Adaptive Large Neighborhood Search (ALNS)

Transcription

1 Aaptive Large Neighbrh Search (ALNS)

2 Cnsier a general integer prgramming prblem Min x X Z n f ( x) n where X R. Aaptive Large Neighbrh Search i lcal search prceure ( ALNS ) Lcal search prceure: Simulate annealing Tabu Search i several simple algrithms t mify the current slutin

3 Cnsier a general integer prgramming prblem Min x X Z n f ( x) n where X R. Aaptive Large Neighbrh Search i lcal search prceure ( ALNS ) Lcal search prceure: Simulate annealing Tabu Search i several simple algrithms t mify the current slutin

4 Simulate Annealing (SA) Initialize Select an initial slutin x Select an initial temperature TP Let x = x = x Let iter = niter = an TP = TP X Repeat until stpping criterin satisfie iter = iter 1 ; niter = niter 1 minriter = ; Value = f ( x) Repeat until minriter = minritermax minriter = minriter 1 Generate ranmly x ' V N( x) If f = f ( x ' ) f ( x) < then x = x ' else Generate ranm number r (, 1] f If r < e TP then x = x ' If f ( x) < f ( x) then x = x, an niter = If f ( x) < Value then niter = TP = α TP If iter = iter max r niter = niter max then stpping criterin is satisfie x is the best slutin generate

5 Simulate Annealing (SA) Initialize Select an initial slutin x Select an initial temperature TP Let x = x = x Let iter = niter = an TP = TP X Repeat until stpping criterin satisfie iter = iter 1 ; niter = niter 1 minriter = ; Value = f ( x) Repeat until minriter = minritermax minriter = minriter 1 Generate ranmly x ' V N( x) If f = f ( x ' ) f ( x) < then x = x ' else Generate ranm number r (, 1] f If r < e TP then x = x ' If f ( x) < f ( x) then x = x, an niter = If f ( x) < Value then niter = TP = α TP If iter = iter max r niter = niter max then stpping criterin is satisfie x is the best slutin generate

6 Simulate Annealing (SA) Initialize Select an initial slutin x Select an initial temperature TP Let x = x = x Let iter = niter = an TP = TP Repeat until stpping criterin satisfie X iter = iter 1 ; niter = niter 1 minriter = ; Value = f ( x) Repeat until minriter = minritermax minriter = minriter 1 Generate ranmly x ' V N( x) If f = f ( x ' ) f ( x) < then x = x ' else Generate ranm number r (, 1] f If r < e TP then x = x ' If f ( x) < f ( x) then x = x, an niter = If f ( x) < Value then niter = TP = α TP If iter = iter max r niter = niter max then stpping criterin is satisfie x is the best slutin generate Destry heuristic selects q variables x i an these variables have n value assigne Repair heuristic assigns a feasible value t these q variables x i accring t the values f the ther variables that remain at their current values Several simple algrithms t mify the current slutin x : Destry the current slutin x an Repair t get a new feasible slutin x

7 Outline f the ALNS framewrk Cnstruct a feasible slutin ; set x x = x Repeat until stp criteria is met i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π If, then : r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair i If x is acceptable accring t the lcal search prceure, then x : = x i Upate the scres π an π i f x < f x x = x r Return x

8 Outline f the ALNS framewrk Cnstruct a feasible slutin ; set x x = x Repeat until stp criteria is met i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estr y an repair Aaptive layer: If x π i is acceptable accring t the lcal search prceure, then x : = j Prbabilistic chices n xt specify i Upate the scres π an π π i r the intervals in the rulette wheel selectin i If f x < f x, then x : = x Return x

9 Outline f the ALNS framewrk Cnstruct a feasible slutin ; set x x = x Repeat until stp criteria is met i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair Scre: i If x is acceptable accring t the lcal search Past prceure, perfrmance then t cntribute x : = x i Upate the scres π an π uring the slutin prcess i f x < f x x = x If, then : r Return x

10 Outline f the ALNS framewrk Cnstruct a feasible slutin ; set x x = x Repeat until stp criteria is met Accepte slutins New slutin rejecte i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair If x Scre upate cess: i is acceptable accring t the lcal search prceure, then x : = x π i: the scre btaine at i Upate the scres π an π r the current iteratin i If f x < f x, then x : = x Then Return x Scre upate: ( equally ivie between an ) New best slutin Nt previusly visite slutins ( ) π : = ρπ 1 ρ π i i i r

11 Outline f the ALNS framewrk Cnstruct a feasible slutin ; set x x = x Repeat until stp criteria is met Destry heuristics wrking well with the chsen repair heuristics i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair i If x is acceptable accring t the lcal search Nising prceure, r ranmizatin then x : i= n heuristics: x i Upate the scres π an π i If, then : Return x f x < f x x = x r Heuristic selectin: Fast repair heurristics t cnstruct full slutin given a partial slutin T btain a prper iversificatin Avi stagnating search prcesses where the estry an repair neighbrhs wul perfrm the same mificatins

12 Outline f the ALNS framewrk Cnstruct a feasible slutin x; set x = x Repeat until stp criteria is met i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair Destry peratrs O : i If x is acceptable accring t the lcal Ranm search prceure, remval then x : = x i Upate the scres π an π Wrst r critical remval i If, then : Return x f x < f x x = x r ( ) Relate remval Small remval Histry-base remval

13 Outline f the ALNS framewrk Cnstruct a feasible slutin ; set x x = x Repeat until stp criteria is met i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair Repair neighbrhs ( O ): i If x is acceptable accring t the lcal search prceure, then x : = x Greey appraches i Upate the scres π an π r Regret heuristics (least ba chice) i If f x < f x, then x : = x Apprximatin algrithms Branch-an-bun algrithms Return x

14 Outline f the ALNS framewrk Cnstruct a feasible slutin x; set x = x Repeat until stp criteria is met i Chse a estry peratr O accring t its scre π i Chse a repair peratr O accring t its scre π r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair Cuple neighbrhs : i If x is acceptable accring t the lcal search Fr prceure, each Othen x : = x i Upate the scres π an π i f x < f x x = x If, then : r assciate a subset K O Return x

15 Variant: Large Outline Multiple-Neighbrh f the ALNS framew Searc rh k LMNS Cnstruct a feasible slutin ; set x x = x ( ) Repeat until stp criteria is met i Chse a estry peratr O ranmly accring t its scre π i Chse a repair peratr O accring ranmly t its scre π If, then : r r i Generate a new slutin x frm the current slutin x using the heuristics t estry an repair i If x is acceptable accring t the lcal search prceure, then x : = x i Upate the scres π an π i f x < f x x = x r Return x

16 Reference D. Pisinger, S. Rpke, "A General Heuristic fr Vehicle Ruting Prblems", Cmputers &Operatins Research 34 (27),