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Decoding• A maximal empty rectangular space (ERS) is an emptyrectangular space not contained in any other ERS.• ERSs are generated and updated using the DifferenceProcess of Lai and Chan (1997).• When placing a rectangle, we limit ourselves only tomaximal ERSs. We order all the maximal ERSs and placethe rectangle in the first maximal ERS in which it fits.• Let (x[i], y[i]) be the coordinates of the bottom leftcorner of the i-th ERS.LION 7 ✤ January 2013BRKGA
Decoding• A maximal empty rectangular space (ERS) is an emptyrectangular space not contained in any other ERS.• ERSs are generated and updated using the DifferenceProcess of Lai and Chan (1997).• When placing a rectangle, we limit ourselves only tomaximal ERSs. We order all the maximal ERSs and placethe rectangle in the first maximal ERS in which it fits.• Let (x[i], y[i]) be the coordinates of the bottom leftcorner of the i-th ERS.i-thERSLION 7 ✤ January 2013(x[i], y[i])BRKGA
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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
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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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Example of restart strategy for BRK
Page 115 and 116: 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 and 146: Constrained orthogonal packing• r
Page 147 and 148: 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: Decoding• Simple heuristic to pac
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
Page 197 and 198: Some remarksWe have extended this t
Page 199 and 200: Literature• BRKGAs have been appl
Page 201 and 202: Telecommunications• Routing: Eric
Page 203 and 204: Scheduling• Job-shop scheduling:
Page 205 and 206: Network optimization• Concave min
Page 207 and 208: Packing• 2D orthogonal packing: G
Page 209 and 210: Transportation• Tollbooth assignm
Page 211 and 212: Automatic parameter tuning• GRASP
Page 213 and 214: Software• C++ API: Toso and R. (2
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