Application of Genetic Algorithms for the Design of Large-Scale Reverse

Application of Genetic Algorithms for the Design of Large-Scale Reverse

Logistic Networks in Europe’s Automotive Industry

Ralf Schleiffer, Jens Wollenweber, Hans-Juergen Sebastian University of Technology Aachen, Templergraben 64, 52072 Aachen, Germany

E-mail: {schleiffer, wollenweber, sebastian}@or.rwth-aachen.de

Florian Golm, Natasha Kapoustina Ford Research Center Aachen, Suesterfeldstrasse 200, 52072 Aachen, Germany

E-mail: {nkapoust, fgolm}@ford.com

Abstract

After providing a brief overview on Europe’s

legislative situation with regard to the demanded

recovering and recycling of end-of-life vehicles the

paper characterizes the specifications of a modeling approach describing the design of a recycling

network including a large variety of different

cooperating actors set at diverse positions on a

reverse supply-chain. It advances with a discussion

of requirements a suitable optimization approach

needs to fulfill. Then it proceeds by addressing the optimization approach that has been chosen and it

concludes by providing results for an ideal network

design.

1 Introduction

At present there are between seven million and

nine million end-of-life vehicles (ELVs) of classes

M1 and N11, which annually have to be cleared

within the European Community.2 According to the

German consortium Arbeitsgemeinschaft Altauto it is

estimated that between 1.1 million and 1.9 million of

these vehicles will emerge in Germany only [1]. In

order to handle the associated environmental

problems the European regulation for ELVs became

effective in April 1998, and with it the voluntary

pledge regarding the environmentally sound

management of ELVs within the framework of the

1 The class M1 contains vehicles for passenger transport with a

maximum of eight seats, not including the driver’s seat. The class N1 accumulates vehicles for good transport with a

maximum permissible weight of up to 3.5 tons. 2 All terms are used according to the English translation of their

definition in the German law concerning the recovery of end-of-

life vehicles, version three, 7 August, 2001 (Gesetz über die

Entsorgung von Altfahrzeugen, Lesefassung Artikel 3, Stand 7. August 2001).

closed substance cycle and waste management act

signed by all parties involved in Germany’s

automotive industry, including those companies that

collect, recycle, recover and dispose ELVs. The goal

of both, the European regulation and the voluntary

pledge, is the environmentally sound utilization and

disposal of ELVs of classes M1 and N1.

The current version of Germany’s Gesetz ueber

die Entsorgung von Altfahrzeugen (AltfahrzeugG)

from 21 June 2002 obligates all manufacturers of

vehicles belonging to any of the two classes, to take

back vehicles of their brand at approved acceptance

or collection facilities free of charge for the vehicle’s

current owner. In particular, it is demanded that such

a facility exists in a reasonable distance3 for each

owner. Apart from the establishment of such an area-

wide network of facilities to retract ELVs the ELV

ordinance defines ratios to be complied with during

the subsequent treatment of ELVs and stripped

vehicles. Thus, it is demanded that by 1 January 2006

at least eighty-five percent of a vehicle’s weight have

to be recovered, reused and utilized, and that at least

eighty percent of a vehicle’s weight has to be

recovered and recycled.4 A further aggravation of

these ratios is intended for 2015.

Specialists estimate that from twenty-five up to

thirty percent of the total costs of the recycling

network to be installed occur in the domain of

logistics. Apart from pure transport costs, which must

be beard by the manufacturer at last, this fact implies

an increasing burden for the transportation network,

which is hardly accessible at present, as well as a

clear raising of transport-caused emissions. In the

terms of both, economical and ecological aspects, the

exhaustion of the potentials for possible reductions of

3 A reasonable distance is a distance that is below 50 km. 4 Weight percentages always refer to the accumulated weight of all

ELVs per year.

Proceedings of the 37th Hawaii International Conference on System Sciences - 2004

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the necessary transportation for the common welfare

compatible recycling of ELVs is thus urgently

required. Above all the methods and approaches of

reverse supply-chain-management combined with

operations research modeling techniques provide the

necessary capabilities to find solutions for this

problem, which due to a suitable network design

contribute to the reduction of transportation,

measured in terms of cost, mileage and

environmental pollution, and which actively support

the inter-modal transfer of goods. In order to achieve

the goals of a high-standing and environmentally

compatible treatment of ELVs, an integrated network

design, that apart from facilities to take back, to

dismantle and to utilize ELVs also covers shredding

enterprises and facilities for the subsequent treatment

of valuable material, is inevitable.

In the following we will report on the

development of a decision-support tool. The tool

enables the generation of networks that consist of

locations where owners can return their vehicles

(acceptance and collecting), dismantling facilities,

shredding companies, post-shredding facilities,

enterprises for the reuse, recycling, recovery and

disposal of material of ELVs, and that helps reducing

the number of required transports to a minimum.

2 Facility Location Problem

The task that can be derived from the brief

explanation above is to solve some sort of a facility

location problem. The simplest form of this problem

is that given a set of facility locations and a set of

customers who are served from these facilities it has

to be decided, which facilities should be used, and

which customers should be served from which

facility, so as to minimize the total cost of all

customers, measured as a weighted sum of all

distances between a customer and the facility that

serves him (figure 1).

Figure 1. Facility location problem (example)

2.1 Customers and availability of ELVs

The circles in figure 1 represent the location of

customers, who are in our case the last owners of

ELVs. It is not necessary that this is a one-to-one

relationship, i.e. that each circle represents exactly

one last owner. The area shown can as well be

considered as being divided into N∈n distinct and

non-overlapping two-dimensional zones iZ , ni ≤≤1 ,

which unification is the whole area under

consideration and where n equals the number of

occurrences of the circles. In our approach these

zones relate directly to a considered European

country’s zip-code areas. The ELVs within the

borders of zone iZ add up to the total supply of

ELVs N∈is in iZ . Historic data about cleared cars

is taken to forecast this number of ELVs per zone.

The forecasted numbers can be modified within the

tool to validate optimization results with different

scenarios.

We assume that is is concentrated at a single

point ( )iii Zyx ∈, , which is the geographical centroid

of iZ , determined by the surrounding traverse. This

assumption limits the granularity of our approach to

the granularity of zip-code areas.

Supposing that a facility jF , N∈≤≤ mj1 with

unconstrained capacity is located at the point

( )n

i

ijj Zyx1

~,~

=

∈ and having ( ) R∈ji FZdist , denote the

distance between zone iZ and facility jF , we model

transport cost ic per distance unit as a non-

monotonic and non-continuous function

( )( ) R∈jii FZdistc , that is equal to zero for all

transports within a user-defined “reasonable

distance”, R∈rd , and that for distances higher than

this point of discontinuity gradually decreases. Then

the total transport cost of taking back zone iZ ELVs

by facility jF is ( )jiii FZdistcs ,⋅⋅ and the problem to

be solved is to minimize the sum of all these cost

over all zones iZ and over all facilities jF , i.e.

( )= =

⋅⋅⋅n

i

m

j

ijjiii sFZdistcs1 1

,min where [ ] R⊂∈ 1,0ijs is a

real-valued decision-variable indicating the portion

of zone iZ ELVs served by facility jF . Note that

modeling ijs as a real-valued variable instead of

choosing a binary form enables allowing each zone

iZ to be served by multiple facilities. Hence we

assume that the portion of customers of zone iZ

forwarding their ELV to facility jF decreases with

increasing distance as long as ( ) rdFZdist ji ≤, .


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Otherwise we assume that the accordant ELV is

collected by the vehicle’s manufacturer and shipped

to that facility jF for which ( ) ( )jiji FZdistFZdist ′< ,,

for all { } jmj FFFF /,,1∈′ . Therefore it is clear

that as m and as rd increase the inbound cost for

the union of all facilities jF converges to zero. And

thus with rd fixed a lower bound for m so that

( ) 0,1 1

=⋅⋅⋅= =

n

i

m

j

ijjiii sFZdistcs can easily be

determined.

Up to here we considered only those facility

location problems in which a set of alternative

locations are provided and the task is to assign zones

to facilities and vice versa. With regard to the

problem to be solved here it is necessary to extend

the description above inasmuch as both, finding a

concrete position for each facility and determining

the number of facilities to be located, m , become

part of the problem. Thus our specific job consists of

three tasks, namely to determine the number of

facilities to be located, to identify positions for new

facility locations and to assign zones which are

served by these facilities. When m is large this

general problem is NP-hard. Then it becomes

relevant to use approximation algorithms or to

investigate important restricted situations where

polynomial time-algorithms can be applied.

2.2 Reverse supply chain

Making the problem more realistic requires

placing it within a broader context and to consider

also facilities kS , N∈≤≤ ok1 served by those

facilities jF which locations have to be determined.

Note that it is not necessary that each kS is served by

exactly one facility jF . kS being served by multiple

jF are an option, too.

Now, if m is low, e.g. because of facility

installment expenses, the number of ELVs, which

have to be picked up at the last owners place of

residence increases for the reason that for a growing

number of zones the reachability criterion cannot be

met. In these cases it might be cheaper to set up

further facilities for the retraction of ELVs, to group

collected ELVs for transshipment and then to

transport these groups to the closest facility jF , than

it is to pick up each ELV at the last owners home and

to ship it directly to the closest facility jF . In order

to model this we introduced an additional stage in the

reverse supply-chain, namely retraction facilities, lO ,

N∈≤≤ pl1 , which number and which locations have

to be determined. Noting { }pOO ,,:1

=O we obtain

an supply-chain consisting of the links

SFOZ →→→ .

At all the specific problem definition considered

here can be classified as a multi-level facility

location problem striving to minimize the overall cost

of the recycling system as well as its environmental

impact. Thus transport cost between an ELV’s

location and the place where it is taken back and

inserted into a recycling process are only partially

assigned to the vehicle manufacturer.

ELVs that have been returned to an approved

retraction facility are shipped to a dismantler where

the intrinsic dismantling and recycling processes

commence.

The choice of transport means is of crucial

influence on the transport cost that arise per vehicle

until it arrives at a dismantler’s place. Thus

transportation with trucks, railway and inland

waterway ship are regarded, whereby apart from the

criteria of transport costs and environmental impact

also flexibility, speed, availability, load factor and

transport-chain integration are considered.

In this transport phase the following assumptions

are made:

• If either a retraction company or a dismantler is

reachable within a pre-defined distance from a last

owner’s place of residence the last owner is

responsible for returning his ELV and to pay for

the transport.

• If neither a retraction company nor a dismantler is

located within a pre-defined distance from a last

owner’s place of residence the shipment of the

ELV from the last owner’s residence to either a

retraction company or a dismantler is paid by the

vehicle manufacturer.

• Transports between retraction facilities and

dismantlers are typically performed by truck.

Optionally the ELVs are picked up at the retraction

company, transported to a transshipment place,

transferred to another transshipment place, picked

up by truck and delivered to a dismantler. The

transport between two transshipment points is

always performed by mass transportation, i.e. by

rail or inland waterway ship, with regard to the

availability of the corresponding transport mode.5

At a dismantler various parts of the ELV are

removed. The disassembly depth as well as the

capacity of a dismantler influence the disassembly

cost per vehicle. Of special importance is also a

possible spatial separation of the single disassembly

processes. For instance the evaluation and the

dewatering of an ELV and in consequence also the

5 Due to small distances within the concrete German recycling network only transport by truck is considered.


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disposal of fluids can take place at another place than

the actual disassembly. Here it is assumed that

evaluation, dewatering and disassembly of an ELV is

performed at a single location, though the model can

be extended to cover other options as well.

After the disassembly process is completed the

single components of the ELV are transported to

further facilities where these components are

recovered, recycled, re-used or disposed. These

facilities are considered as the customers of

dismantling enterprises. They include shredders,

distribution centers for used parts, disposal sites and

specific facilities for the recycling of fuel, used oil,

brake fluid, steel, aluminum, copper, other types of

metal, plastic, glass, rubber and batteries.

If information about additional facilities becomes

available our approach allows including these within

the optimization. Thus the specific facility location

problem considered here can be described as

determining the optimum number of retraction and

dismantling companies together with their locations

and their ideal capacities with regard to an optimal

selection of customers and post-dismantling

enterprises out of given sets of customers and

facilities, given a number of ELVs which are

reasonably distributed over the set of retraction and

dismantling companies due to the location of these.

The description above illustrates the need to

consider the input parameters of the multi-level

facility location problem. These input parameters

contain

• the location of the single parties involved in the

recycling process, including current owners of

ELVs (the distribution of ELVs),

• the characteristics of single ELVs, e.g. brand and

age,

• the availability of alternative transport modes

between these parties (including multi-modal

transport options), as well as mode-specific travel

distances, travel times and cost,

• the characteristics, capacity restrictions and cost of

single facilities

• and the political, the regulative and the economical

situation in single European countries.

3 Problem Solving Approach

The objective of our search for solutions to the

problem considered here is not the identification of a

globally optimal network design. This follows

straight away from the circumstance that due to the

constantly expanding availability of information

those networks which are today identified as optimal

need not be considered optimal anymore when new

information, e.g. a new group of customers is

included tomorrow. Therefore the focus of the

approach here is to provide decision-makers with

further insights into the problem structure by

identifying characteristics of good solutions. In

particular the idea is to dynamically generate and

identify solutions with regard to the decision-maker’s

objectives and optimization criteria. Currently these

criteria are the minimization of overall costs N∈cost

and environmental impact 8R∈tenvironmen whereby

a modification of these objectives is possible at any

time. Hence the objective function can be written as

( )tenvironmencostmin with denoting a

conjunction to be specified later.

3.1 Decision variables

Decision variables are the number of dismantling

facilities mm ≤ and the number of retraction

facilities pp ≤ , their locations, their capacities and

the topology of the network, i.e. the selection of

customers served by the dismantling facilities and the

transshipment points used throughout the single

paths.

Working on the basis of (five-digit) zip code

areas simplifies the problem in a notable way

because both the number of dismantling facilities that

can optionally be placed and the number of locations

for these facilities is bounded. The same is true with

regard to retraction facilities. Note that all other sets

considered here contain a finite number of elements,

too.

3.2 Optimization strategy

As experience has shown, during the process of

designing suitable and better performing recycling

networks it is best to present more and more

acceptable and reasonable designs to the decision-

makers in order to enable them to develop their

requirements to find network designs of high

performance to all considered obligations. Genetic

algorithms (GA), a family of stochastic optimization

techniques, are expected to do so. They give

multiple, acceptable and near optimum solution

candidates in large, possibly unstructured and non-

smooth search spaces. This means that decision-

makers are offered the possibility to examine these

candidates and that their judgment on these gives

them more information to develop their preference

statements with regard to various requirements.

3.2.1 Real-life evolution

The foundation of evolutionary theory comes up

with the synthesis of Darwinian theory and

Mendelian genetics. It is the idea that all species

share the same descent and that distinct species arise


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through adaptive processes [7]. During many

generations those species that are best adapted to

their environment survive while other species die out

and others in turn emerge newly because of cross

breeding and mutation.

In natural systems any individual member of any

species can be described by its genetic package. The

genetic code is contained in a sequence of discrete

units called chromosomes. Each chromosome is a

linear structure of genes, which can be of different

types and which includes information. Pairing

disrupts the structure of these chromosomes as genes

are added, deleted or exchanged and thus pairing

might be a source of mutation that passes on from

generation to generation. So during the reproduction

process it is natural that the offspring’s genetic code

becomes a mixture of the parents’ genetic make-up.

And it is also natural that during this process several

chromosomes mutate.

Such mutations of single genes usually have very

little effect but in some instances they rise to protein

whose function is severely disrupted. Sometimes the

changes in the genes are of such a negative influence

that the individual has difficulties to live under the

current environmental conditions. And in this case it

has a high probability to die and not to participate in

the reproduction process that forms a future

generation. Nevertheless it can also happen, that the

genetic changes confer benefits, that they lead to

desirable characteristics, which enable the individual

to survive more efficiently than others of its species

do. In this case the individual’s chance to spread its

genetic code through following generations

increases.

Indeed, all present species have probably

inherited a whole host of mutations that have

occurred among their ancestors. Inheritance of

parents’ genes and mutation lead only to slight

changes in the genetic code, as even mutated

offspring is recognizable similar to its parents.

Therefore the evolutionary change is gradual. Solely

at certain specified times the slow tempo of genetic

evolutionary adaptation is punctuated by a fairly

rapid speeding up because of environmental changes.

3.2.2 Artificial GA evolution

The way in which genetic algorithms explore the

search space can be considered analogous to the

basic evolution theory for nature.

The fundamental principles for building an

artificial system that reproduces and mimics the

workings of evolution date back to the biologist

Barricelli who formulated an article about artificial

methods to realize evolutionary processes [3].

Although he intended to support the comprehension

of natural happening, his work was not far away from

the ideas on which Holland’s genetic plans were

building [21, 22]. Barricelli proposed selection,

mutation, and production of offspring, either with or

without replacement of a parent, and crossing to be

essential for the evolutionary processes. In a

traditional genetic algorithm individuals are

represented by their chromosomes, which are

encoded by binary bit strings of fixed and equal

length. They are initialized by chance in order to

establish a first generation. Then, based on an

objective function, a fitness test is performed and

every individual is given a fitness score relative to

the fitness of all other individuals. Next a new

population, consisting of the offspring of the actual

one is created. To do so, individuals are randomly

selected out of the current generation. The

probability for such a selection of any individual as a

parent for the next generation is modeled in

proportion to the individual’s fitness score, so that

the probability that better fitting individuals are

chosen for mating is higher than the probability to

chose worse competitors. Once it is determined how

often any individual is selected, so to say how often it

mates, each individual is as often paired with

randomly selected other individuals as it itself has

been selected. For each pair one position on the bit

string is randomly chosen and the two individuals

swap all bits either left or right from the selected

position. Afterwards every bit of the two new

chromosomes is inverted with a probability that is

close to zero. The resulting new chromosomes form

the offspring. After pairing has been performed for

all selected individuals the offspring forms the next

generation that then itself is exposed to a fitness test

starting the reproduction process again. This

procedure is repeated until a pre-defined number of

generations is reached. And if some pre-conditions

are met the final generation consists of a large

number of individuals, which are highly adapted to

their environment.

In order to understand the way in which

adaptation works, it is essential to return to the

genetic description of individuals. Any chromosome

on its own is only a piece of information. But in

combination with other chromosomes a new source

for information becomes available. Reflecting on

similarities among the genetic strings this new

information can be expanded inasmuch as patterns in

the strings, so called schemes, can be made out.

Such a schema is a string over the original

alphabet that is extended by a “don’t care” symbol as

a placeholder for a not specified letter. It is of the

same length as a chromosome describing an

individual. Therefore a schema categorizes a class of

similar genetic strings. Especially highly fit schemes

with short-defining length, so called building blocks,


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are propagated from generation to generation

increasing the number of well adapted individuals

[17]. These building blocks are the lock opener for

Holland’s schema theorem that explains that the

expected amount of fit schemes increases from

generation to generation, since remarkably fit

individuals are more likely to pass their genes to the

next generation than less fit ones do.

Pairing, since it disrupts schemes, is the source

for variation and for innovation. In particular

mutation and crossover are those operators on which

variation is build. Crossover generates new schemes

by combining components of successful ones based

on the knowledge of past adaptation. The schemes

obtained are excerpts of the information included in

the original chromosomes. The other operator,

mutation, acts rather blind. Variation through

mutation leads to chromosomes with new and

unknown schemes for which success cannot be

guaranteed.

In particular a GA exhibits several major

characteristics that make it exceptionally attractive:

• One of its great advantages is the easy applicability

to changes in a stated problem. A GA is a kind of

self-learning algorithm that can incorporate new

information as characteristics of the search space

change, i.e. if changes occur it can go on using

what it has already learned and can apply the

changes during the following generations in the

sense that if a point in the search space has been

proved useful in the past, it is possible that it will

be useful in a new, but similar situation.

• Furthermore a GA’s type of search gives it the

capability to escape from local optima, whereas

greedy algorithms may not, i.e. if one individual

becomes trapped on a local optima others avoid

this area of the search space.

• The next advantage to be pointed out is that there is

no need for the computation or even the existence

of derivatives. In fact, continuity of the functions is

not required. So a GA can work with a much

broader class of functions than most other

algorithms.

For further details on genetic algorithms refer to

[9, 10, 11]. For a discussion of different operators

and coding schemes refer to [27, 28, 30, 32] and the

references given therein.

3.3 The GA used for network optimization

In our approach we apply a binary coded GA,

which coding scheme reflects a one-gene-one-facility

correspondence on a linear string, which sub-strings

represent different types of facilities. The first

generation is created by chance to produce a large

diversity among the individual networks, i.e. for

every single gene we perform a random experiment,

and decide with a fifty-to-fifty chance whether it is

seeded with “ 0 ” or with “1 ”. Due to the danger of

pre-mature convergence no user-definable

initialization, e.g. heuristic or stochastic a-priori

determination of network alternatives [6, 14], is

applied.

Once the first generation is fully initialized we

perform a fitness proportionate reproduction, i.e.

roulette wheel selection, in combination with an

elitist strategy that copies a pre-defined percentage of

the best networks into the next generation [9, 18].

This enables the search to focus more deeply on

those regions that are already identified as relatively

good in relation to others and to secure that the best

solutions do not get lost within the reproduction

phase. However this strategy can be a source for too

fast convergence towards a possible super network,

i.e. one with much higher fitness than the others in its

population. Options to avoid this are the application

of a ranking system with the expected number of a

network’s offspring depending on its rank according

to its fitness value, and being independent of the

fitness’ magnitude [2], the usage of linear

normalization [9], of fitness scaling [31] or of sharing

functions [16, 26, 25], all three reducing the

difference between the expected number of offspring

among the individuals of one generation, and the

limitation of the number of accepted offspring per

individual depending on the age of the generation.

Applying these methods not only a dominant

individual contributes to the gene pool of future

generations but the others do so as well.

In the version reported here our GA applies

independent single gene mutation, limited to a

Hamming distance of one between the network

before and after mutation,6 and one-point crossover.

The initial probabilities that a network participates at

crossover and at mutation have been set to forty and

to twenty percent, respectively. In addition a fight

operator, evolving on the survival-of-the-fittest

Darwinian principle, and working similar to a

simulated annealing strategy has been implemented.

The probability for its usage is initialized with five

percent.

Naturally, as further insights into the optimization

problem become available there are options to

improve the encoding of the networks’ phenotypes as

well as the choice of operators, which obviously

depend on one another. However, today it is not clear

whether the objective function is smooth and regular

so that networks with reasonable fitness are similar to

each other in terms of their Hamming distance, or

6 Although this limits the GA’s fast exploration of various regions

of the search space it secures that highly fit schema are not heavily destroyed by mutation.


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whether we are dealing with an objective function

with many local optima and one global optimum that

might be isolated on a flat surface, i.e. an impulse

function problem. Thus there is the hope that future

experience with the specific problem will guide us in

the development of even better performing operators

and coding schemes.

3.4 Strategy variables

The selection of parameter settings for the GA’s

strategic variables is of fundamental concern for the

performance of the algorithm as it influences the

balance7 between exploring the search space and

exploiting the best network alternatives that are

already obtained. A poor choice of values for these

variables can direct the search as well to premature

convergence in sub-optimal solutions as it can lead to

a random walk [20]. For instance increasing the

crossover rate raises the chance to recombine

desirable schema but at the same time it raises the

probability to destroy them, too. Similar twofold is a

rise in the mutation rate. On one hand lost genetic

information is replaced but on the other hand there is

the risk of resulting in a random walk.

During recent years researchers spent quite a lot

of work on finding ideal settings for a GA’s strategy

variables. In his Ph.D. thesis De Jong suggested a

population size between fifty and one hundred

individuals in combination with crossover and

mutation rates of sixty and of one-tenth percent

respectively, to be most efficient in his test problems.

These settings were mainly used by many other

researchers because determining settings that are

optimal for a particular problem is often more

difficult and computationally more expensive than

solving the problem with worse parameter

adjustments [8]. Eleven years later, Grefenstette

came up with a parameter set that outperformed De

Jong's choice by improving the average fitness per

generation in the same test problems that were used

by De Jong. Grefenstette's findings that became the

new standard settings very soon were a population

size of thirty individuals, a crossover rate of ninety-

five percent and a mutation rate of one percent. In

1985 Goldberg offered a theoretical study on

specifying fixed population sizes a priori. He

computed an optimum value with regard to the

expected number of new schemata per individual and

defined it depending on the length of the genetic

strings as ringgenetic_stlength_of_.. ⋅⋅ 2102651 , causing immense

exponentially increasing population sizes.

Controversy to this finding, Schaffer’s empirical tests

7 E.g. selection and fight put pressure on the individuals by

reducing the genetic variation while crossover and especially mutation are inclined to persevere it.

came to the result that a population size between

twenty and thirty individuals, a crossover rate of

between seventy-five and ninety-five percent and a

mutation rate between a half percent and one percent

were best to gain an optimal performance on average

fitness.

Unfortunately, fixing strategic variables to

optimal values depends strongly on the

characteristics of the unknown search space and what

it even more important: it depends on the actual stage

of the search process [19], since first high quality

regions have to be identified and afterwards these

need to be inspected [4]. From this it follows

immediately that it is best to modify their values

dynamically.

Now, the idea to dynamically reorganize the

relation between strategic variables' values is not

new. It was already published during the late 80's.

Since then many articles arose, dealing with the

construction of self-adapting values, based on control

mechanisms and on knowledge bases. Davis

suggested applying an operator in proportion to the

performance of the offspring it produces. His

suggestion was to calculate the fitness of an offspring

whenever an operator has been used and to increase

the frequency of using this specific operator if the

fitness of the offspring produced is higher than that

of the currently best member of the population [8]. In

1993 Xu and Vukovich suggested fuzzy rules for

altering strategic parameters [34]. Following

Goldberg’s suggestion to include online population-

sizing techniques [16] they focused on rules to

determine the crossover rate depending on the actual

population size and on the age of the generation.

In the course of the optimization described here

we use a very similar approach as we dynamically

modify the setting of strategic variables based on a

set of rules, which are comparable to those identified

by Lee and Takagi [23], except that we keep the

population size constant.

3.5 Pre-optimization

From the explanation given above it is obvious

that the length of each genetic string characterizing

an individual recycling network can turn out to be

that large that the obtainment of beneficing solutions

in reasonable time becomes questionable. In order to

reduce the number of optional locations for

dismantling facilities, which need to be examined in

more detail within the optimization process, a pre-

optimization stage has been introduced. This first

“first-order” analysis is used as an approximation to

the real problem. It assumes continuous locations for

a pre-given number of nm ≤< ~1 dismantling


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facilities, whereas the specifications of zones and of

ELVs supplied in these zones remain unchanged.

In this stage only inbound transport cost at a

dismantling facility are considered and hence the task

is to minimize the term ( )= =

⋅⋅⋅n

i

m

j

ijjiii sFZdistcs1

~

1

,

with the interpretation of the variables being the

same as in chapter 2.1. To do so we group the single

representative supply points into a number of m~

clusters, each represented by a two-dimensional

vector giving the location of an associated facility, so

that the sum of weighted distances is minimized.

The Fuzzy-C means algorithm, based on the

iterative minimization of an objective function is one

of the best known and best performing fuzzy

clustering algorithms [5, 33, 35]. It is used in a wide

field of applications and it excellently fits to the task

at hand [24, 31].

The grade, to which zone iZ ELVs belong to

each cluster, i.e. the portion of zone iZ ELVs that are

shipped to each dismantler jF , mj ~1 ≤≤ , is

represented by a mn ~× fuzzy membership matrix

( ) [ ] mnmns

~~

1,0 ×× ⊂∈ Rτ , where 0N∈τ refers to the

number of iterations. This matrix is referred to as a

fuzzy partition. The component situated in the i th

row and in the j th column of ( )τs is ( ) [ ] R⊂∈ 1,0τijs

denoting the above mentioned grade. Obviously for

each column in ( )τs it has to be ensured that

( ) 1

~

1

==

m

j

ijs τ so that every zone’s ELVs are completely

shared among the facilities.

Although there are other possibilities, like using

a-priori knowledge about the relation between supply

points, we create the initial fuzzy partition by chance.

Once the first partition is available the iteration is

initiated with the computation of cluster centers, i.e.

the location of dismantling facilities

( )( )( )( )

( )( )=

=

⋅=+

n

i

ij

n

i

iij

j

s

Zs

F

1

1:1α

α

τ

ττ , mj ~1 ≤≤ . It commences by

updating ( )1+τs , i.e. by calculating

( ) ( )( )( )( )

1

~

1~

1

2

~ ,1

,1:1

−

=

−

++

=+m

j ij

ij

ijZFdist

ZFdists

α

ττ

τ , +∈ Rα . Note

that the closer the value of α is to one, the more

crisp the partition will be ( ( ) ϑτα 1→→ijs , { }1,0∈ϑ ) and

the higher its value is the more fuzzy the partition

can be expected ( ( )m

sij ~1

∞→→

ατ ). This procedure is

iterated until a pre-defined stop criterion is reached.

In a next step the network that has been obtained

is evaluated in terms of transport cost, environmental

impact, average distance, entropy and partition

coefficient. Before further usage it can be modified

manually by the decision-makers, e.g. it can be

combined with additional, already existing, licensed

and non-licensed dismantling facilities. The network

of dismantling facilities generated this way defines

the set F as it is employed during GA optimization.

4 Results and next steps

The step before starting the optimization of a

recycling and recovery network is to verify the

efficiency of the chosen problem solving approach.

In this paper we focus on the possible benefit of the

pre-optimization stage. In order to verify the

usefulness of this stage two different experiments are

chosen to compare results which are obtained with

and without the pre-optimization.

In the first experiment we start with the pre-

optimization stage and use the results of this stage as

input data for the GA optimization. Therefore the GA

in this experiment can chose only from the subset of

facilities which is given by the pre-optimization. In

the second experiment only GA is used to optimize

the recycling network, so that it is able to chose from

the complete set of facilities. In both experiments the

GA optimizes recycling networks with no retraction

facilities and with a subset of the most important

second level facilities kS . Both experiments use the

same GA optimization parameters and have the same

total time for optimization. This means that the time

which is taken in experiment 1 for the pre-

optimization is given as additional time for the GA in

experiment 2.

In table 1 is a short description of the two

experiments:

Table 1. Experiment descriptionExperiment 1 Experiment 2

Selection of dismantler

subset Fpre with Fuzzy

C-means algorithm

No pre-optimization

GA optimization with

input Fpre and second

level facilities kS

GA optimization with all

dismantlers jF and

second level facilities kS

There are three scenarios with different numbers

of ELV zones iZ , different numbers of possible

facilities jF and different numbers of possible

second level facilities kS . Because of the GA´s

stochastic characteristics every experiment is

repeated three to five times (depending on the

scenario size) to get an average solution for each


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combination of scenario and experiment. Table 2

gives the main parameters of each scenario, table 3

shows the results of all optimization runs.

Table 2. Scenario description

Scenario 1 2 3

No. of ELVs 20,041 25,434 132,508

No. of ELV zones iZ 712 2,384 8,351

No. of facilities jF 103 157 873

No. of second level

facilities kS

3 9 39

Table 3. Results summary

Average cost Experiment 1 Experiment 2

Scenario 1 100% 116%



It is not possible to show absolute cost values,

therefore results of experiment 1 are normalized to

100% and results of experiment 2 are set in relation

to experiment 1 results.

In all scenarios experiment 1 finds better

solutions than experiment 2 and the gap increases

with the scenario size. While the average gap

between experiment 1 and experiment 2 is low in

scenario 1 (16%), it increases to 34% in scenario 2

and reaches 233% in scenario 3. As a conclusion of

this experiments it is obvious that the pre-

optimization stage is important for the optimization

approach, especially for large scenarios.

The optimization runs also show that it is

necessary to adapt the chosen GA parameters

especially for large scenarios. In Table 4 the cost

reduction from the first GA iteration8 to the final

result is shown.

Table 4. Cost reduction during GA

Cost reduction Experiment 1 Experiment 2

Scenario 1 52% 64%

Scenario 2 56% 43%

Scenario 3 51% 1%

In both experiment 1 and 2 the GA improves the

starting solution cost for scenario 1 and 2. In scenario

3 experiment 1 reduces the starting solution costs in

8 The starting iteration always contains several reasonable solutions.

the same way as in scenarios 1 and 2, only the GA

for experiment 2, scenario 3 is not able to improve

the starting solution significantly. The reason for this

poor performance is the large number of possible

facilities jF in this scenario in combination with too

a low mutation parameter. Further experiments have

shown that the GA performance can be improved by

increasing the mutation parameter.

In the future further research to improve the

optimization results will be done. Main tasks will

focus on adapting the GA parameters, splitting the

linear GA string into substrings, one for each facility

type, and reduce total scenario size by aggregating

the ELV zones into a few number of ELV clusters.

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