A hybrid particle swarm optimization algorithm for optimal task assignment in distributed system

-()(0/1 )

- CMin-TAPHPSO (LingoGAHPSO)Comment

standard formmatrix-vector (Max or Min)Z = c1x1+ c2x2 + + cnxnZ = CXa11x1+a12x2++a1nxn=b1a21x1+a22x2++a2nxn=b2..am1x1+am2x2++amnxn=bmAx=b

x10,x20,,xn0x0

b10,b20,,bm0b0

Simplex method George B. Dantzig (Convex set)(Extreme point)()

()

-

(Max)Z = 5x13x2x3Z = 5x13x2x32x14x2x3 8x12x22x3 102x1x2 6xj 0 , j = 1,2,3.2x14x2x3s18x12x22x3s2102x1x2s36x1, x2, x3 0s1, s2, s3 0

(Pivot row)(Entering variable)(Pivot column)(x1, x2 x3, s1, s2, s3) (3, 0, 2, 0, 3, 0)

G. Minty V. Klee 1971, n, 2n 1

(Elliott wave theory)Ralph Nelson ElliottElliott

() = *1.618 0.3820.50.618 ()

Travelling salesman problem

ModelObjective:Min ij cij xij

Constraints:i xij = 1 for all j.j xij = 1 for all i.xij {0, 1} for all i, j

Cost(distance)2 -> 3 -> 1 - >5 -> 400000

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Travelling salesman problemThree Types of TSP Problem

Symmetric TSP ( )

Assymmetric TSP ( )

Euclidean TSP ()

Branch & boundIs a way of finding optimal solution in different kinds of optimization problem

Break the problem into smaller ones and solve them

Can be used in Integer Programming, discrete optimization problem and combinatorial optimization problem

Branch & boundThree Basic Steps

Branching: Divide feasible solutions into smaller subsets

Bounding: Computes the lower and upper bound of a subset

Fathoming: Stops searching when a subset cant have an optimal solution

Branch & bound (Knapsack problem example)Knapsack problem ()

Branch & bound (Knapsack problem example)

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(W)(P)115Kg4524Kg1310Kg649Kg9000

Monkey searchIs an agent-based metaheuristic to solve global optimization problem

The monkey explores trees and learns which branched lead to better food sources

Is not an exact replica of the behavior of monkeys real life

Monkey search

Task assignment problem (TAP) (Cost-minimization task assignment problemCMin-TAP) (Reliability-maximization task assignment problemRMax-TAP) (Cost-Reliability Analysis task assignment problemCRA-TAP)

Task assignment problem (TAP)Task interaction graph (TIG)

Processor interaction graph (PIG)

The objective of task assignment is to find an optimal assignment of the task such that the total cost is minimized while at the same time, all the resource constraints are satisfied.

|E| = 5

Existing Methods for solving TAPMathematical ProgrammingInteger linear programming, branch and boundProviding exact solutions but could be extremely time-consuming for solving large scaled problems.Meta-heuristics:Genetic algorithms (GA)Simulated annealing (SA)Proving approximate solution with reasonable time.

Notations

Problem formulationInteger quadratic programming

Integer linear programming [3,4]

HPSO algorithm

Proposed Hybrid PSO (HPSO) for solving TAPParticle representation

Fitness evaluation

Task 5 is assigned to processor 2.

Exact solution using Lingo|E| calculates the number of existing communication demands in the TIG.

*Experimental Results Optimal parameterization (HPSO)Swarm size = 80

*Experimental Results Optimal parameterization (GA)Population size = 80Crossover rate = 0.7 Mutation rate = 0.1

Approximate solutions using GA and HPSO

Convergence analysisGbest analysis (r, n, d)=(20,12, 0.8) for ? times.

Probi(s) j s Entropyj j entropy ()entropy ()

Convergence analysis

Worst-case analysisProvide a guarantee of solution quality.(r, n, d)=(20,12, 0.8) for 1000 times normal distribution

Conclusion We have proposed a hybrid particle swarm optimization (HPSO) algorithm which finds a near-optimal task assignment with reasonable time. HPSO outperforms the GA.The convergence and the worst-case analyses of the HPSO have been empirically conducted.

CommentHPSOGAhybrid(local search)HPSOPSO

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