Chao Fang
email: chaofang@cityu.edu.hk

Franck Marle
email: franck.marle@ecp.fr

Dealing with project complexity by matrix-based propagation modelling for project risk analysis

Keywords: risk interaction, complexity, design structure matrix, risk propagation, project risk management

Dealing with project complexity by matrix-based propagation modelling for project risk analysis

Chao Fang, Franck Marle

Introduction

Project risk management (PRM) is crucial to the success of projects. Unexpected conditions or planning errors may lead to delays, over-costs and other failures which can undermine the successful realization of the project. We refer to such potential events which may affect positively or negatively project activities and project results as project risks. Risk analysis is an indispensable activity for project management, even more for complex projects dealing with large stakes and involving interdependent tasks and organizations (Eckert et al. 2004, Kloss-Grote and[START_REF] Kloss-Grote | How to measure the effectiveness of risk management in engineering design projects? Presentation of rmpass: A new method for assessing risk management performance and the impact of knowledge management-including a few results[END_REF]. In classical PRM, risk analysis is used for evaluating and prioritizing risks, essentially with respect to their probability and impact [START_REF] Chapman | The controlling influences on effective risk identification and assessment for construction design management[END_REF]Ward 2003, PMI 2008). The outputs of the analysis support decision-making, e.g., in terms of planning response actions and allocating resources.

Projects are facing a growing complexity, in both their structure and context. In addition to the organizational and technical complexities described by Baccarini [START_REF] Baccarini | The concept of project complexitya review[END_REF], project managers have to consider a growing number of parameters (e.g., environmental, social, safety, and security) and a growing number of stakeholders, both inside and outside the project. The existence of numerous and diverse elements which are strongly interrelated is one of the main characteristics of complexity [START_REF] Corbett | Tackling industrial complexity Cambridge: Cambridge: Institute for Manufacturing[END_REF][START_REF] Chu | Theories of complexitycommon denominators of complex systems[END_REF][START_REF] Jones | A new status index derived from sociometric analysis[END_REF]. The complexity of project leads to the existence of a network of interdependent risks [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF], where complex phenomena may occur, hard to anticipate and hard to keep under control. For instance, there might be propagation from one -upstream‖ risk to numerous -downstream‖ risks; on the other side, a -downstream‖ risk may arise from the occurrence of several -upstream‖ risks which may belong to different categories.

The extreme case of this propagation behaviour is the chain reaction phenomenon or the -domino effect‖. Another phenomenon is the loop, namely a causal path that leads from the initial occurrence of an event to the triggering of subsequent consequences until the initial event occurs once more. An example of loop is that one initial risk, project schedule delay, may have an impact on a cost overrun risk, which will influence a technical risk, and then propagate to and amplify the original risk of schedule delay.

Many risk management methods and associated tools have now been developed.

They are usually based on two concepts: probability and impact, assessed by qualitative or quantitative approaches. Many of these methods independently evaluate the characteristics of risks, and focus on the analysis of individual risks. Risks are usually listed and ranked by one or more parameters [START_REF] Williams | A classified bibliography of recent research relating to project risk management[END_REF][START_REF] Baccarini | The risk ranking of projects: A methodology[END_REF][START_REF] Raz | Use and benefits of tools for project risk management[END_REF]. Generally, these methods do not take into account the subsequent influence of risks and cannot represent the interrelation between them. We can also cite the creativity-based or the expertise-based techniques, like expert judgement using Delphi, affinity diagram, peer interviews or risk diagnosis methodology (RDM) [START_REF] Kawakita | The original KJ method Tokyo: Kawakita Research Intitute[END_REF][START_REF] Kerzner | Project management: A systems approach to planning, scheduling and controlling New York[END_REF][START_REF] Keizer | From experience: Applying the risk diagnosing methodology[END_REF].

To comprehensively understand a risk, it is helpful to identify its causes as well as its effects. Several methods include this principle, but they still concentrate on a single risk for simplifying the problem [START_REF] Carr | A fuzzy approach to construction project risk assessment and analysis: Construction project risk management system[END_REF]Tah 2001, Heal and[START_REF] Heal | Modeling interdependent risks[END_REF].

For instance, failure modes and effects analysis (FMEA) consists in a qualitative analysis of dysfunction modes followed by a quantitative analysis of their effects, in terms of probability and impact [START_REF] Stamatis | Failure mode and effect analysis: FMEA from theory to execution Milwaukee[END_REF]; fault tree and cause tree analyses determine the conditions which lead to an event and use logical connector combinations [START_REF] Pahl | Engineering design -a systematic approach[END_REF][START_REF] Ferdous | Fault and event tree analyses for process systems risk analysis: Uncertainty handling formulations[END_REF]. These methods are unable to model complex interactions among different risks. Few specific methods are able to model risk correlations with a network structure. Several papers on the application of the Bayesian belief network (BBN) have appeared in recent years in the field of project risk management [START_REF] Fan | BBN-based software project risk management[END_REF][START_REF] Lee | Large engineering project risk management using a Bayesian belief network[END_REF][START_REF] Trucco | A Bayesian belief network modelling of organisational factors in risk analysis: A case study in maritime transportation[END_REF]), which could model risk interrelations, from multiple inputs to multiple outputs. Nevertheless, BBN demands oriented links, is inherently acyclic, and hence does not easily model the loop phenomenon. These methods are thus not always applicable for practical purpose and fail in some cases to represent the real complexity of the interdependencies among risks.

In the last decade, a number of studies have focused on the modelling of complex systems such as critical infrastructures from the standpoint of network theory, to understand how the network underlying the system influences its behaviour, and eventually its characteristics of stability and robustness to faults and attacks [START_REF] Zio | From complexity science to reliability efficiency: A new way of looking at complex network systems and critical infrastructures[END_REF]. Some network centrality measures have been applied, for the first time, to large-scale engineering design and product development networks by Braha and Bar-Yam in (Braha and Bar-Yam 2004b, Braha and Bar-Yam 2004a[START_REF] Braha | Complex engineered systems: Science meets technology[END_REF][START_REF] Braha | The statistical mechanics of complex product development: Empirical and analytical results[END_REF]. These works have provided two interesting results. First, they have shown by analysing large scale product design and development networks that many of these measures are highly correlated. Second, they have shown that the robustness and stability of complex engineering systems is closely linked with the existence of hubs, and that the network behaviour is sensitive to its structure. In addition, eigenstructure or eigenvector analysis has been used for identifying key features in the engineering design iterations in [START_REF] Smith | Identifying controlling features of engineering design iteration[END_REF] and for exploring some hiding information in the complex product development projects in [START_REF] Yassine | Information hiding in product development: The design churn effect[END_REF].

Since all these measures have been proven to be relevant for the analysis of complex networks, some of them are tailored and applied in this work, for the first time, in the area of project risk management. With a view to exploring the interrelationships among project risks, a matrix-based approach for building the project risk network and analysing risk propagation behaviour is proposed. Classical project risks lists are used as the inputs of the network model. They usually only take into account the negative aspects of risks. Existing methods like the Design Structure Matrix (DSM) for dependency modelling are employed to identify and evaluate risk interactions. An eigenstructure analysis is used with the goal of measuring the importance of risks in the network. A risk propagation model is introduced to calculate risk propagation and thus to re-evaluate risk probability and criticality taking into account risk interactions. The aim is to provide project managers with an improved quantitative risk analysis method and thus to support them in making more reliable risk response planning decisions.

The paper is organized as follows. Section 2 describes the process of building the project risk network. Section 3 performs an eigenstructure analysis based on the risk network for measuring risk importance in terms of influence in the network. Section 4 presents a matrix-based risk propagation model for re-evaluating risk characteristics and updating risk prioritisation. An example of an application to a real project in the construction industry is presented in Section 5 to illustrate the proposed method. We conclude the paper in Section 6 with a discussion of the proposed approach and perspectives.

Modelling of project risk network

There are a number of classical methods for identifying individual project risks based on analogy (Riek 2001, Smith and[START_REF] Smith | Proactive risk management. Controlling uncertainty in product development[END_REF], on heuristics [START_REF] Kerzner | Project management: A systems approach to planning, scheduling and controlling New York[END_REF][START_REF] Chapman | The controlling influences on effective risk identification and assessment for construction design management[END_REF] or analytically [START_REF] Shimizu | Reliability problem prevention method for automotive components-development of GD'3' activity and drbfm method for stimulating creativity and visualizing problems[END_REF]. The resulting project risk list serves as an input for studying risk interactions in order to build the project risk network.

Nomenclature

Identification of risk network

Identification is the first step of determining the dependency relationship between the identified risks. The design structure matrix (DSM) method introduced by Steward [START_REF] Steward | The design structure matrix: A method for managing the design of complex systems[END_REF]) has proven to be a practical tool for representing and analyzing relations and dependencies among system components (Browning 2001, Danilovic and[START_REF] Danilovic | Managing complex product development projects with design structure matrices and domain mapping matrices[END_REF]. For our study, we use the concept of DSM with risks, in the context of project management. There are three ways to identify risk interactions. First, the formal interrelations between project objects such as tasks, actors and product components facilitate identifying the interrelations between the risks related to these objects. For instance, the project schedule gives information about task-task sequence relationships.

This helps to identify the correlation between two risks of delay for these tasks. A component-component relationship (functional, structural or physical) means that the risks, which may be related to product functions, quality, delay or cost, can be linked, since one problem on one component may have an influence on another (e.g., budget limits). This may be modelled using DSM, which are square matrices linking homogeneous natures of objects. Second, in a similar way, the domain mapping matrix (DMM) introduced by Danilovic and Browning [START_REF] Danilovic | Managing complex product development projects with design structure matrices and domain mapping matrices[END_REF] are helpful in identifying risk interactions across different domains of the project. This may be modelled in only one matrix, using the combination of square DSM and rectangular DMM, called multiple-domain matrix (MDM) introduced by Lindemann and coworkers [START_REF] Lindemann | Structural complexity management: An approach for the field of product design[END_REF]. Third, there may be relationships between risks related to the same object, like for instance risk of delay and risk of overcost for a task.

Whatever the nature (or type) of dependency between two objects (homogeneous or heterogeneous) or two attributes of the same object, it is transformed into a single interaction between two risks. These risks may be related to the same object or not, and may be of the same nature or not. Risk interaction is thus considered as the existence of a possible precedence relationship between two risks. We define the Risk Structure Matrix (RSM), which is a binary and square matrix with RSM ij = 1 when there is a link from R j to R i . It does not address concerns about the probability or impact assessment of this interaction. We put a sanity check between R i and R j . Suppose we know that the actor who owns R i declared R j as a cause. If the actor who owns R j did not declare R i as a consequence, then there is a mismatch. This means that the two actors are not always mutually aware of their interaction, and this is the case in reality.

Each mismatch is studied and solved by asking simultaneously to both actors if there is or not a potential dependency relationship between the two risks they own respectively, like the analogous works by Sosa and co-workers about the interactions between project actors [START_REF] Sosa | The misalignment of product architecture and organizational structure in complex product development[END_REF].

Assessment of risk network

In the assessment step, we not only evaluate risk characteristics such as risk probability and risk impact, but also assess the strength of risk interactions, which is interpreted as transition probability between risks.

Risk impact may be assessed on a qualitative scale (ordinal or cardinal scale with 5 or 10 levels for instance) or on a quantitative scale (financial loss for instance).

Risk impact is assessed by classical methods, based upon a mix of previous experience and expert judgement.

For the probability assessment, we make a distinction between the probability of a risk to be triggered by another risk inside the network and its probability caused by external events or risks which are outside the system. Spontaneous probability can be interpreted as the evaluated likelihood of a risk, which is not the effect from other activated risks inside the system. On the other hand, transition probability is the evaluation of direct cause-effect relation between two risks. For the example in Figure 1, Risk 5 occurs only in accordance with its spontaneous probability; and Risk 6 may arise from both its spontaneous probability and the transition probability between Risk 5 and Risk 6.

Qualitative scales are often used to express probability with 5 to 10 levels (e.g., very rare, rare, unlikely, etc.) which correspond to non-linear probability measures (e.g., 10 -4 , 10 -3 , 10 -2 , etc.). Logarithmic scales have been used by statisticians for decades [START_REF] Fleiss | Statistical methods for rates and proportions[END_REF]. They allow to distribute probabilities unevenly. In practice, they devote more space to small values, imposing a compressed, logarithmic mapping. Based on this principle, we can use, for example, the following equation for converting qualitative scales into quantitative measures of risk probability:

() 10 q p     (1)
where p indicates the quantitative probability measure, q indicates the qualitative scale value, with parameters 0, 0   which are set by experience and are case-dependent.

A numerical structure matrix can provide more detailed information than a binary one about the risk network for assisting decision-making. Thus the RSM can be converted into the Risk Numerical Matrix (RNM) through assessing the risk interaction.

RNM ij is defined as the strength value of the cause-effect interaction from R j to R i . With regard to the project risk network, values in the RNM can also be interpreted as the transition probability between risks. For example, if the element RNM(4,3) is equal to 0.25, then the probability of Risk 4 originating from Risk 3 is considered to be 25%

under the condition that risk 3 is activated.

Eigenstructure analysis

In mathematics, eigenvalues and eigenvectors are related concepts in the field of linear algebra, which describe characteristics of a matrix. Analysing these eigenstructure properties gives important information about the adjacency matrix and its related network. The mathematical expression of eigenstructure decomposition is as follows: if

M is a square matrix, a non-zero vector v is an eigenvector of M if and only if there is a scalar λ such that

Mv v   (2)
The scalar λ is said to be the eigenvalue of M corresponding to v.

Bonacich suggested that the eigenvector of the largest eigenvalue of an adjacency matrix could make a good network centrality measure [START_REF] Bonacich | Factoring and weighting approaches to status scores and clique identification[END_REF]. For example, the variant of the eigenvector centrality measure is used in Google's PageRank ® algorithm [START_REF] Page | The pagerank citation ranking: Bringing order to the web[END_REF]. Another closely correlated centrality measure is the Katz centrality measure [START_REF] Jones | A new status index derived from sociometric analysis[END_REF]. In our study, we perform eigenstructure analysis on the risk network with the intention of exploring importance measurement of project risks within the network context.

Let x i denote the score of the i-th node, i.e., the measure of the importance of Risk i. We use the square matrix A to represent the RNM of the weighted risk network.

Hence, ()

ij ij A a RNM  and 0 1 if there is an edge joining node to 0 otherwise ij ij a j i a        (3)
For the i-th node, let its score x i be proportional to the sum of the scores of all the nodes which are directly connected to it. Here we take into account both the input and output links, i.e., both the immediate predecessor and successor risks of Risk i in the network. Thus, we get the following equation:

( ) ( ) 1 1 1 1 () N i ij j ji j ij ji j j IN i j OUT i j x a x a x a a x                (4)
where IN(i) is the set of nodes that are direct predecessors of the i-th node and OUT(i)

is the set of nodes that are direct successors of the i-th node. In this way, the importance score of R i is equal to the average importance score of all its neighbour risks. Then we can reformulate the Eq. ( 4) as:

1 () T x A A x   (5)
where A T is the transpose matrix of A, and then as the eigenvalue equation:

() T A A x x   (6)
In general, there will be many different eigenvalues λ for which an eigenvector solution exists. However, in linear algebra, the Perron-Frobenius theorem, proved by Perron and Frobenius (Perron 1907), [START_REF] Frobenius | Über matrizen aus nicht negativen elementen: Königliche Akademie der Wissenschaften[END_REF], asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. Usually, the Perron-Frobenius theorem applies to our case of risk network and the matrix A+A T .

We define the i-th element x i of the eigenvector corresponding to the largest eigenvalue λ * as the eigenvector centrality of R i in the risk network. Eigenvector centrality is a measure of the importance of a node in the risk network. In this sense, eigenvector centrality calculates not only direct connections but also indirect long-term propagations. Thus the complete risk network is taken into account. Mathematically, eigenvector centrality is closely related to the influence measures, such as those proposed in [START_REF] Jones | A new status index derived from sociometric analysis[END_REF][START_REF] Taylor | Influence structures[END_REF][START_REF] Friedkin | Theoretical foundations for centrality measures[END_REF][START_REF] Hadi | A new measure of overall potential influence in linear regression[END_REF]). The idea is that even if a node influences directly only one other node, which subsequently influences many other nodes (who themselves influence still more others), then the first node in that chain is highly influential [START_REF] Borgatti | Centrality and network flow[END_REF]. For calculating the risk eigenvector centrality, besides the output links of a risk which contribute to its impact measure in the network, we also incorporate its input links for measuring its importance in terms of probability. That is why the matrix A+A T is based on for the proposed eigenstructure analysis with regard to project risk management.

Risk propagation model

In the project risk network, the nodes (risks) are assessed in terms of spontaneous probability and impact; the edges (risk interactions) are assessed as the transition probability from one risk to another. In the matrix-based risk propagation model, we assign the original risk probability evaluated by classical methods without considering interactions to each risk as its spontaneous probability. The assessed values in the RNM are used as transition probability between the related risks. Some assumptions are made in order to calculate risk propagation in the network:

(1) A risk may occur more than one time during the project (as witnessed in practical situations). Risk frequency is thus accumulative if arising from different causes or if arising several times from the same cause.

(2) The structure and values of RNM do not vary during the analysis time. In other words, there is no added or removed risk, and the transition probability between risks will not change during the analysis.

Hence, the RNM can be regarded as similar to the stochastic matrix or transition matrix used to describe the transition of a Markov chain [START_REF] Buzacott | Stochastic models of manufacturing systems[END_REF]Shanthikumar 1993, Latouche et al. 1999). This principle has been applied to industrial engineering, for example, Smith and Eppinger introduced a work transformation matrix based on the DSM method to model the engineering design iteration process [START_REF] Smith | Identifying controlling features of engineering design iteration[END_REF]. However, different from conventional stochastic matrix, in our model the RNM is a square matrix where all entries are nonnegative real numbers and inferior to 1, but the sum of each row or column is not necessarily equal to 1. A foundation work with respect to modelling risk interactions for propagation analysis has been presented by the authors in [START_REF] Fang | Modelling risk interactions to re-evaluate risks in project management[END_REF].

Suppose there are N identified risks in the network. We use vector s to represent the spontaneous probability of risks and A still denotes the RNM of transition probabilities. P(R) is the vector of risk probability or frequency after propagation analysis.

We define one step of risk propagation as that the occurrence of one risk R i may propagate to impact the risks that have direct cause-effect relationship with it (i.e., R j if 0 ji RNM  ). For the example of risk network in Figure 1, R5 can propagate to trigger R1 directly within one step; it may also influence R1 in two steps of propagation through R6 or R4, and within three steps through the chain R5->R6->R4->R1, and so on. As vector s also represents the initial vector of risk probabilities, after m steps of propagation, the probability vector of risks propagated from the initial state is thus equal to m As  . If we only consider m steps of propagation and according to the assumption of accumulative risk frequency, the re-evaluated risk probability vector can be obtained by the following equation:

1 1 0 ( ) ( ) ( ) m m m i i i i i i P R s A s I A s A s               ( 7
)
where I is the N-order identity matrix. In the limit of infinite propagation steps in the project development: 0 ( ) lim( )

m i m i P R A s     (8)
Multiplying both sides of Eq. ( 8) by (I -A), and then we get that

1 0 ( ) ( ) ( ) ( ) ( ) m im i I A P R I A A s I A s             (9)
It is not guaranteed that the infinite power of the matrix A would converge to 0, as in the following equation:

1 lim 0 m m A    (10)
Some research papers established sufficient conditions for the convergence of infinite product of matrix, e.g., in [START_REF] Thomason | Convergence of powers of a fuzzy matrix[END_REF][START_REF] Daubechies | Sets of matrices all infinite products of which converge[END_REF][START_REF] Bru | Convergence of infinite products of matrices and inner-outer iteration schemes[END_REF][START_REF] Holtz | On convergence of infinite matrix products[END_REF]). In the project risk network, for example, if a risk is involved in several loops and the sum of the products of all the transition probabilities along these loops is greater than 1, the risk propagation process does not converge. This type of risk propagation is not likely to occur in practice and is outside the scope considered by our model.

Nevertheless, since A is the risk numerical matrix which is usually sparse and composed of transition probabilities at small values less than 1, usually the condition of Eq. ( 10) is satisfied. Thus, risk probability can be re-evaluated by the following equation:

1 ( ) ( ) P R I A s     (11)
Moreover, it is possible to predict the consequences of the occurrence of one or several initial risks using this model. We assign, for instance, 100% to the spontaneous probability of R i , while all the other risks have 0% initial values. That is to say, the initial vector s = I i , where I i is the i-th column of the identity matrix I. We can then anticipate the occurrence of the rest of the network, and thus evaluate the global consequences of R i .

Criticality is an important indicator used for prioritizing risks and usually defined as the product of risk probability and impact. Similar to risk probability, we can refine risk criticality by integrating all the potential consequences in the network of a given risk. Giving R i with its re-evaluated probability (risk frequency), we redefine its criticality by:

1 ( ) ( ) ( ) i n i j R j j C R G R P R    (12)
where C(R i ) is the criticality of R i ; G(R j ) is the original evaluated impact (G for gravity) of R j ; and () i Rj PR denotes the probability of R j as the consequence of P(R i ). According to Eq. ( 11), the re-evaluated risk criticality is expressed by:

1 ( ) ( ) ( ( )) Ti ii C R G I A I P R       (13)
The vector of risk criticalities can be calculated by the following equation:

1 ( ) ( ) .* ( ) T C R I A G P R     (
( ) ( ) ( ) c i a i b i  (15)
The re-evaluation of risk characteristics such as probability and criticality enables us to update the risk prioritization results and then to design new risk response actions.

Application to a tramway infrastructure project

In this study, we implement the proposed approach to a real large project, aimed at building the infrastructure and associated systems of a tramway. This project takes place in a city with a population of 750 000. Both the classical project risk analysis and the proposed matrix-based risk propagation analysis are carried out.

The project includes the construction and implementation of tramway, equipments, and civil work, with 10 years duration and hundreds of millions € of budget. The leading company is a designer and manufacturer of trains, which recently extended its scope by proposing -turn key‖ projects, including not only the trains, but also the complete infrastructure around the trains. The project thus comprises:

 The construction of a depot to stock trains and to execute their control and maintenance;

 The installation of tracks throughout the city, over land with many steep slopes;

 The delivery of the corresponding trains, including redesign activities if the current version does not fit with the city's specific requirements;

 The establishment of a traffic signalling operating system, which gives priority to the tramway so as to guarantee travel time performance levels.

Building the project risk network

The first step is to build the project risk network. An original project risk list has been The direct assessment by experts is used on this case study for evaluating the strength of risk interactions. The assessment of the potential risk interactions was performed on a 10-level Likert scale (Likert 1932, Maurer and[START_REF] Maurer | A comparison of likert scale and traditional measures of self-efficacy[END_REF], thanks to the high expertise of interviewees. In particular, this step requires the participation of several experts involved in the project since it necessitates a very wide overview of the project elements and stakes. In the end, the RNM of the project was obtained. It is displayed in Figure 3. Various gray scales are used to indicate the strength levels of the risk interactions.

Figure 3. RNM of the tramway project

Classical project risk analysis

As we can see in the classical project risk list (Table 1), basic characteristics of project risks have been assessed by the project manager and associated experts, including qualitative probability (or likelihood) and impact (or severity) scales, as well as criticality (aggregation of probability and impact). In classical project risk analysis, risks are considered most important if having both high probability and impact. The results of this type of analysis are used by the project manager for risk response planning. Resources are firstly allocated to manage the risks prioritized with high criticality.

In Figure 4, the classical project risk analysis results are displayed in a risk probability vs. impact diagram, where each risk identified in Table 1 is represented by a dot. The limits between different criticality levels should be defined a priori, before the risk assessment. For example, risks can be categorized into several levels of criticality, such as critical, high, moderate and low risks. In Figure 4, we have highlighted the topten risks (displayed by their IDs) according to their criticality value.

The project is based on a contract including many contractual terms involving financial penalties in case of failure, whether on time or quality aspects.

Results using the proposed methods

In Table 2, we consolidate the risk analysis results by the risk propagation model and the eigenstructure analysis. They are compared with the original risk estimates obtained by classical methods. Conducting the eigenstructure analysis by Matlab software, we get the unique largest real eigenvalue λ * = 1.4866 of the RNM. The eigenvector centralities of each risk are also given in Table 2. 2 of the risk network structure, many other risks lead to these nodes which are closely related to financial performance.

Consequently, they have high frequency to occur during the project, and even more than one time (accumulation).

Risk criticalities can also be refined by the risk propagation model. As a result, the risk rankings have also changed. Seen from Several risks have greatly risen with respect to their rankings. For example, the ranking of R10 (Travel time performance) has increased from No.43 to No.10 with an upgrade of +33. Moreover, in the new prioritisation results, the value gap between risks becomes different from that in the classical method. For example, R43 and R37 are two risks with high criticalities and R43 is ranked superior to R37. After re-evaluation by modelling risk propagation, R43 is still regarded more critical than R37, but the gap between their relative criticality values becomes much larger, shown in Table 3. This is the opposite for R43 and R7, where R7 is still behind after re-revaluation, but closer.

The shift of priorities reflects the intensity of risk interactions in the network.

The eigenvector centrality provides a measurement of risks in terms of their position and importance in the network. As we can see in in the top-ten list because it has direct contacts with some key nodes such as R2, R7 and R43, which enhance the measure of its influence in the network.

Discussion

This paper has presented a matrix-based method for modelling risk interactions and then re-evaluating risks in terms of various indicators. A realistic application to a tramway implementation project is performed with the involvement of the project manager and the team of experts.

The DSM method is used in the field of PRM for identifying risk interactions.

The parameters in the constructed project risk network are assessed so that we are able to analyse the risk propagation behaviour in a quantitative manner. An eigenstructure analysis is performed based on the risk network. The eigenvector centrality provides a measure of the importance of risks which reflects the influence of a risk taking into account both its direct and indirect connections with the other risks in the network. The introduced risk propagation model enables us to anticipate the risk propagation in the risk network. The risk prioritisation has been updated in terms of criticality ranking. The underestimation of some risks in classical methods is due to the neglect of the complexity-related information. Thus, the proposed model provides the project manager with more and new insights on risks and their role in the risk network. Several critical risks identified by classical project risk analysis are confirmed by our analysis to play an important role in the risk network. But in addition, some new risks are highlighted as key elements by this analysis, which are supplement results to classical analysis. This information enables complementary mitigation actions suitable for the particular type of risks to be planned by the project manager. These actions are included in the global risk response plan, as part of the project plan, to adequately manage the required resources, budget and time. Without this analysis, the project manager would probably not have decided to plan actions on these specific risks.

It is shown that the refined risk criticality analysis results are closely correlated with the eigenvector centrality-based risk prioritisation. The risks with high eigenvector centrality are also ranked as top risks in the refined risk criticality analysis. For example, in the case study presented, night risks (R2, R7, R10, R12, R16, R18, R37, R43, R55) are the same in the top-ten risk list respectively based on these two measures. It is due to the fact that the risks with high eigenvector centrality act as -hubs‖ which play the role of key passages for risk propagation. This is helpful for the project manager in giving priority to certain risks and in designing more effective response actions. For example, paying attention and making effort to prevent the source risks like R7 and R16 may help avoid many problems arising at later stages; corrective or protection actions are often designed for the critical accumulation risks such as R2, R43 and R55 to reduce losses; a mix of strategies, e.g., preventive and confinement actions can be applied to transition risks in the network like R10, R12, R18 and R44 for mitigating the risk propagation and thus reducing the global risk exposure. It should be noted that these findings coincide with the conclusions drawn by Braha and Bar-Yam in (Braha andBar-Yam 2004a, Braha and[START_REF] Braha | The statistical mechanics of complex product development: Empirical and analytical results[END_REF] on the analysis of large production development networks.

The proposed approach can be used as a supplement to the classical project risk analysis. It can support project managers in making subsequent managerial decisions, such as risk response planning and allocating available budget or resources. Our approach encourages people to meet together and communicate/ coordinate better, in order to manage potential interactions and potential propagations. It underlines the need for cooperation and transversal communication within the project team. Fieldwork proves us that it does indeed improve communication between people, since this approach does not seek the identification of responsibility and/or accountability, but the identification of propagation chains and human interfaces. Allocating resources and conducting actions on these key risks or interactions can then be efficient for mitigating the propagation phenomena and reducing the overall risk exposure.

As a next step, some other network centrality measures in literatures will be used to analyse and prioritise risks, and the results will be compared with the refined risk criticality analysis and eigenstructure analysis with respect to project risk management. The uncertainties of estimated inputs on risks and risk interactions will be modelled and their propagation in the risk network should be calculated, e.g., though a sensitivity analysis. The concept of robustness, e.g., the traditional static robustness and the dynamic robustness raised in (Braha andBar-Yam 2004a, Braha and[START_REF] Braha | The statistical mechanics of complex product development: Empirical and analytical results[END_REF], will also be investigated in terms of the project risk network.

Figure 1

 1 Figure 1 gives an example to show the use of such a RSM to represent the

Figure 1 .

 1 Figure 1. Illustration of RSM (adapted from (Fang and Marle 2012))

  It assigns relative centrality scores to all nodes in the network based on the principles: (1) connections to more nodes contribute more to the score of the node; (2) connections to high-scoring, namely important nodes, contribute more to the score of the node; (3) higher strength of the connection to other nodes contributes more to the score of the node.

  14) Here the symbol -.*‖ represents the array multiplication or the Hadamard product (Johnson 1974) of matrices. For example, the Hadamard product .* c a b  of two vectors a = [a 1 , a 2 , … , a n ] and b = [b 1 , b 2 , … , b n ] is still an n-order vector and its elements are defined as:

Figure 2 .

 2 Figure 2. Project risk network structure of the tramway project

  Almost every problem is potentially transformed directly or indirectly into an additional cost and then a profit loss. It is thus not surprising to see that R2 (Liquidated damages on intermediate milestone and delay of progress payment threshold), R37 (Rejection of Extension Of Time), R43 (Return profit decrease) and R55 (Available cash flow decrease) are among the most critical risks. Other risks with high criticality are generally related to the final delivery, like R12 (Operating certificate), or some big parts of the project, like R18 (Civil Work delay) and R40 (Operating Centre installation).

Figure 4 .

 4 Figure 4. Display of classical project risk analysis results

Table 2 .

 2 Project risk analysis results of the tramway projectBased on the risk analysis results in Table2, risks are prioritised by different indicators. The top-ten risks are listed in Table3.

Table 3 .

 3 Risk prioritisation results according to different indicators

	Ranking	Risk Criticality by Classical Method	Re-Evaluated Risk Criticality by Risk Propagation Model	Eigenvector Centrality based on RNM
		Risk ID	Value	Risk ID	Value	Risk ID	Value
	1	R43	3.048	R2	23.4	R2	0.5553
	2	R37	2.667	R43	11.4	R10	0.3248
	3	R55	2.667	R55	9.1	R43	0.2980
	4	R2	2.465	R18	6.8	R12	0.2971
	5	R3	1.905	R12	5.7	R7	0.2384
	6	R40	1.541	R37	5.4	R18	0.2235
	7	R12	1.524	R16	4.5	R55	0.2075
	8	R18	1.524	R7	4.3	R16	0.1996
	9	R29	1.524	R41	2.8	R44	0.1746
	10	R7	1.314	R10	2.7	R37	0.1726
	In our risk propagation model, risk probability can be re-evaluated through Eq.
	(11) into risk frequency. As we can see in Table 2, the frequency of risks has increased
	to varying extent taking into account the interactions with other risks in the network.
	Some risks have high risk frequency which is greater than one. As mentioned in the
	assumption of the model, this is consistent with the reality that one risk may occur more
	than once during the project. The examples are R2 (Liquidated damages on intermediate
	milestone and delay of progress payment threshold), R43 (Return profit decrease) and
	R55 (Available cash flow decrease). Shown in Figure

  Table 3 and the column of ranking shift in Table 2, some important risks based on classical risk analysis remain in high

	positions, such as R43, R2, R55, R37 (Risk of partial rejection of our request for
	extension of time)), R7 (Traffic signalling, priority at intersections), R12 (Operating
	certificate delay), and R18 (Civil Work delay & continuity). On the other hand, some
	risks have dropped out of the top-ten rankings, such as R3 (Vehicle storage in another
	city), R40 (OCS installation), R29 (Additional poles overcost for Tramway Company).

Table 3

 3

	, most of the critical

Acknowledgement

The work described in this paper started when the first author was with ECP. It was completed and partially supported by a grant from City University of Hong Kong (Project No.9380058).

assessment, and decision-making in project and multi-projects context. He has a MSc of Ecole Centrale Lyon

provided by the project manager and the expert team, which contains 42 project risks.

The risk list has been updated when performing the risk interaction identification. Some new risks have been added into the list, for two reasons: some were a consequence or cause of other risks already present in the initial list; others were seen as intermediary risks which were useful to explain the link between two or more risks of the initial list.

Thus, the resulting project risk list contains 56 identified risks at the main level, with their name, domain and risk owner information, as shown in Table 1.

Identification of the risk interactions defines the structure of the project risk network. Using the DSM-based method described in Section 2.1, we get the RSM and the corresponding risk network structure (shown in Figure 2).


Hadi Jaber

Franck Marle
email: franck.marle@centralesupelec.fr

Marija Jankovic

Improving Collaborative Decision-Making in New Product Development Projects using Clustering Algorithms

Keywords: Organizational decision processes, project teams, organization design, technical organizations, project and R&D management, New Product Development

Numerous decisions have to be made in early design processes. Often times they involve many actors, with the difficulty that they are shared across numerous parallel collaborative groups, for coordination and meeting scheduling reasons. This paper aims at facilitating collaborative decision-making process by grouping actors according to the relationships they have due to their assignment to decisions. Clusters of actors are proposed in order to provide decision-makers with a temporary and complementary organization designed for making efficiently simultaneous collaborative decisions. This approach has been illustrated through actual data in new product development project in the automotive industry.

Introduction

Importance of early design stages decisions has been underlined for many years [START_REF] Bellut | Les processus de la décision[END_REF]- [START_REF] Hansen | A Mapping of Design Decision Making[END_REF]. In complex system design, management of collaborative decision-making is characterized by many decisions impacting numerous product-and project-relatedparameters. Multi-domain nature of these processes needs involvement of a wide range of actors, like project manager, system engineer, technical engineers, purchasers, architect engineers, product planners, supply chain managers and quality engineers. During early complex system design stage, decision owners need to manage decision-making process and establish temporary decision teams, identifying relevant experts in the project. In many cases, these teams are not properly established; hence, many actors participate in a large number of meetings but fail to be efficiently related to the decision-making outcome and impact. This may involve loss of efficiency and additional risks in communication and coordination between actors [START_REF] Browning | Adding Value in Product Development by Creating Information and Reducing Risk[END_REF], [START_REF] Browning | Modeling impacts of process architecture on cost and schedule risk in product development[END_REF].

We propose in this paper complementary temporary teams mutualizing the maximum number of collaborative decisions to be managed in each cluster. The aim of this approach is thus to support the management of NPD projects by providing more consistent and fewer collaborative teams, to make the process more adaptive and less risky [START_REF] Lévárdy | An adaptive process model to support product development project management[END_REF]. It permits to enhance communication and coordination in complex system design, underlined as important in [START_REF] Morelli | Predicting Technical Communication in Product Development Organizations[END_REF], by increasing within-cluster interaction values while reducing the time load due to meetings. With increase of within-cluster value, actors can better follow-up on decision impacts and manage design results. Section 2 introduces how data are gathered using matrixbased approach. Section 3 describes the clustering strategy. Section 4 presents results of an analysis based upon one actual complex system design project in automotive industry.

Discussion about managerial implications is introduced in Section 5. Conclusions and perspectives are drawn in Section 6.

Modeling the relationships between actors due to the decisions they contribute to

The matrix-based modeling

The Dependency and Structure Modeling approach has proven to be a practical tool for representing and analyzing relations and dependencies among system components. This methodology has been created and developed by Steward, Eppinger and Browning [START_REF] Steward | The Design Structure System: A Method for Managing the Design of Complex Systems[END_REF]- [START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF].

A DSM (Design Structure Matrix) is a square matrix, representing interactions between its elements. A DSM is a square matrix, with the rows and columns identically labeled and ordered, and where the off-diagonal elements indicate relationships between the elements.

Depending on the number and location of identified relationships, elements may be [START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF], [START_REF] Thompson | Organizations in action[END_REF]:

• Dependent (e.g. sequential if temporality is a parameter of the relationship),

• Independent (or parallel),

• Coupled,

• Conditionally connected (contingent relationship).

A DSM can be binary or numerical, with qualitative or quantitative assessment.

A DMM (Domain Mapping Matrix) is a rectangular matrix, displaying relationships between two types of elements of different domains [START_REF] Danilovic | Managing complex product development projects with Design structure matrices and domain mapping matrices[END_REF], [START_REF] Akao | Quality Function Deployment[END_REF]. For instance, the Actor-Decision matrix (AD) is built by modeling affiliation relationships between actors and decisions. This data is an input of the approach.

Actor relationships or dependencies are important because they affect the efficiency of team communication, thus directly influencing design process outcomes. Hepperle et al. use DSM principles to analyze communication dependencies between actors in product development [START_REF] Hepperle | Analyzing communication dependencies in product development using the design structure matrix[END_REF]. They propose a Communication Grid Method based on the identification of the network structure underlying the communication relationships between these actors. Finally, in this paper, one DSM and 4 DMM are defined and used:

• The Actor-Decision Matrix, called AD, usually known as Responsibility Assignment or Affiliation Matrix (RAM), defined as a DMM.

• The Decision-Group Matrix, called DG. DG is a way of grouping decisions introduced in [START_REF] Jankovic | Collaborative Decision in design project management : a particular focus on automotive industry[END_REF], as a first attempt to take into account collaborative decisionmaking processes. It is also a DMM.

• The Actor-Group Matrix, called AG, is based on the previous collaborative decisionmaking processes. AG is a DMM obtained by combining AD and DG matrices.

• The Actor-Actor Matrix, called AA. It represents the relationships between actors, on which clustering will be applied. It is an organization-related DSM, which has been the object of several works [START_REF] Lorsch | Managing Group and Intergroup Relations[END_REF]- [START_REF] Sosa | The Misalignment of Product and Organizational Structures in Complex Product Development[END_REF]. AA is a matrix obtained from AD, as introduced in Sect. 2.2.

• The Actor-Cluster Matrix (called AC) is resulting organization using our approach. It is the improved version of collaborative decision-making groups AG. AC is a DMM obtained by clustering Actor-Actor AA matrix.

Global approach

The initial organization is made of 13 groups, called G k . The 70 decisions D j are affiliated to one and only group G k , forming the DG matrix. The 64 actors A i are assigned to decisions, forming the AD matrix, with the relation AD ij =1 iff actor A i is assigned to decision D j . As a consequence, the AG matrix is built as the product of AD and DG. This means that AG ik =0iff the actor A i does not belong to the group G k . Otherwise, AG ik represents the number of decisions included in group G k owned by actor A i . The AG matrix is obtained as follows using Eq. ( 1) and ( 2):

AG = AD * DG (1) ∀ 𝑖 ∈ 1. . 𝑁𝐴 , 𝑗 ∈ 1. . 𝑁𝐺 , 𝐴𝐺 𝑖𝑘 = 𝐴𝐷 𝑖𝑗 * 𝐷𝐺 𝑗𝑘 𝑁𝐷 𝑘=1 (2)
Where NA=number of actors, ND = number of decisions, NG = number of groups.

The existing organization AG serves as a comparison point with proposed clusters AC. AC is the result of the clustering of the AA matrix, itself obtained by multiplying the matrix AD by its transpose DA=AD T , using the Eq. ( 3):

AA = AD*DA (3)
This product is obtained as follows (Eq. ( 4)):

∀ 𝑖, 𝑗 ∈ 1. . 𝑁𝐴 , 𝐴𝐴 𝑖𝑗 = 𝐴𝐷 𝑖𝑘 * 𝐷𝐴 𝑘𝑗 𝑁𝐷 𝑘=1 (4)
This matrix enables direct interactions between actors to be analyzed. For instance, direct interactions in the US Senate have been analyzed by Bartolomeiet al [START_REF] Bartolomei | Engineering Systems Multiple Domain Matrix: an Organizing Framework for Modeling Large-Scale Complex Systems[END_REF].

Several proposals are obtained for AC, running simultaneously several algorithms with several configurations. The final recommendation is made considering the relevance of the clusters (within-clusters total value, cluster size, cluster density, number of clusters), in order to keep the algorithmic solution applicable to real-life project.

This generation of several alternatives enables comparisons and sensitivity analysis. Finally, the most relevant complementary organizational configuration AC is compared to the existing one AG, and implemented if judged better and applicable. Details of the clustering approach will be given in Section 3.

The automotive project case study

Vehicle development projects are very long and complex, with the participation of between 1500 and 2000 project members. Usually, this type of project can take between two to four years when concurrent engineering is used as a basic organizational hypothesis. Early design stages can be long as 8 to 10 months. The overall early design stage integrates 70 decisions organized into 13 collaborative decision-making processes. The data gathering process represents a result of several working groups integrating 30 cross-domain project members.

Some of these processes are: innovation integration process, manufacturing and supply chain feasibility and scheduling, design style, economic optimization, and purchasing.

Collaborative decisions integrate members from different domains. There are in total 64 different actors participating in the 13 collaborative decision-making processes. Fig. 1 shows affiliations of actors to decisions (AD) and of decisions to groups (DG). Some of these actors contribute to the process, even if not assigned to collaborative decisions. Fig. 1 permits to simultaneously visualize the three matrices used as inputs in our approach, respectively AD, DG and AG. Several members, like the project manager (A 51 ), participate in almost all groups (their rows in AG being almost filled), whilst some other actors participate to fewer groups like A 1 (the sum of non-null cells in AG first row being equal to 1). Maximum number of decision-makers in one group is 21 (the sum of non-null cells in AG columns). Defining the groups can be difficult to decide and to implement. There are two main parameters that need to be discussed: 1) the size of the group, i.e. the number of actors one wants to put in one group, and 2) the number of groups, i.e. the total number of groups that one wants to coordinate in one project. Indeed, it is very time-consuming for people, with intertwined meetings and decisions and potential issues like meeting sequence. The network of direct connections between actors due to their assignment to their respective decisions, is shown hereafter in Fig. 2.

Fig. 2. The network of direct relationships between actors (AA)

Fig. 2 is a graph representation of the matrix AA calculated using Eq. ( 4). A graph consists of nodes and edges, respectively actors and relationships between pairs of actors. The weight on the edge between two actors represents the number of common decisions assigned to these two actors. The size of the node (and its color) is proportional to the number and the weight of its direct edges, the darkest and the biggest node corresponding to the actor whohas the highest value of connected weighted edges. The network is composed of very interrelated parts, difficult to cut into disjunctive clusters. This is the object of the following Section to define an adequate algorithm to propose clusters tailored to decision-makers requirements and constraints.

Clustering approach

General terms

In this section, we note G (V, E) a graph where V is the list of nodes, and E is the list of edges in the graph. Lines that connect two nodes and thus define a relationship between them are called edges. We also note partition of the graph P=(C 1 , ... , C k ). Actors A i will be assigned to clusters C j , forming the matrix AC. The most common approach to this problem in the literature has been to ignore edge direction and apply methods developed for community discovery in undirected networks, but they discard potentially useful information contained in the edge directions. In this paper, we selected algorithms while extending clustering objective function and methodology to directed graph. Measures are extended by considering edge directionality as inherent network characteristics, like the directed version of modularity (clustering objective function) used by Leicht and Newman [START_REF] Leicht | Community structure in directed networks[END_REF].

Resolution strategy

A clustering approach has been developed in a similar context forming clusters of heterogeneous elements (project risks). The Risk-Risk Matrix (RR) had been modeled to capture risk interactions. Then, a risk clustering approach had been applied in order to enhance efficiency in risk management by capturing possible risk propagation paths [START_REF] Marle | Project Risk Management Processes : Improving Coordination Using a Clustering Approach[END_REF]- [START_REF] Marle | Forming risk clusters in projects to improve coordination between risk owners[END_REF].

In this case, actors groups were formed indirectly due to the fact that they own risks within the same cluster. In our case, actors are directly grouped together because of the decisions they are contributing to.

A cluster may contain similar elements, with a particular element called centroid [START_REF] Filippone | A survey of kernel and spectral methods for clustering[END_REF]. On the opposite, some works focus on edges that are least central or most "between" clusters, and remove them from the original graph in order to build the strongest clusters with the remaining edges [START_REF] Girvan | Community structure in social and biological networks[END_REF]- [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF]. Newman et al. are co-authors of numerous works in the field of finding community structures in complex networks [START_REF] Leicht | Community structure in directed networks[END_REF], [START_REF] Clauset | Hierarchical structure and the prediction of missing links in networks[END_REF]- [START_REF] Clauset | Structural Inference of Hierarchies in Networks[END_REF]. Specific DSM-related clustering techniques have been developed and implemented in industrial applications like IGTA (Idicula-Gutierrez-Thebeau Algorithm) for clustering Component-DSM [START_REF] Idicula | Planning for Concurrent Engineering[END_REF]- [START_REF] Thebeau | Knowledge Management of System Interfaces and Interactions for Product Development Process[END_REF], or the DSM-based algorithm of Borjesson and Holtta-Otto [START_REF] Borjesson | A Module Generation Algorithm for Product Architecture based on Component Interactions and Strategic Drivers[END_REF]. In the field of product architecture modularity, Hsuan and Gassmann proposed a mathematical model to analyze, then optimize this clustering objective [START_REF] Hsuan | Managing modularity of product architectures: toward an integrated theory[END_REF]. Moreover, several works focus on clustering rectangular matrices [START_REF] Li | A matrix-based clustering approach for the decomposition of design problems[END_REF], [START_REF] Li | Methodical Extensions for Decomposition of Matrix-Based Design Problems[END_REF]. An example of the applications of clustering DSM is the clustering of organizational units performing overlapped activities in order to reduce complexity of coordination in a product development project [START_REF] Yang | An Overlapping-based Design Structure Matrix for Measuring Interaction Strength and Clustering Analysis in Product Development Project[END_REF].

Instead of selecting a single algorithm and optimizing in the space of possibilities, our resolution strategy will be based on 3 well-known algorithms, developed in different contexts [START_REF] Leicht | Community structure in directed networks[END_REF], [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF], [START_REF] Borjesson | A Module Generation Algorithm for Product Architecture based on Component Interactions and Strategic Drivers[END_REF]. This provides the benefits of each of these algorithms, which may offer either large clusters or dense clusters or balanced clusters, etc.

An innovation of this work is thus to assembly a solution from pieces of solutions obtained in different ways and using different problem configurations. There is no universally optimal configuration of clusters, but it depends on the judgment of the decision maker. Clustering aims then at defining the best data set partitioning for given parameters. The solution is strongly dependent upon the decision-maker.

We have tested a wide spectrum of clustering algorithms, and we decided to use the three algorithms which are most adapted to our case study. These algorithms can be either parameterized or unsupervised, which is useful when no prior knowledge is provided. Such parameters can be the number of desired clusters, the maximal size of clusters, and allowance The idea is that the more often pairs of actors are proposed together in the different configurations, then the more robust the decision of putting them together in the final solution is. The third stage is the post processing of the obtained results. This is done by combining extractions of particular clusters or pieces of clusters from different solutions. This combination is based on the quality indicators and the frequency analysis of the results (the number of times the couple of actors were put together). A hybrid solution, that meets best the needs of the decision maker, is built using a mix of clusters from all configurations.

Cluster validity

Cluster validation is a major issue in cluster analysis; in fact, much more attention has to be paid to cluster validity issues (checking the quality of clustering results). However, it must be emphasized that the results obtained by these methods are only tools at the disposal of the expert in order to evaluate the resulting clustering.

We define INTRA(C i ) in Eq. ( 5) as the sum of edges included in cluster C i (noted W i ), divided by the total sum of edges in the matrix AA, denoted TW (for Total Weight).

INTRA(C i ) = W i / TW (5)
The term INTRA has been chosen to reflect the notion of intra-cluster interdependencies, obtained as the sum of intra-cluster edges. To obtain the W i , we create the matrix CC as the following product using Eq. ( 6):

CC = CA * AA * AC (6)
This is obtained in two stepsfollowing Eq. ( 2) as the product of CA by the product of AA and AC:

CC = CA * (AA * AC) (7)
The W i are the diagonal cells of CC.

However, the implementation of the i -th cluster C i requires the use of a certain number of actors. This is why we moderate the raw performance of the clustering algorithm by the managerial efficiency, counting the Number of Actors involved in C i , called NA(C i ), as described in Eq. (8):

P(C i ) = INTRA (C i ) / NA(C i ) (8)
Moreover, we also consider the interdependency value between two clusters C i and C j , called

INTER(C i , C j ).
INTRA<<INTER).
The final performance index is then calculated as in Eq. ( 9):

P'(C i ) = 1000* INTRA (C i ) / (NA(C i )*INTER(C i )) (9)
The factor 1000 is added to give results in an easily readable range (to estimate and to compare). These indicators permit comparison between proposed clustered configurations against each other; and afterwards against initial configuration AG, both in terms of organizational efficiency (P index) and in terms of role given to the actors (P' index). For instance, if a cluster is always proposed whatever the algorithm and whatever the configuration, then one can be confident to put it in the final proposed configuration.

Results: proposing new groups of interrelated actors

Initial organization into collaborative decision-making processes had been a proposition in order to enhance project management and gain time. Several previous works have accounted for this organization proposition. In the follow-up of three projects the overall gain for the early design stages was estimated up to 20% which is an interesting improvement with regard to duration of this phase. This organization was based upon collaborative decision-making model and in particular information modelling. However, one of the aspect that was important and that has been discussed with design team is to take into account information propagation.

Due to the time, these configurations have not been implemented in projects but have been considered as important. In view to this observation, authors have started to look into different clustering techniques and related organizations that are discussed henceforth.

In this case, no constraint was initially applied on cluster size for first runs, then several maximal sizes have been introduced to test the sensitivity of the solutions (in a range from 7 to 13). It showed a great stability in the proposed configurations.

Fig. 3 shows the final clustering results for AA. Firstly, whatever the algorithm and the configuration, there is systematically a 9-actor cluster which is proposed. This cluster C 1 is not only strongly connected (high INTRA within-cluster interactions) but also connected to the rest of the network.

Secondly, two clusters are proposed as a discussion group with the actors of C 1 , for INTERconnection reasons. Indeed, the number and strength of interdependencies are far higher between C 2 and C 1 than within C 2 for instance (idem for C 3 ). Fig. 3 shows that 21 actors are in this situation, respectively 11 actors in C 2 and 10 in C 3 . These clusters C 2 and C 3 could also be implemented as working groups, but their density INTRA is judged too low.

Thirdly, a 16-actor cluster C 4 is proposed where interactions are mainly within the cluster, and not outside. This is a cluster built for INTRA-connection reason (called Intra-only on Fig. 3). The performance indicators P and P' are calculated using Eq. ( 8) and (9) for the clusters C i , i=1 to 4 and the groups G i , i=1 to 13, as summarized in Table The clustering of actors implies grouping decisions, due to the assignments of these actors.

For C 1 , there are two types of decisions: the ones when they are leaders, inviting other actors as guests; and the ones when they are guests. In the first type,we propose to split the corresponding 51 decisions associated to C 1 into two subgroups. This means two series of meetings, but subgroups contain respectively 32 and 19 decisions; so a single group could also have been judged as practically intractable in view to number of decisions and number of participants. C 4 involves a 6-decision group involving 16 actors.

The 13 remaining decisions may be considered using three strategies:

• A single group, with less consistence and the only advantage to propose less meetings to people, but knowing that their connections are lower than in the other groups,

• 13 singletons, meaning that each decision is managed independently with its affiliated actors. This increases the number of assignments but may also increase the efficiency of decision-making for each decision,

• The current organizational groups, knowing that decisions of seven groups 1, 3, 4, 5, 11, 12 and 13 are present. This means running 7 series of meetings.

With regard to these strategies, we have the choice for the 13 remaining decisions between 1, 7, and 13 groups. There are slight differences, but the performance of the overall configuration is always far better than the initial organization; hence the result is not sensitive to this choice.

The most important resulting fact is that with a 9-actor group plus occasional guests, 51 decisions (on a total of 70) can be managed in a coordinated and collaborative way; and with an extra 16-actor group, 6 additional decisions are grouped consistently. The managerial implication of implementation of one of these scenarios is discussed in Section 5.

Implication of these groups on early stage new product development management

Main challenge observed in industry is to manage these collaborative decision-making processes with regard to efficiency (especially time loss that actors have been experiencing) and coordination. The proposed approach highlights several strategies with the possibility of implementing one or more clusters, involving multiple decisions to be made with coordination. However, whatever the chosen scenario, it is always far more performant than the initial configuration AG, since the latter had the major disadvantage to assign many actors to almost every group. The constraint of disjunction between clusters really permits to focus for a given actor on less groups, reducing drastically the number of assignments and meetings compared to AG.

Comparing clustering results for AA with the initial organization AG enables interesting remarks to be done. First, cluster C 1 is very important in terms of INTRA and INTERvalues. It is constituted of important members of the project (e.g. project manager, prototype design manager, technical project manager, design process manager). However, this 9-actor group was not currently formed in the existing project organization.

Second, the other implemented cluster C 4 permits to group 16 actors with less assignments, which is a gain compared to AG. For the rest of the actors of C 2 and C 3 , it is not mandatory to make them participate to groups which would be not dense enough to deserve the effort. They are simply guests when some specific meetings require their presence, which can be more efficient and easier to coordinate.

Third, the proposed groups are composed of permanent members and temporary guests.

Members of C 1 need to participate to other decision groups (but far less than in AG) and actors outside of C 1 need to participate punctually to some C 1 meetings. The proposed decision groups corresponding to actors' clusters are as follows: a 32-decision DG 1 and a 19decision DG 2 managed by C 1 , a 6-decision DG 3 managed by people of C 4 , and a last 13decision DG 4 whose leadership has to be defined. By counting the total number of actors involved in each proposed decision group, we find the following results showed in Table 2. We believe that this approach can effectively support management of NPD projects and related teams. The proposed modeling approach can also be discussed in relation to the group organization. Initial organization stemmed from the cultural and historical development of the company. That organization is built around 13 collaborative decision-making processes that are domain oriented. However, possible matrix-based visualizations and modeling can be a support to constructively decide possible organizations. Moreover, it brings flexibility and agility to the organization with regards to NPD objectives (for example if there is a new technology integration that might be extremely risky).

Another important point is clustering and proposition of different organizations in view to different NPD objectives, like enhancing communication, or geographical proximity, etc. As the approach proposes different clustering techniques with regard to different criteria, possible team organizations can be examined at the very beginning of the project in order to adapt the organization with the objective of maximizing the possibility for project success.

Conclusion

In this paper, a methodology is introduced to propose groups of actors involved in numerous collaborative decisions. These groups are formed using an original approach combining several classical clustering algorithms. The first results show different reasons to group actors and different roles of these actors in the network structure and behavior. Particularly, a strong cluster of 9 actors has been identified. Two strategies of recommendations can be done after this analysis. Groups can be formed, for two main reasons, INTRA-and INTER-connections.

Moreover, recommendations for simplifying assignments can also be proposed, with two main strategies: reassigning people in order to have less actors involved in decision chains, and reducing redundancy (the same couple of actors assigned to the same couple of

Fig. 1 .

 1 Fig. 1. The affiliation matrices (AD, DG and AG).

  for clusters to overlap (to produce non-disjoint clusters). Furthermore, we have created a framework that allows to enter clustering parameters (forming what we call a configuration), then calculate and operate efficiently and ergonomically the input data. The solutions provided by these algorithms are automatically processed, to give local and global quality indicators of clusters. Moreover, it helps building the final solution from pieces of one or more proposed solutions to assemble the best solution corresponding to the expectations of the decision maker. We propose a 3 stage clustering process for clustering numerical DSMs.The first stage is information gathering, about input data and parameters definition. The second stage consists in running each algorithm several times with several problem configurations. Afterwards, one obtains a number of clustered solutions, with quality indicators for each solution and for each cluster in the solution. In addition, a frequency analysis is done to indicate the number of times that each couple of elements (actors in our case study) were put together in a clustered solution.

Fig. 3 .

 3 Fig. 3. The clusters for AA

  It is defined as the sum of edges for the couples of nodes where one belongs to C i and the other one belongs to C j . This represents the amount of inter-clusters interactions.

It corresponds to the non-diagonal cells of the matrix CA*AA*ACpreviously introduced in Eq. (6). For a given C i , we define the INTER(C i ) as the total INTER(C i , C j ) values for all the C j . The meaning of INTER is to compare relatively INTER and INTRA in order to determine whether actors in the cluster should be leaders (if INTRA>>INTER) or guests (if

Table 1 .

 1 . For memory, the groups G i form the existing configuration AG and the aim of the clustering is to propose an improved version AC composed of clusters C i . Comparison between proposed clusters C i and initial groups G i in terms of Two elements may be highlighted with these results. First, the initial organization is far from efficient due to multiple assignments of actors to multiple groups. They are involved in numerous meetings, but with many connections outside the groups, which is a factor of loss of time and potential loss of coordination.Second, clusters C 1 and C 4 are dense enough in terms of INTRA to justify their implementation. C 4 is less dense than C 1 in terms of P indicator, but is also quite independent from other clusters, meaning that P'(C 4 ) index is almost as good as P'(C 1 ). C 2 and C 3 are not dense enough to be created as a collaborative group. However, their INTER value with C1 is high enough to justify the integration of their actors in some C 1 meetings as guests.The question is now how to implement clusters C 1 and C 4 , and particularly what to do with the rest of the decisions and actors.

	performance indicators
	Id	INTRA	NA	P	INTER	P'
	C 1	1724	9	191.5	486	394.1
	C 2	58	11	5.3	412	12.8
	C 3	8	10	0.8	74	10.8
	C 4	126	16	7.9	25	315
	G 1	1988	21	94.7	16908	5.6
	G 2	1956	13	150.4	16976	8.8
	G 3	2056	20	102.9	16979	6.1
	G 4	1652	11	150.2	15752	9.5
	G 5	896	13	68.9	11839	5.8
	G 6	416	6	69.3	8667	8
	G 7	1490	9	165.6	15053	11
	G 8	616	6	102.7	10041	10.2
	G 9	2074	13	159.5	17212	9.3
	G 10	1594	12	132.8	15450	8.6
	G 11	40	5	8	2909	2.7
	G 12	1164	13	89.5	13442	6.7
	G 13	174	6	29	5566	5.2

Table 2 .

 2 Composition of decision groups within actors' clusters

	Decision group Nb of decisions Members	Guests
	DG1	32	9 (C1)	14
	DG2	19	9 (C1)	13
	DG3	6	16 (C4)	3