Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Giuseppe Toscani
email: giuseppe.toscani@unipv.it

FAST DIFFUSION EQUATIONS: MATCHING LARGE TIME ASYMPTOTICS BY RELATIVE ENTROPY METHODS

Keywords: May 4, 2011. 2010 Mathematics Subject Classification. 35B40, 35K55, 39B62 Fast diffusion equation, porous media equation, Barenblatt solutions, Hardy-Poincaré inequalities, large time behaviour, second moment, asymptotic expansion, intermediate asymptotics, sharp rates, optimal constants. ,

A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.

Introduction and main results

Consider on R d the fast diffusion equation [START_REF] Arnold | Entropies and equilibria of many-particle systems: an essay on recent research[END_REF] ∂v ∂τ

+ ∇ • v ∇v m-1 = 0
for some m ∈ (m c , 1) with m c := (d -2)/d. Assume that the initial data is a given nonnegative function v 0 in L 1 (R d ). It is well known (see for instance [START_REF] Barenblatt | On some unsteady motions of a liquid and gas in a porous medium[END_REF]) that the large time behavior of the solution is captured by the Barenblatt solutions given for any (τ, y) ∈ R + × R d by

B(τ, y) := (1 + τ ) -1 m-mc D + 1 2 d (m-mc) (1 + τ ) - 2 d (m-mc ) |y| 2 -1 1-m +
where D > 0 is determined by the condition

v 0 L 1 (R d ) = B(τ, •) L 1 (R d ) .
Using entropy methods, it has been established in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF] how a linearized problem involving the relative entropy and the relative Fisher information determines the best rate of convergence towards the Barenblatt solution. The note [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] is devoted to a refinement of the estimates in which the dependence on v 0 is clarified and the precise value of the best possible rate of convergence is computed in terms of a spectral gap of the linearized operator associated to the relative entropy and the relative Fisher information for all values of m < 1. By taking advantage of the translation invariance, it is moreover possible to impose that the solution evolves in the orthogonal of the eigenspace associated to the first non-zero eigenvalue of the linearized operator for m ∈ (m 1 , 1) with m 1 := (d -1)/d, thus providing an improved rate of convergence. The corresponding conserved quantity is the center of mass, while the generators of the eigenspace are the derivatives of the Barenblatt solution with respect to each of the coordinates. There is no other conserved quantity known, so that further improvements cannot be achieved directly by this method.

One may however notice that the eigenspace corresponding to the second non-zero eigenvalue in the range m ∈ (m 1 , 1) is generated by the infinitesimal dilation of the Barenblatt solution. It is therefore natural to try to adjust the Barenblatt solution by a scaling. This can be done by taking a time-dependent change of variables where the scale is determined by the solution itself and not anymore by its asymptotic, self-similar behavior, thus providing improved convergence rates. Asymptotically, we will recover the self-similar profile, but with a better matching. There is a price to pay: the rescaled equation has a time-dependent coefficient, which converges to a constant. From the point of view of the entropy -entropy production inequality, however, nothing is changed, which is the main observation of this paper.

Let m 1 := d/(d+ 2). In the range m ∈ ( m 1 , m 1 ), the infinitesimal dilation of the Barenblatt solution generates the eigenspace corresponding to the first non-zero eigenvalue. Our time-dependent change of variables therefore improves on the rate of convergence for any m ∈ ( m 1 , m 1 ), and also for any m ∈ (m 1 , 1) if the center of mass is chosen at the origin.

The reader interested in understanding the heuristics of our approach is invited to go directly to Section 2. In the remainder of this section, we will give a precise statement of our main result and some additional references.

Define the mass and the center of mass of v 0 respectively by

M := R d v 0 dy and x 0 := 1 M R d y v 0 dy .
Consider the family of the Barenblatt profiles

(2) B σ (x) := σ -d 2 C M + 1 σ |x| 2 1 m-1 ∀ x ∈ R d
where σ is a positive real parameter and

C M := M M * -
Let us recall the definition of m c , m 1 , m 1 and introduce the exponents m 2 and m 2 , for later use:

m c = d -2 d , m 1 = d -1 d , m 1 = d d + 2 , m 2 := d + 1 d + 2 and m 2 := d + 4 d + 6 , which are such that m c < m 1 < m 2 < 1, m c < m 1 < m 2 and, if d ≥ 2, m 1 ≤ m 1 and m 2 ≤ m 2 .
For later purpose, it is also convenient to define

(3)

K M := R d |x| 2 B 1 dx = (1 -m) m 1 m -m 1 M C M for any m ∈ ( m 1 , 1). Notice that K M is finite if m > m 1 .
If v is a solution of (1), consider the time-dependent scale R(τ ) defined by ( 4)

d log R dτ = 2 K M R d |y -x 0 | 2 v(τ, y) dy d 2 (m-mc) , R(0) = 1 .
The justification of such a choice for τ → R(τ ) will be made clear in Section 2. Note that, in view of the fact that the quantity R d |y -x 0 | 2 v(τ, y) dy is non-decreasing in time along the solution of the fast diffusion equation ( 1) and because of its asymptotic behaviour, the time-dependent scale R(τ ) is such that log R(τ ) is increasing from zero to infinity. We also define σ as a function of τ by the condition

σ(τ ) := 1 K M R(τ ) 2 R d |y -x 0 | 2 v(τ, y) dy .
As a consequence, the equation for R can be rewritten as

(5) 2 σ -d 2 (m-mc) = R 1-d (1-m) dR dτ
and we can define for any x = y/R(t) ∈ R d the Barenblatt type solution B by

B(τ, y) := 1 R(τ ) d B τ, y R(τ ) where B(τ, x) := B σ(τ ) (x) .
The difference of B with the Barenblatt solution B is that B depends on R d |y| 2 v(τ, y) dy, so that they are only asymptotically equivalent, as we shall see later. The point is that B provides a better asymptotic matching than B. Our goal is indeed to measure the rate of convergence of v towards B. For this purpose, it is convenient to change variables and study the rate of convergence of u defined by

u(τ, x) = R(τ ) d v (τ, x 0 + R(τ ) x)
towards B. Let us consider the relative entropy of J. Ralston and W.I. Newmann defined in [START_REF] Newman | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I[END_REF][START_REF] Ralston | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II[END_REF] by

E(τ ) := 1 m -1 R d u m -B m -m B m-1 (u -B) dx . Theorem 1. Assume that m ∈ ( m 1 , 1), d ≥ 2.
Let v be a solution of [START_REF] Arnold | Entropies and equilibria of many-particle systems: an essay on recent research[END_REF] with initial datum v 0 ∈ L 1 + (R d ) such that v m 0 and y → |y| 2 v 0 (y) are integrable. With the above notations, we have

lim sup τ →∞ R(τ ) γ(m) E(τ ) < ∞ , where R(τ ) ∼ τ 1 d (m-mc ) as τ → ∞ and γ(m) =        ((d-2) m-(d-4)) 2 4 (1-m) if m ∈ ( m 1 , m 2 ] , 4 (d + 2) m -4 d if m ∈ [ m 2 , m 2 ] , 4 if m ∈ [m 2 , 1) .
Once a relative entropy estimate is known, it is possible to control the decay rate of u -B in various norms, for instance in L q (R d , dx) for q ≥ max 1,

2 d (1-m) 2 (2-m)+d (1-m)
, or in C k , by interpolation. Up to a change of variables, this also allows to prove decay rates of v -B. See [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF] for more details.

Compared to the results of [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], an improvement for any m > m 1 has been obtained. The values obtained in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] for γ(m) are indeed

γ(m) = 2 d m -2 (d -2) if m ∈ ( m 1 , m 1 ] , 2 if m ∈ [m 1 , 1) ,
except that in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] the scale R(τ ) is determined by the self-similar Barenblatt solutions (both scales are anyway equivalent as τ → ∞: see Lemma 8). Also see Figure 2 at the end of this paper for more details on m → γ(m) in the setting of [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] compared to the results of Theorem 1.

Compared to other methods, it may look surprising that the scale R(τ ) and, as a consequence, the coefficient σ both depend on the solution v of (1). Asymptotically, as τ → ∞, R(τ ) is equivalent to the scale given by the selfsimilar change of variables, but what has been gained is a better matching with the closest Barenblatt solution. The family of the Barenblatt solutions is globally invariant under scaling and, among all such solutions, there is one which is closer to our solution of the evolution equation: the one with the same second moment.

Convergence results of a suitably rescaled flow associated to (1) towards an asymptotic profile has been established in [START_REF] Friedman | The asymptotic behavior of gas in an n-dimensional porous medium[END_REF] for m > m c (also see for instance [START_REF] Vázquez | Asymptotic behaviour for the porous medium equation posed in the whole space[END_REF]) and in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Daskalopoulos | On the extinction profile of solutions to fast diffusion[END_REF] for m ≤ m c . Getting rates of convergence beyond a simple interpolation between mass and uniform estimates has required the use of the relative entropies introduced by J. Ralston and W.I.

Newman in [START_REF] Newman | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I[END_REF][START_REF] Ralston | A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II[END_REF]. First results in this direction have been achieved in [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF] using the entropy / entropy-production method of D. Bakry and M. Emery (also see [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF] for general diffusions and [START_REF] Arnold | Entropies and equilibria of many-particle systems: an essay on recent research[END_REF] for an overview) and in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] using sharp Gagliardo-Nirenberg interpolation inequalities. F. Otto made the link with gradient flows with respect to the Wasserstein distance in [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF], and D. Cordero-Erausquin, B. Nazaret and C. Villani gave a proof of the corresponding Gagliardo-Nirenberg inequalities using mass transportation techniques in [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities[END_REF]. The condition m ≥ m 1 was a strong limitation to these first approaches. Gagliardo-Nirenberg inequalities indeed degenerate into a critical Sobolev inequality for m = m 1 , while the displacement convexity condition also requires m ≥ m 1 . For m < m 1 , various limitations appear. To work with Wasserstein's distance, it is crucial to have second moments bounded, which amounts to request m > m 1 for the Barenblatt profiles; see for instance [START_REF] Denzler | Phase transitions and symmetry breaking in singular diffusion[END_REF][START_REF]Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology[END_REF]. Linearization of entropy estimates around the Barenblatt profiles has been considered in [START_REF] Carrillo | Poincaré inequalities for linearizations of very fast diffusion equations[END_REF][START_REF] Lederman | On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass[END_REF] for m ∈ (m c , 1). In a certain sense, this is also the strategy in [START_REF] Denzler | Phase transitions and symmetry breaking in singular diffusion[END_REF][START_REF]Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology[END_REF]. Integrability of the Barenblatt profiles means m > m c . This condition has been removed in a series of recent papers (see [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF][START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF]) together with a clarification of the strategy of linearization of the relative entropies, at least from the point of view of functional inequalities. In this paper, we shall however restrict m to the interval ( m 1 , 1), for spectral reasons that are explained in Section 3 and for the second moment to be well defined. For m ≤ m 1 , even with an appropriate definition a relative second moment, our method gives no improvement on the convergence rates because of the presence of the continuous spectrum.

Rescalings and convergence towards Barenblatt solutions, or intermediate asymptotics, has not been the only issue of large time asymptotics. We can for instance quote [START_REF] Toscani | A central limit theorem for solutions of the porous medium equation[END_REF] for a study (in the porous media case) of the time evolution of the second moment, and [START_REF] Carrillo | Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering[END_REF][START_REF]Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology[END_REF][START_REF] Mccann | Second-order asymptotics for the fast-diffusion equation[END_REF][START_REF] Cáceres | Kinetic approach to long time behavior of linearized fast diffusion equations[END_REF] for the search of improved convergence rates when moment conditions are imposed in the framework of Wasserstein's or other Fourier based distances. The question of improved rates has been precisely formulated in [START_REF]Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology[END_REF], and solved in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] in the weighted L 2 framework that we shall use in this paper, as far as the first moment (position of the center of mass) is concerned. The main contribution of this work is to explain how improvements based on the second moment can also be achieved.

This paper is organized as follows. In Section 2, we explain how faster convergence results can be achieved by introducing an appropriate timedependent rescaling, which is given by (4) and not by the explicit dependence of the Barenblatt solutions. Improved Hardy-Poincaré inequalities are established in Section 3, using the spectral results of [START_REF] Denzler | Phase transitions and symmetry breaking in singular diffusion[END_REF][START_REF]Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology[END_REF] and the spectral equivalence found in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]. The large time behaviour of the solution is studied in Section 4. The proof of Theorem 1 is then completed in Section 5. Further considerations on the case d = 1 and the limiting regime as m → 1 - are presented in the last section.

The relative entropy approach

The result of Theorem 1 is easy to understand using a time-dependent rescaling and the relative entropy formalism. Define the function u such that [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] v(τ, y

+ x 0 ) = R -d u(t, x) , R = R(τ ) , t = 1 2 log R , x = y R
where v is a solution of (1) with initial datum

v 0 ∈ L 1 + (R d ).
A simple computation shows that u has to be a solution of [START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF] ∂u ∂t

+ ∇ • u σ d 2 (m-mc) ∇u m-1 -2 x = 0 t > 0 , x ∈ R d ,
with initial datum u 0 = v 0 (we assume that R is chosen such that R(0) = 1) and σ given by

2 σ -d 2 (m-mc) = R 1-d (1-m) dR dτ ,
which is nothing else than [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. By virtue of the definition of R(τ ), the new time t = 1 2 log R(τ ) increases monotonically from 0 to +∞. Consequently, the old and new times can be uniquely related, and τ can be expressed in terms of t through the inverse function of R, so that τ = R -1 (e 2t ). Using this transformation and with a slight abuse of notations, we shall consider from now on σ as a function of t. It is important to notice that, as long as dσ dt = 0, the Barenblatt profile B σ is not a solution of ( 7), but we may still consider the relative entropy

F σ [u] := 1 m -1 R d u m -B m σ -m B m-1 σ (u -B σ ) dx .
Let us briefly sketch the strategy of our method before giving all details.

If we consider a solution of [START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF] and compute the time derivative of the relative entropy, we find that (8)

d dt F σ(t) [u(t, •)] = dσ dt d dσ F σ [u] |σ=σ(t) + m m -1 R d u m-1 -B m-1 σ(t) ∂u ∂t dx .
Here comes the main difference with previous works. As we shall see below in the proof of Lemma 2 (also see Remark 1), the first term of the right hand side in (8) involves

d dσ F σ [u] |σ=σ(t) = m D(σ) 1 -m R d |x| 2 B σ dx - R d |x| 2 u dx |σ=σ(t)
where D(σ) := d dσ σ d (mc-m)/2 . When taking a time-dependent rescaling based on the self-similar variables, one finds that σ is constant in t, so that dσ dt = 0 and the term dσ dt d dσ F σ [u] does not contribute. In our approach, σ depends on t but can still be chosen so that this term does not show up either. It is indeed enough to require that (9)

R d |x| 2 B σ dx = R d |x| 2 u dx ,
which amounts to ask that R solves the ordinary differential equation ( 4), to obtain that d dσ F σ [u] = 0. This will be justified in the first step of our method, below (see Lemma 2).

In a second step, we shall use the fact that

(10) d dt F σ(t) [u(t, •)] = - m σ(t) d 2 (m-mc) 1 -m R d u ∇ u m-1 -B m-1 σ(t) 2 dx .
From there on, the computation goes essentially as in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]. For completeness, we will briefly reproduce it. However, with our choice of σ, we gain an additional orthogonality condition which will be explicitly stated in the third step of the method: see Lemma 3. This orthogonality condition is the crucial point (see Corollary 7) for improving the rates in Theorem 1, compared to the results of [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]. Now let us give further details.

First step: choice of the scaling parameter. For a while, we do not need to take into account the dependence of σ in t. The main idea of this paper is indeed to choose σ in terms of u by minimizing σ → F σ [u], so that

d dσ F σ [u] = 0 .
Lemma 2. For any given u ∈ L 1 + (R d ) such that u m and |x| 2 u are both integrable, if m ∈ ( m 1 , 1), there is a unique σ = σ * > 0 which minimizes σ → F σ [u], and it is explicitly given by

(11) σ * = 1 K M R d |x| 2 u dx .
For σ = σ * , the Barenblatt profile B σ * satisfies [START_REF] Cáceres | Kinetic approach to long time behavior of linearized fast diffusion equations[END_REF].

In [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF], K M is the constant which has been defined in [START_REF] Bartier | Improved intermediate asymptotics for the heat equation[END_REF]. The condition m > m 1 guarantees that B m σ is integrable and

K M = R d |x| 2 B 1 dx is finite. Proof. We have to minimize e(σ) := (1 -m) R d B m σ dx + m R d B m-1 σ u dx ,
which corresponds to the two σ-dependent terms in the expression of

F σ [u]. Using the fact that B m-1 σ (x) = σ d (1-m)/2 (C M + |x| 2 /σ) and R d u dx = M , it is easy to see that e(σ) = (1 -m) σ d 2 (1-m) M C M + K M + m σ d 2 (1-m) M C M + R d |x| 2 σ u dx
Collecting terms, we get

e(σ) = σ d 2 (1-m) M C M + (1 -m) K M + m σ d 2 (1-m)-1 R d |x| 2 u dx .
Hence, by optimizing on σ > 0, we find that σ = σ * is given by

σ * = m (m -m c ) (1 -m) M C M + (1 -m) K M R d |x| 2 u dx .
Using (3), we observe that

m (m -m c ) (1 -m) M C M + (1 -m) K M = 1 K M ,
which proves (9) and therefore [START_REF] Carrillo | Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities[END_REF].

Remark 1. To prove (9) directly, we may notice that σ * is determined by the condition

0 = e ′ (σ) = m (1 -m) R d B m-2 σ dB σ dσ (B σ -u) dx .
Hence we obtain

0 = m R d (B σ -u) dB m-1 σ dσ dx = m R d (B σ -u) C(σ) C M + D(σ) |x| 2 dx ,
where

C(σ) = d dσ σ d (1-m)/2
, and D(σ) = d dσ σ d (1-m)/2-1 . Taking into account that both B σ and u have the same mass M , we get [START_REF] Cáceres | Kinetic approach to long time behavior of linearized fast diffusion equations[END_REF].

The dependence of σ in t when u is a solution of (7) has not been taken into account yet. The choice of Lemma 2 determines an ordinary differential equation for R in terms of R d |x| 2 u(t, x) dx. Undoing the time-dependent rescaling [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], this equation is exactly [START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF]. With the choice R(0) = 1, we recall that u 0 = v 0 .

As already mentioned, the choice of σ in Lemma 2 has a major interest. If we consider a solution of ( 7) and compute the time derivative of the relative entropy, we find that the first term of the right hand side in [START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF]

drops so that d dt F σ(t) [u(t, •)] = m m -1 R d u m-1 -B m-1 σ(t)
∂u ∂t dx and we are back to the usual computations in self-similar variables.

Second step: the entropy / entropy production estimate. According to the definition of B σ , we know that

2 x = σ d 2 (m-mc) ∇B m-1 σ .
Using [START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF], we obtain

m m -1 R d u m-1 -B m-1 σ(t) ∂u ∂t dx = - m σ(t) d 2 (m-mc) 1 -m R d u ∇ u m-1 -B m-1 σ(t) 2 dx ,
thus proving [START_REF] Carrillo | Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering[END_REF]. Let w := u/B σ and observe that the relative entropy can be written as

F σ [u] = m 1 -m R d w -1 - 1 m w m -1 B m σ dx .
Define the relative Fisher information by

I σ [u] := R d 1 m -1 ∇ (w m-1 -1) B m-1 σ 2 B σ w dx .
For a solution u of ( 7), we find that

d dt F σ(t) [u(t, •)] = -m (1 -m) σ(t) d 2 (m-mc) I σ(t) [u(t, •)] ∀ t > 0 .
As in [5, Lemma 3] (also see [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]) we can estimate from below and above the entropy

F σ [u] by (12) h m-2 R d |f | 2 B 2-m σ dx ≤ 2 m F σ [u] ≤ h 2-m R d |f | 2 B 2-m σ dx where f := (w -1) B m-1 σ , h 1 (t) := inf R d w(t, •), h 2 (t) := sup R d w(t,
•) and h := max{h 2 , 1/h 1 }. The fact that h is bounded for any t > 0 is easy to prove by the Maximum Principle if h(0) is finite. See for instance [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF] for more details. Even if h(0) is infinite, h is anyway bounded for any t > 0, large enough, when m ∈ (m c , 1); see [START_REF] Bonforte | Global positivity estimates and Harnack inequalities for the fast diffusion equation[END_REF]Theorem 1.2]. By [5, Corollary 1], we also know that lim t→∞ h(t) = 1.

According to [5, Lemma 7] (also see [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]) , the generalized Fisher information satisfies the bounds (13)

R d |∇f | 2 B σ dx ≤ [1 + X(h)] I σ [u] + Y (h) R d |f | 2 B 2-m σ dx where h -1 1 h 2 (2-m) 2 ≤ h 5-2m =: 1 + X(h) , d (1 -m) ( h 2 h 1 ) 2 (2-m) -1 ≤ d (1 -m) h 4 (2-m) -1 =: Y (h) . Notice that X(1) = Y (1) = 0.
Third step: orthogonality conditions. To obtain decay rates of t → F σ(t) [u(t, •)] is now reduced to establish a relation between

R d |f | 2 B 2-m σ dx and R d |∇f | 2 B σ
dx. This is the purpose of the next section, but before let us make a few additional observations on the properties of f = B m-2 σ u -B m-1 σ . Lemma 3. Let u be a solution of [START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF] and f = B m-1 σ (u/B σ -1) where σ = σ(t) is defined by [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. With these notations, the function f has the following properties, for any t > 0 :

(i) Mass conservation:

R d f (t, x) B 2-m σ dx = 0 if m > m c . (ii) Position of the center of mass: R d x f (t, x) B 2-m σ dx = 0 if m > (d -1)/(d + 1). (iii) Conservation of the second moment: R d |x| 2 f (t, x) B 2-m σ dx = 0 if m > m 1 .
Notice that Property (ii) has already been used in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]Theorem 7] to obtain improved rates of convergence for m ∈ (m 1 , 1). Property (iii) is new and arises from the fact that the change of variables ( 6) is chosen in Lemma 2 by imposing a condition on the moment and not according to the self-similar variables corresponding to the Barenblatt solutions. Restrictions on m are such that mass, first and second moments are well defined for B σ . As in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], such conditions could be relaxed by considering moment conditions on f only, or relative moment conditions; we will however not pursue in this direction, as such an approach does not improve on the rates of convergence.

Proof. It is straightforward to rewrite the conservation of mass along evolution into Property (i).

Equation ( 1) being independent of y, it is clear that R d y v(τ, y) dy = R d y v 0 (y) dy for any τ > 0. As a consequence, the center of mass of u is located at x = 0: R d x u(t, x) dx = 0, and so we get Property (ii).

Finally, Property (iii) is a direct consequence of (9).

Improved Hardy-Poincaré inequalities

When M = M * and σ = 1, with the notations of Section 1, the quantities

R d |f | 2 B 2-m σ dx and R d |∇f | 2 B σ dx involve various powers of (1 + |x| 2
). On R d , we shall therefore consider the measure dµ α := µ α dx, where the weight µ α is defined by µ α (x) := (1 + |x| 2 ) α , with α = 1/(m -1) < 0, and study the operator

L α,d := -µ 1-α div [ µ α ∇• ]
on the weighted space L 2 (dµ α-1 ). The operator L α,d is such that

R d f (L α,d f ) dµ α-1 = R d |∇f | 2 dµ α .
Notice that in the range m ∈ (m c , 1), that is for α ∈ (-∞, -d/2), dµ α is a bounded positive measure. If additionally m > m 1 , that is for α < -(d + 2)/2, then dµ α-1 is a bounded measure and R d |x| 2 dµ α is also finite.

Let α * := -(d-2)/2. Based on [START_REF] Denzler | Phase transitions and symmetry breaking in singular diffusion[END_REF][START_REF]Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology[END_REF][START_REF] Blanchet | Hardy-Poincaré inequalities and applications to nonlinear diffusions[END_REF][START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], we have the following result.

Proposition 4. The bottom of the continuous spectrum of the operator

L α,d on L 2 (dµ α-1 ) is λ cont α,d := (α -α * ) 2 .
Moreover, L α,d has some discrete spectrum only for α < α * . For d ≥ 2, the discrete spectrum is made of the eigenvalues

λ ℓk = -2 α (ℓ + 2 k) -4 k k + ℓ + d 2 -1 with ℓ, k = 0, 1, . . . provided (ℓ, k) = (0, 0) and ℓ + 2k -1 < -(d + 2 α)/2. If d = 1, the discrete spectrum is made of the eigenvalues λ k = k (1 -2 α -k) with k ∈ N ∩ [1, 1/2 -α].
Let α * := -(d -2)/2. The following result has been established in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF].

Corollary 5 (Sharp Hardy-Poincaré inequalities). Let d ≥ 2. For any α ∈ (-∞, α * ) ∪ (α * , 0), there is a positive constant Λ α,d such that (14) Λ α,d R d |f | 2 dµ α-1 ≤ R d |∇f | 2 dµ α ∀ f ∈ L 2 (dµ α-1 )
under the additional condition

R d f dµ α-1 = 0 if α < α * . Moreover, for d ≥ 3, the sharp constant Λ α,d is given by Λ α,d =        1 4 (d -2 + 2 α) 2 if α ∈ -d+2 2 , α * ∪ (α * , 0) , -4 α -2 d if α ∈ -d, -d+2 2 , -2 α if α ∈ (-∞, -d) .
For d = 2, Inequality (14) holds for all α < 0, with the corresponding values of the best constant Λ α,2 = α 2 for α ∈ [-2, 0) and Λ α,2 = -2 α for α ∈ (-∞, -2). For d = 1, ( 14) holds, but the values of Λ α,1 are given by

Λ α,1 = -2 α if α < -1/2 and Λ α,1 = (α -1/2) 2 if α ∈ [-1/2, 0).
The constant Λ α,d is determined by the spectral gap and corresponds either to the lowest positive eigenvalue, λ 1,0 or λ 0,1 , or to the bottom of the continuous spectrum, λ cont α,d := 1 4 (d + 2 α -2) 2 (see Fig. 1). With additional orthogonality conditions, one improves on the spectral gap in the range for which discrete spectrum exists. A first result in this direction has been achieved in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] for solutions with center of mass at the origin. Here we give a refined version of it, by going to the next order, that is, by considering functions with zero moments up to order two.

Corollary 6 (Improved Hardy-Poincaré inequalities). Under the assumptions of Corollary

5, if α < -(d + 2)/2 and f ∈ L 2 (dµ α-1 ) is such that R d f dµ α-1 = 0 , R d x f dµ α-1 = 0 and R d |x| 2 f dµ α-1 = 0 , then (14) holds for any d ≥ 3 with Λ α,d =        1 4 (d -2 + 2 α) 2 if α ∈ -d+6 2 , -d+2 2 , -8 α -4 (d + 2) if α ∈ -(d + 2) , -d+6 2 , -4 α if α ∈ (-∞, -(d + 2)] . For d = 2, Λ α,2 = α 2 for α ∈ [-4, -2) and Λ α,2 = -4 α for α ∈ (-∞, -4).
The constant Λ α,d is now determined either by the lowest of the two eigenvalues, λ 1,1 or λ 2,0 , or by the bottom of the continuous spectrum, λ cont α,d (see Fig. 1), since the components corresponding to the eigenspaces associated to λ 1,0 and λ 0,1 have been removed.

, A crucial observation is that we can scale Inequality [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities[END_REF].

λ 01 = -4 α -2 d λ 10 = -2 α λ 11 = -6 α -2 (d + 2) λ 02 = -8 α -4 (d + 2) λ 20 = -4 α λ 30 λ 21 λ 12 λ 03 λ cont := 1 4 (d + 2 α -2) 2 α = -d α = -(d + 2) α = -d+2 2 α = -d-2 2 α = -d+6 2 α 0 α = - √ d -1 -d 2 α = - √ d -1 -d+4 2 α = - √ 2 d -d+2 2 (d = 5) Essential spectrum of Lαd , Lαd Spectrum of , αd m c = d-2 d m 1 = d-1 d m 2 = d+1 d+2 m 1 = d d+2 m 2 = d+4 d+6 Spectrum of (1 -m) L 1 (m-1) d (d = 5) Essential spectrum of 1 2 4 6 / , (1 -m) L 1 (m-1) d / , m Figure
Corollary 7. Let M > 0 and d ≥ 2. With B σ defined by (2), let f be a function in L 2 (B 2-m σ dx) such that R d (1, x, |x| 2 ) f B 2-m σ dx = (0, 0, 0) and ∇f ∈ L 2 (B σ dx) .
Then the inequality

Λ α,d R d |f | 2 B 2-m σ dx ≤ σ d 2 (m-mc) R d |∇f | 2 B σ dx
holds with Λ α,d as in Corollary 6.

Proof. The proof relies on a simple change of variables. From Corollary 6, we know that

Λ α,d R d |g| 2 1 + |x| 2 α-1 dx ≤ R d |∇g| 2 1 + |x| 2 α dx
for any g ∈ L 2 (dµ α-1 ) satisfying the conditions of Corollary 6. Then Corollary 7 holds for f such that f (x) = g(x/ √ C M σ ) for any x ∈ R d , which concludes the proof.

Estimates on the second moment

Up to now, we have not determined the behavior of R(τ ) as τ → ∞, nor the fact that σ has a finite, positive limit as t → ∞. These properties can be deduced for instance from [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF]. For the convenience of the reader, let us give some details. As in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF], consider the standard change of variables

(τ, y) → t = 1 2 log R 0 (τ ), x = y R 0 (τ )
based on the self-similar behavior of the Barenblatt solution B(τ, y), where

R 0 (τ ) := (1 + 2 d (m -m c ) τ ) 1 d (m-mc ) .
With the above change of variables, if v is a solution of (1), then the function U (t, x) := v(τ, y) solves

∂U ∂t + ∇ • U ∇U m-1 -2 x = 0 t > 0 , x ∈ R d ,
which is nothing else than [START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF], except that here σ is replaced by 1. It has been established in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF] that

lim t→∞ e 2 c t F 1 [U (t, •)] = 0
for some positive constant c. By Hölder's inequality, we know that

R d U m dx = R d U B m-1 1 m B m (1-m) 1 dx ≤ R d U B m-1 1 dx m R d B m 1 dx 1-m
, thus proving that

F 1 [U ] ≥ R d B m 1 dx + m 1 -m R d U B m-1 1 dx - 1 1 -m R d U B m-1 1 dx m R d B m 1 dx 1-m . With k(t) := R d U B m-1 1 dx R d B m 1 dx = R d |x| 2 U dx + M C M K M + M C M ,
this can be rewritten as As a consequence, we observe that

F 1 [U ] ≥ R d B m 1 dx k m -1 -m (k -1) m -1 ∼ |k(t) -1| 2 as t → ∞ ,
R d |x| 2 U (t, x) dx -K M = O e -c t as t → ∞ ,
Undoing the change of variables, we find that

1 R 0 (τ ) 2 R d |y| 2 v(τ, y) dy -K M = O R 0 (τ ) -c/2
as τ → ∞ , which, by definition of σ, gives

σ = 1 R(τ ) 2 (1 + 2 d (m -m c ) τ ) 2 d (m-mc ) 1 + O R 0 (τ ) -c/2
as τ → ∞, where τ → R(τ ) is given by (4). Using [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF], this means that 1 R(τ )

dR dτ = 2 (R 0 (τ ) 2 σ) -d 2 (m-mc) = 1 d (m -m c ) 1 τ 1 + O τ -ε
with ε = min{1, c/2} > 0. With these estimates, we can prove the following result.

Lemma 8. With the notations of Sections 1 and 2,

R(τ ) ∼ τ 1 d (m-mc) as τ → ∞
and, as a function of t, t → σ(t) is positive, decreasing, with

lim t→∞ σ(t) =: σ ∞ > 0 .
More precisely, we know that for some

C ∞ > 0, R(τ ) = C ∞ τ 1 d (m-mc ) (1 + o(1)) as τ → ∞ and σ ∞ = (2 d (m -m c )) 2 d (m-mc ) /C 2 ∞ .
However, the value of σ ∞ in terms of v 0 is not known.

Proof. The asymptotic behaviors of R and σ are direct consequences of the above computations. We only have to prove the monotonicity of σ. According to Lemma 2, we know that

σ(t) = 1 K M R d |x| 2 u(t, x) dx .
Using [START_REF] Bonforte | Special fast diffusion with slow asymptotics: Entropy method and flow on a riemannian manifold[END_REF] and integrating by parts, we compute

dσ dt = 1 K M d dt R d |x| 2 u(t, x) dx = 2 d (1 -m) m K M σ d 2 (m-mc) R d u m dx -4 σ .
On the other hand, by Lemma 2, we have

R d B m-1 σ(t) u(t, x) dx = R d B m σ(t) dx , so that F σ(t) [u(t, •)] = R d u m -B m σ(t) m -1 dx = 2 m d (1 -m) 2 K M σ d 2 (1-m) - 1 1 -m R d u m dx .
Hence we have proved that

dσ dt = -2 d (1 -m) 2 m K M σ d 2 (m-mc) F σ(t) [u(t, •)] ≤ 0 ,
which completes the proof.

Remark 2. We may notice that the second moment converges to a well defined value:

lim t→∞ R d |x| 2 u(t, x) dx = K M σ ∞ .
This is not the one which is usually found by considering the time-dependent rescaling corresponding to the self-similar Barenblatt solutions.

Proof of Theorem 1

We are now ready to resume with the relative entropy estimates and conclude the proof of Theorem 1. Using ( 12), ( 13) and Corollary 7, we find as in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF]

that (15) F σ [u] ≤ h 2-m [1 + X(h)] 2 Λ α,d -σ Y (h) m σ d 2 (m-mc) I σ [u]
as soon as 0 < h < h * := min{h > 0 : Λ α,d -sup t∈R + σ(t) Y (h) ≥ 0}. Two differences with [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] arise: Λ α,d has been improved in Corollary 7, to the price of a factor σ, which however plays no role because it also appears in the computation of d dt F σ(t) [u(t, •)]. The m factor is present because F σ has not been normalized as in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], and also because the equation for v is not the same.

As in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], uniform relative estimates hold, according to [7, Inequality (5.33)]: for some positive constant C, we have

0 ≤ h(t) -1 ≤ C F σ(t) [u(t, •)] 1-m d+2-(d+1)m .
Summarizing, we end up with a system of nonlinear differential inequalities, with h as above and, at least for some t * > 0 large enough,

d dt F σ(t) [u(t, •)] ≤ -2 Λ α,d -σ(t) Y (h) 1 + X(h) h 2-m (1 -m) F σ(t) [u(t, •)]
for any t > t * . Gronwall type estimates then show that lim sup

t→∞ e 2 (1-m) Λ 1/(m-1),d t F σ(t) [u(t, •)] < ∞ .
Notice as in [START_REF]Asymptotics of the fast diffusion equation via entropy estimates[END_REF] that for some constant c > 0, lim t→∞ e c t (h(t) -1) = 0, so that the fact that the quotient I σ , the value of m → γ(m) is given by the curve of Case 1 when no assumptions are made on the initial data. The curve of Case 2 corresponds to the case where the center of mass is chosen at the origin, as in [START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], while the curve of Case 3 corresponds to the exponent found in Theorem 1.

Concluding remarks

The case of the heat equation, i.e. m = 1, is not covered in our approach. However, we may pass to the limit as m → 1 -in Corollary 7, for the special choice 1/σ = 2 (1-m). Both weights B 2-m σ and B σ converge to the Gaussian weight, so that the conditions of Corollary 7 become R d

(1, x, |x| 2 ) f e -|x| 2 /2 dx = (0, 0, 0) and ∇f ∈ L 2 (e -|x| 2 /2 dx) .

By requiring the orthogonality with respect to all Hermite polynomials up to order two, we achieve the improved Poincaré inequality where λ ℓk are defined in Proposition 4 and α = 1/(m -1). See [START_REF] Bartier | Improved intermediate asymptotics for the heat equation[END_REF] for more details on improved convergence rates of relative entropies in case m = 1.

For completeness, let us extend our results to the case d = 1, which is very simple. Eigenvalues of L α,1 are ordered uniformly with respect to α, according to Proposition 4. Let α < -1/2 and consider f ∈ L 2 (dµ α-1 ) such that

1 .

 1 Figure 1. Spectrum of L α,d as a function of α (left), and spectrum of (1 -m) L 1/(m-1),d as a function of m (right), for d = 5.

  |k(t) -1| = 0 .

Remark 3 .

 3 [u]/F σ [u] in[START_REF] Daskalopoulos | On the extinction profile of solutions to fast diffusion[END_REF] is not estimated exactly by (2 Λ α,d )/(m σ) plays no role for the rate of convergence. This completes the proof of Theorem 1 with γ(m) = (1 -m) Λ 1/(m-1),d (see Fig.2). Exactly as in[START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF] Corollary 3], explicit estimates of the constants can be obtained. If v 0 /B 1 is uniformly bounded from above and from below by two positive constants, an estimate of lim sup τ →∞ R(τ ) γ(m) F σ(t)[u] in Theorem 1 can be given in terms of v 0 by computing a Gronwall estimate.

Figure 2 .

 2 Figure2. For d = 5, the value of m → γ(m) is given by the curve of Case 1 when no assumptions are made on the initial data. The curve of Case 2 corresponds to the case where the center of mass is chosen at the origin, as in[START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], while the curve of Case 3 corresponds to the exponent found in Theorem 1.

2 R

 2 d |f | 2 e -|x| 2 /2 dx ≤ R d |∇f | 2 e -|x| 2 /2 dx .Compared with the results of[START_REF] Bonforte | Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities[END_REF], nothing is gained, aslim m→1 - (1 -m) λ 01 = lim m→1 - (1 -m) λ 20 = 4 ,

R 1 m+1

 1 f dµ α-1 = 0 , R x f dµ α-1 = 0 and R |x| 2 f dµ α-1 = 0 .Then[START_REF] Cordero-Erausquin | A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities[END_REF] holds withΛ α,d = (α -1 2 ) 2 if α ∈ -5 2 , -1 2 , -6 (α + 1)) if α ∈ (-∞, -52 ] . Theorem 9. Assume that d = 1, m ∈ (1/3, 1). Let v be a solution of[START_REF] Arnold | Entropies and equilibria of many-particle systems: an essay on recent research[END_REF] with initial datum v 0 ∈ L 1 + (R) such that v m 0 and |y| 2 v 0 are integrable. With the above notations, we have lim sup τ →∞ R(τ ) γ(m) E(τ ) < ∞, where R(τ ) ∼ τ

(1-m) d (m-mc )

Acknowledgments. The authors acknowledge support both by the ANR-08-BLAN-0333-01 project CBDif-Fr (JD) and by MIUR project "Optimal mass transportation, geometrical and functional inequalities with applications" (GT). The warm hospitality of the laboratory Ceremade of the University of Paris-Dauphine, where this work have been partially done, is kindly acknowledged. The authors thank two anonymous referees for their careful reading of the paper.


Nuno Fernandes

Octávio

Keywords: DISPATCHING, DYNAMIC PRODUCTION order release, simulation order release, set-up time, simulation

In this paper, we report a simulation study on the role of sequence-dependent set-up times in decision making at the order release level of a workload controlled make-toorder flow-shop. The study evaluates the potential for set-ups savings, dependent on the level of workload in the shop, for two alternative strategies, namely considering set-up times centrally, within the release decision or locally, within the dispatching decision. These strategies are compared and assessed on the basis of two main performance measures namely time in system and standard deviation of the job lateness. Results indicate that the local strategy, which has been traditionally adopted in practice and in most of the studies dealing with sequence-dependent set-up times, does not always give the best results. The release frequency and the shop workload appear critical to the selection of the strategy to adopt, strongly influencing system performance.

Introduction

Workload Control (WLC) is a production, planning and control (PPC) concept that has received much attention in recent years [START_REF] Thurer | Workload control release mechanisms: from practice back to theory[END_REF]. It is particularly appropriate for jobbing and flow-shops in the make-to-order (MTO) sector of industry [START_REF] Haskose | Performance analysis of make-to-order manufacturing systems under different workload control regimes[END_REF]. WLC applies the basic principles of input/output control [START_REF] Plossl | Capacity Planning and Control[END_REF] to keep the length of queues on the shop floor at appropriate levels.

The aim is to achieve short, stable and predictable shop flow times towards meeting the promised delivery dates. This requires limiting and balancing workload on the shop floor to avoid temporary overloading or underloading of machines. When workloads are balanced the queues on the shop tend to be stable. Stable queues lead to predictable shop flow times, which can be used to determine the planned release times of orders.

Order release is described as an essential decision function and a core parte of WLC [START_REF] Missbauer | Models of the transient behaviour of production units to optimize the aggregate material flow[END_REF]. It determines the type, amount and time point of release of new orders into the shop [START_REF] Qi | An efficient new job release control methodology[END_REF]. For this propose, an order release mechanism is used, in combination with a pre-shop pool. Orders that arrive to the production system are gathered in this pool and are only released if they fit the workload norms, or bounds, of the required machines or capacity groups. This means that the decision to release an order is based on its influence on the current workload of the shop floor. As pointed out by [START_REF] Land | The performance of workload control concepts in job shops: Improving the release method[END_REF], the pre-shop pool acts as a shop floor buffer against the dynamics of the incoming flow of orders, e.g.

reducing perturbations due to order cancelation and allowing later ordering of raw materials, among other benefits.

WLC conceptualises a shop floor as a queuing system. Any released order (job) enters the shop and goes to the first machine of its routing. It waits in the queue if the machine is busy. Once processed in a machine, the job is moved to the next machine of its routing where it again waits until processing starts. WLC acts to ensure that workload at each machine do not exceed its norm, which is established with basis on the maximum acceptable flow time at each machine. Limiting in this way the workload means that, a limit on the number of jobs that can join the queue of a machine is imposed. Therefore, at times, jobs are not released because the workload norms of the required machines would be exceeded. Thus, [START_REF] Haskose | Performance analysis of make-to-order manufacturing systems under different workload control regimes[END_REF] considered the existence of finite buffers at machines. The shop flow time of a job is the sum of the set-up time, the processing time and the queuing time at each machine on the job's routing. Set-up time refers to the time required to prepare a machine to perform a job operation, such as cleaning, adjusting and changing tools and fixtures. Set-up times are dependent on both, the job to be processed and the one that had been processed immediately before. Most WLC literature assume that set-up time is either nonexistent or consider it as part of the processing time of the operation. While this may be acceptable for scheduling in some production environments, in many others sequence-dependent set-up times need to be taken into consideration separately. In this situation, shop performance cannot be effectively improved without the aid of appropriate scheduling procedures which take set-up times into account [START_REF] Kim | Impact of sequence-dependent set-up time on job shop scheduling performance[END_REF].

From the perspective of the WLC concept, essentially two alternative strategies can be considered to deal with sequence-dependent set-up times:

considering them centrally, i.e. within the release decision, or locally, i.e. within the dispatching decision.

The first strategy is concerned with the role of set-ups in scheduling jobs on one or more machines to optimize certain objectives. Although different scheduling problems dependent on set-up times have been investigated in the literature, e.g. [START_REF] Allahverdi | A review of scheduling research involving set-up considerations[END_REF][START_REF] Allahverdi | A survey of scheduling problems with set-up times or costs[END_REF][START_REF] Cheng | A review of flowshop scheduling research with set-up times[END_REF][START_REF] Liu | Scheduling flexible flow shops with sequencedependent setup effects[END_REF][START_REF] Norman | Scheduling flowshops with finite buffers and sequence-dependent setup times[END_REF], few studies have been reported on dynamic jobbing and flow-shops with sequence dependent set-up times. Two examples are the works by [START_REF] Kim | Impact of sequence-dependent set-up time on job shop scheduling performance[END_REF] and recently by [START_REF] Vinod | Simulation-based metamodels for scheduling a dynamic job shop with sequence-dependent set-up times[END_REF]. Both studied dynamic job shops with sequence-dependent set-ups using computer simulation. Kia et al. (2009) have also recently investigated dispatching rules for sequence-dependent setup times in a dynamic flexible flow line. These studies showed that set-up-oriented dispatching rules were very effective on improving shop performance, when compared with ordinary rules such as shortest processing time (SPT) or first-in-firstout (FIFO). The difference in performance between these two groups of rules, ordinary and set-up-oriented, was emphasised as shop load and set-up to processing time ratio increased.

The second strategy is concerned with the role of set-ups in decision making at the higher planning levels of the WLC system. Until now, this topic has hardly received attention in the literature. A remarkable exception is the work of Missbauer 1997), which examined the functional relationship between work-in-process (WIP) and total set-up time, in order to establish the suitable level of WIP on the shop floor.

This paper reports an investigation into the implications of sequencedependent set-up times in decision making at the order release level of a workload controlled make-to-order flow-shop. In particular, it attempts to show if orders should be sequenced in the pre-shop pool or on the shop floor in order to reduce the number of set-ups and improve system performance. As long as the avoided set-up time is greater than the time the orders wait in the pre-shop pool, due to set-up based order release, time in system of jobs is likely to be reduced. However, the objective of workload balancing within the release decision, required by the WLC approach, may conflict with the strategy of reducing set-ups. The impact of this on the shop performance is here evaluated through a simulation study. The results of the study will contribute for better decision in choosing between the two above referred control strategies, to deal with sequence-dependent set-up times in practice.

The remainder of this paper is organized as follows. Section 2 presents and discusses the experimental design of the simulation study referred. Section 3 is focused on the analysis of the results from simulation experiments and in Section 4 concluding remarks and directions for future research work are put forward.

Simulation Study

Simulation model and production system configuration

To investigate the effects of the two alternative control strategies discussed in the previous section, a simulation study using Arena ® software [START_REF] Kelton | Simulation with arena, 3 rd Ed[END_REF]), was set-up. A dynamic flow-shop is considered under the following assumptions:

1. The shop has six machines, M1 to M6, all equal in terms of capacity.

2. Each job has six operations each of one processed on each of the six machines in the same order, starting on machine M1 and ending on machine M6.

3. A machine can only perform one operation at a time on any job and an operation of a job can be performed by only one machine at a time.

4. Each machine is continuously available and there are no breakdowns.

5. Operations are processed without pre-emption.

6. Job processing cannot be started at a machine before it is finished at the previous one. 7. The transportation time between machines is assumed to be zero.

8. Set-up time of each job on each machine is sequence-dependent. 9. Each machine has a limited buffer capacity, i.e. a limit to the workload allowed to be released to the machine; nevertheless, no restriction is imposed to the movement of released jobs from a machine to the next, after processing.

10. Orders arrive continuously to the production system. By production system we mean the pre-shop pool plus the shop floor.

Due dates of orders are set externally and known upon arrival. Four types of jobs are considered, each of which with an equal probability of being assigned to an arriving order. The inter-arrival time of orders follows a negative exponential distribution, with a mean that results in a machine utilisation rate of 90% when ordinary rules are used for both, order release and dispatching (section 2.2). The mean inter-arrival time (ν) of orders, is given by the following equation [START_REF] Yu | Bio-inspired scheduling for dynamic job shops with flexible routing and sequence-dependent set-ups[END_REF]Ram 2006, Vinod and[START_REF] Vinod | Simulation-based metamodels for scheduling a dynamic job shop with sequence-dependent set-up times[END_REF]:

. .

p g m µ µ ν = U (1)
Where µ p is the mean time per operation, including set-up, µ g is the mean number of operations per order, U is the shop utilization and m is the number of machines in the shop.

Orders arrive at the production system (see figure 1) over time and flow directly into the pre-shop pool. At release time t, orders in the pool are selected for eventual release according to a priority rule i.e. the selection rule for release. An order is released only if, as a consequence of this, the accounted workload of each machine in its routing does not exceed its workload norm. If one or more workload norms are exceeded the order must wait in the pool until, at least, the next releasing period. Once an order is released the workload of each machine in the order routing is updated with the workload contribution of the selected order. This procedure is repeated until all orders in the pool, at release time t, have been considered for release.

Therefore, only a subset of orders currently waiting in the pool is released each time order release is activated, i.e. at release time t.

[Insert figure 1]

Different methods to update workload at machines upon order release are presented in literature. [START_REF] Breithaupt | The workload control concept: theory and practical extensions of Load Oriented Order Release[END_REF] make a review of these methods. The adjusted aggregated load method, which has been shown to perform well in flowshops [START_REF] Oosterman | The influence of shop characteristics on workload control[END_REF], is adopted in this work. The underlying idea of this method is that the accounted workload of an operation to a machine is a function of the sequencing position of the operation on the orders' routing. The workload contribution is obtained by dividing the order operation time in a machine by the operation position number in the orders' routing and added up to the machine accounted workload.

After release, priority dispatching rules are used to control the progress of the (1994) consider that 20% represents a realistic set-up time and provides an environment that will differentiate the performance of sequencing rules without giving undue advantage to set-up-oriented rules. Jobs of the same type can be processed with the same machine setting, i.e. no set-ups are required for the same type of job. (or time) of an order is determined by backward scheduling from the due date using the planned lead times in all machines of the orders' routing. These were established through some pilot simulation runs. SLRD, on the other hand, selects an order for release which is of same type of the order that had just been released. When there is no such an order, another is selected using LRD rule.

Experimental Design

The releasing period length (T) determines the time interval between order release activations, i.e. between releasing times and therefore determines the releasing frequency. The value of T influences the amount and of work that is released into the shop each time order release is activated. For T equal to zero the continuous timing convention is in place and for T greater than zero we say that a discrete time convention is applied [START_REF] Bergamaschi | Order review and release strategies in a job shop environment: A review and a classification[END_REF]. Using the former, order release may occur at any time during the system operation; Using the latter order releases may occur only at periodic intervals of length T. [START_REF] Land | Parameters and sensitivity in workload control[END_REF] explains that the choice of an appropriate period between releases is a delicate decision. A long release period results in increased opportunities to find orders in pool that fit workload norms, and therefore may lead to a better load balancing. However, it also delays orders in the pool -on average an order has to wait T/2 before being released into the shop floorwhich may increase the time jobs spend in the system.

The releasing period T was tested at seven levels, namely: 0, 2, 4, 6, 8, 10 and 20 hours. These different levels allow us to understand the influence of the release period and were chosen after a previous preliminary study for obtaining enough points to represent the pattern of change of performance curves.

Workload norms (WLN) levels are deterministic parameters, setting the maximum workload that can be released from the pool to each machine and, therefore, setting the maximum shop workload. To determine the best performing WLN levels it is common practice in simulation studies (e.g. Thuerer et al. 2009[START_REF] Henrich | Semi-interchangeable machines: implications for workload control[END_REF][START_REF] Land | Parameters and sensitivity in workload control[END_REF][START_REF] Oosterman | The influence of shop characteristics on workload control[END_REF][START_REF] Land | The performance of workload control concepts in job shops: Improving the release method[END_REF] assuring that the release of orders to the shop floor was not constrained by load limits.

In such a situation, immediate release under the continuous timing convention and unrestricted periodic release under the discrete timing convention, takes place. Then WLN levels were stepped down, in order to gradually increase the level of restriction to the release of orders i.e. from unrestricted to highly restrictive values. Nine WLN levels were tested. Since machines have identical characteristics, i.e. utilization, operation processing times, stream of arriving orders and average flow times, WLN levels were set identical for all machines. According to [START_REF] Oosterman | The influence of shop characteristics on workload control[END_REF] the adjusted aggregated load method, which is used in this study, allows for identical workload norm levels, independently of the machines' position within the order's routing.

Performance measures

The primary measure of the system performance is time in system. Time in system is the time an order or job spends waiting in the pre-shop pool plus the shop flow time. It provides a measure of the speed of the jobs through the whole system and is directly related to the percentage of late jobs. Reducing time in system has a beneficial impact on reducing the overall response time to customers.

Shop flow time is also recorded. This refers to the time that elapses between job release and job completion. It helps evaluating the performance of the shop floor operation. Reducing the shop flow time has also intrinsic benefits. In particular, reduces WIP and, therefore, minimizes the costs of tied-up capital.

As an indicator of timing performance, the standard deviation of the job lateness is used. It indicates how close the completion times of jobs are to their planned due dates. The mean job lateness was also recorded.

Experimental Results

During simulation runs, data were collected under system steady-state. The length of each run was for 125,000 simulated hours including a warm-up period of 25,000 hours. The average values of 100 independent replications are presented as results. The statistical analysis was performed using the paired Student t-test with a 95% confidence level. [Insert Table 2] Figure 2 shows time in system performance for each control strategy under continuous timing convention, i.e. the releasing period length is set to zero (T = 0).

This means that an order release may occur at any time during system operation. A point on a curve is the result of simulating a control strategy, i.e. A1, A2 or A3, at a specific workload norm level. Series of simulations experiments with decreasing workload norms levels, from unrestricted to highly restricted levels, were performed.

Thus, time in system performance is indicated for different levels of workload norm tightness. Note that the shop flow time is used as an instrumental variable that indicates the level of tightness of the WLN: the lower the shop flow time, the higher the WLN tightness. In this figure, time in system is plotted against shop flow time,

showing the relative performance of the control strategies.

[Insert figure 2]

Performance curve of strategy A1 is based on the use of ordinary rules at both release and dispatching decisions. It uses the LRD rule at release and the FIFO rule at dispatching. The curve has its right end point at a shop flow time of 27.9 hours. This is the result of releasing jobs immediately to the shop floor, i.e. jobs do not wait in the pre-shop pool. Tighter WLN, first leads to slightly lower values of time in system and after a certain point, i.e. the point of minimum time is system, represented by a square mark on the curve, time in system increases markedly. The minimum value of time in system is achieved for a shop flow time of 26.9 hours. Since time in system is the sum of the pool time and the shop flow time, this means that waiting times in the pool are increasing more than waiting times on the shop floor are decreasing. Thus, to avoid deterioration of time in system, WLN cannot be set excessively tight. Table 3 shows the simulation results with the 95% confidence intervals on the mean, for the two following "points":

1. the right end point of each strategy, representing an non-controlled release situation that results from unrestricting WLN. 2. the point of minimum time in system of each strategy, representing the minimum time in system that results from an appropriate WLN level (found empirically).

Note that for performance curve of strategy A3, the right end point and the point of minimum time in system are coincident and refers to the unrestricted WLN level.

The mean job lateness is also recorded in Table 3. It was observed, through pilot simulation runs, that the behaviour of time in system is very similar to that of mean job lateness, i.e. good results in terms of time system mean good results in terms of mean job lateness.

[Insert Table 3]

Performance curve A2 and A3 are based on the use of set-up-oriented rules for order release (A2) and for order dispatching (A3). Figure 2 shows that the right end points of A1 and of A2 are coincident. This happens because under immediately release differences in strategies are expressed only at dispatching. Since A1 and A2 use the same dispatching discipline, i.e. FIFO, the same performance value would be expected, as observed. Under immediately release, the right end point of A3 is placed rather differently in the figure, as expected, because it uses a different dispatching rule, i.e. SIMSET. This explains the behaviour of the three strategies at these extreme points.

Tightening of WLN, under A3, results in deterioration of time in systems due to fewer opportunities for set-up savings. Nevertheless, at each level of norm tightness, control strategy A3, based on set-up-oriented dispatching, clearly outperforms control strategy A1, based on FIFO dispatching, for time in system measure. Results show that set-up-oriented control strategies are very effective on improving system performance, i.e. time in system. This is in line with previous findings, e.g. [START_REF] Kim | Impact of sequence-dependent set-up time on job shop scheduling performance[END_REF]. It is also noteworthy that strategies A1 and A2 have an approximate behaviour in terms of the StDev of lateness. In fact, the timing performance of set-up oriented releasing (A2) is similar to, or better than, the timing performance of strategy A1, across the whole range of norm tightness. Next, based on figures 4 and 5, we compare control strategies A1 to A3 under the influence of the release period length, for the two "points" above referred. Thus, figure 4 shows time in system for different releasing periods under unrestricted WLN and figure 5 shows the time in system for different releasing periods for the WLN level that results in the minimum time in system. In both figures we can observe that performance curves A2 and A3 cross each other, meaning that the releasing period length, which determines the releasing frequency, influences the relative performance of the control strategies. It is also shown that strategies A2 and A3 always perform better than strategy A1. It is worth pointing out here that it has been observed in previous studies (e.g. [START_REF] Land | Parameters and sensitivity in workload control[END_REF], that under unrestricted WLN an increase in the releasing period leads to an increase in the average time in system. This is in fact shown in figure 4 for control strategies A1 and A3. However, for control strategy A2, this is verified only for long releasing period lengths. This happens because a moderate increase of the release period length favours set-up savings, which result in a decrease of the time in system. However, increasing the release period length above a certain value, set-up savings no longer compensate for the pool delay of orders and time in system starts to increase.

[Insert figures 4 and 5]

Results show that, particularly, for large release period lengths or for situations with low levels of workload on the shop floor, set-up oriented dispatching does not seems to be as effective as set-up oriented releasing, in terms of time in system performance. In situations of non-controlled workload or looser WLN, set-up oriented dispatching becomes attractive in terms of the time in system performance, but performs comparatively worse in terms of the timing performance, measured by the standard deviation of lateness. This leads to conclude that the answer to our research question is not independent from the level of workload in the shop, the release period length and performance measure considered.

Conclusions

Sequence-dependent set-up times may lead to major set-up savings if appropriate production control strategies are used. This can have a major influence on manufacturing system performance. This paper reports a simulation study of such influence in the scope of the Workload Control concept. Results show that the release frequency, as function of the release period length, and the shop workload are critical for the performance of the control strategies studied. Therefore, they need to be conveniently tuned in the order release procedure to be adopted. Results also show that, considering set-up times locally within the dispatching decision, as traditionally has been done, does not always result in the best performance. The degree of influence of set-up times on the control strategies performance may certainly vary with the set-up time values adopted or necessary in real situations. This, however, was not object of study in this research.

Future research work will explore other case study settings, including the influence of set-up time to processing ratio, and seek to make adjustments to the traditional release procedures in order to account for sequence-dependent set-up times in a more effective manner.

F

Figure 2

 2 Figure 2 also shows that, as WLN becomes tighter, performance curves A2 and A3 cross each other. This means that the level of norms' tightness influences the decision for considering set-up time centrally or locally. Loose WLN, lead to a high level of shop floor WIP, i.e. longer queues of jobs, which, in turn, result in increased opportunities for set-up-oriented dispatching towards optimization of job sequencing with respect to set-ups. Thus, when set-up-oriented rules are used within the dispatching decision, curve A3, the lowest total set-up time and the lowest time in

[

FiguresFigure 1 :

 1 Figures

Figure 2 :Figure 3 :

 23 Figure 2: Time in system performance for different control strategies.

Figure 4 :Figure 5 :

 45 Figure 4: Time in system at unrestricted workload norms for different release period lengths.

  jobs through the shop floor. Operations processing times are stochastic following a 2-Erlang distribution with a mean of 0.75 hours per job. Set-up times are deterministic

and equal to 0.15 hours (i.e. 20% of the mean operation processing time). For simplicity, we assume the same set-up time for each job type. In the literature, set-up times have been typically set between 20% and 40% of mean processing time

(e.g. kia et al. 2009[START_REF] Vinod | Simulation-based metamodels for scheduling a dynamic job shop with sequence-dependent set-up times[END_REF][START_REF] Kurz | Comparing scheduling rules for flexible flow lines[END_REF]

.

Kim and Bobrowski

Table 1

 1 finished to be processed on the same machine. When there is no such a job, another is selected using the FIFO rule.Two selection rules for releasing are considered: the ordinary Latest Release Date (LRD) rule and the set-up oriented Similar set-up and Latest Release Date (SLRD) rule. According to the LRD the highest releasing priority is given to the more urgent order i.e. that which has the lowest latest release date. The latest release date

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summarises the four experimental factors and associated levels studied: (1) dispatching rule; (2) selection rule for releasing; (3) releasing period length; and (4) workload norm levels.

[Insert table 1]

Two types of dispatching rules were tested on the shop floor: the ordinary FIFO and the set-up-oriented SIMilar Set-up (SIMSET). Since WLC reduces the length of queues on the shop floor, it has been suggested in the literature

[START_REF] Bechte | Theory and practice of load-oriented manufacturing control[END_REF]

that WLC allows for the use of a simple dispatching rule such as FIFO. With this rule jobs are processed in the order they arrive at a machine, i.e. the highest priority is given to the job which is waiting most in queue. No consideration is given to set-up time savings. SIMSET, on the other hand, gives the highest priority to the job with the smallest set-up time, i.e. selects a job of the type of the one that just

Table 2

 2 relevant for this study, and therefore was not considered. The strategies have different implications for shop floor control and performance. While control strategy A1 gives no importance to savings in set-up time, control strategy A2 considers set-up times within the order release decision, and control strategy A3 considers them within the dispatching decision.

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shows control strategies A1 to A3 that result from combining selection rules for order releasing with dispatching rules. One of such combinations is not

Table 1 .

 1 Experimental factors and levels.

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Table 2 .

 2 Control strategies by combining dispatching and selection rules for releasing.

	Control strategy	Shop flow time	Time in system	Mean lateness	StDev of lateness
	A1 (unrestricted WLN)	27.899 (±0.10)	27.899 (±0.10)	3.402 (±0.10)	15.469 (±0.10)
	A1 (WLN=10 hours)	26.857 (±0.09)	27.853 (±0.12)	3.354 (±0.12)	15.204 (±0.10)
	A2 (unrestricted WLN)	27.899 (±0.10)	27.899 (±0.10)	3.402 (±0.10)	15.469 (±0.10)
	A2 (WLN=5 hours)	22.007 (±0.04)	25.037 (±0.07)	0.538 (±0.07)	16.335 (±0.14)
	A3 (unrestricted WLN)	22.581 (±0.07)	22.581 (±0.07)	-1.920 (±0.07)	27.305 (±0.15)

Table 3 .

 3 Control strategies performance results.

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