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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
123451753122610113338322141511725201534
12345100001200100310000401000500010
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)
(W)(P)(P / W)115Kg45000300024Kg100002000310Kg60000500049Kg90001000
224411356150101101020-1044105606150-817-45621911169150110101017560111150
(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
*(Basic variable) (Nonbasic variable) (Pivot operation)(Basic solution) (Basic feasible solution)(Nondegenerate)*(Relative profit) cjzj cjzj (cjzj0)(Entering variable) cjzj (Pivot column)(Leaving variable) ()(Minimum ratio rule)(Pivot row) (Gauss-Jordan) (Canonical equation) () *(2-2)xn+ixjj =1,2,,n xn+ii =1,2,,mSii=1,2,,mZjCjZj 3 2 3 17*REREElliot****(yijkl)ikjkyijkl=1**