Grouping problems and their applications Grouping Genetic Algorithm (GGA) Evolutionary Strategy (ES) Grouping Evolution Strategy Experimental Results A.H. Kashan Grouping Evolutionary Strategy (GES) 2 Partitioning a set (V) of n items into a collection of mutually disjoint subsets (groups, Vi) such that: D V Gi , Gi G j , i j. i 1 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) 3 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 Individual 2: 1 2 3 1 2 {2, 5}{1, 4}{3} {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) 4 Problem representation: 2 , 5 4, 1 3 ≡ Item Part B A C B A Group Part : A B C The Mutation: elimination of some existing groups, insert the missing items by a problem depended heuristic A.H. Kashan Grouping Evolutionary Strategy (GES) 5 A.H. Kashan Grouping Evolutionary Strategy (GES) 6 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 A.H. Kashan 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 Grouping Evolutionary Strategy (GES) 7 Xti = xti1, xti2, ..., xtid a solution of current population Yti = yti1, yti2, ..., ytid an offspring obtained via mutation Zd = t Nd (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 ps is the % of successful mutations, 0.8 c 1 A.H. Kashan Grouping Evolutionary Strategy (GES) 8 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: A.H. Kashan 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 Grouping Evolutionary Strategy (GES) 9 The main steps of (1 + )-GES: Initialization Initial solution generation Obtain offspring via NSG No Finish A.H. Kashan Yes Has the termination criteria been satisfied? Selection of best individual Grouping Evolutionary Strategy (GES) 10 Solution representation: solution X with DX groups as a structure whose length is equal to the number of groups Xi: 2, 3, 5 1, 7 6, 9, 4 9, 10 The first solution is generated randomly A.H. Kashan Grouping Evolutionary Strategy (GES) 11 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 (Y tid) 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) 12 Similarity measure: t t |Y X t t id d| Jaccard ' s Similarity (Yid ,X d ) t |Yid X dt | Distance/Dissimilarity measure: t t |Y X t t id d| Jaccard ' s Distance (Yid ,X d ) 1 t |Yid X dt | Then, Gaussian mutation operator in GES is introduced as follows: Distance ( yidt , xdt ) z d A.H. Kashan Grouping Evolutionary Strategy (GES) 13 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: Distance ( yidt , xdt ) Beta d ( t , t ), d 1,..., D, i 1,..., A.H. Kashan Grouping Evolutionary Strategy (GES) 14 Fixing the value of t at a constant level 1, we only consider t as the endogenous strategy parameter Then, Distance ( yidt , xdt ) Beta d ( t , ), d 1,..., D, i 1,..., Ultimately, number of inherited items by each group of new solution is: nidt Distance ( y , x ) 1 t Beta d ( t , ) |x d| t id t d nidt (1 Beta d ( t , ))|x dt | A.H. Kashan Grouping Evolutionary Strategy (GES) 15 Inheritance Phase: Post assignment Phase: A.H. Kashan Grouping Evolutionary Strategy (GES) 16 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) 17 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) 18 GES Problem GGA min num of bins 56 Time (Sec) 517.4 HARD0 56 Time (Sec) 103.7 HARD1 57 110.0 57 473.4 57 HARD2 57 105.8 57 446.1 57 HARD3 56 102.9 56 432.3 56 HARD4 57 110.5 58 452.9 58 HARD5 56 105.1 57 483.8 57 HARD6 57 104.0 57 440.4 57 HARD7 55 107.4 55 431.2 55 HARD8 57 106.2 57 465.7 57 HARD9 56 102.9 57 485.3 57 Average 56.4 105.8 56.7 462.8 56.7 A.H. Kashan Bins Bins 56 Grouping Evolutionary Strategy (GES) 19

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