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GAs and random keys• Introduced by Bean (1994)for sequencing problems.• Individuals are strings ofreal-valued numbers(random keys) in theinterval [0,1).S = ( 0.25, 0.19, 0.67, 0.05, 0.89 )s(1) s(2) s(3) s(4) s(5)LION 7 ✤ January 2013BRKGA
GAs and random keys• Introduced by Bean (1994)for sequencing problems.• Individuals are strings ofreal-valued numbers(random keys) in theinterval [0,1).• Sorting random keys resultsin a sequencing order.S = ( 0.25, 0.19, 0.67, 0.05, 0.89 )s(1) s(2) s(3) s(4) s(5)S' = ( 0.05, 0.19, 0.25, 0.67, 0.89 )s(4) s(2) s(1) s(3) s(5)Sequence: 4 – 2 – 1 – 3 – 5LION 7 ✤ January 2013BRKGA
Page 1 and 2: Biased random-key geneticalgorithms
Page 3 and 4: Summary• Metaheuristics and basic
Page 5 and 6: MetaheuristicsMetaheuristics are hi
Page 7 and 8: Genetic algorithmsLION 7 ✤ Januar
Page 9 and 10: Genetic algorithmsIndividual: solut
Page 11 and 12: Genetic algorithmsGenetic algorithm
Page 13 and 14: Genetic algorithmsProbability of se
Page 15 and 16: Crossover and mutationmutationaComb
Page 17 and 18: Crossover and mutationmutationaComb
Page 19 and 20: Evolution of solutionsLION 7 ✤ Ja
Page 27 and 28: Encoding with random keys• A rand
Page 29 and 30: Encoding with random keys• A rand
Page 31 and 32: Encoding with random keys: Sequenci
Page 33 and 34: Encoding with random keys: Sequenci
Page 35 and 36: Encoding with random keys: Subsetse
Page 37 and 38: Encoding with random keys: Assignin
Page 39 and 40: Encoding with random keys: Assignin
Page 41: GAs and random keys• Introduced b
Page 45 and 46: GAs and random keys• Mating is do
Page 53 and 54: GAs and random keysInitial populati
Page 55 and 56: GAs and random keysAt the K-th gene
Page 57 and 58: GAs and random keysAt the K-th gene
Page 59 and 60: GAs and random keysEvolutionary dyn
Page 61 and 62: GAs and random keysEvolutionary dyn
Page 63 and 64: Biased random key genetic algorithm
Page 65 and 66: How RKGA & BRKGA differRKGAboth par
Page 67 and 68: How RKGA & BRKGA differRKGAboth par
Page 69 and 70: Paper comparing BRKGA and Bean'sMet
Page 71 and 72: Pr(t BRKGA≤ t RKGA) = 0.740Probab
Page 73 and 74: Pr(t BRKGA≤ t RKGA) = 0.999set co
Page 75 and 76: Pr(t BRKGA≤ t RKGA) = 0.733set co
Page 77 and 78: set k-coveringproblem: scp41-2Pr(t
Page 79 and 80: Pr(t BRKGA≤ t RKGA) = 0.881set k-
Page 81 and 82: Pr(t BRKGA≤ t RKGA) = 0.847set k-
Page 83 and 84: Observations• Random method: keys
Page 85 and 86: Observations• Random method: keys
Page 87 and 88: Framework for biased random-key gen
Page 89 and 90: Decoding of random key vectors can
Page 91 and 92: solutionNetwork monitor location pr
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Randomized heuristic iterationcount
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In most of the independent runs, th
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In most of the independent runs, th
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However, some runs take much longer
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However, some runs take much longer
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Probability that algorithm will tak
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Probability that algorithm will sti
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Restart strategies• First propose
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Restart strategy for BRKGA• Recal
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Example of restart strategy for BRK
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Specifying a biased random-key GA
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Specifying a biased random-key GA
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Specifying a biased random-key GAPa
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kgaAPI: A C++ API for BRKGA• Effi
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kgaAPI: A C++ API for BRKGA• Effi
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An example BRKGA:Packing weightedre
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Constrained orthogonal packing• G
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Constrained orthogonal packing• G
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Constrained orthogonal packing• r
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Constrained orthogonal packing• r
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ObjectiveAmong the many feasible pa
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ObjectiveAmong the many feasible pa
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ApplicationsProblem arises in sever
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Hopper & Turton, 2001Instance 4-2 6
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Hopper & Turton, 2001Instance 4-2 6
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BRKGA forconstrained 2-dimorthogona
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Encoding• Solutions are encoded a
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Decoding• Simple heuristic to pac
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Decoding• Simple heuristic to pac
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Decoding• A maximal empty rectang
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132 4BL can run into problems eveno
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433412RTPS: 1-2-4-312RTPS: 1-2-3-42
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Decoding• If LB is used, ERSs are
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1BL2BL3LB4BLERS[1]LION 7 ✤ Januar
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2BL3LB4BL1BLERS[1]LION 7 ✤ Januar
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3LB4BL1BL2BLLION 7 ✤ January 2013
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3LB4BLERS[2]1BL2BLLION 7 ✤ Januar
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4BL3LBERS[1]1BL2BLLION 7 ✤ Januar
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4BL4 does fitin ERS[2].3LBERS[2]1BL
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Experimental resultsLION 7 ✤ Janu
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Design• We compare solution value
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Design• We compare solution value
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Number of best solutions / total in
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New BKSfor a 100 x100doublyconstrai
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Some remarksWe have extended this t
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Literature• BRKGAs have been appl
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Telecommunications• Routing: Eric
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Scheduling• Job-shop scheduling:
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Network optimization• Concave min
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Packing• 2D orthogonal packing: G
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Transportation• Tollbooth assignm
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Automatic parameter tuning• GRASP
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Software• C++ API: Toso and R. (2
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