Scheduling of Flexible manufacturing systems: an antcolony optimization approach

By:By:

R Kumar, M K Tiwari, R Kumar, M K Tiwari,

and R Shankarand R Shankar

Tom WillesTom Willes

19 November 200819 November 2008

ME 482ME 482

Paper FunctionPaper Function

FMS systems consist of numerous CNC FMS systems consist of numerous CNC machines able to perform many different machines able to perform many different functions on different parts. functions on different parts. Due to this flexibility, a given operation could be Due to this flexibility, a given operation could be performed on a number of different machines, performed on a number of different machines, allowing for many different routings.allowing for many different routings.Among the many available routings, the best Among the many available routings, the best available machine for the process should be available machine for the process should be selected. selected. This creates a very difficult scheduling problem.This creates a very difficult scheduling problem.This paper attempts to use an ant colony This paper attempts to use an ant colony optimization (ACO) approach to selecting the optimization (ACO) approach to selecting the best route for each part.best route for each part.

ImportanceImportance

In a high production flexible manufacturing system, minor changes in the routing of a part can greatly affect the production of the system.

It is important that this scheduling and routing be optimized for the best throughput possible.

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ParametersParametersc c small positive constantsmall positive constantG G set of arcs connecting all possible set of arcs connecting all possible

combinations of nodescombinations of nodesGGkk set of all nodes still to be visited by set of all nodes still to be visited by ant kant kj j a joba jobj’ j’ maximum number of jobs available maximum number of jobs available from from time 0 onwardstime 0 onwardsJ J set of jobsset of jobsk k an antan antm m a machinea machinem’ m’ maximum number of machinesmaximum number of machinesM M set of machinesset of machinesn n a nodea nodeN N total number of operationstotal number of operationsNC NC counter for number of iterationscounter for number of iterationsN_maxN_max maximum number of iterations maximum number of iterationso o an operationan operationoojj an operation of job jan operation of job jo’o’jj maximum number of operations for maximum number of operations for job jjob jOj Oj operation set of job joperation set of job jO’ O’ set of all operationsset of all operationsp p randomly generated quantityrandomly generated quantity

pp++bestbest best makespanbest makespan

pp++iteriter optimal makespan for an iteration iteroptimal makespan for an iteration iter

ppj,o,mj,o,m processing time of operation o of job j on processing time of operation o of job j on machine mmachine m

p0 p0 parameter used to attain quick convergence parameter used to attain quick convergence of the algorithmof the algorithm

Pk Pk makespan by the kth antmakespan by the kth antPPkk

ilil (t) (t) transition probability of moving from node I transition probability of moving from node I to to node l for ant knode l for ant kq q random numberrandom numberQ Q positive constantpositive constantSSkk set of nodes allowed at the next step by ant set of nodes allowed at the next step by ant kkt t timetimetabutabukk list of nodes travelled by ant klist of nodes travelled by ant kTTkk tabu list of ant ktabu list of ant kα α factor that controls the importance of the trailfactor that controls the importance of the trailβ β factor that controls the importance of factor that controls the importance of visibilityvisibilityηηklkl visibility from node k to node lvisibility from node k to node lρρ coefficient that represents the evaporation of coefficient that represents the evaporation of

the trail between time t and t + number of the trail between time t and t + number of iterations iterations

ττklkl trail level from node k to node ltrail level from node k to node lΤΤilil(0)(0) initial pheromone trail on edge ilinitial pheromone trail on edge ilΔΔttkk

ilil quantity per unit time of pheromone trail laid quantity per unit time of pheromone trail laid on the edge (on the edge (ii, , ll) by the kth ant between time ) by the kth ant between time

t t and t + number of iterationsand t + number of iterations

The Work of an Ant ColonyThe Work of an Ant Colony

Each time an ant goes from point A to point Each time an ant goes from point A to point B, it lays a chemical on the ground called B, it lays a chemical on the ground called pheromone pheromone along its path.along its path.Future ants needing to get from point A to Future ants needing to get from point A to point B then follow this pheromone trail. point B then follow this pheromone trail. Occasionally, an ant will stray just a bit from Occasionally, an ant will stray just a bit from the original pheromone trail. the original pheromone trail. Each ant follows the path with the most Each ant follows the path with the most pheromone.pheromone.As more and more ants follow this trail, it As more and more ants follow this trail, it may shift as the pheromone of lesser may shift as the pheromone of lesser traveled paths evaporate. traveled paths evaporate. Eventually, the trail of ants follows the Eventually, the trail of ants follows the optimal path between point A and point B. optimal path between point A and point B.

Ants find shortest path videoAnts find shortest path video

http://http://video.google.com/videoplay?docidvideo.google.com/videoplay?docid=-4748362485426843791&hl=en=-4748362485426843791&hl=en

The MathThe Math

ρρ is an evaporation coefficient (0 < is an evaporation coefficient (0 < ρρ < 1)< 1)ΔΔττilil is the quantity per unit time of is the quantity per unit time of the pheromone trail laid by ant the pheromone trail laid by ant kkppkk

ilil(t) is the probability that ant (t) is the probability that ant kk goes from node goes from node ii to node to node llSSk k is the set of nodes to which and is the set of nodes to which and kk can gocan goηηil il is the visibility from node is the visibility from node ii to node to node ll, a heuristic value, a heuristic valueαα and and ββ are parameters to control are parameters to control the relative importance of the trail the relative importance of the trail versus visibilityversus visibility

More MathMore Math

This term is the probability of This term is the probability of finding new pathsfinding new pathsIt prevents a very quick It prevents a very quick convergence of the algorithmconvergence of the algorithmAt lower iterations, the At lower iterations, the probability that an ant will stray probability that an ant will stray from a previous path to find a from a previous path to find a new one is greater.new one is greater.At higher iterations, the ant is At higher iterations, the ant is more likely to follow the most more likely to follow the most traveled path, the best path.traveled path, the best path.

Program SummaryProgram Summary

Step 1: Initialize ProgramStep 1: Initialize ProgramStep 2: Breaks iteration counting loop (2-7)Step 2: Breaks iteration counting loop (2-7)Step 3: Breaks ant counting loop (3-6)Step 3: Breaks ant counting loop (3-6)Step 4: Breaks node selection loop (4-5)Step 4: Breaks node selection loop (4-5)Step 5: Node Selection (random)Step 5: Node Selection (random)Step 6: Ant counterStep 6: Ant counterStep 7: Updating pheromone trail, probability Step 7: Updating pheromone trail, probability value, and best makespan thus farvalue, and best makespan thus farStep 8: Output resultStep 8: Output result

Step 1: Initialization

1. Represent the problem using a weighted directed graph.2. Randomly distribute ants on the nodes.3. Set t = 0; /*{time counter}*/4. Set NC = 1; /*(number of iterations/number of

cycles counter)*/5. Set τil(0) = c; on each node /* τil(0) is the initial pheromone trail on the edge il and c is a small positive constant*/6. Set Δτil = 0; on each node /* Δτil is the increase in

the trail level on edge il */7. Set tabuk = 0; /* tabuk gives the list of nodes traversed

by ant k */8. Set p0 = 0;

Steps 2-4Steps 2-4

Step 2. If NC > N_max go to Step 8, else go to Step 3 /*N_max is the maximum number of iterations*/

Step 3. If m > m_max go to Step 7 else go to Step 4 /*m gives the ant number and m_max stands for the maximum number of ants*/

Step 4. If tabuk > tabukmax go to Step 6 else go to Step 5

/*tabukmax gives the maximum number of nodes to

be visited by ant k*/

Step 5: Node SelectionStep 5: Node Selection1. Generate random number p (0 ≤ p ≤ 1).2. If p ≤ p0 then go to Step 5(3) else go to Step 5(4).3. Generate random number q (q E Sk), select = q, go to Step 5(10).4. Compare probabilities of possible outgoing nodes.5. Choose the node having the highest probability (pk

il).6. Generate a random number q (0 ≤ q ≤ 1).7. If q ≥ pk

il then go to Step 5(8) else go to Step 5(9).8. Generate a random number q (q E Sk), select = q, go to Step 5(10).9. Choose the node with the highest pk

il value, select = 1.10. Choose the node select as the next node to move to.11. Add select to tabuk, delete it from Gk and Sk and go to Step 4.

Steps 6-7Steps 6-7

Step 6. m = m + 1, go to Step 3.

Step 7: updating

1. Find p+iter ; /* p+

iter is the optimal makespan for iteration*/

2. If p+iter < p+

best then p+best = p+

iter; /* p+best is the best

makespan*/

3. Update τil (t).4. Empty all tabu lists (i.e. tabuk).5. NC = NC + 1.

6. p0 = loge(NC) / loge(N_max), go to Step 2.

Step 8Step 8

Output pOutput p++bestbest

This is the best makespan found in all the This is the best makespan found in all the iterations of the algorithm.iterations of the algorithm.

Simple Numerical IllustrationSimple Numerical Illustration

Setup times ignoredSetup times ignored

5 jobs5 jobs

4 operations on each job4 operations on each job

3 machines3 machines

Traditional method gives a Traditional method gives a throughput time of 439throughput time of 439

This ACO method gives a This ACO method gives a throughput time of 420throughput time of 420

Another ExampleAnother Example

Setup times includedSetup times included

2 jobs2 jobs

2 operations on each job2 operations on each job

3 machines3 machines

4 tools for each machine4 tools for each machine

CPU timeCPU time

The parameter The parameter ρρ doesn’t seem to have a strong doesn’t seem to have a strong effect on CPU time, but starts to have some effect on CPU time, but starts to have some effect at very high iterations.effect at very high iterations.The effect of the number of iterations appears to The effect of the number of iterations appears to be nearly linear, as we would anticipate.be nearly linear, as we would anticipate.

Iterations and best makespanIterations and best makespan

It appears as though the algorithm is able It appears as though the algorithm is able to converge on the best makespan at to converge on the best makespan at higher iterations.higher iterations.

Algorithm’s convergenceAlgorithm’s convergence

ConclusionsConclusions

“The value of the makespan for the problem with 400 iterations comes out to be 439 by the method proposed by Lee and Dicesari [36] and Yim and Lee [37], whereas for the same problem the present method gives 420, which is lower than the value obtained by Lee and Dicesari [36]. Thus the superiority of the present method is proved over the PN-based method proposed by Lee and Dicesari [36].”

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Conclusions (continued)Conclusions (continued)

I think…I think…The authors failed to prove their method better The authors failed to prove their method better than others in use.than others in use.

They failed to prove the value of this model.They failed to prove the value of this model.

The authors were creative in their attempt to apply The authors were creative in their attempt to apply an ant model to a FMS.an ant model to a FMS.

Conclusions (continued)Conclusions (continued)

All they did was find a lower makespan. All they did was find a lower makespan. Is it attainable in a real FMS?Is it attainable in a real FMS?

What adjustments would make this more What adjustments would make this more practical for industrial use? practical for industrial use?

It would be more valuable if the output was a It would be more valuable if the output was a matrix showing the trail of the best makespan, the matrix showing the trail of the best makespan, the final ant trail.final ant trail.

BYU 31 UofU 27BYU 31 UofU 27