A combination metaheuristic-simulation for solving a transportation problem in a hospital environment *

Transcription

1 International Conference on Industrial Engineering and Systems Management IESM 2011 May 25-27, 2011 METZ- FRANCE A combination metaheuristic-simulation for solving a transportation problem in a hospital environment * Virginie ANDRE a,b, Nathalie GRANGEON a, Sylvie NORRE a, Frédéric PHILIPPE b a LIMOS, CNRS UMR 6158 Université Blaise Pascal, Clermont-Ferrand Aubière, France b CHRU de Clermont Ferrand 58 Avenue Montalembert Clermont-Ferrand Abstract This paper deals with the scheduling of logistic flows in a hospital environment. This work is based on a study proposed by the CHRU of Clermont-Ferrand, which wants to organize the transport of meals, linen and medicines for internal and external hospitals. The objective is to schedule the activities under precedence constraints, release dates and due dates. Moreover, the realization of each activity requires containers. The number of available containers is limited and known. Some steps require in addition to the containers, production line, loading bay, unloading bay and cleaning area. We have modeled our problem as a PDP-TW with resource constraints. We propose to solve it with a combination metaheuristic - simulation. The simulation model allows to carry the complexity due to the use of multiple resources and the metaheuristic allows to solve the scheduling problem. The results are given for the instances corresponding to the study proposed by the CHRU. Key words: combination, metaheuristic, simulation model, pickup, delivery 1 Introduction The CHRU of Clermont-Ferrand sees lot of modifications: the building of new hospital, the reorganization of the care units, the closing down of a hospital, the centralization of the meal production and services to the external hospital. In this study, we work on the transport of meals, linen and medicines. Before the reorganization, 5 drivers and 3 vehicles were used for the 41 activities per day. In the future, the number of drivers and vehicles must allow to realize 86 activities per day. The drivers work according to a planning. This planning is called a shift defined by a beginning date, an end date and a mealtime, so a shift is composed of two slots. Some other resources such as the production lines, loading bays, unloading bays and cleaning areas are available in limited number. These resources must be taken into account for the preparation and the loading, before the delivery, and for the unloading and the cleaning, after the pickup. In a week, the number of activities per day fluctuates but the weeks are similar. The objective is to propose a decision making aid tool which allows to share the activities for a day between the drivers and to schedule these activities under precedence constraints, release dates and due date, for a given number of resources and given slots for each drivers. * This paper was not presented at any other revue. Corresponding author: N. Grangeon. addresses: {andre,grangeon,norre}@moniut.univ-bpclermont.fr, fphilippe@chu-clermontferrand.fr

2 IESM 2011, METZ FRANCE, May A study concerning only meal transport has previously been done [2]. Many simplifying assumptions were considered and the number of transport activities was weak. Proposed method cannot be used for the whole problem. This first study allows to underline the difficulty to take into account the whole resources. Moreover, the sharing of some resources by many products needs to consider all products at the same time. We propose to solve it with a combination between a metaheuristic and a simulation model. The simulation model allows to consider the complexity due to the use of multiple resources and the metaheuristic allows to solve the scheduling problem. The problem is described in the next section. Then, in the third section, we will present the PDP-TW, give a brief state of the art and explain the link with our problem. In the fourth section, the proposed approach will be described. The simulation model, the solution encoding, the neighboring system and the metaheuristics will be presented. Before the conclusion, the proposed approach is tested on many instances provided by the CHRU. 2 Problem statement The problem consists in organizing the transport of several kinds of products (for example: linen, meals, medicines). A kind of product is delivered in a specific container. The number of available containers for a kind of product is known and limited. A production site fills up a container, then a vehicle delivers it in a consumption site that uses it and this container takes back to the production site for cleaning. For example, the central production unit (CPU) produces the meals; a vehicle delivers it to the care units then, after consumption, this container comes back to the CPU for cleaning. The linen is prepared by the laundry, delivered to the care units then, after purpose, come back to the laundry for washing. The medicines are delivered by the pharmacy to the care units and the empty containers come back to the pharmacy for cleaning. This process can proceed over two days: the return of the container to the production site is realized the day after the delivery. Each activity needs a driver and a vehicle. A driver works according to a shift composed of two slots. Each slot has a timetable. The drivers eat at the depot, so, they must come back to this site. A driver can drive whatever vehicle. There are several kinds of vehicle (un-refrigerated, refrigerated ). A kind of product cannot be transported by any kinds of vehicle but a kind of vehicle can transport many kinds of product (for example meal can only be transported by refrigerated vehicle, un-refrigerated vehicles can transport medicines or linen). When not used, drivers and vehicles are in a specific site named depot. Each site (production or consumption) has one or several loading bay(s) and one or several unloading bay(s). Each production site has one or several cleaning area (which permit to wash the container before they are used again) and one or several production lines (which permit to fill up the container). The number of resources in each site is known and limited. Each production line and each cleaning area is opened according to a planning. Let N transport activities to be realized, a transport activity concerns either a transport of clean container or a transport of dirty container. A transport activity is defined by an origin site and a destination site. Fig. 1. A transport of clean container and a transport of dirty container For a transport of clean container, the origin is a production site and the destination is a consumption site (Figure 1). A transport of clean container is composed of five steps which require a container and have a known duration: Step 1. The filling of the container which requires a production line, Step 2. The loading of the container in a vehicle which requires a driver and a loading bay in the production site, Step 3. The transport of the container by the vehicle from the production site to the consumption site which requires a driver, 2

3 IESM 2011, METZ FRANCE, May Step 4. The unloading of the container from the vehicle at the consumption site which requires an unloading bay in the consumption site and a driver, Step 5. The consumption of the container in the consumption site. For a transport of dirty container, the origin is a consumption site and the destination is a production site (Figure 1). A transport of dirty container is composed of five steps which require a container and have a known duration: Step 1. The routing of the container to the loading bay in the consumption site, Step 2. The loading of the container in a vehicle which requires a driver and a loading bay in the consumption site, Step 3. The transport of the container by the vehicle from the consumption site to the production site which requires a driver, Step 4. The unloading from the vehicle of the container at the production site which requires an unloading bay in the production site and a driver, Step 5. The cleaning of the container in the production site which requires a cleaning area. When two successive steps require the same kind of resource, the used resource is the same. Moreover, the step 1 of each activity is defined by a release date and the step 4 can have a due date. There are precedence constraints between two activities: for example, the transport of the dirty linen can only be done after the transport and the consumption of the clean linen. Empty transports have also to be realized in the following cases: Between two successive transport activities assigned to the same driver and the same vehicle if the destination of the first activity is different of the origin of the second one: an empty transport must be realized in order to convey the vehicle and the driver from the destination of the first activity to the origin of the second one, Before the first transport activity assigned to a driver: an empty transport must be realized in order to convey the vehicle and the driver from the depot to the origin of the first activity, After the last activity assigned to a driver: an empty transport must be realized in order to convey the vehicle and the driver from the destination of the last activity to the depot, Between two transport activities assigned to the same driver which require two distinct vehicles; two empty transports must be realized: the first one in order to convey the vehicle from the destination of the first activity to the depot and the second one in order to convey the new vehicle from the depot to the origin of the second activity. An empty transport requires a vehicle and a driver. Constraints are the following: Release date of transport activities, Precedence constraints between transport activities, Incompatibility between kinds of products and kinds of vehicles, Timetable of drivers, production lines and cleaning areas, For each kind of resources, the number of used resources at a given time must be lower or equal to the number of available resources, Some delivery activities have due date for step 4. The objective is to assign the activities to the slots and to schedule these activities while satisfying precedence constraints, release dates and due dates with a given number of slots and vehicles and minimizing the number of unrealized activities, the lateness of activities and the hours of overtime. 3 State of the art The nearest problems of the literature are the Pickup and Delivery Problem with time windows (PDP-TW) [12] and the static Dial-a-Ride problem (DARP). 3.1 Static DARP and PDP-TW statement There is a variant of the PDP-TW; this is the Dial-a-Ride problem (DARP). It exists two kind of DARP: the static DARP where the requests are known and the dynamic variant where the routing is done in real time and new requests popup dynamically. The main differences between the static DARP and the PDP-TW concern the context (person or patient transport for the static DARP and product transport for the PDP-TW), the load (for the first case, the load is an integer and the second case the load is a real) and the constraint of the travel duration in a vehicle (only for the static DARP). In this paper, we consider the static DARP and the PDP-TW. 3

4 IESM 2011, METZ FRANCE, May Let a set of n customers, each customer formulating a transport activity for a loading that must be transported between an origin and a destination. Each pickup on the origins and each delivery on the destinations must be realized in a given time window. Let G(S,E) be a complete and directed graph, where S={S i i=0,,2n} is the vertex set and E= {(S i,s j ) i=0,,2n,j=0,,2n,i j} is the edge set. The loading is defined by i and the unloading by n+i. In this graph, vertex s 0 corresponds to the depot; the other vertices represent the set of customer. Each customer is defined by: a non-negative quantity d i, a duration necessary to realize the service at the vertex: loading duration (resp. unloading duration) for the demand if S i is a loading site (resp. unloading), a time window [e i ;l i ], where e i is the lower bound and l i is the upper bound of the time window of the vertex S i where the loading (or unloading) is authorized. Each edge (S i,s j ) has a distance c ij and a transport duration which respects triangle inequality. Let m be the vehicle set, each vehicle v has capacity q v (v=1,,m), the objective is to determine a route R={R v,v=1,,m}, with the minimum duration that satisfy the following constraints: each customer is visited exactly once, each vertex S i (i=0,,2n) is visited exactly once, the loadings must respect the vehicle capacity, the pickup site for a customer must be visited before the delivery site, the departure date for each vehicle is greater or equal to e 0 and the return date is lower or equal to l 0, for each vertex, the service must be realized in the time window. 3.2 Some papers for the PDP-TW and the static DARP To evaluate a solution for these problems, there are many objective functions. [12] give the most common criteria: Duration (D): the duration includes the travel times, waiting tines, loading and unloading times and the break times; Completion time (CT): the completion time is the time that the service at the last location is completed; Travel time (TT): this criterion is defined by the time spent on actual traveling between different locations. Length route (LR): this is the total distance traveled between the different locations; Client inconvenience (CI): this criterion is measured in terms of pickup time deviation (between the effective pickup date and the desired pickup date) and of delivery time deviation (between the effective delivery date and the desired delivery date); Number of used vehicle (NV): this criterion is always used in DARP systems; Number of customers served (NCS): a customer is served when the pickup and the delivery have been realized; Routing cost (RC): this cost can be stated in minutes, hours, or kilometers; Lateness (L): the delay is measured between the desired due date and the effective end date; Time window violation (TWV): this criterion is the number of non-respected time window. The different studies consider many constraints such as: (1) Time windows: concern the dates for the pickup and the delivery, related to the transportation requests; (2) Route duration: related to the customers travel duration; (3) Ride time: all the vehicles, assigned to the same depot have the same timetable; (4) Vehicle availability: each vehicle has its timetable. Table 1: Characteristics of the different papers Papers Environment Problem Criteria (1) (2) (3) (4) Method [1] Hospital patients DARP Min RC and CI Tabu search [3] Hospital patients DARP Max NCS and min CI Simulated annealing [4] Transport DARP Min RC Genetic algorithm [5] Transport DARP Min RC Tabu search [7] Hospital products PDP-TW Min L and CI Genetic algorithm [8] Hospital patients DARP Min RC and CI Tabu search [9] Transport DARP Min RC, RT and TWV Genetic algorithm [10] Transport PDP-TW Min RC Tabu search [11] Hospital patients DARP Min NV and CI Genetic algorithm [14] Hospital patients DARP Min RC Tabu search [15] Public transit DARP Min NV and CI Genetic algorithm [13] Transport PDP-TW Min NV, RC and CI Tabu search 4

5 IESM 2011, METZ FRANCE, May The table 1 shows the objective functions, the constraints, the environment and the different methods used to solve the problem. The most used constraint is the time windows and the second is the ride time. The criteria are the minimization of the routing cost and the customers inconvenience. Many authors study this problem. Usually, a solution is represented by a set of vectors corresponding to a list of vertices (pickup and delivery) assigned to a vehicle. For instance, for the problem shown in Figure 1, a solution of this problem is represented as following: A, C, B, D. The often used neighborhood system are: moving a request in the same route and moving a request to another route. [14] presents a study in a hospital environment, and specially the organization of transport of product, where the author has to organize the transports from the production site to the consumption site and from the consumption site to the production site. The vehicle comes back to the depot. The authors propose genetic algorithm and use the Partially Mapped Crossover [6] which consists in recopying a segment of a parent P2 relative in a solution E1 child. Once the authors have constructed routes with this approach, they use a simulation model to test the routes with stochastic travel durations. 3.3 Comparison between the PDP-TW and our problem The Table 2 presents a comparison between the characteristics of the PDP-TW and those of our problem. Table 2: Comparison between the PDP-TW and our problem Characteristics PDP-TW Our problem Activities N requests / activities N activities Time window at the origin Time window at the destination [ei;li], i=1,2n for closed or soft time windows [e i ; ], i=1,n Release date for the 1 st step [0;l n+i ], i=1,n Due date for the end of the 4 th step Fleet Homogeneous or heterogeneous Heterogeneous Depot Single Single Resource availability Time window on the depot Timetable for drivers, lines and cleaning areas Capacity q j 1, j=1,m q j =1, j=1,m Load d i 1,i=1,n d i 1,i=1,n Transport duration Known Known Empty transport between the depot and the first activity Yes Yes Empty transport between the destination and the depot Yes Yes Empty transport between two activities Yes Yes Empty transport between an activity and the depot and between the depot and the next activity No Yes Precedence constraints between activities No Yes Mobile resources Vehicles, drivers and Vehicles containers Fixed resources Bays, production line None and cleaning areas Our problem seems to be similar to a PDP-TW with simplifying assumptions (d i = q j = 1), but it integrates additional constraints: precedence constraints between activities and many kinds of resource available in a limited quantity. 4 Proposed approach The objective is to propose a decision making aid tool which allows to share the activities for a day between the drivers and to schedule these activities under precedence constraints, release dates and due date, for a given number of resources and given slots for each driver. A solution is an assignment of the activities to the slots of the drivers and their schedule. We propose a combination of a metaheuristic and a simulation model as described by figure 2. The simulation model allows to consider the complexity due to the use of multiple resources and the metaheuristic allows to solve the scheduling problem. For each assignment created by the metaheuristic, the simulation model gives an evaluation of objective function. As the processing time of the simulation model is 5

6 IESM 2011, METZ FRANCE, May quite long, we prefer individual based methods to population based methods. So, we propose to use metaheuristics based on simulated annealing which builds, for a given number of drivers and their slots and given activities of transport, a good schedule. A schedule is called good if: All the activities are realized. Some activities may not be realized when one of the required resources is not available or the precedence constraints are not respected. The hours of overtime are less than 10 minutes for each driver. A driver has hours of overtime when he finishes his work after his slots. The lateness are reasonable (depends on the activity). An activity is late when the fourth step ends after its due date. 4.1 Metaheuristic Fig. 2. Principle of the proposed approach We have examined the different literature methods to solve the problem, but, as d i = q j = 1, these methods do not seem to be appropriate to our problem (Figure 3). In our problem, a vehicle must be unloaded before being loaded. This means that the solution for our problem is represented as following: 1, 2, where 1 is the request to transport one load from A to B and 2 the request to transport one load from C to D (Figure 3). So, we must consider the activities instead of the vertices. Problem Solution Load diagram Fig. 3. Example for our problem In Figure 4, we take into account 5 kinds of resources: the vehicle, the driver, the bays (bays at A, B, C and D), the production line and the containers. 6

7 IESM 2011, METZ FRANCE, May Fig. 4. Gantt diagrams for the uses of the resources Figure 4 shows the use of the vehicle, the driver, the production line and the containers. Firstly, the driver and the vehicle realize an empty transport from the depot to A and wait the container (colored in grey). In the same time, the first container is prepared at the production line. The second production line cannot begin the filling in the same time than the first. This production line is closed according to its planning (colored in black).then the container is loaded in the vehicle using a loading bay, the vehicle and the driver transport the container from A to B and deliver it to B. During this activity, the second container is prepared and waits a vehicle and a driver at C. Figure 5 gives the principle of the iterated local search. But we also consider local search (i.e. fig.5 without steps 9, 10 and 11). 1. Let σ an initial solution 2. σ* σ 3. While necessary 4. Choose uniformly and randomly σ in the neighboring system of σ 5. If H( ) H( )then If H( ) H( *) then σ* End if 8. End if 9. If the solution has not been improved since a certain number of iteration then 10. Apply the neighboring system many times and accept the obtained solution σ each time 11. End if 12. End while Fig. 5. Principle of the iterated local search Solution encoding: A solution is represented by a set of vectors: = { 1, 2,, M }, where i is the sequence of activities assigned to slot j, j=1,2m, j = { 1,j, 2,j,, NBj,j } where NB j is the number of activities assigned to slot j, j=1,2m such 2M as NB N and M is the number of available shifts. j j 1 Comparison of two solutions: We consider three criteria: H1( ) the sum of the squared delay for the activities which are not realized the studied day: we consider that an unrealized activity is completed the following day, so, the delay of each unrealized activity is one day, H2( ) the sum of the squared hour of overtime: the hour of overtime is obtained by the difference between the effective end of work and the end date of the slot, H3( ) the sum of the squared lateness: the lateness is the difference between the effective end of the step 4 and the desired due date. The three criteria, expressed in squared minutes, can be hierarchically ordered and we consider the hierarchical objective function: H( ) = H1( ) + 1.H2( ) + 2.H3( ) with 1 and 2 are chosen according to the granted significance of H2( ) and H3( ). 7

8 IESM 2011, METZ FRANCE, May Neighboring system: The neighboring system consists in moving an activity to a new position either among the activities assigned to the same slot or among the activities assigned to another slot. If the activity is linked to another activity by a precedence constraint, we verify that this constraint is respected in the case of these two activities are assigned to the same driver. The principle algorithm of the neighboring system is given by Figure Choose randomly and uniformly a slot j 1, j 1 {1,2m} 2. Choose randomly and uniformly a slot j 2, j 2 {1,2m} (j 1 can be equal to j 2 ) 3. Choose randomly and uniformly a position i 1, i 1 {1, NB j1 } 4. Let a = i1,j1, the activity in position i in j1 5. Delete the activity a from j1 6. Choose randomly and uniformly a position i 2, i 2 {1, NB j2} (if j 1 = j 2 then i 1 i 2 and i 2 such as precedence constraints are respected) 7. Insert a at a position i 2 in j2 Fig. 6. Principle of the neighboring system 4.2 Simulation model In a first time, we have developed a discrete event simulation model by using the Witness software. This simulation model allows to point out the interactions between entities and the complexity of the scheduling problem. Figure 7 gives an extract of the input data. For each activity, we give the type of activity (transport of clean container TCC or transport of dirty container TDC), the origin, the destination, the release date, the type of vehicle and the due date. With the origin and the destination, the required resources for each activity (production line, kind of container, bays, cleaning area) are deduced. The type of activity and its origin allow to determine the type of production line, in the case of the transport of clean container. For each slot, we give the timetable and assign a list of activities to realize. No Activity Origin Destination Release Kind of date vehicle Due date Pred. Driver s slot TCC Laund. NHE 6:30 am Tr_Norm 2:30 pm 6:00 am 12:30 am 50 TDC NHE CPU 1:00 pm Tr_Frigo 25 12:00 am 2:00 pm 51 TDC GM CPU 1:00 pm Tr_Iso_gd 26 Activity allocation 52 TDC GM CPU 1:00 pm Tr_Iso_gd TDC NHE CPU 1:00 pm Tr_Frigo TDC GM CPU 1:00 pm Tr_Iso_gd TDC Clem Laund. 1:30 pm Tr_Norm Activity characteristics Driver s activities Fig. 7. Input data for the simulation model For example, the delivery activity (49) starts from the laundry to the hospital Estaing (NHE), the filling can begin at half past six, the transportation requires a normal truck and the container have to be unloaded before half past two. The driver works according to two slots, the first one begins at 6:00am, finishes at 12:00am and the second slot begins at 12:30am and finishes at 2:00pm. During the first slot, the driver realizes the activities in this order: 52, 50, 53, 51, 54. Each driver realizes the activities according to its assigned sequence. An activity can only begin when all resources are available, when the release date is reached, when all the preceding activities in the sequence are realized and when the precedence constraints are satisfied.. The list of activities is never altered. The model: determines the completion time of each step of the activities, deduces the number of activities which are unrealized, deduces the lateness with the completion time and the due date, computes the busy rate for each driver, computes the number of hours of overtime. Considering the great processing times and the wish to combine the simulation model with a metaheuristic, we have built a second discrete event simulation model in Java programming language. The model determines the same performance criteria than the previous one and presents many sheets for the managers. The sheets are used 8

9 IESM 2011, METZ FRANCE, May to provide the results according to two views: table of values and Gantt chart. The Gantt chart shows for each resource the steps of the transport activities (filling, loading, unloading, transport, cleaning ), the waiting periods, the inactivity periods The first simulation model has been used to validate the second one. The processing time for the simulation model built with Witness is 5 seconds, when the processing time for the other simulation model is less than 1 second. 5 Results We will present the results in two parts. Firstly, we will consider dedicated drivers which transport either meals or linen and medicines. Then, we consider no dedicated drivers (the drivers can realize transports for any kind of product). 5.1 Dedicated drivers Firstly, we have tested our approach on the instances given in [2] where only the transport of meals is considered. The Table 3 presents the results for the exact method (linear programming) and the results for our method (local search). M is the number of shifts, the number of slots is equal to 2M. H2 is the sum of the lateness given in minutes and H3 is the sum of the hours of overtime given in minutes. In the two studies, the assumptions are the same except that drivers do not realize the loading and the unloading. For the linear model, the first instance is modeled by 5104 variables and 7759 constraints; the second one, with 3 drivers, is modeled by 4620 variables and 7052 constraints. The results given by the two methods are very similar. Among the solutions, the solution obtained by the linear programming and 4 shifts has been chosen by the CHRU. The processing time is less than 1 minute for the iterated local search with iterations. Table 3: Results for the transport of meals Method N M Unrealized activities Sum of lateness H2 # activities with lateness Sum of hour of overtimes H3 # shifts with overtime Linear programming Local search Linear programming Local search Then, we have studied the transport of linen and medicines. Firstly, a solution with 5 drives has been proposed by the transport manager and has been evaluated by the simulation model, and then our method has been successively applied with 5, 4.5 and 4 shifts. For each instance, we have chosen 1 = 50 and 2 = 50, which means that, we look for solutions with no unrealized activities. Table 4: Results for the transports of linen and medicines Scenarios N M Unrealized activities H1 Sum of # activities with lateness H2 lateness Sum of hour of overtime H3 # shifts with overtime Manager The solution proposed by the transport manager, has one unrealized activity, 5 activities with lateness, 135 minutes for the sum of lateness but no hour of overtime. Despite the lateness (for one activity) and the hours of overtime, the solution with 4.5 shifts has been chosen by the CHRU because the lateness can be negotiated with the consumption site and the hours of overtime are not so large. To conclude about the dedicated drivers, a total of 8.5 shifts (17 slots) is required to realize the set of 61 activities according to the CHRU. 5.2 No dedicated drivers In this part, we consider the whole activities (22 transports of meal and 39 transports of linen or medicines). Tables 5a and 5b present the results obtained with the local search and the iterated local search with

10 IESM 2011, METZ FRANCE, May iterations and iterations without improvement for the iterated local search. In table 5a, 1 = 100 and 2 =1. In table 5b, 1 = 1 and 2 = 100. In this case, the processing time is less than 5 minutes. Table 5: Results for the different kinds of product Local search Iterated local Iterated local Local search search search Sc. N M H1 H2 H3 H1 H2 H3 Sc. N M H1 H2 H3 H1 H2 H Table 5a Table 5b The two methods allow to obtain feasible solutions in the most of the case. We obtain a solution with 8 drivers (Table 5a) and 7 drivers (Table 5b), which is not possible to obtain with dedicated drivers. But, the CHRU prefers the solution with dedicated drivers, because, in the solutions with no dedicated drivers, the drivers switch from a vehicle to another, too many times. 6 Conclusion and further works In this paper, we worked on a real-life problem, modeled by a PDP-TW with resource constraints. Because of the great number of resources, available in a limited quantity, we proposed a simulation model to compute many performance criteria for a solution (activity assignment and sequence). This model has been combined with a metaheuristic in order to obtain a good schedule of the transport activities. The computational experiment, with the instances given by the CHRU showed that it is better to consider non dedicated slots. The CHRU has implemented our solution during March Their first observation is that the transport durations depend on the travel period in the day and some activities may be late because of some delay. So, a first improvement of the simulation model will consider the dependent travel durations. A second improvement concerns the optimization problem. In the future, we will consider the problem of determining the number of slots and their timetable, supposed as known in this paper. The proposed combination will be integrated in a decision aid tool which will help the manager to build the timetables for the next modifications of the transport activities of the CHRU. 7 References [1] Majid Aldaihani, Maged M. Dessouky, Hybrid Scheduling methods for paratransit operations, Computers & Industrial Engineering, 45:75-96, 2003 [2] Virginie André, Nathalie Grangeon, Sylvie Norre, and Frédéric Philippe, Dimensionnement et ordonnancement de livraisons repas, Société Française de Recherche Opérationnelle et d Aide à la Décision ROADEF 09, 2009 [3] John Baugh, Gopal Krishna, Reddy Kakivaya, John Stone, Intractability of a dial a ride problem and a multiobjective solution using simulating annealing, Engineering Optimization, 2 :91-123, 1998 [4] Alberto Colorni, Marco Dorigo, Francesco Maffioli, Vittoria Maniezzo, Giovanni Righini, Marco Trubian, Heuristics from nature for hard combinatorial optimization problems, International Transactions in Operational Research, 1:1-21, 1996 [5] Jean-François Cordeau, Gilbert Laporte, A tabu search heuristic for the static multi-vehicle dial a ride problem, transportation Research, Part B, 37: , 2003 [6] David E. Goldberg, Genetic algorithms in search, optimization and machine learning, Addison-Wesley, 1989 [7] Yannick Kergosien, Algorithmes de tournées de véhicules pour l optimisation des flux de produits et de patients dans un complexe hospitalier, Thèse de doctorat, Université de Tours, 2010 [8] Emanuel Melachrinoudis, Ahmet B. Ilhan, Hokey Min, A Dial-A-Ride problem for client transportation in a health care organization, Computers & Operations Research, 34: , 2000 [9] Rene Munch Jorgensen, Jesper Larsen, Kristin Berg Bergovinsdottir, Solving the dial-a-ride problem using generic algorithms, Journal of the operational Research Society, 58: , 2007 [10] William P. Nanry, J. Wesley Barnes, Solving the pickup and delivery problem with time windows using the reactive tabu search, Transportation Sciences, 34: , 20007[11] Brahim Rekiek, Alain Delchambre, 10

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