A New Solution Approach for Grouping Problems Based

A New Solution Approach for Grouping Problems Based on Evolution Strategies Ali Husseinzadeh Kashan Spring 2010

Agenda Grouping problems and their applications Grouping Genetic Algorithm (GGA) Evolutionary Strategy (ES) Proposed Grouping ES Experimental Results Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 2

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, various timetabling problems, cell formation problem, etc. Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 3

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 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 Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 4

Grouping Genetic Algorithm (GGA) Group encoding: A B 2 , 5 4, 1 C 3 Item Part ≡ B A C B A Group Part : A B C The Crossover: the general pattern Parent 1 7, 2 5, 3 4, 1 6 Child 1: 7, 2 4, 1 5, 36 6 Parent 2 3, 2 4, 1 5, 6 7 3, 52 3, 5, 63 4, 1 Child 1: 2, 7 The Mutation: eliminate some existing groups; insert the missing items by a problem depended heuristic Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 5

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 = Xt 1 , Xt 2 , . . . , Xtμ Repeat until (Termin. Cond satisfied) Do Mutation: create a set Qt = Yt 1 , Yt 2 , . . . , Yt of solutions via mutation Yti d = Xtikd + Zd ; d = 1, . . . , D, i = 1, . . . , New population ( t +1): the μ best of the μ+ candidate solutions in t Q t are selected. Replace the current best solution if it is better than the best solution found so far Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 6

Evolutionary Strategy (ES) Xti = xti 1, xti 2, . . . , xtid a solution of current population Yti = yti 1, yti 2, . . . , 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 l l l = / a if ps > 1/5 = . a if ps < 1/5 = if ps = 1/5 where ps is the % of successful mutations, Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 7

Evolutionary Strategy (ES) Difficulty with developing the grouping version of ES: ES owns a Gaussian mutation to produce new real-valued solution vectors during the search process. To introduce GES, we should develop a new comparable mutation which works based on the role of groups, while keeping the major characteristics of the classic ES mutation. The paper is going to cover this issue. Originally, ES has been introduced for optimizing non-linear functions in continuous space. But grouping problems are all discrete. We will show we can keep the new mutation in continuous space while using the consequences in discrete space. Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 8

Grouping Evolutionary Strategies Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 9

GES: Initial Solution representation: solution X with DX groups as a structure whose length is equal to the number of groups Xi: 2, 3, 5 Xi 1 1, 7 Xi 2 6, 9, 4 9, 10 Xi 3 Xi 4 The first solution is generated randomly Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 10

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 (Y tid) By reshaping (1) in the form of Yti d - Xtd = Zd, Substitution of “-” operator with an appropriate one in grouping problem Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 11

GES: New Solution Generation Similarity measure: Distance/Dissimilarity measure: Then, Gaussian mutation operator in GES is introduced as follows: Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 12

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: l l 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: Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 13

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, the number of inherited items by each group of new solution is: Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 14

GES: New Solution Generation Inheritance Phase: Xt: 7 1 5 10 2 11 9 4 8 3 6 12 ntid: 2 3 1 Y t: Post assignment Phase: Missed Items: 1 5 11 9 6 12 Yt: 7 10 2 4 8 3 Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 15

GES: New Solution Generation Two type of constructive heuristic: l l First-fit Best-fit Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 16

GES: Experimental Results one-dimensional bin packing problem: l l l 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 Ali Husseinzadeh Kashan Grouping Evolutionary Strategy (GES) 17

GES: Experimental Results GES Problem GGA min num of bins 56 Time (Sec) 517. 4 HARD 0 56 Time (Sec) 103. 7 HARD 1 57 110. 0 57 473. 4 57 HARD 2 57 105. 8 57 446. 1 57 HARD 3 56 102. 9 56 432. 3 56 HARD 4 57 110. 5 58 452. 9 58 HARD 5 56 105. 1 57 483. 8 57 HARD 6 57 104. 0 57 440. 4 57 HARD 7 55 107. 4 55 431. 2 55 HARD 8 57 106. 2 57 465. 7 57 HARD 9 56 102. 9 57 485. 3 57 Average 56. 4 105. 8 56. 7 462. 8 56. 7 Ali Husseinzadeh Kashan Bins Grouping Evolutionary Strategy (GES) 56 18