecise information [START_REF] Dubois | Possibility Theory: An Apraoch to Computerized Processing of Uncertainty[END_REF].

Definition 3: A possibility distribution  on the set of interpretation  is a mapping from  to [0,1] which satisfies the normalization condition: : ( ) 1

      .
By analogy with the mathematical expectation of a random variable, the expectation associated to a possibility distribution is also defined in [START_REF] Dubois | The mean value of fuzzy numbers, Fuzzy Sets and Systems[END_REF].

MD  is defined by: * * ( ) ( ( ) ( )) 2 MD E E     
.

Aggregating Possibility Distributions with a Choquet Integral

Given elementary possibility distributions 1 2 , ,..., n    , the aggregated possibility distribution F  using an operator F can be computed using the extension principle of Zadeh (1965) which states that:

1 2 1 2 , ,..., / ( , ,..., ) 1 1 2 2 ( ) sup min( ( ), ( ),..., ( ))

n n F x x x F x x x x n n x x x x      
To facilitate notations, we denote 1 2 ( , ,..., )

F n F      .
The direct computation of the aggregated possibility distribution by this formula may be extremely time consuming, but in case of intergral [START_REF] Grabisch | Agrégation multicritère. Utilisations de la logique floue, Hermes Grabisch[END_REF] proposes a much more practical way to compute the aggregated distribution

C 
 based on the hyperplan-linearity of the Choquet integral (the Choquet integral in (4) behaves as a weighted mean in each hyperplan

(1) (2) ( ) / ... n H x x x       
). The computation only needs to be processed on a finite set of particular points.

IDENTIFYING THE BEST AND MOST FEASIBLE CONFIGURATION

The Precise Assessment Problem

On one hand, there are the customer's needs and the manufacturer's aspirations on the other hand. A set of criteria is associated to the customer's needs and a capacity  captures his priorities over the coalitions of criteria. In other words, any configuration   can be assessed through its results on attributes, e.g., 1 2 ( , ,..., )

n x x x x      , as 1 1 2 2
( ) ( ( ), ( ),..., ( ))

customer n n U x C u x u x u x      
where the (.) i u 's are utility functions related to the criteria. The higher () customer U x  , the better  matches the customer's needs.

Otherwise the preference model of the manufacturer is related to the effort the configuration requires. The more difficult is the achievement of the configuration, the less attractive the cost/benefit ratio for the manufacturer. The difficulty of achievement related to the configuration  is computed from a capacity '

 that captures the conjoint impediments over the coalitions of criteria. Hence, ' 1 1 2 2 ( ( ), ( ),..., ( ))

nn C d x d x d x    
assesses the perceived difficulty related to the configuration  by the manufacturer, where the (.) i d 's are functions that measure impediments to the achievement of the requirements related to criteria. The preference model of the manufacturer is then given by ' 1 1 2 2 ( ) 1 ( ( ), ( ),..., ( ))

manufacturer n n U x C d x d x d x       .
Finally, the overall score, the configuration receives is:

( ) ( ( ), ( ))

customer manufacturer U U x U x     (5)

The uncertain assessment problem extension

Now that we have introduced the basic assessment of a configuration when it is described by precise characteristics 12 ( , ,..., )

n x x
x , let us introduce the configuration selection problem in real design conditions. In product design, the selection of a satisfactory solution is iterative, i.e., designers first determine a class of solutions then refine the solution into this class.

Ex: designers infer from the requirements expressed by some customer that this latter searches for a robot in a class of autonomous robot equipped with a rolling base with tracks, a fork and a set of exteroceptive sensors. Indeed, we assume that at this preliminary design stage a configuration is composed of one type of rolling base, one type of sensor equipment and one type of gripper device and represents a class of robot. The designers have numerous potential candidates (the possible instances of such a robot) to the customer's requirements. To each of the robot requirement a specific attribute is associated. From their set of potential offers, the designers only know the attributes' ranges of values he can offer to the customer with such a class of robot. They have to get more precise information to refine their proposal. Mathematically speaking, at this early design stage, the values (resp. impediments to performance reaching) of any attribute are known in imprecise and uncertain way and are assumed to be represented as a possibility distribution.

Let consider that  is a finite set of configurations. To each configuration   is associated its performance vector x  , and the possible range of values related to attribute i is provided by the expert designers as a possibility distribution i   Imoussaten et all (2014).. As it has been seen previously, a utility scale () i i ux  and a difficulty () i i dx   scale are then related to the i th attribute by the expert designers for each configuration (whereas the (.) i u 's only depend on the customer's needs, the (.) i d  's depend on the configuration  ). In this way, two further possibility distributions are constructed for the configuration  on attribute i such that

i i i u     and i i i d      whith i /g ( ) ( ) max ( ( )) i i y y x i g x y       where (.) (.) i i g u  or (.) (.) i i g d  
. Note that this step could also be the result of a statistical analysis from a database of past settings.

Then the assessment of a configuration in uncertain environment can then be extended as: (7)

Basic Principles of the Preference Elicitation Step

Without getting into the specifics of computation, this section provides the key points for indirect identification of the parameters of the aggregation operator that provides the overall utility U . This method called MACBETH has initially been proposed for identifying the weights of a weighted average mean Bana e Costa (1994) and then extended to the case of the Choquet integral [START_REF] Labreuche | [END_REF]. We simply give the principles of indirect identification method. Let us consider a finite set X. When the elements of X can be ranked w.r.t. to their attractiveness, they define an ordinal information. It means that a number () Ux can be associated to any element x of X such that:

, :

[ ( ) ( )] , :[ ( ) ( )] x y X xPy U x U y x y X xIy U x U y         ()
where relation P « is more attractive than » is asymmetric and non-transitive and relation I « is as attractive as » is an equivalence relation. The () Uxvalues define an ordinal scale.

Based upon this first level of information P and I, an interval scale can then be built. The second step consists in evaluating the difference of intensity of preference between elements of X i.e. ( ) ( )

U x U y 

. A system of inequalities is deduced from these two steps. The resolution of this system provides an interval scale.

APPLICATION

Presentation of the Application

The robotic challenge Robafis is organized annually by the French association of Systems Engineering AFIS to promote Systems Engineering practice in engineers' schools. The scope of the Robafis_2013 challenge was to build an autonomous mobile robot able to compete with other robots and using some imposed materials. Each robot was limited to a 3 0.3m cube and had to achieve the following mission as quickly as possible: to grasp and transport some various coloured spheres between several stock devices spread over a plan playground. Dark lines were drawn on the playground to guide the robot between stock devices. The autonomous robot was built from five sub-systems: a gripper device, sensors equipment, a rolling base, a control device and a battery. The programmable control device and the battery were imposed to the competitors. The robot configurations depend upon the design options and on the skills of each competitor. The design alternate choices are resumed in Table 1. There are three principles for designing the gripper device, three for the rolling base and two for the sensors equipment. Thus, there are 18 configurations to be compared, corresponding to the 3x3x2 admissible configurations. For the sake of computation illustration, let us compare for instance the five configurations: (G 2 ,R 1 ,S 1 ), (G 3 ,R 3 ,S 2 ), (G 1 ,R 2 ,S 2 ), (G 1 ,R 1 ,S 1 ), (G 2 ,R 2 ,S 1 ).

Data of the Application

Four criteria are considered to decide between configurations: the robot cost (1), the robot average speed capacity (2), the reliability of the robot (3), and the maintainability of the robot (4) (see Table 1). Taking into account the competition rules, Tables 2 gives examples of customer preferences (the competitors) and feasibility difficulty estimation modeled by two capacities based on the extension of the MACBETH method to the Choquet integral as presented in section 3.3. 3 gives the MD position indicators of the aggregated distributions for the five considered configurations. According to the results of Table 3, the configuration (G 1 , R 1 , S 1 ) is among the least satisfying and the most difficult one and should be rejected. The most satisfying configuration (G 2 , R 2 , S 1 ) is among the most difficult ones. If the manufacturer knows that, according to his/her workload the acceptable feasibility expectation must be upper than 0.53 to do the job, the (G2,R1,S1) configuration should be chosen but the customer satisfaction expectation could be only 0.23 and so the risk not to satisfy the customer is high. Otherwise (G 2 , R 2 , S 1 ) can be chosen.

Results and Comments

CONCLUSIONS

This paper has proposed a possibilistic approach to better manage the balance between strategic ambition and manufacturing realism in the case of the design or the improvement of complex systems. The approach takes into account the operational, physical or preferential relationships between performance expressions and the uncertainty of preliminary design context. The aim was to guide the designer to find and implement the most satisfying solutions without being exposed to the risks of choosing infeasible solutions. At this aim, fuzzy models of the expected performance and of the ability to achieve have been defined. The proposed method has been illustrated in the case of the design of an autonomous robot. In future research work we expect to widen our method (using [START_REF] Imoussaten | Identifying priority lines of improvement. Application to tourism data[END_REF] results established in a probabilistic context) to define on which set of criteria an existing product has to be improved in priority given the preference and feasibility models.

Definition 4 :

 4 Given a possibility distribution two distribution functions are associated to  and are named respectively upper distribution * F and lower distribution * associated to  is imprecise and is then defined in Dubois et al. (1987) by the interval: for sake of simplicity, it may be useful to define a position indicator which allows reducing the possibility distribution into a unique precise representative value. Indeed, it clearly facilitates comparing possibility distributions. Definition 6: The position indicator of a possibility distribution  denoted by ()

  possibility distribution representing the degree of possibility of the global satisfaction (resp. difficulty) of the customer (resp. manufacturer

Fig. 1 ,

 1 Fig.1, shows the kinds of inputs needed for each attribute and configuration. For instance: the possibility of the maintainability attribute values (time in minute to repair the robot) taking into account past experiences and state of the

  Fig. 1. i   and i d  for the maintainability attribute (xi: time to repair the robot in minute) and configuration (G 2 , R 2 ,S 1 ).

  Fig. 2. Possibility distributions i   and customer U 

  Fig. 3. Possibility distributions i   and manufacturer D  for

Table 1 . Design options

 1

	Gripper device Rolling base	Sensors equipment
	G 1 : fork taking	R 1 : with 4	S 1 : 2 color sensors (marks on
	sphere from	wheels	ground detection), one light
	below		sensor (sphere detection)
	G 2 : gripper	R 2 : 2 wheels	S 2 : 1 color sensor (marks
	taking sphere	drive + 1 free	detection), 1 color sensor
	laterally	wheel	(sphere detection)
	G 3 : grapnel	R 3 : with
	taking sphere	tracks
	over

Table 2 . Competitors preferences and difficulty of achievement

 2

Table 3 . Configurations global indicator (

 3

	( MD U  cust	.	)	,	1		man MD D  (	.	)	)
	Configurations	( MD U  cust	.	)	1		man MD D  (	.	)
	G 2 ,R 2 ,S 1			0,52				0,53
	G 2 ,R 1 ,S 1			0,23				0,83
	G 3 ,R 3 ,S 2			0,21				0,62
	G 1 ,R 1 ,S 1			0,21				0,4
	G 1 ,R 2 ,S 2			0,19				0,66
	Table


Diadie Sow

Abdelhak Imoussaten

Pierre Couturier

Jacky Montmain

A qualitative approach to set achievable goals during the design phase of complex systems

Keywords: multi criteria decision support, criteria interaction, conceptual design, performance management, qualitative reasoning

The problem addressed in this paper is "how to set ambitious targets when improving or designing a product while these targets remain within the reach of the manufacturer". Thus, improvements to be focused on are those which both have a significant positive impact on product performance and correspond to operational changes properly under control by the manufacturer. While some approaches in the literature have already addressed each of the two issues of the improvement problem, few deal with both of them at the same time. In this paper we investigate a qualitative approach that conciliates both points of view as an optimization problem. The notion of interaction between any two objectives to be simultaneously satisfied is central in our framework. An illustrative example related to the design phase of autonomous robot is provided.

Introduction

Competition in industry is becoming increasingly intense; therefore, in order to satisfy fluctuating demand and customers' increasing expectations, deal with the competition and remain or become market leader; industries must focus on searching for sustainable advantages. The survival of a company is heavily dependent on its capacity to identify new customer requirements and develop new products [START_REF] Shen | An integrated approach to Innovative Product Development using Kano's model and QFD[END_REF]. However, dealing with new products or improving existing ones in today's technology-driven market presents significant risks. Many constraints must be taken into account into the design or improvement phases [START_REF] Baykasoglu | Prediction and multi-objective optimization of high strength concrete parameters via soft computing approaches[END_REF][START_REF] Ng | A critical analysis of current engineering design methodologies from a decision making perspective[END_REF]. The number of components, functions, and interactions within complex products/systems is becoming so substantial that responding to the requirements of customers, and moreover at lower cost is anything but obvious [START_REF] Moulianitis | A model for concept evaluation in design--an application to mechatronics design of robot grippers[END_REF].

To remain successful, manufacturers must adapt to an unprecedented rate of change in their processes and practices. Such a challenge requires large forecasting capacities. Industrials must be able to produce challenging but achievable goals. The corresponding optimization problem to be solved for the industrial manufacturer is how to design new products or improve products according to customers' requirements at the limit of what is technically feasible as he is aware of his available enterprise-level skills.

Defining achievable targets is a matter of situation awareness to relevantly manage the balance between strategic ambition and manufacturing realism. Thus, improvements to be made as a priority, i.e. targets that must be raised considerably, are those which allow both significant positive impacts on product/system performance but also correspond to actions that are derived from the expertise of the manufacturer. Industrials are thus supposed to relevantly make a prediction about the positive impact one improvement or another one might have on the global performance of the product/system but simultaneously be aware of the actions they can actually undertake with regard to their skill [START_REF] Hauser | The house of quality[END_REF]. Forecasting the performances of a new product is far from obvious: it may be confusing to associate preferential interactions between goals and behavioral influences actions may have on these goals. This is further complicated by the qualitative characteristic of the knowledge used to do this forecasting: impacts of changes in the system/product configuration upon the expected performances cannot generally be properly quantified. Defining achievable targets is thus a risky and not deterministic process. The two aspects of the problem that this paper addresses when, improving a product or a system are:

• Assessing which performances should be improved first to maximize customer expectations [START_REF] Labreuche | Determination of the criteria to be improved first in order to improve as much as possible the overall evaluation[END_REF]; • Identifying the configurations of the system that should best fulfill these expected goals [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF].

Sequential management of these two steps may lead to non-convergent situations: expected goals do not necessarily match feasible ones. Identifying achievable goals necessitates a conjoint management of these two steps. Moreover, the problem is all the more complicated because knowledge regarding actions-and-goals relationships is mostly imprecise, especially in the preliminary stage of product design [START_REF] Antonsson | Imprecision in Engineering Design[END_REF]. Two challenges are hereafter considered when improving a complex system: at the strategic level, which changes in system outputs would bring actual improvements that would best fulfill customers' expectations, and at the operational level, how system configuration adjustments have to be carried out in order to achieve these targets. This paper is based on this semantic distinction to organize approaches in the literature and finally propose a unified framework. These two problems have been extensively studied in the literature. Nevertheless few works integrate both of them. The first point of view focuses on defining strategic targets without actual feasibility considerations [START_REF] Labreuche | Determination of the criteria to be improved first in order to improve as much as possible the overall evaluation[END_REF][START_REF] Keeney | Decisions with Multiple Objectives: Preferences and Value Tradeoffs[END_REF], which may lead to focus on unachievable goals. Conversly, the second point of view focuses on the capacity to achieve specific goals while not taking into account the commitment to do it [START_REF] Giorgini | Reasoning with Goal Models, 21 st Int. Conf. on Conceptual Modeling (ER02)[END_REF][START_REF] Felix | Relationships between goals in multiple attribute decision making[END_REF][START_REF] Montmain | Preference and causal fuzzy models for manager's decision aiding in industrial performance improvement[END_REF], which may lead to focus on meaningless goals for the strategic development of the industrial manufacturer. Some attempts seem nevertheless to be aware of the necessity to deal with both aspects when improving and designing systems [START_REF] Clivillé | Feasible Optimization[END_REF][START_REF] Fleurey | Modeling and validating dynamic adaptation[END_REF]. The MAUT-like formalism in [START_REF] Clivillé | Feasible Optimization[END_REF] is attractive, but hides the difficulty of establishing the behavioral model. In [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF], we proposed to identify the coalition of criteria to be improved first using the index value defined in [START_REF] Labreuche | Determination of the criteria to be improved first in order to improve as much as possible the overall evaluation[END_REF] while being the least difficult to be reached; both identifications are merged in a unique optimization problem, unlike to [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF] where they are sequentially processed. This sequencing might lead to divergent cases where strategic targets never met feasible ones. The model in [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF] avoids this misleading process, but is limited to deal with quantitative knowledge. Yet the difficulty is still compounded by the only imprecise or incomplete data and knowledge that are generally available in the conceptual phase of design. In this paper, we propose to reformulate the problem defined in [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF] in a qualitative framework. Adequate qualitative operators as Sugeno integral and median operator will replace quantitative ones: Choquet integral and weighted average. Section 2 poses the problematic and the unified approach with the related optimization problem and proposes redefining the two sub problems in a qualitative framework. Section 3 proposes an illustration with a case study and discusses the results. Finally Section 4 gives some conclusions and perspectives.

2

Modeling policy aspirations and capacity to act

Problem characterization and notations

In order to study a complex system, we should characterize it by a set of parameters. For example, an autonomous robot can be characterized by several parameters: type of rolling base, size, engine type, embedded energy capacity etc. Each configuration of these parameters is an instantiation of a robot. To check if there is an improvement when changing the configuration, the decision maker has to consider the objectives that have to be achieved by the robot in terms of autonomy, reliability, mobility, cost etc. The possible configurations perform different outputs that more or less satisfy the objectives. In a general context, let a system be characterized by its ters , , … . Let Γ be the set of all possible values of the vector ( , , … ). A system is then defined by a configuration Γ. Improving a system is to make it evolve from a configuration Γ to a configuration ′ Γ which gives a better satisfaction of the objectives that have been fixed for the system under cost constraints (money, risk, time etc.). Let us denote = {1,2, . . }, the set of criteria. The system performance is evaluated by its elementary performance measures , , … . An action is associated with each change of parameter . The set of actions is denoted by . As there are operational constraints over the set of actions, some of them cannot be performed together: they are said to be mutually exclusive. We define an action plan denoted as a subset of non-exclusive actions in A. We are searching among the available action plans , the ones that better improve the customer's satisfaction and that are compliant with the capacity of the manufacturer to apply them. The greater the number of individual performances to be improved, the more difficult it is to find the proper action plan. Optimistically, one can seek for an action plan that improves all objectives associated to the system. But, it can be more relevant for the decision maker to improve a subset of criteria of that leads to reasonable satisfaction degree when the improvement is written-down to its achievability. For ⊆ , let ( ) denote the degree to which improving criteria of seems achievable considering the available actions; and ( ) the expected degree of satisfaction the improvement on criteria in should provide. In [START_REF] Imoussaten | Modélisation et pilotage de la phase de délibération dans une décision collective-vers le management d'activités à risques[END_REF], we proposed an optimization problem (1) in a multi-criteria aggregation framework to identify the set of criteria * that maximizes the expected degree of satisfaction under achievability constraints:

max #⊆$ ( ) %&' ()* *+ ( ) ≤ ( ) (1)
The problem can be seen as a multi-criteria decision making problem subject to feasibility. However, feasibility has little to do with explicit operational constraints. ( ) rather assesses how much confidence there is that the actions the manufacturer intends to commit will give satisfaction. The idea is to make comparable the degree of confidence in achieving one goal and the expected degree of satisfaction this goal would provide.

In [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF], ( ) and ( ) were defined in a quantitative setting. In this paper we propose to redefine these quantities in a qualitative setting. Indeed, the system's response to configuration changes and the decision-maker expectations are generally only imprecisely and incompletely known during the conceptual design phase. Expected benefits and achievability of targets are assessed through adequate operators in this qualitative framework.

Qualitative characterization of the capacity to achieve a subset of criteria

The qualitative approach involved in this section has been originally proposed in [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF][START_REF] Imoussaten | Modélisation et pilotage de la phase de délibération dans une décision collective-vers le management d'activités à risques[END_REF] and is summarized in the following. It models the impacts that actions have on the system's performances. It covers three sub problems:

• how to characterize the impact of one parameter change on a given performance?

• how to merge the impact of several parameters' changes on a given performance?

• how to assess the overall impact a configuration may have on a set of criteria?

Let -( . ) denote the degree to which the action . may support performance -. As soon as -( . ) >0, the action . contributes to the satisfaction of the criterion /. Let 0 -( . ) be the degree to which the action . may distract the performance -. As soon as 0 -( . ) >0, the action . harms the criterion /. The qualitative action-performance relationship has then to be extended to action plans. The major difficulty is that for a given action plan and a criterion /, several actions in may affect -positively and several other actions in may affect -negatively. What is then the resulting effect of on the performance -? To answer this question, standard multi-criteria approaches cannot be applied since both positive and negative impacts on performance level are considered here. Indeed, the estimation of the merged impact of an action plan naturally depends on the system behavior, as well as on the designer's/operator's decisional behavior: a pessimistic attitude (whereby a risk aversion position will focus attention on the lowest positive merged impacts of the action plan) vs. an optimistic attitude (whereby risk acceptance will focus attention on the most highly positive merged impacts). In both cases, we suppose that the merged negative impacts cannot be under rated. For example, from an optimistic point of view, the merged effect can be defined as:

-( ) = max 1∈3 4 5 (1 ) -( ) and 0 -( ) = max 1∈3 4 6 (1 ) 0 -( ), 1 While in a pessimistic approach:

-( ) = min Then, the degree to which an action plan may affect a subset of criteria is yet to be defined. Let ⊆ be a subset of criteria and an action plan. The resulting degree to which should contribute to the improvement of while not deteriorating the other criteria in \ can be assessed, from a pessimistic point of view by: More precisely # ( ) >0 if any criterion in is improved by , whereas no criterion in \ is distracted 2 .

% # ( ) = min -∈# -( ) /> ∀ ∈ \ @ . ( ) >
Finally, max 1 # ( ) characterizes the highest degree to which criteria in I can be expected to be improved considering all possible action plans. Hence, the quantity %( ) = max 1 # ( ) is considered here as the assessment of the capacity to achieve improvement of .

1 Considering 0 -( ) = min 1∈3 4 6 (1 ) 0 -( ), should be too permissive. 2 In fact we believe that only the pessimistic point of view is a reasonable attitude to compute % # ( ) since a conjunctive aggregation is necessary in order to guarantee that all criteria of I are expected to be improved.

2.3

Qualitative characterization of the commitment to achieve a subset of criteria

In this sub-section, we are interested in determining the criteria that should be improved first to satisfy as much as possible the decision-maker expectations. For this purpose, we build a value function F defined on the set of all the criteria subsets of , and that estimates the expected overall satisfaction the improvement of criteria in would provide, knowing that the initial performances are given by the vector of performances G = ( G , G , … , G ). In our qualitative multi-criteria framework, the overall satisfaction related to any vector of partial performances ( , , … , ) is modeled as the qualitative aggregation of the -'s performance on the i th criterion by the Sugeno integral. This operator both allows modeling the importance of criteria but also the preferential interactions among them. Let us start by recalling some notations concerning the Sugeno integral.

Let L be an ordered qualitative scale with 1 H its highest value and 0 H its lowest value. A function I: 2 K → M is called a fuzzy measure if it satisfies the following conditions:

1. I(∅) = 0 H 2. I( ) = 1 H

I is monotonic non decreasing for inclusion, i.e., for any , N ⊆ ⊆ N ⇒ I( ) ≤ I(N).

The monotonicity of I means that the weight of a subset of criteria cannot decrease when new criteria are added to it (see [START_REF] Keeney | Decisions with Multiple Objectives: Preferences and Value Tradeoffs[END_REF][START_REF] Giorgini | Reasoning with Goal Models, 21 st Int. Conf. on Conceptual Modeling (ER02)[END_REF] for further details). Let I be a fuzzy measure on N taking values in the scale M. Let consider a function >: ⟶ M, then the Sugeno integral of > with respect to I, denoted R (>) is given by: R (>) = max

S-S (min(> T(-) , IU T(-) V),

where W is a permutation on such that > T( ) ≤ > T( ) ≤ ⋯ > T( ) , and T(-) = {W(/), W(/ + 1), … W( )}. This integral is monotone, and obviously presents a compromise behavior. It searches the importance exceeding a certain level, and then performs a compromise deal between the selected values. Another point of view is to see this combination as the disjunction of conjunctions. All the criteria, their interactions and their importance, are taken into account to assess the aggregated score of a given vector of elementary ratings [START_REF] De Campos | A unified approach to define fuzzy integrals[END_REF][START_REF] Dubois | The use of the discrete Sugeno integral in decision-making: a survey[END_REF][START_REF] Grabisch | The symmetric Sugeno integral[END_REF]. Let us consider that the overall performance of the system is evaluated from criteria performances using a Sugeno integral operator with respect to the fuzzy measure I: R ( ) = max S-S (min( T(-) , IU T(-) V) where -is the i th performance with a value in the same ordinal scale M = {0 H , … , 1 H } as the capacity I. By considering the set of criteria and the initial vector of performances G , several vectors of performances Z are possible improvements on . If G = ( G , G , … , G ), then any Z = ( Z , Z , … , Z ) such that ∀/ ∈ , - Z > [ G is a possible improvement. Let us note \ A+]* ^ the set of all these vectors such that Z is obtained from Z_ by increasing each - Z_ one step up (steps are the levels of L) for all the criteria in I. In other words, similarly to [START_REF] Labreuche | Determination of the criteria to be improved first in order to improve as much as possible the overall evaluation[END_REF], only improvements obtained by progressing from G to 1 # along the diagonal of the | |-cube are considered (where | |denotes the cardinal of ) in order to be sure to improve all criteria in . To each expected improvement Z is associated its overall performance R ( Z ). Then, the expected overall performance in this qualitative framework of performance can be assessed by the median of the R ( Z ) a %. The median provides how worth it is improving criteria on (the median is an associative qualitative compensatory aggregation operator [START_REF] Arif | AWFO (un opérateur d'agrégation) pour la reconnaissance des formes. 15e congrès francophone de Reconnaissance des Formes et Intelligence Artificielle[END_REF]):

F ^( ) = \(b( R ( G ), R ( ), … R ( c1d )), where c1d = (1 # , K\# G ) .
Note that the cardinal number of \ A+]* ^ may be odd or even. Since in the last case the median consists of finding a value between the two middle points, such a choice depends on the decision maker's attitude. A pessimistic attitude would take first middle point as the median, while an optimistic one would take the second point.

Optimization problem in the qualitative setting

Note that by construction, ⊆ a ⟹ % # ( ) ≥ % # g ( ), ∀ , and therefore %( ) ≥ s(I a ). In the other hand, ⊆ a ⟹ F ^( ) ≤ F ^( ′) for a given initial performance vector G . Hence these two functions are respectively non-increasing and nondecreasing with respect to (with respect to the inclusion relationship), and so coincide at an optimal given * solution of the optimization problem [START_REF] Shen | An integrated approach to Innovative Product Development using Kano's model and QFD[END_REF]. Hence, the problem of searching the subset of criteria that are both profitable and achievable is equivalent to the optimization problem (1) with s(I) and F ^( ) as arguments. However precautions are necessary to properly process the inequality constraint of problem (1).

In the case where these functions take their values in different ordered qualitative scales, a third ordered qualitative scale can be introduced where values have the semantic of satisfaction degree or of possibility degree in order to draw the comparison of the inequality in [START_REF] Shen | An integrated approach to Innovative Product Development using Kano's model and QFD[END_REF]. Solving this problem may be extremely hard with 2 |3j| action plans and 2 |K| subsets of criteria to search. The first thorny exponential problem comes from the computation of %( ): heuristics have already been introduced in a branch and bound algorithm in [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF] to compute %( ) in a reasonable time. Further heuristics are needed to solve (1) while computing the minimum of %( )′s values; they will be addressed in our future works. We give here just an overview of two heuristics. The first heuristic will be based on the inverse monotonicity of %( ) and F ( )( ⊆ a ⟹ %( ) ≥ s(I a ) and, ⊆ a ⟹ F ^( ) ≤ F ^( ′) ); a second heuristic will introduce lower and upper bounds for %( ). In the following study case | | and | l| are small enough to perform exhaustively all the required computations.

Design Application

The following example has no validation ambition but aims at illustrating the proposed approach. The robotic challenge Robafis is organized annually by the French association of Systems Engineering AFIS to promote Systems Engineering practice in engineers' schools 3 . The scope of the challenge is for instance (Robafis_2013) to build an autonomous mobile robot able to compete with other robots and using some provided and imposed materials. Each robot is limited to a 0.3 n \ n cube and has to achieve the following mission as quickly as possible: to grasp and transport some various colored spheres between several stock devices spread over a plan playground. Some dark lines are drawn on the ground to guide the robot between stock devices. The autonomous robot was broken down into four sub-systems: a gripper device, sensors equipment, a rolling base, a control device. The programmable control device type is also imposed to the competitors. The physical alternatives (the possible robot configurations) depend on the design options and on the skills of each competitor.

The preference model depends on the strategy followed by each competitor. The solution principles are resumed in Table 1. There are three principles for designing the gripper device, three for the rolling base and two for the sensors equipment. Thus, there are 18 actions plans to be compared, corresponding to the 3x3x2 admissible configurations.

In this example an action is the choice of one element of the configuration, and an action plan is a configuration.

Data of the application

Four criteria are considered to decide between configurations: the sparsity of used components (Cr1), the robot speed capacity (Cr2), the reliability of the robot (Cr3), and the maintainability of the robot (Cr4). Taking into account the competition rules, Table 2 provides an example of competitors' preferences between the criteria modeled by a qualitative capacity function I: 2 K → M where N = {1,2,3,4}, and L = {0, a, b, c, d, e = 1 H }; 0 means: not important at all and e means very important. Table 3 presents, for each of the four criteria, the positive impact (denoted by "+") or the negative impact (denoted by "-") of each elementary technical choice on the satisfaction level of each criteria. Furthermore a confidence degree in the fulfillment of such an impact is defined on the same qualitative scale L. It must be emphasized here that the approach differs from just aggregating subcriteria of multiple options and choosing the best one since actions may have positive or negative impacts on the criteria (it is a bipolar problem) and that Table 3 results from an analysis of a behavior model of the robot and describes the confidence the designers have in the impacts the actions may have on each partial performance.

Computation of t u v

Let's consider some given initial performance vectors: G = (0 0 0 0), p G w = (0 a 0 a), p G x = (0 d b c). The Sugeno integral for each of these initial performance vectors are respectively: R (p G ) = 0, R (p G w ) = a, R (p G x ) = d with respect to the capacity defined in table 2. Hence, the values of the F ^4 for each initial performance vectors are given in the Table 4:

Table 4. F ^4 ( ) values for different initial performances G 4 F 4 / {1} {1,2} {1,3} {1,4} {1,2,3} {1,2,4} {1,3,4} {1,2,3,4} F ^( ) a b b b c c c c F ^w ( ) a b b b c c c c F ^x (I) c c d c d c d d F 4 / {2} {2,3} {2,4} {2,3,4} {3} {3,4} {4} F ^( ) a c c c b b a F ^w ( ) a c c c b b a F ^x ( ) c d c d d d c
It can be easily checked in Table 4 that ⊆ a ⟹ F ^4 ( ) ≤ F ^4 ( ′) and F ^4 depends on the initial performance: the higher the initial global performance value R ( G 4 ), the higher F ^4 ( ). As the configuration to be selected is the first design choice, we assume that the initial performance vector is the null vector (p G ). Then, with such an initial vector, the Sugeno integral properties, the median operator and the ordinal scale L imply that F z ^ cannot exceed the value ).

Computation of {(|)

The values of the %( )′% for each subset of criteria are computed with an optimistic attitude (see section 2.2) and are given in Table 5: Considering the initial performance p G , %( ) takes its values on {), b, (}. The only valid coalitions * that satisfy the inequality constraint of (1) are the coalition {2,4} and {1,2,4 } which happen to maximize, in this particular case both %( ) and F w ^( ).

Thus, according to the competitors' preference model defined in Table 4, the best strategy to follow is to build a robot with high speed and easily maintainable. Consid-ering Table 3, the three configurations that improve * and do not degrade the performance on / * are : a 1 ={G 1 ,R 1 ,S 2 }, a 2 = {G 1 ,R 2 ,S 2 } and a 3 ={G 1 ,R 3 ,S 2 }. The three rolling bases are acceptable at this stage of the design. A sensibility analysis has to be undergone in order to consolidate this configurations choice. Such analysis should be included in a wider risk analysis that should be the scope of further research work.

Conclusion

Dealing with new products or improving existing ones in today's technology-driven markets present important risks. In the case of complex industrial systems, the number of constraints and goals rapidly becomes inextricable. Manufacturers are faced with an unprecedented rate of changes in their processes and practices. This challenge requires large forecasting capacities to produce challenging but achievable goals. In [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF], we have proposed a model in a quantitative framework of performance to better manage the balance between strategic ambition and manufacturing realism. Nevertheless, the forecasts are yet compounded by the only imprecise or incomplete knowledge generally available in the conceptual phase of design. In this paper, we have proposed to reformulate the problem defined in [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF] in a qualitative framework. This qualitative approach allows lifting the constraint encountered in [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF]. It allows qualitative reasoning with a gradient like approach, which seems more consistent with the available knowledge during design phases. The case study of a design autonomous robot shows the necessity of considering simultaneously know-how and ambition in design project to set achievable goals. Identifying achievable goals necessitates a conjoint management of feasibility and ambition in their definition contrarily to [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF] where they were sequentially managed. In future works, we will propose another point of view to this conjoint control issue. The idea is to integrate the capacity %( ) in the computation of F # in such a way that improvements are all the less credible than they require high level of know-how whereas improvements are all considered equally probable in the models of [START_REF] Montmain | Multi-criteria improvement of complex systems[END_REF] and [START_REF] Sow | Between policy aspirations and capacity to act: how designing achievable goals[END_REF].

  1∈3 4 5 (1 ) -( ) and 0 -( ) = max 1∈3 4 6 (1 ) 0 -( ) where - 9 ( ) = {a ∈ ap : -( ) >0} and - < ( ) = {a ∈ ap : 0 -( ) >0}.

  0 . ( ) +A . < ( ) = ∅C 0 +*ℎ(AF/%( , (respectively max -∈# -( ) /> ∀ ∈ \ @ . ( ) > 0 . ( ) +A . < ( ) = ∅C 0 +*ℎ(AF/%( from an optimistic point of view).

Table 1 .

 1 Solution principles

	Gripper device	Rolling base	Sensors equipment
	G1: Fork (taking the	R1: Four wheels	S1: two color sensors for following the
	sphere from below)	rolling base	dark lines, one light sensor for recog-
				nizing the sphere color
	G2: Lateral gripper	R2: Two wheel	S2: one color sensor for following the
	(pinching	laterally	drive and one free	dark line, one color sensor for recog-
	the sphere)		wheel rolling base	nizing the sphere color
	G3: Grapnel (taking	R3: rolling base
	over the sphere)	with tracks

Table 2 .

 2 Competitors' Preferences: the qualitative capacity function values I( )

	I	I( )	I	I( )	I	I( )	I	I( )
	∅	0	{C4}	a	{C1,C3}	b	{C1,C3,C4}	d
	{C1}	a	{C1,C4}	b	(C2,C3)	d	{C2,C3,C4}	d
	{C2}	a	{C2, C4}	c	(C1,C2,C3)	e	{C1,C2,C3,C4}	e
	{C1,C2}	b	{C1,C2,C4}	d	{C3}	b	{C3, C4}	b
			Table3. Impacts of configurations
			Number of Pieces Speed Reliability Maintainability
			Cr1		Cr2	Cr3	Cr4
	G1: fork	+e		-a	-b	+e
	G2: gripper	-b		-a	+d	-a
	G3: graspel	-d		-a	+c	-d
	R1: 4wheels	+e		-c	-d	+c
	R2: 3 wheels	-c		+d	+c	-b
	R3: trackers	-d		+c	-c	-d
	S1: 2+1	-c		+e	+c	-c
	S2: 1+1	+b		-d	-c	+c

Table 5 .

 5 %( ) values

	s / I %( )	{1} e	{2} e	{1,2} e	{3} d	{1,3} d	{2,3} d	{1,2,3} d	{4} e
	s / I {1,4} {2,4} {1,2,4} {3,4} {1,3,4} %( ) e e e c c	{2,3,4} {1,2,3,4} c c

(http://www.robafis.fr/RobAFIS/Bienvenue.html).


T V Tran

F Moussouni

S Brisset

P Brochet

Adapted Output Space-Mapping Technique for a Bi-Objective Optimization

Keywords: Space-mapping, Multi-objective optimization, 3D finite element model, Transformer. I

Multi-objective optimizations by means of 3D finite element models result in very high computation burden. To have an affordable computation cost, the output space-mapping technique is applied with a new method where the scalar correction of the model outputs is replaced by a set of corrective functions. This method is used for the bi-objective optimization of a transformer and allows finding the complete Pareto optimal set in less than two days on a laptop.

II. OPTIMIZATION PROBLEM

The safety isolating transformer is a one-phase step-down transformer. It uses grain-oriented E-I laminations. The primary and secondary windings are both wound around the frame surrounding the central core (Fig. 1).

It has been selected as a simple practical example capable to provide a quantitative evaluation of the savings possible with the proposed procedure.

The bi-objective optimization problem of a safety isolating transformer contains 7 design variables. There are three parameters a, b, c for the shape of the lamination, one for the frame d, two for the section of conductors S 1 , S 2 , and one for the number of primary turns n 1 . The geometrical parameters and section of conductors are shown in Fig. 1.

The bi-objective optimization problem is expressed as:

V V I I C T C T s.t. M ir co tot ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Δ ≤ °≤ °≤ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - μ η (1)
There are 6 non-linear inequality constraints in this problem. The copper and iron temperatures T co , T ir should be less than 120°C and 100°C, respectively. The magnetizing current I μ /I 1 and drop voltage ∆V 2 /V 2 should both be less than 10%. Finally, the filling factors of both coils f 1 and f 2 should both be lower than 0.5. The objective functions are to maximize the efficiency η and to minimize the total mass M tot of iron and copper materials. All electric and thermal quantities are computed at full-load. Digital Object Identifier inserted by IEEE

A. Coarse and fine models

The multi-physical phenomena within the transformer are electric, magnetic and thermal. They are modeled by a lumped-mass (coarse) model (LM) and a 3D finite element (fine) model (3D FE) [START_REF] Tran | A Benchmark for Multi-Objective, Multi-Level and Combinatorial Optimizations of a Safety Isolating Transformer[END_REF].

The LM model is built with the assumption that the voltage drop due to the magnetizing current is neglected. Therefore the maximal induction depends on the primary voltage. The computational time of the coarse model is very short (50 ms on an Intel Pentium M 2.13 GHz laptop).

The 3D FE magneto-thermal model is built with the assumption that all magnetic and electric quantities are sinusoidal. The iron loss is computed with Steinmetz formula described in [START_REF] Tran | A Benchmark for Multi-Objective, Multi-Level and Combinatorial Optimizations of a Safety Isolating Transformer[END_REF] and the leakage inductances are calculated with the magnetic energy. Full-load and no-load simulations are used to compute all characteristics of the safety isolating transformer. The 3D FE model with magneto-thermal coupling requires a very expensive computational time (about 2 hours on an Intel Pentium M 2.13 GHz laptop).

B. Designer's dilemma

Generating 100 optimal solutions of the Pareto optimal set by means of the 3D FE model requires 10 000 hours while 5 minutes only are needed with the LM model. In fact, as solving 3D FE model is very expensive only six points are computed and considered as a reference set (Fig. 2). The points are found by using the mono-objective OSM technique to reduce the computation time. The LM model is used to build an extended Pareto optimal set in a short time. It is obvious that this last optimal set is far from the reference set.

A first solution to this dilemma is to interpolate the reference set. Unfortunately, no information on the design parameters can be obtained because the interpolation can only be made in the objective space.

Therefore, the authors propose to adapt OSM technique for bi-objective optimization problems in order to generate an extended and accurate Pareto optimal set in less than 100 hours.

III. ADAPTED SPACE-MAPPING TECHNIQUE

A. Bi-objective optimization algorithm

According to the state of the art, the ε-constraint method is a useful approach to build a Pareto optimal set [START_REF] Murano | Multiobjective Optimization of Electro-optic Modulators by Using the ε-constraint Method[END_REF]. Moreover, this method may reach Pareto optimal solutions in the nonconvex region. The ε-constraint method consists to transform the multi-objective problem in a single-objective problem. Among the objectives, one is kept and the others are transformed in inequality constraints:

0 and s.t. min ≤ ≤ = ≠ ∈ (x) k (x) f (x) f x f j i j i X x * ε (2)
where i f is the objective kept,

B. Space-mapping techniques

Space mapping techniques aim to use both coarse and fine models to reduce the computational time and increase the accuracy of the obtained solution [START_REF] Choi | A New Design Technique of Magnetic Systems using Space Mapping Algorithm[END_REF]- [START_REF] Encica | Aggressive Output Space-Mapping Optimization for Electromagnetic Actuators[END_REF].

In general, the coarse computationally cheaper model is

denoted by m R z c ∈ ) ( with n R Z z ⊂ ∈
, and the fine

computationally expensive model is denoted by m R x f ∈ ) ( with n R X x ⊂ ∈
. In practical engineering, the coarse and fine model spaces are often the same, i.e. x represents the solution of a given optimization problem:

0 min arg ≤ - = ∈ (x) k s.t. y f(x) x f X x * (3)
where m R y ∈ denotes a vector of design specifications and can be zeros in the case of a minimization. In practice, solving (3) is very expensive. Therefore, the faster optimization problem based on the coarse model is preferred:

0 ) ( . . ) ( min arg * ≤ - = ∈ z k t s y z c z c Z z (4)
One approach of SM techniques, called Output Space-Mapping (OSM) [START_REF] Encica | Aggressive Output Space-Mapping Optimization for Electromagnetic Actuators[END_REF] consists to modify the coarse model by

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = Θ ∈ + ) ( ) ( ) , ( ) , ( min arg ) ( ) ( ) ( ) (
≤ - = + + ∈ + j c j X x x k t s y x c x j θ θ (6)
After that, a simulation with

) 1 ( + j
x by the fine model is performed to compute the fine responses ) (

) 1 ( + i x f and ) ( ) 1 ( + i x k f
. The algorithm stops when all the corrective coefficients are unchanged from one iteration to the next one.

C. Corrective spline functions

A corrective coefficient is introduced for each objective and constraint therefore the corrected coarse model has the following expression:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ) ( ) ( )). ( Diag( ) , ( ) , ( x k x c x k x c c c ε θ θ θ (7)
In the case of a mono-objective optimization only a set of p scalar coefficients is searched (p is the number of objectives and constraints computed with the fine model).

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = + ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( j f j c j j x k x k x f x c j θ (8)
where the indices f and c denote the fine and coarse model respectively.

In the case of a multi-objective optimization, the p corrective coefficients' values must be changed for each solution from the Pareto set, i.e. for each value of the threshold value ] , [ max min ε ε ε ∈ . So that these p coefficients are replaced by p corrective functions of ε:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = + M ) ( ) ( ) 1 ( ε ε θ i j S (9)
where S(ε) is a spline cubic interpolation function. The spline cubic functions avoid the oscillations that appear in the polynomial interpolation approach when the order is high.

Since the mapping functions are defined, they are used by the multi-objective algorithm to correct the coarse model and to obtain a new Pareto optimal set.

The following section presents the proposed algorithm that combines OSM and ε-constraint algorithm.

D. Proposed algorithm

At the beginning of the algorithm (j = 0), all the corrective coefficients are initialized to unity:

I = ) 0 ( θ (10)
At each iteration, a multi-objective optimization is performed by using the ε-constraint method and the corrected coarse model in order to obtain a new Pareto optimal set in a short time: values 30 takes and

)) ( , ( 0 )) ( , ( )) ( , ( min max min ) ( 2 ) ( ) ( 1 ε ε ε ε ε θ ε θ ε θ ≤ ≤ ≤ ≤ ∈ j j c j X x x c x k s.t. x c (11)
Then, (2 j +1) solutions *

x belonging to the Pareto optimal set are chosen in order to compute the responses of the fine model )

( * i x f and ) ( * i f x k
. Note that only 2 j-1 points are new.

To establish the corrective functions at the next iteration, the new points are at the center of the intervals, in the same way as the Dichotomy method.

If the following condition is checked, the algorithm stops:

τ ε θ ε θ ε θ ε θ ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + )) ( , ( )) ( , ( )) ( , ( )) ( , ( ) ( * ) ( * ) 1 ( * *
) 1 ( j c j j c x k x c x k x c j (12)
where τ is the required accuracy. If the corrected coarse model responses are not close enough to the fine model ones, the algorithm continues by updating the corrective functions (8) with the new points.

To summarize, the algorithm carries out the following main steps: 0. initialization: j = 0, θ (0) =I 1. build a Pareto optimal set by using the εconstraints method and the corrected coarse model c(x, θ (j) ) to solve (11) 2. choose 2 j + 1 points on the Pareto optimal set. 3. evaluate the fine model with the chosen points to compute f(x) and k f (x) 4. update the p corrective functions θ (j+1) (ε). 5. if (12) stop the algorithm else j = j + 1, go to 1.

IV. APPLICATION CASE

The bi-objective optimization problem of the safety isolating transformer presented in section II is solved with the proposed algorithm.

For this optimization problem, two constraints (filling factors f 1 , f 2 ) among six and one objective (total mass M tot ) among two are not evaluated by the 3D FE model. Indeed, both filling factors and the total mass are analytically computed. So that, five responses η, T co , T ir , I μ /I 1 , and ∆V 2 / V 2 need 3D FE computation, i.

f f V V I I C T C T s.t. tot ir co ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Δ ≤ °≤ °≤ - ε η μ (13)
At initialization (j = 0), the coarse model is not corrected yet and the corrective functions are all equal to one. Two points at min j ε and max j ε are computed with the fine model to test the stop criteria and build the corrective functions for the next iteration. At iteration 1, the corrective functions are now linear (Fig. 4) and a new optimal set is obtained. In Fig. 4, the corrective function is given only for the efficiency of the transformer, i.e. θ 1 .

At each iteration, one new point is added at the middle of each couple of points, resulting in 2 j-1 new points amongst a total of (2 j + 1) points, i.e. 3D FE model evaluations.

Fig. 3 shows the Pareto optimal set obtained by using the corrective functions. 30 points are given at each iteration.

At the end of the optimization, a Pareto optimal set very close to the reference set is found.

Finally, only 17 3D FE simulations are needed to obtain an accurate Pareto optimal set. The computational time of the proposed algorithm is approximately 34 hours (1.4 day) on an Intel Pentium M 2.13 GHz laptop. The reference Pareto optimal set requires 60 hours (2.5 days) on the same machine.

To have an accurate Pareto optimal set by using the 3D FE model only, approximately 10,000 hours (14 months) are required. All the computed points are shown with circles in Fig. 4.

V. CONCLUSION

The output space-mapping technique is adapted to provide a practical way to build an accurate Pareto optimal set in biobjective optimization using full 3D coupled finite element model and keeping the computational time in an acceptable limit. In addition to the 3D FE model, a simpler one provided with corrective coefficients has been used. Cubic spline interpolation functions are used to predict the values of the corrective coefficients. By using these corrective functions, the ε-constraints bi-objective optimization algorithm gives an extended and accurate Pareto optimal set in less than 2 days.

TFig. 1 .

 1 Fig.1. Bi-objective optimization problem of a safety isolating transformer

f

  ≠ are the other objectives, f k are the ordinary constraints, and j ε are the threshold values. By varying j ε between min j ε and max j ε , the whole Pareto optimal set may be found. Two mono-objective optimizations are computed to determine the upper and lower threshold values. min j ε is found by minimizing i j f ≠ and max j ε is found by minimizing i f . Using the ε-constraint method with the coarse model, a Pareto front with 100 points is quickly built and shown in Fig. 2.

  The non-linear constraints computed by the coarse and the fine models are

Fig. 2 .,

 2 Fig.2. Pareto optimal set using the coarse and fine models 2 4 6 8 10 12 14 16 0.03

.

  e. only five values of corrective coefficients [θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ] are computed for each threshold value ] The losses 1-η are c 1 in (11) and the total mass M tot is c 2 . Therefore, the bi-objective optimization problem (1) becomes a set of 30 mono-objective optimization problems with one additional constraint on the total mass M tot :

Fig. 4 .

 4 Fig.4. Corrective functions for the efficiency at each iteration.

Fig. 3 .

 3 Fig.3. Pareto optimal set obtained at each iteration


Éric Lichtfouse
email: eric.lichtfouse@dijon.inra.fr

Sen Dou

Cyril Girardin

Micheline Grably

Jérôme Balesdent

Françoise Behar

Mireille Vandenbroucke

Unexpected 13 C-enrichment of organic components from wheat crop soils: evidence for the in situ origin of soil organic matter

Keywords: carbon-13, humic acids, humin, soil lipids, plant alkanes, soil alkanes, pyrolysis of humic acids

Various organic constituents extracted from wheat and from soil organic matter have been analysed for their carbon content, their absolute concentration and their stable carbon isotope ratios. Most organic subfractions from plants, or soil, are 13 C-depleted by up to 9.4‰ relative to bulk organic matter, mainly as a result of their higher lipid content. Furthermore, soil organic constituents are unexpectedly 13 C-enriched by +1.5‰ to +4.3‰ relative to homologous plant constituents. Indeed, the selective preservation of plant lignin and lipids, following incorporation into the soil biomass, should have led to the accumulation of 13 C-depleted compounds. Hence, these results favour the in situ formation of soil organic matter either by recondensation of small molecules or by selective preservation of biopolymers from soil microorganisms.

INTRODUCTION

The decay of plants leads to the preservation of a few percents only of plant carbon into soils, the bulk being rapidly biodegraded then returned to the atmosphere as CO 2 . Although many bulk chemical properties of soil organic matter are well-known, there is still a lack of understanding of the precise origin, and pathways, of transformation of soil organic substances, notably at the molecular level (Schnitzer and Khan, 1978). The formation of humic substances is believed to occur by two major processes [START_REF] Hayes | Structures of Humic Substances: the Emergence of "forms[END_REF]: 1) the partial degradation and selective preservation of resistant macromolecules from plants or soil organisms and 2) the polycondensation of small molecules resulting from biomass degradation, e.g. the reaction of amino acids with carbohydrates [START_REF] Maillard | Synthèse des matières humiques par action des acides aminés sur les sucres réducteurs[END_REF][START_REF] Maillard | Identité des matières humiques de synthèse avec les matières humiques naturelles[END_REF]. Carbon isotope ratios have been used to define the origin and transformation of organic substances from soils and sediments [START_REF] Nissenbaum | Chemical and isotopic evidence for the in situ origin of marine humic substances[END_REF], Collister et al., 1992, Lichtfouse and Collister, 1992[START_REF] Macko | Early diagenesis of organic matter in sediments. Assessment of mechanisms and preservation by the use of isotopic molecular approaches[END_REF], Collister et al., 1994a, Lichtfouse et al., 1994ab, Lichtfouse, 1995). In this note we show, using isotopic comparisons of various soil and plant organic substances, that the major part of soil organic matter where wheat has been grown is unlikely to be derived by the selective preservation of the plant's lipids or lignin, and thus must have been formed within the soil.

EXPERIMENTAL Plant and soil sampling

Soils, cropped solely with plants following the C 3 photosynthetic pathway (O' Leary, 1981), were cored through the 0-35cm ploughed horizon in September 1987 at La Minière, near Versailles, France, after 3 years of wheat cropping (1984)(1985)(1986). The soil is a dystric eutrochrept, silty loam [START_REF] Balabane | Input of fertilizer-derived labelled N to soil organic matter during a growing season of maize in the field[END_REF]. The 0-35 cm ploughed horizon comprises 11.4% of sand (50-2000 µm), 73.1% of silt (2-50 µm) and 15.5% of clay (0-50 µm). Two additional soil samples were cored at Boigneville, Essone, France, in 1990 and1993 after respectively 20 and 23 years of wheat cropping. These soils have very similar characteristics to the soils from the La Minière field [START_REF] Balesdent | Effect of tillage on soil organic carbon mineralization estimated from 13 C abundance in maize fields[END_REF]. Nine cores, taken at various locations, were well-mixed, dried at 20°C then sieved to 2000 µm. Wheat leaves, stems and roots (Triticum aestivum) were washed with distilled water then freeze-dried at -20°C. All samplings and further analyses were repeated three times.

Organic matter fractionation

Dried soils, leaves, stems and roots were finely ground and extracted ultrasonically (30 min., 30°C, 3x) with CHCl 3 -MeOH (3/1 v/v). Extracts were fractionated into neutral and acid fractions by passage through silica gel impregnated with KOH followed by HCO 2 H acidification [START_REF] Mccarthy | A rapid quantitative method for the separation of free fatty acids from other lipids[END_REF]Duthie, 1962, Collister et al., 1994a). Neutral fractions were fractionated into hydrocarbon-ester, ketone, alcohol and polar fractions by silica-gel thin layer chromatography, using CH 2 Cl 2 as developer and 1,2:3,4-dibenzanthracene, friedelin and cholesterol as reference compounds (modified after Lichtfouse et al., 1994c). Neutral fractions were fractionated into alkane-alkene, aromatic and ester fractions by silica-gel thin layer chromatography, with n-hexane as developer and using n-octacosane, 1-phenyldodecane, 2-methylphenanthrene and 1,2:3,4-dibenzanthracene as reference compounds. Individual n-alkanes from the alkane-alkene fractions were identified by gas chromatography-mass spectrometry, and co-elution with pure standards, and quantified by gas chromatography fitted with a flame ionization detector and using 5α-androstane as internal standard.

Ground dry soils were extracted 48 h. at 20°C under N 2 with 0.1 M NaOH to give an insoluble fraction (humin and minerals) and a soluble fraction (humic and fulvic acids). Humic acids were separated from the soluble fraction by precipitation at pH 1.5 with 2.4 M HCl (3x). Humic acids were further purified by centrifugation at pH 7 to remove most clay particles, dialysed for 10 days against distilled water then freezedried at -20°C. Humic acids were pyrolysed at 550°C in an open gold tube under continuous argon flow to give pyrolysates and residues.

Isotope analysis

Isotopic compositions, and carbon contents, were measured on a Carlo Erba NA 1500 elemental C and N analyser coupled to a VG Sira 10 mass spectrometer: precision 0.03‰, overall deviation 0.1‰ (3 replicates). Isotopic analyses of individual n-alkanes were carried out under a continuous helium flow using an HP 5890 gas chromatograph coupled to a CuO furnace (850°C) and a cryogenic trap (-100°C): this system was coupled to a VG Optima mass spectrometer. Ion currents were monitored continuously (m/z = 44, 45 and 46: precision 0.1‰, overall deviation 0.3‰, 3 replicates). Carbon isotope compositions are expressed in per mil. relative to the Pee Dee Belemnite Nitrogen content in parenthesis. n.d.: not determined standard: δ 13 C = [( 13 C/ 12 Csample/ 13 C/ 12 Cstd) -1] x 10 3 , where 13 C/ 12 Cstd = 0.0112372.

RESULTS AND DISCUSSION

Mass balance

Yields, carbon contents and isotope values of various organic fractions from wheat and wheat crop soils from the La Minière field are reported in Table 1. Humin and humic acids amount respectively to 664 and 63 mg carbon per g of soil organic carbon, and are thus, quantitatively, the most abundant organic components of the soil. Soil and wheat organic extracts, and their subfractions, occur in minor amounts. Nethertheless, they are probably highly involved in soil transformation processes because they represent the "free" part of the organic matter. Moreover, due to their high lipid content, they should be selectively preserved in soils because lipids are more resistant to biodegradation than carbohydrates and amino acids. However, the carbon contents of organic extracts from wheat, averaging 23.6 mg C per g of bulk C, are higher than that from soil (10.4 mg C/g bulk C). Carbon concentrations of individual nalkanes are also higher in wheat leaf than in soil organic matter: 0.041 vs. 0.019 mg of carbon per g of bulk carbon, respectively, for the C 31 n-alkane. Also, the decrease of soil-vs. plant carbon concentration of organic extracts, and n-alkanes, suggests that the selective preservation of lipids is probably not a major pathway in the formation of soil organic matter because a higher content of lipids should have remained in the soils. This view is strenghthened by consideration of carbon isotope ratios, as discussed in the following section.

Carbon isotopic compositions

Carbon isotope ratios of wheat and soil organic components range respectively from -36.7‰ to -27.79‰ and from -35.7 ‰ to -25.59‰ (Table 1, Fig. 1). These values are typical of the organic carbon from plants fixing CO 2 by the C 3 photosynthetic pathway [START_REF] Nissenbaum | The distribution of the stable carbon isotope ( 13 C/ 12 C) in fractions of soil organic matter[END_REF][START_REF] Deines | The isotopic composition of reduced organic carbon[END_REF][START_REF] O'leary | Carbon isotope fractionation in plants[END_REF], Collister et al., 1994b, and refs. therein). With the exception of humin and humic acid pyrolysates, all organic subfractions are depleted in 13 C, reaching a value of -9.4‰ relative to bulk organic carbon. This 13 C-depletion can be attibuted mainly to the higher lipid content of those organic subfractions relative to the bulk organic matter [START_REF] Deines | The isotopic composition of reduced organic carbon[END_REF].

Plant lipids and lignin are 13 C-depleted relative to bulk tissue, whereas amino acids, cellulose and hemicellulose are usually 13 C-enriched [START_REF] Deines | The isotopic composition of reduced organic carbon[END_REF][START_REF] Benner | Depletion of 13 C in lignin and its implications for stable carbon isotope studies[END_REF][START_REF] Schleser | δ 13 C pattern in a forest tree as an indicator of carbon transfer in trees[END_REF]. Since lipids and lignin are more resistant to biodegradation relative to other plant constituents, and since the major part of plant carbon is rapidly recycled to the atmosphere, then the accumulation of plant matter into soil by selective preservation of plant lipids, or lignin, versus polysaccharides should have led to a notable shift of the isotope compositions of soil organic components toward 13 Cdepleted values. For instance, the selective preservation of aliphatic plant components, as a major process, should have given δ 13 C values as low as -36‰ for soil organic matter, as suggested by the δ 13 C values of n-alkanes (Fig. 1). However, we do not observe a depletion but rather a 13 C-enrichment of +1.5‰ to +4.3‰ for soil versus wheat organic components: +1.5-3.8‰ (average +2.6‰) for wheat parts vs. bulk soils, +2.1-4.3‰ (average +3.0‰) for wheat part extracts vs. soil extract. Humin also shows a notable 13 C-enrichment of +2.2‰ to +4.0‰ (average +3.0‰) versus bulk wheat parts. This result is of great significance because humin is the major soil fraction, amounting here to 66% of soil organic C and, most important, humin is considered to be the most stable organic component of soils (Schnitzer andKhan, 1978, Hayes et al., 1989, andrefs. therein). Moreover, it should be noted that a 13 C-enrichment * of about +2‰, in deeper horizons versus surface litter, is commonly observed with increasing depth in forest soil profiles [START_REF] Volkoff | Carbon isotopic fractionation in subtropical Brazilian grassland soils. Comparison with tropical forest soils[END_REF][START_REF] Melillo | Carbon and nitrogen dynamics along the decay continuum: plant litter to soil organic matter[END_REF][START_REF] Martin | Estimate of organic matter turnover rate in a savanna soil by 13 C natural abundance. measurements[END_REF][START_REF] Balesdent | Site-related δ 13 C of tree leaves and soil organic matter in a temperate forest[END_REF]. Hence, all these lines of evidence demonstrate that the selective preservation of plant lipids and lignin is not a major pathway of organic matter accumulation into soils. Two other pathways can thus be considered.

Firstly, soil organic matter may be formed by condensation of small molecules from plant degradation products or from soil biomass. The rapid biodegradation of 13 C-enriched plant polysaccharides [START_REF] Benner | Depletion of 13 C in lignin and its implications for stable carbon isotope studies[END_REF] followed by condensation of carbohydrate monomers with amino acids [START_REF] Maillard | Synthèse des matières humiques par action des acides aminés sur les sucres réducteurs[END_REF], which should be 13 C-enriched, is in good agreement with this hypothesis.

Soil organic matter may alternatively be stored by selective preservation of resistant biopolymers from soil microorganisms consuming 13 C-enriched plant polysaccharides. Indeed, it has been shown recently that various microorganisms, such 5 as microalgae, cyanobacteria and bacteria, are able to biosynthesize cell wall aliphatic biopolymers which are extremely resistant to acid and alkaline hydrolysis [START_REF] Chalansonnet | Cyanobacterial resistant biopolymers. Geochemical implications of the properties of Schizothrix sp. resistant material[END_REF][START_REF] Berre | Occurrence of non-hydrolysable, macromolecular, wall constituents in bacteria. Geochemical implications[END_REF][START_REF] Derenne | Non-hydrolysable macromolecular constituents from outer walls of Chlorella fusca and Nanochlorum eucaryotum[END_REF][START_REF] Leeuw | A review of macromolecular organic compounds that comprise living organisms and their role in kerogen, coal, and petroleum formation[END_REF]. Moreover, the occurrence of these resistant biopolymers in soil is strongly supported by the detection of notable amounts of aliphatic structures within humic acids and humin by NMR studies [START_REF] Hatcher | Aliphatic structure of humic acids; a clue to their origin[END_REF][START_REF] Hatcher | Geochemistry of humin[END_REF][START_REF] Buddrus | Isolated paraffinic methyl groups in humic substances[END_REF].

Whatever pathway is followed during soil organic matter formation, we have shown, on both isotopic and quantitative grounds, that the major part of crop soil humic substances must be formed in situ, either by recondensation of relatively small molecules or by selective preservation of soil biopolymers. Though the formation of soil humic substances by selective preservation of plant lignin and lipids is not excluded, this process is likely to give rise to minor amounts only of crop soil organic carbon.

Figure 1 .

 1 Figure 1. Carbon isotope ratios of various organic components from wheat and wheat crop soils. All data are from the La Minière field, except values of bulk soil carbon from the Boigneville field (-25.92‰, -25.78‰). Note the 13 C-depletion of most soil subfractions versus bulk soil carbon. Note also the 13 Cenrichment of soil components versus wheat components.

Table 1 .

 1 Chemical composition of wheat and wheat crop soil organic components from the La Minière field. δ 13 C values of bulk carbon from the Boigneville field are -25.92‰ and -25.78‰ after wheat cropping during 20 and 23 years, respectively.

	WHEAT	YIELD	CARBON	CARBON	δ 13 C	Δδ a
		(µg/g dry weight)	(weight %)	(mgC/gCtotal)	(‰)	(‰)
	_________________________________________________________________
	Bulk leaf		44.99	1000	-29.61	0
	Bulk stem		42.85	1000	-28.47	0
	Bulk root		34.36	1000	-27.79	0
	Leaf organic extract	26480	73.20	43.08	-33.19	-3.58
	Stem organic extract	7900	52.30	9.64	-30.95	-2.48
	Root organic extract	8568	72.90	18.18	-31.64	-3.85
	Leaf C 25 n-alkane	0.8	85.23	0.002	n.d.
	Leaf C 27 n-alkane	4.6	85.26	0.009	-35.9	-6.3
	Leaf C 29 n-alkane	16.5	85.29	0.031	-36.5	-6.9
	Leaf C 31 n-alkane	21.7	85.32	0.041	-36.7	-7.1
	Leaf C 33 n-alkane	7.6	85.34	0.015	-36.7	-7.1
	_________________________________________________________________
	SOIL	YIELD	CARBON	CARBON	δ 13 C	Δδ a
		(µg/g soil dry weight) (weight %)	(mgC/gCtotal)	(‰)	(‰)
	_________________________________________________________________
	Bulk soil		0.95 (0.09) c	1000	-26.28	0
	Humin b		0.63 (0.06) c	663.6	-25.59	+0.69
	Humic acids	1812	32.78 (3.89) c	62.52	-27.45	-1.17
	Humic acids pyrolysate	89.8	62.15	5.87	-26.04	+0.24
	Humic acids pyr. residue	1028	37.14	40.19	-27.64	-1.36
	Organic extract	161.2	61.00	10.35	-28.89	-2.61
	Acid fraction	38.0	n.d.		-29.24	-2.96
	Neutral fraction	57.0	n.d.		-29.40	-3.12
	C 25 n-alkane	0.02	85.23	0.002	-34.6	-8.3
	C 27 n-alkane	0.05	85.26	0.005	-34.2	-7.9
	C 29 n-alkane	0.16	85.29	0.014	-35.7	-9.4
	C 31 n-alkane	0.21	85.32	0.019	-35.7	-9.4
	C 33 n-alkane	0.06	85.34	0.005	-35.7	-9.4
	_________________________________________________________________

a fraction minus bulk.

b Analyses were carried out on humin including mineral matrix. c

Acknowledgements-The Institut National de la Recherche Agronomique is gratefully acknowledged for a post-doctoral fellowship for Dr. Sen Dou. We thank Clemence Bosc, Université Pierre et Marie Curie, Dr. May Balabane, I.N.R.A. Versailles and Dr. Claude Largeau, École Nationale Supérieure de Chimie de Paris for helpful discussions. For a critical and helpful review, we are grateful to Prof. John Hayes, Indiana University.