Emmanuel Zimmermann

Tsegay Tesfay Mezgebe

Hind BRIL E L Haouzi

Philippe Thomas
email: philippe.thomas@univ-lorraine.fr

Rémi Pannequin
email: remi.pannequin@univ-lorraine.fr

Mélanie Noyel
email: mnoyel@acta-mobilier.fr

Multicriteria Decision-making Method for Scheduling Problem Based on Smart Batches and their Quality Prediction Capability

Keywords: Proactive decision-making, Analytic hierarchy process (AHP), Multicriteria decision method, Quality prediction, Smart products, Rework

The scheduling problem in manufacturing companies with high rework rates remains a complex research area to date. This paper presents a new approach for manufacturing scheduling that combines a predictive schedule with a proactive multicriteria decision-making method based on smart batches and their quality prediction capability. Each batch embeds an algorithm that allows it to predict its quality out of the next workstation. As soon as a batch determines that its process is too hazardous, a collaborative rescheduling decision, using the analytic hierarchy process (AHP), is initiated with its peer. This article details the proposed approach along with the AHP structure and presents the considered decision problem. A simulation model inspired by a lacquering-robot case study is described to validate this proposition. Then, the results of different scenarios are presented and discussed, highlighting the impact of social myopia on smart batches.

Introduction

Scheduling problems have been addressed by some of the most important studies in manufacturing control management. Since 1960, many exact and heuristic approaches have been proposed in the literature to achieve the best outcomes, such as maximum profit or lowest cost [START_REF] Hanssmann | A Linear Programming Approach to Production and Employment Scheduling[END_REF][START_REF] Held | A Dynamic Programming Approach to Sequencing Problems[END_REF]. Classical techniques that determine globally optimized schedules are not very useful when facing disturbances, which is notably the case for companies suffering from high rework1 rates. This can be explained by the fact that as soon as a need for rework appears, the global schedule becomes obsolete because of the lack of agility of centralized/predictive solutions. To address this issue, two main paths could be explored: 1) robust scheduling methods and 2) distributed production scheduling methods. Motivated by the framework of Industry 4.0, the current focus is mainly on the second path. In the existing requirement for dynamic decision, the distribution of decisional ability to enable the system's self-adaptation to changing situations is seen as a great improvement vector for real-time problem-solving and adaptability enhancement [START_REF] Mezgebe | CoMM: a consensus algorithm for multi-agent-based manufacturing system to deal with perturbation[END_REF]. However, such systems suffer from the phenomenon of social myopia, where decisional entities in distributed reactive decision-making cannot balance their local objectives with the system's global objective [START_REF] Rey | Reducing myopic behavior in FMS control: A semi-heterarchical simulation-optimization approach[END_REF]. Based on the need for distributed decision processes and their inherent risk of social myopia, we identified a holonic architecture-based elaboration of the hybrid control system as the most promising approach owing to its ability to couple predictive and reactive approaches to achieve global optimization while limiting myopia [START_REF] Cardin | Coupling predictive scheduling and reactive control in manufacturing hybrid control architectures: state of the art and future challenges[END_REF][START_REF] Zimmermann | A hybrid manufacturing control based on smart lots in a disrupted industrial context[END_REF].

In this paper, the scheduling problem of a lacquering-robot-based work center in a wood processing company, Acta Mobilier2 , is considered. To be profitable in terms of both material consumption and the use of available time, the production consists of large batches. On the shop floor level, after lacquering, batches are directed to an oven for 7 h of drying. However, any non-quality work done during lacquering is only visible after the product is dried. Hence, if a poor-quality batch appears, it is only detected when leaving the oven 7 h after completion of the lacquering. Consequently, because it is very likely that the entire batch must be reworked, its process time is doubled and lengthened by another lacquering and drying process. Therefore, in previous studies, a prediction model of non-quality production risk was developed based on the use of a neural network (NN) [START_REF] Noyel | Reconfiguration process for neuronal classification models: Application to a quality monitoring problem[END_REF]. Starting from this NN and its subsequent improvements [START_REF] Thomas | Using a Classifier Ensemble for Proactive Quality Monitoring and Control: the impact of the choice of classifier types, selection criterion, and fusion process[END_REF], we propose to use it as a means of triggering and reordering as well as for criterion selection involved in the analytic hierarchy process (AHP) decision-making process.

Further, a hybrid control system is proposed in this paper, coupling predictive scheduling and a proactive multicriteria decision method focused, in particular, on non-quality production risk. This is an extension of the work of [START_REF] Zimmermann | Using analytic hierarchical process for scheduling problem based on smart lots and their quality prediction capability[END_REF] with significant improvements in problem modeling and including the presentation of a statistical study evaluating social myopia (number of active/smart batches in the decision-making process). This evaluation approach can be used to identify the optimal number of smart batches that provides a good trade-off between algorithm performance and results quality.

The rest of the paper is organized as follows: Section 2 presents research related to the contribution of hybrid manufacturing control systems to industrial real-time scheduling issues. Section 3 explains the proposed hybrid manufacturing control system coupling a predictive schedule with a proactive decisional process-based smart batch using the AHP method. Section 4 presents an industrial application by considering the proposed approach for the case study of a lacquering robot in the Acta Mobilier company. A simulation model is presented to validate the possible advantages of the proposed method. Experiments comparing the AHP-based decision-making process with actual decisional rule practices in the company are described. Finally, the impact of myopia on smart batch analysis is evaluated. The last section presents the conclusions and future prospects.

Related work

Today's globally volatile market is compelling manufacturing industries to adapt and react to different factors, such as requirements for highly flexible and value-added activities, highquality products, or shorter product life cycles. These new stakes have made centralized predictive schedulers obsolete, aggravating the flow disturbances due to non-quality in production processes. To meet these new requirements of high quality and flexibility, companies can either invest in state-of-the-art technologies and production tools or adapt their control architectures. Concerning the first option, we can consider the NN-based setup to limit non-quality as proposed by [START_REF] Noyel | Reconfiguration process for neuronal classification models: Application to a quality monitoring problem[END_REF]. The model depends on a multitude of both internal and external factors, such as actual workstation setups, production range of the next product to be processed, and environmental factors such as air humidity or temperature. However, the model proved to be insufficient because the considered system was working at technological limits. The second option, which could be combined with the first one, would be to try to predict the impact of non-quality products on production processes. For example, considering the same case study, [START_REF] Noyel | Reconfiguration process for neuronal classification models: Application to a quality monitoring problem[END_REF] proposed a model based on a NN with the capability to predict the risk of non-quality for a particular product at a particular time. Using this capability, dynamical rescheduling can be performed to prevent non-quality, independently of technological limits.

This second option is supported by recent developments concerning hybrid control architectures. Most recent studies (such as [START_REF] Pach | ORCA-FMS: a dynamic architecture for the optimized and reactive control of flexible manufacturing scheduling[END_REF][START_REF] Barbosa | Dynamic self-organization in holonic multi-agent manufacturing systems: The ADACOR evolution[END_REF][START_REF] Indriago | H2CM: A holonic architecture for flexible hybrid control systems[END_REF][START_REF] Quintanilla | A modeling framework for manufacturing services in service-oriented holonic manufacturing systems[END_REF]Jimenez J. F. et al., 2017;[START_REF] Zimmermann | A hybrid manufacturing control based on smart lots in a disrupted industrial context[END_REF][START_REF] Valckenaers | ARTI reference architecture-PROSA revisited[END_REF][START_REF] Mezgebe | CoMM: a consensus algorithm for multi-agent-based manufacturing system to deal with perturbation[END_REF] have already proven the adequacy of such architectures to address resilience and adaptability issues for production flows. In their work, [START_REF] Zimmermann | A hybrid manufacturing control based on smart lots in a disrupted industrial context[END_REF] proposed a hybrid architecture based on the product driven system concept, where an instance of the quality prediction NN model would be embedded on each product or batch, assuming that an optimized schedule was provided to workstations. This approach combines both global and local scheduling, the global one being established with no consideration for any rework problem and local scheduling being established for every work center, which follows its own optimization and is able to recalculate itself depending on the situation. Notably, as variant external factors such as atmospheric pressure or air humidity had an important impact on production in the studied case, products were able to decide to alter their schedule. Another characteristic of the architecture proposed by [START_REF] Zimmermann | A hybrid manufacturing control based on smart lots in a disrupted industrial context[END_REF] is that products in the workstation queue must normally be processed following the optimal global schedule. Still, to prevent non-quality, local rescheduling might be performed under peculiar conditions by the products, and thanks to the risk prediction process.

To model and implement this hybrid architecture, many collective decision-making approaches executed in multi-agent systems have been introduced. Some of them are inspired by social attitudes such as consensus [START_REF] Mezgebe | CoMM: a consensus algorithm for multi-agent-based manufacturing system to deal with perturbation[END_REF] and majority voting [START_REF] Pitt | Voting in Multi-Agent Systems[END_REF], or from the domain of game theory [START_REF] Parsons | Game theory and decision theory in multi-agent systems[END_REF]. Others seek to imitate biological behaviors such as swarm robotics [START_REF] Schmickl | Get in touch: cooperative decision making based on robot-to-robot collisions[END_REF], bee swarms [START_REF] Teodorovic | Transport modeling by multi-agent systems: a swarm intelligence approach[END_REF], and ant colonies [START_REF] Xiang | Ant colony intelligence in multi-agent dynamic manufacturing scheduling[END_REF]. In addition, the use of the AHP offers advantages to decision-makers for prioritization. The AHP relies on the establishment of a hierarchical structure to deal with complex decision-making and is able to consider all the criteria appearing in the decision-making process. The method is typically structured with several levels; some are mandatory: the global objective (level 0) and alternatives (last level). Between these two ends, there is a variable number of potential levels (named sub-criteria). The AHP method uses pairwise comparisons of same-level elements whose relative importance is evaluated at a higher level; a value scale is used to express the degree of importance of one element compared to another.

The adequacy of the AHP method for solving complex business decisions has already been well established in previous studies [START_REF] Cheng | Analytic hierarchy process (AHP)[END_REF][START_REF] Vaidya | Analytic hierarchy process: An overview of applications[END_REF][START_REF] Saaty | Decision making with the analytic hierarchy process[END_REF][START_REF] Liu | The voting analytic hierarchy process method for selecting supplier[END_REF][START_REF] Bian | Identifying influential nodes in complex networks based on AHP[END_REF][START_REF] Moslem | Analysing stakeholder consensus for a sustainable transport development decision by the fuzzy AHP and interval AHP[END_REF]. [START_REF] Büyüközkan | Digital competency evaluation of lowcost airlines using an integrated IVIF AHP and IVIF VIKOR methodology[END_REF] stated that the AHP [START_REF] Saaty | The analytic hierarchy process[END_REF] is the most utilized multicriteria decision-making method. Likewise, the AHP has been used for scheduling problems in the last two decades. Notably, [START_REF] Momoh | Optimal generation scheduling based on AHP/ANP[END_REF] proposed an application of the AHP and analytic network process to prioritize, schedule, and optimize power unit price allocations. Later, [START_REF] Chan | A hybrid genetic algorithm for production and distribution[END_REF] coupled an AHP to a genetic algorithm to solve the production-distribution problem, while [START_REF] Azadeh | Integration of DEA and AHP with computer simulation for railway system improvement and optimization[END_REF] combined the AHP to data envelopment analysis in computer simulations to find optimum alternatives with multiple quantitative and qualitative variants. Moreover, the works of [START_REF] Louati | Multicriteria decision making service for an intelligent control of individuals flow in restricted area[END_REF] on people flow control based on isoarchic architecture with decision-making using multicriteria analysis and Kubler et al. (2016) regarding applications of fuzzy AHP can also be considered. From these works, it can be concluded that the AHP is one of the most used methodologies for complex decision-making considering both qualitative and quantitative variables. Nonetheless, designing control systems using the AHP to avoid and/or minimize the impact of reworks in real manufacturing processes remains an underexplored research area to date.

Proposition

The proposed hybrid control system was designed with the aim of reducing the non-quality while limiting the consumption of raw materials. To ensure this objective, [START_REF] Zimmermann | Using analytic hierarchical process for scheduling problem based on smart lots and their quality prediction capability[END_REF] proposed a hybrid manufacturing system that combines a smart batches-based distributed control system and centralized schedule optimization. Consequently, the proposal here is to embed the batches (minimal product agent entities) with their own instance of the NN quality prediction model, to trigger it before setting up a batch on the workstation. If the risk of poor quality "" is weak (under a certain threshold), then the batches are processed following the forecasted global schedule (Fig. 1). Otherwise, if the NN determines that the risk exceeds the accepted threshold for the next batch to be produced, then the system enters a collective decision with the other batches queued at the workstation (Fig. 2). The purpose of this negotiation is to determine the most suitable batch to be produced by considering the different criteria of the workstation. Accordingly, five criteria were selected in this proposition: 1) risk of non-quality, determined by the developed NN, 2) critical ratio (depending on customer due date), 3) balance between different product family flows, 4) adjustment times induced by the batch change, and 4) nervousness of a system (which is represented by the number of modifications made compared to original planning). The last criterion aims to limit logistical disruptions at the workstation. (i.e., the supply of lacquer, panels, or any other components necessary for machine operations).



In the proposed AHP-based collaborative decision-making process, batches can negotiate to decide which one is going to be processed next. During the implementation of the AHP [START_REF] Saaty | The analytic hierarchy process-What it is and how it is used[END_REF]Shameen et al. 2020), each batch is compared with others in the queue, criterion by criterion, to build comparison matrices. Fig. 3 presents the 3-layer structure of the AHP used in this study. Level 0 represents the objective of the problem, consisting of the selection of the best batch (present in the queue) to be produced. Level 1 includes the five criteria previously described, which influence the selection in Level 0. Level 2 compares the different alternatives (five batches) of the problem, as it occurs: the batches present in the queue. However, the number of alternatives/batches to be tested can have an impact on the overall performance of the process. This problem will be studied in Section 4 (myopia). Four steps in the process must be followed. First, the criteria and alternatives matrices must be created. Then, the criteria weight vector should be computed, followed by the calculation of the alternatives weight vector. Finally, an inconsistency problem must be considered. These steps are described in the following sections.

Interpretation 1 𝐷𝐷(𝑘) ∈ ]𝐷𝐷(𝑙) -𝛼; 𝐷𝐷(𝑙) + 𝛼[ 3 𝐷𝐷(𝑘) ∈ ]𝐷𝐷(𝑙) -2𝛼; 𝐷𝐷(𝑙) -𝛼] 5 𝐷𝐷(𝑘) ∈ ]𝐷𝐷(𝑙) -3𝛼; 𝐷𝐷(𝑙) -2𝛼] 7 𝐷𝐷(𝑘) ∈ ]𝐷𝐷(𝑙) -4𝛼; 𝐷𝐷(𝑙) -3𝛼] 9 𝐷𝐷(𝑘) ∈ ]-∞; 0, 𝐷𝐷(𝑙) -4𝛼]
Balance of product flows: For companies with several product flows, furnisher-workstations must provide work alternatively to different customer workstations. As avoiding starvation is one of the schedule objectives, any schedule change might have an important impact on production flows, if done carelessly. In the studied case, only two production flows were considered: shiny and matt. Hence, the following formula was established:

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑘) = 𝑙𝑜𝑎𝑑_𝑠ℎ𝑖𝑛𝑦±𝑙𝑜𝑎𝑑_𝑘 𝑙𝑜𝑎𝑑_𝑠ℎ𝑖𝑛𝑦+𝑙𝑜𝑎𝑑_𝑚𝑎𝑡𝑡+|𝑙𝑜𝑎𝑑_𝑘| (3)
where 𝑙𝑜𝑎𝑑_𝑠ℎ𝑖𝑛𝑦 and 𝑙𝑜𝑎𝑑_𝑚𝑎𝑡𝑡 represent the shiny and matt workloads at the computation time. Then, 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑘) is the ratio of the shiny workload plus or minus (plus if batch 𝑘 is shiny, minus if matt) the impact of batch 𝑘 on the total load, and β is an objective value set at 50% (0.5). We then express the deviation from the target as follows:

𝑔𝑎𝑝(𝑘) = |𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑘) -𝛽| (4)
Then, we can estimate which batch, k or l, generates the most important impact on the global workload by calculating 𝐷𝑅 = 𝑔𝑎𝑝(𝑘)+𝜀 𝑔𝑎𝑝(𝑙)+𝜀

(5)

with ε being a small value. Let also 𝜇 be the threshold set to 10%.

Interpretation 1 𝐷𝑅 ∈ ]1 -𝜇; 1 + 𝜇[ 3 𝐷𝑅 ∈ ]1 -2𝜇; 1 -𝜇] 5 𝐷𝑅 ∈ ]1 -3𝜇; 1 -2𝜇] 7 𝐷𝑅 ∈ ]1 -4𝜇; 1 -3𝜇] 8 9 𝐷𝑅 ∈ ]0; 1 -4𝜇]
Nervousness: Let 𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) be the number of position switching made with respect to the original schedule.

Each time two batches are swapped, a switching counter is incremented by 1: 𝑆𝑤𝑖𝑡𝑐ℎ(𝑘)+= 1, with 𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) ∈ [0; 𝑛] and 𝑛 equal to the number of batches (5). Let δ be a threshold value set to 1. Non-quality risk:

Let 𝑅𝑖𝑠𝑘(𝑘) be the value computed by the NN, and φ be a threshold set up to 10%. 𝑅𝑖𝑠𝑘 represents a computation made with an NN using external parameters (such as atmospheric pressure and humidity) and internal parameters (such as color or thickness of the batch) to determine the probability of a batch having a quality defect during processing [START_REF] Noyel | Reconfiguration process for neuronal classification models: Application to a quality monitoring problem[END_REF].

Interpretation 1 𝑅𝑖𝑠𝑘(𝑘) = 𝑅𝑖𝑠𝑘(𝑙) 3 𝑅𝑖𝑠𝑘(𝑘) ∈ ]𝑅𝑖𝑠𝑘(𝑙) -𝜑; 𝑅𝑖𝑠𝑘(𝑙)] 5 𝑅𝑖𝑠𝑘(𝑘) ∈ ]𝑅𝑖𝑠𝑘(𝑙) -2𝜑; 𝑅𝑖𝑠𝑘(𝑙) -𝜑] 7 𝑅𝑖𝑠𝑘(𝑘) ∈ ]𝑅𝑖𝑠𝑘(𝑙) -3𝜑; 𝑅𝑖𝑠𝑘(𝑙) -2𝜑] 9 𝑅𝑖𝑠𝑘(𝑘) ∈ ]0; 𝑅𝑖𝑠𝑘(𝑙) -3𝜑]
Setup time:

In this study, the workstation was considered as disposing of two types of setups: one for shiny products and one for matt ones. Any batch change entails a setup time. However, we must differentiate same-type batch changes (𝑏 𝑘𝑙 (𝑖) = 1) and different-type batch changes (𝑏 𝑘𝑙 (𝑖) = 5). In the first case, we can consider the setup time to be very low compared to the setup time in the second case. In this way, 𝑝𝑟𝑜𝑔(𝑘) is the setup of batch 𝑘 and 𝑙𝑎𝑠𝑡_𝑝𝑟𝑜𝑔 is the setup of the last batch that has been processed on the workstation. Step 2: Computing the vector of weights of criteria:

Let A be the pairwise comparison matrix of criteria. A satisfies the following constraint:

𝑎 𝑖𝑗 . 𝑎 𝑗𝑖 = 1 𝑎𝑛𝑑 𝑎 𝑖𝑖 = 1 (6)
The values of 𝑎 𝑖𝑗 are chosen among the possibilities given in Table 1, according to the interpretation of experts in the work. In this study, all the criteria were considered equivalent to each other, and therefore all 𝑎 𝑖𝑗 values were set to 1. The study led in the company so far did not allow sufficient work expertise to determine which criterion should be prioritized. However, the model could easily be improved as soon as a work expert gives his or her interpretation.

Once the matrix is built, it is possible to derive from A the normalized pairwise comparison matrix Anorm by making equal to 1 the sum of the entries on each column. Finally, the criteria weight vector w (that is an m-dimensional column vector) is built by averaging the entries in each row of Anorm.

Step 3: Computing the vector of weights of alternatives:

The same work must now be done to model the links between the different alternatives and each of the criteria. In addition, for each criterion, an 𝑛 × 𝑛 pairwise comparison matrix 𝐵 (𝑖) should be established and must have the same properties as A: 𝑏 (𝑖) 𝑘𝑙 . 𝑏 (𝑖) 𝑙𝑘 = 1 and 𝑏 (𝑖) 𝑘𝑘 = 1

(7)
The AHP applies to each matrix 𝐵 (𝑖) the same procedure as for the pairwise comparison matrix A, that is, it divides each entry by the sum of the entries in the same column and then averages the entries in each row, thus obtaining the score vectors 𝑠 (𝑗) , j=1,..., m. The vector contains the scores of the evaluated options with respect to the j th criterion. Once the weight vector w and score matrix S are computed, the AHP obtains a vector v of global scores by multiplying S and w.

Step 4: Inconsistency problem:

As explained previously, the 𝑎 𝑖𝑗 values of matrix A were set to 1. In this special case, the consistency problem of matrix A is irrelevant. However, the matrices 𝐵 (𝑖) comparing the alternatives/batches will be different depending on the characteristics of the evaluated batches. Therefore, consistency must be evaluated on the score matrix 𝑆 for each run of the process. To do so, the CI is calculated as follows:

𝐶𝐼 = 𝑥-𝑛 𝑛-1 (8)
where n is the number of comparisons and 𝑥 is the average of the values of the vector whose 𝑖 th element is the ratio of the 𝑖 th value of 𝐵 (𝑖) • 𝑆 to the corresponding value of 𝑆 is necessary. Thus, the consistency ratio is

𝐶𝑅 = 𝐶𝐼 𝑅𝐼 (9)
where RI is a random evaluation value [START_REF] Saaty | The analytic hierarchy process-What it is and how it is used[END_REF]. If the CR is below 0.1, the inconsistency is acceptable, and the best pretender is evaluated and selected. Otherwise, the original batch will be kept according to the optimization made by the global schedule. In fact, if the CR is above 0.1, the risk of inconsistency is beyond the acceptable threshold, generating small confidence in the choice of the new batch to process. In this case, it is preferred to keep confidence in the original schedule rather than risk a hazardous choice. Normally, if the CR is above 0.1, an interaction with the decision maker is required to revise the judgment. Until now, this assumption remains a proof of concept, as the method is not very elaborated. However, the objective is to have a fully automated process without the need for human contribution during execution. However, it seems possible to automatize the process of matrix correction as soon as CR > 0.1. The pseudocode of the AHP process is derived from that proposed by [START_REF] Saaty | The analytic hierarchy process-What it is and how it is used[END_REF]. The main difference with the standard AHP lies in the determination of the matrices 𝐵 (𝑖) to compare the alternatives; each run of the process requires the elaboration of new pairwise comparison matrices, as each batch has its own surface, color, due date, and setup. In addition, for each computation, the environment is different: workflows and non-quality risks are continuously evolving, depending on external or internal parameters.

Algorithm of the AHP:

Inputs: n: number of criteria and number of alternatives A: 𝑛 × 𝑛 criteria matrix Outputs: Return a number between 1 and n corresponding to the selected batch 𝑆 = 𝑛 × 𝑛 //score matrix 𝐵 (𝑖) = 𝑛 × 𝑛 //pairwise comparison matrix for criterion i. 𝑤 = 1 × 𝑛 //weight vector

𝐴 ̅ = ( 𝑎 11 ̅̅̅̅ … 𝑎 𝑛𝑛 ̅̅̅̅̅ ⋮ ⋱ ⋮ 𝑎 𝑛1 ̅̅̅̅̅ ⋯ 𝑎 𝑛𝑛 ̅̅̅̅̅ ) 𝑤𝑖𝑡ℎ 𝑎 𝑖𝑗 = 𝑎 𝑖𝑗 ∑ 𝑎 𝑙𝑗 𝑛 𝑙=1 𝑤 𝑗 = ∑ 𝑎 𝑗𝑙 𝑛 𝑙=1
𝑛

For 𝑖: = 1 to 𝑛 𝐵 (𝑖) = ( 1 … 1 ⋮ ⋱ ⋮ 1 ⋯ 1 )
For 𝑗: = 1 to 𝑛 For 𝑘: = 1 to 𝑛 𝑏 (𝑖) 𝑗𝑘 = 𝑐𝑜𝑚𝑝𝑎𝑟𝑒_𝐴𝐻𝑃( 𝑖, 𝑗, 𝑘) // is elaborated dynamically.

𝑏 (𝑖) 𝑘𝑗 = 1 𝑏 (𝑖)

Case study: Industrial applicability

Description of the industrial context

As highlighted in Section 3, the proposition deals with the specific case of a lacquering robot in the Acta Mobilier company. This company suffers from a high rework rate, above 30%, owing to technological limitations and the high degree of product quality expectations from customers. The lacquering robot feeds two separate product flows:

• "Matt" products, directly delivered to the shipment station.

• "Shiny" (or "high gloss" in Fig. 4) products, which are processed by the polishing machine before joining the other product type in the shipping station (Fig. 4).

Validation approach

The Intelligent Manufacturing Systems-Network of Excellence (IMS-NoE) has shown the relevance of simulation-based benchmarking in estimating the efficiency of plant-wide control and organizational issues before their practical deployment into complex manufacturing systems [START_REF] Cavalieri | Benchmarking the performance of manufacturing control systems: Design principles for a web-based simulated testbed[END_REF][START_REF] Valckenaers | A benchmarking service for the manufacturing control research community[END_REF]. For this reason, a Python-based simulation model was developed to represent the lacquering workshop and its queue. In the simulation, the queue was set to five batches. These batches participate in the decision-making process for the best pretender. This has been modeled by five entering queues, numbered from q1 to q5, with the capacity to process one single batch. These five queues can collaborate in the case of non-quality risk detection, as presented in Section 3. Every time a batch is completed on the workstation, a quality check is performed. If the predicted quality satisfies the acceptability range, it leaves the queue, and a new product enters. If not, it puts itself back into the queue.

The sequence for product entry in the queue is determined by the optimal predictive schedule.

In the normal state, queue q5 is supposed to be processed next. Fig. 5 illustrates this scenario: batches B1 to B5 are sorted according to the predictive global schedule and without poor-quality detection. B1 is the first batch to be processed if no poor-quality risk is detected. Each time a batch enters a machine, the others move to the next queue. Then, according to the product information, the robot realizes the corresponding program, and the batch leaves the machine with the remaining time reduced by drying time (7 h). Note that the processing time is proportional to the global panel surface in the batch. When the batch exits the system, a random number is generated to determine whether it is defective. If defective, it returns to the input queue to be reworked. Otherwise, it exits the machine and goes to one of the two possible downstream queues: high gloss or matt (Fig. 4).

Fig. 5. Flow in a stable environment [START_REF] Zimmermann | Using analytic hierarchical process for scheduling problem based on smart lots and their quality prediction capability[END_REF] Before entering the lacquering robot, batch B1 in q5 launches the computation of its embedded NN to determine its non-quality risk. If the risk is higher than the threshold of 25%, as established in Section 3, the AHP-based application is recalled, addressing the procedure for selecting the best replacement. The algorithm presented in Section 3 is then launched to determine which batch best corresponds to the five optimization criteria. Accordingly, Fig. 6 shows a scenario where the NN of B1 has evaluated a very heavy risk of potential non-quality for the initial batch entry order in the decisional-based AHP process and, as a result, rescheduling puts B3 in the first position to achieve better overall quality.

Fig. 6. Flow in case of non-quality risk detection [START_REF] Zimmermann | Using analytic hierarchical process for scheduling problem based on smart lots and their quality prediction capability[END_REF] The variables of this new model are listed below, and Table 7 presents a summary of their numerical values. o Variables related to production range:

• "Area" is the total surface of the products composing the batch, expressed in m².

• "Program" is the number of the program the machine should use to realize its operation on a batch (for example: 2:2 means that program number 2 will lead to process the two wood panel sides with shiny paint). • "Due date" is the batch due date, expressed in hours.

• "Number of pass" expresses the number the batch processing loops on the robot. o Variables related to the workstation:

• The four programs (P1, P2, P3, and P4) with their own duration times are expressed in h/m². • The two possible setup times, expressed in hours:

• S0 if the last batch and next batch program are in the subset {P1, P2} or in the subset {P3, P4}. • S1 elsewhere (corresponding to a finishing change). o "Defect happening" counts the number of defectives batches produced.

Experimental protocol and discussions

To validate the proposed method, simulation experiments were conducted under several distinct conditions, with and without the use of the proposed AHP-based approach and considering a constant number of decision participants. For each case, a set of one thousand simulations were performed. The means and standard deviations (std) for the FIFO and AHP cases are presented in Table 8. Hence, to draw the benefits of the AHP method over the others, a global cost function (expressed in euros) described as the weighted sum of the following three KPIs (also expressed in euros) is compared:

o Lateness represents the number of overdue hours multiplied by the penalty cost. It is an approximate way to express the cost of the lateness, which is complex to represent, because a product in overdue implies a heavy impact on the company brand-building, a loss of customer confidence and, in some cases, a complementary truck departure. o The rework cost, which is the cost implied by the fact of a re-working batch on the robot and is taken directly from the Acta Mobilier company actual data.

o The workforce cost is calculated by multiplying the cost in man-days of two operators (the minimal number of human operators needed to pilot the robot) by a makespan.

Hence, Table 8 highlights that the AHP algorithm provides a better solution for a nonoptimized dataset. For example, the rework cost in the AHP is 25% lower than that in the FIFO scheduling approach, for a global reduction of 26%.

Table 8. Comparing FIFO and AHP in an actual case

The first case studied is an actual example of the production of a lacquering robot without any optimization approach. The production of a team (eight working hours per day) was used in this study. Then, a simulation test with an optimal schedule having an equal balance of the gloss and matt flows is used to determine the benefit of the proposal placed in the optimal possible case. Hence, from Table 9, it can be concluded that when a data set is already scheduled to create a good balance between the two product flows to be supplied, the AHP method still offers a rework improvement of 21% for a global enhancement of 20%.

Table 9. Equal distribution condition

Then, the following two tests were conducted to determine if the solution remained effective when one of the decision criteria was defective.

a) Under an extreme load distribution (only high-gloss products or only matt products), which also impacts the minimization of the setup time (see Table 10). Although the system faces this situation, the result obtained using the AHP is still improved by 18% compared to that of the FIFO method for only high gloss and 25% for only matt. b) As soon as all products set the same due date with an optimized schedule, the benefits brought by the AHP are almost exclusively on the rework cost and to some extent on the lateness, as indicated in Table 11. At this time, the improvement is recorded as 23%.

Social myopia of smart batches analysis

In Subsection 4.3, the performance of the proposed method should be closely linked to its settings (e.g., weights of the criteria and number of actors in the entire decision-making process). However, validation was performed considering a constant number of batches as decisional participants (one product representing one batch). Now, the focus will be on estimating the impact of the number of decision process participating entities on overall quality. Therefore, to measure the impact of social myopia, the number of decisional products (i.e., batches in the queue) is set as a variable. To do so, the simulation was re-launched, running on the data set corresponding to a real production week for the lacquering robot. The weights of all the criteria were set to 1, and only the number of products involved in the decision-making process varied.

Accordingly, simulation tests were performed considering 1, 3, 5, 7, 10, and 20 batches. The two extreme values (i.e., 1 batch and 20 batches) were used to obtain extreme values, even if they were irrelevant. Indeed, one single batch can only communicate with itself, implying that no real decision-making process can be performed. If the lacquering robot does not operate in pull flow with its upstream station, this situation rarely occurs. Likewise, it is highly unlikely for this workstation to have a queue with more than 10 products, except in the case of extreme congestion caused by machine failure. Figures 7 and8 summarize the results obtained for the above-mentioned situations (where m is the number of batches in the queue). As expected, the results are the poorest when a single batch is involved in the decision process. From Figure 9, it can be noted that for more than five batches in the queue, the average results stagnate as the computing time increases. Hence, in view of the results and having knowledge of the evolution of the usual flow at the entrance of the lacquering robot, the choice of five communicating batches for decision-making under standard operating conditions seems to be the most relevant. To better analyze the results, a one-dimensional variance analysis (ANOVA) was used to measure the effect of the change in modality of 'myopia' on the number of reworks, cumulative delay, and time calculation. To apply ANOVA, some recommendations must be verified.

-A single variability factor (myopia) with six modalities (m = 1, m = 3, m = 5, m = 7, m = 10, and m = 20) -Random and independent draws -Observations supposed without errors.

The ANOVA test involved comparing two different modalities of the factor. It generates a p-value that must be compared to the value of the risk  that we have to choose (here taken at 5%). The two hypotheses are as follows:

-H0: The two considered modalities of the factor studied have the same impact on the variable. -H1: The two considered modalities of the factor studied have different impact.

Accordingly, the result of the test is obtained by comparing the p-value with the value of the chosen risk . That is, if the p-value is greater than risk  (p > 0.05), then H0 is accepted with a risk, if the p-value is less than risk  (p < 0.05), then H0 is rejected.

From Table 11, we can observe that H0 is systematically rejected when modalities 1 and 3 are at stake. Consequently, choosing one of these two modalities has a significant impact on the result. When comparing modalities 5, 7, 10, and 20, H0 is systematically accepted (except for case 5-10, which still gives a value close to the threshold). Therefore, choosing between modalities 5, 7, 10, and 20 does not have any impact on the number of reworks. Thus, setting modality at 5 seems to be the best compromise regarding the number of reworks.

In addition, relative to the impact of myopia on delay in Table 12, we can notice that H0 is systematically rejected when modalities 1 and 3 are at stake. Hence, choosing one of these two modalities has a significant impact on the result. When comparing modalities 5, 7, 10, and 20 sets, H0 is systematically accepted and choosing between modalities 5, 7, 10, and 20 has no different impact on the delay. Thus, setting this modality at 5 also seems to be the best compromise in terms of delay.

Finally, corresponding to the impact of myopia on the time calculation in Table 13, it can be noticed that H0 is rejected for all combinations except for the two modality pairs, 1-3 and 5-7.

Conclusions and outlooks

In this work, we have demonstrated the interest of using a hybrid manufacturing control system based on a predictive schedule and the AHP multicriteria decision method as a decision support tool in the context of a company suffering from a high rework rate. The consideration of the number of batches that participate in the decision (social myopia) is proven to be important and has a considerable impact on the quality and calculation time. The AHP method is used dynamically when disturbances occur or when the smart batches using the embedded NN algorithm estimate a high risk of non-quality. The analysis of myopia allows us to consider an evolutionary number of batches participating in the decisions, depending on the quality of the results. The real number of batches in the queue of the workstation should be considered.

Even if the simulation results show interesting improvement of different KPIs, there are still many issues and complementary scenarios to explore. Future work might, for example, use a larger dataset from the company to evaluate the behavior of the shop floor over a longer period. Furthermore, harvesting more data from the simulations could lead to a self-adapting pairwise comparison matrix constructed by an active learning approach. In addition to costs, the profiles of the high gloss and matt loads in the case study should be analyzed. Further, complements should be provided to the model to make it even more realistic. For instance, the consumption speed of the two output queues should be added to the model. To go further, a complementary study of the weight to give to each criterion must be led with the company experts after the application practical feedback. The fact that other workstations also feed them and that different types of reworks with different process times may occur (from simple correction to total reproduction) should also be considered.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Flow chart for poor-quality risk evaluation, inspired from Zimmermann et al. (2018)

Fig. 3 .

 3 Fig. 3. The AHP model structure of the studied case Step 1: Relative importance of the criteria and their indicators

Fig. 4 .

 4 Fig. 4.: Flow chart of different product flows for the lacquering robot in Acta Mobilier[START_REF] Zimmermann | Using analytic hierarchical process for scheduling problem based on smart lots and their quality prediction capability[END_REF]

Fig. 7 .

 7 Fig. 7. Impact of social myopia on delay (tardiness) (in hours)

Fig. 9 .

 9 Fig. 9. Impact of social myopia on computing time (in seconds)

Table 1

 1 In the case of companies delivering weekly customers, switching two batches with the same due date has no significant impact on production schedule and performance. Considering that, let α be the threshold set here to 4 h (for a semi-8-h work shift). Let 𝐷𝐷(𝑘) and 𝐷𝐷(𝑙) be the due dates of batches 𝑘 and 𝑙.

	Table 1. Values of the decision criteria
	Numerical value	Interpretation

lists the comparison scales used in the AHP process. This comparison between two batches i and j is done level by level, starting with the criteria level, then with the alternatives level. We describe the transposition of this comparison scale to the five criteria mentioned below.

Table 2 .

 2 Values of the alternatives regarding criterion Due date Value of 𝑏 𝑘𝑙

	(𝑖)

Table 3 .

 3 Values of the alternatives regarding criterion Balance of product flows

	Value of 𝑏 𝑘𝑙	(𝑖)

Table 4 .

 4 Values of the alternatives regarding criterion Nervousness

	Value of 𝑏 𝑘𝑙	(𝑖)	Interpretation
	1		𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) = 𝑆𝑤𝑖𝑡𝑐ℎ(𝑙)
	3		𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) ∈ ]𝑆𝑤𝑖𝑡𝑐ℎ(𝑙) -𝛿; 𝑆𝑤𝑖𝑡𝑐ℎ(𝑙)]
	5		𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) ∈ ]𝑆𝑤𝑖𝑡𝑐ℎ(𝑙) -2𝛿; 𝑆𝑤𝑖𝑡𝑐ℎ(𝑙) -𝛿]
	7		𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) ∈ ]𝑆𝑤𝑖𝑡𝑐ℎ(𝑙) -3𝛿; 𝑆𝑤𝑖𝑡𝑐ℎ(𝑙) -2𝛿]
	9		𝑆𝑤𝑖𝑡𝑐ℎ(𝑘) ∈ ]0; 𝑆𝑤𝑖𝑡𝑐ℎ(𝑙) -3𝛿]

Table 5 .

 5 Values of the alternatives with respect to criterion Non-quality risk

	Value of 𝑏 𝑘𝑙	(𝑖)

Table 6 .

 6 Values of the alternatives regarding criterion Setup time

	Value of 𝑏 𝑘𝑙	(𝑖)	Interpretation
	1		𝑝𝑟𝑜𝑔(𝑘) = 𝑝𝑟𝑜𝑔(𝑙)
	5		𝑝𝑟𝑜𝑔(𝑘) ≠ 𝑝𝑟𝑜𝑔(𝑙) and 𝑝𝑟𝑜𝑔(𝑘) = 𝑙𝑎𝑠𝑡_𝑝𝑟𝑜𝑔

Table 7 .

 7 Summary of variables of the new model

	Products
	Area Float	[2.5:40]
		1:1high-gloss side
	Program Integer
		2:2high-gloss sides

Table 10 .

 10 Extreme load distribution condition

	Real case	mean Lateness	std	mean Rework	std	mean Operating cost std	mean global cost std
	FIFO	389.8	549.0	5346.5	4175.0	161.5	25.4	5897.8	4449.2
	AHP	175.8	350.5	4025.1	3472.7	149.7	20.5	4350.6	3619.4
	Equal	Lateness		Rework		Operating	cost	global	cost
	distribution	mean	std	mean	std	mean	std	mean	std
	FIFO	0.0	0.0	1791.5	1036.5	89.8	13.9	1881.3	1048.5
	AHP	0.0	0.0	1417.1	876.1	82.4	10.0	1499.5	884.4

Table 11 .

 11 Same due-date condition

Table 11 .

 11 ANOVA test results for reworks

				Myopia 1 3	5	7	10	20
					1			-	0	0	0	0	0
	of reworks			3 5				-	0 -	0 0.1691	0 0.0457	0 0.1755
	Number			7						-	0.4085	0.8351
					10							-	0.6017
	Table 12. ANOVA test results for delay/tardiness
				Myopia 1 3	5	7	10	20
	Cu	mu	lati	ve	del 1	ay/	tar	din -	ess 0	0	0	0	0

Table 13 .

 13 ANOVA test results for time calculation

See the works of[START_REF] Jamal | Optimal manufacturing batch size with rework process at a single-stage production system[END_REF] and[START_REF] Wee | A production model for deteriorating items with stochastic preventive maintenance time and rework process with FIFO rule[END_REF] to define and clarify the meaning of rework.

Acta Mobilier is a French company in the wood industry.

Acknowledgements

Authors would like to thank the company, for the case study and provided dataset. They also thank the reviewers for their contributions, which helped to improve significantly this paper.


Chloé Cherrier

Catherine Potard

Alice Richard

Emmanuel Rusch

Robert Courtois

Chloe Cherrier

Parental attachment and physical intimate partner violence in young adults: Mediational role of dysfunctional attitudes

Keywords: physical intimate partner violence, dysfunctional attitudes, parental attachment style, young adults

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Intimate partner violence (IPV) is a worldwide public health issue that can take different forms: psychological, physical and sexual. Physical intimate partner violence (PIPV), associated with other forms of violence, has serious consequences on mental and physical health. It is defined as "the intentional use of physical force with the potential for causing death, disability, injury, or harm" [START_REF] Saltzman | Intimate partner violence surveillance: Uniform definitions and recommended data elements, version 1.0[END_REF] in a romantic relationship. It can be used by the intimate partner as an inappropriate strategy to keep control of the relationship and/or the partner. PIPV victims are in a subjugated relationship and do not always have the personal resources to get out of it. In France, women are the main PIPV victims, with 154,000 cases identified in 2018 [START_REF] Miprof | La lettre de l'observatoire national des violences faites aux femmes[END_REF]. According to the National Survey of Violence against Women in France (2001), 20% of victims are 20-24 years old. Young adult victims of IPV are exposed to a risk of re-victimization in adulthood (e.g., [START_REF] Smith | A longitudinal perspective on dating violence among adolescent and college-age women[END_REF]. Young adulthood is a time of discovering oneself and others and of identity building, and is thus a crucial developmental period.

Many studies have examined the vulnerability, risk and protective factors of PIPV [START_REF] Stith | Intimate partner physical abuse perpetration and victimization risk factors: A meta-analytic review[END_REF], and one of the most widely recognized links is parental attachment style. Bowlby's theory (1980) is based on the fact that all human beings continually try to develop bonds of attachment with those around them in order to survive. Three styles of parental attachment have traditionally been described: secure, insecure-avoidant and insecure anxious-ambivalent [START_REF] Ainsworth | Patterns of attachment: A psychological study of the strange situation[END_REF]. A secure attachment to parents enables young adults to form positive interpersonal relationships. Young adults with insecure attachment to their parents find it difficult to establish intimate relationships with others and are unable to find the right emotional distance with their partners (too close or too distant). Furthermore, the family environment during childhood determines how individuals construct their representations of the self, of others and of the world around them. These representations are known as internal working models [START_REF] Pietromonaco | The Internal Working Models Concept: What Do We Really Know about the Self in Relation to Others[END_REF]; they may change throughout life, but are predefined during childhood and are linked to the type of parental attachment.

There are some similarities between dysfunctional attitudes (DA) and distorted working models. Beck's cognitive theory (1976) postulates that childhood maladaptive beliefs will be retained in adulthood and will lead to DA when negative or stressful events occur. DA have often been studied as a component of depressive self-schemas and are defined as negative, excessive and inflexible if-then statements concerning the self [START_REF] Clark | Scientific Foundations of Cognitive Theory and Therapy of Depression[END_REF]. [START_REF] Weissman | The Dysfunctional Attitude Scale: A Preliminary Investigation[END_REF] developed the Dysfunctional Attitude Scale (DAS) to evaluate these maladaptive self-schemas. [START_REF] Bouvard | Etude de validation sur une population française de l'échelle d'attitudes dysfonctionnelles de Weissman et Beck (DAS forme A)[END_REF] carried out a factor analysis of the DAS items and obtained four factors which they termed seeking the esteem of others, the capacity to oppose others, independence from others, and seeking approval of others. These types of DA are strongly associated with sociotropy or autonomy personality traits [START_REF] De Graaf | Measuring dysfunctional attitudes in the general population: The dysfunctional attitude scale (form A) revised[END_REF]. Sociotropy is characterized by excessive investment in interpersonal relationships; sociotropic individuals need encouragement and attention from others (cf. the DAS dimensions of seeking the esteem of others and seeking approval by others). By contrast, autonomy is characterized by independence, self-control and achievement (cf. the DAS dimensions of the capacity to oppose others and independence from others). DA related to sociotropy are linked to negative self-beliefs, and those related to autonomy are negative beliefs of others [START_REF] Otani | Relationship of negative and positive core beliefs about the self with dysfunctional attitudes in three aspects of life[END_REF]. Previous studies have also supported the link between DA and parental attachment styles [START_REF] Fuhr | Attachment, dysfunctional attitudes, self-esteem, and association to depressive symptoms in patients with mood disorders[END_REF][START_REF] Roelofs | The mediating role of early maladaptive schemas in the relation between quality of attachment relationships and symptoms of depression in adolescents[END_REF]. Insecure attachment has been found to predict later development of DA in adolescents [START_REF] Zhou | Change in dysfunctional attitudes and attachment in interpersonal psychotherapy for depressed adolescents[END_REF] and young adults [START_REF] Bosmans | Attachment and symptoms of psychopathology: Early maladaptive schemas as a cognitive link[END_REF][START_REF] Hankin | Adult attachment dimensions and specificity of emotional distress symptoms: Prospective investigations of cognitive risk and interpersonal stress generation as mediating mechanisms[END_REF]. Insecure attachment to the mother seems more predictive of DA than insecure attachment to the father [START_REF] Ingram | Individual differences in dysfunctional automatic thinking and parental bonding: Specificity of maternal care[END_REF]. Insecure attachment to the same sex parent may increase DA [START_REF] Ingram | Individual differences in dysfunctional automatic thinking and parental bonding: Specificity of maternal care[END_REF][START_REF] Otani | Parental overprotection increases sociotropy with gender specificity in parents and recipients[END_REF][START_REF] Otani | Affectionless control by the same-sex parents increases dysfunctional attitudes about achievement[END_REF], but there is no consensus on this point. [START_REF] Otani | Parental overprotection increases sociotropy with gender specificity in parents and recipients[END_REF] found that women with insecure attachment to their mother showed greater sociotropy than the others, but in a later study [START_REF] Otani | Affectionless control by the same-sex parents increases dysfunctional attitudes about achievement[END_REF], they found higher levels of autonomy. For men, no significant association between insecure parental attachment and any specific type of DA was found in the earlier study [START_REF] Otani | Parental overprotection increases sociotropy with gender specificity in parents and recipients[END_REF], but in the second study [START_REF] Otani | Affectionless control by the same-sex parents increases dysfunctional attitudes about achievement[END_REF], men with insecure attachment to the father reported higher autonomy. [START_REF] Miljkovitch | L'attachement dans la relation de couple : une continuité de l'enfance[END_REF] showed that parental attachment and internal working models are lifelong and therefore continue throughout adulthood, notably affecting intimate relationships. Adults who are insecure in their relationships retain their childhood working models. However, secure adults tend to adjust their representations to those of their partner. If these "encoded schemas" are negative, they can be the source of various mental health problems [START_REF] Bosmans | Attachment and symptoms of psychopathology: Early maladaptive schemas as a cognitive link[END_REF], such as depression [START_REF] Hankin | Adult attachment dimensions and specificity of emotional distress symptoms: Prospective investigations of cognitive risk and interpersonal stress generation as mediating mechanisms[END_REF][START_REF] Otani | Marked differences in core beliefs about self and others, between sociotropy and autonomy: Personality vulnerabilities in the cognitive model of depression[END_REF]. However, to our knowledge, very few studies have explored the relationships between IPV victimization, especially PIPV, and DA. Kaygusuz (2013) showed that young adults who have negative beliefs about themselves and others are more likely to have nonharmonious and conflicting interpersonal relationships. [START_REF] Dye | Anger, irrational beliefs, and dysfunctional attitudes in violent dating relationships[END_REF] investigated PIPV perpetration in a student population, and its links in particular with DA. No significant difference in DA was observed between the violent and non-violent groups. The authors suggested that future research should investigate whether cognitive distortions are present during affect-inducing partner conflict situations.

The aim of the present study was to examine the mediating role of DA between parental attachment and PIPV in young adults. We hypothesized that insecure attachment styles would lead individuals to inappropriate beliefs about themselves and others, resulting in DA that could be a factor of PIPV vulnerability. More specifically, we explored whether the four dimensions of DA (related to sociotropy or autonomy) are mediation variables in the relationship between attachment to the mother or father and PIPV victimization.

METHODS

Procedures and participants

The current study had a cross-sectional design. A self-administered online questionnaire designed for young adults aged 18-30 years old was distributed through various networks (university, sports clubs, professional, etc.). The inclusion criteria were for participants to have been involved at least once in their life in a romantic relationship and to have been in contact with at least one parent. The sample consisted of 915 participants including 84.8% (n = 776) women, with an average age of 23.59 years ± 3.35, and average level of education of 15.07 years ± 2.70 (corresponding to a bachelor's degree).

Measures

The Revised Conflict Tactics Scale 2 (CTS2; [START_REF] Straus | The Revised Conflict Tactics Scales (CTS2)[END_REF]translated by Lussier, 1998) was used to measure the level of conflict and abuse between partners. We only used the physical assault subscale and the form for victims (39 items). We used a dichotomous scoring method (yes (1): victim of at least one act of PIPV, no (2): never been a victim of PIPV). The Inventory of Parent and Peer Attachment (IPPA; [START_REF] Armsden | The inventory of parent and peer attachment: Individual differences and their relationship to psychological well-being in adolescence[END_REF] translated by [START_REF] Vignoli | Validation of a brief measure of adolescents' parent attachment based on Armsden and Greenberg's three-dimension model[END_REF] measures young adults' attachment to their mother and father. It has 28 items measuring three dimensions of attachment: communication, trust, and alienation. Based on these dimensions, the scale identifies three attachment styles [START_REF] Vivona | Parental attachment styles of late adolescents: Qualities of attachment relationships and consequences for adjustment[END_REF]. In the present study, only secure and insecure styles were studied. Insecure-avoidant and insecure anxious-ambivalent styles were combined in the insecure category. The Dysfunctional Attitude Scale (DAS; [START_REF] Weissman | The Dysfunctional Attitude Scale: A Preliminary Investigation[END_REF], translated by Bouvard et al., 1994) evaluates the level of DA, based on Beck's cognitive theory. We used form A of the scale, which has 29 items investigating 4 factors: DA1 Seeking the esteem of others; DA2 The capacity to oppose others; DA3 Independence from others; DA4 Seeking approval of others. DA1 and DA4 refer to the personality dimension of sociotropy, and DA2 and DA3 to autonomy [START_REF] De Graaf | Measuring dysfunctional attitudes in the general population: The dysfunctional attitude scale (form A) revised[END_REF]. A high score indicates greater cognitive bias.

Statistical analysis

Data were analyzed using SPSS version 25 and the process module developed by [START_REF] Hayes | Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach[END_REF]. Participants were divided into two groups: victims of PIPV and non-victims of PIPV. Independent sample t-tests and chi-square tests were used to compare the characteristics of the participants in the two groups.

A series of logistic (top-down using the likelihood ratio) and linear regressions were conducted, based on the four steps proposed by [START_REF] Baron | The Moderator-Mediator Variable Distinction in Social Psychological Research. Conceptual, Strategic, and Statistical Considerations[END_REF], to examine whether DA mediate the relationship between parental attachment and PIPV. Next, a bootstrap analysis was performed to demonstrate the indirect (mediated) effect of DA. In this analysis, if the 95% confidence interval (CI) for the indirect effect does not include zero, then it can be concluded that the indirect effect is significant at the 5% level. As the dependent variable was dichotomous, the effect size of the mediation could not be analyzed [START_REF] Mackinnon | Mediation Analysis[END_REF]. Models of attachments to the mother and the father were tested separately.

RESULTS

Descriptive and bivariate statistics

Table 1.

Comparison of independent variables between victims of PIPV groups

Data are presented as % or mean ± standard deviation; PIPV: physical intimate partner violence; DA total: Total score of dysfunctional attitudes; DA1: Seeking the esteem of others; DA2: The capacity to oppose others; DA3: Independence from others; DA4: Seeking approval of others.** p < .01, *** p < .001.

Victims of PIPV represented 21.1% (n = 206) of the sample. There was no significant difference between the non-victim group and victims of PIPV group in age, sex or level of education. More victims than non-victims reported an insecure style of attachment to the mother (53.5%, 𝚇 2 = 13.27, p < .001), and to the father (52.4%, 𝚇 2 = 12.35, p < .001).

Victims also had higher DA scores than non-victims, overall (t = -3.20, p < .01) and on the 4 dimensions (DA1: t = -2.70, p < .01; DA2: t = -2.79, p < .01; DA3: t = -3.27, p < .01; DA4: t = -2.76, p < .01) (

Mediation analysis

The mediating models remained significant after adjusting for age, sex and education. The four dimensions of DA were included in a top-down likelihood-ratio logistic regression model, with separate models for attachment to the mother and attachment to the father. Attachment to the mother (Figure 1). The first step, the direct path from attachment to the mother, was a significant association with PIPV (OR = 1.850, 95% CI = 1.311 to 2.611, p < .001). In the second step, there was a significant association between attachment to the mother and DA1 (β = .110, p < .01). In the third step, higher scores on DA1 were associated with PIPV victimization (OR = 1.284, 95% CI = 1.090 to 1.513, p < .01). In the final step, when DA1 was included in the regression model, the association between attachment to the mother and PIPV decreased (OR = 1.759, 95% CI = 1.241 to 2.490, p < .01), indicating a partial mediation effect of DA1. Bootstrap confidence intervals showed that the indirect effect of DA1 (β = .059, 95% CI = .013 to .123) in the association between attachment to the mother and PIPV victimization was significant, indicating partial mediation of the effect of attachment to the mother on PIPV victimization through DA1. More precisely, path analysis showed that an insecure style of attachment to the mother was positively associated with DA1, which in turn increased the risk of being a victim of PIPV. Attachment to the father (Figure 2). The first step, the direct path from attachment to the father was a significant association with PIPV (OR = 1.874, 95% CI = 1.292 to 2.717, p < .01). In the second step, there was a significant association between attachment to the father and DA2 (β = .172, p < .001). In the third step, higher scores on DA2 were associated with PIPV victimization (OR = 1.245, 95% CI = 1.069 to 1.450, p < .01). In the final step, when DA2 was included in the regression model, the association between attachment to the father and PIPV decreased (OR = 1.728, 95% CI = 1.185 to 2.519, p < .01), indicating partial mediation. Bootstrap confidence intervals showed that the indirect effect of DA2 (β = .091, 95% CI = .026 to .173) in the association between attachment to the father and PIPV victimization was significant, indicating partial mediation of the effect of attachment to the father on PIPV victimization through DA2. More precisely, path analysis showed that an insecure style of attachment to the father was positively associated with DA2, which in turn increased the risk of being a victim of PIPV.

DISCUSSION, LIMITATIONS AND IMPLICATIONS

The current study examined the mediating role of DA in the relationship between parental attachment and PIPV victimization. In our sample, 21.1% of the participants had already been victims of at least one act of PV by one of their partners. The prevalence of victims of PIPV varies from 9% to 49% across studies (WHO, 2010). In this study, we were interested to see whether there was a history of PIPV in the life of young adults, in order to determine whether negative beliefs that developed during childhood could be a vulnerability factor for PIPV later in life. The results showed that participants' sex was not associated with PIPV victimization, as both women and men had experienced PIPV, highlighting the fact that physical victimization is not specific to women. These results support those of previous studies [START_REF] Straus | Dominance and symmetry in partner violence by male and female university students in 32 nations[END_REF]. They also show that insecure attachment to the father or mother increased the risk of being a victim of PIPV by 1.9 compared to securely attached participants. Parental attachment is an important vulnerability factor for PIPV victimization [START_REF] Velotti | Exploring relationships: A systematic review on Intimate Partner Violence and attachment[END_REF]. Young adults with insecure parental attachment have a greater propensity to have DA as a whole [START_REF] Hankin | Adult attachment dimensions and specificity of emotional distress symptoms: Prospective investigations of cognitive risk and interpersonal stress generation as mediating mechanisms[END_REF]. When DA was included in the meditational model, only DA1 (sociotropy) -Seeking the esteem of othersfor attachment to the mother and DA2 (autonomy) -The capacity to oppose others -for attachment to the father increased the model's significance. These results are similar to those in the literature, showing that insecure attachment to the mother is related more to sociotropy [START_REF] Otani | Parental overprotection increases sociotropy with gender specificity in parents and recipients[END_REF][START_REF] Otani | Affectionless control by the same-sex parents increases dysfunctional attitudes about achievement[END_REF], and that attachment to the father is related more to autonomy [START_REF] Otani | Affectionless control by the same-sex parents increases dysfunctional attitudes about achievement[END_REF]. Indirect paths of both mediation models were significant using the bootstrap method. Insecure attachment to the mother would lead to more DA1 (sociotropy), with a 1.3 increase in the risk of being a victim of PIPV at least once in a lifetime. Similarly, insecure attachment to the father would lead to more DA2 (autonomy), increasing by 1.2 the risk of being a victim of PIPV at least once in a lifetime. These mediations are partial. These first results in the field of IPV can be compared with results obtained in studies of mental health disorders, such as depression [START_REF] Hankin | Adult attachment dimensions and specificity of emotional distress symptoms: Prospective investigations of cognitive risk and interpersonal stress generation as mediating mechanisms[END_REF][START_REF] Otani | Marked differences in core beliefs about self and others, between sociotropy and autonomy: Personality vulnerabilities in the cognitive model of depression[END_REF].

These findings suggest that approval and dependence patterns formed during young adulthood may be induced by insecure relationships with the mother, which may in turn be related to victimization by an intimate partner. By contrast, greater autonomy, notably the capacity to oppose others, is likely to be induced by insecure relationships with the father, in turn related to physical victimization by an intimate partner. These findings are in accordance with those of [START_REF] Stith | Intimate partner physical abuse perpetration and victimization risk factors: A meta-analytic review[END_REF], who suggested that young adults' personal characteristics could represent vulnerabilities increasing the risk of being a victim of PIPV. Based on the current results, DA could compromise the ability of young adults to have harmonious romantic relationships, because they will tend to be drawn to partners who fit their dysfunctional internal working models, heightening their vulnerability to PIPV experiences.

Limitations. Several limitations should be borne in mind. First, the sample was composed solely of young French adults, which may reduce the generalizability of the findings. Secondly, self-report questionnaires reflect a subjective perception and may be influenced by social desirability, and a multi-method approach would be preferable (e.g., semi-structured interviews). Furthermore, the scoring method used to evaluate PIPV was dichotomous and it was therefore not possible to test the size of the indirect effects. Participants were mainly female, and it is possible that the lack of difference in PIPV experiences between women and men was biased. Considering the relational nature of partner violence in young adults, it also seems important to take into account the potential relationship dynamics involved in the participants' experiences, such as dyadic adjustment. In this study, we focused only on PIPV victims, but it would be interesting to study other forms of violence and the bi-directionality of PIPV among young adults. Finally, the crosssectional nature of the data restricted our ability to test causal relations and to understand the mechanisms behind the path leading from emotional insecurity to IPV. It would be useful to replicate this research in a different cultural context, to examine whether these interactions are culturally specific. Additional research is needed to determine cognitive and emotional vulnerability factors in the risk of PIPV among insecurely attached individuals.

Implications. No published studies have investigated the link between PIPV and DA. Young adults with insecure internal working models will tend to have DA and to turn to partners with the same profile, thus increasing the risk of dysfunctional relationships. The current study provides an exploratory basis for appropriate prevention or intervention programs for young adults. For example, while it does not seem possible to act on the attachment of young adults to their parents, we can try to act on their beliefs and representations in an attempt to deconstruct them. Universal prevention programs to strengthen life skills such as cognitive skills (e.g. critical thinking, problem solving and decision-making) or even emotional skills (e.g. being aware of oneself, of one's history and one's limits), could be moderated and reduce dysfunctional beliefs or attitudes built up since childhood. Thus, this research provides some initial ideas for interventions to prevent IPV, and particularly PIPV, in young adults.

CONCLUSION

Insecure parental attachment and DA can be factors of vulnerability to PIPV. Further analysis of the results shows that dysfunctional sociotropic attitudes appear to partially mediate attachment to the mother and PIPV victimization among young adults, while dysfunctional autonomous attitudes appear to partially mediate attachment to the father and PIPV.

  sociotropy) for attachment to the mother and DA2 (autonomy) for attachment to the father significantly improved the fit of the model.

.

  Figure 1. Adjusted mediation model of dysfunctional attitudes between attachment to mother and PIPV β = standardized coefficient; OR= odds ratio; ** p < .01, *** p < .001. PIPV: physical intimate partner violence; DA1: Seeking the esteem of others.

Figure 2 .

 2 Figure 2. Adjusted mediation model of dysfunctional attitudes between attachment to father and PIPV

Table 1 )

 1 . The young adults with an insecure style of attachment to

		Non-victims of PIPV	Victims of PIPV
	Independent variables	n	M ± SD or %	n	M ± SD or %	𝚇 2	t
	Age	709	23.58 ± 3.38 206 23.63 ± 3.28		-.18
	Sex	709		206		2.19
	Male	101	14.2	38	18.4
	Female	608	85.8	168	85.6
	Education (years)	709	15.17 ± 2.66 206 14.75 ± 2.83		1.95
	Attachment to mother 593		172		13.27***
	Secure	368	62.1	80	46.5
	Insecure	225	37.9	92	53.5
	Attachment to father	531		142		12.35***
	Secure	338	63.7	70	47.6
	Insecure	193	36.3	72	52.4
	DA total	709	3.19 ± 1.00	206	3.43 ± .88		-3.20**
	DA1	709	3.41 ± 1.09	206	3.65 ± 1.01		-2.70**
	DA2	709	2.68 ± 1.20	206	2.94 ± 1.14		-2.79**
	DA3	709	3.12 ± 1.19	206	3.39 ± .99_		-3.27**
	DA4	709	3.07 ± 1.28	206	3.35 ± 1.19		-2.76**

Data are presented as mean ± standard deviation; PIPV: physical intimate partner violence; DA total: Total score of dysfunctional attitudes; DA1: Seeking the esteem of others; DA2: The capacity to oppose others; DA3: Independence from others; DA4: Seeking approval of others. * p < .05.

AUTHOR(S) INFORMATION

Full name: Chloe Cherrier Institutional affiliation: EE 1901 QualiPsy, University of Tours, France Institutional address: Faculté Arts et Sciences Humaines, 3 rue des Tanneurs, BP 4103 -337041 Tours cedex 1 Short biographical sketch: Chloe Cherrier is a PhD student in psychology in the Quality of Life and Psychological Health unit (QualiPsy). Her research interest focuses mainly on IPV among young adults and particularly the links with parental attachment and cognitive skills. She also works in the community FRAPS Centre-Val de Loire (Fédération Régional des Acteurs en Promotion de la Santé / regional federation of health promotion Actors) and conduct educational programs on emotional and sexual life with young people. Tours cedex 1 Short biographical sketch: Robert Courtois is an associate professor / senior lecturer in psychopathology and clinical psychology. He is also a part-time hospital psychiatrist (Psychiatric University Clinic, University Hospital of Tours, France). He manages the CRIAVS Centre-Val de Loire (Centre de ressources pour les intervenants auprès des auteurs de violences sexuelles / regional resource center for people working with sexual offenders).

Full name: Catherine Potard

Institutional affiliation: EA 4638 Laboratoire de Psychologie des Pays de la Loire (regional Psychology Research Laboratory), University of Angers Institutional address: Université d'Angers, Maison de la Recherche Germaine Tillion, 5 bis, boulevard Lavoisier, 49045 Angers Cedex 1, France Short biographical sketch: Catherine Potard is currently Associate Professor of Psychology at the University of Angers, principal member of the regional psychology research laboratory (EA 4638) and associate member of the Emerging Team QualiPsy (EE 1901). She is also a psychologist in the Resource Centre for people working with perpetrators of sexual violence -CRIAVS -Centre-Val de Loire (University Hospital of Tours, France). Her research topics focus on the study of psychological risk, vulnerability, and protective factors in the development and maintenance of risk behaviors and interpersonal violence during developmental transition periods such as adolescence and early adulthood.