The 2 Dimensional Cutting Stock Problem

Published Date: 02 Nov 2017

K.Ramesh* and N.Baskar †

* Department of Mechanical Engineering, M.I.E.T Engineering College, Gundur,

Trichy, Tamilnadu, India. Pin: 620007.

† Department of Mechanical Engineeering, M.A.M College of Engineering, Siruganur, Tiruchirappalli,Tamilnadu,India. Pin: 621 105.

*[email protected], † [email protected]

Abstract

The 2 Dimensional cutting stock problem is a common problem uncounted in the in sheet metal industries, lock industries, textile industries, etc.Here the problem is to reduce the waste in order to increase the profit. This problem is also called the general two dimensional problem or NP hard problems. This type of problem can be analyzed with varies heuristic approach with the combination of GA.Here we have analyzed the problem by using the GA with sample combination of the parts. The optimum value is found by the above technique and the concept has to be compared with standard available software like NEST MASTER 2010, the results are obtained using the trial & Error method.

1. INTRODUCTION

In Mass production industries small in efficiencies will lead to huge wastage while arranging the parts into the sheet. There are the different algorithms or approaches are currently used for the

Key word: Strip, Sheet Metal, Optimization, Genetic Algorithm,

arrangement of different parts into the

master sheet. The objective is to minimize the wastage or maximize the utilization of parts into sheet.

Many Industries, the experienced labours are decided to generate the optimum layout ,but the layout or may not be the optimum with time consuming is the major factor.Obvisouly it is the challenging task to obtain efficient in a reasonable time . Many researches were carried out to develop the methods for blank layout basically from the mathematical model to genetic algorithm.

Sheet metal stamping or blanking is the operation of cutting a flat surface from sheet metal, the metal is punched out from the sheet is called ‘blank’ and left out metal is scrap. The way in which arranging the different blanks or parts to be cut is otherwise is called the nesting of sheet .The aim of this process is to minimize the scrap and reduce the consuming time since 1960.

The same process is carried out manually with help of skilled worker to develop the good layout for spending more time to achieve the optimum result. This process is classified into one dimensional strip layout and two dimensional strip layout

Most of the modern manufacturing companies are developed from the manual method to semi automatic or manual method with help of computer technique [ adomwicz &albona1] to reduce the time and increase the cost saving. The facility layout problems are developed with the constraints like facilities, sequence, width [Al-hakim2].

The manual method further developed by many researchers using mathematical model,here limited variables associated with the problem can able to be solved[WilliamW.chow3].The linear programming problem (LP)can used only the compact layouts [paul et el4][Eisemann et al5 ].Similarly Dynamic programming problem developed by[Haims& freeman6] .Eventhough the huge modification can be arrived in this area ,the optimum results from the step by step procedure is called Heuristic approaches will give hands to researchers get the feasible solution[Nee &venkatesan7].This technique is further classified into two types (i)Regular (ii)Irregular[Albona&Saruppo8]

The Heuristic process is mixed with specific algorithm is a meta heuristic algorithm for best searching sequence of the shapes to handle into the sheets, but the process involves more number of variables. The small in efficiencies will lead to high wastage with time consuming is called NP hard problem[T.J.Nye9],this type problem will fall on nontraditional Technique can be solved with help of various techniques like GA[Freeman&Shapira10][Bortfeldt et al11 ]

[Jakobs12] ,SA[Jain et al13],TB,ACO,NN,etc

GA is the recent optimum technique tool in the sheet metal industries, they use only three operators like crossover, mutation, interactive [Islier et al14 ].But it affects the diversity of the problem however it may affect the good solution. But there are number of nesting software’s are available with the limited condition will not satisfying the global targets.

2. INTRODUCTION TO GENETIC ALGORITHM

Genetic algorithm is a part of evolutional computing technique in the area of artificial intelligence since 1960’s.This algorithm was invented by John Holland and developed by his teammates. This algorithm consists of cells which a set of chromosomes has called population or otherwise called as a string having a genes.It encodes a particular type of problem. Each genes having a value called allele. The whole set of chromosome is called genome, considering a regular set of genes in genome is called geno type. It has a mental & physical characteristic like appearance colour, approach, intelligence etc.

The reproduction is nothing but a recombination of genes after crossover, mutation operation. Genes from parents form in some way to whole new chromosome After it is mutated(A bit of chromosome has changed ).then fitness of the chromosome can be determined by the life of the organism.

The solution of the given problem is looking for two extreme limit ie maximum or minimum in the search space (It can be known by the time of solving a problem) but search is complicated.we do not know where to start & search.This type of problem can be solved by Simulated Algorithm,Tabu search,Genetic algorithm.Basically this type of problem gives us a good solution but not a real optimum value.

2.1. BASIC STEPS IN GA

STEP 1-START- Generate the random population of n chromosome

STEP 2-FITNESS-Evaluate the fitness function of each chromosome in the population

STEP 3-NEW POPULATION-Create a new population for repeating the following steps, until get the new one

STEP 4-SELECTION-Select the two parent chromosomes in the population according to fitness function

STEP 5-CROSSOVER-with crossover probability PC(85% to 95 %),crossover the parent to form a new chromosome called as offspring

STEP 6-MUTATION-with mutation probability pm (0.5%to1%) mutate the new chromosome (offspring)to get a fine tuned.

STEP 7-ACCEPTING-place the new chromosome in the population

STEP 8-REPLACE-use new generated population for further execution of the problem.

3.PROPOSED GENETIC ALGORITHM

Many optimization problems from the industrial world, in particular the manufacturing systems are complex in nature and quite hard to solve by conventional optimization techniques. Simulating the natural evolutionary process of human being results in stochastic optimization techniques called evolutionary algorithms, which can often the outperform the conventional optimization methods when applied to difficult real world problem.

Genetic algorithms are search algorithm based on the mechanics of natural selection and natural genetics. Basically, these algorithms are different from normal optimization techniques and search procedures in four different ways

1. GAs work with a coding of parameter set not the parameter themselves.

2. GAs search from a population of points, not a single point.

3. GAs use pay-off values (objective function) information, not derivatives

4. GAs use probilistic transition rules not deterministic rules

3.1.CHOROMOSOME REPRESENTATION

Binary string coding is the most classical approach used by GA researchers, because its simplicity and tractability. Nowadays the direct manipulation value chromosome raised some considerable interest .This type of GA closely referred as real coded GA.Here problems can be solved by the above said one.

3.2. EVALUATION OF OBJECTIVE

As far as the objective function is concerned, the total utilization of the parts in the required sheet. Here the arrangement or the sequence of the parts with respective orientation will be converted into coding; it can be solved by the formula. n

The effective utilization O(s) = ∑ (Ai /As)

I=1

As –Area of the Sheet

Aj –Area of the pattern

3.3. INITIAL POPULATION

This is the first step of the genetic algorithm process. It is created by generating the random variables from time to time.

3.4. SELECTON MECHANISM

The selection process has been done after creating the initial population .In order to get good offspring proficient parent selection mechanism is necessary. This process is used to determine the number trials for one particular string used in reproduction. The chance of selecting one chromosome as a parent should be directly proportional to the number offspring produced. Generally various methods are used for selection processes viz,proportionate selection method, roulette wheel selection ,stochastic universal sampling etc.

Three measures of the performance of the selection process are bias, spread and efficiency. Bias defines the obsolete difference between the actual and expected selection probabilities of individuals. Spread is the range in the possible number of trials that individuals may achieve. The efficiency is related to overall time complexity of the problem

In most practice, a proportionate selection approach is adopted as the selection procedure. It belongs to the fitness proportional selection and can select a new population with respect to the probability distribution based on the fitness value. This process constructed as follows:

1. Calculate the fitness value for each chromosome

2. Calculate the total fitness value for the population

3. Calculate the selection probability for each chromosome

4. Calculate cumulative probability for each chromosome.

The following relation gives the probability of selecting the ith string

Pi = Fi / Favg

Where

Favg –Average value of the fitness function

Fi -The fitness function of the ith string

Pi - The probability selecting the ith string. Using the above selection procedure a new population is created.

3.5. CROSS OVER OPERATION

How ever the selection process will not produce any new chromosome, but the crossover can perform. Exchange of genetic material among the chromosome is termed as cross over. Various techniques have been devoted for cross over operation.

Traditionally, GA researchers set the number of cross over points as one or two. These cross over points are randomly chosen and the segments of the chromosome between them are exchanged

Chromosome before cross over

2 215 5 36 4 78 3 205 1 96

3 125 4 89 1 13 1 124 3 59

Chromosome after cross over

2 215 5 36 4 78 1 124 3 59

5 125 4 89 2 13 3 205 1 96

3.6. MUTATION OPERATION

Mutation involves flipping a bit of chromosome after finishing the cross over operation. Here the parameter Pm is used to control the mutation operation is called the probability of mutation. Although the mutation is a secondary operation, it will restore lost genetic material. The crossover cannot regenerate a bit at same position, while the mutation could perform. In order to subject the string for mutation, a random number is generated from 1 to the length of the chromosome.

The number of strings to be mutated depends upon the mutation probability Pm and it is identified the formula

Length of the chromosome X Population size (Pm) =Number of strings to be mutated

Generally the mutation probability lies between the limit 0 to 1 to be precise, the value lies in the range 0.001 to 0.05.since we are using the real coded GA.The minimum number of strings mutated will be two and its sum.

3.6.1. TYPES OF MUTATION

1. Single point mutation

Chromosome Before mutation

2 215 5 36 4 78 1 124 3 59

5 125 4 89 2 13 3 205 1 96

Chromosome after mutation

2 215 1 36 4 78 5 124 3 59

5 125 4 89 2 13 3 205 1 96

2. Two point mutation

Chromosome Before mutation

2 215 5 36 4 78 1 124 3 59

5 125 4 89 2 13 3 205 1 96

Chromosome after mutation

2 215 5 205 4 78 1 124 3 59

5 125 4 89 2 13 3 036 1 96

Advantage of the first method over the second method is the probability of two bits to be mutated is 1.But in the second method, there may be a chance that the mutation points lie on bits, which have same numerical value.

4. EXPERIMENTAL ANALYSIS

Intel core i 3 system with 4GB Ram,

500 GB hard disk was used to generate data’s. The software like C soft is used as front Engine and ACAD 2006 is used as back end. Initially we have analyzed 3parts with their orientation and samples of results was given in the fig1-14.By analyzing these configuration we will come to conclusion that the combination of the parts 8 9 10 (Fig 8) gave the better utilization efficiency. The same process can be analyzed by different combination in the formula of m! X n! X no of rotations.

Fig.1 (combination of part 1 2 3)

Fig.2 (combination of part 2 3 4)

Fig.3 (combination of part 3 4 5)

Fig.4 (combination of part 4 5 6)

Fig.5 (combination of part 5 6 7)

Fig .6(combination of part 6 7 8)

Fig.7 (combination of part 7 8 9)

Fig.8 (combination of part 8 9 10)

Fig.9 (combination of part 9 10 11)

Fig.10 (combination of part 10 11 12)

Fig.11 (combination of part 11 12 13)

Fig.12 (combination of part 12 13 14)

Fig.13 (combination of part 13 14 15)

Fig.14(combination of 26 parts using NESMATER 2010)

5. CONCLUSION

The metal stamping process was optimized by GA using the computer technology, which can be optimized the layout of two dimensional cutting stock problem. This algorithm may give the better solution only on the local optimum. If the global variables has been modified to achieve the efficient results. The gab of the parts gets closed by applying the specific heuristic algorithm because of the genetic algorithm process attains the ultimate criterion.