Approches basées sur les métaheuristiques pour la gestion de flotte en temps réel

Transcription

1 Approches basées sur les métaheuristiques pour la gestion de flotte en temps réel Frédéric SEMET LAMIH, UMR CNRS, Université de Valenciennes

2 Motivation Réseau terrestre (GSM) Telecommunication GPS laptop / PC RF RS232 Switch IP LAN Embarked System Capteurs

3 Dynamic Vehicle Dispatching Local area Wide Area Routing Courrier services Dial-a-ride Less-than-truckload trucking No routing Emergency services Truckload trucking

4 In real-time context : - Allocation problem - Reallocation problem Allocation problem : Which vehicle must be sent to answer a task? Reallocation problem : Vehicle routing model or Location model + additional constraints

5 Local search method Main Idea : Stat from an initial solution (feasible?) Apply local changes to the current solution while these moves lead to an improvement of the objective function. Problem : These methods stop in a local optimum. The solution quality and the computing time rely on how rich is the neighborhood of the current solution.

6 Basic principle of Tabu search Allow moves leading to no improvement of the current solution to not be blocked in a local optimum Problem : How avoids cycling? Solution: Keep a record of the search and avoid to visit again solutions previously considered : tabu moves

7 Solution space, neighborhood and tabu list Tabu search extends classical local search method. The search is performed within a solution space including all possible solutions. Tabu list : Short term memory Several possibilities : List of last encountered solutions List of last moves performed (forbid inverse transformations) List the main features of the solutions or of the transformations

8 Maximize f(s) on a given domain. Simple tabu search s : current solution f* : value of the best known solution s* : best known solution T : tabu list N(s) : neighborhood of s (solutions visited from s by applying a simple tranformation) : "feasible" subset of N(s) (no tabu or satisfying the aspiration criterion). N (s) /* Initialization */ Select (build) an initial solution s 0. s:= s 0 ; f* := f(s 0 ) ; s* = s 0 ; T : = ; /* Search */ While the stopping criterion is not satisfied s : = arg max ( f ( s')) s' N ( s) If f(s) > f* then s* = s; f* := f(s) ; Declare tabu the current move (in T);

9 Real-time management of ambulances

10 Double covering constraints All the demand covered by an ambulance within r 2 (15) minutes A proportion a (90%) of the demand covered within r 1 (7) minutes

11 Double covering constraints Objective: Allocate ambulances to potential location sites to maximize the demand covered by 2 ambulances within r 1 minutes in a such way that : (1) the double covering constraints are satisfied ; (2) p ambulances are deployed.

12 Main Features of the Tabu Search Algorithm Solution space : Solution s specified by the number of ambulances located at each vertex of W may be infeasible / covering constraints. Objective function : with : M 1 >> M 2 >> 1. F(s) = f(s) - M 1 f 1 (s) - M 2 f 2 (s)

13 Initial solution : Solve the linear relaxation of the ILP Upper bound on the optimal value. Construct an initial solution using a rounding procedure. Tabu status : When an ambulance is moved from v n+j to v n+j : (j,j) is tabu for Θ 1 iterations.

14 Neighbourhood structure : Generate a set N(s) of neighbour solutions : Generate N(s) aspiration chains Aspiration chain: Move a number of ambulances to improve F(s)

15 Diversification : When the best known solution f* was not improved for Θ 2 iterations: 5 closest neighbours of v n+j all vertices of W not among the 5 closest neighbours Stopping rule : (f* 0.99 (initial upper bound)) or (f* was not improved for Θ 3 iterations)

16 Step 1 : (Initialization) Solve the linear relaxation - Build an initial solution Step 2 : (Neighbour solution) Generate the best neighbour of the current solution (invoke if necessary the diversification phase) Step 3 : (Incumbent update and stopping rule) Update the current solution and the tabu status Udpate the best known solution if an improvement occurred Goto step 2 while the stopping criterion is not satisfied

17 Redeployment Problem Ambulance location model + additional constraints Additional constraints? Move only a limited number of ambulances Do not move always the same vehicles Avoid repeated round trips between 2 location sites Avoid long trips Avoid assignments near the end of a shift Take into account the breaks of the paramedics...

18 Parallelization strategy Optimization : Given a current deployment : 1 - For each site, select which ambulance will be assigned to a call. 2 - Compute the relocation decisions associated with the future assignment of this ambulance. 3 - Store this relocation strategy. Data updates : Relocations strategies when an ambulance is again available.

19 Simulation 4 types of calls : - urgent call - one ambulance : 80% - urgent call - two/three ambulances : 3% - less urgent call : 10% - pending call : 7% Radii: - urgent call : r 1 = 7 min. - less urgent call : r 2 = 15 min. Frequency of calls : - population. - day period. Ambulances : - number of ambulances / day period. - speed : zone (3) day period.

20 Demand points and waiting sites in Montreal and Laval Demands points Waiting sites

21 Each simulation : 7 hours on 16 Ultra Sparc Station 142 calls and 54 ambulances on average. Average results on 6 simulations : Response times to calls : - All calls covered in less than 15 min. - Average response time for less urgent calls : 9 min % of urgents calls covered in less than 7 min. - Average response time for urgent calls : 3 min. 30 sec. Relocation strategies : - Ambulances moved in 38% of cases % of relocations involve less than 5 ambulances ambulances are moved on average % of relocations strategies were computed. - When not computed, max. available time : 32 sec.

22 Real-time management of trucks TESS Project PEPSAT

23 Routes planning Determine m routes such that : 1. The route of each vehicle starts and ends at given locations 2. A request is assigned to a unique vehicle 3. The volume capacity and the carrying capacity are respected 4. The pick-up and delivery locations are visited within their time-windows 5. The total cost is minimized and the number of requests served is maximized

24 Tabu search Moves Route of vehicle i Route of vehicle j Fictitious route Route of vehicle i Route of vehicle i Fictitious route Solution space : Solution s specified by the sequence of locations served by each vehicle may be infeasible / capacity constraints / time-window contraints / assignment constraints.

25 2 requests : Move evaluation - Compatibility matrix

26 Compatibility matrix - Move evaluation Compatibility matrix M: m ij = if request i and j cannot be served on the same route visiting i+ before if request i and j can be served on the same route visiting i+ before if request i and j can be served on the same route if j+ is visited after j+ j+ i Move evaluation: Assume that vehicle k serves requests 1 and 2 and that 1 + is served before 2 + Request i can be served by vehicle k and i + is visited between 1 + and 2 + if : m 1i m i2 0

27 Cost insertion matrix - Objective function Cost insertion matrix : Given request i not served by vehicle k, we keep in memory : - the cost of the route of vehicle k when request i is served - the capacity constraint and time-window constraint violations - the positions of i + and i - Objective function : f(s) = c(s) + α q(s) + β w(s) + ϕ c (s) + p(s) Cost associated with an unserved request = cost of service on a specific route a, b and ϕ are dynamically updated during the search

28 Tabu list - Aspiration function Tabu list : When request i is removed from the route of vehicle k, it is forbidden to reinsert i in the route of k for a given number of iterations. Aspiration function : The tabu status can be overridden when the insertion of a request in a route improved the best known solution (all requests have to be assigned).

29 Diversification Keep in memory the number of visited solutions in which a request is served by a given vehicle. At a given iteration, a solution s not improving the current solution s (f(s ) >f(s)) is penalized according to the frequencies associated with the assignments of requests to vehicles.

30 Computational results 28 random instances (Testbed of Li & Lim) : - 50 requests - 10 vehicles Class Instances Improved Best known Average solutions lc % lr % lrc %

31 Computational times Average Minimum Maximum LAMIH LL Times are reported in seconds

32 From static to dynamic route planning Dynamic route planning Planning on a rolling horizon At time t different events can happen :. A set of requests was completed. Only the pick-up parts of some request were performed. A new set of requests have to be considered. Requests have not to be considered anymore. Vehicles were delayed. A new vehicle is available. A vehicle is not available anymore

33 From static to dynamic route planning What cannot be changed :. Requests for which only the pick-up parts were performed cannot be moved. All actions realized during the decision sphere The decision sphere is a time interval starting at t ending at t+ t devoted to to the computation and the evaluation of a new transportation planning. The positions of vehicles known at t have to be estimated at t+ t

34 From static to dynamic route planning The tabu search algorithm is run given the status of the vehicle at at t+ t The current solution is reevaluated regarding :. the vehicle availability. the delays of vehicles New requests / requests assigned to vehicle not available Fictitious route It may exist no feasible solution regarding: - the assignment constraints - the time-windows constraints

35 Conclusion - Challenging problems : specialization in an industry sector take into account cost structure better integration - New methods - Real-time aspects