Elodie Robert
email: elodie.robert@volvo.com

Christophe Berenguer

Keomany Bouvard

Hoceane Tedie

Romain Lesobre

Dynamic sequential decision making for missions and maintenances scheduling for a deteriorating vehicle

Keywords: Stochastic deterioration, maintenance planning, mission scheduling, dynamic sequential scheduling approach

This article proposes an approach to define sequentially a schedule for both missions and maintenance operations for a deteriorating vehicle. Both activities are jointly scheduled to adapt to the real vehicle usage and improve the hauler productivity by integrating the constraints directly in the decision-making process. The monitoring information regarding the vehicle health state, the failure occurrences and the new missions to integrate in the schedule are considered as opportunities to update the initial schedule. The method is composed of three steps: a stochastic modelling of the vehicle deterioration evolution integrating the missions effects, a definition of the maintenance policy and schedule structure and the dynamic scheduling algorithm to define and update the schedule.

INTRODUCTION 1.Context and background: motivation for the work

To deliver satisfying transport solutions, the manufacturers of commercial heavy vehicles have understood the necessity to increase their offers in terms of services. The development of an efficient maintenance management system has become a key for success to enhance the solution quality. The main objective is to guarantee the vehicle availability for the missions to be performed. Therefore, optimizing the maintenance planning according to the vehicle usage is a way to reach this goal.

The customers for the different brands of the Volvo Group can subscribe to a maintenance contract for the vehicle. This contract is based on the vehicle configuration and the operating conditions specified by the purchaser. From these information, a maintenance planning is created. Maintenance intervals and the different operations to do at each planned stop are defined to improve the vehicle availability and guarantee its performances at a specified level. This maintenance planning is static because the maintenance intervals are never updated during the vehicle life. Moreover, as the vehicle health state is never considered, the total maintenance cost is impacted by unplanned stops generating high immobilization costs.

In this framework, some research works have been led in the Volvo Group to improve the maintenance planning by integrating the components deterioration in the decision-making process and grouping the maintenance operations in a relevant way to reduce the maintenance costs [START_REF] Bouvard | Condition-based dynamic maintenance operations planning & grouping. Application to commercial heavy vehicles[END_REF]. A maintenance policy has also been developed to ensure, at some risk, the autonomy of a multi-component system on given operation periods [START_REF] Lesobre | A usage-informed preventive maintenance policy to optimize the maintenance free operating period for multicomponent systems[END_REF]. However, none of these approaches consider the variability of the hauler activity with the different missions the vehicle has to perform. For many haulers, mission scheduling and maintenance planning are independently defined. It can lead to non-optimal schedules insofar as the maintenance planning is not adapted to the real usage of the vehicle. Jointly optimizing the maintenance and missions scheduling becomes relevant to improve the haulers productivity while adapting to their operational constraints. By doing so, the preventive maintenance operations are scheduled and completed without impacting the vehicle availability and the missions execution order is adapted to the constraints. The generated schedule can evolve with time according to the occurring disruptions such as new missions to schedule, monitoring information regarding the vehicle health state or a failure.

State of the art

Scheduling both missions and maintenance operations at the same time while considering the possibility to have planning update according to disruption occurrences raises two main challenges: the way to generate a schedule integrating both activities and the rescheduling strategy.

The literature identifies two classes of static approaches to jointly schedule maintenance and production. The first one is a two-stage strategy that schedules the production tasks at first and uses it as a constraint to integrate preventive maintenance operations in the schedule. Benbouzid et al. [START_REF] Benbouzid | Résolution du problème de l'ordonnancement conjoint maintenance/production par la stratégie séquentielle[END_REF] develop an approach that firstly schedules the production tasks and uses the production tasks order as a constraint to integrate periodic maintenance slots in the initial schedule. The objective is to find the best maintenance periodicity to balance the maintenance cost and the risk of the machine unavailability. Cassady et al. [START_REF] Cassady | Integrating preventive maintenance planning and production scheduling for a single machine[END_REF] suggest a similar method to schedule jobs and maintenance operations for a single vehicle to minimize the total expected completion time to perform the jobs. The second class deals with an integrated strategy that simultaneously schedules both activities. Feng et al. [START_REF] Feng | Joint optimization of flowshop sequence-dependent manufacturing cell scheduling and preventive maintenance[END_REF] propose an integrated strategy to minimize the costs generated by jobs tardiness and maintenance operations, either preventive or corrective. In a previous contribution [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF], a method was developed to jointly schedule both activities to minimize the total maintenance cost. Among all the existing approaches, two solving methods are applied. It is either an exact method [START_REF] Cassady | Integrating preventive maintenance planning and production scheduling for a single machine[END_REF] that tries out every scheduling possibility or a heuristic method such as genetic algorithm [START_REF] Benbouzid | Résolution du problème de l'ordonnancement conjoint maintenance/production par la stratégie séquentielle[END_REF], [START_REF] Feng | Joint optimization of flowshop sequence-dependent manufacturing cell scheduling and preventive maintenance[END_REF], [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF]. Larger size scheduling problems justify the emergence of heuristic methods as considering all the possible schedules is impossible in a satisfying computation time.

The variety of approaches comes from the chosen hypotheses in terms of optimization criterion, maintenance policy and deterioration model. The optimization criteria are either cost-related by considering maintenance and/or production costs [START_REF] Feng | Joint optimization of flowshop sequence-dependent manufacturing cell scheduling and preventive maintenance[END_REF], [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF] or time-related when considering the completion times to do the jobs [START_REF] Benbouzid | Résolution du problème de l'ordonnancement conjoint maintenance/production par la stratégie séquentielle[END_REF], [START_REF] Cassady | Integrating preventive maintenance planning and production scheduling for a single machine[END_REF]. Multi-objective optimization is also applied to optimize both maintenance and production costs as well as the completion time [START_REF] Da | Integrated preventive maintenance and production scheduling optimization on uniform parallel machines with deterioration effect[END_REF]. The preventive maintenance strategy differs from one approach to another. It can either bring the system back to an as good as new state [START_REF] Cassady | Integrating preventive maintenance planning and production scheduling for a single machine[END_REF], [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF] or consider that a maintenance operation can be imperfect [START_REF] Feng | Joint optimization of flowshop sequence-dependent manufacturing cell scheduling and preventive maintenance[END_REF]. Finally, the deterioration affects the system in different ways i.e. its age [START_REF] Feng | Joint optimization of flowshop sequence-dependent manufacturing cell scheduling and preventive maintenance[END_REF] or its health state [START_REF] Cassady | Integrating preventive maintenance planning and production scheduling for a single machine[END_REF], [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF]. The time to complete a task can also be influenced due to the deterioration state [START_REF] Cassady | Integrating preventive maintenance planning and production scheduling for a single machine[END_REF]. However, the possibility to update the schedule or reschedule the remaining tasks according to real time information or events is never considered.

Rescheduling consists in updating an existing schedule in response to disruptions or other changes. It includes new jobs, machine failures and machine repairs [START_REF] Vieira | Rescheduling manufacturing systems: a framework of strategies, policies, and methods[END_REF]. Vieira et al. [START_REF] Vieira | Rescheduling manufacturing systems: a framework of strategies, policies, and methods[END_REF] define the scope of rescheduling research in production according to three axes: the rescheduling environment, the strategy and the methods. The environment is either static with a finite set of jobs to do or dynamic. The rescheduling strategies are divided into two groups: the dynamic and predictive-reactive strategies. For the dynamic strategy, there is no schedule but dispatching rules or a control strategy to dispatch the tasks while for the predictive-reactive strategy, a schedule is generated and updated in response to disruptions. Finally, the methods are either to generate a schedule or to repair it. The schedule repair methods are in fact only needed in predictive-reactive approaches.

There are two major challenges when initiating rescheduling. It is necessary to estimate if the rescheduling is worth it or if doing a schedule update will be too expensive in terms of resources and changes. Then, thinking about the strategy to trigger rescheduling is unavoidable. By doing periodic rescheduling, the events occurring between two rescheduling points are ignored. If it is combined with an event-driven rescheduling method to consider the disruptions that are significant enough, the rescheduling approach can be attractive [START_REF] Aytug | Executing production schedules in the face of uncertainties: A review and some future directions[END_REF]. The rescheduling problem is mostly studied without considering maintenance. Hoogeveen et al. [START_REF] Hoogeveen | Rescheduling for new orders on a single machine with setup times[END_REF] consider the rescheduling problem to schedule sets of jobs to reduce the production costs and limit disruptions by avoiding too many schedule updates. The few researches investigating the rescheduling problem while considering maintenance often use simple maintenance strategy. Wang et al. [START_REF] Wang | Integrated rescheduling and preventive maintenance for arrival of new jobs through evolutionary multi-objective optimization[END_REF] study the rescheduling problem in response to the arrival of new jobs in a single machine layout where preventive maintenance should be determined. The objective is to optimize both the total completion time to do the jobs as well as the maintenance and production costs. The processing time of each job is affected by the deterioration and the allocation of more resources on a job can reduce its processing time. However, only one maintenance slot has to be scheduled. Its effect on the machine deterioration varies according to its duration.

Contribution of this paper

The previous contributions detailed in section 1.2 either limit the joint missions and maintenance scheduling problem to a static one or consider rescheduling options but only in a limited framework either with just the jobs to do and the new ones to add or without the maintenance strategy to adapt the production scheduling to the system health state.

In this paper, a predictive-reactive approach is proposed to jointly schedule the available missions and the maintenance operations for a deteriorating vehicle. An initial joint schedule is generated based on the available missions at the initial stage. To do so, the deterioration properties of the vehicle as well as the maintenance and production costs are considered to determine the best schedule. Then, different events such as failures, health state information or new missions can trigger schedule updates.

The approach encompasses three parts. The first one is the stochastic modelling of the vehicle deterioration evolution that integrates the different deterioration effects from the different missions. The second part deals with the definition of the maintenance policy and the schedule structure. The final part explains the dynamic scheduling strategy to optimize the initial static schedule defined with the missions characteristics and the vehicle state. Sequential rescheduling occurs according to the available information: a failure or a vehicle deterioration measurement or a new mission to add to the mission pool. Numerical examples are then presented to illustrate the behaviour and performance of the method. The results show the benefits of using a sequential approach that takes into account all the available on-line information on the vehicle state and operating requirements.

PROBLEM STATEMENT

A vehicle has an initial set of missions to complete. The severity of the missions differs from one to another because the operational conditions change. Indeed, the topography, the road type and quality have a direct impact on the deterioration evolution and it has to be considered when modelling the vehicle usage.

As the vehicle deteriorates over time, maintenance operations have to be integrated when scheduling the missions execution. From there, a joint schedule for missions and maintenance operations is initially defined. This schedule evolves according to the events occurring during the schedule completion. A failure of the vehicle, an information regarding its current health state or additional missions that have to be integrated in the schedule justify the necessity to update the initial schedule.

The objective is then to develop an approach to use the different event and real-time information to define sequentially a schedule for both missions and maintenance activities for a deteriorating vehicle. The main challenges lie in the way to reorganize the missions when new ones have to be added in the schedule and to evaluate if an event is significant enough to initiate a rescheduling. Figure 1 describes the reactive strategy when events occur. At the beginning, an initial schedule is set to schedule the missions M 1 , M 2 , M 3 and M 4 as well as maintenance operations MA. At t 1 , a new mission M 5 is requested and has to be added to the schedule. A schedule update is generated to react to this event and M 5 will be performed after M 4 . At t 2 , a measurement of the current health state is available. The question is to know whether a rescheduling is needed. If the failure risk is too high before the next maintenance operation, it can be relevant to reschedule. An update is then set up and occurs at the end of M 4 . Then, the missions M 5 and M 3 are performed but a failure happens during M 3 . A corrective maintenance is necessary to repair the vehicle as well as a rescheduling. The maintenance operation MA scheduled at the end of M 3 in the updated schedule (2) is postponed after M 1 .

Done M 4 M 1 M 1 M 1 M 1 M 2 M 2 M 2 M 2 M 3 M 3 M 3 M 3 M 4 M 4 M 5 M 4 M 5

METHODOLOGY

This section describes the dynamic sequential decision method to schedule the missions and maintenance operations operating for a deteriorating vehicle.

Deterioration modelling

The vehicle health state evolves over time according to its activity and the different usage conditions it is submitted to. The effect of the different missions can be related to the road type and road conditions as well as the driver type. It is then very difficult to quantify exactly how each of these parts affects the deterioration evolution. In the Volvo Group, usage severity levels are defined by the mix of several conditions of use as topography or activity type.

The deterioration phenomenon is assumed gradual and depends on the usage conditions. That is why, a stochastic process has been chosen to model the vehicle deterioration evolution. The Gamma process has been favoured due to its capacity to represent gradual deterioration phenomena for industrial systems [START_REF] Lesobre | A usage-informed preventive maintenance policy to optimize the maintenance free operating period for multicomponent systems[END_REF], [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF]. It is one of the most used process when it comes to model the deterioration of mechanical components. It has already been applied in a previous work [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF].

Each mission is characterized by a mission duration, earnings when the mission is achieved, a starting deadline representing the date before which the must have started, a unitary delay cost and the influence on the vehicle deterioration. The unitary delay cost corresponds to the cost charged for each delay unit. It then reduces the final gain earned after the mission execution.

To illustrate the different severity levels of the missions, it is assumed that the vehicle deterioration trend depends on the severity level. So, the Gamma process modelling the deterioration has varying parameters. Each mission is then associated with a pair of parameters to take into account the missions characteristics when estimating the vehicle lifetime. Thanks to this model, the developed maintenance strategy is adapted to the vehicle usage.

Maintenance model and schedule structure

The adopted maintenance model is a deterioration-threshold failure model [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF]. When the deterioration exceeds a threshold L, the vehicle fails. The objective is then to avoid failures during missions by scheduling maintenance operations at the right time. Thanks to the deterioration model presented in the part 3.1, the failure probabilities and the remaining useful life can be estimated to decide whether the health state is enough to dispatch the vehicle on a mission or if a maintenance operation is needed. It is an essential information to consider when building a joint schedule for missions and maintenances as well as for the rescheduling strategy. It is assumed that each maintenance operation restores the vehicle to an "as good as new" health state.

The schedule structure is defined as follows. Missions are grouped into blocks and the bocks are separated by a maintenance operation. To perform the 5 missions described in Figure 1, the final

schedule is π = {(M 2 , M 4 )(M 5 , M 3 , M 1 )}.
It is composed of two blocks and at each block end, a maintenance is performed.

The maintenance strategy enable to know how to build the schedule. But, an optimization criterion is necessary to schedule at best missions and maintenance activities.

Optimization criterion

To schedule both missions and maintenance operations, the optimization criterion C has to consider the maintenance costs as well as the missions characteristics. It is then composed of two parts: the total maintenance cost C m considering both preventive and corrective maintenance and the production gain generated by the missions achievement. For a schedule π composed of n missions, this criterion represents the operating incomes generated by π (Eq.1).

C(π) = n ∑ i=1 g m (i) -c d (i) -C m (1)
The raw gain and the delay cost generated by the mission i are respectively denoted g m (i) and c d (i).

The production gain for the mission i is then equal to g m (i)c d (i).

The delay cost c d (i) for the mission i is based on the estimated delay time t d (i) and the unitary delay cost C ud (Eq.2). This time represents the difference between the starting deadline d max (i) defined in the mission characteristics and E t s (b) the average time for the beginning of the block b containing the mission i (Eq.3). It is assumed that all the missions in the block b start at E t s (b) . This assumption is sufficient to orientate the scheduling algorithm towards the schedule maximizing C and enables to save some computation time. If E t s (b) is smaller than the starting deadline, there is no delay in the mission execution.

c d (i) = t d (i) ×C ud (2)
t d (i) = max{0, E t s (b) -d max (i)} (3)
To estimate E t s (b) , it is necessary to know the number of previous preventive maintenances, the missions scheduled before the block b and the average number of failures that could occur in the previous blocks.

E t s (i) = t 0 + (k -1)d p + k-1 ∑ b=1 d c E N f (b) + N m (b) ∑ i=1 d(i) (4)
where t 0 is the initial time for the schedule,

C m = N b ∑ b=1 C 0 +C f N f (b) ∑ i=1 P f (b, i) (5)
A preventive maintenance, occurring at the end of each block, costs C 0 while a corrective maintenance costs C f . C f is assumed to be greater than C 0 to include the immobilization costs. P f (b, i) is the probability to have at least i failures in the block b. While P f (b, i) is greater than 1%, N f (b), the number of considered failures for the block b, is incremented.

The dynamic scheduling algorithm defines the missions and maintenance schedule by maximizing the criterion C.

Dynamic scheduling algorithm

The dynamic scheduling algorithm aims at populating the blocks of missions to optimize the operating incomes considering both the maintenance costs and the gains generated by the missions completion. It optimizes the initial schedule with the missions characteristics and the vehicle state. Then, sequential rescheduling occurs according to the available information such as a failure occurring in a block, a real-time deterioration measurement and new requested missions.

An initial schedule is generated based on the genetic algorithm developed by Robert et al. [START_REF] Robert | A predictive approach to jointly schedule missions and maintenances for a deteriorating vehicle[END_REF]. It is a static method that schedules the missions initially available as well as the maintenance operations to optimize the criterion defined in the section 3.3. The deterioration model for the vehicle is considered to fill in the blocks and a maximum failure probability P max is used as a condition on the block filling process. If the probability P f (b, 1) to have at least one failure in the block b is greater or equal to P max , the block is considered as not feasible because too risky.

This initial schedule is implemented and during its execution, different real-time event can occur and trigger possible updates:

• A failure happens during a mission. A corrective maintenance operation is performed to restore the vehicle health state to an as good as new state. Then, a rescheduling is initiated on the remaining missions. A corrective maintenance is a major event to consider when scheduling the next missions and preventive maintenance slots as it strongly affect the current vehicle health state. The first scheduled mission is the undone part of the unfinished mission. The static algorithm is then adapted to consider the unfinished mission position in the schedule.

• A new mission is requested. It has to be added to the schedule as soon as possible because of the starting deadline characterizing the mission. As a mission cannot be interrupted once it has started, a new mission can only available at the end of the already scheduled missions. A rescheduling is then applied to integrate the mission in the schedule. This rescheduling affects the remaining missions and the preventive maintenance slots already scheduled.

If a deterioration measurement is available at the end of the mission completed just before the rescheduling order, this information is integrated in the rescheduling and conditions the first block filling for the new schedule.

If there is no deterioration measurement available, the rescheduling is performed normally.

• No new mission is requested but a deterioration measurement is recorded.

If the mission is at a block end, the possible gains generated by a rescheduling with respect to the current schedule are estimated. If they are greater than the rescheduling limit condition C lim , the rescheduling is adopted. Otherwise, the current schedule is kept.

When the mission is not at a block end, the first question is to know whether the block can be finished or if the failure risk is too high with respect to the maximum failure probability P max . If it is impossible to finish the block, a rescheduling is performed. Otherwise, the possible gains driven by a rescheduling are estimated. If they are significant enough, a rescheduling is applied.

The described strategy is represented by Figure 2. The way to fill in the blocks can slightly vary according to the information available. A failure during a mission, caused by an excess of the deterioration D with respect to the failure threshold L, implies that the first mission to do in the new schedule is the failed mission remaining part. When the deterioration level D is known, it conditions the way to compute the failure probability of the new schedule first block. The failure threshold not to be exceeded becomes L -D.

This method is a predictive-reactive one insofar as a schedule exists and is updated according to the occurring events. Sequential rescheduling enables to adapt at best the schedule to the vehicle health state and usage. are respectively denoted α and β . The raw gain earned when a mission is completed without delay is g m = 5000 and the unitary delay cost is C ud = 50 for all missions.

For the static method with no rescheduling, the 6 first missions are scheduled to optimize the criterion defined in the section 3.3. Once, all these missions are performed, the 12 others are scheduled and performed.

Regarding the occurring events, 4 new missions are available at the end of the missions 1, 3 and 5 and deterioration information are available at the end of the missions 1, 2, 5, 6, 8, 10, 12, 13, 14, 17 and 18.

Results

The initial schedule based on the 6 first missions is identical for both methods. It is defined by

π i = {(4, 2)(3)(5, 1)(6)}.
Permutations inside the blocks and between the blocks can happen. The only difference is in the final schedule. Indeed, with the dynamic sequential method, the schedule evolves according to the occurring events and the decisions are made as reactions to these events.

For each method, 250 simulations are generated to compare both the operating incomes generated by the schedules and the computation time necessary to obtain the different schedules. Let π ds (Eq.6) and π s (Eq.7) be examples of final schedules respectively for the dynamic sequential method and the static one.

Figure 3 shows that the dynamic sequential method is more profitable than the static one by about 6200. The benefit represents the costs of two corrective maintenance operations. However, the dynamic sequential method is more time-consuming than the static method. Indeed, the static method requires about 10s to compute the schedule while the dynamic sequential one needs about 50s, namely 5 times longer. Figure 4 shows the distribution of the total rescheduling number n t according to the simulations made with the dynamic sequential method. On average, 5 updates are performed to schedule the 18 missions.

For each simulation, it can be divided into 4 parts according to the events causing the schedule updates. The parameters n f , n d , n m and n md respectively denote the rescheduling number caused a failure during a mission, an available deterioration information, new requested and a mix of both new requested missions and deterioration information. For each cause, the rescheduling strategy is different. The total rescheduling number n t is then defined by:

n t = n f + n d + n m + n md (8)
Figure 5 illustrates the distributions of the rescheduling numbers n f , n d , n m and n md . Among the schedule updates, 2 always happen. It is due to new requested missions and deterioration information available at the end of missions 1 and 5. Likewise, one update always occurs owing to new requested missions. It takes place at the end of the mission 3.

The benefits generated by the dynamic sequential method comes from the schedule updates performed when events occur (failures, new missions) or information on the vehicle health state are retrieved. It is considered that failures and new missions necessarily involve updates of the current schedule. Indeed, a failure implies that the current schedule was not exactly well designed for the vehicle use. This event also enables to know that after the corrective maintenance, the deterioration state is back to 0 and it is a piece of information that can be used to schedule the remaining missions accordingly. Likewise, arrivals of new missions means that the schedule must be updated as soon as possible because each mission is characterized by a starting deadline. If new urgent missions are requested, rescheduling is compulsory to prevent potential delays on the new missions execution. The only considered flexibility in terms of rescheduling is when an information regarding the vehicle health state is available. The notion of rescheduling limit condition is then introduced. This limit condition C lim enables to decide whether or not a rescheduling is cost effective. If C lim is not overstepped, no rescheduling is applied. C lim is fixed at C 0 /2 for the dynamic method results presented in Figure 3. The operating incomes increase when C lim decreases (Figure 6). It is assumed that no cost is induced when a rescheduling happens. By considering it, a balance between the rescheduling costs and potential gains could be identified and these gains could be compared with the cost necessary to retrieve monitoring health state information. However, as the starting deadline is part of the mission characteristics, adding a rescheduling cost may not be relevant. The rescheduling impact is already considered in the operating incomes through the delay costs to limit the rescheduling disruption.

CONCLUSION

In this paper, a dynamic sequential method to schedule both missions and maintenance operations for a deteriorating vehicle has been proposed. The schedule is updated when events such as failures or new missions occur and when deterioration information are available. The criterion enabling to optimize the obtained schedules is based on the operating incomes taking into account the maintenance costs, the gains earned when the missions are completed and the delays when performing the missions. The interest of rescheduling according to the events is illustrated through a comparison with a method that do not consider any information to update the initial schedule. Rescheduling improves the operating incomes at the expense of the computation time and the rescheduling limit condition also plays its part to increase the operating incomes.

Figure 1 :

 1 Figure 1: Principle to schedule missions and reschedule according to the occurring events (new mission M 5 , deterioration measure, failure)

  d p and d c respectively denote the preventive and corrective maintenance durations, d(i) is the duration of the mission i, E N f (b) is the average number of failures in the block b and N m (b) is the number of missions in the block b. The maintenance cost C m (Eq.5) depends on the number of blocks N b composing the schedule as well as the number of failures considered in each block b, denoted N f (b).

Figure 3 :

 3 Figure 3: Convergence analysis of the operating incomes: static VS dynamic sequential

Figure 4 :Figure 5 :

 45 Figure 4: Distribution of the total rescheduling number n t

2 Figure 6 :

 26 Figure 6: Operating incomes according to the rescheduling limit condition C lim

Table 1 :

 1 Parameter definitions

	Parameter	Definition	Value Parameter	Definition	Value
	C 0	Preventive maintenance cost	1000	C f	Corrective maintenance cost	3000
	d p	Preventive maintenance duration	2	d c	Corrective maintenance duration	4
	L	Failure deterioration threshold	100	P max	Maximum failure probability	0.1

Table 2 :

 2 Mission characteristics

	Mission N o Duration (h)	α	β	Failure probability Starting deadline
	1	21	0.1335 0.1	0.0021	90
	2	21	0.1837 0.1	0.0087	12
	3	8	0.3959 0.1	0.0035	32
	4	8	0.3285 0.1	0.0015	2
	5	2	1.3254 0.1	0.0016	43
	6	3	1.3206 0.1	0.0099	110
	7	3	1.0150 0.1	0.0030	45
	8	10	0.4216 0.1	0.0132	48
	9	13	0.2465 0.1	0.0037	70
	10	44	0.1041 0.1	0.0195	200
	11	19	0.1937 0.1	0.0070	150
	12	9	0.4043 0.1	0.0067	98
	13	13	0.2077 0.1	0.0017	190
	14	3	0.8863 0.1	0.0016	160
	15	5	0.9130 0.1	0.0192	165
	16	3	0.8177 0.1	0.0012	128
	17	22	0.1303 0.1	0.0023	165
	18	6	0.6972 0.1	0.0127	145

  The schedule π ds only have 9 blocks while π s have 13 ones. Then, π s performs 4 more maintenance operations. It explains why the gains generated by π ds are equal to 80550 while the ones for π s are 67600.

	π ds = {(4, 2, 5)(3, 8, 7)(6)(9, 1, 16)(12, 18)(11, 14)(15)(13, 17)(10)}	(6)
	π s = {(4, 2)(3)(5, 1)(6)(7)(8)(9, 16)(15)(12)(14, 13)(18)(17, 11)(10)}

APPLICATION EXAMPLE

The application example analyses the reactions of the dynamic sequential algorithm according to the occurring events. The obtained schedules are simulated to estimate the gains generated by the schedules and the results are compared with a static method for which no rescheduling can happen when a schedule is defined.

Parameters & missions definitions

The different parameter necessary for the simulations are defined in Table 1.

The initial list of missions is composed of 6 missions. The next 12 missions are progressively added in the mission pool and have to be integrated in the schedule. Table 2 describes the characteristics for the 18 missions to schedule. The shape and scale parameters for the deterioration Gamma process


Anahita Basirat
email: anahita.basirat@univ-lille2.fr

Cédric Patin
email: cedric.patin@univ-lille3.fr

Caroline Moreau
email: caroline.moreau@chu-lille.fr

Relationship between Perception and Production of Intonation of French in Parkinson's Disease

Keywords: prosody production, prosody perception, Parkinson's disease, sensorimotor interaction, French intonation

Parkinson's disease (PD) causes impairments in both perception and production of speech. The link between perception and production disorders in PD is, however, not well known. In this study, we examined whether there is a relationship between perception and production of intonation in French in PD individuals. Fifteen PDs and fifteen age-matched controls discriminated questions and statements in auditory-only, visual-only and audiovisual modalities. They were also asked to produce these utterances while their voice was recorded. Participants had to process intonation cues to discriminate between yes-no questions and statements in the perception task and to mark these utterances in the production task. Our results showed that PDs marked their questions and statements less than controls: the F0 rise at the end of the questions was smaller in PDs and contrary to controls, no F0 fall was observed at the end of the statements produced by PDs. Importantly, PDs who were more impaired in marking questions and statements were poorer at discriminating between these utterances in the perception task. These findings suggest that dysprosody in PD weaken the performance of PD individuals in processing prosody cues during speech perception.

Introduction

Parkinson's disease (PD) is the second most common neurodegenerative disorder [START_REF] De Lau | Epidemiology of parkinson's disease[END_REF] and causes significant impairments of speech production. Prosody is one of the aspects which is strongly impaired in PD: parkinsonian speech is characterized by less variable fundamental frequency (F0) and intensity, and longer pause times (for a review, see [START_REF] Goberman | Acoustic analysis of parkinsonian speech i: Speech characteristics and l-dopa therapy[END_REF]). Dysprosody has a strong impact on communicative abilities of PDs as listeners have difficulties to detect the intention of PD individuals when it is conveyed by prosody cues. For example, contrastive stress or yes-no questions produced by PDs are less accurately recognized by listeners [START_REF] Pell | The impact of parkinsons disease on vocal-prosodic communication from the perspective of listeners[END_REF]. In addition to prosody production impairment, it has also been suggested that PDs may suffer from prosody perception [START_REF] Pell | On the receptive prosodic loss in parkinson's disease[END_REF], but the amount of deficiency differs greatly between patients, probably because of specific brain lesions [START_REF] Lloyd | Comprehension of prosody in parkinson's disease[END_REF].

Although disorders in both perception and production of speech have been reported in PD, the link between these deficits is not well understood. It is well known that perception and production systems of speech are closely interrelated (e.g. [START_REF] Galantucci | The motor theory of speech perception reviewed[END_REF][START_REF] Kreiner | The missing link in the embodiment of syntax: Prosody[END_REF][START_REF] Schwartz | The perception-for-action-control theory (PACT): A perceptuo-motor theory of speech perception[END_REF]), and this throughout the lifespan. For example, if the movement of a relevant articulator (e.g. tongue tip) is constrained when 6-month-old infants listen to speech sounds, their abil-ity to discriminate phonemes that involve this articulator decreases [START_REF] Bruderer | Sensorimotor influences on speech perception in infancy[END_REF]. In adults, motor learning influences the way speech sounds are perceived: disturbances to jaw movements while repeating words (motor learning period) modify the way adults classify vowels after the motor learning period [START_REF] Nasir | Auditory plasticity and speech motor learning[END_REF]. Speech perception is thus influenced by modifications in speech production system. Based on these findings about the influence of production system on the speech perception, we hypothesize that speech production impairments in PD would impact the speech perception of PD individuals. If this is true, speech perception disorder observed in PD could be understood as a consequence of the impairment of the speech production system.

It has also been suggested that speech perception deficit in PD individuals may also impact their speech production [START_REF] Keyser | The relationship between speech production and speech perception deficits in parkinson's disease[END_REF]. For example, the external auditory feedback (i.e. feedback from one's own voice during speech production) is less monitored by PDs than controls [START_REF] Mollaei | Sensorimotor adaptation of speech in parkinson's disease[END_REF]. This impairment, in long term, may affect speech motor representations and thus worsen speech production [START_REF] Arnold | Pathomechanisms and compensatory efforts related to parkinsonian speech[END_REF]. These findings suggest that the disorders in speech perception and production in PD are interrelated even if the precise nature of this relation remains to be clarified.

The goal of the current study is to examine the relationship between prosody perception and prosody production in PD. We focused on the intonation difference between yes-no questions and statements in French of France. As intonation cues are conveyed by both auditory and visual speech [START_REF] Srinivasan | Perceiving prosody from the face and voice: Distinguishing statements from echoic questions in english[END_REF], we also examined the ability of PDs in visual speech processing. Based on the above-mentioned literature, we expected that there would be a positive correlation between the performance of PDs in perception and production of prosody.

Materials and methods

Participants

Fifteen individuals with idiopathic PD in their "on" state of medication (4 females) and fifteen one-to-one age-matched controls (11 females) participated in this study. All participants were native French speakers of France. Their mean age was 67 years (SD = 9). They did not have hearing problems by selfreport and all had normal or corrected-to-normal vision. The mean disease duration of PD individuals 1 was 11 years (SD = 5). The PD individuals have not had deep brain stimulation surgery. The study was conducted in accordance with the Helsinki Declaration and the ethical guidelines of the Department of Speech and Language Therapy of University of Lille. Before testing, all participants were informed about the study by a written document and signed a consent statement.

Materials and stimuli

Twelve French sentences were selected from [START_REF] Patel | Processing prosodic and musical patterns: a neuropsychological investigation[END_REF]. A male native French speaker of France produced each sentence twice, once as a statement and once as a yes-no question (e.g. "il parle Franc ¸ais." vs. "il parle Franc ¸ais ?"). Thus, no morphosyntactic cue was available to distinguish between questions and statements. The speaker was recorded audiovisually by a camera in front of him. The head, neck, and shoulders of the speaker were visible in the videos. The videos were edited using Adobe Premier Pro. The sound level was adjusted to 70 dB using Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF].

Experimental design and procedure

During the experiment, the participants performed a prosody perception task followed by a prosody production task. All stimuli were presented using OpenSesame [START_REF] Mathôt | Opensesame: An opensource, graphical experiment builder for the social sciences[END_REF].

Prosody perception task

In the perception task, participants received an auditory-only (A), an audiovisual (AV) and a video-only (V) block. In each block, ten pairs of sentences, i.e. ten questions and ten statements, were presented to participants through headphones (in A and AV modalities) and on the computer screen (in AV and V modalities). Two other pairs were repeated twice in order to remove any response bias. These pairs were not included in data analyses. The order of the stimuli presented in each block was random. During the experiment, participants were asked to classify each sentence as "question" or as "statement" by pressing F or J keys. The keys were counterbalanced across participants. Before the experimental blocks, all participants received a training block (one question-statement pair) in the auditoryonly, audiovisual and visual-only modalities. This was done to familiarize them with the experimental procedure. The order of the experimental blocks was counterbalanced across participants.

Prosody production task

In the production task, participants were asked to produce all the sentences used in the prosody discrimination task, once as "question" (question block) and once as "statement" (statement block). Due to the fatigability of PDs, no distractor was included. In each trial, one of the sentences was selected in a randomized order. The selected sentence was written on the computer screen in front of the participants. All productions were recorded on the computer using a headset microphone. The participants received a training block (one question-statement paire) before the experimental blocks. The order of the experimental blocks was counterbalanced across participants.

Data Analyses

Prosody perception task

The number of correct and false responses was calculated for each speech modality (A, AV and V) and each type of sentences (question and statement). The ability of participants to discriminate between questions and statements was measured using d-prime as suggested in the signal detection theory [START_REF] Macmillan | Signal detection theory[END_REF]. D-prime is the difference between the z-score of the hit rate and the z-score of the false alarm rate. A greater d-prime in this experiment represents a better ability to discriminate between questions and statements based on prosody. A repeatedmeasures Analysis of variance (ANOVA) was performed using Group (PD/control) as a between-subject factor and Modality (A/AV/V) as a within subject factor. Student's t tests were used for post hoc comparisons.

Prosody production task

For each produced sentence, the mean F0 of the first and the last vowel was extracted using Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF]. This was done as yes-no questions and statements are typically marked in French by respectively a F0 rise and a F0 fall at the end of the utterance [START_REF] Di Cristo | Intonation in french[END_REF][START_REF] Delais-Roussarie | Intonational phonology of french: Developing a tobi system for french[END_REF]. Vowels were automatically detected using the Praat script Prosogram [START_REF] Mertens | The prosogram: Semi-automatic transcription of prosody based on a tonal perception model[END_REF] when possible, or were subjectively selected using the signal and the second formant as cues. Measurements were made using semitones, a scale that helps reducing gender differences [START_REF] Simpson | Phonetic differences between male and female speech[END_REF]. Thus, four measures per participant and per question-statement pair were obtained: initial F0 and final F0 for the question and initial F0 and final F0 for the statement. Mean F0 was then averaged across questions and statements per participant. A repeated-measures ANOVA was performed using Group (PD/control) as a between-subject factor and Type (question/statement) and Position (initial/final vowel) as within subject factors. Student's t tests were used for post hoc comparisons.

Relationship between perception and production tasks

Our main goal was to test whether there is a relationship between the performance of PDs in prosody perception and prosody production tasks. In the perception task, we used dprimes which represent the ability of participants to distinguish different prosodies. For prosody production, we focused on the F0 changes throughout utterance production. We measured the performance of each participant by calculating the difference between the F0 rise throughout their questions and the F0 fall throughout their statements (i.e., [final F0 -initial F0] in questions -[final F0 -initial F0] in statements). This represents to what extent participants marked questions and statements. For the sake of brevity, this value is called ∆F0. We tested the correlation between d-primes and ∆F0s (i.e. perception-production correlation) using Pearson correlation coefficient.

Results

Prosody perception task

Figure 1 shows the averaged d-primes in auditory-only, audiovisual and visual-only modalities for PDs and controls. The ANOVA yielded a significant effect of Modality (F (2, 56) = 18.92, p < 0.001): The d-primes in the visual-only modality was smaller than the d-primes in the auditory-only modality (t(29) = 5.19, p < 0.001) and the d-primes in the audiovisual modality (t(29) = 4.45, p < 0.001). There was no difference between the auditory-only modality and the audiovisual modality (t(29) = 0.19, p = 0.85). The main effect of Group and the Group x Modality interaction were not significant (respectively, F (1, 28) = 1.06, p = 0.31 and F (2, 56) = 0.20, p = 0.82).

Prosody production task

Figure 2 shows the mean F0 of initial and final vowels in the questions and statements for PDs and controls. Significant results of the ANOVA performed on F0s are reported in table 1. As expected from the literature, there was a significant interaction between Position and Type. Importantly, PDs and controls performed differently in the production task as shown by significant Group × Type and Group × Position × Type interactions. Post hoc analyses showed that there was a F0 rise in the questions for both PDs and controls (respectively, t(14) = 5.80, p < 0.001 and t(14) = 10.29, p < 0.001). However, this rise was smaller for PDs compared to controls (t(28) = 3.34, p < 0.01). Moreover, there was a significant difference between PDs and controls in producing statements (t(28) = -3.60, p < 0.01): there was a F0 fall for controls (t(14) = -6.41, p < 0.001) but not for PDs (t(14) = -1.38, p = 0.19).

It is worthwhile to remind that there were more females in the control group than in the PD group (11 versus 4). To ensure that the difference between PDs and controls in the prosody production task was not due to the gender difference, we compared age-matched subsets of PDs and controls with exactly the same number of females and males in each group: we analyzed all possible age-matched subsets with four females and four males with PD and four females and four males without PD (thirty six subsets in total). We focused on ∆F0s (i.e., [final F0 -initial F0] for questions -[final F0 -initial F0] for statements, see Data Analyses). When all participants were included, the ∆F0 was 6.07 semitones for PDs and 13.37 semitones for controls (see Figure 2). The same trend has been observed when the age-matched subsets of 4 females and 4 males with PD and 4 females and 4 males without PD were analyzed: across subsets, the mean ∆F0 was 4.54 semitones for PDs and 11.99 semitones for controls. Student's t tests showed that the difference between PDs and controls was significant for all subsets (mean t(14) = 3.60, mean p = 0.006).

Relationship between perception and production of prosody

Figure 3 shows the performance of each participant in the perception (vertical axis) and production (horizontal axis) tasks.

For the perception task, the d-primes in the auditory-only, audiovisual and visual-only modalities are shown. For the production task, ∆F0s are shown. The correlation between these measures for PDs and controls are presented in table 2. For PDs, the correlations were strong (A: 0.68, AV: 0.64, V: 0.73) and significant (see table 2). No significant correlations were observed for controls (A: 0.32, AV: 0.14, V: -0.02).

Discussion

Numerous studies report that both speech perception and speech production are impaired in PD. However, the link between these deficits is not well understood. Our goal was to investigate whether there is a relationship between perception and production of prosody in PD. We examined the ability of PDs and agematched controls in a prosody perception task followed by a prosody production task. Our main findings are discussed below.

In the prosody perception task, participants were asked to discriminate between questions and statements which were pro- duced by a typical speaker. As expected, prosody discrimination in auditory-only and audiovisual modalities was better compared to visual-only modality. The results showed that the performance of PDs did not significantly differ from that of controls. This was true for auditory-only, visual-only and audiovisual modalities. It has been suggested that only some PD individuals suffered from prosody perceptual deficits, probably because of specific brain lesions [START_REF] Lloyd | Comprehension of prosody in parkinson's disease[END_REF]. Although it is necessary to be cautious when interpreting a null result, our finding suggests that, in line with previous studies, PD does not systematically lead to abnormal perception of auditory and visual prosody cues.

In the prosody production task, we tested the performance of participants in marking questions and statements by respectively F0 rise and F0 fall at the end of these utterances. Our results revealed significant differences between PDs and controls. Although PDs marked questions by rising the F0, the F0 rise was smaller in PDs compared to controls. In statements, no F0 fall was observed for PDs contrary to controls. Our results are in line with previous studies showing that intonation production is impaired in PDs [START_REF] Skodda | Intonation and speech rate in parkinson's disease: General and dynamic aspects and responsiveness to levodopa admission[END_REF][START_REF] Ma | Intonation contrast in cantonese speakers with hypokinetic dysarthria associated with parkinsons disease[END_REF]. Further research is required in order to establish if this absence of F0 fall in statements for the PD speakers is linked to an absence of Final Lowering [START_REF] Liberman | Intonational invariance under changes in pitch range and length[END_REF], or if any other downtrend phenomenon such as declination [START_REF] Cohen | On the anatomy of intonation[END_REF], is involved. Moreover, it would be helpful to examine the influence of sentence length and structure, and production task (ex. reading versus narration) on this result.

The main goal of our study of was to examine whether there is a relationship between prosody production and prosody perception in PD. The results revealed a significant positive correlation between the performance of PDs in question-statement discrimination (d-prime) in the perception task and the amount of F0 difference between questions and statements (∆F0) in the production task. This was true for the auditory-only, visualonly and audiovisual modalities. Although the relationship between speech perception and speech production in PD, especially in loudness processing [START_REF] Keyser | The relationship between speech production and speech perception deficits in parkinson's disease[END_REF][START_REF] Clark | Loudness perception and speech intensity control in parkinson's disease[END_REF], has been discussed in the literature, the nature of this link remains debated. In fact, as discussed in the introduction, this link can been understood in two ways [START_REF] Bruderer | Sensorimotor influences on speech perception in infancy[END_REF][START_REF] Nasir | Auditory plasticity and speech motor learning[END_REF][START_REF] Keyser | The relationship between speech production and speech perception deficits in parkinson's disease[END_REF][START_REF] Mollaei | Sensorimotor adaptation of speech in parkinson's disease[END_REF][START_REF] Arnold | Pathomechanisms and compensatory efforts related to parkinsonian speech[END_REF]: (1) Speech production disorders influence speech perception system and cause modifications in speech perception, (2) Auditory perceptual deficits impact speech motor representations and thus lead to speech production impairments. Our findings are consistent with the first mechanism. In fact, although PDs did not differ from controls in prosody discrimination, those who were poorer at marking questions and statements also showed poorer performance in prosody perception. This suggests that production impairments weakened their performance in discriminating between questions and statements. In other words, substantial disorders in prosody production impact mechanisms of auditory and visual prosody perception. This interpretation is consistent with the fact that prosody perception and prosody production were not correlated in controls as they did not suffer from any speech production disorder. To further investigate this issue, it would be helpful to carry out a longitudinal study in order to assess perception and production of prosody in PD starting from early stages of the disease.

Conclusion

In summary, we investigated the performance of PDs in the perception and production of intonation in French using questions and statements. We observed that PDs marked their questions and statements less than controls. Although PDs did not differ from controls in discriminating between questions and statements perceptually, those who were poorer at marking questions and statements showed poorer performance in the discrimination task. This suggests that deficits in prosody production in PD impact prosody perception mechanisms.

Figure 1 :Figure 2 :

 12 Figure 1: Prosody discrimination Task. Mean D-prime in auditory-only, audiovisual and visual-only modalities for PDs and controls. Error bars indicate standard errors of the mean.

Figure 3 :

 3 Figure 3: Performance in the prosody production (horizontal axis) and prosody perception (vertical axis) tasks in the auditory-only (top), audiovisual (middle) and visual-only (bottom) modalities for PDs (left) and controls (right).

Table 1 :

 1 Significant results of ANOVA for the prosody production task.

	Effect	F(1,28)	p
	Position	60.58	<0.001
	Type	112.25	<0.001
	Position × Type	127.15	<0.001
	Group × Type	10.84	<0.01
	Group × Type × Position	17.95	<0.001

Table 2 :

 2 Correlation between the performance of PDs and controls in the production and the perception tasks in auditory-only (A), audiovisual (AV) and visual-only (V) modalities.

	Group	Speech modality Pearson's r	t(13)	p
	PDs	A	0.68	3.33	< 0.01
	PDs	AV	0.64	2.99	< 0.05
	PDs	V	0.73	3.82	< 0.01
	Controls	A	0.32	1.24	= 0.24
	Controls	AV	0.14	0.53	= 0.61
	Controls	V	-0.02	-0.07 = 0.95

This data was not available for one of the PD individual.

Acknowledgements

We thank the subjects for their participation in the experiment and Marie Behaegel, Mathilde Bourdon, Vincent Deffaux, Germain Hannebique, Jules Fumel and Sébastien Serlet for their help in preparing the stimuli and/or collecting data.