A Grouping Genetic Algorithm Using Linear Linkage Encoding for Bin ...

A Grouping Genetic Algorithm Using Linear Linkage Encoding for Bin Packing  ¨ ur Ulker, ¨ ¨ Ozg¨ Emin Erkan Korkmaz, and Ender Ozcan

Yeditepe University, Department of Computer Engineering, Kadikoy/Istanbul Turkey {oulker,ekorkmaz,eozcan}@cse.yeditepe.edu.tr

Abstract. Linear Linkage Encoding (LLE) is a representation method proposed for grouping problems. It has already been used in solving data clustering, graph coloring and timetabling problems based on multiobjective genetic algorithms. In this study, this novel encoding scheme is investigated on bin packing again using a genetic algorithm. Bin packing benchmark problem instances are used to compare the performance of traditional recombination operators and custom made LLE crossover operators which are hybridized with parametrized placement heuristics. The results denote that LLE is a viable candidate for bin packing problem whenever appropriate genetic operators are chosen.

1

Introduction

Most of the grouping problems, such as, data clustering, graph coloring, or bin packing require partitioning of a set items into n mutually disjoint subsets [8]: V = V1 ∪ ... ∪ Vk ∪ ... ∪ Vl ∪ ... ∪ Vn

and Vk ∩ Vl = Ø,

where

k = l. (1)

In different problems this partitioning process is subject to a different set of constraints. Various meta-heuristics such as simulated annealing [13], tabu search [10], genetic algorithms (GAs) [12], etc. have been applied to solve grouping problems. GAs derived from the population genetics and the Darwinian theory of evolution are powerful tools commonly used in search and optimization for solving complex problems ([4], [11], [12], [16]-[19]). In spite of the satisfactory performance of the traditional GAs on many NP-hard optimization problems, unfortunately, the same achievement is not usually observed on grouping problems. This is because many evolutionary algorithms do not address the dynamics of a grouping problem: how to handle the groups. The commonly used representations usually suffer from redundancies due to the ordering of groups. Moreover the genetic material representing groups might easily be disrupted by the genetic operators and/or by the rectification process after the operators are applied. Therefore a genetic algorithm requires special operators for solving a grouping problem. 

The author is currently on leave of absence and working as a research fellow in the ASAP group at the University of Nottingham.

G. Rudolph et al. (Eds.): PPSN X, LNCS 5199, pp. 1140–1149, 2008. c Springer-Verlag Berlin Heidelberg 2008 

A Grouping Genetic Algorithm Using LLE for Bin Packing

1.1

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Bin Packing Problem

Bin packing is a combinatorial NP-hard grouping problem in which items of different sizes has to be packed into a minimal number of bins of fixed capacity C. In classical one dimensional bin packing problem [2], a sequence of S = (I1 , I2 , ..., Ik , ...In ) items, each with a size s(Ik ) ∈ (0, 1] are packed into a minimum number of unit-capacity bins (partition  them into a minimum number m of subsets B1 , B2 , ..., Bl , ...Bm such that Ik ∈Bl s(Ik )  1, 1  l  m). There are exact and approximate methods proposed for solving bin packing. The exact methods guarantee that the optimal result will be achieved, while the approximate approaches provide satisf actory approximations. Martello and Toth’s branch-and-bound reduction algorithm [15] (MTP) is an exact approach which is used as the basic reference in most comparative studies of bin packing. MTP although slow (for large instances) generally gives excellent results. The MTP procedure attempts to find bins dominating all others. After such a bin is found, the problem is reduced by removing the dominating bin. In order to prevent an exponential search, only dominating bins of at most three items are taken into account. Constructive heuristics and (meta-)heuristics constitute approximate approaches. In the first fit heuristic (FF), an item Ik is placed in the first (lowest indexed) partially-filled bin Bj into which it could fit (capacity(Bl + s(Ik )  1). If this is not possible, a new bin containing Ik as the first item is created. A variant of first fit is first fit decreasing (FFD) in which items are first sorted in decreasing weight and then items are picked up one by one beginning with the largest item and each item is placed into the first bin that can accommodate it. In the best fit heuristic, an item Ik is placed in the partially filled bin Bl with the highest level (level(Bl )  1 − s(Ik )) and ties are broken in favor of lower index bins. Similar to FF, Best Fit has a decreasing variant, in which items are again sorted in decreasing order and placed into the best-filled bin that can accommodate it. Although best-fit decreasing is slightly more complicated than FFD, surprisingly it cannot beat FFD. Both heuristics have the worst case performance of 11 9 Opt+4 where Opt is the number of bins in the optimal solution 6 [2]. Dosa [5] proved that the tight bound for FFD is 11 9 Opt + 9 . Falkenauer [7] uses a Hybrid Grouping Genetic Algorithm (HGGA) which is heavily modified to suit the structure of the grouping problems. His genetic algorithm works with whole bins rather than with individual items. In HGGA representation, a standard chromosome representing the IDs of the items are augmented with a group part, encoding the groups on a one gene for one group basis. The important point with the genetic operators is that they work on the group part of the chromosome, the standard item part is just used to identify which items form which group. Falkenauer used a strategy similar to the domination criterion of Martello and Toth to place the eliminated bins (free items). Free items are swapped with non-free items (items currently placed within bins) such that the bins will consist of few large items rather than many small items. Items that cannot be placed with this replacement strategy are re-inserted into the solution using a first-fit heuristic. Mutation works also similarly; it destroys

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a few bins from the element and reinserts the missing items using the mentioned local search procedure. Linear Linkage Encoding (LLE) is proposed as a novel representation scheme for grouping problems [6]. LLE uses a link-based structure for objects within the same group. Genetic operators work on the encodings by altering the links. It is reported that the performance of LLE is superior to number encoding (NE) which is the most common encoding scheme used in grouping problems. Unlike NE, LLE does not require an explicit bound on the number of groups that can be represented in a fixed-length encoding. The greatest strength of LLE is that the search space is reduced considerably. There is a one to one correspondence between the encodings and the solutions when LLE is used. Du et al. [6] utilized LLE for data clustering, while Ulker et al. [21] experimented with LLE on graph coloring and timetabling problems. A multi-objective genetic algorithm is used in both studies. In this paper, these previous studies on LLE is extended. Bin packing is chosen as a testbed for investigating LLE further using a single objective genetic algorithm. The performance of different genetic operators are compared based on LLE.

2

Linear Linkage Encoding for Grouping Problems

Number Encoding (NE) is the most widely used representation in grouping problems. In this scheme, each gene is reserved for an object and the value of the gene indicates the group ID that the corresponding object belongs to. If six objects are to be grouped, the individual 234212 is a valid chromosome in NE and it encodes the solution where first object is in group 2, second in 3 and so on. Other chromosomes exist that represent exactly the same solution. 123141 is such an example where the order of the groups is different. However, this ordering proposed by the representation is irrelevant in terms of building the solution. In [20], it is denoted that NE is against the minimal redundancy principles for encoding scheme. Group Encoding (GE) is an alternative representation for grouping problems. In this scheme, the objects in the same group are placed into the same partition set. The sets are separated from each other by using special markers in the chromosome. For instance, the above example can be represented as (1, 4, 6)(2)(3)(5) in GE. However, the ordering redundancy still holds. For instance, (2)(3)(5)(1, 4, 6) would represent the same solution. A special representation is used in [9] for the problem. This representation augments standard number encoding with a group part. However, the same redundancy problem exists for this augmented form, too. The difference from group encoding is that traditional search operators work on the group part of the encoding. The redundancy that exists in above representational schemes is due to the symmetric structure of search spaces in grouping problems. Many researchers have proposed symmetry breaking methods to prune redundant search spaces [3] [1]. Linear Linkage Encoding (LLE) is a new representation scheme proposed

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to eliminate the symmetry problem in grouping problems. The encoding scheme [6] (LLE) was first proposed to solve the clustering problem. A multi-objective genetic algorithm (MOGA) is used in this application. In [6], it is denoted that LLE can reduce the search space considerably. In LLE, a gene is reserved for each object just like NE. However, the value of a gene is interpreted as a link from one object to another object of the same group. With n objects, any partition set on them can be represented using a chromosome of length n. Two objects are in the same group if either one can be reached from another through the links. When a gene value is equal to its own index, then it is considered as the last element of a group (ending node). The following two constraints are imposed on LLE links: – The value of each gene is greater than or equal to its index but less than or equal to n. – No two genes can have the same value except if one of them is an ending node. In LLE, the objects of a group form a linear path ending with a self referencing last item. In [14], it has been proved that a one to one mapping has been obtained between the possible partitions and the chromosomes of LLE (Figure 1). The genetic operators may disturb the two constraints mentioned above. A rectification process would be needed in order to recover the chromosome in such a case. The process is quite straightforward and the details are presented in [6]. 1

1

2

2

3

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4 2

3

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5

6

3

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5

6

Fig. 1. A LLE chromosome and the partition it represents

The non-redundancy advantage of LLE diminishes if a crossover operator causes huge jumps on the search space. Traditional crossovers like 1PTX or UX can easily destroy the building blocks and hence result in huge changes on the partition represented. It is important to preserve the order of the groups as much as possible during a genetic operation. Therefore, two different ordering mechanisms which assign group IDs to the groups are investigated within the context of LLE. These ordering mechanisms are based on the cardinality of the groups and the lowest index number in each group. In Cardinality Based Ordering, the group ID is determined based on the group cardinality (set size). The group with the highest cardinality is assigned group ID 1, the second highest will be identified as group 2, and so on. In Lowest Index Ordering, the IDs are assigned to groups based on the the smallest index in each group.

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A Grouping Genetic Algorithm for Bin Packing Problem

The GA used in this study utilizes LLE as the representation scheme. The individuals are initialized using the FF heuristic and the resulting individuals are converted into the LLE form. In order not to have the same individual for the whole population, a random permutation of the items are fed into the FF heuristic. At the end of the initialization it is ensured that the capacity of each bin is not exceeded. As a mate selection method, tournament selection is used. Some preliminary experiments are performed and it is observed that using higher selection pressure yields better results. The performances of different crossover operators, including the traditional ones (1PTX, UX) are compared. A non-traditional smart mutation is utilized. A trans-generational GA replaces the current generation with the next one by keeping only the best individual from the current population. A straightforward fitness function would just take the inverse of the number of bins. However, as pointed by Falkenauer [9] as well, such a fitness function will result a very unfriendly fitness landscape in which many combinations with one more bin than optimal solution will have the same fitness value. Instead, the function proposed by [9] is used in this study: N (Fi /C)2 (2) f (s) = i=1 N where, N is the number of bins, Fi is the fill of bin i and C is maximum bin capacity for a given solution s. 3.1

Crossover Operators

Ulker et al. [21] reports that Lowest Index Max Crossover (LIMX) performs well on different graph coloring and timetabling problems. In LIMX, a single child is generated using two individuals with two aims: transmit large groups to preserve Cardinality Based Ordering, and to transmit groups beginning with lowest index number (to preserve Lowest Index Ordering). Beginning with the lowest index number (vertex) which has not been assigned first we calculate the length of the links (path length) in both parents. Already assigned vertices are not counted in this link length calculation. This allows finding the largest set in parents beginning with the lowest index number. Then the links (and thus vertices) are transmitted to the child from the parent based on the link-lengths. Then, the next unassigned lowest index number is found and the process is repeated until all vertices are assigned. For example, assume that {(1, 3, 6), (2, 4), (5)} and {(1, 2), (3, 4, 5, 6)} are selected as mates. The starting lowest index is 1. (1, 3, 6) is longer than (1, 2), so (1, 3, 6) is copied to the child. Then, the indices (1, 3, 6) are deleted from the mates: {(2, 4), (5)} and {(2), (4, 5)}. Now, the current lowest index is 2. (2, 4) is larger than (2) so it is transmitted to the child and deleted from the mates: (5) and (5). Finally, (5) is copied to the child as the last group and a new offspring is generated as {(1, 3, 6), (2, 4), (5)}.

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A modified uniform crossover (MUX) method is utilized on the clustering problem with LLE in [14]. In MUX, instead of the actual values like in UX, the value of the ending node of the group in which the item belongs to is passed to the offspring. This ensures groups were not separated. It is observed that this crossover tends to combine the groups having the same ending node in both parents. MUX is the second operator tested in bin packing domain. 3.2

Smart Mutation

A smart mutation that targets no overfilled bins during the packing is utilized within the grouping GA. In mutation, k randomly chosen bins are destroyed and the contents of these bins are redistributed to rest of the bins. The performance of GA is observed for various values of the parameter k, referred to as mutation rate. Furthermore, due to the nature of the crossover operators, the resulting individual may have bins whose capacities are exceeded and thus may need an additional repair procedure apart from the usual LLE rectification mechanism. This repair procedure checks all of the bins and removes randomly selected items from the over-filled bins until the capacity is not exceeded anymore. These removed items are combined with the items from the previously deleted bins. Then, all of them are reinserted into the rest of the bins using a heuristic. If it is not possible to insert some items without exceeding the capacity of existing bins, then a new bin is created. FF and FFD heuristics are used as repair heuristics. Like previous algorithms of Falkenauer [7] and reduction algorithm of Martello and Toth [15], a procedure based on the domination criterion has been adopted. When an excess item is to be inserted back to the solution, first it is compared with the items already present in the bins. An excess item replaces an item in the bin while not causing an overfill in the bin and the replaced item is put into the excess items list. This procedure increases the number of well-filled bins. In the first-fit heuristic, the lowest-index based ordering is used. Due to the nature of the bin packing instances, cardinality based ordering will be no different than random ordering, therefore it is ignored.

4

Experimental Results

Two sets of test instances provided by Falkeanuer [7] are used in the experiments. In the first set, the maximum bin capacity is set to 150 and each integer item is randomly generated using a uniform distribution between 20 and 100. Falkenauer reports that this distribution gives most difficult instances for the method proposed by Martello and Toth [15]. In the second set, the item sizes are drawn from the range (0.25, 0.50) to be packed into bins with maximum capacity 1. In these instances, a well-filled bin must contain one large item and two small items. That is why Falkenauer referred them as ’triplets’. Falkenauer [7] points out a similarity between triplets and 3SAT which is considered as the most difficult kSAT problem. In order to preserve the difficulty of the problem, the generated instances have known local optima with maximum bin capacities

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of 1000. 20 different instances of triplets containing 60, 120, 249 and 501 items are then generated. The experiments are performed on Pentium IV, 2 GHz Linux machines with 256 Mb memory. The population size, tournament size and crossover rate are fixed as 100, 10 and 1.00, respectively. The first fit (FF) and first fit decreasing (FFD) heuristics are used as repair mechanisms. Mutation rate is the number of bins destroyed in an element. Different experiments destroying 1, 2 and 4 bins are carried out. For consistency with the previous experiments of Falkenauer [7] and Martello [15], each instance in a set is tested only once. Hence, as a total of twenty runs are performed. The performances of four crossover operators ( 1PTX, LIMX, UX and MUX) are tested using FF and FFD with mutation rates of 1, 2 and 4 providing a total of 6 combinations for each crossover. In order to compare in a fixed heuristic and mutation rate setting, a ranking mechanism that takes ties into account is implemented. The ranking method takes into consideration the success ratio (number of times the optimal solution is found), the mean number of bins found and the mean number of generations. The t-tests are also carried out to ascertain if there are statistically significant differences between different settings. If one mutation instance (combination of repair heuristic and mutation rate) is better than all the others then it is given a ranking of 1, and if it is worse than all others, it is given a ranking of 6. The average ranking of each mutation instance over the problems for each crossover is presented in Figure 2. It is observed that crossover performance differs based on the repair heuristic utilized. Although, there is no significant performance variance among the choice of repair heuristic and the mutation rate, considering the overall performance of each, the best repair heuristic turns out to be FF. 1PTX performed best in a FF and the mutation in which 2 bins are destroyed (FF2). LIMX operated best when used with FF1 which is closely followed by FFD4. For UX, FFD2 performed best while for MUX FFD1 is best and closely followed by FFD2. Table 1 provides a comparison of the results for different approaches, including the grouping GA with the best setups for each crossover (1PTX - FF2, LIMX FF1, UX - FFD2, MUX - FFD1). The results clearly show that the proposed algorithm is superior to Martello and Toth’s [15] reduction algorithm in all problem 6

avg. rankings

5

FFD1

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FFD2 FFD4

3

FF1

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FF2

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FF4

0 1PTX

LIMX

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MUX

Fig. 2. Performance comparison of crossovers using different repair heuristics

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Table 1. Mean number of bins obtained in best setups for each crossover, T heo denotes theoretical minimum lower bound on the number of bins Instance Set U120 U250 U500 U1000 T60 T120 T249 T501 AVG

Theo 1PTX FF2 LIMX FF1 UX FFD2 MUX FFD1 HGGA [7] M&T [15] 49.05 49.05 49.10 49.05 49.05 49.15 49.15 101.55 101.70 101.80 101.75 101.65 101.70 102.15 201.20 201.30 202.35 201.50 201.30 201.20 203.40 400.55 400.65 417.50 401.45 401.05 400.55 404.45 20.00 20.95 21.00 21.00 21.00 20.10 21.55 40.00 41.00 41.00 41.00 41.00 40.00 44.10 83.00 84.00 84.00 84.15 84.05 83.00 90.45 167.00 168.00 168.85 169.80 169.20 167.00 181.85 132.79 133.33 135.70 133.71 133.54 132.84 159.64

sets. Falkenauer’s HGGA [7] remains however the best algorithm in terms of overall solution quality especially for the more difficult triplet instances. However, for the uniform distribution instances, LLE with 1PTX is very competitive, in fact for smaller instances it provides some packings that cannot be found with HGGA. For the triplet instances, LLE with 1PTX is consistently one bin short of the optimal packing. The results presented in Table 1 are also evaluated using some statistical tests. Each approach (1PTX - FF2, HGGA, M&T, etc.) is ranked according to its performance for each problem instance. The ANOVA test over the average ranks shows that performances of 1PTX - FF2, HGGA and M&T are not equal to each other within a confidence interval of of 99.99%. Then, t-tests are used to make pairwise comparisons between 1PTX - FF2 and the others. It has been observed that 1PTX - FF2 and HGGA generate significantly better performances as compared to M&T within a confidence interval of of 99.99%. HGGA seems to perform better as compared to 1PTX - FF2, however, this difference in performance is not statistically significant under the t-test. In terms of mean number of bins and number of generations, 1PTX usually provides the best results especially for difficult large triplet instances. LIMX performs somewhat inconsistently. It sometimes gives results close to 1PTX on some large triplet instances. However it lags far behind especially in the larger uniform distribution instances. UX and MUX behave consistently and yield an acceptable performance in all test setups. LIMX and the other crossover operators are not particularly suitable for the bin packing instances tested because for all of the instances average number of items per group is quite small (2 to 3 for uniform distribution instances and 3 for triplet instances). Preservation of large groups is not important due to the small average size of the bins and the low epistasis between items. Because of small and low epistasis bins, 1PTX has a good performance as the likelihood of destruction of well-filled bins is lower.

5

Conclusions

In this study, the performance of LLE has been tested on bin packing. Several crossover operators that can be used with LLE have been investigated. Unlike

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graph coloring [21], the performance of the traditional crossovers shines when used in bin packing. They were able to generate very competitive results close to hybrid grouping genetic algorithm (HGGA) of Falkenauer. They match HGGA in uniform distribution instances and for smaller instances outperform them. However for most difficult triplet instances, 1PTX, the best performing crossover in the test setup, was constantly one bin short of the optimal solution. This problem probably requires enhancements in the mutation operator and this is one of the future research directions. Crossovers utilized in graph coloring domain (e.g., LIMX) cannot perform competitively on bin packing instances. This is an expected result since in bin packing it is not crucial to preserve very large sets during generations. The most difficult test instances usually require packing few items to a bin (2 to 3 in this case). This is why ordering of groups based on cardinality does not make sense as it will essentially be no different than random ordering. Hence, traditional crossover operators perform quite well in this domain. Linear linkage encoding is a viable candidate for solving grouping problems especially if the number of groups is not known beforehand. In such problems, the search is performed on a smaller search space than other encodings such as number encoding. New operators that will make better use of this representation awaits research. Moreover, other approaches for solving grouping problems may also utilize LLE as their representation method, such as, hyper-heuristics [18]. Acknowledgments. This research is funded by TUBITAK (The Scientific and Technological Research Council of Turkey) under the grant number 105E027.

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