Keywords: Industrial production system, Discrete event system, buffers, decoupling points, performances 

Trying to answer industrial challenges regarding production planning and control, Ptak and Smith propose a new method called Demand Driven MRP. Their basis is to cope with variability instead of facing it. Amongst the different key points of the method, buffer location choice is not clearly defined. We aim to investigate its impact on the industrial performance. This study will rely on a design of experiments led with a discrete event simulator. It includes an in vitro case study with 7 different buffers placements and three types of variability (customer, supplier and process) with 3 levels each. The results show that the choice concerning buffer placement can affect the performance up to 15 point of OTD and 100% of Working capital. They also demonstrate that there is no buffer placement that would allow dealing with all types of variability and levels. Buffer placements policy has to be adapted to variability context to reach a high level of performance.

Study of buffer placement impacts on Demand Driven MRP performance

Stephanie Bayard*, Frederic Grimaud*, Xavier Delorme*  1. INTRODUCTION Industrial production planning and control becomes harder and harder to manage as the economic environment is more and more tumultuous. The acronym VUCA for Volatile, Uncertain, Complex and Ambiguous is often used to define this new context like the new geopolitical conditions after the cold war. It also signs that traditional tools and strategies can be less effective than before because of those deep changes. This is what [START_REF] Ptak | Orlicky's Material Requirement Planning[END_REF] consider when they make the analogy into the industrial world. They emphasize that traditional tools, such as MRP2 created in the mid 70's by [START_REF] Orlicky | Material requirements planning: the new way of life in production and inventory management[END_REF], cannot be effective into the new competitive world while variability is too high. They think that variability causes disruptions into the planning systems and can lead to counterproductive decisions such as safety stock, safety lead times for example [START_REF] Ptak | Precisely Wrong : why conventional planning system fail, 1st ed[END_REF]. Those protections can worsen the situation because they disturb the real demand signal that leads to complicate planning decision process. This is why Ptak and Smith created their new theory called Demand Driven MRP (DDMRP) to answer those new constraints. They wanted to simplify production and planning control. Amongst other tools, they namely use stock buffers allowing decoupling points within the bill of materials. Decoupling points are supposed to dampen variability effects. If some heuristics are given to decide buffer locations, they are not really detailed in the method and [START_REF] Miclo | Challenging the "Demand Driven MRP" Promises : a Discrete Event Simulation Approach[END_REF] showed that it can vary from one expert to another facing the same case. The present study aims to evaluate buffer location impact on the system performances by using a design of experiment led with a discrete event software.

We will first explain the key points of DDMRP and buffers location challenges, then the research process with design of experiments and the study case, then the results and finally conclusion and perspectives.

DDMRP AN ORIGINAL APPROACH

Ptak and Smith want to simplify the planning and production control, so they have been developing their concept since 10 years. We will first explain the theory, then its reported results, and then the buffer location challenge.

Demand Driven Adaptive Enterprise basics

Ptak and Smith' theory is still evolving and DDMRP is now considered as the first step of a whole system called Demand Driven Adaptive Enterprise (DDAE) [START_REF] Ptak | The demand driven adaptive enterprise (DDAE): surviving, adapting, and thriving in a vuca (volatile, uncertain, complex, and ambiguous) world[END_REF]. Its first step, DDMRP, uses inventory buffers strategically placed and managed through a new method. Then Demand Driven Operating Model integrates the use of capacity and time buffers in addition to control points. Next, the authors recommend using a Demand Driven Sales and Operations Planning (DDS&Op) as a tactical reconciliation step and finally an Adaptive S&Op as a strategic layer. All details can be found in the dedicated literature [START_REF] Ptak | Orlicky's Material Requirement Planning[END_REF][START_REF] Miclo | Challenging the "Demand Driven MRP" Promises : a Discrete Event Simulation Approach[END_REF][START_REF] Ptak | Precisely Wrong : why conventional planning system fail, 1st ed[END_REF][START_REF] Ptak | The demand driven adaptive enterprise (DDAE): surviving, adapting, and thriving in a vuca (volatile, uncertain, complex, and ambiguous) world[END_REF].

The present study only focuses on DDMRP as most firms experienced this first step. [START_REF] Bayard | DMRP vs MRP2 : impacts of choice on financial and customer performance facing different demand profiles[END_REF] showed that out of 60 industrial cases studied between November 2018 and November 2019 only 3 companies were above DDMRP i.e. inventory buffers. Experts' and users interviews, done during this period, also highlighted expectations concerning DDMRP set up especially upon buffer locations choice.

DDMRP tools

DDMRP cannot be sum up as only the use of strategically placed buffers. It includes different others innovations and tools.

First, buffers sizing is totally renewed. Instead of having only a whole buffer it is now divided into 3 coloured zones: the red one is the safety, the yellow one the average consumption during the average lead time and the green zone that represents the art of the ordering system. Fig. 1. Buffer zone calculations and purposes" [START_REF] Ptak | Precisely Wrong : why conventional planning system fail, 1st ed[END_REF] Those zones are exploited to determine whether or not the planner has to refill the buffer. The DDMRP system will calculate the Net Flow Equation (NFE) and positions it into the buffer zones. The NFE is determined by the following formula NFE = On-hand + on-order-qualified actual order demand With:

-on-hand: stock physically available, -the on-order: total quantity that has been ordered but not received -qualified actual order demand: sales orders past due + sales orders due today + qualified spikes. Qualified spikes are "cumulative daily demand within a qualifying time window that threatens the integrity of the buffer" (Ptak and Smith, 2018b). This last point is subject to adaptation.

An order is triggered only if the NFE is situated into the yellow or the red zone. The order quantity, to supply or to make, is determined by the difference between the NFE and the top of the green zone, the maximum quantity in the buffer. Once the order is launched, DDMRP also determines its relative priority by indicating a ratio between the NFE and the top the zone. All orders are displayed with this same indicator allowing quick decision making concerning production planning and control.

The innovation is to consider only real orders in the NFE calculation, the parameters are dynamically adjusted and the priority indicators simplify decision making in the planning process.

DDMRP results

Interesting results have been shown both on the academic and on the industrial worlds. Those industrial results were partially confirmed by the benchmark led by [START_REF] Bayard | DMRP vs MRP2 : impacts of choice on financial and customer performance facing different demand profiles[END_REF].

Decoupling points and buffers

One central concept of DDMRP is the decoupling point created while buffering an item.

Decoupling points is defined by the APICS as "the locations in the product structure or distribution network where inventory is placed to create independence among processes or entities".

Decoupling point is not a new idea and its history can be traced back to the concept of order penetration point (OPP) defined by [START_REF] Sharman | The rediscovery of logistics in[END_REF]. Different other authors work on this concept and especially on its optimal positioning [START_REF] Hoekstra | Integral logistic structures: developing customer-oriented goods flow[END_REF][START_REF] Olhager | Strategic positioning of the order penetration point[END_REF]. Yet in those studies, the decoupling point was considered as the point where the production flow shifted from pushed to pull thanks to the customer order arrival. The choice of its location relies on the customer tolerance lead-time.

Fig. 2. The customer decoupling point [START_REF] Olhager | Strategic positioning of the order penetration point[END_REF] DDMRP is a method relying on a multiple decoupling points system. From the first definition of a unique OPP, the concept of decoupling points has evolved into multiple decoupling points. This notion was first evocated by [START_REF] Swaminathan | Managing Broader Product Lines through Delayed Differentiation Using Vanilla Boxes[END_REF] and was adopted in supply chains context since beginning of 2000's. The target is to create the independence into the process or supply chains. If the unique decoupling point positioning rely on the customer tolerance lead-time, multiple decoupling points location become more challenging.

Therefore, before discussing the buffer locations methods, it is interesting to evaluate its influence on performance facing variability. To achieve this goal, we will develop a research methodology.

Green Zone The data table includes in a defined format all information needed to run the simulation. This includes item data, resources data, schedules, times, bill of materials… It can be compared to a company items database.

The design of experiments table includes the parameters that can be updated during the experiment. Information is then transferred into the planning process simulator which will generate production and supply orders for each reference considering their own parameters and the conditions of the experiment and for the length of the simulation. A spreadsheet is generated for each experiment.

Performance measurement

To evaluate the global performance of the planning process, 2 types of performances are evaluated through 2 KPIs:

-Customer performance through the On Time Delivery (OTD) in percentage (1). It has to be maximized.

-The financial performance: with the total Working Capital (WC) (2). It has to be minimized.

DESIGN OF EXPERIMENT

To be able to evaluate impact of buffer location on production and planning system performance, we developed a design experiments led on an in vitro case. We will first present the case, then the design of experiments and finally the results.

The case study

The case we chose is the item FPB of the ABC Company developed in Ptak and Smith [START_REF] Ptak | Demand driven material requirements planning (DDMRP): version 3[END_REF]. This item is simple, parameters are known and buffer location is discussed by the authors. Figure 4 represents the bill of materials of item FPB.

The example originally offers 3 final products but our goal is to get a clear understanding of buffer placement effect without considering multiple cases of usage of an item. This why only one final product is considered. We choose FPB amongst the others because its average usage is 100 pieces per week. This figure allows a quick analysis of results highlighting links between demand changes and items behaviour. To simplify the analysis, we also choose to set all bill of material links at 1 unit opposite to some coefficients 3 used in the original case. We want to avoid some leveraging phenomenon due to multiple links in the BOM.

Fig. 3. Bill of material of item FPB

Other parameters such as lead times and unit costs come from the authors. As the original example is limited to planning operations, we had to create manufacturing data respecting industrial reality and fitting given lead times. We calculated process lead times allowing an 80% loading rate based on the average weekly consumption and planning lead times. This calculation highlighted the need for multiple resources for all manufactured items except FPB. We assume to use a dedicated set of resources and a single step per manufactured item to avoid prioritization issues. Therefore, the industrial organization can be assimilated to a job shop. We also consider the added value of the production process. We use a cost of 70€ per working hour which is an average given for traditional mechanical industry in France. Main parameters are summed up in table 2.

The design of experiments

Based on this in vitro case, the design of experiments was conducted to evaluate the impact of buffer location upon global performances in different variability contexts as shown in figure 5.

The buffer placement factor (F1) embeds 7 values : based on 3 main policies and all their combinations: final product (F), made items (M) in this case items 201 and 207 to synchronize the flow after the buffers-and purchased items (P). Table 3 sums up the buffer positions for each value of F1 value. Buffer sizing is made as recommended by Ptak and Smith and we assume to choose the maximum calculation result for the green zone as recommended. Upon this assumption, the validation step of the model showed that the size of the generated orders does not fit with the industrial organization. Namely, for an average demand of 100 FPB per week, we have less but larger orders. This led to seize a resource for a much longer lead-time than expected and generated multiple shortages. In real industrial world, planners would take decisions to adjust the production organization. Therefore, we decided to create transfer orders allowing to dispatch job orders between available resources of the set dedicated to the item. This phenomenon highlighted the need for an additional task of scheduling while the green zone sizing generates orders much larger than the average demand.

Variability is considered through 3 angles: demand side, supply side and process side. For all of them we define 3 levels from low (1) to high (3) of variability defined by uniform low to apply to the average demand for the demand, to the lead time for the supplier variability and to the process time for the process. We apply the following laws to the different levels of variability level 1, UNIF(0.95,1.05), level 2 UNIF (0.85,1.15) and level3 UNIF(0.5,1.5). A global variability level is calculated by aggregating the 3 levels of the 3 variability factors therefore this global variability indicator can vary from 3 to 9. Due to the combination of factors and levels, the design of experiments embeds 189 experiments. For each of them, 50 replications of 78 weeks were run.

Results

Our goal is to evaluate if buffer placement can affect global performance under different variability types and levels.

Confidence intervals are globally good. For the OTD, 67% of them are equal or under 5% with a maximum at 15% concerning low results. Even with the upper bound, there is no overlapping of intervals in the same set of parameters. For the Working Capital, only 16% of them are over 5% with a We will first study the average results of each buffer placement choice over the whole design of experiments. We will consider in this synthesis the KPI OTD in percentage and the working capital(WC) basis 100 the overall average WC represented by the green point . As shown in the figure 6, there is no ideal solution.

We can see that the most effective regarding OTD is the buffer placement 7 (policy FMP) that involves the 3 types of buffers. The average OTD reaches 99.5% with a confidence interval under 5% at its maximum level. However, this good performance is reached with an average WC 20% over the average performance. Two other buffer placement choices presenting the same profile (high OTD -high WC) are the 1 (F) and the 5 (FP). They both involve a buffer on the final product explaining a high working capital. The less performant choice is the number 2 with buffer only on the sub-assembly level. The OTD is poor with a WC next to the average. Therefore, this option would be to be unsuitable in the industrial world. The choices numbers 3 (P), 4 (FM) and 6 (MP) are closed regarding the OTD, next to 92%. But number 3 and 6 (with no buffer on the final product) seem to be better as they present a lower WC than solution 4 which includes buffer on FPB. They can be good trade-off for companies who do not target 100% OTD.

Those results are the ones given on average whatever the level of variability. If we consider the level of global variability, results can be refined. As shown in table 4, the global variability indicator consists of the sum of the level (1 for low, 2 for medium or 3 for high) of the 3 types of variability (customer, supplier and process). In the case of low global variability (Index 3 or 4), the best option is probably the choice 6 (MP) with an OTD at 100 and an improvement of the WC next to 20% compared to average. Buffering at all levels such as in choice 7 (buffer at all levels) leads to a large increase of the working capital without reaching a perfect OTD. Buffering only the final product seems to be unsuitable in this case with a large increase of the WC with a lower OTD. At the opposite with the high variability index (7 to 9), the choice 7 offers the best trade-off with the best OTD and WC under the average. In the medium situation (5 and 6), the choice is harder as no option combines very high OTD and low working capital.

In our case with DDMRP, in a context of low variability, it is not necessary to buffer too many items. It also shows that buffering the final product is not the good answer.

Nevertheless, it seems that the higher the variability is, the better the choice 7 (FMP) is. Multiplying buffers allows to diminish buffers size, therefore the WC and improves the OTD. Choice 3 (P) offers the best WC but never the best OTD. It appears as not a very recommendable solution.

The first results were given for a global variability index. We can also analyse results per type of variability considering the two other variability types as low.

Concerning customer variability, it appears that the best solution varies with the increasing variability. With low variability, the best solution is buffer placement 6 (MP) with a 100% OTD and a WC -19 points. For level 2 and 3 of the variability, concerning the OTD (100%) is the choice 7 with buffers at all levels and a WC -11points. Nevertheless, this solution is not suitable with a low variability because it leads to a +91% increase of the WC. Surprisingly, we notice that buffering only the final product is never the best solution with a lower OTD and a higher WC.

If we consider supplier variability, the best choice regarding the OTD is also policy number 6 (MP) with buffers on purchased items and sub-assemblies. This allows a 100% OTD and a 19% of the WC whatever the degree of variability. The worst solution is the number 4 with buffer on the final product and on sub-assemblies (OTD Fig. 6. OTD vs WC synthesis 74.4 and WC +15pts). Buffering only on final product is much more effective concerning the OTD (95.4 %) with a higher increase of the working capital (+26 points). Finally, buffering only purchased items (3) can be a good alternative with a 98.3 OTD and a working capital divided by 2.

Regarding process variability, the situation is not clear as there no dominating solution. If the variability is low, then policy 6 (MP) is the best solution with a decrease of the working capital (-19points) but results are deteriorated for a high level of variability (95.9 OTD). For the higher level of variability, this is no real ideal solution. Buffers on all levels (solution 7) gives a good OTD (99.3%) but doubles the working capital. It can be suitable for a company that can afford it. Otherwise, a good trade-off is to buffer the final product and the purchased items ( 5).

It appears that there is no ideal buffer placement that would allow combining good OTD and low working capital in all variability types and levels.

CONCLUSION AND PERSPECTIVES

DDMRP is a relatively new method generating interest of the industrial world. The first academic results and industrials testimonies globally confirm announced results. However, research and experts also showed a need for deeper investigation especially on parameters fine tuning.

Buffer placement is one of the elements that is not really documented. If the rules exist, it is not easy to choose and experts do not always agree on the same case. The purpose of this research was to evaluate if buffer placement has an impact or not on DDMRP performances. We use a design of experiments based on Arena and multiple variability contexts.

Results show that there is not an absolute best solution.

Amongst overall results, buffer placement at all levels (Final product sub assembly and purchased items) seems to reach the best average OTD with 99.5% but with a greatly high working capital. However, detailed results per level of global variability demonstrated that this solution is not the best one in case of low variability context. In the same vein, if we focus on a single type of variability, results are more balanced and different trade-off can be considered. Nevertheless, in our case, buffering only the final product or only the sub-assemblies seems be ineffective in all tested variability types and levels. With our assumptions, choosing the wrong buffer placement can lead to a loss of performance of almost 15 points regarding OTD and 100% regarding the WC.

Therefore, we can conclude that buffer placement seems to have an impact on global performance. Those results have to be confirmed through other design of experiments embedding various bills of materials, more or less complex. We also consider that the long lead times used in this case can impact results. Further studies with different lead times could be led to extend them. We also highlighted that, in our case, the orders size generated by DDMRP requires an additional stage of collocating work orders in the short term scheduling. This observation has to be investigated to validate if it can be extended to all DDMRP situations.

Fig. 4 .

 4 Fig. 4. Research framework

Fig

  Fig. 5. Design of experiments scheme

Table 1

 1 below shows the good results with industrial sector risen by the study of[START_REF] Hudelmaier | The Demand Driven Radar[END_REF] concerning 3 main Key Performance Indicators (KPI) and published on the official website of Demand Driven Institute.

	Table 1 : DDMRP results in Industrial Manufacturing
	(Hudelmaier et al., 2019)
		Median	Best In Class
	Service Level increase	+17pts	+45pts
	Lead Time Reduction	-60%	-85%
	Inventory Reduction	-26%	-54%

  The discrete event simulator based on Arena automatically launches the whole design of experiments. It plays the different experiments reading the files. Arena is only the player, no decision is taken in this stage except delaying orders in case of shortage. It calculates a set of predetermined KPI and exports them to the results spreadsheet. After centralization, results can be treated as needed for each design of experiments. The model was validated by following classical steps recommended by[START_REF] Sargent | Verification and validation of simulation models[END_REF].

Table 2

 2 Main Items Data

	Item	Lead Time (weeks)	Unit Cost	Unit operation time (hours)	Nb of Resource
	FPB	1	500	0,3	1
	101	3	500	0,9	2
	207	3	290	0,9	3
	310	2	290	0,6	3
	411P	30	200
	410P	30	90
	201	5	210	1,6	5
	301	4	110	1,2	4
	401P	10	110
	302P	12	100

Table 3

 3 Buffer position for each F1 value

	Factor 1 Value	1	2	3	4	5	6	7
	Buffer placement policy	F M P FM FP MP FMP
	FPB B			B	B		B
	101
	207		B		B		B	B
	310
	411P			B		B	B	B
	410P			B		B	B	B
	201		B		B		B	B
	301
	401P			B		B	B	B
	302P			B		B	B	B

Table 4

 4

			Global performance through level of variability
	F1	Policy	KPI	3	4	Global Variability Index 5 6 7	8	9
	1	F	OTD 95,4 95,8 95,7 95,6 95,4 94,9 94,4
			WC	126	126	126	127	127	127	128
	2	M	OTD 75,1 83,4 85,0 86,8 85,8 85,0 81,8


S Himmiche

A Aubry

P Marangé

J-F Pétin

A framework for robust scheduling under stochastic perturbations

Keywords: Robust scheduling, Robustness Framework, Industry 4.0

Industry 4.0 reflects a new area for production workshops. This new concept aims to bring more flexibility and agility to the production shopfloor. In this context, the scheduling problem is an important issue. In fact, a proposed schedule has to guarantee a high performance level able to take into consideration the several changes and perturbations occurring in the workshop. The proposed approaches usually aims to find a robust schedule capable of optimizing some criteria with considering a perturbation. With the new requirements of the workshop and the importance of decision making when implementing the schedule, it is essential to extend the robust scheduling problem to be adaptable to the needs of the decision maker in evaluating several properties. We propose, in this paper, a robust scheduling framework based on a robustness specification. The paper gives also the using of this framework in a decision making context.

INTRODUCTION

The evolution of the manufacturing towards Industry 4.0 is impacting the production workshops that are undergoing a gradual transformation based on new information technologies and communication. It is linked to the evolution of cyber-physical systems (CPS) in general and in particular cyber-physical production systems (CPPS). According to [START_REF] Monostori | Cyber-physical systems in manufacturing[END_REF], a cyber-physical production system production is composed of autonomous and cooperative entities and subsystems that interconnect according to the context, within and through all levels of production, from the machine to the logistics networks. This type of systems is characterized by an increased degree of autonomy, flexibility and agility. the increasing of communications with the environment allows for product customization. Processes are allowing the collection of historical data related to the follow-up of the workshop and its performances. In this context, the scientific community relates many challenges. Among these challenges, [START_REF] Monostori | Cyber-physical systems in manufacturing[END_REF] identified six main research issues.

• Autonomous systems adaptable to changes that may occur in the environment ; • Cooperative production systems: development of learning algorithms, of consensus and distributed detection ; • Identification and prediction of dynamic systems ;

• Robust scheduling for the management of disturbances that may occur in the future produced during its execution; • Merging real and virtual systems ; • Human-Machine Interaction.

The scheduling of production workshops is an integral part of these challenges [START_REF] Vaidya | Industry 4.0 -A Glimpse[END_REF] with emphasis on the notion of perturbations induced by a greater variability of demand and flexibility of the production environment. Thus, the fourth challenge introduces robust scheduling as a necessary condition for development of Workshops implementing the concept of the industry 4.0. The present article is focused on this challenge. Indeed, the implementation of workshops 4.0 or cyber-physical production systems is based in particular on a technological part related to data processing: big data and data science . Indeed, this part allows to obtain an important flow of data and thus of information to build models of production workshop behavior. The place of data in the implementation of Workshop 4.0 is a significant opportunity. In fact, the information collected can constitute an important support for decision making in the Workshop. Taking the perturbations into account, in the scheduling issue, is therefore important in order to consider the real environment of the production system. Scheduling under perturbations is, however, not a new research problem. Indeed, this question has been addressed in depth in the field of Operational research [START_REF] Billaut | Flexibility and Robustness in Scheduling[END_REF]. Existing literature proposes to build robust schedules that guarantee particular performances and this despite the occurrence of perturbations. In [START_REF] Bidot | A theoretic and practical framework for scheduling in a stochastic environment[END_REF], a classification of existing approaches is given to consider the stochastic environment of production. They cover two complementary issues: the problem of searching for an optimal solution and evaluating the performance of the scheduling solution in the presence of perturbations. The proposed resolution approaches are often dedicated to a specific scheduling problem because depending on the type of production workshop, specific perturbations considered and performance indicators applied. This paper focus on the notion of performance that may be evaluated in the case of scheduling under perturbations. In this context, we propose a specification of the robustness performance based on existing indicators. The specification will then allow the presentation of several robustness problems corresponding to the decision maker requirements. Actually, the proposed specification is based on the use of perturbations data to evaluate the performance of a schedule in the aim of helping the decision maker to implement a resistant schedule to perturbations. To do so, this paper is organized as follows. First, the necessary background to define the problem of scheduling under perturbations is presented in Section 2. Then the performance of robustness is presented in Section 3 as a good candidate for evaluating the impact of perturbations on the schedule. Then, the notion of service level is presented as a metric for robustness evaluation. In section 4, a framework gathering several robustness problems is presented based on the specification adopted for robustness evaluation. The use of the framework defined is illustrated in Section 5 allowing the resolving of several robustness problems. This section will highlight the importance of the framework in decision making process. Some conclusions and perspectives are detailed in Section 6.

BACKGROUND

In general, the scheduling production problem is based on, first, the definition of a workshop and the constraints that should be considered. The workshop is defined by a set J of jobs j that are processed by a set R of resources r (NbR). Each job j is defined by a set of operations o jk to be executed for completing the job. An execution duration d jkr is associated to the operation o jk on the resource r. The manufacturing workshops have different characteristics that may lead to precedence constraints between operations of the same job. For instance, the Flow Shop and the Job Shop have some precedence constraints between the operations of each job (o jk is thus the k th operation to be executed). For the Open Shop, the operations of a job can be executed in any order. When generating the schedule, an optimization criterion is considered. In this paper, we will consider the total completion duration (named Makespan) C max . A schedule s (a solution) is an allocation of the operations to the resources and a sequencing of the operations on each resource (defining a starting and a completion date). Regarding a particular solution schedule, we can know: (i) the sequencing of the operations on the resources, (ii) The reference Makespan C ref max that does not consider the perturbations. In a deterministic context, the workshop data is considered stable and certain. In reality, some perturbations may affect the generated schedule when occurring in the workshop. Indeed, two types of perturbations are identified : i) Uncertainties and ii) Hazards. On one hand, the uncertainties are defined as the difference between planned and actual information available (as uncertainties on the demand, or durations ). On the other hand, the hazard is defined as the occurrence of uncontrollable events (resource breakdown, urgent order, etc.). In order to integrate the perturbations into the scheduling problem, various approaches can be used for their description [START_REF] Li | Process scheduling under uncertainty: Review and challenges[END_REF]: scenario based method, fuzzy based method and stochastic method.

In this paper, we focus on the stochastic description of perturbations, where each perturbation is considered as a random variable that follows some probability distribution. In the context of Industry 4.0, it can be assumed that historical data are available in the workshop such that relevant and sufficiently precise probabilistic models of perturbations can be recovered. How to build these models is out of scope of this paper. Methods coming from data science are good candidates for building these models. So, in the following of this paper, we will consider that these perturbation models are known.

Two complementary strategies can be used to take into account perturbations in a scheduling process [START_REF] Billaut | Flexibility and Robustness in Scheduling[END_REF]):

• The first scheduling approach is called proactive: the scheduling is calculated offline based on the a priori knowledge of the perturbations trying to give to the solution some desired quality against these predictive perturbations. • The second scheduling approach is called reactive: the scheduling is computed online according to the current available information and regarding the occurrence of events.

In the context of this paper, we are interested in the first strategy. The performance of a given solution schedule faced to perturbations can be measured either by a performance of flexibility or by a performance of robustness [START_REF] Billaut | Flexibility and Robustness in Scheduling[END_REF]. Flexibility performance is defined as the degree of freedom the schedule has during its execution to respond to actual conditions. Robustness performance is defined as the ability of a schedule to maintain some performances despite the various perturbations.

As part of our work, we are interested in assessing the robustness of a solution schedule faced to perturbations.

ROBUSTNESS AS A PERFORMANCE FOR SCHEDULE UNDER PERTURBATIONS

Robustness performance has been integrated into production management over the last century. Its definition has not stopped evolving. In (V.J. Leon et al., 1994), a schedule is robust if being insensitive to unpredictable perturbations on the shopfloor while considering the deployed control policy. In [START_REF] Sevaux | A genetic algorithm for robust schedules in a one-machine environment with ready times and due dates[END_REF], the authors argue that one way to take into account stochastic data of a scheduling problem is to find solutions that are robust. In [START_REF] Billaut | Flexibility and Robustness in Scheduling[END_REF] robustness is presented as a qualifier that refers to the ability to tolerate approximations on assumptions, models or data. In [START_REF] Davenport | Slackbased techniques for robust schedules[END_REF], the authors define a robust schedule as being able to absorb a certain level of unexpected events (hazards) without having to resort to re-scheduling. More recently, another definition is proposed by [START_REF] Ivanov | Schedule robustness analysis with the help of attainable sets in continuous flow problem under capacity disruptions[END_REF]: a schedule that is able to achieve the planned performance despite perturbations is said to be robust. These different formulations converge towards the same idea of robustness: the performance, quality or capability that a schedule must maintain or guarantee despite the various perturbations and variations emanating from the production system or its environment. In the following, we will focus on the concept of robustness as performance assessment to identify and address a corpus of relevant robustness issues and to provide the appropriate response elements to contribute to the identified robustness challenge, in particular by [START_REF] Monostori | Cyber-physical systems in manufacturing[END_REF] for the implementation of Industry 4.0. This paper aims to answer the following questions. First, how to specify robustness performance? Second, how to structure robustness problems from this specification?

A definition of robustness

To integrate the notion of robustness into the problem of scheduling under perturbations, it is important to specify the measure of this performance. In the literature, this measure depends on the context of the problem and the performance objective to be achieved. For instance, in [START_REF] Goren | Robustness and stability measures for scheduling: single-machine environment[END_REF], the expectation of the makespan is used as a measure of robustness. In [START_REF] Goren | Robustness and stability measures for scheduling: single-machine environment[END_REF], maximum regret is adopted as the measure. In [START_REF] Beck | Job shop scheduling with probabilistic durations[END_REF], robustness is quantified by the probability that a scheduling performance reaches a given level. As soon as the perturbations are modeled as random variables, it seems natural enough to associate robustness with a probability measure to evaluate the risk that the schedule performance is too degraded. For production scheduling, the measure of probability allows to quantify the capacity of a given solution to maintain performance in a perturbed context. It is up to the decision-maker to define the performance criteria of the scheduling, for example, the makespan, a cost or any other classic scheduling criteria (maximum delay, work in progress, etc.). In this context, (Dauzère-Pérès et al., 2010) introduces the notion of service level for the measurement of scheduling robustness. This service level measures the probability that the performance of a schedule is lower (or higher) than a given value. The notion of service level can then be considered as a robustness specification . To assess the service level, the following elements are considered. The schedule s to be evaluated, the optimization criterion ( here the total completion time (makespan) denoted as C max ), the set of perturbations to be considered P ert and the deadline d to be satisfied by s despite the perturbations. Regarding these elements, the service level is defined as follows.

Definition 1. Considering a schedule s, a set of predicted perturbations P ert, and a deadline d to be satisfied, the service level RL(s, P ert, d) is defined as the probability P r that the makespan C max of s is less than or equal to the deadline d, despite the set of perturbations P ert.

This definition can be formally translated into the expression 1.

RL s, P ert,

d = P r C max (s, P ert) ≤ d (1)
According to the service level, it is possible to qualify if a schedule is robust as follows.

The threshold RL lim allows to define the accepted risk of not meeting the deadline. In fact if RL lim = 95% would mean that the decision maker would accept that the deadline would not be met in 5% of cases only.

A FRAMEWORK FOR ROBUSTNESS ANALYSIS

The definition of robustness highlights two complementary but competing aspects of the robustness: stability and sensitivity . They can be seen as two facets of the same issue (robustness). Stability refers to the set of perturbations P ert that is supported by a given schedule without degrading to much its performance. This can be considered as the risk that is covered by the schedule. In this case, the tolerated performance degradation is fixed by the deadline d.

On the other hand, sensitivity refers to the deadline d itself, which allows to measure at what point my schedule is sensitive to perturbations defined by P ert: i.e. how much the makespan is deviated due to the perturbations fixed by P ert. The following definitions are then given for these two aspects. First,the stability measures the variations allowed at the inputs of the problem (measured by P ert) without causing too much deviation in the output of the problem (constrained by d). Second, the sensitivity measures the variations generated on the output of the problem (measured by d when the problem inputs vary (constrained by P ert).

Intuitively, we understand that these two notions are concurrent. If we try to increase stability (P ert contains "more" perturbations) then the risk is to increase also the sensitivity ( d also increases). If we try to decrease the sensitivity (decrease d) then the risk is to degrade the stability (P ert must contain "less" perturbations). Thus, the robustness definition is fundamental because it highlights several robustness issues depending on whether the decision-maker is more interested in stability or sensitivity, or even looking for a compromise between the two. The objective of this section is to highlight and formalize this set of issues defining a framework for robust scheduling. Actually, when considering the specification of robustness given in equation ( 2), it is possible to define several robustness problems. These problems can relate to stability and/or sensitivity analysis. In fact, these problems are defined according to the knowledge of the various input parameters of the problem: s, P ert, d and RL lim .

Each input may or may not be known by the decision maker defining a relevant robustness problem.

The identified robustness problems can be classified into two categories:

• the evaluation problems, which assess the robustness of a known schedule s according to the different parameters of the problem. • the optimization problems, which aim to compute the schedule s * that optimizes one or more parameters of the defined problem.

Only the evaluation problems are presented in this paper with considering that an initial schedule s is known and needs to be evaluated. In fact, for each evaluation problem, an optimization problem can be associated that searches Evaluation problems enable the expression of different characteristics that can be satisfied by the schedule. Based on the various parameters, the specification of the robustness is adapted to express the different problems (Table 1). Actually, by changing all parameters, eight robustness problems can be identified and then formalized. For each problem, we define the known parameters (input) and the searched parameters (output). Problems are sorted according to the number of characteristics to be evaluated.

For the first evaluation problem (EV 1 ), the aim is to know if a given solution s can be qualified as robust regarding the definition in equation ( 2).

The second problem EV 2 aims to evaluate the service level.

The next two defined problems focus on the properties of sensitivity and stability. The sensitivity analysis (EV 3 ) aims to assess the minimal deadline that the schedule can satisfy with considering the perturbations and the defined robustness threshold. The expected answer is here the value of the minimal deadline obtained d. The stability analysis (EV 4 ) aims to assess the maximum set of perturbations that can be handled by the schedule s with satisfying a known deadline and a defined robustness threshold. The expected answer is here the maximal set of perturbations P ert.

The last defined problems are concerned by the compromise between two or more caracteristics. In fact, the problem defined in EV 5 expresses the compromise between the sensitivity property and the service level. It means that the specification will look for the minimal deadline d satisfied by s with the maximal service level that can be reached RL. The second compromise expressed is between the stability and the service level (EV 6 ). This compromise aims to find the set of maximal perturbations P ert and the maximal perturbations RL. The problem EV 7 express the compromise between the sensitivity and the stability. It actually allows to obtain a kind of balance between the minimal deadline reached d when the maximal perturbations are considered P ert. The last defined problem EV 8 analyses the global compromise between the three properties: sensitivity, stability and the service level.

The defined framework is based on the service level expression. Therefore, the resolution of each problem can be based on the assessment of the service level RL(s, P ert, d) (EV 2 ). Thus, to solve the other problems, a first step would be to assess RL and a second step to compare it to the threshold RL lim .

The goal of the next section is to show how this framework can be applied to a representative use case.

APPLICATION

The purpose of this application is to explore the use of the framework defined in an industrial scenario. The scenario presented allows to illustrate decision situations that may occur in a production workshop. The presented scenario is based on a workshop with a production unit in charge of product assembly and a packaging unit in charge of finalizing the order of the customer (Figure 1 ). In the assembly shop two pre-existing production lines ensure the manufacturing of products. In order to expand its market, the company decides to open a third assembly line to support new customer needs. The production line is a Flow Shop type with two resources and 3 new jobs (representing 3 new products). The scheduling problem related to this application is denoted in the literature as F 2||C max (F for flow shop, 2 for the number of resources and C max for the makespan to be minimized). This Flow Shop line integrates two resources r 1 and r 2 . 3 jobs has to be completed on this line. Each job is defined by a pair of operations o jk . Each operation must be executed on a resource respecting a reference execution duration d ref jkr established by the product design department. The reference execution times are listed in Table 2. From these problem data, six deterministic schedules (corresponding to the 6 possible permutations in the set of 3 jobs) are determined (Figure 2). Among these schedules, the optimal one for the deterministic problem corresponds to the schedule s 3 with a makespan of 44 hours. Three scenarios are described to exploit the proposed robustness framework. We will show how the appropriate use of the framework provides relevant information to the decision-maker to help him make the right choice in its workshop. The scenarios are described as follows:

(1) When the new flowshop line is implemented, the decision-maker does not have an historical data of the perturbation such it is not possible to have a relevant model of the uncertainties on execution durations. The decision-maker want to choose the schedule that allows to absorb the largest deviation on these durations. (2) After six months of production, the production data history now enables to have information on the realised execution durations. The decision-maker needs now to choose the schedule having the best behavior faced to the uncertainties on execution durations. (3) The products manufactured on the production line are sent to the the packaging unit. The decision maker would like to provide to the packaging workshop manager a realistic delivery date of these products despite the uncertainties on the execution duration.

To solve these scenarios, we make the assumption that the problems presented in the framework have been implemented in a decision making process allowing the assessment of the service level RL s, P ert, d .

Scenario 1: Maximum deviation supported

The decision-maker is interested here in the uncertainties that may impact the execution durations of operations.

The objective is to identify and implement the scheduling that will allow the absorption of a maximum set of perturbations P ert while satisfying a defined robustness threshold RL lim and a deadline d. Thus, obtaining this maximum set of perturbations means performing a stability analysis (EV 4 ). We assume that each execution duration can be defined as Table 3. Stability analysis

d
The schedule s 3 is able to absorb a deviation of 37% while the two other schedules absorb only 12%. The decisionmaker will obviously takes the decision to implement the schedule s 3 for more safety.

Scenario 2: Choosing the most robust schedule

After retrieving historical data, the decision-maker may wish to reconsider the choice of s 3 in order to execute the most robust scheduling when facing actual perturbations. The six deterministic schedules (Figure 2) are then reevaluated taking into consideration the available information on the realized execution durations. We consider again that each execution duration can be defined as We assume that the decision-maker decides to implement s 4 .

Scenario 3: Determination of the delivery date

Following the determination of robust scheduling (s4), the decision-maker must inform the packaging unit of the delivery date of the products despite the uncertainties. This delivery date must be the most accurate possible to avoid creating stock upstream of the packaging workshop or conversely to create a lack of jobs in this one. For the schedule s 4 , the decision-maker wants to know which deadline d he can promise to the person in charge of the packaging workshop despite the perturbations (with the same conditions as in scenario 2). In order to determine this deadline, it is sufficient to carry out a sensitivity analysis (EV 3 ). The input parameters of the problem are therefore, the schedule s 4 , the perturbation of uncertainty on execution durations with the same parameters as previously and a service level threshold equal to RL lim . When applying the EV 3 problem, we can find the minimal deadline reached is d = 53 hours for s 4 . This represents the deadline that the decision maker can propose.

CONCLUSION AND PERSPECTIVES

This paper proposes a formalization of robustness specification for evaluating the schedule facing perturbations. The robustness measure is then based on service level as robustness metric. From this specification, this paper define a framework gathering several relevant problems of robustness. In fact, instead of simply with trying to evaluate and then maximize a single performance of robustness, we have shown in this paper that the robust scheduling problem can be enriched. Indeed, robust scheduling is a multi-faceted issue that can be approached considering different points of view (stability, sensitivity, service level). It seems important to us, since the final objective is to help the decision maker, to be able to preserve these different points of view in a robust scheduling framework. This is the goal of the presented framework.

We then illustrated how these problems can effectively be useful to the decision maker's concerns in a real-world context.

The proposed framework allows to contribute to the concept of industry 4.0 and specially to the robust scheduling challenge. When implementing such framework in a decision process, it can give to the decision maker a constant information on the schedule to be implemented in the workshop. When using the data recovered form the data science unit in the workshop. The decision process can be continuously adapted to the upcoming changing in the perturbations considered. A perspective of this work is therefore to implement this framework in an industrial context in order to create a complete decision making process. This perspective opens the doors for several research questions going from data processing to the implementation of the process in a flexible and agile environment. Furthermore, the proposed framework has been defined in the context of scheduling, but this framework can be more general. Indeed, it would be interesting to generalize this framework whatever the problem, the criterion and the perturbation taken into account.

  Fig. 1. Application scenario

Fig. 2 .

 2 Fig. 2. Deterministic schedules for the problem F 2||C max

  jkr = d ref jkr + ∆d jkr where ∆d ref jkr ∈ -δ × d ref jkr ; +δ × d ref jkr . Here, ∆ is the fluctuation generated by the perturbation on the reference execution duration of operation d ref jkr and δ is the percentage of this variation. Thus, solving (EV 4) means finding P ert = max {δ} such that RL s, P ert, d ≥ RL lim . We consider that the service level threshold is fixed to RL lim = 95% and the deadline is fixed at d = 51 hours which means that the decision maker authorize 7 hours (one day) more than the shortest makespan (44 hours). Results, for several schedules s i whose makespan is initially under d, are given in table 3.

  Definition 2. A schedule s, that has to satisfy a deadline d despite some perturbations P ert, is qualified as robust if its robustness level RL(s, P ert, d) is greater than or equal to a service level threshold RL lim .

This definition can be formally translated into the expression 2.

RL s, P ert, d ≥ RL lim

Table 1 .

 1 Robustness problems frameworkthe solution s * that optimizes the criterion defined in the evaluation problem.

	ID	Input parameters	Output parameters	Type of problem	Specification
	EV 1	s, P ert, d, RL lim	Booleen : True or False	Robustness qualification	Is s robust? RL s, P ert, d ≥ RL lim ?
	EV 2	s, P ert, d	RL s, P ert, d	Service level analysis	What is the value of RL s, P ert, d ?
	EV 3	s, P ert, RL lim	d		Sensitivity analysis	What is the minimal value of d denoted as such that: RL s, P ert, d ≥ RL lim ?	d
	EV 4	s, d, RL lim	P ert		Stability analysis	What is the maximum set P ert denoted as P ert such that: RL s, P ert, d ≥ RL lim ?
	EV 5	s, P ert	RL lim ,	d	Compromise analysis sensitivity/service level	What is the couple RL lim , d , such that: RL s, P ert, d ≥ RL lim ?
	EV 6	s, d	RL lim , P ert	Compromise analysis stability/ Service level	What is the couple RL lim , P ert , such that: RL s, P ert, d ≥ RL lim ?
	EV 7	s, RL lim	d, P ert		Compromise analysis sensitivity/stability	What is the couple P ert, d , such that: RL s, P ert, d ≥ RL lim ?
	EV 8	s	d, P ert, RL lim	Compromise anlysis sensitivity/stability/Service level	What are d, P ert, RL lim , such that: RL s, P ert, d ≥ RL lim ?

Table 2 .

 2 Flow Shop conception data

	Job Operation d ref jkr (hour)
	1	o 11	12
	1	o 12	12
	2	o 21	4
	2	o 22	16
	3	o 31	20
	3	o 32	8

  d jkr = d ref jkr + ∆d jkr where ∆d ref jkr ∈ -δ × d ref jkr ; +δ × d ref jkr .The historical date allow to evaluate δ = 20%. Moreover, the decision maker accept a deviation of the reference makespan such that the accepted deadline is fixed to d = 110%C ref max . The goal is to identify, among the six deterministic schedules, the most robust one. So we evaluate for each of them, its service level (EV 2). The results are given in table 4.

	Schedule C ref max	d	RL
	s 1	48	53 97%
	s 2	56	61 95%
	s 3	44	48 93%
	s 4	48	53 97%
	s 5	60	66 97%
	s 6	56	61 94%

Table 4 .

 4 Robustness evaluation for 6 schedules Three schedules (s 1 , s 4 , s 5 ) have the best and same service level (97%). The decision-maker can decide to choose the one whose Makespan C ref max is the lowest making a compromise between optimality (makespan) and robustness (service level). Here s 1 and s 4 have the same compromise.


Mohammad Amin Yazdani

Amirhosssein Khezri
email: amir_hossein.khezri@ensam.eu

Lyes Benyoucef
email: lyes.benyoucef@lis-lab.fr

A Linear Multi-Objective Optimization Model for Process and Production Planning Generation in a Sustainable Reconfigurable Environment

Keywords: Sustainability, RMS, production planning, process planning, multi-objective optimization

According to the growing rate of customer's options' in a hotly competitive market, companies must be, simultaneously, cost and time-efficient and environmentally harmless. Thus, reconfigurable manufacturing system (RMS) is proposed to cope with these new challenges. This paper addresses the problem of multi-objective process and production planning generation in a sustainable reconfigurable environment. For economic sustainability, the total production as well as process costs are considered, where for social sustainability, the workers' health conditions are addressed. For environmental sustainability, the hazardous wastes of machines and their harmful gases emission are considered. First, a linear mixed integer model is proposed. And an adapted version of the Lp-metric approach is used to solve the problem second. Finally, to illustrate the applicability of our approach, an illustrative numerical example is presented and the results analysed.

INTRODUCTION

Nowadays, manufacturing is recognized as an important driver of economic development. Industries need to meet the constantly growing worldwide demand for capital and consumer goods while considering the associated environmental, economic, and social. Mostly, environmental factors have been believed as important to the survival of manufacturing companies. More specifically, the emission of liquid hazardous waste, greenhouse gas (GHG) and etc., are classified as environmental threats needed to be considered [START_REF] Beekaroo | Developing a sustainability index for Mauritian manufacturing companies[END_REF]. Hence, next generation manufacturing systems must be adapted with the necessity of rapid product development, as well as complicated environments issues (social, economic and environment). In this regard, new or enhanced vast of products can be manufactured, delivery time and cost can be cut, and productivity can be ensured [START_REF] Vyatkin | Now that's smart![END_REF].

As new generation of manufacturing systems, reconfigurable manufacturing system (RMS) was introduced. It considers numerous aspects like mass customization, globalization, social changes and unstable periodic market changes. The term reconfiguration has been defined as the "capability of the system to allow manufacturer rapidly adjusting production capacity in response to changing market demands". The adjustment can be comprised in add, remove, or modify specific process capabilities, software, controls, or machine structure, etc. The concept of RMS is based on a combination of previous manufacturing systems exited, flexible manufacturing system (FMS) and dedicated manufacturing system (DMS), respectively. It combines the FMS high flexibility with the DMS high productivity due to its six main features: scalability, convertibility, diagnosability, modularity, integrability and customization [START_REF] Koren | Design of reconfigurable manufacturing systems[END_REF].

Sustainable manufacturing comprises of several facets of production that have a strong interaction together-this collection, composed of product, process, and system-level in the manufacturing environment. Furthermore, sustainable manufacturing is used for continuous improvement of human life in such terms as satisfaction and other lifestyle matters. Nowadays, a vast number of constraints related to the economical and social aspects of sustainability can be seen in the communities. RMS-due to its flexibility and characteristics-can increase the system sustainability and responsiveness to satisfy the market needs. Moreover, utility function of the workers, energy consumption of machines, harmful gas emission and liquids can be considered as the social, environmental and economical affairs of the sustainability in this paper.

Currently, RMS offers several aspects and faces many new challenges. It is a very active research field with rich contributions to problems such as process planning, production planning, layout optimization, design, and production schedule [START_REF] Benyoucef | Reconfigurable Manufacturing Systems: From Design to Implementation[END_REF]. In this paper, a new approach toward the sustainable RMS is considered. For the social aspects, workers' body health is considered, where environmental aspects comprise hazardous liquid waste and gas emission (Carbon dioxide CO2, Methane CH4, Nitrous oxide N2O and etc.) during the production. A novel combination of the process planning and production planning issues in the sustainable area is proposed. For the production planning, it is crucial to satisfy the market demands by considering the limits of the whole system. Where, for the process planning, the best production sequences of several products are considered. The aim is to minimize the total production as well as process costs, and minimize the harmful effects of the gases and liquids on the human body in a sustainable environment.

The rest of the paper is organized as follows. Section 2 reviews some recent works dedicated to sustainable manufacturing, process planning and production planning. Section 3 describes the studied problem and its mathematical formulation. Section 4 presents the Lp-metrics method used to overcome the multi-objective functions. Section 5 shows an illustrative numerical example and analyses the obtained results. Section 6 illustrates some sensitivity analyses related to changes of demands and weights in Lp-metrics. Section 7 concludes the paper with some future work directions.

RELATED WORKS

Sustainabilty in manufacturing systems

First, it is necessary to define the concept of sustainability in manufacturing. The U.S. Department of Commerce defines sustainable in manufacturing as "the creation of products which aims to minimize negative environmental impacts, save energy and natural resources, being safe for employees, society, and being economically sound (Bogu, 2014). Moreover, it is possible to consider the manufacturing sustainability as a separate concept that can combine with other area of working. In manufacturing systems, not only does it paramount to use optimization methods, but also it is probable to enter processes and products into account to acquire a sustainable environment [START_REF] Jayal | Sustainable manufacturing: Modeling and optimization challenges at the product, process and system levels[END_REF]. Hence, bringing the sustainability into the metrics components as well as concentration on the energy usage, global climate change, and so on can pay the way for an enterprise to be sustainable [START_REF] Reich-Weiser | Metrics for sustainable manufacturing[END_REF]. [START_REF] Dornfeld | Moving towards green and sustainable manufacturing[END_REF] proposed the measurement of sustainability metrics by using methods and tools that give rise to better understanding of the green and sustainability in the manufacturing process. The assessment of energy saving and maintenance play decisive roles in making the usage of the more apparent. The optimal maintenance level is the goal of maintenance, and the amount of energy that can be saved during the switching is the objective of the energy system in this problem. The implementation of real-time is for acquiring producing in resource and energy effective ways [START_REF] Huang | A Maintenance and energy saving joint control scheme for sustainable manufacturing systems[END_REF].

Process planning in manufacturing systems

State of the art assigned to the process planning contains a myriad of papers. Therefore, some definitions are considered as groundwork and basement of the manufacturing systems and RMS. In each manufacturing process planning a great link between the characteristic of products and the utility of the system can be seen. It means that, we can contemplate it as a bridge between product and resources. Moreover, for any product, a process plan leads to finding the best sequence of operations and discover all machines paths that give rise to produce it as well [START_REF] Chaube | An adapted NSGA-2 algorithm based dynamic process plan generation for a reconfigurable manufacturing system[END_REF]. In order to achieve a reconfigurable process plan, it is needed to think about such paramount matters' relationship as product features, process plan components, and all manufacturing system elements. In some conventional process planning generation methods, some contradictions were found with the scheduling of the operations. [START_REF] Khezri | Sustainable multi-objective process plan generation in RMS through modelling energy consumption[END_REF] used augmented ε-constraint to optimize three objectives, respectively, the total production costs, the total completion time, and the amount of energy consumption during the manufacturing process. The main contribution of that paper is the consideration of sustainability in producing a single-unit in a reconfigurable environment. Besides, [START_REF] Touzout | Multi-objective sustainable process plan generation in a reconfigurable manufacturing environment: exact and adapted evolutionary approaches[END_REF] proposed two meta-heuristic namely NSGA-II and AMOSA, and a MILP model. Greenhouse gases and the amount of harmful gas emissions, as well as the cost of manufacturing, are considered. [START_REF] Haddou-Benderbal | Modularity assessment in reconfigurable manufacturing system (RMS) design: an archived multi-Objective simulated annealing-based approach[END_REF] introduced three objectives comprising the maximization of the system modularity, and the minimization of the system completion time and costs. They used simulated annealing (SA) based approach to solve the problem

Production planning in manufactuing systems

For the controlling and process of the production in area of scheduling, capacity and output, production planning can be useful in the manufacturing systems. Several reviews are existed in this field [START_REF] Pereira | Tactical sales and operations planning: A holistic framework and a literature review of decision-making models[END_REF]. In. the case of reconfigurable environment, product quality and product quantity in the processing time can play the barrier and limits of the final work. This kind of restriction is seen in the machines constraints of the RMS [START_REF] Jacob | Integrating product function design, production technology optimization and process equipment planning on the example of hybrid additive manufacturing[END_REF].

From the above literature review, we can conclude that the combination of process and production planning in a sustainable environment is an interesting and new topic for RMS. Moreover, some new approaches, like the Lp-metrics method can be used to solve the problem more efficiently.

PROBLEM STATEMENT AND MATHEMATICAL FORMULATION

Problem statement

In this paper, reconfigurable machines are able to manufacture multiple products types. Raw materials will come and be processed by several processes and production plans to answer market demands. During the production time, there can be some changes such as configurations chances. Moreover, other orders can intervene to answer the market demands to find the most profitable and cost-effective process and production plans during one-time horizon. There is no shortage of raw materials and the final products have no inventory costs. In our case, for each product, the process plan is a matrix with n columns (number of operations) and three rows (operations, machines and configurations). Table 1 presents two examples of process plans that should be read from left to right, column by column. For example, to manufacture product P1, the operation Op2 has to be realised by machine M1 using configuration H3.

The only work of the workers is to check the machines working and their changes according to the input information, and they don't have any other impacts on the production process. However, they have some sustainability matters. Workers are influenced by the harmful gases of the production, which can affect their health conditions during the working hours. Each machine has its liquid usage in several operations and positions during manufacturing. After finishing the work, there will be some harmful liquids for the environment that are considered in the sustainable objective function as well. The whole concept of the work is aimed to reduce the total process and production costs, and the sustainability objective function.

Table 1. Optimum sequences of each product

Product 1 Operations Op1-Op2-Op4-Op3-Op5 Machines M1-M-M1-M2-M2 Configurations H2-H3-H2-H1-H1 Product 2 Operations Op1-Op3-Op2-Op4-Op5 Machines M2-M2-M2-M1-M1
Configurations H3-H1-H1-H2-H2

1) Minimization of the total process and production costs.

The cost function comprises the cost of process planning such as machine usage and changes during the process and production planning, such as the amount of final product production to answer the market demands.

2) Minimization of the sustainability function that contains the amount of destructive environmental and social components.

It has two different aspects, the first one is environmental, and the second one is social.

Environmental sustainability concerns the emission of the harmful gases and remnants of the toxic and harmful liquid from the machines working. The social sustainability indicator contains the effects of the gas emissions on the worker's body that can be conceived as their working utility function as well.

In Fig. 1, the input and output of the problem are illustrated. The whole system inputs are operators/workers, raw materials, process information and the liquid of the machines. The production process is under both social and market pressures. The outputs of the system are information, social effects of the products and process on the community and workers, final products, wastes and emissions such as carbon footprint, liquid, materials and parts.

Model assumptions

The proposed model is based on the following assumptions:  Workers check the function of machines during the working process and changing;

 Workers have exposed the harmful gases during the checking works and usage of the fuels and liquids by the machines;  Multiple products can be produced;  Each product can follow several process plan;  Each product have several configuration types in the described RMS;  The demand of the market is deterministic;  During the process, no failure will happen;  It is necessary to produce more than the markets' demand;  There is no cost for the storage of the products;  During the operations, the harmful gases emit and effects workers bodies' health.

Mathematical formulation

In the following, we describe more in details the proposed multi-objective model:

Sets i , i

Sustainability function

The objective functions and constraints are defined as follows:

Min 𝑍 1 = ∑ ∑ ∑ ∑ 𝑥 𝑖𝑗ℎ 𝑝 × 𝑃𝑇 𝑖𝑗𝑝 𝐻 ℎ=1 𝑃 𝑝=1 𝐽 𝑗=1 𝐼 𝑖=1 × 𝑃𝐶 𝑖𝑗𝑝 + + ∑ ∑ ∑ ∑ 𝑣 𝑗𝑗 ′ 𝑚𝑝 × 𝑇𝑀 𝑗𝑗 ′ 𝑚 × 𝐶𝑀 𝑗𝑗 ′ 𝑚 𝑃 𝑝=1 𝑀 𝑚=1 𝐽 𝑗 ′ =1 𝐽 𝑗=1 + ∑ ∑ 𝑞 𝑝 × 𝑄𝐶 𝑝ℎ 𝐻 ℎ=1 𝑃 𝑝=1 + ∑ ∑ 𝐴𝑆 𝑝ℎ * 𝐴𝐶 𝑝ℎ 𝐻 ℎ=1 𝑃 𝑝=1 (1)
Min 𝑍 2 = 𝑤 𝑆𝑜 𝑠 𝑆𝑜 + 𝑤 𝐸𝑛 𝑠 𝐸𝑛 (2) Subject to,

∑ ∑ 𝑥 𝑖𝑗ℎ 𝑝 𝑃 𝑝=1 𝐼 𝑖=1 = 1 ∀𝑗 = 1 … 𝐽 ∀ℎ = 1 … 𝐻 (3) ∑ ∑ 𝑥 𝑖𝑗ℎ 𝑝 𝑃 𝑝=1 𝐽 𝑗=1 = 1 ∀𝑖 = 1 … 𝐼 ∀ℎ = 1 … 𝐻 (4) ∑ 𝑥 𝑖𝑗ℎ 𝑝 × 𝑃 𝑖 𝐻 ℎ=1 ≤ ∑ ∑ ∑ 𝑥 𝑖 ′ 𝑗 ′ ℎ ′ 𝑝 𝐻 ℎ ′ =1 𝐽-1 𝑗 ′ 𝐼 𝑖 ′ =1 ∀𝑖 = 1 … 𝐼 ∀𝑗 = 1 … 𝐽 ∀𝑝 = 1 … 𝑃 (5) ∑ 𝑦 𝑗ℎ 𝑚𝑝 𝐻 ℎ=1 = 1 ∀𝑗 = 1 … 𝐽 ∀𝑚 = 1 … 𝑀 ∀𝑝 = 1 … 𝑃 (6) 𝑦 𝑗ℎ 𝑚𝑝 ≥ 𝑥 𝑖𝑗ℎ 𝑝 ∀𝑖 = 1 … 𝐼 ∀𝑗 = 1 … 𝐽 ∀𝑚 = 1 … 𝑀 ∀𝑝 = 1 … 𝑃 ∀ℎ = 1 … 𝐻 (7) ∑ 𝑥 𝑖𝑗ℎ 𝑝 + 𝑥 𝑖𝑗-1ℎ 𝑝 𝐼 𝑖=1 ≤ 𝑣 𝑗𝑗 ′ 𝑚𝑝 + 1 ∀𝑗, 𝑗 ′ = 2 … 𝐽 ∀𝑚 = 1 … 𝑀 ∀𝑝 = 1 … 𝑃 ∀ℎ = 1 … 𝐻 (8) ∑ ∑ 𝑥 𝑖𝑗ℎ 𝑝 𝐽 𝑗=1 𝐼 𝑖=1 × 𝐴𝑆 𝑖𝑝ℎ × 𝐻 = 𝑞 𝑝 ∀𝑝 = 1 … 𝑃 ∀ℎ = 1 … 𝐻 (9) 𝑞 𝑝 = 𝐷 𝑝 ∀𝑝 = 1 … 𝑃 (10) 𝑞 𝑝 × 𝑇 ℎ ≤ 𝐶𝐴𝑃 ℎ ∀𝑝 = 1 … 𝑃 ∀ℎ = 1 … 𝐻 (11) 𝐴𝑆 𝑝ℎ -𝐴𝑆 𝑝ℎ ′ + (1 -𝑣 𝑗𝑗 ′ 𝑚𝑝 ) × BM -1 ≥ 0 ∀𝑗, 𝑗 ′ = 1 … 𝐽 ∀𝑚 = 1 … 𝑀 ∀𝑝 = 1 … 𝑃 ∀ℎ, ℎ ′ = 1 … 𝐻 (12) ∑ 𝑞 𝑝 × 𝑇𝑃 𝑝 ≤ 𝑃𝐶𝐴𝑃 𝑝 (
𝑥 𝑖𝑗ℎ 𝑝 Є {0,1} 𝑦 𝑗ℎ 𝑚𝑝 Є {0,1} 𝑣 𝑗𝑗 ′ 𝑚𝑝 Є {0,1} ∀𝑖 = 1 … 𝐼 ∀𝑗 = 1 … 𝐽 ∀𝑚 = 1 … 𝑀 ∀𝑝 = 1 … 𝑃 ∀ℎ = 1 … 𝐻 (17)
Objective function (1) details the total cost, where objective function (2) presents the sustainability matters. Constraint (3) ensures every operation is done according to process plan. Constraint (4) indicates that each operation is processed once for products in each configuration. Constraint (5) states that each operation is processed if all of the predecessors operations are already finished. Constraint (6) illustrates that each configuration contains machines and products. Constraint (7) ensures that if operation Opi performed at position p, machine Mm and configuration Hc are required. Constraint (8) shows that if there is a change in the machines between the positions or not during the production. Constraint (9) claims that the number of final products is equal to the production sides. Constraint (10) indicates that it is crucial to answer all the demands in the proposed system. Constraint (11) introduces the production capacities. Constraint (12) uses to break the loops in the production site of RMS. Constraint (13) indicates that the amount of production time should not exceed the available time. Constraint ( 14) illustrates the environmental impacts of the production. Constraint (15) considers the limitation of total required liquid in during the production process. Constraint ( 16) indicates the social impacts of the production.

MULTI-OBJECTIVE PROPOSED APPROACH

There are such useful multi-objective approaches, as εconstraint, goal programming, compromise programming and reference point method [START_REF] Romero | Goal programming, compromise programming and reference point method formulations: linkages and utility interpretations[END_REF]. Various usage of the compromise programming leads to be opted, regarding its performance in solving multi-objective optimization problems. In this paper, the compromise programming is applied to solve the aforementioned mathematical model. There are a couple of benefits to implementing the compromise programming in this problem that makes it more suitable than other approaches. First of all, this approach considers the preference of business owners and decisionmakers to choose the relative weights of objectives according to their business. It means the weights of the objective functions are not just an input like goal programming, by more flexible to solve the problem. The decision-maker has significant impacts on the final results. Furthermore, it is easier to consider several weights for the objective functions that make it easier to solve the problem than to find a specific weight for each one of them separately.

For the proposed model, for both cost and sustainability objectives, the model should be solved separately as the Lpmetrics method [START_REF] Babbar | A multi-objective mathematical model integrating environmental concerns for supplier selection and order allocation based on fuzzy QFD in beverages industry[END_REF]. The optimum values of the cost objective function 𝑍 1 and sustainability objective function 𝑍 2 are considered respectively, 𝑍 1 * and𝑍 2 * .

Min Lp-metric:

𝑍 3 = 𝑤𝑒𝑖𝑔ℎ𝑡 1 × 𝑍 1 -𝑍 1 * 𝑍 1 * + 𝑤𝑒𝑖𝑔ℎ𝑡 2 × 𝑍 2 -𝑍 2 * 𝑍 2 * (18)
In constraint ( 18), 0≤ ∑ 𝑤𝑒𝑖𝑔ℎ𝑡 𝑖 2 𝑖=1

≤1 is the relative weight of each objective function. The decision-maker is the one who chooses between the several considered weights according to his preference, called as Lp-metrics coefficients. In the first step, by considering this single objective function and our constraints, a single-objective mixed-integer linear programming model can be achieved from the mathematical formulation. Furthermore, the problem is solved using Gams software.

ILLUSTRATIVE EXAMPLE AND DATA ANALYSIS

In this section, the presented mathematical formulation is implemented in GAMS 30.3.0 and test problems are solved by a laptop computer with Intel(R) Core(TM) i7-6700HQ CPU 2.60GHz and 12GB RAM. By implementing the model in the GAMS solver, the optimum values of 𝑍 1 and 𝑍 2 can be obtained.

Solving a test problem

For this test problem, five different operations can be processed in five positions by machines M1 and M2 using three different configurations. The system wants to answer market demands of products P1 and P2 according to Table 3. By assumption, wso and wen are equal to 0.5 in the objective function Z2 for both products. Moreover, P1 and P2 have the same number of operations (Op1, Op2, Op3, Op4 and Op5) related by the precedence graph of Fig. 2. However, Op1, Op2, Op3, Op4 and Op5 of P1 are different from Op1, Op2, Op3, Op4 and Op5 of P2. The differences are due to the using machines and configurations for each product.

Fig. 2. Operations precedence graph

By solving the test problem, we can discover the optimum way of production of each product between the multiple possible paths of the process for each one. In Table 1, the best sequences of each product depicted. The obtained results are completely logical. For example, product one is on the machine M2 and configuration H1 during operation Op3 and just one change has happened to the machines like product P2. It happens because in the nature of the mathematical model, some extra costs are considered for the extra changes during the process, and the sequence is preferred to do by the minimum amount of changes between machines.

After solving this scenario, we can see the payoff of the problem in Table 2. To obtain such a table, it is essential to consider each objective separately and solve the problem. After 25 times of running the problem, the best amount of cost function is equal to 29000€ and the amount of harmful effects of liquid and gases are 1.5 hazardous units (H.u).

Pareto frontier solutions

In Fig. 3, the obtained Pareto frontier solutions using Lpmetrics method are demonstrated. Pareto frontier solutions include 25 points. In the aforementioned Pareto, the best amount of the cost function is 29000€ while the sustainable function is 40 (H.u), and the best amount of sustainable function happens when it is equal to 1.5 (H.u), and at that point, the amount of cost function is 66987€.

Considering the impacts of different parameters on the proposed model, sensitivity analyses are conducted. Rely on the performances of the objective functions values and parameters this data analysis is depicted in the next section. In Table 3 and Fig. 4, the changes of the 𝐷 𝑝 are shown. This parameter plays a decisive role in the whole problem. Hence the changes of this parameter can have an unpredictable on the changes of the objective function; however, the whole cost will raise, or we can explain that it will have the reverse effect on the optimum amount of the objective. Moreover, it is possible to predict the final optimum of the objective function by this parameter approximately. Fig. 4. LP-metrics behaviour according to the changes in demand

Changes of weights in Lp-metrics

The sensitivity analysis of weights let the decision-maker to choose more wisely with better information about the effects of each objective function on the problem. To make it clear, the decision-maker is the one who decides the effect of each objective function on the whole problem.

In Table 4 and Fig. 5, we can see the different amount of the Lp-metrics objective function by the changes of the weights. It is important to note that the changes in weights have a great impact on the 𝑍 3 . Therefore, a three-dimensional graph is plotted to show the changes more clearly. Accordingly, the weight of the cost function can have more effects on the 𝑍 3 , and they both have ascending rates in most cases. Consequently, the optimum value of the Lp-metric objective is obtained when the decision-maker chooses the weights, respectively, 0.9 for the cost function and 0.1 for the sustainable function. According to the used data, the results show that when the first objective function weight is at a high level, the total problem can be more optimal. It means that the decision-maker should focus on the cost aspects more than the other, which is rational. Nevertheless, in real life, decisionmakers should consider a balance between different objective functions because of the rules that exist to save the environmental aspects. Therefore, maybe the value of weights between 0.4 and 0.6 can have better impacts according to objective functions' value. Besides, it can show the closeness of the presented model and final solving procedure, making it so close to reality and reliable for industrial cases and case studies.

Performance of the algorithm

To discover the capability of the proposed algorithm, the results of the GAMS software have been compared with the results of the Lp-metrics in the case of small and medium scales. In our case, an instance of the problem is represented by I-J-P-M-H where I is the number of operations, J the number of positions, P the number of products, M the number of machines and H the number of configurations. The results are depicted in Table 5. As it seen, by solving the problem in 6 runs, the performances of the LP-metrics are better than Gams when the size of the instances increases.

Table 5. CPU times of LP-metrics vs Gams Lp-metrics Demand reconfigurable environment. The sustainability indicators include the harmful liquids and gases emission during the production and its effects on the worker's body. A linear multi-objective mixed integer model is proposed, where two objectives are minimized namely the harmful amount of gas and liquid emissions, as well as the total process and production costs. Moreover, an Lp-metrics based method is proposed to solve the problem.

For future work directions, meta-heuristic algorithms such as AMOSA, NSGA-II, etc., can be adapted and compared. Moreover, the integration of presented RMS with stochastic programming or inventory management can be another interesting extension of the model.

Fig. 1 .

 1 Fig.1. Involved streams in manufacturing process(Khezri et al., 2010)

'

  Indices of operations ∈ {1, …, I} j , j' Indices of positions in the sequences ∈ {1, …, J} p, p' Indices of units ∈ {1, …, P} m ,m' Indices of machines ∈ {1, …, M} h , h' Indices of configurations ∈ {1, …, Cycle time of configuration h 𝐶𝐴𝑃 ℎ Maximum capacity of the configuration h PCAP Production capacity 𝐷 𝑝 Market demand of product p in the considered period H Number of configurations in the considered period TPp Total production time of product p 𝐴𝐶 𝑝ℎ Assignment cost of product p to sequence h 𝐶𝑀 𝑗𝑗 ′ 𝑚 Cost of changing from position j to position j' by machine m 𝑇𝑀 𝑗𝑗 ′ 𝑚 Time of changing from position j to position j' by machine m 𝑃𝑇 𝑖𝑗𝑝 Processing time of operation i of product p at position j 𝑃𝐶 𝑖𝑗𝑝 Processing cost of operation i of product p at position j 𝑃𝑅 𝑖 Set of predecessors of operation i 𝑄𝐶 𝑝ℎ Production cost of product p in configuration h 𝐸𝐿 𝑖,𝑗 Harmful liquid remnants of the operation i at position j 𝐸𝐺 𝑖,𝑗 Harmful gases emission of the operation i at position j 𝑙 𝑖,𝑗 Required liquid for operation i at the position j L Total available liquid EF The effect of harmful gases on the human body during the working hours 𝑤 𝑆𝑜 Related weight of social sustainability 𝑤 𝐸𝑛 Related weight of environmental sustainability

Fig. 3 .

 3 Fig. 3. The obtained Pareto frontier 6. SENSIVITY ANALYSIS

Fig. 5 .

 5 Fig. 5. Lp-metrics behaviour illustration in change of weights

  AND PERSPECTIVESIn this paper, for the first time, the whole sustainability pillars like social, environmental, and economic are combined with the process and production planning generation in 0

Table 2 .

 2 Pay off table of objective functions

	𝒁 𝟏	𝒁 𝟐
	29000	40
	66987	1.5

Table 3 .

 3 Objective functions' behaviours considering change in demand

	𝑫 𝒑	𝒁 𝟏	𝒁 𝟐	Lp-metrics
	20 k	3130 1.5 4.44089E-16
	25 k	3879 1.6 -0.039
	30 k	3804 4.3 0.011102
	35 k	5232 7.1 -0.029
	40 k	5157 9.9 6.66134E-16
	45 k	6735 12.7 -0.027
	50 k	6585	18	-1.1102E-16

Table 4 .

 4 Lp-metrics behaviour analysis in change of weights

	Weight 1	Weight 2	Lp-metrics
	1	0	2.22E-15
	0.9	0.1	3.33E-16
	0.8	0.2	2.89E-15
	0.6	0.4	1.55E-15
	0.5	0.5	2.22E-16
	0.4	0.6	1.11E-15
	0.2	0.8	4.44E-16
	0.1	0.9	4.44E-16
	0	1	2.22E-16