Permutation distance measures for memetic algorithms with population management

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1 for memetic algorithms with population management for M PM M PM Marc Sevaux University of Valenciennes, CNS, UM 8530, LMIH-SP Le Mont Houy - Bat Jonas 2, F Valenciennes cedex 9, France marc.sevaux@univ-valenciennes.fr 2 University of ntwerp, Faculty of pplied Economics Prinsstraat 13, B 2000 ntwerp, Belgium kenneth.sorensen@ua.ac.be MIC 2005 Sixth Metaheuristics International Conference Vienna ugust 2005 eversal

2 M PM: memetic algorithms with population management for M PM M PM eversal

3 M PM: memetic algorithms with population management for M PM M PM eversal

4 M PM: memetic algorithms with population management for M PM M PM eversal

5 M PM: memetic algorithms with population management for M PM M PM eversal

6 M PM: memetic algorithms with population management for M PM M PM eversal

7 M PM: memetic algorithms with population management for M PM M PM eversal

8 M PM: memetic algorithms with population management for M PM M PM eversal

9 M PM: memetic algorithms with population management for M PM M PM eversal

10 M PM: memetic algorithms with population management for M PM M PM eversal

11 The total weighted tardiness single machine scheduling problem for M PM n jobs One machine No preemption elease date r j Processing times p j Due dates d j Weights w j Completion time C j Tardiness T j = max(0, C j d j ) Objective: minimize total weighted tardiness (1 r j w j T j ) N P-Hard in a strong sense M PM eversal

12 for M PM Simple M PM for total weighted tardiness single machine scheduling problem (1 r j w j T j ) Linear ordered crossover Pairwise adjacent interchange local search General pairwise interchange mutation Sawtooth population management parameter Develop a normalized version of each distance measure (d [0, 1]) Examine each of the and compare performance M PM eversal

13 for M PM (absolute) (relative) Longest Common Subsequence eversal Kendall s Tau emark: only permutations of [1 n] M PM eversal

14 The Exact Match Distance S. onald. More distance functions for order-based encodings. In Proceedings of the IEEE Conference on Evolutionary Computation, pages , IEEE Press, New York, 1998 Hamming distance for permutations { n 0 if S(i) = T (i) d em (S, T ) = x i where x i = 1 otherwise i=1 Normalized: ˆd em (S, T ) = 1 n d em Example: S [1,2,3,4,5] d em = 5 T [2,4,5,1,3] ˆdem = 1 for M PM M PM eversal

15 The Distance S. onald. More distance functions for order-based encodings. In Proceedings of the IEEE Conference on Evolutionary Computation, pages , IEEE Press, New York, 1998 Spearman s footrule Total deviation of all items d dev (S, T ) = Normalized: ˆd dev (S, T ) = n i j where T (j) = S(i) = k k=1 { 2/n 2 d dev (S, T ) if n is even 2/(n 2 1) d dev (S, T ) Example: S [1,2,3,4,5] d dev = 10 T [2,4,5,1,3] ˆd dev = 10 2/24 =.833 if n is odd for M PM M PM eversal

16 The Squared Distance Spearman s rank correlation coefficient Sum of squares of deviations d sdev (S, T ) = Normalized: n (i j) 2 where T (j) = S(i) = k k=1 ˆd sdev (S, T ) = 3 n 3 n d sdev(s, T ) Example: S [1,2,3,4,5] d dev = 22 T [2,4,5,1,3] ˆd dev = 22 3/120 =.55 for M PM M PM eversal

17 The Distance V. Campos, M. Laguna, and. Martí. Context-independent scatter search and tabu search for permutation problems. INFOMS Journal on Computing, 17: , 2005 bsolute position of the items is important Differences between indices of items Normalized: ˆd (S, T ) = d (S, T ) = n S(i) T (i). i=1 { 2/n 2 d (S, T ) if n is even 2/(n 2 1) d (S, T ) Example: S [1,2,3,4,5] d = 10 T [2,4,5,1,3] ˆd = 10 2/24 =.833 if n is odd for M PM M PM eversal

18 The Distance V. Campos, M. Laguna, and. Martí. Context-independent scatter search and tabu search for permutation problems. INFOMS Journal on Computing, 17: , 2005 elative position of the items is important Number of times S(i + 1) does not follow S(i) in T Normalized: ˆd (S, T ) = 1 n 1 d (S, T ) Example: S [1,2,3,4,5] d = 3 T [2,4,5,1,3] ˆd = 3/4 =.75 for M PM M PM eversal

19 The Distance for M PM.. Wagner and M.J. Fischer. The string-to-string correction problem. Journal of the ssociation for Computing Machinery, 21: , 1974 Three edit operations (add, remove, substitute) Minimum number of edit operations required to transform S into T Or: minimum cost of edit transformation of S into T Normalized: divide by n Example: S [1,2,3,4,5] d edit = 4 (emove and add 1 and 3) T [2,4,5,1,3] ˆd = 4/5 =.8 M PM eversal

20 The Longest Common Subsequence Distance for M PM Subsequence: sequence of characters that appears in the same order in the string Example: [3, 5, 7] is a subsequence of [1, 2, 3, 4, 5, 6, 7] : longest subsequence that appears in both strings distance: n Normalize: ˆd lcs = d lcs n 1 Example: S [1,2,3,4,5] d lcs = 2 ( = [2,4,5]) T [2,4,5,1,3] ˆd lcs = 2/4 =.5 lternative: longest common substring M PM eversal

21 The eversal Distance for M PM. Caprara. Sorting permutations by reversals and eulerian cycle decompositions. SIM Journal on Discrete Mathematics, 12(1):91 110, 1999 Minimum number of substring reversals required to transform S into T eversal Important in molecular biology But: N P-hard: Too bad: 2 opt M PM eversal

22 Kendall s Tau M. Kendall and J. Dickinson Gibbons. ank Correlation Methods. Oxford University Press, New York, 1990 Number of pairwise adjacent permutations required to transform S into T where z ij = Normalized: d τ (S, T ) = n 1 n i=1 j=i+1 z ij { 1 if S(i) < S(j) and T (i) > T (j) 0 otherwise ˆd τ (S, T ) = 2 d τ (S, T ) n 2 n Example: S [1,2,3,4,5] d τ = 6 T [2,4,5,1,3] ˆd τ = 6 2/20 =.6 for M PM M PM eversal

23 of for M PM 20 instances, 100 jobs 5 minutes per run Complexity vg. it. O(n) O(n) O(n) O(n 2 ) O(n 2 ) O(n 2 ) Longest common subsequence O(n 2 ) Kendall s tau O(n 2 ) M PM eversal

24 Ò Ó «ª Ô ƒ YX Õ ± ˆ Š a` Œ cb ¹ Üed Ž Ýgf»º )( S } ÑÐ Fraction over best-found distance distance distance distance distance distance Longest common subsequence Kendall s tau ~ ÖØ ²³ µ Z[\] ^_ ÙÛÚ! "# $% &' ÞàßàáÛâàã TUVW hijk ¼½¾ lmàá š noâã pqäå äøåàæûçàè œ rsæç tuèé ÊË *+,-./ :; + <= + >? + BC + DE + FG + HI + žÿ JK + vwxy LM + ÌÍÎÏ NO + z{ PQ Data file é

25 for M PM Simple work better Tentative explanation Local search is very fast but not very powerful Simple allow for a lot more iterations Better cannot compensate Perspectives Better local search mechanisms Different problems Interplay between population management strategy and distance measure... M PM eversal

26 for memetic algorithms with population management for M PM M PM Marc Sevaux University of Valenciennes, CNS, UM 8530, LMIH-SP Le Mont Houy - Bat Jonas 2, F Valenciennes cedex 9, France marc.sevaux@univ-valenciennes.fr 2 University of ntwerp, Faculty of pplied Economics Prinsstraat 13, B 2000 ntwerp, Belgium kenneth.sorensen@ua.ac.be MIC 2005 Sixth Metaheuristics International Conference Vienna ugust 2005 eversal

Permutation distance measures for memetic algorithms with population management

MIC2005: The Sixth Metaheuristics International Conference??-1 Permutation distance measures for memetic algorithms with population management Marc Sevaux Kenneth Sörensen University of Valenciennes, CNRS,