A Hybrid Approach Based on Multi-Objective Simulated Annealing and Tabu Search to solve the Dynamic Dial a Ride Problem Lazhar Khelifi1, Issam Zidi1,2
Kamel Zidi1,2, Khaled Ghedira1
(1) University of Tunis, SOIE-Management Higher Institute, 41, Rue de la liberté, Cite Bouchoucha Le Bardo 2000, Tunisie
[email protected] [email protected]
(2) University of Gafsa, Faculty of Sciences of Gafsa, Zarroug, Gafsa 2112, Tunisie
[email protected] [email protected]
Abstract—This paper describes an application of a Hybrid Algorithm based on the Multi-Objective Simulated AnnealingTabu Search (MOSA-TS) to solve the dynamic Dial a Ride Problem (DRP). In fact, different versions of the dynamic Dial k Ride Problem are found in every day practice; transportation of people in low-density areas, transportation of the handicapped and elderly persons and parcel pick-up and delivery service in urban areas. The problem is to affect every new passenger request to one of the vehicles and to design a new route and schedule for this vehicle. This affectation must be done in real time. In this work, we offer our contribution to the study and to solve the dynamic DRP in the application using a multi agent system based on Hybrid Algorithm: Multi-Objective Simulated Annealing -Tabu Search (MOSA-TS). Keywords— Heuristics; Dial a Ride Problem, multi agent system; Multi-Criteria Optimization; Multi-Objective Simulated Annealing -Tabu Search (MOSA-TS) Algorithm;
I.
INTRODUCTION
In the research of transport systems, the DRP problem is largely known [1] [2]. It consists on searching the optimum way to transport a set of passengers which are territorially distributed in different locations, considering diverse constraints, for example the vehicle capacity and the time windows (TW) which are time intervals where a passenger can be picked or delivered on the respective location. Different versions of the dynamic Dial-A-Ride problem are founded in the literature. In the static version of the Dial-ARide Problem, it is customary to collect the requests for transportation the day before the beginning of the service. But in the dynamic version of the problem, the vehicle routing and scheduling must be done in real time so that the customers demand for the fast service is satisfied. So the problem is to assign every new passenger request to one of the vehicles and to design a new route and schedule for this vehicle. This assignment must be done in real time. DRP is a NP-hard problem [3]. So the exact methods are not able to solve such problem in a reasonable time, especially as the problem size is important [1]. In this case, we often use methods that find approximate solutions in a reasonable time
by applying heuristics and meta-heuristics, such as those based on genetic algorithms, simulated annealing, tabu search etc…[4][5][6]. In addition, it is a multi-criteria problem. So we need a multi-objective method to solve this latter. The multi-objective methods has a rather different aspect to scalar-objective one. In scalar-objective method the goal is to find one global optimum, on the other hand, the multi-objective methods must find a set of solutions. In our case, we use the Hybrid Algorithm Multi-Objective Simulated Annealing-Tabu Search (MOSA-TS) to solve the dynamic multi-criteria DRP. The remainder of the paper is organized as follows. The second part of this communication is devoted to the literature review of the dynamic DRP. In the third, we present our mathematical model for the DRP. Developed approach is described in part four. In the fifth section, the simulation results are presented. This is followed by conclusion and some perspectives for this work. II.
THE DIAL A RIDE PROBLEM (DRP)
A. A Dynamic DRP example In Fig. 1, we present a simple example of a situation for a vehicle in the dynamic context; the vehicle must provide service to both requests known in advance and new requests. Thus, +i and -i represent respectively the points of departure and destination of the customer i. Requests known in advance clients are represented by black nodes. The new requests are represented by white nodes. The time windows of pickups and deliveries points are represented by the values in brackets. The solid lines represent the planned route for the vehicle. The thick arrow indicates the position of the vehicle. The current route planned is as follows: pick-up of client 1, delivery of client 1, pick-up of client 2, delivery of client 2. When the new request is received at t=t1, the problem is to insert new departure and arrival points of the request 3 on the route minimizing objectives of the DRP. Fig. 1 represents at the same time a solution for the DRP.
978-1-4799-0313-9/13/$31.00 ©2013 IEEE 227
[11:30,12] -3
[12, 12:30]
[12, 12:30]
-2
[11,11:15]
[11,11:15]
+2
-2
[10:45, 11]
+2
+3
[10, 10:30]
-1
[8, 9]
+1
[10, 10:30]
-1
V1
t0
[8, 9]
+1
V1
t
t1
Figure 1. Schematic representation of a Dynamic DRP.
B. Literature Review DRP (Dial a Ride Problem) is an extension of the PDP (Pickup & Delivery Problem) where the transport of goods is replaced by the transport of persons [7]. Several versions of the DRP have been studied over the past 30 years. In the paper [6], we find a more detailed presentation of the state of the art of this problem. In this section, we give a brief literature review of the dynamic DRP. The majority of research has been focusing on the static DRP but Wilson et al, 1971 have solved the dynamic one [8]. Naba et al, have solved a dynamic DRP using a distributed scheduling algorithm [9]. A parallel algorithm for the dynamic DRP was developed by Attanasio et al [10], in his algorithm when a new request arrives; each of the parallel threads inserts the request randomly in the current solution and runs the tabu search for obtaining a feasible solution. Coslovich et al, [11] proposed an algorithm for a dynamic DRP; the algorithm maintains a repository of feasible solutions that are compatible with the partial routes already performed. When a new request arrives, an efficient insertion algorithm attempts to insert it in at least one of the solutions in the repository. The request is accepted only if the insertion algorithm succeeds. Xiang et al, [12] have studied a sophisticated dynamic DRP in which travel and service times have a stochastic component. Beaudry et al, [13] have developed a two-phase algorithm for solving a complex dynamic DRP arising in the transportation of patients in hospitals, such as insertion and tabu search. The algorithm succeeded in reducing the waiting times for patients while using fewer vehicles in computational experiments on real data from a German hospital. Zidi et al, [14] presented a new approach based on the algorithm of Multi-Objective Simulated Annealing (RSMO) for solving the dynamic Dial-a-Ride Problem (DRP). Aggregation and routing were performed by the local search procedure and neighborhood structure, their
approach has yielded remarkable results. But, RSMO algorithm has disadvantages in multi-objective optimization which are summarized in its inability to find a set of solutions. However RSMO can do the same work by repeating tests because it converges to the global optimum with a probability distribution. The Multi-objective Simulated Annealing-Tabu search algorithm (MOSA-TS) was not used to solve the dynamic DRP in the previous works mentioned in the state of the art. For this reason and the advantages of this algorithm we have chosen to apply it to solve this problem. Indeed the MOSA-TS algorithm uses the domination concept and the annealing scheme for efficient search, as well as a tabu list keeps the latest best solutions accepted. III.
MATHEMATICAL FORMULATION OF DRP
The DRP has been modeled mathematically in several research works. It is generally modeled by a multi-objective mathematical program. In this section, we present the mathematical modelling of our DRP. This model is characterized by two main objectives. The first one is economic, and the second is the quality of service given to travelers. In this work, we solve a multi-objective DRP using the MOSA-TS algorithm in the dynamic context. In the following part, we present our mathematical formalization of the problem. x Variables of DRP n: number of transport requests, A={1,...,n}: pickup locations , D={n+1,... ,2n}: delivery locations, M: set of vehicles depots , N=D U A U M: the set of all nodes in the graph, Request i: consist of pickup i and delivery n+i, Vi: set of nodes visited by a transport demand i, V: set of vehicles, Qv: capacity of a vehicle v, [ai bi]: time window of pickup point of demand, [ai+n bi+n]: time window of delivery point of demand,qi: number loaded onto vehicle at node i. qi=qn+i, Tijv: travel time from i to j with the vehicle v, Taiv: arrival time for the request i with the vehicle v, Tsiv: start time of service for the request i with the vehicle v, NSVi: the number of stations visited by a trans demand i of a client, Liv: the load of vehicle v after visiting node i, Cijv=CijCv: cost of travel from i to j with the vehicle such that Cv is the cost of using vehicle v, Xijv: decision variable of the problem , Xijv=1: if the vehicle v takes a direct path from i to j, else , Xijv = 0. x The Multi-objective function The Multi-objective criterion: Economic criterion+ Service Quality criterion Economic criterion
ECO
¦ ¦ ¦ X ijvCijv
(1)
RT NSV
(2)
iN jN vV
Service Quality criterion
SQ RT: Ride Time
NSV: Number of Stations visited by all transport demand
228
With
¦ ¦ (Taiv Tsiv )
RT
(3)
iD vV
¦ NSVi
NSV
iD
(4)
x Mathematical Model Minimize F ( X ijv )
(5)
Subject to
(11): A vehicle v must satisfy the time window of Delivery location i + n. (12): Ensures that the number of passengers passed on a path (i, j) by a vehicle v is conserved. (13): The number of passengers in the vehicle v after visiting i is higher than that collected in i and less than the maximum capacity of vehicle. (14): Ensures that the actual loads of the vehicles are set to zero at the depots. (15): Guarantees that decision variables Xijv will be binary. x Degree of dynamism The degree of dynamism, dod, of a model instance, is:
¦ ¦ X ijv 1
i D
(6)
v V , i D
(7)
j A D, v V
(8)
v V ,(i, j ) N
(9)
dod
vV jN
¦ X i jv ¦ X j ,n i ,v 0
jD A
jD A
¦ X ijv ¦ X jiv
iN
0
iN
X i jv (Tsiv Ti jv Tsjv ) d 0
i N , v V
(10)
v V ,(i, j ) N
(11)
X i jv ( Liv qvj L jv ) 0
i D, v V
(12)
qiv d Liv d Qv
i D, v V
(13)
m M , v V
(14)
ai d Tsiv d bi ai n d Taiv d bi n
Lmv
0
Xijv >0;1
i N , j N , v V (15)
x Description of constraints (5): The multi-objective function of the DRP taking into account the quality of service given to passengers. (6): Each customer will be assisted once, for just a vehicle. (7): A delivery place will always be in the same route that its respective pickup place. (8): The flow contention (everything that enters is the same to everything that leaves). (9): Ensures that the arrival time at location j must be later than the sum of departure time from location i and travelling time tijv between the locations if that leg is to be part of the route. For example, if vehicle v traverses arc (i,j), where j is a pickup node after service, then its departure time from node j is equal to the departure time from the previous node i plus travel time tijv. (10): A vehicle v must satisfy the time window of pickup point i.
ndyn
(16)
n
n: the total number of customer requests. ndyn: the number of dynamic requests. IV.
DEVELOPED APPROACH
The dynamic DRP is a multi-criteria problem [14]. So we need a multi-objective method to solve the problem. In this research we develop a hybrid approach based on the Multi Objective Simulated Annealing and Tabu Search (MOSA-TS) algorithm to solve the dynamic DRP. Indeed the MOSA-TS algorithm is characterized by a better exploration of the search space because the algorithm used a tabu list to store the solutions, so prohibiting moving to new configurations on the search space previously visited. Our approach is composed by 2 major’s phases. The first phase is the assignment of transport demands to vehicles. The second phase is the route planning for each vehicle. The initial solution of the MOSA-TS algorithm is generated by a distribution heuristic. In the second phase (route planning) we use the local search structure of the MOSA-TS algorithm to generate the best itinerary for each vehicle. Indeed, our approach benefits from the multi-agent techniques that have opened an efficient way to solve diverse problems in terms of cooperation, conflict, negotiation and concurrence within a society of agents. Our approach based on the multi-agent system developed in this research is composed by 3 major’s agents. The first one is client agent who sends a request to the Dispatching centre agent. The dispatching centre agent broadcasts this request for all the vehicle agents. The vehicle agent makes an optimization process based on the MOSA-TS algorithm. The dispatching centre agent collects all the responses from vehicles and it chooses the best one using the domination concept. A. Optimization with the Multi-Objective Simulated Annealing -Tabu Search algorithm (MOSA-TS): Agent vehicle In this research we propose an approach based on the Hybridization of the Multi-Objective Simulated Annealing (MOSA) algorithm and the tabu search (TS) algorithm. MOSA-TS use the domination concept and the annealing scheme for efficient search [15].
229
Indeed, MOSA inherent with good convergence property but lacking with keeping previously generated solution. However, due to lack of memory, search may oscillate around the local minimum. Therefore, incorporating a memory; one of the characteristics of TS leads to chance of getting prominent solution in the solution space. So, in our approach for not fall in the risk of recycling, we affected to the Multi-Objective Simulated Annealing algorithm a fundamental element in the tabu search: the tabu list which is a flexible memory [16], which allows keeping the last one better solution accepted. This is intended to prevent blockages in local minimum, by prohibiting the looking back too quickly to configurations already visited. But if the size of the tabu list k is too high, then unvisited configurations will be unfairly inaccessible and the ability of the method to exploit the neighborhood will be reduced. Conversely, if the size of the tabu list k is too low, then the method is very likely to be trapped in a local minimum. The tabu list therefore avoids all cycles of length less than or equal to k. In addition tabu list can prohibit attractive moves, so it is necessary to use algorithmic devices that will allow revoking tabu criterion of some moves. These are called aspiration criteria [16]. The simplest and most commonly used aspiration criterion consists in allowing a move, even if it is tabu, if it results in a solution with an objective value better than that of the current best-known solution. The pseudo-code of implemented MOSA-TS algorithm is described in Fig. 2. 1. Initialize parameters (Initial temperature T0, tabu list L,...) 2. CREATE (an initial solution S using the distribution heuristic method); 3. CREATE (any neighbor S' to S using neighborhood Structure); 4. IF S’ belongs in L, go to step 5; ELSE, go to step 6; 5. IF S ' does not qualify the aspiration Criterion, go to step 3; 6. IF C(S’) dominates C(S) Update the solution vector current. S ← S’; Put S’ in L ; Update the optimal solution; Go to Step 8; 7. IF C(S’) not dominate C(S) Take (z Є [0,1]); IF z
