Benoît Perthame

Edouard Ribes
email: edouard.augustin.ribes@gmail.com

Karim Touahri
email: touahrikarim91@gmail.com

Delphine Salort

A model for cost efficient Workforce Organizational Dynamics and its optimization

Keywords: Workforce planning, Flexible workforce, Structured equations, Cost optimization, Asymptotic analysis

This paper presents a workforce planning model scalable to an entire hierarchical organization. Its main objective is to design a cost optimal target which leverages flexible workforce solutions while ensuring an efficient promotional flux. The value of this paper lies in its proposal of an adequate flexibility rate using various solution types and in its discussion about external hiring ratios. The mathematical structures of the models are analyzed and numerical simulations illustrate the theoretical background.

Introduction

To our knowledge, human resource planning has mainly focused on providing insights on hiring and training decisions. This has boiled down to questions of workforce planning & knowledge acquisition [START_REF] Bordoloi | Human resource planning in knowledge-intensive operations: A model for learning with stochastic turnover[END_REF][START_REF] Burger | On a Boltzmann mean field model for knowledge growth[END_REF]. According to a recent review [START_REF] Bergh | Personnel scheduling: A literature review[END_REF], these topics are becoming more and more workforce centric, as firms try to integrate employee preferences in their decisions. This naturally raised the question of career and employee lifecycle within the company and leads to the question of employee promotion strategy [START_REF] Klehe | Career adaptability, turnover and loyalty during organizational downsizing[END_REF][START_REF] Komarudin | Balancing desirability and promotion steadiness in partially stochastic manpower planning systems[END_REF][START_REF] Henderson | Stochastic optimal control of internal hierarchical labor markets[END_REF]. The question of the employee lifecycle is complex, because it depends not only in the company purpose and market landscape but also in the individual employee dynamics. We have therefore investigated those dependencies within 3 complementary papers, and this paper is the first one of the series. More precisely, the main objective of this article is to present a new partial differential equation model, perform its theoretical and numerical studies, in order to address the description and cost optimization of workforce lifecycle from a company standpoint. In this perspective, we use the notion of workforce flexibility and notably the use of temporary contracts to yield cost efficiency, which is standard among the operation research literature (see [START_REF] Qin | Workforce flexibility in operations management[END_REF] for a recent review). However, in the present article, we integrate a new continuous formalism which allows to integrate mixed approaches in complex environment. The second paper focuses on turnover prediction and on identification of endogenous and exogenous parameters that influence voluntary turnover decisions ( [START_REF] Ribes | Employee turnover prediction and retention policies design: a case study[END_REF]). The turnover rates are also used in the model data in section 4. Finally the third paper, under preparation, adopts a mean field game approach and tries to reconnect company policies with individual aspirations.

The article is organized as follows. In section 2, we build a dynamic organization representation based upon recent researches in this field [START_REF] Komarudin | Balancing desirability and promotion steadiness in partially stochastic manpower planning systems[END_REF] by taking into account the time spent by personnel in a job qualification class. While a simple model, which already use the idea of a waiting time before promotion has been used, with a simple fixed time delay model in [START_REF] Henderson | Stochastic optimal control of internal hierarchical labor markets[END_REF], in our model, we integrate a more accurately control of this notion of waiting time, by adopting a continuous approach. In section 3, we determine theoretically the best strategy to adopt in order to minimize the total labor cost. To this, we assume that the cost can be modulated via the recruitment of temporary contracts. Indeed, the possible cost burden of having an organization which promotion waiting times are important will be the center of this investigation. More precisely, a cost optimal solution leveraging a flexible workforce will be proposed and fully characterized. In section 4, we illustrate our theoretical results via numerical simulations. We first implement the organizational model with a strategy which allows to reduce the external hiring with specific rules. In a second part, we test, via genetic algorithms, the optimal strategy to minimize the labor cost. We show that a mix of "floaters" and "temporary workers"(see [START_REF] Wild | Manpower capacity planning -a hierarchical approach[END_REF] and [START_REF] Harper | Modelling the size and skill-mix of hospital nursing teams[END_REF] for examples of the individuals approaches), can be used to efficiently design a full organizational footprint. Finally, in section 5, we finish our article by a conclusion an perspectives.

Important legal remarks

The findings and opinions expressed in this paper are those of the authors and do not reflect any positions from any company or institution. Finally, please bear in mind that to protect confidentiality numbers have been disguised in a way that preserves the same analysis and conclusions as the actual case study.

Organization dynamics

We begin with the main assumptions and model we use in order to describe the dynamic of the workforce when employees may be promoted on several hierarchical jobs. We discuss several situations with respect to the possibility of external hiring and we propose a theoretical study of this model, in order to determine the well-posedness of the related system of equations, the possible stationary states and the necessity of external hiring, depending upon parameter combinations, in order to to obtain a well-posed problem. This model is the basis of the study of the optimization cost.

Organization model

We represent a company by a set of L jobs ordered in terms of level of responsibility from j = 1 to L. We denote by ρ j (t, s) the number of workers which at time t, have seniority s in their job class. We assume that:

• in each job j ∈ {1, .., L}, the total number of workers in the job category j across seniority categories is constant over time and equal to N j > 0.

• in each job j, workers can leave the company. We assume that worker turnover follows an exponential law with parameter µ j .

• workers can also stay in the same job category j and wait to be promoted from job j to j + 1 .The promotion can only occur if they have spent a time superior to τ j ≥ 0 in their job j . The associated promotion rate is defined as P j .

• the company can also hire externally in each job j with a rate h j . We denote by h j (t)N j the number of external workers recruited externally for the job j.

These assumptions can be formalized using the following balance law for workers in each job j = 1, ..., L, which takes into account the seniority increase with time, as well as turnover and promotion:

∂ t ρ j + ∂ s ρ j + µ j ρ j + I s≥τ j P j (t)ρ j (t, s) = 0, s ≥ 0, t ≥ 0, (1)
The next boundary condition describes the flux of workers in class j = 1, ...L that newly arrive either by external hiring or internal promotion:

ρ j (t, s = 0) = h j N j + P j-1 (t)A j-1 (t), (P 0 = 0, P L = 0), (2)
with the notation

A j (t) = +∞ s=τ j ρ j (t, s)ds. (3)
Notice that for the first job class of job, only external hiring is possible. Thus the above condition holds with P 0 = 0. Also, for the last class, promotion is no longer possible and thus P L = 0. To simplify the setting, we assume that retirement is included in the average departure rate µ j . The initial state is denoted by ρ 0 j (s) > 0 and solutions remain positive. We recall that this type of system, called renewal equations, is standard and well understood [START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Gurtin | Non-linear age-dependent population dynamics[END_REF][START_REF] Farkas | On the stability of stationary age distributions[END_REF], and has previously been used for workforce planning purposes [START_REF] Doumic | Toward an integrated workforce planning framework using structured equations[END_REF].

As the number of workers in the job category j is assumed constant, the following constraint appears:

N j = +∞ 0 ρ j (t, s)ds > 0 is given. ( 4
)
Integrating equation ( 1), we find a consistency condition on the boundary

ρ j (t, s = 0) = µ j N j + P j A j (t). ( 5
)
Combining the boundary condition (2) and the constraint (5) leads to ∀j = 1, ..., L:

µ j N j + P j A j (t) = h j (t)N j + P j-1 A j-1 (t), (P 0 = P L = 0) (6)
At this stage, it has to be noticed that, because we have imposed the L constraints that the N j 's are constant, we also need L Lagrange multipliers to fulfill the equality [START_REF] Henderson | Stochastic optimal control of internal hierarchical labor markets[END_REF]. These are to be chosen among the 2L parameters P j and h j and thus some degrees of freedom remain. In a first pass, a perfect internal labor market will be considered. This means that no external hiring will exist except at the first job layer j = 1 and that the L free parameters are h 1 , P j for j = 1, ...L -1. In a second pass, a proportion of external hires among the various job layers will be introduced. It will be shown that it results from a simple generalization of the first case.

Organization behavior -no external hiring

Under the preliminary assumption of a perfect internal labor market, the company's recruitment policy prohibits external recruitment except for the first job layer. This means that h j = 0 for all j ≥ 2. Therefore, the constraint (6) becomes:

µ j N j + P j A j (t) = P j-1 A j-1 (t), for j = 2, ..., L, µ 1 N 1 + P 1 A 1 (t) = h 1 (t)N 1 . (7)
Promotional fluxes Assuming that these relation can be solved backward (that is A j (t) > 0, for all t ≥ 0 and j ∈ {1, ..., L -1}), the equation [START_REF] Ribes | Employee turnover prediction and retention policies design: a case study[END_REF] imposes that for the job level j = L:

µ L N L = P L-1 A L-1 (t)
This means that the promotion flux P L-1 A L-1 is independent of time. This can be further cascaded down the organization layers with j = L -1 to get:

µ L-1 N L-1 + P L-1 A L-1 = P L-2 A L-2 ,
and thus:

µ L-1 N L-1 + µ L N L = P L-2 A L-2 .
Iterating the same process shows that:

L l=j µ l N l = P j-1 A j-1 , ∀j, 2 ≤ j ≤ L.
Therefore, the promotion fraction P j (t) only depends upon A j (t) thanks to the relation:

P j (t) = C no j+1 A j (t) , C no j := L l=j µ l N l . (8)
Departing from equations ( 1) and ( 2), this generates the following system of L nonlinear equations:

∂ t ρ L + ∂ s ρ L + µ L ρ L = 0, ρ L (s = 0, t) = µ L N L , (9)
and for 1 ≤ j ≤ L -1,

∂ t ρ j + ∂ s ρ j + µ j ρ + C no j+1 A j (t) I s≥τ j ρ j = 0, ρ j (s = 0, t) = L l=j µ l N l . (10)
The boundary condition for j = 1 then states that h 1 N 1 = L l=1 µ l N l . The interpretation is clear: the total number of hired workers at job level j = 1 should compensate the total number of workers leaving the company.

Well-posedness of equation (1), assuming (7) Conditions on the initial data are now computed. This introduces a well posed problem for equation [START_REF] Bordoloi | Human resource planning in knowledge-intensive operations: A model for learning with stochastic turnover[END_REF], when the parameters are chosen as in [START_REF] Ribes | Employee turnover prediction and retention policies design: a case study[END_REF]. This leads us to describe the set of initial data such that A j (t) > 0, for all t ≥ 0 and j ∈ {1, ..., L -1}. The following proposition therefore holds:

Proposition 2.1 Let ρ 0 j (s) ∈ L 1 (R + ) such that for all t ≤ τ j e -tµ j τ j t ρ 0 j (s -t)ds + 1 µ j (1 -e -µ j t ) L l=j µ l N l < N j , (11)
Assuming that for all j = 1, ..., L -1, the model parameters are such that:

A j := µ j N j e -µ j τ j -(1 -e -µ j τ j )C no j+1 µ j > 0. ( 12
)
Then, equation (10) is well posed. We have A j (t) > 0 for all t ≥ 0 and A j (t) = A j for all t ≥ τ j .

Remark 2.2 Let us mention that proposition 2.1 implies that equation ( 10) is linear as soon t ≥ τ j because A j then becomes constant.

Proof of proposition 2.1 Estimating (11) with t = 0 is equivalent to choosing an initial data such that A j (0) > 0. Moreover, as long as A j (t) > 0, using the explicit formula for the solution of equation ( 10) via the method of characteristics, we find that for t ≤ τ j ,

ρ j (t, s) = ρ 0 j (s -t)e -µ j t if τ j ≥ s ≥ t and ρ j (t, s) = L l=j+1 µ l N l e -µ j s if s ≤ t ≤ τ j .
Using the mass conservation, we obtain that for t ≤ τ j :

A j (t) = N j - τ j 0 ρ j (t, s)ds = N j -e -tµ j τ j t ρ 0 j (s -t)ds - 1 µ j (1 -e -µ j t ) L l=j µ l N l ,
and so A j (t) > 0 because of condition [START_REF] Perthame | Transport equations in biology[END_REF]. Now, for t ≥ τ j , with the method of characteristics, we obtain that :

ρ j (t, s) = L l=j µ l N l e -µ j s if s ≤ τ j
which implies that :

A j = µ j N j e -µ j τ j -(1 -e -µ j τ j ) L l=j+1 µ l N l µ j , ∀t ≥ τ j .
Stationary states and convergence of the solution of Equations ( 9)- [START_REF] Harper | Modelling the size and skill-mix of hospital nursing teams[END_REF] The following proposition expresses the fast convergence to a steady state Proposition 2.3 Assume that for all j ∈ {1, ..., L -1}, we have A j > 0. Then, there exists a unique stationary state of Equations ( 9)- [START_REF] Harper | Modelling the size and skill-mix of hospital nursing teams[END_REF] given by

ρ L (s) = µ L N L e -µ L s , ρ j (s) = e -µ j s+B j (τ j -s) -   L l=j µ l N l   .
Moreover, for initial data chosen as in proposition 2.1, there is a constant C such that for all j ∈ {1, ..., L} and t ≥ max j τ j +∞ 0

|ρ j (t, s) -ρ j (s)|ds ≤ Ce -µ j t +∞ 0 |ρ j (0, s) -ρ j (s)|ds.
Proof of proposition 2.3 Using that for t ≥ τ j , A j (t) = A j , the stationary state of equation ( 1) is the stationary state of the following linear equation:

∂ t ρ j + ∂ s ρ j + µ j ρ + B j I s≥τ j ρ j = 0, ρ j (s = 0, t) = L l=j µ l N l (13)
with B j := L l=j+1 µ l N l A j which leads to the formula of the stationary states given in Proposition 2.3.

To prove asymptotic convergence, we set m j = ρ j (t, s) -ρ j (s). We have for all j ∈ {1, ..., L}

∂ t |m j | + ∂ s |m j | + µ j |m j | = 0.
Integrating this equation and using the Gronwall lemma, we find the result of convergence which ends the proof of Proposition 2.3.

Organization behavior -with external hiring

We remove the constraint h i = 0 and allow the external organization to source itself externally. We adapt the method developed in the previous subsection to this situation with external sourcing and we show that, with a more general condition on parameters, it yields similar results. Assuming again that for all j, A j (t) > 0, we introduce the hiring ratio α j > 1, for job class j according to the below equation:

µ j N j + P j A j (t) = P j-1 A j-1 (t) + h j .N j = α j P j-1 A j-1 (t), 2 ≤ j ≤ L, (14)
This gives us the external hiring rate h j (t) through the relation:

α j = 1 + h j (t)N j (t) P j-1 A j-1 (t) . (15)
In other words, α j is an interesting parameter for the establishment of human resources policies as α j -1 represents the ratio of external over internal hires at level j.

Promotional fluxes Following the argument in the previous subsection, we infer from ( 14), successively for j = L and for j = L -1,

µ L N L = α L P L-1 A L-1 , µ L-1 N L-1 + α -1 L µ L N L = α L-1 P L-2 A L-2 .
Iterating this process for 2 ≤ j ≤ L -1, leads to

µ j N j + L l=j+1 µ l N l .( l k=j+1 1 α k ) = α j P j-1 A j-1 ,
that is also written

C j := L l=j µ l N l .( l k=j 1 α k ) = P j-1 A j-1 ,
or also

P j (t) = C j+1 A j (t) , j = 1, ..., L -1.
This generates the following systems of L nonlinear equations:

∂ t ρ L + ∂ s ρ L + µ L ρ L = 0, ρ L (s = 0, t) = µ L N L , (16)
and for 1 ≤ j ≤ L -1,

∂ t ρ j + ∂ s ρ j + µ j ρ + C j+1 A j (t) I s≥τ j ρ j = 0, ρ j (s = 0, t) = µ j N j + C j+1 (= α j C j ). ( 17
)
Study of equations ( 16)- [START_REF] Scrucca | GA: A package for genetic algorithms in R[END_REF] The structure of equations ( 16)-( 17) are exactly the same as equations ( 9)- [START_REF] Harper | Modelling the size and skill-mix of hospital nursing teams[END_REF]. Hence, in this section, we just state our results without proofs since they follow exactly the preceding section. Given α j ≥ 1, we begin with conditions on the initial data to have a well posed problem for equations ( 16)- [START_REF] Scrucca | GA: A package for genetic algorithms in R[END_REF]. The following proposition holds:

Proposition 2.4 Let ρ 0 j ∈ L 1 (R + ), 1 ≤ j ≤ L, be such that for all t ≤ τ j e -tµ j τ j t ρ 0 j (s -t)ds + 1 µ j (1 -e -µ j t )(µ j N j + C j+1 ) < N j , (18)
Assume that for all j = 1, ..., L -1, the model parameters are such that:

A j := µ j N j e -µ j τ j -(1 -e -µ j τ j )C j+1 µ j > 0. ( 19
)
Then equation ( 10) is well posed. We have A j (t) > 0 for all t ≥ 0 and

A j (t) = A j , ∀t ≥ τ j .
Remark 2.5 Let us mention that, given (τ j ) j∈{1,..,L-1} , (µ j ) j∈{1,..,L} and (N j ) j∈{1,..,L} , we can always find (α j ) j∈{1,..,L} such that for all j ∈ {1, .., L}, A j > 0. In the same way, given an initial data in L 1 such that for all j ∈ {1, .., L}, A j (0) > 0, we can always find (α j ) j∈{1,..,L} such that A j (t) > 0 for all t ≥ 0.

Proposition 2.6 Assume that for all j ∈ {1, ..., L -1}, we have A j > 0. Then, there exists a unique stationary state of equations ( 16)- [START_REF] Scrucca | GA: A package for genetic algorithms in R[END_REF] given by:

ρ L (s) = µ L N L e -µ L s , ρ j (s) = e -µ j s- C j+1 A j (s-τ j ) + (µ j N j + C j+1 ) .
Moreover, for initial data chosen as in Proposition 2.4, there is a constant C such that for all j ∈ {1, ..., L} and t ≥ max j τ j +∞ 0

|ρ j (t, s) -ρ j (s)|ds ≤ Ce -µ j t +∞ 0 |ρ j (0, s) -ρ j (s)|ds.
Minimal external hiring A company may prefer to promote internally to capitalize on its own workforce and provide its employees with better careers opportunities to minimize their turnover rate.

To do so, the procedure is to choose the 'minimal values' of the vector (α 1 , ..., α L ) so as to impose the constraint [START_REF] Lucasius | Understanding and using genetic algorithms. Part 1. concepts, properties and context[END_REF]. This can be performed by a descending algorithm. Departing from A L-1 > 0, we find :

C L := µ L N L α L < µ L-1 N L-1 e -µ L-1 τ L-1 (1 -e -µ L-1 τ L-1 ) , α min L = max 1, (1 -e -µ L-1 τ L-1 )µ L N L µ L-1 N L-1 e -µ L-1 τ L-1 .
This allows to compute α min L-1 because we impose A L-2 > 0, which is

C L-1 := 1 α L-1 µ L-1 N L-1 + µ L N L α L < µ L-2 N L-2 e -µ L-2 τ L-2 (1 -e -µ L-2 τ L-2 ) ,
Hence the formula for

α min L-1 = max 1, 1 -e -µ L-2 τ L-2 µ L-2 N L-2 e -µ L-2 τ L-2 µ L-1 N L-1 + µ L N L α min L .
3 Organization cost structure and flexible workforce

In the previous section, we have shown that it is possible to control the dynamic of an entire organization in a continuous framework. The aim of this section is to deliver a theoretical setting to understand how to make the labor cost efficient and optimal. The problematic being that, for a given position, permanent workers in an organization have an increasing salary with seniority. As such, it may prove detrimental to have them waiting for a promotion. Flexible workforce solutions will therefore be discussed to minimize the overall labor cost burden of the organization. To this, we first study the impact of the use of temporary workers in the labor cost and then integrate a mixed approach of "floaters" and temporary workers.

Leverage temporary workers to speed up the organization

General problem statement Using the notations in section 2, we consider that the population N j in each job j ∈ 1, ..., L can be divided in two subcategories:

• The permanent population N p j > 0 which evolves within the organization with the same dynamics as described earlier and which compete for promotion. The permanent population distribution across seniority level at steady state will be referred to as ρ p j (s).

• The temporary population N t j > 0 which is used by the organization as a buffer to meet its workload requirements and will not be used for promotion.

We denote by p j ∈ [0, 1] the proportion of the population N j in level j under a permanent contract. We may write

N j = N p j + N t j = (1 -p j )N j + p j N j . (20)
The organization cost structure is supposed to obey the following rules at each level j:

• the cost of each position w p j (s) > 0 in the permanent population N p j is growing with seniority s,

• the cost of each temporary contract w t j is assumed to be constant. We also assume that w t j > w p j (0) to account that a temporary workforce may come at a premium over a freshly hired permanent employee.

The overall operating cost of the organization Cost org is therefore defined as the sum of the operating costs Cost j at each level j,

Cost org = L j=1 Cost j (21)
with

Cost j = (1 -p j )N j w t j + ∞ 0 ρ p j (s)w p j (s)ds. ( 22
)
Computing the organization cost According to Proposition 2.6 and Equation ( 17), we have

• ρ p L (s) = µ L N L p L .e -µ L s , • ρ p j (s) = (µ j N j + C j+1 )e -µ j s+ C j+1 A j (τ j -s) - for 1 ≤ j < L,
where

C j (p j , ..., p L ) := L l=j µ l N l .p l . l k=j 1 α k , A j = µ j N j .p j .e -µ j τ j -(1 -e -µ j τ j )C j+1 µ j .
Therefore, we may define the optimal cost organization by:

Cost opt org = min (p 1 ,...,p L )∈[0,1] L L j=1 (1 -p j )N j w t j + ∞ 0 ρ p j (s)w p j (s)ds . ( 23
)
Note that, similarly to the previous condition [START_REF] Lucasius | Understanding and using genetic algorithms. Part 1. concepts, properties and context[END_REF], we need to impose a limitation of the p j in order to satisfy A j > 0, namely

p j > (1 -e -µ j τ j )C j+1 µ j N j .e -µ j τ j =: p min j ( 24
)
where p min j depends only upon p j+1 , ..., p L .

Solution Approach

In order to derive explicit formulae, we now assume that there is no external hiring (α j ≡ 1) and that the cost of permanent employees grows exponentially with a limited rate r (compared to attrition) w p j (s) = w 0 j e rs , r < µ j , Then, we can obtain explicitly the cost at level j with formula which can be implemented within a standard optimization algorithm, see Section 4.2.

Proposition 3.1 The following expressions holds for the costs Cost j ,

Cost j = (1 -p j )N j w t j + w 0 j (µ j N j p j + C j+1 ) µ j -r 1 - µ j C j+1 e rτ j (µ j -r)α j C j + rC j+1 e µ j τ j , 1 ≤ j < L, (25)
Cost L = (1 -p L )N L w t L + w 0 L α L C L µ L -r . ( 26
)
Proof. These formulas are derived as follows for j < L. Note that we do not copy the calculation for the case j = L. Departing from the formula for ρ p j (s) in Section 3,

1 w 0 j ∞ 0 ρ p j (s)w p j (s)ds = (µ j N j p j + C j+1 ) ∞ 0 e -µ j s+rs+(r+ C j+1 A j )(τ j -s) -ds = (µ j N j p j + C j+1 )[ 1-e -(µ j -r)τ j µ j -r + e -(µ j -r)τ j µ j -r+ C j+1 A j ]
Therefore, since µ j A j = e -µ j τ j α j C j -C j+1 , we have

µ j -r w 0 j ∞ 0 ρ p j (s)w p j (s)ds = (µ j N j p j + C j+1 ) 1 -e -(µ j -r)τ j + (µ j -r)
A j e -(µ j -r)τ j e -µ j τ j α j C j -r A j = (µ j N j p j + C j+1 ) 1 + e -(µ j -r)τ j -(e -µ j τ j α j C j -r A j )+(µ j -r) A j e -µ j τ j α j C j -r A j = (µ j N j p j + C j+1 ) 1 -e -(µ j -r)τ j C j+1 e -µ j τ j α j C j -r A j .

We end up with the expression:

µ j -r w 0 j ∞ 0 ρ p j (s)w p j (s)ds = (µ j N j p j + C j+1 ) 1 - µ j C j+1 e rτ j µ j α j C j -rα j C j + rC j+1 e µ j τ j .
This formula immediately yields (25).

Optimality conditions

The specific form of the cost in formula (26) has an ascending property because Cost j only depends on p j , ...p L . This allows us to write:

Cost opt org = min p L Cost L + min p L-1 Cost L-1 + min p L-2 [Cost L-2 ... + min p 1 Cost 1 ...] .
Notice that this structure is conflicting with the descending structure for the minimal values of the α j . For this reason analytical formula for a global minimizer in (p j , α j ) j=1,...L is not possible. We exemplify the choices of optimal proportions of permanent workers p j 's in the next paragraph, assuming the α j 's are known.

A specific case of optimality conditions Computing the optimal conditions remains rapidly very tedious, at least theoretically. But rather than providing directly numerical simulation results (that are in any cases developed in the end of this paper), we will restrict our strategy of leverage of temporary workers only in the first job or the two first jobs to derive and discuss closed formulas.

Case 1: Allow contractors only in the lowest hierachical level. In this setting, the cost function depends only on the variable p 1 because p 2 = p 3 = ... = p L = 1.. Because only C 1 , among the C j 's, depends on p 1 , from (25), we then have to tackle the minimums with respect to the variable p 1 of the function

Cost 1 (r, p 1 ) = (1 -p 1 )N 1 w t 1 + w 0 1 α 1 C 1 µ 1 -r 1 - µ 1 C 2 e rτ 1 (µ 1 -r)α 1 C 1 + rC 2 e µ 1 τ 1 .
We have

Cost 1 (r, p 1 ) = N 1 -w t 1 + w 0 1 µ 1 µ 1 -r 1 - rµ 1 (C 2 ) 2 e (r+µ 1 )τ 1 ((µ 1 -r)α 1 C 1 + rC 2 e µ 1 τ 1 ) 2 • We deduce that Cost 1 (r, p 1 ) = 2N 1 w 0 1 rµ 2 1 (C 2 ) 2 e (r+µ 1 )τ 1 ((µ 1 -r)α 1 C 1 + rC 2 e µ 1 τ 1 ) 3 > 0 on ]0, 1[.
Hence, Cost 1 is a strictly convex function with respect to the variable p 1 and there exists a unique minimum p * 1 of Cost 1 on [0, 1]. Moreover, if p * 1 ∈]0, 1[, we have Cost 1 (r, p * 1 ) = 0. We can further analyze the optimal choice of p 1 .The optimality condition Cost 1 (r, p 1 ) = 0 gives us :

1 - µ 1 -r µ 1 w t 1 w 0 1 = rµ 1 (C 2 ) 2 e (r+µ 1 )τ 1 (µ 1 -r)p 1 µ 1 N 1 + C 2 (re µ 1 τ 1 + µ 1 -r) 2 < r µ 1 e (r-µ 1 )τ 1
because we have assumed from (24) that p 1 µ 1 N 1 > (e µ 1 τ 1 -1)C 2 (a relation written as p 1 > p min 1 ). Therefore:

• when w t 1 is low the optimal cost is achieved with the smaller tenure rate p * 1 = p min 1 which allows to ensure internal promotion for level 2 (see (24)),

• when w t 1 is large, the minimal cost is also reached by saturating the constraint, but now on the other extreme p * 1 = 1 which means that there is no temporary workers, • the middle range where the optimal hiring rate satisfies p min 1 < p * 1 < 1 occurs when the parameters are such that:

rµ 1 (C 2 ) 2 e (r+µ 1 )τ 1 (µ 1 -r)µ 1 N 1 + C 2 (re µ 1 τ 1 + µ 1 -r) 2 < 1 - µ 1 -r µ 1 w t 1 w 0 1 < r µ 1 e (r-µ 1 )τ 1 .
Case 2: Allow contractors only in the two first levels. In this setting, we assume that p 3 = ... = p L = 1. This case is much more complex to deal with, for instance, conversely to the previous case, convexity of the cost function cost org is lost, even in the simpler situation where L = 2. Hence, here, we restrict our analysis to the derivation of the relations leading to a critical point, that is ∇Cost org = 0.

As before, because only C 1 , among the C j 's, depends on p 1 , from (25), we obtain the first order optimality condition ∂Costorg ∂p 1

= 0 with the relation:

(µ 1 -r)N 1 w t 1 = w 0 1 µ 1 N 1 1 - rµ 1 (C 2 ) 2 e (r+µ 1 )τ 1 ((µ 1 -r)α 1 C 1 + rC 2 e µ 1 τ 1 ) 2 . ( 27
)
The next first order optimality condition ∂Costorg ∂p 2 = 0 is written

N 2 w t 2 = µ 2 N 2 w 0 1 µ 1 -r 1 -µ 1 (C 2 +α 1 C 1 )e rτ 1 (µ 1 -r)α 1 C 1 +rC 2 e µ 1 τ 1 + α 1 C 1 µ 1 C 2 e rτ 1 (µ 1 -r)+re µ 1 τ 1 (µ 1 -r)α 1 C 1 +rC 2 e µ 1 τ 1 2 + w 0 2 µ 2 -r 1 - µ 2 C 3 e rτ 2 (µ 2 -r)α 2 C 2 +rC 3 e µ 2 τ 2 + α 2 C 2 (µ 2 -r)µ 2 C 3 e rτ 2 (µ 2 -r)α 2 C 2 +rC 3 e µ 2 τ 2 2 ,
which,using the first condition, can be simplified as

N 2 w t 2 = µ 2 N 2 w t 1 µ 1 + w 0 1 µ 1 -r - µ 1 α 1 C 1 e rτ 1 (µ 1 -r)α 1 C 1 +rC 2 e µ 1 τ 1 + α 1 C 1 µ 1 C 2 e rτ 1 re µ 1 τ 1 (µ 1 -r)α 1 C 1 +rC 2 e µ 1 τ 1 2 + w 0 2 µ 2 -r 1 - µ 2 C 3 e rτ 2 (µ 2 -r)α 2 C 2 +rC 3 e µ 2 τ 2 + α 2 C 2 (µ 2 -r)µ 2 C 3 e rτ 2 (µ 2 -r)α 2 C 2 +rC 3 e µ 2 τ 2 2 . ( 28
)
These two relations ( 27), (28) define the possible critical points p 1 , p 2 of the cost function in this case.

Add floaters to account for task specificities

In the previous subsection, it was shown that the use of a temporary workforce can be optimized across all organizational levels to yield more cost efficiency by reducing the organization idleness and its associated costs. We now complete our model by adding a final layer to achieve what we expect to be a realistic description of an entire organization.

Final problem statement Building on all the previous sections, assume that a company is actually the sum of K business units. In each business unit k ∈ [1, K], the workforce is organized around the same level structure j ∈ [1, L] and we assume it is now structured in 3 parts (see Figure 1)

• a k business unit specific permanent population in level j noted N p,k j which follows the same dynamics as described earlier and which compete for promotion within the business unit.

• a temporary population N t,k j which is used by the business units as buffer to meet its workload requirement and is not used for promotion.

• a permanent "floater" population N f loat j that is not business unit specific and does not compete for promotion. The introduction of "floaters" is of interest because it represents another type of workforce, where workers tend to be generalists that ripe the benefits of an organizational knowledge without competing for promotion. It also illustrates how an organization can benefit from transversal moves on top of vertical ones. Noting p k j ∈ [0, 1] the proportion of the population N k j at level j in business unit k that is under a permanent contract and g k j ∈ [0, 1] the proportion of floaters, the following arises:

N k j = N k j p k j permanent + N k j .g k j floaters + N k j .(1 -p k j -g k j ) temporary , p k j + g k j ≤ 1, N p,k j := N k j p k j , (29)
and we may define for j ∈ [1, L],

N f loat j := K k=1 N k j .g k j . (30)
From a cost standpoint, the following rules at each level j and business unit k apply:

• the cost of each position w p,k j (s) > 0 in the permanent population N p,k j is growing with seniority s,

• the cost of each temporary contract w t,k j is assumed to be constant. We also assume that w t,k j > w p,k j (0) to account that a temporary workforce may come at a premium over a freshly hired permanent employee,

• the cost of the "floaters" w f loat j (s) > 0 is growing with seniority s. This leads to a cost optimization program that is similar to (23),

Cost opt org =min p k j + g k j ≤ 1, k ∈ [1, K], j ∈ [1, L] K k=1 L j=1 N k j .(1 -p k j -g k j ).w t,k j + N k j .g k j .w f loat j (s)e -µ j s ds + w p,k j .ρ p,k j (s)ds . (31)
Figure 1: Representation of a company structure with temporay employees and floaters.

Solution Aapproach

We may analyze this cost functional and introduce the floaters averaged cost w f a j = w f loat j (s)e -µ j s ds and we obtain from (31)

Cost opt org = min

p k j + g k j ≤ 1, k ∈ [1, K], j ∈ [1, L] K k=1 L j=1 N k j . (1 -p k j -g k j )
.w t,k j + .g k j .w f a j + w p,k j .ρ p,k j (s)ds .

We first minimize over g k j with the constraint 0 ≤ g k j ≤ 1 -p k j . Because the functional depends linearly on g k j it is attained either with g k j = 0 or g k j = 1 -p k j depending on the values of w t,k j and w f a j . We find the simplified formula:

Cost opt org = min p k j ≤ 1, k ∈ [1, K], j ∈ [1, L] K k=1 L j=1 N k j . (1 -p k j )
. min(w t,k j , .w f a j ) + w p,k j .ρ p,k j (s)ds .

We are back to a problem, similar to the case of temporary workers in section 3.1, with the additional complexity of the business units.

Algorithms, simulations & results

It is possible to determine numerically the solution of the organizational model ( 1)-( 2) with the size constraint [START_REF] Henderson | Stochastic optimal control of internal hierarchical labor markets[END_REF] for each job class. It allows either to minimize external hiring or to implement practically the above strategies. Either we determine the highest possible internal promotion rates P j (t), j = 1, ..., L -1 and, when there is not enough workers ready to be promoted, minimal external hiring is used to compensate the gap. Or, we impose a proportion of external hiring. The method we use is to solve the system (1)-( 2), after re-writing it under the form ∂ t ρ j + ∂ s ρ j + µ j ρ j + P j (t)ρ j (t, s) = I s<τ j P j (t)ρ j (t, s), ρ j (t, s = 0) = (µ j + P j )N j + P j B j (t), B j (t) = P j (t)

τ j 0 ρ j (t, s)ds, (32)
with the coefficients P j and h j as in (6).

Optimal organizational model (external hirings)

The advantage of this form comes from the explicit-implicit discrete version which is both stable and easy to implement:

   ρ k+1 j,i -ρ k j,i ∆t + ρ k j,i -ρ k j,i-1 ∆s + (µ j + P k j ) ρ k+1 j,i = P k j I s i <τ j ρ k j,i , i ≥ 1, ρ k j,0 = (µ j + P k j )N j + ∆s P k j 0<s i ≤τ j ρ k j,i . (33)
Here we have used a time step ∆t, a s-step ∆s and the superscript refers to time t k = k∆t, the subscript i refers to the seniority s i = i∆s. The only stability condition comes from the CFL restriction on ∆t, namely ∆t ≤ ∆s. We have used here a standard upwind scheme for discretizing the s derivative [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources[END_REF][START_REF] Leveque | Numerical methods for conservation laws[END_REF]. Notice that, assuming by iteration the constraint,

∆s i ρ k j,i = N j , (34)
we find: ∆s i≥1 ρ k+1 j,i -N j + ∆t(µ j + P k j )∆s i≥1 ρ k+1 j,i = ∆tρ k j,0 + ∆t∆s P k j 0<s i ≤τ j ρ k j,i .

This also gives:

∆s i≥1 ρ k+1 j,i + ∆t(µ j + P k j )∆s i≥1 ρ k+1 j,i = N j + ∆t(µ j + P k j )N j ,
the solution of which is ∆s i≥1 ρ k+1 j,i = N j . This means that the discretization (33) preserves the discrete version (34) of the constraint [START_REF] Henderson | Stochastic optimal control of internal hierarchical labor markets[END_REF].

It remains to determine the hiring rates h k+1 j , P k+1 j . To do so, we argue backward on the job index j. After computing ρ k+1 j,i , and departing from j = L and P k+1 L = 0, we impose until j = 1 with

P k 0 = 0, h k+1 j N j + P k+1 j-1 A k+1 j-1 = µ k+1 j N j + P k+1 j A k+1 j , A k+1 j := N j -∆s 0<s i ≤τ j ρ k+1 j,i . (35)
As observed previously, these are 2L -1 parameters for L constraints and several options are possible.

Maximize the internal promotion To achieve this goal, one can choose to take the largest value P k+1 j-1 satisfying (35), imposing however that P k+1 j-1 ≤ P max in order to avoid the subtle interpretation of P k+1 j-1 = +∞. This yields the following formula:

P k+1 j-1 = min P max , µ k+1 j N j + P k+1 j A k+1 j A k+1 j-1
.

If P k+1 j-1 < P max , we choose h k+1 j = 0. If P k+1 j-1 = P max , we choose h k+1 j-1 in order to achieve the equality (35).

Impose a proportion of external hiring We may also choose to impose a given minimal proportion α of external hirin. This means h k+1 j = αP k+1 j-1 + δ k+1 j , with δ k+1 j-1 ≥ 0 nonnegative being used when the α-strategy is not enough to fill the job level j -1. Then, we obtain the formula:

P k+1 j-1 = min P max , µ k+1 j N j + P k+1 j A k+1 j (1 + α)A k+1 j-1
.

If P k+1 j-1 < P max , we choose h k+1 j = αP k+1 j-1 . If P k+1 j-1 = P max , we choose δ k+1 j in order to achieve the equality (35).

Numerical examples

We have tested the algorithm with data describing an internal labor market with a low attrition rate. We concentrate ourselves on its steady state. The Table 1 gives the data N j , τ j and the corresponding maximal internal promotion rate P j . We also compute the proportion of employees ready to be promoted RP j = A j /N j and the average promotion when ready to be promoted

T j = ∆s s i >τ j s i ρ k j,i .
With low attrition, external hiring can be avoided.

Table 2 gives the results with a higher attrition rate and external hiring is needed.

Numerical simulation of a cost optimal structure with flexible workers

Under the proposed organizational model described in section 3.1, it is possible to minimize a company labor cost footprint. The main question behind the minimization problem is about understanding the value of paying a flat premium to outsource some activities versus having to support constant cost increases due to a pay for tenure type of wage scheme. As previously mentioned, this problem is difficult to solve and does not lead to closed formulas. In order to numerically approximate the cost optimal state of the organization, genetic algorithms (referred to as GAs) have been used. GAs consist in search heuristics inspired by the basic principles of biological evolution [START_REF] Scrucca | GA: A package for genetic algorithms in R[END_REF]. In a nutshell, they start with a set of candidate solutions (population), and for each iteration (generation), the better solutions are selected (parents) and used to generate new solutions (offspring). The generation of new solutions can follow a variety of pattern: parent information can be recombined (crossover) or randomly modified (mutation) etc The offspring are then used to update the population (evolution) and replace its weakest members. After each iteration, the overall population gets closer to a minimum. GAs have empirically proven useful to reach global minimum in a number of iterations that is quadratic in the number of sought parameters (see [START_REF] Kumar | Genetic algorithm: Review and application[END_REF] for a review).

In the case at hand, a generic GA implementation in the software R leveraging the genalg package [START_REF] Lucasius | Understanding and using genetic algorithms. Part 1. concepts, properties and context[END_REF] was used. Once natural boundaries were selected for the parameters of interest (namely α j &p j ), a population size of 200 individuals and a total of 250 iterations were fixed with the default implementation of the algorithm. Note that we specified a mutation chance of 10%. The organization parameters that were used are described in Table 3 and echo the previous subsection's simulations. As per wage scheme, a 40% increase per hierarchical level was assumed based on a 35$/h salary at the lowest organizational level (j = 1). For the sake of simplicity, the wage increase rate associated to tenure r will be assumed at 4%. In the developed example, a uniform premium B was assumed across all levels so that w t j = (1 + B).w p j , j = 1, ..., 5.

We have tested three cases, B = 5%, B = 10% and B = 20% which we have compared to the no-contractor scenario, i.e., B = ∞. Results are displayed in the Table 4.

The organizational footprint that is being discussed here can build upon itself internally for a total of 1,13M$/h, but does not reach optimality if no flexibility is considered. Leveraging an internal labor market however leads to a certain degree of stagnation. As such leveraging flexible workforce solutions becomes financially interesting under the previous assumption for a premium B < 20%. The associated financial gain is of course increasing as the contractor premium B gets lower. As such a 5% premium yields more than 12% in savings. Finally note that, in all our experiments, the top of the organization (j = 5) never gets outsourced. However the proportion of external vs internal sourcing (represented by α) increases at level 4 when flexible workforce solutions are cheap.

Conclusion and perspectives

In this paper, a scalable hierarchical organization model was built. Its main features were to account for employee promotional readiness while controlling labor costs. The model shows that organizational inertia may come at a cost depending in its set up. Because of the organization needs and design, employees are indeed potentially waiting for a new internal opportunity while performing the same task. This occurs in situations with low turnover rates and is detrimental for both the company that pay them to stay (henceforth increasing its operational costs) and for the individual.

The proposed solution investigates the notion of temporary labor. Rather than using flexibility to adapt to a volatile workload, the ideas developed in this paper are to optimally contract a part of this organization to create a lean and cost optimal environment for the individuals that belong to the company. The main results shows that it is possible to do so with various flexibility solutions (floaters, temporary workers) and with different model of employee commitment (employee grown internally vs brought from the outside).

If the proposed method is helpful in setting up an organization target, its operationalization would have to be discussed. The primary question that arises lies in the potential use of huge chunks of flexible workers. Activities may not necessarily be available for outsourcing (ex: proprietary methods or processes). Behind this potential core versus non-core activities discussion, the introduction of flexible workers may not call for the same type of workers in the core activities. For instance, for permanent people manager will have to deal with important workforce changes. While coaching their team members, they will have to adapt their efforts between permanent and flexible workers. This organizational method also calls for a robust succession process especially in the promotion attributions that is not described in this paper. We believe this discussion should be skill based and requires the definition of company-wide standards. To this extent, note that a lot of ground has already been covered by principal-agent theoretical approaches (see [START_REF] Prendergast | The provision of incentives in firms[END_REF] for example). Finally another potential operational issue is about performance and productivity. Creating change in the workforce to optimize costs may come at a performance loss if not managed correctly. This calls for caution in potential implementation as the resulting balance may not beneficial to the organization.

 1: (Low turnover) The promotion rates P j , ready to be promoted ratio RP j and waiting for promotion times T j for a company with 5 class jobs of size N j (in thousands of employees), attrition rate µ j = 0.08 per year, and waited to be promoted time τ j in years. Internal hiring is sufficient.


Antoine Blanchard

dir. Joëlle Le Marec

Comment gouverner les sciences ? Telle est l'une des questions posées par les études sociales des sciences et technologies (science studies ou STS), qu'elles posent depuis "la décennie 1970 à la faveur du développement conjoint d'une pluralisation des épistémologues et des rationalités scientifiques, de l'attention aux dimensions sociales du fonctionnement des sciences et d'une mobilisation politique et culturelle interne aux sciences expérimentales et aux sciences de la nature" (p. 11). Comment gouverner les études sociales des sciences ? Dans ce domaine pas véritablement institutionnalisé mais "donnant lieu à de multiples collaborations interdisciplinaires, et à une exploration permanente des frontières entre production académique, action culturelle et politique, et expérimentation institutionnelle" (p. 11), l'agencement des chercheurs, leur pilotage et leur évaluation (pour parler crûment) sont indissociables de leur objet de recherche.

L'ouvrage dirigé par Joëlle Le Marec est issu des premières journées d'étude du cluster 14, centré sur les "Enjeux et représentations de la science, de la technologie et de leurs usages" et créé en 2006 par la région Rhône-Alpes. On reconnaîtra, parmi les contributeurs, les noms de Joëlle Le Marec elle-même, Dominique Pestre (conseiller scientifique du cluster), Florian Charvolin (membre du comité de programme), Dominique Vinck etbonne surprise !deux politistes : Renaud Payre et Gilles Pollet (tous trois membres du comité scientifique). Lors de ces journées, le cluster 14 s'est demandé "comment une communauté de recherche sur les sciences peut-elle développer une réflexion collective sur les dispositifs qui l'organisent, sur le contexte dans lequel elle se développe, sur les rapports à la scientificité qu'elle active, sur les contraintes qu'elle subit et qu'elle se donne ?" (p. 14). Chacune des contributions en tente un aperçu. Dominique Pestre revient sur les principaux apports des études des sciences. Il décrit d'abord le "nouveau régime de production, d'appropriation et de régulation des savoirs et des produits techno-scientifiques" qui s'est mis en place, après les années 1960, "dans le cadre d'une redéfinition globale du monde" (p. 17). Alors que les élites sont myopes face à ces nouvelles réalités, il conclut sur quatre pages en proposant quelques principes normatifs destinés à guider la recherche en STS : "reconnaître la variété des valeurs et des projets humains, comme la variété des moyens et des formes de savoir" ; "réapprendre à dialoguer" et "réapprendre la difficulté et la nécessité des choix en contexte d'intérêts divergents" ; "prendre des risques et 'penser généreusement' les questions les plus complexes" ; "être imaginatif" dans les façons de "'se coltiner' à des questions de valeur et des questions politiques".

Renaud Payre et Gilles Poulet retracent une histoire de la science politique en s'attachant "aux liens existants entre les produits intellectuels et savants et les activités sociales de ceux qui les font advenir" (p. 45). Ils montrent comment la policy analysis est devenue une "science du gouvernement" en se nourrissant de multiples emprunts et échanges avec les mondes de l'administration publique, des think tanks, des fondations, des programmes militaires et d'armement etc. Leur ambition est "de proposer une double réflexivité, du chercheur sur ses objets, ses méthodes, ses disciplines, mais également du chercheur vers le décideur", bien que les leviers de cette action ne sont pas explicités. Dominique Vinck fait de l'anthropologie de laboratoire pour nous raconter comment les chercheurs réagissent aux injonctions contradictoires et au tensions qui traversent la recherche contemporaineentre constat d'un échec personnel, "reconstruction biographique" et "redéfinition de ce qu'ils sont et de ce qui les anime" (p. 75). En une page, l'auteur s'attache à la question de la réflexivité institutionnelle, qu'il interprète comme "le développement d'un savoir partagé et d'une nouvelle vision et visée de l'institution elle-même" capable de mieux accompagner les chercheurs dans leurs usages "des institutions, des normes, des injonctions et des dispositifs qui les visent", pour développer de "nouvelles formes communautaires de socialisation" et des "processus collectifs de différenciation des carrières" (p. 78).

Florian Charvolin, spécialiste de la question des savoirs amateurs en science, montre comment la persistance des sciences à amateurs (astronomie, botanique, entomologie etc.) est un "débordement du cadre mis en place par le capitalisme scientifique du XIXe et XXe siècle" et remet en cause le "professionnalisme salarié" de la recherche scientifique (p. 92). Il montre aussi la part que l'attachement et l'affect ont dans ces sciences de terrain (l'observation d'un oiseau par exemple), et qui "enlève encore un peu de crédit à la coupure stricte entre amateur et professionnel" (p. 88). Enfin, cela ouvre la voie à une réflexion sur la pratique de recherche et l'idée que l'engagement bénévole ne se fait pas "au détriment de la connaissance exacte" (p. 89).

Toutes ces idées sont reprises et prolongées dans le texte ambitieux de Joëlle le Marec qui milite en faveur d'un questionnement des "imaginaires qui sous-tendent les choix des terrains ou des objets" (p. 111) en passant plutôt par l'engagement dans les choix quotidiens des chercheurs que par le commentaire et les textes. Elle promeut la création de communautés de recherches pratiquant la critique les unes des autres et "qui s'informent et se contraignent mutuellement par l'extension de la discussion sur leurs pratiques d'enquêtes et leurs rapports aux terrains" (p. 112), construisant ainsi "un espace d'intersubjectivité autonome par rapport aux lieux et aux temps qui organisent la recherche comme activité de production salariée" (p. 113). L'ouvrage s'ouvre et se referme sur les écrits de Roger Fougères, vice-président du Conseil régional, délégué à l'Enseignement supérieur et à la recherche. Le discours d'un acteur politique, fut il dans l'opposition et à l'origine d'un dispositif aussi unique que celle des clusters de recherche, pâtit souvent d'un manque d'originalité et de profondeur académique, ces textes ne faisant malheureusement pas exception.

La réflexivité institutionnelle qu'annonce l'ouvrage prolonge la réflexivité à laquelle sont habituées les STS, qu'il s'agisse du principe constitutif mis en avant par le "programme fort" de Bloor et Barnes, des réflexions sur la posture axiologique des STS (cf. Dominique Pestre, "L'analyse de controverses dans l'étude des sciences depuis trente ans. Entre outil méthodologique, garantie de neutralité axiologique et politique", Mil neuf cent, n° 25, 2007) ou sur la construction d'objets de recherche à la frontière des sciences dures et des STS (cf. Gilles Tétart et Didier Torny, "« Ça tue parfois mais ce n'est pas dangereux ». Injonction institutionnelle et mobilisation scientifique autour d'un pathogène émergent, Bacillus cereus", Revue d'anthropologie des connaissances, vol. 3, n° 1, 2009). Plutôt qu'un ouvrage définitif sur la question de la réflexivité, le lecteur y trouvera un jalon pour penser la recherche en STS, dans un contexte politique qui compte beaucoup pour les auteurs sorte de complément à l'ouvrage de Jean-Michel Berthelot et coll. : Savoirs et savants. Les études sur la science en France (Paris : PUF, 2005). Surtout, l'ouvrage tourne autour du cluster 14, qui en est à la fois le point de départ et le point de fuite. Malheureusement, ce qui nous est présenté comme relevant "d'une réflexion politique" et témoignant "d'une expérience de réflexivité institutionnelle en cours" (p. 12) ne fait pas toujours suffisamment la preuve des spécificités du cluster 14 comme "large espace communautaire et interdisciplinaire" (p. 12), où s'est imposée une "logique du dialogue" (p. 11) et de la "discussion critique intense et régulière" (p. 12), sans être jamais "entravé par des modèles fonctionnels d'une séparation des registres du cognitif et du social" (p. 12). Où sont les "doctorants de différentes disciplines et différentes équipes de recherche qui réfléchissent ensemble à leur rapport aux contextes de recherche dans lesquels ils sont impliqués" (p. 14) ? Où est l'invention de nouvelles formes de discours scientifique ou de publication qui fassent mieux jouer la réflexivité, la pluridisciplinarité et la pensée en réseau ? Si la juxtaposition des textes trouve sa justification dans l'observation que "la singularité et la pluralité des démarches restent des facteurs essentiels de la qualité scientifiques (sic), et non des faiblesses potentielles dans un contexte de concurrence favorable à une injonction de lisibilité et donc d'homogénéisation des questions et des méthodes" (p. 12), elle déçoit sans doute un peu. À l'image des recherches sur les sciences dont Joëlle Le Marec écrit qu'elles contribuent "à étudier, mais aussi à entretenir et complexifier le feuilletage de plans hétérogènes et contradictoires qui constituent les sciences" (p. 110), l'ouvrage se situe entre plusieurs plans. Portant haut la bannière de la réflexivité institutionnelle et produit d'une nouvelle organisation de la recherche, il donne beaucoup à penser mais peine à convaincre de la mise en pratique de cette réflexivité au sein du cluster 14.