Different Local Search Algorithms in STAGE for Solving Bin Packing Problem

Gholamreza Haffari

Sharif University of Technologyhaffari@ce.sharif.edu

Overview

Combinatorial Optimization Problems and State Spaces

STAGE Algorithm Local Search Algorithms Results Conclusion and Future works

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

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.

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)

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)

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

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

State Space of BPP

a1, a2, a3, a4

a2, a4, a3, a1

a1, a4, a2, a3. . .a1, a2, a4, a3

. . . . . .

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

Local Optimum Solutions

The quality of a local optimum resulted from a local search process depends on a starting state.

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?

Other Features of a State

Other features of a state can help the search process.

(Boyan 1998)

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.

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.

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

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

Diagram of STAGE

(Boyan 98)

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)

Best Improvement HC

Generates all of the neighboring states, and then selects the best one.

…

1

4 7 2

First Improvement HC

Generates neighboring states systematically, and then selects the first good one.

5

4 7

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.

Different LSAs

Different LSAs for solving U250_00 instance

http://www.ms.ic.ac.uk/info.html

Different LSAs, bounded steps

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.

Different LSAs, bounded moves

Some Results

• It is better to search the solution space randomly rather than systematically, by comparing STHC and others.

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.

Questions