Using Multi-chromosomes to Solve a Simple Mixed Integer Problem Hans J. Pierrot and Robert Hinterding Department of Computer and Mathematical Sciences Victoria University of Technology PO Box 14428 MCMC Melbourne 8001, Australia email: hpierrot@acslink.net.au, rhh@matilda.vut.edu.au Abstract. Multi-chromosomes representations have been used in Genetic Algorithms to decompose complex solution representations into simpler components each of which is represented onto a single chromosome. This paper investigates the eects of distributing similar structures over a number of chromosomes. The solution representation of a simple mixed integer problem is encoded onto one, two, or three chromosomes to measure the eects. Initial results showed large dierences, but further investigation showed that most of the dierences were due to increased mutation, but multi-chromosome representation can give superior results. 1 Introduction Multi-chromosome representations are not often used in Genetic Algorithms (GAs), but when they are, each chromosome has been dierent in structure (Juli, 1993; Hinterding & Juli, 1993; Hinterding, 1997; Ronald et al., 1997). To date no work has been done to determine the eect of using multiple chromosomes where each chromosome has the same structure. The eect of using multi-chromosome representation to solve numeric test functions was investigated in Pierrot (1997). Here a number of scalable functions were solved with the parameters evenly distributed over a varying number of chromosomes. Experiments used one, six and twenty four chromosomes for each of the six functions, and also varied the mutation technique. These experiments show that the method of mutation was the main determinant in deciding if multiple chromosomes were eective or not. In this paper a simple mixed integer problem representing a manufacturing task is used to investigate the eects of using multi-chromosome representation for mixed numeric representation problems. The questions addressed here are: { What is the eect of using multiple chromosomes? { Does the order of the parameters have an eect? Should the control variables be grouped together or should the variables for each machine be kept together?
{ What is the eect of having either real or integer count variables? Linear constraint problems are traditionally solved by linear programming using the Simplex method. When some of the constraints are integer in form then the problem is dened as a mixed integer problem. Branch and bound techniques are then applied to satisfy the integer constraints. The computation time increases for each integer constraint added, hence integer constraints are kept to a minimum. Therefore both integer and real quantity variables were tested. The results show that the use of multi-chromosomes can improve the performance of a GA for mixed integer problems. This is attributed to the way that mutation works with this structure. 2 Multi-chromosomes GAs traditionally use only one chromosome in their representation, and generally a uniform representation is used for each gene. This allows the same reproduction operators to be used for all genes. Suitable standard operators can just be plugged in as required to implement a specic approach. Complex problems require complex representations. When a complex structure is encapsulated in a single chromosome then this leads to needing to know the structure of the chromosome to be able to operate on it. For example, Davidor (1991) uses complex gene representation to dene the path of a robot arm. There is one allele in the gene for each link of the robot arm. Each gene represents one position, or arm conguration on the movement path of the robot arm. The chromosome contains as many genes as required to describe a particular path for the tip of the robot arm. Special mutation and crossover operators were needed for this representation. This is an instance of using multi-component chromosomes. That is chromosomes where all the genes are placed in one chromosome and the representation of the genes is either not homogeneous or the gene has a compound representation. This increased complexity in the gene or chromosome structure increases the complexity of the genetic operators and forces the creation of special versions of these operators. One way of overcoming these diculties is to use a multi-chromosome representation. Here an individual would contain more than one chromosome where each chromosome could have its own representation. The genes in an individual chromosome would then have a homogeneous representation. This would result in each chromosome having its own genetic operators. These could then be chosen from the standard operators. An example of this is Juli (1993), who uses multiple chromosomes to represent the structure of pallet loads. The problem is to optimise the loading of trucks, placing pallets in such a way as to minimise unloading time. There are a number of constraints on how the load can be constructed determined by layer content and pallet limits. Initially a single chromosome GA with order based representation was used with disappointing results.
A multi-chromosome GA was then created which carried additional information in the second and third chromosomes, these chromosomes contained information about the position of the pallets on the truck and how many of each dierent pallet type was to be used. This resulted in a better exploration of the problem space and improved the results. Because of the extra information the decoder could build dierent loads even though the information in the rst chromosome was the same. The results for the multi-chromosome GA outperformed the single chromosome GA. Hinterding (1997) used multi-chromosome representation to solve the Cutting Stock Problem with a GA using self-adaptation. One chromosome has the group based parameters for the problem. The second chromosome is used to hold the self-adaptive parameters, and each chromosome has its own reproduction operators. This is needed as the group based chromosome needs to be manipulated in an entirely dierent way to the numeric chromosome which holds the selfadaptation parameters. Ronald (1997) used a multi-chromosome representation to solve a modied Travelling Salesperson Problem. The normal TSP problem was augmented by the need to consider the cost of transport and a penalty for changing from one mode of transport to another. Dierent modes of transport were optimal for dierent distances. Two chromosomes were used. The rst chromosome encoded the standard TSP and used permutation encoding to describe the circuit of the cities. The second chromosome described the method of transport when leaving each city. This separation of the parameters into two chromosomes allowed simple genetic operators to be used for each of the two chromosomes. Using multi-chromosome representation allows the representation of an individual to be decomposed or factored into simple and easy to manipulate components where each chromosome can use simple genetic operators suited to the simple and homogeneous genes in that chromosome. 3 Multi-chromosomes and Mixed Integer Problems A simple mixed integer problem was found representing a manufacturing task. This was coded as a GA using a penalty function for constraint handling. This mixed integer GA was tested using a varying number of chromosomes. The Crosby Company is contracted to produce 500 ttings next week for one of its customers. Crosby has three machines in its machine shop that can produce the ttings, but at dierent variable and xed costs. These costs and weekly production limits are shown in Table 1. The xed cost is incurred only if the machine is set up to produce the tting. The objective is to determine how to produce 500 ttings at minimal cost. This problem is usually solved as a mixed integer problem using a linear program. The control variables which determine if a machine will be used must be integers. The quantity to be produced on each machine can be represented by either an integer or a real variable. If this was to be solved using a linear program then these values would be treated as reals to speed up the solution
Table 1. List of production costs for each machine Machine per unit xed weekly production cost set up cost production limit 1 $1.12 $60 300 2 $1.40 $55 250 3 $1.23 $50 270 process. For this reason it was decided to try using both integers and reals for the quantity variables. The optimal answer to the problem is to set up only machines 1 and 3, and to produce 300 ttings on machine 1, and 200 on machine 3. This gives a minimal cost of $692. 3.1 Representation The initial idea was to use separate chromosomes for the two dierent types of variables; that is one chromosome for the real variables and another for the integer variables. This followed the ideas of Juli (1993) in the truck loading problem. Hence a chromosome would represent all integer or all real variables, but it was realised that by recording the length and the value range for each gene in a chromosome, it did not matter if the gene represented an integer or a xed point real. Some minor changes were needed for the Gaussian mutation operator as it operates on genes (function variables), but the crossover operator was not aected at all. As a result a single chromosome could represent any mixture of integer and real values. In all tests run, the control variables were treated as one bit integers. A value of one meant that the relevant machine was set up and a value of zero meant that the machine was not set up. The number of units produced was represented by a real value encoded using 10 bits, or by an integer value using 9 bits. Further tests were run changing the order of the variables to see if having the control variables next to the count variables was better than having all control variables together and then all the count variables. The control variables for machine 1, 2 and 3 are f1, f2 and f3 respectively and the count variables for the three machines are x1, x2 and x3. Figures 1 & 2 show how the variables were distributed across the chromosomes for the various test cases. The common parameter settings for the GA are shown in Figure 3. A new individual is produced by either crossover or mutation but not both. For the initial tests each chromosome was mutated once when the mutation function was invoked and Poisson based mutation was not used. This means that only one gene was mutated in each chromosome.
one chromosome f1 f2 f3 x1 x2 x3 chrom. 1 chrom. 2 f1 f2 f3 x1 x2 x3 chrom. 1 chrom. 2 chrom. 3 f1 f2 f3 x1 x2 x3 Fig. 1. Representation: chromosomes with parameters grouped one chromosome f1 x1 f2 x2 f3 x3 chrom. 1 chrom. 2 f1 x1 f2 x2 f3 x3 chrom. 1 chrom. 2 chrom. 3 f1 x1 f2 x2 f3 x3 Fig. 2. Representation: chromosomes with parameters intermixed 3.2 Gaussian Mutation Gaussian mutation is used for all of the experiments and is implemented by adding Gaussian noise to the value of the gene similar to that used in Evolution Strategies. The dierence being that there is only one rate and not one for each function parameter as in Evolution Strategies. We use a default std. dev. parameter of 0.1. 3.3 Evaluation Function The evaluation function had to perform two tasks. One, calculate the cost of producing the ttings given by the the function: (f1 60) + (f2 55) + (f3 50) + (x1 1:12) + (x2 1:40) + (x3 1:23) Here f1 is set to one if machine 1 is in use and zero if not in use. Similarly f2 is one if machine 2 is in use and f3 is one if machine 3 is in use. x1 represents the number of units being produced by machine 1, x2 represents the number of units produced by machine 2 and x3 represents the units produced on machine 3. Population size of 50 No duplicates allowed Binary Gray encoding replacement rate of 90% crossover rate of 60% Gaussian mutation number of evaluations = 12000 Fig. 3. GA parameters used for the mixed integer problem
The second task is to compute the penalty function to model the constraints in the problem. The constraints are: x1 300 x2 240 x3 270 x1 + x2 + x3 500 M = 300 x1 M f1 x2 M f2 x3 M f3 f1; f2; f3 2 f0; 1g The rst three constraints are easily met when they are dened as real values within a dened range. If they are integer values, they can hold any value between 0 and 511. The evaluation function returns a tness value which is the sum of the cost and the penalty functions. 3.4 The Penalty Function The penalty method used is a simplication of the method of Homaifar et al. (1994). They suggest that a family of intervals should determine appropriate penalty values. A penalty coecient should be attached to each level and multiplied by the error value. A single level was employed in this GA for each constraint. In most cases the GA rst nds a local optimal solution using all three machines. Sometimes it gets stuck there, otherwise it will nd a solution with machine 2 turned o and then approaches the solution near the true minimal cost of $692 when machine 1 produces 300 units and machine 3 produces 200 units. 4 Results The results shown in Tables 2 & 3 give the average results from batches of 40 runs. The gures show how many times in the batch of 40 runs the cost was reduced to be less than 700 (optimal cost is 692). All other runs gave an answer near the local optima of 747. The plot in Figure 4 shows the way the results are distributed in a typical test. The results have been sorted into ascending sequence to better show how results are distributed. Two sets of tests were run using the 1, 2 or 3 chromosome representations, and using real or integer values for the count variables. One set had the three control parameters rst followed by the variables for the number of units produced. The other set of tests had each control variable next to the matching number of units produced. These representations are shown in Figures 1 & 2. It can be seen in Table 2 that more chromosomes give a better result. Using integers gives a better result in the one and two chromosome cases. It can also be seen that having related genes close together does not give a clear improvement. Therefore the position of each gene is not particularly important for this specic problem. What are the dierences between the one chromosome and three chromosome representation? The rst is the way that crossover takes place: in the one
Plot of results from initial test of 2 chromosomes x 40 observations 30 20 10 0 680 700 720 740 760 780 cost Fig. 4. Distribution of results (40 runs) chromosome representation this is standard two point crossover; in the three chromosome representation crossover takes place on each chromosome, hence it is like six point crossover. The other dierence is the eect of mutation. In multi-chromosome representations each chromosome is mutated, hence with the three chromosome representation three genes are mutated, while only one gene is mutated in the single chromosome representation. To verify this further tests were run. The multi-chromosome representations were tested with only one chromosome being mutated. Six point crossover and higher levels of mutation were performed on the single chromosome representation. To increase the mutation rate, Poisson mutation was enabled. Now a Poisson probability function was sampled to determine how many genes would mutate. The mutation tests were run with a mean () of 1, 2 and 3. Table 2. Summary of results of mixed integer tests Rep. No. Control then Paired control and Chroms count variables count variables real one 3 6 integer one 13 10 real two 23 22 integer two 28 31 real three 26 38 integer three 37 37
The results of these tests showed that six point crossover gave a small improvement to the eectiveness of the single chromosome representation. Increasing the mutation rate by using Poisson mutation had a larger eect and combining the two was even more signicant. However this still did not equal the results for the three chromosome representation. Using multi-chromosome representations with only one chromosome being mutated did not give good results. The three chromosome result was very close to the six point crossover result for the single chromosome representation as expected. To reduce mutation in the multi-chromosome representations Poisson mutation was used and the mean mutation rate divided by the number of chromosomes. When reducing mutation in this manner the results for the two chromosome representation are the best. The three chromosome results are not as good as those obtained when Poisson mutation is not used. This would indicate that a steady mutation of one variable per chromosome gives a better result than the average of one mutated variable per chromosome. This is caused by the way that mutation takes place when Poisson mutation is used. Each time that mutation is invoked a Poisson random number is generated, for low values of, a signicant number of samples will be zero, and a chromosome identical to the parent chromosome will be produced. Table 3. Results: varying crossover for integer representation Mutation Poisson 2.7 No Poisson No. chrom. % Crossover % Crossover 0 20 40 60 0 20 40 60 1 38 36 32 30 28 24 15 12 2 36 39 35 33 40 36 37 28 3 24 35 37 36 39 39 40 37 As mutation appears to have such a major eect on the problem further runs were conducted varying the crossover rate to decrease crossover and therefore increase the proportion of the population produced by mutation. Table 3 shows the results for the integer representations with the best values found for (the Poisson mean) as well as the results when Poisson mutation is not used. Using one chromosome representation the best results are obtained when Poisson mutation is used. For multi-chromosome representations the best results are obtained when Poisson mutation is turned o, hence a regular mutation rate of 1 mutation per chromosome is used. There also appears to be a pattern in the crossover rate. The one chromosome representation appears to do best at the crossover rate of 20% whereas the multi-chromosome representation appears to do best at a crossover rate of 40%. This is most likely due to multi-chromosome representations using crossover more eectively and the eect of a steady muta-
tion rate when Poisson mutation is not being used in this representation. The results for the real variables are similar to the above numbers though in most cases lower in value than for the integer tests. Up to this point the cost of 700 has been used as a cut o in deciding which representation is the better. Using this method the multi-chromosomes are better by a small margin. The close results obtained in the nal tests when the mutation rate was increased prompted a closer look at what happened at cut o points dierent to a cost of less than 700. An examination of a range of cut o values from 700 down to 692.1, the perfect score, showed that reducing the cut o down to 697.1 made no dierence to the numbers for the best result in each representation. count difference 10 9 8 7 6 5 4 3 2 1 plot of differences between n-chrom and 1-chrom int 2-1 int 3-1 692.1 692.7 693.3 693.9 694.5 695.1 695.7 696.3 696.9 697.5 cutoff for accepting answer 698.1 698.7 699.3 699.9 Fig. 5. Dierences in results for multi and single chroms. To get a better feel for what was happening the best answer for each of the three representations for both integer and real quantity variables were determined and plotted. This showed that there was a marked variance between the dierent representations. To highlight these a plot of the dierence between the one chromosome and the multi-chromosome representation was produced. The graph in Figure 5 shows the dierence between the one chromosome representation and the Multi-chromosome representations for the integer tests. The comparison between the one and two chromosome representations is labelled int 2{1 and the comparison between the one and three chromosome representations is labelled int 3{1. These represent the dierences between the best number of answers for each cut o value. Examination of this plot shows that making the cut o too close to the actual optimum gives a non typical picture. Values from 692.3 through 694.5 show a signicant advantage to using a multi-chromosome representation. The results for the real quantity variables were similar but only went as high as 693.5 before falling o dramatically.
5 Conclusions and Discussion This investigation looked at a simple mixed integer problem as a vehicle for determining if a multi-chromosome representation would bring any benets. We show that using multiple chromosomes can give superior results to a single chromosome representation for mixed integer problems. Contributing factors to these improvements are:- { the increase in crossover points which is similar to a six point crossover when comparing the three chromosome conguration to the single chromosome conguration. { the increased mutation provided by mutating each chromosome. While these factors account for some of the improvement, the multi-chromosome representation still has real benets as far as simplicity and eectiveness are concerned. The other questions investigated appear to have the following results:- { the order of the genes does not appear to have any clear cut impact on the solution. { the use of integer variables for the count variables appear to give improved results. This is in contrast to the traditional linear program where it is more ecient to code variables as reals to avoid the extra computation of the branch and bound process. References Davidor, Y.: Genetic Algorithms And Robotics { A Heuristic Strategy For Optimization. Singapore: World Scientic Publishing, 1991. Hinterding, R.: Self-adaptation using Multi-chromosomes. In: Proceedings of the 4th IEEE International Conference on Evolutionary Computation. IEEE Press. 1997, pp 87{91. Hinterding, R., & Juli, K.: A Genetic Algorithm for Stock Cutting: An exploration of Mapping Schemes. Technical Report 24COMP3. Department of Computer and Mathematical Sciences, Victoria University of Technology, Victoria Australia, 1993. Homaifar, A., Lai, S. H. Y., & Qi, X.: Constrained Optimization via Genetic Algorithms. Simulations, Vol. 62, 1994, pp. 242{254. Juli, K.: A multi-chromosome genetic algorithm for pallet loading. In: Proceedings of the Fifth International Conference on Genetic Algorithms. 1993, pp. 476{73. Pierrot, H. J.: An investigation of Multi-chromosome Genetic Algorithms. Masters Thesis, Victoria University of Technology, Melbourne, Australia, 1997. Ronald, S., Kirkby, S., & Eklund, P.: Multi-chromosome Mixed Encodings for Heterogeneous Problems. In: Proceedings of the 4th IEEE International Conference on Evolutionary Computation. IEEE Press. 1997, pp 37{42. This article was processed using the LaT E X macro package with LLNCS style