28 April 2004 Javier Faulín & Israel Gil 1 DESCRIPTION OF THE ALGACEA-2 ALGORITHM IN THE ROUTING OPTIMIZATION IN CVRP Javier Faulín Department of Statistics and Operations Research Public University of Navarra. Pamplona, NA SPAIN Israel Gil Ramírez Department of Production. Guardian Glass Navarra. SPAIN

28 April 2004 Javier Faulín & Israel Gil 2 Contents The Vehicle Routing Problem New algorithm ALGACEA-2 (Acronym of Savings Algorithm with Bounded Entropy (SABE Algorithm), Second Version) Conclusions

28 April 2004 Javier Faulín & Israel Gil 3 The Vehicle Routing Problem The new algorithm ALGACEA-2 has been conceived for solving VRPs in a constructive and intuitive way. ALGACEA-2 solves specifically Capacitated Vehicle Routing Problems (CVRPs). It is a minimisation problem in distances and vehicles having only one depot, a set of customers to visit and a vehicle fleet limited in number and in capacity.

28 April 2004 Javier Faulín & Israel Gil 4 The Vehicle Routing Problem The Clarke and Wright’s Savings Algorithm takes into account the saved distance if two customers independently served unifies their routes using the same delivery vehicle. 1 i j S = d 1,i +d 1,j – d i,j

28 April 2004 Javier Faulín & Israel Gil 5 The Vehicle Routing Problem The Monte Carlo techniques assume that each potential node has a concrete probability to be joined to a specific route. The choice is randomly performed.

28 April 2004 Javier Faulín & Israel Gil 6 The Vehicle Routing Problem The Monte Carlo techniques in the VRP arena has initially been developed by Buxey (1979) and Fernandez & Mayado (2000) giving the following probabilities of introducing a node in a route respectively:

28 April 2004 Javier Faulín & Israel Gil 7 New Algorithm ALGACEA-2 ALGACEA-2 is a generalisation of the Buxey’s algorithm, assigning probabilities of insertion to each node in each couple of nodes. The probability of joining a concrete node in a specific place between nodes i and j is:

28 April 2004 Javier Faulín & Israel Gil 8 New Algorithm ALGACEA-2 This means that it is necessary to manage a probability matrix as the following one:

28 April 2004 Javier Faulín & Israel Gil 9 New Algorithm ALGACEA-2 We will control the balance in probability matrix using an Entropy function.

28 April 2004 Javier Faulín & Israel Gil 10 New Algorithm ALGACEA-2 The tests showed the existence of an Entropy threshold below which the probability distribution is appropriate for building outstanding routes. Entropy Information Level Bounded Entropy Zone

28 April 2004 Javier Faulín & Israel Gil 11 New Algorithm ALGACEA-2 The maximum level of Entropy varies with the number of customers to deliver. Several tests were carried out, generating the following upper bounds for Entropy: nEnt.n n n n 11110,66210,35310,22410,17 20,95120,64220,32320,22420,17 30,9130,61230,29330,22430,16 40,87140,58240,26340,21440,16 50,85150,55250,25350,20450,15 60,83160,52260,24360,20460,15 70,8170,47270,24370,19470,14 80,76180,44280,23380,19480,14 90,73190,41290,23390,18490,13 100,69200,38300,22400,18500,13

28 April 2004 Javier Faulín & Israel Gil 12 New Algorithm ALGACEA-2 ALGACEA reckons up the matrix Entropy in each step of the algorithm using the following weighting coefficients  successively : [ ] It is shown that this choice policy involves better outcomes in the nodes insertion than using a fixed 

28 April 2004 Javier Faulín & Israel Gil 13 New Algorithm ALGACEA-2 The Savings Algorithm only finishes a route when the delivery vehicle has been filled up. But, this could cause problems:

28 April 2004 Javier Faulín & Israel Gil 14 New Algorithm ALGACEA-2 But the most convenient delivery would have been the following one:

28 April 2004 Javier Faulín & Israel Gil 15 New Algorithm ALGACEA-2 ALGACEA could finish a specific route according to the load level of the delivery vehicle in relation to a Beta (9,1) distribution. The average load level obtained using this procedure is around 90%. E (x)= 0,9 Var (x)= 8,

28 April 2004 Javier Faulín & Israel Gil 16 Conclusions ALGACEA-2 was tested in three cases: PLIGHT-1, Solomon I and Solomon II. The ALGACEA-2 outcomes were compared to the solutions given by other algorithms: CWS Algorithm, FGMS Method, Sweep Algorithm, Buxey’s Algorithm and GRASP Procedure

28 April 2004 Javier Faulín & Israel Gil 17 Conclusions Comparison of the final solutions for several algorithms in the optimization of the PLIGHT-1 Case.

28 April 2004 Javier Faulín & Israel Gil 18 Conclusions Comparison of the final solutions for several algorithms in the optimization of the Solomon I Case.

28 April 2004 Javier Faulín & Israel Gil 19 Conclusions Comparison of the final solutions for several algorithms in the optimization of the Solomon II Case.

28 April 2004 Javier Faulín & Israel Gil 20 Conclusions ALGACEA-2 improves the remaining methods in the majority of cases, whether we use time or distance ALGACEA-2 is simpler in calculations than the GRASP methods, the Buxey’s algorithm and the FGMS procedure. ALGACEA-2 usually involves more routes than other methods, but without a great increase in number of vehicles.

28 April 2004 Javier Faulín & Israel Gil 21