Nicolas Durrande
email: durrande@emse.fr

David Ginsbourger

Olivier Roustant

Laurent Carraro

ANOVA kernels and RKHS of zero mean functions for model-based sensitivity analysis

Keywords: Gaussian process regression, global sensitivity analysis, Hoeffding-Sobol Decomposition, SS-ANOVA

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ANOVA kernels and RKHS of zero mean functions for model-based sensitivity analysis

Introduction

Let f be a real-valued function defined over D ⊂ R d . We assume that f is costly to evaluate and that we want to study some global properties of f such as the influence of a variable x i on the output f (x 1 , . . . , x d ). As the number of evaluations of f is limited, it may be unaffordable to run sensitivity analysis methods directly on f . Thus, it can be helpful to replace f by a mathematical approximation for performing such studies [START_REF] Marrel | Calculations of sobol indices for the gaussian process metamodel[END_REF]. In this article we propose a class of models that is well suited for performing global sensitivity analysis. First, we present some background in sensitivity analysis, interpolation in Reproducing Kernel Hilbert Space (RKHS), and on the class of ANOVA kernels. Then we construct RKHS of zero mean functions and derive a subclass of ANOVA kernels that is well suited for the ANOVA representation of the best predictor and for global sensitivity analysis. The use of this new class of kernels is illustrated throughout the article on classical test functions from the sensitivity analysis literature.

Sensitivity analysis

The purpose of global sensitivity analysis is to describe the inner structure of f and to analyse the influence of each variable or group of variables on f [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]. A traditional approach is to study the variance of f (X) where X is a random vector over D. Hereafter, we assume that f ∈ L 2 (D, µ), where

D = D 1 × • • • × D d is a
Cartesian product space of sets D i ⊂ R and where µ = µ 1 ⊗ • • •⊗ µ d is a product of probability measures over D i . The measure µ describes the variability of the inputs and we define X as a random vector with probability distribution µ. Note that although this framework is often considered in sensitivity analysis, the product structure of µ implies that the components of X are independent.

For d = 1, any g ∈ L 2 (D, µ) can be decomposed as a sum of a constant plus a zero mean function [START_REF] Gu | Smoothing spline ANOVA models[END_REF],

g = D g(s)dµ(s) + g - D g(s)dµ(s) .
The two elements of this decomposition are orthogonal for the usual L 2 scalar product so we have a geometric decomposition of L 2 (D, µ):

L 2 (D, µ) = L 2 1 (D, µ) ⊥ ⊕ L 2 0 (D, µ) (1)
where L 2 1 (D, µ) denotes the subspace of constant functions and L 2 0 (D, µ) the subspace of zero mean functions: L 2 0 (D, µ) = {g ∈ L 2 (D, µ) : D g(s)dµ(s) = 0}. Similarly, if d > 1, the space L 2 (D, µ) has a tensor product structure [START_REF] Krée | Produits tensoriels complétés d'espaces de Hilbert[END_REF]

L 2 (D, µ) = d i=1 L 2 (D i , µ i ). (2)
Using Eq. 1 and the notation L 2 B (D, µ)

= d i=1 L 2 Bi (D i , µ i ) for B ∈ {0, 1} d we obtain L 2 (D, µ) = d i=1 L 2 1 (D i , µ i ) ⊥ ⊕ L 2 0 (D i , µ i ) = ⊥ B∈{0,1} d L 2 B (D, µ). (3)
A key property is that two subspaces L 2 B and L 2 B ′ are orthogonal whenever B = B ′ . Given an arbitrary function f ∈ L 2 (D, µ), the orthogonal projection of f onto those subspaces leads to the functional ANOVA representation [START_REF] Efron | The jackknife estimate of variance[END_REF][START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates[END_REF] (or Hoeffding-Sobol decomposition) of f into main effects and interactions:

f (x) = f 0 + d i=1 f i (x i ) + i<j f i,j (x i , x j ) + • • • + f 1,...,d (x), (4)
where f 0 is the orthogonal projection of f onto the space of constant functions L 2 {0} d (D, µ) and where f I (I ⊂ {1, . . . , d}) is the orthogonal projection of f onto L 2 ind(I) (D, µ) with ind(I) i = 1 if i ∈ I and ind(I) i = 0 if i / ∈ I. By construction, the integral of f I with respect to any of the variables indexed by i ∈ I is zero. This decomposition gives an insight into the influence of each set of variables x I = {x i , i ∈ I} on f . For the constant term, the main effects, and the two-factor interactions, one gets the classical expressions [START_REF] Gu | Smoothing spline ANOVA models[END_REF]:

f 0 = D f (x)dµ(x) f i (x i ) = D-i f (x)dµ -i (x -i ) -f 0 f i,j (x i , x j ) = D -{i,j} f (x)dµ -{i,j} (x -{i,j} ) -f i (x i ) -f j (x j ) -f 0 (5)
with the notations D -I = i / ∈I D i and µ -I = i / ∈I µ i . As one can see from Eq. 5, the calculation of any f I requires having recursively computed all the f J 's for J ⊂ I, which makes it cumbersome (if not practically impossible) to get high order interactions.

Recall the random vector X with distribution µ. L 2 (D, µ)-orthogonality between any two terms of the decomposition implies that the variance of the random variable f (X) can be decomposed as

var(f (X)) = d i=1 var(f i (X i )) + i<j var(f i,j (X i,j )) + • • • + var(f 1,...,d (X)). (6)
For the subsets I ⊂ {1, . . . , d}, the global sensitivity indices S I (also called Sobol indices) are then defined as

S I = var(f I (X I )) var(f (X)) . (7)
S I represents the proportion of variance of f (X) explained by the interaction between the variables indexed by I. The knowledge of the indices S I is very helpful for understanding the influence of the inputs, but the computation of the f I 's is cumbersome when evaluation of f is costly since they rely on the computation of the integrals of Eq. 5. Following [START_REF] Marrel | Calculations of sobol indices for the gaussian process metamodel[END_REF], it can then be advantageous to perform the sensitivity analysis on a surrogate model m approximating f .

Optimal approximation in RKHS

The class of functional approximation techniques considered in this work, commonly referred to as Kriging or Gaussian process regression in contemporary statistical learning settings, can be seen as methods of optimal interpolation in RKHS. f is here assumed to be known at a set of points X = {X 1 , . . . , X n } with X i ∈ D. Given a RKHS H of real-valued functions over D with reproducing kernel K(., .), we want to select a function of H that is a good approximation of f . Two options are usually considered [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF]: the first one is to solve an interpolation problem, and the second a regularization problem.

The interpolation approach consists of finding a function h ∈ H that satisfies h(X i ) = f (X i ) for i = 1, . . . , n (for which we will use the vectorial notation h(X ) = f (X )). As there might be an infinite number of function of H that satisfy this criterion, the optimal interpolator is defined as [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF]:

m(x) = argmin h∈H (||h|| H : h(X ) = f (X )) . (8)
When the Gram matrix K of general term

K i,j = K(X i , X j ) is invertible, we obtain m(x) = k(x) t K -1 F (9)
where F = f (X ) is the column vector of observations and k(.) is the column vector of functions (K(X i , .)) 1≤i≤n .

For the regularization approach, the best predictor m is defined to be the minimizer of:

n i=1 (h(X i ) -F i ) 2 + λ||h|| 2 H , (10)
where the parameter λ is a positive number tuning the trade-off between the norm of m and the distance to the observations. The solution of this minimization problem is m(x) = k(x) t (K + λI) -1 F.

From the probabilistic point of view, the regularization corresponds to approximating f based on noisy observations : f obs (X i ) = f (X i ) + ε i where the ε i are independent random variables with distribution N (0, λ) [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF].

As it appears in Eqs 9 and 11, m and m are a linear combinations of the K(X i , .). These basis functions and the influence of the parameter λ are illustrated in Figure 1 on a toy example for two popular kernels:

k b (x, y) = min(x, y) and k g (x, y) = exp -(x -y) 2 , ( 12
)
known respectively as the Brownian and the Gaussian covariance kernels.

The interpolation problem can be seen as a particular case of a regularization problem where λ → 0 [START_REF] Gu | Smoothing spline ANOVA models[END_REF]. For the sake of clarity, we focus hereafter on the best interpolator m. However the general case of the regularization framework is addressed in Section 4.

The kernel K can be any symmetric positive definite (s.p.d.) function [START_REF] Vangeepuram | Reproducing Kernel Space Embeddings and Metrics on Probability Measures[END_REF] and it has to be chosen in practice. It is customary to arbitrarily select K from usual s.p.d. functions (e.g. Matérn, or power-exponential kernel families) according to some prior knowledge about f such as its regularity. However, this choice has great impact on the resulting model (e.g. Figure 1). If a kernel depends on some parameters, a traditional approach is to estimate them by maximum likelihood to ensure the kernel is adapted to the data [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF].

In the next section we focus on a particular family of s.p.d. functions called ANOVA kernels, which are designed to offer a separation of the terms with different orders of interaction. The main contribution of this paper, which is presented in the third section, deals with a special case of ANOVA kernels tailored for an improved disentanglement of multivariate effects.

ANOVA kernels and a candidate ANOVA-like decomposition of the best predictor

The name "ANOVA kernel" has been introduced in the literature of machine learning by Stitson et Al. [START_REF] Mark | Support vector regression with ANOVA decomposition kernels[END_REF] in the late 90s. These kernels allow control of the order of interaction in a model and enhance their interpretability [START_REF] Gunn | Structural modelling with sparse kernels[END_REF][START_REF] Berlinet | Reproducing kernel Hilbert spaces in probability and statistics[END_REF].

For D = D 1 × • • • × D d ,
they are constructed as a product of univariate kernels 1 + k i , where 1 stands for a bias term and the k i 's are arbitrary s.p.d. kernels

on D i × D i (1 ≤ i ≤ d): K AN OV A (x, y) = d i=1 (1 + k i (x i , y i )) = 1 + I⊂{1,...,d} i∈I k i (x i , y i ). ( 13
)
As shown on the right hand side of Eq. 13, ANOVA kernels can also be seen as a sum of separable kernels with increasing interaction orders [START_REF] Duvenaud | Additive Gaussian processes[END_REF].

Denoting by 1 i and H i the RKHS of functions defined over D i with respective reproducing kernels 1 and k i , K AN OV A is the reproducing kernel of the space [START_REF] Aronszajn | Theory of reproducing kernels[END_REF][START_REF] Berlinet | Reproducing kernel Hilbert spaces in probability and statistics[END_REF]

H AN OV A = d i=1 (1 i + H i ) = 1 + I⊂{1,...,d} H I . ( 14
)
with H I = i∈I H i ⊗ i / ∈I 1 i . From Eq. 13 the particular structure of K AN OV A allows us to develop the n × 1 vector k(x) of Eq. 9 as follows:

k(x) = 1 + I⊂{1,...,d} i∈I k i (x i ) (15)
where ⊙ denotes a element-wise product:

i∈I k i (x i ) j = i∈I k i (x i , x j ).
From this relation, we can get the decomposition of the best predictor m given in Eq. 9 onto the subspaces H I :

m(x) = 1 t K -1 F + I⊂{1,...,d} i∈I k i (x i ) t K -1 F (16)
Noting

m 0 = 1 t K -1 F and m I (x) = i∈I k i (x i ) t K -1 F, we obtain an
expression for m which is similar to its ANOVA representation:

m(x) = m 0 + d i=1 m i (x i ) + i<j m i,j (x i,j ) + • • • + m 1,...,d (x 1,...,d ). ( 17
)
In this expression, the m I have the useful feature of not requiring any recursive computation of integrals. However, this decomposition differs from the ANOVA representation of m since the properties of Eq. 4 are not respected. Indeed, the m I of Eq. 17 are not necessarily zero mean functions and any two terms of the decomposition are generally not orthogonal in L 2 (D, µ). This point is illustrated numerically on a test function at the end of the second example. It can be seen theoretically if the k i are Ornstein-Uhlenbeck kernels. In this case, it is known that 1 i ⊂ H i [START_REF] Antoniadis | Analysis of variance on function spaces[END_REF]. As all the H I contain 1 they are obviously not orthogonal.

An alternative to avoid this issue is to consider RKHS such that for all i, H i is L 2 -orthogonal to the space of constant functions 1 i (i.e. the H i are spaces of zero mean functions for µ i ). This construction ensures that the decomposition of Eq. 16 has the properties required in Eq. 4 so we benefit from the advantages of the two equations: the meaning of Eq. 4 for the analysis of variance; and the simplicity of computation for the m I 's from Eq. 16. This approach has been applied to splines under the name of Smoothing Spline ANOVA (SS-ANOVA) [START_REF] Wahba | Smoothing spline anova for exponential families, with application to the wisconsin epidemiological study of diabetic retinopathy[END_REF][START_REF] Gu | Smoothing spline ANOVA models[END_REF]. However, SS-ANOVA is based on a decomposition of the inner product of a spline RKHS and it cannot be immediately applied for other RKHS. In the next section we introduce a general method for extracting RKHS of zero mean functions from any usual RKHS. As this approach is not limited to splines, it can be seen as an extension of the SS-ANOVA framework.

RKHS of zero mean functions

Decomposition of one-dimensional RKHS

Let H be a RKHS of functions defined over a set D ⊂ R with kernel k and µ a finite Borel measure over D. Furthermore, we consider the hypothesis:

H 1. (i) k : D × D → R is µ ⊗ µ-measurable. (ii) D k(s, s)dµ(s) < ∞.
Note that when is µ a probability measure, any bounded kernel (e.g. Gaussian, power-exponential, Matérn) satisfies the condition (ii). Thus H1 is not a very restrictive hypothesis.

Proposition 1. Under H1, H can be decomposed as a sum of two orthogonal sub-RKHS, H = H 0 ⊥ ⊕ H 1 where H 0 is a RKHS of zero-mean functions for µ, and its orthogonal H 1 is at most 1-dimensional.

Proof. From H1, the integral operator

I : H → R, h → D h(s)dµ(s) is bounded since |I(h)| ≤ D | h, k(s, .) H |dµ(s) ≤ ||h|| H D k(s, s)dµ(s) (18)
According to the Riesz representation theorem, there exists a unique R ∈ H such that ∀h ∈ H,

I(h) = h, R H . If R(.) = 0, then all f ∈ H are centered functions for µ, so that H 0 = H and H 1 = {0}. If R(.) = 0, then H 1 = span(R)
is a 1-dimensional sub-RKHS of H, and the subspace H 0 of centered functions for µ is characterized by

H 0 = H ⊥ 1 .
For all x ∈ D the value of R(x) can be calculated explicitly. Indeed, recalling that k(x, .) and R are respectively the representers in H of the evaluation functional at x and of the integral operator, we get:

R(x) = k(x, .), R H = I(k(x, .)) = D k(x, s)dµ(s). (19)
The reproducing kernels k 0 , k 1 of H 0 and H 1 satisfy k = k 0 + k 1 . Let π denote the orthogonal projection onto H 1 . Following [4, theorem 11] we obtain

k 0 (x, y) = k(x, y) -π(k(x, .))(y) = k(x, y) -D k(x, s)dµ(s) D k(y, s)dµ(s) D×D k(s, t)dµ(s)dµ(t) (20)
Example 1. Let us illustrate the previous results for the kernels k b and k g introduced in Eq. 12. As previously, we choose D = [0, 5] and define µ as the uniform measure over D. Following Proposition 1, k b and k g can be decomposed as a sum of two reproducing kernels:

k b (x, y) = k b,0 (x, y) + k b,1 (x, y) k g (x, y) = k g,0 (x, y) + k g,1 (x, y). ( 21
)
Figure 2 represents sections of k b,i (x, y) and k g,i (x, y) for various values of y.

We observe from this figure that k b,0 (., y) and k g,0 (., y) are zero mean functions. Moreover, k b,0 (., y) and k b,1 (., y) (respectively k g,0 (., y), k g,1 (., y)) are orthogonal for the inner product induced by k b (resp. k g ) but they are not orthogonal for L 2 (D, µ).

2.2. Remarks on the numerical computation of k 0 and k 1 In the previous example, the integrals of Eq. 20 have been approximated using a Riemann sum operator I R , and we discuss here the influence of this approximation. Let k0 be the approximation of k 0 using Riemann sums in Eq. 20. By definition of a RKHS, the evaluation is linear continuous on H so the operator I R is also linear continuous and the pattern of the previous proof is still valid when I is replaced by I R . The function k0 corresponds in fact to Proposition 1 applied to the averaging operator I R . We thus have the following result:

Corollary of Proposition 1: The function k0 is the reproducing kernel of the subspace of H of zero mean functions for I R . This ensures that the approximation of Eq. 20 by Riemann sums is still a s.p.d. function. For all x ∈ [0, 5] we thus have I R ( k0 (x, .)) = 0 and I(k 0 (x, .)) = 0, but I R (k 0 (x, .)) and I( k0 (x, .)) are generally not equal to zero.

Note that this remark about the Riemann sum is not valid for Monte Carlo approximations of integrals. Indeed, Monte Carlo integration has random output and the use of such methods in Eq. 20 leads to a random function that cannot be interpreted as a kernel (for example the obtained function is not symmetric with probability one).

The kernel k b can be used to compare the error induced by replacing I by I R since the integrals of Eq. 20 can be computed analytically. For I R based (f) k g,i (., 4)

Figure 2: Representation of the kernels k b,i (., y) and k g,i (., y) for y = 0, 2, 4 and i = 0, 1. The dashed lines correspond to k b,1 , k g,1 and the solid lines are for k b,0 and k g,0 . The original kernels k b (., y) and kg(., y) can be retrieved by summing the two curves.

on a set of 100 points equally spaced over [0, 5], the maximum distance we observed between k b,0 and kb,0 is lower than 0.02 so the two functions would be indistinguishable in Figure 2. Hereafter, the integrals will always be computed using Riemann sum approximations and we will not differentiate k 0 and k0 any more.

Generalization for multi-dimensional RKHS

The former decomposition of one-dimensional kernels leads directly to the decomposition of tensor product kernels if the k i 's satisfy H1:

K(x, y) = d i=1 k i (x i , y i ) = d i=1 (k i 0 (x i , y i ) + k i 1 (x i , y i )). ( 22
)
Since k i 0 (x i , y i ) is a kernel of a one-dimensional RKHS, its effect is similar to the constant function 1 in an ANOVA kernel. This decomposition highlights the similarity between usual tensor product kernels (power-exponential, Brownian, Matérn...) and ANOVA kernels.

A new class of kernels for sensitivity analysis

We now propose a special class of ANOVA kernels:

K * AN OV A (x, y) = d i=1 (1 + k i 0 (x i , y i )) = 1 + I⊂{1,...,d} i∈I k i 0 (x i , y i ), (23)
where the k i 0 are obtained by decomposing kernels as in the previous section. We now develop the use of such kernels for the ANOVA representation of the best predictor and for the computation of global sensitivity indices.

3.1. ANOVA representation of m Proposition 2. If m is a best predictor based on K * AN OV A , m I = i∈I k i 0 (x i ) t K -1 F (24)
is the term of the functional ANOVA representation of m indexed by I. Hence, the decomposition of m given by Eq. 17 coincides with its functional ANOVA representation (Eq. 4).

Proof. The kernels k i 0 are associated with RKHS H i 0 of zero-mean functions, so we have

1 i ⊥ L 2 H i 0 . The underlying RKHS associated with K * AN OV A is H * AN OV A = d i=1 1 i ⊥ ⊕ H i 0 ( 25
)
where ⊥ stands for the L 2 inner product. The result follows.

Unlike usual ANOVA kernels, the class of K * AN OV A ensures that the terms m I are mutually orthogonal in the L 2 sense.

Example 2: ANOVA representation. In order to illustrate the use of the K * AN OV A kernels, we consider an analytical test function called the g-function of Sobol [START_REF] Saltelli | Sensitivity analysis[END_REF]. This function is defined over [0, 1] d by

g(x 1 , . . . , x d ) = d k=1 |4x k -2| + a k 1 + a k with a k > 0. ( 26
)
We consider here the case d = 2 and we set a 1 = 1, a 2 = 2. Starting from a one-dimensional Matérn 3/2 kernel

k m (x, y) = (1 + 2 |x -y|) exp(-2 |x -y|), (27)
we derive the expression of K * AN OV A using Eq. 20 and 23:

K * AN OV A (x, y) = 2 i=1     1 + k m (x i , y i ) - 1 0 k m (x i , s)dµ(s) 1 0 k m (y i , s)dµ(s) 1 0 k m (s, t)dµ(s)dµ(t)     .
(28) We then build the optimal interpolator m ∈ H * AN OV A based on the observation of g at 20 points of [0, 1] 2 (which result from a LHS-maximin procedure [START_REF] Santner | The design and analysis of computer experiments[END_REF]). According to Proposition 2, the function m can be split as a sum of 4 submodels m 0 , m 1 , m 2 and m 12 which are represented in Figure 3. We observe numerically that the mean value of m 1 , m 2 and m 12 is lower than 10 -14 (in absolute value), corroborating that these functions are zero-mean. Similarly, the value of the numerical computation of the scalar products between any two different functions of the set {m 0 , m 1 , m 2 , m 12 } has the same order of magnitude.

As a comparison, we consider on this test function the usual ANOVA kernel

K AN OV A (x, y) = (1 + k m (x 1 , y 1 )) × (1 + k m (x 2 , y 2 ))
. For the corresponding model, the mean value of m 1 , m 2 and m 12 is typically in the range [-0.1, 0.3] and a scalar product between m I and m J in [-0.01, 0.03] for I = J. (a)

g x 1 x 2 (b) m x 1 x 2 (c) m 0 x 1 x 2 (d) m 1 x 1 x 2 (e) m 2 x 1 x 2 (f) m 12

Computation of sensitivity indices

We have seen that the terms m I of the ANOVA representation of m can be obtained naturally when using kernel K * AN OV A . As the sensitivity indices are based on those terms, they also benefit from the structure of K * AN OV A and they can be computed analytically and efficiently: Proposition 3. The sensitivity indices S I of m are given by:

S I = var(m I (X I )) var(m(X)) = F t K -1 i∈I Γ i K -1 F F t K -1 d i=1 (1 n×n + Γ i ) -1 n×n K -1 F (29)
where Γ i is the n × n matrix Γ i = Di k i 0 (x i )k i 0 (x i ) t dµ i (x i ) and 1 k×l is the k × l matrix of ones.
Proof. The numerator is obtained by direct calculation. Using that k i 0 is a vector of zero mean functions and the independence of the X i we have:

var(m I (X I )) = var i∈I k i 0 (X i ) t K -1 F = F t K -1 cov i∈I k i 0 (X i ) K -1 F = F t K -1 i∈I cov k i 0 (X i ) K -1 F = F t K -1 i∈I Di k i 0 (x i )k i 0 (x i ) t dµ i (x i ) K -1 F. (30)
For the denominator, we obtain similarly

var(m(X)) = F t K -1 i∈I Di 1 n×1 + k i 0 (x i ) 1 n×1 + k i 0 (x i ) t dµ i (x i ) K -1 F -F t K -1 1 n×n K -1 F. (31)
We then use the property that the k i 0 (x, .) are zero mean functions again to obtain Di

1 n×1 + k i 0 (x i ) 1 n×1 + k i 0 (x i ) t dµ i (x i ) = 1 n×n + Γ i .
A similar property is given in [START_REF] Chen | Analytical variance-based global sensitivity analysis in simulation-based design under uncertainty[END_REF] for usual separable kernels but the computation of S I require computation of all S J for J ⊂ I. This recursivity induces a propagation of the numerical error when the integrals in Γ i are approximated numerically. Here, the particular structure of K * AN OV A avoids any recursivity and each index is given directly by Proposition 3.

Example 3: Computation of sensitivity indices. As previously, we consider the g-function as a test function to illustrate Proposition 3. One particular asset of g is that the sensitivity indices S I can be obtained analytically:

S I = i∈I 1 3(1+ai) 2 d k=1 1 + 1 3(1+a k ) 2 -1 . ( 32
)
The value of the dimension is set to d = 5 and the coefficients are arbitrarily chosen to be (a 1 , a 2 , a 3 , a 4 , a 5 ) = (0.2, 0.6, 0.8, 100, 100). With these values, the variables X 1 , X 2 and X 3 explain 99.99% of the variance of g(X).

Now, we compare the indices given by Proposition 3 for three different kernels versus the analytical value given by Eq. 32. The kernels we use are the Matérn 3/2 kernel k m introduced in Eq. 27, the Gaussian kernel k g of Eq. 12 and the kernel kb (x, y) = 1 + min(x, y). In the latter, the value 1 is added to the kernel of the Brownian motion to relax the constraint of nullity at the origin.

The Design of Experiment (DoE) is a 50 point maximin LHS over [0, 1] 5 . As each DoE leads to a particular model and a different value of the sensitivity indices, computation of the indices was repeated for 50 different DoEs. The obtained mean values and the standard deviations are gathered in Table 1.

As for example 1, we studied for the model based on kb the error due to the approximation of the integrals in Eq. 29 by Riemann sums based on 100 points per dimension. The error on the final values of the indices does not appear to be significant since it is typically around 10 The computational accuracy of the global sensitivity indices is judged to be satisfactory for kb and k m . On the other hand, the model based on k g performs significantly worse than the others. This can be explained by the unsuitability of the Gaussian kernel for the approximation of a function that is not differentiable everywhere. This example highlights the importance of choosing a kernel suited to the problem at hand. Although the indices are correctly estimated with kb and k m , the sum of the indices presented in Table 1 is respectively equal to 0.94 and 0.90 for these kernels (whereas we obtain 1 for the first line). It implies that the associated models present variations for some sets of indices that are not represented in this table and eventually the sparsity in the variance structure of the g-function is not perfectly captured by the model. As the kernels K * AN OV A are ANOVA kernels, methods such as SUPANOVA [START_REF] Gunn | Structural modelling with sparse kernels[END_REF] or hierarchical kernel learning [START_REF] Bach | High-dimensional non-linear variable selection through hierarchical kernel learning[END_REF] can be applied for detecting this sparsity. The combination of K * AN OV A and SUPANOVA present similarities with the COSSO [START_REF] Lin | Component selection and smoothing in multivariate nonparametric regression[END_REF] for splines but such developments are out of the scope of this article.

Generalization to regularization problems

We have considered for now the interpolation point of view and the expression of the best predictor given in Eq. 9. However, all the results can immediately be generalized to regularization problems. Indeed, the proofs of Propositions 2 and 3 are still valid if we replace the Gram matrix K by K + λI so we have the following results.

Corollary of Proposition 2:

The terms of the ANOVA representation of the regularization best predictor m

(x) = k(x) t (K + λI) -1 F are mI = i∈I k i 0 (x i ) t (K + λI) -1 F. (33)
Corollary of Proposition 3: The expression of the sensitivity indices of m is

S I = F t (K + λI) -1 i∈I Γ i (K + λI) -1 F F t (K + λI) -1 d i=1 (1 n×n + Γ i ) -1 n×n (K + λI) -1 F . ( 34
)
Example 4. The aim of this last example is to illustrate two points: the corollaries we have just stated and the use of a non-uniform measure. We consider here a simple test function introduced in [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]:

f : R 2 → R x → x 1 + x 2 2 + x 1 x 2 . (35)
The input space R × R is endowed with a measure µ 1 ⊗ µ 2 where µ i is the standard normal distribution. With these settings, the ANOVA representation of f is f

(x) = 1 f0 + x 1 f1 + x 2 2 -1 f2 + x 1 x 2 f1,2 (36)
and the sensitivity indices S 1 , S 2 and S 1,2 are respectively equal to 1/4, 1/2, 1/4. As we want to illustrate here a regularization issue, we assume that the observations of f are affected by a noise of observation f obs = f + ε where ε is a specific realization of a Gaussian white noise with variance λ.

The DoE is a 20 point LHS maximin over [-5, 5] 2 and we consider the following kernel over R 2 × R 2 :

K * AN OV A (x, y) = 200 (1 + g 0 (x 1 , y 1 )) (1 + g 0 (x 2 , y 2 )) (37)
where g 0 is obtained by decomposing a Gaussian kernel k g (x, y) = exp -x-y 10 2 .

The value 200 is a scale parameter roughly estimated from the variance of the observations at the DoE.

Figure 4 represents the model and main effects we obtained for λ = 1. It appears that despite the small number of design points, the submodels m1 and m2 give a good approximation of the analytical main effects f 1 and f 2 . Table 2 gives the values of the sensitivity indices for different levels of noise. As var(f (X)) = 4 and var(ε(X)) = λ, the variance of the noise we introduced is up to four times larger than the variance of the signal we want to extract. The computation of the indices appears to be very robust for this example, even for a low signal-to-noise ratio.

For λ = 0, the model retrieves the analytical value of the parameters. When λ increases, the standard deviation of the estimates increases and the index S 2 tends to be slightly underestimated. One can note that the analytical value is at the most one standard deviation away from the estimated mean.

Concluding remarks

We introduced in this article a special case of ANOVA kernels with particular univariate kernels so that the L 2 -orthogonality to constants is respected. This new class of kernels offers good interpretation properties to the models since their functional ANOVA decomposition can be obtained analytically, without the usual recursive integral calculations. Finally, we showed that this natural decomposition of the best predictor benefits the computation of the global sensitivity indices. The use of Riemann sums for the numerical computation of the integrals has been discussed and the error of approximation has shown to be negligible in various examples. Up to calculation or tabulation of the integral of univariate kernels, replacing the usual ANOVA kernels with the ones proposed here may be done at negligible cost in applications, with substantial benefits for the model interpretability and global sensitivity analysis studies.

The parameter estimation issue for K * AN OV A has not been raised yet in this article. This is however an important point for the practical use of those kernels. The use of the likelihood theory has been considered, but many points such as the links between the optimal parameters for K and the optimal parameters for the associated K * AN OV A need to be studied in detail. Finally, since the pattern of the proof of Proposition 1 can be applied to any bounded operator on H, lines of future research include focusing on other operators than the integral operator I, for example for building RKHS respecting orthogonality to a family of trend basis functions.

Figure 1 :

 1 Figure 1: Examples of models obtained with kernels k b and kg with D = [0, 5]. The set X = {1, 2.5, 4} and the observations F = (-0.5, 0.75, 0.5) t are chosen arbitrarily. The left panels represent the basis functions on which the model is based and the right panels show the obtained models m and m for two values of λ.

Figure 3 :

 3 Figure 3: Representation of the g-function, the model m and all the submodels on [0, 1] 2 . The z scale is the same on all graphs.

Figure 4 :

 4 Figure 4: Representations of the model and two submodels for λ = 1. The dots correspond to the observations and for panels (b) and (c), the dashed lines stand for the analytical main effects given in Eq. 36.

Table 1 :

 1 Mean value and standard deviation of the sensitivity indices for the g-function.

	-5 .

Table 2 :

 2 Mean value and standard deviation of the sensitivity indices for the test function f for various values of the noise parameter λ.

Acknowledgments

The author would like to thank the reviewers for their valuable comments. Their constructive remarks contributed significantly to the actual version of the article. We are also grateful to James Hensman and Ben Youngman for all their remarks and comments.


Eric Pairel
email: eric.pairel@univ-savoie.fr

Ephraim Goldschmidt

Benjamin Vayre

Boukar Abdelhakim

Maurice Pillet

Éric Pairel

Ephraïm Goldschmidt

Process plans and manufacturing dimensions for the steering of machining: The Copilot-Pro methodology

Keywords: Automatic lathe, Screw machining, Process plan, Manufacturing dimensions, tool correction, Copilot-Pro

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Introduction

One of the daily issues of the setters of machining is to correct the shape and location of the cutting tools in order that workpieces meet the tolerances specified by the designers. The technique commonly used is to machine a first workpiece, then to extract it from the machine tool in order to measure it and finally to compute the corrections to the tools.

The dimensions which are measured on the workpiece do not correspond to the dimensions specified by the designer of the part. For instance, in order to adjust the final location of the tool 1 (see figure 1), the setter will measure the manufacturing dimension CF1 instead of the dimensions CE1 and CE2 specified by the designer. Indeed it is easier to calculate the correction of tool 1 by measuring CF1 than by measuring CE1 and CE2 which also depend on the location of tool 2. However, this method has several disadvantages: • The setter will tend to measure CF1 before making the surface 2. He will therefore mount and unmount the workpiece twice: Once for CF1 and a second time for CF2, which lengthens the setting times. • The tolerances of these manufacturing dimensions should be much smaller than the ones of the design dimensions CEi. On this example, CF1 and CF2 must have a tolerance about 0.1mm (±0.05mm) in order that the tolerance about 0.2mm (±0.1mm) of the CE2 is guarantied, which is a division by two of the tolerances on this example. • The conformity of the workpieces cannot be declared from these manufacturing dimensions because one would take the risk to reject workpieces which meet the design dimensions. For instance, CF1 and CF2 may have a deviation of 0.07mm and therefore be out of the tolerance of ±0.05mm, whereas neither CE1 nor CE2 are out of their tolerances. Indeed, in this case, CE1 has got a deviation of 0.07mm and CE2 none.

Nevertheless this method, historically attributed to Wade [START_REF] Wade | Tolerance control in design and manufacturing[END_REF], [START_REF] Wade | Tolerance control[END_REF], has largely been developed [START_REF] Johnson | Index tolerance chart simplies production[END_REF][START_REF] Mooney | How to adjust tolerance charts[END_REF][START_REF] Gadzala | Dimensional control in precision manufacturing[END_REF][START_REF] Ngoi | The apparent path tracing approach to tolerance charting[END_REF][START_REF] Irani | Tolerance chart optimization[END_REF][START_REF] Whybrew | A graph-theoretic approach to tolerance charting[END_REF][START_REF] Britton | An industrial implementation of computer-aided tolerance charting[END_REF][START_REF] Britton | CATCH: Computer-aided tolerance charting[END_REF][START_REF] Ngoi | Computer-aided tolerance charting: a 'black box' approach[END_REF][START_REF] Xue | Identifying tolerance chains with a surface-chain model in tolerance charting[END_REF][START_REF] Ji | CAATC -a computer-aided angular tolerance charting system[END_REF].

The main objective of the Copilot-Pro methodology [START_REF] Goldschmidt | Gammes et cotation pour le réglage des machines-outils de décolletage[END_REF], presented in this paper, is to minimize the number of extracted workpieces from the machine. In this example, the workpiece, to be measured, will be extracted from the machine only once the two machining operations are performed. This objective has several advantages: • The setting time, which include measuring time, is minimized.

• The number of setting workpieces is greatly reduced especially when the extracted workpiece can no longer be re-introduced in the machine. This is particularly the case in screw machining that requires the sawing of the workpiece from the bar in which it is machined (see figure 5). • The manufacturing dimensions to measure, have got tolerances wider than those of the Wade's dimensions. They may even correspond, in some cases, to the tolerances of the design dimensions which permit, in this case, to verify the conformity of the workpieces without the risk of rejecting good parts [START_REF] Pairel | The pilot dimensions method: reconciling steering and conformity in workshops[END_REF].

In this example, the workpiece will be extracted after the machining of surfaces 1 and 2. The setter will be able to directly measure the design dimensions CE1 and CE2 which will allow to decide of the conformity of the workpiece and to calculate the corrections, c i , to do on the tools through the following relationships, in which e CEi are the deviation of the dimension CEi :

c 2 = -e CE1
(1) c 1 = e CE2 -e CE1 [START_REF] Wade | Tolerance control[END_REF] These relationships are difficult to determine mentally in front of the machine, but it can be done in advance and provided to the setter who has then only to use them. We do not detail, in this paper, this aspect of the Copilot-Promethodology which has already been presented [START_REF] Pairel | Copilot Pro®: A full method for the steering of the machining[END_REF], but the technique used to group the machining operations before the extraction of a workpiece to measure the manufacturing dimensions and to correct the tools.

The conditions that permit to group the manufacturing operations, before the exit of a workpiece from the machine to measure its dimensions and to correct the tools, are not trivial. They are related to the workin-progress of workpieces in the production flow and to the time constraints between operations especially when there are roughed surfaces to measure before the finishing operations are performed. We study these different conditions in the following sections before the presentation of the manual method for determining these groupings of manufacturing operations and the manufacturing dimensions for the correction of the tools.

Manufacturing phases and stages of the setting and monitoring plans Copilot-Pro

We consider that the general case of a manufacturing process consists of manufacturing stations, after which the workpieces are stored or carried, and within which there are one or more workpiece holders enabling the achievement of one or more manufacturing operations (see figure 2) 1 .

To set the manufacturing tools, it is necessary to measure at the start of production, and periodically during it, a workpiece just being machined (or multiple workpieces to reduce the random deviations by averaging the measured deviations). The least expensive is to measure and set the tools accordingly, just before it enters in a stock in order to do so in hidden time respected to production. However if the measured workpiece has a geometrical deviation greater than a defined acceptance limit, all the work-in-progress workpieces, from the manufacturing operation on responsible for this deviation and that probably also have the same geometrical deviation, will have to be scrapped. This is the first, of the two reasons, why it is not possible, in general, to wait until the end of the manufacturing process, to measure a workpiece and to set its tools accordingly. The second reason is that at the end of the process, the setter of the last manufacturing operations, may not have the control of the setting of the first operations, either because they are performed by another person, or because they have already been made on all workpieces.

So we have to divide chronologically the manufacturing process into manufacturing phases, at the end of which the measuring of workpieces will allows to set the manufacturing operations that they contain. We define the manufacturing phase as the grouping of a set of manufacturing operations, performed on one or more workpiece holders, and which can be jointly set. The addition of a workpiece holding, and the manufacturing operations performed in this workpiece holder, in a phase already containing the previous workpiece holders, should be done taking into account the following two criteria: • Can we jointly control the setting of the new operations with the operations already in the phase? • Can we accept the work-in-progress increasing of workpieces that would result 1 ?

In fact, even if all the manufacturing operations of a phase can be set, this will require measurement stages before the end of the phase because some surfaces will then disappear. This is especially true for rough surfaces of which the setting is nevertheless important because it determines the quality of the finished surface and the lifetime of the finishing tool.

However the introduction of intermediate measuring stages leads to, in most cases, increased cycle time of the measured workpiece. In order for this to not be the case, the intermediate measurement should be done at the storage or the carrying of the workpiece between workpiece-holders. However, the grouping in a single phase of two workpiece-holders, separated by a carrying or stock, is rarely possible. It is therefore in the just-in-time flow of the manufacturing operations that these measuring stages should be introduced to measure the surfaces that disappear. The increase in cycle time that would result is often not acceptable in industry. This has therefore led to define two Copilot-Pro manufacturing plans for any manufacturing process: • The monitoring plan for which a single measuring stage is defined at the end of each manufacturing stage and therefore can not set the tools of surfaces that disappear during the phase. This plan will be used with high frequency when the production is in progress. • The setting plan in which intermediate measuring stages are defined in order to be also able to set the tools of which the surfaces disappear, including roughed surfaces. These intermediate measuring stages leading, in general, to an increase in cycle time, the setting plan will be used with a lower frequency than the monitoring plan and mostly requires a production stop in order to have time to extract the workpiece out of the machine and to measure it. It will also be used for initial setting of all tools before starting production.

To limit the number and duration of the intermediate measuring stages of the setting plan, we are going to seek to achieve the most manufacturing operations possible before having to do a measuring stage.

1. In classical productions, which are push flows, the Copilot-Pro manufacturing phases correspond to manufacturing stations because those last end with stocks. We find here the commonly accepted sense of the term of the manufacturing phase. But the Copilot-Pro definition, more general and more precise, allows to consider more important grouping of operations, especially in just-in-time flows. These groupings of operations before the measuring stages, are called the manufacturing stages. In the case of the monitoring plan, there is only one single manufacturing stage, followed by a single measuring stage and a single setting stage in a manufacturing phase, while for the setting plan, there may be several manufacturing stages, each followed by a measuring stage, and completed at the end of the phase, by a setting stage. The figure 3 illustrates this hierarchical organization of a plan in phases, stages and manufacturing operations, and their chronological sequences.

Manufacturing process taken in example

We consider the example of a part, of which the design drawing is given in figure 4, machined from a free-cutting bar on a automatic lathe, also called screw machine, according to the manufacturing process shown in figure 5.

The design dimensions of the workpiece are named CE1, CE2 and CE3 (see figure 5. The diameter dimensions are not indicated in order to simplify the presentation). These dimensions have got tolerances even if they are not indicated here. At any time, the location of the second workpiece holding (operation 50 on figure 5), which is performed by the secondary spindle of the machine, is well known relatively to the location of first workpiece holding, which is performed by the main spindle. This is why the workpiece is not laid on its plate surface in the second workpiece holder. The secondary spindle pinches the workpiece before the sawing off (operation 40), then moves back (operation 50) to permit the achievement of the operations 60 and 70.

Setting plan of the example process

On the manufacturing process presented in the previous section, the setter can simultaneously control all settings of the tools. In addition the work-in-progress of workpieces is acceptable. If it is a single spindle lathe, it is about of two workpieces: One on the main spindle, the other on the secondary spindle. The two workpieces holders, and the manufacturing operations they allow to perform, can be placed in the same phase.

However, at the end of the manufacturing phase, the rough surfaces will be gone (operations 20 and 60). The monitoring plan does not allow us to adjust the location of these surfaces. We are therefore going to determine the manufacturing stages and the measuring stages of the setting plan necessary to set all the manufacturing operations.

To group the largest number of manufacturing operations before performing a measuring stage, we are going to discuss the constraints between operations, in relation with their grouping into a single manufacturing stage.

The most obvious constraint is that no operation of a stage in the setting plan, should remove a surface already done in the stage otherwise the operation which has machined this last surface could not be set. In particular, a roughing operation should be performed in a manufacturing stage anterior to the one containing the finishing operation. This constraint is the first in table 1. As the manufacturing stages are not yet established, this constraint will be introduced between the two operations. Graphically, we represent by an arc in thick line, oriented from the roughing operation towards the finishing operation (see arcs between operations 20 and 30 on the one hand, and 60 and 70 on the second hand, in figure 6).

On the other hand, there are technological constraints that impose a chronological order between operations (some operations can only be achieved if others have been before). These constraints, three in number, are explained in table 1, numbered 2, 3 and 4. They require that the first operation, in chronological order, is in an anterior or identical stage to the one containing the second operation. These constraints are represented graphically by a dashed arc oriented from the first operation to the second operation in the chronological order (see arcs in dashed lines on figure 6).

A last constraint is related to the Copilot-Pro method of determining the manufacturing dimensions presented below. If, in a phase, a workpiece holding, uses a surface not manufactured or laid on in the phase, then it should open a new manufacturing stage. This is obviously the case of the first workpiece holding of the phase but it can also be the case for further workpiece holders in the phase even if it is uncommon.

In order that this workpiece holding opens a new stage, we should introduce constraints between the last manufacturing operations of the previous workpiece holders and this workpiece holder, in the aim to indicate that those operations should be placed in stages anterior to the one containing this new workpiece holding. This constraint is number 5 in the table. It occurs only very rarely and is not present on the process presented in this article. The table below lists the five constraints that seem necessary and sufficient, until now, for the grouping of operations in manufacturing stages 1 .

Table 1: Constraints between operations for their grouping in manufacturing stages.

By systematically applying these constraints to the operations of the manufacturing process used as an example, we obtain the graph of figure 6, which then determines the grouping of operations in manufacturing stages. Indeed, one finds that the operation 40 can be performed without having achieved the operation 30 (since there is no arc between them). The first stage will be made up of the operations 10, 20 and 40 which are connected by a constraint of kind "anterior or identical stage". The machined part will have the form shown on the first drawing of the figure 9.

1. These constraints are said automatic. Other constraints, which can not be determined automatically from the description of the process, can be introduced. For example to indicate that it is not possible to perform a groove in a bore as the bore has not been achieved. These constraints are of type "anterior or identical". Indeed the groove may be machined in the same stage as the bore.

# Constraints

Representations on the graph of constraints 1

If an operation (like a finishing) removes the surface achieved by another (like a roughing), this last one must be in an anterior stage to the one containing the first a .

a. This constraint has been highlighted by Ephraïm Goldschmidt [START_REF] Goldschmidt | Gammes et cotation pour le réglage des machines-outils de décolletage[END_REF]. We have generalized it for every operation removing a surface made by another operation.

Thick arc from the roughing operation towards the finishing operation.

2

Any workpiece holding must be made in an anterior or identical stage to those containing the manufacturing operations performed in this workpiece holding b .

b. This constraint has been also highlighted by Ephraïm Goldschmidt [START_REF] Goldschmidt | Gammes et cotation pour le réglage des machines-outils de décolletage[END_REF].

Dashed arc from the workpiece holding towards the manufacturing operations performed in this workpiece holding.

3

A surface must be machined in an anterior or identical stage to those containing a workpiece holding using this surface c . c. This constraint has been expressed by B. Vayre [START_REF] Vayre | Algorithme de génération des étapes de mesurage et de réglage pour Copilot-Pro®[END_REF].

Dashed arc from the machining operation of the surface towards the workpiece holders using this surface.

4

The sawing off must be machined in an anterior or identical stage to the one containing the next workpiece holding if it exists d .

d. This constraint had not been introduced into any work before this article. If it is not respected, there is a risk of getting a stage containing machining operations performed before and after a sawing off, while this last one is in a subsequent stage, which is not technically possible.

Dashed arc from the sawing off operation towards the following workpiece holding.

5

The latest manufacturing operations, performed in the first workpiece holders of a manufacturing phase, must be performed in the anterior stages to the one containing a workpiece holder using a surface neither machined nor laid on in this phase e .

e. This constraint has been highlighted by B. Vayre [START_REF] Vayre | Algorithme de génération des étapes de mesurage et de réglage pour Copilot-Pro®[END_REF]. If this constraint is not respected, there is a risk of placing the new workpiece holding in the stage containing the previous workpiece holders and then to introduce manufacturing dimensions between, on the one hand, surfaces made by the manufacturing operations of the previous holders and, on the other hand, the surface used by the new workpiece holding. But the location of this last surface is unknown and uncontrolled in the previous workpiece holders.

Thick arcs from the latest manufacturing operations performed in the first workpiece holders of a manufacturing phase towards the workpiece holder using a surface neither performed nor laid on in this phase.

Fig. 6. Graph of constraints between manufacturing operations.

The operation 30 should be performed in a second stage with the operations 50 and 60 which are connected by constraints of kind "anterior or identical stage". The workpiece after the first stage can not be reused for the second stage. It will be necessary to make a second workpiece with the operations 10, 20, 30, 40, 50 et 60. In other words the new manufacturing stage adds operations to those made in previous stages. Finally a third stage will be needed to perform the operation 70.

The workpieces coming from the three manufacturing stages are represented in figure 9 1 .

Manufacturing dimensions of the setting plan

Next, we need to identify each surface. The identifier of a surface begins with a capital letter, corresponding to the direction of the dimension which locates it, followed by a serial number and a lowercase letter if it is a rough surface (see figure 7).

To determine the manufacturing dimensions, we create a table in which, for each stage in line, we place the X character for each machined surface and the O character for each holding surface (or both if the surface is first machined then placed on a workpiece holder. See figure 8).

Since for each new stage, the setter will have made the operations of the previous stage, the surfaces of these operations remain after the new stage (unless it was the roughed surface of the finished surface machined in the stage). This information is indicated by the I character. For example, Z1 is made in stage 1.1 and is not removed by any following operations. So this surface is present on the workpiece in stages 1.2 and 1.3. This is indicated by the I character in these stages (see figure 8).

The manufacturing dimensions that can be measured and controlled by the setting of tools, are those which link two surfaces present in the same stage, that is to say surfaces identified by the X, O or I characters. Thus, at the end of stage 1.1, any dimension between Z1, Z3a and Z4 can be measured and controlled by the setter. In contrast, the dimension between Z2 and Z2a, for example, is never measurable (in none of the lines, the two surfaces are present simultaneously).

To search for all the manufacturing dimensions that need to be measured at the end of each stage, we build a second table, under the first, to represent the surface linked by design dimensions and stock removal dimensions (see figure 8). The stock removal dimensions correspond to the thickness given for finishing operations. They are also toleranced. For each design dimension and stock removal dimension, we are looking, in the first table, for the shortest path between the surfaces which the dimensions connect. Thus, for the design dimension CE1, CE2 and CE3, are obtained manufacturing dimensions CF Z1Z4 , CF Z1Z3 and CF Z2Z4 measurable at the end of stage 1.3 (or even earlier for CF Z1Z4 , CF Z1Z3 ).

For the stock removal dimensions CM1 and CM2, the surfaces they connect are never simultaneously present in the stages. They should therefore be measured indirectly by a chain of at least two manufacturing dimensions. We can use, for instance, the previous manufacturing dimensions, CF Z2Z4 and CF Z1Z3 :

CM1 = -CF Z2Z4 + CF Z4Z2a (3) CM2 = -CF Z1Z3 + CF Z1Z3a (4)
This choice can result in reduced tolerances of the dimensions CF Z2Z4 and CF Z1Z3 , especially if the tolerances of the stock removal dimensions, are of the same order of magnitude as those of the design dimensions. Other chains would be possible to prevent this decrease, however, they would introduce additional manufacturing dimensions. So, by choosing this solution, we can specify the state of the workpiece after each manufacturing stage and the manufacturing dimensions to be measured, on three manufacturing designs (figure 9).

The measurement of the deviation of these manufacturing dimensions will permit to set all the tools [START_REF] Pairel | Copilot Pro®: A full method for the steering of the machining[END_REF]. One of the tools should be fixed, that is to say, considered without deviation, to calculate the deviations of others. Otherwise an infinite number of solutions are possible. In general, in screw machining, it is the sawing off tool which is fixed because it is very close to the spindle.

Monitoring plan of the example process

The monitoring plan is used during the production. The measurement of the workpiece is done only at the end of each manufacturing phase. As some rough surfaces have then disappeared, some manufacturing dimensions of the setting plan are no longer measurable. In our example, these are the manufacturing dimensions CF Z1Z3a and CF Z2aZ4 . Only the manufacturing dimensions, of the surfaces remaining at the end of the manufacturing phases, remain, that is to say, for the example, the dimensions CF Z1Z4 , CF Z1Z3 and CF Z2Z4 which correspond, in this example, to the design dimensions (it is not always the case). However, their tolerances must be re-calculated because they are no longer used to control the stock removal dimensions CM1 and CM2. They will have, in this example, directly the tolerances of the design dimensions. The manufacturing dimensions to measure in this monitoring plan are only a part of the manufacturing dimensions of the setting plan. This has the advantage of being able to use the same process of measurement at the end of the manufacturing phases for the setting plan as for the monitoring plan.

Influence of intermediate stocks on the plans and on the manufacturing dimensions

By way of comparison, and of explanation of the Copilot-Promethodology, we will now determine the setting plan and the manufacturing dimensions of a variant of the previous manufacturing process. This variant consists in producing the part in two times: First operations from 10 to 40 are performed and, after an intermediate stock, operations from 50 to 70 are performed (see figure 10). For this variant, the holding 2 is different. The workpiece must be located on the flat surface Z2 (see figure 10).

Due to the intermediate stock, we consider that it is not possible to wait for the end of the process to set the tools. The work-in-progress, which may be put off, is too big. The setting and monitoring plans will therefore consist of two manufacturing phases.

By performing the constraint graph, it is possible to determine that each phase will consist of two manufacturing stages. We have therefore, for this variant, two manufacturing phases and consequently two setting stages, instead of a unique, and four manufacturing stages, each one followed by a measuring stage, instead of three. Moreover, by representing the design dimensions and the stock removal dimensions in a table under the setting stages, we observe that, for this variant of the process, the design dimension CE3 can no longer be measured directly because the surfaces that it connects do not exist simultaneously in any manufacturing stage (see figure 11).

Several sets of manufacturing dimensions are possible. When seeking for a set with a minimum number of manufacturing dimensions, we can obtain those ones given from equation 5 to equation 9. For the most part of them, they use the Z3 surface, common to the two manufacturing phases. Those manufacturing dimension can be indicated on the manufacturing drawings (see figure 12). Compared with the initial process, we see that it is better to have a single manufacturing phase rather than two. Indeed, the design dimension CE3 can no longer be measured even in the monitoring plan. The tolerances on the two manufacturing dimensions which replace CE3 should have smaller tolerances, especially if the stock removal dimensions have tolerances of the same order of magnitude as the design dimensions.

Conclusion

We have presented a method to arrange the manufacturing operations in order to minimize the number of measuring stages necessary to set the cutting tools during the manufacturing of a part. This method is part of a comprehensive methodology, called Copilot-Pro for the preparation of the manufacturing and the setting of the tools in production or in a production halt. It streamlines the operations of measuring workpieces and setting tools that are too often left up the setters in industry, leading to many scrapped workpieces, long times of production halt and lower quality for the batches of workpieces.

If the manual preparation, as presented in this paper, can be engaged on workpieces with few machined surfaces, the use of software for more complex workpieces is essential. We are currently developing a software implementing this innovative method.

Fig. 1 .

 1 Fig. 1. Design dimensions specified by the designer, CEi, and manufacturing dimensions used by the setter, CFi.

Fig. 2 .

 2 Fig. 2. General diagram of a manufacturing process according to the hierarchical model "Station / Workpiece-holders / Operations" of Copilot Pro.

1 Fig. 3 .

 13 Fig. 3. Hierarchical organization of the Copilot-Pro plans and typical chronological process of phases, stages and operations.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Design drawing of the part.

1 .Fig. 7 .

 17 Fig. 7. Identification of the machined surfaces.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Manufacturing stages of the setting plan and table of the design dimensions and stock removal dimensions.

Fig. 10 .

 10 Fig. 10. Manufacturing process with an intermediate stock CE1 = CF Z1Z3 + CF Z3Z4(5) CE2 = CF Z1Z3[START_REF] Ngoi | The apparent path tracing approach to tolerance charting[END_REF] CE3 = CF Z2Z3 + CF Z3Z4[START_REF] Irani | Tolerance chart optimization[END_REF] CM1 = CF Z2aZ3 -CF Z2Z3[START_REF] Whybrew | A graph-theoretic approach to tolerance charting[END_REF] CM2 = CF Z3Z4 -CF Z3aZ4[START_REF] Britton | An industrial implementation of computer-aided tolerance charting[END_REF]

Fig. 11 .

 11 Fig. 11. Setting plan for the variant manufacturing process: Two manufacturing phases of two setting stages each one.

Fig. 12 .

 12 Fig. 12. Manufacturing drawings of the setting plan for the process with an intermediate stock (the machined surfaces are highlighted with blue thick lines).

The description of the process according to the hierarchical model "Process / Station / Workpiece holding / Manufacturing Operation" is the formatted input data of the Copilot-Pro method. Stocks and carriers systems are associated with stations that they precede.


Sylvain Kubler

Alexandre Voisin

William Derigent

Éric Rondeau

André Thomas

A fuzzy analytic hierarchy process for group decision making: application for embedding information on communicating materials

Keywords: Fuzzy AHP, group decision making, Intelligent product, Product Lifecycle Management, Data Dissemination

It is not uncommon today to use intelligent products for ensuring an information continuum all along the product lifecycle. However, it is not that easy to identify what information should be stored on the product, in particular because of the large number of persons who are concerned by it. This paper proposes to use fuzzy AHP to select contextsensitive information from the database, that must be stored on the product. Through our approach, points of view from different actors are aggregated thanks to the fuzzy logic.

I. INTRODUCTION

In recent years, some authors have argued the use of intelligent products in the context of PLM (product lifecycle management) [START_REF] Wong | The Intelligent Product Driven Supply Chain[END_REF]. Indeed, a product moves through numerous companies through its lifecycle and technical, semantic and organizational interoperability between companies is not always ensured, thereby contributing to information loss. Considering the product as an information vector (in which data can be stored) should contribute to improved interoperability throughout the product lifecycle (PLC). However, most of the time, products only provide a network pointer to a linked database (e.g. a RFID tag) [START_REF] Wong | The Intelligent Product Driven Supply Chain[END_REF]. Moreover, this kind of product is still limited (e.g. risk of tag damage, small memory capacity,. . . ).

As a result, we propose a new concept referred to as communicating material which considers the material as intrinsically communicating. Our recent works [START_REF] Kubler | Prototyping of a communicating textile[END_REF], [START_REF]Information dissemination process for contextaware products[END_REF], [START_REF]Storage of information on manufactured products using "communicating materials[END_REF] published on this concept try to solve related research questions (product manufacturing, data processing,. . . ). Among them, the automatic selection of relevant information from external databases to the material itself is an interesting topic. As a result, a data dissemination process is proposed to identify what the relevant information to users is and where it should be stored: on databases or on the products themselves? This process is applied each time a user wants to store information on the product and consists of two main steps :

1) selecting relevant information from the database that should be stored/replicated on the product [START_REF]Information dissemination process for contextaware products[END_REF], 2) storing information on the product and, subsequently, retrieving it. A communicating textile prototype is designed in [START_REF] Kubler | Prototyping of a communicating textile[END_REF] 1 and a set of tools are developed in [START_REF]Storage of information on manufactured products using "communicating materials[END_REF] to communicate with it. 1 A huge quantity of RFID µtags are spread in the textile.

This paper focuses on the first process step. The information relevance is dependent upon a variety of factors such as the user concerns, the product environment, etc. An approach is developed in [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF] to select context-sensitive information by matching the context with data, which uses a Logical Data Model (LDM). Figure 1 gives insight into a part of a LDM, where one entity corresponds to a relational table as depicted with the entity/table Material. The attributes listed in each entity correspond to the table columns and each row is referred to as a tuple. For instance, Material has 3 attributes and 4 tuples. One data item corresponds to one table cell (cf. Figure 1).

The model developed in [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF] assesses the relevance of all data items, from all tables. Their model uses the notion of priorities which are numerical values (either supplied or generated via observation and experimentation) assigned using a multifaceted evaluation of criteria. The higher the relevance value, the higher is the probability that this data item will be stored on the product. For instance, the data items T Mat{3,3} in Figure 1 is the most relevant compared to the others with a value of 0.40. The approach proposed in [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF] has been applied in our previous work [START_REF]Information dissemination process for contextaware products[END_REF]. However, this approach does not offer possibilities to take into account several opinions and may turn out to be inappropriate to our context. Indeed, at a given moment of the PLC, several experts are concerned by selecting information that must be stored on the product, therefore generating different points of view. All points of view are legitimate and must be taken into account. Accordingly, this paper proposes an approach which combines the use of the analytic hierarchy process (AHP) with the fuzzy logic. AHP enables to handle the multi-criteria decision making (MCDM) problem and fuzzy logic enables to take into account multiple points of view.

Section II briefly introduces fuzzy AHP. Section III introduces the fuzzy set semantic and representation used in our approach. Finally, section IV details the stages that compose our fuzzy AHP method (a case study is used as a "red line" to make the understanding easier).

II. RELATED WORK

AHP [START_REF] Saaty | The Analytic Hierarchy Process[END_REF] deals with MCDM problems and is now used extensively in numerous sectors [START_REF] Vaidya | Analytic hierarchy process: An overview of applications[END_REF] by means of decision matrices expressing the expert points of view. However, classical AHP does not integrate an aggregation mechanism to allow the expression of multiple points of view. Therefore, to solve this issue, some works propose to use the fuzzy logic in conjunction with AHP.

Fuzzy set theory was introduced by [START_REF] Zadeh | Fuzzy sets[END_REF] to deal with problems involving vagueness of human thinking. In a classical set A, an element belongs entirely or not to A. In a fuzzy set Ã, an element can have different degrees of membership to Ã.

The earliest work in fuzzy-AHP appeared in [START_REF] Van Laarhoven | A fuzzy extension of saaty's priority theory[END_REF]. [START_REF] Chang | Applications of the extent analysis method on fuzzy ahp[END_REF] introduced a new approach for handling fuzzy AHP, with the use of triangular fuzzy numbers for pairwise comparison scale and the use of the extent analysis method (FEAHP). In the literature, many works implement FEAHP. However, authors generally use the same membership functions (i.e. triangular) without thinking about the meaning of the information that must be modeled. Moreover, calculations in FEAHP are applied on particular points/parts of membership functions which can lead to information loss (i.e., loss of the form of the membership function).

In our study, we choose to use fuzzy AHP, on the one hand, in preserving the form of membership functions2 and, on the other hand, by explaining clearly the semantic of each fuzzy set. To do so, specific tools/representations are used and are introduced in the next section.

III. FUZZY SET SEMANTIC AND REPRESENTATION

A solution that allows to preserve the form of membership functions consists in using the α-cut representation. Section III-A introduces such a representation. The fuzzy set semantic used in our approach relies on two types of group consensus which are detailed in section III-B.

A. α-cut representation

In a fuzzy set Ã, an element can have different degrees of membership to Ã. For each element of the referential x, a degree of membership to the fuzzy set is assigned, noted µ Ã(x). Thus, when µ Ã(x) = 1, x belongs entirely to Ã; when µ Ã(x) ∈]0; 1], x belongs more or less to Ã. In our study, trapezoidal intervals are considered as depicted in Figure 2 As with crisp numbers, arithmetical operations can be defined to manipulate fuzzy sets. One technique consists in using a representation of fuzzy sets in the form of crisp intervals, called α-cuts [START_REF] Zadeh | Fuzzy sets[END_REF]. Then, operations are applied on these intervals. An α-cut level of à (α ∈]0, 1]) corresponds to the crisp set A α such as :

A α = {x ∈ R| µ Ã(x) ≥ α} α ∈]0, 1] (1)
by definition A 0 = support( Ã)

Figure 2(b) depicts the trapezoidal membership function µ Ã(x) by using 4 α-cut levels: α = {0, 0.33, 0.66, 1}. The use of crisp intervals makes it possible to preserve a calculation simplicity, but the membership function must be sampled as many times as α-cut levels. In our study, we use 11 α-cut levels: α ∈ {0, 0.1, 0.2, . . . , 0.9, 1}.

B. Group consensus

Two evaluation cases are identified in our study. Let us consider the evaluation of a student by his teachers:

• case (a) : we seek to evaluate the students' skills in Mathematics by different teachers and exams. Thus we get several measures of a same object, • case (b) : we seek to evaluate the students' skills by teachers of Mathematics, English and Sport. The purpose here is to find a representation between several points of view on a same object. The aggregation of different decision maker's point of view is thus performed according to the case, (a) or (b):

• consensus (a) : we propose to aggregate the crisp points of view and no information should be neglected. Accordingly, evaluation are aggregated via a fuzzy interval where all points of view form the kernel. However, we cannot rule out that the target value is located beyond the kernel (a new expert could give a new point of view outside the interval boundaries).

We propose therefore to extend the support through a slope. 2) collection and aggregation of information for determining: i) fuzzy sets of alternatives with respect criteria, ii) the relative criteria importance, 3) creation of the fuzzy judgment matrix à based on the fuzzy sets of alternatives with respect to criteria, 4) computation of the fuzzy performance matrix H by synthesizing à with the relative criteria importance, 5) alternative ranking. This step needs to aggregate the multi-criteria performance of alternatives in a fuzzy vector R and then, to rank alternatives. To make the understanding easier, a scenario is considered in the rest of the paper whose parts are preceded by the symbol "➫". This scenario relies on the piece of LDM introduced in Figure 1.

A. Stage 1 : AHP structure

Our MCDM problem is broken down into the hierarchical structure depicted in Figure 5. The alternatives are the data items (cf. level 3) which must be assessed and ranked in term of relevancy (cf. level 1).

B. Stage 2 : Information collection + Aggregation

In this section, we provide information about the way to evaluate each alternative with respect to each criterion and the way to obtain the criteria importance.

1) Enumeration: Through this criterion, users may enumerate information they judge important to store on the product. To do so, each decision maker p enumerates attributes from tables (p ∈ {1, . . . , Q} with Q the total number of decision makers). Let t a table from the LDM and v an attribute of t. If v is enumerated by p, the enumeration score s p (v, t) = 0.5 or 1 (depending on the preference intensity), otherwise 0 (see equation 1). Decision makers may come from different areas (production manager, shipping manager, etc.) and thereby, may want to store different information on the product. Accordingly, consensus (b) is used (cf. Figure 3(a)) and the set of points of view (noted V) is aggregated through a classical interval in Equation 3. Finally, the score of a data item l ∈ v with respect to C e , noted Φe (l), is equal to s(v, t).

s(v, t) = [min(V) max(V)], V = {s 1 (v, t), ..., s Q (v, t)} (3)
➫ Three experts enumerate respectively attributes from Material (i.e. ID_Material, Description and value) as in Figure 6 (M is the abbreviation of Material). Expert 1 judges "useful" information from attribute ID_Material: s 1 (ID_Material,M) = 1 and "useless" information from attribute Description. Aggregation of the three expert's point of view, regarding each attribute, is performed based on Equation 3. Equation 4 details the calculation of s(value, M) which is depicted in Figure 6. Fuzzy sets related to T Mat {3, 1}, T Mat {3, 2}, T Mat {3, 3} with respect to C e , noted φe (T Mat {3, 1}), φe (T Mat {3, 2}), φe (T Mat {3, 3}), are respectively equal to s(ID_Material, M), s(Description, M) and s(value, M) because they respectively belong to these three attributes. 2) Contextual: The enumerated adjustment gives users the freedom to specify information they want to store on the product. However, they could omit important information. Indeed, they might not be aware of all the data needed by the downstream actors (along the PLC). As a result, a new criterion referred to as contextual (C r ) is integrated in order to moderate and to balance C e . First, let us note that a multitude of information systems exist over the PLC (e.g. ERP, PDM, MES) which are not concerned by the same data (i.e. the same entities from the LDM).

s(value, M) = [min(1, 1, 0) max(1, 1, 0)] = [0, 1] (4)
s 1 (ID_Material,M) = 1 s 1 (Description,M) = 0 s 1 (value,M) = 1 s 2 (ID_Material,M) = 1 s 2 (Description,M) = 0.5 s 2 (value,M) = 1 s 3 (ID_Material,M) = 1 s 3 (Description,M) = 0 s 3 (value,M) = 0

Fuzzy opinion aggregation

D 1 G =     G 1 G 2 G 3 G 4 G 1 1 1 1 5 7 G 2 1 1 1 3 5 G 3 5 3 1 7 G 4 1 7 1 5 1 7 1     D 2 G =     G 1 G 2 G 3 G 4 G 1 1 3 1 3 5 G 2 1 3 1 1 3 5 G 3 3 3 1 7 G 4 1 5 1 5 1 7 1    

Fuzzy opinion aggregation

Equation 6 The idea is to identify specific "entity groups" through the LDM according to, for instance, the information systems and, therefore, to evaluate their importance over the PLC.

+ Table I G 1 G 2 G 3 G 4 G 1 G = G 2 G 3 G 4
In our study, each decision maker p performs pairwise comparisons between entity groups as in Equation 5, with Z the number of groups defined trough the LDM. The importance of entity group i over entity group j evaluated by the decision maker p is noted s p ij . This evaluation is based on the 1 to 9 point scale from [START_REF] Saaty | The Analytic Hierarchy Process[END_REF]: {1, 3, 5, 7, 9}. s p ij =1 means that groups i and j are equal in importance and s pij =9 means that group i is strongly favored j.

Experts who proceed to evaluations come from domains related to the information systems involved in PLC/PLM and have a same point of view on the criterion. As a result, it is necessary to aggregate the matrices D p G | p = {1, . . . , Q} based on a consensus (b). All elements s p ij are aggregated in a fuzzy set noted sij through rules from Table I 3 . We assume that experts agree on the sense of the importance (G i > G j or G i < G j ). Matrix G in Equation 6 synthesizes all fuzzy sets. Finally, the relative importance of each group G i is computed in Equation 7 on the basis of 11 α-cuts (one α-cut is noted λ α G (G i )) and gives rise to a fuzzy set λG (G i ). These importances are synthesized in a fuzzy vector ΛG as in Equation 8. If a data item l is contained in a table included in G i , its score with respect to C c , noted Φc (l), is thus equal to λG (G i ).

D p G =       G1 . . . GZ G1 s p 11 . . . s p 1Z . . . . . . . . . . . . GZ s p Z1 . . . s p ZZ       , s p ij = 1 i = j s p ji -1 i = j (5) G =      G1 . . . GZ G1 s11 . . . s1Z . . . . . . . . . . . . GZ sZ1 . . . sZZ      (6) λ α G (Gi) = Z j=1 G α (i, j) Z k=1 Z j=1 G α (k, j) ∀ α = {0, 0.1, 0.2, . . . , 1} (7)
ΛG = [ λG(G1), λG(G2), . . . , λG(Gi), . . . , λG(GZ )]

➫ four entity groups are defined in the LDM. Two experts provide the pairwise comparisons in Figure 7. Expert 1 highly favors G 1 over G 4 (s 1 14 = 7) and expert 2 gives an importance slightly below (s 2 14 = 5). Aggregation of D 1 G and D 2 G is then performed based on rules from 3 we limit ourselves to extend the support to the graduation immediately superior or inferior of the Saaty scale because we consider five scale points and one graduation represents 25% of the scale.

G i > G j min(s p ij ) = 1, max(s p ij ) = 1 sij = [1 1 1 3] min(s p ij ) = 1, max(s p ij ) = 9 sij = [1 1 9 9] min(s p ij ) = 9, max(s p ij ) = 9 sij = [7 9 9 9] min(s p ij ) = 1, max(s p ij ) = x ∀ x = 3, 5, 7 sij = [1 1 x(x + 2)] min(s p ij ) = x, max(s p ij ) = 9 ∀ x = 3, 5, 7 sij = [(x -2) x 9 9] min(s p ij ) = x 1 , max(s p ij ) = x 2 ∀ x 1 , x 2 = 3, 5, 7 sij = [(x 1 -2)
x 1 x 2 (x 2 + 2)] Fuzzy sets sji are deducted from fuzzy sets sij

min(s p ij ) = x 1 max(s p ij ) = x 2 ∀ x 1 , x 2 = 1, 3, 5, 7, 9 sji = [ 1 9 1 x 2 1 x 1 1]
Table I. Equation 9details the aggregation computation related to elements s 1 14 and s 2 14 whose the resulting fuzzy set s14 is depicted in Figure 7 (zoom 1). Finally, the relative importance of each group G i is computed via Equation 7and are synthesized in ΛG . We can see that information from G 3 is the most important at this stage of the PLC 4 , follows up by G 1 , G 2 and G 4 . If we assume that Material ∈ G 1 , therefore, fuzzy sets of T Mat {3, 1}, T Mat {3, 2}, T Mat {3, 3} with respect to C c , noted φc (T Mat {3, 1}), φc (T Mat {3, 2}), φc (T Mat {3, 3}) are equal to λG (G 1 ).

s 1 14 = 7, s 2 14 = 5 Table I ---------------→ [(x 1 -2) x 1 x 2 (x 2 +2)] s14 = [(5 -2) 5 7 (7 + 2)] (9)
3) Data size: This criterion aims to favor the storage of information on the product according to the data size. Since products are often memory-constrained, data relevance should decrease when data size increases. Such a behavior can be obtained via Equation 10, with d the size of a data item l and k p a constant adjusted by the expert p. Figure 8 shows two functions considering two different k p . It can be observed that the smaller k p is, the bigger the data authorized to be stored on the product is (data with sizes > 60 bytes are neglected when k = 1.08). As previously, experts may have difference points of view on this criterion and may adjust differently k p . Let us consider an expert 1 who is interesting to a high quantity of tuples (e.g., expert 1 enumerates many attributes) and an expert 2 who is interesting to a small quantity. As a result, expert 1 may intend to store on the product small data items to get a "complete/diversified" view of the quantity, while expert 2 wants to authorize the storage of big data items (i.e. k 1 > k 2 ). Accordingly, the consensus (a) is used for aggregating the different k values in a fuzzy set k as in Equation 11. Finally, the fuzzy set of a data item l with respect to C s , noted Φs (l), is computed in Equation 12.

(k p ) -d with k p ∈ [1; +∞] d ∈ N (10)
D 1 ρ =   C e C c C s C e 1 7 7 C c 1 7 1 3 C s 1 7 1 3 1   D 2 ρ =   C e C c C s C e 1 5 7 C c 1 5 1 5 C s 1 7 1 5 1  

Fuzzy opinion aggregation Equation 6 + Table I

k = [min(V) max(V)], V = {k 1 , ..., k Q } (11) φs(l) = k-d = max(V) -d min(V) -d (12)
➫ a unique expert 1 adjusts the coefficient k p at 1.08 to neglect data items > 60 bytes. The fuzzy set resulting of the aggregation is equal to a singleton set of 1.08 as detailed in Equation 13 and illustrated in Figure 9. The fuzzy set of T Mat {3, 1} with respect to C s , noted Φs (T Mat {3, 3}) is computed in Equation 14 5 and illustrated in Figure 9. This figure also provides the fuzzy sets Φs (T Mat {3, 1}) and Φs (T Mat {3, 2}) which depends on both data item sizes. 4) Criteria importance: Experts specify the criteria importance via pairwise comparisons as in C r . The relative importance of each criterion, noted λρ (C x ), is computed based on the same rules (Table I) and equations (6 to 8).

➫ 2 experts perform pairwise comparisons in Figure 7 and the relative importance of criteria are synthesized by Λ ρ . We can see that C e is the most important criterion (i.e. experts take the liberty of choosing information to store on the product at the expense of the others criteria). 5 T Mat {3, 3} is a character string "15mm" and 1 ASCII character is coded on 1 byte. That is why the power is equal to "4" in Equation 14.

C. Stage 3 : Fuzzy judgment matrix Ã

After getting the fuzzy set of each alternative l (l ∈ {1, . . . , n}) with respect to each criterion x (x ∈ {C e , C c , C s }), noted φx (l), the fuzzy judgment matrix à is created as depicted in Figure 11. At this stage, only scores of alternatives with respect to criteria are taken into account, without considering the relative importance of criteria. As a result, we synthesize the fuzzy matrix à with the criteria importance in a fuzzy performance matrix H. The performance score hx (l) consists in multiplying the fuzzy set φx (l) by the criterion importance itself λ ρ (C x ). This multiplication is carried out based on their α-cut levels as detailed in Equation 15. Finally, the matrix H is obtained as depicted in Figure 12, in which we highlight the multiplication he (T Mat {3, 1}).

hx(l) = φ α x (l) • λ α ρ (Cx) ∀ α = {0, 0.1, . . . , 1} (15)

E. Stage 5 : Alternative ranking

This stage ranks alternatives according to their fuzzy sets but, until now, 1 alternative has 3 fuzzy sets (cf. Figure 12). As a result, it is necessary to proceed to the multicriteria aggregation and then, to compare alternatives.

In our approach, the multi-criteria aggregation consists, for each alternative, in summing the 3 fuzzy sets based on their α-cuts as detailed in Equation 16. Figure 13 illustrates the summing operation r(T Mat {3, 1}).

r α (l) = x∈{e,c,m,s} h α x (l) ∀ α = {0, 0.1, . . . , 1} (16)
r 0 r 0 r 0.1 r 0.1 r 0.2 r 0.2 r 0.3 r 0.3 r 0.4 r 0.4
r 0.5 r 0.5 r 0.6 r 0.6 r 0.7 r 0.7 r 0.8 r 0.8 r 0.9 r 0.9 ∆Yr 0.0 ∆Yr 0.1 ∆Yr 0.2 ∆Yr 0.3 ∆Yr 0.4 ∆Yr 0.5 ∆Yr 0.6 ∆Yr 0.7 ∆Yr 0.8 ∆Yr 0.9

x ⋆ = 0.966 In our study, the approach based on the center of gravity (CoG), originally proposed by Yager [START_REF] Yager | On a general class of fuzzy connectives[END_REF], is used for ranking alternatives according to their α-cut levels. This technique computes, for each fuzzy number, an index noted x ⋆ which is used to compare and to rank alternatives. This index is a crisp value located on the xaxis and is computed via Equation 17. However, since our approach uses α-cut representations, the integral functions are therefore discretized and are an approximation of the result. We propose to compute the superior approximation result as in Equation 17, where m is the number of α-cuts (i.e., α = { 0 α0 , . . . , 0.8 αm-2 , 0.9 αm-1 }), r α and r α respectively indicate the minimal and the maximal values of the α-cut and ∆ Y r α corresponds to the level difference between the α-cut and α + 1-cut levels. All these notations are detailed in Figure 13 with regard to r(T Mat {3, 3}). It can be observed that the CoG of T Mat {3, 1} (x ⋆ = 1.089) is higher than the CoG of T Mat {3, 2} (x ⋆ = 0.672) and T Mat {3, 3} (x ⋆ = 0.966). As a result, T Mat {3, 1} gets a better ranking (cf. podium in Figure 13) and is stored in priority on the product.

Finally, the list of data items is ranked and is stored on the communicating material according to its available memory space (thanks to the process step 2). In our application, around 500 data items are ranked (representing ≈ 30 Mbytes). Our textile can only embed 4 Mbytes and only the first 70 data items can therefore be stored on it.

V. CONCLUSION For years, the use of intelligent products have been used to create an information continuum all along the product life cycle. However, it is not that easy to identify what information should be stored on the product. As a result, we propose a data dissemination process to select contextsensitive information from databases to the products. The information relevance depends on many factors leading to a multi-criteria decision making (MCDM) problem. To handle the MCDM problem, we propose an approach based on fuzzy AHP which is particularly interesting compared to conventional approaches (e.g., FEAHP) because, the membership function forms are preserved through the use of α-cut representations and, the fuzzy set semantic is analyzed in view of our application (two consensus are proposed to aggregate multiple points of view).

Figure 1 .

 1 Figure 1. View of a Logical Data Model (LDM) and a relational table

  (a), noted [a, b, c, d]. Intervals [b, c] and [a, d] are respectively called the "kernel" and "support" of µ Ã(x).

Figure 2 .

 2 Figure 2. Trapezoidal fuzzy sets: [a, b, c, d]

Figure 3 (

 3 a) illustrates the construction of the membership function on the basis of three distinct points of view noted D 1 , D 2 and D 3 , • consensus (b) : as in consensus (a), no information should be neglected when aggregating points of view. The x-axis is a common semantic of points of view. Information complexity is also represented by an interval but, in opposition to consensus (a), the support is not extended because the addition of a new evaluation consists in adding a new point of view (e.g. add the Philosophy).

Figure 3 (Figure 3 .

 33 Figure 3. Aggregation of multiple points of view via 2 consensus

Figure 4 .

 4 Figure 4. Fuzzy AHP consisting of 5 stages

Figure 5 .

 5 Figure 5. General architecture of the hierarchy

  attribute) 0.5 enumerated (interesting attribute) p = {1, . . . , Q} 0 not enumerated (useless attribute)(2)

Figure 6 .

 6 Figure 6. Fuzzy opinion aggregation related to Enumeration

Figure 7 .

 7 Figure 7. Pairwise comparisons between entity groups carried out by a group of experts, aggregation and computation of their relative importance

Figure 8 .

 8 Figure 8. Adjustment of k p performed by a decision maker p

Figure 10 .Figure 9 .

 109 Figure 10. Pairwise comparisons between criteria carried out by a group of experts, aggregation and computation of their relative importance

  k = [min(1.08) max(1.08)] = [1.08 1.08]

Figure 11 .

 11 Figure 11. Fuzzy judgment matrix à D. Stage 4 : Fuzzy performance matrix H

Figure 12 .

 12 Figure 12. Fuzzy performance matrix H

2 3

 2 Tex tile wh i...

Figure 13 .

 13 Figure 13. Alternative ranking according to the CoG

  x ⋆ = µ Ã (x) × x dx µ Ã (x) dx ≈ α={α 0 ...α m-1 } r α + r α 2 • ∆ Y r α α={α 0 ...α m-1 } ∆ Y r α(17)

  Three criteria are defined at level 2: Enumeration, Contextual and Data Size which are respectively abbreviated C e , C c , C s and are detailed in the next stage.

	Level 1		Data item ranking (relevance)
	Level 2	Enumeration	Contextual	Data size
	Level 3	T Mat{3,1}	T Mat{3,2}

Table I RULE

 I DEFINITION OF FUZZY NUMBERS FOR PAIRWISE COMPARISON

	Condition	Membership function
	Fuzzy sets sij with

We think it is sensible to preserve the form of fuzzy sets when ranking alternatives.

The more the surface of the membership function form, the more the fuzzy set is important in our case.