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A New Solution Approach for Grouping Problems Based on Evolution Strategies

A New Solution Approach for Grouping Problems Based on Evolution Strategies. By: A. H. Kashan. Agenda. Grouping problems and their applications Grouping Genetic Algorithm (GGA) Evolutionary Strategy (ES) Grouping Evolution Strategy Experimental Results.

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A New Solution Approach for Grouping Problems Based on Evolution Strategies

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  1. A New Solution Approach for Grouping Problems Based on Evolution Strategies • By: A. H. Kashan

  2. Agenda • Grouping problems and their applications • Grouping Genetic Algorithm (GGA) • Evolutionary Strategy (ES) • Grouping Evolution Strategy • Experimental Results A.H. Kashan Grouping Evolutionary Strategy (GES)

  3. Grouping Problems • Partitioning a set (V) of n items into a collection of mutually disjoint subsets (groups, Vi) such that: • Partition the members of set V into D (1≤ D ≤ n) different groups where each item is exactly in one group • Ordering of groups is not relevant • well-known problems as grouping problems: • graph (vertex/edge) coloring, bin packing, batch-processing machine scheduling, line-balancing, timetabling, cell formation, vehicle routing etc. A.H. Kashan Grouping Evolutionary Strategy (GES)

  4. Grouping Genetic Algorithm (GGA) • Two main representation schemes: • Number encoding: each item is encoded with a group ID, for example 2 1 3 2 1 • Redundancy: example, • Individual 1: 2 1 3 2 1 {2, 5}{1, 4}{3} • Individual 2: 1 2 3 1 2 {1, 4}{2, 5}{3} • Group encoding: items belonging to the same group are placed into the same partition, for example {2, 5}{1, 4}{3} • Search operators can work on groups rather than items • Groups are the meaningful building blocks of solutions A.H. Kashan Grouping Evolutionary Strategy (GES)

  5. Grouping Genetic Algorithm (GGA) Group Part Item Part • Problem representation: • The Mutation: elimination of some existing groups, insert the missing items by a problem depended heuristic A B C : ≡ A.H. Kashan Grouping Evolutionary Strategy (GES)

  6. The Crossover: the general pattern A.H. Kashan Grouping Evolutionary Strategy (GES)

  7. Evolutionary Strategy (ES) • Darwin’s theory: the most important features of the evolution process are inheritance, mutation and selection • Main steps of (μ+)-ES: • Initial solutions:  t= Xt1 , Xt2 , ..., Xtμ  • Repeat until (Termin.Cond satisfied) Do • Mutation: create a set Qt =  Yt1 , Yt2 , ..., Yt  by using mutation • New population  t +1: the μ best of the μ+ candidate solutions in  t  Q t • Replace the current best solution if it is better than the best solution found so far Yti d = Xtikd + Zd ; d = 1,...,D, i = 1,..., A.H. Kashan Grouping Evolutionary Strategy (GES)

  8. Evolutionary Strategy (ES) • Xti = xti1, xti2, ..., xtid a solution of current population • Yti = yti1, yti2, ..., ytid an offspring obtained via mutation • Zd= tNd (0, 1) •  t:distance of an offspring candidate solution from the parent •  t is varied on the fly by the “1/5 success rule” • This rule resets  t after every k iterations by •  =  / c if ps > 1/5 •  =  . c if ps < 1/5 •  =  if ps = 1/5 • where psis the % of successful mutations, 0.8  c  1 A.H. Kashan Grouping Evolutionary Strategy (GES)

  9. Evolutionary Strategy (ES) • Difficulty with developing the grouping version of ES: • New Mutation Scheme: • Producing new real-valued solution vectors during search process using Gaussian mutation in ES • Developing a new comparable mutation based on the role of the groups, while keeping the major characteristics of the classic ES mutation • Discrete Search Space: • ES is suitable for optimizing non-linear continuous functions but grouping problems are all discrete. • We will show how we can keep the new mutation in continuous space while using the consequences in discrete space A.H. Kashan Grouping Evolutionary Strategy (GES)

  10. Grouping Evolutionary Strategy • The main steps of (1 + )-GES: Initialization Initial solution generation Obtain offspring via NSG No Has the termination criteria been satisfied? Selection of best individual Finish Yes A.H. Kashan Grouping Evolutionary Strategy (GES)

  11. GES: Initial Solution • Solution representation: solution X with DX groups as a structure whose length is equal to the number of groups Xi: • The first solution is generated randomly Xi1 Xi2 Xi3 Xi4 A.H. Kashan Grouping Evolutionary Strategy (GES)

  12. GES: Initial Solution • Yti d = Xtd + Zd ; d = 1,...,D, i = 1,..., (1) • The key idea is to use appropriate operators in the place of arithmetic operators • Indeed, we have to determine how many items of current groups (X td) must be inherited by the new groups (Ytid) • By reshaping (1) in the form of Yti d - Xtd = Zd, • Substitution of “-” operator with an appropriate one in grouping problem A.H. Kashan Grouping Evolutionary Strategy (GES)

  13. GES: New Solution Generation • Similarity measure: • Distance/Dissimilarity measure: • Then, Gaussian mutation operator in GES is introduced as follows: A.H. Kashan Grouping Evolutionary Strategy (GES)

  14. GES: New Solution Generation • Zd values are unrestricted in sign but the range of distance measure is only real values in [0, 1] • Appropriate source of variation: • With 0 and 1 as the lower and upper bound of candidate PDF • With flexible PDF that provides different chances for getting a specific value in [0, 1] by means of some controllable parameter(s) • The new mutation operator of GES: A.H. Kashan Grouping Evolutionary Strategy (GES)

  15. GES: New Solution Generation • Fixing the value of  t at a constant level  1, we only consider  t as the endogenous strategy parameter • Then, • Ultimately, number of inherited items by each group of new solution is: A.H. Kashan Grouping Evolutionary Strategy (GES)

  16. GES: New Solution Generation       7 5 9 • Inheritance Phase: Xt: 1 10 2 11 4 8 3 6 12 ntid: 2 3 1       Yt: • Post assignment Phase: 1 11 Missed Items: 12 5 6 9       Yt: 10 3 7 2 4 8 A.H. Kashan Grouping Evolutionary Strategy (GES)

  17. GES: New Solution Generation • Two type of constructive heuristic: • First-fit • Best-fit • Comparison of the best solution out of  new obtained solution with the current solution (X t) • 1/5-success rule: increase if the observed estimate of the success probability exceeds a given threshold (Pt) during G successive iterations and vice versa. A.H. Kashan Grouping Evolutionary Strategy (GES)

  18. GES: Experimental Results • one-dimensional bin packing problem: • set of n items, • size of jth item is sj, • objective is to pack all items into the minimum number of bins (groups) of capacity B • Comparisons: The GGA proposed by Falkenauer (a steady-state order-based GA and its overall procedure) • Benchmark: ten problem instances via the URL: http://www.wiwi.uni-jena.de/Entscheidung • Implementation: MATLAB 7.3.0, Pentium 4, 3.2 GHz of CPU, 1 GB of RAM A.H. Kashan Grouping Evolutionary Strategy (GES)

  19. GES: Experimental Results A.H. Kashan Grouping Evolutionary Strategy (GES)

  20. Thanks for your attentions

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