An Ant Colony Optimisation Algorithm for the Set Packing Problem

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1 An Ant Colony Optimisation Algorithm for the Set Packing Problem Xavier GANDIBLEUX 1,2, Xavier DELORME 2 and Vincent T KINDT 3 (1) LINA - Laboratoire d Informatique de Nantes Atlantique Universite de Nantes 2 rue de la Houssiniere BP92208, F Nantes cedex 03 FRANCE Xavier.Gandibleux@lina.univ-nantes.fr (2) LAMIH - Recherche Opérationnelle et Informatique Université de Valenciennes et du Hainaut-Cambrésis Le Mont Houy, F Valenciennes cedex 09 FRANCE Xavier.Delorme@univ-valenciennes.fr (3) LI - Laboratoire d Informatique Polytech Tours 64 avenue Jean Portalis, F Tours FRANCE Tkindt@univ-tours.fr ANTS 04 Fourth International Workshop on Ant Colony Optimization and Swarm Intelligence Bruxelles, Belgium, September 05 08, 2004

2 An ACO Algorithm for The SPP (slide #2) ANTS 2004, September 05 08, 2004 The Set Packing Problem (SPP) Given a finite set I = {1,...,n} of items {T j },j J = {1,...,m}, a collection of m subsets of I a packing is a subset P I such that T j P 1, j J which 2 Max z(x) = X 3 c i x i i I X t i,j x i 1, j J i I (SPP) 6 x i {0,1}, i I t i,j {0,1}, i I, j J Strongly NP-Hard (Garey and Johnson 1979) Node Packing Problem : P i I t i,j = 2, j J

3 An ACO Algorithm for The SPP (slide #3) ANTS 2004, September 05 08, 2004 The literature about the SPP Theoretical results and Exact algorithms : polyhedral theory, facets (Padberg 1973) a Branch & Cut algorithm (Nemhauser 1999, Rossi 2001) a lagrangian relax. with subgradient opt. (Rönnqvist 1995) use a general solver (as LINDO) (Kim 1997) Heuristics and meta-heuristics : GRASP with a path-relinking technique, a learning process and a dynamic tuning of parameters (Delorme 2004) Practical applications : a cutting stock problem (Rönnqvist 1995) a ship scheduling problem (Kim 1997) bounds for a RCPSP (Mingozzi 1998) a ground holding problem (Rossi 2001) a railway planning problem (Zwaneveld 1996, Delorme 2003)

4 An ACO Algorithm for The SPP (slide #4) ANTS 2004, September 05 08, 2004 A railway planning problem Planning the construction or reconstruction of infrastructures Capacity of one component / junctions of a rail system Junction Pierrefitte-Gonesse, north of Paris

5 An ACO Algorithm for The SPP (slide #5) ANTS 2004, September 05 08, 2004 ACO principles for the SPP

6 An ACO Algorithm for The SPP (slide #6) ANTS 2004, September 05 08, 2004 A constructing solution process alternating exploration mode and exploitation mode with more diversification at the beginning of the solution process, and more intensification at the end (SACO, Stuztle 1998) : probability evolves along the solution process application of a local search on each solution built by an ant (memetic algorithm, Murata 1995) including a territory disturance strategy inspired by a warming up strategy, well-known for the SA

7 An ACO Algorithm for The SPP (slide #7) ANTS 2004, September 05 08, 2004 Construction of a solution (1/3) x 1 f 1 f 2 f 3 f 4 f 5 x 2 x 3 x 4 x 5 x 5 x 1 x 4 x 2 x 3 x 1 x 2 x 3 x 4 x

8 An ACO Algorithm for The SPP (slide #8) ANTS 2004, September 05 08, 2004 Construction of a solution (2/3) x 1 x 1 f 1 f 2 x 2 f 2 x 2 f 3 x 3 x 3 f 4 x 4 f 4 x 4 f 5 x 5 x 5 x 5 x 1 x 4 x 2 x 3 x 1 x 2 x 3 x 4 x x 1 x 2 x 3 x 4 x

9 An ACO Algorithm for The SPP (slide #9) ANTS 2004, September 05 08, 2004 Construction of a solution (3/3) x 1 x 1 x 1 f 1 f 2 x 2 f 2 x 2 x 2 f 3 x 3 x 3 f 4 x 3 f 4 x 4 f 4 x 4 x 4 f 5 x 5 x 5 x 5 x 5 x 1 x 4 x 2 x 4 x 2 x 42 x 3 x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x

10 An ACO Algorithm for The SPP (slide #10) ANTS 2004, September 05 08, 2004 Outline of the algorithm

11 An ACO Algorithm for The SPP (slide #11) ANTS 2004, September 05 08, 2004 elaboratesolgreedy( sol ) ; localsearch( sol ) ; copysol( sol, bestsolknown ) initpheromones( φ ) ; iter 0 while not( isfinished?( iter ) ) do resettozero( bestsoliter ) for ant in 1... maxant do if isexploitation?(ant, iter, iteronexploit, maxiter ) then elaboratesolutiongreedyphi( φ, solution ) else elaboratesolutionselectionmethod( φ, solution ) end if localsearch( sol ) if performance( sol ) > performance( bestsoliter ) then copysol( sol, bestsoliter ) if performance( sol ) > performance( bestsolknown ) then copysol( sol, bestsolknown ) end if end if end for managepheromones( φ, bestsolknown, bestsoliter ) ; iter++ end while

12 An ACO Algorithm for The SPP (slide #12) ANTS 2004, September 05 08, The elaboratesolutiongreedyphi procedure I t I ; x i 0, i I t while (I t ) do i bestvalue(φ i, i I t ) x i 1 ; I t I t \ {i } ; I t I t \ {i : j J, t i,j + t i,j > 1} end while - - The elaboratesolutionselectionmode procedure I t I ; x i 0, i I t P log 10 (iter)/ log 10 (maxiter) while (I t ) do if ( randomvalue(0,1) > P ) then i roulettewheel(φ i, i I t ) else i bestvalue(φ i, i I t ) end if x i 1 ; I t I t \ {i } ; I t I t \ {i : j J, t i,j + t i,j > 1} end while

13 An ACO Algorithm for The SPP (slide #13) ANTS 2004, September 05 08, 2004 elaboratesolgreedy( sol ) ; localsearch( sol ) ; copysol( sol, bestsolknown ) initpheromones( φ ) ; iter 0 while not( isfinished?( iter ) ) do resettozero( bestsoliter ) for ant in 1... maxant do if isexploitation?(ant, iter, iteronexploit, maxiter ) then elaboratesolutiongreedyphi( φ, solution ) else elaboratesolutionselectionmethod( φ, solution ) end if localsearch( sol ) if performance( sol ) > performance( bestsoliter ) then copysol( sol, bestsoliter ) if performance( sol ) > performance( bestsolknown ) then copysol( sol, bestsolknown ) end if end if end for managepheromones( φ, bestsolknown, bestsoliter ) ; iter iter + 1 end while

14 An ACO Algorithm for The SPP (slide #14) ANTS 2004, September 05 08, The managepheromones procedure for i in 1... ncol do φ i φ i rhoe if (bestsoliter.x[i] = 1) then φ i φ i + rhod end if end for -- Pheromone evaporation - - Pheromone deposition - - Territory disturbance if isstagnant?( bestsolution, iterstagnant ) and isexistsphinul?( φ ) and isrestartenable?( iter, lastiterrestart, maxiteration ) then for i in 1... ncol do φ i φ i 0.95 log10(iter)/log10(maxiteration) -- Disturb the pheromones end for for i in 1... random(0.0,0.1 ncol) do φ random(1,ncol) random(0.05,(1.0 iter/maxiteration) 0.5) end for - - Offset on the pheromones with low level for i in 1... ncol do if φ i < 0.1 then φ i φ i + random(0.05,(1.0 iter/maxiteration) 0.5) end if end for end if

15 An ACO Algorithm for The SPP (slide #15) ANTS 2004, September 05 08, 2004 Illustration: pb500rnd WSPP : pb_500rnd15 seed : 0.50 GR : 1059 LS : Restart! 1 1 Pheromones - after the restart phi(x) phi(x) z(x) BestGlobal BestIteration AllSolutions #iterations x #ants Pheromones - before the restart variable x variable x

16 An ACO Algorithm for The SPP (slide #16) ANTS 2004, September 05 08, 2004 Illustration: z(x) for pb500rnd WSPP : pb_500rnd15 seed : 0.50 GR : 1059 LS : 1110 ACO : z(x) BestGlobal BestIteration AllSolutions #iterations x #ants

17 An ACO Algorithm for The SPP (slide #17) ANTS 2004, September 05 08, Pheromones corresponding to the first best solution known for this instance phi(x) WSPP : pb_500rnd15 seed : 0.50 GR : 1059 LS : 1110 ACO : variable x z(x) Pheromones phi BestGlobal BestIteration AllSolutions #iterations x #ants 0.6 phi(x) variable x

18 An ACO Algorithm for The SPP (slide #18) ANTS 2004, September 05 08, 2004 Contribution of the pheromone matrix 1150 WSPP : pb_500rnd15 seed : 0.50 GR : 1059 LS : 1110 ACO : z(x) BestGlobal BestIteration AllSolutions #iterations x #ants To see contribution of the pheromone matrix it would have been beneficial to note what was the performance of the ACO algorithm when alpha - weight for the pheromone information is set to zero Management of pheromone information disabled

19 An ACO Algorithm for The SPP (slide #19) ANTS 2004, September 05 08, 2004 Numerical experiments

20 An ACO Algorithm for The SPP (slide #20) ANTS 2004, September 05 08, 2004 Parameters maxiter 200 Number of iterations determined a priori maxant 15 Number of ants for each iteration phiinit 1.0 Initial pheromone assigned to a variable rhoe 0.8 Pheromone evaporation rate rhod phiinit * (1.0 - rhoe) Pheromone deposition rate phinul Level of pheromon considered as zero iteronexploit Percentage of iterations when Exploitation mode is activated iterstagnant 8 Declare the procedure stagnant when no improvement is observed

21 An ACO Algorithm for The SPP (slide #21) ANTS 2004, September 05 08, 2004 Experiments information Numerical instances variables; constraints; WSPP/USPP Experimental environment PC Pentium III; 800 MHz; 628 Mb; Debian Cplex version GRASP (Delorme 2004) Ada language; gnat 3.14 ACO C language; gcc and O3

22 An ACO Algorithm for The SPP (slide #22) ANTS 2004, September 05 08, 2004 Data set 1: Instances variables: 100/300/500 constraints density of matrix between 2.0% and 3.1% USPP/WSPP [1,20] 200 variables: 200/600/1000 constraints density of matrix between 1.0% and 2.6% USPP/WSPP [1,20]

23 An ACO Algorithm for The SPP (slide #23) ANTS 2004, September 05 08, 2004 Inst. 0/1 Solution GRASP ACO Opt. CPUt(s) Avg CPUt(s) Avg CPUt(s) 100r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r

24 An ACO Algorithm for The SPP (slide #24) ANTS 2004, September 05 08, 2004 Inst. 0/1 Solution GRASP ACO Opt. CPUt(s) Avg CPUt(s) Avg CPUt(s) 100r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r

25 An ACO Algorithm for The SPP (slide #25) ANTS 2004, September 05 08, 2004 Data set 2: Instances variables: 500/1500/2500 constraints density of matrix between 0.7% and 2.3% USPP/WSPP [1,20] 1000 variables: 1000/5000 constraints density of matrix between 0.58% and 2.65% USPP/WSPP [1,20]

26 An ACO Algorithm for The SPP (slide #26) ANTS 2004, September 05 08, 2004 Inst. 0/1 Solution GRASP ACO Best known Avg CPUt (s) Avg CPUt (s) 500r01 323* r02 24* r03 776* r04 61* r r06 8* r r r r r r12 33* r13 474* r14 37* r * r16 88* r17 192* r18 13* r r r * r400 48* r * r600 15* r r *

27 An ACO Algorithm for The SPP (slide #27) ANTS 2004, September 05 08, 2004 Inst. 0/1 Solution GRASP ACO Best known Avg CPUt (s) Avg CPUt (s) 500r01 323* r02 24* r03 776* r04 61* r r06 8* r r r r r r12 33* r13 474* r14 37* r * r16 88* r17 192* r18 13* r r r * r400 48* r * r600 15* r r *

28 An ACO Algorithm for The SPP (slide #28) ANTS 2004, September 05 08, 2004 ACO solution vs best known solution ratio ACO / best known solution Avg ACO Max ACO instances

29 An ACO Algorithm for The SPP (slide #29) ANTS 2004, September 05 08, 2004 Conclusion Perspectives Even not sophisticated, ACO can find a good quality of solutions (including on large instances) ACO can find very good solutions, but it reacts differently according to the instances (convergence sometimes low?, more iterations?) A local search must be implemented for the USPP Faced with GRASP, this ACO version needs generally more CPUt, and sometimes takes a huge CPUt ACO, in its current version, is a good starting point. A more efficient version can be expected in integrating more input (coming from the ACO field or other metaheuristics) in the algorithm

Ant Colony Optimization: an introduction. Daniel Chivilikhin

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