1. Bin Packing Optimization
Rachel Jackson
Kirklyn Fields

2. Overview Introduction Algorithms Results Q/A What is bin packing?
Why 380?
Applications
Algorithms
Exhaustive Search
First Fit Decreasing
Genetic Algorithm
Results
Q/A
Overview

3. Introduction Bin Packing Problem:
Objects of different sizes must be packed into a finite number of containers, or “bins”
Each bin has a finite volume. Must minimize the number of bins used
Computational complexity NP-Hard Problem.
Decision problem of deicing if a certain number of bins is optimal, is NP-Complete
Two Versions
Online: items arrive one at a time
Offline: all items are given up front
Introduction

4. Problem with lots of applications, that requires us to design our own implementations of common algorithms we will study.
Applications:
Loading files into memory
Job scheduling
TV advertisement slots
Packing by cost/weight
Why 380?

5. Formal Problem Statement
Given a set of bins , B with the same size S, and a list of j items each with sizes 𝒘 𝟏 .. 𝒘 𝒏 to pack inside of those bins, find the minimum number of bins ,K ,such that the sum of items within each bin is <= S.
Mixed Integer Programming formulation:
Minimize 𝐊= 𝒊=𝟏 𝒏 𝑩 𝒊
Subject to
𝒋=𝟏 𝒏 𝑩(𝒘 𝒋 ) ≤𝑺
Where K is the sum of bins B, needed to pack j items, subject to the sum of j items must be less than the size of each bin S.
Formal Problem Statement

6. Brute Force Brute Force Approach to Combinational Logic
Focused on constraint Satisfaction
Selects the Solutions that user asks for
Algorithm
Take a Set of Items
Find all possible combinations
Check If each combinations passes the constraint check
Pass the solutions to the User

7. Brute Force

8. First Fit Decreasing Greedy Approximation Time complexity O(n log n)
Algorithm:
Initially all bins empty beginning with 0 bins
Sort items to be packed in to decreasing order.
For each item:
Attempt to place the item in the first bin that can accommodate the item
If no bin is found:
Create a new bin
First Fit Decreasing uses at most (4M + 1)/3 bins if the optimal is M.
First Fit Decreasing

9. First Fit Decreasing Results
Example with bin size = 30
First Fit Decreasing Results
First Fit
Decreasing
Total
Bins 10 20 8 28 7
Full 1 6 13 3 16 19 2 21 22
Done
2 Bins

10. First Fit Decreasing Results

11. Hybrid Grouping Genetic Algorithm
Overview
Initialize a population of possible solutions (50) using first fit
For a specified amount of time(in our case 50 generations):
Select two parents
Generate offspring through crossover
Mutation
Grade fitness
Hybrid Grouping Genetic Algorithm

12. Hybrid GA (cont) 𝑠 𝑖 = sum of the ith bin
Fitness = 𝑖=1 𝑛 ( 𝑆 𝑖 𝑐 ) 𝑘 𝑛
N = number of bins
𝑠 𝑖 = sum of the ith bin
c = capacity of the bin
k = 2 (can be any constant greater than1)
Should maximize fitness score.
Hybrid GA (cont)

13. Crossover Select random parents Clone parents
Hybrid GA (cont)
Crossover
Select random parents
Clone parents
Chose two cross over points
Swap bins between parents and clones
Remove duplicates and add any unused elements from parents into children

14. Hybrid GA (Cont) Mutation - 50% Select bins at random to be destroyed.
This allows for reinsertion of items within those bins

15. GA Results

16. First Fit Decreasing, and Genetic Algorithms are as expected faster than brute force
However they do not always find the optimal solution
Many variations of problem that can be applied to many real world situations.
Results/ Conclusion

17. Q&A Q:What is the bin packing problem?
A: Given a set of items pack them into the smallest amount of bins possible.
Q: Describe the Frist Fit Decreasing Algorithm
A: given a set of n items, sort them in decreasing order. Until all items are packed:
Attempt to pack each item into the first available bin, if the item will not fit, create a new bin and pack that item.
Q: What is the time of complexity of First Fit Decreasing/?
A: O(nLogn)
Q&A

18. Fin
Questions?