Amir Aly
email: amir.aly@ensta-paristech.fr

Adriana Tapus
email: adriana.tapus@ensta-paristech.fr

Speech to Head Gesture Mapping in Multimodal Human-Robot Interaction

Keywords: Coupled HMM, audio-video signal synchronization, signal mapping

In human-human interaction, para-verbal and nonverbal communication are naturally aligned and synchronized. The difficulty encountered during the coordination between speech and head gestures concerns the conveyed meaning, the way of performing the gesture with respect to speech characteristics, their relative temporal arrangement, and their coordinated organization in a phrasal structure of utterance. In this research, we focus on the mechanism of mapping head gestures and speech prosodic characteristics in a natural human-robot interaction. Prosody patterns and head gestures are aligned separately as a parallel multi-stream HMM model. The mapping between speech and head gestures is based on Coupled Hidden Markov Models (CHMMs), which could be seen as a collection of HMMs, one for the video stream and one for the audio stream. Experimental results with Nao robot are reported.

I. INTRODUCTION

Robots are more and more present in our daily lifes and the new trend is to make them behave more natural so as to obtain an appropriate social behavior and response. The work described in this paper presents a new methodology that allows the robot to automatically adapt its head gestural behavior to the user's profile (e.g. the user prosodic patterns) and therefore to produce a personalizable interaction. This work is based on some findings in the linguistic literature that show that head movements (e.g., nodding, turn taking system) support the verbal stream. Moreover, in human-human communication, prosody express the rhythm and intonation of speech and reflect various features of the speakers. These two communication modalities are strongly linked together and synchronized. Humans use gestures and postures as a communicative act. McNeill in [START_REF] Mcneill | Hand and mind : what gestures reveal about thought[END_REF] defines a gesture as a movement of the body synchronized with the flow of speech. The mechanism of the human natural alignment of the verbal and non-verbal characteristic patterns based on the work described in [START_REF] Eyereisen | Gestures and Speech: Psychological Investigations[END_REF] shows a direct relationship between prosody features and gestures/postures, and constitute an inspiration for our work.

Recently, there has been a growth of interest in socially intelligent robotic technologies featuring flexible and customizable behaviors. Based on the literature in linguistics and psychology that suggests that prosody and gestural kinematics are synchronous and therefore strongly linked together, we posit that is important to have a robot behavior that integrates this element. Therefore, in this paper, we describe a new methodology for speech prosody and head gesture mapping for human-robot social interaction. The gesture/prosody modeled patterns are aligned separately as a parallel multi-stream HMM model and the mapping between speech and head gestures is based on Coupled Hidden Markov Models (CHMMs). A specific gestural behavior is estimated according to the incoming voice signal's prosody of the human interacting with the robot. This permits to the robot to adapt its behavior to the user profile (e.g. here the user prosodic patterns) and therefore to produce a personalizable interaction.

To the best of our knowledge, very little research has been dedicated to this research area. An attempt is described by the authors in [START_REF] Michalowski | A dancing robot for rhythmic social interaction[END_REF] that present a robotic system that uses dance so as to explore the properties of rhythmic movement in general social interaction. Most of the existing works are related to computer graphics and interactive techniques. A general correlation between head gestures and voice prosody had been discussed in [START_REF] Munhall | Visual prosody and speech intelligibility: Head movement improves auditory speech perception[END_REF], [START_REF] Kuratate | Audio-visual synthesis of talking faces from speech production correlates[END_REF]. The emotional content of the speech can also be correlated to some bodily gestures. In [START_REF] Valbonesi | Multimodal signal analysis of prosody and hand motion: Temporal correlation of speech and gestures[END_REF], it is discussed the relation between voice prosody and hand gestures, while [START_REF] Quek | Gesture cues for conversational interaction in monocular video[END_REF] discusses the relation between the verbal and semantic content and the gesture. In [START_REF] Graf | Visual prosody: facial movements accompanying speech[END_REF], which is somehow closed to the discussed topic on this research, presents the relation between prosody changes and the orientation of the head (Euler angles). Moreover, authors in [START_REF] Sargn | Analysis of head gesture and prosody patterns for prosody-driven head-gesture animation[END_REF], proposed a mechanism for driving a head gesture from speech prosody.

Our work presents a framework for head gesture and prosody correlation for an automatic robot gesture production from interacting human user speech. The system is validated with the Nao robot in order to find out how naturalistic will be the driven head gestures from a voice test signal with respect to an interacting human speaker. The rest of the paper is organized as following: section II presents the applied algorithm for extracting the pitch contour of a voice signal; section III illustrates the detection of head poses and Euler angles; section IV describes speech and gesture temporal segmentation; section V presents the speech to head gesture coupling by using CHMMs; section VI resumes the results obtained; and finally, section VII concludes the paper.

II. PROSODIC FEATURES EXTRACTION

In human-robot interaction applications, the human voice signal can convey many messages and meanings, which should be understood appropriately by the robot in order to interact properly. Next, we describe the methodology used for pitch extraction. Talkin [START_REF] Talkin | A robust algorithm for pitch tracking[END_REF] defined the pitch as the auditory percept of tone, which is not directly measurable from a signal. Moreover, it is a nonlinear function of the signal's spectral and temporal energy distribution. Instead, another vocal characteristic, the fundamental frequency F 0, is measured as it correlates well with the perceived pitch. Voice processing systems that estimate the fundamental frequency F 0 often have 3 common processes: (1) Signal Conditioning; (2) Candidate Periods Estimation and (3) Post Processing.Signal preconditioning process is concerned by removing interfering signal components like noise and DC offset, while post processing process chooses the more likely candidate period in order to precisely estimate the fundamental frequency F 0. Talkin in [START_REF] Talkin | A robust algorithm for pitch tracking[END_REF] developed the traditional (NCC) method in order to estimate reliably the voicing periods and the fundamental frequency F 0 by considering all candidates simultaneously in a large temporal context. This methodology uses two pass normalized cross correlation (NCC) calculation for searching the fundamental frequency F 0 which reduces the overall computation load with respect to the traditional (NCC) method. The procedures of the algorithm are illustrated in Figure 1.

In this work, we choose to express the characterizing vector of the voice signal in terms of the pitch and the intensity of the signal.

III. HEAD POSE ESTIMATION

During social human-robot interaction, robots should be able to estimate the human head pose. This can help the robot to understand the human focus of attention and/or the meaning of the spoken message. The authors in [START_REF] Chutorian | Head pose estimation in computer vision: A survey[END_REF] present a survey Fig. 2: Detecting the face rectangle that contains all salient points on the different existing algorithms for head poses estimation and detection.

Our methodology used for the detection of head poses is Viola and Jones algorithm [START_REF] Viola | Robust real-time face detection[END_REF]. After extracting the head region, the eyes are detected by the valley points' detection algorithm [START_REF] Wong | A robust scheme for live detection of human faces in color images[END_REF]. After detecting the location of eyes, it is possible to detect the location of the other salient points of the face using the geometry of the face [START_REF] Wong | An efficient algorithm for human face detection and facial feature extraction under different conditions[END_REF].

For example, if the distance between the two eyes points (1&3) equals to D (see Figure 2), and point 2 is the midpoint between the eyes, then the mouth point 4 is located at a distance = 1.1D downwards from point 2.

The X-Y coordinates of the rectangle surrounding the salient points of the face (points 5, 6, 7, and 8) (see Figure 2) could be precised as following:

• The difference between the Y -coordinates of points (5&1 or 3&7) = 0.2 * 1.8D

• The difference between the X-coordinates of points (5&1 or 3&7) = 0.225 * 1.8D

After calculating the coordinates of points (5, 7), the coordinates of points [START_REF] Valbonesi | Multimodal signal analysis of prosody and hand motion: Temporal correlation of speech and gestures[END_REF][START_REF] Graf | Visual prosody: facial movements accompanying speech[END_REF] are directly calculated based on the vertical distance between points (7&8 or 5&6), which is equal to 1.8D.

One of the problems that may appear when detecting the surrounding rectangle of the facial salient points is the rotation of the head clockwise and counterclockwise (see Figure 3). Therefore, the (X, Y ) coordinates of the eyes has to be rotated first to (X -, Y -) before following the previous steps in order to precise the points of the surrounding rectangle, because the above metioned relations are valid when the eyes coordinates are in the same plane of the face (i.e., if the face is rotated, the coordinate of the eyes have also to be located in the rotated plane). The direction of rotation will be detected by calculating the slope (i.e., rotation angle θ) of the line passing by the two eyes using their (X, Y ) coordinates. The rotation of the axes is described by the following equations:

X -= Xcosθ -Y sinθ (1)
Y -= X sin θ + Y cos θ (2)
After calculating the coordinates of a face salient point in the rotated (X -, Y -) plane while the head is rotating clockwise or counterclockwise, it is important to make the inverse rotation each time to detect the location of this point in the (X, Y ) plane. The pose of the head is calculated in terms of Euler angles: Pitch, Yaw, and Roll. These angles are calculated using the previously detected 8 salient facial points and their relative positions based on the geometry and symmetry of faces (Figure 2) as following:

Y aw i = b 1 [ (P 2 x,i -C1 x,i ) + (P 2 x,i -C2 x,i ) 2D eyes 0 - (P 2 x,0 -C1 x,0 ) + (P 2 x,0 -C2 x,0 ) 2D eyes 0 ] + b 2 [ (P 4 x,i -C1 x,i ) + (P 4 x,i -C2 x,i ) 2D eyes 0 - (P 4 x,0 -C1 x,0 ) + (P 4 x,0 -C2 x,0 ) 2D eyes 0 ] (3)
P itch i = b 3 [ (P 2 y,i -C3 y,i ) + (P 2 y,i -C4 y,i ) 2D eyes 0 - (P 2 y,0 -C3 y,0 ) + (P 2 y,0 -C4 y,0 ) 2D eyes 0 ] + b 4 [ (P 4 y,i -C3 y,i ) + (P 4 y,i -C4 y,i ) 2D eyes 0 - (P 4 y,0 -C3 y,0 ) + (P 4 y,0 -C4 y,0 ) 2D eyes 0 ] (4)
where:

• P 2 x,i , P 4 x,i : the x coordinates of the midpoint between eyes, mouth point, respectively (see Figure 2), in frame i of the video. • P 2 y,i , P 4 y,i : the y coordinates of the midpoint between eyes, mouth point, respectively (see Figure 2), in frame i of the video. • P 2 x,0 , P 4 x,0 : the x coordinates of the midpoint between eyes, mouth point, respectively (see Figure 2), in frame 0 which is the reference frame in the video (1st frame). • P 2 y,0 , P 4 y,0 : the y coordinates of the midpoint between eyes, mouth point, respectively (see Figure 2), in frame 0 which is the reference frame in the video (1st frame). • C1 x,i : the x coordinates of the center point between point 5 and point 6 (Figure 2), in frame i.

• C2 x,i : the x coordinates of the center point between point 7 and point 8 (Figure 2), in frame i.

• C3 y,i : the y coordinates of the center point between point 5 and point 7 (Figure 2), in frame i.

• C4 y,i : the y coordinates of the center point between point 6 and point 8 ( The calculation of Roll angle is straightforward, it depends on the coordinates of the midpoint between eyes (point 2) in frame i with respect to the reference frame [START_REF] Yip | Pose determination of human head using one feature point based on head movement[END_REF], and it is clear that the value of Roll angle in the first reference frame equals to 0.

Roll i = tan -1 ( P 2 y,i -P 2 x,i ) -tan -1 ( P 2 y,0 -P 2 x,0 ) (5)

IV. SPEECH AND HEAD GESTURE SEGMENTATION

The mapping between speech and head gestures is done by using the Coupled Hidden Markov Models (CHMMs), which could be seen as a collection of HMMs, one for the video stream and one for the audio stream. The advantage of this model over a lot of other topologies is its ability to capture the dual influences of each stream on the other one across time (see Figure 6). In the beginning, speech and head gestures streams are aligned separately as a parallel multi-stream HMM model.

The mapping between speech and head gestures is performed in 2 main steps: (1) the first is modeling the gesture sequence and the associated voice prosody sequence (in terms of their characteristic vectors) into two separate HMMs; (2) then after training both models, a correlation between the two HMMs is necessary so as to estimate a final head gesture states sequence given a speech test signal.

The HMM structure used in analyzing gestures (and similarly voice prosody) is indicated in Figure 4. It is composed of N parallel states, where each one represents a gesture composed of M observations. The goal of the transition between states S EN D to S ST ART is to continue the transition between states from 1 to N (e.g., after performing gesture state 1, the model transfers from the transient end state to the start state to perform any gesture state from 2 to N in a sequential way and so on). In order to be able to model gestures/prosody, it is necessary to make a temporal segmentation of the video

A. Speech Temporal Segmentation

Speech is segmented as syllables presented by the states from 1 to N as indicated in Figure 4. The segmentation is performed by intersecting the inflection points (zeros crossing points of the rate of change of the curve) for both the pitch and intensity curves, beside the points that separate between the voiced and unvoiced segments of the signal (see Figure 5 for an example of pitch and intenisty curves). When comparing the two curves together, 5 different trajectory states can result [START_REF] Ringeval | Automatic intonation recognition for the prosodic assessment of language impaired children[END_REF] (see Table II).

The goal is to code each segment of the signal with its corresponding pitch-intensity trajectory class (e.g., a voice signal segment coding could be: 5, 3, 4, 2, etc.). This segmental coding is used as label for CHMM training. The next step corresponds to segmenting the voice signal with its corresponding trajectory labeling into syllables. Arai and Greenberg in [START_REF] Arai | The temporal properties of spoken japanese are similar to those of english[END_REF] defined the average duration of a syllable as 200 ms and this duration can increase or decrease according to the nature of the syllable as being short or long. Practical tests proved that within a syllable of duration varying from 180 ms to 220 ms, the average number of trajectory classes in its corresponding pitch and intensity curves is around 5. Therefore, given the voice signal with its segments coded by the corresponding pitch-intensity trajectory labels, each 5 segments of the signal will create a syllable state (from 1 to N ) and the corresponding 5 labels will be the observations M within the syllable state.

B. Gestures Temporal Segmentation

The average duration for making gestures, in general, varies between 0.1 to 2.5 seconds according to the speed and the performed gesture as being pointing or head gesture for example. In case of head gestures, the average duration of performing a gesture will be limited to 0.4 seconds [START_REF] Nickel | Real-time recognition of 3d-pointing gestures for human-machine-interaction[END_REF][START_REF] Moubayed | Effects of visual prominence cues on speech intelligibility[END_REF]. In our case, the camera used to capture the gestures had the ability of capturing 30 frames/second, and therefore we can estimate to 12 frames the average number of frames sufficient to characterize a gesture.

Similarly to the speech temporal segmentation (see Section IV-A), gesture temporal segmentation is performed by comparing the 9 trajectory classes according to the sinusoidal evolution of the extracted angles curves. However, the mechanical characteristics of our platform (NAO robot) are limited only to pitch and yaw movements, therefore introducing only 5 trajectory classes (see Table III). In the context of the CHMM model each group of 12 frames will form a complete gesture state from 1 to N , and the corresponding coding labels will constitute the observations within the gesture state.

V. SPEECH TO HEAD GESTURE COUPLING

A typical CHMM structure is shown in Figure 6, where the circles present the discrete hidden nodes/states while the rectangles present the observable continuous nodes/states, which contain the observation sequences of voice and gestures characteristics. According to the sequential nature of gestures and speech, the CHMM structure is of type lag-1 in which couple (backbone) nodes at time t are conditioned on those at time t -1 [START_REF] Rabiner | A tutorial on hidden markov models and selected applications in speech recognition[END_REF][START_REF] Rezek | Coupled hidden markov models for biosignal interaction modelling[END_REF][START_REF] Rezek | Estimation of coupled hidden markov models with application to biosignal interaction modelling[END_REF]. A CHMM model λ C is defined by the following parameters:

π C 0 (i) = P (q C 1 = S i ) (6)
a C i|j,k = P (q C t = S i |q audio t-1 = S j , q video t-1 = S k ) (7) b
C t (i) = P (O C t |q C t = S i ) (8)
where C ∈ {audio, video} denotes the audio and visual channels respectively, and q C t is the state of the coupling node in the c th stream at time t [START_REF] Nean | A coupled hidden markov model for audio-visual speech recognition[END_REF][START_REF] Liang | Speaker independent audio-visual continuous speech recognition[END_REF].

The training of this model is based on the maximum likelihood form of the expectation maximization (EM) algorithm. Supposing there are 2 observable sequences of the audio and video states O = {A 1..N , B 1..N } where A 1..N = {a 1 , • • • , a N } is the set of observable states in the first audio sequence, and similarly B 1..N = {b 1 , • • • , b N } is the set of observable states in the second visual sequence, and S = {X 1..N , Y 1..N } is the set of states of the couple nodes at the first audio chain and the second visual chain respectively [START_REF] Rezek | Coupled hidden markov models for biosignal interaction modelling[END_REF][START_REF] Rezek | Estimation of coupled hidden markov models with application to biosignal interaction modelling[END_REF]. The expectation maximization algorithm finds the maximum likelihood estimates of the model parameters by maximizing the following function [START_REF] Rezek | Estimation of coupled hidden markov models with application to biosignal interaction modelling[END_REF]:

f (λ C ) = P (X 1 )P (Y 1 ) T ∏ t=1 P (A t |X t )P (B t |Y t ) (9)
P (X t+1 |X t , Y t )P (Y t+1 |X t , Y t ), 1 ≤ T ≤ N
where:

• P (X 1 ) and P (Y 1 ) are the prior probabilities of the audio and video chains respectively The training of the CHMM differs from the standard HMM in the expectation step (E) while they are both identical in the maximization step (M) which tries to maximize equation 9 in terms of the expected parameters [START_REF] Penny | Gaussian observation hidden markov models for eeg analysis[END_REF].The expectation step of the CHMM is defined in terms of the forward and backward recursion. For the forward recursion we define a variable for the audio and video chains at t = 1:

α audio t=1 = P (A 1 |X 1 )P (X 1 ) (10)
α video t=1 = P (B 1 |Y 1 )P (Y 1 ) (11)
Then the variable α is calculated incrementally at any arbitrary moment t as follows:

α audio t+1 = P (A t+1 |X t+1 ) ∫ ∫ α audio t α video t P (X t+1 |X t , Y t )dX t dY t ( 12
)
α video t+1 = P (B t+1 |Y t+1 ) ∫ ∫ α audio t α video t P (Y t+1 |X t , Y t )dX t dY t (13)
Meanwhile, for the backwards direction there is no split in the calculated recursions which can be expressed as follows:

β audio,video t+1 = P (O N t+1 |S t ) = ∫ ∫ P (A N t+1 , B N t+1 |X t+1 , Y t+1 ) P (X t+1 , Y t+1 |X t , Y t )dX t+1 dY t+1 (14)
After combining both forward and backwards recursion parameters, an audio signal will be tested on the trained model, generating a synthesized equivalent gesture that most likely fit the model. The generated gesture sequence is determined when the change in the likelihood is below a threshold.

VI. EXPERIMENTAL RESULTS

The experimental testbed used in this study is the humanoid robot Nao developed by Aldebaran Robotics. For the training and testing, we used the MVGL-MASAL gesture-speech Table IV depicts the confussion matrix between the original and synthesized gesture labels trajectories. The confussion matrix reveals that the trajectory state 5 in which there would be no change in the Y aw and P itch angles is the dominant trajectory class. This can be a result of the smoothing processes and/or of the precision of Euler angles extracting algorithm; however this will not cause unnaturaleness when the robot and the human are interacting in long conversations.

After calculating the score of similarity between the trajectory labels of the original and the synthesized signals, it is important to generate the corresponding Y aw and P itch curves for the head motion and compare them to the original curves by calculating the total average root mean square (RMS) error between the corresponding curves points. The RMS errors found between the generated Y aw and P itch curves with respect to the original curves are 10% and 12% respectively.

In fact, the obtained score 62% and the RMS errors between the original and the synthesized curves can be considered a reasonable result, because the duration and the surrounding environement conditions of the test video and the training videos set were similar. Also, the speaker's tonality in all training and test videos were similar. However, we don't know yet the score we will obtain in real applications where the robot will be tested under different condiditions. The performed head gestures could differ in the amplitude or the direction from one person to another without hindering the transfer of the meaning of the gesture message between interacting humans and similarly, between the interacting robot and human. Figures 7 and8 show a comparison between a part of the original and synthesized pitch and yaw curves (after being smoothed by a median filter) of the test video from the database. A video of the speech-gesture mapping system with Nao

VII. CONCLUSIONS

This research focuses on synthesizing head gestures based on speech characteristics (e.g., pitch and intensity of the signal). Our mapping system is based on the Coupled Hidden Markov Model (CHMM) that tries to find a coupling joint between audio and visual sequences. The audio sequence is composed of parallel states presenting the syllables and each syllable is composed of a specific number of observations (M=5, in our case). Meanwhile, the video sequence has the same parallel construction where the states present the gestures and each state is composed of another specific number of observations determined experimentally (M=12, in our case).

After training the CHMM on audio-visual sequences from a database, and when a test audio signal is generated, the system tries to find a corresponding sequence of gestures based on its own experience learnt during the training phase. The generated gesture sequence is the sequence that achieves the maximum likelihood estimation with the speech test signal. Our system shows a score of 62%, which measures the similarity between the original gesture sequence labels and the synthesized gesture sequence labels, over a test video of 8 minutes. This can be considered a good score. The proposed system is able to generate appropriate robot head gesture from speech input, which allows it to produce an automatic natural robot behavior that is almost completely absent from present-day human-robot interactions. Further work will focus on creating a triadic alignment between the speech, head gestures, and hand gestures in different human-robot interactional contexts that will allow the robot to interact naturally under different conditions.

Fig. 1 :

 1 Fig. 1: Pitch Tracking

Fig. 3 :

 3 Fig. 3: Tracking Salient Face Details in Different Cases of Rotation: (a) Rotation clockwise; (b) Rotation counterclockwise

Frame1

Figure 2 )

 2 , in frame i. The regression values b 1 , b 2 , b 3 , and b 4 are constants throughout all the video frames. They are calculated by fixing the absolute values of Y aw and P itch angles in the second and third frames (according to empirical test) as shown in Table I. The substitution of the second and third values of P itch and Y aw in the equations 3 and 4, leads directly to the computation of the values of the constants b 1 , b 2 , b 3 , and b 4 .

Fig. 4 :

 4 Fig. 4: HMM structure for gesture and prosody analysis Trajectory Class Trajectory State 1 pitch ↑ & intensity ↑ 2 pitch ↑ & intensity ↓ 3 pitch ↓ & intensity ↑ 4 pitch ↓ & intensity ↓ 5 Unvoiced segment

Fig. 6 :

 6 Fig. 6: Coupled Hidden Markov Model CHMM lag-1 Structure

Fig. 5 :

 5 Fig. 5: Speech, Pitch and Intensity Curves (The red parts in the voice signal are the unvoiced parts, while blue parts are the voiced parts of the signal. The black points depict the inflection points of the signal, while green points represent the separating points between the unvoiced and the voiced segments.)

  Confussion matrix of the original and synthesized trajectories' classes Turkish database[START_REF] Sargn | Analysis of head gesture and prosody patterns for prosody-driven head-gesture animation[END_REF]. The database is composed of 4 videos of different durations that go from 6 to 8 minutes. It contains the audiovisual information of different subjects instructed to tell stories to children audience. We use one part of the database for the training of the models and the other part for the testing. The audio signals are extracted and then they are processed in order to extract the relevant prosodic characteristics. The proposed speech to gesture mapping methodology was tested on the database using cross validation algorithm. The was trained on the audio/visual sequences of 3 videos from the database, and then tested on the audio sequence of the 4th video. The corresponding generated gestures are compared to the natural gesture sequence in the video of test and an average score of 62% was found in terms of the similarity of trajectory classes.

Fig. 7 :Fig. 8 :

 78 Fig. 7: View of the original (blue curve) and synthesized (red curve) Pitch angles of a part of the test video

TABLE I :

 I Yaw and Pitch Initial Angles (Frames 1-3) Used for Calculation of Regression Values

TABLE II :

 II

Voice Signal Segmentation Labels content to detect the M number of observations in each state and the total number of states N .

TABLE III :

 III Gesture Segmentation Labels


Vidal Ludovic-Alexandre

Marle Franck

Bocquet Jean-Claude

Measuring project complexity using the Analytic Hierarchy Process

Keywords: Project, Complexity, Evaluation, Multi-criteria, Analytic Hierarchy Process (AHP)

Project complexity is ever growing and needs to be understood and measured better to assist modern project management. The overall ambition of this paper is therefore to define a measure of project complexity in order to assist decision-making. A synthesised literature review on existing complexity measures is proposed in order to highlight their limitations. Then, we identify the multiple aspects of project complexity. We then propose a multi-criteria approach to project complexity evaluation, through the use of the Analytic Hierarchy Process. In the end, it permits to define a relative project complexity measure. Complexity scales and subscales are defined in order to highlight the most complex alternatives and their principal sources of complexity within the set of criteria and sub-criteria which exist in the hierarchical structure. Finally, a case study within a start-up firm in the entertainment industry is performed. Conclusions and research perspectives are given in the end.

Introduction

A project is a temporary and unique endeavour undertaken to deliver a result. This result is always a change in the organization, whatever it is in its processes, performance, products or services. This transformation consists then in a gap between a start and a final state. Time and resources are consumed to produce results, which may be deliverables and/or performance improvement and/or resource improvement (skills, knowledge). Each project is unique because there is always at least one of the following parameters that changes: targets, resources and environment. This makes making project management an even more complex process.

For all practical purposes, lots of studies have been done, based on statistical calculations or surveys. Limits and lacks have been detected in research as well as in industry about the project predictability. Namely, usual parameters (time, cost and quality) are clearly not sufficient to describe properly the complete situation at a given time. As a whole, the conclusion of these studies is that current methods have shown their limits, since they cannot face anymore the stakes of ever growing project complexity, which results in damages or failures for the projects [START_REF] Williams | The need for new paradigms of project complexity[END_REF], [START_REF] Whitty | And then came Complex Project Management[END_REF]. In other words, project ever growing complexity is an ever growing source of project risks. Identifying existing project complexity sources and levels of project complexity has thus become a crucial issue in order to assist modern project management. The main objective is then to build up a project complexity index so it can be used as an indicator, notably when facing the issue of project selection in multi-project environments.

Measuring project complexity: a literature review 2.1. Project complexity definition

Research works on the concept of complexity have been conducted for years. The difficulty is that there is actually a lack of consensus on what project complexity really is. As Sinha et al. [START_REF] Sinha | A complexity index for the design process -International Conference on Engineering Design[END_REF] underline it, "there is no single concept of complexity that can adequately capture our intuitive notion of what the word ought to mean". Complexity can be understood in different ways, not only in different fields, but has also different connotations within the same field (Morel on some additional works (Baccarini, 1999), [START_REF] Edmonds | Syntactic measures of complexity -Thesis of the University of Manchester for the degree of doctor of philosophy in the faculty of arts[END_REF], (Marle, 2001), [START_REF] Austin | Modelling and managing project complexity[END_REF], [START_REF] Vidal | Project complexity understanding and modelling to assist project management -PMI Research Conference[END_REF]: "project complexity is the property of a project which makes it difficult to understand, foresee and keep under control its overall behaviour, even when given reasonably complete information about the project system."

Existing project complexity measures

As far as this research work is concerned, a literature review on existing (project) complexity measures was performed. A total of 42 measures were listed, notably thanks to the works of Edmonds [START_REF] Edmonds | Syntactic measures of complexity -Thesis of the University of Manchester for the degree of doctor of philosophy in the faculty of arts[END_REF], [START_REF] Latva-Koivisto | Finding a complexity measure for business process models -Research report Helsinki University of Technology, Systems Analysis[END_REF] or [START_REF] Nassar | Developing a complexity measure for schedules[END_REF]. If interested, one should directly refer to them for more information on complexity measures and formulations. As a whole, there are basically three kinds of project complexity measures which can be found in the literature.

The first group gathers measures which correspond to the computational complexity of some project management issues, such as the sequencing and scheduling problem [START_REF] Akileswaran | Complexity of the project sequencing problem[END_REF].

The second group gathers measures which are related to a model of the project structure as a graph (task graph, organisation graph, etc…). For instance, we can think of:



The Coefficient of Network Complexity (CNC) defined by Kaimann (Kaimann, 1974) applies to both PERT and precedence networks. They can also apply to any model of a project as a graph. In the case of PERT networks, the CNC is equal to the quotient of activities squared divided by events.



The cyclomatic number defined by Temperley [START_REF] Temperley | Graph Theory and Applications -England[END_REF] is given in equation (1). S is the cyclomatic number, A is the number of arcs, N is the number of nodes.

S = A -N + 1

(1)



Arguing that complexity measures such as CNC are imperfect since they take redundant arcs into account and therefore show that the system is more complex than it actually is, [START_REF] Nassar | Developing a complexity measure for schedules[END_REF] define the following measure:

Cn = 100 × (Log(a/(n-1))/Log[(n 2 -1)/4(n-1)])% if n is odd Cn = 100 × (Log(a/(n-1))/Log[n 2 /4(n-1)])% if n is even (2)
The third group gathers more holistic measures such as systems thinking oriented measures or informational measures. For instance, we can think of:



The traditional static entropic measurement of complexity by the Shannon information [START_REF] Shannon | The mathematical theory of communication -Bell Systems[END_REF] Sha = -Σ log2 (p(ni))

(3)  Even though not calculated in the book, complexity indexes may be deduced from the areas of spider charts used by [START_REF] Haas | Managing complex projects: a new model -Vienna, VA : Management Concepts[END_REF], who described project complexity using a complex systems thinking approach to identify several aspects of complexity (such as team composition and performance, cost/duration or political sensitivity/multiple stakeholders).

This approach is particularly adapted to the issue of project selection.

Limits of existing project complexity measures

Existing measures have shown their limits for several reasons. First, some limits have been highlighted about the reliability of such measures. Second, these measures are often non intuitive for the final users and thus give results which are difficult to communicate on. Finally, these measures mainly refer to a model of the project system.

For the first group of computational complexity measure, the main drawback is that they do not focus on the complexity of the project system in itself. Indeed, with such measures, it can only be assessed given the bias of a specific issue like scheduling.

For the second group of graph-based complexity measures, some of them lack of reliability since counterexamples can be found. For instance, some graphs and networks were sharing the same CNC but were very different considering their easiness to be managed. One of the main reasons for this lack of reliability is that these measures refer to a single aspect of (project) complexity, essentially in terms of interdependencies. Moreover, measures such as the CNC, the cyclomatic number or the one proposed by Nassar and Hegab refer in essence to an existing network or graph. Such graphs are specific models of the project system, which restrict the view and understanding of project complexity. For instance, a project can be modelled thanks to different WBS (Work Breakdown Structure), PERT networks or Gantt charts, depending on the detail level, willingness of the project manager, etc… Applying such measures to these kinds of elementary models of the project systems cannot properly account for a measure of project complexity since they are in essence relative to the model. This is less the case with approaches based on systems thinking such as the one proposed by Haas.

However, for the last group of holistic complexity measures, such measures are sometimes difficult to calculate for non-skilled users, which make it all the more complex to perform and analyse them. Moreover, in the case of the Shannon number, the mathematical formulation does not permit a reference to real project complexity factors. Both the identification of important complexity sources and possible actions for complexity handling/reduction are not facilitated.

As a whole, in order to overcome the limits of existing measures, this paper aims at defining a systems thinking oriented index, which is as far as possible:

 Reliable, meaning the user can be confident with the measure.

 Intuitive and user-friendly, meaning it should be easily computed and implemented, and that users must understand why it assesses project complexity.

 Independent of the project models, so that the measure is an evaluation of project complexity and not an evaluation of the complexity of a given project model.

 Able to highlight project complexity sources when building up the measure, so that the user can analyse more properly project complexity and thus make his decisions with a better vision of the problem.

Research methodology

In order to do so, we propose to carry out the following methodology:

 Performing a broad state of the art to understand the multiple aspects of project complexity through the construction of a project complexity framework and refining it thanks to an international Delphi study (Part 4).

 Using the Analytic Hierarchy Process to build up the corresponding project complexity index (Part 5).

 Testing the whole approach on a case study to draw conclusions and identify future research perspectives (Parts 6 and 7).

4. Building up a project complexity framework using the Delphi methodology

Identifying complexity factors using a systems-thinking approach

This paper claims for the use of systems thinking to encompass all the aspects of projects.

Basically, our systems thinking-based approach is notably based on or at least consistent with the works of [START_REF] Boulding | General Systems Theory[END_REF], (Von Bertalanffy, 1972), [START_REF] Simon | The Sciences of the artificial -Cambridge The[END_REF], (Le Moigne, 1990), [START_REF] Penalva | La modélisation par les systèmes en situations complexes -Thèse en sciences de Parix XI Orsay préparée au Laboratoire d'Informatique appliquée de Commissariat à l'Energie Atomique[END_REF], [START_REF] Williams | Modelling complex projects[END_REF], [START_REF] Heylighen | Complexity and Philosophy -Complexity[END_REF], [START_REF] Bocquet | How to build a design product and its end-product system? An original approach called SCOS[END_REF]. This paper considers the following definition of a system: "a system can be defined as an object, which, in a given environment, aims at reaching some objectives (teleological aspect) by doing an activity (functional aspect) while its internal structure (ontological aspect) evolves through time (genetic aspect) without losing its own identity" [START_REF] Vidal | Modelling project complexity -International Conference on Engineering Design[END_REF]. According to this definition, a project can be considered as a system. As a whole, a very large number of possible project complexity factors can be identified, when structuring a literature review using the four aspects of systems thinking (which permits that the identification of these factors is quite robust). This list was constituted, knowing that some factors were gathered under a common denomination in the end. The final list is composed of 70 project complexity factors.

Structuring the framework

It appeared that in order to make more sense both for academic and industrial practitioners, instead of using the 4 aspects of systems thinking to structure the framework, another typology was needed. That is why we used the following structure for the framework:

 First, there is a categorization into two columns, splitting factors using Baccarini's traditional dichotomy (technical / organizational) (Baccarini, 1999)  Then, there is a categorization into four rows, using four families of factors, the denomination of which was commonly used in fieldwork and in the literature (since these denominations have widely been used in research articles for instance): size, variety, interdependencies and context-dependence.

This framework aims at being a reference for any practitioner to identify and characterize some aspects of its project complexity, so that he can understand more efficiently the stakes of its project complexity management.

Refining the framework using an international Delphi study

With this framework, project complexity appears as a multiple aspects or multiple criteria characteristic of a project. Measuring it should thus require the consideration of these multiple aspects thanks to the proper use of a pertinent multi-criteria decision-making methodology now that the multiple aspects of project complexity have been identified. However, according to Baker and al. [START_REF] Baker | Guidebook to Decision-Making Methods[END_REF], criteria used in multi-criteria decision making methods should be:

 able to discriminate among the alternatives and to support the comparison of the performance of the alternatives  complete to include all goals,  operational and meaningful,

 non-redundant,  few in number.
Here, the multiple aspects are not few in number and implementing it could quickly be impossible in fieldwork. That is why we called for a refining of this framework thanks to an international Delphi study which was carried out on academic and industrial practitioners. This Delphi study makes the point of ongoing publications in which detailed results of this study will be found.

However, a few words can be said about it thanks to Figure 1, which draws a synthesis of our approach and results. The Delphi survey was conducted thanks to BCC electronic mail sent to 19 international academic and 19 industrial experts in project management, who were varied in terms of geographic location and gender.

The Delphi survey group size appears to be very different in the literature. However, it is often recommended to have a group between 9 and 18 participants in order to draw some relevant conclusions and avoid at the same time difficulty to reach consensus among experts. We argue, such as in (Okoli and Pawlowski, 2004), that an experts categorization should be made properly The questionnaire was divided into eight sections, following the structure of the project complexity framework. Participants were asked to evaluate the contribution of each factor to project complexity on 5-level Likert scales. We left them the possibility not to answer, to answer they did not know, or to add any other complexity factor which would have been missing.

Figure 1. Overall process during the international Delphi study

Of the first 18 proposed project complexity drivers after the panelists' evaluation (considered as essential), only 2 of them (11.1%) are of a technological type (Figure 2). Organizational complexity thus seems to be the greatest source of complexity for projects and project management today.

Project managers should thus focus on organizational issues when tackling and dealing with complexity. This also appears to be legitimate when discussing with industrials facing their project day-to-day life. Of these first 18 project complexity drivers, 11 of them belong to the family of project interdependencies (61.1%), making it the most contributive family of project complexity drivers, before context-dependence (16,7%), variety (16,7%) and size (5.6%). This also appears to be consistent with former works of the academic literature and with the industrials' feelings about complexity in discussions. This is also enlightened by the number of tools and works that have been developed to try to better catch project interactions and interdependencies, such as interactions model [START_REF] Marle | Modèle d'informations et méthodes pour aider à la prise de décision en management de projets -Thèse en Génie Industriel de l[END_REF], or Design Structure Matrices (Steward, 1981).

Figure 2. Synthesis of Delphi results for each criterion, sorted by decreasing mean value

The 18 essential factors constitute the refined project complexity framework, which can be found hereunder in Figure 3.

Figure 3. Refined project complexity framework

This refined project complexity framework constitutes the basis of project complexity evaluation problem, which we propose to address thanks to the use of a proper multi-criteria decision methodology.

5. Using the Analytic Hierarchy Process (AHP) to assess project complexity

Choosing the AHP

In general, decision-making is the study of identifying and choosing alternatives based on the values and preferences of the decision-maker. Making a decision implies that some alternatives are to be considered and that one chooses the alternative(s) that possibly best fits with the goals, objectives, desires and values of the problem. Choosing the most suitable multi-criteria methodology is in itself a multi-criteria choice.

From [START_REF] Gershon | Model choice in multi-objective decision making in natural resource systems[END_REF], [START_REF] Deason | A multi-objective decision support system for water project portfolio selection[END_REF] and [START_REF] Tecle | Choice of multicriteria decision making techniques for watershed management[END_REF], six criteria and 30 methods are identified for this problem. Indeed, regarding the issue of project complexity evaluation, preferences are given to intuitiveness, pertinence, notoriety and adequacy to the project management context.

Moreover, the method should be able not only to rank the alternatives, but also to propose scorings in order to facilitate the transition to the calculation of a complexity index.

Simultaneously, the literature study enabled to assess each of the 30 considered methods according to the criteria. Finally, a single synthesis criteria method, the Analytic Hierarchy Process (AHP), obtains the best score. This can notably be understood since [START_REF] Al-Harbi | Application of the AHP in project management[END_REF] showed that AHP has numerous applications in the project management field [START_REF] Bea | Multicriteria decision making applied to project outsourcing -International Conference on Engineering Design[END_REF][START_REF] Gourc | Vers un modèle conceptuel du risque pour le pilotage et la conduite des activités de biens et de services[END_REF]. For instance, [START_REF] Ahmad | Software Project Management Tools: Making a Practical Decision Using AHP -SEW[END_REF] used the AHP to select the most appropriate software project management tool. The authors argue that "the AHP provides a flexible, systematic, and repeatable evaluation procedure that can easily be understood by the decision-maker in selecting the appropriate software project management tool". Finally, the reader should note that the AHP also has many applications in different contexts which all underline the user-friendliness and intuitiveness of the methodology [START_REF] Lin | Improving AHP for construction with an adaptive AHP approach -Automation in Construction[END_REF][START_REF] Gerdsri | Applying the Analytic Hierarchy Process (AHP) to build a strategic framework for technology roadmapping[END_REF]Chiu and Chen, 2008).

The Analytic Hierarchy Process (AHP)

The Analytic Hierarchy Process (AHP) was developed by Thomas Saaty [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF], [START_REF] Saaty | The Analytic Hierarchy Process: Planning, Priority Setting[END_REF], [START_REF] Saaty | How to make a decision: the analytic hierarchy process[END_REF]. It is a multi-criteria decision-making method which permits the relative assessment and prioritization of alternatives. The AHP is based on the use of pair-wise comparisons, which lead to the elaboration of a ratio scale. The AHP uses a model of the decision problem as a hierarchy, consisting of an overall goal, a group of alternatives, and a group of criteria which link the alternatives to the goal.

With the refined project complexity framework, an AHP hierarchical structure can be built. The overall goal (objective) is the ranking of alternatives (projects or project scenarios or areas)

according to their complexity level, meaning the AHP score which is obtained in the end catches and aggregates the importance of complexity factors on each alternative. First level criteria (intermediary goals) correspond to the four groups of project complexity factors (project size, project variety, project interdependencies and project context-dependence). Second level criteria then correspond to the factors of the refined framework. Moreover, if the structure remains stable, rejected criteria after the Delphi study or new criteria can be added, or existing criteria can be omitted to better cling to a specific context.

Proposing a project complexity index

Calculations are then performed thanks to this hierarchical structure (for more detail, see [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF] or [START_REF] Saaty | The Analytic Hierarchy Process: Planning, Priority Setting[END_REF]. A global score is obtained for each alternative, this score underlining the relative value of the goal (here complexity) within the set of alternatives studied. Given this score, a relative measure of project complexity is proposed. Let S(i) be the priority score of alternative Ai obtained thanks to AHP calculations (0 ≤ S(i) ≤ 1). We propose that the relative complexity of alternative Ai, given the specific context of the set of alternatives, can be expressed as the following ratio:

1 0 )) ( max( ) (     i i CI i S i S CI
A relative project complexity scale between 0 and 1 can be built thanks to this approach (this index permits to classify projects / project scenarios / project areas according to their global score regarding the main project complexity sources). Subscales can then be defined in the same manner to focus on specific aspects of project complexity and highlight how a project is complex regarding interdependencies or context for instance.

Case study 6.1. Introduction

The case study takes place within a start-up firm, the main activity of which is the production of stage musicals in France. Staging musicals or theatre plays are definitely projects, as underlined by [START_REF] Lehner | The staging model: The contribution of classical theatre directors to project management in development contexts[END_REF]: it refers to all artistic, technical and organizational processes which permit to stage a musical. Project start corresponds to the idea and choice of the show to stage. The methodology and measure proposed in this research work proved to be helpful in this case study. First, as shown in Figure 4, a first table was built to analyse the situation of the firm regarding project complexity. The relative weight of each sub-criterion can be evaluated, which gives information on where projects are likely to be more complex within the firm. Project managers should pay particular attention to the project complexity factors which get the best relative score (last column in Figure 4). On the contrary, some aspects of project complexity (low scores) may potentially be neglected at a first sight. This piece of information thus permits one to concentrate more efficiently on the principal factors of project complexity under a given firm and project environment.

In the end, final results are obtained and permit to realise a ranking of projects according to a complexity scale / index (from 0 to 1), as shown on Figure 5. It can be noted that two projects (P6 and P1) appear to be much more complex than the others, then Project 2 appears to be significantly more complex than the other ones.

Figure 5. Relative project complexity index in the case study First, the existence of a numerical relative evaluation of project complexity within a project portfolio appears to be promising since it permits to know which projects are to be the most complex ones, but also how complex projects are. Moreover, this global ranking according to the relative project complexity index we propose is all the more interesting in this case study since the employees who were interviewed had made an a priori ranking which was different. In that a priori ranking, P3 was ranked second and P7 was ranked third, whereas P6 was only placed fourth. With this numerical assessment of complexity and this ranking given, discussions were held with the participants, and communication around the notion of complexity was facilitated.

They started to share their experience on complexity factors and realized that the difference with the a priori ranking they had done was mainly due to some communication and psychological barriers they had. For instance, P7 was a priori ranked by them third because the majority of them did not know where to find the skills and competence for the design of a specific special effect. Four of them had thus ranked this project as one of the most complex ones (two of them had even ranked it as the most complex). But when performing the pair-wise comparisons, the fifth employee, who had ranked a priori P7 as the less complex, said that he had already worked with such special effects and knew who could design them easily. The others changed their minds.

Such example in our case study proves both the necessity to facilitate and promote communication in order to manage complex projects more efficiently and the benefits obtained with the project complexity refined framework and index we propose.

Finally, before performing sensitivity analyses (which make the point of ongoing publications), we claim for the use of specific numerical and graphical comparisons of projects which obtain close scores. For instance, in this case, project 6 obtains a global score of 0.308 when project 1 obtains a score of 0.292, which makes in the end a small difference (around 5%). In order to assist the decision-makers with their decisions, a closer look is to be done over the two projects. When realizing that the score of interdependencies (the main factor) is 0.185 for project 6 and 0.142 for project 1 (difference of around 25%), the people who were interviewed in this case study definitely evaluated project 6 more complex that project 1. Indeed, when analysing closer why such a difference was obtained, the participants underlined notably a greater specifications interdependence and interdependence of information systems for project 6. These specific interdependencies seemed all the more difficult to handle for the participants, which led them to the conclusion to reject project 6 at the time of this study. More precise comparisons can even be performed when descending to the level of sub-criteria and comparing projects on 0 to 1 relative subscales.

In the end, the team chose to launch project 1, which appeared to be a comprise between different factors: complexity, expected returns (financial, image, experience,…), etc... This practical conclusion about this case study underlines the fact that the project complexity measure we propose can notably be in the end one of the criteria used in the project selection problem within a portfolio. As a whole, the measure permitted both to highlight some important complexity sources (which underlines how it is helpful to understand better project complexity) and to facilitate communication around these notions, which was not done or not properly done before. We thus hope that, more than a criteria in the problem of project selection, this measure is also a tool to promote discussions about complexity due to a better understanding and visualisation of it.

Conclusions and perspectives

As a whole, this paper elaborates an AHP-based methodology and measure to evaluate relative project complexity. The works proposed here answer the problem which was set after the literature review on existing (project) complexity measures. Indeed, as shown theoretically and validated with a first case study, the project complexity index proposed here permits to overcome to a great extent the limits of existing ones as it is:

 Reliable, since the final users are confident with the results, measures and scales which are proposed. During the case study, no numerical result was ever challenged by the participants of the case study. Moreover, no other complexity factor was suggested by the participants even though they had the opportunity do so. On the contrary, the results permitted to identify more precisely project complexity sources which were consciously or unconsciously felt without being clearly mentioned or stated.

 Intuitive and user-friendly. The users understand the construction of the measure and scales, and why they do measure project complexity or the level of complexity regarding a given criterion, sub-criteria, or set of (sub-)criteria. This results in the end in a facilitation of communication on project complexity and project complexity factors.

 Globally independent of the project models which are used for project management. Indeed, no reference to project management tools or models (Work Breakdown Structure, PERT networks, Gantt charts, risk lists, etc…) was ever made during the construction of the measure or during the case study. However, the refined project complexity framework is the basis of the measure. One should not forget that this framework is in essence a specific model, not of a project, but of what project complexity stands for. As a consequence, the measure which is proposed refers theoretically to this project complexity model. However, the final user is free to change some aspects (project complexity factors) in the AHP hierarchical structure proposed in this study.

 Able to highlight project complexity sources when building up the global complexity scale and the subscales (as one can understand where the global score of each alternative comes from).

However, some limitations do appear in this work and offer perspectives for future research on project complexity evaluation.

 First, the case study which was carried out offers interesting insights on our approach (both for the obtained results, which helped the decision-makers in the firm, and for processes which were followed to implement this approach). However, its context was somewhat quite specific due to the sector of this firm. In particular, there was a low project management maturity level in the firm, with very few project management processes formalised, which implied some difficulties or hesitations when performing some pair-wise comparisons. This led us to think of two improvements. The first one is theoretical and consists in the introduction of fuzzy numbers (fuzzy-AHP for instance). The second one is more practical and consists in performing another case study to test the robustness of our approach in other contexts. Indeed, we aim at conducting a case study in a firm which has more background in project management (for instance in a civil engineering or automotive firm).

 A second observation is that the Analytic Hierarchy Process has received some criticisms on the fact that rankings can vary when adding or subtracting an alternative to the set of alternatives on which the study is performed [START_REF] Holder | Some comments on the analytic hierarchy process[END_REF]. We thus recommend the users to give specific attention to the step when the set of alternatives to be compared is selected. For instance, projects on which the final users have too few information or data may not be selected first as the quality of pair-wise comparisons may be considerably reduced. Finally, even with

  before undertaking the Delphi survey in order to build up the most representative panel. As for them, Skulmoski et al. require different aspects for the participants to be selected in the Delphi survey panel (Skulmoski et al., 2007):  Sufficient knowledge and experience about the survey issues,  Capacity, willingness and time to participate,  Good communication skills.Our prospective panel was then constituted of 38 experts, 19 of them being industrial practitioners (identified notably thanks to professional or scholar social networks) and 19 being academics (identified notably thanks to their publications), and at the same time 19 being men, 19 being women. The questionnaire was introduced by a page explaining the overall purpose and structure of the survey as well as the experts anonymity conditions at each stage of the study.

Figure 4 .

 4 Figure 4. Overall criteria and sub-criteria weights: project complexity factors comparison

Figure 1 .

 1 Figure 1. Overall process during the international Delphi study

Figure 2 .

 2 Figure 2. Synthesis of Delphi results for each criterion, sorted by decreasing mean value

FigureFigure 4 .Figure 5 .

 45 Figure 3. Refined project complexity framework

[START_REF] Morel | Through the looking glass of complexity : the dynamics of organizations as adaptive and evolving systems[END_REF]. As for us, we propose the following definition for project complexity based

Project end is generally considered as the first performance or the first week of performances (the activities which follow are often considered as the core day-to-day running activities of a theatre production firm). In the case studied, the firm has a portfolio of 7 projects of musicals to be staged in 2010-2012. The managers of the firm, who have both artistic and industrial backgrounds, recently launched their activity. They are assisted by several possible investors and partners.

They wonder which show(s) they should produce first. In this case, the achievement of the first performance is considered as part of the project. Project 1 is the production of a French adaptation of a Broadway musical, whereas projects 2 to 7 are original creations. Projects 1, 2, 6 and 7 include some special effects (notably pyrotechnical ones for some of them). Projects 1 to 6 require detailed work on costume and set design (notably with advanced mechanical structures and machinery for some of them).

As for them and as for us, project complexity appears here as one of the criteria which should be considered before making a decision on this issue. Other criteria may notably be linked with the overall project performance regarding the values creation processes of the project (notably in terms of profit, image, etc…). A global project selection process may then be defined after this study in order to include all these factors in a multi-criteria approach to select the best project.

We focus only on project complexity evaluation for the moment. Currently, in the context of this firm, the importance of project complexity as a factor for selection is all the more true than the lack of experience in start-up firms implies even greater difficulties to properly handle project complexity. As a consequence, a proper evaluation of relative project complexity appears to be really necessary.

Results

We carried out our research thanks to interviews of some possible future project team members (5 participants) following the AHP evaluation process, given our hierarchical structure. These people were asked to perform an a priori ranking of the projects in terms of complexity. This a priori ranking was necessary to highlight the possible differences between their initial perception and the results obtained. these recommendations, as for any decision-making process and tools, great caution should be taken, notably thanks to careful sensitivity analyses.

 Finally, future research is going to explore the possibility to extend this model to an ANP (Analytic Network Process) model in order to integrate the correlation between the different project complexity factors (as suggested when calculating some Spearman's rank correlation coefficient). Indeed, Taliscali and Eran (Taliscali and Ercan, 2006) say that their results suggest that "the ANP model represents reality as well as reliability better than the AHP model" due to the better integration of the interactions which exist between criteria. We think that in our case, exploring the possibility of using the ANP may be interesting since in essence, the criteria and sub-criteria of our structure are not independent.

However, the number of required judgement elicitations is likely to increase and become tougher when dealing with the interrelation between criteria or subcriteria.

Acknowledgements

The authors would like to thank people of the start-up for their support to this work.