Agathe Métaireau
email: agathe.metaireau.etu@univ-lille.fr

Rabin Kumar Sahu
email: rsahu@vekia.fr

Simon Delecourt
email: sdelecourt@vekia.fr

Alexandre Gerussi
email: agerussi@vekia.fr

Manuel Davy
email: mdavy@vekia.fr

Multi-periodic joint replenishment planning method

Keywords: Joint replenishment, Multi-period planning, Inventory control, Integer Linear Programming

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INTRODUCTION

A large part of the research conducted on inventory replenishment planning focuses on single-item control. In many references, the target is to find the best policy and its parameters. The final goal is to reach optimal or near-optimal inventory control for given situations and different objectives. There are however several industrial contexts where treating items separately is not the best strategy. In these contexts, there exist interactions between items. Fixed ordering cost or all-unit and incremental discounts are examples of this aspect. In such contexts, the problem cannot be reduced to several single-item sub-problems, because the constraints apply to the whole order. In this paper, we will only consider first-order interactions between the items, which can happen in various situations, in addition to the ones we mentioned above.

As this type of problem cannot be decomposed, there is a need to develop strategies to jointly replenish the items under control. Moreover, reaching allunit or incremental discounts sometimes requires ordering more than the optimal quantity we compute for one coverage period, and considering several periods is a way to reach the discounts with relevant ordering quantities, which is an incentive to consider multi-periodic replenishment planning.

As a result, we will consider in this article a multi-item multi-period replenishment problem, with a fixed cost constraint and for various types of allunit discounts. The multi-periodic joint replenishment problem has already been addressed in the literature for different contexts. Some authors calculate the Economic Order Quantity for this context, like [START_REF] Das | Multi-item inventory model with quantitydependent inventory costs and demand-dependent unit cost under imprecise objective and restrictions: A geometric programming approach[END_REF], who use geometric programming and gradient-based non-linear programming on capacitated systems and [START_REF] Mousavi | Optimizing multiitem multi-period inventory control system with discounted cash flow and inflation: Two calibrated metaheuristic algorithms[END_REF], who show the performance of a genetic algorithm with a mix of incremental and all-unit discount. [START_REF] Mandal | Multi-item multiperiod optimal production problem with variable preparation time in fuzzy stochastic environment[END_REF] use fuzzy techniques and genetic algorithms to address the optimal production control problem with the demand rate being stock-dependent. [START_REF] Rezaei | A deterministic, multi-item inventory model with supplier selection and imperfect quality[END_REF] use genetic algorithms in the context of defective items with supplier selection. [START_REF] Gao | Study of optimal order policy for a multiperiod multi-raw material inventory management problem under carbon emission constraint[END_REF] use dynamic programming to find a multi-raw material control policy under carbon emission constraints. [START_REF] Al-E-Hashem | Multi-product multi-period Inventory Routing Problem with a transshipment option: A green approach[END_REF] developed a Mixed-Integer Linear Program (MILP) to solve an inventory routing problem with deterministic demand and transshipment allowed.

The multi-period joint replenishment problem is NP-Hard ( [Sahu, 2020]). Therefore, it is impossible to solve it exactly in a reasonable time, and it is necessary to solve it approximately. In this article, we tried to reduce the solution space of the problem we address by modeling. In this model, the orders we can place are represented by objects we call "ordering blocks", as defined in [Sahu, 2020]. The optimization aims to find the best combination of blocks for the considered horizon. It enabled us to formulate the problem we want to solve as an Integer Linear Program (ILP) and to solve it using a classic branchand-bound solver. As there are multiple cases of allunit discounts in industrial replenishment contexts, the ILP has been written to be generic and applicable to a large set of discount types.

The ILP allows an easy formulation of the problem. As free and commercial solvers already exist, there is no need to develop more complicated heuristics to solve ILPs. As a result, the method we offer is easily implementable in real multiperiodic jointreplenishment contexts. Moreover, the optimization tool we develop is intended to be a module assisting a single-item single-period replenishment optimization engine. Therefore, it is suitable to be integrated with already existing optimization tools.

The remainder of the paper is organized as follows. We will first describe the approach we developed and the notations we use in section 2. Then, we will present the ILP in section 3. We will next describe the experimental protocol we put in place to validate the model and present the numerical results in section 4. Finally, we will provide a conclusion in section 5.

DEVELOPED APPROACH AND NOTATIONS

In this section, we will first detail the problem we address and give some examples of application in industrial replenishment contexts. Then, we will explain the approach we put in place to model and solve it.

Details of the Problem and Use Cases

As we stated before, we focus on constraints concerning several items at the same time. It implies a need to design an optimization method that takes into account all the items simultaneously. We search to plan the replenishment for groups of items jointly, on several periods for a finite horizon, considering joint costs and discounts. In this paper, we con-sider only all-unit discounts. This means the discount is applied on all the items ordered once the breakdown is reached. We apply a direct grouping strategy, as defined in [START_REF] Bastos | A systematic literature review on the joint replenishment problem solutions: 2006-2015[END_REF], and consider only groups of items that have the same cycle time.

There are two major categories of multi-item constraints we consider: fixed ordering cost and all-unit discount. They are defined as follows.

Fixed ordering cost : It is a cost applied to the whole order once we order something (i.e. when the total quantity ordered is strictly superior to 0). For example, a supplier can add a transport cost to the ordering cost, and this cost is only applied if something is ordered. This cost is also called in the literature "Major Ordering Cost".

All-unit discount : It is a discount applied on all the items ordered (i.e. on the total ordering cost) once a given breakdown is reached. There can be one or several breakdowns with corresponding discounts. The breakdowns can also concern different values, such as the total ordering cost or the total quantity ordered, but also the total weight or volume of the order for example. A practical example of the all-unit discount is the Franco, where the supplier charges a constant penalty cost if the total ordering cost is lower than a given breakdown.

Figures 1 and2 are graphical examples of the two notions explained above.

Modeling Approach and Definitions

As the problem is NP-Hard, we can't expect to find the optimal solution to large-scale instances in a reasonable time. Therefore, we have to define a process to find good-quality solutions in a suitable delay. In this article, we choose to reduce the search space by modeling. First, we constrain the time steps at which an order can be placed. It seems to be a reasonable choice, as the ordering dates are frequently constrained in real-life replenishment contexts.

We define the notion of coverage period as a time interval [t 1 ,t 2 ] for which an order can be placed at t 1 and no order can be placed until t 2 . In other words, this is the smallest possible interval between two orders.

We also choose to constrain the quantities that can be ordered. At the beginning of every coverage period, the decision-maker can only choose among a set of possible ordering quantities. These quantities are the optimal ordering quantities for t coverage periods, with t going from 1 to the end of the planning horizon. They are calculated before the multi-periodic joint replenishment optimization.

As a result, we represent the possible orders as an object called "block". Every block contains several attributes. First, it has a length, which is the number of coverage periods it encompasses. It has then a quantity to order, which is the precomputed optimal quantity for the block's length, and a cost, which is the total cost incurred for ordering the optimal quantity. The block's cost involves the purchase cost, the holding cost, the lost-sale or backorder cost, and any relevant cost for the problem under study. A block can also contain another attribute named value, which represents the value compared to the discount breakdown. This value can be the purchase cost, the volume, the weight, etc. Figure 3 represents all the possible blocks for one item and a time horizon of three coverage periods.

Notations

In this section, we describe the notations we use in the remainder of the article. All the notations we will T -Time horizon N -Size of the item set I -Number of cost breakdowns K -Fixed ordering cost B nt 1 t 2 -Block corresponding to the order of item n from coverage period starting at t 1 and ending at t 2 q nt 1 t 2 -Quantity to order for the block B nt 1 t 2 C nt 1 t 2 -Cost incurred by the block B nt 1 t 2 V nt 1 t 2 -Value of the block B nt 1 t 2 to be compared to the breakdowns l i -Breakdown i f i -Fixed cost or discount for a total order value under the breakdown i u i -Cost or discount per unit for a total order value under the breakdown i

Solution Approach

The goal of the multi-periodic joint replenishment optimization method we develop in this article is to find what quantity of each item to order at every time step of the planning horizon while minimizing the total cost. This total cost, which we call TC in this section, includes the block's cost and the costs and discounts incurred by the multi-item constraints.

Let P NT be the set of possible block combinations for an item set of size N and a planning horizon T . As an example, a representation of the set P 22 is provided in Figure 4. Given N and T , each possible block combination p in P NT has a total cost TC(p), which is the sum of the costs of the blocks selected in p and the corresponding multi-item costs and discount. To find the best replenishment planning in this context, we are looking for the block combination p * with the minimum cost, as described in the equation 1.

p * = arg min p∈P NT TC(p) (1)
In the next section, we will describe the methodology we use to find p * .

OPTIMIZATION METHODOLOGY

An intuitive solution to solve the problem we described above would be to generate every possible block combination, compute their respective total cost, and choose the one with the lowest total cost. However, the number of possible block combinations for a time horizon T and an item set of size N is 2 N×T -1 . It means the number of possible block combinations quickly increases with the growth of both T and N. For example, if T = 5 and N = 5, there are 1 048 576 possibilities, which illustrates how the brute force approach rapidly becomes intractable.

To tackle this, we chose to solve the problem by using an Integer Linear Program (ILP). This optimization method has the advantage of being easy to formulate, implement, and solve using commercial or free solvers.

We will present the formulation of the ILP in the next sections. Section 3.1 lists the data used in the ILP, Section 3.2 details its variables. Section 3.3 is dedicated to the objective. Section 3.4 lists all the constraints and provide some explanation of them. Section 3.5 details the domains of the variables.

Data

T -Time horizon N -Set of items I -Number of breakdowns K -Fixed ordering cost M -Suitably large number ∀ n ∈ N, ∀t 1 ∈ 1, T , ∀t 2 ∈ 1, T , C nt 1 t 2 -Cost of the block B nt 1 t 2 ∀ n ∈ N, ∀t 1 ∈ 1, T , ∀t 2 ∈ 1, T , V nt 1 t 2 -Value of the block B nt 1 t 2 ∀ i ∈ 1, I , l i -i-th breakdown ∀ i ∈ 1, I + 1 , f i -Fixed cost
or discount corresponding to a value ordered over l i-1 and under l i ∀i ∈ 1, I + 1 , u i -Cost or discount per unit corresponding to a value ordered over l i-1 and under l i

Variables

∀ n ∈ N, ∀t 1 ∈ 1, T , ∀t 2 ∈ 1, T , x nt 1 t 2 = 1 i f the block B nt 1 t 2 is selected 0 otherwise ∀t ∈ 1, T , η t1 = 1 i f the value ordered is 0 0 otherwise ∀t ∈ 1, T , η t2 = 1 i f the value ordered is strictly between 0 and l 2 0 otherwise ∀t ∈ 1, T , ∀ i ∈ 3, I η ti = 1 i f the value ordered is between l i-1 and l i excluded 0 otherwise ∀t ∈ 1, T , η tI+1 = 1 i f the value ordered is greater than l I 0 otherwise ∀t ∈ 1, T , ∀ i ∈ 1, I + 1 y ti = Value ordered at t i f η ti = 1 0 otherwise 3.3 Objective min ∑ n∈N T ∑ t 1 =1 T ∑ t 2 =1 C nt 1 t 2 x nt 1 t 2 + T ∑ t=1 K(1 -η t1 ) + T ∑ t 1 =1 T ∑ t 2 =1 I ∑ i=1 (u i y ti + f i η ti ) 3.4 Constraints ∀ n ∈ N, ∀t 1 ∈ 1, T , ∀t 2 ∈ 1, T , t 1 x nt 1 t 2 ⩽ t 2 x nt 1 t 2 (2) ∀ n ∈ N, T ∑ t 1 =1 T ∑ t 2 =1 (t 2 -t 1 + 1)x nt 1 t 2 = T (3) ∀ n ∈ N, T ∑ t=1 x ntT = 1 (4) ∀ n ∈ N, ∀t ∈ 1, T -1 , T ∑ t 1 =1 x nt 1 t ⩽ T ∑ t 2 =1 x nt+1t 2 (5) ∀t ∈ 1, T , M(1 -η t1 ) ⩾ ∑ n∈N T ∑ t 2 =1 V ntt 2 x ntt 2 (6) ∀t ∈ 1, T , (1 -η t1 ) ⩽ ∑ n∈N T ∑ t 2 =1 V ntt 2 x ntt 2 (7) ∀t ∈ 1, T , ∀ i ∈ 2, I + 1 , l i-1 η ti ⩽ ∑ n∈N T ∑ t 1 =1 V ntt 2 w ntt 2 (8) ∀t ∈ 1, T , ∀ i ∈ 2, I , M(1 -η ti ) ⩾ ∑ n∈N T ∑ t 2 =1 V ntt 2 x ntt 2 + ε (9) ∀t ∈ 1, T , I+1 ∑ i=1 η ti = 1 (10) ∀t ∈ 1, T , ∀ i ∈ 1, I + 1 , y ti ⩾ ∑ n∈N T ∑ t 2 =1 (V ntt 2 x ntt 2 ) -M(1 -η ti ) (11) ∀t ∈ 1, T , ∀ i ∈ 1, I + 1 , y ti ⩽ ∑ n∈N T ∑ t 2 =1 V ntt 2 x ntt 2 (12) ∀t ∈ 1, T , ∀ i ∈ 1, I + 1 , y ti ⩽ Mη ti (13)
Constraint 2 ensures that we select only blocks for which the starting coverage period is lower than or equal to the ending coverage period. Constraint 3 ensures the entire planning horizon is covered. Constraints 4 and 5 ensure that the blocks we select do not overlap. Constraints 6, 7, 8, and 9 rule the value of η ti according to its definition. If the total value ordered can be lower than 1, it is necessary to add a "+ε" at the end of the right part of constraint 7 to ensure its good functioning. The "+ε" at the end of the left part of constraint 9 allows having the behavior of a strict inequality. Constraint 10 ensures only one η ti is selected each period. Constraints 11, 12, and 13 rule the value of y ti according to its definition.

Domains

∀ n ∈ N, ∀t 1 ∈ 1, T , ∀t 2 ∈ 1, T , x nt 1 t 2 ∈ {0, 1} ∀t ∈ 1, T , ∀ i ∈ 1, I , η ti ∈ {0, 1} ∀t ∈ 1, T , ∀ i ∈ 1, I , y ti ∈ N
The domain of y ti is consistent with the unit of V . We consider the case where V only takes integer values, but if V takes continuous values, the Integer Linear Program becomes a Mixed Integer Linear Program (MILP).

EXPERIMENTATIONS AND COMPUTATIONAL RESULTS

To evaluate the interests of multiperiodic joint replenishment planning and the performance of the method we developed, we chose to compare it with a singleperiod single-item optimization method. We used a single-period single-item ordering engine as a baseline for our experimentations. This engine uses a sample-based optimization approach for stochastic replenishment planning. Given a set ov demand samples, this engine aims to find the optimal quantity to order on a fixed coverage period. It will minimize the so-called "coverage period cost", which is the expected cost on the whole period, and includes the holding cost, the lost-sale cost, and an potential single-item fixed cost. The optimal quantity is found by using an enumerativesearch heuristic, described in [Sahu, 2020].

Both order engines are compared on the total cost performance criterion, which includes the purchase cost, the inventory cost, the shortage cost, and the multi-item cost. In the experiments, the multi-item constraint is a Franco penalty, as described in the section 2.1, so the multi-item cost includes only eventual Franco penalties.

To conduct experiments, we generated twenty different instances, each containing ten items. We assume perfect forecast, i.e. we assume that we perfectly know the demand distribution of the items. In every instance, all the demand distributions of the items are stationary Poisson distribution, and we assume all the unmet demand is lost. The parameters of the instances we used are detailed in the tables 1 and 2. The Integer Linear Program has been implemented

Instance n°λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 λ 9 λ 10 1 10 10 10 10 10 10 10 10 10 10 using the PuLP library in Python, and solved using the COIN-OR Branch and Cut (CBC) solver.

To compare both replenishment processes, we used the following experimental protocol.

1. Generate demand samples for every item of the instance and every period of the planning horizon.

2. Given initial inventory levels, launch the singleitem single-period order engine. Simulate the order and demand process and update the inventory levels.

3. Record the costs associated with the simulated period.

4. Repeat steps 1. and 2. on the whole planning horizon.

5. Compute the total cost for the single-item singleperiod ordering simulation.

6. Given the same initial inventory levels, launch the multi-period multi-item order engine. Record the replenishment planning.

7. Simulate the order and demand process using the same demand samples on the entire planning horizon and record the costs associated with every simulated period.

8. Compute the total cost for the multi-period multiitem ordering simulation.

This protocol was used on the twenty instances described above. It allowed us to compare the different costs incurred by both optimization methods, and The values displayed in table 3 are defined as follows. For a given cost type, let C j SPSI and C j MPMI be respectively the total cost incurred by the single-period single-item approach for the instance j and the total cost incurred by the multi-period multi-item approach on the same instance. Then, let S j = TC Regarding the cost comparison, we obtained an overall cost reduction of 8.70% by using the multiperiod multi-item approach. We also observe higher purchase and inventory costs, which can be explained by the fact that this order engine tends to order more than the single-period single-item one. The cost reduction comes from the lost sale cost and the multiitem cost. As a result, the proposed multi-period multi-item order engine allows to optimize the total cost regarding the multi-item constraints, as it was designed for, but it also allows to reduce lost sales, and therefore to increase the service level.

CONCLUSION AND FUTURE RESEARCH

In this paper, we present a method to optimize inventory replenishment by taking into account multi-item constraints. We develop a multi-period multi-item optimization tool to answer this problem. As the multiperiodic joint replenishment planning is an NP-Hard problem, we approximate the problem by modeling, to obtain good solutions in a reasonable time. To reduce the search-space, we introduced the concept of ordering blocks. This approximation seems to be relevant because it is in line with industrial ordering processes, and still adaptable to various situations. We wrote an Integer Linear Program to find the optimal combination of blocks by minimizing the sum of the single-item costs (purchase, inventory, and shortage costs) and the multi-item costs (fixed ordering cost, Franco costs, discounts). This Integer Linear Program has been developed in the purpose to be adaptable to various all-unit discounts. This multi-period multi-item approach was compared with a single-period single-item order engine. Using twenty generated instances, the performance of both methods was evaluated based on the costs obtained by simulation using a sample of the demand. We chose to restrict our study to stationary Poisson demand and to a single Franco constraint, but the model allows to treat various demand types and multiitem constraints. We showed that the multi-period multi-item order engine we developed incurred a reduction of the total cost on all the instances, especially on the multi-item costs and on the shortage cost. Therefore, this method allows better management of multi-item constraints and discounts and an increase in the service level.

For further research, the model can be tested with different discount types, like a fixed ordering cost or a fixed discount, as we only tested with a Franco here. It would also be interesting to evaluate the performances of the method in case of the imperfect forecast, to see how the demand uncertainty impacts the relevancy of this approach. The model can also be extended or adapted to tackle incremental discounts, which were excluded in this research.

Figure 1 :

 1 Figure 1: Representation of the cost function in case of a fixed ordering cost. The dark blue curve is the case without fixed ordering cost.

Figure 2 :

 2 Figure 2: Representation of the cost function for an all-unit discount with three breakdowns and three corresponding fixed discounts. The dark blue curve is the case without any discount.

Figure 3 :

 3 Figure 3: Possible blocks for one item and a time horizon of three coverage periods.use are listed below. The subscripts t refer to the time steps, n to the items, and i to the cost breakdowns.T -Time horizon N -Size of the item set I -Number of cost breakdowns K -Fixed ordering cost B nt 1 t 2 -Block corresponding to the order of item n from coverage period starting at t 1 and ending at t 2 q nt 1 t 2 -Quantity to order for the block B nt 1 t 2 C nt 1 t 2 -Cost incurred by the block B nt 1 t 2 V nt 1 t 2 -Value of the block B nt 1 t 2 to be compared to the breakdowns l i -Breakdown i f i -Fixed cost or discount for a total order value under the breakdown i u i -Cost or discount per unit for a total order value under the breakdown i

Figure 4 :

 4 Figure 4: Representation of the set of possible block combination for two items and a planning horizon of two periods

Figure 5 :

 5 Figure 5: Comparison of the purchase costs given by the single-period single-item and multi-period multi-item approaches on all the instances.

Figure 6 :

 6 Figure 6: Comparison of the inventory costs given by the single-period single-item and multi-period multi-item approaches on all the instances.

S

  and S % be their respective averaged values on all the instances.

Figure 7 :

 7 Figure 7: Comparison of the lost sale costs given by the single-period single-item and multi-period multi-item approaches on all the instances.

Figure 8 :

 8 Figure 8: Comparison of the Franco costs given by the single-period single-item and multi-period multi-item approaches on all the instances.

Table 1 :

 1 Details of the instances used for the experiments. λ i represents the mean of the Poisson distribution for the demand of the i-th item of the instance. The global parameters are the purchase cost p = 5, storage cost h = 0.1 and lost sale cost w = 10.

		2	13 25 14 15 41 12 20 34 19 34
		3	39 44 13 21 15 47 31 14 29	8
		4	45 39 20 11 11 9	9 10 42 10
		5	24 30 14 34 42 35 24 35 40 46
		6	18 17 14 32 22 23 7 11 11 19
		7	47 33 50 25 48 38 16 11 18 33
		8	39 47 41 26 31 28 13 32 47 36
		9	30 6 17 18 34 50 46 48 50 10
		10	36 26 14 18 48 11 44 21 15 16
		11	24 41 42 37 22 26 45 7 42 49
		12	50 46 48 43 16 13 46 6 38 42
		13	20 48 13 20 41 48 15 36 6	48
		14	21 34 18 47 29 31 44 44 34 44
		15	38 44 22 25 38 42 9 49 5	30
		16	22 11 13 36 15 20 35 43 31	7
		17	37 38 12 18 45 49 47 30 46 31
		18	44 15 18 8 45 30 10 21 36 41
		19	19 31 42 40 37 50 5 16 23 22
		20	11 43 43 19 19 46 25 7	7	31
	ranco Instance n°F threshold	Franco penalty
	1	800	276.35
	2	2838	654
	3	2250	753
	4	2250	581.25
	5	2150	939
	6	1700	500
	7	2400	920.5
	8	2943	943
	9	2150	899.5
	10	2250	711
	11	2889	963
	12	2150	1019.95
	13	2350	867
	14	2050	999
	15	2650	878.7
	16	2250	668
	17	3078	1025.95
	18	1800	766
	19	2484	828
	20	1550	722.45

Table 2 :

 2 Franco parameters of the instances.

Table 3 :

 3 Averaged savings obtained with the multi-period multi-item approach compared to the single-period singleitem approach.


Frédéric Paran
email: paran@emse.fr

Didier Graillot
email: graillot@emse.fr

Djamel Mimoun
email: mimoun@emse.fr

Games, Role-Playing, Tools and Models as a Learning Process to Simulate Groundwater Management Negotiation

How to make relevant decisions to preserve the quality of the hydrosystem and human wellbeing? This work assists in answering this question at a local level. A negotiation support simulator for a regional project is proposed which includes the numerous actors involved in water resource management projects. This simulator combines negotiation support tools and role-playing game for gravel extraction in an alluvial aquifer. During the simulation, each participant acts out the role of specific entity with their own objectives and stakes. Together, the players have to decide around the negotiating table, where is the best location for a new gravel pit location in the territory? During the negotiation process, they have to defend their interests and to accept the territory as a multidimensional entity (biophysical and human), to achieve a balanced solution. For this endeavor, each player may use different tools to design a spatial argumentation from GIS territorial representation. For example, to show the impact of the gravel pit on the aquifer, a player can create a resource for persuasion with a physical model (MODFLOW) or a biological model (subterranean ecology). The simulation familiarizes players with the complexity of territorial arbitration and negotiation support tools (GIS, physical and biological models). Furthermore, players are aware of the difficulty to produce more ecological negotiations and consider human aspects in conservation of water resources.

INTRODUCTION

Aquifers are very important water resources for human uses (e.i. supply, industry, irrigation, watering, or leisure activities) and for hydrosystem biodiversity. In addition, aquifers are in danger because they are treated as invisible resource. For [START_REF] Roux | Vulnérabilité des aquifères aux pollutions[END_REF], this vulnerability is composed of two criterions:

1) Vertical propagation speed of pollution to the aquifer; 2) Regeneration speed of water quality.

The aim of this work consists in simulating groundwater management negotiation, accompanied by a mediator, to reduce this vulnerability using an example of gravel extraction in an alluvial aquifer. This simulation makes use of an on line/web simulator composed simply of html pages, role-playing and negotiation support tools (e.i. physical model). Its design is based on the negotiation learning process developed by De [START_REF] De Carlo | Teaching "concertation": the acceptance of conflicts and the experience of creativity using La Francilienne CD Rom[END_REF], notion of spatial representations proposed by [START_REF] Lardon | Modèles spatiaux pour le développement territorial : vue d'ensemble d'une démarche en cours[END_REF] and the concept of spatial argumentation [START_REF] Paran | Représentations territoriales pour la gestion équilibrée d'un patrimoine écologique et anthropique dans le domaine de l'eau[END_REF] available on line : http://www.agora21.org/entreprise/sommaire7.html ; [START_REF] Paran | Un territoire en 3 dimensions (T3D) et un acteur en 4 dimensions (A4D) pour une argumentation spatiale adaptée[END_REF].

Objectives of this negotiation simulation consist in design a commonly arrived decision to know:

-Where is the best new gravel pit location in the territory? -What is the best method of rehabilitation after gravel extraction? -Who is the more qualified to manage the site after rehabilitation?

This paper presents the method used to simulate groundwater management negotiation. It focuses on the use of physical model to design spatial argumentation for decision making. First results are discussed at the end of this paper.

Agreement protocol

In parallel

METHOD TO SIMULATE GROUNDWATER MANAGEMENT NEGOTIATION

This section locates and presents the contested region defining its hydrogeologic characteristics. Then simulator architecture is described focusing on tools, and especially the physical model, for spatial argumentation.

Hydrogeologic View of the Area for Simulation

The study area was an alluvial aquifer located in the sedimentary plain of the Forez which is part of the Loire River catchment, France. This sedimentary plain has an area of 750 km². Average altitude is 340 m and the plain is crossed from south to north by the Loire River [START_REF] Duclos | Géologie et minéralisation uranifères de la Plaine tertiaire du Forez[END_REF]. The Forez aquifer is essentially fed by rainwater and is drained by the Loire River. The general groundwater flow direction is from southwest to northeast. The aquifer consists of quaternary alluvial deposits (gravels, sands, clays) and the surface area and volume of the aquifer are 10 km² and 18 km 3 , respectively. The lower confining bed which consists of Oligo-Miocene impermeable claystone is located at a depth of 6 m below the soil surface. Although the Loire River is generally considered as one of the last seminatural large rivers in France, the Forez plain has been highly modified by human activities including river embankment, intensive agriculture, and in particular gravel extraction [START_REF] Ulmer | Expertise écologique de l'Ecozone du Forez[END_REF]. Gravel pits either formed artificial ponds fed by groundwater or they were filled in with demolition materials.

Simulator Structure and Role-Playing Game to Simulate Negotiation (figure 1.)

Technically, the simulator is designed and functions like a web-site and is written in html language with interactive links (figure 2.). By visiting the simulator each participant can find all necessary information to know what to do and create context for negotiation:

1) A short description of the game development stage by stage;

2) The territorial context, location, the gravel industry and gravel rehabilitation techniques are presented;

3) A glossary which defines all the key words used in the simulator.

To begin virtual negotiation participant have to choose their role to play. A game helps to gives all information and particulars about: 1) The 14 actors implied, like gravel industrialist, nature protector, fisher, hunter, famer, mayor or administration (e.i. name, activities, documentation, goals, stakes, territorial perceptions, or social relationships);

2) The 4 gravel sites in bargaining (e.i. location, profitability, impacts on groundwater, ecosystem, sociosystem or landscape).

This information is deliberately fragmentary to force each participant to meet each other and interact as well as use different tools. If the simulator gives knowledge and information on a computer, role-playing is the way to simulate negotiation in a room around a table. During the game, each participant has to write in a log book. It is very important to a trace of what happened during the simulation and analyze its development. Furthermore, there are debriefings to take stock of the situation stage by stage and teaching sessions to help participant and answer any theoretical questions.

The role-playing session is organized in several stages supervised by a mediator. First, each one has to make a preliminary public declaration to present the actor he (or she) plays and their position. Secondly, they begin to design an argumentation, based on meeting with the other actors or experts (e.i. hydrogeologist, ecologist or sociologist) and using tools, to defend his best gravel site and his best rehabilitation (e.i. wildlife reserve, refilled for agriculture, water storage or leisure activities). This resource persuasion is improved with the news of two declarations which take account of other actor's reactions. Thirdly, a final negotiation meeting is organized to decide, if possible, the final choice and write an agreement protocol with the help of a mediator.

Tools to Design Spatial Argumentation

Here, the argumentation is not only verbal but needs to be spatial. For this, each participant can use different tools, with the help of an expert, to design this argumentation and especially GIS (Geographic Information System) to represent reasons cartographically. Different tools are available to the player. Physical model (ModFlow) for a better understanding of the functioning of aquifer and simulate the future gravel pit's impact on it, biological model (subterranean invertebrate) and chemical model to refine this understanding with a different perspectives, multi-criteria analysis (MCA) for a valuation of the ecologic potential of the future gravel pit and sociological model for a better comprehension of actors.

RESULTS: NEGOTIATION AND SPATIAL ARGUMENTATION… …Design with Physical Groundwater Flow Model

A finite-difference groundwater flow model (ModFlow type) was used with data from the following parameters; water level measurements in multiple wells, permeability values obtained from resistivity measurements and pumping tests, and estimates of rainfall, evaporation, and infiltration. The model provided piezometric levels for cells in a grid every 50m. The model was calibrated by modifying permeability until the calculated piezometric levels and the measured piezometric levels were similar (average error: 24cm). Details of the groundwater flow model were provided by [START_REF] Mimoun | Spatialisation de l'information : une aide à l'analyse hydraulique et paysagère développée lors de la réhabilitation de sites post-industriels, Cas des réaménagements des gravières en eau en milieu alluvionnaire[END_REF].

With results of ModFlow, it is possible to design spatial argumentation on GIS maps making easier the choice of the new gravel site. The figure 3 gives two examples of spatial argumentation:

1) Argument 1 shows that site with strong hydraulic gradient must be avoided because it adversely impacts on aquifer flow;

2) Argument 2 shows that sites with wells upstream must be avoided too because wells could be pump out. Physical model can refine these arguments by simulating the impacts of futures gravel sites on hydraulic gradients and piezometric levels.

…Design by Another Physical Way

It is possible to design many physical representations and argumentations based on hydraulic objects (e.i. wells, piezometers, limnimeters, gravel pit, refilled gravel pit, stream), topography, geology (e.i. substratum level, alluvium nature), hydrogeologic attributes (e.i. permeability, gradients, flow speed, aquifer thickness, vadoze thickness). For example, we can spatialize aquifer pollution vulnerability in function of its vadoze thickness, quarry gravel richness in function of alluvium permeability.

DISCUSSION

Finite-difference groundwater flow models with regular grid are not the only existing models. For example, we can use finite-difference groundwater flow models with irregular grid which conforms better to hydrologic object shape [START_REF] Mimoun | Spatialisation de l'information : une aide à l'analyse hydraulique et paysagère développée lors de la réhabilitation de sites post-industriels, Cas des réaménagements des gravières en eau en milieu alluvionnaire[END_REF] or analytic elements models which represent better hydrologic shape and field reality [START_REF] Dauvergne | Comparison between finite different methods and AEM for groundwater of a local aquifer in France (Loire)[END_REF]. It could be very interesting for a better understanding of model results by actors.

But sometime, physical models are not sufficient for understanding aquifer functioning when they don't reflected a reality adapted to actors. Indeed, physical models present limits regarding their calibration (what is data precision?), their interpretation and significance (can the all actors understand these models?). Then it must be necessary to design another kind of argumentation for better representation of hydrosystem complexity and vulnerability:

-Firstly, a biological model based on subterranean invertebrates could be mobilized. These organisms are indicators of aquifer thickness and permeability [START_REF] Paran | Représentations territoriales pour la gestion équilibrée d'un patrimoine écologique et anthropique dans le domaine de l'eau[END_REF]. Then, a spatial argumentation for horizontal and vertical aquifer vulnerability and potential quarry gravel richness could be represented and designed. As we can use different physical models we can use different biological models based on different organisms (e.i. benthic invertebrates, macrophytes).

-Secondly, groundwater physico-chemical models could provide compelling reasons too [START_REF] Paran | Représentations territoriales pour la gestion équilibrée d'un patrimoine écologique et anthropique dans le domaine de l'eau[END_REF]. For example, we could show on map farming pollution with nitrates and its influence on gravel pit lakes water quality (e.i. eutrophication) in case of leisure activities rehabilitation; or impacts of refilled gravel pits on groundwater quality in wells.

-Thirdly, sociological models [START_REF] Paran | Représentations territoriales pour la gestion équilibrée d'un patrimoine écologique et anthropique dans le domaine de l'eau[END_REF] could be use for a better understanding of actor territorial representations of hydrologic objects (e.i. aquifer, gravel pits lakes, river) and relationships (e.i. actors in conflict, in cooperation).

CONCLUSION

Finally, this work shows one way to mobilize physical model in a learning process to simulate groundwater management and design spatial argumentation with territorial representation like maps from GIS. This work takes all its direction by associating different kinds of complementary representations to build argumentation, because a territory, in our case groundwater, is multidimensional and made of physical, ecological and human dimensions. Moreover, maps are not the only way for argumentation, comprehension, and conciliation of typically complex environmental decisions. We can imagine other representations like 3D views, movies, or landscape diagram in accordance with sensitivity of the different territorial actors [START_REF] Lardon | Modèles spatiaux pour le développement territorial : vue d'ensemble d'une démarche en cours[END_REF].

In this manner, our work is a contribution helping to make relevant decisions to preserve the quality of the hydrosystem and human wellbeing. This work assists in answering this question at a local level, in a context of alluvial gravel extraction, and proposes a negotiation support system for a regional project including numerous actors involved in water resource management projects. The simulation familiarizes players with the complexity of territorial arbitration and negotiation supports tools (GIS, physical and biological models). Furthermore, players are aware of the difficulty to reach a more ecologic negotiation and human conservation of water resources.

Figure 2 .

 2 Figure 2. Negotiation simulator home page (only in French at present)

Figure 1 .

 1 Figure 1. Simulator structure and role-playing game to simulate negotiation

Figure 3 .

 3 Figure 3. Spatial argumentation: examples in ellipses with number 1 and 2 (Paran, 2007)