Rostand Tayong

Thomas Dupont

Philippe Leclaire

On the variations of acoustic absorption peak with particle velocity in micro-perforated panels at high level of excitation

The acoustic behavior of micro-perforated panels ͑MPP͒ is studied theoretically and experimentally at high level of pressure excitation. A model based on Forchheimer's regime of flow velocity in the perforations is proposed. This model is valid at relatively high Reynolds numbers and low Mach numbers. The experimental method consists in measuring the acoustical pressure at three different positions in an impedance tube, the two measurement positions usually considered in an impedance tube and one measurement in the vicinity of the rear surface of the MPP. The impedance tube is equipped with a pressure driver instead of the usual loudspeaker and capable of delivering a high sound pressure level up to 160 dB. MPP specimens made out of steel, dural and polypropylene were tested. Measurements using random noise or sinusoidal excitation in a frequency range between 200 and 1600 Hz were carried out on MPPs backed by air cavities. It was observed that the maximum of absorption can be a positive or a negative function of the flow velocity in the perforations. This suggests the existence of a maximum of absorption as a function of flow velocity. This behavior was predicted by the model and confirmed experimentally.

I. INTRODUCTION

Micro-perforated panels ͑referred to as MPPs͒, when associated with air cavities, are of great interest in noise reduction applications such as Helmholtz resonators. They are robust and easy to manufacture and they can be used in hostile temperature and pressure environments. Several models were developed in the linear regime to describe their surface impedance, their absorption coefficient and their transmission loss factor. [1][START_REF] Dupont | Acoustic properties of lightweight micro-perforated plate systems[END_REF][START_REF] Hersh | Helmholtz resonator impedance model. Part I: Nonlinear behavior[END_REF] In the case of high sound pressure excitation, it is thought that the jet formation ͑vorticity͒ at the opening modifies significantly the absorption mechanisms. By measuring the velocity in the aperture with the help of a hot wire, Ingard and Ising [START_REF] Ingard | Acoustic nonlinearity of an orifice[END_REF] showed that at high sound pressure levels the flow separates at the outlet orifice, forming a high velocity jet. During the inflow half-cycle, the incident flow at the inlet of the orifice is essentially irrotational but highly rotational ͑in form of jetting͒ after exiting from the outlet orifice. The acoustic particle velocity is increased sharply as the wave is squeezed into the minute perforations. The nonlinear regime implies that the acoustic properties ͑mainly the im-pedance͒ are dependent upon the acoustic particle velocity either in front of the panel or into the aperture. During the other half of the cycle, the flow pattern is reversed. Cummings 5,[START_REF] Cummings | Transient and multiple frequency sound transmission through perforated plates at high amplitudes[END_REF] works on perforated plate lead to development of a one-dimensional numerical time-domain model to simulate the nonlinear resistance of a perforated plate. From his work, it is revealed that when there is no mean flow, the fraction of the energy absorbed depends non-linearly on the oncoming sound and high-amplitude acoustic waves are required before this mechanism of sound absorption is effective. Cummings [START_REF] Cummings | Transient and multiple frequency sound transmission through perforated plates at high amplitudes[END_REF] main interest was focusing on the acoustic transmission loss through the perforated plate. Kraft and co-workers [START_REF] Chandrasekharan | Experimental study of acoustic impedance of MEMS-based micro-perforated liners[END_REF][START_REF] Kraft | Acoustic treatment design scaling methods[END_REF][START_REF] Kraft | Acoustic treatment design scaling methods. Volume 4: Numerical Simulation of the Nonlinear Acoustic Impedance of a Perforated Plate Single-Degree-of-Freedom Resonator Using a Time-Domain Finite Difference Method[END_REF] proposed a model for MPP for high sound pressure. Their model was derived from the measurements of single degree-of-freedom liners over a wide range of sound pressure levels. It gives a prediction for the combined linear and nonlinear resistance and reactance for a face sheet. The nonlinear parameter which is considered is the discharge coefficient defined as the ratio of the actual discharge ͑discharge that occurs and which is affected by friction as the jet passes through the orifice͒ divided by the ideal discharge ͑without friction͒. The discharge coefficient value varies between 0.5 and 0.9. Kraft and co-workers [START_REF] Chandrasekharan | Experimental study of acoustic impedance of MEMS-based micro-perforated liners[END_REF][START_REF] Kraft | Acoustic treatment design scaling methods[END_REF][START_REF] Kraft | Acoustic treatment design scaling methods. Volume 4: Numerical Simulation of the Nonlinear Acoustic Impedance of a Perforated Plate Single-Degree-of-Freedom Resonator Using a Time-Domain Finite Difference Method[END_REF] used a fixed discharge coefficient of 0.76 in their model assuming constant nonlinear behavior regardless of the flow conditions. Moreover, their resistive and reactive part of the plate impedance are independent of frequency essentially assuming constant orifice mass. Melling 10 derived an expression for the resistive impedance in the nonlinear regime. The discharge coefficient used by Melling is valid only for the case where the panel thickness is smaller than the orifice diameter. However, this discharge coefficient is highly frequency dependent. Later on, Maa [START_REF] Maa | Micro-Perforated Panel at High Sound Intensity ͑Inter Noise[END_REF] showed that the acoustic non linearity of apertures is an external phenomenon i.e., the internal impedance is independent of the sound intensity. He suggested a nonlinear impedance term expression for MPP with small open area ratio. This expression is the ratio of the acoustic velocity inside the aperture divided by the product of sound speed and open area ratio. More recently, Hersh et al. [START_REF] Hersh | Helmholtz resonator impedance model. Part I: Nonlinear behavior[END_REF] derived a model extended to multiple orifices via the open area ratio assuming continuity of volume velocity and no interactions between the holes. The discharge coefficient and some other parameters of the model were determined empirically.

This work is aimed at describing the acoustic absorption coefficient variations for microperforated panels ͑in the absence of mean flow͒ at high level of excitation. It uses a relatively simple linear model ͑Maa's model͒ to end up describing the high sound level variations of the absorption coefficient involving parameters that are easier to estimate than the discharge coefficient. Since the main focus of the work is the high sound level case, the choice of this relatively simple model was preferred. The proposed resulting model based on dimensional analysis and Forchheimer's law sets the use of two dimensionless parameters and a limit Mach number which are shown to critically determine the resistive part of the sample under high sound pressure levels. A number of authors have studied porous materials with more complicated pore shapes with more elaborated models ͑for example, Champoux and Stinson 12 ͒. Other authors [START_REF] Horoshenkov | The acoustic properties of granular materials with pore size distribution close to log-normal[END_REF] also investigated the acoustical properties of porous materials with a pore size distribution. In our case, the pore geometry is simple ͑cylinders with uniform diameter͒ and so we keep a simple model in order to focus on the nonlinear aspect of the propagation.

In this paper, it is noted that depending on the described dimensionless parameters that are found, the resulting absorption coefficient will either solely decrease with the increase of Mach number or in a first phase increase to a maximum level before decreasing in a second phase with the increase of Mach number. The first section deals with the theoretical analysis of Maa's MPP model in the linear regime. Section II presents the derivation of the proposed model. In this section, a dimensionless parameter involved in the MPP behavior is introduced. An experimental and theoretical analysis shows that the maximum of absorption can be a positive or a negative function of the flow velocity in the perforations, suggesting the existence of a maximum of absorption as a function of flow velocity. This behavior was predicted by the proposed model and confirmed experimentally. The final section describes the experimental setup, and offers comments on the results.

II. LINEAR REGIME MODEL

The main mechanism of absorption in the linear regime is the conversion of the acoustical energy into heat. In this regime ͑low sound pressure and velocity amplitudes͒, if the dimensions of the MPP ͑diameter of holes, holes separation, thickness͒ are small with respect to the impinging acoustic wavelength, and if the aperture dimension ͑diameter of holes͒ is of the order of the viscous and thermal boundary layers thicknesses, the major part of the acoustical energy is dissipated through viscous and thermal effects. The model proposed in this paper does not account for the presence of mean flow.

A. Maa's linear model

In every straight cylindrical duct, the sound field expression of progressive waves is written as p = P + e -jt , ͑1͒

where P + denotes the waves amplitude in the positive direction, is the angular frequency. Based on the theory and equations of acoustical propagation in short and narrow circular tubes, Maa 1 derived an equation of aerial motion given for one perforation by

j 0 u - r ץ ץr ͫ r ץ ץr u ͬ = ⌬p h , ͑2͒
where ⌬p is the pressure drop across the tube, h the length of the tube ͑which corresponds to the thickness of the MPP͒, the dynamic viscosity, 0 the density of air, r the radial coordinate, and u the particle velocity in the perforation. By solving the equation with respect to the velocity, the specific acoustic impedance of the short tube defined as the ratio of ⌬p to the average velocity ͗u͘ over a cross-sectional area of the tube is given by

Z perf = ⌬p ͗u͘ = j 0 h ͫ 1 -2 x ͱ -j J 1 ͑x ͱ -j͒ J 0 ͑x ͱ -j͒ ͬ -1 , ͑3͒
where x = d ͱ 0 / 4 is a constant defined as the ratio of ori- fice diameter d to the viscous boundary layer thickness of the air in the orifice, J 0 and J 1 the Bessel functions of the first kind of orders 0 and 1, respectively. An approximation of the above equation on the Bessel functions and valid for narrow tubes was given by Maa 1 as

Z perf = 32h d 2 ͱ1+ x 2 32 + j 0 h ͫ 1 + 1 ͱ 3 2 + x 2 /2 ͬ . ͑4͒
Due to the end radiating effects at the aperture, an end correction factor should be taken into account twice ͑once for each end͒. This correction is important when the perforation diameters are greater or of the order of the plate thickness. [START_REF] Ingard | Acoustic nonlinearity of an orifice[END_REF] The radiation impedance for an opened cylindrical tube is that of a piston in a baffle and is given by Morse [START_REF] Morse | Vibration and Sound ͑McGraw-Hill[END_REF] ͑for the case k 0 d Ӷ 1͒ and written as

Z ray = ͑k 0 d͒ 2 2 + j 8k 0 d 3 , ͑5͒
where k 0 is the wave number. The radiation impedance should be doubled accounting for each aperture side. The effect of the vibration of the air particles on the baffle in the vicinity of the aperture increases the thermo-viscous frictions. To take this effect into account, Rayleigh [START_REF] Rayleigh | Theory of Sound II ͑Macmillan[END_REF] proposed an additional factor on the resistive part of the tube impedance. This resistance is given by

R S = 1 2 ͱ 2 0 .

͑6͒

Under a certain number of assumptions, a process of homogenization can be applied, providing an expression of the impedance for multiple perforations. The minimal distance between perforations must be greater than their diameters and smaller than the wavelength and so it is possible to consider that there is no interactions between the apertures and the absorption mechanism is dominant. The MPP must be thinner than the wavelength to ensure the continuity of the velocities on both sides of the plate. Within these assumptions, the visco-thermal interactions between the fluid and the solid are taken into account through a viscosity correction function. 1 The total impedance of the MPP is then given by the impedance of one perforation divided by the open area ratio :

Z MPP = Z perf + 2Z ray + 4R S . ͑7͒
The factor 4 in the previous expression accounts for the fact that the resistance R S given by Rayleigh is too small in practice. Ingard 16 so as Allard [START_REF] Allard | Propagation of Sound in Porous Media[END_REF] also observed this behavior in their models.

III. MODEL FOR THE IMPEDANCE OF MPPS AT HIGH SOUND PRESSURE LEVELS

A. Variation of the impedance with MPP geometrical parameters

From experimental measurement using hot wires, Ingard and Ising 4 have observed that over a half period of the propagation of the high-amplitude sound wave, the incident flow is irrotational while the outflow is a highly rotational jetting. During the other half period the flow pattern is reversed. According to Ingard, [START_REF] Ingard | On the theory and design of acoustic resonators[END_REF] air current loses energy at the inlet due to friction on the panel surface along which part of the air current has to move when it is squeezed into the small area of the tube. They have also observed that the orifice resistance varies linearly with particle velocity. From the semi-empirical approach of Ingard, [START_REF] Ingard | On the theory and design of acoustic resonators[END_REF] we investigated a model based on dimensional analysis [START_REF] Buckingham | On physically similar systems: Illustrations of the use of dimensional equations[END_REF] in order to study the influence of the MPP geometrical parameters. This model is found to be consistent with including the Forchheimer nonlinear flow regime in the linear model. This approach was used by Auregan and Pachebat [START_REF] Auregan | Measurement of the nonlinear behavior of acoustical rigid porous materials[END_REF] for the study of nonlinear acoustical behavior of rigid frame porous materials.

Strictly speaking, the characteristic impedance is not defined for nonlinear wave propagation. However, if the harmonic distortion is not too high ͑first harmonic at much higher amplitude than the following harmonics͒, it is possible to define an impedance. Ingard and Ising [START_REF] Ingard | Acoustic nonlinearity of an orifice[END_REF] proposed an expression where the real part of the impedance is given by

Re͕Z MPP ͖ NL = Au + B, ͑8͒
where Re͕Z MPP ͖ is the MPP resistance, u the velocity in the perforation, A and B are constants. It is found that the variation of the reactance with u is fairly moderate. Melling 10 explained that this remark about the nonlinear reactance ͑imaginary part of the impedance͒ is considered valid for thin plates. He stated that the reactance tends to an asymptotic limiting value ͑this would be confirmed by our experimental results͒ of approximately one-half the linear regime value.

Dividing u by the speed of sound c 0 , the Mach number M = u / c 0 is introduced and this expression can be written in a normalized form as

Re͕z MPP ͖ NL = aM + b, ͑9͒
where z MPP is the normalized impedance, a and b are now dimensionless parameters that will depend on the MPP geometrical features and on the fluid constants. This was already noticed by Maa 11 in a recent work in which he observed that a is inversely proportional to the open area ratio while Hersh [START_REF] Hersh | Helmholtz resonator impedance model. Part I: Nonlinear behavior[END_REF] observed that the resistance is dependent on the ratio h / d. From the dimensional analysis, [START_REF] Buckingham | On physically similar systems: Illustrations of the use of dimensional equations[END_REF] it is found that the variation seems to be a more complicated combination of these two behaviors:

a = Kͩ d h ͪ l ͩ h 0 c 0 ͪ m n , ͑10͒
where K, l, m, n are constants and is the dynamic viscosity.

The constant K is related to the shape of the aperture edge. Such behavior has been confirmed by Atig et al. [START_REF] Atig | Termination impedance of openended cylindrical tubes at high sound pressure level[END_REF] experimental results. They studied the influence of the edges shapes on the resistive part of the orifice impedance of a cylindrical tube. It is revealed from their work that depending on the shape of the aperture edge ͑rounded edge of different radii͒, the amplitude of the nonlinear losses is crucially influenced. Although all the parameters having an influence on the value of K are not fully understood, both the edges shape and the type of material ͑as it will be observed later͒ seem to affect the value of K.

B. Constant determination from the Forchheimer nonlinear flow model in the perforations

The study of rigid frame porous materials at high sound pressure levels by Auregan and Pachebat 19 makes use of Forchheimer's law, [START_REF] Mcintosh | Nonlinear wave propagation through rigid porous materials. I: Nonlinear parametrization and numerical solutions[END_REF] which states that for high Reynolds number ͑Re larger than unity͒, the flow resistivity increases linearly with the Reynolds number. Since the Reynolds number is proportional to the aperture diameter, the resistivity ͑also seen as the real part of the impedance per unit thick-ness͒ is directly proportional to the diameter. In fact, from expression ͑10͒, by considering l = m = 1 and n = -1, one easily finds the result given by Auregan and Pachebat [START_REF] Auregan | Measurement of the nonlinear behavior of acoustical rigid porous materials[END_REF] if the viscous characteristic length is taken equal to the perforation radius ͑case of cylindrical pores͒.

Finally, the model for the coefficient of the real part of the MPP impedance is

Re͕z MPP ͖ NL = aM + b, ͑11͒ a = Kͩ d h ͪͩ h 0 c 0 ͪ -1 .

͑12͒

a can be further simplified as

a = K dc 0 , ͑13͒
where is the kinematic viscosity. It is found that the coefficient b is related to dimensionless coefficient ␦ of Auregan and Pachebat [START_REF] Auregan | Measurement of the nonlinear behavior of acoustical rigid porous materials[END_REF] through the relation

b = ͑1 + ␦͒Re͕z MPP ͖ linear ͑14͒
where ␦ is characteristic of three regimes: the linear regime ͑␦ =0͒, the weakly nonlinear regime ͑␦ 0͒, and a transition regime between them. The weakly nonlinear regime is asymptotically linear as M increases and therefore, b corresponds to the intercept of the straight line with the vertical axis ͑b has no real physical interpretation as for M = 0, the asymptotic behavior does not hold͒. In this study, both K and ␦ ͑or b͒ were determined from a linear fitting.

IV. ABSORPTION COEFFICIENT OF A MPP BACKED BY A CAVITY

Under the same assumption on harmonic amplitudes, it is possible to define a cavity impedance for high sound pressure levels and it is given by the usual expression ͑normalized form͒:

z c = -j cot͑k 0 D c ͒, ͑15͒
where D c is the air gap thickness and the normalized surface impedance of the total system ͑MPP coupled to an air cavity͒ is given by the superposition of the two impedances: 1,4

z s = z MPP + z c . ͑16͒
The reflection coefficient is given by the usual formula

R = z s -1 z s + 1 , ͑17͒
and the acoustic absorption coefficient can be calculated from

␣͑͒ = 1 -͉R͉͑͒ 2 .

͑18͒

When the real and the imaginary part of the total impedance z s are separated, another expression of the absorption coefficient can be given by

␣ = 4 Re͕z s ͖ ͑1 + Re͕z s ͖͒ 2 + ͑Im͕z s ͖͒ 2 .
͑19͒

V. MAXIMUM ABSORPTION AS A FUNCTION OF MACH NUMBER

By inserting the expression of the real part of the MPP impedance in the equation for the absorption coefficient we have

␣ = 4͑aM + b͒ ͑1 + aM + b͒ 2 + ͑Im͕z s ͖͒ 2 . ͑20͒
The maximum of absorption is obtained for Im͕z s ͖ =0:

␣ M = 4͑aM + b͒ ͑1 + aM + b͒ 2 .

͑21͒

This expression is now differentiated with respect to M in order to study the variations with the Mach number ͑and implicitly with the incident sound pressure level͒

␣ץ M ץM = 4a͑1 -aM -b͒ ͑1 + aM + b͒ 3 . ͑22͒
This result shows that a critical value of the Mach number exists for which the absorption coefficient is extremum ͑Fig.

1͒. Since a 0, this last expression provides a limit Mach number M c given by

M c = 1 -b a . ͑23͒
From the study of the variations of the absorption coefficient function, it is found that the peak of absorption ͑maximum of absorption with respect to frequency͒ increases with the Mach number, reaches a maximum as the Mach number approaches its critical value and then decreases for M increasing beyond the critical value M c . It is worth noticing that Maa [START_REF] Maa | Micro-Perforated Panel at High Sound Intensity ͑Inter Noise[END_REF] predicted this behavior. In the present article, we propose a more refined model, we define a critical value for the Mach number and include experimental data. Evidently, this behavior ͑also observed by Umnova et al. [START_REF] Umnova | Behavior of rigid-porous layers at high levels of continuous acoustic excitation: Theory and experiment[END_REF] for rigid-porous layers͒ can be observed if the sample resistance is less than the medium resistance and practically very often if the critical value is above the linear/nonlinear regime limit. Indeed, the experimental results will show that in some cases, a value for M c will not be identified if it is located in the linear range. In this case the MPP absorption peak will only decrease with the increasing sound pressure level.

It is also worth noticing that ␣ M ͑M c ͒ = 1 for any MPP with M c located in the nonlinear domain. This result shows potential applications in the design of MPPs.

VI. EXPERIMENTS

A. MPP samples

The measurements are performed on strong copolymermade MPP, steel-made and dural-made MPP ͑Fig. 2 and Table I͒. All the sample panels have an external diameter of 100 mm and all the holes are well separated from each other ͑no interactions between the apertures͒ and are evenly distributed over the panel area. The mounting conditions of the MPP inside the tube are closer to the clamped conditions than to the simply supported conditions.

B. Impedance tube and data acquisition

A schematic of the impedance tube used is shown in Fig. 3. It is a rigid circular plane-wave tube with a diameter of 100 mm. Plane wave propagation is assumed below the cutoff frequency of the impedance tube ͑about 1.7 KHz͒. At the left hand side, a compression driver JBL model 2450J is mounted as the source of excitation. A transition piece provides a continuity transition between the circular section of the compression driver and the circular cross section of the plane-wave tube. At the right hand side of the tube, a soundproof plunger is used as the rigid backing wall. By moving the plunger along the longitudinal axis of the tube, one is able to create an air cavity behind the MPP sample. The MPP sample is mounted between the speaker and the plunger. Three 1 / 4Љ microphones are used to perform the signal detection. Two microphones are used to calculate the surface impedance of the sample by the standard impedance tube measurement technique. [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. I. Theory[END_REF] The third microphone ͑reference micro in Fig. 3͒ acts as a reference microphone to get the level of pressure at the sample surface. When performing high sound pressure measurements inside an impedance tube, it is important to check the resulting standing wave. In fact, up to a certain level, depending on the frequency excitation and the sample parameters, the standing wave seems to saturate and therefore the linear propagation hypothesis would no longer be valid. Effects of bifurcation may occur. These phenomena of saturation and bifurcation were observed and shown by Maa and Liu. [START_REF] Maa | Nonlinear standing waves: Theory and experiments[END_REF] A first excitation is done with a white noise signal in order to have a general view of the absorption coefficient curve and locate the ab-sorption peak position. The result is used to determine the resonant frequency and to perform a second excitation ͑sine excitation͒ at frequencies concentrated around the absorption peak͑s͒. The amplitude of the source is adjusted such that the sound pressure level measured by the reference microphone is set at the desired level. The sound pressure level is varied from 90-155 dB at the face of the MPP monitored using the reference microphone. A phase and amplitude calibration method is used to correct the transfer function between the measurement microphones. The measurement are carried out at high sound pressure levels. However, assuming the planewave hypothesis, one can measure the pressures and velocities on any section of the tube using the two microphone method. See for instance Dalmont [START_REF] Dalmont | Acoustic impedance measurement. Part I: A review[END_REF] for more details. The acoustic velocity on the panel surface is given by

u = j p 1 Z 0 H cos͑k 0 l 1 ͒ -cos͑k 0 l 2 ͒ sin͑k 0 s͒ , ͑24͒
where l 1 = s + l 2 as in Fig. 3. H is the ratio of pressures measured on two points of the tube. Z c is the characteristic impedance of air, k 0 is the wave number, p 1 is the pressure on microphone 1, and l 1 ͑resp. l 2 ͒ is the distance from microphone 1 ͑resp. 2͒ to the panel sample. The values of the velocity shown in the experimental results are the viscous peak corresponding particle velocity.

VII. RESULTS AND COMMENTS

A. Comparison between theory and experiment for ␣ M as a function of M

In this section, the measurements are performed taking a single MPP with an air cavity behind and a rigid wall. The dimensionless parameters K and ␦ used to fully determine expression ͑9͒ are given in Table II. Figure 4 shows the comparison between experimental results and the present model simulations for the resistance as a function of the Mach number for all the MPP samples. The air cavity depth is 50 mm. The experiments and the present model are in good agreement for the high excitation levels. This result points out the fact that for high sound pressure levels, the dependency of the resistance and the Mach number is linear. The slope is different from one MPP to another, revealing therefore the fact that this slope, in the same measurement conditions, depends on the MPP geometrical parameters. This result also shows that the value of constant K seems to be related to the type of material ͑1.43ϫ 10 -3 for the copolymer samples, 1.56ϫ 10 -3 for the steel sample, 2.2ϫ 10 -3 for the dural sample͒, and as already mentioned, to the shape of the aperture edge. Figure 5 shows the comparison between experimental results and the present model simulations for the maximum absorption coefficient versus the Mach number for MPP1, MPP2, and MPP3 samples. The air cavity depth is 50 mm. The experiments and the simulation are in fairly good agreement. This confirms the fact that depending on the value of the limit Mach number with the increase of sound levels, the absorption peak will in a first phase rise up to a maximum value before, in a second phase, decreasing. It should be pointed out that for the three cases presented, as already mentioned in the theoretical part, the maximum of ␣ M cannot be predicted theoretically because the limit Mach number M c is below the linear/nonlinear limit regime. The limit Mach number is observable experimentally on MPP1 and MPP3 ͑the values of the maximum are given in Table II͒ in a region where the proposed model is not valid and therefore the maximum cannot be predicted theoretically. For MPP2, the maximum could neither be observed experimentally nor predicted. In fact, if this limit Mach number is low, the absorption peak will solely decrease with the increase of sound pressure levels. Figure 6 shows the comparison be-tween experimental results and present model for another MPP ͑MPP4͒. This example illustrates the case where the maximum of ␣ M is observable theoretically, because the critical Mach number is predicted in the domain of validity of the model. The present model prediction is in good agreement with the experimental results except for Mach numbers below this critical Mach number because the maximum of ␣ M is predicted in a region where the model is valid while the maximum of ␣ M in Fig. 5 was predicted at very low Mach numbers, where the model is not valid. In fact, the maximum of ␣ M will be observable theoretically for low values of b ͑or ␦͒ ͑see Table II͒.

B. Absorption coefficient as a function of frequency

Figure 7 shows the comparison between experimental and simulations results ͑Maa, [START_REF] Maa | Micro-Perforated Panel at High Sound Intensity ͑Inter Noise[END_REF] Hersh et al., [START_REF] Hersh | Helmholtz resonator impedance model. Part I: Nonlinear behavior[END_REF] agreement except for the Hersh high sound model which does not fit well with the experiment results. Figures 8͑a͒ and8͑b͒ show the comparison between experimental and simulations results of the surface impedance ͕resistance ͓Fig. 8͑a͔͒ and reactance ͓Fig. 8͑b͔͖͒ of MPP1 at 145 dB in front of the panel ͑reference microphone͒ in the ͓200-1600 Hz͔ frequency range for an air cavity depth of 50 mm. The present model and the measurements are in very good agreement. In Fig. 8͑a͒, Maa's model underestimates the nonlinear resistance whereas Hersh's model agrees well with the experimental results except around the structural response frequency. In Fig. 8͑b͒, the models are in good agreement with the experimental results except for Hersh's model. Figure 9 shows the comparison between experimental and simulation results ͑Maa, [START_REF] Maa | Micro-Perforated Panel at High Sound Intensity ͑Inter Noise[END_REF] Hersh et al., [START_REF] Hersh | Helmholtz resonator impedance model. Part I: Nonlinear behavior[END_REF] and the present model͒ for the absorption coefficients of MPP3 at 145 dB in front of the panel ͑reference microphone͒ in the ͓200-1600 Hz͔ frequency range for an air cavity depth of 40 mm. The present model and the measurement results are in very good agreement. Figures 10͑a͒ and10͑b͒ show the comparison between experimental and simulations results for the surface impedance ͕resistance ͓Fig. 10͑a͔͒ and reactance ͓Fig. 10͑b͔͖͒ of MPP3 at 145 dB in front of the panel ͑reference microphone͒ in the ͓200-1600 Hz͔ frequency range for an air cavity depth of 40 mm. In Fig. 10͑a͒, for the resistance part, below 800 Hz, the present model and the measurement results are in very good agreement. Yet beyond 800 Hz, the agreement is not good. This may imply considering a certain highfrequency-dependence of the nonlinear parameter for a more accurate prediction. However, as mentioned, for relatively high-frequency, the absorption seems to be much more influenced by the imaginary part of the impedance. Maa's model underestimates the measurement results while Hersh's model tendency is good compared to the measurement results. In Fig. 10͑b͒, for the reactance part, the present model and Maa's model both agree accurately with the measurement results.

VIII. CONCLUSION

A model for the high levels of excitation of MPP when backed by an air cavity has been proposed and tested experimentally. Two dimensionless parameters and a limit Mach number were found and used. These are suitable to predict the impedance of the system at high sound pressure levels. The results and the analysis showed that the model and the experiments are in good agreement. The theoretical work revealed the fact that with the increase of sound pressure level ͑or Mach number͒, the viscous absorption peak will in a first phase rise to a maximum value and then decrease in a second phase. Experimentally, it was noticed that this latter result can clearly be observed if the value of the limit Mach number is above the linear regime limit of the MPP. If the limit Mach number is below the linear regime limit, the absorption peak will only decrease with increasing sound pressure level. Finally, it was shown that MPP are very sensitive to the incident sound pressure levels. Thermal effects around the MPP system tends to play an important role with the increase of sound intensities. Further work will consist in investigating these effects and to properly understand their influence on the MPP impedance for high levels of excitation. It may also be interesting to investigate the apertures interactions when submitted to relatively high sound pressure amplitudes.

ACKNOWLEDGMENTS

The support grant for this work was provided by the Conseil Régional de Bourgogne.

APPENDIX: MODEL INCLUDING THE STRUCTURAL BEHAVIOR OF THE PANEL

In the absorption test of the copolymer micro-perforated panel, an additional sound absorption peak was found around the frequency of 900 Hz due to the panel vibration first mode. Theoretical and experimental studies were done by Ford and McCormick [START_REF] Ford | Panel sound absorbers[END_REF] and also Frommhold et al. [START_REF] Frommhold | Acoustic performance of membrane absorbers[END_REF] on panel absorbers without perforations. From these studies, it was found that this additional absorption peak is due to the panelcavity resonance. Ford and McCormick 26 gave an expression of the structural impedance of such system ͑membrane coupled to an air cavity͒. This expression is considered and used in this paper when the structural resonant frequency of the panel is not beyond the frequency range of study ͑200-1600 Hz͒. Only the first structural mode is considered since the other structural frequency modes are always out of the frequency range studied. Since the MPP backed by an air cavity system can be considered analogous to an electrical circuit, it is well assumed that the normalized acoustic impedance of the whole system Z s is given by

Z s = Z MPP Z vib Z MPP + Z vib + Z c , ͑A1͒
where Z MPP is the MPP specific impedance. Z c is the air cavity impedance. Z vib is the structural impedance of an air cavity backed membrane ͑without perforations͒ well described by Ford and McCormick [START_REF] Ford | Panel sound absorbers[END_REF] as

Z vib = D rig B mn h 4 + jM p A mn + 1 j ͩ D rig B mn h 4 + 0 c 0 2 D c ͪ, ͑A2͒
where A mn and B mn are the modal constants, M p the mass per unit area of panel, h is the lateral dimension ͑thickness of the panel͒, D c is the air cavity depth, and D rig the bending stiffness of the panel. Where

D rig = Eh 3 12͑1 -p 2 ͒ , ͑A3͒
where E is the Young modulus ͑Table I͒, = 10%, and p = 0.3. Since our MPP mounting conditions are rather close to clamped-clamped, A mn = 2.02 and B mn = 2640 ͑see Table I for the other parameters͒. An important remark to mention in this section is about Lee et al. [START_REF] Lee | Sound absorption of a finite flexible micro-perforated panel backed by an air cavity[END_REF] observations that if the forcing frequency ͑viscous peak͒ is higher than the first structural resonant ͑structural peak͒, then the structural vibration degrades the absorption. And in contrast, if the forcing frequency is lower than the first structural resonant frequency, the structural vibration enhances the absorption.

FIG. 1 .

 1 FIG. 1. Absorption coefficient at the resonance frequency ͓␣ M in expression ͑21͔͒ versus the acoustical Mach number in front of the MPP.

  Figure7shows the comparison between experimental and simulations results ͑Maa,[START_REF] Maa | Micro-Perforated Panel at High Sound Intensity ͑Inter Noise[END_REF] Hersh et al.,[START_REF] Hersh | Helmholtz resonator impedance model. Part I: Nonlinear behavior[END_REF] and the present model͒ for the absorption coefficients of MPP1 at 145 dB in front of the panel ͑reference microphone͒ in the ͓200-1600 Hz͔ frequency range for an air cavity depth of 50 mm. The simulation of the present model and the measurement are in very good agreement. The first peak ͑around 564 Hz͒ is the perforation absorption peak and the second peak ͑around 900 Hz͒ is the result of the panel absorption peak resulting from the structural response of the panel coupled to an air cavity. This structural response is described in the Appendix. The presented models and the measurements are in fairly good

FIG. 6 .

 6 FIG. 6. Experimental results of the maximum of absorption coefficient as a function of the Mach number in front of the panel. Air cavity depth of 50 mm.

FIG. 8 .

 8 FIG. 8. Comparison between the surface impedances ͓͑a͒ resistance and ͑b͒ reactance͔ of MPP1 at 145 dB ͑U = 0.325 m / s͒ on the reference microphone. Air cavity depth of 50 mm.

FIG. 10 .

 10 FIG. 10. Comparison between the surface impedances ͓͑a͒ resistance and ͑b͒ reactance͔ of MPP3 at 145 dB ͑U = 0.337 m / s͒ on the reference microphone. Air cavity depth of 40 mm.

1D

  . Y. Maa, "Potential of micro-perforated panel absorber," J. Acoust. Soc. Am. 104, 2861-2866 ͑1998͒.

TABLE I .

 I Micro-perforated panel characteristics.

	h	d		Density	Young modulus
	͑mm͒	͑mm͒	͑%͒	͑kg/ m 3 ͒	͑Pa͒	Material
	MPP1 2.2	1.0 2.2	900	0.49ϫ 10 10	Strong copolymer
	MPP2 2.2	1.0 0.8	900	0.49ϫ 10 10	Strong copolymer
	MPP3 2.0	0.7 1.94	7800	21ϫ 10 10	Steel
	MPP4 2.0	1.0 1.6	2700	0.69ϫ 10 10	Dural

TABLE II .

 II Dimensionless parameters of the microperforated panels.

	FIG. 2. Perforated panel sample.
		M c	K	␦
	MPP1	0.18ϫ 10 -3	1.43ϫ 10 -3	2.024
	MPP2	0.22ϫ 10 -3	1.43ϫ 10 -3	3.868
	MPP3	Not observable	1.56ϫ 10 -3	0.956
	MPP4	9.41ϫ 10 -5	2.2ϫ 10 -3	0.8

FIG. 3. Schematic of the impedance tube used for the measurements.


Denis Bouyssou
email: bouyssou@lamsade.dauphine.fr

Marc Pirlot
email: marc.pirlot@fpms.ac.be

An axiomatic approach to TACTIC

Keywords: Conjoint measurement, TACTIC, outranking methods

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Résumé

Beaucoup de méthodes de surclassement reposent sur l'utilisation du principe de concordance / non discordance. Ce principe conduit à déclarer qu'une action en surclasse une autre si la coalition des critères appuyant cette proposition est « suffisamment importante » (condition de concordance) et si aucun critère ne la « rejette fortement » (condition de non discordance). Ce mode de comparaison est simple et relativement naturel. Il est bien connu qu'il peut conduire à des relations binaires non transitives. Ce texte utilise des techniques issues de la théorie du mesurage conjoint pour proposer une analyse axiomatique des relations pouvant être obtenues par application du principe de concordance / non discordance. Cette analyse met l'accent sur la caractéristique cruciale de ces relations : elles induisent sur chaque critère une relation de comparaison d'écarts de préférence très pauvre. On se concentrera sur les méthodes de surclassment, telles que TACTIC, visant à bâtir une relation interprétée comme une préférence stricte. L'objectif est de donner à ces méthodes une base axiomatique solide fondée sur la théorie du mesurage conjoint.

Mots-clés : Mesurage conjoint, TACTIC, méthodes de surclassement.

Introduction

A central problem in MCDM (Multiple Criteria Decision Making) is to build a preference relation on a set of alternatives evaluated on several attributes, taking into account preferences expressed on each attribute and inter-attribute information (such as weights). The classical way of doing so [START_REF] Keeney | Decisions with multiple objectives: Preferences and value tradeoffs[END_REF]) is to use a value function. This leads to define a function v associating a real number v(x) to each alternative x and to declare that x is better than y if and only if v(x) ≥ v(y). The number v(x) depends upon the evaluation x 1 , x 2 , . . . , x n of x on the n attributes, most often through an additive aggregation such that:

x P y ⇔ n i=1 v i (x i ) > n i=1
v i (y i ).

(A) Roy and Bouyssou (1993, p. 233-237) have detailed the possible difficulties associated with such an approach. In particular, it requires a detailed analysis of the tradeoffs between the various attributes through a time-consuming and cognitively demanding interaction with the decision-maker. Facing such difficulties, the analyst may use a less demanding comparison procedure between alternatives. One such procedure, based on the concordance / nondiscordance principle (henceforth, the CNDP), was first proposed by [START_REF] Roy | Classement et choix en présence de points de vue multiples (la méthode ELECTRE)[END_REF] and underlies most of the well-known ELECTRE methods [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF]. ELECTRE methods build a reflexive preference relation interpreted as an "at least as good as" relation between alternatives. The CNDP is also at work in outranking methods, such as TACTIC [START_REF] Vansnick | On the problems of weights in MCDM (the noncompensatory approach)[END_REF], that lead to an asymmetric relation interpreted as "strict preference". We focus here on the latter type of methods. It should be noted that, contrary to the ELECTRE methods, TACTIC focuses on building a strict preference relation without distinguishing indifference from incomparability. However, the analysis below can easily be extended to cover outranking methods closer to ELECTRE1 in which such a distinction is made. Let x and y be two alternatives evaluated on several attributes. The CNDP leads to comparing these two alternatives along the following lines:

• compare the evaluations of x and y on attribute i and decide whether attribute i favors x, favors y or favors none of x and y. Repeat this operation for each attribute. This defines three disjoints subsets of attributes: those favoring x, those favoring y and those for which none of the two alternatives is favored,

• compare the set of attributes favoring x with the set of attributes favoring y in terms of "importance",

• for each attribute in the set of attributes favoring y, investigate whether this attribute "strongly" favors y,

• declare that "x is preferred to y" if the set of attributes favoring x is "more important than" the set of attributes favoring y and if there is no attribute strongly favoring y.

This way of comparing alternatives has a definite "ordinal" flavor and does not require a much detailed analysis of tradeoffs. The price to pay is that it does not always lead to preference relations, henceforth called strict outranking relations (SOR), having "nice" transitivity properties. On a practical level, this means that using such relations to elaborate a recommendation, e.g., through the selection of a subset containing "good" alternatives, is not easy and requires the use of specific techniques (see, e.g., [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF]. On a theoretical level, such relations are quite different from the transitive structures usually studied in conjoint measurement (see [START_REF] Krantz | Foundations of measurement[END_REF][START_REF] Wakker | Additive representations of preferences: A new foundation of decision analysis[END_REF], e.g. those representable using the additive model (A). This probably explains why outranking methods have not been much investigated at an axiomatic level. The present paper concentrates on this last aspect. Adopting a framework for conjoint measurement tolerating intransitive preferences proposed in Bouyssou and Pirlot (2002b) will allow us to propose an axiomatic characterization of SOR using axioms that will emphasize their main specific feature, i.e. the very crude way in which they isolate various levels of "preference differences" on each attribute. This extends the results in [START_REF] Bouyssou | Ordinal aggregation and strict preferences for multi-attributed alternatives[END_REF] and Bouyssou and Pirlot (2005c) to include the possibility of discordance. This paper is not the first attempt to analyze the CNDP from a theoretical perspective and two earlier attempts at doing so should be mentioned. The work of [START_REF] Bouyssou | Noncompensatory and generalized noncompensatory preference structures[END_REF] on TACTIC has the same objective as the present paper. It uses a weakening of a noncompensation condition introduced in [START_REF] Fishburn | Noncompensatory preferences[END_REF]. As detailed in Bouyssou and Pirlot (2005a), the use of such a condition is not entirely satisfactory since it is quite strong and does not allow to take all forms of concordance conditions into account (in particular, it excludes the possibility to include an additive threshold in the concordance condition of TACTIC). The work of [START_REF] Greco | Axiomatic basis of noncompensatory preferences[END_REF] is also related to the present paper. They axiomatically study a particular class of reflexive outranking relations built using the CNDP. As in the present paper, the CNDP is analyzed through its consequences on the induced relations comparing preference difference on each attribute (this idea was developed independently in Bouyssou and Pirlot, 2002a). Their analysis is not conducted within the framework of a general conjoint measurement model and, therefore, is not well-suited to analyze the specific features of outranking relations within a larger class of relations. Furthermore, it only deals with a very specific type of outranking relations, in which the concordance condition is close to the one used in ELECTRE I and uses some axioms that, in our view, are not especially attractive.

This paper is organized as follows. We introduce our setting in Section 2. Strict outranking relations are defined in Section 3. Our general framework for conjoint measurement allowing for nontransitive preferences is presented in Section 4. Section 5 characterizes SOR without discordance effects. Section 6 extends these results to allow for discordance. A final section discusses our results and presents directions for future research. Proofs are gathered in an appendix.

Definitions and Notation

In this paper we consider a set X = n i=1 X i with n ≥ 2. Elements of X will be interpreted as alternatives evaluated on a set N = {1, 2, . . . , n} of attributes. When i ∈ N , we denote by X -i the set j∈N \{i} X j . With customary abuse of notation, (x i , y -i ) will denote the element w ∈ X such that w i = x i and w j = y j for all j ∈ N \ {i}.

We use P to denote an asymmetric binary relation on X interpreted as a strict preference relation between alternatives. The symmetric complement of P is denoted by I (i.e. x I y ⇔ [Not[x P y] and Not[y P x]]).

We say that attribute i ∈ N is influent (for P) if there are x i , y i , z i , w i ∈ X i and a -i , b -i ∈ X -i such that (x i , a -i ) P (y i , b -i ) and Not[(z i , a -i ) P (w i , b -i )] and degenerate otherwise. It is clear that a degenerate attribute has no influence whatsoever on the comparison of the elements of X and may be suppressed from N . In order to avoid unnecessary minor complications, we suppose henceforth all attributes in N are influent.

Strict outranking relations

The following definition, building on Bouyssou and Pirlot (2005c), [START_REF] Fargier | Modélisation des préférences par une règle de concordance généralisée[END_REF] and [START_REF] Greco | Axiomatic basis of noncompensatory preferences[END_REF], formalizes the idea of a SOR, i.e., of a strict preference relation that has been obtained comparing alternatives by pair using the CNDP.

Definition 1 (Strict outranking relations) Let P be an asymmetric binary relation on X = n i=1 X i . We say that P is a SOR if there are: • asymmetric binary relations P i and V i on each X i such that

V i ⊆ P i (i = 1, 2, . . . , n),
• a binary relation £ between disjoint subsets of N that is monotonic w.r.t. inclusion, i.e. such that for all A, B, C, D ⊆ N with

A ∩ B = ∅ and C ∩ D = ∅, A £ B C ⊇ A and B ⊇ D ⇒ C £ D,
such that, for all x, y ∈ X,

x P y ⇔ [P (x, y) £ P (y, x) and V (y, x) = ∅] ,

where P (x, y) = {i ∈ N :

x i P i y i } and V (y, x) = {i ∈ N : y i V i x i }.
Hence, when P is a SOR, the preference between x and y only depends on the comparison in terms of "importance" of the subsets of attributes favoring x or y in terms of the asymmetric relation P i (concordance) and on the absence of any attribute such that y i V i x i (non-discordance). It is useful to interpret x i P i y i as "x i is strictly preferred to y i ", x i V i y i as "x i is strongly preferred to y i " and A £ B as "the coalition A of attributes is more important than the coalition B of attributes".

The main objective of this paper is to characterize SOR within a general framework of conjoint measurement, using conditions that will allow us to isolate their specific features. A well-known example of an outranking method leading to a SOR is TACTIC [START_REF] Vansnick | On the problems of weights in MCDM (the noncompensatory approach)[END_REF]. This method builds an asymmetric preference relation P on X letting, for all x, y ∈ X,

x P y ⇔     
i∈P (x,y) w i > ρ j∈P (y,x) w j + ε and Not[y j V j x j ], for all j ∈ P (y, x) where P i and V i are semiorders on X i such that V i ⊆ P i , w i > 0 is the weight assigned to attribute i ∈ N and ρ > 1 and ε ≥ 0 are thresholds.

Defining £ letting, for all A, B ⊆ N such that A ∩ B = ∅:

A £ B ⇔ i∈A w i > ρ j∈B w j + ε,
it is easy to see that P as defined in TACTIC is a SOR. Simple examples inspired by Condorcet's paradox show that the relation P in TACTIC is not always transitive and may even have cycles. The main differences between a preference relation obtained with TACTIC and a SOR are that, in a SOR, it is neither supposed that the importance relation between coalitions can be represented additively using weights nor that P i and V i are strict semiorders.

A general framework for nontransitive conjoint measurement

This section follows the analysis in Bouyssou and Pirlot (2002b) using asymmetric relations instead of reflexive relations. We envisage here binary relations P on X that can be represented as:

x P y ⇔ F (p 1 (x 1 , y 1 ), . . . , p n (x n , y n )) > 0, (M )
where p i are real-valued functions on X 2 i that are skew symmetric (i.e. such that p i (x i , y i ) = -p i (y i , x i ), for all x i , y i ∈ X i ) and F is a real-valued function on n i=1 p i (X 2 i ) being odd (i.e. such that F (x) = -F (-x), abusing notation in an obvious way) and nondecreasing in all its arguments.

The intuition underlying model (M ) is that functions p i measure (positive or negative) preference differences and that the function F synthesizes these measures. Indeed, the analysis of model (M ) is based on relations comparing preference differences on each attribute induced by P. Definition 2 (Relations * i and * * i ) Let P be a binary relation on a set X = n i=1 X i . We define the binary relations * i and * * i on X 2 i letting, for all x i , y i , z i , w i ∈ X i ,

(x i , y i ) * i (z i , w i ) ⇔ [for all a -i , b -i ∈ X -i , (z i , a -i ) P (w i , b -i ) ⇒ (x i , a -i ) P (y i , b -i )], (x i , y i ) * * i (z i , w i ) ⇔ [(x i , y i ) * i (z i , w i ) and (w i , z i ) * i (y i , x i )] . The definition of * i suggests that (x i , y i ) * i (z i , w i
) can be interpreted as saying that the preference difference between x i and y i is at least as large as the preference difference between z i and w i . Indeed, as soon as

(z i , a -i ) P (w i , b -i ), (x i , y i ) * i (z i , w i ) implies (x i , a -i ) P (y i , b -i ).
However, the definition of * i does not imply that the two "opposite" differences (x i , y i ) and (y i , x i ) are linked. This is at variance with the usual intuition concerning preference differences and motivates the introduction of the relation * * i . We have may not be complete. Interesting consequences obtain when this is the case. This motivates the introduction of the following two conditions.

(x i , y i ) * * i (z i , w i ) when both (x i , y i ) * i (z i , w i ) and (w i , z i ) * i (y i , x i ) hold. This implies that * * i is reversible, i.e. (x i , y i ) * * i (z i , w i ) ⇔ (w i , z i ) * * i (y i , x i ).
Definition 3 (Conditions ARC 1 and ARC 2) Let P be a binary relation on a set X = n i=1 X i . This relation is said to satisfy:

ARC 1 i if (x i , a -i ) P (y i , b -i ) and (z i , c -i ) P (w i , d -i )    ⇒    (x i , c -i ) P (y i , d -i ) or (z i , a -i ) P (w i , b -i ), ARC 2 i if (x i , a -i ) P (y i , b -i ) and (y i , c -i ) P (x i , d -i )    ⇒    (z i , a -i ) P (w i , b -i ) or (w i , c -i ) P (z i , d -i ), for all x i , y i , z i , w i ∈ X i and all a -i , b -i , c -i , d -i ∈ X -i . We say that P satisfies ARC 1 (resp. ARC 2) if it satisfies ARC 1 i (resp. ARC 2 i ) for all i ∈ N .
Condition ARC 1 i suggests that either the difference (x i , y i ) is at least as large as the difference (z i , w i ) or vice versa. Condition ARC 2 i suggests that the preference difference (x i , y i ) is linked to the "opposite" preference difference (y i , x i ). Taking

x i = y i , z i = w i , a -i = c -i and b -i = d -i shows that ARC 2 implies that P is independent, i.e. that (x i , a -i ) P (x i , b -i ) implies (y i , a -i ) P (y i , b -i ), for all i ∈ N , all x i , y i ∈ X i and all a -i , b -i ∈ X -i .
The consequences of ARC 1 i and ARC 2 i on our two relations comparing preference differences on each attribute are noted below.

Lemma 1 1. ARC 1 i ⇔ [ * i is complete], 2. ARC 2 i ⇔ [for all x i , y i , z i , w i ∈ X i , Not[(x i , y i ) * i (z i , w i )] ⇒ (y i , x i ) * i (w i , z i )], 3. [ARC 1 i and ARC 2 i ] ⇔ [ * * i is complete].
4. In the class of asymmetric relations, ARC 1 and ARC 2 are independent conditions.

These two conditions allow to characterize model (M ) when X is finite or countably infinite (see Bouyssou and Pirlot, 2003, Theorem 1).

Theorem 1 Let P be a binary relation on finite or countably infinite set X = n i=1 X i . Then P has a representation (M ) iff it is asymmetric and satisfies ARC 1 and ARC 2.

It should be observed at this point that model (M ) is sufficiently general to contain as particular cases most conjoint measurement models, when interpreted in terms of an asymmetric binary relation, including the classical additive model (A) (see [START_REF] Krantz | Foundations of measurement[END_REF][START_REF] Wakker | Additive representations of preferences: A new foundation of decision analysis[END_REF], and the additive difference model (see [START_REF] Tversky | Intransitivity of preferences[END_REF]. We show below that SOR form a subclass of the binary relations having a representation in model (M ).

Strict concordance relations

In Bouyssou and Pirlot (2005c), we showed that strict concordance relations, i.e., SOR in which all relations V i are empty, are exactly the binary relations having a representation in model (M ) with all functions p i taking at most three distinct values. We briefly recall here the main points in this analysis.

Definition 4 (Conditions MAJ 1, MAJ 2) Let P be a binary relation on a set X = n i=1 X i . This relation is said to satisfy:

MAJ 1 i if (x i , a -i ) P (y i , b -i ) and (z i , a -i ) P (w i , b -i ) and (z i , c -i ) P (w i , d -i )            ⇒    (y i , a -i ) P (x i , b -i ) or (x i , c -i ) P (y i , d -i ), MAJ 2 i if (x i , a -i ) P (y i , b -i ) and (w i , a -i ) P (z i , b -i ) and (y i , c -i ) P (x i , d -i )            ⇒    (y i , a -i ) P (x i , b -i ) or (z i , c -i ) P (w i , d -i ), for all x i , y i , z i , w i ∈ X i and all a -i , b -i , c -i , d -i ∈ X -i . We say that P satisfies MAJ 1 (resp. MAJ 2) if it satisfies MAJ 1 i (resp. MAJ 2 i ) for all i ∈ N .
Intuitively, the role of MAJ 1 i is to limit the number of possible equivalence classes of * i and, hence, * * i . Indeed, suppose that we have

(x i , a -i ) P (y i , b -i ) and Not[(y i , a -i ) P (x i , b -i )]
. This means that the preference difference between y i and x i is not larger than the preference difference between x i and y i . In a strict concordance relation, this is only possible if x i P i y i . But this means that no preference difference is larger than the preference difference between x i and y i . Hence if (z i , c -i ) P (w i , d -i ), we must have that (x i , c -i ) P (y i , d -i ), as required by MAJ 1 i . The interpretation of MAJ 2 i is dual: if the preference difference between y i and x i is not larger than the preference difference between x i and y i , then it is the smallest possible preference difference. The impact of these two conditions is made precise in the following:

Lemma 2 ARC 1 i , ARC 2 i , MAJ 1 i and MAJ 2 i hold iff the relation * * i is a weak order having at most three distinct equivalence classes.

Relations P for which all relations * * i have at most three distinct equivalence classes are intuitively quite close from strict concordance relations. In such relations, the preference between x and y only depends on the subsets of attributes favoring x or y in terms of the asymmetric relation P i . It does not depend on preference differences between the various levels on each attribute besides the distinction between levels indicated by P i . It is not difficult to show that, if a relation P has a representation in model (M ) and is such that all * * i have only three distinct equivalence classes, it will have a representation in model (M ) in which all functions p i take at most three distinct values. Using such a representation, the relations P i defined letting x i P i y i ⇔ p i (x i , y i ) > 0 are indeed asymmetric and capture all the information contained in the relations * * i . This is at the heart of the characterization of strict concordance relations proposed in Bouyssou and Pirlot (2005c). We have:

Theorem 2 Let P be a binary relation on X = n i=1 X i . Then P is a strict concordance relation iff it is asymmetric and satisfies ARC 1, ARC 2, MAJ 1 and MAJ 2. Furthermore, conditions ARC 1, ARC 2, MAJ 1 and MAJ 2 are independent in the class of all asymmetric relations on X.

We have shown in Bouyssou and Pirlot (2005a) that this approach to concordance relations was more general than the one based on the use of a "noncompensation" condition as in, e.g., [START_REF] Bouyssou | Noncompensatory and generalized noncompensatory preference structures[END_REF] or [START_REF] Fargier | Modélisation des préférences par une règle de concordance généralisée[END_REF].

Strict outranking relations

Theorem 2 characterizes SOR in which discordance plays no role. In order to allow for possible discordance effects, some of the conditions used in Theorem 2 have to be weakened. It is not difficult to see that a SOR always satisfies conditions ARC 1, ARC 2, MAJ 1. Hence, condition MAJ 2 is the natural candidate for such a weakening. Indeed, condition MAJ 2 i may be violated due to discordance effects. Suppose that (x i , a -i )

P (y i , b -i ), (w i , a -i ) P (z i , b -i ), (y i , c -i ) P (x i , d -i ), with x i P i y i , w i P i z i while Not[x i V i y i ] and w i V i z i . It may happen that we have Not[(y i , a -i ) P (x i , b -i )]. But, since w i V i z i , it is also impossible that (z i , c -i ) P (w i , d -i
) and MAJ 2 i is violated. It is easy to convince oneself that such violations may only occur if w i V i z i . Therefore, if we require that (z i , e -i ) P (w i , f -i ), for some e -i , f -i ∈ X -i , we are sure that this will never happen. This is precisely what condition MAJ 3 i requires.

Definition 5 (Condition MAJ 3) Let P be a binary relation on a set X = n i=1 X i . This relation is said to satisfy MAJ 3 i if

(x i , a -i ) P (y i , b -i ) and (w i , a -i ) P (z i , b -i ) and (y i , c -i ) P (x i , d -i ) and (z i , e -i ) P (w i , f -i )                    ⇒    (y i , a -i ) P (x i , b -i ) or (z i , c -i ) P (w i , d -i ), for all x i , y i , z i , w i ∈ X i and all a -i , b -i , c -i , d -i , e -i , f -i ∈ X -i . We say that P satisfies MAJ 3 if it satisfies MAJ 3 i for all i ∈ N .
It is clear that MAJ 3 i weakens MAJ 2 i . The intuition behind this condition is that allowing for discordance effects possibly creates two new equivalence classes of * * i . When * * i has five equivalence classes, the lowest equivalence class will contain the pairs (x i , y i ) such that y i V i x i . On the contrary, the highest possible class will contain the pairs (x i , y i ) such that x i V i y i (they will not play a role that is different from the pairs in the second equivalence class, i.e., the pairs (z i , w i ) such that z i P i w i but Not[z i V i w i ]). Condition MAJ 3 i formalizes an idea exposed in Bouyssou and Pirlot (2002a). It is similar in spirit to a condition used in [START_REF] Greco | Axiomatic basis of noncompensatory preferences[END_REF].

In presence of the other conditions, going from MAJ 2 i to MAJ 3 allows to go from a relation * * i having at most three equivalence classes to a relation * *

i than can have up to five, the last equivalence class modeling discordance effects. The main result in this paper says that weakening MAJ 2 to MAJ 3 in Theorem 2 is exactly what is needed to characterize SOR. We have:

Theorem 3 Let P be a binary relation on X = n i=1 X i . Then P is a SOR iff it is asymmetric and satisfies ARC 1, ARC 2, MAJ 1 and MAJ 3. Furthermore, conditions ARC 1, ARC 2, MAJ 1 and MAJ 3 are independent in the class of all asymmetric relations on X.

As we shall see below, it is not difficult to strengthen this result imposing additional conditions that will imply that both P i and V i are strict semiorders.

If we neglect the question of the additive representation of the importance relation between coalitions of attributes using weights, the above theorem may therefore be seen as giving an axiomatic characterization of the binary relations that can be obtained with TACTIC. Within the framework of model (M ), their main distinctive characteristics lie in conditions MAJ 1 and MAJ 3 implying that, on each attribute, only very few classes of preference differences are taken into account.

Discussion

The main contribution of this paper was to propose a characterization of SOR within the framework of model (M ) that includes many different types of aggregation models as particular cases. Our characterization has emphasized the main specific feature of SOR with model (M ), i.e., the option not to distinguish a rich preference difference relation on each attribute: only five distinct classes of preference differences on each attribute, with the lowest class playing a very particular role, are allowed in a SOR. This is in line with the common view of outranking methods as being "more ordinal" than the additive model (A).

Theorem 3 can be seen as giving a sound theoretical foundation to outranking methods aiming at building a strict preference relation using the CNDP, as in the TACTIC method [START_REF] Vansnick | On the problems of weights in MCDM (the noncompensatory approach)[END_REF]. The results in this paper can be extended in several directions.

First, our definition of SOR does not require the relations P i and V i to possess any remarkable property besides asymmetry and the fact that V i ⊆ P i . This is at variance with what is done in most outranking methods. (remember, e.g., that in TACTIC both P i and V i are strict semiorders). Following the analysis in [START_REF] Bouyssou | Additive difference' models without additivity and subtractivity[END_REF], it is not difficult to tackle this case replacing in model (M ), the terms p i (x i , y i ) by terms ϕ i (u i (x i ), u i (y i )) where u i are real-valued functions on X i , ϕ i are real-valued functions on u i (X i ) 2 that are skew symmetric and nondecreasing in their first argument. Adding conditions MAJ 1 and MAJ 3 to the conditions underlying such a model allows to characterize SOR in which P i and V i are strict semiorders.

Second, we restricted our attention here to an asymmetric relation P interpreted as strict preference. It is not difficult to extend our analysis to reflexive relations, interpreted as "at least as good as" relations, for which: x S y ⇔ [S(x, y) ¤ S(y, x) and V (y, x) = ∅], where S is a reflexive binary relation on X, S i is a complete binary relation on X i , ¤ a relation on 2 N that is compatible with set inclusion and S(x, y) = {i ∈ N : x i S i y i } and V i is included in the asymmetric part of S i . Such an analysis, requires to distinguish "indifference" from "incomparability" and raises interesting duality questions. It is conducted in Bouyssou and Pirlot (2005a) when all relations V i are empty. This is generalized in Bouyssou and Pirlot (2005b) to cope with discordance effects.

Appendix: proofs Lemma 1 and Theorem 1

See Bouyssou and Pirlot (2003, Lemma 4) and Bouyssou and Pirlot (2003, Theorem 1).

Lemma 2

The proof of Lemma 2 will use the following definitions and results.

Definition 6 Let P be a binary relation on a set X = n i=1 X i . This relation is said to satisfy:

U C i if (x i , a -i ) P (y i , b -i ) and (z i , c -i ) P (w i , d -i )    ⇒    (y i , a -i ) P (x i , b -i ) or (x i , c -i ) P (y i , d -i ), LC i if (x i , a -i ) P (y i , b -i ) and (y i , c -i ) P (x i , d -i )    ⇒    (y i , a -i ) P (x i , b -i ) or (z i , c -i ) P (w i , d -i ), for all x i , y i , z i , w i ∈ X i and all a -i , b -i , c -i , d -i ∈ X -i . We say that P satisfies U C (resp. LC) if it satisfies U C i (resp. LC i ) for all i ∈ N . Lemma 3 1. U C i ⇔ [Not[(y i , x i ) * i (x i , y i )] ⇒ (x i , y i ) * i (z i , w i ), for all x i , y i , z i , w i ∈ X i ]. 2. LC i ⇔ [Not[(y i , x i ) * i (x i , y i )] ⇒ (z i , w i ) * i (y i , x i ), for all x i , y i , z i , w i ∈ X i ]. 3. [ARC 1 i , ARC 2 i , U C i and LC i ] ⇒ [ * *
i is a weak order having at most three equivalence classes].

Proof

See Bouyssou and Pirlot (2003, Lemma 11). P Lemma 4 Let P be a binary relation on X = n i=1 X i . If * * i is a weak order having at most three equivalence classes then P satisfies ARC 1 i , ARC 2 i , U C i and LC i .

Proof

The necessity of ARC 1 i and ARC 2 i follows from Part 3 of Lemma 1. Suppose that U C i is violated so that (x i , a -i )

P (y i , b -i ), (z i , c -i ) P (w i , d -i ), Not[(y i , a -i ) P (x i , b -i )] and Not[(x i , c -i ) P (y i , d -i )]. This implies (x i , y i ) * i (y i , x i ) and (z i , w i ) * i (x i , y i ). Using ARC 2 i , we easily obtain (z i , w i ) * * i (x i , y i ) * * i (x i , x i ) * * i (y i , x i ) * * i (w i , z i ), so that * * i has five equivalence classes. The proof with LC i is similar. P Lemma 5 1. U C i ⇒ MAJ 1 i , 2. LC i ⇒ MAJ 2 i , 3. [ARC 1 i and MAJ 1 i ] ⇒ U C i , 4. [ARC 2 i and MAJ 2 i ] ⇒ LC i .

Proof

Parts 1 and 2 are obvious since MAJ 1 i (resp. MAJ 2 i ) amounts to adding a premise to

U C i (resp. LC i ). Part 3. Suppose that U C i is violated so that (x i , a -i ) P (y i , b -i ), (z i , c -i ) P (w i , d -i ), Not[(y i , a -i ) P (x i , b -i )] and Not[(x i , c -i ) P (y i , d -i )], for some x i , y i , z i , w i ∈ X i and some a -i , b -i , c -i , d -i ∈ X -i .
We distinguish two cases.

• If (z i , a -i ) P (w i , b -i ), then (x i , a -i ) P (x i , b -i ), (z i , a -i ) P (w i , b -i ) and (z i , c -i ) P (w i , d -i ) imply, using MAJ 1 i , (y i , a -i ) P (x i , b -i ) or (x i , c -i ) P (y i , d -i ), a contradiction.

Proof (Theorem 2)

The necessity of ARC 1, ARC 2, MAJ 1 and MAJ 2 follows from Lemma 6 and Lemma 2. Sufficiency follows from Lemma 2 and Lemma 7.

The following examples show that, in the class of asymmetric relations, conditions ARC 1, ARC 2, MAJ 1 and MAJ 2 are independent.

Example 1 (ARC 1, MAJ 1, MAJ 2, Not[ARC 2 i ]) Let X = {a, b}×{x, y}.
Consider the asymmetric relation P on X containing only the two relations (a, x) P (b, y) and (a, y) P (b, x). We have, abusing notation:

• (a, b) * 1 [(a, a), (b, b), (b, a)] and • [(x, y), (y, x)] * 2 [(x, x), (y, y)]. ARC 2 2 is violated since Not[(x, x) * 2 (x, y)] and Not[(x, x) * 2 (y, x)].
It is clear that ARC 1 and ARC 2 1 hold. It is easy to see, using Parts 1 and 2 of Lemma 3, that U C and LC hold. Hence MAJ 1 and MAJ 2 hold in view of Parts 3 and 4 of Lemma 5.

Q Example 2 (ARC 1, ARC 2, MAJ 1,Not[MAJ 2 i ])
Let X = {a, b} × {x, y, z} and P on X be identical to the strict linear order (abusing notation in an obvious way):

(a, x) P (a, y) P (a, z) P (b, x) P (b, y) P (b, z), except that (a, z) I (b, x). It is easy to see that P is asymmetric. We have, abusing notation, x), (y, y), (z, z), (y, x), (z, y)] * 2 (z, x). Using Lemma 1, it is easy to check that P satisfies ARC 1 and ARC 2. It is clear that U C 1 , LC 1 and U C 2 hold. This shows that MAJ 1 1 , MAJ 1 2 and MAJ 2 1 hold. MAJ 2 2 is violated since we have (a, x) P (a, y), (a, x) P (a, z), (a, y) P (b, x) but neither (a, y) P (a, x) nor (a, z) P (b, x).

• (a, b) * 1 [(a, a), (b, b)] * 1 (b, a) and • [(x, y), (x, z), (y, z)] * 2 [(x,
Q Example 3 (ARC 1, ARC 2, MAJ 2, Not[MAJ 1 i ])
Let X = {a, b} × {x, y, z} and P on X be identical to the strict linear order (abusing notation in an obvious way):

(a, x) P (b, x) P (a, y) P (b, y) P (a, z) P (b, z), except that (b, x) I (a, y). It is easy to see that P is asymmetric. We have, abusing notation:

• (a, b) * 1 [(a, a), (b, b)] * 1 (b, a) and • [(x, z), (y, z)] * 2 (x, y) * 2 [(x, x), (y, y), (z, z)] * 2 [(y, x), (z, x), (z, y)].
Using Lemma 1, it is easy to check that P satisfies ARC 1 and ARC 2. It is clear that U C 1 , LC 1 and LC 2 hold. This shows that MAJ 1 1 , MAJ 2 1 and MAJ 2 2 hold. MAJ 1 2 is violated since (a, x) P (a, y), (a, x) P (a, z), (b, x) P (a, z) but neither (a, y) P (a, x) nor (b, x) P (a, y).

Q Example 4 (ARC 2, MAJ 2, MAJ 1, Not[ARC 1 i ])
Let X = {x, y, z} × {a, b, c} × {p, q, r} and P on X be empty except that: (x, a, r) P (y, b, p), (z, a, r) P (x, b, p), (x, a, p) P (y, b, q), (z, c, r) P (x, a, p), (y, a, r) P (x, b, p), (x, a, r) P (z, b, p), (x, a, r) P (x, b, p), (y, a, r) P (y, b, p), (

P (y, b, p), (x, a, q) P (y, b, r), (x, a, r) P (y, b, q). It is easy to check that P is asymmetric. We have, abusing notation:

• (x, y) * 1 [(x, x), (y, y), (z, z), (y, x), (x, z), (y, z), (z, y)] and (z, x) * 1 [(x, x), (y, y), (z, z), (y, x), (x, z), (y, z), (z, y)], while (x, y) and (z, x) are incomparable in terms of * 1 . • (r, p) * 3 [(p, p), (q, q), (r, r), (p, q), (q, p), (p, r), (q, r), (r, q)].

• (a, b) * 2 [(a,
For j ∈ {2, 3}, ARC 1 j , ARC 2 j , U C j and LC j are clearly satisfied. On attribute 1, it is easy to check that ARC 2 1 is satisfied, while ARC 1 1 is violated. Condition LC 1 holds so that MAJ 2 1 holds. It remains to check that condition MAJ 1 1 holds. Consider the first premise of MAJ 1 i (x i , a -i ) P (y i , b -i ). One of the two possible conclusions of MAJ 1 i is that (y i , a -i ) P (x i , b -i ).

Since (x, y) * 1 (y, x) , it is clear that MAJ 1 1 cannot be violated if x i = y and y i = x. Similarly, since (z, x) * 1 (x, z), MAJ 1 1 cannot be violated if

x i = x and y i = z.
Given the structure of * 1 , there are several cases to examine. Suppose first that (x i , y i ) in the first premise is equal to (x, y). If the first premise is taken to be (x, a, r) P (y, b, p) then MAJ 1 1 cannot be violated since we have (y, a, r) P (x, b, p). Now if any other premise is taken, there will not be any (z i , w i ) satisfying the second premise. A similar reasoning shows that if (x i , y i ) is equal to (z, x), then MAJ 1 1 cannot be violated. Suppose now that (x i , y i ) in the first premise is distinct from (x, y), (y, x), (z, x) and (x, z). Then the only possibility is take a -i = (a, r) and b -i = (b, p). Therefore MAJ 1 1 cannot be violated because, for all α, β ∈ X 1 , we have (α, a, r) P (β, b, p) and (β, a, r) P (α, b, p). Hence, MAJ 1 1 holds.

Q P

Theorem 3

The proof will use the following results.

Lemma 8 Let P be a binary relation on X = n i=1 X i . If P is a SOR then it satisfies ARC 1, ARC 2, MAJ 1 and MAJ 3.

Proof

[ARC 1] Let £, P i , V i be a representation of P as a SOR. Suppose that (x i , a -i ) P (y i , b -i ), (z i , c -i ) P (w i , d -i ). This implies that Not[y i V i x i ] and Not[w i V i z i ]. Suppose that y i P i x i . The definition of a SOR implies that (z i , a -i ) P (w i , b -i ). If x i P i y i , the definition of a SOR implies (x i , c -i ) P (y i , d -i ). If x i I i y i and z i S i w i , the definition of a SOR implies (z i , a -i ) P (w i , b -i ). If x i I i y i and w i S i z i , the definition of a SOR implies (x i , c -i ) P (y i , d -i ).

[ARC 2] Suppose that (x i , a -i ) P (y i , b -i ), (y i , c -i ) P (x i , d -i ). This implies that Not[y i V i x i ] and Not[x i V i y i ].

Suppose that x i P i y i . If z i P i w i , we know that Not[w i V i z i ] and the definition of a SOR leads to (z i , a -i )

P (w i , b -i ). If w i P i z i or w i I i z i , we know that Not[z i V i w i ]. The definition of a SOR leads to (w i , c -i ) P (z i , d -i ).
The proof is similar if we suppose that y i P i x i .

Suppose now that x i I i y i . If z i P i w i or z i I i w i , we know that Not[w i V i z i ] and the definition of a SOR implies that (z i , a -i ) P (w i , b -i ). If w i P i z i , we know that Not[z i V i w i ] and the definition of a SOR implies that (w i , c -i ) P (z i , d -i ).

[MAJ 1] Suppose that (x i , a -i ) P (y i , b -i ), (z i , a -i ) P (w i , b -i ) and (z i , c -i ) P (w i , d -i ). This implies that Not[y i V i x i ] and Not[w i V i z i ].

If y i P i x i , we know that Not[x i V i y i ] so that we have (y i , a -i ) P (x i , b -i ), using the definition of a SOR. If x i P i y i , we know that Not[y i V i x i ] so that we have (x i , c -i ) P (y i , d -i ), using the definition of a SOR. If x i I i y i , the definition of a SOR implies (y i , a -i ) P (x i , b -i ).

[MAJ 3] Suppose that (x i , a -i ) P (y i , b -i ), (w i , a -i ) P (z i , b -i ), (y i , c -i ) P (x i , d -i ) and (z i , e -i ) P (w i , f -i ). By construction, we know that Not[

x i V i y i ], Not[y i V i x i ], Not[z i V i w i ] and Not[w i V i z i ]. If y i P i x i , the definition of a SOR implies (y i , a -i ) P (x i , b -i ). If x i I i y i , the definition of a SOR implies (y i , a -i ) P (x i , b -i ). If x i P i y i , the definition of a SOR implies (z i , c -i ) P (w i , d -i ).
P Lemma 9 Let P be a binary relation on X = n i=1 X i . If P satisfies ARC 1, ARC 2, MAJ 1 and MAJ 3, then, for all x i , y i , z i , w i , r i , s i ∈ X i , 1.

(x i , y i ) * i (y i , x i ) ⇒ (x i , y i ) * i (z i , w i ). 2. [(x i , y i ) * i (y i , x i ) * i (z i , w i )] ⇒ (r i , s i ) * i (z i , w i ). Furthermore, we have Not[(z i , a -i ) P (w i , b -i )], for all a -i , b -i ∈ X -i .

Proof

Part 1 follows from Lemma 3, since ARC 1 i and MAJ 1 i imply U C i .

Part 2. Suppose that, for some x i , y i , z i , w i , r i , s i ∈ X i , we have (x i , y i ) * i (y i , x i ) * i (z i , w i ) and (z i , w i ) * i (r i , s i ). This implies (x i , a -i ) P (y i , b -i ), Not[(y i , a -i ) P (x i , b -i )], (y i , c -i ) P (x i , d -i ), Not[(z i , c -i ) P (w i , d -i )] and (z i , e -i ) P (w i , f -i ), for some a -i , b -i , c -i , d -i , e -i , f -i ∈ X -i .

Using ARC 2, we know that (w i , z i ) * i (x i , y i ) so that (w i , a -i ) P (z i , b -i ). Using MAJ 3 i , (x i , a -i ) P (y i , b -i ), (w i , a -i ) P (z i , b -i ), (y i , c -i ) P (x i , d -i ), and (z i , e -i ) P (w i , f -i ) imply (y i , a -i ) P (x i , b -i ) or (z i , c -i ) P (w i , d -i ), a contradiction. Note that the contradiction is obtained as soon as (z i , e -i ) P (w i , f -i ), for some e -i , f -i ∈ X -i . This proves the second part of the assertion. P

Lemma 10 Let P be a binary relation on X = n i=1 X i satisfying ARC 1, ARC 2, MAJ 1 and MAJ 3. Define the relation P i on X i defined letting, for all x i , y i ∈ X i ,

x i P i y i ⇔ (x i , y i ) * i (y i , x i ).
We have:

1. P i is asymmetric and nonempty, 2. z i P i w i and x i P i y i imply (z i , w i ) ∼ * i (x i , y i ), 3. x i I i y i and z i I i w i imply (x i , y i )

∼ * i (z i , w i ) ∼ * i (y i , x i ) ∼ * i (w i , z i ) ∼ * i (a i , a i ),
4. z i P i w i and x i I i y i imply (z i , w i ) * i (x i , y i ) and (x i , y i ) * i (w i , z i ).

Proof

Part 1. We have

x i P i y i iff (x i , y i ) * i (y i , x i ). Since * i is asymmetric, it follows that P i is asymmetric. The influence of i ∈ N implies that there are x i , y i , z i , w i ∈ X i such that (x i , y i ) * i (z i , w i ). If (z i , w i ) * i (y i , x i ), we obtain (x i , y i ) * i (y i , x i ) so that x i P i y i . Suppose that (y i , x i ) * i (z i , w i ). It is impossible that (z i , w i ) *
i (x i , x i ) since, using the fact that * i is a weak order, this would imply (x i , y i ) * i (x i , x i ) and (y i , x i ) * i (x i , x i ), violating ARC 2. Hence we have (x i , x i ) * i (z i , w i ). Using ARC 2, this implies (w i , y i ) * i (x i , x i ) so that (w i , y i ) * i (z i , w i ) and w i P i z i . Hence, P i is not empty.

Part 2. Suppose that z i P i w i and x i P i y i . Using lemma 9, we have (z i , w i ) * i (x i , y i ) and (x i , y i ) * i (z i , w i ) so that (z i , w i ) ∼ * i (x i , y i ). Part 3. Suppose that x i I i y i and z i I i w i . Using the definition of P i and ARC 1, this implies (x i , y i ) ∼ * i (y i , x i ) and (z i , w i ) ∼ * i (w i , z i ). The conclusion follows from ARC 2.

Part 4. Suppose that z i P i w i and x i I i y i , so that (x i , y i ) ∼ * i (y i , x i ) and

(z i , w i ) * i (w i , z i ). If (x i , y i ) * i (z i , w i ), ARC 2 i implies (w i , z i ) * i (y i , x i ), a contradiction. Similarly if (w i , z i ) * i (x i , y i ), ARC 2 i implies (y i , x i ) * i (z i , w i ), a contradiction. P
Lemma 11 Let P be a binary relation on X = n i=1 X i satisfying ARC 1, ARC 2, MAJ 1 and MAJ 3. Define the relation V i on X i letting, for all x i , y i ∈ X i ,

x i V i y i ⇔ [(z i , w i ) * i (w i , z i ) * i (y i , x i ) for some z i , w i ∈ X i ] .
We have:

1. V i is included in P i . 2. z i V i w i and x i V i y i imply (w i , z i ) ∼ * i (y i , x i ). 3. z i P i w i , Not[z i V i w i ], x i P i y i and Not[x i V i y i ], imply (w i , z i ) ∼ * i (y i , x i ). 20 4. z i V i w i , x i P i y i , Not[x i V i y i ] imply (y i , x i ) * i (w i , z i ). Proof Part 1. We have x i V i y i iff (z i , w i ) * i (w i , z i ) * i (y i , x i ). Using ARC 2 i , (w i , z i ) * i (y i , x i ) implies (x i , y i ) * i (z i , w i ), so that (x i , y i ) * i (y i , x i ) and x i P i y i . Part 2. Suppose that x i V i y i and z i V i w i . By definition, we have (a i , b i ) * i (b i , a i ) * i (y i , x i ) and (c i , d i ) * i (d i , c i ) * i (w i , z i ) Using lemma 9, we know that, for all a -i , b -i ∈ X -i , we have Not[(y i , a -i ) P (x i , b -i )] and Not[(w i , a -i ) P (z i , b -i )], so that (y i , x i ) ∼ * i (w i , z i ).
Part 3. Suppose that z i P i w i and x i P i y i By definition, we have

(x i , y i ) * i (y i , x i ) and (z i , w i ) * i (w i , z i ). Suppose that (y i , x i ) * i (w i , z i ). This would imply (x i , y i ) * i (y i , x i ) * i (w i , z i ), contradicting the fact that Not[z i V i w i ]. Similarly it is impossible that (w i , z i ) * i (y i , x i ). Hence, we have (y i , x i ) ∼ * i (w i , z i ). Part 4. Suppose that z i V i w i , x i P i y i and Not[x i V i y i ]. We have (a i , b i ) * i (b i , a i ) * i (w i , z i ) and (x i , y i ) * i (y i , x i ). Supposing that (b i , a i ) * i (y i , x i ) would contradict the fact that Not[x i V i y i ]. Hence, we have (y i , x i ) * i (b i , a i ), so that (y i , x i ) * i (w i , z i ). P
Proof (Theorem 3) Necessity follows from Lemmas 8. We show sufficiency. Define the relation P i and V i on X i as in Lemmas 10 and 11. We have shown that P i is asymmetric and nonempty and that V i is included in P i . Consider any two disjoint subsets A, B ⊆ N and let:

A £ B ⇔ [x P y, for some x, y ∈ X such that P (x, y) = A and P (y, x) = B].

Suppose that x P y. Using Part 2 of Lemma 9, we know that V (y, x) = ∅. By construction, we have P (x, y) £ P (y, x).

Suppose now that V (y, x) = ∅ and P (x, y) £ P (y, x) and let us show that we have x P y. By construction, P (x, y) £ P (y, x) implies that there are z, w ∈ X such that z P w, P (x, y) = P (z, w) and P (y, x) = P (w, z).

For all i ∈ N such that z i I i w i , we have, by construction, x i I i y i so that, using Lemma 10, (x i , y i ) ∼ * i (z i , w i ). For all i ∈ N such that z i P i w i we have x i P i y i so that, using lemma 10, (x i , y i ) ∼ * i (z i , w i ).

For all i ∈ N such that w i P i z i , we have y i P i x i . By hypothesis, we have Not[y i V i x i ]. Because z P w, we have Not[w i V i z i ]. Using lemma 10, we know that (x i , y i ) ∼ * i (z i , w i ). Hence, we have (x i , y i ) ∼ * i (z i , w i ), for all i ∈ N so that z P w implies x P y.

It remains to show that £ is monotonic. Suppose that A £ B, so that, for some x, y ∈ X, P (x, y) = A, P (y, x) = B and x P y. Since x P y, we know that Not[y i V i x i ], for all i ∈ N . Suppose that C ⊇ A, B ⊇ D and

C ∩ D = ∅. Let E = C ∩ B, F = C \ [A ∪ B], so that C = A ∪ E ∪ F . Let G = B \ [D ∪ C], so that B = D ∪ F ∪ G. Let H = N \ [C ∪ B]
. Since P i is nonempty, choose on each i ∈ N , a i , b i ∈ X i such that a i P i b i .

Consider the following alternatives:

A E F D G H x x i x i x i x i x i x i y y i y i y i y i y i y i z x i a i a i x i a i x i w y i b i b i y i a i y i For all i ∈ A∪D ∪H, we clearly have (z i , w i ) ∼ * i (x i , y i ). For all i ∈ E ∪F , we have z i P i w i so that, using Lemma 9, (z i , w i ) * i (x i , y i ). For all i ∈ G, we have y i P i x i and z i I i w i . Using Lemma 10, this implies (z i , w i ) * i (x i , y i ). Hence, for all i ∈ N , we have (z i , w i ) * i (x i , y i ). Therefore, x P y implies z P w, so that we obtain C £ D.

It is clear that MAJ 2 i implies MAJ 3 i . Hence, in view of Theorem 2, the independence of the conditions will be established if we give an example of relation satisfying ARC 1, ARC 2, MAJ 1 and MAJ 3 i on all but one attribute.

Example 5 (ARC 1, ARC 2, MAJ 1, Not[MAJ 3 i ]) Let X = {x, y, z} × {a, b} × {p, q} and P on X be empty except that: (x, a, p) P (x, b, p), (x, a, p) P (x, b, q), (x, a, p) P (y, a, p), (x, a, p) P (y, a, q), (x, a, p) P (y, b, p), (x, a, p) P (y, b, q), (x, a, p) P (z, a, p), (x, a, p) P (z, a, q), (x, a, p) P (z, b, p), (x, a, p) P (z, b, q), (x, a, q) P (x, b, q), (x, a, q) P (y, a, q), (x, a, q) P (y, b, q), (x, a, q) P (z, a, q), (x, a, q) P (z, b, q), (x, b, p) P (y, b, p), (x, b, p) P (y, b, q), (x, b, p) P (z, b, p), (x, b, p) P (z, b, q), (x, b, q) P (y, b, q), (x, b, q) P (z, b, q), (y, a, p) P (x, b, p), (y, a, p) P (x, b, q), (y, a, p) P (y, b, p), (y, a, p) P (y, b, q), (y, a, p) P (z, a, p), (y, a, p) P (z, a, q), (y, a, p) P (z, b, p), (y, a, p) P (z, b, q), (y, a, q) P (x, b, q), (y, a, q) P (y, b, q), (y, a, q) P (z, a, q), (y, a, q) P (z, b, q), (y, b, p) P (z, b, p), (y, b, p) P (z, b, q), (y, b, q) P (z, b, q), (z, a, p) P (x, b, q), (z, a, p) P (y, b, p), (z, a, p) P (y, b, q), (z, a, p) P (z, b, p), (z, a, p) P (z, b, q), (z, a, q) P (y, b, q), (z, a, q) P (z, b, q).

It is easy to check that P is asymmetric. It is not difficult to see that we have, abusing notation,

• [(x, y), (x, z), (y, z)] * 1 [(x, x), (y, y), (z, z), (y, x), (z, y)] * 1 (z, x).

• (a, b) * 2 [(a, a), (b, b)] * 2 (b, a) and

• (p, q) * 3 [(p, p), (q, q)] * 3 (q, p). This shows that ARC 1, ARC 2 and MAJ 1 hold. It is easy to see that MAJ 2 2 and MAJ 2 3 hold so that MAJ 3 2 and MAJ 3 3 are satisfied. Condition MAJ 3 1 is violated since (x, a, p) P (y, a, p), (x, a, p) P (z, a, p), (y, a, p) P (x, b, p) and (z, a, p) P (x, b, q) but neither (y, a, p) P (x, a, p) nor (z, a, p) P (x, b, p). Q P

  a), (b, b), (c, c), (b, a), (b, c), (c, b), (a, c), (c, a)],

At least if the "weak preference" zone that is used is some ELECTRE methods is neglected. This zone that models an hesitation between strict preference and indifference is not easy to analyze from a theoretical point of view unless one has recourse to nonclassical logic as in[START_REF] Tsoukiàs | From concordance/discordance to the modelling of positive and negative reasons in decision aiding. In: Aiding Decisions with Multiple Criteria: Essays in Honour of[END_REF].

• If Not[(z i , a -i ) P (w i , b -i )]. Using ARC 1 i , (x i , a -i ) P (x i , b -i ) and

(z i , c -i ) P (w i , d -i ) imply (z i , a -i ) P (w i , b -i ) or (x i , c -i ) P (y i , d -i ).

Hence, we must have (x i , c -i ) P (y i , d -i ), a contradiction.

Part 4. Suppose that LC i is violated so that (x i , a -i )

We distinguish two cases.

Hence, we must have (z i , c The proof of Theorem 2 uses the following results.

Lemma 6 Let P be a binary relation on X = n i=1 X i . If P is a strict concordance relation then all relations * * i are weak orders having at most three equivalence classes.

Proof

See Bouyssou and Pirlot (2003, Lemma 6). P Lemma 7 Let P be an asymmetric binary relation on X = n i=1 X i . If all relations * * i are weak orders having at most three equivalence classes then P is a strict concordance relation.

Proof

See Bouyssou and Pirlot (2003, Lemma 7). P