multiple knapsack problem

8/6/2019 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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multiple knapsack problem

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