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Artificiel Bee Colony (ABC) Algorithme Isfahan University of Technology Fall 2010 1 Elham Seifossadat Faegheh Javadi

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Job shop scheduling problems are considered to be a member of a large class of intractable numerical problems known as NP-hard. Job shop scheduling is concerned with finding a sequential allocation of competing resources that optimizes a particular objective function. Isfahan University of Technology Fall 2010 2

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3 A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 3 A finite set J of n jobs to be processed on a finite set M of m machines. Each job Ji must be processed on every machine and consists of a chain of mi operations Oi1, Oi2,…,Oim which have to be scheduled in a pre-determined given order. Oij is the jth operation of job Ji which has to be processed on a machine Mx for a processing time period of τij without interruption and preemption.

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Isfahan University of Technology Fall 2010 4 4 A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Each machine can process only one job and each job can be processed by only one machine at a time. The longest duration in which all operations of all jobs are completed is referred to as the makespan Cmax.

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Isfahan University of Technology Fall 2010 5 5 A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Ai be the set of ordered pairs of operations constrained by the precedence relations for each job Ji. For each machine Mx, the set Ex describes the set of all pairs of operations to be performed on the machine. For each operation Oij, let its earliest possible process start time be Tij.

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Isfahan University of Technology Fall 2010 6 A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING

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Isfahan University of Technology Fall 2010 7

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 8

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING The challenge is to adapt the self-organization behavior of the colony for solving job shop scheduling problems. There are two major characteristics of the bee colony in searching for food sources: waggle dance and forage (or nectar exploration). Isfahan University of Technology Fall 2010 9

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING waggle dance A forager fi on return to the hive from nectar exploration will attempt with probability p to perform waggle dance on the dance floor with duration D = di A, it will also attempt with probability ri to observe and follow a randomly selected dance. Isfahan University of Technology Fall 2010 10

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 11 Profitability rating for a forager:

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 12 The bee colonys average profitability rating:

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 13 The dance duration:

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 14 Adjusting Probability of Following a Waggle Dance:

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Forage (Nectar Exploration) Isfahan University of Technology Fall 2010 15 A population of L foragers is defined in the colony. When a forager is at a specific node, it can only move to next node that is defined in a list of presently allowed nodes, imposed by precedence constraints of operations.

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 16 A forager chooses the next node from the list according to the state transition rule:

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 17 The rating ρij of the edge (directed) between node I and node j is given by:

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 18

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING The performance of the honey bee colony scheduling approach is studied by evaluating them on the following 82 job shop problem instances. The sizes of these problems range from 6 to 50 jobs and 5 to 20 machines. Isfahan University of Technology Fall 2010 19

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 20

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A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING Isfahan University of Technology Fall 2010 21

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Bee Colony Optimization (BCO) Isfahan University of Technology Fall 2010 22

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Bee Colony Optimization (BCO) There are two alternating phases (forward pass and backward pass) constituting single step in the BCO algorithm. The hive is an non-natural object, with no precise location and does not influence the algorithm execution. Isfahan University of Technology Fall 2010 23

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Bee Colony Optimization (BCO) In each forward pass, every artificial bee visits NC solution components, creates partial solution, and after that returns to the hive. Isfahan University of Technology Fall 2010 24

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Bee Colony Optimization (BCO) Isfahan University of Technology Fall 2010 25

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Bee Colony Optimization (BCO) In the backward pass, all artificial bees share information about the quality of their partial solutions. Having all solutions evaluated, each bee decides with a certain probability whether it will stay loyal to its solution or not. Isfahan University of Technology Fall 2010 26

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Bee Colony Optimization (BCO) Isfahan University of Technology Fall 2010 27

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Bee Colony Optimization (BCO) Isfahan University of Technology Fall 2010 28

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Bee Colony Optimization (BCO) When all solutions are completed the best one is determined, it is used to update global best solution and an iteration of the BCO is accomplished. At this point all B solutions are deleted, and the new iteration could start. The BCO runs iteration by iteration until a stopping condition is met. Isfahan University of Technology Fall 2010 29

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Bee Colony Optimization (BCO) B - The number of bees in the hive; NC - The number of constructive moves during one forward pass. Isfahan University of Technology Fall 2010 30

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Bee Colony Optimization (BCO) (1) Initialization: an empty solution is assigned to each bee; (2) For each bee: // (the forward pass) (a) Set k = 1; // (count constructive moves in the forward pass) (b) Evaluate all possible constructive moves; (c) Choose one move using the roulette wheel; (d) k = k + 1; If k NC Goto step (b). (3) All bees are back to the hive; // (backward pass starts) (4) Evaluate (partial) objective function value for each bee; (5) Each bee decides randomly whether to continue its own exploration and become a recruiter, or to become a follower; (6) For each follower, choose a new solution from recruiters by the roulette wheel; (7) If solutions are not completed Goto step 2; (8) Evaluate all solution and find the best one; (9) If the stopping criteria is not met Goto step 2; (10) Output the best solution found. Isfahan University of Technology Fall 2010 31

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Bee Colony Optimization (BCO) Loyalty decision Isfahan University of Technology Fall 2010 32

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Bee Colony Optimization (BCO) Isfahan University of Technology Fall 2010 33 Recruiting process

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Scheduling Independent Tasks by BCO Let T = {1, 2,..., n} be a given set of independent tasks, and P = {1, 2,...,m} a set of identical machines. The processing time of task i (i = 1, 2,..., n) is denoted by li. Isfahan University of Technology Fall 2010 34

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Scheduling Independent Tasks by BCO Isfahan University of Technology Fall 2010 35 Probability pi that specific bee chooses task i was equal:

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Scheduling Independent Tasks by BCO Isfahan University of Technology Fall 2010 36 Probability pj that specific bee chooses processor j was calculated as:

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Scheduling Independent Tasks by BCO Isfahan University of Technology Fall 2010 37

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References A BEE COLONY OPTIMIZATION ALGORITHM TO JOB SHOP SCHEDULING- Chong, Low, Sivakumar, and Gay-Proceedings of the 2006 Winter Simulation Conference. Bee Colony Optimization: The Applications Survey - DUˇSAN TEODOROVI´C TATJANA DAVIDOVI´C and MILICA ˇSELMI´C- ACM Transactions on Computational Logic,2011. Isfahan University of Technology Fall 2010 38

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