Marc Baboulin
email: marc.baboulin@inria.fr

Serge Gratton
email: serge.gratton@enseeiht.fr

Rémi Lacroix
email: remi.lacroix@lri.fr

Alan J Laub
email: laub@ats.ucla.edu

Efficient computation of condition estimates for linear least squares problems

Keywords: linear least squares, condition number, statistical condition estimation, variancecovariance, GPU computing, MAGMA library moindres carrés linéaires, conditionnement, estimateur statistique de conditionnement, variance-covariance, calculs sur GPU, bibliothèque MAGMA

Linear least squares (LLS) is a classical linear algebra problem in scientific computing, arising for instance in many parameter estimation problems. In addition to computing efficiently LLS solutions, an important issue is to assess the numerical quality of the computed solution. The notion of conditioning provides a theoretical framework that can be used to measure the numerical sensitivity of a problem solution to perturbations in its data. We recall some results for least squares conditioning and we derive a statistical estimate for the conditioning of an LLS solution. We present numerical experiments to compare exact values and statistical estimates. We also propose performance results using new routines on top of the multicore-GPU library MAGMA. This set of routines is based on an efficient computation of the variance-covariance matrix for which, to our knowledge, there is no implementation in current public domain libraries LAPACK and ScaLAPACK.

Introduction

We consider the overdetermined linear least squares (LLS) problem

min x∈R n Ax -b 2 , (1)
with A ∈ R m×n , m ≥ n and b ∈ R m . Assuming that A is full column rank, Equation (1) has a unique solution x = A + b where A + is the Moore-Penrose pseudoinverse of the matrix A, expressed as A + = (A T A) -1 A T . We can find for instance in [START_REF] Björck | Numerical Methods for Least Squares Problems[END_REF][START_REF] Golub | Matrix Computations[END_REF][START_REF] Higham | Accuracy and Stability of Numerical Algorithms[END_REF] a comprehensive survey of the methods that can be used for solving efficiently and accurately LLS problems.

The condition number is a measure of the sensitivity of a mapping to perturbations. It was initially defined in [START_REF] Rice | A theory of condition[END_REF] as the maximum amplification factor between a small perturbation in the data and the resulting change in the problem solution. The perturbations are measured using metrics, for example norms. Namely, if the solution x of a given problem can be expressed as a function g(y) of a data y, then if g is differentiable (which is the case for many linear algebra problems), the condition number of g at y can be defined as (see e.g. [START_REF] Geurts | A contribution to the theory of condition[END_REF])

κ(y) = max z =0 g (y).z z . (2)
From this definition, κ(y) is a quantity that, for a given perturbation size on the data y, allows us to predict to first order the perturbation size on the solution x. Associated with a backward error [START_REF] Wilkinson | Rounding errors in algebraic processes[END_REF], condition numbers are useful to assess the numerical quality of a computed solution. Indeed numerical algorithms are always subject to errors although their sensitivity to errors may vary. These errors can have various origins like for instance data uncertainty due to instrumental measurements or rounding and truncation errors inherent to finite precision arithmetic. LLS can be very sensitive to perturbations in data in particular when the right-hand side is too far from the column space (see [20, p. 98]). It is then crucial to be able to assess the quality of the solution in practical applications. It was shown in [START_REF] Golub | Note on the iterative refinement of least squares solution[END_REF] that the 2-norm condition number cond(A) of the matrix A plays a significant role in LLS sensitivity analysis. It was later proved in [START_REF] Wedin | Perturbation theory for pseudo-inverses[END_REF] that the sensitivity of LLS problems is proportional to cond(A) when the residual vector is small and to cond(A) 2 otherwise. Then [START_REF] Geurts | A contribution to the theory of condition[END_REF] provided a closed formula for the condition number of LLS problems, using the Frobenius norm to measure the perturbations of A. Since then many results on normwise LLS condition numbers have been published (see e.g. [START_REF] Arioli | A partial condition number for linear least-squares problems[END_REF][START_REF] Björck | Numerical Methods for Least Squares Problems[END_REF][START_REF] Eldén | Perturbation theory for the least squares problem with linear equality constraints[END_REF][START_REF] Gratton | On the condition number of linear least squares problems in a weighted Frobenius norm[END_REF][START_REF] Grcar | Adjoint formulas for condition numbers applied to linear and indefinite least squares[END_REF]).

It was observed in [START_REF] Higham | A survey of componentwise perturbation theory in numerical linear algebra[END_REF] that normwise condition numbers can lead to a loss of information since they consolidate all sensitivity information into a single number. Indeed in some cases this sensitivity can vary significantly among the different solution components (some examples for LLS are presented in [START_REF] Arioli | A partial condition number for linear least-squares problems[END_REF][START_REF] Kenney | Statistical condition estimation for linear least squares[END_REF]). To overcome this issue, it was proposed the notion of "componentwise" condition numbers or condition numbers for the solution components [START_REF] Chandrasekaran | On the sensitivity of solution components in linear systems of equations[END_REF]. Note that this approach must be distinguished from the componentwise metric also applied to LLS for instance in [START_REF] Baboulin | Using dual techniques to derive componentwise and mixed condition numbers for a linear functional of a linear least squares solution[END_REF][START_REF] Cucker | On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems[END_REF]. This approach was generalized by the notion of partial or subspace condition numbers where we study the conditioning of L T x with L ∈ R n×k , k ≤ n, proposed for instance in [START_REF] Arioli | A partial condition number for linear least-squares problems[END_REF][START_REF] Baboulin | A contribution to the conditioning of the total least squares problem[END_REF] for least squares and total least squares, or [START_REF] Cao | A subspace error estimate for linear systems[END_REF] for linear systems. When L is a canonical vector e i , it is equivalent to the condition number of the ith component, while when L is the identity matrix, it is the same as the classical condition number mentioned above. The motivation for computing the conditioning of L T x can be found for instance in [START_REF] Arioli | A partial condition number for linear least-squares problems[END_REF][START_REF] Baboulin | Computing the conditioning of the components of a linear least squares solution[END_REF] for normwise LLS condition numbers.

Even though condition numbers provide interesting information about the quality of the computed solution, they are expected to be calculated in an acceptable time compared to the cost for the solution itself. Computing the exact (subspace or not) condition number requires O(n 3 ) flops when the LLS solution x has been aready computed (e.g., using a QR factorization) and can be reused to compute the conditioning [START_REF] Arioli | A partial condition number for linear least-squares problems[END_REF][START_REF] Baboulin | Computing the conditioning of the components of a linear least squares solution[END_REF]. This cost is affordable when compared to the cost for solving the problem (O(2mn 2 ) flops when m n). However statistical estimates can reduce this cost to O(n 2 ) [START_REF] Gudmundsson | Small-sample statistical estimates for matrix norms[END_REF][START_REF] Kenney | Small-sample statistical condition estimates for general matrix functions[END_REF]. The theoretical quality of the statistical estimates can be formally measured by the probability to give an estimate in a certain range around the exact value.

This paper is organized as follows. In Section 2 we first summarize some existing results for the condition numbers of the LLS solution or its components. For each of these quantities, we propose practical algorithms and evaluate the computational cost. More specifically in Section 2.1 we derive a new expression for the statistical estimate of the conditioning of x. Then in Section 3 we present numerical experiments to compare the LLS conditioning with their corresponding statistical estimates. We also propose performance results for the computation of these quantities using new routines on top of the MAGMA [START_REF] Tomov | Towards dense linear algebra for hybrid GPU accelerated manycore systems[END_REF] parallel library. For the exact values, these routines are based on the computation of the variance-covariance for which, to our knowledge, there is no routine in the public domain libraries LAPACK [START_REF] Anderson | LAPACK Users' Guide[END_REF] and ScaLAPACK [START_REF] Blackford | ScaLAPACK Users' Guide[END_REF], contrary to the NAG [START_REF]The Numerical Algorithms Group[END_REF] library. Our implementation takes advantage of the current hybrid multicore-GPU architectures and aims at being integrated into MAGMA.

Notations A ∈ R m×n r means that A is a m-by-n matrix of rank r. The notation • 2 applied to a matrix (resp. a vector) refers to the spectral norm (resp. the Euclidean norm ) and • F denotes the Frobenius norm of a matrix. The matrix I is the identity matrix and e i is the ith canonical vector. The uniform continuous distribution between a and b is abbreviated U(a, b) and the normal distribution of mean µ and variance σ 2 is abbreviated N (µ, σ 2 ). cond(A) denotes the 2-norm condition number of a matrix A, defined as cond(A) = A 2 A + 2 . The notation | • | applied to a matrix or a vector holds componentwise.

Condition estimation for linear least squares

In Section 2.1 we are concerned in calculating the condition number of the LLS solution x and in Section 2.2 we compute or estimate the conditioning of the components of x. We suppose that the LLS problem has already been solved using a QR factorization (the normal equations method is also possible but the condition number is then proportional to cond(A) 2 [6, p. 49]). Then the solution x, the residual r, and the factor R ∈ R n×n of the QR factorization of A are readily available (we recall that the Cholesky factor of the normal equations is, in exact arithmetic, equal to R up to some signs). We also make the assumption that both A and b can be perturbed, these perturbations being measured using the weighted product norm (∆A, ∆b

) F = ∆A 2 F + ∆b 2 2 .
In addition to providing us with simplified formulas, this product norm has the advantage, mentioned in [START_REF] Gratton | On the condition number of linear least squares problems in a weighted Frobenius norm[END_REF], to be appropriate for estimating the forward error obtained when the LLS problem is solved via normal equations.

Conditioning of the least squares solution

Exact formula We can obtain from [START_REF] Baboulin | Computing the conditioning of the components of a linear least squares solution[END_REF] a closed formula for the condition number of the LLS solution as

κ LS = R -1 2 R -1 2 2 r 2 2 + x 2 2 + 1 1 2 . ( 3
)
This equation requires mainly to compute the minimum singular value of the matrix A (or R), which can be done using iterative procedures like the inverse power iteration on R, or more expensively with the full SVD of R (O(n 3 ) flops). Note that R -T 2 can be approximated by other matrix norms (see [19, p. 293]).

Inria

Statistical estimate Similarly to [START_REF] Cao | A subspace error estimate for linear systems[END_REF] for linear systems, we can estimate the condition number of the LLS solution using the method called small-sample theory [START_REF] Kenney | Small-sample statistical condition estimates for general matrix functions[END_REF] that provides statistical condition estimates for matrix functions. By Taylor's theorem, the forward error ∆x on the solution x(A, b) can be expressed as

∆x = x (A, b).(∆A, ∆b) + O( (∆A, ∆b) 2 F ). (4)
The Then, as mentioned in Equation ( 2), the condition number of x corresponds to the operator norm of x (A, b), which is a bound to first order on the sensitivity of x at (A, b). We now use [START_REF] Kenney | Small-sample statistical condition estimates for general matrix functions[END_REF] to estimate ∆x 2 by

ξ(q) = ω q ω n |z T 1 ∆x| 2 + • • • + |z T q ∆x| 2 , (5)
where z 1 , • • • , z q are random orthogonal vectors selected uniformly and randomly from the unit sphere in n dimensions, and ω q is the Wallis factor defined by

ω 1 = 1, ω q = 1 • 3 • 5 • • • (q -2) 2 • 4 • 6 • • • (q -1) for q odd, ω q = 2 π 2 • 4 • 6 • • • (q -2) 1 • 3 • 5 • • • (q -1)
for q even. ω q can be approximated by

2 π(q-1 2 )
. It comes from [START_REF] Kenney | Small-sample statistical condition estimates for general matrix functions[END_REF] that if for instance we have q = 3, then the probability that ξ(q) lies within a factor α of ∆x 2 is

P r( ∆x 2 α ≤ ξ(q) ≤ α ∆x 2 ) ≈ 1 - 32 3π 2 α 3 . (6)
For α = 10, we obtain a probability of 99.9%.

For each i ∈ {1, • • • , q}, using Equation ( 2) we have the first-order bound

|z T i ∆x| ≤ κ i (∆A, ∆b) F , (7)
where κ i denotes the condition number of the function z T i x(A, b). Then using ( 5) and ( 7) we get

ξ(q) ≤ ω q ω n q i=1 κ 2 i 1 2
(∆A, ∆b) F .

Since on the other hand we have

∆x 2 ≤ κ LS (∆A, ∆b) F ,
then we will consider that

κLS = ω q ω n q i=1 κ i 2 1 2 (8)
is an estimate for κ LS .

RR n°8065

We point out that κLS is a scalar quantity that must be distinguished from the estimate given in [START_REF] Kenney | Statistical condition estimation for linear least squares[END_REF] which is a vector. Indeed the small-sample theory is used here to derive an estimate of the condition number of x whereas it is used in [START_REF] Kenney | Statistical condition estimation for linear least squares[END_REF] to derive estimates of the condition numbers of the components of x (see Section 2.2). Now we can derive Algorithm 2.1 that computes κLS as expressed in Equation ( 8) and using the condition numbers of z T i x. The vectors z 1 , • • • , z q are obtained for instance via a QR factorization of a random matrix Z ∈ R n×q . The condition number of z T i x can be computed using the expression given in [START_REF] Baboulin | Computing the conditioning of the components of a linear least squares solution[END_REF]) as

κ i = R -1 R -T z i 2 2 r 2 2 + R -T z i 2 2 ( x 2 2 + 1) 1 2 . ( 9
)
The accuracy of the estimate can be tweaked by modifying the number q of considered random samples. The computation of κLS requires computing the QR factorization of an n × q matrix for O(nq 2 ) flops. It also involves solving q times two n × n triangular linear systems, each triangular system being solved in O(n 2 ) flops. The resulting computational cost is O(2qn 2 ) flops (if n q). Algorithm 2.1 Statistical condition estimation for linear least squares solution (SCE_LLS)

Require: q ≥ 1, the number of samples Generate q vectors z 1 , z 2 , ..., z q ∈ R n with entries in U(0, 1) Orthonormalize the vectors z i using a QR factorization

for j = 1 to q do Compute κ j = R -1 R -T z j 2 2 r 2 2 + R -T z i 2 2 ( x 2 2 + 1) 1 2 end for Compute κLS = ωq ωn q j=1 κ 2 j with ω q = 2 π(q-1 2 )

Componentwise condition estimates

In this section, we focus on calculating the condition number for each component of the LLS solution x. The first one is based on the results from [START_REF] Baboulin | Computing the conditioning of the components of a linear least squares solution[END_REF] and enables us to compute the exact value of the condition numbers for the ith component of x. The other is a statistical estimate from [START_REF] Kenney | Statistical condition estimation for linear least squares[END_REF].

Exact formula By considering in Equation ( 9) the special case where z i = e i , we can express in Equation ( 10) the condition number of the component x i = e T i x and then calculate a vector κ CW ∈ R n with components κ i being the exact condition number for the ith component expressed by

κ i = R -1 R -T e i 2 2 r 2 2 + R -T e i 2 2 ( x 2 2 + 1) 1 2 . ( 10
)
The computation of one κ i requires two triangular solves (R T y = e i and Rz = y) corresponding to 2n 2 flops. When we want to compute all κ i , it is more efficient to solve RY = I and then compute Y Y T , which requires about 2n 3 /3 flops.

Statistical condition estimate

We can find in [START_REF] Kenney | Statistical condition estimation for linear least squares[END_REF] three different algorithms to compute statistical componentwise condition estimation for LLS problems. Algorithm 2.2 corresponds to the algorithm that uses unstructured perturbations and it can be compared with the exact value given in Equation [START_REF] Cucker | On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems[END_REF]. Algorithm 2.2 computes a vector κCW = (κ 1 , • • • , κn ) T containing the statistical estimate for the κ i 's. Depending on the needed accuracy for the statistical estimation, the number of random perturbations q ≥ 1 applied to the input data in Algorithm 2.2 can be Inria adjusted. This algorithm involves two n × n triangular solves with q right-hand sides, which requires about qn 2 flops. Algorithm 2.2 Componentwise statistical condition estimate for linear least squares (SCE_LLS_CW)

Require: q ≥ 1, the number of perturbations of input data for j = 1 to q do Generate S j ∈ R n×n , g j ∈ R n and h j ∈ R n with entries in N (0, 1)

Compute u j = R -1 (g j -S j x + Ax -b 2 R -T h j ) end for Let p = m(n + 1) and compute vector κCW = q i=1 |uj | qωp √ p with ω q = 2 π(q-1 2 )
3 Numerical experiments

In the following experiments, random LLS problems are generated using the method given in [START_REF] Paige | LSQR: An algorithm for sparse linear equations and sparse least squares[END_REF] for generating LLS test problems with known solution x and residual norm. Random problems are generated as [A, x, r, b] = P (m, n, ρ, l) such that A ∈ R m×n , r 2 = ρ and cond(A) = n l . The matrix A is generated using

A = Y D 0 Z T , Y = I -2yy T , Z = I -2zz T
where y ∈ R m and z ∈ R n are random unit vectors and D = n -l diag(n l , (n -1) l , (n -2) l , • • • , 1).

We have x = (1, 2 2 , ..., n 2 ) T , the residual vector is given by r = Y 0 v where v ∈ R m-n is a random vector of norm ρ and the right-hand side is given by b = Y DZx v . In Section 3.1, we will consider LLS problems of size m × n with m = 9984 and n = 2496.

Accuracy of statistical estimates

Conditioning of LLS solution

In this section we compare the statistical estimate κ LS obtained via Algorithm 2.1 with the exact condition number κ LS computed using Equation (3). In our experiments, the statistical estimate is computed using two samples (q = 2). For seven different values for cond(A) = n l (l ranging from 0 to 3, n = 2496) and several values of r 2 , we report in Table 1 the ratio κLS /κ LS , which is the average of the ratios obtained for 100 random problems. The results in Table 1 show the relevance of the statistical estimate presented in Section 2.1. For n ≥ 1 2 the averaged estimated values never differ from the exact value by more than one order of magnitude. We observe that when l tends to 0 (i.e., cond(A) gets close to 1) the estimate becomes less accurate. This can be explained by the fact that the statistical estimate κ LS is based on evaluating the Frobenius norm of the Jacobian matrix [START_REF] Gudmundsson | Small-sample statistical estimates for matrix norms[END_REF]. Actually some additional experiments showed that κ LS /κ LS evolves exactly like R -1 2 F / R -1 2 2 . In this particular LLS

R -1 2 F / R -1 2 2 = 1 + (n/(n -1)) 2l + (n/(n -2)) 2l + • • • + n 2l /n 2l = n k=1 1 k 2l .
Then when l tends towards 0, R -1

F / R -1 2 ∼
√ n, whereas this ratio gets closer to 1 when l increases. This is consistent with the well-known inequality

1 ≤ R -1 F / R -1 2 ≤ √ n. Note
that the accuracy of the statistical estimate does not vary with the residual norm.

Componentwise condition estimation

Figure 1 depicts the conditioning for all LLS solution components, computed as κ i /|x i | where the κ i 's are obtained using Equation [START_REF] Cucker | On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems[END_REF]. Figures 1(a) and 1(b) correspond to random LLS problems with respectively cond(A) = 2.5 • 10 3 and cond(A) = 2.5 • 10 9 . These figures show the interest of the componentwise approach since the sensitivity to perturbations of each solution component varies significantly (from 10 2 to 10 8 for cond(A) = 2.5 • 10 3 , and from 10 7 to 10 16 for cond(A) = 2.5 • 10 9 ). The normalized condition number of the solution computed using Equation ( 3) is κ LS / x 2 = 2.5 • 10 3 for cond(A) = 2.5 • 10 3 and κ LS / x 2 = 4.5 • 10 10 for cond(A) = 2.5•10 9 , which in both cases greatly overestimates or underestimates the conditioning of some components. Note that the LLS sensitivity is here well measured by cond(A) since r 2 is small compared to A 2 and x 2 , as expected from [START_REF] Wedin | Perturbation theory for pseudo-inverses[END_REF] (otherwise it would be measured by cond(A) 2 ).

In Figure 2 we represent for each solution component, the ratio between the statistical condition estimate computed via Algorithm 2.2, considering two samples (q = 2), and the exact value computed using Equation [START_REF] Cucker | On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems[END_REF]. The ratio is computed as an average on 100 random problems. We observe that this ratio is lower than 1.2 for the case cond(A) = 2.5 • 10 3 (Figure 2 (a)) and close to 1 for the case cond(A) = 2.5 • 10 9 (Figure 2 (b)), which also confirms that, similarly to κ LS in Section 3.1.1, the statistical condition estimate is more accurate for larger values of cond(A). 3.2 Computing least squares condition numbers on multicore-GPU architectures

Variance-covariance matrix

In many physical applications, LLS problems are considered using a statistical model often referred to as linear statistical model where we have to solve

b = Ax + , A ∈ R m×n n , b ∈ R m ,
where is a vector of random errors having expected value E( ) = 0 and variance-covariance V ( ) = σ 2 b I. In statistical language, the matrix A is called the regression matrix and the unknown vector x is called the vector of regression coefficients. Following the Gauss-Markov theorem [START_REF] Zelen | Linear estimation and related topics[END_REF], the least squares estimates x is the linear unbiased estimator of x satisfying

x = arg min x∈R n Ax -b 2 ,
with minimum variance-covariance equal to

C = σ 2 b (A T A) -1 . (11)
The diagonal elements c ii of C give the variance of each component xi . The off-diagonal elements c ij , i = j give the covariance between xi and xj . Then instead of computing condition numbers (which are notions more commonly handled by numerical linear algebra practitioners) physicists often compute the variance-covariance matrix whose entries are intimately correlated with condition numbers κ i and κ LS mentioned previously. When the variance-covariance matrix has been computed, the condition numbers described in Section 2 can be easily obtained. Indeed, we can use the fact that

R -1 2 2 = C 2 σ 2 b , R -T e i 2 2 = cii σ 2 b , and R -1 R -T e i 2 = Ci 2 σ 2 b
where C i and c ii are respectively the ith column and the ith diagonal element of the matrix C. Then by replacing respectively in Equations ( 3) and [START_REF] Cucker | On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems[END_REF] we

κ LS = C 1/2 2 σ b ((m -n) C 2 + x 2 2 + 1) 1/2 , (12)
and

κ i = 1 σ b ((m -n) C i 2 2 + c ii ( x 2 2 + 1)) 1/2 . (13)
Note that, when m > n, 1 m-n r

Implementation details

To our knowledge, there is no existing routine in public domain libraries LAPACK [START_REF] Anderson | LAPACK Users' Guide[END_REF], ScaLA-PACK [START_REF] Blackford | ScaLAPACK Users' Guide[END_REF], PLASMA [START_REF]PLASMA Users' Guide, Parallel Linear Algebra Software for Multicore Architectures[END_REF], MAGMA [START_REF] Tomov | Towards dense linear algebra for hybrid GPU accelerated manycore systems[END_REF] to compute the variance-covariance matrix or LLS condition numbers. We propose an implementation for the MAGMA library (release 1.2.1) which is a dense linear algebra library for heterogeneous multicore-GPU architectures with interface similar to LAPACK. We developped a set of routines that compute the following quantities:

-Variance-covariance matrix C.

κ LS , condition number of x.

κ CW , vector of the κ i 's, condition numbers of the solution components.

-κLS , statistical estimate of κ LS .

-κCW , vector of the statistical estimates the κ i 's.

The variance-covariance computation requires inverting a triangular matrix and multiplying this triangular matrix by its transpose (similarly to the LAPACK routine DPOTRI [1, p. 26] that computes the inverse of a matrix from its Cholesky factorization). These operations use a block algorithm which, for the diagonal blocks, is performed recursively. The recursive part is performed by the CPU for sake of performance while the rest of the algorithm is executed on the GPU.

The computation of the exact condition number κ LS from the variance-covariance using Equation [START_REF] Geurts | A contribution to the theory of condition[END_REF] involves the computation of the spectral norm of C which is generally computed via an SVD. However, since A is a full rank matrix, C is symmetric positive definite and its singular values coincide with its eigenvalues. Then we use an eigenvalue decomposition of C which is faster than an SVD because it takes into account the symmetry of C. The tridiagonalization phase is performed on the GPU while the subsequent eigenvalue computation is performed on the CPU host.

The statistical estimates computed via Algorithms 2.1 and 2.2 require the generation and orthonormalization of random vectors followed by 2 triangular solves. The generation of the random vectors and the triangular solves are executed on the GPU. However the orthonormalization is applied to small matrices (due to the small number of samples) and thus is performed on the CPU because this procedure would not take advantage of the GPU.

Performance results

In this section we present performance results for computing the variance-covariance matrix and LLS condition numbers. The tests have been achieved on a multicore processor Intel Xeon E5645 (2 sockets × 6 cores) running at 2.4 GHz (the cache size per core is 12 MB and the size of the main memory is 48 GB). This system hosts two GPU NVIDIA Tesla C2075 running at 1.15 GHz with 6 GB memory each. MAGMA was linked with the libraries MKL 10.3.8 and CUDA 4.1, respectively, for multicore and GPU.

We show in Figure 3 the CPU time to compute LLS solution and condition numbers using 12 threads and 1 GPU. We observe that the computation of the variance-covariance matrix and of the components conditioning κ i 's are significantly faster than the cost for solving the problem with respectively a time factor larger than 3 and 2, this factor increasing with the problem size. The κ i 's are computed with the variance-covariance matrix using Equation [START_REF] Golub | Matrix Computations[END_REF]. The time overhead between the computation of the κ i 's and the variance-covariance computation comes from the computation of the norms of the columns (routine cublasDnrm2) which has a nonoptimal implementation. As expected, the routines that compute statistical condition estimates outperform the other routines. Note that we did not mention on this graph the performance for computing κ LS using Equation [START_REF] Geurts | A contribution to the theory of condition[END_REF]. Indeed this involves an eigenvalue decomposition of the variance-covariance matrix (MAGMA routine magma_dsyevd_gpu), which turns out to be much slower than the LLS solution (MAGMA routine magma_dgels3_gpu) in spite of a smaller number of arithmetic operations. Even though the theoretical number of flops for computing κ LS is much smaller than for computing x (O(n 3 ) vs O(mn 2 )), having an efficient implementation on the targetted architecture is essential to take advantage of the gain in flops.

We can illustrate this by comparing in Figure 4 the time for computing an LLS solution and its conditioning using LAPACK and MAGMA. We observe that MAGMA provides faster solution and condition number but, contrary to LAPACK, the computation of the condition number is slower than the time for the solution, in spite of a smaller flops count. This shows the need for improving the Gflop/s performance of eigensolvers or SVD solvers for GPUs but it also confirms the interest of considering statistical estimates on multicore-GPU architectures to get fast computations.

Conclusion

In this paper we studied the condition number of an LLS solution and of its components. We summarized the exact values and statistical estimates for these quantities. We also derived an expression for another statistical condition estimate for the LLS solution. In numerical experiments we compared the statistical estimates with the exact values. We proposed a new implementation for computing the variance-covariance matrix and the condition numbers using the library MAGMA. The performance results that we obtained on a current multicore-GPU system confirm the interest of using statistical condition estimates. Subsequently to this work, new routines will be proposed in the next releases of LAPACK and MAGMA to compute the variance-covariance matrix after a linear regression.

  notation x(A, b) means that x is a function of the data A and b and x (A, b) is the derivative of this function. x (A, b).(∆A, ∆b) denotes the image of (∆A, ∆b) by the linear function x (A, b).

9 (Figure 1 :

 91 Figure 1: Componentwise condition numbers of LLS (problem size 9984 × 2496)

Figure 2 :

 2 Figure 2: Comparison between componentwise exact and statistical condition numbers

2 2

 2 is an unbiased estimate of σ 2 b [6, p. 4].

Figure 3 :

 3 Figure 3: Performance for computing LLS condition numbers with MAGMA

Figure 4 :

 4 Figure 4: Time for LLS solution and condition number

Table 1 :

 1 Ratio between exact and statistical condition numbers (q = 2)

	cond(A)	n 0	n	1 2	n 1	n	3 2	n 2	n	5 2	n 3
	r 2 = 10 -10 57.68 3.32 1.46 1.19 1.10 1.03 1.07
	r 2 = 10 -5 57.68 3.33 1.45 1.18 1.07 1.09 1.05
	r 2 = 1	57.68 3.36 1.45 1.19 1.19 1.05 1.15
	r 2 = 10 5	57.68 3.33 1.24 1.04 1.05 1.05 1.02
	r 2 = 10 10	57.68 1.44 1.07 1.09 1.00 1.01 1.07
	problem we have

RR n°8065

Inria

rue Honoré d'Estienne d'Orves Bâtiment Alan Turing Campus de l'École Polytechnique 91120 Palaiseau Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399

Acknowledgements: We are grateful to Jack Dongarra and Stanimire Tomov (Innovative Computing Laboratory, University of Tennessee) for technical support in using the multicore+GPU library MAGMA. We also want to thank Julien Langou (University of Colorado, Denver) for his collaboration on the LAPACK kernels for condition number computations.


Saint Gilles Neubert
email: gilles_neubert@esc-saint-etienne.fr

Blandine Carine Dominguez
email: carine.dominguez@univ-st-etienne.fr

B Ageron

Inter-organizational alignment to enhance IT driven services innovation in a supply chain: the case of RFID

Keywords: Strategic Alignment Model, Inter-Organizational Alignments, RFID, Supply Chain Management Strategic Alignment Model, Inter-Organizational Alignments, RFID, Alignment Perspective, Supply Chain Management

(Enterprise) Inter-organizational alignment to enhance IT driven services innovation in a supply chain: the case of RFID

Introduction

Today, globalization and technological innovations call for improved organizational adaptability and more flexible and advanced systems relative to manufacturing, logistics, engineering, information and process technology [START_REF] Momme | Framework for outsourcing manufacturing: strategic and operational applications[END_REF].

While focusing on their core competencies, companies have developed interaction with a network of partners that are incorporated in the decision making process, leading to increased dependency between business organizations and greater supply chain complexity. Information Technology and Information Systems (IS/IT) supporting the organization and its upstream and downstream partners, have been recognized as critical factors in Supply Chain improvement [START_REF] Koh | Managing uncertainty in ERP-controlled manufacturing environments in SMEs[END_REF]Saad, 2006, Neubert et al., 2004). As pointed out by [START_REF] Gunasekaran | Information systems in supply chain integration and management[END_REF], today it is impossible to have an effective supply chain without IT/IS which is like a nervous system for Supply Chain Management [START_REF] Gunasekaran | Information systems in supply chain integration and management[END_REF]. The growing rate of goods exchange, especially in logistics-intensive industries that require reactivity and flexibility, has led to increasingly sophisticated needs in supply chain channels. Today, most improvements in companies' processes and interactions are supported by IT driven services innovation. Recently, Radio

Frequency Identification (RFID), a wireless automatic identification and data capture technique, has been emerging as a new wave of inter-organizational systems. This technique is expected to radically transform interdependent supply chain business processes and Supply Chain practices [START_REF] Srivastava | Radio frequency ID technology: the next revolution in SCM[END_REF][START_REF] Tajima | Strategic value of RFID in supply chain management[END_REF][START_REF] Fosso Wamba | A contingency model for creating value from RFID supply chain network projects in logistics and manufacturing[END_REF]. Besides the technological issues related to what is considered as a disruptive innovation (Lefebvre et al., 2006, Vail and[START_REF] Vail | Disruptive innovation offers far-reaching solutions[END_REF], the literature shows the importance of organizational transformation to gain full benefit from this new IT [START_REF] Asif | Integrating the supply chain with RFID: a technical and business analysis[END_REF][START_REF] Cannon | RFID in the contemporary supply chain: multiple perspectives on its benefits and risks[END_REF][START_REF] Delen | RFID for better supply chain management through enhanced information visibility[END_REF]. The fit between IT and business, i.e. the alignment between IS/IT and strategies, is considered as one key for firm competitiveness and performance [START_REF] Avison | Using and validating the strategic alignment model[END_REF]. Most of the research has considered strategic alignment as a standalone problem, i.e. between one company's strategy and its own information system, often based on the Strategic Alignment Model (SAM) developed by [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF]. So far, this question has not been empirically extended to the inter-organizational level. This gap leads us to ask how do companies, embedded in a supply chain, adapt their information systems and organizations to reach their strategic goals with their suppliers and customers.

This paper addresses the issue of inter-organizational alignment in the context of RFID implementation in a jewellery supply chain consisting of a middle-sized retailer (MSR) and its logistic services provider (LSP). Based on a longitudinal description of the project from 2006 until the present, we highlight the main actors and their expectations, the business processes and their transformations during the project, the information flows and the IT infrastructure that support RFID.

The paper is organized as follows. First, we review past research on SAM in Supply Chain (SC). Then, we describe the research methodology and the case study we carried out. Finally, we discuss our results and the implications of our work for further research on inter-organizational strategic alignment.

Strategic alignment in Supply Chains

For many years, researchers have drawn attention to the importance of alignment between business and IT. Several studies have been conducted on this issue according to the activity sector [START_REF] Baets | Some empirical evidence on IS Strategy alignment in banking[END_REF] conducted a survey in the banking industry, [START_REF] Bush | The alignment of information systems with organizational objectives and strategies in health care[END_REF], in the health care sector), to the country (Teo and Ang, 1999, studied IT/business alignment in Singaporean companies) or to the company size [START_REF] Hussain | IT Alignment in Small Firms[END_REF], focused their research on SMEs). In 1993, Henderson and Venkatraman, influenced by research from MIT, proposed one of the most widely cited of all alignment models: the Strategic Alignment Model (SAM) [START_REF] Chan | IT alignment: What have we learned[END_REF]. As this model is relevant both in academic and professional literature [START_REF] Avison | Using and validating the strategic alignment model[END_REF] we use it as a framework to support analysis of inter-organisational alignment in our RFID case study. In this section, the question of alignment practices in the context of supply chain will be discussed. [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF] built the SAM on two propositions: first, the economic performance of a company depends on the strategic fit between strategy and the organizational and technological infrastructures that are deployed; second, strategic alignment relies on a dynamic process of adjustment between strategy and functional integration (organization and IT infrastructure). The SAM aims at understanding the dynamic fit between four related key domains (see Figure 1): business strategy, IT strategy, organization and IT infrastructure. As the business environment is constantly changing and IT innovations are abundant, strategic alignment should not be seen as a state of being, but as a process of change and continuous adaptation over time to optimize performance.

Strategic Alignment Model Background

In the model, a distinction is also drawn between the external perspective (strategy) and the internal focus (infrastructure and processes).

Figure 1: Strategic Alignment Model [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF] defined interaction perspectives between IT and business activities. Each perspective is made up of three domains: the anchor, the pivot, and the impacted domain. The anchor is considered the domain that is the strongest area of the business. The change that the business is to undergo is directed based on this perspective. The pivot designates the domain of the enterprise that will serve as the support for change through re-alignment. The domain of impact is the area that will be directly affected through the changes made in the pivot domain through the re-alignment [START_REF] Henderson | Strategic Alignment: A model For Organizational Transformation Via Information Technology[END_REF][START_REF] Luftman | Transforming the Enterprise: The Alignment of Business and Information Technology Strategies[END_REF][START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF][START_REF] Papp | Assessing strategic alignment in real-time[END_REF].

There are a total of eight individual alignment perspectives (see Table 1)

resulting from the analysis of strategic fit and functional integration simultaneously.

Strategy Execution: this perspective considers the business strategy as the driver of both organization design choices and the logic of the IT infrastructure. The anchor domain is business strategy. The pivot domain is the business infrastructure that needs to be changed. The resulting domain of impact is the information technology infrastructure. Technology Potential: this perspective is also driven by business strategy but the pivot is information technology strategy. The information technology infrastructure is thus the domain of impact.

Competitive Potential: the anchor is information technology strategy and the pivot is business strategy, while organization infrastructure is the impacted domain. This perspective focuses on how emerging information technologies can influence and enable new business strategies.

Service Level: in this perspective the anchor is information technology strategy, the pivot information technology infrastructure, and the area of impact is organizational infrastructure. The focus of this perspective is the manner in which information technology can improve how products and services are delivered.

Organization IT Infrastructure: this perspective results in process improvements from information technology and the application of value to the business processes The anchor is organization infrastructure, the pivot is information technology infrastructure, and the impacted domain is information technology strategy.

IT Infrastructure Strategy: the focus of this perspective is the improvement of information technology strategy based on the implementation of emerging and existing information technology infrastructures. The anchor is information technology infrastructure, which drives the pivot, information technology strategy, and thus impacting business strategy IT Organization Infrastructure: the anchor of the seventh perspective is also IT infrastructure, with the pivot being organizational infrastructure and the impact area being business strategy. Although similar to IT infrastructure strategy, IT in this perspective is the driving force and architect by which the visions and processes are carried out Organization infrastructure strategy: this final individual perspective exploits the capabilities to enhance new products and services, influence strategy, and develop new relationships. Business infrastructure is the anchor, and this perspective enables enhancement to business strategy (the pivot) thus changing the IT strategy (affected area)., Table 1: Alignment perspectives (from [START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF] Papp (1995) also identified four fusion perspectives which combine two of the individual perspectives ( Even if the SAM appears to be the reference in strategic alignment, it has been broadly criticized. [START_REF] Ciborra | De Profundis ? Deconstructing the Concept of Strategic Alignment[END_REF] argues that alignment is not the right way to consider organizations that are mainly driven by improvisation and tinkering with the resources at hand. [START_REF] Smaczny | Is an alignment between business and information technology the appropriate paradigm to manage IT in today's organisations?[END_REF] asserted that alignment, as a geometrical and mechanical approach, is not the appropriate paradigm for managing IT in today's organisations. Finally, [START_REF] Palmer | The performance impacts of quick response and strategic alignment in specialty retailing[END_REF], using a basic alignment model and strategy descriptors, have shown that there is no association between fit and performance.

On the contrary, other authors have defended the SAM. [START_REF] Lederer | Co-ordination of information systems plans with business plans[END_REF] have suggested that alignment increases the likelihood of developing systems more critical to the organisation and of obtaining top management support for IS. Besides, the application and analysis of alignment will facilitate a more competitive and profitable organisation (Galliers, 1991, Luftman andPapp, 1994). Literature shows a debate on the "capacity" of the model to tackle the role of IT in business performance, and as written by [START_REF] Coltman | E-business strategy and firm performance: a latent class assessment of the drivers and impediments to success[END_REF], one single model can probably not explain the relationship between IT, environment, structure, feasibility, managerial beliefs and performance. Nevertheless, other authors have shown an alternative view of alignment by extending the focus beyond strategic planning and strategy fit to include issues such as knowledge sharing among groups of humans [START_REF] Nelson | The Contribution of Shared Knowledge to IS Group Performance[END_REF], the social dimension of information alignment [START_REF] Reich | Factors that Influence the Social Dimension of Alignment Between Business and Information Technology Objectives[END_REF], or the level of use of IT and not just the nature of IT adopted by companies both in supplier and buyer organizations [START_REF] Fimbel | Alignement stratégique. Synchroniser les systèmes d'information avec les trajectoires et manoeuvres des entreprises[END_REF].

In the same vein, [START_REF] Galliers | Reflections on Information Systems Strategizing[END_REF] and Sledgianowski and Lufman (2005) argue that the alignment issue has to be extended and integrated enterprise-wide in a supply chain perspective. In this perspective [START_REF] Luftman | Assessing Business-IT alignment[END_REF] proposes a model of five levels of strategic alignment maturity:

1. Initial/Ad Hoc Process: organizations lack communications between the IT and the business or have a poor communications infrastructure. levels), leverage IT assets on an enterprise-wide basis to extend the reach (the IT extrastructure) of the organization into the supply chains of customers and suppliers.

Committed

Partnerships with external partners become an important issue of strategic alignment.

Nevertheless, this issue highlights key questions that are still unsolved [START_REF] Avison | Using and validating the strategic alignment model[END_REF]. First, it points out the link between the capacity of companies to align themselves and their level of performance. Secondly it underlines how companies align with one another. Finally it stresses how alignment can be measured.

Seeking answers to these questions can lead to a better understanding of the factors and connection modes that companies can set up to operationalize their alignment.

Information Technology and performance in the Supply Chain

It is now assumed that Information Technology is a key issue in better coordinating enterprises for supply chain performance [START_REF] Arshinder | Supply chain coordination: Perspectives, empirical studies and research directions[END_REF] but despite a large number of publications on IT alignment (for a detailed literature review on SAM, see [START_REF] Chan | IT alignment: What have we learned[END_REF], few authors have examined the impact of alignment in the context of supply chain. This lack of attention to the interorganizational context can be explained by the fact that alignment is merely "an engineering formal structure process alignment". Thus, it totally ignores business partnerships and the increased need for collaboration in the context of distributed supply sources.

This collaboration between partners has been covered extensively in the strategic management literature [START_REF] Bowersox | The strategic benefits of logistics alliances[END_REF][START_REF] Hanman | Benchmarking your firm's performance with best practice[END_REF][START_REF] Laseter | Balanced Sourcing: Cooperation and Competition in Supplier relationships[END_REF][START_REF] Gilmour | Benchmarking supply chain operations[END_REF][START_REF] Bowersox | Integrated supply chain logistics makes a difference[END_REF] and especially in the domain of Supply Chain Management. One of the main features is that vertical-process integration from suppliers to customers can be performed through inter-firm strategic alliances. In order to achieve vertical process integration, many enterprises are changing their organisation to break down both intra and inter enterprise barriers and the literature indicates that integration can support business processes at two different levels [START_REF] Romano | Co-ordination and integration mechanisms to manage logistics processes across supply networks[END_REF], intra-company integration and inter-company integration.

McDonald (1991) developed a model in which he examined external impacts of alignment on customers, suppliers and markets. Similarly, [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF] have recognized that alignment must be both internal and external to an organization and suggest that organizations must align their business and IT strategies with industry and technology forces. Based on this recommendation, [START_REF] Galliers | Reflections on Information Systems Strategizing[END_REF] also mentions the need to pay attention to alignment with main customers and suppliers all along the supply chain. [START_REF] Rey | Beyond words: testing alignment among interorganizational performance measures[END_REF] have extended Henderson and Venkatraman's test of co-alignment in the context of interorganizational relationships. By capturing successful contractual relationships between supplier chain partners, they observe that effective cooperation relies on business partners' need to align their performance measures (Yeung et al., 2006, Kaplan and[START_REF] Kaplan | Alignment using the Balanced Scorecard to create corporate strategy[END_REF]. They conclude that alignment of inter-organizational relationship performance measures depends on the fit between the objectives of the contract and the partners.

With the intention of streamlining their supply chain processes and controlling costs, leading retailers around the world are relying more on the use of information technologies, and as noticed by Fosso [START_REF] Fosso Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF] the use of automatic identification and data collection is becoming one of the means to support intra and inter organizational business processes and information exchange with their partners.

In recent years, RFID, an emerging supply chain technology classified as a wireless automatic identification and data capture technology, has come to be viewed as an Inter-Organizational Information System (IOIS) [START_REF] Curtin | Making the most out of RFID technology: A Research Agenda for the Study of the Adoption, Usage and Impact of RFID[END_REF]. But introduction of cross-company IT systems has always been a difficult and time consuming task [START_REF] Goebel | Adoption of cross-company RFID: An empirical analysis of perceived influence factors[END_REF]. Among other things, asymmetric costs and benefits, different attitudes to risk and capabilities across supply chain participants can complicate the adoption and efficient use of inter-organizational information systems.

This observation is related to the issue of supply chain alignment while managing a new IT implementation project between organisations. The next paragraph will give more details on RFID as an IOIS. product, which opens new possibilities in managing goods throughout their life cycle (Cao et al., 2007). Emerging technologies such as RFID enable lifecycle actors to gather and analyse product lifecycle information, and make efficient decisions without spatial and temporal constraints [START_REF] Jun | A framework for RFID applications in product lifecycle management[END_REF]. RFID could bring value in various areas such as business process reengineering (Tzeng et al., 2008), improving service quality, saving costs [START_REF] Ferrer | When is RFID right for your service ?[END_REF], enhance visibility and accuracy in the supply chain [START_REF] Véronneau | RFID benefits, costs, and possibilities: the economical analysis of RFID deployment in a cruise corporation global service supply chain[END_REF], reduced shrinkage, reduced material handling, increased data accuracy, faster exception management, improved information sharing [START_REF] Tajima | Strategic value of RFID in supply chain management[END_REF], improved customer service and logistics resource-management [START_REF] Sarac | A literature review on the impact of RFID technologies on supply chain management[END_REF]. In their paper, Huang et al. (2008) proposed a detailed list of benefits typically reported in the literature. They identified recent trends in RFID-based application, such as smart object, part fabrication, product assembly, JIT, mass customization, maintenance or Product Lifecycle Management.

New business challenges with RFID

For these reasons, RFID is seen as a key driver for logistics innovation. It can add value along the entire supply-chain and related logistical operations and business relationships for more effective business process design. It provides a means of tracking supplier items from the supplier, through the distribution network, to the point of consumption. [START_REF] Angeles | RFID Technologies: Supply-chain applications and implementation issues[END_REF] explain that the role of RFID in the retail supply chain is to streamline inventory management by providing highly detailed views of product shipments and inventory including information like data on product location, product characteristics, and product inventory levels. The four main processes of supply chain that can be directly concerned with or affected by RFID are [START_REF] Fosso Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF]: put away and replenishment, order filling, shipping, and product and asset tracking. If technical issues are beginning to be solved, business issues such as business value, inter-organisational relationships, and business knowledge development emerge as particularly salient. The supply-chain possibilities presented by RFID can be considered as transformational (Niederman et al., 2007). Environment scanning is a key step [START_REF] Avila | Understanding and classifying information system alignment approaches[END_REF] in helping companies identify the processes where RFID can create more value. This constitutes an important external signal for the SAM model that can lead enterprises to redefine their strategy and restructure their organisations.

The major benefits of RFID implementation will come from solutions across the whole supply chain. From an inter-organisational point of view, taking into account all the partners is difficult because there are many disputes regarding sharing costs and benefits among the manufacturers, logistics providers, distributors and retailers [START_REF] Visich | Enhancing Product Recovery Value in Closed-loop Supply Chains with RFID[END_REF]. Despite these difficulties, IT strategy, business infrastructure and IT infrastructure need to be coordinated to better realize the benefits from IT investments. This is the essence of the strategic alignment concept [START_REF] Hua | Applying the strategic alignment model to business and ICT strategies of Singapore's small and medium-sized architecture, engineering and construction enterprises[END_REF].

The long-term impact of RFID may go well beyond simply improving supply chain efficiency. [START_REF] Krotov | RFID as a Disruptive Innovation[END_REF] reported some examples of RFID applications that rely on wireless, automatic identification, but are not merely concerned with identification. At the core of these applications is the idea that objects (advertisement displays or exhibition stands) can seamlessly interact with humans, creating new types of customer services.

The impact of RFID on business strategy should not be underestimated. It could lead to unexpected new service development and business innovation.

Implementing RFID can be seen by companies as a new way of doing old things. One of the objectives of this paper is to highlight in a case study how this project was not only an ICT issue. The impact on the business strategy and on the processes of the members of the supply chain has to be forecasted and analysed to understand the alignment process, inside the retailer organisation but also within the supply chain.

Case study description and methodology

The need for empirical analysis in the context of alignment research has been outlined

by [START_REF] Chan | IT alignment: What have we learned[END_REF], who observed that "research on strategic alignment is too mechanistic and fails to capture real life". This was perhaps the inspiration for the recent upsurge in empirical research on strategic alignment. The majority of these studies have focused on quantitative methodologies and very little focus on describing the complexity of RFID implementation projects in a supply chain as did those studies developed by [START_REF] Loebbecke | Piloting RFID along the supply chain: a case analysis[END_REF][START_REF] Loebbecke | Use of innovative content integration information technology at the point of sale[END_REF]. We conducted a single-case study to attain indepth insight and understanding into this contemporary phenomenon [START_REF] Yin | Case Study Research: Design and Method[END_REF]. Our case study method addresses strategic and operational aspects. This section will present the methodology adopted in conducting the case study.

Methodology overview

Our case study was performed with a major French retailer that we will call MSR (middle-sized retailer). The choice of this particular case is largely explained by the fact that:

The retailing sector is a pioneer in the implementation of RFID technology.

Previous research has been conducted in this sector in other countries (in Germany with Kaufhof and the USA with Wal-Mart), but no research has been done in France.

The chosen company is an early adopter in terms of RFID technology projects. The jewellery supply chain with RFID has never been studied before.

Finally, the project under study successfully passed the pilot phases and was implemented in more than 80 jewellery stores Data was collected at different supply chain levels including the corporate level, retailing centres, Logistic Service Providers, and from other partners.

Qualitative data was gathered by using interviewing techniques. We conducted 5 semi-structured interviews with major actors involved in the project.

Case study description

The company

Founded in 1898, the MSR is a leading food retailer, active in more than 10 000 stores in multiple retail formats (hypermarkets, supermarkets, urban markets, and discount markets). Its operating performance, which has grown steadily over the past several years, now exceeds the industry average with revenues of nearly 25 KE in 2007. This is the result of the Group's unique position in the market and its ability to anticipate changing lifestyles and consumer practices. The company offers a different approach to retailing, one tailored to meet each consumer's specific expectations. Moreover, the company sought to be responsible not only with its customers, but also with its social and economic stakeholders. In order to meet these values, the MSR is forward thinking and innovative in several domains (for example, the company's urban network uses a river transportation system). The RFID technology project is a new challenge that the company has to deal with. It was launched in 2006 with the objective of improving sales productivity in the stores and the reliability of the supply processes.

Actors involved in the RFID project

Not all the actors of the MSR jewellery supply chain are involved in the RFID project.

The participants include:

• Suppliers of jewellery: more than 60 worldwide jewellery suppliers deal with the MSR which then distributes the products. • Group MSR: there are 4 different headquarters-based departments involved in the retailer's project: 1) the IT Department (called IDT), 2) Purchasing, 3) Supply Chain, and 4) the Textile Business Unit that is responsible for both textile and jewellery products.

• The stores of the MSR: In 2009 the company operated 80 stores mainly in France. They are physically located inside the MSR hypermarkets in France.

They sell jewellery like watches, rings, earrings, and bangles directly to final customers, like traditional jewellery stores. The sales policies of these stores are completely independent from their hosting hypermarkets.

• The LSP warehouse: LSP is a logistic services provider that receives jewellery from suppliers, stores the products, prepares orders, prints RFID tags according to the type of product, and delivers the goods to the stores. Once a year, the MSR and the LSP jointly participate in the warehouse inventory. LSP is in charge of these services both for the MSR and another chain of a jewellery store. To avoid problems between the two retailing companies, LSP has physically divided the warehouse into two separate and distinct areas where the products are handled.

• The printing company: this company is equipped to produce paper labels from paper rolls and to integrate RFID smart chips in the paper rolls. The printing company then cuts up the rolls to make specific paper tags that are affiliated with the individual items of jewellery to describe them (weight, number of carats etc…).

• The smart chip producer: this company develops UHF smart chips for the MSR group to help them manage their supply chain. They also participated in the creation of specific RFID paper tag readers, as these materials did not initially exist in their offerings. The reader can be considered as the hardware aspect of the RFID project.

Physical processes and information flows.

The LSP warehouse receives products from suppliers each day, and controls them for the MSR Group. Once the products have been controlled, they are tagged by LSP and stored for future delivery. Once a week, the MSR's stores receive shipments from LSP. After delivery, the sales personnel of the stores control the products with the RFID Reader and store or display the products according to the product status (in stock or on order). Currently, there are three main processes in this supply chain that link the jewellery suppliers and the LSP warehouse, and then the LSP warehouse and the "Boutique Or" stores: 1) Daily reception of products from 60 worldwide jewellery suppliers delivering to LSP, 2) Weekly deliveries to each store, from LSP and 3) LSP's annual inventories.

The IT infrastructure.

The IT infrastructure of the MSR jewellery supply chain was specifically developed for the RFID project and is independent from GOLD, the retailer's main ERP system.

There are three main computer IS that support the RFID project:

• ERP GOLD: The MSR Group's main ERP that aims to centralize all key data.

At present, it is not integrated with SINEX

• SINEX: This is the IS that was developed for LSP. SINEX deals with both inventories and day to day receipt of jewellery from suppliers. LSP also physically houses the hardware like the tag printers and RFID readers. This material belongs to the MSR.

• In-house software in the MSR's stores: there are composed of different systems that are not integrated one with another. Each IS is dedicated to a specific task: collection, receipts, inventories, and editing tags when necessary (a paper tag has broken or when prices are changed through promotions).

Case study analysis

We used Fosso, [START_REF] Fosso Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF]'s framework that that we adapted and completed to illustrate our case study on RFID implementation. Fosso Wamba's framework is specifically developed for RFID projects and is composed of a sequence of three main phases that describe RFID project implementation:

• Opportunity seeking,

• Pilot project and validation and

• RFID project deployment.

Phase 1: Opportunity seeking

This initial phase deals with opportunities to implement an RFID project in a specific value chain. It is composed of six steps.

Phase 1: Opportunity seeking

Step 1: Primary motivation (Why?)

Step 2: Analysis of the product value chain (What and why?)

Step 3: Identification of the critical activities (Which?)

Step 4: Mapping of the network of firms supporting the PVC (Who and with whom?)

Step 5: Mapping of intra-organizational processes for the identified opportunities as they are (How specifically within the organization?)

Step 6: Mapping of inter-organizational processes for the identified opportunities as they are (How specifically between the organizations?)

Phase 2: Pilot project and validation

The objective of this second phase is to develop a pilot project to validate the feasibility of the RFID implementation. With respect to the inter-organizational alignment problem, the goal is to identify the business opportunities that could come Phase 2: Pilot project and validation

Step 7: Evaluation of RFID Network opportunities in the PVC with respect to the product (level of granularity), to the firms involved in the network, and to the specific activities in the PVC

Step 8: Evaluation of potential RFID Network applications including scenario building and process optimization ("As could be") (HOW within and between organizations?)

Step 9: Mapping of intra-and inter-organizational processes integrating RFID technology

Step 10: Validating business and technological processes integrating RFID technology with key respondents. Feasibility analysis and evaluation of the challenges including ERP and middleware integration and process automation

Phase 3: RFID project deployment

The last phase concerns the validation of the pilot project and deployment of the technology over the whole supply chain.

Phase 3: RFID project deployment

Step 11: Proof of concept (POC) with the pilot project: evaluation including ERP and middleware integration and process automation at all supply chain member levels and decision to go for the beta test replicating POC scenarios in real-life

Step 12: Pilot replicating and evaluation of anticipated VS realised benefits and impacts of RFID. Appropriation by the different organisations involved and their staff members

Inter-organizational alignment dynamics during the RFID project

The project was launched with the idea of inserting an innovative IT in a supply chain, namely introducing RFID tags in the MSR's SC.

SAM according to steps 1, 2, 3: a competitive potential perspective

Step 1: The MSR's primary motivation was to seek opportunities presented by RFID technology to improve processes. An external IT vendor proposed running a preliminary study on RFID implementation (Step 1) for the MSR Information Technology Department (ITD).

Step 2: A manager at the MSR, based in the ITD department, became interested in developing the idea and ultimately took on responsibility for the project. He pushed the idea to the MSR's corporate department that officially agreed to start the RFID project in January 2007. Step 3: The MSR decided to first experiment with RFID technology implementation on their jewellery activities which are under the textile business unit. Choice of this Product Value Chain (PVC) was motivated because the jewellery activity has its own value chain that is fully independent from the food supply chain (from suppliers to points of sale). In terms of information systems, this jewellery supply chain is also independent from the group's main ERP with no direct integration. Additionally, this independence is reinforced by the fact that the MSR works with an external logistic services provider (LSP) for jewellery products. They decided to use RFID for the logistics activities of the value chain.

Next table give a summary of the mapping between the first step of the project and the strategic alignment model.

Fosso Wamba et al. ( 2007)

SAM [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF].

Step 1 : Primary motivation Step2 : Product value chain

Step 3 : Critical activities

Scanning IT environment

Business Strategy decision

Analyses of business impact Table 3: Mapping the first steps of Alignment in the RFID project.

SAM according to steps 4, 5, 6

Step 4: The project involved different partners in the Supply Chain: the MSR's headquarters, its 80 points of sale (stores of the MSR), and a single logistic services provider (LSP). The PSR designated one specific store for a pilot project and contacted the corporate division of LSP to ask them to collaborate in the RFID project for the jewellery supply chain they were in charge of. Identification of the activities and mapping of intra and inter organisation infrastructure concerns was done using the SAM. SAM [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF].

Step 4: Mapping of the network

Step 5: Mapping of intra organizational processes

Step 5: Mapping of inter organizational processes Inter organizational impact of IT implementation Identification of the processes that will need to fit with the new IT within the organization Identification of the processes that will need to fit with the new IT between the organizations Table 4: Mapping of AS IS processes and activities in the actual supply chain

SAM according to steps 7, 8, 9, 10

Service innovation through disruptive technology calls for multidisciplinary skills and therefore cannot be achieved through a single R&D department. Competencies must be gathered from different actors to reach the goal. Moreover, a jewellery product flow is characterised by small products that are handled manually, low level of demand, high value unit, with items made with metal components (gold, silver etc.) that are not always readable using RFID tags.

Steps 7 and 8: The MSR's IT Strategy Department started to explore the environment to find potential partners that could participate in the project of adapting the needed IS and developing the RFID technology. An initial RFID application scenario was proposed. The objective was to define the possible ways of using RFID, within and between organisations. For the MSR and its partners, the following challenges and applications were identified:

• Producing new labels with RFID Tags The MSR negotiated with the label producer and requested they adapt the size/format of their labels according to the type of jewellery and to include the RFID tag. As the label producer was not equipped to insert the chips in the labels, The MSR asked the Tag producer to accomplish this task. This alignment concerns the strategic domain of 3 companies, the Software Service Provider (SSP), the Label producer and the Tag producer who all agreed to develop new products for the project.

The last step of this phase consisted of validating all the business processes, applications and technological developments that were required for the RFID Project.

From the RFID technology side, this step included development of specific RFID readers adapted to very small products, such as jewellery, that were labelled with RFID tags.

From the Information System side, software modifications integrating RFID applications into legacy systems were required, including wireless middleware to SAM [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF]

SAM according to steps 11 and 12

In January 2008 the MSR's pilot project was launched in a single point of sale testing different kinds of tags. The process under scrutiny was automated receipt of orders at this point of sale. For the Logistics Service Provider, the pilot process consisted of relabelling all products delivered to this specific point of sale after product picking was performed at the warehouse. The objectives of steps 11 and 12 were to validate the development and modifications performed during the pilot project and then propose the final B2B organisation.

For the MSR, this step included updating software and designing a new RFID reader better adapted to the project.

For the Logistics Service Provider, these steps included reengineering the business sourcing process by attaching the product tag upon receipt of the product, instead of at the moment of product delivery (as in the pilot project). It also included As indicated in Table 3, IT strategy acts as an enabler of the project. In this case study, the MSR's IT Manager explored how IT could enable new or enhanced business strategies with corresponding organizational implications [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF]. The challenge in this particular project was identifying the appropriate Value Chain upon which to initially test implementation of this technological innovation. This paragraph will highlight different perspectives of alignment [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF] according to the different phases [START_REF] Fosso Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF] of the project.

Accordingly to the phase 1 in Fosso Wamba et al ( 2007), the IT manager identified and interpreted the IT environment trends to assist the business managers in understanding the potential opportunities and threats from an IT perspective. We identified the role of the IT manager as being a catalyst and IT Strategy as the anchor Domain. Then Business strategy Domain made an evaluation of the potential application of implementing RFID. As they perceived RFID as a disruptive innovation

, not yet used in the company, they made an evaluation of possible opportunities and threats to minimize risk. They decided on which value chain to experiment the project, Jewellery, and to limit for instance the RFID use to logistics activities. Business Strategy is the pivot domain.

Once the value chain was chosen, those processes involved in its logistics activities were identified for modification. Organization infrastructure is therefore the impact domain. This is perspective called "Competitive Potential" in Henderson and Venkatraman's proposal (see column 1 on Table 7).

As the logistics activities were chosen for RFID implementation, MSR need the cooperation of other organization involved in the Jewellery Supply Chain:

Logistic services provider (LSP), an external supplier in the value chain, and Stores. In step 4 of the project, the MSR's IT manager, in charge of the project, asked the LSP's top management to be a partner in the RFID project. Once again, the LSP's Business Strategy made an evaluation of the impact of the project on its strategy and business unit and decided to accept, but to limit the RFID experiment to the MSR's

Chain and not to implement it on other supply chain. The impact on the LSP's

Organisation was then analysed.

This Second alignment sequence concerned both the MSR and the LSP. The anchor domain is the MSR's IT strategy, the pivot domain is the LSP business strategy, and the LSP organisation infrastructure is the impacted domain (see column 2 on Table 7). This is an inter-organizational competitive potential alignment perspective.

Next sequence, third one, appears during phase 2: pilot project and validation [START_REF] Fosso Wamba | Integrating RFID Technology and EPC Network into a B2B retail supply chain: a step toward intelligent business processes[END_REF]. To have an entire Supply Chain for this phase, the MSR's IT strategy had to find a Pilot Store that would agree to participate and modify its organisation to fill the requirements of the RFID technology. This is a multisite competitive potential alignment perspective (see column 3 on Table 7 ). The RFID technology has been identified by IT strategy as a driver to examine the changes needed in the underlying Information System and its technology. As described earlier by [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF], the role of the IT Strategy Manager was then to determine the best allocation of scarce resources both within the organization and in the IT marketplace (label development, tag creation, readers and printers, IS, etc.). The IT Strategy department looked in the environment for some skills and competencies on RFID technology: specific labels suitable for the Jewelry supply chain, very small RFID tag development, IT/IS integration, printers, RFID writers and readers.

Furthermore, as described in the previous section, there were gaps in the business processes that needed to be closed to enable and exploit the improved technology infrastructure identified at different levels of the supply chain. This included the MSR in terms of replenishment, its Stores for reception and inventory control, and the Logistic Service Provider for product reception and labeling. When the different partners involved in the project agreed on the IT project development, the potential changes on processes at the LSP and within the 80 MSR's stores were examined. This was not a linear process because RID technology is not mature. For different reasons due to the technology development, the tags and labels were modified, and consequently the organisation and processes were changed. This fourth sequence of alignment is a Service Level Perspective in an inter organizational context.

It was initiated with the pilot project and deployed within the 80 MSR's jewellery stores and also concerned the Logistic Service Provider. As previously noted, in this project, the main driver of process and IS alignment at LSP was the IT department from Casino (Column1 Table 8). This is chiefly due to the competency of the MSR's IT department to manage the project, even at its supplier location. This is equally true of the MSR's stores (Column 2 Table 8) that lack an IT department and rely directly upon the MSR for IT strategy.

An inter organisational Service Level Perspective

A Multisite Service Level Perspective business strategy and IT infrastructure. This combination was described by [START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF] as Organisation infrastructure fusion.

The first pivot that was handled was the business strategy. This is not because the business strategy domain was the weakest one, as suggested by [START_REF] Coleman | Strategic alignment: analysis of perspectives[END_REF], but rather because RFID is a disruptive innovation. In this project, even if the short term impact on the business of each project partner was low, a strategic business agreement was needed to launch this first RFID project because of potentially high long term impact. Initially the eventual business strategy changes resulting from use of RFID were not identified. Nevertheless, the choice was made pursue the project involving logistics activities. For future applications, once RFID competency and skills are acquired and the IT infrastructure is established, further projects could take on more business strategy oriented goals.

For the other players, namely the software provider and the tag printing company, the RFID and IT marketplace project led to product innovation via the creation of new integrated software (SSP), new labels and new RFID tags. Gunasekaran et al. (2004) have suggested that "the alignment between information models and supply chain models or objectives needs further investigation". These new research challenges call for:

Conclusion

• identification of critical IT success factors for an integrated and aligned supply chain

• attaining alignment between IT and business models in the supply chain context

• an empirical test of the SAM in the context of inter-organizational relationships.

To address these challenges, we conducted research using a jewellery supply chain in an inter-organizational context. This case study provides answers to the dynamics of alignment in a jewellery supply chain and underlines the main difficulties observed during a pilot project and implementation phases. It also highlights evolving benefits sought by the retailer and its suppliers.

The main conclusions to be drawn from this study are:

• There is not a single strategic alignment but a combination of intra and inter organizational strategic alignments between the two supply chain partners (MSR and LSP). Following [START_REF] Papp | Assessing strategic alignment in real-time[END_REF], the nature of alignment sequences are larger than those highlighted by [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF]. Our case showed two main alignment sequences: the Competitive Potential in phase 1 of the project followed by the Service Level in phase 3. The phase 2 of the project split the Competitive Potential within two sub alignment sequences: a multisite competitive potential alignment perspective and an inter-organizational competitive potential alignment. These combinations of alignments correspond to the Organisation infrastructure fusion [START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF].

• The alignments are incremental and are the result of interactions between different types of alignment in the supply chain. Each intra organizational alignment impacts the other intra organizational alignments of the partners in the same supply chain. The quality of "local" alignment (intra organizational alignments, which can be multisite) impacts the global supply chain (inter organizational alignments).

• The value identified at the beginning of the project by the IT Strategy department was used as an incentive to change logistic business processes on a single value chain (jewellery product). This explains the initial alignment sequence form: Competitive Potential.

• Additional value has been created and exploited by the different actors of the RFID project even if apparently no companies have created and sustained competitive advantage [START_REF] Tajima | Strategic value of RFID in supply chain management[END_REF] This research introduces several questions. First, this case questions the impact of the different sequences of alignment during the pilot phase on the deployment phase of a project. Second, the context of the relationships between the different companies can change the sequences of alignments: here the MSR imposes the project to the LSP without an initial discussion on long-term strategic potential of RFID in their own business models (Gunasekaran et al., 2004). Third, [START_REF] Papp | Assessing strategic alignment in real-time[END_REF] invites to consider the future interactions to facilitate long-range planning and strategy formulation. How can the different alignment sequences help us to imagine the future evolutions of this project in order to fully benefit from RFID technology?

Our research shows certain limits concerning methodology. We focus on a single case study and our findings are contextual and cannot be generalized. Moreover, this RFID project was complex due to its emerging nature and included many different actors from different companies or services, leading to multiple goals and objectives. Strategy Execution: this perspective considers the business strategy as the driver of both organization design choices and the logic of the IT infrastructure. The anchor domain is business strategy. The pivot domain is the business infrastructure that needs to be changed. The resulting domain of impact is the information technology infrastructure.

List of Figures and Tables

Technology Potential: this perspective is also driven by business strategy but the pivot is information technology strategy. The information technology infrastructure is thus the domain of impact.

Competitive Potential: the anchor is information technology strategy and the pivot is business strategy, while organization infrastructure is the impacted domain. This perspective focuses on how emerging information technologies can influence and enable new business strategies.

Service Level: in this perspective the anchor is information technology strategy, the pivot information technology infrastructure, and the area of impact is organizational infrastructure. The focus of this perspective is the manner in which information technology can improve how products and services are delivered.

Organization IT Infrastructure: this perspective results in process improvements from information technology and the application of value to the business processes The anchor is organization infrastructure, the pivot is information technology infrastructure, and the impacted domain is information technology strategy.

IT Infrastructure Strategy: the focus of this perspective is the improvement of information technology strategy based on the implementation of emerging and existing information technology infrastructures. The anchor is information technology infrastructure, which drives the pivot, information technology strategy, and thus impacting business strategy IT Organization Infrastructure: the anchor of the seventh perspective is also IT infrastructure, with the pivot being organizational infrastructure and the impact area being business strategy. Although similar to IT infrastructure strategy, IT in this perspective is the driving force and architect by which the visions and processes are carried out Organization infrastructure strategy: this final individual perspective exploits the capabilities to enhance new products and services, influence strategy, and develop new relationships. Business infrastructure is the anchor, and this perspective enables enhancement to business strategy (the pivot) thus changing the IT strategy (affected area).,

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  RFID technology is derived from a World War II manufacturing automation technique allowing aircrafts to identify themselves to other friendly aircraft and commanders on the ground. From 1998 to 2003, the Massachusetts Institute of Technology was a leader in retail initiatives, and in 1999 they established the Auto-ID Centre. The Auto-ID Centre at MIT officially closed in late 2003, transferring its intellectual property, including RFID, to Electronic Product Code Global where RFID achieved greater importance than ever before. At present EPC Global standards only cover the deployment of RFID in the supply chain. The technology gained attention when Wal-Mart required its top suppliers to adopt RFID technology, as did the U.S. Department of Defence.Despite these projects, RFID is still considered as a disruptive IT with a low level of maturity and a lack of standardisation. Companies know little about this technology, how it can be used, and where it can be applied to improve which processes.RFID offers several advantages for the handling and management of objects in the end-to-end supply chain: an increased visibility of product flows and at the same time a reduced effort for manual object identification and a unique identity of each alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 11

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  from this new technology and to propose process and IT reconfiguration to fit with RFID. It has four steps. alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 18

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Fosso

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•

  alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 22 Tag editing: Labelling the products with RFID tags at the warehouse and at the points of sale • Tag reading: Using RFID for the reception of products at the points of sale and for inventory control The next step consisted of defining the "B2B" Intra and Inter organizational processes integrating the new RFID technology. This network not only concerned the MSR and LSP, but also integrated new actors with knowledge and competencies to produce the RFID technology. LSP agreed to collaborate on the project and then aligned its operational processes to fit MSR's requirements (receiving, warehousing, order picking and delivery) and its IT infrastructure (investing in a smart RFID printer, RFID readers, adapting their ERP System and developing interfaces with the MSR's IS) (steps 9-10).

  alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 23 compare the original order to those items received, specific software to write the RFID tags and to print the labels. Fosso Wamba et al.(2007)

Fosso

  alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 24 development of an integrated software able to create the tag directly from the MSR's IS. The final step in the implementation process was deployment of the solution to all MSR's stores. In April 2009 the MSR deployed the solution to the 80 points of sale, thus implementing the RFID technology and its relevant software, and training the individuals who would be using the system. Step 11 and 12, RFID project deployment. RFID implementation resulted in performance improvements in the SC main logistic activities including: • Increased productivity at the point of sale thanks to time savings and reduced human error during reception and inventory control • Improved warehouse inventory control • Greater jewellery traceability along the entire supply chain 4.5. Inter-organizational alignment sequences during project phases alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 25

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  RFID pilot project and supply chain Business Process alignment alignment to enhance IT driven services innovation in a supply chain: the case of RFID Page 27

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Figure

  Figure 1: Strategic Alignment Model

  alignment to enhance IT driven services innovation in a supply chain: the case of

Table 2 :

 2 Table2). Fusion perspectives (from[START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF]

	Organization Strategy Fusion
	Combination of IT Infrastructure Strategy Perspective and IT
	Organizational Infrastructure Perspective
	Organization Infrastructure Fusion
	Combination of Competitive Potential Perspective and Service Level
	Perspective
	IT Strategy Fusion
	Combination of Organizational Infrastructure Strategy Perspective F and Organizational IT Infrastructure Perspective o r IT Infrastructure Fusion
	Combination of Strategy Execution Perspective and Technology P e Potential Perspective
	e r
	R
	e
	v i e
	w
	O n l
	y

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Table 5 :

 5 Inter organisational Alignment within all B2B partners.

	.

Table 8 :

 8 RFID project deployment and supply chain Business IT and Business Process alignmentAs shown in Table7and Table8this alignment process combines two perspectives that have one anchor, the IT Strategy domain, and two pivot domains:

•

  . The success of the RFID project will lead the MSR to develop a new project with this technology in another product value chain. For the MSR's stores, reception and inventory have become quicker and leave more time for selling. Finally, several product innovations have been developed by the software provider and the tag printing company who are able to propose new products for new markets. Ultimately, the impact of the RFID project on the MSR's global strategy was very limited because the MSR's Strategy Department decided to restrict the project to the logistics activities of a single Value Chain. Other opportunities for RFID, such as marketing innovation or adding new customer services were not pursued in the initial application of this disruptive innovation. During the deployment phase, IT Strategy led the project but could not exploit all the potential of the RFID technology to improve sales activities.• Impact on the logistics business process and organisation of the LSP was also analysed. The LSP has other clients in addition to the MSR, and as such,

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

evaluation of the impact of RFID technology implementation on its other business units was explored. The possibility for the LSP to deploy this technology to other customers and to develop new business from this emerging competitive advantage was considered, but not chosen by LSP's strategy department.

Table 1 :

 1 Alignment perspectives (from[START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF]

	Organization Strategy Fusion
	Combination of IT Infrastructure Strategy Perspective and IT
	Organizational Infrastructure Perspective
	Organization Infrastructure Fusion
	Combination of Competitive Potential Perspective and Service Level
	Perspective F IT Strategy Fusion o Combination of Organizational Infrastructure Strategy Perspective r and Organizational IT Infrastructure Perspective
	P IT Infrastructure Fusion e r Combination of Strategy Execution Perspective and Technology e Potential Perspective
	R
	e
	v i e
	w
	O n l
	y

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Table 2 :

 2 Fusion perspectives (from[START_REF] Papp | Determinants of Strategically Aligned Organizations: A Multiindustry, Multi-perspective Analysis[END_REF]

	Fosso Wamba et	SAM (Henderson
	al. (2007)	and Venkatraman,
		1993).
	Step 1 : Primary	Scanning IT
	motivation	environment
	Step2 : Product	Business Strategy
	value chain	decision
	Step 3 : Critical	Analyses of business
	activities	impact

Table 3 :

 3 Mapping the first steps of Alignment in the RFID project.

Table 8 :

 8 RFID project deployment and supply chain Business IT and Business Process alignment

Step 5: Six business processes were identified by the MSR for the Product Value Chain under examination: 1) purchasing, 2) replenishment at the point of sale, 3) demand management (Warehouse replenishment), 4) IT integration, 5) selling, and 6) invoicing. These business processes could either take place at the headquarters, or at the store. The activities involved in the value chain for both partners are: reception of the jewellery at the warehouse, picking at the warehouse, inventories at the warehouse, reception of jewellery at the point of sale, inventorying jewellery at the point of sale.

Step 6: Inter organisational process concerns:

• Delivery (from suppliers to the warehouse): the LSP warehouse regularly sends orders to jewellery suppliers that are located worldwide. Suppliers then deliver the orders to LSP's warehouse.

• Ordering (from point of sale to the warehouse): Depending on their sales, each MSR's shop individually and independently sends its order to LSP's warehouse via the group's ERP.

• Delivery (from the warehouse to the point of sale): LSP prepares the requested jewellery and sends it to the shops based on the order received from each MSR's shop. Fosso Wamba et al. ( 2007) SAM [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF].

Step 4: Mapping of the network

Step 5: Mapping of intra organizational processes

Step 5: Mapping of inter organizational processes

Inter organizational impact of IT implementation

Identification of the processes that will need to fit with the new IT within the organization Identification of the processes that will need to fit with the new IT between the organizations Table4: Mapping of AS IS processes and activities in the actual supply chain Fosso Wamba et al. ( 2007) SAM [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF] (2007) SAM [START_REF] Henderson | Strategic alignment: leveraging information technology for transforming organizations[END_REF].

Step 11: Evaluation including ERP and middleware integration and process automation at all supply chain member levels

Step 12: Pilot replicating and appropriation by the different organisations involved and their staff members