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An Evolutionary Strategy Approach for Efficient Grouping Problems Solution

This paper presents a new methodology for addressing grouping problems using Grouping Evolutionary Strategy (GES) and Grouping Genetic Algorithm (GGA). The core focus is on partitioning a set of items into disjoint groups while maintaining efficiency in populations. Key concepts include different representation schemes, mutation and crossover tactics, and the applicability of GES in handling discrete search spaces. Experimental results demonstrate the efficacy of this approach in various well-known grouping problems, including timetabling and vehicle routing.

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An Evolutionary Strategy Approach for Efficient Grouping Problems Solution

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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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