Michael Goldman

SELF-SIMILAR MINIMIZERS OF A BRANCHED TRANSPORT FUNCTIONAL

We solve here completely an irrigation problem from a Dirac mass to the Lebesgue measure for a two dimensional analog of a functional previously derived in the study of branched patterns in type-I superconductors. The minimizer we obtain is a self-similar tree.

Introduction

For a measure µ on R × (a, b) such that µ = µ t ⊗ dt with µ t = i ϕ i δ X i for a.e. t ∈ (a, b) for some (pairwise distinct) X i ∈ R, we consider the branched transportation type functional (see Section 2 for a more precise definition)

(1.1) E(µ) := b a ♯{ϕ i = 0} + i ϕ i | Ẋi | 2 dt,
where Ẋi denotes the time derivative of X i (t). This is indeed a branched transportation problem since by the Benamou-Brenier formula [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF], the second term is exactly the length of the curve t → µ t measured in the Wasserstein metric, while the first term forces concentration and thus branched structures.

Our main focus is the irrigation problem of the Lebesgue measure from a Dirac mass. By this we mean that we want to minimize the cost (1.1) under the condition that the starting measure µ a is a Dirac mass and that µ t converges (weakly) to the Lebesgue measure as t goes to b. This implies that the measure µ t must infinitely refine as t → b. We will also be interested in the case when both the initial and final measures are the Lebesgue measure. More generally, for two given measures µ ± of equal mass, we study the following Dirichlet problem

(1.2) min µ {E(µ) : µ a = µ -, µ b = µ + } ,
where the boundary condition is understood in the sense that µ t ⇀ µ -as t → a and µ t ⇀ µ + as t → b.

Our main result is a full characterization of the minimizers of (1.2) in the case µ -is a Dirac mass, µ + is the Lebesgue measure restricted to an interval of length µ -(R) and b-a is large enough. In order to fix notation, since the problem is invariant by translations, we may assume that a = 0, b = T , µ -= ϕδ X and µ + = dx [-ϕ/2, ϕ/2] for some T, ϕ > 0 and 1991 Mathematics Subject Classification. 49N99, 35B36.

Figure 1. The optimal configuration µ * X ∈ R. As will be apparent below, up to rescalings and shears, we may further normalize to X = 0 and ϕ = 1, so that

µ 0 = δ 0 and µ T = dx [-1/2, 1/2].
In this case, as will become clearer in the proof, the threshold value T = 1/4 naturally appears. In order to state our main theorem, let us define for t ∈ [0, 1/4], the dyadically branching measure µ * t (see Figure 1). For k ≥ 0, let t k := 1 4 1 -1 2 3k/2 be the branching times. We define recursively µ * t in the intervals [t k-1 , t k ]. Let X 0 1 = 0 and µ * 0 = δ 0 . Assume that µ * t is defined in [0, t k-1 ] and that µ * t k-1 = 2 -(k-1) 2 k-1 i=1 δ X k-1 i . For t ∈ [t k-1 , t k ] and 1 ≤ i ≤ 2 k , we now define X k i (t). For this, let us divide [-1/2, 1/2] in 2 k intervals of equal size and let X k i be the barycenter of the i-th such interval i.e. X k i := -1 2 + i-1 2 k + 1 2 k+1 . We then let

X k i (t) := t -t k-1 1 4 -t k-1 X k i -X k-1 ⌈i/2⌉ + X k-1 ⌈i/2⌉ ,
and µ * t := 2 -k 2 k i=1 δ X k i (t) . Notice that with this definition, for every t ∈ [0, 1/4), k ∈ N and 1 ≤ i ≤ 2 k , the mass at X k i (t) is irrigating the interval (X

k i -2 -(k+1) , X k i + 2 -(k+1)
) and X k i is moving at constant speed towards X k i (and would reach it at time T = 1/4 if there were no further branching points). Our main theorem is the following Theorem 1.1. For T = 1/4, µ 0 = δ 0 and µ T = dx [-1/2, 1/2], µ * is the unique minimizer of (1.2). Moreover, if T ≥ 1/4, the unique minimizer of (1.2) is given by µ t = δ 0 for t ∈ [0, T -1/4] and µ t = µ * t-(T -1/4) for t ∈ (T -1/4, T ), with

As a consequence, we obtain the following corollary (see Lemma 3.1 for the exact definitions of the rescaling and shear) Corollary 1.2. For X ∈ R, T, ϕ > 0 with T ϕ -3/2 ≥ 1/4, the unique minimizer of (1.2) with µ 0 = ϕδ X and µ T = dx [-ϕ/2, ϕ/2] is given by a suitably sheared and rescaled version of the optimal measure for X = 0, ϕ = 1 and T = T ϕ -3/2 . Moreover

E(µ) = ϕ 3/2 1 2 - √ 2 + T + ϕ T |X| 2 .
As an application of Corollary 1.2, we will further derive a full characterization of symmetric (with respect to t = 0) minimizers in the case a = -b = -T , µ ± = dx [-1/2, 1/2] and T ≥ 1/4 (see Theorem 4.1).

The proof of Theorem 1.1 is based on the tree structure of the minimizers of (1.2) (see Proposition 2.7) which together with invariance by scaling and shearing (Lemma 3.1) leads to a recursive characterization of the minimizers. Indeed, if we let

E(T ) := min{E(µ) : µ 0 = δ 0 , µ T = dx [-1/2, 1/2]}, then (see (3.4)) (1.3) E(T ) = min N i=1 ϕ i =1 N i=1 ϕ 3/2 i E(T ϕ -3/2 i ) + 1 12T 1 - N i=1 ϕ 3 i .
This formula reflects the fact that if at level T , the minimizer branches into N pieces of respective masses ϕ i then up to rescaling and shearing, each of the subtrees solves the exact same problem as the original one (connecting a Dirac mass to the Lebesgue measure). In particular, if we define T * to be the first branching time, meaning that if T > T * then the minimizer of E(T ) cannot branch for a time T -T * , we may use that T * ϕ

-3/2 i ≥ T * to obtain E(T * ϕ -3/2 i ) = E(T * ) + T * (ϕ -3/2 i -1)
and then rewrite (1.3) in the purely analytical form (see (3.13))

(1.4)

E(T * ) -T * T * = min N i=1 ϕ i =1 (N -1) + 1 12T 2 * 1 -N i=1 ϕ 3 i 1 -N i=1 ϕ 3/2 i .
The idea of the proof of Theorem 1.1 is to use (1.4) to prove that T * = 1/4 and that the corresponding minimizer has exactly two branches of mass 1/2 at time zero. Once this is proven, the conclusion is readily reached thanks to the recursive nature of the problem.

In order to prove that T * = 1/4 and N = 2, we introduce for fixed N ≥ 2 the quantity

α N := inf ϕ i ≥0 1 -N i=1 ϕ 3 i 1 -N i=1 ϕ 3/2 i : N i=1 ϕ i = 1 .
By (1.4), if at time T * the minimizer has N branches then since for N i=1 ϕ i = 1 there holds 1

-N i=1 ϕ 3/2 i ≤ 1 -N -1/2
, we have the lower bound

E(T * ) -T * T * ≥ N -1 1 -N -1/2 + α N 12T 2 * .
In Proposition 3.11, we use this together with an upper bound on E(T * ) given by a dyadically branching construction to obtain both that T * ≤ 1/4 and that a lower bound on α N gives a corresponding upper bound on N (see (3.15)). These lower bounds on α N are obtained in Lemma 3.13 using a computer assisted proof whose details are given in Appendix A. This excludes that N ≥ 3. The case N = 2 is finally studied in Proposition 3.14 where we prove that T * = 1/4 and that the mass splits in half.

The variational problem (1.2) may be seen as a two dimensional (one for time and one for space) analog of the three dimensional (one for time and two for space) problem derived in [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF] as a reduced model for the description of branching in type-I superconductors in the regime of very small applied external field. We refer the reader to [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF] for more precise physical motivations and references. In this regime, the natural Dirichlet conditions appearing are µ ± = dx [-1/2, 1/2]. Let us point out that in the three dimensional model, the term ♯{ϕ i = 0} is replaced by i ϕ 1/2 i . This is in line with the interpretation of the first term in (1.1) as an interfacial term penalizing the creation of many flux tubes. That is, if we are in (1 + d)-dimensions, it is proportional to the perimeter of a union of d-dimensional balls of volume ϕ i (which is 2

√ π i ϕ 1/2 i if d = 2 and 2♯{ϕ i = 0} if d = 1
). As already alluded to, the second term in (1.1) may be interpreted as the Wasserstein transportation cost [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF] of moving such balls. In many models describing pattern formation in material sciences, branching patterns similar to the one observed here are expected. However, it is usually very hard to go beyond scaling laws [START_REF] Kohn | Surface energy and microstructure in coherent phase transitions[END_REF][START_REF] Zwicknagl | Microstructures in low-hysteresis shape memory alloys: Scaling regimes and optimal needle shapes[END_REF][START_REF] Choksi | Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy[END_REF][START_REF] Conti | Branched microstructures in the Ginzburg-Landau model of type-I superconductors[END_REF]. In some cases, reduced models have been derived [START_REF] Otto | Domain branching in uniaxial ferromagnets: asymptotic behavior of the energy[END_REF][START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF][START_REF] Conti | Deformation concentration for martensitic microstructures in the limit of low volume fraction[END_REF] but so far the best results concerning the minimizers are local energy bounds leading to the proof of asymptotic self-similarity [START_REF] Conti | A lower bound for a variational model for pattern formation in shape-memory alloys[END_REF][START_REF] Viehmann | Uniaxial Ferromagnets[END_REF]. Our result is thus the first complete characterization of a minimizer in this context. Of course, this was possible thanks to the simplicity of our model (one dimensional trees in a two dimensional ambient space). We should however point out that our result is not fully satisfactory since we are essentially able to study only the situation of an isolated microstructure (due to the constraint T ≥ 1/4) whereas one is typically interested in the case T ≪ 1 where many microstructures are present and where the lateral boundary conditions have limited effect i.e. one tries to capture an extensive behavior of the system. As detailed in the final section 4, we believe that even in the regime T ≪ 1, every microstructure is of the type described in Corollary 1.2.

As pointed out in [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF], the functional (1.1) bears many similarities with so-called branched transport (or irrigation) models [START_REF] Xia | Interior regularity of optimal transport paths[END_REF][START_REF] Maddalena | A variational model of irrigation patterns[END_REF][START_REF] Bernot | Optimal transportation networks[END_REF] (see also [START_REF] Brancolini | Path functionals over Wasserstein spaces[END_REF][START_REF] Brasco | An equivalent path functional formulation of branched transportation problems[END_REF] for a formulation reminiscent of our model). Also in this class of problems, there has been a strong interest for the possible fractal behavior of minimizers. Besides results on scaling laws [START_REF] Brancolini | Optimal energy scaling for micropatterns in transport networks[END_REF] and fractal regularity [START_REF] Brancolini | Fractal regularity results on optimal irrigation patterns[END_REF], to the best of our knowledge, the only explicit minimizers exhibiting infinitely many branching points have been obtained in [START_REF] Paolini | An example of an infinite steiner tree connecting an uncountable set[END_REF] for a Dirac irrigating a Cantor set and in [START_REF] Marchese | An optimal irrigation network with infinitely many branching points[END_REF] in an infinite dimensional context. In particular, the optimal irrigation pattern from a Dirac mass to the Lebesgue measure is currently not known for the classical branched transportation model. One important difference between our model and branched transportation is that in our case, minimality does not imply triple junctions nor conditions on the angles between the branches.

The organization of the paper is the following. In Section 2, we recall the definition and the basic properties of the functional E. Then, in Section 3 we prove Theorem 1.1. In the final Section 4, we give an application of Theorem 1.1 to the irrigation of the Lebesgue measure by itself and state an open problem. Appendix A contains the computer assisted computation of α N .

Notation In the paper we will use the following notation. The symbols ≃, , , ≪ indicate estimates that hold up to a global constant. For instance, f g denotes the existence of a constant C > 0 such that f ≤ Cg. We denote by H 1 the 1-dimensional Hausdorff measure. For a Borel measure µ, we will denote by supp µ its support.

Acknowledgment

I warmly thank F. Otto for many useful discussions and constant support as well as A. Lemenant for a careful reading of a preliminary version of the paper. The hospitality of the Max Planck Institute for Mathematics in the Sciences of Leipzig, where part of this research was carried out is gratefully acknowledged. Part of this research was funded by the program PGMO of the FMJH through the project COCA.

The variational problem and main properties of the functional

In this section we first give a rigorous definition of the energy E(µ) and then prove that the boundary value problem (1.2) has minimizers which are locally given by finite union of straight segments so that the representation (1.1) makes sense. Definition 2.1. For a < b we denote by A a,b the set of pairs of measures µ ≥ 0, m with m ≪ µ, satisfying the continuity equation

(2.1) ∂ t µ + ∂ x m = 0 in R × (a, b)
and such that µ = µ t ⊗ dt where, for a.e.

t ∈ (a, b), µ t = i ϕ i δ X i for some ϕ i ≥ 0 and X i ∈ R. We denote by A * a,b := {µ : ∃m, (µ, m) ∈ A a,b } the set of admissible µ. Further, we define E : A a,b → [0, ∞] by (2.2) E(µ, m) := b a ♯{supp µ t }dt + R×(a,b) dm dµ 2 dµ
and (with abuse of notation)

E : A * a,b → [0, ∞] by (2.3) E(µ) := min{E(µ, m) : m ≪ µ, ∂ t µ + ∂ x m = 0}.
Equation (2.1) is understood in the sense of distributions (testing with test functions in C ∞ c (R × (0, T ))). Contrary to [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF], we use free boundary conditions instead of periodic ones but this makes only minor differences. In the sequel we will only deal with measures µ of bounded support. In this case, because of (2.1), µ t (R) does not depend on t. Let us point out that for such measures, the minimum in (2.3) is attained thanks to [1, Th. Let us also notice that by the Benamou-Brenier formula [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], we have for every measure µ, and every t, t ′ ∈ (a, b),

(2.4) W 2 2 (µ t , µ t ′ ) ≤ E(µ)|t -t ′ |
, where the 2-Wasserstein distance between two measures µ and ν of bounded second moment with µ(R) = ν(R) is defined by

W 2 2 (µ, ν) := min R×R |x -y| 2 dΠ(x, y) : Π 1 = µ, Π 2 = ν ,
where the minimum is taken over measures on R × R and Π 1 and Π 2 are respectively the first and second marginal of Π. In particular for every measure µ with E(µ) < ∞, the curve t → µ t is Hölder continuous with exponent one half in the space of measures (endowed with the metric W 2 ) and the traces µ a and µ b are well defined.

Given two measures µ ± on R with µ + (R) = µ -(R) and bounded support, we are interested in the variational problem

(2.5) inf {E(µ) : µ a = µ -, µ b = µ + } .
Let us first notice that if L > 0 is such that supp µ -∪ supp µ + ⊆ [-L/2, L/2], then we may restrict the infimum in (2.5) to measures satisfying supp µ t ⊆ [-L/2, L/2] for a.e. t ∈ (a, b). Indeed, if µ is admissible with µ t = i ϕ i δ X i then letting X i := min(L/2, |X i |)signX i and then µ t := i ϕ i δ X i , we get that µ is admissible and has lower energy than µ (i.e. the energy decreases by projection on [-L/2, L/2]). From now on we will only consider such measures.

As in [10, Prop 5.2] (to which we refer for the proof), a simple branching construction shows that any pair of measures with equal flux may be connected with finite cost.

Proposition 2.2. For every pair of measures µ

± with supp µ ± ⊆ [-L/2, L/2] and µ + (R) = µ -(R) = ϕ, there is µ ∈ A * a,b such that letting b -a = 2T , µ a = µ -, µ b = µ + and E(µ) T + ϕL 2 T .
If µ + = µ -, then there is a construction with

E(µ) T + T 1/3 ϕ 1/3 L 2/3 .
From this, arguing as in [10, Prop. 5.5], we obtain that Proposition 2.3. For every pair of measures µ ± with bounded support and µ + (R) = µ -(R), the infimum in (2.5) is finite and attained.

We now give some regularity results for minimizers of (2.5). These can be mostly proven as in [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF] so we state them without proof. Let us first recall the notion of subsystem.

Proposition 2.4 (Definition of a subsystem). Given a point (X, t) ∈ [-L/2, L/2] × (a, b) and µ ∈ A *
a,b with E(µ) < ∞, there exists a subsystem µ ′ of µ emanating from (X, t). By this we mean that there exists µ ′ such that

(i) µ ′ ≤ µ i.e. µ -µ ′ is a positive measure, (ii) µ ′ t = aδ X , where a = µ t (X), (iii) if m is such that E(µ) = E(µ, m), then ∂ t µ ′ + ∂ x dm dµ µ ′ = 0.
In particular, (ii) implies that (µ t -µ ′ t ) ⊥ δ X in the sense of the Radon-Nikodym decomposition. We call µ + := µ ′ R × (t, b) the forward subsystem emanating from (X, t) and µ -:= µ ′ R × (a, t) the backward subsystem emanating from X. Lemma 2.5 (No loops). Let µ be a minimizer for the Dirichlet problem (2.5), t ∈ (a, b). Let X 1 , X 2 be two points in the line {(x, t) : t = t}. Let µ 1 and µ 2 be subsystems of µ emanating from (X 1 , t), resp. (X 2 , t). Let (X + , t + ) be a point with t + > t and (X -, t -) a point with t -< t, and such that µ 1 and µ 2 both have Diracs at both X + and X -with nonzero mass. Then X 1 = X 2 .

As in [START_REF] Conti | A branched transport limit of the Ginzburg-Landau functional[END_REF], a consequence of this lemma is that we have a representation of the form

(2.6) µ = i ϕ i 1 + | Ẋi | 2 H 1 Γ i
where the sum is countable and Γ i = {(X i (t), t) : t ∈ [a i , b i ]} with X i absolutely continuous and almost everywhere non overlapping.

Another consequence is that if there are two levels at which µ is a finite sum of Diracs, then it is the case for all the levels in between. Since E(µ) < ∞ implies in particular that ♯{ϕ i = 0} < ∞ for a.e. t ∈ (a, b), this means that µ t is in fact a locally finite (in time) sum of Dirac masses and thus, the sum in (2.6) is finite away from the initial and finite time.

For measures which are concentrated on finitely many curves, we have as in [10, Lem. 5.9], a representation formula for E(µ).

Lemma 2.6. Let µ = N i=1 ϕ i √ 1+| Ẋi | 2 H 1 Γ i ∈ A * a,b with Γ i = {(X i (t), t) : t ∈ [a i , b i ]
} for some absolutely continuous curves X i , disjoint up to the endpoints. Every ϕ i is then constant on [a i , b i ] and we have conservation of mass. That is, for z := (x, t), letting

I -(z) := {i ∈ [1, N] : t = b i , X i (b i ) = x} I + (z) := {i ∈ [1, N] : t = a i , X i (a i ) = x}, there holds i∈I -(z) ϕ i = i∈I + (z) ϕ i . Moreover, m = i ϕ i √ 1+| Ẋi | 2 Ẋi H 1 Γ i and
(2.7)

E(µ) = i b i a i 1 + ϕ i | Ẋi | 2 dt.
In particular, this proves that for minimizers, formula (2.7) holds (where the sum is at most countable). By a slight abuse of notation, for such measures we will denote

E(µ) = b a ♯{ϕ i = 0} + i ϕ i | Ẋi | 2 dt.
We gather below some properties of the minimizers Proposition 2.7. A minimizer of the Dirichlet problem (2.5) with boundary conditions µ ± satisfies (i) Each X i is affine.

(ii) There is monotonicity of the traces in the sense that for every t ∈ (a, b), if µ t = i ϕ i δ X i with X i ordered (i.e. X i ≤ X i+1 ) and if µ i,+ is the forward subsystem emanating from X i , then the traces

µ i,+ b satisfy supp µ i,+ b = [x + i , y + i ] with y + i ≤ x + i+1 .
The analogous statement holds for the backward subsystems. (iii) If µ -= ϕδ X then µ has a tree structure. (iv) If µ -= µ + , then letting a = -T and b = T , there exists a minimizer which is symmetric with respect to the t = 0 plane. For every such minimizer, the number of Dirac masses at time t is minimal for t = 0.

Proof. Item (i) follows from fixing the branching points and minimizing in X i . The other points are simple consequences of Lemma 2.5.

The monotonicity property (ii), is analogous to the monotonicity of optimal transport maps in one space dimension [START_REF] Villani | Topics in optimal transportation[END_REF].

Irrigation of the Lebesgue measure by a Dirac mass

In this section we consider (1.2) with a = 0 and b = T , µ -= ϕδ X and µ + = dx [-ϕ/2, ϕ/2]. We will denote

E(T, ϕ, X) := min{E(µ) : µ 0 = ϕδ X , µ T = dx [-ϕ/2, ϕ/2]}.
For simplicity we let E(T, ϕ) := E(T, ϕ, 0) be the energy required to connect the Lebesgue measure to the centered Dirac mass and E(T ) := E(T, 1). The following lemma shows that understanding E(T ) is enough for understanding E(T, ϕ, X). Lemma 3.1. For every T, ϕ, X, there holds,

(3.1) E(T, ϕ, X) = E(T, ϕ) + 1 T ϕ|X| 2 . Moreover, if µ t = i ϕ i X i (t) is optimal for E(T, ϕ), then letting X i (t) := (1-t T )X +X i (t), μt = i ϕ i X i (t) is optimal for E(T, ϕ, X).
Furthermore, we have

(3.2) E(T, ϕ) = ϕ 3/2 E(T ϕ -3/2 ).
In addition, if µ t = i ϕ i X i (t) is optimal for E(T ϕ -3/2 ) then letting t := ϕ 3/2 t, ϕ i := ϕϕ i , and X i := ϕX i ( t), then μt = i ϕ i δ X i is optimal for E(T, ϕ).

Proof. For µ t = i ϕδ X i admissible for E(T, X), we define μt := i ϕ i δ Xi , where Xi (t) := (1 -t T )X + X i (t). Then, μt is admissible for E(T, ϕ, X) and

E(μ) = T 0 ♯{ϕ i = 0} + i ϕ i | Ẋi - 1 T X| 2 dt (3.3) = T 0 ♯{ϕ i = 0} + i ϕ i | Ẋi | 2 dt + |X| 2 T 2 T 0 i ϕ i dt -2 X T T 0 i ϕ i Ẋi dt.
For ε > 0, thanks to Lemma 2.6 and the fact that X i (0) = 0, we have

T -ε 0 i ϕ i Ẋi dt = i ϕ i X i (T -ε).
Furthermore, testing the weak convergence of µ t to dx as t → T , with the function x, we get lim

ε→0 i ϕ i X i (T -ε) = ϕ/2 -ϕ/2 xdx = 0.
Finally, since by Hölder's inequality applied twice and

i ϕ i = ϕ, T T -ε i ϕ i | Ẋi | ≤ ϕ 1/2 T T -ε ( i ϕ i | Ẋi | 2 ) 1/2 ≤ ϕ 1/2 ε 1/2 T T -ε i ϕ i | Ẋi | 2 1/2 , we get T 0 i ϕ i Ẋi dt = lim ε→0 T -ε 0 i ϕ i Ẋi dt + T T -ε i ϕ i | Ẋi |dt = 0.
Combining this with i ϕ i = ϕ and (3.3), we get

E(μ) = E(µ) + 1 T ϕ|X| 2 ,
from which the first part of the proposition follows noticing that the map µ → μ is oneto-one between admissible measures for E(T, X) and admissible measures for E(T, ϕ, X).

The second part follows simply by using the rescaling t := ϕ 3/2 t, ϕ i := ϕϕ i , and X i := ϕX i ( t).

Remark 3.2. Using the same type of rescalings as the one leading to (3.2), it is not hard to prove that T → E(T ) is a continuous function.

As a consequence of the monotonicity of the support and of the previous lemma, we can derive the following fundamental recursive characterization of E(T ). Lemma 3.3. For every T > 0, (3.4)

E(T ) = min N i=1 ϕ i =1 N i=1 ϕ 3/2 i E(T ϕ -3/2 i ) + 1 12T 1 - N i=1 ϕ 3 i ,
so that in particular the energy does not change if we reorder the ϕ i .

Proof. Let ϕ 1 , .., ϕ N be the fluxes of the branches leaving from the point (0, 0) (if it is not a branching point then N = 1). Up to relabeling, we may assume that the ϕ i are ordered i.e. ϕ 1 corresponds to the first branch, ϕ 2 to the second and so on. By the monotonicity of the traces (Proposition 2.7), the N branches are independent and the mass from the first branch will go to [-1/2, -1/2+ϕ 1 ], the second branch will go to [-1/2+ϕ 1 , -1/2+ϕ 1 +ϕ 2 ] and so on. Let X i be the centers of the intervals of length ϕ i i.e. X i = -1 2 + j<i ϕ j + ϕ i 2 . We first prove that From (3.1) and (3.2) we have

(3.5) E(T ) = min N i=1 ϕ i =1 N i=1 E(T, ϕ i , X i ) = min N i=1 ϕ i =1 N i=1 ϕ 3/2 i E(T ϕ -3/2 i ) + 1 T N i=1 ϕ i |X i | 2 ,
so that we are left to prove that for every (ϕ i ) N i=1 with N i=1 ϕ i = 1, there holds

(3.6) N i=1 ϕ i |X i | 2 = 1 12 1 - N i=1 ϕ 3 i .
We prove this by induction on N. For N = 1, there is nothing to prove. For N = 2, since ϕ 2 = 1 -ϕ 1 , the left-hand side of (3.6) is equal to

ϕ 1 -1 + ϕ 1 2 2 + (1 -ϕ 1 ) ϕ 1 2 2 = 1 4 ϕ 1 (1 -ϕ 1 ),
which is equal to the right-hand side of (3.6). Assume now that (3.6) holds for N -1. Let then ϕ = ϕ 1 + ϕ 2 and X = X 1 + ϕ 2 2 . By the induction hypothesis,

N i=3 ϕ i |X i | 2 + ϕ|X| 2 = 1 12 1 - N i=3 ϕ 3 i -ϕ 3 so that N i=1 ϕ i |X i | 2 = 1 12 1 - N i=1 ϕ 3 i + 1 12 (ϕ 3 1 + ϕ 3 2 -ϕ 3 ) + ϕ 1 |X 1 | 2 + ϕ 2 |X 2 | 2 -ϕ|X| 2 .
We are thus left to prove that

(3.7) 1 12 (ϕ 3 1 + ϕ 3 2 -ϕ 3 ) + ϕ 1 |X 1 | 2 + ϕ 2 |X 2 | 2 -ϕ|X| 2 = 0.
By definition of ϕ, X and since

X 2 = X 1 + ϕ 1 +ϕ 2 2
, we have

1 12 (ϕ 3 1 + ϕ 3 2 -ϕ 3 ) = - ϕ 1 ϕ 2 4 (ϕ 1 + ϕ 2 ) = - ϕ 1 ϕ 2 ϕ 4 .
Analogously we can compute

ϕ 1 |X 1 | 2 + ϕ 2 |X 2 | 2 -ϕ|X| 2 =ϕ 1 |X 1 | 2 + ϕ 2 |X 1 + ϕ 2 | 2 -ϕ|X 1 + ϕ 2 2 | 2 =ϕ 1 |X 1 | 2 + ϕ 2 |X 1 | 2 + ϕ 2 ϕX 1 + ϕ 2 ϕ 2 4 -ϕ|X 1 | 2 -ϕϕ 2 X 1 - ϕϕ 2 2 4 = ϕ 2 ϕ 4 (ϕ -ϕ 2 ) = ϕ 1 ϕ 2 ϕ 4 .
Adding these two equalities we get (3.7) which concludes the proof of (3.6).

Before going further, let us point out that for T, t > 0 using as test configuration for E(T + t), δ 0 in [0, t] extended by the minimizer of E(T ) in [t, T + t], we obtain

(3.8) E(T + t) ≤ E(T ) + t.
This together with (3.4), motivates the introduction of the largest branching time:

(3.9)

T * = inf{T : E(T + t) = E(T ) + t, ∀t ≥ 0}.
By definition of T * and (3.8), we see that for every ε > 0, E(T * -ε) + ε > E(T * ) which means that every minimizer of (3.4) for T = T * must have N > 1 branches (there must be branching at time zero). We will also need the following simple lemma.

Lemma 3.4. Let X ∈ R, T, ϕ > 0 with T ϕ -3/2 > T * . And let µ t be a minimizer for E(T, ϕ, X). Letting X(t) := (1 -t T )X, for t ∈ [0, T -ϕ 3/2 T * ] there holds, µ t = ϕδ X(t) . Proof. Since T ϕ -3/2 > T * , by definition of T * , in [0, T ϕ -3/2 -T * ], every minimizer of E(T ϕ -3/2 ) is of the form δ 0 . Therefore, by (3.1) and (3.2), if

X(t) := (1 -t T )X, then for t ∈ [0, T -ϕ 3/2 T * ], µ t = ϕδ X(t) .
We can now state the main result of this section. Proposition 3.5. We have T * = 1/4 and if ϕ 1 , .., ϕ N are optimal in (3.4) for T = T * , then N = 2 and ϕ 1 = ϕ 2 = 1/2. Moreover,

E(1/4) -1/4 = 1 2 - √ 2 .
The proof of this proposition will consist of the remaining part of this section. Before doing so, let us see how it implies Theorem 1.1. In the proof, we will use the following notation Definition 3.6. For µ ∈ A * a,b with µ t = i ϕ i δ X i and X ∈ R, let S X (µ) be the measure defined by (S X (µ)) t := i ϕ i δ X i +X .

Proof of Theorem 1.1. By definition of T * , for T ≥ 1/4, we have E(T ) = E(1/4)+(T -1/4) and if µ is a minimizer for E(T ), then it coincides with δ 0 in [0, T -1/4] and with (a translated version of) a minimizer for E(1/4) in [T -1/4, T ]. Therefore, it is enough to prove that for T = 1/4, the only minimizer of E(1/4) is given by µ * .

Let µ be such a minimizer and let us prove by induction that µ = µ * . Recall first that we defined

t k = 1 4 1 -1 2 3k/2 . Assume that µ t = µ * t for t ∈ [0, t k-1 ] and that µ t k-1 = 2 -(k-1) 2 k-1 i=1 δ X k-1 i for some ordered X k-1 i ∈ R. By monotonicity of the support, in [t k-1 , 1/4], each of the forward subsystems µ +,i emanating from X k-1 i must be of the form µ +,i t = S X k-1 i (µ i t-t k-1 ) where µ i is a minimizer of E( 1 4 -t k-1 , 2 -(k-1) , X k-1 i -X k-1 i ) = E( 1 4 (2 -(k-1) ) 3/2 , 2 -(k-1) , X k-1 i -X k-1 i
). By (3.2) and Proposition 3.5, every minimizer of

E( 1 4 (2 -(k-1) ) 3/2 , 2 -(k-1) , X k-1 i -X k-1 i
) must branch into two pieces of equal mass. Thus, we can further decompose µ i = µ i,1 + µ i,2 where µ i,1 = S -2 -(k+1) (ν i,1 ) with ν i,1 a minimizer for E( 14 (2

-(k-1) ) 3/2 , 2 -k , 2 -(k+1) + X k-1 i -X k-1 i
) and similarly for µ 2 i . Let

Y k 2i-1 (s) := -2 -(k+1) + (1 - s 1 4 -t k-1 )(X k-1 i -X k 2i-1
) and

Y k 2i (s) := 2 -(k+1) + (1 - s 1 4 -t k-1 )(X k-1 i -X k 2i ), Since X k-1 i -2 -(k+1) = X k 2i-1 and X k-1 i + 2 -(k+1) = X k 2i , by Lemma 3.4, for s ∈ [0, t k -t k-1 ], µ i s = 2 -k (δ Y k 2i-1 (s) + δ Y k 2i (s)
) and thus letting

X k 2i-1 (t) := t -t k-1 1 4 -t k-1 (X k 2i-1 -X k-1 i ) + X k-1 i and X k 2i (t) := t -t k-1 1 4 -t k-1 (X k 2i -X k-1 i ) + X k-1 i ,
we finally obtain as claimed that for t

∈ [t k-1 , t k ], µ +,i t = 2 -k (δ X k 2i-1 (t) + δ X k 2i (t) ).
We may start investigating the properties of T * .

Lemma 3.7. There holds 0 < T * < ∞. As a consequence, the infimum in (3.9) is attained.

Proof. We first observe that for every T > 0, by (2.4)

(3.10) E(T ) ≥ T + W 2 2 (δ 0 , dx [-1/2, 1/2]) T .
Let us prove that T * < ∞. Let T ≥ 1 and µ t be a minimizer for E(T ). By (3.8), for every

T ≥ 1, E(T ) ≤ E(1) + (T -1)
. By the no-loop condition, if µ t has its first branching at time t 0 then in [t 0 , T ] it has at least two branches and thus E(T ) ≥ 2(T -t 0 ). Putting these two inequalities together we get t 0 ≥ T -(E(1) -1). Letting T 1 := E(1) -1, which is positive by (3.10), and assuming that T ≥ T 1 , this implies that before T -T 1 , no branching may occur. Hence, for T ≥ T 1 ,

E(T ) = E(T 1 ) + (T -T 1 ), that is T * ≤ T 1 .
We now prove that T * > 0. By (3.10), if T * = 0, for every T 1 ≤ T ,

E(T ) = E(T 1 ) + T -T 1 ≥ T + W 2 2 (δ 0 , dx [-1/2, 1/2])
T 1 which letting T 1 → 0 would give a contradiction to E(T ) < ∞. The fact that the infimum in (3.9) is attained follows by continuity of T → E(T ).

The next result is a form of equipartition of energy which will be used to prove that

T * ≤ 1/4. Lemma 3.8. If ϕ 1 , .., ϕ N > 0 are such that E(T * ) = N i=1 ϕ 3/2 i E(T * ϕ -3/2 i ) + 1 12T * 1 - N i=1 ϕ 3 i , then (3.11) T * (N -1) = 1 12T * 1 - N i=1 ϕ 3 i .
Proof. By (3.6), if we denote by X i the barycenters of the intervals of length ϕ i , it is enough to prove that (3.12)

T * (N -1) = 1 T * N i=1 ϕ i |X i | 2 .
Let µ t = i ϕ i δ X i be optimal for T * . By definition of T * , the fact that T * ϕ -3/2 i > T * and Lemma 3.4, there is ε > 0 such that for t ∈ [0, ε], X i (t) = t T * X i . For ε > ε > 0, we are going to construct a competitor for E(T * -ε).

In [ε -ε, T * -ε], let Y i (t) := X i (t + ε) and in [0, ε -ε], Y i (t) := 1 T * ε ε-ε tX i so that Y i (ε -ε) = X i (ε). Therefore, E(T * -ε) ≤ E(T * ) -Nε + 1 T 2 * i ϕ i |X i | 2 (ε -ε) ε 2 (ε -ε) 2 - 1 T 2 * i ϕ i |X i | 2 ε = E(T * ) -N - 1 T 2 * i ϕ i |X i | 2 ε + o(ε 2 ).
Hence,

E(T * -ε) -E(T * ) -ε ≥ N - 1 T 2 * i ϕ i |X i | 2 + o(ε).
Using (3.8) and letting ε → 0, we get

N - 1 T 2 * i ϕ i |X i | 2 ≤ 1.
We similarly define a competitor for

E(T * + ε) by letting Y i (t) := X i (t -ε) in [ε + ε, T * + ε] and Y i (t) := 1 T * ε ε+ε tX i in [0, ε + ε] and get E(T * + ε) ≤ E(T * ) + Nε + 1 T 2 * i ϕ i |X i | 2 (ε + ε) ε 2 (ε + ε) 2 - 1 T 2 * i ϕ i |X i | 2 ε = E(T * ) + N - 1 T 2 * i ϕ i |X i | 2 ε + o(ε 2 ).
From this we infer that

E(T * + ε) -E(T * ) ε ≤ N - 1 T 2 * i ϕ i |X i | 2 + o(ε).
By definition of T * and by continuity of E,

E(T * + ε) -E(T * ) ε = 1 and therefore N - 1 T 2 * i ϕ i |X i | 2 ≥ 1,
which conclude the proof of (3.12).

Remark 3.9. Notice that (3.11) is compatible with N = 2, ϕ 1 = ϕ 2 = 1/2 and T * = 1 4 . Using the characterization (3.4), we show another characterization of E(T * ) which has the advantage of not being recursive anymore. Proposition 3.10. There holds

(3.13) E(T * ) -T * T * = min N i=1 ϕ i =1 (N -1) + 1 12T 2 * 1 -N i=1 ϕ 3 i 1 -N i=1 ϕ 3/2 i .
Moreover, if ϕ 1 , .., ϕ N are minimizers for E(T * ) then they also minimize the right-hand side of (3.13) and vice-versa.

Proof. Let ϕ i be the optimal fluxes for (3.4). By definition of T * , we have for every ϕ i with

i ϕ i = 1 (since ϕ -3/2 i ≥ 1) E(T * ) ≤ N i=1 ϕ 3/2 i E(T * ϕ -3/2 i ) + 1 12T * 1 - N i=1 ϕ 3 i = N i=1 ϕ 3/2 i (E(T * ) + (T * ϕ -3/2 i -T * )) + 1 12T * 1 - N i=1 ϕ 3 i = E(T * ) N i=1 ϕ 3/2 i + T * N i=1 (1 -ϕ 3/2 i ) + 1 12T * 1 - N i=1 ϕ 3 i . Therefore E(T * ) -T * ≤ (E(T * ) -T * ) N i=1 ϕ 3/2 i + (N -1)T * + 1 12T * 1 - N i=1 ϕ 3 i ,
and then

E(T * ) -T * T * ≤ (N -1) + 1 12T 2 * 1 -N i=1 ϕ 3 i 1 -N i=1 ϕ 3/2 i
with equality for ϕ i = ϕ i .

For N ≥ 2, we introduce a quantity which will play a central role in our analysis. Let

α N := inf ϕ i ≥0 1 -N i=1 ϕ 3 i 1 -N i=1 ϕ 3/2 i : N i=1 ϕ i = 1 .
We now prove that T * ≤ 1/4 and that a lower bound on α N gives an upper bound on N.

Proposition 3.11. There holds

T * ≤ 1 4 .
Moreover, the number N > 1 of branches of the minimizer for T * , satisfies

(3.14) √ N( √ N + 1)T * + α N 12T * ≤ √ 2 2( √ 2 -1) .
As a consequence,

(3.15) √ N ≤ 1 2 -1 + 1 + 6 α N ( √ 2 -1) 2 1/2 .
In particular, since α N ≥ 1, this gives N ≤ 6.

Proof. We start with the upper bound T * ≤ 1 4 . Let ϕ 1 , .., ϕ N be optimal for E(T * ). Then, by (3.13) and (3.11),

E(T * ) -T * T * = 2(N -1) 1 -i ϕ 3/2 i .
Using that for every ϕ 1 , .., ϕ N with

ϕ i = 1, (3.16) 1 1
-i ϕ 3/2 i ≥ 1 1 - √ N -1
we get

E(T * ) -T * T * ≥ 2(N -1) √ N √ N -1 .
Since the right-hand side is minimized for N = 2 (among N ∈ N, N ≥ 2), we have

(3.17) E(T * ) -T * T * ≥ 2 √ 2 √ 2 -1 .
We proceed further by proving an upper bound for the left-hand side of (3.17). For every T > 0, we can construct the self similar competitor for which every branch is divided into two branches of half the mass at every branching point (which is at

T k = T (1 -( 1 2 ) 3k/2
). Let E(T ) be its energy. Then, arguing as in (3.4), we have

E(T ) = 2 1 2 3/2 E(T ) + 2 T - 1 2 3/2 T + 1 16T that is ( E(T ) -T ) 1 - 1 √ 2 = T + 1 16T from which we get E(T ) -T T = √ 2 √ 2 -1 1 + 1 16T 2 .
Since by definition E(T ) ≤ E(T ), we get

(3.18) E(T ) -T T ≤ √ 2 √ 2 -1 1 + 1 16T 2 .
For T > 1 4 , the right-hand side is strictly smaller than 2 √ 2 √ 2-1 hence by (3.17), we cannot have T = T * . This gives the upper bound.

We now turn to (3.14). For this, we notice that

(3.19) E(T * ) -T * T * = N -1 1 -i ϕ 3/2 i + (1 -i ϕ 3 i ) 12T 2 * (1 -i ϕ 3/2 i ) ≥ N -1 1 -N -1/2 + α N 12T 2 * .
Since T * ≤ 1/4, by definition of T * (recall (3.9)), E(1/4) -1/4 = E(T * ) -T * , combining (3.19) with (3.18) for T = 1/4 and

N -1 1 -N -1/2 = √ N( √ N + 1),
yields (3.14). We finally derive (3.15). For this, multiply (3.14) by

T * to obtain that √ N( √ N + 1)T 2 * - √ 2 2( √ 2 -1) T * + α N 12 ≤ 0.
This implies that the polynomial

√ N( √ N + 1)X 2 - √ 2 2( √ 2-1) X + α N
12 has real roots (and that T * lies between these two roots) so that

∆ = 1 2( √ 2 -1) 2 - α N 3 √ N ( √ N + 1) ≥ 0,
which is equivalent to

( √ N ) 2 + √ N - 3 2α N ( √ 2 -1) 2 ≤ 0.
Since the largest root of this polynomial (in the variable √ N ) is given by 1 2 (-1 + (1 + Estimate (3.15) shows that by obtaining a good lower bound on α N , we may exclude that N ≥ 3. This is the purpose of the next lemma whose proof is essentially postponed to Appendix A. Lemma 3.13. For 3 ≤ N ≤ 6, (3.20)

α N > 6 ( √ 2 -1) 2 (2 √ N + 1) 2 .
As a consequence, the number of branches at T = T * equals two. We now conclude the proof of Proposition 3.5 by studying the case N = 2.

Let first

T -:= 1 4 (1 -(1 - 4 3 (2 - √ 2) 1/2 ).
Then 1/4 ≥ T * ≥ T -. Indeed, (3.14) for N = 2 (recalling that α 2 = 2) may be seen to be equivalent to

(3.21) T 2 * - 1 2 T * + 1 6 √ 2( √ 2 + 1) ≤ 0.
In particular, T * has to lie between the two roots of the right-hand side of (

E(T * ) -T * = E(1/4) -1/4 ≤ √ 2 2( √ 2 -1)
.

Using (3.13), we obtain for ϕ optimal for T * , the bound

(3.22) T * 1 + 1 4T 2 * ϕ(1 -ϕ) 1 -ϕ 3/2 -(1 -ϕ 3/2 ) ≤ √ 2 2( √ 2 -1)
.

The next proposition shows the reverse inequality which concludes the proof of Proposition 3.5. (3.23)

T 1 + 1 4T 2 ϕ(1 -ϕ) 1 -ϕ 3/2 -(1 -ϕ) 3/2 ≥ √ 2 2( √ 2 -
λ ∈ [2T -, 1/2] and ϕ ∈ [0, 1/2], (3.24) 1 + 1 λ 2 ϕ(1 -ϕ) ≥ √ 2 λ( √ 2 -1) (1 -ϕ 3/2 -(1 -ϕ) 3/2 ).
It will be more convenient to work with a := 3

√ 2λ 2( √ 2-1) . Letting a -:= 3 √ 2 √ 2 -1 T -= 3 2(2 - √ 2) (1 -(1 - 4 3 (2 - √ 2)) 1/2 ) ≃ 1.36, we are reduced to a ∈ [a -, 3 √ 2 4( √ 2-1) ]. Inequality (3.24) then reads (3.25) L(a, ϕ) := 1+ 9 2a 2 ( √ 2 -1) 2 ϕ(1-ϕ) ≥ 3 a( √ 2 -1) 2 (1-ϕ 3/2 -(1-ϕ) 3/2 ) =: R(a, ϕ).
Let us first notice that L(a, 0) = 1 > 0 = R(a, 0) and that for ϕ = 1 2 , (3.25) reads,

1 + 9 8a 2 ( √ 2 -1) 2 ≥ 3 a √ 2( √ 2 -1) ,
which always holds true (in terms of λ, this amounts to 1 + 1 4λ 2 ≥ 1 λ ). Moreover, the inequality above is strict if a < 3 √ 2 4( √ 2-1) . We are going to study the variations (for fixed a) of L(a, ϕ) -R(a, ϕ). By differentiating, this is equivalent to study the sign of

(3.26) D(ϕ) := 1 -2ϕ -a((1 -ϕ) 1/2 -ϕ 1/2 ). Let X := ϕ 1/2 . For a ∈ [a -, 3 √ 2 4( √ 2-1) ] and X ∈ [0, 1/ √ 2]
, since 1 -2X 2 + aX ≥ 0, the sign of (3.26) is the same as the sign of

P (X) := (1 -2X 2 + aX) 2 -a 2 (1 -X 2 ) = 4X 4 -4aX 3 + 2(a 2 -2)X 2 + 2aX + (1 -a 2 ).
Since P has roots {±1/ √ 2}, we can factor it to obtain

P (X) = 2(X 2 - 1 2 )(2X 2 -2aX + (a 2 -1)). For a > √ 2, 2X 2 -2aX + (a 2 -1) has no real roots and therefore, for a ∈ [ √ 2, 3 √ 2 4( √ 2-1) ], P is negative inside [0, 1/ √ 2] and thus ∂ ϕ L -∂ ϕ R ≤ 0 implying that (3.27) min ϕ∈[0,1/2] L(a, ϕ) -R(a, ϕ) = L(a, 1/2) -R(a, 1/2) ≥ 0, with strict inequality if a < 3 √ 2 4( √ 2-1) or ϕ = 1/2. This proves (3.24) for a ∈ [ √ 2, 3 √ 2 4( √ 2-1) ]. If now a ∈ [a -, √ 2
], besides ±1/ √ 2, P has two more roots (3.28)

X ± := a ± √ 2 -a 2 2 . For a ∈ [a -, √ 2], 0 ≤ X -≤ 1/ √ 2 ≤ X + ,
and thus P is negative in [0, X -] and positive in [X -, 1/ √ 2] from which,

(3.29) Ψ(a) := min ϕ∈[0,1/2] L(a, ϕ) -R(a, ϕ) = L(a, X 2 -) -R(a, X 2 -). Let us now prove that for a ∈ [a -, √ 2), Ψ ′ (a) ≤ 0. We first compute Ψ ′ (a) = ∂ a L(a, X 2 -) -∂ a R(a, X 2 -) + 2X -∂ a X -(∂ ϕ L(a, X 2 -) -∂ ϕ R(a, X 2 -)). By minimality of X -, ∂ ϕ L(a, X 2 -) -∂ ϕ R(a, X 2 -) = 0 so that Ψ ′ (a) = ∂ a L(a, X 2 -)-∂ a R(a, X 2 -) = 3 a 2 ( √ 2 -1) 2 1 -X 3 --(1 -X 2 -) 3/2 - 3 a X 2 -(1 -X 2 -) . A simple computation shows that X 2 -= 1 2 (1 -2a √ 2 -a 2 ) so that Ψ ′ ≤ 0 is equivalent to 1 2 √ 2 (1 -2a √ 2 -a 2 ) 3/2 + 1 2 √ 2 (1 + 2a √ 2 -a 2 ) 3/2 + 3 4a (1 -a 2 (2 -a 2 )) ≥ 1. This indeed holds since for a ∈ [a -, √ 2], 1 2 √ 2 (1 -2a √ 2 -a 2 ) 3/2 + 1 2 √ 2 (1 + 2a √ 2 -a 2 ) 3/2 + 3 4a (1 -a 2 (2 -a 2 )) ≥ 1 2 √ 2 ((1 -2a -2 -a 2 -) 3/2 + 1) + 3 4 √ 2 (1 -a 2 -(2 -a 2 -)) ≃ 1.02 > 1.
Therefore, Ψ ′ ≤ 0 and thus for a

∈ [a -, √ 2
]. We were unfortunately not able to prove this fact which would have yield a more elegant proof of (3.23).

Applications and open problems

In this section we use Theorem 1.1 to characterize the symmetric minimizers of ), the unique symmetric minimizer of (4.1) is equal in [0, T ] to S -1/4 (µ 1 ) + S 1/4 (µ 1 ), where µ 1 is equal to the unique minimizer of E(T, 1/2) given by Corollary 1.2 in [0, T ] and to its symmetric in [-T, 0]. For T > 4(2 √ 2 -2), it is given by the unique minimizer of E(T ) given by Theorem 1.1 in [0, T ] and to its symmetric in [-T, 0]. Proof. By Proposition 2.7, we know that a symmetric minimizer exists. Let µ be such a minimizer. Thanks to the symmetry, we can restrict ourselves to study its structure in [0, T ]. We let

E + (µ) := T 0 ♯{supp µ t } + i ϕ i | Ẋi | 2 dt.
Let µ 0 = N i=1 ϕ i δ X i . We first claim that X i = X i , where as before, X i = -1 2 + j<i ϕ j + ϕ i 2 . Indeed, applying the same shear as in (3.1), we obtain by minimality of µ,

E + (µ) ≥ E + (μ) + 1 T N i=1 ϕ i |X i -X i | 2 ≥ E + (µ) + 1 T N i=1 ϕ i |X i -X i | 2 ,
where the inequality could arise from a decreasing of the number of branches after the shear. This proves the claim. For i = 1, .., N, let µ +,i be the forward system emanating from (X i , 0). Then, by monotonicity of the traces (Proposition 2.7), µ +,i T = dx [X iϕ i /2, X i + ϕ i /2] and thus, by the no-loop property Moreover, µ +,i = S X i (µ i ) where µ i is some minimizer of E(T, ϕ i ). Let now T ≥ 1/4. Since ϕ i ≤ 1, we have ϕ For fixed N, this is minimized by ϕ i = 1/N so that 2-2) , this concludes the proof. As already explained in the introduction, this theorem is not completely satisfactory. Indeed, physically, the most significant case is T ≪ 1 (where many microstructures should appear), which is not covered by Theorem 4.1. However, if we could prove the following conjecture,

E + (µ) = 1 2 - √ 2

Conjecture

For T < T * every minimizer of E(T ) branches at time zero (or equivalently E(T -ε) < E(T ) -ε for ε small enough), then the picture would be almost complete. Indeed, in that case, arguing as in the proof of Theorem 4.1, we would have that every symmetric minimizer µ with µ 0 = N i=1 ϕ i δ X i would satisfy (4.2). Now for 1 ≤ i ≤ N, let ϕ i,1 , .., ϕ i,N i be the N i branches starting from (0, X i ). As in the proof of (3.4), we would have

E(T, ϕ i ) = N i k=1 E(T, ϕ i,k , X i,k ),
where X i,k := -1 2 + j<k ϕ i,j + ϕ i,k

2 . Since the minimizer corresponding to E(T, ϕ i,k , X i,k ) cannot branch at time zero, we would have (if the conjecture holds) that ϕ -3/2 i,k T ≥ 1/4 so that Corollary 1.2 applies and the structure of the minimizers would be fully determined. Let us point out that our conjecture would be for instance implied by the convexity of T → E(T ).

8. 3 . 1 ]

 31 . Moreover, the minimizer is unique by strict convexity of m → R×(a,b)

Proof.

  By inverting the relation between α N and N in (3.15), it is readily seen that (3.20) and (3.15) are incompatible. Therefore, proving the lower bound (3.20) directly excludes the possibility of having N > 2 branches. Since the proof of (3.20) is basically based on a reduction of the problem defining α N to a union of one dimensional optimization problems which can be solved by a computer assisted proof, we postpone it to Appendix A.

Proposition 3 . 14 .

 314 For T ∈ [T -, 1/4] and ϕ ∈ [0, 1],

(4. 1 ) 1 . 4 . 1 .

 1141 min{E(µ) : µ ±T = ϕ/Ldx [-L/2, L/2]}, at least for T large enough. By rescaling, it is enough to consider ϕ = L = Theorem For T ∈ [

  N -1/2 + NT. The function x → 1 2-√ 2 x -1/2 +xT is minimized by x opt = 2T (2 -√ 2) -2/3 . Since x opt < 2

  3.21) which yields T * ≥ T -(recall that the bound 1/4 ≥ T * was already derived in Proposition 3.11). Now, if T * ≤ 1/4 then E(1/4) = E(T * ) + 1/4 -T * so that using (3.18) for T = 1/4, we get

  Proof. By symmetry we may assume that ϕ ∈ [0, 1/2]. Letting λ := 2T , (3.23) is equivalent to show that for

1) , with equality if and only if T * = 1 4 and ϕ = 1/2.

  Remark 3.15. From (3.21), one could infer the simpler bound T * ≥ we still have (3.28) and (3.29). Numerically, it seems that Ψ is decreasing not only in [a -, √ 2] but actually on the whole [1, √

	which ends the proof of (3.23).	], by (3.27) √ 2) > 0, Ψ(a) ≥ Ψ(
	to a ≥ 1. For a ∈ [1,	√	2],	3 √	1 √ 2(	2+1) which leads

α N ( √ 2-1) 2 ) 1/2 ), we have obtained (3.15). Remark 3.12. From the proof of the previous proposition, one could also get a lower bound for T * . We will make use of this fact later on to study the case N = 2.

N 3 4 5 6 α N > 1.76 1.4 1.17 1.01 Table 1. Critical values of α N .

Appendix A We finally prove (3.20).

Lemma A.1. For 3 ≤ N ≤ 6, (A.1)

Proof. In Table 1, the values of the right-hand side of (A.1) are given for N ∈ [START_REF] Brancolini | Path functionals over Wasserstein spaces[END_REF][START_REF] Brasco | An equivalent path functional formulation of branched transportation problems[END_REF]. Since we are not able to prove these bounds analytically, we resort to a computer assisted proof.

From Table 1, we see that we want to compute α N with a precision of 10 -2 .

Step 1 (Computation of α 2 ):

We start by computing α 2 . We claim that

Since 3/2 , in order to prove that the minimum is attained for ϕ = 0, it is enough to show that for ϕ ∈ [0, 1/2],

Since the two expressions agree for ϕ = 0, by differentiating, we are left with the proof of

Since the polynomial,

and thus considering X = ϕ 1/2 , we see that (A.3) holds and thus (A.2) is proven.

Step 2 (Lower bound for α N ): Consider now N ≥ 3. Since

i ϕ i = 1} (the only problem could arise when all the ϕ i 's but one go to zero but then it is easy to see that

), the minimum is attained. If the minimum is attained at the boundary then α N = α N -1 . Since the values in Table 1 are decreasing with N and since α 2 = 2, in that case by a simple induction we would be over. Otherwise, we claim that for every N, the optimal ϕ i may take only two values: ϕ repeated k ∈ [1, N] times and 1-kϕ N -k repeated N -k times. Indeed, fix i = j ∈ [1, N] and for |ε| small enough, define φi := ϕ i + ε, φj := ϕ j -ε and φk := ϕ k for k = i, j. By minimality, the derivative in zero of

k is equal to zero. From this, we get the condition

N has only one positive root, this means that ϕ i may take at most two values, proving the claim. We now claim that we can further reduce to k = N -1 and ϕ ≤ 1/N i.e. N -1 'small' fluxes and a 'large' one. Indeed, if ϕ i takes the values ϕ < ψ := 1-kϕ N -k , then from the discussion above, we must have

we can use that for ϕ i = ϕ j = ψ, the second order derivative of (A.4) is positive at ε = 0 to obtain that ψ 3/2 ≤ α N /8 which would give a contradiction. Therefore, (A.5)

where

We are thus left to estimate a finite number of one dimensional functions.

For 3 ≤ N ≤ 6, we want to estimate min ϕ∈[0,N -1 ] f N (ϕ) with a precision of 10 -2 . For this we will compute the values of f N for a sufficiently fine discretization of [0, N

in order to get a precision of 10 -2 on inf I f N , it is enough to use a discretization step δ ≤ 10 -2 Λ -1 . We are thus naturally led to estimate sup |f ′ N |. This is a tedious but rather elementary computation which we include for completeness. We can compute

we need to be a little careful. A Taylor expansion shows that lim

which is always strictly bigger than the critical values computed in Table 1. Hence, if we can find η > 0, such that in [0, η], f ′ N is positive, we will have

From (A.6), we see that f ′ N is of the same sign as

Since for every ϕ ≥ 0,

and

we can first infer that

We may now bound from above f N (ϕ). Notice that regarding the numerator,

In order to bound from below the denominator, we first claim that for ϕ ≤

Indeed, letting

we have Ψ N (0) = 0 and

≤ 0, we have Ψ ′ N ≤ 0 and thus also Ψ N ≤ 0, proving (A.9). From this we get that for ϕ ≤

Putting (A.8) and (A.10) together, we get that for ϕ ≤

Inserting this back into (A.7), we get that for ϕ ≤

x, and since ϕ ≤ 8 1+16(N -1) 2 implies 5/6ϕ 1/2 ≤ 1/7, we have

In particular, if ϕ ≤ 1 108(N -1) , then h N ≥ 0. To sum up, we have proven that

.

Letting η := 1 540 (which corresponds to

In particular, we need to study min g N . Taking the derivative, we obtain

Therefore, g ′ N is zero only if ϕ = 1/N so that g ′ N is first positive and then negative and g N attains its minimum on the boundary so that min

Injecting this into (A.12), we obtain We can thus take δ := 10 -6 < 10 -2 sup

Using a simple Scilab code, we find the values of α N given in Table 2. We see that the values we find are well above the critical values given in Table 1.

Remark A.2. Although we are not able to prove it, the numerics show that for N ≥ 4, the minimum in α N is attained for equidistributed masses i.e. ϕ = 1/N. Remark A.3. Arguing as in Step 2, it could have been proven that also for the original minimization problem E(T ), the optimal masses may take at most two distinct values.


El-Awady Attia
email: elawady.attia@ensiacet.fr

Le Lann

Attia El-Awady

Jean-Marc Le-Lann

Affectation flexible des effectifs : Vers un modèle de gestion des compétences Workforce flexible allocation: Towards a model of skills management

Keywords: Ressources humaines, Affectation, Flexibilité, évolution d'expérience Human resources, allocation, flexibility, experience evolution

Parmi les stratégies favorisant la réactivité des entreprises, le développement de la polyvalence tient une place prometteuse. Nous présentons dans cet article la planification de projets multi-périodes avec problème d'affectation d'effectifs, en prenant en compte deux axes de flexibilité. Le premier est la modulation du temps de travail, qui permet à l'entreprise, pour un effectif constant, de bénéficier de fluctuations appréciables des capacités de travail. Le deuxième levier de flexibilité est la polyvalence des opérateurs, avec une vision dynamique de leurs compétences : les évolutions de ces compétences sont suivies dans le temps en fonction de la pratique (ou de la non-pratique). Dans cette approche, les durées des tâches d'un programme d'activités ne sont pas connues à l'avance, et dépendront des efficacités des acteurs sollicitéset donc de leurs affectations antérieures. Nous présentons ici les caractéristiques de ce problème, ainsi qu'une approche de sa résolution, basée sur les algorithmes génétiques.

Introduction

Les ressources humaines sont un des gisements de flexibilité les plus prometteurs [START_REF] Golden | Towards a definition of flexibility: in search of the Holy Grail?[END_REF], et en développant cette flexibilité, l'entreprise accroît sa réactivité [START_REF] Duquenne | Characterization and modelling of flexible resources allocation on industrial activities[END_REF][START_REF] Franchini | Workload control of human resources to improve production management[END_REF][START_REF] Edi | Affectation flexible des ressources dans la planification des activités industrielles: prise en compte de la modulation d'horaires et de la polyvalence[END_REF][START_REF] Davis | Influence of workload imbalances on the need for worker flexibility[END_REF]. Dans une telle optique, l'exécution d'une tâche donnée requiert un certain nombre de compétences, et chaque mission correspondant à une compétence peut être réalisée par un ou plusieurs opérateurs dont chacun peut pratiquer une ou plusieurs compétences. La littérature distingue une nature « homogène » des compétences (la performance est la même pour toutes les compétences) avec des applications : on peut Axx-2 citer la technologie de production et développements de logiciels [START_REF] Li | Scheduling projects with multi-skilled personnel by a hybrid MILP/CP benders decomposition algorithm[END_REF][START_REF] Drezet | A project scheduling problem with labour constraints and time-dependent activities requirements[END_REF], les centrales de production d'énergie [START_REF] Eitzen | Multi-Skilled Workforce Optimisation[END_REF], ou la production alimentaire [START_REF] Franchini | Workload control of human resources to improve production management[END_REF]. Par ailleurs, une nature « hétérogène » des qualifications, dans laquelle chaque compétence a une performance spécifique, a été étudiée pour les centres de services [START_REF] Valls | Skilled workforce scheduling in Service Centers[END_REF], les technologie de l'information [START_REF] Heimerl | Scheduling and staffing multiple projects with a multi-skilled workforce[END_REF], la conception et le développement de produits [START_REF] Hlaoittinun | Managing the Competencies of Team Members in Design Projects through Multi-period Task Assignment[END_REF], ou la sélection de projets dans un portefeuille [START_REF] Gutjahr | Competence-driven project portfolio selection, scheduling and staff assignment[END_REF]. Cette nature hétérogène est particulièrement justifiée dans le domaine du génie des procédés où les savoir-faire sont primordiaux ; les opérateurs sont souvent polyvalents : chacun maîtrise une compétence de base, et peut exercer une ou plusieurs compétence(s) additionnelle(s), avec pour chacune une quantification de ses performances [START_REF] Duquenne | Characterization and modelling of flexible resources allocation on industrial activities[END_REF][START_REF] Edi | Affectation flexible des ressources dans la planification des activités industrielles: prise en compte de la modulation d'horaires et de la polyvalence[END_REF]. La nature des différentes compétences peut y être structurelle et procéduraleune combinaison d'activités physiques et mentales, où la capacité de décision s'ajoute au geste. En ce qui concerne ce type de compétences, l'augmentation de la productivité par la pratique est un facteur considérable de réduction des efforts en fil du tempsréciproquement, le phénomène d'oubli par manque de pratique est également remarquable, [START_REF] Bailey | Forgetting and the learning curve: a laboratory study[END_REF]. Cet article est organisé de la manière suivante : la section 2 discute la caractérisation et la modélisation du modèle, la section 3 décrit l'approche de résolution proposée. La section 4 présente et discute quelques résultats, à la fin et dans section 5, nous concluons notre travail.

Caractérisation et modélisation du modèle

j k i j k i a k i k i EE d     k  nk i, et w i,k 0, (1)

Relations temporelles entre tâches

Nous présentons les relations temporelles généralisées sous leur forme standard, en les exprimant comme des liens début-début, [START_REF] Demeulemeester | Project scheduling: a research handbook[END_REF] ; Les relations entre deux tâches i et c vont se traduire par des contraintes liant leurs dates de débuten intégrant un délai l i,c qui peut également exprimer la présence d'un décalage temporel (avance ou retard) entre tâches [START_REF] Valls | Skilled workforce scheduling in Service Centers[END_REF] :

dd i +l i,c ≤ dd c ,
(2)

La durée du projet

Nous supposons que le projet est astreint à une durée contractuelle « L », à laquelle peut se greffer un « délai de grâce » . Si le résultat est fourni au client avec un retard supérieur à , des pénalités de retard sont comptées ; réciproquement, on évitera d'achever le travail plus tôt que (L  ) pour éviter des coûts de stockage. Par conséquent, il est nécessaire que la durée réelle LV du projet se situe dans l'intervalle :

L -  (3)

Caractérisation de la polyvalence et qualification des acteurs

Dans la littérature, on trouve de nombreuses formes de la modélisation de la qualification d'opérateurs [START_REF] Duquenne | Characterization and modelling of flexible resources allocation on industrial activities[END_REF][START_REF] Gutjahr | Competence-driven project portfolio selection, scheduling and staff assignment[END_REF][START_REF] Hlaoittinun | Managing the Competencies of Team Members in Design Projects through Multi-period Task Assignment[END_REF][START_REF] Heimerl | Scheduling and staffing multiple projects with a multi-skilled workforce[END_REF][START_REF] Valls | Skilled workforce scheduling in Service Centers[END_REF]. Nous adoptons le modèle de [START_REF] Duquenne | Characterization and modelling of flexible resources allocation on industrial activities[END_REF] pour lequel la performance d'un acteur a dans l'exercice d'une compétence k est appelée efficacité ( a,k ) : si  a,k =1, il exécutera toute mission dans un délai correspondant à l'estimation « standard » ; mais si le travail total ( a,i,k ) de l'acteur a sur la tâche i pour la compétence k est supérieur à la charge de travail (w i,k ) qui caractérise cette tâche, l'efficacité sera calculée comme le rapport ( a,k = w i,k / a,i,k ). Lors de l'affectation des acteurs, il faut vérifier qu'aucun acteur ne peut être affecté sans avoir un niveau minimum de qualification θ min,k :

θ min,k ≤ ( a,k(nddi,k)   a,i,k,j ≤ 1,  a,  k,  j (4)
où les ( a,i,k,j ) sont les variables d'affectation :  a,i,k,j = 1 : si l'acteur a est affecté à la mission de compétence k liée à la tâche i pendant le jour j. Et  a,i,k,j = 0 : sinon.

On doit également vérifier que le travail des acteurs est suffisant pour équilibrer la charge requise w i,k :

k i, , w , 1 ) , ( , , , , , , , , , , 
                  k i ER a df dd j k i dd n k a j k i a j k i a k i k i k i    (
 a,k(n  ddi,k) = 1/[1+(1/ a,k (ini)  1)(n) b ] (6) Où b = log (r a,k )/log (2) ; les deux facteurs [r a,k et  a,k (ini)] sont respectivement la vitesse d'apprentissage
de l'acteur a dans la compétence k, et son efficacité initiale au début de la simulation.

Régression

L'efficacité se dégrade en absence de pratique. Selon le modèle de [START_REF] Wright | Factors Affecting the Cost of Airplanes[END_REF] et les travaux de Jaber et Bonney (1996), l'érosion de l'efficacité dépend des périodes d'interruption () et du nombre de répétitions du travail avant cette période (n eq ), toutes exprimées en jours équivalents de 7 heures :

f a,k = 1/[1+(1/ a,k (ini) -1)( n eq ) b-f ( n eq +) f ] (7)
où f est le niveau d'efficacité d'acteur après une période d'interruption (). La pente de la courbe de « désapprentissage » f est donnée par :

f = -b (b+1)  log (n eq )/log (+1) (8)
: est le rapport entre deux périodes ( = Tb / Ta ) : une durée ininterrompue Ta d'exercice d'une compétence donnée, et la durée Tb nécessaire pour que, si la compétence en question n'est plus pratiquée, l'efficacité décroisse jusqu'à revenir à sa valeur initiale (Figure 1).

Estimation des paramètres

Cfaba =           k PR ac ac ab a ab k Ed Ed PR / 1 1 pour chaque a  PR k , et 0   

Caractérisation des calendriers de travail

Selon l'annualisation du temps de travail, tout acteur effectue une quantité fixée d'heures de travail par an : le plan de charge de chacun peut alors présenter des fluctuations quotidiennes, hebdomadaires, qui doivent respecter quelques contraintes : les durées maximales du travail (normal et supplémentaire) quotidien, hebdomadaire, moyen sur une période de douze semaines consécutives, et annuel (Pour plus de détails sur la modélisation de ces contraintes voir [START_REF] Edi | Affectation flexible des ressources dans la planification des activités industrielles: prise en compte de la modulation d'horaires et de la polyvalence[END_REF]).

Fonctionobjectif

Nous attendons cinq objectifs de la réalisation du projet : quatre termes de coûts (f 1 ,…, f 4 ) à minimiser, et une fonction de profit virtuel (f 5 ) à maximiser :

Minimiser: 5 4 3 2 1 f f f f f f      (10)
Le terme (f 1 ) représente le coût réel du travail exécuté dans les heures normales, à un coût horaire "U a " :

            A a S S s s a a FW SW U f 1 , 1  (10-a)
Le terme (f 2 ) représente un coût virtuel associé à la perte de flexibilité de travail des acteurs à la fin de l'horizon de simulation. Il est calculé en fonction du taux d'occupation moyen de l'acteur par rapport à la durée standard de travail par semaine "C s0 ", et vise à préserver la flexibilité future de l'entreprise :

                A a S S s s a a Cs S UF f FW SW 1 , 2 1 ) 0 /(  (10-b)
Le terme (f 3 ) est le coût des heures supplémentaires, qui s'ajoute à celui des heures normales (f 1 ) ; il est déterminé en appliquant un facteur multiplicatif "u" au coût horaire standard :

             A a S S s s a a FW SW HS u U f 1 , 3 (10-c)
Le terme (f 4 ) représente une pénalité associée à un achèvement d'un travail en dehors de sa plage de tolérance [START_REF] Vidal | Optimisation des plans de charge pour un flow-shop dans le cadre d'une production en Juste A Temps: 2-Formulation mathématique[END_REF], comme vu en 2.1.3 :

                                        L LV L L LV L LV L LV UL f f f LV L j Si, 0 1 1 ) ( 3 1 4 (10-d)
Enfin le profit virtuel (f 5 ) est lié à l'évolution globale des compétences des acteurs dans l'entreprise. L'équation (10-e) exprime la différence d'efficacité des acteurs entre le début et la fin du projet. Une constante (U k ) exprime l'importance de ce facteur et nous permet de traduire cette différence en termes de bénéfice pour l'entreprise :

                     K k A a at e j n k a A a at e n k a n) Dat e de f i k(n a k k NA U f 1 1 début ) de D ( , 1 début ) de D ( , , 5 /    (10-e)

Résolution du problème par codage des priorités

Durées « élastiques » (non prédéfinies) des tâches et vision dynamique des compétences font que la description de ce problème aboutit à un modèle non linéaire. Nous nous sommes orientés vers la recherche de solutions approchées qui peuvent être obtenues avec des méthodes heuristiques comme les algorithmes génétiques. L'AG que nous proposons utilise le codage de priorités : les chromosomes rassemblent trois groupes de variables de décision : quel ordre de traitement des tâches? Quels acteurs affecter en priorité? Et avec quelle capacité de travail quotidien? Les individus représentés par ces chromosomes sont ensuite évalués par référence à la fonction-objectif, afin de calculer pour chacun son adaptation. La figure 2

Résultats et discussions

Nous avons appliqué cette méthode à un exemple simple de 10 tâches, 10 acteurs, et 4 compétences, tiré du manuscrit de Edi (2007). Ce problème a été résolu avec satisfaction de toutes les contraintes, avec différentes importances, exprimées par le coût fictif f 5 , accordées à la culture de la polyvalence. Ces simulations ont logiquement mis en évidence le risque, si l'entreprise vise systématiquement un coût minimal de réalisation du projet, de voir péricliter les compétences secondaires des opérateurs : ces derniers sont toujours préférentiellement affectés sur leurs compétences principales (celles qui aboutissent au temps d'exécution minimal), et subissent une dégradation de leur efficacité dans leurs compétences secondaires par phénomène d'oubli. L'entreprise doit donc se poser la question : même si un programme en cours ou à exécuter à court terme est réalisable en ne mobilisant que des acteurs pleinement compétents, quel est le surcoût à accepter pour cultiver la polyvalence et préserver la réactivité future ?

Conclusion et perspectives

Nous avons achevé la caractérisation et la modélisation des différents aspects de notre problème d'affectation flexible et multi-périodes des ressources humaines en prenant en compte la modulation du temps de travail, la polyvalence des acteurs, et les phénomènes d'apprentissage et d'oubli. L'ensemble, solveur de scenarii, peut être utilisé comme un outil d'aide à la décision pour gérer et développer les compétences de l'entreprise, et donc sa capacité à réagir à des événements imprévisibles. Dans cet esprit, les développements futurs de ce travail consisteront à élaborer des « matrices de passages » permettant de quantifier la facilité ou la difficulté de migrations de compétences entre différents métiers ou entre différents niveaux de qualification dans le même métier.

  PR k : est le cardinal de l'ensemble PR k .-Calculer les paramètres de l'acteur débutant : r ces procédures de similarité, on peut évaluer les paramètres de l'acteur pour n'importe quelle compétence k. Par conséquent nous pouvons choisir les acteurs les plus appropriés à la polyvalence.

2.1 Caractérisation du projet 2

  .1.1 La durée des tâches Un projet est décomposé en I tâches uniques et originales. Pour chaque tâche i I , on dispose de durées de réalisation : minimale (D i

					min ), maximale (D i max ) et standard (D i ) : la durée réelle de la tâche (d i ) doit
	vérifier D i min  d i  D i max . Une des caractéristiques de la tâche est le nombre des compétences (nk i )
	nécessaires à son exécution. Pour chaque compétence k  nk i , une charge de travail w i,k , en heures, est
	définie. Ainsi, à chaque compétence k de la tâche i une durée réelle d i,k (en jours) est associée. Le temps
	d'exécution de la tâche i tâche est donc d i = max (d i,k ), i, et  k  nk i . Il est donc nécessaire de
	déterminer chacune des durées d i,k , : elles sont déduites de l'efficacité équivalente des opérateurs affectés
	(EE i,k ) et de leur capacité journalière de travail ( a,i,k,j ) :
	,		w	,	,
	,	,	,	,	,

  Le groupe des praticiens : là également, nous supposons que l'entreprise a des données récentes, permettant de déterminer les caractéristiques nécessaires, même si l'efficacité n'est pas de 1. Le groupe des novices et débutants : il n'existe aucune donnée relative au fonctionnement de ce groupe.

		0,85
		0,80
		0,75
		0,70
	Éfficacité	0,55 0,60 0,65
		0,50
		0,45
		Apprentissage
		0,40
		Dégradation
		0,35
		1 27 53 79 105 131 157 183 209 235 261 287 313 339 365 391 417 443 469 495
		Équivalents-jours de 7 h
	Figure 1. L'effet de l'apprentissage et de l'oubli sur l'efficacité des acteurs
	La procédure est la suivante :
	-Choisir un ensemble (PR
	Nous supposerons que l'entreprise a un profil détaillé de chaque opérateur, qui décrit les différents
	domaines de connaissances, la scolarité, les compétences principales, les compétences secondaires, etc.
	Chacun de ces paramètres a un score prédéfini. Afin de déterminer leurs paramètres ( a,k (ini), r a,k , et Tb),
	nous utilisons la distance euclidienne à partir des groupes d'experts et praticiens.

, r et Tb Pour estimer l'efficacité initiale  a,k (ini), la vitesse d'apprentissage (r a,k ), et la période de « désapprentissage » Tb des acteurs, nous les classons en trois catégories : Le groupe des experts : c'est le cas simple ; leur efficacité est à la valeur nominale, et l'entreprise dispose des données historiques qui permettent d'apprécier  a,k (ini), r a,k et T b . k ) d'acteurs praticiens et experts dans la compétence k,

-Pour chaque acteur de a  PR k , on peut lire ( a,k (ini), r a,k , et Tb) dans la base des données de production, -Calculer la distance euclidienne (Ed) entre l'acteur débutant (ab) et chaque acteur a  PR k , -Calculer le facteur de correction « Cf » pour l'acteur a, (équation 9). La personne la plus appropriée sera celle dont le facteur de correction est le plus élevé.

  représente une procédure de mise en oeuvre de cet AG :

	L'Algorithme Génétique:
	{Générer la population initiale "g=1" avec la taille IP
	Évaluer la population g=1
	SI les critères d'arrêt ne sont pas satisfaits, répéter
	{Sélectionner certains individus de g pour les mettre dans g+1
	Croisement de certains individus de g et les mettre dans g+1
	Régénérer des certains nouveaux individus et leur mettre dans g+1
	Mutation de certains individus choisis parmi g+1
	Affectation des acteurs selon les décisions des chromosomes
	Évaluer la population g+1
	G = g+1}}
	Figure 2. L'Algorithme Génétique
	Axx-6

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