E Ricardo
email: ericardo@ipfn.técnico.ulisboa.pt

F Da Silva

S Heuraux

A Silva1

The effect of plasma turbulence on the plasma position reflectometry measurements at DEMO

Introduction and overview

A variety of plasma diagnostics will be implemented on DEMO to provide the plasma state with necessary accuracy to the control system. In the present devices, the plasma position and shape are measured with the magnetic diagnostics. One of the major issues of its implementation in DEMO is the large integrator drifts that can occur during the long pulses due to the high levels of radiation [START_REF] Biel | Diagnostics for plasma control -From ITER to DEMO[END_REF]. This can lead to a wrong plasma position measurement, putting all the operation at risk.

The prime candidate to complement or substitute the magnetic measurements is Microwave reflectometry [START_REF] Biel | Diagnostics for plasma control -From ITER to DEMO[END_REF][START_REF] Silva | First assessment of microwave diagnostics for DEMO[END_REF]. This diagnostic is capable of measuring electron density profiles in the edge. With the separatrix density is known, its position can thus be inferred [START_REF] Santos | Reflectometry-based plasma position feedback control demonstration at ASDEX Upgrade[END_REF]. This diagnostic is known by its reduced access, robustness and reliability, fulfilling the diagnostic requirements to the operation with the necessary resolution.

The DEMO Plasma Position Reflectometer

The DEMO plasma position reflectometer (DEMO PPR) consists in a system of multi-reflectometers distributed poloidally with the purpose of measuring the plasma position and shape. Since DEMO will operate in a pre-defined scenario, each reflectometer of the system can be optimized according to the expected parameters.

In this work we evaluate the effects of turbulence in the measurement performance of GAP13 (HFS) using REFMULF, a FDTD code for reflectometry simulations [START_REF] Da Silva | Introducing REFMULF, a 2D full polarization code and REFMUL3, a 3D parallel full wave Maxwell code[END_REF].

Measurements are influenced by: • poloidal view • emission angle • antenna assembly • operation plasma scenario • any plasma deformation (plasma displacement, turbulence, etc…) Previous studies:

• at least 15 measurements are required for a reliable [START_REF] Ricardo | Assessment of a multi-reflectometers positioning system for DEMO plasmas[END_REF] reconstruction of the separatrix • the maximum number of measurements is limited by spatial constraints [START_REF] Ricardo | Assessment of a multi-reflectometers positioning system for DEMO plasmas[END_REF] • 100 virtual positions were defined • it has been shown that the error and the power can be optimized at each location [START_REF] Ricardo | Assessment of a multi-reflectometers positioning system for DEMO plasmas[END_REF]. Most of them are optimized aligning the emission perpendicularly to the separatrix.

• The system is stable for plasma displacements of 15cm, different scenarios and antenna assemblies

The effect of plasma turbulence

Results

Future R&D

This study demonstrates the order of turbulence amplitude supported by the system before losing the position information or the signal in a plasma with similar characteristics. In order to prove the reliability of the entire DEMO plasma position reflectometer system, it is necessary • to consider a higher number of samples to improve the statistics;

• to consider other spectrums of turbulence, including profiles from gyrokinetic simulations; • doing the same procedure to the other positions of the system; • to evaluate a more realistic power amplitude with 3D codes; For each amplitude, 400 different turbulent plasmas were generated with a random spectral phase 𝜙(𝑖, 𝑗) profile. The obtained signals were analyzed using an automatized IQ detection method. The statistical analysis of the data describes the reliability of the reflectometer.

  By sweeping the frequency and measuring the O-mode round-trip delay, the plasma density profile [𝑟 𝐹 , 𝑛 𝐹 ] is reconstructed with This model assumes a plane wave being reflected in a density profile varying along the propagation direction. The simulations predict if the measurement error at the separatrix (accuracy of this model) is in the required range of [-1,1] cm. model simulation The turbulence affects the round-trip time delay and the received power. The plasma density is written in the form The turbulence is modulated with a Kolmogovov like spectrum 𝑆, using 𝑘 𝑘𝑛𝑒𝑒 = 1/𝜌 𝑠 ~8 𝑐𝑚. The perturbation dn 𝑒 is defined as: random In this study we assume a frame centered at the separatrix, fixed spectrum and varying turbulence amplitude (1-10%). Simulation setup Antenna setup box Region of interest 3% 10% Before the oversized waveguide there is a taper to a fundamental one 10 cm cavity A great part of the electric is reflected to the antenna A considerable part of the signal is scattered, the detected amplitude is low Plasma-Wall reflections Simulated frames 1-10% 𝑑𝑛(𝑥, 𝑦)/𝑅𝑀𝑆(𝑑𝑛 𝑥, 𝑦 )

  This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. IST activities also received financial support from "Fundação para a Ciência e Tecnologia" through project UID/FIS/50010/2019.

(

  calculated with 𝐵(𝑟 𝑠𝑒𝑝 ) and 𝑇 𝑒 ~3 -4 𝐾𝑒𝑉)

  figure


Nicola Berti
email: nicola.berti.5@studenti.unipd.it

Christian Artigues
email: artigues@laas.fr

Olga Battaïa
email: olga.battaïa@isae-supaero.fr

Romain Guillaume
email: romain.guillaume@irit.fr

Daria Battini
email: daria.battini@unipd.it

Heuristic approaches for scheduling manufacturing tasks while taking into account accumulated human fatigue

Keywords: Scheduling, human factors, fatigue, constructive heuristics, optimization

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Decision-making process behind the scheduling of manufacturing tasks plays a fundamental role in industrial environment. Nowadays its impact on the system performance is more and more investigated due to the availability of manufacturing data [START_REF] Pinedo | Scheduling: theory, algorithms, and systems[END_REF]. The main purpose of scheduling is to organize the manufacturing process in the best way that a performance function is optimized and existing constraints of the manufacturing environment are respected [START_REF] Pinedo | Planning and scheduling in manufacturing and services[END_REF]. Such performance indicators can be expressed in different ways such as flow time, work in progress and throughput [START_REF] Digiesi | The effect of dynamic worker behavior on flow line performance[END_REF], but also makespan, total weighted completion time, maximum lateness, and the total number of tardy jobs [START_REF] Ferjani | A simulation-optimization based heuristic for the online assignment of multi-skilled workers subjected to fatigue in manufacturing systems[END_REF]. However, most of them are profit/cost oriented. Only recently new models were proposed in order to optimize the working conditions such as to reduce physical workload [START_REF] Mossa | Productivity and ergonomic risk in human based production systems: A job-rotation scheduling model[END_REF][START_REF] Otto | Reducing physical ergonomic risks at assembly lines by line balancing and job rotation: A survey[END_REF]. It shows an important gap in scheduling literature where the impact of scheduling decisions on human performance and health is usually ignored [START_REF] Lodree | Taxonomy for integrating scheduling theory and human factors: Review and research opportunities[END_REF].

To fill this gap, a significant effort on modelling of human factors has to be made. This study is dedicated to the integration of one of such factors, human accumulated fatigue in scheduling decision-making process. The objective is to assign (and schedule) a set of given tasks to a number of human operators taking into account their fatigue and their need to have a break.

The remainder of this article is organized as follows. Section 2 presents the existing mathematical models used to evaluate human fatigue in manufacturing. Section 3 provides the new optimization problem we introduce and a small example of the considered problem. Section 4 presents the algorithms that we have developed to solve the introduced optimization problem. Section 5 reports the results of the computational experiments. Finally, Section 6 provides conclusions and suggestions for future research. [START_REF] Lodree | Taxonomy for integrating scheduling theory and human factors: Review and research opportunities[END_REF] stated that in scheduling literature human factors are often ignored although their impact on the performance of the manufacturing system is considerable. The authors even claimed that the scheduling research and the human factors research literature represent two disjoint sets because of the rarity of collaborations between researchers of these communities. However, [START_REF] Grosse | Incorporating human factors into decision support models for production and logistics: current state of research[END_REF] showed that recently the number of studies that try to integrate human factors into decision support models have increased because of the awareness on the impact of these factors on the final performance of the manufacturing systems. In particular, such human factors as physical ergonomic risks are more and more integrated in such optimization problems as assembly line balancing and job rotation as revealed a recent survey [START_REF] Otto | Reducing physical ergonomic risks at assembly lines by line balancing and job rotation: A survey[END_REF].

STATE-OF-THE-ART

Regarding the human fatigue, some studies have already been conducted to identify fatigue causing factors which reduce the performance of workers, and consequently, the productivity of the manufacturing system. In the following, we overview the mathematical models the most frequently used for evaluating fatigue in the literature on manufacturing systems. [START_REF] Eilon | On a mechanistic approach to fatigue and rest periods[END_REF] presents two formulas modeling the loss in production due to fatigue and the gain in production rate due to a recovery time. They found the net gain (call it p) of production between these two conditions. Thus, a rest period is worthwhile if the gain exceeds the loss, then they found the maximum value of p. They also found at what point in time it is most beneficial to introduce a rest period.

Gentzler [START_REF] Gentzler | Analytical and computer modeling of work scheduling[END_REF] aim to maximize the function W (x) which is

W (x) = φ(x)
x + θ where φ(x) is the amount of work produced in one work period, x is the length of time of a work period and the length of a rest period and θ is a constant long enough to ensure that the work rate of the next work period starts at v(0), the unfatigued state. Setting W (x) = 0 states that the optimal time to introduce a rest period is when the average work rate of a cycle is equal to the instantaneous work rate.

Maximum endurance time (MET) is the duration for which a specific body posture (or muscular effort) can be sustained by a worker before his/her capability limits are reached. In [START_REF] Jaber | Incorporating human fatigue and recovery into the learningforgetting process[END_REF], it is a function of the level of the force being applied i.e., f M V C , which is a fraction of the muscle's maximum voluntary contraction (MVC) when performing a specific task. M ET = β 0 • e β•f M V C or power forms, e.g., M ET = α 0 • f -α M V C where α 0 , α, β 0 and β are model-specific parameters and MET is measured in minutes. The model used in [START_REF] Jaber | Incorporating human fatigue and recovery into the learningforgetting process[END_REF] to evaluate fatigue is as follows:

F i+1 (t) = R(τ i ) + (1 -R(τ i ))(1 -e -λ(tn-ti) ) (1)
• R(τ i ) is the residual fatigue carried forward into cycle i + 1; • λ is a fatigue exponent, specifying fatigue rate; • t n is the production time of the cycle i; • t i is determined by projecting the value of R(τ i ) on the fatigue curve as: [START_REF] Glock | An integrated cost and worker fatigue evaluation model of a packaging process[END_REF] use the description of fatigue accumulation above for developing a biomechanical model to estimate the expected fatigue-recovery parameters in manual packaging process. Moreover, the authors perform a sensitivity analysis of the fatigue parameter λ evaluating how the value of this parameter changes for modifications of %M V C by 10% and 20% (plus/minus).

t i = -ln(1 -R(τ i )) λ (2)
The study of [START_REF] Perez | Discrete event simulation as an ergonomic tool to predict workload exposures during systems design[END_REF] proposes the following behavior of fatigue accumulation while trying to simulate a dynamic process using static MET models:

F t = f F t-1 + ∆t M ET i + ∆t R j (3)
The fatigue accumulation after task i (F i ) is calculated as follows:

F i+1 = (F i-1 + f F i ) • (1 -f R i ) (4)
• F i is the fatigue rate after task i ;

• f F i is the fraction of fatigue contribution per task i ;

• f R i is the fraction of recovery received after task i ;

• R i is the recovery time (in seconds) needed after task i to bring fatigue down to 0.

In [START_REF] Givi | Modelling worker reliability with learning and fatigue[END_REF], the fatigue accumulation in the "learning-forgetting-fatigue-recovery model" (LFFRM), is modelled by an exponential model of the following form:

F (t i ) = R(t i-1 ) + (1 -R(t i-1 )(1 -exp(λt i ))) (5) where F (t) is the accumulated fatigue over time t i , R(t i-1 )
is the residual fatigue after a break carried along cycle i -1, and λ is the fatigue index describing the severity of the work performed, t i is the length of the current production cycle and t i-1 is the length of the previous one. At time t = 0, fatigue is zero and as time increases, fatigue converges asymptotically to 1. A recovery function can be also defined. [START_REF] Fruggiero | The role of human fatigue in the uncertainty of measurement[END_REF] propose the following model to evaluate the amount of physical stress L t accumulated by a worker L t after time t.

L t = f i • t i - L max RA i • M ET i • b i (6)
where f i is the amount of physical stress for executing task i of duration t i , b i is the break time following task i, L max is the maximum fatigue index, RA i (Rest Allowance) required for task i as a fraction of M ET i .

Ferjani et al. ( 2017) associate a penalty coefficient d j to each machine j to model the difficulty of work on that machine, such that 0 ≤ d j ≤ 1. This coefficient d j expresses the speed of fatigue accumulation. The value of G i (θ) characterizes how the level of fatigue of a worker increases. G i (θ) = 1-e -dj •θ where θ is the duration. F i (t i ) is the worker initial fatigue at time t i . ∆ ij (t i , t) expresses the increase of fatigue generated during a new task on machine j, between its beginning t i and the current time t, which can be expressed as follows:

∆ ij (t i , t) = (1 -F i (t i ))(1 -e -dj (t-ti) ) (7
) As a consequence, from time t to t i , the level of fatigue is updated as follows:

F i (t) = F i (t i ) + ∆ ij (t i , t) (8
) Since the aim is to correct the theoretical processing times to take the fatigue level into account, they suggest that these processing times increase according to a logarithm function, i.e. T kj (t i ) represents corrected processing time of the task waiting in the k-th position on machine j to which the worker i can be assigned at the instant t i , as follows:

T kj (t i ) = T kj (1 + δd j (ln(1 + F i (t i )))) (9)
The parameter δ represents the influence of fatigue. Typically, it allows a maximum degradation time to be considered, given that when it is zero then fatigue does not influence the task duration. They also develop an heuristic to dynamically assign workers to machines so as to reduce the mean flowtime. The proposed approach takes the impact of fatigue into consideration. [START_REF] Visentin | Fatigue accumulation in the assignment of manual material handling activities to operators[END_REF] measure the physical fatigue through the energy expenditure rate. Considering their exponential trends, the fatigue F (t w ) and the recovery R(τ ) related to a certain task i are modelled as follows:

F (t w ) i = Ėwi + ( ĖR -Ėwi ) • e -λtw i (10)
The value of F (t w ) i is linked to:

• the duration of task w is t w ;

• the energy expenditure rate at rest ĖR (The energy expenditure rate at rest ĖR , as stated by [START_REF] Battini | Ergo-lot-sizing: An approach to integrate ergonomic and economic objectives in manual materials handling[END_REF], is 1.86 Kcal/min for a standard worker); • the maximum energy expenditure rate Ėwi ; • λ, which is the parameter of fatigue accumulation of the specific operator.

If after task i the operator has to perform consecutively another task i + 1 without taking a rest, the fatigue is accumulated. Moreover, the task can have a higher or a lower value of Ėwi than the previous one. Considering this, the trend of fatigue accumulation for this second task is formulated as:

tw i 0 F (t w ) i = Ėwi • t wi +                  +(F (t w ) i-1 -Ėwi ) • e -λ•tw i -λ + (F (t w ) i-1 -Ėwi ) λ if Ėwi ≥ F (t w ) i-1 +(F (t w ) i-1 -Ėwi ) • e -µ•tw i -µ + (F (t w ) i-1 -Ėwi ) µ if ĖWi < F (t W ) i-1 (11)
F (t w ) i : fatigue reached after the end of the first task. Parameters λ and µ are related to the personal characteristics of the operator, which respectively indicate how the fatigue is accumulated (λ) and alleviated (µ). Moreover, with this RA (rest allowance) it is possible to calculate the total time the operator needs to execute all tasks including the time necessary for the recovery:

t tot = n i=1 t wi + k τ r k + RA • ( n i=1 t wi + k τ r k ) (12)
where τ r k is the time necessary to reach 1.86 Kcal/min. The value of RA is estimated as a percentage of the working time the operator needs to stay at ĖR i.e. at total recovery.

Baucells and Zhao (2018) present a model for marginal productivity decreasing because of fatigue, where Y 0 ≥ 0 is the initial fatigue level, and γ > 0 is the fatigue recovery rate:

Y (t) = Y 0 • e -γt + t 0 e -γ(t-s) dX(s) (13)
The literature analysis shows that except the heuristic developed in [START_REF] Ferjani | A simulation-optimization based heuristic for the online assignment of multi-skilled workers subjected to fatigue in manufacturing systems[END_REF] and used to minimize the flowtime, no scheduling procedure takes into consideration the fact that operators' performances often decline because of the fatigue generated during task execution.

This study proposes a set of constructive heuristics capable of taking this into account while scheduling a set of tasks with the objective to minimize the makespan. The optimization problem is formally defined in the next section.

PROBLEM DEFINITION

We consider the problem of scheduling of a given set of tasks in a manufacturing cell. The set of operators with their personal characteristics (λ and µ) are known. The set of tasks w ∈ W to be scheduled with their energy expenditure rate E w is known as well. Index i is used to enumerate the tasks performed in sequence by the same operator. The operators perform their tasks independently.

The typical scheduling constraints have to be respected:

• All tasks have to be scheduled;

• Each task has to be scheduled once;

• Each operator can perform one single task per time;

• Preemption of tasks is forbidden.

The objectives considered is this study are the makespan and the fatigue accumulation by the operators. Each time an operator accumulates the fatigue to the level of rest allowance, a break is imposed which duration has an impact on the final makespan. The fatigue models used are discussed here below.

Fatigue evaluation

We use a model derived from the proposition of [START_REF] Visentin | Fatigue accumulation in the assignment of manual material handling activities to operators[END_REF], where fatigue is accumulated after each task i performed according to eq. 14.

tw i 0 F (t w ) i = tw1 0 F (t w ) 1 + • • • + tw i tw i-1 F (t w ) i (14)
The fatigue caused by the first task after a total recovery break is calculated with formula 15: 16) is used to evaluate the accumulation of the fatigue after task i taking into account the difference between the energy expenditure rate of the current and previous tasks.

tw1 0 F (t w ) 1 = Ėw1 •t w1 +( ĖR -Ėw1 )• e -λ•tw 1 -λ + ( ĖR -Ėw1 ) λ (15) Then, formula (
tw i tw i-1 F (t w ) i = Ėwi • t wi +                  +( Ėwi-1 -Ėwi ) • e -λ•tw i -λ + ( Ėwi-1 -Ėwi ) λ if Ėwi ≥ Ėwi-1 +( Ėwi-1 -Ėwi ) • e -µ•tw i -µ + ( Ėwi-1 -Ėwi ) µ if Ėwi < Ėwi-1 (16)
The main difference between the model used in [START_REF] Visentin | Fatigue accumulation in the assignment of manual material handling activities to operators[END_REF] and our model is between equations ( 11) and ( 16). In our model, the fatigue is calculated for each task i and the accumulated fatigue is evaluated by ( 14). The value of accumulated fatigue is generated by the sum of the different contributions of tasks performed by the same operator. It is a key variable of our model used in order to guarantee the fairness in task distribution among operators.

Each time an operator accumulates the fatigue to the level of rest allowance, a break is imposed which duration is calculated according to formula:

τ r k = -ln ( ĖR ) + ln F (t w ) k µ (17)
It represents the time necessary to reach 1.86 Kcal/min from the value of energy the operator reaches when the threshold is exceeded. While the operator is restoring energy until the value of ĖR , a small amount of fatigue has to be considered in the total value of accumulated fatigue. This value is calculated with the formula:

tw k +τr k tw k R(τ ) k = F (t w ) k -µ •e -µ•tw k •(e (-ln F (tw) k )+ln ( ĖR ) -1)
(18) This amount of fatigue "accumulated during a break" can be considered as negligible, a fixed value is added each time an operator needs a break, for a precise estimation of the total accumulated fatigue.

In the next section, we present the constructive heuristics that we developed in order to find a compromise between the total makespan and the distribution of the fatigue among the operators. In the next section, we study the performances of these rules in a numerical experiment.

CONSTRUCTIVE ALGORITHMS

COMPUTATIONAL EXPERIMENTS

Example of solution

Figure 1 presents a short example of scheduling of 12 tasks and 5 operators following the rule 1 for task selection and rule 1 for operator selection. Parameters of tasks are given in Table 1. The personal characteristics of 5 operators are given in Table 2 as well as the makespan and total accumulated fatigue obtained as results of scheduling. The maximal difference in the accumulated fatigue among the operators is equal to 358.85-127.01=231.84.

Instance generation

Fifteen problem instances have been randomly generated for this numerical experiment. Each problem instance contains 24 tasks to be assigned to 5 operators. Each operator has different values of parameters λ and µ expressing their personal characteristics, they are reported in Table 2. The values of Ėw have been randomly generated on interval [2-8] Kcal/min. The duration of tasks has been also generated randomly on interval [5-20] minutes. These 15 problem instances were solved with heuristics obtained by all combinations of task and operator selection rules, i.e. with 8 × 9=72 heuristic rules. The obtained results are reported in Tables 34567.

Results and analysis

Table 3 reports Table 3. Rules that provided the best makespan / accumulated fatigue

We can notice that the best rules for task selection are T2 and T6 for makespan and T1 and T5 for accumulated fatigue. These rules give the priority to the most longest (T1, T5) or to the most difficult tasks (T2, T6). The best rules for operator selection are O2 and O8 both for makespan and accumulated fatigue (they select as the first criterion the operator with minimum makespan and the minimum accumulated fatigue, respectively). Rule O9 performed well only for makespan.

Table 4 reports, for the rules that provided at least one best solution ether for makespan or accumulated fatigue: the number of best obtained values (NBM and NBF) respectively for makespan and accumulated fatigue and maximal, average and minimal values of the gap from the best values for makespan and accumulated fatigue.

We can notice that different heuristic rules are the most efficient for different problem instances: the maximal frequency of best makespan is 4 and accumulated fatigue 3. These values are reached by the T202 combination (in green), which gives priority to the most difficult task and assigns it to the operator with the minimal makespan. However, the average gaps are quite small for all reported combinations. We can also notice that the majority of heuristic rules were capable of providing both best values Table 4. Gaps from the best values of makespan and accumulated fatigue of makespan and accumulated fatigue. This means that the working conditions of workers can be improved with no or little impact on total completion time due to a fairer task distribution among the workers. solved. The best values are highlighted in green. One sees again the good results of the T2O2 combination (along others). Especially O8 and O9 rules that considers the selection of the operator with the minimum accumulated fatigue as well as T5 (that selects the most difficult task as the secondary criterion, the main criterion being the max task duration) and T6 (variant of T2 with max task duration instead of min task duration as the secondary criterion) perform also well.

Table 8 reports the maximal, average and minimal values of differences in the accumulated fatigue among all operators in the same problem instance. Not surprisingly, the rules that lead to the minimum of maximal fatigue accumulated lead also to the fairest distribution of fatigue.

Table 7. Difference between Max value of Accumulated Fatigue and Min value of Accumulated Fatigue

Multi-start heuristic

Other results have been provided thanks to the creation of a multi-start heuristic which has been obtained from the initial model. A random selection criteria is used to choose an operation selection rule and an operator selection rule every time a new task is assigned in order to find better performance results in terms of makespan reduction and lower value of maximum accumulated fatigue. Random task selection have been carried on for 1000 iterations for each of 15 instances already used previously.

Then, it is presented a comparison between multi-start heuristics: one which can select randomly among 24 (4×6) best selection rules already founded before in the article (4 best rules to select operation and 6 best rules to select operator) and the other which can randomly select among all 72 available rules (8 × 9).

Table 8. Multi-start heuristic results with 72 rules available to be selected Table 9. Multi-start heuristic results with 24 rules available to be selected The conclusion is that the model which run with only 24 best selection criteria provides better results than the model which run with all 72 rules of selection.

In order to show that multi-start heuristic, which is allowed to randomly choose every time a new task has to be assigned among 24 possibilities, can provide better results than the heuristic initially presented in the article which has to follow the same criteria until the assignment of all tasks, following table are presented to demonstrate that for every single parameter (makespan, accumulated fatigue and difference in accumulated fatigue) multi-start heuristic is able to present at least one better solution in almost all instances. In this study, we propose a new scheduling problem where human fatigue affecting operator performance is taken into consideration while assigning and scheduling a given set of tasks to a set of operators of a manufacturing cell. All operators work in parallel and execute the tasks independently each from other. All tasks are characterized by a certain level of difficulty having a different impact on each operator taking into account his/her personal characteristics. Each time the fatigue level exceeds a defined level, a break is imposed to the operator at the end of the task. Its duration contributes to the total makespan. In order to solve this scheduling problem with the objectives to minimize the makespan and the maximal accumulated fatigue, 72 constructive heuristics have been developed and evaluated through a numerical experiment. The obtained results showed that the best task selection rules mix the most longest and the most difficult task selection. However, this approach has to be further developed in several directions. First of all, the heuristic rules can be integrated in a metaheuristic or hyperheuristic approach. Secondly, the impact of variability of personal characteristics of operators and difficulty of their exact measurement should be evaluated and this uncertainty should be integrated in the problem model.

Algorithm 1

 1 Calculate M akespan max op, task -read input data L is the list of unscheduled tasks, initialized with the set of given tasks while L is not empty do j ← select Task (L, op, task) o ← select Operator (j, op, task) Schedule (j, o, L, op, task), Update L end while return M akespan ∀o ∈ op return AccumulatedF atigue ∀o ∈ op The pseudo-code of the general algorithm is represented by Algorithm 1. This algorithm uses 8 different rules for selecting next task. All of them use two criteria, if a tie occurs according to the first criterion, the second one is applied to select a task. (1) Max task duration & Min task energy (2) Max task energy & Min task duration (3) Min task duration & Min task energy (4) Min task energy & Min task duration (5) Max task duration & Max task energy (6) Max task energy & Max task duration (7) Min task duration & Max task energy (8) Min task energy & Max task duration Algorithm 1 uses 9 different rules for selecting next operator: the criteria are applied in the specified order. (1) Min operator makespan & Min value of λ (2) Min operator makespan & Max value of λ (3) Min number of tasks already assigned & Min operator makespan & Min value of λ (4) Min number of tasks already assigned & Min value of λ & Min operator makespan (5) Min number of tasks already assigned & Min value of fatigue & Min value of λ (6) Min number of task already assigned & Min operator makespan & Max value of λ (7) Min value of fatigue & Min Number of tasks already assigned & Min value of λ (8) Min value of accumulated fatigue & Min number of tasks already assigned & Max value of λ (9) Min value of accumulated fatigue & Min number of tasks already assigned & Min value of λ & Max value of µ

Figure 1

 1 Figure1shows the obtained schedule of 12 tasks including the breaks required by Rest Allowance when the energy threshold is exceeded (value > 4.3). Final short breaks show how long should be the rest to return the energy expenditure rate to the initial value of 1.86 Kcal/min.

  the rules that provided the best values of makespan (BestM) and accumulated fatigue (BestF) for P1-P15 problem instances solved. The rules are encoded in the following way: T1O1 corresponds to the first rule of task (T) selection and to the first rule for operator (O) selection. The best results summarized for task and operator selection rules are also reported.

Table 1 .

 1 Task and Operators' parameters

	Task	1	2	3	4	5	6
	tw	15.0	8.7	12.6 17.5 6.8 10.5
	Ėw	3.2	5.3	2.7	7	5.2	6.1
	Task	7	8	9	10	11	12
	tw	9.7	13.6 18.3 15.9 7.7 16.5
	Ėw	4.8	4.1	2.6	8.1	6.3	4.9
	Operator	λ	µ	Makespan Accumulated fatigue
	1	4.0 1.5	39.78		358.85
	2	3.5 2.0	38.32		127.01
	3	2.5 1.5	41.22		299.47
	4	6.0 2.5	39.28		315.73
	5	4.5 1.0	43.01		158.52

Table 5

 5 reports the maximal, average and minimal values of makespan among all problem instances solved with the heuristic rules proposed.

Table 5 .

 5 Makespan obtained

Table 6 .

 6 Max values of Accumulated FatigueTable 6 reports the maximal, average and minimal values of accumulated fatigue among all problem instances

Table 10 .

 10 Makespan comparison between 72 heuristics and multi-start heuristic

Table 11 .

 11 Accumulated fatigue comparison between 72 heuristics and multi-start heuristic

Table 12 .

 12 Difference in accumulated fatigue comparison between 72 heuristics and multi-start heuristic 6. CONCLUSION


Youssef Bouanan
email: youssef.bouanan@ims-bordeaux.fr

Mathilde Forestier
email: mathilde.forestier@gmail.com

Judicael Ribault
email: jribault@gmail.com

Gregory Zacharewicz
email: gregory.zacharewicz@ims-bordeaux.fr

Bruno Vallespir
email: bruno.vallespir@ims-bordeaux.fr

DEVS-BASED FRAMEWORK FOR MESSAGE DISSEMINATION IN MULTI-LAYER NETWORKS

Keywords: Social Simulation, DEVS Formalism, Social Network Diffusion Process

Many research efforts focus on the definition of models and techniques to simulate and predict the reaction of individuals face to an information. In this context, great benefits could derive from the exploitation of individual's personality and cultural values in the diffusion model process. In this paper we describe a new architecture for agent-based model using the DEVS (Discrete EVent System Specification) framework and show how this architecture is flexible and can serve to simulate the dissemination process. In more details, we define a set of models of individuals characterized by a set of state variables to represent the behavior of an individual and the mesh between the individuals within a multi-layer social network. Then, we introduce the platform architecture, specifically designed to simulate message propagation in a multilayer network. In the end, a military scenario of message diffusion during a stabilization phase is used to test our DEVS models on the platform and the relevancy of the simulation results.

INTRODUCTION

Nowadays, many researchers are interested in developing new and more efficient systems for social simulation. The issues explored include psychology, organizational behavior, sociology, political science, economics, anthropology, geography, engineering, archaeology and linguistics (takahashie et al, 2007;[START_REF] Bouanan | Using DEVS and CELL-DEVS for Modelling of Information Impact on Individuals in Social Network[END_REF]. In the field of military defense, modeling the human operators in a system is complex because it has to represent the behaviors in critical situations. So, military simulation systems have to support detailed analysis of individual and organizational performance. The SICOMORES project seeks to create a given ethno-cultural community in an artificial society framework. It aims to formalize and simulate the propagation of the psychological effects in a multi-layer social network.

Social networks play an important role in studying the propagation of information, innovation, ideas, and influence among its members. When information appears -for example, the use of cell phones among students, the adoption of a new information system within the enterprise, or the rise of a political movement in an unstable society: it can either die out quickly or makes significant inroads into a population. Network diffusion process allows us to understand the dynamics and the propagation of information in social network. These networks can consist of individuals, group of person or organizations. The interactions could be done by physical contact, collaboration, religious meeting or some forms of verbal or written communication depending on the situations. This network diffusion process is used to raise numerous research issues such as contamination minimization problem or influence maximization problem (IMP). Agent-based modeling approach is used to model complex systems composed of interacting and autonomous 'agents'. Agents have behaviors, often described by simple rules and interactions with other agents, which in turn influence their behaviors. By modeling agents individually, the full effects of the diversity that exists among agents in their attributes and behaviors can be observed as it gives rise to the behavior of the system as a whole. This paper begins by representing the agent-based model approach, The DEVS formalism and the dissemination process in social networks. Then, it provides model of individuals with DEVS characterized by a set of attributes and it presents our architecture to simulate the diffusion process in a multi-layer network using DEVS. At last, the final part concerns our experiments and the conclusion.

BACKGROUND AND RELATED WORK

Agent-Based Social Simulation

Agent-based social simulation (ABSS) consists of social simulations that are based on agent-based modeling, and implemented using artificial agent technologies [START_REF] Davidsson | Agent based social simulation: A computer science view[END_REF]. ABSS is a scientific discipline concerned with simulation of social phenomenon, using computer-based multi-agent models. In these simulations, people or group of people are represented by agents. Different agent-based platforms have been used for implementing agent-based social simulation [START_REF] Tobias | Evaluation of free Java-libraries for social-scientific agent based simulation[END_REF]. NetLogo is the highest-level platform, providing a simple yet powerful programming language, built-in graphical interfaces, and comprehensive documentation. MASON, Repast, and Swarm are "framework and library" platforms, providing a conceptual framework for organizing and designing Agent-Based Models (ABMs) and corresponding software libraries. Most of the simulation frameworks are oriented towards applications in the field of artificial intelligence and are not sufficiently dedicated for social scientific applications. In our study we develop a new library based on DEVS formalism in which agents can be modeled as 'free' and complex objects that represent such as institutions as social scientific models do.

DEVS Formalism

The DEVS formalism for modeling and simulation [START_REF] Zeigler | Theory of modeling and simulation: integrating discrete event and continuous complex dynamic systems[END_REF] is based on discrete events modeling. It tends to represent more mathematical oriented notations consisting of sets and functions in algebraic representation. The sets in DEVS formalism specify the potential states, inputs, and outputs of the model; the functions describe the transitions either from outside input or from the time expiration of the state.

The DEVS formalism provides a framework with mathematical concepts based on the sets theory and systems theory to describe the structure and the behavior of a system. With DEVS, there is an explicit separation between a model and its simulator. Once a model is defined, it is used to build a simulator (i.e. a device able to execute the model instructions). Two kinds of models are proposed: the atomic models, which describe behavior, and the coupled models, which describe a hierarchy. The DEVS formalism supports an open approach to formalism extension, allowing the researcher to explore new extended or specialized formalism [START_REF] Zeigler | DEVS formalism and methodology: unity of conception/diversity of application[END_REF]. These extensions facilitate the development of models for various applications in many different domains such as biology, engineering, and sociology. For example, Barros proposed the dynamic structure DEVS (DSDEVS) formalism which allows changes in model structure during execution [START_REF] Barros | The dynamic structure discrete event system specification formalism[END_REF]. Chow and Zeigler proposed the parallel DEVS (P-DEVS) for parallel execution benefits (Show and Zeigler 1996). From a network modelling perspective, Uhrmacher proposed Multi-Level-DEVS (ml-DEVS) which supports an explicit description of macro and micro level [START_REF] Uhrmacher | Combining micro and macro-modeling in devs for computational biology[END_REF]) and Wainer proposed the cell-DEVS formalism which is a combination of cellular automata and DEVS that allows the implementation of cellular models with timing delays [START_REF] Wainer | N-dimensional Cell-DEVS models[END_REF]. There are a large number of DEVS based simulators. It is difficult to compare the performance or the implementation of these tools. The DEVS standardization group lists on his web site the most used DEVS tools known by the DEVS community (DEVS 2015). Among these platforms, the proposed DEVS models in this study is implemented in the VLE (Virtual Laboratory Environment) because it supports multimodeling simulation and analysis by using recent developments in the theory of modeling and simulation proposed by Zeigler. It is also possible to perform statistical analysis of results thanks to a plug-in that allows communication between VLE and R [START_REF] Quesnel | VLE: a multimodeling and simulation environment[END_REF].

Diffusion process in social network

Social network consists of interconnected individuals linked by informal patterned flows of information and communication that are described as social ties [START_REF] Tenkasi | Social networks and planned organizational change the impact of strong network ties on effective change implementation and use[END_REF]. Some modern examples of social networks include online social networks, where vertices are user accounts and edges represent a relationship between accounts (e.g., friendship, coworkers), and communications networks, where vertices represent email addresses or phone numbers, and edges represent e-mails sent or phone calls between vertices. Social networks play an important role in studying the propagation of information, innovation, ideas, and influence among its members. An idea will appear -for example, the use of cell phones among students, the adoption of a new information system within the enterprise -and it can either die out quickly or make significant inroads into the population. The network diffusion processes have a long history in social sciences [START_REF] Rogers | Diffusion of innovativeness[END_REF]. With the advent of sufficient storage and computational power, this network diffusion process became an emerging research area in computer science [START_REF] Domingos | Mining social networks for viral marketing[END_REF]. Propagation models are designed to reproduce phenomena observed in social networks with applications in viral marketing, spread of disease and diffusion of ideas and innovations. Most models proposed recently are extensions from the independent cascade (IC) [START_REF] Goldenberg | Talk of the network: A complex systems look at the underlying process of word-of-mouth[END_REF] and the linear threshold models (LT) [START_REF] Granovetter | Threshold models of collective behavior[END_REF]. The two models characterize two different aspects of social interaction. The IC model focuses on individual (and independent) interaction and influence among friends in a social network. The IC models can also be identified with the so-called Susceptible/Infective/Recovered (SIR) model for the spread of disease in a network [START_REF] Bailey | The mathematical theory of infectious diseases and its applications[END_REF]). The LT model focuses on the threshold behavior in influence propagation, which we can frequently relate to; when enough of our friends bought a new phone, played a new computer game, or used new online social networks, we may be converted to follow the same action.

Human behaviour

Human behaviour modelling as individuals in groups and in societies is the subject of several fields of research: social science, economics, epidemiology and military service because it has such an important role in many aspects of daily life. Scientific literature abounds in heterogeneous and highly specialized, theoretically founded concepts of human cognition, emotion and other behaviour aspects. A few related works have provided DEVS models of human behaviour that we use with slight modifications: [START_REF] Seck | Use of a dynamic personality filter in discrete event simulation of human behavior under stress and fatigue[END_REF]) present a DEVS based framework for the modelling and simulation of human behaviour with the influence of stress and fatigue; [START_REF] Faucher | PSYOPS and CIMIC operations: from concepts to G-DEVS models[END_REF]) proposed a first approach using G-DEVS formalism for Civil-Military Cooperation actions (CIMIC) and Psychological actions (PSYOPS), which are actions of influence that take precedence over combat.

CONTRIBUTION

In this section, we propose an agent-based model for the information diffusion in a Multi-layer network.

Problem statement

Several research studies attempted to model and simulate the influence diffusion in a social network. However, the focus is on one layer network (one link or relationship between persons) e.g., the message disseminates only within friends. In addition, they did not consider the individual and cultural factors on diffusion process; the message disseminates in the same way between two friends or two coworkers. Actually, the way how people communicate is highly dependent of the person who people are talking to. In the frame of SICOMORES project, we aim to provide solutions to artificially generate multi-layer social network of realistic population and simulate the effects of information on population with a propagation algorithm of the effects across networks. The challenge is to improve the realism of socially intelligent agents and take into account the impact of individual's personality and cultural values in the diffusion model process. The SICOMORES system can be useful in the domain of military simulation for training and education.

DEVS-based Social Agent Model

Framework There are several methods to study the information propagation in social network, such as complex network analysis [START_REF] Ma | Complex Network Analysis[END_REF], cellular automata [START_REF] Goldenberg | Talk of the network: A complex systems look at the underlying process of word-of-mouth[END_REF]) and agent based modeling [START_REF] Smith | Reconsidering models of influence: the relationship between consumer social networks and word-of-mouth effectiveness[END_REF]. Based on our needs, we use agent-based models and simulations for this study. Computational models and simulations, especially agent based ones, have been widely used to study a variety of social, organizational and natural phenomena. Agent-based models are capable of simulating macro-level structures resulting from micro-level interactions of heterogeneous agents within complex systems. We use a low-level language (DEVS) to define a set of models of individuals characterized by a set of state variables to construct their behaviors and the mesh between the individuals within an MSN. Then, we develop an architecture and a methodology that promotes modularity and reusability of these models as a package. We implement these models in a virtual laboratory based on DEVS formalism (VLE). This simulator has been tested and validated with several scenarios [START_REF] Quesnel | The Virtual Laboratory Environment-An operational framework for multi-modelling, simulation and analysis of complex dynamical systems[END_REF]).

In the context of this study, we need modularity and reusability. We have to perform the simulations by varying the inputs but also the algorithms of the agents. For example, diffusion algorithms may change depending on the targeted population, the type of message or network on which the message travels. In order not to have to fully develop a model for each combination of algorithms we want to set in the agent, we decided to opt for a modular approach. This approach needs some methodology to be carrying through. Section 3.2.1 presents the architecture of simulation and methodology used. Section 3.2.2 describes the proposed solution in terms of modular and reusable model design.

Architecture and methodology

Figure 1 presents the simulation architecture we have developed.

Figure 1: Agent simulation architecture

Pre-simulation: Before making a simulation, we build a new experiment and develop the simulation models we need if they do not already exist in the repository. An experiment includes one or several groups. A group includes many individuals. A graph represents all relations between individuals. We opted for the storage of information in a database so that each experiment is easily accessible and re-playable. The repository contains all the models available to execute the simulations, i.e., all the servers and proxies that will be used by the generator to produce the simulation model. Simulation: The simulation starts with the experiment to simulate, as well as servers and proxies to use. In this way, for the same experiment, we can test different behavior algorithms. The generator connects to the database to retrieve all the information from the experience to execute. The generator instantiates agents and related links. The DEVS simulation is executed using VLE and produces a result set in the form of a file. Post-simulation: The R script post-processes the simulation result. The result file is used to visualize the course of the simulation and to develop an analysis and conclusion. The analysis can lead to a new simulation, i.e., makes a new pre-simulation, simulation and postsimulation.

DEVS model

Figure 2 presents the solution we have developed to take advantage of a low-level framework allowing the development of algorithms that take into account the specificity of each individual. This solution also considers our needs for modularity and reusability. Our simulation is based on a static multi-layer graph (1). Specifically, we generate a Multi-layer Social Network (MSN). Our MSN models individuals with attributes (such as gender, age and opinion) and several relations linking them (such as family relation or friendship relation). The idea behind the use of an MSN is to assume that people do not communicate in the same way according to people with whom they are talking, e.g., families or friends. In the second phase of the build run (2), we set the architecture based on a proxy/server framework to model the several layers of relations based on DEVS specifications. Each node of the MSN is modeled by a server-node and has as much proxies that there are layers. The node a in the multi-layer graph becomes the servernode S a in phase (2) and detains P a0 to P an proxies. The sever-node contains the individual behavior, i.e., how an individual receives an information and how this information can change his state. The proxies associated to a sever-node contain the rules of message dissemination for each layer. Finally, the phase (3) corresponds to the runtime: using an action scenario of simulation, and a simulation engine we simulate the propagation of information in the multi-layer social network. Each agent is created at run-time by a generator that has access to the experiment data. This generator initializes each individual, i.e., each server and proxy, and binds the server with all of its proxies. Then, each proxy is bounded to several proxies depending on the graph between individuals.

Formalization of human behaviour

In the agent-based model, individuals or group of individuals are represented as agents. Each agent is described by a set of attributes:

• Static attributes: gender, social status, religion, age class, ethnicity, leadership and language. • Dynamic attributes (variables): opinion, interest, un/satisfied-needs.

Static attributes are intrinsic or unchanged parameters, i.e., time has no effect on them. Dynamic attributes evolve with time or events. For example, individuals can be reached or not by the information depending on its opinion and the social network configuration. We use DEVS specifications to describe the human behavior. As we presented in section 1.1, DEVS is a well-defined formalism which has numerous advantages over other formalism in the modeling of complex dynamic system. The following notations describe the server model. We do not describe detailed operations of this model; we just explain the role and interface of the server model (Server-Node).

Figure 3: DEVS specification of individual model

Figure 3 describes the message influence on the individual behavior and potentially its dissemination using the DEVS specifications. The first phase is used to configure and initialize the agent's attributes. Then, when the agent is in the "IDLE" phase and if it receives an external event from another agent on port "In_info" (In_info? Packet), it goes into phase "phase_0". Here, the agent calculates the trust factor α(i,j) between him and the sender depending on the sender's religion, language and age class. If the trust factor is higher than a predefined threshold, the receiver goes into phase "phase_1" else it returns to "IDLE". Then, if the message strength is still strong enough, the receiver goes into phase "phase_2" else it returns to "IDLE". This message creates an impact on the individual, and eventually its behavior depending on the agent's opinion and the relationship between him and the sender. After a period, processing-time, the receiver transmits the message on its ego-network. After the contact between receiver and sender, the receiver's variables (opinion, interest, satisfaction) change according to the message content, the sender and cultural factors.

Diffusion process in multi-layer social network

The multi-layer social network is defined as a graph G=(V,E,L) where V is a set of nodes (DEVS model), L is a set of labels (also called layers), E is a set of labeled edges (DEVS coupling relation), i.e., the set of triples (u,v,d) where u,vÎV are nodes and dÎL is a label. Each node represents an individual with characteristics who can share information on one of its networks L. Each individual who receives a message and does not have enough interest in it becomes "passive'' and the message do no spread longer. In this regard, the state of the nodes can belong to one of these three categories: (type A) informed node: the node has received a message and spread it to its neighbors; (type B) uninformed node: the node has not received any message, (type C) passive node: the node has received a message but does not spread it.

Four conditions can cause the end of the simulation: (1)

The individual who receives the information is a type B;

(2) The strength of the message to be propagated falls bellow a given threshold; (3) All nodes are a type C; (4) It has been a long time (higher than a given threshold) since the scenario occurred.

EXPERIMENT

Overview

We illustrate the concepts presented in the paper by modeling and simulating the spread of information in a multi-layer social network. The context of the experiments is motivated by confidential training scenarios provided by the French military. The goal of this scenario is to predict the reactions of people facing events. In more details, the military missions aim to reduce local support for the insurgents by convincing people that it is not in their interest to support or join an insurgency. Using the data described in the scenario, we generate a population of 100 individuals connected on three layers: family, friends and neighbors. These three layers represent the primary groups defined by [START_REF] Litwak | Primary group structures and their functions: Kin, neighbors, and friends[END_REF]. The following are types of characteristics that may describe an individual: ethnicity, religion, gender, social status, etc. Related to the concepts of opinion change described by [START_REF] Friedkin | Social influence networks and opinion change[END_REF]) and at the contact of a receiver with an emitter, the receiver calculates the trust factor α(i,j). This mechanism for opinion change has been validated using an approach based on three issues [START_REF] Friedkin | Social influence networks and opinion change[END_REF]. The α(i,j) represents the trust between agents based on their attributes. The trust factor has value α(i,j)=1 if the two agents i and j are similar (they have same religion, same ethnicity and same class of age) and α(i,j)=0, otherwise. In the first case, agents receive the message and their variables (opinion, interest and satisfaction) change over time depending on some rules; for simplicity, the receiver agent adopts the emitter's opinion. Then, it is able to propagate the message to its neighboring agents. In the other case (α(i,j)=0), the receiver will enter the informed state and propagate the message, if the emitter is a lead agent. In the all other cases, the message is ignored and the receiver will not propagate anything.

Settings

To test our architecture, we drive three experiments with the same population but with different algorithms of message propagation (Table 1). In the first experiment, we do not consider the type of the message or the type of the relationship between the emitter and the receiver to implement rules in the proxies (one graph with different connections). In this case, the uninformed agents receive the message from their neighbors without calculating the trust factor. The trust factor between all the agents is equal to one. In the second experiment, we take into account the trust factor between agents. In this experiment, the receiver calculates the trust factor between him and the emitter before accepting the packet. The diffusion process depends on the strength of the message and on the trust factor.

Tableau 1: Description of simulated experiments

Experiments Description

Experiment 1

Trust factor between all agents is equal to one Experiment 2

The diffusion process depends on the trust factor but not on the type of connections Experiment 3 Each proxy has some specific rules to filter the information

In the third experiment, we use our multi-layer architecture by implementing specific rules to the type of message and to the relationship between different agents. We block the diffusion process in the friends and neighbors layer. We assume that at night and in the absence of communication channels the message will be spread only by face to face between family members.

Result and discussion

The results show that both the network structure and the node attributes (individual and cultural factors) have an impact on the proportion of informed agents and their opinion. Figure 4 presents the initialization phase of the simulation using the generator as explained in Figure 2 in Section 3.2.2. Each node in the network represents an agent with different attributes. These entities may be uninformed individuals or information sources. Source nodes initiate the original message. The info-sources are chosen randomly. Each node is colored according to its status: green for informed nodes, red for passive nodes and white for uninformed nodes. At t 0 , all nodes are uninformed except for the three info-sources: nodes V3, V24 and V70. As we are working with an MSN, we have three kinds of link: family relationships (red links), friendship relationships (green links) and neighborhood relationships (yellow links).

Figure 4: Initializaton of the simulation

Figure 5 shows the simulation results of the first experiment. The green nodes represent the agent who received the message and is interested by it. Red nodes represent the agent who received the message and not interested by it. White nodes represent the agent who did not receive the message. In this case, the diffusion process depends only on the strength of the message. The process is ended when the strength of the message becomes less than zero. The notion of interest used here does not refer to the receiver's emotion causes by the message. At this stage, we use a function that determinates if the agent is interested in the message or not by comparing closeness of the emitter's and receiver's characteristics (e.g., religion, ethnicity, age, language, social status). In the Figure 7 we observe that only the nodes connected by the family relationship are the informed nodes because we add some rules in the other layers to block and filter the diffusion process. We can explain this result by sending an urgent message at night and in the restriction of the use of communication channels (phone and internet). The different experiments show the flexibility of our Server/proxy architecture based on DEVS. We easily manage the diffusion process according to numerous parameters. We can simulate more realistic phenomena based on cultural and human factors. In addition, we can exploit our architecture to study the information diffusion in a dynamic network. Our architecture separates the individual behavior modeling from the networks structure modeling. It allows tuning easily the rules of diffusion or the structure of the network as soon as we have to change the context of the study or to test different human's behavior models. Finally, the human behavior algorithms implemented in the models presented in this paper are quite simple but however innovatively used by mixing social sciences studies and multi-layer social networks. The behavior algorithms used in the models were validated in social science studies but in a different context [START_REF] Friedkin | Social influence networks and opinion change[END_REF]. The network topology is coming from [START_REF] Mitchell | Social networks in urban situations: analyses of personal relationships in Central African towns[END_REF][START_REF] Lazega | Réseaux sociaux et structures relationnelles[END_REF]). The VLE simulation platform was also validated [START_REF] Quesnel | The Virtual Laboratory Environment-An operational framework for multi-modelling, simulation and analysis of complex dynamical systems[END_REF][START_REF] Ramat | Virtual laboratory environment"(VLE): a software environment oriented agent and object for modeling and simulation of complex systems[END_REF]. Nevertheless, because we merged human behavior and networks structure in a DEVS M&S environment for the first time, the complete framework hasn't been validated yet. We have started to confront the first scenarios models and simulation results to military experts. We have received a positive feedback but we agree that this feedback lies on simple and limited number of models; so it does not permit to validate the genericity of these models of networks and individual behavior. We find only two ways to validate our models. The first, already started, will consist in facing military expert with several sets of input scenarios and results obtained by simulation. The second issue is to set historical situations in models, run them, observe and compare output to historical facts. We are at the moment working on both aspects.

CONCLUSION AND PERSPECTIVES

We presented in this paper a new simulation framework feature for MSN based on the DEVS formalism. This low-level framework allows a full control of the agents' models; it promotes also modularity and reusability. This new architecture is fully dedicated to simulate, as transparently as possible, the propagation of information within a population network. In details, the mechanisms implemented in the models are, at the moment, simple but they are easily upgradeable by modelers. In addition, we assumed that relationships between people are too complex to be modeled by a unique network. Furthermore, information diffusion is dependent of the category of relationship between the people who communicate. So, the DEVS-based agent framework with the Server/Proxy architecture has been proposed to separate individual behavior in Server and perception of the environment in Proxy. It is flexible and sensitive to changes in the environment. In the last section, we managed two experiments showing (1) the importance of managing each relationship separately in the information diffusion process and (2) the simplicity to modify the diffusion rules for each relationship using our Sever/Proxy architecture. We are aware that, even if the different technics employed were previously validated in their respective domain, the models and simulation results proposed in the paper have not been validated. Nevertheless, we already confronted the first results to expert in the domain and will continue in that way. Beyond this, the perspective of our work seems promising to facilitate the setup and run of network structured DEVS models. This framework can be used to solve the problem of influence maximization (IM); find a set of β initially activated nodes (info-sources) with the maximum number of activated nodes after the time step t by launching several simulations with different nodes. Then, we will compare the final results to select the nodes that maximum influence or spread the information. As perspective, we are integrating progressively more variables to describe more accurately human behavior including social and psychological feelings and health. The inner idea is to obtain, using simulation, a better understanding of how information can affect behavior of people and population. We are also looking to adapt our model to other domains such as marketing, logistics and manufacturing.

Figure 2 :

 2 Figure 2: Conceptuel overview

Figure 5 :Figure 6 :

 56 Figure 5: Result of the first experiment

Figure 7 :

 7 Figure 7: Result of the experiment 3

ACKNOWLEDGMENTS

This work is supported by the SICOMORES Project N° 132936073 funded by French DGA (Direction générale de l'armement). It involves the following partners: IMS university of Bordeaux, LSIS university of Marseille and MASA Group.

BRUNO VALLESPIR is full professor at University of Bordeaux, IMS laboratory. He is member of several international working groups (IFIP, IFAC), he participated to 5 European projects, has directed more than 20 PhD students and written more than 120 papers in journals and conferences. His email address is bruno.vallespir@ims-bordeaux.fr.