Guillaume Martin
email: gmartin@mines-albi.fr

Matthieu Lauras

Pierre Baptiste

Dynamical multi-parameter sizing of DDMRP buffers in finite capacity flow-shops

The DDMRP (Demand Driven Material Requirements Planning) methodology uses buffer stocks to (i) maintain a high level of service, (ii) stop the spread of uncertainty and (iii) adapt to market changes. According to theory, the size of these buffer stocks should be defined regularly. This sizing involves several parameters and policies to update them, but very little information is available on this subject. We aim to help practitioners choose sizing policies, while maximizing the performance of a given workshop. We have developed an experimental design to compare many combinations of flow-shops and bottleneck constraints, taken from industrial usecases, and using discrete event simulation. The results show that (i) different degrees of dynamism are needed depending on the performance metric chosen by practitioners, (ii) completely dynamic control does not systematically lead to better performance, and (iii) contrary to what the existing literature on DDMRP suggests, varying buffer sizes may be less effective than fixed ones for an important part of use-cases.

Introduction

Over the last decades, supply chain and material management methods faced increasingly complex problems: more volatile markets, increasing uncertainty both from demand and supply, complexity from high degrees of product customization or network of actors [START_REF] Bennett | What VUCA really means for you[END_REF]. The Demand Driven Material Requirements Planning method [START_REF] Ptak | Demand driven material requirements planning[END_REF]) is a materials management framework that is part of the various recent contributions that were developed to deal with this problem. By building on top of previous successful methods such as MRP, Kanban or Theory of Constraints and adding a few innovations, the DDMRP claims the ability to reach higher performance with simpler management [START_REF] Miclo | Demand driven MRP: assessment of a new approach to materials management[END_REF].

Using DDMRP requires two main steps: (i) creating a network of buffers to help mitigate uncertainties and (ii) implementing a dynamic adaptation of the sizes of these buffers. Creating the network of buffers (where to place them and what types to use) is the first crucial step of a successful DDMRP implementation, but is outside the scope of this paper. It has been previously addressed in [START_REF] Ptak | Demand driven material requirements planning[END_REF] for general principles and in [START_REF] Rim | Strategic inventory positioning for MTO manufacturing using ASR lead time[END_REF] with an optimization approach or [START_REF] Pekarčíková | Demand driven material requirements planning. Some methodical and practical comments[END_REF] with a consideration for product types. The research work presented in this paper focuses on the second step of the implementation: the sizing of all the physical buffers once placed. work presented in this paper aims to answer two questions: (i) what are the main parameters that could be tuned dynamically for DDMRP buffers while maintaining sufficient workshop performance and (ii) what policies or combinations of policies for parameter update give the best performance, and for which variation of flow-shop? We restricted our study perimeter to flow-shop variations as they make up a considerable part of implementations and appear in our industrial use-case.

Section 2 reviews previous work on dynamical parameter setting inside and outside the DDMRP methodology. Section 3 details the characteristics of our use-case and how we designed our experiments. Section 4 summarizes the simulation results statistically and produces a set of recommended policy choices. Lastly, Section 5 details where to go from now with the newly acquired knowledge.

Literature review

The literature review is split into two sections. In 2.1, we review the past works on DDMRP that may inform us on how sizing works, and on which parameters or policies influence the sizing. In 2.2, we review the parallels between DDMRP sizing and sizing as defined in other planning methods that preceded DDMRP. Finally, 2.3 concludes on our research questions. These sections should help us identify the research gaps for three questions:

1. what are the parameters influencing the dynamic sizing of a DDMRP physical buffer? 2. which of these parameters are supposed to be dynamically updated according to the DDMRP literature? and, 3. do other fields in literature close to the DDMRP contain elements on dynamical update of these parameters?

Sizing the DDMRP buffers

Most works on DDMRP use or challenge guidelines given by [START_REF] Ptak | Demand driven material requirements planning[END_REF], which serves as a basis for the body of knowledge on the method. They identify a set of equations that, once summed, add up to form the final size of a buffer. Appropriately sized buffers allow to decouple demand from supply, while avoiding having too much in stock. These equations call for four mandatory parameters and one additional parameter:

• the Average Daily Usage (ADU), equivalent to the daily demand of a buffered reference, • the Decoupled Lead Time (DLT),1 defined as the longest manufacturing lead time between two consecutive buffers, • the Variability Factor (VF), a coefficient between 0 and 1 proportional to demand instability, • the Lead Time Factor (LTF), a coefficient between 0 and 1 inversely proportional to the buffered reference's DLT, • finally, the Minimum Order Quantity (MOQ), which is an optional parameter accounting for the minimum lot size.

As recently emphasized by [START_REF] Azzamouri | Demand driven material requirements planning (DDMRP): A systematic review and classification[END_REF], the founders of the DDMRP gave only a few details on parameter definition. In [START_REF] Ptak | Demand driven material requirements planning[END_REF], they set the parameters as follows:

• they recommend using moving averages over the past demands to compute the ADU of buffered references. The value of the average window is left to the end user however. [START_REF] Ptak | DDMRP. Demand driven material requirements planning[END_REF] gave an update and proposed to use forecasts in conjunction with past demands in determining the ADU. ADU is the only parameter that is explicitly said to vary over time.

• they set the DLT directly from the reference data-sheet, instead of computing it based on historical data. • they recommend values for VF and LTF according to a three level scale. • if possible, they recommend against using MOQ as performance of DDMRP may be hindered by fixed lot sizes.

As a result, in [START_REF] Ptak | Demand driven material requirements planning[END_REF], only the ADU is explicitly said to be dynamically controlled. Most of the recent contributions specific to the DDMRP follow these recommendations: [START_REF] Kortabarria | Material management without forecasting: From MRP to demand driven MRP[END_REF], [START_REF] Miclo | Demand driven MRP: assessment of a new approach to materials management[END_REF] or Velasco Acosta, Mascle, and Baptiste (2019), for example, all study a DDMRP instance under an evolving ADU only. For the remainder of this paper, we will refer to any rule or function to dynamically set a parameter value as a "policy".

Definition 2.1 (Policy). A policy is any rule or function that returns the new value for a dynamically set parameter. Some authors have tested alternatives parameter setting policies. [START_REF] Dessevre | Decoupled lead time in finite capacity flowshop: a feedback loop approach[END_REF] is the only known source using a variable DLT in a DDMRP context. They evaluate the impact of using different policies to set the value from the previous registered cumulative lead times. They review the impacts of three policies on the performance of a flow shop: (i) keeping the DLT constant thus following other researchers, (ii) using exponential smoothing with different smoothing factors and (iii) setting the DLT as a percentile of registered cumulative lead times for a specific reference in the flow shop. After submitting the flow shop to a perturbation in demand, they conclude that controlling both parameters dynamically is possible but can lead to spiraling effects when bottleneck load is near capacity. But, contrary to the rules from [START_REF] Ptak | Demand driven material requirements planning[END_REF], to limit the complexity of their simulation, they do not change the value of the LTF with the value of the DLT. [START_REF] Lee | A mathematical safety stock model for DDMRP inventory replenishment[END_REF] propose an alternative to using the VF parameter. They define the lower zone of the buffer through coefficients of variation of demand and lead time. They achieve lower inventories than [START_REF] Ptak | Demand driven material requirements planning[END_REF] with their policy, with high client satisfaction. They also define the DLT as the "average lead time" without further details on how it was measured.

Through their study of the literature, [START_REF] Azzamouri | Demand driven material requirements planning (DDMRP): A systematic review and classification[END_REF] showed a tendency to use different types of simulations in order to obtain new scientific results on the DDMRP. The most prominent one is Discrete Event Simulation (DES), that allows to model complex workshops and specific DDMRP behaviors. Recent works can be divided into two categories: those that compare performance of different material management methods with DDMRP on the same workshop and those that compare alternative ways to control the DDMRP, still on the same workshop. [START_REF] Shofa | Effective production planning for purchased part under long lead time and uncertain demand: MRP vs demanddriven MRP[END_REF] use DES to compare MRP and DDMRP in a problem with long purchasing lead times and shows that DDMRP achieves lower average inventory. [START_REF] Miclo | Demand driven MRP: assessment of a new approach to materials management[END_REF] compare MRP, Kanban and DDMRP and show that DDMRP globally outperforms the other two in an assembly workshop. Lastly, [START_REF] Thürer | Production planning and control in multi-stage assembly systems: an assessment of Kanban, MRP, OPT (DBR) and DDMRP by simulation[END_REF] compare different production planning and control methods on a multi-stage assembly use-case. They refine results from [START_REF] Miclo | Demand driven MRP: assessment of a new approach to materials management[END_REF] and give insights to choose between different material management methods. [START_REF] Dessevre | Decoupled lead time in finite capacity flowshop: a feedback loop approach[END_REF] and [START_REF] Lee | A mathematical safety stock model for DDMRP inventory replenishment[END_REF] belong to the second category. They both evaluate different policies to control the DDMRP on the same use-case. [START_REF] Dessevre | Decoupled lead time in finite capacity flowshop: a feedback loop approach[END_REF] use a complete DES software, while [START_REF] Lee | A mathematical safety stock model for DDMRP inventory replenishment[END_REF] use spreadsheet simulation.

Table 1 compares how far policies are detailed in all the previously cited works. It shows that most of the contributions detail how the ADU parameter is set over time; most of the time however, authors assume or elude the policies to set the rest of the parameters. For the remainder of this paper, we define the term "policy" as any function able to propose a new value for a parameter by taking into account either a past time series of data or the present state of the system. It follows the definition in the works of [START_REF] Koulouriotis | Simulation optimisation of pull control policies for serial manufacturing lines and assembly manufacturing systems using genetic algorithms[END_REF].

Dynamic sizing for related parameters outside the DDMRP

The DDMRP methodology inherits from several materials management theories [START_REF] Ptak | Demand driven material requirements planning[END_REF], it is then fair to turn to these to search for dynamical update policies which could help practitioners size their DDMRP buffers dynamically. We identified two main candidate fields: literature from MRP and from card and token controlled production, such as kanban or CONWIP. In these, we aim to find parallels to our dynamical buffer sizing problem. We also consider close parameter definitions between fields. This subsection is organized thematically to better see parallels between the different parameter types. [START_REF] Dolgui | Supply planning under uncertainties in MRP environments: A state of the art[END_REF] and [START_REF] Yeung | Parameters affecting the effectiveness of MRP systems: a review[END_REF] define the list of parameters impacting the performance of MRP workshops as:

• the size of safety stocks,

• the length of the safety and planned lead times,

• the lot sizing policies,

• the horizon of the Master Production Schedule (MPS),

• and the planning horizon of the production orders. Among these five, three are relatively close to DDMRP parameters:

• the safety stock size parameter can be closely linked to the definition of the lower zone of DDMRP buffers (see [START_REF] Ptak | Demand driven material requirements planning[END_REF] or [START_REF] Lee | A mathematical safety stock model for DDMRP inventory replenishment[END_REF]), • the lot sizing policy is related to the MOQ parameter,

• and the definitions of safety and planned lead times are close to the definition of the DLT.

Defining the MOQ is, in itself, optional in a DDMRP environment [START_REF] Ptak | Demand driven material requirements planning[END_REF]. It is also a problem of its own, as shown by the numerous papers on the topic 7288 results on the Web Of Science only with the request ("lot sizing" OR "lotsizing" OR "lot-sizing" OR "order quantit*" OR "batch size" OR "lot size*" on July, 30, 2021). For these two reasons, the dynamic lot sizing problem will not be considered further in the paper. Lastly, the MPS planning horizon is very specific to the MRP environment and has no counterpart in a DDMRP setting for the moment. This leaves us with two topics to explore.

Defining safety stocks.

Earlier works on the definition of safety stocks in an MRP environment can be found in [START_REF] Orlicky | Material requirements planning: the new way of life in production and inventory management[END_REF], where they are sized according the standard deviations of errors between real and forecast demand. [START_REF] Inderfurth | How to protect against demand and yield risks in MRP systems[END_REF] extends these works by including a risk factor, either for unstable demands or yields. The risk factor is established according to the uncertainties and acts as a multiplication factor of the safety stock. [START_REF] Grubbström | A net present value approach to safety stocks in a multi-level MRP system[END_REF] proposes to use safety stock for end products only. The size of these safety stocks is defined as the difference between the sum of production orders and the sum of products consumed by the customers. For this reference, optimal is decided for the sizes that assure the best margin in the last period. [START_REF] Caridi | Buffering against lumpy demand in MRP environments: a theoretical approach and a case study[END_REF] consider safety stocks in an MRP environment for irregular demands. They recommend to use two types of safety stocks: one for day-to-day variance sized without taking spikes into account, and one for unusual demand spikes. Lastly, [START_REF] Axsäter | Inventory control[END_REF] gives several approaches using either optimization or probabilistic measures to decide the safety stock size.

Setting lead times.

When setting lead times, we emphasize the difference between two concepts. Firstly, practitioners may want to estimate the completion date of each individual order with relative degrees of precision, which is called lead time in most systems and is defined as the time between order creation in the system and order delivery. This must be done for each order and may also be referred to as "lead time quotation" in the literature. This, however, is not what is needed in order to define the DLT. DLT definition calls for the second concept: setting lead time as a management parameter in order to size the system [START_REF] Hopp | Factory Physics[END_REF]. As a result, we limit our literature review to setting lead times as management constants and not producing a lead time quotation.

In MRP environments, this concept has two sides: (i) defining the planned lead times and, if necessary, (ii) defining the safety lead times. [START_REF] Yano | Setting planned leadtimes in serial production systems with tardiness costs[END_REF] defines optimal planned lead times in a twostage production environment, with stochastic production and supply times. The works are also extended for the 𝑁 stage problem under the hypothesis of independence of lead times for the different stages. The article uses minimal storing costs and late orders as optimality criteria. Findings include the fact that lead time variance is different at each stage and that, lower in the chain, lead time variation is a combination of variations from the higher stages. The article recommends using a safety lead time on the stage with the most variance and to define the planned lead time afterwards. [START_REF] Melnyk | Implementation of material requirements planning: safety lead times[END_REF] use the same decomposition of planned lead time and safety lead time. They observe that acceptable values for the safety lead times are between two and three times the variation of lead times. They also raise the alert of increasing work-in-process generated by an overestimated safety lead time. They recommend using such a safety lead time for components lower in the bill of materials, especially if they are shared between numerous references. [START_REF] Buzacott | Safety stock versus safety time in MRP controlled production systems[END_REF] and Molinder (1997) compare using safety stocks and safety lead times in a simulated workshop. They establish several strategies to choose between any of the two: for [START_REF] Buzacott | Safety stock versus safety time in MRP controlled production systems[END_REF], we can use safety lead times if forecast quality is good enough. For [START_REF] Molinder | Joint optimization of lot-sizes, safety stocks and safety lead times in an MRP system[END_REF], if there is high variability on demand and little on lead time, using safety stocks is a better choice. In case of high variability for both, use safety lead times. [START_REF] Hnaien | MRP parameterization under lead times uncertainties: Case of multilevel serial production systems[END_REF] or Dolgui, Ammar, [START_REF] Dolgui | A state of the art on supply planning and inventory control under lead time uncertainty[END_REF] propose a series of cases for determining optimal planned lead times in an MRP environment with stochastic lead times. The series progressively details the use case of a production stage by continuously reducing the hypotheses. The only hypothesis kept is the knowledge of the distribution of the lead times and their discrete nature. The search for optimal values of the planned lead times is then obtained through discrete optimization. [START_REF] Axsäter | Planning order releases for an assembly system with random operation times[END_REF] approaches the problem of defining planned lead times in a multi-stage workshop. The workshop is decomposed into individual stages and the author applies the solution given by [START_REF] Yano | Setting planned leadtimes in serial production systems with tardiness costs[END_REF] for the single stage to each of them. Conclusions shift towards the difficulty of finding exact numerical solutions to large problems and the need to use heuristics to solve it.

The works from [START_REF] Altendorfer | Influence of lot size and planned lead time on service level and inventory for a single-stage production system with advance demand information and random required lead times[END_REF] deal with defining both lot sizes and planned lead times in a single stage shop. Authors provide a decision framework to set both parameters with a desired service level. We recall the warnings from [START_REF] Altendorfer | Influence of lot size and planned lead time on service level and inventory for a single-stage production system with advance demand information and random required lead times[END_REF], also found in [START_REF] Sridharan | Master production scheduling make-to-stock products: a framework for analysis[END_REF], on the nervousness induced by too many production parameters changes, however.

Major works on lead time definition also borrow from queuing theory. Central results can be found in Hopp and Spearman (2011), for example. Most of them derive from the now famous Little's law [START_REF] Little | A proof for the queuing formula: L=𝜆 W[END_REF] applied to production lines: Work in process = Cycle time * Throughput For example, [START_REF] Hopp | Factory Physics[END_REF] consider planned lead time as the probability of the actual lead time to be under a certain value. That value can be obtained as a quantile of the lead times distribution. It could also be set using normal distributions or t distributions if appropriate [START_REF] Hopp | Factory Physics[END_REF]. [START_REF] Hopp | Factory Physics[END_REF] also recommend aggregating the flows when considering a multi-product system. It is done by expressing throughput in costs of goods sold (euros per day) and work in process in euros for example.

Literature synthesis

Analysis of the works on the DDMRP shows that authors agree on five parameters to define the buffer size, with varying degrees of importance. The majority of authors evaluate the DDMRP's performance and, consequently, follow the policies prescribed by [START_REF] Ptak | Demand driven material requirements planning[END_REF]. Few authors have challenged these policies and successfully proposed variants.

The DDMRP methodology claims better performance by regularly updating the buffer sizes through these parameters. Most authors only assume a varying ADU; some include a varying DLT. It appears that no work has been done on combining more than one varying parameter, set by more than one policy.

In the meantime, literature shows that, outside works specific to the DDMRP, there are numerous policies that could be used to set the DDMRP parameters dynamically. As a whole, previous works also show the interest of using simulated workshops to establish new results.

Motivated by the gaps in the literature, we focus our study on dynamically controlling the Average Daily Usage (ADU) and the Decoupled Lead Time (DLT) of a DDMRP workshop, as they are the two most influential parameters for setting buffers sizes (see next section for detailed formulas). We will carry the experiment on simulated versions of several types of workshops to (i) have more general results and (ii) set the appropriate level of workshop complexity and still be able to analyze simulation outputs. This design of experiment approach supported by simulation is found in previous DDMRP works such as [START_REF] Miclo | Demand driven MRP: assessment of a new approach to materials management[END_REF] (see 3.3 for more details). Our main objective is to identify to what extent can a DDMRP workshop be dynamically controlled by the ADU and DLT parameters, and what situations may be disadvantageous.

Dynamically sized DDMRP model

We base our study on the use-case of dental prosthesis plant, which manufactures semi-finished blanks. Section 3.1 details the main model assumptions behind our work and the plant capacities. In this section, we also detail the future block factors of our experiments: the various types of shop structures in between stages and the different bottleneck stresses. These block factors help generalize our study outside the use-case by synthesizing specific characteristics into more universal types of problems. In Section 3.2, we add dynamical control policies to our model and explain all possible policy definitions we will test. Section 3.3 explains our design of experiments and the metrics we used in the study.

Shop characteristics and assumptions

Our study is based on an industrial use-case with well defined buffer stages. Stages are enclosed between DDMRP buffers with several parallel machines in between. The plant produces 142 finished products overall, all from only 2 types of raw materials. Following Thürer et al.

(2020), we did not build a complete simulation of the 142 individual routings and constructed a smaller aggregated model with same features as the original. We also assumed raw material buffers to be finite and refilled according to stochastic lead times (exponential distribution using the reference's DLT as parameter). Different shop structures are found in the global production process, as will be detailed further in the paper.

In our use-cases, work centers have stochastic set-up and processing times. We used the industrial data to adapt triangular distributions for each product-machine pair. All work centers have parallel machines, with varying capacities (details in Table 2), with perfect yield. Work centers are opened 24 h a day, 5 days a week. In line with the industrial use-case, we assumed batch moves and sequential batch production at the work centers. All the work centers follow the queue discipline recommended by [START_REF] Ptak | Demand driven material requirements planning[END_REF]: highest priority is given to the order with the lowest buffer penetration ratio (buffer level over buffer size in percentage).

Demand data were aggregated for 10 final products over 1000 worked days from the industrial dataset. As shown in Fig. 1, demand knows two types: (i) relatively stable as for Product 2 on the left, or (ii) with various degrees of lumpiness as for Product 5 on the right. Such data patterns were already encountered in previous DDMRP works. See Pekarčíková et al. (2019) or [START_REF] Dessevre | Decoupled lead time in finite capacity flowshop: a feedback loop approach[END_REF], for example.

Shop structure variations

Along the complete industrial process, three shop structures can be found at various stages. We isolated each of them into a separate buffer stage, in order to assess policy choices for each structure. These structure are close to the ones found in previous works on lead time evaluation, such as [START_REF] Öztürk | Manufacturing lead time estimation using data mining[END_REF]. We named the structures: type A, type V and type O.

Type A workshop is shown in Fig. 2 on the left. In a type A workshop, all product routes coincide at the last work center. Note that this is not an assembly work center however. Type V workshop is shown in Fig. 2, at the center. In this type of workshop, all routes start at the first work center and then diverge. Lastly, type O workshop is shown in Fig. 2, on the right. In a type O workshop, all routes start at the first work center and end at the last one. Whatever the structure, routes are assigned a single branch and material flow is not shared between branches.

Bottleneck factors

Much like the works of [START_REF] Thürer | Production planning and control in multi-stage assembly systems: an assessment of Kanban, MRP, OPT (DBR) and DDMRP by simulation[END_REF], we include several levels of bottleneck occupation in our workshops. The motivation for this choice is twofold: (i) first, the industrial stages that motivated the study have several bottlenecks either because of high occupation of a limited resource work center or because common work centers are clogged by the sequential batch production discipline of the plant and, (ii) we wish to study the system under increasing load levels and refine our recommendations of the best policies for dynamical parameter update.

We consider three levels of bottleneck occupation, noted Low, Medium and High. Corresponding occupation ratios are given in Table 3. Table 4 also defines which machine is the bottleneck in each workshop structure.

ADU and DLT dynamical policies

As explained in 2.3, we wish to investigate a two-parameter dynamical update of a DDMRP workshop. These two parameters relate to: either (i) following the changes in demand with the ADU, or (ii) controlling the internal flow of materials with the DLT. The Eqs. (1) to (4) dictate the standard way of defining zone sizes [START_REF] Ptak | Demand driven material requirements planning[END_REF]) (we use "standard" to remind the reader that the majority authors use these equations, see [START_REF] Lee | A mathematical safety stock model for DDMRP inventory replenishment[END_REF] for alternatives). The buffer's total size is the sum of all zones. For each parameter, we selected three policies, defined hereafter.

Green zone = 𝑀𝑎𝑥(𝐴𝐷𝑈 * 𝐷𝐿𝑇 * 𝐿𝑇 𝐹 , 𝑀𝑂𝑄)

(1)

Yellow zone = 𝐴𝐷𝑈 * 𝐷𝐿𝑇 (2) Red zone base = 𝐴𝐷𝑈 * 𝐷𝐿𝑇 * 𝐿𝑇 𝐹 (3)
Red zone safety = 𝐴𝐷𝑈 * 𝐷𝐿𝑇 * 𝐿𝑇 𝐹 * 𝑉 𝐹 (4)

ADU update policies

The first policy is the constant ADU policy: the parameter is set at the beginning of the experiment and never updated. This level serves as a baseline for comparisons. In this setting, ADUs are selected randomly between 50 and 200 units per day. This distribution is based on the average demands from the industrial use-case. For the rest of the paper, this policy is called "Constant ADU" and encoded as "ADU0" in the results analysis.

The second policy is a moving average over past demand. This is the policy recommended by [START_REF] Ptak | Demand driven material requirements planning[END_REF]. It uses the previous 30 open days to determine the ADU using a moving average function. The 30 days window was determined based on [START_REF] Ptak | Demand driven material requirements planning[END_REF]: window equivalent to twice the longest lead time in the available data, 15 days in our use-case. We call this policy "Simple moving average" for the rest of the study and encode it as "ADU1".

The last policy is a moving average including forecasts, as proposed in [START_REF] Ptak | DDMRP. Demand driven material requirements planning[END_REF]. This policy takes the last 30 open days, adds a 30-day forecast and takes the average of these 60 data points as the new ADU. It is said to better counteract disturbances in demand [START_REF] Ptak | DDMRP. Demand driven material requirements planning[END_REF]. We refer to this policy as "SMAFC" for Simple Moving Average with ForeCasts. We encode it as "ADU2".

DLT update policies

As in the previous paragraph, the first policy is a constant policy for the DLT. DLTs are selected randomly from the available data-sheets containing 5, 10 or 15 days. This will also serve as a comparison baseline and is called "Constant DLT" and encoded as "DLT0".

The second policy is extracted from Hopp and Spearman (2011) as there is no mention of a dynamical policy for DLT in [START_REF] Ptak | Demand driven material requirements planning[END_REF]. For each reference, it returns the 95th percentile of the registered manufacturing times, with a weekly period. This policy is called "Percentile DLT", and given the code "DLT1".

The last policy is a modification of the previous one. We call it "Unique DLT" and define it as an extension of the previous policy where all references receive the same DLT value, equal to the 95th percentile of all the registered manufacturing times. Is it proposed in order to evaluate if controlling a stage of the workshop with a single dynamic DLT suffices to ensure good performance. It is also motivated by the fact that slowest references with long lead times may hinder global flow in the workshop [START_REF] Goldratt | The goal: excellence in manufacturing[END_REF][START_REF] Schragenheim | Drum-buffer-rope shop floor control[END_REF]. This last policy is encoded as "DLT2".

Design of experiments and metrics

In order to evaluate the multiple combinations of policies and workshop layouts, we created a Design Of Experiments (DOE) revolving around Discrete Event Simulation (DES). Several other papers have already showed the efficiency of this approach (see [START_REF] Georgiadis | Dynamic drum-buffer-rope approach for production planning and control in capacitated flow-shop manufacturing systems[END_REF] and [START_REF] Zhang | Simulation-based optimization of dispatching rules for semiconductor wafer fabrication system scheduling by the response surface methodology[END_REF], or specifically to DDMRP Miclo et al. (2019) and [START_REF] Thürer | Production planning and control in multi-stage assembly systems: an assessment of Kanban, MRP, OPT (DBR) and DDMRP by simulation[END_REF]). Our work borrows heavily on the methods of [START_REF] Georgiadis | Dynamic drum-buffer-rope approach for production planning and control in capacitated flow-shop manufacturing systems[END_REF] which used simulation to help practitioners choose between different parameter policies in a drum-buffer-rope workshop. Using computer simulation is especially beneficial whenever analytical forms of the model are not available or hardly tractable, or when it is necessary to run multiple iterations of a simulation to account for randomness effects. This is exactly our situation as DDMRP methods involve multiple non-linear equations and our workshops have stochastic lead times and processing times.

The chosen DOE is a full factorial design, containing two policy factors and two blocking factors. Factors are arranged as follows:

• Blocking factor 1 is the workshop structure,

• Blocking factor 2 is the bottleneck occupation,

• Policy factor 1 is the ADU update policy, • Policy factor 2 is the DLT update policy.

We summarize the DOE with the experiment setup values in Table 5. We consider three levels for each factor and, as it is a full factorial DOE, we run all possible combinations of these four factors. The factor named "Workshop structure" goes over the three types of workshops described in 3.1.2 . The factor named "Bottleneck occupation" takes into account the occupation/capacity ratio in the workshop, also described in 3.1.2. The factor named "ADU Policy" regroups our three ADU control policies. As the names are somewhat long, we gave each level a code, used in the following figures. Lastly, the factor named "DLT Policy" gathers our three DLT control policies. These levels were also given a specific code for easier reading.

Each experiment is evaluated against four metrics adapted from Miclo et al. ( 2019) or [START_REF] Thürer | Production planning and control in multi-stage assembly systems: an assessment of Kanban, MRP, OPT (DBR) and DDMRP by simulation[END_REF]. The first metric is the average fill rate, defined as the average percentage of orders directly satisfied from the DDMRP buffers. Fill rate was chosen because our use-case was Make-To-Stock environment.

The next two metrics evaluate inventory performance. The first is average finished goods inventory defined as the average number of end items in the physical buffers, thus excluding work-in-process. The second is average work-in-process inventory of all buffered references.

The last metric evaluates lead time performance. We consider average throughput time, where we collect completion dates of all production orders minus the release date at the first work center. 2Each individual experiment is replicated 100 times with an individual random seed. Each simulation is run for 1000 days and metrics are computed only after a warm-up period of 300 days.

Results, recommendations and discussion

We separate the results into two analyses. In Section 4.1, we build trees showing which factors and levels explain the most change in performance for all our metrics. Section 4.2 then details only the best combinations of policies for each of our simulation scenarios. Section 4.3 discusses the interest of this contribution for managing DDMRP workshops and the limits of our results.

Conditional inference trees of factor importance

For each of the four metrics, we conducted a 4-way analysis of variance, including all interactions. Each ANOVA revealed statistically significant main and interactions effects at a 1% confidence level (𝑝 ≤ 0.01). To further investigate the effects of each factor, we build conditional inference trees ("ctrees" ) for each metric [START_REF] Hothorn | A lego system for conditional inference[END_REF]. ctrees are grown by performing recursive univariate splits of our factors but do so by using the split with the smallest 𝑝-value (hence most statistically significant), instead of maximizing the explained variation (such as classic decision trees). Trees have been cropped to a depth of 4 to limit the place of the figures and because further splits were less significant at the 1% level. Each of the subsequent figures contains a ctree, the distribution of the output values at the leaves, along with the means. Distributions in a box-plot help the reader evaluate dispersion of the results for the leaves. For level definitions, please refer to Section 3.2. All metrics were rescaled to the [0, 1] scale in order to compare similar ranges, avoiding uselessly high values but keeping the same distributions.

Fig. 3 shows the tree for the Fill Rate. The most important feature is the type of bottleneck occupation in this case. This is revealed by the three zones delimited by:

• left-hand side of the plot and the "Low" bottleneck condition with better Fill Rates between 20% and 90%, • right-hand side of the plot, further divided into "Medium" (Fill Rate between 13% and 31%) and "High" (between 7% and 17% Fill Rate) bottleneck occupation ratios. This is no surprise for the Fill Rate, considering the link between sizing the DDMRP physical buffers and queuing theory. This confirms that little can be done in a DDMRP workshop regarding sizing under heavy bottleneck constraints, and that solutions must be searched elsewhere, in capacity increase or load balancing for example. In most branches, the decision on the ADU policy is encountered before the decision on the DLT policy. Across bottleneck levels and workshop types, using variables ADU policies such as ADU1 or ADU2 prove to bring better results for the Fill Rate, without any clear distinction of which method outperforms the other. When in presence of a "Low" bottleneck level, using the DLT2 policy shows better results on average. For "Medium" and "High" levels, the fixed DLT0 policy seems better. Fig. 4 gives results for the average Finished Goods Inventory. Bottleneck occupation is partly as important as previously, with a "Low" level governing the left-hand part of tree and other levels the righthand part. However in this case, the choice of DLT policy is more impacting. In all branches, choosing the fixed DLT0 policy ensure lower finished goods inventory, while variable DLT policies result in higher and more varied inventory levels. The choice of ADU policy has less consequences, with a fixed ADU ensuring lower inventory on average. The distribution of workshop types in the tree shows that in some case of type A, inventories are lower. This might be explained by the first work center helping to "starve" the line in this set-up, thus avoiding build-up of material down the line.

Fig. 5 shows the impacts for WIP. This time the choice of DLT policy is the most important: by choosing either a fixed DLT (DLT0) or a unique variable DLT for the stage (DLT2), WIP is greatly reduced, as shown on the left-hand side of the tree. Performance on this side is equivalent for all levels of bottleneck occupation and workshop types. These branches suggest that using DLT0 enables good and stable results, while using DLT2 may allow to reach areas of very good performance if one is willing to account for instability. The variable DLT1 policy gives interesting results in the Low bottleneck and Type V structure, but it is otherwise outperformed, probably because it creates extremely unstable systems.

Effects of factors on average Throughput Time appear in Fig. 6. Here, choosing the fixed DLT0 policy has major consequences: it leads to stabler and lower throughput times across all cases, as shown on the left branches. The exception is the "Low" bottleneck occupation and "Type V" workshop having both ADU and DLT as variable parameters is beneficial to the average throughput time. As in the previous paragraph, we can suspect that a small error in the DLT parameter may have great consequences in the system's stability whereas, for this metric, the choice of ADU appears much lower in the tree.

Recommendations on policy choices

We conclude our results with a set of recommendations for practitioners. These recommendations are built using the simulation results. We selected the combination with the best average for each cell. We also confirmed by a one-way ANOVA for each metric that these means were significant at the 1% level. The recommendations are gathered in Table 6. We now go over our six proposed policies and evaluate how they fare performance-wise.

The "Constant ADU" policy ADU0 appears in several cells of Table 6, especially when not considering Fill Rate as the main metric. The "Simple Moving Average" ADU1 and "SMAFC" ADU2 policies appear about as many times but are in majority in the Fill Rate. This may be explained by two aspects of the experiments:

• as each metric is considered independently of others in our study, there is no compromise between all metrics. As a result, a very good combination fill rate-wise may behave poorly on another metric. Consequently, stable combinations of fixed parameter policies may be more efficient than variable ones, especially for finished goods or WIP if the parameters are already close to "optimal". • it is possible that the random setting of parameters during initialization was already very favorable. Data came from an industrial dataset where practitioners already some experience on setting the parameters according to their specific environment.

The overall distribution of the 5 combinations in Table 6 is: (i) 36% for (ADU0, DLT0), (ii) 28% for (ADU1, DLT0), (iii) 25% for (ADU2, DLT0), (iv) 8% for (ADU1, DLT1) and (v) 3% for (ADU2, DLT1). In more than a third of the scenarios, the fully constant solution over 1000 days is the best (but never if Fill Rate is the metric). The combination (ADU1, DLT0), recommended in [START_REF] Ptak | Demand driven material requirements planning[END_REF] is the second most frequent and scattered across the four metrics. Fully variable solutions are recommended in only four scenarios (11%). The "Constant DLT" DLT0 clearly outperforms the other and is present in a majority of cells. Very few cells, and only for the Fill Rate metric, call for the "Percentile DLT" DLT1 policy. The "Unique DLT" DLT2 is absent of the recommendations, despite being a major factor in the ctrees, particularly for average Throughput.

Discussion

Our main objective is to identify to what extent can a DDMRP workshop be dynamically controlled by the ADU and DLT parameters, and what situations may be disadvantageous. This study's objective was to understand to what extent a DDMRP workshop could be dynamically controlled using more parameters than the ADU usually mentioned in the literature. As analytical models were not easily tractable, we had to resort to computer simulation to back up our design of experiments. While, this limits the universal character of our contribution, we introduced blocking factors to take into account multiple workshop types and machine occupation scenarios.

Our results show that, in many cases (36%), stable control is actually a better solution for stable inventory levels and throughput time. This finding goes against the claims of DDMRP being able to adapt to changes and still maintain good performance. Still, in a bit less than a third of the remain cases, the solution proposed by [START_REF] Ptak | Demand driven material requirements planning[END_REF] with only a varying ADU is the best one. Contrary to previous claims from [START_REF] Dessevre | Decoupled lead time in finite capacity flowshop: a feedback loop approach[END_REF], dynamically controlling both the ADU and DLT seems to be reserved to a few select cases, and solely when the fill rate is the preferred metric. No clear reason appears for the moment as to why some cases within the fill rate column in Table 6 tolerate multiple dynamic control better than others. Authors from [START_REF] Dessevre | Decoupled lead time in finite capacity flowshop: a feedback loop approach[END_REF] had, however, warned against controlling both parameters in cases where it would introduce instability.

Three aspects then stand out when considering the reach of our findings: the use of independent metrics, the non-sharing of work centers and the trouble of analyzing deep interactions.

We chose to measure performance on four distinct metrics to be able to explain the factor effects independently. This, however, prevented us from having a global (and possibly weighted) view on an aggregated metric. Further studies could be done with the same DOE to refine our analysis. This could prove important as practitioners may look for balanced choices rather than one-sided solutions. This could also help in resolving the discrepancies in policy choice when preferring the fill rate versus the other metrics.

Our use-case required that, outside of the bottleneck work centers, all other work centers were only shared by specific products. This allowed us to define clear branches in the workshop structures. This, however, works against the generalization of our findings; the study could be applied to general job shops, for example.

Lastly, we need to address the analysis complexity limit of our study. The 4-way ANOVA showed meaningful interactions up to the fourth degree, preventing us from drawing conclusions on the main effects of factors. We chose ctrees as a way to conduct multiple comparison of all the groups and extract the most import factor choices. Having limited the depth to the fourth level, we loose precision of analysis and the leaves of our trees do not contain the same number of samples. The maximum meaningful depth for the trees according to our data was 7, but it severely hampered reading.

Conclusions

We challenged the possibility of controlling two parameters dynamically for several configurations of DDMRP workshops. We tested different combinations of policies to set both the Average Daily Usage (ADU) and Decoupled Lead Time (DLT), while considering different workshop types and bottleneck occupation levels in order to generalize our conclusions at maximum. The variety of our study is mainly motivated by the fact that DDMRP settings come in various shapes and sizes and no two evaluations are really the same. We aimed to start bridging that gap.

Results suggest that two parameter setting (ADU and DLT) may be reserved to very few cases, particularly when considering evaluation metrics independently from each other. This calls for further investigation using a mixture of metrics, either from weighting or from goal programming, for example. Results also hint at the fact that fixed DLT parameters may be sufficient when work centers are not shared. However, in more general workshop structures, this hypothesis may be relaxed, again calling for some more experiments. Lastly, and maybe more surprisingly, we found that a third of cases had better performance when statically sized instead of dynamically, which goes against common claims of the DDMRP.

We conclude that, for the many scenarios we considered, it may be sufficient to control the sizes of the DDMRP buffers only through the ADU parameter. A good number of cases even suggest keeping it constant, thus simplifying the problem to the accurate sizing of Theory of Constraints or Queuing Theory approaches.

Our study is another step towards understanding the limits and intricacies of applying the DDMRP to real cases. Our findings suggest new avenues for research. Can we derive general recommendations for job shops instead of flow-shops? would composite metrics alter our conclusions? or are there better policies for DDMRP parameter? are some of the still unanswered questions in our field.

CRediT authorship contribution statement

Guillaume Martin: Ideas, Methodology, Software, Validation, Formal analysis, Writing. Matthieu Lauras: Ideas, Resources, Data curation, Supervision, Writing -review & editing. Pierre Baptiste: Ideas, Supervision, Writing -review & editing.

Fig. 1 .

 1 Fig. 1. Two types of demands encountered in the dataset.

Fig. 2 .

 2 Fig. 2. Types of workshops.

Fig. 3 .

 3 Fig. 3. Conditional tree for factors impacting Fill Rate.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Conditional tree for factors impacting Finished Goods Inventory.

Fig. 6 .

 6 Fig. 6. Conditional tree for factors impacting Throughput Time.

Table 1

 1 Author vs. Parameter in DDMRP works, sorted by date (oldest).

	Reference	Parameter
		ADU	DLT	VF	LTF	MOQ
	Ptak and Smith (2016)	1	p	1	1	p
	Ptak and Smith (2018)	n	p	1	1	p
	Kortabarria et al. (2018)	1	-	-	-	-
	Shofa et al. (2018)	1	-	-	-	-
	Miclo et al. (2019)	1	-	-	-	-
	Velasco Acosta et al. (2019)	1	-	-	-	-
	Dessevre et al. (2019)	1	n	-	-	-
	Lee and Rim (2019)	1	p	-	-	-
	Thürer et al. (2020)	1	-	-	-	-
	-not addressed or supposed to follow Ptak and Smith (2016).
	1 single policy.

𝑝 partially addressed. 𝑛 several policies.

Table 2

 2 Work center capacities.

	Work center	0	1	2	3	4	5
	Capacity	6	3	3	3	3	2

Table 3

 3 Bottleneck occupation ratios.

	Bottleneck level	Low	Medium	High
	Occupation ratio	50%	80%	95%
	Table 4
	Workshop VS. bottleneck position.
	Workshop type	Type A	Type V	Type O
	Bottleneck machine	Last	First	First

Table 5

 5 Experiment setup values.

	Factor type	Factor name	Factor level (Factor Code)
	Block factor	Workshop structure	Type A	Type V	Type O
		Bottleneck occupation	Low	Medium	High
	Policy factor	ADU policy	Constant ADU (ADU0)	Simple Moving Average (ADU1)	SMAFC (ADU2)
		DLT policy	Constant DLT (DLT0)	Percentile DLT (DLT1)	Unique DLT (DLT2)

Table 6

 6 Policy recommendation based on best simulation results.

	Bottleneck Workshop Fill	Finished	WIP	Throughput
	occupation type	rate	goods	inventory	time

Our study will focus on these two bold parameters, as shown by the synthesis of the literature further in the paper.

Following the methods from[START_REF] Thürer | Production planning and control in multi-stage assembly systems: an assessment of Kanban, MRP, OPT (DBR) and DDMRP by simulation[END_REF], we considered average lead time too, defined as the difference between order completion and pool entry dates. However results were exactly the same as average throughput time. This metric is consequently left out of the study.

Data availability

The data that has been used is confidential.


V Simic

S Boileau
email: boileau@glvt-cnrs.fr

Laurent Bouteiller

L Gallez

P Merlin

GAS BARRIER AND ADHESION OF INTERPENETRATING POLYMER NETWORKS (IPNs) BASED ON POLY(DIURETHANE BISMETHACRYLATE) AND DIFFERENT EPOXY-AMINE NETWORKS

Keywords: Interpenetrating polymer networks (IPNs), polymethacrylate network, epoxyamine network, gas barrier properties

Gas barrier and adhesion of interpenetrating polymer networks based on poly(diurethane bismethacrylate) and different epoxy-amine networks

Introduction

Flexible packaging films for oxygen sensitive products (such as foods and electronics) are generally made of multilayer structures. Each of these layers brings a specific property to the material such as high oxygen barrier and good adhesion to a substrate. The current trend is to reduce the number of these layers while keeping the same mechanical and technical characteristics [1]. The development of more "ecological" systems (absence of halogenated compounds such as poly(vinylidene chloride) and of solvents) is also a current trend. Among all the possibilities of combining different properties (polymer blends, copolymerization…), our attention was focused on the use of interpenetrating polymer networks (IPNs). IPNs are combinations of two cross-linked polymers (each made via independent reaction processes) held together by permanent entanglements and obtained under special methods of preparation [START_REF] Sperling | Interpenetrating Polymer Networks and Related Materials[END_REF]. The absence of cross-reactions between the two networks is an important condition to allow morphology control. So, a system which combines a free-radical polymerization of a bis(methacrylate) compound with the polyaddition of amines on epoxy resins was chosen. This type of system has not yet been extensively studied [START_REF] Dubuisson | [END_REF][4][5][6].

Generally, organic polymers which exhibit high oxygen barrier properties present crystalline domains and a compact structure [7]. So, IPNs with high crosslink densities of both networks and which have the highest possible degree of interpenetration would be the best candidates.

A bis(methacrylate) bearing urethane groups able to establish strong interchain cohesion through hydrogen bonding interactions was used as the monomer and plurifunctional amines were used as hardeners for epoxy resins.

In the present paper, the results of feasibility investigations on such systems are reported. Gas barrier properties of IPNs as a function of the curing sequence are also examined.

Experimental Part

Reagents N,N'-Bis (2-methacryloyloxyethyl) -1,9 nonylene biscarbamate (DiUdiMA-Aldrich) was used as the monomer. The methacrylate network formation was photo-induced instantaneously by UV irradiation with 2,2-dimethoxy-2-phenylacetophenone (1% by weight -Aldrich) as the photo-initiator. Triethylenetetramine (TETA-Kodak), 1,3-bis ( The crosslinking reaction of epoxides by amines was carried out at room temperature.

Addition of a surfactant (Triton X100-Aldrich) was necessary when DGGOA was used as the epoxide to improve film wetting. All compounds were used as delivered. The formula of these precursors is listed in Table 1. For the study of crosslinking mechanisms, triethylene glycol dimethacrylate (TEGDMA-Aldrich) was used as potentially photo-curable monomer and a monofunctional amine : n-octyl amine (nOCTA-Sigma) was used as a curing agent for NPG100.

Preparation of films

Formulations were cast on polypropylene films (OPP, 30 m thick), which had been activated by a Corona discharge treatment previously, in order to improve their wettability. A second OPP film was laminated on the top. Samples were irradiated with a medium pressure mercury lamp (120W/cm) after different waiting times. The conveyor belt speed (0.10 m/s) and the number of exposures set to 3, were selected to achieve the highest polymer conversion. The light intensity received by samples corresponds to 12 J.cm -2 .

Instrumentation

IR spectra were recorded on thin samples coated onto OPP films held between two KBr discs with a Perkin Elmer FTIR 1760 spectrophotometer. 1 H NMR spectra were recorded in CDCl 3 on a Bruker AC 200 MHz spectrometer. Oxygen transmission rates (OTR) of coated samples were determined using a Mocon Ox-tran 2/20 test equipment according to the standard method ASTM D3985. The test cell was composed of two chambers separated by the film.

Films were subjected to 100% oxygen on one side and to nitrogen with 3% hydrogen on the other side. Nitrogen/hydrogen gas containing oxygen which transferred through the film was conducted to the coulometric sensor, and measurements were obtained when a steady state was reached. The relative humidity (RH) of the two gases was controlled by a humidifier. All the reported results have been automatically normalized to 1 atmosphere. Peel tests were recorded on a Lloyd L500 instrument. Measurements were performed according to the ASTM standard method on 15 mm wide samples and with a peel speed of 200mm/min.

OTR measurements and peel tests were performed by Dow Corning.

Glass transition temperatures of the coatings were measured on a differential scanning calorimeter (DSC, Perkin Elmer Model 4). The temperature was scanned from -150 to +150°C at a heating rate of 20 °C.min -1 . The evolution of sample turbidity as a function of reaction time was followed on a 554 Perkin Elmer UV spectrophotometer. Refractive indexes were measured at 20°C with a Prolabo universal refractometer.

Quantitative analysis of infrared spectra

The conversion, "Conv", of the reactive groups (epoxy or C=C double bonds of the bismethacrylate monomer) in the various investigated systems was determined in the usual way : Conv = (C 0 -C t )/C 0 and by assuming that the Beer-Lambert relation is valid : Conv = (A 0 -A t )/A 0 where C is the concentration, A represents the absorbance, and the subscripts 0 and t denote reaction times zero and t, respectively.

The analyzed peaks were the oxirane ring deformation at 910 cm -1 or 859 cm -1 for the epoxyamine network and the double bond deformation at 1637 cm -1 for the methacrylate network.

The intensity of each peak was related to a reference at 1190 cm -1 , the intensity of which remains unchanged during all the curing process.

Results and Discussion

IPNs based on a methacrylate network (prepared according to a free-radical process) interpenetrated with an epoxy-amine network (cured at room temperature without the addition of catalyst) were studied. At first, the absence of cross-reactions between components of each network was carefully examined because non-interfering reaction routes are an essential condition for a good structure control.

Evaluation of cross-reactions between the two networks.

The Michaël addition of amines on activated C=C double bond is a well-known reaction which could reduce the crosslinking density of both networks and create covalent bonds between the two networks. As a result such reactions could affect gas barrier performances of the material (scheme 1). In order to investigate the presence of inter-crosslinking reactions, for the simplicity of its 1 H NMR spectrum (figure 1, (1)). n-OctA was used rather than a plurifunctional amine to avoid the formation of a network. Its 1 H NMR spectrum is given as

a reference in figure 1-(2).
It can be seen from examination of the last spectrum of figure 1 :

the appearance of a doublet at 1.1 ppm (c') corresponding to the methyl protons of the Michaël addition product on TEGDMA.

-the decrease of the peak at 1.3 ppm corresponding to NH 2 protons () of n-octyl amine -the appearance of new peaks between 2.4 and 3 ppm (see assignments on figure 1)

the appearance of triplets (d') and (e') due to a different chemical environment for these protons after the Michaël addition.

-the decrease of the terminal vinyl protons ((a) and (b)) of TEGDMA at 6.0 and 5.5 ppm respectively.

All these observations are consistent with the occurrence of Michaël addition. Furthermore, it is possible to determine a degree of conversion for this reaction by comparing the intensities of peaks a and b in the mixture with the theoretical intensity expected for those peaks. The theoretical intensity expected for TEGDMA protons was calculated from the intensity of methyl protons of n-octyl amine knowing the [C=C]/[NH 2 ] initial ratio. Thus, after one night at room temperature 40% of TEGDMA has reacted with the amine.

As shown above, the addition of the amine on TEGDMA (C=C) double bonds occurs, but in the IPN, this reaction competes with the epoxy crosslinking process. This is why all the components necessary to build up the final "IPN" (i.e. TEGDMA, nOCTA and NPG 100

([C=C]/[NH 2 ] = 1, ([NH 2 ]/[epoxy] = 1
)) were mixed together and the resulting mixture was analyzed by 1 H NMR spectroscopy after one night reaction at room temperature (figure 2).

Since We have shown in this part the absence of interfering mechanisms in the bis (methacrylate)/epoxy-amine system. As the two networks can be prepared independently, a control of IPN morphology as well as an adjustement of its properties for the desired application could be expected.

The first expected property is a low oxygen permeability. So, a highly interpenetrated polymer network, with the highest crosslinking density is required in order to decrease the gas diffusion through the polymer membrane. The second desired property is a good adhesion of IPN films on a polymeric substrate (for instance, oriented polypropylene sheets) the gas barrier performances of which are insufficient for the application but has good mechanical properties. Epoxy resins were chosen to promote adhesion. As networks have to be also compact (with the lowest free volume between the chains) plurifunctional compounds of a rather small size or compounds able to establish interchain interactions such as hydrogen bonds to increase the cohesion inside the material were chosen. Thus, the following system : diUdiMA/TETA-DGGOA (see Table 1) was tested.

The IPN is formed in situ between two polypropylene sheets : epoxy-amine reaction evolves at room temperature whereas the methacrylic network formation is initiated by irradiation at given times. As the methacrylate network is instantaneously formed by photo-polymerization we can modulate the moment of its formation as a function of the degree of conversion of epoxy-amine crosslinking reaction.

DiUdiMA/TETA-DGGOA system

First of all, kinetics for epoxy-amine crosslinking process at room temperature in the diUdiMA/TETA-DGGOA system were followed by IR spectroscopy in order to irradiate the system at pertinent times. Examples of FTIR absorption spectra, collected at different times during the curing process are presented in figure 3. The conversion was determined according to the decrease of the peak height at 910 cm -1 corresponding to the deformation of the oxirane ring. The peak at 1190 cm -1 , the intensity of which remains unchanged during all the curing process, was used as a reference. The curing rates for the poly(diUdiMA)/TETA-DGGOA system (polydiUdiMA formed immediately after the film formation) and TETA-DGGOA single network curing process were also followed (figure 4). As shown in figure 4, the epoxy-amine curing rate is slower in 50% wt. diluted system (with diUdiMA or polydiUdiMA) than in the bulk. Moreover, in dilute systems, the formation of the polymethacrylate network does not affect the reaction rate.

Visual examination of the IPN mixture reveals that after a few hours, phase separation occurs due to the molecular weight increase of the epoxy-amine network. Such a behavior is attractive for studying the influence of the IPN curing sequence (varying with the moment at which irradiation was induced) on its gas barrier properties. In fact, three significant events play a role in the development of IPN morphology and have to be considered [8] :

-the phase separation -the gelation of the epoxy-amine network -the gelation of the methacrylate network

The relative position of these three events along reaction time will influence the IPN structure. Homogeneity can be achieved by choosing a system in which crosslinking occurs faster than microphase separation.

To determine at which time the demixion occurs, the change of sample turbidity was followed by UV-Vis spectroscopy. The evolution of sample absorbance at 600 nm vs reaction time is reported in figure 5. The time of phase separation determined at the onset of turbidity is about 7 hours of reaction.

Gelation time of the epoxy-amine network was determined by dissolving small amounts of the reacting mixture, collected at given intervals of time, in methylene chloride. The gelation point was assumed to be reached when the samples became insoluble (at around 7.75h-8h of reaction). Gelation time of the methacrylate network is the moment of irradiation since the photopolymerization is very fast.

Then, a series of thin (9-13 m) films based on diUdiMA/TETA-DGGOA system was prepared. The methacrylate network formation was induced by UV irradiation at different times of the epoxy-amine curing process. These tests were carried out with a view to determine the influence of the moment of irradiation on film properties.

Three different situations were considered:

-films irradiated beyond 7h of epoxy-amine reaction. IPNs presenting a very poor level of interpenetration are expected as the phase separation occurred.

-films irradiated after a short waiting time (t<~2h). As epoxy and amine are still fluid enough, it can be assumed that, after photopolymerization of the methacrylate, diffusion towards microdomains is still possible and thus could lead to demixion.

-films irradiated after an intermediate waiting time (~2h<t<7h). In this case, one can suppose that diffusion is limited by the increase of viscosity due to epoxy-amine curing process, and thus a better network interpenetration is expected.

Films based on diUdiMA (photocured at different times)/TETA-DGGOA system were prepared and morphology studies were performed by using DSC and optical miscroscopy.

Unfortunately, it was impossible to detect glass transitions of IPNs and all films were transparent whatever the moment of their irradiation.

At the same time, oxygen permeability of laminates was evaluated. OTR results obtained for these laminates are reported in table 2. Measurements were performed at 50 and 80 % of relative humidity (RH). As OTR values depend strongly on coating thickness (e), a correction has been done in order to normalize the results. OPP contribution on laminates permeability has also been subtracted according to:

1 OTR(Lam inate)  2 OTR(OPP)  e OTR(corr) OTR corr    OTR Lam inate  * OTR OPP   OTR OPP   2OTR Lam inate   * e
where OTR(OPP sheet of 30µm thickness) is equal to 1620 cc/m 2 .day at 50% RH and 1545 cc/m 2 .day at 80% RH ; e denotes sample thickness.

Consequently, OTR(corr) corresponds to the permeability of a 1 m thick IPN layer.

In order to evaluate the contribution of each network, single network laminates were also prepared and their oxygen permeability was measured. The evolution of corrected OTR values for films irradiated after different waiting times in the case of diUdiMA/TETA-DGGOA system is reported in figure 6. Firstly, one can notice that at 50% RH, IPN behavior in terms of gas barrier performance is intermediate between those of each single network or (polydiUdiMA network being the most permeable). Moreover, one can notice that similar OTR values are obtained whatever the moment of irradiation.*** When the RH value is increased to 80%, higher OTR values are obtained but a similar trend as before is observed.

The best OTR results are obtained for the TETA-DGGOA single network. The high crosslinking density inside the cured layer can be responsible for the observed O 2 barrier effect.

In the case of IPNs, a good adhesion between the different layers has been observed. Films were always torn during peel tests. This means that the adhesive strength is higher than

DiUdiMA/BY16871-NPG100 system

This system is different from the previous one since turbidity does not increase during the epoxy-amine addition. Phase separation occurrence is not excluded before irradiation but the corresponding domain size is smaller than visible light wavelengths (400-700 nm). Indeed, refractive indexes of the two networks were measured and were found to be equal to 1.485 and 1.455 at 20°C for the diUdiMA and the NPG100-BY16871 mixture, respectively. In comparison, the system which exhibited an obvious phase separation (TETA-DGGOA) presented a comparable refractive index difference: 1.485 for diUdiMA and 1.524 for the TETA-DGGOA mixture.

Different films were prepared and irradiated at different times after film formation. Results of OTR measurements on these laminates are collected in table 3. In the same manner as before, corrected OTR values as a function of photo-curing time are reported in figure 7. It is clear from these results that BY16871 has a very "negative" effect on oxygen permeability performances, since OTR values are extremely high. One can also notice that similar OTR values are obtained whatever the irradiation time. Moreover, no significant difference between measurements at 50% and 80% RH is observed. Peel tests on these laminates were performed. A good adhesion between sheets was observed since the substrate was always torn during the measurements.

DiUdiMA/ D 5 amine-NPG100 system

DiUdiMA/ D 5 amine-NPG100 system is an initially clear formulation which remains transparent until the end of epoxy-amine reaction. Same experiments as before were performed on this system. O 2 permeability results for this system are reported in table 4. As shown in figure 8, the same OTR values were obtained whatever the instant of irradiation.

Moreover, films are little affected by RH increase from 50 to 80%. One can also notice that film gas barrier performances are better than those obtained with the other aminofunctional siloxane (BY16871). The substrates were always torn during peel tests.

The gas permeability through a membrane depends both on diffusivity (i.e. the mobility of polymer chains, the size of the free volume or the size of gas molecules) and solubility of the gas in the polymer. Tg values of networks (table 5) reflect the differences of crosslinking density which exist between all these materials. If we compare networks containing aminofunctional siloxane (BY16871 and D 5 amine) the difference observed for gas barrier properties could be assigned to a difference of network compactness.

Conclusion

Based on the above investigation it can be concluded that cross-reactions in the IPNs prepared from a free-radically photocured bis(methacrylate) compound and epoxy resins Reacting mixture after 1 night at RT Table 3. Oxygen permeability measurements on diUdiMA/BY16871-NPG100 system irradiated at different time after films formation and corresponding single networks.

C C H 2 C CH 3 O O NH CH 2 O C NH 9 C CH 2 O O O O C 2 C CH 2 H 3 C CH 2 O 2 N,N'-Bis(2-methacryloyloxyethyl) -1,9 nonylene biscarbamate Aldrich DMPA C O OCH 3 OCH 3 C 2,2-Dimethoxy-2-phenylacetophenone Aldrich DGGOA O CH 2 O CH 2 N O O CH 2 N,N-Diglycidyl-4-glycidyloxyaniline Aldrich NPG100 O CH 2 CH 2 C CH 3 CH 3 O CH 2 CH 2 O O Neopentyl glycol diglycidyl ether Aldrich TETA H CH 2 NH CH 2 H 2 N 3 Triethylenetetramine Kodak BY16-871 Si Si H 3 C H 3 C O H 2 C C H 2 C NH 2 CH 3 CH 3 CH 2 CH 2 CH 2 NH 2 H 2 1,3-Bis(3-aminopropyl)tetramethyldisiloxane Dow Corning D 5 amine CH NH CH 2 CH 2 NH 2 Si H 3 C CH 2 CH 3 5 O N-[3-(Dimethoxymethylsilyl)-2-methylpropyl)-1,2- ethylenediamine] hydrolyzate Dow Corning Scheme 1. + NH 2 R C CH 3 O H 2 C OR' NH CH 2 CH CH 3 R C O OR'

Michaël addition (a) Epoxy-amine reaction (b) Free-radical polymerization of the methacrylate

C CH 3 O H 2 C OR' n + R°C CH 3 O OR' CH 2 R n-1 C CH 3 O OR' CH 2 °+ NH 2 R O R' NH CH 2 CH R OH R' NH CH 2 CH R OH R' + O R' N CH 2 CH R OH R' CH 2 CH OH R'
CH 3 Ha Hb c C O O CH 2 CH 2 OCH 2 CH 2 OCH 2 CH 2 O H 3 C C O d e f f H H NH 2 CH 2 CH 2 CH 3 6     (1) (2) (3)

  -2-methylpropyl]-1,2-ethylenediamine hydrolyzate (D 5 amine -Dow Corning) were used as hardeners. N,N-Diglycidyl-4-glycidyloxyaniline (DGGOA-Aldrich) and neopentyl glycol diglycidyl ether (NPG100-Aldrich) were used as epoxides. The [NH]/[epoxy] ratio was chosen to obtain the highest glass transition temperature (Tg). The same weight proportion (50/50) of both networks was used in all experiments. The composition of the different IPNs was as follows: for diUdiMA/TETA-DGGOA: 50/10.4-39.6; diUdiMA/BY16871-NPG100: 50/18-32; diUdiMA/ D 5 amine-NPG100: 50/18.7-31.3.

  TEGDMA signals remain unchanged (figure 2-(2)), only the epoxy-amine reaction occurs in this case because it is faster than Michaël addition (figure 2-(3)).

6N/ 15mm .

 15mm Two other amines were tested: 1,3-bis (3-aminopropyl) tetramethyldisiloxane (BY16871) and N-[3-(dimethoxymethylsilyl)-2-methylpropyl-1,2-ethanylenediamine] hydrolyzate (D 5 amine). The triepoxy compound (DGGOA) was replaced by a difunctional one (NPG100) for miscibility reasons in the case of D 5 amine and partial wetting in the case of BY16871. It will be possible with these new formulations to compare the influence of crosslinking density on IPN gas barrier properties.

(

  cured with plurifunctional amines) are absent. Thin films containing a diUdiMA/TETA-DGGOA interpenetrating polymer network as the central layer present rather good gas barrier properties. The introduction of BY 16-871 in IPNs dramatically drops laminate gas barrier performances. When D 5 amine is used as the hardener an intermediate gas barrier behavior is observed. In none of the studied systems an influence of the curing sequence (modulated by the moment of irradiation) on IPN gas barrier properties has been shown. This result clearly indicates whether the moment of irradiation does not affect IPN structures or it has no effect on IPN properties. Finally, good adhesion properties between the different layers are observed in all examined systems.

Figure 1 . 1 H

 11 Figure 1. 1 H NMR spectra of (1) TEGDMA (2), n-octyl amine, (3) TEGDMA+n-

Figure 2 . 1 HFigure 3 .

 213 Figure 2. 1 H NMR spectra of (1) NPG100, (2) NPG100 +n-octyl amine after 1 night at

Figure 4 .Figure 5 .

 45 Figure 4. Conversion (Conv) versus time plots for the epoxy-amine crosslinking

Figure 6 .

 6 Figure 6. Oxygen permeability measurements on diUdiMA/TETA-DGGOA

Figure 7 .

 7 Figure 7. Oxygen permeability measurements on diUdiMA/BY16871-NPG100 films in

Figure 8 .

 8 Figure 8. DiUdiMA/D 5 amine-NPG100 system. Gas barrier

Figure 1 . 1 H

 11 Figure 1. 1 H NMR spectra of (1) TEGDMA, (2) n-octyl amine, (3) TEGDMA+n-

Figure 2 . 1 H

 21 Figure 2. 1 H NMR spectra of (1) NPG100, (2) NPG100 +n-octyl amine after 1

Figure 3 .

 3 Figure 3. FTIR absorbance spectra in the 2000-600 cm -1 wavenumber range,

Figure 4 .

 4 Figure 4. Conversion (Conv) versus time plots for the epoxy-amine crosslinking

Figure 5 .Table 2 .

 52 Figure 5. Evolution of diUdiMA/TETA-DGGOA absorbance at 600nm vs

Figure 6 .

 6 Figure 6. Oxygen permeability measurements on diUdiMA/TETA-DGGOA

Figure 7 .Table 4 .

 74 Figure 7. Oxygen permeability measurements on diUdiMA/BY16871-NPG100

Figure 8 .

 8 Figure 8. DiUdiMA/NPG100-D 5 amine system. Gas barrier performances vs

Table 1 .

 1 The reagents for IPN preparation

	Abbreviation	Formula	Source
	DiUdiMA

Table 2 .

 2 Oxygen permeability measurements on DiUdiMA/TETA-DGGOA system and corresponding single networks, performed at 50 and 80% RH. The methacrylate polymerization was induced by irradiation at different times after films formation (called "waiting time").

	Sample reference Waiting time	Sample	OTR	OTR	OTR corr	OTR corr
		(h)	thickness	50% RH	80% RH	50% RH	80% RH
			(m)	cc/m 2 /day	cc/m 2 /day	cc/m 2 /day	cc/m 2 /day
	151 to	0	12.8	80	83	1136	1190
	151 to+1h	1	13.6	34	33	483	469
	151 to+3h30	3.5	17.5	59	82	1114	1606
	151 to+5h	5	13.1	79	119	1147	1843
	151 to+6h30	6.5	15.4	43	57	699	948
	151 to+8h08	8	15	36	47	565	751
	151 to+8h30	8.5	11.6	74	116	945	1584
	150 to+15h30	15.5	16.5	39	43	676	751
	150 to+18h30	18.5	14.5	46	55	707	859
	150 to+22h	22	13.3	43	50	604	711
	150 to+25h	25	11.5	43	53	522	654
	150 To+39h15	39.25	10.8	45	49	515	565
	150 To+42h10	42.17	10	59	74	636	819
	150 To+47	47	11.7	53	64	664	817
	1.5% by weight of triton X100 was added in the formulation to improve wetting and films were laminated at
	50°C.
	Sample reference	Sample	OTR
		thickness	50% RH
		(m)	cc/m

2 /day OTR 80% RH cc/m 2 /day OTR corr 50% RH cc/m 2 /day OTR corr 80% RH cc/m 2 /day Adhesion (N/15 mm)

	coating	1800 2000						50% RH 80% RH OTR corr PolydiUdiM A at 50% RH OTR corr TETA-DGGOA at 50% RH
	a 1  m thick	1200 1400 1600
	(cc/m2/day) for	600 800 1000
	Corr	400
	OTR	200
		0
		0	5	10	15	20	25	30	35	40	45	50
						Waiting time (h)
	TETA-DGGOA *	8.5		13.9		15		120	130		Substrate tears
	Poly (diUdiMA)	9.4		144		155		1646	1823		1.15
	* formulation was cast on OPP films after1 hour of reaction in the pot in order to obtain the same
	coating thicknesses as those obtained for IPNs. 1% of triton X100 was added to improve wetting and
	the film was laminated at 50°C.

Table 3 .

 3 Oxygen permeability measurements on diUdiMA/BY16871-NPG100 system irradiated at different time after films formation and corresponding single networks.

	Sample reference	Waiting time (h)	Sample Thickness e (µm)	OTR 50% RH cc/m 2 /day	OTR 80% RH cc/m 2 /day	OTR corr 50% RH cc/m 2 /day	OTR corr 80% RH cc/m 2 /day
	90 (t o )	0	12	435	445	11283	12618
	90 (t o + 2h)	2	11.3	437	461	11389	12944
	90 (t o + 4 h)	4	11.5	445	461	12092	13173
	90 (t o + 8h)	8	9	508	518	13386	14188
	100A (t o + 21h)	21	11	438	446	11146	11628
	90 (t o + 24h)	24	10	492	497	13581	13969
	90 (t o + 29h)	29	/	465	477	/	/
	100A (t o + 51h)	51	13	381	400	9786	10799
	90 (t o + 55h)	55	/	461	491	/	/
	100A (t o + 67h)	67	12.6	391	424	9989	11861
	Sample reference	Sample Thickness e (µm)	OTR 50% RH cc/m 2 /day	OTR 80% RH cc/m 2 /day	OTR corr 50% RH cc/m 2 /day	OTR corr 80% RH cc/m 2 /day	Adhesion (N/15 mm)
	86 Poly(diUdiMA) network	9.4	144	155	1646	1823	1.15
	107 NPG100-BY16871 network	9.5	596	559	21468	19280	Substrate tears

Table 4 .

 4 DiUdiMA/D 5 amine-NPG100 system. O 2 transmission rates measured for samples irradiated after different waiting times.

	Sample reference Waiting time (h)	Sample thickness (m)	OTR 50% RH cc/m 2 /day	OTR 80% RH cc/m 2 /day	OTR corr 50% RH cc/m 2 /day	OTR corr 80% RH cc/m 2 /day
	154 to	0	9.8	194	199	2500	2628
	154 to+2h	2	10.1	154	156	1921	1975
	154 to+3h30	3.5	12.7	135	150	2058	2365
	154 to+4h30	4.5	11.1	151	171	2060	2439
	154 to+5h30	5.5	9.5	157	175	1850	2150
	154 to+6h30	6.5	11.1	145	161	1961	2258
	154 to+7h30	7.5	11.9	140	165	2014	2497
	154 to+24h	24	10.4	148	177	1884	2389
	154 to+26h	26	8.15	179	205	1873	2275
	154 to+30h	30	10	144	164	1752	2083
	154 to+48h	48	12.7	131	152	1983	2404
	Sample reference		Sample Thickness e (µm)	OTR 50% RH cc/m 2 /day	OTR 80% RH cc/m 2 /day	OTR corr 50% RH cc/m 2 /day	OTR corr 80% RH cc/m 2 /day	Adhesion (N/15 mm)
	86 Poly(diUdiMA) network	9.4	144	155	1646	1823	1.15
	180 NPG100-D 5 amine network	8.9	455	516	9871	12735	Substrate tears

Table 5 .

 5 Glass transition temperatures (Tg) of the different networks

	Network	Tg (°C)
	Poly(diUdiMA) network	not detected
	TETA-DGGOA network	not detected
	BY16871-NPG100 network	-10
	D 5 amine-NPG100 network	+33

Table 1 .

 1 The reagents for IPN preparation

Table 5 .

 5 Glass transition temperatures (Tg) of the different networks

ACKNOWLEDGEMENTS

The authors wish to thank R. Bischoff for initiating this research project.