Application of Genetic Algorithm for Rectangular Object Layout Optimization*

Xiaozhen Mi 1, Xiaodong Zhao 1, Wenzhong Zhao 1, Wenhui Fan 2

1Mechanical Engineering department, Dalian Jiaotong University, Dalian, P.R. China mxz@djtu.edu.cn

2Automation department, Tsinghua University, Beijing, P.R. China fanwenhui@tsinghua.edu.cn

Abstract

Rectangular Object Layout is not a simple NP (nondeterministic polynomial) optimization problem because of the practical production rules, such as guillotine. Under some situations, the guillotine is even more important than the pure using ratio of metal sheets. To improve the using ratio of stock sheets and to reduce the production costs, the genetic algorithm is adopted to optimize rectangular object layout on stock sheets. The purpose of this paper is to construct the model of genetic algorithm and design the genetic operators. Combined with the lowest horizontal line-search algorithm, Genetic Algorithm model is applied into rectangular object layout optimization. Results show that the model in the paper can satisfy not only the practical production requirements of guillotine, but also the requirement for production convenience. In this way, user can get optimal layout results effectively and quickly and a higher material using ratio at the same time for normal production practice.

Keywords: Genetic Algorithm, Genetic Operators, Rectangular Object Layout, Guillotine.

1. Introduction

The optimization of rectangular object layout is to put rectangular objects of different sizes in definite stock sheets and get the higher using ratio of material sheets with no interference and not beyond the boundary [1]. According to different technical and production requirements of companies, rectangular objects can be placed horizontally or vertically. Sometimes objects may be put at any angles for adapting certain performance requests or sizes. Because of too many choices for layout solution on stock sheets, problems are complicated usually. In fact, practices need an effective and easier method to prepare materials, to organize production, to manage their supply chains in order to get higher material using ratio and lower costs.

The Rectangular Object Layout is not a simple NP

(nondeterministic polynomial) optimization problem [2].Many researchers are engaged in this area and bring forward many approximate solutions[3], such as the sequent order algorithm, the matching algorithm for surplus rectangle and so on. The basic strategy of approximate solution is to select the best solution at current situation, so it is hard or impossible to find out a optimal solution. In the early of 1960s, Gilmore and Gomory[4~7]published four papers and put forward to the layout problem. During 1980s there ware many research results [8~11] on rectangular layout and solved some problems of rectangular parts and irregular parts layout. Segenriech and Braga put forward to a Genetic algorithm solution for multi parts layout problem[12].Genetic algorithm has the powerful ability to search globally and find out optimal solution with bigger probability. But unilateral pursuing for higher ratio and no concerning practical requirements from industry could not reach higher ratio and lower costs in production. It may take more human-hours to cut down material costs, and at last people got opposite results.

Combining the lowest-horizontal-line searching algorithm, the paper builds a genetic algorithm model and tries to solve rectangular object optimal layout. Main requirements from industry are guillotine and same-parts-together rule. Leaving safe distances for operating and special fiber directions or performances for metal parts are practical requirements too. Later one is represented by the texture of parts in this paper, that is, parts must be put at special directions with the drawing fiber of metal sheets. Optimization can help to improve the using ratio of stock sheets, reduce preparing time and standardize process management. Fig. 1 shows the different results of layout.

In Fig.1, layout solution a) and b) will give different production costs because same parts when putting together need much less operating time for cutting. Guillotine is a particular requirement from shear machine when cutting rectangular sheet parts. Shear machines can only cut straight lines through the whole sheet width, so layout c) is an ideal theoretical solution and not suitable for practical manufacturing. The correct and useful result should be layout d).

The objective of this paper is to propose an optimized solution for rectangular object layout by gene *Sponsored by State Key Technologies R&D Program

(2006BAF01A01)

_____________________________________978 -1-4244-1651-6/08/$25.00 © 2008 IEEE

theory and the lowest-horizontal-line search algorithm, which gives not only a higher material using ratio but also satisfies requirements in industry. The building process of genetic algorithm model for rectangular object optimal layout, including gene coding, decoding, population initialization, crossover operator, mutation operator, fitness function and operator selection, is in chapter 2. Chapter 3 gives a case study. Chapter 4 is conclusion parts.

2. Genetic algorithm model for rectangular object optimal Layout

The genetic algorithm is a global optimization method. It imitates the natural process of gene evolution and reconstruction. Parameters of the problem are coded and called "gene". Some genes build up a "chromosome" and then selections, crossovers, and variance operator similar to nature can be carried through. After many repeated operations one can find out a optimal solution [13,14,15]. Usually the evolutionary generation is about 100~ 500.

2.1 Gene coding The first key problem to be solved is the coding in

order to apply genetic algorithm. It is a key step for designing genetic algorithm too. Coding decides not only the arrangement form of an individual chromosome, but also the decoding from genotype to individual performance type in search space, and the designing of genetic operators. The coding influences the performance and the efficiency of the genetic algorithm directly.

Because stock sheets and parts are both rectangle, parts can be laid down mainly in horizontal or vertical directions in order to improve the using ratio of stock sheets. So the coding of each gene can be positive or negative. Provision in this paper is: positive serial numbers of rectangle parts stand for parts placed in horizontal direction, namely long sides of parts parallel

with X direction; negative serial numbers of the rectangle parts stand for parts placed in vertical direction, namely the long sides of the parts parallel with Y direction. For example, the chromosome (3, -1, 2, -4, 5) means that part No.3 is placed horizontally first; Then part No.1 vertically; part No.2 horizontally; part No.4 vertically; Finally part No.5 horizontally (in Fig. 2).

b. Same parts put togethera. Same parts not together

c. Guillotine unsatisfied d. Guillotine satisfied

Parts Unused Cutting Line

Fig. 1 Industry requirements for layout

Concerning characteristics of rectangular object optimal layout problem, we adapt the integer coding. Each rectangle part will be placed unify to carry on serial number first. Serial numbers for rectangle parts can be positive or negative, decided by the directions being placed horizontally or vertically. Each part should have a gene coding. If the total number of rectangle parts is n, then n serial numbers of rectangle parts constitute into a chromosome. The length of each chromosome is same as the total part number. In a chromosome, the coding of each gene should be corresponded to a serial number of rectangle parts. The arrangement of all gene codes consists of a chromosome.

2.2 Decoding Decoding helps people to confirm the conversion

method from the individual genotype to individual phenotype. In our case, this process is to convert chromosomes to a layout chart. On the basis of the lowest-horizontal-line search algorithm [16], steps are as follows:

1. Suppose that the initial highest contour is the bottom of stock sheets.

2. Whenever putting a part Pi onto stock sheet, try to select a lowest segment of the selection horizon from the highest contour. If there are several segments, we select the very left side and test whether the width of that line is larger than or equal to the width of the spare parts:

(1) If the width of that line is larger than the width of spare parts, then place that part in this position and renew the highest contour of the spare parts in the meantime.

(2) Otherwise, search a spare part that can be placed backward from the position of spare parts and exchange positions of these two spare parts in the meantime. If there is no one, promote the lowest horizon to the lower

3 24

5

XFig. 2 Part arrangements and

chromosome of genes

Y

height and renew the highest contour. Namely: For P= {P1, P2 …Pi…Pj…, Pn}, 1< Pi<Pj<n, if the part Pi’s width is larger than the width of the lowest horizon, then search in {Pi+1 …Pj…Pn}. If the part Pi’s width is smaller than the width of the lowest horizon, then place part Pj. At the same time, P= {P1, P2, …Pi…Pj… Pn} is renewed into P= {P1, P2…Pj…Pi… Pn}.

3. Repeat the 2nd step until the spare part placement finish.

4. Repeat above-mentioned process until all spare parts placement finish.

2.3 Initialization population Initialization population plays a key rule on the

convergence speed and solution quality of genetic algorithm. A simple method to generate the initialization population is random generating. This method is easy to be realized but can not assure the quality of initialization population. Due to neglecting data characteristics of stock sheets and spare parts, it is difficult to acquire a better initialization population with the random method. It is not easy to acquire satisfactory results even taking longer time. We recommend the heuristic rule used when generating initialization population.

The scale of initialization population plays certain influences on the performance of genetic algorithm. The bigger the scales of initialization population, the higher the diversity of the individual, then the algorithm is not easy to go into a risk of getting a partial solution. But bigger scales of initialization population may increase calculation time so to decrease the performance of genetic algorithm. If scales of initialization population are too small, the genetic algorithm’s search space is limited and as a result searching stops, probably, at the immaturity stage. In order to avoid this phenomenon, we must keep the community diversity, that is, scales of the community can't be too small. The scale of the community is between 30 and 200. An inadequate scale of the community will influence on comprehensive performance of genetic algorithm [17].

2.4 Crossover operator Crossover operator plays a core role in genetic

algorithm. Crossover means exchanging part structure of parents’ individual to generate new individual. Crossover is a main method to generate new individuals and decides the ability for global search in genetic algorithm. Crossover probability is between 0.4 and 0.99.

In chromosomes coded by integral sequences, the most difficult thing for crossover is that there will bring illegal genes after exchanging parts of gene fragments between two chromosomes, namely there will contain repeated gene in the chromosome. In the process of chromosome crossover, we need to make sure that not

only chromosome crossover occurs, but also no repeated gene appears after chromosome crossover.

The method for chromosome crossover is as follows: First, choose a fix crossover position and then keep the front part of genes fixed. Genes behind the cross point are crossover. Compare uncrossed genes between two chromosomes. Remained genes of chromosome are put into two array P[] and Q[] with original sequence after throwing away the same gene in two gene fragments. During operating the cross gene fragment of the chromosome genes not equal to array P[] and Q[] will exchange directly. The gene equal to array P[] and Q[] will be changed into genes in array P[] and Q[] and then exchange.

For example, two chromosomes to be crossover are: A = {-3, 5, 7, -1, 6, 8, -2, 4}, B= {5, -2, 4, -7, 8, 6, 3, -1}, and steps for the chromosome crossover are as follows:

(1) Firstly, set a crossover position and choose the fifth position is the point of intersection (the very left of each chromosome is 1, from left to right, the position serial number increases one by one in order). The arrow stands for the point of intersection;

part of fixed part of crossover

3,5,7, 1,6, 8,-2,4Apart of fixed part of crossover

5, 2, 4, 7,8, 6,3,-1B(2) Compare the fixed part of chromosome A and

B chromosome. Both A and B have gene 5 and 7. Then 5 and 7 are throwing away from fixed part of chromosome gene fragments A. Remaining genes of A chromosome are put into array P[]={-3, -1, 6}by original sequence. In the same way, 5 and 7 are throwing away from fixed part of chromosome gene fragments B, and the remaining gene of B chromosome are put into array Q[]={-2, 4, 8} by original sequence.

(3) Compare the crossover part of A chromosome and B chromosome. Each has different part: A chromosome has 8, -2, 4 and B has 6, 3, -1. Different part are then substituted respectively for the gene of P[] and Q[]. At this time, the crossover gene of A chromosome are changed into {-3, -1, 6}, and the B are changed into {-2, 4, 8}.

(4) After three step above, A chromosome and B chromosome are changed into:

Two new chromosomes will be generated after crossover operation:

A’ -3 5 7 -1 6 -2 4 8B’ 5 -2 4 -7 8 -3 -1 6

The implementing method of crossover operation is: set the number of rectangle parts is n, cross position is a random count in [1, n]. Set the cross probability in advance and crossover operation occurs according to the following situations:

Produce a random R between 0 and 1 If (R is smaller than cross probability)

{Crossover operation are carried on between two chromosomes in the population from random choice

}ElseCrossover operation didn’t occur.

2.5 Mutation operator From the point of genetic algorithm, the solution

evolution mainly depends on the choosing mechanism and crossover strategy. The mutation is an assisted method to generate new individuals and decides the local search ability of genetic algorithm. Combined crossover operator with mutation operator, we can accomplish the global search and the local search in the searching space. It will result in an optimization process with good searching performance by genetic algorithm. Mutation probability is between 0.4~ 0.99. Generally the mutation operator adopts the rotating mutation and the position mutation. In our case, the number of rectangle parts is n.

The thought for rotating mutation is producing a random num between 1 and n, and then the number is set to the opposite.

For example: Producing a random num 5,For Chromosome A= {-3, 5, 7, -1, 6, 8, -2, 4}

The result of the rotating mutation is: A'= {-3, 5, 7, -1, -6, 8, -2, 4}

The thought for position mutation is producing two random number num1 and num2 between 1 and n, and then exchange two positions each other.

For example: Producing two random num1 and num2 is 3 and 6,

For Chromosome A= {-3, 5, 7, -1, 6, 8, -2, 4} The result of the position mutation is: A'= {-3, 5, 8, -

1, 6, 7, -2, 4}

2.6 Fitness function The definition of fitness function applies a great

influence on the efficiency of solving problem. There are two methods to define fitness function: One method mainly considers the biggest height of the sheet layout and tries to make the height of the sheet layout lower. It can be expressed by the reciprocal of maximum height. The other method is considering reusing of wasting sheet and tries to improve the utilization ratio by reusing the waste part of sheet (between the highest outline of sheets and the highest contour of the sheet layout). Suppose the total area of the rectangle part is area1. Two kinds of fitness functions are defined according to different stock sheets [18].

(1) The width is definite and length is infinite For the situation of definite width and infinite length,

the height of layout is generally used to evaluate the quality of layout. The fitness function is as follows:

1areaFitnessh w

1

h——the maximum height of stock sheet w——the width of stock sheet (2) Both width and length are definite The fitness function is as follows:

12

areaFitnessarea h w

2

area2——the total area not including the last one stock sheet

h——the maximum height of the last stock sheet w——the width of the stock sheet Here the fitness function means the utilization ratio

of stock sheet

2.7 Operator selection In the process of genetic and natural evolution of

biology, species with more adaptive ability have more opportunities to be inherited to the next generation, and species with lower adaptive ability have fewer opportunities to be inherited to the next generation. The genetic algorithm imitates this process to select operator (or called Reproduction Operator) so to keep the survival of the fittest species in the community. Individuals with higher adaptability have higher probability to be inherited to the next generation; individuals with lower adaptability have smaller probability to be inherited to the next generation. The operator selection of genetic algorithm is used to decide how to select individuals in the community to inherit to the next generation.

The operator selection adopts elitist strategy [19]. The individuals with the highest adaptability in the current community don’t participate in crossover and mutation operation, but substitute the individuals with the lowest adaptability after crossover and mutation operation. The operation process is as follows:

(1) Find out the individuals with highest adaptability and the lowest adaptability in the current community.

(2) If the adaptability of the best individual in the current community is higher than the best individual up to now, the best individual in the current community becomes the new best individual up to now.

(3) The new best individual up to now then substitutes the worst individual in the current community.

This strategy makes sure that the best individual will not be broken by crossover and mutation operation. It is very important for assurance of genetic algorithm’s convergence.

3. Case study

According to the models above, we develop a system for rectangular object optimal layout on platform VB.NET. Part parameters for layout are in

Tab.1. The length of stock sheets is 1480 mm; the width is 1250 mm. When the texture of a part is ‘No’ in Tab.1, this part can be placed on sheets at any directions and has no any special requirement for metal performances. If the texture of a part is ‘Yes’, this part can only be placed on sheets at a specific direction, say, at a right angle with X direction. Table 1. Part Parameters in layout

The length of chromosome is 8 (equal with the total part number). Four kinds of situations are calculated and analyzed. Under each situation, we can get best layout and corresponding results.

1) The Generation of evolution changes and other parameters remain unchanged. The scale of initial population is 80. The crossover probability is 0.6. The mutation probability is 0.001. Under this situation, the larger the generation of evolution, the higher the using ratio of material. At the same time, the time of calculating is longer. Results are in Fig.3. In the paper 40-90 is recommended for rectangular part layout problem.

2) The scale of initial population changes and other parameters remain unchanged. The Generation of evolution is 50. The crossover probability is 0.6. The mutation probability is 0.001. With the increasing of initial population, the using ratio of material increases

and the calculating time increases too. It is unnecessary for the scale of initial population more than their permutations and combinations. Results are in Fig.4. In the paper 60-100 is recommended for rectangular part layout problem.

3) The crossover probability changes and other parameters remain unchanged. The scale of initial population is 80. The Generation of evolution is 50. The mutation probability is 0.001. The using ratio of material reaches the optimal when the crossover probability is about 0.6. Results are in Fig.5. In the paper 0.5-0.8 is recommended for rectangular part layout problem.

4) The mutation probability changes and other parameters remain unchanged. The scale of initial population is 80. The Generation of evolution is 50. The crossover probability is 0.6. The using ratio of material reaches the optimal when the mutation probability is about 0.001. Results are in Fig.6. In the paper 0.5-0.8 is recommended for rectangular part layout problem.

The optimized layout is in Fig.7. In Fig.7, letters represent parts (Tab.1) and numbers behind letters represent the number of parts. In this case, material utilization ratio is up to 96.92%. Parameters are: the scale of initial population=80, the generation of evolution=50, the crossover probability=0.6, the mutation probability=.001. The most important is that the high ratio is easy to fulfill in production.

Part Name

Part width (mm)

Part length (mm)

Part number Texture

A 62 589 20 NoB 125 500 50 NoC 242 300 18 NoD 42 250 35 NoE 76 89 89 NoF 54 105 65 NoG 45 50 45 NoH 120 120 30 No

Fig. 5 Results when the crossover probability changes

The crossover probabilityTh

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Fig.3 The results when the generation of evolution changes

The generation of evolution

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Fig.6 Results when the mutaionprobability changes

The mutation probability

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Fig.4 Results when the initial population changes

The initial population

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4. Conclusion

Genetic algorithm is an important method in the modern intelligent computation because of its strong global searching ability. Genetic Algorithm model proposed in the paper is very efficient for rectangular object layout optimization. Results can not only satisfy the practical production requirements of guillotine, but also the requirement for production convenience, and can be used for practical mass production in cutting stock.

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Fig.7 Optimal results for the case

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