Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm

19 Pages·2006·0.1 MB·English

Solving the Maximum Cardinality
Bin Packing Problem with a
Weight Annealing-Based Algorithm
Kok-Hua Loh
University of Maryland
Bruce Golden
University of Maryland
Edward Wasil
American University
10th ICS Conference
January 2007
Outline of Presentation
(cid:1)
Introduction
(cid:1)
Concept of Weight Annealing
(cid:1)
Maximum Cardinality Bin Packing Problem
(cid:1)
Conclusions
1
Weight Annealing Concept
(cid:1)
Assigning different weights to different parts of a combinatorial problem to
guide computational effort to poorly solved regions.
(cid:2) Ninio and Schneider (2005)
(cid:2) Elidan et al. (2002)
(cid:1)
Allowing both uphill and downhill moves to escape from a poor local
optimum.
(cid:1)
Tracking changes in objective function value, as well as how well every
region is being solved.
(cid:1)
Applied to the Traveling Salesman Problem. (Ninio and Schneider 2005)
(cid:2) Weight annealing led to mostly better results than simulated annealing.
2
One-Dimensional Bin Packing Problem (1BP)
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Pack a set of N = {1, 2, …, n} items, each with size t , i=1, 2,…, n,
i
into identical bins, each with capacity C.
(cid:1)
Minimize the number of bins without violating the capacity
constraints.
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Large literature on solving this NP-hard problem.
3
Outline of Weight Annealing Algorithm
(cid:1)
Construct an initial solution using first-fit decreasing.
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Compute and assign weights to items to distort sizes according
to the packing solutions of individual bins.
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Perform local search by swapping items between
all pairs of bins.
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Carry out re-weighting based on the result of the previous
optimization run.
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Reduce weight distortion according to a cooling schedule.
4
Neighborhood Search for Bin Packing Problem
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From a current solution, obtain the next solution by swapping items between bins
with the following objective function (suggested by Fleszar and Hindi 2002).
p
∑
Maximize f = (l )2
i ∑qi
i=1 l = t bin load i
i j
j=1
p = number of bins
q = number of items in bin i
i
t = size of item j
j
f = (5+3)2 + 22 = 68 f = (5+3+ 2)2 =100
new
5
Neighborhood Search for Bin Packing Problem
(cid:1)
Swap schemes
(cid:2) Swap items between two bins.
(cid:2) Carry out Swap (1,0), Swap (1,1), Swap (1,2), Swap (2,2) for all pairs of bins.
(cid:2) Analogous to 2-Opt and 3-Opt.
(cid:1)
Swap (1,0) (suggested by Fleszar and Hindi 2002)
Bin α Bin β Bin α Bin β
f = (1+3+ 4)2 + (4+ 4)2 =128 f = (3+ 4)2 +(1+ 4+ 4)2 =130
new
(cid:1)
Need to evaluate only the change in the objective function value.
∆f = (l −t )2 +(l +t )2 −l2 −l2
l = total load of bin α
α j β j α β
α
t = size of item i
i
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Neighborhood Search for Bin Packing Problem
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Swap (1,1)
(f = 162) ( f = 164)
new
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Swap (1,2)
(f = 162) ( f = 164)
new
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Weight Annealing for Bin Packing Problem
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Weight of item i
w = 1 + K r
i i
 C −l 
residual capacity r =  i 
i
 C 
C = capacity
l = load of bin i
i

(cid:1)
An item in a not-so-well-packed bin, with large r ,
i
will have its size distorted by a large amount.
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No size distortions for items in fully packed bins.
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K controls the size distortion, given a fixed r .
i
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Weight Annealing for Bin Packing Problem
(cid:1)
Weight annealing allows downhill moves in a maximization
problem.
(cid:1)  200−l 
Example
C = 200, K= 0.5 , w =1+0.5 i 
i
 200 
w =1.025 w =1.087
1 2
Transformed space f = 70126.3 Transformed space f = 70132.2
new
Original space f = 63325 Original space f = 63125
new
(cid:2) Transformed space - uphill move
(cid:2) Original space - downhill move
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