Migrations and HIV transmission as experienced by people living with HIV in Lao PDR

Keywords: HIV, vulnerability, PLWHIV, migration, Lao PDR, Thailand

The aim of this paper is to assess the interactions between migration paths and HIV infection risks among and from the views of people living with HIV in Lao PDR.

Twenty-four in-depth interviews with eight interviewees were carried out by four trained members of the Association of People Living with HIV (APL+), in 2017. The qualitative analysis was done manually to explore the influence of social determinants through migratory paths on the evolution of individual lifestyles, working and living environments, access to information, prevention and healthcare that contribute to HIV infection risks.

Low levels of economic and education lead to migration and also increase risky practices and vulnerability to HIV infection of migrant workers. Insufficient awareness and access to prevention against HIV/STIs in their home country increase risky behaviours during migration. Moreover, inadequate referral system for HIV treatment, stigma, and family relationships contribute to influence the life courses of migrants' post-HIV infection.

Raising awareness and providing access to information on HIV/STIs, both for internal and external migrant workers are crucial in order to diminish the prevalence of HIV infection among the target audience. A reinforcement of the existing collaborative health measures between Lao PDR and Thailand is needed to enhance migrants' access to healthcare services.

Introduction

Lao PDR has a low prevalence of HIV infection with 0.3% among the population aged 15-49 years, although, in the absence of any intervention plan, projections show an increasing rate in 2020 (ASEAN & Secretariat, 2014;UNAIDS, 2016). The majority of reported HIV cases (86%) were from border provinces (Directory of LAO CSO 2014, 2014). HIV prevalence rates in Laos bordering countries are estimated as follows: Cambodia (0.5%), Vietnam (0.3%) and Myanmar (0.7%) [START_REF] Who | GHO | By category | Prevalence of HIV among adults aged 15 to 49 -Estimates by country[END_REF]. Thailand, with a rate of 1.4% has the highest prevalence in the area (HIV / rapid assessment, 2008). Leaving one's territory is not a new phenomenon in Lao PDR. Since the opening of the country to the market economy from 1990 onwards, migratory phenomena have taken place due to the government's relocation policies, the economic attractiveness of certain regions (road construction, hydroelectric and mining projects, agri-food industry) [START_REF] Phetchanpheng | Interrogating dislocated masculinities and HIV vulnerabilities a case study of Vietnamese migrant workers in Southern Laos[END_REF].

There is scientific evidence that migration often leads workers to harsh living conditions, unstable financial situations as well as low-paying jobs, insecurity, etc. [START_REF] Alemayehu | Perceived quality of life among Visceral Leishmaniasis and HIV co-infected migrant male workers in Northwest Ethiopia: a qualitative study[END_REF][START_REF] El-Bassel | The Silk Road Health Project: How Mobility and Migration Status Influence HIV Risks among Male Migrant Workers in Central Asia[END_REF]. Most migrants experience poor health during their migration as a consequence of their limited access to the healthcare system in their destination country [START_REF] El-Bassel | The Silk Road Health Project: How Mobility and Migration Status Influence HIV Risks among Male Migrant Workers in Central Asia[END_REF][START_REF] Rai | Migration as a risk and a livelihood strategy: HIV across the life course of migrant families in India[END_REF][START_REF] Saggurti | Migration, Mobility and Sexual Risk Behavior in Mumbai, India: Mobile Men with Non-Residential Wife Show Increased Risk[END_REF]UNDP, 2015;[START_REF] Webber | Facilitators and barriers to accessing reproductive healthcare for migrant beer promoters in Cambodia, Laos, Thailand and Vietnam: A mixed methods study[END_REF].

Lao PDR is both a migrant worker source and destination country since approximately 11,000 Vietnamese, 300,000 Chinese and 2,000 Thai migrant workers were reported in the country in 2013 (HIV / rapid assessment, 2008;ILO ASEAN TRIANGLE project, 2015). However, many Lao migrant workers are seeking job opportunities, mainly in Thailand and also in other countries [START_REF] Moh Lao | National HIV and AIDS Strategy and Action Plan 2016-2020[END_REF]. In 2013, 926,000 Lao migrants went to Thailand (ILO ASEAN TRIANGLE project, 2015).

The Centre for HIV/AIDS/STIs (CHAS), working under the Lao Ministry of Health in partnership with donors and international organisations implemented various activities to prevent and control HIV/AIDS. However, access to HIV testing, prevention, and care services remain limited, especially in rural areas (UNAIDS, 2016). In addition, Civil Society Organizations (CSOs) play a major role in fighting HIV/AIDS and supporting directly PLWHIV (People Living With HIV) to improve their health status and to help them reach a better quality of life.

The influence of migration on HIV infection is well documented. However, in Laos, little is known about the intimate experiences of migrant people that make them vulnerable to HIV infection. From 2016 to 2018, we conducted the project entitled "Migration, mobility, and HIV/STI vulnerabilities. An Interdisciplinary and Community Based Participatory Research in Lao PDR". Our epidemiological research of 664 people living with HIV from all antiretroviral (ARV) sites emphasises that around 32% of these are migrant workers and 44% were infected in Thailand. For this article, we mobilise especially the data collected within the sub-study that aims to assess the interactions between migration and HIV infection among, and from the view of people living with HIV. The specific objectives of the study were: (1) to explore the difficulty of living and working during and after migration and its relation to HIV vulnerability; and (2) to investigate the change of one's lifestyle through migratory paths and its impact on HIV infection.

Methodology

The results presented here are the findings from ethnographic interviews conducted as part of the project "Migration, Mobilities and HIV/STI Vulnerabilities, an Interdisciplinary and Community Based Participatory Research in Lao PDR". This research aimed to analyse how health with social and migratory paths are interlaced for migrants in Lao PDR. The results presented here are the findings of a sub-study led by representatives of VVIH associations of people in Laos and trained by the IRD in the practice of research. It explores the effects of HIV infection on the life course and examines how and to what extent HIV positive status leads to migration.

Information collected included socio-demographic characteristics, conditions of migration and life in Lao PDR, relational, sexual, and reproductive history, and health-care pathways. It also included the migrants' living conditions, their economic resources (from their work, from a spouse or other family member, state allowance, none) and housing situation (had their own housing, housed by family, associations or hotel's workers, no stable housing). Four sexual partnership types were investigated:

• stable, / A stable partnership was defined as a partnership that lasted at least a year • casual, casual partnership as one lasting less than a year or reported as short transactional partnerships / as one that was accepted, but unwanted, to obtain housing, clothes, food, papers, or money.

• Paid sex defined as reporting paying someone to have sex

Interviewers' selection process

Four representatives of the CSOs Lao Pha and LNP Plus living with HIV have been involved in the research process as interviewers. They originated from four provinces: Savannakhet, Champasack, Huaphanh, and Luang Prabang. A two-day training session was organised by the research team in Huaphanh and Savannakhet Provinces to train and select the interviewers out of a group of twelve participants.

Data Collection

Within each site selected for the study, interviewers identified people to participate in the study.

The selection criteria for the interviewees were defined as PLWHIV aged above 20 years, living with HIV, taking ARV medicine for more than 5 years, and having had a migration experience. Each interviewer had to choose 2 people from its own network, meeting the selection criteria.

Participants were subsidised 50.000 LAK per interview (150,000 in total). They gave oral information on the study. When the person agreed to participate in the study, the interviewer obtained his oral consent. Interviews were conducted in face-to-face mode and mostly at home of the interviewee in order to ensuring privacy. Each interviewer involved in the data collection followed a common interview guide during indepth and open-ended interview. Each interviewer conducted 6 interviews (3 interviews with 2 informants) in Lao language.

The question guide contained open-ended questions to facilitate participants' free expression of their lives and experiences as migrant workers including their partners before and after HIV infection. Participants were asked to recall their experiences chronologically, with references to main events or moments of their life such as marriages, sexual relationships, reproductive events, household changes, professional experiences. They also had to relate their illness experiences, their experiences of migrant life, and trips home. In the end, various questions were also raised about their experiences with HIV healthcare services and treatment.

Analysis Approach

All recorded interviews were transcribed and translated into English. Data analysis was done by the first author. Both Lao and English transcripts were reviewed and compared to avoid any missing point and misunderstanding in the translation. Coding was done manually with several readings to validate the analysis.

The analysis framework was inspired by social determinants of health approaches and literature reviews on migration and HIV [START_REF] Burris | Stigma, ethics, and policy: A commentary on Bayer's "Stigma and the ethics of public health: Not can we but should we[END_REF][START_REF] Lyttleton | Intimate economies of development: mobility, sexuality and health in Asia[END_REF][START_REF] Rai | Migration as a risk and a livelihood strategy: HIV across the life course of migrant families in India[END_REF][START_REF] Saggurti | Migration, Mobility and Sexual Risk Behavior in Mumbai, India: Mobile Men with Non-Residential Wife Show Increased Risk[END_REF]. As shown in Figure 1, it explored influencing social determinants contributing to HIV infection risk as migratory paths, changing of individual lifestyles, working and living environments, access to information as well as prevention and healthcare.

Figure 1: The Detailed Analysis Framework for migration and the risk of HIV infection [START_REF] Burris | Stigma, ethics, and policy: A commentary on Bayer's "Stigma and the ethics of public health: Not can we but should we[END_REF][START_REF] Lyttleton | Intimate economies of development: mobility, sexuality and health in Asia[END_REF][START_REF] Rai | Migration as a risk and a livelihood strategy: HIV across the life course of migrant families in India[END_REF][START_REF] Saggurti | Migration, Mobility and Sexual Risk Behavior in Mumbai, India: Mobile Men with Non-Residential Wife Show Increased Risk[END_REF])

Findings

The scientific investigation highlighted the complex set of interactions between HIV and a person's life trajectory through the key mechanisms of migration. The results will be presented by introducing each of the social determinants that were identified as the factors of migrant workers' HIV vulnerability, as well as their interaction with the evolution of migratory paths "before", "during" and "after" migration. In other terms, the data contribute to giving insights on the life of the interviewees before they initiate the process of migration "Before Migration", to explore how and to which extent they experimented their new life as a migrant worker "During Migration" and to document what happened to them at the time they went back and settled again in their village "After Migration".

Socio-Demographics

The samples comprise eight interviewees, 3 men and 5 women, aged between 23 and 56 years. More than half were from low-income families and had low levels of education (below secondary school). Three people were based in the south of Lao PDR and Vientiane Capital. Five interviewees were initially from and based in Huaphanh and Luang Prabang Provinces. They had moved from their hometowns to Vientiane Capital for work and/or study.

Individual Lifestyle

The social structures of working conditions and living status shape an individual's exposure to risk.

Our interviewees felt that lifestyles in Lao PDR and in Thailand were clearly different, according to their social status.

Before Migration

Some interviewees already had high-risk sexual practices before migrating to Thailand or Vientiane Capital. These include unstable partners, multiple partners, and men looking for sexual services without protection. They explained that they were too young to have sustainable relationships.

"… After I received my salary, I used to go to the sex worker shop and the money was spent in only 2 days. I slept with a sex worker twice without condoms…" (HP 1 (M, 31 yrs.))

During migration

Our informants reported many HIV/STI risky practices during migration including working in the sex trade, having girlfriends who were sex workers, paying for sex, multiple partners, unprotected sex with sex workers, drug use and needle sharing (for instance for tattoos).

Moreover, we also found that condom use was very low during this period. Many participants chose intentionally and/or unintentionally not to use a condom while having sexual intercourse.

Unintentionally: "… I slept with unknown people many times. When drunk, I often forgot to use a condom…" (CPS 4 (F, 27 yrs.)).

Intentionally: "…I asked the girl not to put the condom on and I felt better. From that time, I would rather not use a condom …" (CPS 3 (M, 36 yrs.)).

The data also reveals that some migrants were addicted to non-injectable drugs (such as amphetamines). Many of them shared needles for tattoo making with their friends as it was cheaper than going to a real tattoo shop.

"…. When there were no customers, we used drugs (amphetamine) with girls in the shop. At that time, we followed the trend of tattooing our hand or waist and used the same injection needle…"(CPS 4 (F, 27 yrs.)).

After Migration

Depending on their health status upon their return, some people ceased any practice related to highrisk behaviour. Others came home as healthy individuals or at least without symptoms and pursued their life with high-risk behaviours.

"… I worked here, I smoked, I took ya baa (amphetamine) with my flatmates. But I did not buy it in Thailand because it was too expensive…" (SVK 1 (F, 33 yrs.))

Working environment

The majority of interviewees had only primary school levels of education which makes it difficult for them to find good jobs in the Lao PDR, and in another country.

Before Migration

Many interviewees were farmers or had no jobs before moving to Thailand or to Vientiane Capital. Some of them went to work in other provinces in garment factories and/or construction companies.

« … After farming, I went back to Thailand in 1987 because there was no work, and I did not know any jobs in the Lao PDR… " (VTE 2 (M,48 yrs)

During Migration

Working conditions had changed since there is a larger variety of jobs with no specific requirements available in Thailand. Seven out of eight interviewees were in Thailand as illegal migrant workers. They applied for jobs they considered well paid so they could have enough financial support for both their living expenses there and to secure their homecoming. Moreover, as those migrants usually require the help of a smuggler to cross the borders illegally, a large amount of their income was used to pay it back.

"… During the first month, my salary was 2,000 baht but the Thai smuggler took all of my money for the meal fee during the journey and transportation costs. The Thai and Lao smugglers split my money…" (CPS 3 (M, 36 yrs))

After Migration

Upon returning from Thailand, interviewees' health status became a cause of discrimination in the working environment. It also prevented them from finding a new job.

Living Environment and Stigma and Discrimination

PLWHIV usually experience stigma and discrimination. Following UNAIDS, we will define stigma as the feeling of being treated differently, usually in a pejorative way from the rest of one's community (UNAIDS, 2011).

Before Migration

Two people were infected before their migration; thus, the stigmatisation experience was identified during this period. The stigma was felt at both the healthcare settings and the workplace; this obviously affected adherence to their ARV treatment.

"… I convinced him my partner to get treatment, but he did not go because he was very shy and afraid that he would lose face and be discriminated against by people especially since he was working at a public institution …" (VTE 1 (F, 40 yrs.)).

During Migration

One migrant living with HIV lost her job because many people in the factory knew about her HIV status, including her boss. She was fired because they were afraid, she would infect others.

"…he said that he could no longer accept me because the staff was afraid of this…" (CPS 1 (F, 48 yrs.)).

After Migration

Most participants decided to disclose their HIV status to their friends, colleagues, and family members. However, community stigma and discrimination had a strong effect over their lives. Some of them even attempted suicide, others moved to another province in order to hide their status or escape the gaze of their neighbours and friends.

"…there was gossip by others when they saw me, and it made me feel as if they didn't want me to live anymore. I decided to buy paracetamol. I thought of committing suicide, I had the entire bottle, around 40 or 50 tablets …." (VTE 2 (M, 48 yrs.)).

Access to Prevention, Information, and Healthcare

Working and living conditions could either ease or limit our informant access to healthcare. The ARV medicine was not given to them, due to their illegal status as a migrant worker.

Before migration

Two PLWHIV members of the study had an HIV-positive status and received the ARV treatment before migrating to Thailand.

"…in 2005 the doctor admitted me to take antiviral medicine (ARV) medicine… " (CPS 1 (F, 48 yrs.))

During migration

Opportunities for accessing prevention, information, and healthcare were limited. Because of their illegal migration status, accessing preventive healthcare information in the host country was difficult. One of the participants' applications for receiving ARV treatment was rejected in a Thai public hospital. As she had no passport, she couldn't get the ARV under the agreement between Thailand and Laos on ARV access for legal migrant workers.

"…If you would like to get the treatment here, you cannot if you do not have your passport or ID card… " (CPS 4 (F,27 yrs.) This woman then came back to Lao PDR to request a passport, then went back to Thailand where she could both pursue her job activities and get ARV treatment.

Conversely, one migrant worker in the Lao PDR who moved from their hometown to Vientiane Capital had better opportunities to receive HIV awareness and regular HIV testing at a healthcare facility.

"… While I was selling sex in the shop, I was going to get the blood test very often. The project's team was coming to educate us every 3 months… » (HP 3 (F, 23 yrs.))

After Migration

Five interviewees could not work anymore and had no access to treatment so they came back home being HIV positive and ill. However, here they also faced some difficulties in accessing healthcare services, such as long distances from their home to the hospital, reliance on traditional medicine, self-treatment, or even shyness and sometimes shame to receive the ARV treatment.

"… When I was in Savannakhet, it was very difficult because of the accommodation, sometimes I did not go but I needed to go, sometimes I stayed overnight in a relative's house and sometimes overnight in a hotel… » (CPS 1 (F, 48 yrs.))

Discussion on the Method and Limitations

The main strength of our study is the methodology that is still rarely found in Asian Pacific countries while it is found in the Republic of Congo and Mali [START_REF] Cissé | Factors Associated With Hiv Voluntary Disclosure To One's Steady Sexual Partner In Mali: Results From A Community-Based Study[END_REF][START_REF] Kieto | Factors associated with HIV voluntary disclosure of people living with HIV to their steady sexual partner in the Democratic Republic of the Congo: results from a community-based participatory research[END_REF]. Such methodology is very relevant for conducting qualitative studies on sensitive issues, where research objectives are not always well understood and where informants may feel uneasy talking and sharing biographical data. In Lao PDR, research remains tightly controlled and subject to authorisation from various administrative levels. Sometimes people are contacted upstream and presented to the researcher during his field visit as an observer of the study. Such unexpected changes conducted us to various adaptations in the way we conducted the research. This made it difficult, therefore, to create the bonds of trust and closeness indispensable to the practice of ethnography, the foundation of our discipline. Its principles combine a long immersion in the field, a consideration of the context of enunciation of the collected discourses and a constant positioning of the anthropologist in his field. However, involving pear researchers to collect data among their own social networks still allowed to collect intimate perspectives and sensitive data. This method is the most relevant to explore a new, or little understood, topic, to discover new information and expand existing understanding and to get the perspective of one individual without interference from others.

From our experiences, we observed that there are both positive and negative impacts of having non-researchers collecting the data: on one hand, the interview bias may have been minimised at some point as the interviewers would not lead the discussion to the expected scientific result. On the other hand, even if the questions and probes were framed in the interview guide, the testimonies given were sometimes too broad and went beyond the study's objectives. The option of having the interviewees known to the interviewer may have either limited or increased the amount of information that the interviewee shared. The limited sample, the impossibility to conduct field observations, and the limited experience of the interviewers are the other limitations.

Conclusions and Recommendations

These data relate the living conditions in rural Laos and the poverty that leads migration to Thailand for work. Interviews show the social and domestic violence, the practices of selling sexual services and the use of illicit products. It reveals the links between migrations and various social forms of vulnerability among migrants. It also shed light on the cycle of migration with multiple departures and returns to the home country. These multiple processes can expose migrants to more risks of HIV contamination.

First, the data shows how low economic family status, low-paid jobs and lack of employment in the country-of-origin cause migration. Poverty forces individuals to seek higher wages and a better quality of life, through internal or external migration. This issue has also been documented recently in a study examining the main reasons for women to migrate to Thailand (Kanchana [START_REF] Thilakoun | Life trajectories of migrant women living with HIV in Champasak Province, Lao PDR (Master Thesis)[END_REF]. Women with a low level of education and early marriages are more vulnerable to becoming sex workers. Safe economic opportunities for women with low education are rare and low-paid jobs and lack of employment opportunities in Laos cause migration to bordering countries such as Thailand. Supporting alternative work opportunities for women in Laos and align work pays across the region would be an adequate response to unsafe migrations.

Secondly, high-risk behaviours and unsafe sex practices were frequent during migration. We found that most of the interviewees who went to Thailand migrated at a very young age (<18 years old). They were mostly not well prepared to live in a new environment and to limit inherent risks. Discovering new experiences, they were more likely to have high-risk practices such as noninjectable drug use, sharing needles for tattoos, unsafe sex in a commercial sex context, etc. In addition, the migrants' low level of education matched with limited awareness of HIV/STI transmission and prevention. Also, being far away from their families, our informants experimented freedom and various opportunities to engage with sex workers and multiple partners. This observation tends to confirm the work of Chris Lyttleton who found that there is a significant association between travelling and the HIV risk among men who meet these conditions. The latter are more likely to engage in multiple and unstable relationships and intercourse as the distance from their families gives them the opportunity to do so [START_REF] Lyttleton | Any Port in A Storm: Coming to Terms with HIV in Lao PDR[END_REF]. Recently we also observed a similar pattern among rural young female migrants who supplement their income by working as beer promoters in Vientiane [START_REF] Sychareun | Lifestyles, Sexuality and Gender: Vulnerability to STIs and Unplanned Pregnancy among Female Migrant Beer Promoters in Lao PDR[END_REF].

Since there are increased sexual risks during migration, the lack of HIV and STI awareness for migrant workers makes it even worse although it could be addressed by an intervention plan.

Finally, we see how illegal migrant worker status limits access to Thai healthcare services. In Thailand, a health insurance scheme exists but it requires specific documents such as a residence permit and/or a labour contract from employers, which only a few can provide [START_REF] Sychareun | Informal workers and access to healthcare: a qualitative study of facilitators and barriers to accessing healthcare for beer promoters in the Lao People's Democratic Republic[END_REF][START_REF] Webber | Facilitators and barriers to accessing reproductive healthcare for migrant beer promoters in Cambodia, Laos, Thailand and Vietnam: A mixed methods study[END_REF]. A study in Champasack province found that legal migrants can access the so-called 30 BATH healthcare insurance system, but this does not cover HIV treatment [START_REF] Thilakoun | Life trajectories of migrant women living with HIV in Champasak Province, Lao PDR (Master Thesis)[END_REF]. Lao PDR and Thailand developed a referral system in 2016 called NAPHA (National Access to ARV Program for PLWHIV and AIDS) for both documented and undocumented migrant workers to provide them with better access to care and treatment (UNAIDS, 2016). However, the system does not always appear efficient. We observed that many of our informants, who couldn't provide required documentation had to come back to Lao PDR to continue their ARV treatment. Despite agreement between neighbouring countries related to ARV treatment access, migrant workers face difficulties to access the HIV prevention and treatment in their hosting country.

Being illegal migrant can sometimes appear to be more convenient but working conditions as illegal migrant workers in Thailand prevent them from accessing prevention, information and healthcare systems which seem like a possible improvement to the health system. Once contaminated by HIV, stigma and discrimination were observed both during and after migration. People hesitated to get treatment due to stigma and discrimination in the workplace and in their own network. A referral system between the two countries would allow a continuity of treatment during the migration cycle by reducing the stress to start new treatment procedures in each country and also help prevent more infections.

Acknowledgments

This study was comducted as part of the "Migration, Mobilities and HIV/STI Vulnerabilities" research project. Funding for the study was provided by Expertise France/French Red Cross (Grant number 13INI207), PI. Pascale Hancart Petitet We thank the study participants as well as the Lao and French Red Cross for their coordination and administrative support for running the project as well as this study.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

PHP was the principal investigator of the research study, ZP analysed the data and wrote the article with the support of PHP. PHP, XX and SP conducted training for the interviewers. SP monitored the data collection. All authors read and approved the final manuscript.

Author

Compliance with Ethical Standards

The protocol of the MigrLao study received ethical clearance from the National Ethical Committee of the Lao Ministry of Health on the 9th of January 2017. The research protocol for the study on socialisation processes was submitted and validated by the University of Rennes 2 and research authorisations were given by the Lao Front for National Construction in Vientiane in January 2005. All participants gave oral consent prior to interviews.


est destinée au dépôt

I. INTRODUCTION

In the European Union, road freight transport must comply with several legal requirements including speed limits and access restrictions for example; the optimization framework could precompute and easily include these requirements because they focus only on the distance matrix [START_REF] Goel | Legal aspects in road transport optimization in Europe[END_REF]. Considering the European drivers rules in the Truck Drivers Scheduling is more complex. Different approaches have been used to include respite periods within the vehicle scheduling, considering different regulation rules [START_REF] Archetti | The trip scheduling problem[END_REF]- [START_REF] Goel | Truck driver scheduling in Australia[END_REF]. Several linear formulations for the Truck Driver Scheduling Problem under different regulations have been proposed [START_REF] Goel | A mixed integer programming formulation and effective cuts for minimising schedule durations of Australian truck drivers[END_REF]- [START_REF] Goel | The minimum duration truck driver scheduling problem[END_REF]. Linear formulations of such problems are useful, to evaluate sequences of visits but also as a part a of solution approach designed for routing problems [START_REF] Kok | Optimizing departure times in vehicle routes[END_REF]- [START_REF] Koç | Long-haul vehicle routing and scheduling with idling options[END_REF].

Short respite periods are named breaks and long periods rests (more than 9 hours). A shift is the period starting at the first activity after a rest and finishing at the beginning of the next rest. The working time includes driving periods and service activities at the customer location or the depot (loading/unloading, cleaning the vehicle, etc). Other periods are called periods of availability (POA) or waiting time, meaning that the driver is neither working nor taking a break.

In this paper, the following set of constraints from the EU regulation are considered: R1. A driver cannot drive a total amount of 4.5h without a break of more than 45min referred to as "full break". R2. A full break of 45min can be split into two periods, a first break of at least 15min (and less than 45 min) and a second break of at least of 30min; the breaks should be taken in this order.

R3. If the total working time of a shift is less than 6h, a break of at least 15 minutes has to be scheduled. R4. If the total working time of a shift is between 6h and 9h, a break of 30 min has to be scheduled. R5. If the total working time during a shift is greater than 9h, a break of 45 min is required. R6. The maximal driving time in a shift is 9h that could be extended to 10h twice per week. R7. A shift cannot exceed 24 hours. R8. The minimal rest period is 11h, that could be reduced to 9h three times per week. Therefore, the sum of breaks, POA and working durations is bounded by 13h; 15h if the rest is reduced to 9h. R9. A rest can be taken into two periods, the first an uninterrupted period of at least 3h, and the second a rest of at least 9h. R10. If the shift includes night working time, then the total working time in the shift is limited to 10h.

To the best of our knowledge, different rules have not been previously included in linear formulations, such as: split rests, working breaks and night work [START_REF] Goel | A mixed integer programming formulation and effective cuts for minimising schedule durations of Australian truck drivers[END_REF]- [START_REF] Goel | The minimum duration truck driver scheduling problem[END_REF]. In addition, a previous work have proposed a different assumption on the night work rule, but the night rule of the EU regulation [START_REF] Goel | Legal aspects in road transport optimization in Europe[END_REF]. The paper is organized as follows. A literature review on the Truck Driver Scheuling Problem (TDSP) is provided in Section II. Section III presents the assumptions and the linear formulation. A series of experiments on the MILP are developed in Section III. Finally, in section IV conclusions are given.

II. LITERATURE REVIEW

Different approaches have been used to include breaks and rest periods within the vehicle scheduling, considering different regulation rules. Some of them are interested in verify the feasibility of a given schedule. Archetti and Savelsbergh [START_REF] Archetti | The trip scheduling problem[END_REF] proposed and algorithm with a complexity of 𝑂(𝑛 3 ) to check feasibility of a driver schedule. The sequence of pick-up and delivery request is done considering time windows at the pick-up locations and the United States Hours of Service. Later, Goel and Kok [START_REF] Goel | Truck driver scheduling in the United States[END_REF] under the same regulation rules, developed a method which guarantees to find feasible driver schedules if they exists. This method solves the problem in 𝑂(𝑛 2 ) in the case of single time windows. In the event of multiple time windows, the same complexity could be attained if the gap between subsequent time windows of the same location is at least 10 hours.

Garaix. T. is with Mines Saint-Etienne, Univ Clermont Auvergne, CNRS, UMR 6158 LIMOS, Centre CIS, Departement I4S, F -42023 Saint-Etienne, France (e-mail: garaix@ emse.fr).

A mixed integer programming formulation for the truck drivers scheduling problem considering the European Union drivers rules

Iván Peña-Arenas, Thierry Garaix, Philippe Lacomme, and Nikolay Tchernev Goel [START_REF] Goel | Truck driver scheduling in the European Union[END_REF] presents a breadth-first search (BFS) algorithm to find feasible truck driver schedules if they exists under the working hours regulation for the EU. The experiments use sequences of customers varying from 3 to 12, and the results show that previous methods fail to find feasible solutions half of the time when the size of the instances is large. Another result, is that the possibility of taking breaks in two parts does not bring a big difference for long distance hauling. Considering the Australian standard rules and basic fatigue management, Goel et al [START_REF] Goel | Truck driver scheduling in Australia[END_REF] presented an exact algorithm to solve the truck driver-scheduling problem. In addition, they presented four heuristics, based on removing some of the most computationally expensive steps of the exact method. As a result, the computational time diminishes, albeit, they did not guaranteed feasibility in all cases.

Among linear model formulations of the TDSP, Goel [START_REF] Goel | A mixed integer programming formulation and effective cuts for minimising schedule durations of Australian truck drivers[END_REF] developed a linear model complying with the Australian Heavy Vehicle Fatigue Law, in which the objective function is to minimize the total duration of the schedule. Drivers may only take rest periods before the service at a given location, albeit, they include parking lots or dummy locations with zero working time, to take rest periods after completing the service.

Taking under consideration the Canadian Hours of Service, Goel et al [START_REF] Goel | Truck driver scheduling in Canada[END_REF] find the set of feasible schedules for the Canadian Truck Driver Scheduling Problem (CAN-TDSP) using two heuristics and one enumerative method, the three of them based on a dynamic programming approach. The two heuristic approaches presents a good performance in terms of computational time, while, the enumerative method is not competitive under these criteria. Later, Goel [START_REF] Goel | The Canadian minimum duration truck driver scheduling problem[END_REF] presents a mixed integer programming formulation for the Canadian minimum duration truck driver-scheduling problem (CAN-MDTDSP). The model is solved using a commercial solver CPLEX 12 and compare its results, against an iterative version of the dynamic programming algorithm presented in Goel et al [START_REF] Goel | Truck driver scheduling in Canada[END_REF]. Even though, CPLEX 12 offers better solutions while solving the mixed integer program, the iterative dynamic programming approach requires significantly smaller computational time.

Previous linear formulations considering the EU regulation have been proposed. The minimum duration truck driver-scheduling problem (MD-TDSP) is a generic model adapted to consider both, the EU and the US Hours of Service regulations proposed by Goel [START_REF] Goel | The minimum duration truck driver scheduling problem[END_REF]. Kok et al [START_REF] Kok | Optimizing departure times in vehicle routes[END_REF] present the Vehicle Departure Optimization (VDO) with time-dependent travel times, which is a post-processing step of the VRPTW used as a means to verify the feasibility of a route. Both formulations, assume that breaks and rest periods are taken only at customers or suitable locations, which is not the general case.

Most recent dynamic programming models to generate compliant schedules under EU regulation are proposed by [START_REF] Goel | Legal aspects in road transport optimization in Europe[END_REF] and [START_REF] Tilk | Bidirectional labeling for solving vehicle routing and truck driver scheduling problems[END_REF]. Even though, they consider more rules within their models, they propose a simplified version of the night working rule, assuming that a rest is taken every night, and do not offer optimal solutions for the TDSP.

To sum up, all MILP formulations work under the assumption of taking breaks at customers or suitable locations, meaning that they work under non preemptive assumptions, and do not consider different rules from the EU regulation, particularly the night rule. On the other hand, other algorithms provide feasible schedules considering simplifications of the night working rule or even not considering it at all.

III. MIXED INTEGER LINEAR PROBLEM

A. Data and variables

The initial set of data gives the sequence of locations to visit with the traveling times and the working times at each location. Time window constraints are imposed on the starting time of services at customer locations.

Each activity is modeled through a set of nodes. At each node a processing time and a respite period can be assigned in this order. The number of nodes is bounded according to the type and the duration of the activity, see table I. Figure 1 gives an example of a sequence to evaluate, which has 3 customers and includes the depot at both, first and last position. The set of activities 𝐴 corresponds to the service and transport activities developed at each location. As well, a fixed number of nodes assigned to each activity composes the set of nodes 𝑈.

In order to retrieve the activity 𝑎 related with a node 𝑢, also the first and the last node 𝑢 related with an activity 𝑎, three vectors are defined 𝑎(𝑢), 𝑓𝑢(𝑎) and 𝑙𝑢(𝑎), respectively. All the parameters of the model are listed and the linear constraints are given just after. Note that all durations are in hours. (3)

Sets

Only one type of break could take place at each node is enforced by constraints [START_REF] Goel | Truck driver scheduling in the European Union[END_REF].

∑ 𝑦 𝑢 𝑏 |𝐵| 𝑏=1 ≤ 1 𝑎 = 1, . . , |𝐴| (4)
Constraints ( 5) ensure that service starts in the given time window.

𝑒 𝑎 ≤ 𝑥 𝑓𝑢(𝑎) ≤ 𝑙 𝑎 𝑎 = 1, . . , |𝐴| ; 𝛿 𝑛(𝑎) = 0

(5)

Constraints ( 6) guarantee a full break after 𝐶𝐷𝑇 hours of continuous driving time. Figure 2 depicts this situation. Constraints ( 7)-( 9) enforce full breaks, when the break is greater or equal 45min (𝑏 ≥ 3), and forbid full break if the break is less than 30 min (𝑏 < 2).

𝑑 𝑢𝑣 ≤ 𝐶𝐷𝑇 + 𝑀 ∑ 𝐹 𝑤 𝑣-1 𝑤=𝑢 𝑢, 𝑣 ∈ 𝑈, 𝑢 ≤ 𝑣 (6)
∑ 𝑦 𝑢 𝑏 |𝐵| 𝑏=3 ≤ 𝐹 𝑢 𝑢 = 1, . . , |𝑈| (7)
∑ 𝑦 𝑢 𝑏 |𝐵| 𝑏=2 ≥ 𝐹 𝑢 𝑢 = 1, . . , |𝑈| (8) 1
-𝑦 𝑢 1 ≥ 𝐹 𝑢 𝑢 = 1, . . , |𝑈| (9)
There is not full break at 𝑣 if a break of 15min or 30min is scheduled at 𝑣, and if there is not break less than 45 min between [𝑢, 𝑣[ (as the first part of the split) and a full break is scheduled just before 𝑢. This inequalities [START_REF] Kok | Optimizing departure times in vehicle routes[END_REF] forbids breaks of 30 min to be considered as full break if there is not first part of split break. Note that 𝐹 0 = 1 by definition. In Constraints [START_REF] Tilk | Bidirectional labeling for solving vehicle routing and truck driver scheduling problems[END_REF] and (13) between two rests before 𝑢 and after 𝑣, the sum of breaks has to exceed 30 (45 resp.) min. if the working time exceeds 6 h. (𝑏 𝑢𝑣 = 1) (resp. 9h. and 𝑐 𝑢𝑣 = 1). Constraints ( 16) limit the driving time between 𝑢 and 𝑣 to 𝑀𝐷𝑅 plus the possible extension if a rest precedes 𝑢 and there is not rest until 𝑣. Constraint (17) limit the number of extensions. In addition, note that 𝑦 0 𝑏 = 1 by definition.

𝑀 (2 -∑ 𝑦

𝑑 𝑢𝑣 ≤ 𝑀𝐷𝑅 + 𝐸 𝑢 + 𝑀 (1 -∑ 𝑦 𝑢-1 𝑏 |𝐵| 𝑏=5 + ∑ ∑ 𝑦 𝑤 𝑏 |𝐵| 𝑏=5 𝑣-1 𝑤=𝑢 ) 𝑢, 𝑣 = 1, . . , |𝑈|, 𝑢 ≤ 𝑣 (16) ∑ 𝐸 𝑢 |𝑁| 𝑢=1 ≤ 𝐷𝐸 ( 17
)
As Figure 3 shows, Constraints (18) enforces that between the finishing time plus the break of node 𝑣 and the starting time of node 𝑢 there is less than 𝐷𝐷𝑈 hours, if there is not a rest between 𝑢 and 𝑣 -1. Constraints (23) establish the conditions for a split daily rest. There is a rest of 9h at node 𝑣 and a rest at node 𝑢, if between ]𝑢, 𝑣[ a break of 3h takes place, the rest of 9h taken a v is not a reduced rest, but the first part of a split daily rest.

Constraints ( 27) forbid night work.

𝐶𝑃 𝑢𝑖 + 𝐶𝑃𝑃 𝑢𝑖 = 1 𝑢 = 1, . . , |𝑈|;𝑖 = 1, . . , |𝐼| (27)
EU night rule constraints ( 25)-( 26) and ( 30)-( 32): Night working limits shift working time to 10h.

Constraints (28) enforce the working time between 𝑢 and 𝑣 to be less than 𝑀𝑊𝑁 hours, if between the starting time of 𝑢 and the finishing time of 𝑣 some night work (𝑡 𝑢𝑣 = 1) has been performed and there are less than 𝐷𝐷𝑈 hours elapsed between 𝑢 and 𝑣 (𝑟 𝑢𝑣 = 1). The value of 𝑟 𝑢𝑣 is set by Constraints (32). The value of 𝑡 𝑢𝑣 is set by Constraints (29)-(31) where 𝑡𝑖 𝑢𝑖 means that u starts before the upper bound 𝑈𝐵 𝑖 , finishes after the lower bound 𝐿𝐵 𝑖 and the processing time is greater than 0.

𝑀 * 𝑟 𝑢𝑣 ≥ 𝐷𝐷𝑈 -[𝑥 𝑣 -𝑥 𝑢 ] 𝑢, 𝑣 = 1, . . , |𝑈|; 𝑢 ≤ 𝑣 (31)

C. Assumptions and improvements.

Following the EU rules, we made some assumptions. Each activity can be split into an unlimited number of pieces of any duration. Several breaks or rests can be processed without any working time in between. Break and rest periods durations are not restricted to a limited set of values. Without loss of generality, the schedule starts and ends with a rest at the depot.

We compute an upper bound on the number of nodes per activity depending on its type (service or transport) and duration. If a driving activity is less than 4.5h then optimality requires 1 split of the activity for a rest or no split and 1 additional split for the night, in some cases to cover the night it is necessary to schedule more than one rest or break consecutively. It is easy to state that any break can be scheduled before or after the activity without loss of generality. Similar reasoning gives Table 1 for different values of service and driving activities. The number of nodes is the number of nodes for split plus one. For long activities, some split nodes model rest and break for a shift included in the activity.

We identified the longest possible sequence of breaks in an optimal solution: a break (< 9ℎ) → a rest (≥ 9ℎ) → the first part of split break (3ℎ) → the first part of split break (< 45𝑚𝑖𝑛). The first break is imposed by respite at of the first shift. The two last breaks may initiate split rests and breaks. We can show that such series of breaks and rest do not have to be scheduled in optimal solution only at the beginning of an activity. Therefore, the number of nodes per activity is increased by 3 and the processing times of the three first nodes of an activity are set to zero. For R5 the cumulated duration of break can be reached by two breaks of 20 and 25 minutes. Therefore the duration of break could be optimally set with values different than lower bounds of break durations. An additional continuous variable 𝐷 𝑢 is ranged by break duration interval of the type of break scheduled at node 𝑢. However, minimal respite durations to satisfy constraints usually provide optimal solutions; one example (Instance 41) has been built where optimality is not reach in this case.

Finally, the number of constraints is reduced by limiting the pairs of nodes u and v in the domain of the constraints for which the working or driving time between their activities ensures that the constraint does not apply. For instance, constraint (6) should exclude the nodes for which the cumulated driving time between 𝑎(𝑢) + 1 and 𝑎(𝑣) -1 is greater than 4.5h.

IV. NUMERICAL EXPERIMENTS

A. Instances

We propose a new set of instances PGLT available at https://perso.isima.fr/~igpenaar/Lyon_IEEE_2021/. The total number of instances is 41 and the size of the instances varies from 3 to 15 customers. In addition, we derived one instance from each route of the solutions of the routing problem provided by [START_REF] Goel | Legal aspects in road transport optimization in Europe[END_REF]. This set of 157 instances is named GOEL.

B. Results

All the experiments have been achieved on an Intel® Core™i5-8400 at 2.81 GHz under Windows 10, using C++ and Gurobi 8.1.1. A limit on the computation time has been imposed after two hours.

The first experiment aims to compare EU-Setting (model (1)-( 26); (28)-( 32)) against G-Setting (model ( 1)-( 27)) over the two sets of instances. Note that the solutions obtained with G-Setting on GOEL instances differ from the solutions provided by a heuristic -according to our objective -in [START_REF] Goel | Legal aspects in road transport optimization in Europe[END_REF]. The improvement is more than one hour for 15.3% of the routes for one day to one week.

The results are presented in Table II The second experiment has two treatments. The First execute the MILP under EU-Setting using the completion time (COM) as objective function and compare its results adding the optimal makespan (UBM) as an upper bound constraint (COM+UBM). The second treatment uses the makespan (MKS) as objective function and compare its results adding the optimal completion time (UBC) as an upper bound constraint (MKS+UBC). This experiment only use the PGLT instances, because computational running times using GOEL were excessively high. Table IV For both treatments, in nine instances it was not possible to achieve the same optimal value obtained without the upper bound constraint. The quality of the solution in both cases diminishes 1.7% and 2.7% on average and in the worst case, this reduction could be of 16.9% and 30.0% respectively. In this regard, when minimizing the completion time it is not possible to guarantee a solution with the minimum makespan and the opposite.

V. CONCLUSION

A new MILP formulation for the truck driver scheduling problem under the European Union drivers rules is presented. The model extends previous formulations considering rules like: split daily rests, working breaks and night work. Two different assumptions for the night work rule are considered, in order to compare its results with a previous algorithm in the literature. Results show a good performance obtaining equal solutions in 66.2% of the instances. Different experiments indicates that there is not a significant difference in the solutions between the two assumptions for the night constraint, even though, Goel night assumption is more flexible with the drivers. In addition, the Goel night rule assumption highly improves the total execution time of the model. Finally, the model results could lead to develop handcrafted algorithms in order to solve efficiently the problem.

Figure 1 .

 1 Figure 1. Modeling data structure.

1 )

 1 B. Objective function and constraints.The objective function (1) is to minimize the completion time of the processing time of the last node.𝑀𝑖𝑛 𝑧 = 𝑥 |𝑈| + 𝑝 |𝑈| (Constraints (2) and (3) enforce the sequence between activities and nodes, respectively.𝑥 𝑓𝑢(𝑎) + 𝑐𝑝 𝑎 + ∑ ∑ 𝐵𝐷 𝑏 * 𝑦 𝑢

Figure 2 .

 2 Figure 2. Compulsory full break after CDT hours of driving.

10 )

 10 𝐹 𝑢-1 ) ≥ 𝐹 𝑣 𝑣 = 1, . . , |𝑈|; 𝑢 = 1, … , 𝑣;(Working time between two nodes has to be less than six hours if there is no break in between them, as Constraints[START_REF] Koç | Long-haul vehicle routing and scheduling with idling options[END_REF] apply.𝑤𝑜 𝑢𝑣 ≤ 𝐶𝑊𝑇 + 𝑀 ∑ ∑ 𝐵𝐷 𝑏 *

  Binary indicators 𝑏 𝑢𝑣 and 𝑐 𝑢𝑣 , which are set to one when the working time exceed the their respective limits. 𝑀 * 𝑏 𝑢𝑣 ≥ 𝑤𝑜 𝑢𝑣 -𝑊𝑆𝐵1 𝑢 = 1, . . , |𝑈|; 𝑢 ≤ 𝑣 (14) 𝑀 * 𝑐 𝑢𝑣 ≥ 𝑤𝑜 𝑢𝑣 -𝑊𝑆𝐵2 𝑢 = 1, . . , |𝑈|; 𝑢 ≤ 𝑣 (15)

Figure 3 .

 3 Figure 3. Compulsory rest each DDU hours.Constraints (19) and (20) compute the processing time at node 𝑢.

  Goel night rule constraints (25)-(27): night working is forbidden. Constraints (25) and (26) indicate if a node 𝑢 starts before the upper bound 𝑈𝐵 𝑖 or after the lower bound 𝐿𝐵 𝑖 of interval 𝑖, respectively. 𝑀 * 𝐶𝑃 𝑢𝑖 ≥ 𝑈𝐵 𝑖 -𝑥 𝑢 𝑢 = 1, . . , |𝑈|;𝑖 = 1, . . , |𝐼| (25) 𝑀 * 𝐶𝑃𝑃 𝑢𝑖 ≥ 𝑥 𝑢 + 𝑝 𝑢 -𝐿𝐵 𝑖 𝑢 = 1, . . , |𝑈|; 𝑖 = 1, . . , |𝐼|

  𝑤𝑜 𝑢𝑣 ≤ 𝑀𝑊𝑁 + 𝑀[2 -𝑡 𝑢𝑣 -𝑟 𝑢𝑣 ] 𝑢, 𝑣 = 1, . . , |𝑈|; 𝑢 ≤ 𝑣 (28) 𝑀 * 𝑡 𝑢𝑣 ≥ ∑ 𝑡𝑖 𝑢𝑖 𝑣 𝑤=𝑢 𝑢, 𝑣 = 1, . . , |𝑈|; 𝑢 ≤ 𝑣; 𝑖 = 1, . . , |𝐼| (29) 𝑡𝑖 𝑢𝑖 ≥ 𝐶𝑃 𝑢𝑖 + 𝐶𝑃𝑃 𝑢𝑖 + 𝑁𝑃𝑀 𝑢 -2 𝑢 = 1, . . , |𝑈|; 𝑖 = 1, . . , |𝐼| (30) 𝑀 * 𝑁𝑃𝑀 𝑢 ≥ 𝑝 𝑢 𝑢 = 1, . . , |𝑈|; 𝑖 = 1, . . , |𝐼|

  Activity related with the node 𝑢. 𝑒 𝑎 : Earliest starting time of the activity 𝑎. 𝑙 𝑎 : Latest starting time of the activity 𝑎. 𝐵𝐷 𝑏 = {0.25, 0.50, 0.75, 3, 9, 11}: Minimal break durations. 𝐿𝐵 𝑖 = {0, 24, … , 144}: Upper bound of the night interval 𝑖. 𝑈𝐵 𝑖 = {4, 28, … , 148}: Upper bound of the night interval 𝑖. 𝑊𝐿9: Break duration if working time is less than 9h [30min]. Maximum number of rest reductions per week [3 times]. 𝐷𝐷𝑈: The maximal duration without a rest [24h]. 𝑀𝑊𝑁: Maximum working time if night work is performed [10h]. 𝐹 𝑢 : 1, if at node 𝑢 a full break takes place. 0, otherwise. 𝐸 𝑢 : 1, if the driving time is extended during one hour. 0, otherwise. 𝐻 𝑢 : 1, if a rest of 9h is taken without a previous rest of 3h. 0, otherwise. 𝑏 𝑢𝑣 : 1, if working time between nodes 𝑢 and 𝑣 is greater than 6h. 0, otherwise. 𝑐 𝑢𝑣 : 1, if working time between nodes 𝑢 and 𝑣 is greater than 9h. 0, otherwise. 𝑡 𝑢𝑣 : 1, if working time is performed during the night. 0, otherwise. 𝑟 𝑢𝑣 : 1, if the starting time between two nodes 𝑢 and 𝑣 is less than 𝐷𝐷𝑈 hours. 0, otherwise. 𝐶𝑃 𝑢𝑖 : 1, if the activity at node 𝑢 starts before the upper bound of the interval 𝑖. 0, otherwise. 𝐶𝑃𝑃 𝑢𝑖 : 1, if the activity at node 𝑢 starts after the lower bound of the interval 𝑖. 0, otherwise. 𝑁𝑃𝑀 𝑢 : 1, if there is night work at node . 0, otherwise. 𝑡𝑖 𝑢𝑖 : 1, if the activity at node 𝑢 is developed at the night interval 𝑖. 0, otherwise. 𝑝 𝑢 : Processing time at node 𝑢. 𝑤𝑜 𝑢𝑣 : Working time between nodes 𝑢 and 𝑣. 𝑑 𝑢𝑣 : Driving time between nodes 𝑢 and 𝑣.

	𝑊𝐺9: Break duration if working time is greater than 9h
	[45min].
	𝐶𝑊𝑇: Continuous working time without a break [6h].
	𝑊𝑆𝐵1: Maximal working time in a shift without at least 𝑊𝐿9
	( = 30) minutes of break [6h].
	𝑊𝑆𝐵2: Maximal working time in a shift without at least 𝑊𝐺9
	(= 45) minutes of break [9h].
	𝐶𝐷𝑇: Continuous driving time without a break [4.5h].
	𝑀𝐷𝑅: Regular driving time per shift [9h].
	𝑀𝐷𝐸: Extended driving time per shift [10h].
	𝐷𝐸: Maximum number of driving extensions per week [2
	times].
	𝑅𝐸: Decision variables:
	𝑥 Comfort variables:	𝐴: Driving and service activities. 𝑈: Nodes to schedule breaks. 𝐵 = {15𝑚𝑖𝑛, 30𝑚𝑖𝑛, 45𝑚𝑖𝑛, 3ℎ, 9ℎ, 11ℎ}: Types of break. 𝐼 = {[0,4], [24,28], … , [144,148]}: Night intervals within a
		week.
		Parameters:

𝑀: A big number. 𝑓𝑢(𝑎): First node related to activity 𝑎. 𝑙𝑢(a): Last node of activity 𝑎. 𝑐𝑝 𝑎 : Duration of activity 𝑎. 𝛿 𝑢 : 1, if node 𝑢 is driving; 0, otherwise. 𝑎(𝑢): 𝑢 : Starting time of process at node 𝑢. 𝑦 𝑢 𝑏 : 1, if at node 𝑢 a break of type 𝑏 takes place after the process. 0, otherwise.

TABLE I

 I

	.	MAXIMUM NUMBER OF NODES BY TYPE OF ACTIVITY AND
			DURATION
		Activity	Duration	Maximum number of nodes
		Service	< 6ℎ	3
			< 13ℎ	4
			Otherwise	𝑐𝑝 𝑎 3	+ 3
		Transport	< 4.5ℎ	3
			< 9ℎ	4
			Otherwise	𝑐𝑝 𝑎 3	+ 3

  and Table III. Table II presents four types of statistics for each of the two set of instances under the two set of settings. And in table III, EU-Settings is the base scenario and its performance is compared against G-Settings by computing three different statistics. From Table II, solutions with both settings are close in average but Table III, shows that EU-Setting allows to find better schedules for one half of instances and the improvements can reach 8.5 hours. Moreover, for PGLT instances, G-Setting is infeasible for 7 instances which are feasible with EU-Setting. Nevertheless, G-Setting is solved faster (almost 100 times) than EU-Setting with our formulations.

TABLE II

 II

			.	EU-SETTING VS. G-SETTING
			PGLT		GOEL
	Statistic	EU-Setting	G-Setting	EU-Setting	G-Setting
	Avg.
	completion	33.7	34.4		103.8	104.2
	time (h)
	Avg. time (s)	CPU	238.7	2.4		119.7	3.5
	Max. time(s)	CPU	7200.0	18.6		7200.0	35.3
	Infeasible	7	14		0	0
	TABLE III.	MILP UNDER EU-SETTING PERFORMANCE
			Statistic	PGLT	GOEL
		Instances improved (%)	46.3	50.3
		Avg. GAP (%)		1.7	1.4
		Maximum difference(h)	3.2	8.5

  presents the results.

TABLE IV .

 IV MILP UNDER EU-SETTING ADDING THE MAKESPAN AND THE COMPLETION TIME AS UPPER BOUNDS

	Statistic	COM+UBM MKS+UBC
	Number of instances
	with different objective	9	9
	function value
	Avg. GAP (%)	1.7	2.7
	Max. GAP (%)	16.9	30.0

Peña-Arenas. I., Lacomme, P. and Tchernev, N. are with Université Clermont Auvergne, LIMOS UMR 6158, 63178 Aubière, France (e-mail: [ivpena, placomme, tchernev]@ isima.fr).


Jean-Baptiste Pomet

A Result on Controllability with Constrained Controls

What if 0 ∈ ∂(conv U) instead of (ii) ? Keeping (i), or even: all solutions of ẋ = X 0 (x) are periodic, assuming U convex for simplicity.

What instead of (iii) ?

Motivation

Control system with state x in smooth M of dimension n:

(Σ U ) : ẋ = X 0 (x) + m k=1 u k X k (x) , u = (u 1 . . . u m ) ∈ U ⊂ R m ,
Theorem (Bonnard, 1981, see also Crouch, Brockett...) System (Σ U ) is controllable if (i) the vector field X 0 is Poisson stable, (ii) the convex hull of the set U is a neighbourhood of 0 in R m , (iii) the family {X 0 , X 1 , . . . , X m } is bracket generating.

Solar sail dynamics

q = -µ r r 3 Newtonnian + ε Γ( n) Solar sail
n: orientation of the sail (control). Approximation:

q = -µ r r 3 + ε u , u ∈ K α cone, α < π 2
Gauss equations (I : orbital elements (1st integrals), ϕ ∈ S 1 = R/2πZ):

İ = ε m k=1 u k F k (I , ϕ) , φ = ω(I ) + ε m k=1 u k f k (I , ϕ) , u ∈ U . Controllable? U convex, bounded. 0 ∈ U, 0 / ∈ interior of U.
Workshop on "Optimal Control Theory" -Rouen -June 22 to 24, 2022 J.-B. Pomet 2/11

Obviously,

İ = ε m k=1 u k F k (I , ϕ) , φ = ω(I ) + ε m k=1 u k f k (I , ϕ) , u ∈ U is a special case of (Σ U ) : ẋ = X 0 (x) + m k=1 u k X k (x) , u = (u 1 . . . u m ) ∈ U ⊂ R m , with X 0 = ω ∂ ∂ϕ : for any x = (I , ϕ), t → exp(tX 0 )(x) is periodic with period T (x) = 2π/ω(x).

Sufficient condition for controllability

Velocity set:

E U 0 (x) = m k=1 u k X k (x), u ∈ U ⊂ T x M .
Pullback along t → exp(tX 0 )(x):

E U τ (x) = m k=1 u k exp(-τ X 0 ) X k (x), u ∈ U ⊂ T x M Theorem (Σ U ) is controllable if (i) t → exp(tX 0 )(x) periodic with period T (x) for all x, (ii) 0 ∈ U, (iii) conv t∈[0,T (x)] E U t (x) is a neighbd of 0 in T x M for all x.
Workshop on "Optimal Control Theory" -Rouen -June 22 to 24, 2022 J.-B. Pomet 4/11

Controllability in prescribed time

(Σ U ) : ẋ = X 0 (x) + m k=1 u k X k (x) , u ∈ U ⊂ R m ,
Accessible sets:

A U T (x 0 ) = {x(T ), with t → (x(t), u(t)) solution of (Σ U ), x(0) = x 0 } . Let t → x(t), [0, t f ] → M be a solution of ẋ = X 0 (x).

Theorem

If U bounded convex, and 0 ∈ U, and

conv t∈[0,t f ] E U t (x(0)) is a neighborhood of 0 in T x(0) M (C) then the set A εU t f (x(0)) is a neighborhood of x(t f ) for any ε > 0.
Idea of the proof Linearize along t → x(t):

δ x = ∂X 0 ∂x (x(t)) δx + m 1 δu k X k (x(t)) (*) or ξ = m 1 δu k exp(-tX 0 ) X k (x(0)) , δx(t) = exp(tX 0 ) (x(0)) ξ(t) .
(*)

with initial condition ξ(0) = 0. Condition (C) is exactly condition for previous linear theorem, hence implies controlability of (*).

Workshop on "Optimal Control Theory" -Rouen -June 22 to 24, 2022 J.-B. Pomet 8/11

Idea of the proof cont'd

x(.) corresponds to control zero, on the border of U, hence it is difficult to use linear approximation.

However, one may built a family of t → x ε (t) and corresponding controls t → u ε (t) that remain in the interior of U and satisfy x ε (0) = x(0), x ε (t f ) = x(t f ). for the linear approximation and for original system.

Then end-point mapping is a submersion around these controls (linear controllability is preserved.

Workshop on "Optimal Control Theory" -Rouen -June 22 to 24, 2022 J.-B. Pomet

This is a 1st order condition

If U is a neighbd of the origin, then (C) is equivalent controllability of the linear approximation, i.e. to Rank{ad j X 0 X k , k ∈ {1, . . . , m}, j ∈ N} = n (at least in the analytic case).

However non classical when U is not a neighbd of 0. Linear time-varying controlability with constraint

Theorem (Schmittendorf Barmish 1980)

is a neighbourhood of the origin if one of the two following equivalent properties hold.

• For p ∈ (R d ) * (line vector),

Application to solar sails

In that case, it is also a necessary condition. Checking condition (C) is more difficult than checking the rank of a family of vector fields: but we manage to set it as convex optimisation problem, computable in that special case. Extensions?

Exploitation: minimum optical requirements

If other brackets that ad j X 0 X k , j ∈ N (1st order approximation) are needed for {X 0 , . . . , X m } to be bracket generating, we see no natural extension in terms of a condition of the same type as (C) to replace rank of Lie brackets when U is not a neighborhood of U.

In that case, conv t∈[0,t f ] E U t (x(0)) ⊂ some nontrivial vector subspace of T x(0) whatever U is.