Taro Kimura
email: taro.kimura@cea.fr

Hofstadter problem in higher dimensions

We investigate some generalizations of the Hofstadter problem to higher dimensions with Abelian

Introduction

The fractal structure of the energy spectrum for the two-dimensional magnetic lattice system, known as Hofstadter's butterfly [1], is one of the most exotic consequences of the quantum property of low dimensionality. Such a fractal nature of the magnetic system can be observed in a simple tightbinding model in the presence of a magnetic field. This Hofstadter problem and its variants have been extensively discussed in various areas of physics and also mathematics. More recently it can be realized even in experimental situations [2,3].

In this paper we extend the Hofstadter problem, which is originally considered in two dimensions, to higher dimensions with not only Abelian, but also non-Abelian, gauge field configurations. So far there have been some attempts to generalize it to the three-dimensional magnetic system [4][5][6][7], and also to that in non-Abelian gauge potential [8][9][10][11][12]. But its generalization to a much higherdimensional system has not yet been studied in the literature. Actually, when we analyze topological matters in the lower-dimensional system, the four-dimensional point of view can be quite useful: topological insulators/superconductors in two and three dimensions [13,14] are deeply connected to the four-dimensional quantum Hall effect (QHE) [15] through the dimensional reduction procedure [16,17]. The formalism discussed in this paper can be applied to arbitrary even-dimensional lattice systems in the presence of a magnetic field. Based on this formalism, we prove that the π -flux state [18,19], which has a gapless excitation in general, is essentially equivalent to the Dirac fermion in arbitrary dimensions. Furthermore, when we apply a non-Abelian gauge field, we can obtain a hierarchical structure in the energy spectrum of the lattice model even in higher dimensions.

This paper is organized as follows. In Sect. 2 we introduce arbitrary even-dimensional lattice models with Abelian gauge field background configuration. We discuss the corresponding Schrödinger equation to the magnetic system, and obtain Harper's equation. We then comment on its connection to the non-commutative torus. We also provide a general proof for the gapless spectrum of the π -flux state, by referring to its equivalence to the naively discretized lattice Dirac fermion. In Sect. 3 we then consider the Hofstadter problem for the non-Abelian gauge field configuration. In particular, PTEP 2014, 103B05 T. Kimura the four-dimensional SU(2) theory is investigated as a fundamental example of non-Abelian gauge theory. We show some numerical results of the model, and discuss the effect of the inhomogeneity of the background flux. Section 4 is devoted to a summary and discussion.

Abelian gauge field models

The first generalization of the Hofstadter problem is formulated in arbitrary even dimensions in the presence of the U(1) gauge field. Before introducing a lattice model, we now consider the following background U(1) field configuration for the continuum theory [20],

F 2s-1,2s = ω s , F μν = 0 for otherwise, (s = 1, . . . , r ). ( 1
)
We then apply the higher-dimensional version of the Landau gauge to this configuration,

A 2s-1 (x) = -ω s x 2s , A 2s (x) = 0. (2)
Here the field strengths ω s are quantized,

ω s = 2π L 2s-1 L 2s n s , n s ∈ Z, (3)
and thus the topological number is given by

Q = 1 2 r • r ! d 2r x μ 1 •••μ 2r TrF μ 1 μ 2 • • • F μ 2r-1 ,μ 2r = r s=1 n s . ( 4
)
This is regarded as the r th Chern number.

To discuss the Hofstadter spectrum, we then realize these configurations on the lattice. The gauge potential (2) is implemented by introducing the link variable, which can be regarded as the Wilson line, U 2s-1 (x) = e -iω s x 2s , U 2s (x) = 1.

(

) 5
Precisely speaking, we have to assign appropriate boundary conditions even for U 2s (x) [20]. In this case the field strength is restricted to the interval (0, 2π) due to the lattice discretization [21,22]. Furthermore, they are characterized by the following fractions,

p s q s ≡ n s L2s-1 L2s . ( 6
)
Here, Li is related to the system size as L i = La where a is the lattice spacing. Note that they satisfy 0 < p i /q i < 1. This fraction plays an essential role in the interesting spectrum of the model we discuss below.

Tight-binding model

The lattice Hamiltonian with this background configuration is defined as

H tight = x,μ c † x+ μU μ (x)c x + c † x U † μ (x)c x+ μ . ( 7
)
This is just the tight-binding hopping model, which describes the non-relativistic particle with the background magnetic field. Introducing the state |ψ = x ψ x c

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Schrödinger equation from the Hamiltonian (7) written in a second quantized form,

2r μ=1 U μ (x -μ)ψ x+ μ + U † μ (x)ψ x-μ = Eψ x . ( 8
)
We then solve Eq. ( 8) by taking a Fourier transformation. Note that the translation symmetry of this model is slightly modified from the usual lattice model due to the background field,

x 2s-1 ∼ x 2s-1 + 1, x 2s ∼ x 2s + q s . ( 9
)
Thus, writing the coordinate as x 2s = q s y s + z s , the wavefunction is Fourier transformed as

ψ x = 1 V k 1 ,...,k 2r exp i r s=1 (k 2s-1 x 2s-1 + q s k 2s y s ) ψz 1 ,...,z r (k 1 , . . . , k 2r ), ( 10
)
where V stands for the effective volume of the system,

V = L 1 • • • L 2r /(q 1 • • • q r ).
The Schrödinger equation ( 8) is rewritten in this basis as

r s=1 ψz+ŝ + ψz-ŝ + 2 cos (k 2s-1 -ω s z s ) ψz = E ψz . ( 11
)
This is the higher-dimensional version of Harper's equation [23]. While a one-dimensional equation is obtained from the magnetic two-dimensional system, we have an r -dimensional equation from the d = 2r theory. Furthermore, the matrix size of this higher-dimensional Harper's equation is N q × N q where N q ≡ q 1 • • • q r . This means that the number of energy bands is just given by N q , and there possibly exist N q -1 energy gaps. To discuss these energy gaps, one has to investigate a transfer matrix and its spectral curve associated with Harper's equation (11). In this case, however, it is impossible to show these energy bands are totally gapped with the same argument as the two-dimensional situation: we cannot apply a naive transfer matrix method, since there are still r directions in Eq. (11). Therefore it is difficult to obtain the corresponding Hofstadter's butterfly by diagonalizing Harper's equation (11): its wings are almost disappearing. Let me comment on the relationship between the non-commutative space and Eq. (11). Indeed, the higher-dimensional tight-binding model, discussed in this section, can be represented in terms of the non-commutative torus. The coordinates of the two-dimensional non-commutative torus T 2 θ are given by

U V = e 2πiθ V U, ( 12
)
with e 2πiθ being the qth root of unity, 2πθ ≡ ω = 2π/q. They can be written in q × q matrix forms,

U = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 1 . . . . . . 0 1 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , V = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 e iω e 2iω
. . .

e (q-1)iω ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 13
)
Based on this description, we can consider the d = 2r -dimensional non-commutative torus:

U s V t = e 2πiθ st V s U t for s, t = 1, . . . , r. ( 14
)
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T. Kimura

In general, θ st is a real symmetric matrix. In the case of (11), it is simply given by a diagonal matrix, 2πθ st = ω s δ st . This means the non-commutativity on the 2r -dimensional torus is introduced to each two-dimensional subspace,

T 2r θ → T 2 θ 1 × • • • × T 2 θ r .
Then the associated Hamiltonian can be written as

H tight = r s=1 U s + U † s + V s + V † s . ( 15
)
Although, precisely speaking, we have to include a factor corresponding to the plane wave, we now omit these factors for simplicity. This representation means that the non-commutative torus operator plays the role of the translation operator with the external magnetic field even in the higher-dimensional case.

π -flux state.

Let us comment on the π -flux state in higher dimensions. 1 For the π -flux state [18,19], all the plaquettes have the same value, P μν = -1, independent of its position x, and directions μ, ν as

P μν (x) = U μ (x)U ν (x + μ)U † μ (x + ν)U † ν (x) = -1. ( 16
)
Such a configuration is realized by the following Z 2 ⊂ U(1) link variables,

U μ (x) = η μ ≡ (-1) x 1 +•••+x μ-1 . ( 17
)
Then the tight-binding Hamiltonian in the second quantized form with this gauge configuration yields

H tight = x d μ=1 η μ c † x c x+ μ + c x-μ . ( 18
)
We now consider the free field theory for simplicity. This is almost the same as the staggered Dirac operator [26-28]

S staggered = x d μ=1 1 2 η μ χx χ x+ μ -χ x-μ . ( 19
)
Actually, by applying the transformation c x → i x 1 +•••+x d c x , the Hamiltonian ( 18) is rewritten as

H tight = x d μ=1 i η μ c † x c x+ μ -c x-μ . ( 20
)
This is just the staggered fermion action (19), up to a constant factor. Note that the staggered Dirac operator is anti-Hermitian while the tight-binding Hamiltonian is Hermitian.

There is no spinor structure in this formulation, but this staggered fermion is directly obtained from the naive Dirac fermion through the spin-diagonalization-see Appendix A for details. The sign factor η μ = (-1) x 1 +•••+x μ-1 is a remnant of the gamma matrix. The staggered fermion enjoys an exact U(1) chiral symmetry, which is generated by x = (-1) x 1 +•••+x d . This symmetry ensures its gapless excitation. Therefore the π -flux state is also generically gapless.

PTEP 2014, 103B05 T. Kimura

Dirac fermion models

We then attempt to extend Hofstadter's problem to the relativistic system. Similar approaches have been seen in the context of graphene [29][30][31][32].

The generic form of the Dirac fermion action is written as

S Dirac = x,μ ψx+ μU μ (x)P + μ ψ x -ψx U † μ (x)P - μ ψ x+ μ ≡ x,y ψx D x,y ψ y . ( 21
)
Here P ± μ stands for the spinor matrix, corresponding to a type of the lattice fermion. 2 The simplest one is given by

P ± μ = 1 2 γ μ . ( 22
)
This is just the naive lattice discretization of the relativistic fermion [34]. 3 We have other choices for the spinor matrix:

• Wilson fermion [34]:

P ± μ = 1 2 (γ μ ± r 1). (23)
• Karsten-Wilczek fermion [36,37]:

P ± μ = 1 2 γ μ (μ = d), P ± μ = 1 2 (γ μ ± irγ d ) (otherwise). ( 24
)
• Boriçi-Creutz fermion [38,39]:

P ± μ = 1 2 ((1 ± ir)γ μ ∓ ir ), = 1 2 d μ=1 γ μ . ( 25
)
All these fermions include a free parameter r . We often consider the case of r = 1 for simplicity. Note that another lattice fermion, which is called the staggered fermion [26][27][28], is essentially equivalent to the naive fermion, and also the π -flux state as discussed in Sect. 2.1. The naive and staggered fermions are transformed to each other by the spin-diagonalization. Thus in this paper we do not deal with the staggered fermion explicitly. The Dirac equation, which corresponds to the Schrödinger equation ( 8), is given by

d μ=1 U μ (x)P + μ ψ x+ μ -U † μ (x -μ)P - μ ψ x-μ = λψ x . ( 26
)
When we write Eq. ( 26) as Dψ = λψ, although this Dirac operator D becomes non-Hermitian in general, one can define an alternative Hermitian operator, H = γ d+1 D. In the following we consider the spectrum of this Hermitian operator as H ψ = Eψ, instead of the original Dirac equation (26).

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T. Kimura

We can also obtain the corresponding Harper's equation by taking a Fourier transformation. In this case, Harper's equation is slightly modified from (11) as

d/2 s=1 γ d+1 P + 2s ψz+ŝ -P - 2s ψz-ŝ + i sin (k 2s-1 -ω s z s ) P + 2s-1 + P - 2s-1 + cos (k 2s-1 -ω s z s ) P + 2s-1 -P - 2s-1 ψz = E ψz . ( 27
)
Here we have 2N q energy bands due to the γ -matrix structure: they appear in a pair for the relativistic theory. In this case, it is again difficult to obtain a fully gapped spectrum since this reduced equation is not one-dimensional, but d/2-dimensional.

Non-Abelian gauge field models

We consider another generalization of the Hofstadter problem in higher dimensions by applying a non-Abelian gauge field as a background configuration. In this paper we concentrate on the fourdimensional model with SU(2) gauge field. A further generalization to higher-dimensional and higher-rank systems seems to be straightforward.

Background configuration

We now introduce the following SU(2) background configuration as a simple generalization of the U(1) background (2),

A 0 = 0, A j = -ω j x 0 σ j for j = 1, 2, 3, (28)
with σ j being the Pauli matrix. Here we define filling fractions

ω j = 2π L 2 n j ≡ 2π p j q j . ( 29
)
From this configuration we compute a field strength,

F 0 j = -ω j σ j , F jk = -ω j ω k x 2 0 i jk σ i . ( 30
)
The total background flux is given by

1 16π 2 L 4 d 4 x μνρσ TrF μν F ρσ = 6π 2 L 6 N L 4 d 4 x x 2 0 = 2π N . ( 31
)
Here this integral is taken over the four-dimensional hypercubic lattice of size L 4 , and we define the flux number N = n 1 n 2 n 3 .

Thus the link variables associated with (28) are given as follows,

U 0 (x) = 1, U j (x) = e -iω j x 0 σ j = 1 cos ω j x 0 -iσ j sin ω j x 0 . ( 32
)
In this case, the translation symmetry of the lattice system yields

x 0 ∼ x 0 + q LCM , x j ∼ x j + 1, (33)
where q LCM is the least common multiple of q 1 , q 2 , and q 3 . This means that the unit cell of this system is also extended in only one dimension as well as the two-dimensional case with U(1) magnetic field. 34) of the flux ω j=1,2,3 = ω 0 ≡ 2π p/q with q = 100, p = 0, 1, 2, . . . 99.

Lattice fermion models

Let us first study the tight-binding model with the non-Abelian gauge field configuration. With a Fourier basis for x 0 = q LCM X 0 + x0 , the Hamiltonian ( 7) gives rise to the corresponding Harper's equation (11),

ψ x0 +1 + ψ x0 -1 + 2 3 j=1 cos k j cos ω j x0 + σ j sin k j sin ω j x0 ψ x0 = E ψ x0 . ( 34
)
This is just as effective a one-dimensional model as the standard two-dimensional Hofstadter problem. The original four-dimensional model is reduced to this due to SU(2) ∼ = S 3 gauge symmetry.

On the other hand, in this case it is also difficult to show its spectrum is totally gapped as is the case for the U(1) background in higher dimensions. Therefore we now study its density of states instead of the original energy spectrum.

Figure 1 shows the density of states for Harper's equation (34) against ω 0 , with homogeneous configuration ω j = ω 0 for j = 1, 2, 3. Although its spectrum is not totally gapped, one can observe a hierarchical structure of the spectrum, which seems fractal at least based on this numerical computation. In order to numerically determine the corresponding fractal dimension, it is necessary to perform the calculation with a larger system size. At the even fraction flux, e.g. ω 0 /(2π) = 1/2, 1/4, . . . , we find a dip in the spectrum. A similar structure is discussed in the ordinary two-dimensional Hofstadter problem at the gapless point of the filling fraction.

In Fig. 2 we show the spectrum in the inhomogeneous flux with fixing ratios of ω j=1,2,3 . When flux in some directions is turned off, e.g. ω = ω 0 (1, 0, 0) and ω 0 (1, 1, 0), the characteristic structure of Hofstadter's butterfly cannot be observed in the spectrum: the gap is completely closed. In the cases of ω = ω 0 (2, 1, 1) and ω 0 (2, 2, 1), there are some gaps in the spectrum and the hierarchical structure. We also find a dip at the gapless point, ω 0 = π .

Let us now comment on the relation to the non-commutative space. We introduce the following non-commutative torus,

U V j = e iω j σ j V j U for j = 1, 2, 3. (35)
Here U is defined as well as (13), while V j=1,2,3 are slightly modified as

V j = diag 1, e iω j σ j , . . . , e (q LCM -1)iω j σ j . ( 36
)
PTEP 2014, 103B05

T. Kimura In this case the non-commutative parameter is matrix valued. Thus, the commutation relations between V j=1,2,3 cannot be written in a simple way. discussed in Sect. 2, the tight-binding Hamiltonian with the external field, corresponding to Harper's equation (34), can be associated with the non-commutative torus. The naive form of the Hamiltonian without momentum dependence is given by

U + U † + 3 j=1 V j + V † j . ( 37
)
However, since the second part is represented as

V j + V † j = diag 2, 2 cos ω j , . . . , 2 cos(q LCM -1)ω j × 1, (38)
we cannot involve a matrix structure in this way. Actually the SU(2) nature of the external field comes through the momentum dependence in (34).

We then investigate the Dirac fermion models as discussed in Sect. 2.2. Applying the background configuration (32), the corresponding Dirac equation is given by

γ 5 P + 0 ψ x0 +1 -P - 0 ψ x0 -1 + 3 j=1
cos k j cos ω j x0 + σ j sin k j sin ω j x0 P + j -P - j + sin k j cos ω j x0 -σ j cos k j sin ω j x0 P + j + P -

j ψ x0 = E ψ x0 . ( 39
)
Here we again show the Hermitian version of the Dirac equation by multiplying by the matrix γ 5 . When we write this Dirac operator in a matrix form, its matrix size is 8q LCM = 2 (color) × 4 (spinor) × q LCM (flux). Here color corresponds to the rank of the gauge flux of SU (2). On the other hand, the number of eigenvalues is given by 4q LCM , since each spectrum is doubly degenerated due to the spinor structure.

PTEP 2014, 103B05

T. Kimura

Summary and discussions

In this paper we have explored some extensions of the Hofstadter problem in higher dimensions.

The first example is formulated with Abelian gauge configuration in d = 2r , giving rise to non-zero topological number. We have shown that half of the gauge potential can be trivial by applying the Landau gauge, thus the corresponding Harper's equation is essentially written as an r -dimensional lattice model. We have also pointed out that the π -flux state is equivalent to the staggered formalism of the relativistic lattice fermion. The latter is directly related to the naive Dirac fermion through the spin diagonalization. This means the π -flux state involves the chiral symmetry, and thus it yields massless excitation in any dimensions.

We have then investigated SU(2) non-Abelian gauge field configuration in four dimensions. We have considered the configuration with one specific direction in four dimensions. In this case, hopping terms for the other three directions are reduced due to SU(2) ∼ = S 3 symmetry of the gauge field. Thus we have obtained the one-dimensional Harper's equation by utilizing the Fourier basis. We have calculated its spectrum numerically, and its hierarchical structure is actually observed.

Let us now comment on possibilities for future work along this direction. The two-dimensional Hofstadter problem is essentially related to the quantum group [29,[40][41][42]]: Harper's equation is directly regarded as Baxter's equation for the one-dimensional U q (sl 2 ) model. Thus it is interesting to explore the corresponding quantum group structure to the generalized Hofstadter problems discussed in this paper. In particular, the non-Abelian version of Harper's equation includes the matrix-valued coefficient. This corresponds to the q-parameter in the two-dimensional case, thus it is natural to investigate a kind of quantum group with matrix-valued q-parameter.

Next is the lattice study with various kinds of lattice fermions, i.e. Wilson [34], staggered [26][27][28], staggered Wilson [43][44][45][46][47], minimal-doubling [36][37][38][39][48][49][50], domain-wall [51][52][53] and overlap fermions [54]. They were originally introduced to tackle the difficulty of the chiral fermion on the lattice, but these formalisms themselves are interesting as statistical lattice models: some of them are actually investigated in the context of condensed-matter physics, for example, graphene, π -flux state, topological insulator/superconductor, and so on. Thus we hope the Hofstadter problem formulated with these lattice fermions is relevant to realistic condensed-matter physics.

It is also definitely interesting to consider the implications of the result obtained in this paper for realistic situations. An important difference between the non-Abelian gauge field and the U(1) magnetic field is whether it breaks the time-reversal symmetry of the system: the SU(2) gauge potential can be applied without breaking the time-reversal symmetry. It implies that, as a consequence of the dimensional reduction of the model discussed in this paper, one can possibly realize the non-Abelian Hofstadter system, for example, in topological insulators whose time-reversal symmetry is not broken. In addition, since there are already experimental techniques to realize the non-Abelian gauge potential and the ordinary Hofstadter system based on the U(1) field, respectively, one can expect that experimental realization of the non-Abelian Hofstadter system might be possible, especially in the cold atomic system.

Fig. 1 .

 1 Fig. 1. Density of states for Harper's equation (34) of the flux ω j=1,2,3 = ω 0 ≡ 2π p/q with q = 100, p = 0, 1, 2, . . . 99.

Fig. 2 .

 2 Fig. 2. Density of states for Harper's equation (34) with the inhomogeneous flux for ω = ω 0 n: (a) n = (1, 0, 0), (b) (1, 1, 0), (c) (2, 1, 1), and (d) (2, 2, 1).

† x |0 , we obtain the corresponding 2/11 at CEA on October 24, 2014 http://ptep.oxfordjournals.org/ Downloaded from

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The author is grateful to T. Misumi for pointing out an essential connection of this argument to that discussed in[START_REF] Grignani | Introduction to Some Common Topics in Gauge Theory and Spin Systems[END_REF]. See also[START_REF] Creutz | [END_REF].4/11at CEA on October

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For example, see[33].

As is well known, this simple lattice discretization scheme has a problem: there exist extra massless modes at low energy, which are called species doublers. To obtain a single chiral fermion we have to implement a much more complicated scheme, e.g. domain-wall fermion, overlap fermion, and so on (see, for example, the recent review in Ref.[35]). However, this problem does not concern our study because it does not affect the spectrum of the Dirac fermion. 5/11 at CEA on October 24, 2014 http://ptep.oxfordjournals.org/ Downloaded from

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Acknowledgements

The author would like to thank M. Creutz, Y. Hidaka, and T. Misumi for useful discussions and comments. The author is grateful to H. Iida for collaboration at the early stage of this work. The author is supported by Grant-in-Aid for JSPS Fellows (No. .

PTEP 2014, 103B05 T. Kimura

Appendix A. Spin diagonalization

We now show that there is an alternative expression for the d-dimensional naive Dirac fermion without spinor matrix structure. It is given by diagonalizing the corresponding γ matrices.

Let us start with the naive Dirac fermion on the lattice,

Then, introducing the field χ x defined as

we can represent the naive lattice fermion (A1) in the following form,

Note that there is no spinor structure in this expression. In other words, the spinor matrix is diagonalized in this basis. This means the naive Dirac fermion can be rewritten in terms of a one-component fermionic field. This is just the staggered formalism of the relativistic fermion [27,28].


M Noyel
email: mnoyel@acta-mobilier.fr

P Thomas

A Thomas
email: andre.thomas@univ-lorraine.fr

P Charpentier
email: patrick.charpentier@univ-lorraine.fr

T Brault
email: tbrault@acta-mobilier.fr

Flow disturbance analysis in workshops with high reworks rate

Keywords: reworks, flow disturbance, indicator, simulation

Companies with high reworks rate have a real problem of flow management. Despite methods in place leading to minimize the number of reworks, they often must implement palliative solutions to manage disturbed flows. We propose in this paper a diagnostic approach based on reworks and flow disturbance indicators to enable them to view the disturbing status of their workshop. This work is a part of a more global work which aim is to develop a flow management control system based on product driven control concept and real-time indicators chosen from those presented in this paper.

Introduction

Some companies present an unavoidable high reworks rate as shown with the lot of works pursued to include reworks in methods such as Economic Production Quantity (EPQ) optimization [START_REF] Chiu | Effects of random defective rate and imperfect rework process on economic production quantity model[END_REF], production run time optimization [START_REF] Chiu | Production run time problem with machine breakdowns under AR control policy and rework[END_REF] or replenishment policy [START_REF] Chiu | Optimal replenishment policy for manufacturing systems with failure in rework, backlogging and random breakdown[END_REF]. Even if there are a lot of methods trying to decrease reworks rate, companies have to dealing with. Quality remains a priority number one, especially when there is an increased of customer demands in a context of difficult mastery of manufacturing process (unstable over time as example). The flow disturbances due to reworks have been analyzed since a long time by many authors [START_REF] Feigenbaum | Quality control: Principles, practice and administration: An industrial management tool for improving product quality and design and for reducing operating costs and losses[END_REF]. But this problem was often studied only on simple or particular cases [START_REF] Love | Rework: a symptom of a dysfunctional supply chain[END_REF][START_REF] Skidmann | Unitary manufacturing cell design with random product feedback flow[END_REF]. Moreover, indicators for analyzing tested situations are often arbitrarily chosen without taking into account the different points of view. However in our meaning, simulation can help to understand each point of view of each indicator token into account. The master objective of this work is to bring to this type of companies a decision support based on several indicators to enable them to better understand the impact of their reworks rate on the complexity of their physical flows on the shop floor.

That leads them to be able to anticipate the impact of quality drifts on flow control problems and to react quickly and well to decrease it thanks to cartography of the real-time state of their workshop. After presenting in more detail the purpose of this paper, possible indicators and their characteristics to handle reworks rate and flow disturbance are listed. Then, the simulation model and the selected test scenarios to compare their behavior rate are explained.

Background and issues

The most relevant approaches to understand and control Quality [START_REF] Garvin | What does product quality really mean[END_REF] is based on manufacturing approach. It combines conformance to customer requirements and the "right the first time" concept to reduce costs and ensure quality. Non-quality products can be the result of a higher than usual customer requirement, which may occur during the conquest of new markets (export ...). In this case, processes are well mastered but a higher quality level is required to reach new customer requirements. Nevertheless, non-quality may also result from unstable manufacturing process or process drift. Many companies working for the luxury market face with this problem because quality requirements lead to be close to technological limitations of the manufacturing process. Because products are expensive, only two reactions are possible to fix the non-quality product detected at the workstation on the existing X workstations: In practice, both cases have sometimes to be considered at the same time but this paper focus on case 2. We call reworks all products repaired by sending it to a previous workstation. If reworks rate is fluctuating, arising disturbances will also be fluctuant and unpredictable, which complicates production control. It is recognized that reworks affect the performance of system productivity [START_REF] Colledani | Impact of quality control on production system performance[END_REF]. COQ (Cost of Quality) method is appeared in the 50s thanks to the concept of the ghost factory which is a workshop parallel to the official workshop with the mission is to repair the defects of the official factory [START_REF] Feigenbaum | Quality control: Principles, practice and administration: An industrial management tool for improving product quality and design and for reducing operating costs and losses[END_REF]. It represents 40% of the production capacity of the official factory. The first standard on the subject appears in France in 1986 (X 50-126) to evaluate the costs of non-quality [START_REF] Abouzahir | Pilotage de l'amélioration des process par les coûts de non-qualité[END_REF]. A statistical analysis relative to one workstation problem points out the oscillatory phenomenon [START_REF] Skidmann | Unitary manufacturing cell design with random product feedback flow[END_REF] that is also mentioned in other works where the aim is to evaluate performance on a workshop with multiproduct type [START_REF] Skidmann | Performance evaluation of a flexible manufacturing cell with random multiproduct feedback flow[END_REF]. The production flow disturbance by reworks is especially important if reworks emerged downstream in the production process [START_REF] Love | Rework: a symptom of a dysfunctional supply chain[END_REF]. So, for the production control, we have to follow and maintain a low reworks level to be sure to simplify the initial piloting problem [START_REF] Noyel | Implantation of an online quality process monitoring[END_REF]. However, even reducing and stabilizing the reworks rate, the resultant disturbance on the flow remains consistent and makes scheduling difficult.To evaluate the reworks rate in a company, an indicator that counts the number of defects is often used. Nevertheless, when reworks rate is growing, the probability to repair several times the same product is growing, too. So how to count these products becomes a critical issue and it is difficult to make a direct link between reworks rate and flow disturbance. As previously said, this paper is a part of a global work which aim is to analyze, compare and identify relevant indicators to measure the reworks impact on workflow disturbance. These indicators are presented in part 3.

•

Indicators determination

Selection of possible indicators

Two kinds of indicators allow to measure reworks rate and product flow disturbance. For reworks, there are global (at the company scale) and local (related to the most stringent workstation, generally the bottleneck) indicators defined bellow:

• !"# $ is the number of reconfigured rooting sheets relative to normal ones. It is independent of the number of disturbed products if we don't work in unit lot size. • % ! is the number of products which have had at least one defect relative to expected products to make. It takes into account the exact number of disturbed products. There is no consideration of products repaired more than once time. There is a weakness due to the fact that some defects may not cause disturbance of the sequence of the manufacturing program but just a longer production time. In the case of a model with lot size unit, % ! = !"# $ because the number of lines reconfigured is directly equal to the number of products on which we have detected at least one defect. Effectively, detection of a defect involves repetition of one or more operations. Whatever the situation: % ! ≤ '. • (")* is the number of products actually processed on the workstation relative to expected products to make.

It is very simple to measure but we have to note that it cannot differentiate if added operations have been made on several products or on the same product. Moreover, (")* is always exceeding 100%.

• +, -. and/or +, /0 is the Operation Ratio that corresponds to the number of real operations compared to what was expected in the routing sheet. As RO is relative to a specific product/lot, it is recommended to take into account the average/mean (AV) and/or standard deviation (SD). RO represents both the rooting sheet disturbance and the level of this disturbance. If any defect exists for one product, its RO is equal to one. It is more difficult to measure than previous indicators.

Similarly, there are possible indicators to measure the workflow disturbance:

• 123 is the work in process (average or max). It is important to note that a well distributed work in process is not necessarily penalizing, even if it is heavy. Note that it is not useful on one workstation because it will be equal to "!* . • "!* is the stock level upstream an important workstation (average or max). • 4 5 is the time to complete all the production. Commonly used in scheduling problems, it is equivalent to the end production date of the last product. Two schedules can give the same 4 5 but with a very different customer service rate.

• 6 ( is the waiting time.

• + 5 is the maximum delay. Depending on the case, be an hour late is sometimes equivalent to be a week late, and sometimes, late penalties are too heavy.

• 7 is the number of late products. • 6 8) * is the out of supply time on the bottleneck workstation. It can be measured only on models with several workstations.

Model presentation

The Petri net presented on Fig. 1 has been implemented in ARENA to analyze the impact of an increase of each reworks indicator on a manufacturing process. Despite the apparent simplicity of this model, model reduction often leads to summarize the problem with a one-or two-workstations model [START_REF] Rabiee | Scheduling a no-wait two-machine flow shop with sequence-dependent setup times and probable rework using robust meta-heuristics[END_REF]. The two machines have the same production rates. There are others similar works in literature that performs a scheduling on a two-machine problem [START_REF] Garvin | What does product quality really mean[END_REF]. The lot size is unitary and each product is directly transferred to the next workstation, not involving additional time corresponding to the pending end of batch before transfer. It avoids the problem of splitting lots to remake only a part of product [START_REF] Skidmann | Unitary manufacturing cell design with random product feedback flow[END_REF]. EDD is chosen in order to allow permutations between reworks and normal products in queues. The due date required for the implementation of the priority rule EDD was determined for each product to represent a "normal" situation with low reworks rate leading to no delay. Probability of reworks occurrence are explained on (Fig. 2) with the following particularity:

9 6 ) (")* 1 → 1 = 6 ) (")* 2 → 1 + 6 ) (")* 2 → 2 6 ) (")* 2 → 1 = 6 ) (")* 2 → 2

Fig. 2. Probability of reworks occurrence

To have different vision of reworks with unitary lots, only % ! , (")* , +, /0 and +, -. are tested. If there is no variability in the model, our indicators are dependents and it is possible to express them with a mathematical formula. But, in reality, variability involves disturbances (ex. Forester Effect). This variability is included in the model by using normal distribution for the product arriving in the first queue, the processing time on workstations and the probability of defect occurrences.

and + 5 are able to differentiate our scenarios (Fig. 8 and9). The FIFO priority rule generates delays even with a very low + ; while the EDD rule generates delays when + ; increases. So it seems to be better to apply EDD at low reworks levels. However, when exceeding a certain reworks level to be determined according to the workshop modeled, either seeking to minimize the maximum delay and keeping the EDD rule, or seeking to minimize the number of late and applying the FIFO. This rule change may be an example of the functions of our support system workflow management. These two indicators seem particularly useful for controlling the system by the rules.

<=>. K. + 5 based on

reworks rate for 2 workstations <=>. W. 7 based on reworks rate for 2 workstations Flow disturbance is also a concept difficult to represent because it exists a lot of point of view. Depending on management objectives, some indicators alone are not appropriate. For example, it is impossible to differentiate the EDD and FIFO rules if we consider only 123 , "!* , 4 5 or 6 1 . The best solution seems to be a combination of indicators. The latter figure shows that the same rule for controlling the workshop is not necessarily effective in all cases of running of the workshop. This therefore shows the usefulness of our support system. Flow management will be able to offer the user the best rule to use based on selected indicators for monitoring the workshop.

Conclusion and outlook

Flows repairs due to reworks disturb production consistently. This is largely confirmed by the literature but rarely studied in depth. To develop a system for assisting the production piloting for companies with high reworks rates, we propose a thorough study on reworks and flow disturbance indicators. There are several ways to control reworks for companies. We must be aware of the different possible points of view and their advantages and drawbacks. In this paper we propose a new indicator, the Operation Ratio that has the advantage of combining the manufacturing program disturbance with the level of disturbance and appears correctly reflect the reworks rate in the workshop. Flow disturbance is also a difficult concept to measure. The usual indicators offer only a single view of the flow disturbance. If we want to have a global view, we need to work on a combination of these different indicators. As perspectives, we have to validate our analysis on a real scale industrial case, which will bring our work to a realistic level. We hope to develop a behavioral mapping of the workshop in order to locate it in real time and make the right decisions on the piloting rules.

──

  Case 1 (Repair) : Sending it to a dedicated workstation leading to: ─ Two new flows created: → et → Maximum number of additional flows: = 2 . . with the number of workstations dedicated to repairs • Case 2 (Rework): Sending it to a previous workstation leading to: ─ Only one new flow created: → Maximum number of additional flows:

Fig. 1 .

 1 Fig. 1. Petri net representing our problem and implemented in ARENA • P1: Wait in queue before workstation 1 • P2: Process on workstation 1 • P3: Wait in queue before workstation 2 • P4: Process on workstation 2 • P5: Finished state

Analysis and results

4.1

Analysis of reworks indicators.

Two scenarios are tested: one and two workstations. All graphs are relatives to the reworks rate + ; applied on workstation(s). 3 products are send through the model. By mathematical definition, % ! 1 + ; and % ! 2 + ; @ 2 + ; (Fig. 3). The dispersion observed with model variability seems to be different between one (only increasing) and two workstations (increase then decrease). This indicator offer a view of the capacity to make product right the first time. By definition, (")* 1 ∑ + ; 1 :

. Dispersion is also due to variability in the model. With two workstations, the formula is already difficult to determine but the same dispersion exist, too. With one workstation as with two workstations, +, /0 is directly equal to (")* (Fig. 5). These indicators offer a view of the volume of load on the workshop. As the number of operations increase in a non-linear mode with the reworks rate, it is important to work to decrease the reworks rate.

<=>. 5. +, -. based on reworks rate for 1 and 2 workstations +, -. offers a view on the predictability on the respect of deadlines. Higher the reworks rate is, more difficult is to predict the number of necessary operations and so the product completion time.

There are many ways for representing the reworks on the workshop. The most appropriate choice to use the indicator may depend on the type of workshop but also the desired view. A combination of several of them can also give a more comprehensive understanding from different perspectives (capacity to make product right the first time, volume of charge on the workshop and predictability on the deadlines respect).

Analysis of workflow disturbance indicators

The same model is used with two workstations but with two other scenarios: EDD and FIFO. The stock upstream the bottleneck machine 1 "!* (Fig. 6) seems to be a good indicator of the disturbance flow as well as work in process 123 (Fig. 7). However, these two indicators are not able to differentiate EDD from FIFO. The similarity between these two indicators is also explained by the one machine model where "!* is equivalent to 123 . Theoretically, + ; increasing tend to get some kind of system saturation. In practice, this still represents more pieces to handle, store, manage simultaneously, which negatively influence all production indicators. 4 5 indicator, commonly used in scheduling, has the same behavior that 6 1 . 4 5 and 6 1 EF are also unable to differentiate our scenarios but 6 1 GEH can. It seems that, in relation to 6 1 GEH , EDD rules application appears more effective than FIFO rules.