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

28 April 2004Javier Faulín & Israel Gil1

DESCRIPTION OF THE ALGACEA-2 DESCRIPTION OF THE ALGACEA-2 ALGORITHM IN THE ROUTING ALGORITHM IN THE ROUTING

OPTIMIZATION IN CVRPOPTIMIZATION IN CVRP

Javier Faulín Department of Statistics and

Operations Research Public University of Navarra.

Pamplona, NA SPAIN

Israel Gil RamírezDepartment of Production. Guardian Glass Navarra.

SPAIN


ContentsContents

The Vehicle Routing ProblemNew algorithm ALGACEA-2 (Acronym of

Savings Algorithm with Bounded Entropy (SABE Algorithm), Second Version)

Conclusions


The Vehicle Routing ProblemThe 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.



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 = d1,i +d1,j – di,j



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.



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:

j ij

ij

j

d

dP

1

1

J

II

II

S

SP

1


New Algorithm ALGACEA-2New Algorithm ALGACEA-2ALGACEA-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:

ji

jiji S

SP

,

,,


New Algorithm ALGACEA-2New Algorithm ALGACEA-2

This means that it is necessary to manage a probability matrix as the following one:

nijii ppp ,,1, ......

nnjnn

nijii

nj

ppp

ppp

ppp

),1(),1(1),1(

,,1,

,1,11,1

......

...............

......

...............

......



We will control the balance in probability matrix using an Entropy function.

nnjnn

nijii

nj

ppp

ppp

ppp

),1(),1(1),1(

,,1,

,1,11,1

......

...............

......

...............

......

i

N

ii ppAH ln)(

1


New Algorithm ALGACEA-2New Algorithm ALGACEA-2The 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


New Algorithm ALGACEA-2New 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:

n Ent. n Ent. n Ent. n Ent. n Ent.

1 1 11 0,66 21 0,35 31 0,22 41 0,17

2 0,95 12 0,64 22 0,32 32 0,22 42 0,17

3 0,9 13 0,61 23 0,29 33 0,22 43 0,16

4 0,87 14 0,58 24 0,26 34 0,21 44 0,16

5 0,85 15 0,55 25 0,25 35 0,20 45 0,15

6 0,83 16 0,52 26 0,24 36 0,20 46 0,15

7 0,8 17 0,47 27 0,24 37 0,19 47 0,14

8 0,76 18 0,44 28 0,23 38 0,19 48 0,14

9 0,73 19 0,41 29 0,23 39 0,18 49 0,13

10 0,69 20 0,38 30 0,22 40 0,18 50 0,13



ALGACEA reckons up the matrix Entropy in each step of the algorithm using the following weighting coefficients successively :

[1 2 4 5 6 8 10 20 50 100]

It is shown that this choice policy involves better outcomes in the nodes insertion than using a fixed



The Savings Algorithm only finishes a route when the delivery vehicle has been filled up. But, this could cause problems:



But the most convenient delivery would have been the following one:


New Algorithm ALGACEA-2New 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)= 0,9Var (x)= 8,18 10-3

10


ConclusionsConclusions

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



Comparison of the final solutions for several algorithms in the optimization of the PLIGHT-1 Case.


ConclusionsConclusions Comparison of the final solutions for several

algorithms in the optimization of the Solomon I Case.


ConclusionsConclusions Comparison of the final solutions for several

algorithms in the optimization of the Solomon II Case.



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.


Documents

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