An Improved Method For Knapsack Problem.

OutlineIntroduction of the Knapsack Problem

ObjectiveAlgorithms

Proposed ImprovementsComparison of Results

Conclusion

An Improved Method For Knapsack

Problem.

Franklin DJEUMOU(WITS), Byron JACOBS(WITS),Dessalegn Hirpa(AIMS), Morgan KAMGA(WITS), Alain

MBEBI(AIMS), Claude MichelNZOTUNGICIMPAYE(AIMS), Blessing OKEKE, MilaineSEUNEU(AIMS), Simphiwe SIMELENE(WITS), Luyanda

NDLOVU(WITS), Joseph KOLOKO (UP)

January 8, 2011Franklin DJEUMOU(WITS), Byron JACOBS(WITS), Dessalegn Hirpa(AIMS), Morgan KAMGA(WITS), Alain MBEBI(AIMS), Claude Michel NZOTUNGICIMPAYE(AIMS), Blessing OKEKE, Milaine SEUNEU(AIMS), Simphiwe SIMELENE(WITS), Luyanda NDLOVU(WITS), Joseph KOLOKO (UP)An Improved Method For Knapsack Problem.


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Conclusion

Introduction of the Knapsack Problem

Objective

AlgorithmsBrute Force MethodGreedy Algorithm Method

Description of the Greedy AlgorithmProblems and Benefits of this Methods

Proposed Improvements

Comparison of Results

Conclusion

Franklin DJEUMOU(WITS), Byron JACOBS(WITS), Dessalegn Hirpa(AIMS), Morgan KAMGA(WITS), Alain MBEBI(AIMS), Claude Michel NZOTUNGICIMPAYE(AIMS), Blessing OKEKE, Milaine SEUNEU(AIMS), Simphiwe SIMELENE(WITS), Luyanda NDLOVU(WITS), Joseph KOLOKO (UP)An Improved Method For Knapsack Problem.


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Conclusion

Introduction

I Knapsack problem consists of finding the best packingconfiguration to maximise benefit while abiding by theweight constraint

I Knapsack Problem cannot be solved in polynomial time



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Conclusion

The Knapsack Problem



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Conclusion

The Knapsack Problem



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Conclusion

Mathematical Formulation

(KP) : maxn∑

i=1

bixi

s.t.n∑

i=1

ωixi ≤ C

xi ∈ {0, 1}



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Conclusion

Objective

I Brief review of existing methods

I Benefits and Pitfalls

I Implement an Algorithm

I Improve optimality while being mindful of time constraints



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Conclusion

Brute Force MethodGreedy Algorithm Method

Brute Force Method

I Enumerates every possible packing configuration

I Choose the best solution

I Optimality is ensured

I Extremely costly in time, for large n



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Conclusion


Brute Force Method

I Enumerates every possible packing configuration

I Choose the best solution

I Optimality is ensured

I Extremely costly in time, for large n



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Conclusion


Greedy Algorithm

I Let ei = pi/wi be the efficiency of item i .

I The greedy algorithm first sorts items in the decreasingorder with respect to their efficiency. i.e item i comesbefore item j if ei > ej .

I It then selects the most efficient item available and placesit in the knapsack, reducing the knapsack’s availablecapacity.



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Conclusion


Greedy Algorithm






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Conclusion


Greedy Algorithm






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Conclusion


Greedy Algorithm






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Conclusion


Greedy Algorithm






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Conclusion


Greedy Algorithm






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Conclusion


Problems and Benefits of this Methods

I The Greedy Algorithm does not solve the problem tooptimality.

I It rather finds a local optimal solution.

I It operates in linear time, which is extremely efficient

I Will occasionally produce the optimal result



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Conclusion









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Conclusion

Proposed Improvements

I Use Genetic Algorithm

I Include the Greedy Solution in the population



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Conclusion

Genetic Algorithm

I Generate Population

I Include Greedy Solution

I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Genetic Algorithm



I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Genetic Algorithm



I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Genetic Algorithm



I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Genetic Algorithm



I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Genetic Algorithm



I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Genetic Algorithm



I Selection

I Crossover

I Mutation

I Next Generation



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Conclusion

Improvement Over Generations



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Conclusion

Time Comparison



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Conclusion

Performance



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Conclusion

Time Comparison



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Conclusion

Time Comparison



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Conclusion

Conclusion

I Genetic Algorithm has a small time cost for a potentialimprovement

I Further improvements can be made to GA by generatinga initially fit population, through small amounts of bruteforce

I The crossover technique can be further optimized forlarge n



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Conclusion

Thank you!!!!

Any question is mostwelcome!!!!



ObjectiveAlgorithms


Conclusion

Thank you!!!!Any question is most

welcome!!!!


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An Improved Method For Knapsack Problem.

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