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

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

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

2

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. All terms are used according to the English translation of their definition in the German law concerning the recovery of end-oflife 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 areawide 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 4

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

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

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).

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 Z i , 1 ≤ i ≤ n , 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 Z i add up to the total supply of ELVs si ∈ N in Z i . 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 si is concentrated at a single point (xi , y i )∈ Z i , which is the geographical centroid of Z i , determined by the surrounding traverse. This assumption limits the granularity of our approach to the granularity of zip-code areas. Supposing that a facility F j , 1 ≤ j ≤ m ∈ N with unconstrained capacity is located at the point

(~x , ~y )∈  Z n

j

j

i =1

i

and having dist (Z i , Fj )∈ R denote the

distance between zone Z i and facility F j , we model transport cost ci per distance unit as a nonmonotonic and non-continuous function ci (dist (Z i , F j ))∈ R that is equal to zero for all transports within a user-defined “reasonable distance”, rd ∈ R , and that for distances higher than this point of discontinuity gradually decreases. Then the total transport cost of taking back zone Z i ELVs by facility F j is si ⋅ ci ⋅ dist (Z i , F j ) and the problem to be solved is to minimize the sum of all these cost over all zones Z i and over all facilities F j , i.e. n

m

(

)

min ¦¦ si ⋅ ci ⋅ dist Z i , F j ⋅ sij where sij ∈ [0, 1] ⊂ R is a i =1 j =1

real-valued decision-variable indicating the portion of zone Z i ELVs served by facility F j . Note that modeling sij as a real-valued variable instead of

Figure 1. Facility location problem (example)

choosing a binary form enables allowing each zone Z i to be served by multiple facilities. Hence we assume that the portion of customers of zone Z i forwarding their ELV to facility F j decreases with increasing distance as long as

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(

)

dist Z i , F j ≤ rd .

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

Otherwise we assume that the accordant ELV is collected by the vehicle’s manufacturer and shipped to that facility F j for which dist (Z i , F j ) < dist (Z i , F j′ )

F j ′ ∈ {F1 ,, Fm }/ F j . Therefore it is clear

for all

that as m and as rd increase the inbound cost for the union of all facilities F j converges to zero. And thus with rd fixed a lower bound for m so that

¦¦ s ⋅ c ⋅ dist (Z , F )⋅ s n

m

i

i

i

j

ij

=0

can

easily

be

i =1 j =1

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 S k , 1 ≤ k ≤ o ∈ N served by those facilities F j which locations have to be determined. Note that it is not necessary that each S k is served by exactly one facility F j . S k being served by multiple F j 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 F j , than it is to pick up each ELV at the last owners home and to ship it directly to the closest facility F j . In order to model this we introduced an additional stage in the reverse supply-chain, namely retraction facilities, Ol , 1 ≤ l ≤ p ∈ N , which number and which locations have

to be determined. Noting O := {O1 ,, O p } we obtain an

supply-chain

consisting

of

the

links

Z→O→F →S .

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

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 cost ∈ N and environmental impact environment ∈ R 8 whereby a modification of these objectives is possible at any time. Hence the objective function can be written as $ denoting a with min (cost $ environment ) conjunction to be specified later.

3.1

Decision variables

Decision variables are the number of dismantling facilities m ≤ m and the number of retraction facilities p ≤ p , 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 decisionmakers 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 nonsmooth search spaces. This means that decisionmakers 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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Proceedings of the 37th Hawaii International Conference on System Sciences - 2004

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

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

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 ninetyfive 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 1.65 ⋅ 2 0.21⋅length_of_genetic_string , 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 populationsizing 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 preoptimization stage has been introduced. This first “first-order” analysis is used as an approximation to the real problem. It assumes continuous locations for ~ ≤ n dismantling a pre-given number of 1 < m

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

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 ~ m

¦¦ s ⋅ c ⋅ dist (Z , F )⋅ s n

is to minimize the term

i

i

i

j

ij

i =1 j =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 Z i ELVs belong to each cluster, i.e. the portion of zone Z i ELVs that are ~ , is shipped to each dismantler F j , 1 ≤ j ≤ m ~ fuzzy membership matrix represented by a n × m ~ ~ n×m n×m s(τ )∈ [0, 1] ⊂ R , where τ ∈ N 0 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 s (τ ) is sij (τ )∈ [0, 1] ⊂ R

denoting the above mentioned grade. Obviously for each column in s (τ ) it has to be ensured that ~ m

¦ s (τ ) = 1 so that every zone’s ELVs are completely ij

j =1

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

¦ ((s (τ )) n

Fj (τ + 1) :=

α

ij

i =1

⋅ Zi

¦ (s (τ )) n

α

)

~ . It commences by , 1≤ j ≤ m

ij

i =1

updating

s (τ + 1) ,

i.e.

ª ~ « m § dist Fj (τ + 1), Z i sij (τ + 1) := «¦ ¨ ~ ¨ « j =1 © dist F~j (τ + 1), Z i ¬

(

(

) ·¸

)¸¹

by 2 α −1

calculating

−1

º » + » , α ∈ R . Note » ¼

that the closer the value of α is to one, the more crisp the partition will be ( sij (τ ) → ϑ , ϑ ∈ {0, 1} ) and α →1

the higher its value is the more fuzzy the partition 1 can be expected ( sij (τ ) → ~ ). This procedure is α →∞ m

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 preoptimization 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 S k . 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 preoptimization 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 description Experiment 1 Experiment 2 Selection of dismantler No pre-optimization subset Fpre with Fuzzy C-means algorithm GA optimization with GA optimization with all input Fpre and second dismantlers Fj and level facilities S k second level facilities S k

There are three scenarios with different numbers of ELV zones Z i , different numbers of possible facilities F j and different numbers of possible second level facilities S k . 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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Proceedings of the 37th Hawaii International Conference on System Sciences - 2004

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

No. of ELVs

20,041

2

3

25,434 132,508

No. of ELV zones Z i

712

2,384

8,351

No. of facilities F j

103

157

873

3

9

39

No. of second level facilities S k

Table 3. Results summary Average cost

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.

References

Experiment 1

Experiment 2

Scenario 1

100%

116%

Scenario 2

100%

134%

Scenario 3

100%

333%

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 preoptimization 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 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 F j in this scenario in combination with too

The starting iteration always contains several reasonable solutions.

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