CS 325- Homework Assignment 8

CS 325- Homework Assignment 8
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Problem 1: (20 pts) In the bin packing problem, items of different weights (or sizes) must be packed into
a finite number of bins each with the capacity C in a way that minimizes the number of bins
used. The decision version of the bin packing problem (deciding if objects will fit into <= k bins) is NPcomplete. There is no known polynomial time algorithm to solve the optimization version of the bin
packing problem. In this homework you will be examining three greedy approximation algorithms to
solve the bin packing problem.
 First-Fit: Put each item as you come to it into the first (earliest opened) bin into which it fits. If
there is no available bin then open a new bin.
 First-Fit-Decreasing: First sort the items in decreasing order by size, then use First-Fit on the
resulting list.
 Best Fit: Place the items in the order in which they arrive. Place the next item into the
bin which will leave the least room left over after the item is placed in the bin. If it does
not fit in any bin, start a new bin.
a) Give pseudo code and the running time for each of the approximation algorithms.
b) Implement the algorithms in Python, C++ or C. Your program named binpack should read in a text file
named bin.txt with multiple test cases as explained below and output to the terminal the number of
bins each algorithm calculated for each test case. Submit a README file and your program to TEACH.
Example bin.txt: The first line is the number of test cases, followed by the capacity of bins for that
test case, the number of items and then the weight of each item. You can assume that the weight of
an item does not exceed the capacity of a bin for that problem.
3
10
6
5 10 2 5 4 4
10
20
4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6
10
4
3 8 2 7
Sample output:
Test Case 1 First Fit: 4, First Fit Decreasing: 3, Best Fit: 4
Test Case 2 First Fit: 15, First Fit Decreasing: 10, Best Fit: 15
Test Case 2 First Fit: 3, First Fit Decreasing: 2, Best Fit: 2
c) Randomly generate at least 20 bin packing instances. Summarize the results for each algorithm.
Which algorithm performs better? How often? Note: Submit a description of how the inputs were
generated not the code used to produce the random inputs.
CS 325- Homework Assignment 8
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Problem 2: (10 pts) An exact solution to the bin packing optimization problem can be found using 0-1
integer programming (IP) see the format on the Wikipedia page.
Write an integer program for each of the following instances of bin packing and solve with the
software of your choice. Submit a copy of the code and interpret the results.
a) Six items S = { 4, 4, 4, 6, 6, 6} and bin capacity of 10
b) Five items S = { 20, 10, 15, 10, 5} and bin capacity of 20
Note: The version of LINDO that you have access to on the OSU server has a limit of 50 integer
variables. Therefore, LINDO will only be able to solve problems with at most 6 items.