Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm
Different Local Search Algorithms in STAGE for Solving Bin Packing Problem
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Different Local Search Algorithms in STAGE for Solving Bin Packing Problem
Gholamreza Haffari
Sharif University of [email protected]
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Overview
Combinatorial Optimization Problems and State Spaces
STAGE Algorithm Local Search Algorithms Results Conclusion and Future works
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Optimization Problems Objective function: F(x1, x2, …, xn)
Find vector X=(x1, x2, …, xn) which minimizes (maximizes) F
Constraints:
g1(X) 0 g2(X) 0 . . . gm(X) 0
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Combinatorial Optimization Problems (COP)
Special kind of Optimization Problems which are Discrete
Most of the COPs are NP-Hard, I.e. there is not any polynomial time algorithm for solving them.
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Satisfiability SAT: Given a formula in
propositional calculus, is there an assignment to its variables making it true?
f(x1, x2, .., xn)
Problem is NP-Complete. (Cook 1971)
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Bin Packing Problem (BPP)
Given a list (a1, a2, …) of items, each of which has a size s(ai)>0, and a bin Capacity C, what is the minimum number of bins for packing items?
Problem is NP-Complete (Garey and Johnson 1979)
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An Example of BPP
a1 a2 a3 a4
b1 b2 b3 b4
Objects list: a1, a2, …, an
Bin’s capacity (bj) is C
Objective function: m
ai < C, aibj, 1j m
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Definition of State in BPP
A particular permutation of items in the object list is called state.
b1 b2 b3 b4
a1 a2 a3 a4
Greedy Algorithm
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State Space of BPP
a1, a2, a3, a4
a2, a4, a3, a1
a1, a4, a2, a3. . .a1, a2, a4, a3
. . . . . .
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A Local Search Algorithm
1) s1) s0 0 : a random start state: a random start state
2)2) for i = 0 to +for i = 0 to +
- - generategenerate new solutions set S from the current new solutions set S from the current solution ssolution sii
- - decidedecide whether s whether si+1i+1 = s’ = s’S or sS or sii
- if a - if a stopping conditionstopping condition is satisfied is satisfied return the return the bestbest solution found solution found
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Local Optimum Solutions
The quality of a local optimum resulted from a local search process depends on a starting state.
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Multi-Start LSA
Runs the base local search algorithms from different starting states and returns the best result found.
Is it possible to choose a promising new starting state?
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Other Features of a State
Other features of a state can help the search process.
(Boyan 1998)
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Previous Experiences
There is a relationship among local optima of a COP, so previously found local optima can help to locate more promising start states.
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Core ideas Using an Evaluation Function to
predict the eventual outcome of doing a local search from a state.
The EF is a function of some features of a state.
The EF is retrained gradually.
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STAGE Algorithm
Uses an Evaluation Function to locate a good start state.
Does local search.
Retrains EF with the new generated search trajectory
Learning Phase
Execution Phase
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Evaluation Function
State Features EF Prediction
EF can be used by another local search algorithm for finding a good new starting point.
Applying EF on a state
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Diagram of STAGE
(Boyan 98)
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Analysis of STAGE What is the effect of using different local
search algorithms?
Local search algorithms: Best Improvement Hill Climbing (BIHC) First Improvement Hill Climbing (FIHC) Stochastic Hill Climbing (STHC)
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Best Improvement HC
Generates all of the neighboring states, and then selects the best one.
…
1
4 7 2
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First Improvement HC
Generates neighboring states systematically, and then selects the first good one.
5
4 7
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Stochastic HC
Stochastically generates some of the neighboring states, and then selects the best one.
The size of the set containing neighbors is called PATIENCE.
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Different LSAs
Different LSAs for solving U250_00 instance
http://www.ms.ic.ac.uk/info.html
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Different LSAs, bounded steps
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Some Results The higher the accuracy in choosing the next
state, the better the quality of the final solution, by comparing STHC1 and STHC2 (PATIENCE1=350, PATIENCE2=700)
Deep paces result in higher quality and faster solutions, by comparing BIHC and others.
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Different LSAs, bounded moves
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Some Results
• It is better to search the solution space randomly rather than systematically, by comparing STHC and others.
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Future works
Using other learning structures in STAGE
Verifying these results on another problem (for example Graph Coloring)
Using other LSA, such as Simulated Annealing.
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Questions