Download - Multiple Knapsack Problem
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y In this problem m knapsacks are given & our aim is same asin single knapsack problem , to fill knapsacks tillmaximum capacity.
y Weight of i object is not constant , it has range 1
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y We have to find a vector x=(x1,x2,.................xn ) such that noknapsack overflows:ni=1wij
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y Genetic Algorithms are a family of computational modelsinspired by evolution.
y These algorithms encode a potential solution to a specificproblem on a simple chromosome-like data structure and apply
recombination operators to these structures as to preservecritical information.
y An implementation of genetic algorithm begins with apopulation of (typically random) chromosomes.
y One then evaluates these structures and allocated reproductive
opportunities in such a way that these chromosomes whichrepresent a better solution to the target problem are given morechances to `reproduce than those chromosomes which arepoorer solutions. The 'goodness' of a solution is typically defined
with respect to the current population.
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y Encoding
y Fitness Evaluation
y Reproduction
y Survivor Selection
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yReproduction operator
yCrossover
yMutation
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y Crossovery Two parents produce two offspringy There is a chance that the chromosomes of the two
parents are copied unmodified as offspringy There is a chance that the chromosomes of the two
parents are randomly recombined (crossover) to formoffspring
y Generally the chance of crossover is between 0.6 and 1.0
y
Mutationy There is a chance that a gene of a child is changedrandomly
y Generally the chance of mutation is low (e.g. 0.001)
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y Two point crossover (Multi point crossover)
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y Mutation
y Generating new offspring from single parent
y Maintaining the diversity of the individualsy Crossover can only explore the combinations of the current gene
pooly Mutation can generate new genes
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y Thus adding the penalty term max{pi} , we getfollowing fitness function to be maximized-
f(x)=ni=1 xi pi. s. max{pi}
Where,y s- the number of knapsacks overfilled .
y The no. of times this term is used , it reflects no. ofknapsacks overfilled.
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y This approach was applied to a series of problems knap15,knap20 & so on given by Petersen , which canbe accessed at Or library by Beasley .
y
In this problem 100 runs are to be filled in 10knapsacks by 15,20 & so on objects respectively forknap15,knap20 problem & so on.
y The value of fitness function produced by each of
these objects containing runs are given . It shows thatwhich runs are feasible to be added & which are not.
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RESULTS OF PROBLEMS KNAP15,20,28,39,50 AS SOLVED
BY GENETIC ALGORITHM
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y In this table only some values are shown becauseothers produce fitness function below minimumoptimal value.
y For example in knap39,some runs produce fitnessfunction value beneath 10,561,so values dont add up to100.
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y We can solve knapsack problem by using geneticalgorithm.
y We allow infeasible strings to participate since theyalso contribute information.
y But we reduce their strength by introducing penalty
term.
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