Vrptw algoritmos geneticos

Alumno: Guevara Hernandez Julio César

Profesora: Dra. Mayra Elizondo C.

Materia: Heurísticos

5/Mayo/2014

1

Contenido

Introducción VRPTW

Formulación

Elección de Meta-heurística.

Elementos del Meta-heurístico.

Algoritmo del GA.

Conclusiones

Referencias

2

10/03/2014 VRP with Time-Windows with AG

1. Introducción VRPTW

COMPLEJIDAD

3

OBJETIVO: Es encontrar las rutas para los vehículos que sirven a

todos los clientes a un mínimo costo (distancia y vehículos), sin

violar las restricciones de las capacidades y tiempos de viaje de los

vehículos, así como las restricciones de las ventanas de tiempo

para cada cliente



2. Formulación

4

3. Elección de Meta-heurística.

El AG y ST, ha mostrado ser

meta-heurísticas con mayor

tiempo computacional pero con

menor costo y cercanía al

optimo local global.(K.C. Tan,

2000)

¡Mejores formas de Resolver el VRPTW son!

5

4. Elementos del Meta-

heurístico El principal uso de de los AG, ha sido por

la realización de combinación de

heurísticas (heurísticos híbridos).

Las técnicas de GA que se han

combinado más, son tales como:

nuevos operadores de cruce y

probabilidad de mutación adaptativa,

para resolver problemas VRPTW con

resultados satisfactorios.

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4.1 Representación de la Meta-

heurística

Se vincula el último cliente visitado en ruta i

con el primer cliente visitado en ruta i+1

para formar una cadena, porque este tipo

de delimitadores en un cromosoma

restringen en gran medida la validez de los

niños producidos por las operaciones de

cruce.

Entidades de la cadena ó cromosomas artificiales. Cadena de longitud N,

donde N son numero

de clientes

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4.2 Creación de una población

La creación de la población inicial “buena” es por medio de la heurística PFIH

(Push Forward Insertion Heuristic). (Solomon,1987), la cual PFIH sirve en la

construcción de la configuración de las rutas de VRPTW, teniendo

aproximaciones a la solución GB (global-best).

Solución Inicial

• Dejar la solución So de PFIH y vecinos aleatorios describan una parte de la población particionada, relacionada con So

• El resto de la población se genera de forma totalmente aleatoria , sin relación con So (permite explorar mas soluciones)

• Fijar parámetros Rand_Ratio, la cual permite establecer la diversidad de la población. También refleja el nivel de confianza de la So

combined a number of GA techniques such as new cross-

over operators and adaptive mutation probability, to solve

VRPTW problems with satisfactory results.

7.1. Chromosome representation

Like in other GA applications, the members of a

population in our GA for VRPTW are string entities

of arti®cial chromosomes. The representation of a

solution we use here is an integer string of length N,

where N is the number of customers in question. Each

gene in the string, or chromosome, is the integer node

number assigned to that customer originally. The

sequence of the genes in the chromosome is the order

of visiting these customers. Using the same example,

the chromosome string that represents the solution in

Fig. 1 is now:

7 ! 8 ! 9 ! 11 ! 12 ! 2 ! 3 ! 1 ! 4 ! 5 ! 6 ! 10

…18†

This representation is unique, and one chromosome can

be only decoded to one solution. It is a 1-to-1 relation.

Note we link the last customer visited in route i with

the ®rst customer visited in route i 1 1 to form one

string of all the routes involved just like in the Tabu

list, but we do not put any bit in the string to indicate

the end of a route now, because such delimiters in a

chromosome greatly restrain the validity of children

produced by crossover operations later. To decode the

chromosome into route con®gurations, we simply insert

the gene values into new routes sequentially, similar to

PFIH. There is a chance that we may not get back the

original routes after decoding, but it is generally assumed

that minimising the number of routes helps in minimising

the total travel cost, therefore, packing a route to its maxi-

mumcapability impliesapotential good solution asaresult.

7.2. Creation of initial population

Under the same assumption that we have made in the

last sections, i.e. the solution from PFIH is reasonably

good and in the vicinity of the GB solution, we create

an initial population in relation to this solution. The

way to do that is by letting the PFIH solution S0 and

its random neighbours ; S[ Nl …S0† describe a portion

of the starting population. The rest of the population is

generated on a totally random basis, unrelated to S0.

The reason for having this mixed population is that: a

population of members entirely from the same neigh-

bourhood cannot go too far from there and hence

give up the opportunity to explore other regions.

The proportion of relevant chromosomes and random

chromosomes is governed by a parameter RAND_RA-

TIO. The higher this ratio, the more diverse the initial

population. This parameter also re ects the con®dence

level of the user to the PFIH solution. If there is a high

chance that global optimum is located in Nl S0), then it

is undoubtedly economical to have a small RAND_RA-

TIO so that the population converges to the optimum

sooner. The total population size is set at 100 and the

number of generations ranges between 500 and 1000, a

compromise between computation time and ®nal result.

Certainly, the more generations the program runs, the

more optimised the solution in most cases, unless the

optimum has already been reached.

7.3. Selection

After we have a population of candidates, we need some

mechanism to select parentsfor mating and reproduction. A

tournament selection mechanism isused for thispurpose. In

tournament selection, two identical copiesof thepopulation

of size N are maintained at every generation. In the begin-

ning, both populationsarearbitrarily ranked. For population

P1, each pair of adjacent chromosomes with indices 2i

and 2i 1 1) in the population are compared. The one with

K.C. Tan et al. / Arti®cial Intelligence in Engineering 15 2001) 281±295 289

Fig. 4. Tournament selection. Tournament selectionprocedure: the®rst timeselectionwill chooseN/2parents, after reordering thepopulation, thesecondtime

selection will choose the other N/2 parents.

Un pequeño

Ratio,

converge mas

rápido al

optimo global

El tamaño total de la población se fija en 100 y el número de generaciones oscila entre 500 y

1000

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4.3 Selección

Selección del

torneo El procedimiento de torneo de

selección: Al comienzo ambas

poblaciones se clasifican

arbitrariamente; la selección por

primera vez elegirá N/2 padres

(el par que tengan el menor

valor de aptitud), después de la

reordenación de la población, la

segunda selección de tiempo

elegirá los otros N/2 madres.

La implicación en este esquema de selección es que se les da prioridad a

cromosomas genéticamente superiores en el apareamiento, pero las entidades

promedio pueden tener alguna posibilidad de ser elegidos también, siempre y

cuando pasan a ser comparados con los cromosomas “peor situados”.

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smaller®tnessvaluequali®esto beapotential parent and let

uscall it fi. After comparing all thepairs in P1, wehaveN/2

`fathers' , namely f0; f1;¼; f…N2 2†=2: Repeat this process for

population P2, and we get m0;m1;¼;m…N2 2†=2; a set of

`mothers' as well. Subsequently, fi and mi are mated, and

totally there are N chromosomes chosen. The procedure is

graphically illustrated in Fig. 4. The implication in this

selection scheme is that genetically superior chromosomes

are given priority in mating but average entities have some

chance of being selected too, provided they happen to be

compared with `worse-off' chromosomes.

7.4. Reproduction

Reproduction is one of the most crucial functions in the

chain of biological evolution. Similarly, reproduction inGA

serves the important purpose of combining the useful traits

from parent chromosomes and passing them on to the

offsprings. An ef®cient and smart reproduction mechanism

is largely responsible for high GA performance. The repro-

ductionof GA consistsof twokindsof operations, crossover

and mutation.

Conventional single/double point crossover operations

are relevant to string entities that are orderless, or of differ-

ent length. They put two integer/binary strings side by side

and makeacut point or two cut points) on both of them. A

crossover is then completed by swapping the portions after

thecut point or between two cut points) in both strings see

Fig. 5). In thecontext of VRPTW, whereeach integer gene

appearsonly oncein any chromosome, such asimpleproce-

dure unavoidably produces invalid offspring that have

duplicated genes in one string. To prevent such invalid

offspring from being reproduced, we de®ne a set of order-

based crossover operators below.

The PMX crossover [12]. The permutation crossover

PMX) method proceeds by choosing two cut points at

random, e.g.

Parent 1: h k c e f d b l a i g j

Parent 2: a b c d e f g h i j k l

The cut-out section de®nes a series of swapping opera-

tions to be performed on the second parent. In the example

case, weswapbwithg, l withhandawith i, andendupwith

the following offspring:

Offspring: i g c d e f b l a j k h

Performing similar swapping on the®rst parent gives the

other offspring:

Offspring: l k c e f d g h i a b j

Heuristic crossover. A random cut is made on two

chromosomes. From thenodeafter thecut point, theshorter

between the two edges leaving the node is chosen. The

process is continued until all positions in the chromosomes

have been considered. Suppose, we have the following

parents:



Assume we choose b to be the ®rst gene in the new

chromosome, we have to ®rst swap b and g in parent 2.

After swapping, if the distance dbl . dbh, where dbl is the

distance between node b and node l in parent 1 and dbh is

thedistancebetween nodeb and nodeh in parent 2, wethen

choose h to be the next node and swap l and h in the ®rst

parent or delete h in the ®rst parent to avoid duplication

later. This process is continued until a new chromosome

of the same length and comprising all the 12 alphabets are

formed. Note that the result varies depending on whether

swapping or deletion is undertaken. We named heuristic

crossover with swapping HeuristicCrossover1 and Heuris-

tic Crossover2 otherwise.

Mergecrossover. Unlikeheuristic crossover, which rear-

rangestheparentsaccording todistancetoproducechildren,

merge crossover operated on the basis of a prede®ned time

K.C. Tan et al. / Arti®cial Intelligence in Engineering 15 2001) 281±295290

Fig. 5. Simplecrossover vs. ordered crossover: demonstrationof singlepoint andorder-basedcrossover operations. Singlepoint crossover needsonly onecut

point for eachchromosome. On theother hand, order-basedcrossover needstwocut pointsand repair of chromosomesmust beperformedafter theoperation.

smaller®tnessvaluequali®estobeapotential parent and let

uscall it fi. After comparing all thepairs in P1, wehaveN/2

`fathers' , namely f0; f1;¼; f…N2 2†=2: Repeat this process for

population P2, and we get m0;m1;¼;m…N2 2†=2; a set of

`mothers' as well. Subsequently, fi and mi are mated, and

totally there are N chromosomes chosen. The procedure is

graphically illustrated in Fig. 4. The implication in this

selection scheme is that genetically superior chromosomes

are given priority in mating but average entitieshave some

chance of being selected too, provided they happen to be

compared with `worse-off' chromosomes.

7.4. Reproduction

Reproduction is one of the most crucial functions in the

chainof biological evolution. Similarly, reproduction inGA

serves the important purpose of combining the useful traits

from parent chromosomes and passing them on to the

offsprings. An ef®cient and smart reproduction mechanism

is largely responsible for high GA performance. The repro-

ductionof GA consistsof twokindsof operations, crossover

and mutation.

Conventional single/double point crossover operations

are relevant to string entities that areorderless, or of differ-

ent length. They put two integer/binary stringsside by side

and makeacut point or two cut points) on both of them. A

crossover is then completed by swapping the portions after

thecut point or between twocut points) inboth strings see

Fig. 5). In thecontext of VRPTW, whereeach integer gene

appearsonly onceinany chromosome, suchasimpleproce-

dure unavoidably produces invalid offspring that have

duplicated genes in one string. To prevent such invalid

offspring from being reproduced, we de®ne a set of order-

based crossover operators below.

The PMX crossover [12]. The permutation crossover

PMX) method proceeds by choosing two cut points at

random, e.g.



The cut-out section de®nes a series of swapping opera-

tions to be performed on the second parent. In the example

case, weswapbwithg, l withhandawith i, andendupwith

the following offspring:

Offspring: i g c d e f b l a j k h

Performing similar swapping on the®rst parent gives the

other offspring:

Offspring: l k c e f d g h i a b j

Heuristic crossover. A random cut is made on two

chromosomes. From thenodeafter thecut point, theshorter

between the two edges leaving the node is chosen. The

process iscontinued until all positions in the chromosomes

have been considered. Suppose, we have the following

parents:



Assume we choose b to be the ®rst gene in the new

chromosome, we have to ®rst swap b and g in parent 2.

After swapping, if the distance dbl . dbh, where dbl is the

distance between node b and node l in parent 1 and dbh is

thedistancebetween nodeb and nodeh in parent 2, wethen

choose h to be the next node and swap l and h in the®rst

parent or delete h in the ®rst parent to avoid duplication

later. This process is continued until a new chromosome

of the same length and comprising all the 12 alphabets are

formed. Note that the result varies depending on whether

swapping or deletion is undertaken. We named heuristic

crossover with swapping HeuristicCrossover1 and Heuris-

tic Crossover2 otherwise.

Mergecrossover. Unlikeheuristic crossover, which rear-

rangestheparentsaccordingtodistancetoproducechildren,

merge crossover operated on the basis of a prede®ned time

K.C. Tan et al. / Arti®cial Intelligence in Engineering 15 2001) 281±295290

Fig. 5. Simplecrossover vs. ordered crossover: demonstrationof singlepoint andorder-basedcrossover operations. Singlepoint crossover needsonly onecut

point for eachchromosome. On theother hand, order-basedcrossover needstwocut pointsandrepair of chromosomesmust beperformedafter theoperation.


2.4 Reproducción. Cruce

La reproducción se realiza con el intercambio de las partes del cromosoma con

corte en el/los punto(s) de cruce Sencillo y Doble

En el contexto de VRPTW, donde cada gen aparece sólo una vez en

cualquier cromosoma, un procedimiento tan simple produce inevitablemente

la descendencia no válida (abortos) que han duplicado los genes en una

cadena. Para evitar tal descendencia no válido de ser reproducido, se

definen un conjunto de elementos de operaciones de cruce.

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Heurísticas de Operaciones de

Cruce

El cruce de permutación

(PMX)

• Método que pasa por la elección de dos puntos de corte al azar,los cambios se realizan en el padre 2.

Heurístico de cruce

• Un corte aleatorio se realizó en dos cromosomas. Desde el nododespués de que el punto de corte, se elige la más corta entre losdos bordes que salen del nodo. Se continúa hasta que se hanconsiderado todas las posiciones en los cromosomas.

Fusionar Cruzado

• Este método viene de precedencia tiempo a menudo se resumena partir de las ventanas de tiempo impuestas por cada nodo. Elautor creó como prioridad de la hora límite de llegada de cadanodo.

La cantidad de padres a combinarse, esta dada por la probabilidad de cruce

(0,1)

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1. PMX: En el caso del ejemplo, intercambiamos b con g, l con h y a con i. Los

cambios se realizan en el Padre 2

Descendencias : i g c d e f b l a j k h

Descendencias : l k c e f d g h i a b j

Ejemplo.

Parent1: h k c e f d b l a i g j

Parent2: a b c d e f g h i j k l

2. HC: Supongamos que elegimos b que es el primer gen en el cromosoma

nuevo, tenemos que primero canje byg en padre 2.

Si distancia dbl>dbh

entonces elegimos h para ser el siguiente nodo, se realiza intercambio l y h en el primer padre

o eliminar h en la primera matriz para evitar la duplicación posterior.

3. FC: Similar a la heurística de cruce, se selecciona un punto de corte al azar

y un primer gen B se elige al azar de los primeros padres y el intercambio se

hace para segunda matriz. El nodo, que viene antes en el precedente tiempo

se convierte en la siguiente gen en el cromosoma nueva.

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2.5 Operación Complementaria:

Mutación

Mutación:Evitar la sobrepoblación homogénea por traer,

(rasgos no relacionados al azar en la población

actual y aumentar la varianza de la población)

Parámetros:

Sólo cambiar de nodo y la secuencia de intercambio se utilizan aquí.

Esta es la probabilidad de mutación , o P mutation , teniendo en valores de [

0,1].

Valores

de P

mutation

Bajos

AltosÓptimos locales

indeseables

Convergencia

demasiado lenta

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precedence. This timeprecedence isoften summarised from

thetimewindowsimposed by each node. Theauthor created

such precedencefrom the latest arrival timeof each node. In

the same example:



Similar to heuristic crossover, a random cut point is

selected and a ®rst gene b is chosen randomly from the

®rst parents and swapping is done to second parent. The

node, which comes earlier in the time precedent

becomes the next gene in the new chromosome. Here,

we de®ne two kinds of merge crossover just like the

heuristic crossover case, MergeCrossover1 and Merge-

Crossover2, to differentiate the swapping and deletion

schemes.

It is noted that both heuristic crossover and merge

crossover produce only one offspring from a pair of

parents. Inspired by the fact that both geographic loca-

tions and time sequences are important in vehicle rout-

ing, we decided to combine the two crossover operators

so that two parents now produce two children, each of

which comes from either the heuristic crossover or the

merge crossover. Naturally, we thus have four combined

operators from the four combinations, namely H1M1,

H1M2, H2M1 and H2M2. Experimental results on the

total ®ve crossover operators including PMX indicated

that H1M2 outperforms the rest especially in clustered

data sets. Not every pair of parents ought to reproduce

in every generation. How many parents are to crossover

is governed by the probability of crossover, a ®x real

number between 0 and 1. In our GA, we set it at 0.77, a

moderate value usually used in other GA implementa-

tion as well. When a couple of parents are determined

not to crossover, they are copied verbatim to the next

generation.

Mutation is a complementary operation to crossover.

The main purpose of mutation is to avoid over homo-

geneous population by bring random, unrelated traits

into the present population and increase the variance

of the population. Several mutation operators have

been proposed in the literature and they are shown in

Fig. 6. Because the chromosomes are of ®xed length in

our implementation, only swap node and swap sequence

are used here. There is another important parameter asso-

ciated with the mutation operations. This is the probability

of mutation, or Pmutation, taking on values from [0,1]. Earlier

research has shown that excessive Pmutation values drives the

GA into convergence sooner than necessary, often resulting

in undesirable local optimal solutions; small Pmutation

produces the opposite result: intolerably slow convergence.

In this project, the author developed a unique adaptive

mutation probability scheme. The scheme adapts Pmutation to

the standard deviation of the population:

S ˆ

••••••••••••••••••XN 2 1

iˆ 0

…xi 2 x†2

N 2 1

vuuuut

…19†

where N is the population size, xi the ®tness value of indi-

vidual i and x is the average®tnessof the population. If Sis

greater or equal to a threshold value MINPOPDEV ˆ 5; a

minimum Pmutation ˆ Pmin ˆ 0:06 is used, otherwise

Pmutation ˆ Pmin 1 0:1 £ …MINPOPDEV 2 S†: Hence, we

can see a minimum mutation probability of 0.06 is always

guaranteed.

7.5. Further improvements

Hill-climbing is a supplementary measure that takes

advantage of local greedy search to improve the chro-

mosomes produced from crossover and mutation proce-

dures. In hill-climbing, a portion of the population are

randomly selected and decoded into their respective

solution form. These solutions then undergo a few itera-

tions of removal and re-insertion operations and are

eventually updated with the new, improved solutions.

Note this is not a complete l -interchange procedure

due to the moderately large time requirement by each

l -interchange. Furthermore, to reduce the complexity

and to prevent from over-reliance on good solutions,

the probability of a chromosome to be selected for

hill-climbing is only set at 0.5.

However, even after hill-climbing, there is still the possi-

bility of degradation of theentirenew generation. To restore

someof thegood chromosomesin theparent generation, the

worst 4% chromosomes in the child generation is sub-

stituted with the best 4% in the parents. Note that the

percentage of recovery should be less than the mutation

rate at any time to have somemutated chromosomesescape

the recovery process and bring the population out of a

premature convergence.

K.C. Tan et al. / Arti®cial Intelligence in Engineering 15 2001) 281±295 291

Fig. 6. Several common mutation operations: demonstration of swap node

and swap sequencemutation operations. Unlikeswap nodeoperation, swap

sequence operation involves reversing sequence of some nodes.Desarrollo de Esquema de probabilidad de mutación adaptativa.

El esquema se adapta P mutation a la desviación estándar de la población:

N :Tamaño_de_Poblacion

ix :El _ logaritmo_de_ la_aplitud _ del _ individuo_ i

x :Es_ la_aplitud _de_ la_media_ poblacional

Si S es mayor o igual a un valor de umbral MINPOPDEV = 5, se utilizara un

mínimo Pmutation = Pmin = 0,06 se utiliza,

De lo contrario Pmutation = Pmin + 0,1 *(MINPOPDEV-S).

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2.6 Mejoras

Hill-climbing es una medida complementaria que se aprovecha de la

búsqueda exhaustiva local para mejorar los cromosomas producidos a partir

de los procedimientos de cruce y mutación. En la Hill-climbing , una parte de

la población son seleccionados al azar y decodifica en su respectiva forma de

solución.

Sin embargo, incluso después de Hill-climbing, todavía existe la posibilidad de

degradación de toda la nueva generación. Para restaurar algunas de las

buenas cromosomas en la generación de los padres, los peores 4 %

cromosomas en la generación niño es sustituido con el mejor 4 % en los

padres.

Remoción Reinserción

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Algoritmo del GA

GA-1:Generar población inicial de cromosomas N (parte de PFIH

y su mutación y en parte de la selección totalmente al azar).

GA-2:

Evaluar el valor de la aptitud en términos de la función

objetivo para cada cromosoma x en la población ,

calcule la aptitud media y desviación estándar, de este

modo establecer la probabilidad de mutación;

GA-3:Crear una nueva población repitiendo los pasos

siguientes hasta que la nueva población se ha

completado;

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• Seleccione dos cromosomas padres de una población mediante la selección de torneo;

Selección

• Con una probabilidad de cruza, cruce los padres para formar una nuevadescendencia. Si no se realizó una cruza, la descendencia es una copiaexacta de los padres;

Crossover (cruza)

• Con una probabilidad de mutación, mutar nuevas crías en un lugar aleatorio ( nodo de sustitución o secuencia de intercambio);

Mutación

• Coloque nuevas crías en una nueva población;

Aceptar

• Realización de Hill-climbing con 1 – vecino de búsqueda First Best;

Hill- climbing

• Reemplazar los peores 4% cromosomas en la nueva población con el 4% de la mejor población de los padres ;

Recuperación

17


GA-4:Actualización de la antigua población con la población

recién generadas

GA-5:

Si se alcanza el número determinado de

generación, detener, llevar a cabo un 2 -

intercambio (GB ) sobre la mejor solución en la

población actual y devuelva la solución

mejorada;

GA-5: En caso contrario vaya al paso 2

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Referencia

- H.F. Wang, 2011; “A genetic algorithm for the simultaneous delivery and

pickup problems with time window”; Computers & Industrial Engineering;

National Tsing Hua University; 2011

- K.C. Tana, 2000; “Heuristic methods for vehicle routing problem with time

windows”; Artificial Intelligence in Engineering National University of

Sinagapore; Singapore,2000

- Nasser A., 2010; “Vehicle routing with time windows: An overview of exact,

heuristic and metaheuristic methods”; Journal of King Saud University ; Al-

Azhar University; April 2010

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Conclusiones20


- Los meta-heurísticos mas representativos para la resolución de VRPTW

hasta el momento ha sido por medio de los AG y Búsqueda Tabú.

- Es importante saber elegir la forma de cruce para que el método tenga

convergencia mas rápido.

- La mutación como la mejora son importantes para obtener las mejores

soluciones y desechar las peores.

- Un mecanismo de reproducción eficiente e inteligente es en gran parte

responsable de alto rendimiento GA

PFIH es un método eficaz para introducir a los clientes en nuevas rutas.

El procedimiento es fácil y sencillo. El método intenta insertar el cliente

entre todas las aristas en la ruta actual. Selecciona el borde que tiene el

costo de inserción adicional más bajo. La verificación de plausibilidad

prueba todas las restricciones, incluidas las ventanas de tiempo y

capacidad de carga. Sólo se aceptarán las inserciones factibles. Cuando la

ruta actual está lleno, PFIH comenzará una nueva ruta y repita el

procedimiento hasta que todos los clientes se dirigen. Por lo general, PFIH

da una razonablemente buena solución factible en términos del número de

vehículos usados. Este número inicial de vehículos proporciona un límite

superior para el número de rutas en la solución .

PFIH sirve el papel de la construcción de la configuración de las rutas de

VRPTW. Es un método eficiente para obtener soluciones factibles . La

información detallada se puede obtener a partir de papel de Salomón.

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Vrptw algoritmos geneticos

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