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Constrained orthogonal packing• r[i] rectangles of type i = 1, ..., N are to be packed in thelarge rectangle without overlap and such that their edgesare parallel to the edges of the large rectangle;• For i = 1, ..., N, we require that:0 ≤P[i]≤ r[i] ≤ Q[i]WH42313112LION 7 ✤ January 2013BRKGA
Constrained orthogonal packing• r[i] rectangles of type i = 1, ..., N are to be packed in thelarge rectangle without overlap and such that their edgesare parallel to the edges of the large rectangle;• For i = 1, ..., N, we require that:0 ≤P[i]≤ r[i] ≤ Q[i]WH42313112Suppose 5≤ r[1] ≤ 12LION 7 ✤ January 2013BRKGA
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Biased random-key geneticalgorithms
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Summary• Metaheuristics and basic
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MetaheuristicsMetaheuristics are hi
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Genetic algorithmsLION 7 ✤ Januar
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Genetic algorithmsIndividual: solut
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Genetic algorithmsGenetic algorithm
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Genetic algorithmsProbability of se
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Crossover and mutationmutationaComb
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Crossover and mutationmutationaComb
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Evolution of solutionsLION 7 ✤ Ja
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Encoding with random keys• A rand
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Encoding with random keys• A rand
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Encoding with random keys: Sequenci
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Encoding with random keys: Sequenci
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Encoding with random keys: Subsetse
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Encoding with random keys: Assignin
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Encoding with random keys: Assignin
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GAs and random keys• Introduced b
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GAs and random keys• Introduced b
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GAs and random keys• Mating is do
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GAs and random keysInitial populati
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GAs and random keysAt the K-th gene
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GAs and random keysAt the K-th gene
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GAs and random keysEvolutionary dyn
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GAs and random keysEvolutionary dyn
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Biased random key genetic algorithm
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How RKGA & BRKGA differRKGAboth par
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How RKGA & BRKGA differRKGAboth par
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Paper comparing BRKGA and Bean'sMet
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Pr(t BRKGA≤ t RKGA) = 0.740Probab
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Pr(t BRKGA≤ t RKGA) = 0.999set co
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Pr(t BRKGA≤ t RKGA) = 0.733set co
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set k-coveringproblem: scp41-2Pr(t
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Pr(t BRKGA≤ t RKGA) = 0.881set k-
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Pr(t BRKGA≤ t RKGA) = 0.847set k-
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Observations• Random method: keys
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Observations• Random method: keys
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Framework for biased random-key gen
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Decoding of random key vectors can
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solutionNetwork monitor location pr
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Randomized heuristic iterationcount
Page 95 and 96: In most of the independent runs, th
Page 97 and 98: In most of the independent runs, th
Page 99 and 100: However, some runs take much longer
Page 101 and 102: However, some runs take much longer
Page 103 and 104: Probability that algorithm will tak
Page 105 and 106: Probability that algorithm will sti
Page 107 and 108: Restart strategies• First propose
Page 109 and 110: Restart strategy for BRKGA• Recal
Page 111 and 112: Example of restart strategy for BRK
Page 123 and 124: Specifying a biased random-key GA
Page 125 and 126: Specifying a biased random-key GA
Page 127 and 128: Specifying a biased random-key GAPa
Page 135 and 136: kgaAPI: A C++ API for BRKGA• Effi
Page 137 and 138: kgaAPI: A C++ API for BRKGA• Effi
Page 139 and 140: An example BRKGA:Packing weightedre
Page 141 and 142: Constrained orthogonal packing• G
Page 143 and 144: Constrained orthogonal packing• G
Page 145: Constrained orthogonal packing• r
Page 149 and 150: ObjectiveAmong the many feasible pa
Page 151 and 152: ObjectiveAmong the many feasible pa
Page 153 and 154: ApplicationsProblem arises in sever
Page 155 and 156: Hopper & Turton, 2001Instance 4-2 6
Page 157 and 158: Hopper & Turton, 2001Instance 4-2 6
Page 159 and 160: BRKGA forconstrained 2-dimorthogona
Page 161 and 162: Encoding• Solutions are encoded a
Page 163 and 164: Decoding• Simple heuristic to pac
Page 165 and 166: Decoding• Simple heuristic to pac
Page 167 and 168: Decoding• A maximal empty rectang
Page 169 and 170: 132 4BL can run into problems eveno
Page 171 and 172: 433412RTPS: 1-2-4-312RTPS: 1-2-3-42
Page 173 and 174: Decoding• If LB is used, ERSs are
Page 175 and 176: 1BL2BL3LB4BLERS[1]LION 7 ✤ Januar
Page 177 and 178: 2BL3LB4BL1BLERS[1]LION 7 ✤ Januar
Page 179 and 180: 3LB4BL1BL2BLLION 7 ✤ January 2013
Page 181 and 182: 3LB4BLERS[2]1BL2BLLION 7 ✤ Januar
Page 183 and 184: 4BL3LBERS[1]1BL2BLLION 7 ✤ Januar
Page 185 and 186: 4BL4 does fitin ERS[2].3LBERS[2]1BL
Page 187 and 188: Experimental resultsLION 7 ✤ Janu
Page 189 and 190: Design• We compare solution value
Page 191 and 192: Design• We compare solution value
Page 193 and 194: Number of best solutions / total in
Page 195 and 196: 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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brkga - AT&T Labs Research

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