Catherine Macombe
email: catherine.macombe@cemagref.fr

Pauline Feschet

Michel Garrabé

Denis Loeillet

Reporting the social indicators to the functional unit for food product. Theoretical contribution regarding the collection of relevant data

Several conceptions of social LCA lend on two often implicit hypothesis i) the source of impacts, the stressor would be either from technical origin, either from social one. It stems from it that relating the quantity of impacts to the quantity of functional unit has to be done through the unit processes, as it is done in ELCA. ii) companies are singly and freely choosing their practice, and even imposing social behaviour. We expect pointing out that we may build another representation. The values of the social indicators may be related proportionally to the functional units, if we handle them at the relevant level. Suggestions about the potential levels for picking up the data will conclude this proposal.

Warning: In the social LCA field, all the authors agree that the conceptual framework (e.g. UNEP, 2009) is far from being comprehensive. Moreover, no one claim that there is a unique or even consensual framework (epistemological, theoretical), to date. So, this paper is not a case study. This paper is a modest contribution towards building a conceptual framework for social LCA. It doesn't provide a list of indicators, it doesn't address the choice of indicators. Because we consider that these steps can't be performed before setting a strong theory of "what really count" among the social impacts of products. And this theory is not available today. However, we can build together parts of the foundations for the social LCA of tomorrow. The modest objectives of this paper are: 1) Discussing two implicit hypothesis underpinning many social LCA case studies 2) Showing that using different hypothesis, we may relate the social indicators (even if we don't provide a list) to the functional unit. Here are the prudent objectives of this paper. We expect it to be seen like a small part of the foundations we claim for.

Introduction

Within the conceptual framework of the Life Cycle Assessment [START_REF] Jolliet | The LCIA midpoint-damage framework of the UNEP/SETAC Life cycle initiative[END_REF], it is worth the indications provided about the impacts to be related to the functional unit. This property means that-up to a point-the quantities of impacts will vary in the same direction and proportionately with the quantities of functional unit. The issue is as critical for the socalled social impacts as it is for the environmental ones. The users need to choose ex-ante between different scenarios able to provide equivalent goods. They therefore require results formulated in proportion with the functional unit. But [START_REF] Reap | A survey of unresolved problems in life cycle assessment. Part 1[END_REF] underpin that most impacts on people are independent of the physical processes that make the product, and more dependent on company behaviour and as such the «relation of the impacts to the product [-] is no longer straightforward » [START_REF] Dreyer | A framework for social life cycle impact assessment[END_REF]. About this critical issue, [START_REF] Kruse | Socioeconomic indicators as a complement to life cycle assessment-an application to salmon production systems[END_REF] have made the distinction between two kinds of inventory indicators, the additive ones and the descriptive ones. The first ones relate to the functional unit (e.g. labour costs). The second may be assessed at each point of the chain, but the authors explain that they cannot be related to the functional unit (e.g. use of child labour). [START_REF] Norris | Social Impacts in Product Life Cycles, Towards life cycle attribute assessment[END_REF] has came up the same difficulty thanks to the « life cycle attribute assessment ». One indicator only amongst the additive ones proposed by Kruse is chosen. It could be the number of work hours spent by each company involved in the product life cycle. The features that Kruse would call "descriptive indicators" are becoming attributes in the new approach. Doing so, researchers can calculate the rate of work hours from local origin involved in one industry providing greenhouse tomatoes in Canada (Andrews et al., 2009).

Both conceptions lend on two often implicit hypothesis, worthy to be discussed: i) The source of impacts, the stressor (the element of pressure) would be either from technical origin, either from social one [START_REF] Parent | Impact assessment in SLCA : sorting the sLCIA methods accrding to their outcomes[END_REF]. It stems from it that relating the quantity of impacts to the quantity of functional unit has to be done through the unit processes, as it is done in ELCA. ii) Companies are singly and freely choosing their practices, like being observant of codes of ethics or not, and even imposing social behaviours.

Our objective is pointing out other hypothesis. Doing so, the values of the social indicators may be related proportionally to the functional units, if we handle them at the relevant scale. Suggestions about the different potential levels for collect will conclude this proposal.

Social impacts are not stemming from unit processes

Stressors causing the social or other impacts are all stemming from social origin. They are depending from numerous social factors, some of them offering drivers to policymakers. The average life expectancy is influenced by drivers on the nationwide scale, like the health policy. Industry may or may not implement policies, embedded in the state policies, for preventing occupational injuries. Companies and moreover groups of companies [START_REF] March | Les organisations. Problèmes psychosociologiques[END_REF] have chosen organisational and technological (Kloepffer, 2003) configurations, the unit processes of today as a result. The unit process is the LCA term to assign one composite body, a layout of diverse resources: human, material and symbolic, gathering people, objects, space, machines, documents, and given the responsibility of doing a particular task [START_REF] Girin | Les agencements organisationnels[END_REF]. In ELCA, the relevant unit processes are special composite bodies. The non-technical parts seem to be cancelled out by the Fordist standardisation, performed in order to secure the return on the machines. The unit processes are time and space stabilized enough for us to build data bases delivering how much X substance is released by the process Y per functional unit. In appearance, everything works as if the machine alone was creating the impacts! But modify the tuning, and the outputs will shift. The machines are not the cause of the social impacts. When Boje (2009) recounts a fatal injury, it is clear that the killer-machine is only one tiny part of the explanation: « social here refers to poverty that would prompt 14-year old Liu pan to work 72 hour weeks on an unsafe machine, at 60% of China's legal minimum wage, to the point of exhaustion »(page 3).

In general, we get only fuzzy ideas about the pathways between the set up of one milk container in one brazilian assentamento (homestead), and the literacy gain which seem to be its output. We feel that many others elements but the milk container itself, take place. The human component of the composite body is re-established. These pathways are neither standardized, nor brought in general use. « It is very difficult to find any consistent differences between different technologies or production routes involved in the production of any given product, simply because the social impacts are so site specific that the variation between sites exceed the variation between technologies or production routes » [START_REF] Weidema | Quantifying Corporate Social Responsibility in the value chain, presentation for the Life Cycle Management Workshop of the UNEP/SETAC Life Cycle Initiative at the ISO TC 207 meeting[END_REF]. By « site specific », we understand "linked with human behaviour". The social impacts are not linkable with each process unit because they are dependent on non standardized human behaviours. Indeed, very numerous human decisions affect the pathway be-tween the unit process and the result. The whole constitutes a "complex system integrating human actors", where the human being's attendance introduces specific types of complexity [START_REF] Girin | Management et complexité : comment importer en gestion un concept polysémique ?[END_REF]. This entails that the future social impact is unpredictable from the unit process state.

Moreover, it is logical to focus on stressors at the level which holds the drivers. Sometimes, the company is not the best relevant level.

Companies comply with institutional isomorphism

Companies are embedded within the social fabric. They create disruptive fields [START_REF] Emery | The causal texture of organizational Environments[END_REF] which bend the fabric, while complying with it, in a common cultural milieu. It is the so-called phenomenon of institutional isomorphism (Di Maggio and Powell, 1983). It means that all the companies belonging to one industry tend to become similar along the time, even to borrow the same or complementary strategies. About 20 developed countries, Maria Gjoldberg (2009) highlights that differences in political-economic background will be reflected in differences in Corporate Social Responsibility (CSR) performance. The countries with the most developed strong CSR practices belong to two groups. The first ones (Switzerland, UK, ND) welcome a high rate of globalized companies, more exposed to the spotlight of watchdogs from NGOs and the media. The seconds ones (Nordic countries) are characterised by close, cooperative and consensual relations between state, business and labour, as well as long-standing tradition for involving civil society in policy making. Her paper concludes by pointing out the need to acknowledge the fundamental interdependence between traditional, "hard" government-corporatist regulation of business responsibilities and "soft" civil regulation of corporate responsibilities." [START_REF] Gjolberg | The origin of corporate social responsibility: global forces or national legacies?[END_REF]. [START_REF] Dreyer | Characterisation of social impacts in LCA. Part 1: Development of indicators for labour rights[END_REF] implicitly acknowledge the influence of more macroscopic levels than the company's one, and the institutional isomorphism phenomenon, because they underpin that the labour rights violation risk depends on the contextual factors surrounding the company. Indeed, they include in those factors the « (1) existence and enforcement of national legislation concerning the issue, and social, cultural, economic and political practices at the location, and (2) the practices of members of the industry ». As an outcome, the number of child labour hours involved in the making of rice in China doesn't depend on the company choice and behaviour, but on the national cultural agreement about it. Even if a Chinese company makes advertisement on its website as being "child labour" free, it can't merely be true if child labour is the rule in the rest of the society (Boje, 2009).

What is the relevant level for picking up data?

Because the social impacts don't stem from the unit processes level, and because of the institutional isomorphism of companies, we speak in favour of assessing the social impact at the sector or industry's levels. The idea is accounting for the evolution of the average practices of the companies making the product in one given country, and not to focus upon the specific unit processes from the company X. At the sector or industry level, one may find a value for the inventory indicators that we may relate to the quantities of functional units. So, the number of hours of child labour by functional unit of the rice sold by the company under scrutiny, and its national suppliers as a whole, may be collected at the industry level. If available, the two indicators values to collect are: 1) the number of children working into the rice industry ; 2) The quantity of rice processed in the country. You may calculate so a national indicator year 2010 of the number of child labour hours per functional unit of rice. Of course, it is a rough assessment, but not so bad, and including all the national companies involved in the chain under scrutiny. [START_REF] Weidema | Quantifying Corporate Social Responsibility in the value chain, presentation for the Life Cycle Management Workshop of the UNEP/SETAC Life Cycle Initiative at the ISO TC 207 meeting[END_REF] advices to filling any data gaps with default data, based on input-output tables extended with social parameters. We argue that these data will be more accurate than data drawn from auditing one company. [START_REF] Jorgensen | Methodologies for social life cycle assessment[END_REF] too emphasizes how the generic data may be worthwhile « the quality of site-specific data is very dependent on the auditing approach and, therefore, not necessarily of high accuracy, and [-] generic data might be designed to take into account the location, sector, size and maybe ownership of a company, and thereby in some cases, give a reasonable impression of the social impacts that can be expected from the company performing the assessed process." Indeed, getting those values don't provide the pathways describing the impacts.

In theory, we are able to calculate the number of hours provided by children for each process unit in the world (for instance for each fruits drying rack) from real world data. All we have to do is to draw up a typology of « drying racks in their social context » (it means taking into account the technique, the place, time, and other social relevant items). And yet, these so precious data will turn out to be very instable along the time, while no one could explain ex-ante the causes of the variations. And so, the final impact measures would be so error-prone, that they would become unusable [START_REF] Lenzen | Uncertainty in Impact and Externality Assessments-Implications for decision making[END_REF]. Indeed, the alternate conversions from former outputs into impact measures are linked with characterisation factors matched with incertitude slot. The longer the chain between the first event (here it is the dryer rank functioning) and the final impact is, the larger the incertitude slot becomes.

The more the data will be picked up at macroscopic level, the more the data will be stable along the time. They will provide a final impact measure as accurate (with narrow incertitude slot) as the causality chain (between the driver and the end-point) will be shorter. Despite they advice us to pick up data at the level of the « companies in which the processes occur », [START_REF] Dreyer | Characterisation of social impacts in LCA. Part 1: Development of indicators for labour rights[END_REF] acknowledge the role of more macroscopic levels : « the translation from performance score [within the company] to risk involves the assessment of the context of the company in terms of geographical location and industry and of the typical level of social impacts that these entail, and interpretation of the company's management effort in the light of this context » (page 247).

If data are lacking at the sector level, it is worth getting them from the immediately inferior level, which is from the industry level or from a group of companies. As one goes along down till the unit processes, data become more and more unstable, because the phenomenon whose data account for, is less and less dependent on these inferior level (the drivers belong to upper levels). However, this idealistic scheme assumes that the drivers are really triggered at each level, initiating with the upper and more general one. National State is assumed to spread its health policy in order to improve the inhabitants' life expectancy, the sectors are assumed to implement their occupational injuries prevention programmes and so on. In case of deficiency from upper levels, the first decision level triggering the drivers will be the relevant one to perform the collect of inventory data.

The table 1 highlights the different levels for picking up the inventory data in the food products field. We provide two examples often quoted in the literature: changes in life expectancy at birth and using child labour. From the bottom of the table (unit processes) it is clear that the company level is the first one where some decisions take place, for instance about the work organisation, which could influence the corresponding impacts. But the stability of the data along the time is worst than at any upper level. So, choosing the rele-vant level for data picking is a compromise between data availability, the best stability along time as possible, and the smallest incertitude slot about the final impacts.

It is worthy of note to set aside efforts in order gathering data of great worth for making up data basis reusable by other researchers.

Table 1 :

 1 Stability of collected data and existence of drivers, according to the level of the collect, within the agri food field.When the former drivers are not triggered, a company alone may handle drivers, depending on the type of company.There may be huge differences according to the type of company (e.g. globalized versus local company) if the former drivers are not triggered. Macombe is member of the ELSA group (Environmental Life Cycle and Sustainability Assessment www.elsa-lca.org ); she thanks all the other members of ELSA for their advice.

	Level for	Time stability	Example 1: Where are the driv-	Example 2: Where are the
	collecting			ers for moving the life expectancy	drivers for using or not child
	data					of workers in rice industry?	labour?
	State level	Very stable	The average life expectancy at birth	It is a cultural issue, so if drivers
						depends on drivers handled at the	exist, they are handled at the Na-
						Nation level.	tion level.
	Agri food	Stable		Some features of the sector (e.g. often	Some features of the sector (e.g.
	sector level			outdoors working conditions) entail	low qualification level needed)
						differences around the average. Some	entail differences. Some drivers
						drivers are handled at this level.	are handled at this level.
	Rice indus-	Stable		Some features of the industry (e.g.	Some features of the industry
	try	level			localisation of the rice industry in	(e.g. localisation of the rice in-
	(companies			remote areas) entail differences	dustry in remote areas) entail
	processing rice)			around the former average. Some drivers are handled at this level. .	differences around the former average. Some drivers are han-dled by this level.
	Group		of	Average stabil-	Depends on the size of the group	Depends on the size of the group
	companies	ity (depends on	within the industry	within the industry
	level within	the size of the
	the industry	rice	group compared with the indus-try size)
	Company	Between weak
	level (e.g.	and	average
	packaging	stability
	plant)
	Agricultural	weak		No relationship	No relationship
	itinerary
	Unit proc-	weak		No relationship	No relationship
	esses level
	C.

Author produced version of the paper presented at LCAfood 2010 VII International conference on life cycle assessment in the agri-food sector Available at : http://www.life-cycle.org/?p=413

Andrews E., Lesage P. et al. (2009): "Life cycle attribute assessment Case study of Quebec Greenhouse Tomatoes." Journal of industrial ecology 13(4), pp 565-578. Boje D.M. (2009): Storytelling, appreciative inquiry and tetranormalization, Proceedings of the International Conference and Doctoral Consortium, Lyon, France, 8-10 June 2009. Di Maggio P.J. and Powell W.W. (1983): The iron cage revisited: institutional isomorphism and collective rationality in organizational fields, American Sociological Review, 48, April, pp 147-160.


Eric Bonjour
email: ebonjour@ens2m.fr-mdulmet@ens2m.fr

Maryvonne Dulmet

Samuel Deniaud
email: samuel.deniaud@utbm.fr

Jean-Pierre Micaëlli
email: jean-pierre.micaelli@insa-lyon.fr

Propagating product architecture decisions onto the project organization: a comparison between two methods

Keywords: product architecture, project organization, design team, Design Structure Matrix, DSM, domain mapping matrix, DMM

Concepts like product architecture and modularity have been introduced in order to limit the effects of technological change on complex product design. Researchers have highlighted that product architectures and design organizations (projects, teams…) are strongly interrelated. However, little research has analyzed this relationship. System architects and design managers need a method that helps them to simulate the mapping of the product architecture onto the project organization by propagating choices and then assessing alternatives. In this paper, two propagation methods are presented and compared. The first one is based on a fuzzy process, which is proposed by the authors. The second one is based on a matrix approach. Both are applied to define new robotized gearbox architectures. A sensitivity analysis is conducted. It is concluded that in new product development situations or in re-engineering projects, system architects could use these methods in the early design stages to forecast the more appropriate design project organization.

Introduction

Modular product design has proved to be efficient to reduce design efforts. According to [START_REF] Ulrich | The Role of Product Architecture in the Manufacturing Firm[END_REF], the product architecture is the mapping of the product functions onto its components. Optimal modular product architectures can be defined as the clustering of components such that the degree of interaction/dependency is maximized within groups (or modules) and minimized between groups (inter-modules) [START_REF] Whitfield | Identifying Component Modules[END_REF]. Modules are commonly described as groups of functionally or structurally dependent components. Research, concerning platform-based product development and product family design [START_REF] Jiao | Product family design and platform-based product development: a state-of-the-art review[END_REF][START_REF] Farrell | A Method to Improve Platform Leveraging in a Market Segmentation Grid for an Existing Product Line[END_REF], has received huge interest over the last decade since it aims at providing methods to identify common modules and generate product variants with distinctive modules (commonality vs. variety, economies of scale vs economies of scope). Modularity has many advantages but few methods exist to partition a product into modules, even in the special case of single complex products. A key feature of product architecture is the degree to which it is modular or integrative [START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF]. [START_REF] Sharman | Characterizing complex product architectures[END_REF] point out that modularity has drawbacks. The inter-module interfaces must allow change to occur within modules without adversely affecting inter-module working. This requires an appropriate definition of interfaces that play the key role of connecting and interacting between components. Few architecting methods have been developed to identify modular product architecture. They use different representations [START_REF] Sharman | Characterizing complex product architectures[END_REF], for instance diagrams [START_REF] Stone | A heuristic method for identifying modules for product architectures[END_REF], networks [START_REF] Sosa | A network approach to define modularity of components in complex products[END_REF] or matrices. The inputs of these methods may be either functional models (Stone et al., 2000, Kurtoglu and[START_REF] Kurtoglu | A graph-based fault identification and propagation Framework for functional design of complex systems[END_REF], or components interactions [START_REF] Pimmler | Integration analysis of product decompositions[END_REF][START_REF] Sosa | Identifying modular and integrative systems and their impact on design team interactions[END_REF][START_REF] Yu | Genetic algorithm for developing modular product architectures[END_REF], or a mapping of functions onto physical components [START_REF] Liu | Transforming Functional Solutions into Physical Solutions[END_REF][START_REF] Bonjour | Mapping Functional Architecture Decisions onto Physical Product Architectures[END_REF]Bonjour et al., , 2009a)), or more complex data intended to take into account key factors of the whole life cycle of the system [START_REF] Gu | Product modularization for life cycle engineering[END_REF].

System architects need formal representations in order to handle interactions between elements in the system [START_REF] Ulrich | Product Design & Development[END_REF]. When couplings between the elements of product domains have not been formally defined, the integration of the teams' contributions is more difficult and requires numerous design iterations. This paper aims at presenting two methods to help design managers to simulate the mapping of the product architecture onto the project organization structure by propagating design choices and then assessing alternatives, early in the system definition stage. First, a concise literature review is presented. Second, an architecture typology and the matrix-based models (DSM and incidence matrix that provide powerful representations of systems architecture) are briefly described. Then two propagation methods between product architecture and organization structure that rely on matrix-based models are shortly presented. The former which is proposed by the authors is based on a fuzzy treatment of matrices and the latter corresponds to a matrix product. These methods are applied to the development of a new robotized gearbox and then compared. Finally, a brief discussion and further research are formulated.

Matrix-based methods for product architecture modeling

Matrix-based product modeling methods represent the product architecture (product elements and their relationships), shown as a matrix. They are being increasingly used in such works [START_REF] Sharman | Characterizing complex product architectures[END_REF] since they can support different research goals: for example, product modularization [START_REF] Whitfield | Identifying Component Modules[END_REF], analysis of technical interactions either within the products or within the project organization [START_REF] Pimmler | Integration analysis of product decompositions[END_REF], and change propagation analysis [START_REF] Clarkson | Predicting change propagation in complex design[END_REF][START_REF] Sosa | Identifying modular and integrative systems and their impact on design team interactions[END_REF]. Two kinds of matrices may support the representation of system architecture:

(1) incidence matrices [START_REF] Chen | A formal two-phase method for decomposition of complex design problems[END_REF], also called Domain Mapping Matrix (DMM) [START_REF] Danilovic | A formal approach for domain mapping matrices (DMM) to complement design structuring matrices (DSM)[END_REF][START_REF] Lindemann | A vision to overcome chaotic 'design for X' processes in early phases[END_REF],

(2) Design Structure Matrices (DSM) [START_REF] Steward | The Design Structure System: A Method for Managing the Design of Complex Systems[END_REF][START_REF] Pimmler | Integration analysis of product decompositions[END_REF].

The matrices in the former category represent relationships between two domains, such as between product functions and components, or between the product domain and the organization domain (Bonjour et al., 2009-b). The matrix in the latter category represents relationships between elements within the same domain. Recently, [START_REF] Eichinger | Extending Design Structure Matrices and Domain Mapping Matrices by Multiple Design Structure Matrices[END_REF] propose to extend DMM and DSM by introducing Multiple Design Structure Matrices.

Incidence matrix (IM)

Incidence Matrices can represent a set of design decisions or relationships between what and how. Some authors use other names such as design matrix in axiomatic design [START_REF] Suh | Principles of Design[END_REF], which pays considerable attention to the relationships between functional requirements and physical design parameters. According to axiomatic design, the decomposition of a design problem follows a "zigzagging" top-down approach between the hierarchies of the functional and physical domains. In [START_REF] Chen | A formal two-phase method for decomposition of complex design problems[END_REF], an interesting decomposition method has been proposed for complex design problems that are represented in an attribute-component incidence matrix. A formal two-phase process has been described to transform this matrix into a block-angular matrix according to a given set of decomposition criteria. [START_REF] Fixson | Product architecture assessment: a tool to link product, process and supply chain design decisions[END_REF] proposes to create a "functioncomponent allocation» matrix. In the cells of this matrix, "percentages of a function can be allocated to components that contribute to this function». [START_REF] Danilovic | Managing complex product development projects with design structures matrices and domain mapping matrices[END_REF] propose an interesting use of Domain Mapping Matrix (DMM). They apply clustering directly on DMM. In the present paper, the authors assume that incidence matrices are of high importance as they ensure the cohesion between the project domains (product, process and organization) [START_REF] Sosa | Aligning process, product, and organizational architectures in software development[END_REF][START_REF] Eppinger | Patterns of product development interactions[END_REF].

2.2

Design Structure Matrix (DSM)

DSM [START_REF] Steward | The Design Structure System: A Method for Managing the Design of Complex Systems[END_REF] represents relationships between elements of the same domain. DSM is now a popular modeling and analysis tool, especially for purposes of decomposition and integration. DSM can be applied on various levels of abstraction to study interactions between functions, between sub-systems or components, between design parameters and between the life-cycle processes [START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF][START_REF] Pimmler | Integration analysis of product decompositions[END_REF]. DSM displays the relationships between elements of a system in a compact and visual format. Hence, DSM is used to identify project domain architectures: the architecture of products [START_REF] Pimmler | Integration analysis of product decompositions[END_REF], the architecture of design process [START_REF] Meier | Design process sequencing with competent genetic algorithms[END_REF] or the decomposition of the projects into different teams [START_REF] Chen | Decomposition of interdependent task group for concurrent engineering[END_REF][START_REF] Sosa | Identifying modular and integrative systems and their impact on design team interactions[END_REF]. DSM is a square matrix with identical elements in rows and columns. Cells along the diagonal have no meaning. Reading across a row reveals what other elements the element in that row provides. Scanning down a column reveals what other elements the element in that column depends on.

From Product Architecture to Organization Structure

In the engineering design field, architecture terminology is often linked to the product. Thus [START_REF] Ulrich | The Role of Product Architecture in the Manufacturing Firm[END_REF] defines product architectures as "the scheme by which the function of a product is allocated to physical components." A key feature of product architecture is the degree to which it is modular or integral. In modular architectures, functions of the product map one-to-one to its physical components. At the other extreme, in integral architectures a large subset of product functions map to a single or small number of components. In real design situations, designers have to make a trade-off between modular and integral architectures. Hence, many products are hybrid [START_REF] Sosa | Designing modular and integrative systems[END_REF]. Their architectures are not fully modular or integral and lie somewhere between the two extremes.

Generally speaking, the notion of architecture is also used for all systems that may be decomposed into smaller inter-related sub-systems, from a functional view and a physical view (IEEE Std 1220™, 2005). Design organization can be considered as a social system that aims at developing a product. The functional view of the organization corresponds to the development process that specifies the goals (or tasks) the design organizational entities (e.g. teams) have to achieve. The physical view of the organization corresponds to all teams that make up the project team and that may be decomposed into smaller groups and individual designers. In complex product development projects, many teams develop the components, or systems, and others are responsible for the integration of these components in the final product. [START_REF] Yassine | Complex concurrent engineering and Design Structure Matrix method[END_REF] call these teams "local development teams" (in charge of sub-systems development) and system teams (product integrators). Previous typologies and works implicitly assume that the most efficient organization structure in case of complex systems development project corresponds to a matching between systems/sub-systems and teams.

To be general, similarly to [START_REF] Sosa | Identifying modular and integrative systems and their impact on design team interactions[END_REF], a modular team is defined as a team whose team members have a lot of information exchanges between one another and that have no (or few) interactions with other design teams. An integrative designer (or design team) is defined as a designer who needs to interact with many other designers or modular teams.

Concerning the component DSM (C DSM) or Organization DSM (O DSM), the purpose is to cluster the elements into modules with relatively high internal interactions and relatively low external interactions. Algorithms for this approach are called clustering algorithms. They reduce the design efforts and the design iterations by appropriate system integration (C DSM) and project coordination (O DSM).

The clustering algorithm

Clustering is a method that, given the mapping of interactions through the use of DSM or graphs, generates modules in a systematic way by optimizing an objective function. In this section, interesting clustering algorithms are briefly reviewed and the clustering algorithm that is used in this work is presented.

Previous works

The goal of a clustering algorithm is to group elements together into clusters. [START_REF] Hartigan | Clustering Algorithms[END_REF] reviewed the basic approach of clustering and discussed different applications of clustering algorithms. The original goal of clustering was to find similarity between elements and group them together based on a threshold of similarity between elements. Recent algorithms have been developed for optimizing modular design of complex products including simulated annealing [START_REF] Fernandez | Integration analysis of product architecture to support effective team colocation[END_REF][START_REF] Thebeau | Knowledge management of system interfaces and interactions for product development processes[END_REF], genetic algorithms [START_REF] Whitfield | Identifying Component Modules[END_REF] or an algorithm based on the real options theory [START_REF] Sharman | Architectural Valuation using the Design Structure Matrix and Real Options Theory[END_REF] originally developed by [START_REF] Baldwin | Design Rules: The Power of Modularity Design[END_REF]. The clustering algorithm used in the present work is based on an algorithm developed by [START_REF] Idicula | Planning for concurrent engineering[END_REF] and improved by [START_REF] Fernandez | Integration analysis of product architecture to support effective team colocation[END_REF] and [START_REF] Thebeau | Knowledge management of system interfaces and interactions for product development processes[END_REF]. This algorithm is a hierarchical bottom-up (aggregation) clustering algorithm. This algorithm has been often used in modularization work or analyzed for enhancing its performance. According to [START_REF] Yu | Genetic algorithm for developing modular product architectures[END_REF], the Thebaud algorithm is likely to be trapped in local optimal solutions and it may fail to accurately predict the formation of "good" clustering arrangements for complex product architectures. Then they propose a new algorithm. They formulate a more complex objective function based on an information theoretic measure of modularity and they use a more robust search strategy, i.e. a genetic algorithm. Although researchers criticize the Thebaud algorithm and propose partial comparisons [START_REF] Yu | Genetic algorithm for developing modular product architectures[END_REF], direct benchmarking is unavailable to compare existing clustering methods.

Principles of the Thebaud clustering algorithm

The Thebaud algorithm is based on an objective function called the Total Coupling Cost. It attempts to capture the following observations in a mathematical form:

• The cost addressing an interaction is proportional to the importance of the interaction.

An important or more frequent interaction requires more attention, more resources, or more work from designers. Thus, a coupling with high value will have a high coupling direct cost. • It is assumed that in a modular architecture, if a given element belongs to a module, it is tightly coupled to other elements in this module and at the same time, this component is weakly coupled to elements belonging to other modules. • For a given element, the cost of being a member of a module increases with the number of modules including a coupling with this element (see Eq. 2). Particularly, an integrative element is an element which should not belong to any module since it has been coupled to many elements belonging to different modules.

For each element in the DSM, the algorithm calculates a coupling cost. Then the sum of the coupling costs for each element gives a total coupling cost. Notations are described in Table 1.

Table 1 Notations for the clustering algorithm

size the size of the DSM, i.e. the number of elements.

DSM(i,j) the coupling value (or strength of interaction) between elements i and j. Note that when i=j, DSM(i,i)=0.

cl_size(k) the number of elements contained in cluster k. pow_cc a parameter that controls the type of penalty assigned to the size of the cluster in the coupling cost.

Equations 1 and 2 show the coupling direct cost for an element. If both elements i and j are in the same cluster k, then the coupling direct cost is calculated through Equation 1. Otherwise, if no cluster contains both i and j, Equation 2 is used where the entire DSM acts as a cluster containing i and j. Equation 3 is the expression for the total coupling cost function that the algorithm attempts to minimize. The parameter pow_cc plays an important role since it penalizes the size of the clusters in the coupling cost function in Eq. 1 and then it affects their number and their size. [START_REF] Dunn | Design Structure Matrices to Improve Decentralized Urban Transportation Systems[END_REF] have recommended to set pow_cc to 1 as a good starting point, or to increase the value as high as 2 if relatively smaller clusters are desired or else to decrease the value as low as 0.5.

In order to enhance the exploration of the whole solution space, Thebaud's algorithm implements two simulated annealing operations that allow the algorithm to reach solutions that it would otherwise have left out. The likelihood of being "trapped" in a sequence of steps that leads to a sub-optimal solution is minimized. Due to the randomness in simulated annealing, several runs are required to reach a satisfying DSM clustering. It is also recommended to change the initial DSM arrangement and to run the algorithm again since, like other algorithms based on simulated annealing, Thebaud's algorithm may be dependent on the initial configuration.

For an interaction between elements i and j that occurs within the cluster k

( ) ( ) ( ) ( ) _ , , , _ () pow cc i j
Coupling Cost e e DSM i j DSM j i cl size k = + × eq 1

For an interaction between elements i and j that occurs outside a cluster

Propagating Product Architecture on Project Organization

The two methods presented in this paper combine DSM and IM as representations and propagation tools of project domain architectures. The main assumption is that one team is responsible for the fulfillment of a single design task. The scope of the present study is limited to the relationships between the product architecture and the organization structure. Their matrix representation corresponds to static DSM. These methods help system architects to predict potential interactions between designers. These interactions are likely to ensure efficient system integration since they are identified by propagating architecture decisions. Then the obtained organization DSM could be used to build teams and minimize the coordination efforts (few interactions between teams).

The propagation method is summarized in four steps. First, identify the product architecture by capturing the Component DSM (C DSM). Second, document the Components vs Organization Incidence Matrix (C-O IM). Third, propagate the product architecture choices through the C-O IM in order to generate the Organization DSM (O DSM). Fourth, identify satisfactory organization structure by applying a clustering algorithm (figure 1). Step 1: Capture the C DSM. By interviewing system architects who design the product architecture, the list of components (C) is identified. Then, the authors proceed to identify the design interfaces. System architects have to fill in a square matrix (C DSM) whose rows and columns are identically labeled with the n components of the product. They have to identify and estimate the coupling value concerning the interaction between two components (if it does not exist, the value is null). Typologies for DSM interactions have been identified by [START_REF] Pimmler | Integration analysis of product decompositions[END_REF]. In this work, the system architect estimates the coupling value by referring to the set of product attributes (or parameters) that the components affect or share. The same idea is proposed in [START_REF] Chen | A formal two-phase method for decomposition of complex design problems[END_REF]. Coupling value is estimated on the scale 0-10.

Step 2: Capture the C-O IM. The assignment relationships between product components and the design organization teams (or designers) are identified. Firstly, the lists of components identified in the product architecture are used. Secondly, the authors survey the system architects to capture the list of design actors involved in the development of the product. Thirdly, similarly to [START_REF] Sosa | Aligning process, product, and organizational architectures in software development[END_REF], this information is documented into a rectangular matrix (C-O IM), whose rows are labeled with the m designers (D) and columns with n components (C). Finally, each IM i,j value is estimated as the level of involvement of 'designer i' in the design of 'component j'. The value is estimated on the scale 0-10 (0 means that 'designer i' is not involved and 10, that he/she is strongly involved). Hence, designers that are involved in the design of the same component or in the design of two components which share interfaces are likely to collaborate together to ensure the product definition and integration. In this work, an assumption is that the sum of involvement levels for each designer must be inferior or equal to 30. This measure is related to the work load.

Step 3: Simulate O DSM starting from C DSM and C-O IM. The organization structure is formalized by means of O DSM. This matrix helps to identify the need for information exchanges between designers. Value 'O DSM u,v ' indicates (or helps to predict) the degree of dependency between 'designer u' and 'designer v'.

Two approaches can be used to calculate the resulting values of the O DSM: either a fuzzy process or a matrix based approach. Concerning the fuzzy process, some rules are defined as the basis of the propagation method. Axioms are formulated in Section 6. Section 7.3 compares the obtained results with the two methods.

Step 4: Identify Organization structure. The Thebaud clustering algorithm is applied on O DSM that has been presented above in section 4.2.

Simulating O DSM starting from C DSM and C-O IM.

This section first presents a propagation method that has been developed in collaboration with the powertrain design office of a French car manufacturer. Section 6.1 formulates the axioms which are at the basis of the proposed method for simulating a new DSM. Since intensity values inside DSM and IM are quite imprecise and subjective, the use of fuzzy logic is relevant [START_REF] Zimmerman | Fuzzy Set Theory and its Applications[END_REF][START_REF] Dubois | Fuzzy sets and systems: Theory and applications[END_REF] to manipulate these values. Section 6.2 describes the fuzzy process that computes potential coupling values between each pair of designers. Section 6.3 presents a simple procedure to aggregate the resulting DSM and filter it in order to ignore meaningless values. Section 6.4 describes the latter method that is a matrix-based approach already used by other researchers.

Axioms

In this section, the axioms that are at the basis of the proposed propagation method are formulated, along with the underlying assumptions.

• Axiom 1 (Fig. 2). Two components (C i , C j ) and two designers (D u , D v ) may be coupled according to two different ways (symmetrical cases). • Axiom 2 (Fig. 3

& L & M & L & M ' X ' Y ' X ' Y 25 & L & M & L & M ' X ' Y ' X ' Y Figure 3 Propagating the (C i , C j ) coupling ' X ' Y & L & M & L & M ' X ' Y
Let's explain the most important axiom, which is Axiom 2. Let us consider that components C i and Cj interact. If designer D u (resp. D v ) is involved in the design of C i (resp. Cj) then he/she is concerned by the definition of Cj (resp. C i ). Designers D u and D v are likely to interact to negotiate their component interfaces and conflicting parameters. Axiom 3 assumes that the higher the coupling value between components C i and C j and the higher the involvement level of the designers D u and D v in the design of these components, the more these two designers are likely to collaborate each other. Axiom 4 handles the case concerning a Component C directly coupled to (D u , D v ), that means that the two designers have to collaborate to the design of this component.

A fuzzy method

A fuzzy propagation method generates n O DSM for each component C i by identifying all the potential interactions between designers related to their involvement in the design of C i or in the design of a component Cj that interacts with C i . Then it applies the rules introduced by the axioms and translated into a fuzzy inference system. This system is described hereafter.

Fuzzification and input membership functions

The "fuzzification" stage corresponds to the transformation of a numerical value through fuzzy variables (input). The structure of the membership functions characterizing the three inputs is chosen by taking into account the architect's reasoning. Four linguistic variables which are Null, Low, Medium and High are defined. The most common membership function, that is, a trapezoidal function, is used. A trapezoidal membership function is used for the fuzzy logic output linguistic variables which are: Null, Weak, Average and Strong (see Figure 4).

Inferences and fuzzy rules

The fuzzy 'if-then' rules are developed to relate input to output variables [START_REF] Zimmerman | Fuzzy Set Theory and its Applications[END_REF]. These rules represent the expert's knowledge about the interactions between input variables and their effects on the output. The inference system approximates the way an architect estimates the coupling between two D. It is based on a set of 20 rules (see Figure 4). This inference system has been implemented with Matlab Toolbox. The objective has been to generate an inference system that is understandable for architects.

Defuzzification and output membership functions

The "defuzzification" stage involves finding a crisp value for the coupling using the output membership function. The aggregated fuzzy output is defuzzified using the "centroid of area" technique. The formula is given in [START_REF] Dubois | Fuzzy sets and systems: Theory and applications[END_REF]. This is the most widely adopted defuzzification method, which is reminiscent of the calculation of expected values of probability distributions. Table 2 summarizes the main features of the fuzzy inference system. The choice of these linguistic variables aims at limiting the influence of low inputs. Tuning the system has been done to choose the more appropriate input-output membership functions and defuzzification method. First, this fuzzy model has been verified with its applications to simple cases whose solutions are known. Second, this fuzzy treatment has been tested by applying it to a past engine design. After few changes to the parameters, this fuzzy model has been validated by the system architects since after DSM clustering, the obtained architecture was judged valid. The visual inspection of the clustered DSM with the minimum coupling cost clearly revealed distinctive clusters. They were meaningful and useful for system architects, who recognize and name them easily.

Aggregate the n C DSM and filter

In order to obtain the aggregated DSM, the average method is used. The intensity of the (D u , D v ) interaction in the aggregated O DSM is computed as follows:

( ) ( ) ( ) 2 n i j v D , u D C , C DSM v D , u D DSM v , u DSM j i ¦¦ = = eq 4
The O DSM (size: m x m) obtained through the average method may be dense, because of the density of the IM. The density of the Component DSM (size: n x n) could not be a problem for predicting the dependencies between the designers. But the clustering algorithm used for determining the "satisfactory" architecture may depend on the density of couplings. Thus, values in the aggregated O DSM may need (but it is not necessary) to be filtered low. The filter aims at reducing meaningless coupling values by converting them to zero:

( ) ( ) 0 j , i DSM then X j , i DSM if = ≤ eq 5
where X is a parameter that is automatically fixed by the algorithm so as to obtain a given DSM density.

The clustering algorithm used for determining the "optimal» architecture may be sensitive to the DSM density (number of values that are not null divided by the total number of values). Actually, in the clustering algorithm, the fact of belonging to a module is determined by the highest coupling values and these intensities are not filtered. A key requirement is to preserve at least one interaction for each element. In the following development of this paper, the filter is adjusted in order to obtain DSM with a maximum density of 70%.

A matrix based method

Several authors use matrix-based methods for representing and managing information regarding project domains and their interactions [START_REF] Danilovic | Managing complex product development projects with design structures matrices and domain mapping matrices[END_REF]. [START_REF] Sosa | Aligning process, product, and organizational architectures in software development[END_REF] maps a product architecture onto an organization structure. He studied the alignment of product architecture and organizational structure by using a matrix product. Using the same formulation as [START_REF] Sosa | Aligning process, product, and organizational architectures in software development[END_REF], that is consistent with the convention used in Section5, denote: [ ] m

, n

T IM O - C transposed matrix
The O DSM is computed as follows:

[ ] [ ] [ ] [ ] m , n T IM O - C . n , n DSM C . n m, IM O - C = m m, DSM O eq 6
Denote C i -Cj the coupling value between two components C i et Cj, and C i -D u , (resp. Cj-D v ) the level of involvement of designer D u (resp. D v ) in the design of component C i (resp. Cj). Note that the dependency value (need for information exchanges) between D u and D v concerning the design of a couple (C i , Cj) may be computed as follows:

( )

) v D j C ).( j C i C ).( i C u D ( C , C v , u DSM O j i - - - = eq 7
That means that the matrix product implicitly assumes that the three variables that have been identified as the inputs of the fuzzy process may be linked by their product.

This section has presented how to simulate O DSM starting from C DSM and C-O IM. For realizing the step 3, two possibilities have been proposed: a fuzzy process and a matrix based approach. The results of the two methods will be compared on the industrial application in section 7.3.

Application to an Industrial Context

In this section, the two methods are applied to the definition of new project architectures in the case of a robotized gearbox design. This last one is an incremental and architectural innovation. It is a hybrid gearbox between manual and automatic gearboxes.

Capturing the initial Product Architecture

The list of components and their interactions are captured by interviewing system architects who have an expertise in robotized gearbox architecture. The intensity of each interaction has been discussed and fixed according to an existing typology of interactions [START_REF] Pimmler | Integration analysis of product decompositions[END_REF]. The robotized gearbox is decomposed into eight subsystems (or components) that are in turn decomposable into more than 100 parts. Figure 5 exhibits a component DSM of the so-called gearbox. The clustering algorithm reveals hybrid architecture, with three modules (or modular subsystems), and two integrative sub-systems. This architecture has been validated by the gearbox designers since they have adopted the same one. The first module (ISC, SYN, ACT) realizes the shifting function (that is linked to a strategic customer requirement: driving pleasure), the second module (ACT, CLU, CIC) realizes power transmission function. The third module contains only one component which is the differential (DIF).

The remaining two components are integrative. They link together all the other modules of the robotized gearbox: IMP (C7) from the inside and the HBX (C8) from the outside.

Capturing the Incidence Matrix

Incidence matrices are the cornerstone of the present research work. We suppose that the building of an appropriate incidence matrix early in the development process is a key activity of system architects. They need to forecast the impact of their architecture decisions on the management of design teams' efforts. System architects should predict required interactions between actors who are involved in the product development and then structure the teams efficiently. The organization responsible for the development of the gearbox is divided into 12 design teams. Each of them is represented by a team leader. There are 8 design teams directly responsible for the development of the 8 components making up the gearbox, 4 function architects responsible for the specification and validation of each system function: gear shifting, coupling, power transmission, strength flow. There is still a project manager responsible for the project management. Two types of interactions are distinguished: technical interactions that directly concern the product design, and coordination efforts that concern the project management. Hence the project manager who has an integrative role from a coordination point of view is removed. The 4 system function architects are surveyed to fill in the Product-Organization incidence matrix. It is a "8 by 12" matrix with, listed in rows, the gearbox components (C) and in columns the development leaders or designers (D) (Figure 6).

Comparison between the fuzzy process and the matrix approach

The fuzzy process and the matrix approach are applied on the gearbox C DSM and C-O IM, and after using the clustering algorithm, the following two architectures (in Fig. 7 and Fig 8) are obtained. First the authors note that the two methods give the same architecture for the O-DSM. They have asked system architects and component development leaders to visualize DSM and inspect their coherence. Even though this approach is informal and subjective, architectures are often issued from repetitive development experiences and may be judged as being satisfactory. The obtained clusters are satisfactory. The architecture reveals the same three clusters (modules or teams). By analyzing these resulting matrices more precisely, the authors note that:

• Module 1: it is the same in the two models. It represents the team designing the differential and the two integrative elements (IMP and HBX): D8, D9, D10, D11, D12.

• Module 3: it is also the same and this team achieves the shifting function: D1, D2, D3.

• Module 2: it corresponds to the team acting on the module achieving the power transmission. It is integral in the case of the fuzzy process. An interaction between D4 and D6 is missing in the case of the matrix approach. Referring to the C-O IM, these designers interact on the clutch internal control only (only one component for three components in this module) but this interaction is required.

Interactions between D8 and D6, or D8 and D3 are missing in the case of the matrix approach ("0" has been added in the cell to highlight the differences). The study achieved in 7.4 can explain this fact.

First discussion

In this example, the fuzzy process has obtained a weak advantage since it succeeds to identify interesting interactions that the matrix approach does not so. These results may be explained. Therefore, the variations of the coupling values are represented according to a 3D curve. Figure 9 represents these functions with the assumptions that Dv-Cj is high (fuzzy process, on the left side) or equals 10 (matrix approach, on the right side). To give a direct comparison between the two functions, a cross section representing F(x,y,z) for x=y=z is made. Two curves are obtained and displayed in Figure 10. A close comparison between these two curves reveals that:

• For low coupling values (0 to 3), the matrix product filters the low values more than the fuzzy process,

• For medium coupling values (4 to 7), the two curves are very different. The fuzzy process gives a step at about 5 whereas the matrix approach tends to lessen (or neglect) the importance of these values. So the fuzzy method is better adapted to take into account the medium coupling values,

• For high coupling values (8 to 10), the two methods give similar results.

These results explain the differences observed in the industrial case. It is rational that more interactions are found in the O DSM obtained by the fuzzy process than in the O DSM obtained by the matrix approach. Fuzzy process Matrix approach ( )

) v D j C ( ) j C i C ( ) i C u D ...( for ... C , C v , u DSM O j i - = - = -

Second discussion

In this section, the interests of these methods are briefly discussed, the comparison results are interpreted and then, some limits concerning this kind of approach are presented.

Interests of the presented methods     By propagating product architecture choices and assignment decisions on the organization domain, a method is proposed to represent and analyzing the sources of complexity (coupling, dependencies) in product development projects. Product architectures can deeply influence design organization structures because product modules are concurrently designed by different design teams. If design managers are not satisfied with the simulated result, they could change the initial matrix IM and simulate a new organization structure. This last one could be judged better for instance, because the teams may match the functional departments. However, the structures generated by these methods are recommendations. Design managers should be aware that the choice of other teams (i.e. other project organization structure) could increase coordination and teams' integration efforts. The proposed methods could be used to enhance the awareness of designers. They will be able to visualize potential required interactions that they would not able to predict before due to the project complexity.

Comparison     The first results presented in this paper indicate that the fuzzy approach is more adapted to predict potential interactions if there are numerous medium values in the input matrices. In addition, the fuzzy treatment is an advantage to deal with input inaccuracies. Thus the fuzzy method is less sensitive than the matrix based method. However, it is clear that other industrial and academic experiments are necessary to decide if one of the two methods outperforms the other one.

Limits of this approach     This paper is complementary to Sosa's work [START_REF] Sosa | Aligning process, product, and organizational architectures in software development[END_REF][START_REF] Sosa | Identifying modular and integrative systems and their impact on design team interactions[END_REF]. Sosa compares potential interactions between designers (expected organization DSM obtained by the matrix based method) and the actual information exchanges (actual organization DSM). According to Sosa, identifying mismatches can help managers to steer their attention to interactions that require special attention and efforts. He determines the potential interactions that were unattended by actual interactions and the actual interactions that were unpredicted by potential interactions. His conclusion is that the propagation methods may help to identify truly unattended interactions between designers that may prevent from integration problems and iterations. But the propagation results may be erroneous. He writes: "truly unpredicted interactions may occur between development actors who interact even though they are not involved in the development of interdependent modules". Some team interactions may be not predicted by propagating design interfaces. Moreover, indirect interactions are possible when two actors are connecting to a third actor who plays a coordination role. In this case, the expected interactions may be managed by a "transitive" flow of information.

Conclusion

The propagation methods presented in this paper compute an O DSM from a given numerical C DSM and a C-O IM. The underlying Organization structure is revealed by means of the Thebaud clustering algorithm. The method proposed in this paper is based on a new fuzzy inference system that generates a DSM. An existing method is based on a matrix product. These methods have been applied to an industrial case and then compared. The proposed method seems to predict potential couplings more efficiently than the matrix based method. A sensitivity study that has been led reveals that the fuzzy process seems to be less sensitive regarding estimation errors in the initial IM. These first results have to be confirmed by further experiments. This paper intentionally applies the method to two project domains. This method may be used similarly to propagate constraints or changes within the product domain (from functional architecture to physical architecture) or from one project domain to another one, for instance, expectation -SF, product -task, task -team. A propagation method should be few sensitive to estimation errors but sensitive to major changes of values explained by architecture changes. Further works will concern the test of the two methods when the goal is to propagate architecture changes and to visualize potential modifications in the other domain.

Figure 1

 1 Figure1Simulating organization structure starting from the product architecture.

  ). If (C i , C j ) and (D u , D v ) are coupled and if C i and C j interact, then D u and D v interact.• Axiom 3. The intensity of the coupling between D u and D v is related to both the coupling value between (C i , Cj) and (D u , D v ) and the coupling value between C i and Cj. • Axiom 4. An assumption is that each C is coupled to itself with an intensity of maximum value, i.e. 10 (the authors adapt the meaning of the diagonal in the DSM).

Figure 2

 2 Figure 2 Coupling a couple of C and a couple of D

Figure 4

 4 Figure 4 Proposed fuzzy process

  whose rows and columns are identically labeled with the n components (C).

  whose rows are labeled with the m designers (D) that make up the Organization and whose columns are labeled with the n components (C).

Figure 5

 5 Figure 5 Component DSM (C DSM) of the robotized gearbox

Figure 6

 6 Figure 6 Gearbox components-organization incidence matrix (C-O IM)

Figure 7 O

 7 Figure 7 O DSM, architecture revealed by the fuzzy process 0 8 9 10 11 12 4 5 6 7 1 2 3 0 0 8 9 10 11 12 4 5 6 7 1 2 3 0

Figure 9

 9 Figure 9 Variations of the coupling value (Du, Dv) according to (Cj, Dv) and (Ci, Cj)

Figure 10

 10 Figure 10 Comparison of elementary contributions with the two methods

Figure 12

 12 Figure 12 Reference architectures after the clustering of the O DSM

Table 2 .

 2 Features of the fuzzy treatment

	And method	Min
	Or method	Max
	Implication	Min
	Aggregation	Max
	Defuzzification	Centroid

Table 3

 3 Results of comparison tests

	T1 Fuzzy process Matrix based Method
	0.2	12	16
	0.4	26	32
	0.6	36	44
	0.8	40	48
	1.0	46	66

Sensitivity analysis

The two methods have been applied to a "didactic case" (Figure 11). Starting from "10 by 10" C DSM and "10 by 10" C-O IM, the O DSM has been computed. The resulting architectures are similar for the two methods. Then a sensitivity analysis is conducted on the obtained architectures. The initial C-O IM is changed, by introducing estimation errors and the obtained architectures are compared. In these simulation experiments, the C DSM values are assumed to be accurate. Estimation errors are introduced into the IM and are modeled as follows: Denote x, y, z three values sampled from the standard uniform distribution. Denote T1, a threshold that will be considered as a variable for the sensitivity analysis. For a given value in the C-O DSM:

• If x > T1, then there is no error. Otherwise, there is an error,

• If y < 0.75 then the absolute value of the error is 1. Otherwise, it is 2.

• If z < 0.5 then the error is negative. Otherwise, the error is positive.

Methodology:

• The reference matrices are defined: C DSM and C-O DSM.

• The reference O DSM is computed and the clustering algorithm is used to reveal the reference architecture for each propagation method (Figure 12).

• The C-O IM values are changed by introducing estimation errors.

• The changed O DSM is computed by the fuzzy process and by the matrix approach.

• The clustering algorithm is used to reveal the architecture in the two cases.

• In each case, the obtained architecture is compared to the reference architecture using comparison criteria: the number of elements which are not in the same cluster.

Table 3 shows the results obtained after 20 experiments for each value of T1. For T1=0.8, 2 components have been placed in another cluster in the case of the fuzzy process (40 / 20 experiments, it is a mean of 2) and 2.4 in the case of the matrix approach. The obtained results show that the fuzzy process is less sensitive than the matrix approach regarding this criterion (elements in the same cluster).