1. Read Addendum after the conclusion.

The radially inertial form of a FL metric

) is important and avoids a mathematical mistake that is too often writtened in papers and books.

The general case

We start from the usual form (2) of the metric, which is expressed in the locally inertial frame for the event E 0 = (today, here). The change of variables (τ, ρ) →

What is our universe now ?

For the century of the formula 15 written by de Sitter

Michel Mizony

Institut Camille Jordan, Umr Cnrs 5208 Automn 2017

For * 1

Résumé

Starting with the Friedmann-Lemaître (FL) metric g F L of an isotropic universe, we give the radialy inertial form of this metric which is a generalized Gullstrand-Painlevé form of metric g GP . Then, for g GP , the equivariant stress-energy tensor is convenient because it has straightforward interpretation in term of velocity and potential. The energy and the entropy are well-defined. For each model for the universe, the osculating manifold is a de Sitter model. Moreover if the universe is the open cone of the future of a point (a big bang event) then this de Sitter model is an open, accelerated, one. So we could easily confront this model with local observations through the osculating de Sitter model, taking into account the observed SNIa and the Hubble parameter H(z) for redshifts z ≤ 2. The recent data about H(z) provide a tool to estimate cosmological parameters for the de Sitter models and their Milne limits ; we find : H o = 65 ± 2 km/s/M pc, Ω o = 0.05 ± 0.02 and an age = 15.2 ± 0.3 Gyr. In other words, our universe contains uniqualy baryonic matter.

mail : mizony@univ-lyon1.fr

Introduction

We describe the Universe by a Friedmann-Lemaître universe model (U, g F L ), i.e. spatially homogeneous and isotropic. Most often, g F L is expressed in its Robertson-Walker form

g F L = ds 2 F L = dτ 2 -R 2 (τ ) (dx 2 + f 2 k (x) dω 2 ), (1)
where f k (x) = x, sin(x), sinh(x) according to the sign k = 0, 1 or -1 of the spatial curvature, and where dω 2 is the element of spherical angle. We will give different expressions of the same metric g F L defined at the event E 0 =(here, today)≡ {τ = τ o , x = θ = φ = 0}.

It is convenient to start from the l ocally inertial form relative to E 0 , defined after defining ρ := R (τ o ) x = R o x, as

ds 2 F L = dτ 2 - R 2 (τ ) R 2 (τ o ) dρ 2 + R 2 (τ o ) f 2 k ( ρ R(τ o ) ) dω 2 . ( 2
)
The slight change with respect to (1) emphasizes that, rigorously, an inertial metric must have the Minkowskian form, and that, in (1), x is an angular coordinate. The model is characterized by a scale factor R(τ ), τ being the cosmic time. We have the Hubble parameter H(τ ) ≡ Ṙ(τ )/R(τ ), the deceleration parameter q(τ ) ≡ -R R Ṙ2 that we assume negative (corresponding to an accelerating universe), and the Einstein equation

H(τ ) 2 + k R(τ ) 2 = 8πGρ(τ ) 3 ≡ Ω(τ ) H 2 (τ ). (3)
We include in the density parameter Ω the contribution of the cosmological constant. Present values of these quantities are written with a zero index : Ω(τ 0 ) = Ω 0 , etc.

In section 2, the generalized Gullstrand-Painlevé form is given and the case of the de Sitter models is developped. The section 2.1 address the global problems, without considering an hypothetic equation of state, but only the entropy equation. The section 2.2 address the de Sitter models. In section 2.3 we address the de Sitter model as an osculating manifold to a flat ΛCDM model. The local observations developped in section 3 underline the usefulness and the suitability of the accelerated de Sitter models and also of the limit case of the Milne's models. After a discussion in section 4, the conclusion ends with revisiting some classical problems.

(τ, r ≡ R(τ ) f k [ ρ R(τo) ])
; with the help of the Einstein's equation [START_REF] Mizony | Cosmological effects in the local static frame[END_REF] this change of variables leads straightforwardly to the form ( [START_REF] Mizony | Cosmological effects in the local static frame[END_REF]) :

g GP = ds 2 = dτ 2 - (dr -H(τ ) r dτ ) 2 1 + (1 -Ω(τ )) H 2 (τ ) r 2 -r 2 dω 2 . ( 4
)
The metric g GP looks like a generalised Gullstrand-Painlevé metric but with an hyperbolic space part (k=-1).

A comoving galaxy is defined by dρ = 0, which implies dr = r H(τ ) dτ . This formulation of the Hubble law remains exact at any time. We note v = r H(τ ) the radial velocity. In the denominator the total energy (1 -Ω(τ )) H 2 (τ ) r 2 gives the potential Φ defined by 2 Φ = Ω(τ ) H 2 (τ ) r 2 so we have the well defined accelerating equation :

γ = dv dτ = ∂v ∂τ + v ∂v ∂r = ∂Φ ∂r + ∂Φ v ∂τ . (5)
The potential Φ being well defined, we can compute the stress-energy tensor ; but the inertial metric g GP has a cross term, so if we want a thermodynamic interpretation of this tensor (cf. Weinberg chap. 2.11) [START_REF] Weinberg | Gravitation and cosmology[END_REF], we must compute the equivariant form of the stress-energy tensor which is not symetric (it is an ordinary mathematical fact which comes from the cross term).

First, the components of the non null equivariant Einstein's tensor G µ ν with v and Φ, and using the identity (5) are given by

G 0 0 = -2 Φ+r ∂ ∂r Φ r 2 G 1 0 = 2 ∂ ∂τ Φ r G 1 1 = -2 vr ∂ ∂r Φ+vΦ+r ∂ ∂τ Φ r 2 v G 2 2 = G 3 3 = - 2 v 2 ∂ ∂r Φ+r ∂ 2 ∂r 2 Φ v 2 +v ∂ ∂τ Φ+rv ∂ 2 ∂r∂τ Φ-r( ∂ ∂r v) ∂ ∂τ Φ rv 2 (6)
We must underline that G 0 1 = 0 is equivalent to ∂v ∂τ + v ∂v ∂r = ∂Φ ∂r + ∂Φ v ∂τ , cf. [START_REF] Mannheim | Alternatives to dark matter and dark energy[END_REF]. We can notice also that the G 1 0 term is related to the entropy, and the conservation law G µ ν;µ = 0 is verified. Otherwise, the non null terms of the equivariant Einstein's tensor G µ ν with the 3 cosmological parameters H(τ ), Ω(τ ) and q(τ ), is even shorter, but it hides the accelerating equation ( 5) :

G 0 0 = -3 H (τ ) 2 Ω (τ ) G 1 0 = H (τ ) 3 r (Ω (τ ) + q (τ )) G 1 1 = G 2 2 = G 3 3 = -H (τ ) 2 (Ω (τ ) -2 q (τ )) (7)
After this study of the geometric tensor, the first member of Einstein's equations, let us pass to the second member of the equations G µ ν = -κ T µ ν where κ = 8πG N /c 4 where G N is the Newton constant ; the non-zero elements are :

T 0 0 = ρ T 1 0 = v(ρ + p) T 1 1 = T 2 2 = T 3 3 = -p, (8)
where ρ denotes a matter-energy density and p a "pressure". Note that p and ρ are always defined as the pressure and energy density measured by an observer in a locally inertial frame that happens to be moving with the fluid at the instant of measurement.

The cross term T 1 0 whose writing follows from the identity

G 1 0 = v (G 0 0 -G 1 1
), would translate to an "energy dissipation or exchange" of this thermodynamic fluid ; the nullity of T 1 0 is equivalent to the constancy of the entropy for a universe model, (cf. [START_REF] Weinberg | Gravitation and cosmology[END_REF] formula 15.6.13).

The fundamental consequence of these three forms of the same tensor is coming from the following query : What does the nullity of the G 1 0 , or T 1 0 terms imply ? We have two answers : first ∂ ∂τ Φ=0 i.e. H 2 (τ ) Ω(τ ) is constant, and second ρ + p = 0. Moreover, if, by analogy with thermodynamic, G 1 0 = 0 translates to a constant entropy then there exits no dissipation because the universe has no exterior part. So it may exists only an exchange between two or more fluids in the universe.

The are two cases : i) there exists only one fluid, moving with the inertial frame and we obtain all the de Sitter universe models, ii) there exists another non co-moving fluid which exchange energy with the comowing one ; this is possible, a radiative fluid is then a convenient choice.

We shall consider the first case in the next subsection, but before, we want to point out that the so called dust fluid, with a null pressure, does not fulfill this entropy condition. It is the case for the ΛCDM models which contain such a fluid. Where is the problem ? Too often a dust fluid is taken, a priori, based on a strong analogy with thermodynamic. It is taken with the covariant tensor coming from the diagonal Robertson-Walker metric ; this form of metric is not inertial, leading to a misunderstanding ; but a Gullstrand-Painlevé like form of this metric is inertial so we can only then apply the thermodynamic analogy with the equivariant stress-energy tensor as a consequence of the general relativity. So, if we have a dust fluid which is comoving then it has a pressure p and a density ρ such that p + ρ = 0 ; we think this fact to come from a basic principle of the general relativity which asserts that the gravitational mass is equal to the inertial one. Indeed the de Sitter expanding models fulfill the Mach principle which asserts that the inertial mass of a body comes from all the content of the universe ; if so, then we have a proof for main results about fluids in general relativity without the analogy with the thermodynamics but with the use of what is inherited from an inertial frame. Starting from another theoretical point of view, the conservation of global energy for the Universe, H. Telkamp [4] found the same result : the open, flat or closed de Sitter models.

The de Sitter case

As it appears that the de Sitter models for isotropic universes models are basic we developp this case but only for de Sitter models coming from a big bang. The study of the de Sitter manifold was made in 1917 ( [START_REF] De | Sitter On Einstein's Theory of Gravitation, and its Astronomical Consequences, Third Paper[END_REF]) and for the model of the expanding universe, de Sitter gives the metric which is very near of the Robertson-Walker form, it was long before Friedman or Lemaître, here his formula 15 with R=1/λ, ω = λ τ , etc. :

ds 2 = dτ 2 - sinh 2 λτ λ 2 (dα 2 + sinh 2 α dω 2 ) , (9)
where the curvature λ is a non-negative real number, so we have an infinity of de Sitter manifold. Each inertial form is given by :

ds 2 = dτ 2 - (dr -r H(τ ) dτ ) 2 1 + r 2 λ 2 sinh 2 λτ -r 2 dω 2 . ( 10
)
These two forms of the same metric are defined on the same open part of a de Sitter manifold which is isomorphic to the SO(1,4)/SO(1,3) manifold, where SO(1,4) is the ten dimensional de Sitter group and SO(1,3) is the Lorentz group, see [START_REF] Mizony | Semi-groupes de causalité et formalisme hilbertien de la mécanique quantique[END_REF] for details. The generalized Gullstrand-Painlevé form [START_REF] Perlmutter | Measurements of Ω and Λ from 42 high-redshift supernovae[END_REF] is introduced in ([3]). For these de Sitter models the function H 2 (τ ) Ω(τ ) is constant, more precisely :

λ 2 = H 2 (τ ) Ω(τ ) = -H 2 (τ ) q(τ ). (11)
Let us take the initial conditions at the event (now,here) for the three cosmological parameters H o , q o , Ω o ; as Ω o = -q o , then the set of de Sitter models is two dimensional. We choose H o and q o as parameters in order to confront these models with observations. The time τ = 0 is the time of the big-bang event, so τ o denotes "now".

The Milne models can be viewed as the limit of these de Sitter models when the curvature λ, tends to 0, H o being fixed ; one Milne model for each value of H o , one de Sitter model for each pair (λ, H o ).

As the redshift z is a useful cosmological observable, we will provide here some formulas. Lemma : Let us take the two parameters (H o , q o = -Ω o ) characterizing one of these models of de Sitter with negative curvature (0<Ω o < 1), we have :

i) The time of the emission of a photon received at τ o with a redshift z :

τ (z) = 1 H o √ -q o arcsinh( √ -q o √ 1 + q o (1 + z) ) (12)
and for z=0, the age of the universe

τ o = 1 H o √ -q o arcsinh( √ -q o √ 1 + q o ) . (13)
ii) The Hubble function z-> H(z) is merely :

H (z) = H o q o z 2 + 2 q o z + z 2 + 2 z + 1 (14)
iii) The angular distance is given by :

dA dS (z) = 1 -H o q o (1 + z) ((1 + z) -1 + (1 + q o )z(2 + z)) . (15)
These are classical formulas [START_REF] Mannheim | Alternatives to dark matter and dark energy[END_REF] formula 237, [START_REF] Mizony | Sur la forme de Painlevé d'une métrique à symétrie sphérique[END_REF] ;

The osculating de Sitter model to a flat ΛCDM one

A flat ΛCDM has two initial parameters : the Hubble parameter H o and the density parameter Ω m of the cocomoving fluid (dust without pressure and dark matter).

For the flat ΛCDM the angular distace is given by an integral formula :

dA ΛCDM (z) = 1 (1 + z) H o z 0 1 Ω m (1 + x) 3 + 1 -Ω m dx . ( 16
)
If we want that dA ΛCDM (z) ≡ dA dS (z) near z=0, then We have also an integral formula for the flat ΛCDM model :

q o = 3 2 Ω m -1.
τ (z) = 1/H o 1/(1+z) 0 1 Ω m /x + (1 -Ω m ) * x 2 dx . ( 17
)
For the figure 2 the same initial values are taken.

The theoretical error is very big for the age because we made an integration for all z and not only for z less than 0.6. Thus the misunderstanding about frames (comoving and inertial) became a big mathematical mistake if we consider the universe model as a whole. The problems are the same for the values coming from the WMAP collaboration ( [START_REF]Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations : Final Maps and Results[END_REF]). Two facts seem weird or puzzling : first the fact that the angular distance is decreasing after z > 1.5 and, second, the very accurate values given by the each team with, for example, ages with more three exact digits.

What confidence can then be given to studies which, based on the a priori of a flat ΛCDM model, give extremely accurate results both with respect to BAO (Baryon Acoustic Oscillations) and to cosmic background radiation because they involve large redshifts ? If we compare the functions z → H(z) for de Sitter models on the one hand and flat ΛCDM models on the other hand, the latter is of order √ z times the former. Another example near the redshift z = 1100 of the CMB : τ (1100) = 0.5 10 6 light years for the standard model and τ (1100) = 16 10 6 light years (32 times more), for the osculating de Sitter universe (but without radiation, so it's just an approximate calculus).

Is it also worth recalling that one of the consequences of the Machian nature of de Sitter models is, ipso facto, no resort to a hypothetical period of inflation is necessary to account for the isotropy of the cosmic background radiation and also for the primordial baryogenesis ( [START_REF] Mannheim | Alternatives to dark matter and dark energy[END_REF]).

The de Sitter models and astronomic observations

For the observations which concern the de Sitter and Milne models, three very interesting papers exist, based on studies of supernovae (SNIa) : in 1998, A. Riess and his team [START_REF] Riess | Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant[END_REF] ; in 2010, F. Farley [START_REF] Farley | Does gravity operate between galaxies ? Observational evidence reexamined[END_REF] and in 2016, V. Lukovic and all [START_REF] Lukovic | Is there a concordance value for H0 ?[END_REF] ; see also in 1999 [START_REF] Perlmutter | Measurements of Ω and Λ from 42 high-redshift supernovae[END_REF]. They pointed that the better values for the Hubble parameter are about from 63 to 65 km/s/Mpc for these models : i) in his historic paper of 1998, A. Riess said : The Hubble constants as derived from the MLCS method, 65.2 ± 1.3 km/s/Mpc, and from the template fitting approach, 63.8 ± 1.3 km/s/Mpc, are extremely robust and attest to the consistency [START_REF] Riess | Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant[END_REF], see also their emblematic figure that highlights many elements ; ii) in 2010, F. Farley gives us a nice proof that the Milne models explain the kinematic face of the SNIa observations and that H o is so around 63 km/s/Mpc ; see also [START_REF] Vishwakarma | A curious explanation of some cosmological phenomena[END_REF] and [START_REF] Nielsen | Marginal evidence for cosmic acceleration from Type Ia supernovae[END_REF]. iii) let us present only the wonderfull figure given by V. Lukovic and all in 2016 : . We have added, on the three panels of this figure, where are the Milne models, using pink color ; the de Sitter models are defined, on these plots, by Ω m = 0 and 0 < Ω Λ < 1. Let us remark that the BAO results are definitely not relevantt for our study, cf. the red line on the (Ω m =, Ω Λ ) plot, bottom left panel. Thus our Ω o must be less than 0.3 and 60 ≤ H o ≤ 68 (top panel), cf. also the paper of Buchert, Coley and others [START_REF] Buchert | Observational challenges for the standard FLRW model[END_REF].

What is now the observational problem ? The comoving density Ω o must be less than 0.3 and, more likely, even near 0 as Farley suggests. For a theoretical reason we have no need for dark energy as explained above via the inertial frame ; we have also no need for dark matter to explain the flat curves of spiral galaxies (a big mathematical mistake about the "exponential disk profile"), this is developed by many astrophysicists, see ( [START_REF] Mannheim | Alternatives to dark matter and dark energy[END_REF]) and a bibliography in ( [START_REF] Mizony | About the rotation curves of spiral galaxies, a program[END_REF]), ( [START_REF] Kroupa | The Dark Matter Crisis : Falsification of the Current Standard Model of Cosmology[END_REF]) and also Y. Sofue ([19]) formula (38). A remark, the conformal gravity and the Einstein gravity are equivalent for the de Sitter universes models. Therefore, the 4% of baryonic matter is allowable.

For the respect of general relativity, we want to test the de Sitter models with other data than these coming from SNIa data. The accuracy of these measures, although not shown here, is small (uncertainties from 5 to 25% for most points), but the number of measures has been well developed in recent years. A first small problem about these data, the smaller value of H(z) is H(0.12) = 68.6, but the functions τ (z) given by 12 and also 17 are increasing, statistics are valid if H o is less than 68.6, stricto sensu.

As the function H(z) is near linear for 0.07 ≤ z ≤ 2, and for the small densities Ω o , we just use the least-square method for some values of H o to compute the values of Ω o . Before for the Mine models we have :

The linear regression is given by H(z) = 59.8 (± 1.2) + 70.0 (± 1.6) z, so a mathematical minimum error around 2% on H(0) ; and for the Milne models by H(z) = 64.32 (± 0.75) (1 + z), so with a 1.2% error. These error bars are independant of the uncertainties on the data. For the de Sitter models we must have Ω o =q o non negative so H(z) ≤ H o (1 * z), the H o Milne model. Moreover Ω o ≥ 0.04 the baryonic density, that gives a lower bound to Ω o and with the data, a lower limit to H o .

These first results from the data about the function z->H(z) confirm those of F. Farley ( [START_REF] Farley | Does gravity operate between galaxies ? Observational evidence reexamined[END_REF]) about the Milne universe and agree with the analysis of V. Lukovic (3).

The results For a fixed H o among the set {63, 65, 67, 69}, by the least-square fit method, the H o -Milne curve is computed (in cyan color on the two following figures), after in black on figures, the development at order two of H(z) at z = 0 is computed also by the least-square fit method, the de Sitter H(z) tangent curve at zero is in green and the H(z)-de Sitter curve such that H(0) = H 0 and q 0 = -0.04, the minimum baryonic density, is in pink. As for H o = 63 km/s/Mpc, the better fit is obtained for Ω o = 0.02, this value is at the borderline and as for H o = 69 km/s/Mpc, the best fit is obtained for Ω o = 0.256, this value is to be rejected because the de Sitter models contain neither dark energy nor dark matter. Even if the case H o = 67 km/s/Mpc is also borderline the figure looks interesting.

H o =64.

Discussion

But, we do not have take care, for this study, of the uncertainties on the data. In a few years, the data will be more numerous and more accurate and so it will be possibe, with more theoretical statistics, to improve this first result.

Also we don't have to pay attention to the contribution of radiation ; even if this latter is tiny for a small redshift, for big redshift as the redshift 1100 for example, it would be necessary. If we added radiation, the scale factor R(τ ) of the FL metric is equal to b + c 1 exp(λ τ ) + c 2 exp(-λ τ ), with relations between the four constants ; R(τ ) appears as a light modidification of the de Sitter metric for z ≤ 2.

The results [START_REF] Kroupa | The Dark Matter Crisis : Falsification of the Current Standard Model of Cosmology[END_REF] rest upon the a priori that, for theoretical reasons, it exists only baryonic matter around 4%, but it could be supposed that it exists a little unseen matter or even a little dark matter, but not in halos around galaxies [START_REF] Kroupa | The Dark Matter Crisis : Falsification of the Current Standard Model of Cosmology[END_REF], perhaps in clusters of galaxies or elsewhere ; so the density Ω o would be bigger and, ipso facto, the Hubble value now H o . This is compatible with the data H o ≤ 67 i.e. Ω o ≤ 0.17, see the table of results and figure 2 above.

But a value of H o as hight as 67 is not compatible with results coming from SNIa based on local methods, cf. for example the works of G. Tammann and B Reindl [START_REF] Tammann | The luminosity of supernovae of type Ia from TRGB distances and the value of Ho[END_REF] who found H o = 63.7 ± 2.3km/s/Mpc or of V. Busti [START_REF] Vinicius | Evidence for a Lower Value for H0 from Cosmic Chronometers Data ?[END_REF] who found H o = 64.9 ± 4.2km/s/Mpc and also of J.-J. Wei, F. Melia and X.-F. Wu [START_REF] Wei | Impact of a Locally Measured H0 on the Interpretation of Cosmic-chronometer Data[END_REF].

"We emphasize here that the CMB estimates are highly model dependent" as Planck team said ( [START_REF]Planck intermediate results. XVI. "Profile likelihoods for cosmological parameters[END_REF] page 30), it is the same for the BAO.

Conclusion

In a first step, we have underlined a theoretical confusion : within a chart radially inertial, there is no need for dark energy. We have also recall why a huge mathematical mystake implies the needness for dark matter to explain the flatness of the curves of rotation for spiral galaxy as many papers said since a long time (25 years). Thus, working in inertial frame instead comobile one the de Sitter models for the Universe appears the good theoretical models. It is well-known that these models are very good for the interpretation of the supernovae (a kinematic effect). The recent data about the Hubble parameter H(z) was the ocasion to confront the de Sitter models with these data. The results are beyond all that could be expected ; no conflict with the SNIa approach. No inflation, no problem of stability, no mystery about all which seems dark, but in conflict with the ΛCDM models. The general relativity go on, even it remains many others problems, but the icing on the cake, the star HD140283 can extend his very long life quietly.

Among the problems to address : i) The BAO and the small fluctuations of the CMB, by using the inertial form of metric with radiation, a difficult problem even for the ΛCDM model [START_REF] Buchert | Observational challenges for the standard FLRW model[END_REF]. ii) The "Pioneer anomaly" which does not come from the dynamics of the universe model [START_REF] Mizony | Cosmological effects in the local static frame[END_REF] but likely from the kinematics of the Milne model as it is also the case for the SNIa. iii) The baryogenesis in this inertial frame for the Universe, and particularely the study of baryogenesis of the "lithium problem". iv) If we want to glue together the quantum mechanics and the general relativity, the invariant de Sitter group is unavoidable ; for this goal, the Lie semigroup of causalty of the de Sitter group is very interesting [START_REF] Dyson | Missed opportunities[END_REF], [START_REF] Mizony | Semi-groupes de causalité et formalisme hilbertien de la mécanique quantique[END_REF], and maybe a good step for a well-posed problematic to address this question.

Addendum : Let us return to the work of de Sitter : On the curvature of space [START_REF] De | Sitter On Einstein's Theory of Gravitation, and its Astronomical Consequences, Third Paper[END_REF] where the main metric (2B) he study is his metric [START_REF] Buchert | Observational challenges for the standard FLRW model[END_REF], i.e. our metric 9. In this wonderful paper W. de Sitter said among others : §7-"If in the future it should be proved that very distant objects have systematically positive apparent radial velocities, this would be an indication that the system B, and not A, would correspond to the truth." §8-"We must then for "ρ o " take the density not within the galactic system, but the average density over a unit of volume which is large cornpared with the mutual distances of the galactic systems. With the numerical data adopted above, this leads to R = 5 10 13 , and there would then be more than a billion galactic systems."

this problem, why ? We shall not confound "plausible" and "probable". Indeed with a bayesian method we test if an hypothesis is possible, plausible. The answer is yes or no, and if yes then the method give us the better parameters for this hypothesis. Suppose now that we have two very different hypothesis which could furnish a good answer, so we obtain two different explanations to interpret the data. For the two cases we don't obtain a probability concerning each hypothesis, we obtain the better possibility for each hypothesis. Each hypothesis may be wrong and at most one is right.

For our problem let us test the four following hypothesis : -a) The ΛCDM H(z) is valid. The least square method is enough to test these hypothesis and the answer is straitghtforward : a) and b) are plausible but a)bis and b)bis are not possible. But J.-J. Wey and others, in a recent paper [START_REF] Wei | Impact of a Locally Measured H0 on the Interpretation of Cosmic-chronometer Data[END_REF], add that two plausible hypothesis are on an equal footing, by a right use of a bayesian method. Now suppose that in two years the astronomers get two new data for two z greater than 2 and obtained by local measures, then we have 3 cases : 1-the new data are in huge favor of the a) hypothesis such that the hypothesis b) became not possible, unlikely ; 2-the new data are in huge favor of the b) hypothesis such that the hypothesis a) became not possible ; 3-the new data don't favor the same hypothesis which is the more likely case. Notice that in the case 1-or 2-we don't have a probability, we have only one possibility, no more.

For the study of the SNIa data it is the same problem, the two hypothesis, flat ΛCDM and de Sitter models, are equally plausible after numerous tests with bayesian methods, c.f. the beautiful paper of J. T. Nielsen and others [START_REF] Nielsen | Marginal evidence for cosmic acceleration from Type Ia supernovae[END_REF] who found -q o = 0.094. As Einstein wanted always a beautiful and simple solution for a problem, we think that a de Sitter model 18 is welcome for the Universe, inside the general relativity theory, because it solve several problem : no useless dark matter, no enigmatic dark energy, no strange inflation, but an understandable inertial mass via the Mach principle, and the mysterious cosmological constant which translates merely the comobile matter ; moreover the kinematic is well posed. For this we were careful to do confusion nor between inertial frame and comobile frame and nor between plausible and likely. We notice that our result, H o = 65 ± 2 km/s/M pc for de Sitter and Milne models, are compatible with their four results ; moreover, they said also that they could not reject the non existence of a transition phenomenon between a non accelerating period and a recent accelerating one for the Universe, a phenomenon which does not exist for de Sitter models.

We could also notice the very recent paper from G. Paturel and others [START_REF] Georges | Hubble law : measure and interpretation[END_REF] who, starting from the study of the redshift z with the frequential point of view z ν , instead the usual z λ , finded the Milne model better to the study of the cepheïds and that give them H o ≈ 63 km/s/M pc, c.f. [START_REF] Sanejouand | A simple Hubble like law in lieu of dark energy[END_REF].

  For example, for the recent Planck ΛCDM model with Ω m = 0.3 and H o = 68, cf. ([7]), we have the de Sitter osculating model today defined by q o = -0.55 and the same Hubble value, here figure 1 :

  One year ago Duan and all give us several recent data about the function z → H(z) ([17]) ; thirty eight data, thanks to astronomers. But eight are coming from the erroneous mathematical use of the BAO. So we have thirty data coming from local measurements, in the table of the values (z,H(z)) : table := [[.7e-1, 69.], [.9e-1, 69.], [.12, 68.6], [.17, 83.], [.179, 75.], [.199, 75.], [.20, 72.9], [.27, 77.], [.28, 88.8], [.352, 83.], [.38, 83.], [.4, 95.], [.40, 77.], [.425, 87.1], [.45, 92.8], [.478, 80.9], [.48, 97.], [.593, 104.], [.68, 92.], [.781, 105.], [.875, 125.0], [.88, 90.0], [.9, 117.], [1.037, 154.], [1.3, 168.], [1.363, 160.], [1.43, 177.], [1.53, 140.], [1.75, 202.], [1.965, 186.]].

Figure 2 :

 2 Figure 2 : for H o = 67 km/s/Mpc.

Figure 3 :

 3 Figure 3 : for H o = 65 km/s/Mpc.

  -a)bis The ΛCDM H(z) is valid and H o ≈ 64 km/s/M pc. -b) The de Sitter H(z) is valid. -b)bis The de Sitter H(z) is valid and H o ≈ 71 km/s/M pc.

7 Annex 2 :

 2 News results about the Hubble parameterThe 2017/11/09 Yu and all[START_REF] Yu | Hubble Parameter and Baryon Acoustic Oscillation Measurement Constraints on the Hubble Constant, the Deviation from the Spatially-Flat Λcdm Model, The Deceleration-Acceleration Transition Redshift, and Spatial Curvature[END_REF] have published a new study on the z → H(z) data, with 31 local measures and 6 others measures with BAO method. They study, by bayesian methods the best plausible H o values with two important options : 1local data or all data, 2-data with z ≤ 2 or not. Here their main results : H o = 66 ± 4 km/s/M pc, H o = 64.3 ± 3.5 km/s/M pc, H o = 65.5 ± 4.8 km/s/M pc, H o = 67 ± 4.4 km/s/M pc.

  3 km/s/Mpc : Milne, Ω o = 0 Age(Milne, 64.3)= 15.2 Gyr H o =65 km/s/Mpc : 0.03<= Ω o <= 0.07 15.2 Gyr <= Age(65)<= 15.4 Gyr H o =67 km/s/Mpc : 0.06<= Ω o <= 0.17 14.9 Gyr <= Age(67)<= 15.5 Gyr H o =69 km/s/Mpc : 0.12<= Ω o <= 0.26 14.8 Gyr <= Age(69)<= 15.6 Gyr

Annex 1 : About bayesian methodsIn several papers, authors work with bayesian methods to study this cluster of 30 points coming from the data z → H(z). Is it necessary ? We think that it is no for


Gianluca D ' Antonio

Joel Sauza Bedolla

Akmal Rustamov

Franco Lombardi

Paolo Chiabert
email: paolo.chiabert@polito.itakmal.rustamov@studenti.polito.it

The role of Manufacturing Execution Systems in supporting Lean Manufacturing

Keywords: Manufacturing Execution Systems, Lean Manufacturing, Information tools

In order to deal with global competition and increased customers expectation, companies must improve the efficiency of their manufacturing processes. Mainly, two approaches are available: the implementation of lean manufacturing practices and the deployment of information tools for data analysis. For a long time, these two strategies have been considered mutually exclusive; recently, it was understood that information concerning the manufacturing process is mandatory for the effective implementation of lean practices. This work aims at showing how Manufacturing Execution Systems (MES) can support the lean manufacturing paradigm into a medium size enterprise. The case study of a company in the supply chain of automotive components is presented to provide evidence about the role of MES and to highlight possible criticalities occurring during its implementation.

Introduction

Today enterprises are driven by a challenging market: on the one hand, companies have to deal with competitors spread all over the world, operating in a variety of different conditions. On the other hand, the expectation of customers is increasing: product with enhanced quality, innovative features and higher customization level must be delivered in shorter times and at a lower cost. Hence, to maintain and improve their competitive advantage, leading organizations in different industrial sectors need to improve process optimization and efficiency.

One of the initiatives that a company may undertake to improve its competitiveness is the implementation of Lean Manufacturing practices; this term has been first introduced by Womack [START_REF] Womack | The machine that changed the world[END_REF] to describe the working philosophy deployed in Japanese companies, with particular concern for Toyota. The essence of this methodology is the elimination of waste and non-productive process, in order to focus on value added operations and produce high-quality products, at the customers demand pace, with little waste.

Another approach is the deployment of automation and Information Technology (IT) tools, which allow to improve process planning and control, as well as to enhance the performance of each step of the manufacturing process. The landscape of the existing software classes and their purposes has been changing over the years, and is still evolving at a high pace. Today, beside the improvement of existing tools, huge efforts are made for the integration and the communication between different information tools and among systems deployed by different companies (for example, among firms belonging to the same supply chain).

For several years, Lean Manufacturing has been considered as opposed to the deployment of IT tools and their integration within and between firms [START_REF] Ward | Impact of Information Technology Integration and Lean/Just-In-Time Practices on Lead-Time Performance[END_REF][START_REF] Powell | MES Support for Lean Production[END_REF]. On one side, the philosophy of lean is "less is better": inventory, variability, material handling, options and choices must be reduced as much as possible to improve the performance of a company. Conversely, the philosophy of IT is "more is better": IT tools allow to better manage large amount of data, with increased flexibility, functions and features [START_REF] Ward | Impact of Information Technology Integration and Lean/Just-In-Time Practices on Lead-Time Performance[END_REF].

According to Ward and Zhou [START_REF] Ward | Impact of Information Technology Integration and Lean/Just-In-Time Practices on Lead-Time Performance[END_REF], the two classes of instruments are complementary both in the concept and in the application: IT tools can be considered a kind of higher-level planning system, while lean practices were related to shop-floor control and execution activities.

Nevertheless, in order to define improvement strategies and assess their impact, the collection and analysis of information is mandatory: the deployment of methodologies for Lean Manufacturing cannot exclude the use of IT tools. Hence, in the last years, IT instruments have been widely adapted, upgraded and expanded to deal with process monitoring and control activities.

The focus of this paper is on the role of Manufacturing Execution Systems (MES) -a class of IT tools -to support the implementation of Lean Manufacturing practices. The remainder of this paper is organized as follows: in Section 2 the Lean Manufacturing paradigm is introduced and some basic knowledge concerning MES is provided. In Section 3, the role of MES to support Lean Manufacturing is discussed. In Section 4, the case study of MES implementation into a medium size enterprise and the support in achieving Lean Manufacturing are presented. In Section 5, conclusive remarks and hints for future developments are discussed.

Background

Lean Manufacturing

Muda is a Japanese word meaning "waste": it is referred to any human activity that needs dedicated resources, but does not create value. Taiichi Ohno, a Toyota executive, introduced the concept of muda in manufacturing, to label all the activities that require resources to take place, but do not add value to the process or to the product [START_REF] Ohno | Toyota Production System: Beyond Large-Scale Production[END_REF]. In particular, he defined seven classes of waste that typically affect a manufacturing process. Namely: (i) Overproduction; (ii) Waiting; (iii) Transport; (iv) Extra processing; (v) Inventory; (vi) Motion; (vii) Defects. These wastes do not add value to the product, hence customers are not willing to pay for them. Manufacturers have to become less wasteful in order to be more profitable and improve their competitiveness. A systemic method to eliminate muda is Lean Manufacturing [START_REF] Womack | Lean thinking: banish waste and create wealth in your corporation[END_REF]. It is an approach inspired by Japanese management methods, in particular by the Toyota Production System. The effectiveness of this approach led to interest among European and American companies. The first attempts in exporting Japanese production methods have been made in the 1970s [START_REF] Womack | The machine that changed the world[END_REF]. At that time, the experience was not successful; according to the prevalent opinion, this was due to cultural differences and to the unique social context of Japan. Nonetheless, in the early 1990s, deeper empirical and theoretical research has been performed and the basis for the first definition of lean manufacturing has been provided [START_REF] Womack | The machine that changed the world[END_REF][START_REF] Houy | ICT and Lean Management: Will They Ever Get Along[END_REF]. This approach is structured in the following 5 principles: (i) Specify value; (ii) Identify the Value Stream; (iii) Flow; (iv) Pull; (v) Perfection.

At the state of the art, the interaction between lean practices and information tools has not yet thoroughly studied. A research performed by Moyano-Fuentes et al [START_REF] Moyano-Fuentes | Impact of use of information technology on lean production adoption: evidence from the automotive industry[END_REF] showed that companies need to increase the degree of use of IT tools in order to implement lean manufacturing practices. Riezebos et al. [START_REF] Riezebos | Lean Production and information technology: Connection or contradiction[END_REF] reviewed the role of IT in achieving lean production over three strategic topics: production control systems, computer aided production systems and maintenance processes. Powell et al. [START_REF] Powell | The concurrent application of lean production and ERP: Towards an ERP-based lean implementation process[END_REF] analyzed ERP implementation processes and proposed a framework for ERP-based lean implementation, with particular concern for Small and Medium Enterprises (SMEs) [START_REF] Powell | Lean production and ERP systems in smalland medium-sized enterprises: ERP support for pull production[END_REF].

Manufacturing Execution Systems

Manufacturing Execution Systems are IT tools commonly deployed in companies involved in traditional manufacturing. A MES enables information exchange between the organizational level, commonly supported by an Enterprise Resource Planning (ERP), and the control systems for the shop-floor, usually consisting in several, different, highly customized software applications [START_REF] Meyer | Manufacturing Execution Systems (MES): Optimal Design, Planning, and Deployment[END_REF]. A schematic of MES positioning in the framework of information tools supporting manufacturing is provided in Fig. 1.

The tasks in charge of a MES were proposed by Mesa International [START_REF]MESA International: MES Functionalities & MRP to MES Data Flow Possibilities[END_REF] in 1997, and then defined through the standards ISA95 [START_REF]ISA95: Enterprise -Control System Integration. Part 1: Models and Terminology[END_REF] and IEC62264 [START_REF]Enterprise-control system integration[END_REF]. A MES has two principal purposes. First, the system has to deal with the top-down data flow: the requirements and the necessities provided by the organizational level must be transformed into an optimal sequence planning meeting such targets. This sequence must be identified by best exploiting the available resources (such as staff, machines, materials, inventory) and taking into account the constraints of the process, such as processing and setup times, and workstations capacity.

The second aim of a MES is to manage the bottom-up data flow. Data concerning process performance and product quality can be gathered at the shop-floor level; the role of MES is to collect such data, analyze it through appropriate mathematical techniques, and extract a synthetic information to provide the business level with an exhaustive picture of the current state of the process. Possibly, the analysis should be performed in real-time, in order to make decisions to control the process with the necessary rapidity. Recently, the development of low-cost, small, easily available sensors led to a great diffusion of monitoring systems to assess product quality and process performance, and to support the improvement of production process.

MES were initially deployed in industries focused in the fields of chemistry and pharmaceutics; then, the spread of such systems increased, but for long time this tool has been considered useful only for large industries. In recent years, it was understood that the benefits provided by a MES can profitably support even smaller companies [START_REF] Gaxiola | Proposal of Holonic Manufacturing Execution Systems Based on Web Service Technologies for Mexican SMEs[END_REF]. According to a recent survey [START_REF]European Commission, Statistics on small and medium-sized enterprises[END_REF], in Europe SMEs represent around the 99% of the enterprises; thus, the potential of MES application is very huge. However, in the implementation of a MES, SMEs and large companies have to face different issues. In Section 4, the attention will be focused on the SME perspective.

According to the last report of Markets and Markets [START_REF]Markets and Markets: Manufacturing Execution System Market by Deployment Type[END_REF], the MES turnover is expected to reach USD 12.6 Billion by 2020 at a CAGR 1 of 10.85% between 2015 and 2020. For a long period, Europe has been the main market for MES; nevertheless, in recent times the Asia Pacific region has superseded Europe as the biggest regional market. North and Latin America are, respectively, the 3rd and the 4th market [START_REF]DKSH: Digitization and the market expansion services industry: driving omni-channel growth, Fourth global market expansion services report[END_REF]

MES support to Lean Manufacturing

As stated in Section 1, for a long time the deployment of IT tools and the lean practices have been considered mutually exclusive. However, recently, the importance of deploying MES to support continuous improvement techniques has been shown. Cottyn [START_REF] Cottyn | Design of a Lean Manufacturing Execution System Framework[END_REF] developed a framework for the integration of lean objectives 1 Compounded Average Growth Rate into MES. He defined an automatic Value Stream Mapping (aVSM) methodology: the aVSM benefits from the information provided by the MES, since it is a rich source of information and historical data useful to define continuous improvement actions. On the other side, MES benefits from aVSM because it does not contain information concerning the value flow. The framework is validated through the case studies of a furniture firm and a food and beverage company.

A further recent work [START_REF] D'antonio | Manufacturing Execution Systems for lean, adaptive production processes[END_REF] aimed at the definition of a methodology for the development of MES-oriented tools to support the path towards Lean Manufacturing. It consists of three steps: (i) identification of the wastes to be eliminated; (ii) identification and description of all the components and resources involved in the process at stake; (iii) description of the analysis technique to be integrated into the MES (data source, mathematical algorithms, information to extracted). The methodology enabled to deal with both the top-down and the bottom-up data flows in charge of MES, and was applied to case studies in heterogeneous fields, such as subtractive and additive manufacturing processes, automated warehousing and automated transport of items through a manufacturing line. The research results provided evidence that the deployment of information-based techniques enables to develop adaptive strategies for waste reduction and achievement of lean manufacturing. The research also showed that an effectively implemented MES is, by itself, a lean tool: appropriate techniques must be implemented to transform the acquired data into the right amount of information, to be delivered at the right person, at the right time; ineffective tools lead to wastes such as useless effort in understanding data or oversized infrastructures for data transmission.

The findings presented in this and the previous Sections encourage the deployment of MES to improve the control of the process and to have a complete and updated picture of the manufacturing flow. In particular, MES can enable the identification of wastes and criticalities, and provide information to support decision aiming at overcoming such issued. The impact of SMEs in the European manufacturing has also been presented. Thus, in the following Section a MES implementation into a SME is presented: the aim of this case study is to show how this system can be implemented into a small size company, the implementation issues that may be faced and the support that it can provide in approaching lean manufacturing.

A case study in a SME

The mission of the industrial partner involved in this project is to design and produce interior components for buses, coaches and trains. Although it is a medium size enterprise, the company is composed of five subsidiaries, located in Europe, Middle East, Asia and America. Namely, the firm produces and assemblies lighting systems, handrails, ceiling panels, pillar coverings and luggage racks. Given this high variety of products, the company deploys a wide range of materials including HPL, plastics (PVC, ABS), metals (mainly aluminum), fiberglass, and textiles. The whole manufacturing process is composed of several operations such as cutting, bending, shaping, turning, milling, welding, assembly, surface coating, painting, and punching. Each product has its own manufacturing process. In this project, the Italian plant has been involved to test the implementation of a MES. This shop floor consists of 21 machines; among them, two are equipped with a PLC.

The need for a MES. Before starting the project, data concerning the production were manually registered on paper sheets by shop-floor operators. Then, the same operators manually entered such data into the company's ERP. The information was used to evaluate performance indicators, such as machines utilization and overall process efficiency. Data concerning failures were collected into specific forms and managed without the support of ERP. This practice resulted in non-standardized data collection and imprecisions; further, in some cases data acquisition was missed.

Although not efficient, this approach for data collection is still common for SMEs. Nonetheless, the lack of appropriate information tools may lead to issues such as missed, redundant, or discordant data. For these reasons, the industrial partner decided to introduce a MES into the company.

MES implementation. The industrial partner had not yet installed any information systems in the shop-floor. Thus, to implement the MES two main issues have been dealt: (i) the acquisition of data; (ii) the infrastructure for data transmission to the central unit.

As stated, the shop-floor is composed of a heterogeneous variety of machines. Among them, only two are equipped with a PLC: so, data concerning work instructions, start and finish time, as well as the error codes related to possible failures are directly taken from the controller; the operator only has to log in the system before starting the operation. However, the majority of the machines is not equipped with a PLC and data must be manually entered. To support a profitable deployment of MES, manual machines have been equipped with a set of tablets connected with the central unit. Such devices are used as an interface between the operators and the MES.

Data transmission is performed through wireless connection: the installation of a cable network would have been more expensive and would have provided a lower flexibility.

MES functionalities. Given the motivations leading to MES installation, the developed system aims at solving issues concerning the traceability of the performed operations.

From the shop-floor operators' perspective, the MES is a digital, standardized support to collect the information that was previously handled manually. Before starting an operation, the operator has to log in the system, by scanning a personal bar code or by entering his ID. Then, he must input the number corresponding to the production order at stake, as well as the work code. This step enables to extract the list of operations to be performed; the operator has to select the specific phase he is dealing with from a drop-down menu and the work instructions are shown. At this stage, all the data necessary to uniquely identify the operation have been introduced in the system. To begin working, the operator clicks a Start button: in this way, the system is aware that some resources are busy and cannot be allocated elsewhere. Similarly, when the operation is finished, a Stop button must be clicked. In the PLCequipped machines, the last two steps are automatically performed. Data concerning the involved resources and the time took by the operation are saved for further analytics. The operator also has to input information concerning issues: a list of possible failures and quality defects is available to shorten the time spent in this task and to standardize the sent information. A free text field is also available for unexpected issues.

From the business level perspective, the interface of the MES enables to take snapshots of the shop-floor state at different time scales. First, the present utilization of the resources can be extracted at any time. Second, analytics concerning arbitrary time spans can be extracted: the average values for the utilization of the equipment, the overall efficiency of the process, the impact of defects and failures can be extracted. Further, the cost of processing a part or a lot can be evaluated. The traceability of the product through its manufacturing process enables to define a product passport [START_REF]European Resource Efficiency Platform: Manifesto & Policy recommendations[END_REF], i.e. a document containing the whole chronology of the manufacturing steps experienced by each part.

The support to Lean Manufacturing. The deployment of the presented MES enables to collect data concerning the events taking place at the shop-floor, the repartition of times, the utilization of the resources and the flow of items through the process. These data are collected through the interface provided to the operators; then, data analysis and aggregation are performed to support the decision making process at the business level. Such decisions must be aimed at the elimination of wastes, in order to focus the efforts on the operations that add value to the product and which a customer is willing to pay for. Given the classification of wastes provided in Section 2, the functionalities of the implemented MES enable to:

-Trace the impact of Defects: whenever a (semi-)finished item does not match the required quality level, the operator inputs an issue message into the MES. The analyses performed by the system enable to identify the classes of product or the operations mostly affected by defects. The tracking capability is an effective tool to support the identification of the ultimate reasons leading to poor product quality. -Trace the impact of Inventory and Waiting: the capability of extracting the instant overview of the shop-floor enables to evaluate the number of items idle into the shop-floor. Inventory minimization is desirable, since idle parts require room to be stocked and represent money already spent which is not yet producing income. Waiting is the counterpart of inventory, since it represents the time that items spend without being processed or transported; this time is mainly due to not synchronized flows of materials. The information collected by the MES enables improved planning of operations and allocation of resources, aiming at minimizing waiting. -Trace the impact of Transport. The chronology of the events experienced by the items enables to identify their path through the shop-floor. Thus, data collected over large time intervals can provide useful hints to re-arrange the layout of the plant, the position of the machines and the placement of warehouses in order to reduce the impact of transport: beside a time waste, transport is also a potential risk, since the items may be damaged.

In this work, the implementation of a MES into a medium size company has been presented. The work has been focused on the improvement of shop-floor data collection, aiming to enhance the quality of process knowledge and to support decision making processes. The system is still in a start-up phase: it has been successfully installed into the company, and beta test have been performed to ensure its correct deployment. The MES is now starting to be heavily used, hence a quantification of the advantages concerning will be available in the next months. The presented case study also represents a pilot test for the company, which consists of five plants spread in different geographical areas. The company aims at developing and setting a complete MES system into the Italian plant; after its validation and the evaluation of benefits, the MES will also be installed into the other plants.

The support that such system can provide to achieve Lean Manufacturing has been described in the previous Section. The current spread and the expected trends for MES diffusion in the next years have been presented in Section 2. Given these two concepts, it can be stated that in future a higher diffusion and support of information tools for setting up and tuning lean practices is expected.

However, the experience gathered through this case study showed that some criticalities may occur. A MES cannot be considered as a plug and play system: a robust system for data acquisition and transmission is necessary. The interaction between machines and MES can be managed by the PLC: the information collected during an operation can be sent to the system and stored. However, in the present case study, the majority of the machines was not equipped with a PLC. Thus, alternative ways to collect data had to be identified. The issue was tackled through the deployment of mobile devices, such as tablets: today, such devices provide a huge calculation capability at a low cost. The heterogeneous fleet of machines presented in this case study is not rare: the majority of companies all over the world can be classified as a small-medium enterprise. For this kind of companies, the turnover of machines occurs slowly, so a low-cost solution for the integration of information tools to trace the events occurring in the shop-floor and to improve process knowledge is desirable. In this case study, the cost necessary to develop the software, to equip the shop-floor with a wireless infrastructure for data transmission and to buy the mobile devices was in the order of 40k€.

  . The major players in the MES market include ABB Ltd. (Switzerland), Andea Solutions (Poland), Dassault Systemes SA (France), Emerson Electric Co. (US), General Electric Co. (US), Honeywell International Inc. (US), Rockwell Automation, Inc. (US), SAP AG (Germany), Schneider Electric SE (France), Siemens AG (Germany), and Werum IT Solutions GmbH (Germany).

Fig. 1 .

 1 Fig. 1. Positioning of MES within an industrial framework.


Dharmendra K Mishra
email: dharmendra-kumar.mishra@univ-lyon2.fr

Aicha Sekhari
email: aicha.sekhari@univ-lyon2.fr

Sebastien Henry
email: sebastien.henry@univ-lyon1.fr

Yacine Ouzrout
email: yacine.ouzrout@univ-lyon2.fr

Traceability in Product Supply Chain: A Global Model

Keywords: Traceability, Product Supply Chain, Global Traceability, Modelling

Products undergo through various processes in their lifecycle from production till use across the complex supply chain. Multiple actors are involved in these processes, generating ample amount of data. All the actors must share these data to make supply chain agile, effective and efficient to enhance the production control process. A traceability system companies use to trace these data. The current traceability system company use, is based on one step up and one step down principle, that is, immediate neighbouring actors share the information among themselves. No all the actors in supply chain share information among themselves which prevents collaboration. But, all these information are needed across the supply chain to enhance the quality of the product. In this work, we propose a global model for traceability to enhance the collaboration among every actors in supply chain with sharing of all the required information. Based on the model, we propose our data model.

Introduction

Products move from various points in supply chain before reaching to the end users. There are multiple actors in the supply chain having multiple roles and objectives. For example, the manufacturers may be buyer of one product and it may be the seller of some other products. Similarly, a supplier may be supplying more than one products manufactured by multiple manufacturers to more than one manufacturers. This results the supply chain management, a complex process. The actors generate much information at every point in supply chain from production till use of the product. These information are needed for the manufacturers to enhance quality control process. There are many examples of hazards happened in the food products. Mad cow disease, horse meat scandal are a few among them. The food and pharmaceutical products are directly related with people's health. A minor contamination with such products causes a big hazards to people. Companies need to recall the product from market after such contamination. They must pay attention to each and every process involved from production till use of the product. They need to know the cause of contamination. They must focus to enhance the processes involved from production till delivery of the products to end users to prevent such happenings in the future. There are some other issues like brand protection, reducing the time to go to market, the companies must focus so as to compete in the market. To achieve all these issues, the actors must need information associated at each point in the supply chain.

As described in the above paragraph, companies are facing with lot many of similar challenges. A timely extraction of information generated at each point is essential for them to cope with the above stated challenges they are facing with. Traceability is a method the companies have been using since a long to trace these data in the supply chain. It is the process of tracking the location of the product in downstream and tracing the process involved during various phases of product development in upstream. As per APICS (American Production and Inventory Control society), traceability is registering and tracking of parts, process and materials used in production by a lot or serial number [START_REF] Alfaro | Traceability as a strategic tool to improve inventory management: A case study in the food industry[END_REF]. In another definition given by GS1 (Global Standards), an organization working in the field of developing various standard, traceability is the ability to follow or study out in detail, or step by step, the history of a certain activity or a process [START_REF]The GS1 Traceability standard: What you need to know[END_REF].

Enhancing the product quality, delivering the product on right time in the market, fighting product counterfeiting to protect the brand are the key challenges companies must have to cope with. A proper trace and track mechanism is quite helpful and essential to cope with these challenges. There are many actors in supply chain separated around the globe having multiple roles and objectives, make the modern day supply chain: a complex system. Tracing all the information at each point is still a challenge. Many works are found in literature in this area. The authors implement traceability system in pharmaceutical supply chain using RFID (Radio Frequency Identification), EPC global (Electronic Product Code) and ebXML (Electronic business using eXtensible Markup language) [START_REF] Barchetti | Impact of RFID, EPC and B2B on traceability management of the pharmaceutical supply chain[END_REF]. I. Charfeddine et. al. proposed an intelligent framework for traceability of containerized goods based on multi agent system and semantic web technology [START_REF] Charfeddine | Intelligent framework for traceability of containerized goods[END_REF]. A traceability system to trace the data is developed in vegetable supply chain [START_REF] Qiao | Research on vegetable supply chain traceability model on two dimensional barcode[END_REF]. The authors use 2D barcode to identify the product's data from cultivating to marketing and use web based technology to extract these information [START_REF] Qiao | Research on vegetable supply chain traceability model on two dimensional barcode[END_REF]. In another work, a system is developed to track the information related with beef slaughter till packaging in the beef supply chain [START_REF] Feng | Development and evaluation on a RFID based traceability system for cattle/beef quality safety in china[END_REF]. A traceability system is developed and implemented in frozen shrimp products and verified and validated to evaluate the system's performance [START_REF] Djatha | An analysis and design of frozen shrimp traceability system based on digital business ecosystem[END_REF]. In another work, need of traceability system is studied to ensure the food safety [START_REF] Aung | Traceability in a food supply chain: safety and quality perspectives[END_REF]. In their work, R. Badia, P. Mishra et. al identify the advancement in technology in the recent traceability system which tracks the information from farm to fork through smart phone [START_REF] Badia | Food traceability: New trends and recent advances, A review[END_REF]. A tracefood framework is developed to exchange the information among actors to solving the proprietary nature of the internal system [START_REF] Stordy | The traceFood framework-Principles and guidelines for implementing traceability in food value chains[END_REF]. But, none of the work guarantee to implement the global traceability system exchanging the information among every actors in global supply chain. In modern traceability system, the actors manage internal traceability using ERP (Enterprise Resource Planning), MES (Manufacturing Execution System) etc. These systems are linked with immediate neighbouring actors. In other word, we can say that output produced by an actor is not linked with input of all the actors. This is a key challenge of modern day supply chain network, which creates a barrier of tracing the cause, when some problem occurs at any point in the network. The purpose of this work is to develop a model to enhance the collaboration among every actors by sharing the traceability data.

The rest of this paper is organized as follow. Section 2 describes the research problem and objectives, A bi-graph model to represent sharing information is described in section 3, proposed data model is described in section 4 and finally section 5 concludes the work.

Research Problem & Objective

As discussed in section 2, SCM (Supply Chain Management) is a complex process. There are multiple actors having multiple roles and objectives. They are separated at multiple points around the globe. Large amount of data is generated at each point through these actors in entire life cycle of the product. The actors must share these data among themselves to make product visible across the supply chain to make it agile and transparent. At one point, among these data, some are very crucial and sensitive, while all the actors are not known to each other in the global supply chain on the other, cause them not to share all these data. But the collaboration must be needed by sharing all the necessary information to address the issues discussed in section 2. This work aims to address the issue formulating a research question: how to make collaboration among multiple stakeholders across global supply chain?

Bi-graph model for the global supply chain

To show the collaboration among multiple stakeholders, we adopt bi-graph model proposed by [START_REF] Li | Bigraph-based modeling and tracing for the food chain system[END_REF]. But, we consider all the stakeholders in global supply chain in contrast to few. In graph theory, bi-graph (Short form of bipartite graph) is a graph, whose vertices are divided into two disjoint independent sets U and V of which there is an edge from every vertices in U to that of in V, such that U∪V = All the vertices in the set and U∩V=φ that is null. There are two entities in the supply chain process: Actors and the Products. There exist a multiple relationship between these two entities. [START_REF] Li | Bigraph-based modeling and tracing for the food chain system[END_REF]. Bi-graph has strong ability of reasoning and logic that attracts researcher from academia and industries to use it [START_REF] Li | Bigraph-based modeling and tracing for the food chain system[END_REF].

Fig. 1. Supply chain process

Bi-graph is a strong tool that can be used to show these relationship. It is used to model many complex network, modern coding theory etc [START_REF] Li | Bigraph-based modeling and tracing for the food chain system[END_REF]. In this work, it helps to show the various relationship among the entities, which is used to trace the required information at every point through an algorithm, as our future work.

We consider a main manufacturing company M1 producing the product by taking sub products, raw materials etc supplied by more than one suppliers. Fig. 1 shows supply chain process of the various actors.

Fig. 2. Bi-graph Model of Supply chain process

The symbol M, S, D, and C stands for manufacturer, supplier, distributer and customer respectively. More than one suppliers supply the product to main manufacturer (M1) manufactured by more than one manufacturers (M2 to Mn). The main manufacturer after its internal process (producing, packaging, etc ) produces the main product and sends them to various customers through various distributers. The customer may be a retailer, a wholesaler or an end user. The relationship is shown by various edges from E1, E2….En.

We model this scenario using the bi-graph technique as shown in figure 2. Let us consider G (A, B, E) be a bi-graph where A= A1, A2…..An represents the actors (Manufacturer, Supplier, Distributer and Customer) shown in dark circle. B= B1, B2…..Bn represents the sub-products, raw materials or finished product shown in white circle. E= E1, E2……En represents the edges connecting the two vertices. The edges represents the relationship between two vertices sets. The relationship is one to one, one to many, many to one and many to many.

The model is applicable to any product manufacturing company. It is necessary to identify the specific point through which the product undergoes. This depends on the type of industry. For example, if we consider a food industry, the product moves from farm to the manufacturer to the distributer or transporter and finally to the end users. Actors produce information through internal process at all of these points. For example, at farm, the farmers do the cultivation, maintains the soil information, bow seeds etc. Similarly, the manufacturer maintains the supplier's information, processing information of raw materials, packaging information etc. It is also necessary to identify the type of the products. The product may be semifinished or the final one.

The traceability system stores all these information produced by actors at each link that is traceable if needed. In the next section, we explain the data model we developed to show the types of data generated by the actors and the relationship between them.

Data Model

The information sharing and management requires the logical representation of data produced by each actors. Fig. 3 shows the data model we propose. It is assumed that there is one main manufacturer who manufactures the final product by assembling or processing the semi-finished products supplied by more than one suppliers. Suppliers in turn supply the semi-finished products manufactured by more than one manufacturers. There are more than one distributer delivering the finished product to many customers. We consider each actors of supply chain and address the issue of complexity. Some works related with data models are found in literature. Thierno et al develop a data model to handle unitary traceability based on IEC 62264 and GS1 global standard [START_REF] Diallo | Using Unitary Traceability for an optimal product recall[END_REF].

Fig. 3. Data Model

The authors aims to represent the internal traceability data to find the root cause and facilitate data exchange. In another work, a data model is proposed to support the traceability process in the food supply chain [13]. Similarly, M. khabbaji et al propose data model to manage the lot traceability on the basis of make order [START_REF] Khabbazi | Data modelling of traceability information for manufacturing control system[END_REF]. The model is said to support to control the material flow in all quality and production process. Some other works [START_REF] Khabbazi | Data modelling of traceability information for manufacturing control system[END_REF][START_REF] Jansen-Vullers | Managing traceability information in manufacture[END_REF], develop internal traceability data model in terms of materials and process data registration. We propose to address the global traceability data model tracing each data at each point from production till the use of the product. It starts with company planner who makes the production plan based on which the order is generated. The production plan also contains the BOM of the product. Each actors are assigned with unique identification number. The actors are linked with city and country to show the geographic distribution. The country and city has unique identification code which act as the foreign key for all the actors. All the actors maintain its own log storing all the necessary information. The manufacturers maintains i/p log that contains date of order, order id and supplier id to show on which date it receives the semifinished product and which supplier is responsible to supply that product. It also maintains o/p log and processing information. The output log contains the sales order and customer information with distributer id. The processing data contains how the products are manufactured, the actors involved in packaging, warehousing etc. The supplier, warehouse and distributer also maintain their own log having the information like date of delivery, date of receiving, storage information, SSCC (Serial Shipping Container Code), customer's information, product lot etc. The product lot has relation with all the actor with reference to country id to track the location of the product with lot id, description, actors involved etc.

The final product contains SGTIN (Serialized Global Trade Item Number), lot number and name of the product having the relation with lot, intermediate product and actual material production.

The data model shown in figure 3 has all the traceable data and their relationship. These data are traced putting lot-id as keyword through a novel algorithm as our future work and implemented in a user friendly web based system based on cloud. We use EPC global network as a standard and use case for the validation.

Conclusion

The product's safety in the global supply chain is still a challenge. Companies need to enhance the product's quality and make it safer to prevent some sort of problem across the chain. To achieve this, a global traceability system is needed to trace all the required information. But, the actors don't share all the information among themselves. All the actors are not well known to each other. They produce various sensitive information which they do not share in the network, which prevents the collaboration among them. In this paper, we propose a bi-graph model to show the relationship among all the stakeholders. Based on this model, we propose a data model to show types of data generated at each point and link between them. The model reduces the supply chain complexity by increasing the collaboration among every actors.


Laureline Plo
email: l.plo@groupe-poult.fr

Vincent Robin
email: vincent.robin@u-bordeaux.fr

Philippe Girard
email: philippe.girard@u-bordeaux.fr

Understanding PLM system concepts to facilitate its implementation in SME: the real case study of POULT

Keywords:

. Our approach was to use formalism based on mind maps to promote exchanges between customer/users and PLM editor. Objective was to facilitate definition of elements that have to be defined, modeled, evaluated and implemented in a software solution to make deployment process more effective. Different real case studies emphasized that our approach have to evolve to be more "customer" centered. In this paper we present one of these real case studies which is in progress. This research collaboration concerns POULT, a manufacturer of biscuits which decided to deploy a PLM solution in few months. We describe the adopted approach to analyze the users' needs and draft the request for proposal which will be send to PLM editors.

Introduction

For SMEs, PLM system is a solution to manage all the processes and associated data generated by events and actions of various lifecycle agents (both human and software systems) and distributed along the product lifecycle phases, from the beginning to the end of life [START_REF] Stark | Product lifecycle management: 21 st century paradigm for product realization[END_REF]. SMEs have some difficulties in the use of PLM systems and do not exploit their full potentiality and a big amount of information and knowledge is being lost or requires a higher human effort to be preserved [START_REF] Kadiri | Current situation of PLM systems in SME/SMI: Survey's results and analysis[END_REF][START_REF] Bruno | A reference ontology to support product lifecycle management[END_REF]. The main problems for SMEs in the exploitation of a PLM system are the lack of models to represent the product lifecycle [START_REF] Bruno | A reference ontology to support product lifecycle management[END_REF] and the approach used by PLM editor to adapt and deploy their solutions for SME [START_REF] Baczkowski | Amélioration du processus de déploiement d'une solution PLM par l'utilisation de cartes heuristiques et de persona: cas LASCOM[END_REF]. This approach is often iterative and incremental and based on models and prototypes (Figure . 1). It offers a bottom-up approach to analyze and model the decisional, functional and organizational structure of the customer but partially ensures consistency between the expectations/needs and the functional aspects of the PLM solution. Moreover, the number of iterations to define the users' expectations is too important and the capitalization of actors' feedback is not sufficiently taken into account. One problem stems from the uniqueness and the diversity of each solution. Other one concerns the fact that views and semantics between the different actors of the project are often different: customers/users are functional and business oriented, while the editor is faced with structural and technological issues. This often leads to a misunderstanding underlying and creates difficulties during the specifications validation step.

Fig. 1.

Iterative and incremental approach to deploy PLM solution [START_REF] Baczkowski | Amélioration du processus de déploiement d'une solution PLM par l'utilisation de cartes heuristiques et de persona: cas LASCOM[END_REF]. This approach is not efficient yet and causes significant delays on the initial planning. To reduce and optimize the analysis of expectations for each new project and to increase the efficiency of the procedure for solutions deployment we proposed to develop a new approach [START_REF] Baczkowski | Amélioration du processus de déploiement d'une solution PLM par l'utilisation de cartes heuristiques et de persona: cas LASCOM[END_REF]. Our objective was to facilitate the preliminary analysis of the system and the characterization of users' expectations and needs. This approach was developed with LASCOM a French PLM editor and was inevitably "editor oriented". In a nutshell, approach optimizes audit and analysis of need for PLM editor (time reduction) and creates a dependency relationship between customer and PLM editor. In this case, SMEs could be reluctant to approach PLM editors and prefer to keep a relative autonomy towards editors. SMEs need time to understand and integrate PLM system and are often suspicious about editors' intentions. To limit harmful effects of a too PLM editor centered approach we propose in this paper an process to enable companies to understand PLM concept, express their needs and define their own specifications in autonomy before calling a PLM editor. Our approach is composed with 4 steps in order to formalize the needs of the company before the first meetings with the PLM editor (figure 2):

First step: understanding the organization of R&D process, to model the existing organization without considering a "PLM editor centered" vision, Second step: understanding the philosophy of PLM approach, to appreciate and estimate possible impacts of the PLM in the company, Third step: considering users in the PLM deployment, to favor acceptation, Fourth step: formalization of the needs and definition of the preliminary specifications of a solution which could be an adapted one for the company.

Expression of customer needs

(audit, analysis of need) In this paper we present our approach through the real case study of Poult, a French producer of biscuits, which decided to improve its research and development process in 2014 and initiated a reflection on the opportunity to use a PLM system. This SME has ever had experiences with different PLM editors and preferred to preliminary work with our research team in order to anticipate and limit difficulties before send a request of proposal to PLM editors again. The paper is structured in order to show our progressive approach to help members of the team "PLM project" (managers of R&D, production, marketing, quality and purchasing departments, R&D assistant, Ms. PLO -PhD student). First section describes the initial vision of the R&D process to be improved. Second section shows how Poult progressively understands PLM concepts and adapts them. Following section presents work in progress which concerns a user centered approach to precisely specify users' needs. Finally, last section concludes and states futures works.

First step: understanding the organization of R&D process

Poult is a French producer of biscuits which decided to improve its research and development process in 2014. R&D process essentially concerns development of new product or evolution of an existing one. Product could be developed for Poult's label or for other customers' labels. Poult controls technical activities to create biscuits but has many difficulties to control and manage data generated during R&D activities. That the reason why managers decided to study the opportunity to deploy a PLM system in the company to allow data management and increase efficiency of R&D phases. As the R&D process was not really formalized first step of our collaboration concerned global identification and definition of principal activities of the R&D process. This process begins when customer sends specifications of the expected biscuit and finishes when the industrial production of the biscuit starts (Table 1). In first time, the vision is quite global because it is relatively unusual for Poult to express its R&D organization.

Second step: understanding the philosophy of PLM approach

Preliminary description of the R&D process (Table 1) is useful to partially understand R&D process but it has to evolve to be more operational with a view to a PLM system deployment. So, second step of our collaboration was to switch from a classical "activities" model to a "product lifecycle" model. As the main problem for SMEs in the exploitation of a PLM system is the lack of models to represent the product lifecycle [START_REF] Bruno | A reference ontology to support product lifecycle management[END_REF], start of this step was not easy. Discussion with project team members leaded to guide our reflection on the use of ontologies to provide an answer to the need for a clear understanding of product lifecycles phases [START_REF] Mckenzie-Veal | Implementing Ontology-based Information Sharing in Product Lifecycle Management[END_REF] and the need of system interoperability [START_REF] Abdul-Ghafour | Integration of product models by ontology development[END_REF]. Several ontologies [START_REF] Jurisica | Ontologies for Knowledge Management: An Information Systems Perspective[END_REF] or researches [START_REF] Lacombe | Contribution à une méthodologie et une modélisation pour accompagner les petites entreprises dans l'étude de leur organisation afin de spécifier leurs besoins et sélectionner une solution ERP[END_REF] exist and have been presented to members of the project with a view to build a common semantic and a share understanding of model. These exchanges allow us to find an agreement and we decide to exploit Bruno et al.'s ontology and UML model [START_REF] Bruno | A reference ontology to support product lifecycle management[END_REF]. Bruno et al. developed in 2015 their reference ontology for PLM which clearly makes appear PLM concepts (Table 2) and relationships between each on them (white classes -Figure 3). POULT chooses to use this ontology because it is based on a serious analysis of existing ontologies and it is considered as understandable and well-adapted to the company thanks to the use of UML.

Our experience with POULT puts in evidence the difficulty to identify ontology and models to present PLM philosophy and concepts to people in charge of a PLM project in a company. Thanks to such an experience, we are working on an approach to define and select possible combinations (ontology + models) depending on the company context to express the philosophy of PLM approach. We will present our propositions in a future paper which will be focused on more conceptual aspects of our research.

Table 2. Description of PLM concepts [5]

Concept Description

Product

A generic term for whatever is produced by a process and serves a need or satisfies a want.

Customer

The reference person of a company that ordered a product.

Project

The term used by a company to indicate the collaboration with a customer for the developing of a product. Each project refers to one product and is connected with one customer. Product component One of the hardware, electronic or software that makes up a product. A component may be subdivided into other components, which combine into sub-assemblies and assemblies to define product. Each product can be made of several components, and the same component can be used by different products.

Material

The material of which a product or a product component is made of.

Physical characteristic

The physical characteristic of a product or product component.

Unit

The unit in which the Physical characteristic is measured.

Activity

An action executed during the product lifecycle of a specific product that can be univocally and unambiguously identified.

File

Electronic data managed and stored as a single object.

Document

A uniquely-identified block of information that can be composed by several files.

Tool

A software tool used to produce a file, for which the name and the version have to be specified.

Resource

An entity that is involved in the execution of an activity. A resource can be of two kinds: Person or Machine.

Role

The term used to define a specific set of skills and responsibilities associated with the employees of the company.

Thanks to Bruno et al.'s propositions we create a common vision of a PLM system between team project members and researchers. Moreover, such description of PLM concepts and ULM class diagram were very helpful for the team which has been able to define its own description of PLM concepts and UML class diagram adapted to Poult. All Bruno et al.'s classes have been conserved (or slightly evolved) and 10 new classes complete the UML class diagram to be closed to the Poult's reality. New classes represent particularity of Poult or food industry. Poult considers a product as a quadruplet [recipe + raw materials + ingredients + packaging] and data associated to this quadruplet are managed separately. So these classes appear in the new version of the diagram. Moreover, food industry has specific rules to ensure quality during all the production process for the final users. These rules oblige Poult to manage precisely specific data (machine characteristic, supplier characteristics, etc.). These classes also complete the UML diagram. Table 3 provides a description of these new classes and Figure 3 shows an UML class diagram of PLM adapted to Poult (added classes in grey).

Table 3. Description of additional PLM concepts adapted to Poult (food industry).

Concept Description

Food-Processing (to complete Product)

R&D department could develop new product (i.e. new biscuits) or new food-processing. So, a class "Foodprocessing" has been created to clearly make appear these two kinds of production.

Production site (to complete Customer)

A product or a food-processing could be ordered by a customer or by Poult's production site. So, a class "Production site" has been created to make appear these two kinds of prime contractor.

Recipe and Recipe component (to complete Product component)

Poult's products are identified thanks to a "recipe" and a recipe is composed with components. That the reason why the class "product component" evolved to be "recipe" and "recipe component" classes. A product has only one recipe which can be made of several components, and the same component can be used by different products.

Raw Material and Ingredient (substitute material)

In food industry the term "raw material" is preferable to "material". Raw material can be made of several "ingredient". Combination of ingredients in a raw material is an important data to respect food regulations.

Packaging

Product is one or more biscuits plus a specific packaging. Poult manages its biscuit and its packaging separately so a specific class has been created for "packaging". Packaging characteristic The different characteristic of a packaging.

Supplier

In food industry, all components have to be controlled. One performance indicator to be controlled is "who supply the component"? "Supplier" and associated characteristics are important data to be managed by PLM. Machine characteristic Food industry rules oblige to manage data about machines used to produce biscuit. Objective is to ensure quality of the process on the whole for final users. So, Poult has to manage data associated to its machines.

Third step: considering users in the PLM deployment

The UML class diagram adapted to Poult shows how project team members have understood PLM. They evolve from a global vision of the R&D process of Poult to a more operational vision which integrates PLM concepts. Thanks to such representation architecture of PLM system appears. As deployment and implementation of an effective solution not only consists in selecting and installing software but also in obtaining the users acceptance, we have to focus now on final users of the PLM system. The understanding of user's needs and expectations is a critical phase for the deployment of such solutions. Nevertheless, much of the traditional research on software project management does not consider users' expectation correctly and neglects to ask project managers about their perspectives and ideas on how to manage projects [START_REF] Rose | Management competences, not tools and techniques: a grounded examination of software project management at WM-Data[END_REF]. This crucial step depends on the ability to understand and consider users' expectations [START_REF] Petter | Managing user expectations on software projects: Lessons from the trenches[END_REF] and to analyze, co-ordinate and control the collaboration between the numerous users concerning by the project: strategic/tactical decision-makers, project managers, designers, experts from different disciplines and with different experiences, external partners, etc. The views and semantics between the different actors of the project are often different. Customers are functional and business oriented, while the editor is faced with structural and technological issues. This often leads to a misunderstanding underlying and creates difficulties during the specifications validation step. To reduce these difficulties, we developed in 2012 an "editor" oriented approach to facilitate exchanges between PLM editors and customers [START_REF] Baczkowski | Amélioration du processus de déploiement d'une solution PLM par l'utilisation de cartes heuristiques et de persona: cas LASCOM[END_REF][START_REF] Baczkowski | Using of the concepts of roles and context in a project management / PLM Solution: The real case study of LASCOM[END_REF]. Our objective was to make easier the preliminary analysis of the system and the characterization of users' expectations and needs. This approach is based on the use of mind map and persona to progressively define users' profiles and expectations.

When a project starts editor initializes a specific map for the customer (Figure 4). Enterprise modelling which regroups data concerning the customer (structural and functional organizations, actors implied in the project, etc.). This branch evolves and is more and more precise according to project evolution. Follow-up of the contributions allows project manager to have a trace of each action realized during the project. The aim of this branch is to be able to follow each evolution of the project in term of "who decided what?" Follow-up of the deliverables regroups all the documents which are produced during the project. This branch helps editor and customer to judge if the progress of the project according to the common deadlines fixed from the start of the project. Software applications definitions is the branch in which we fond all the versions of the software developed during the project.

Each actor of PLM editor which is involved in the project completes his part of the map. Information concerning customer and project lifecycle (activities and associated documents) are in the map and each steps of the software development are capitalized and reusable. Relationships exist between elements of branches to obtain a global and dynamic vision of the structure of the system. Such a simplified representation makes easier relationships between editor and customer [START_REF] Baczkowski | Amélioration du processus de déploiement d'une solution PLM par l'utilisation de cartes heuristiques et de persona: cas LASCOM[END_REF]. A map is associated to one project and to one customer. the PLM is deployed it is easy to maintain solution because editor has customers' data in the map and an evolution of the map is quickly reverberated in the PLM system. Concerning Poult, we have organized interviews of each potential user since few weeks in order to complete the branch "enterprise modeling" (Figure 5). Interviews are in progress and the branch evolves. Other branches cannot be completed, they concern PLM editor and no one is selected yet. When team project members will estimate that they have a clear vision of PLM concepts, of their needs and of the specifications of the solution, they will formalize all of these data (step 4 -figure 2) and send a request for proposal to PLM editors.

Conclusion

Collaboration with LASCOM until 2012 allows us to develop a specific approach to model system in which a PLM solution has to be deployed. Aware about limits of this approach we confront since 2015 our propositions to the reality in a SME, Poult, a French biscuit producer. In this paper, we present the progress of the project with

Fig. 2 .

 2 Fig. 2. Internal process to study relevance of PLM solution deployment in a company.

Fig. 3 .

 3 Fig. 3. ULM class diagram of PLM adapted to Poult.

Fig. 4 .

 4 Fig. 4. Global architecture of a map dedicated to a customer. Four branches structure the map:Enterprise modelling which regroups data concerning the customer (structural and functional organizations, actors implied in the project, etc.). This branch evolves and is more and more precise according to project evolution. Follow-up of the contributions allows project manager to have a trace of each action realized during the project. The aim of this branch is to be able to follow each evolution of the project in term of "who decided what?" Follow-up of the deliverables regroups all the documents which are produced during the project. This branch helps editor and customer to judge if the progress of the project according to the common deadlines fixed from the start of the project. Software applications definitions is the branch in which we fond all the versions of the software developed during the project.

Fig. 5 .

 5 Fig. 5. Preliminary description of the production site of Montauban.

Table 1 .

 1 Preliminary description of the Poult's R&D process activities and tasks.

	Request for
	proposal to PLM editors	1. Understanding the organization of R&D
		process
	4. Formalization of the needs and definition of the preliminary specifications	2. Understanding the philosophy of PLM approach
	3. Considering
	users in the PLM
		deployment

Poult. We describe the progressive and relatively autonomous approach we propose to PLM project team members Poult: 1. Understanding the global organization of the R&D department. 2. Understanding the philosophy of the PLM approach to develop a model adapted to the SME and coherent with PLM concepts. 3. Considering users in the PLM deployment to define precisely specifications and increase the degree of acceptance. Even if this approach seems to be a top-down one it is more complex in reality. As it is very collaborative and participative, it is a hybrid approach between top-down and bottom-up. Team members are implicated but users are very concerned too. Formalisms used to support each step seems to be adapted to users' expectations (SADT for first step -not presented here, Bruno's model of PLM concepts and UML class diagram for second step and mind map for third step). The next step is the definition of the specification to write the request for proposal to PLM editors. As objective is to start PLM deployment in 2017, it looks obvious that the request for proposal will be send during summer 2016.