Spring 2015 Mathematics in Management Science Bin Packing The Problem The Algorithms

Different Scheduling Problem Have collection of idpt tasks (all task times ≤ 10). Instead of asking how long to finish job with 3 processors, instead ask, “What is fewest number of processors need to finish job in under 20 min?” A very different question! Example of a bin-packing problem.

The Bin Packing Problem Assume have: identical “bins” fixed capacity, and objects (“weights”) of various sizes that must be packed in bins. How can we do this most efficiently (i.e., with fewest bins)?

Example A company buys bulk wire in 100 ft spools. They use the wire in a variety of lengths in their product. How should they cut the wire so as to use the fewest spools?

Example A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9ft lengths. How should we cut the boards so as to minimize the cost of material?

Variants Two possibilities: 1. Only know next weight to pack. (Think assembly line.) 2. Have full knowledge of all needed sizes/weights. (E.g., a work order that lists everything that is needed.)

Variants Two possibilities: 1. Only know next wt to pack. 2. Full knowledge all weights. In (2), it makes sense to first sort the weights in a decreasing list. (Why?) Leads to two variants of every bin- packing algorithm.

Variants Two variants of every bin-packing algorithm.

The Bin Packing Problem Pack wts w 1, w 2, …,w n into bins. Each bin has fixed capacity. Use as few bins as possible. A scheduling problem in disguise! unlimited # of procs each with fixed capacity (Goal: minimize # procs)

Dictionary Scheduling task task time processor completion time completing a task # of processors Bin Packing item to pack weight of item bin bin size packing an item # of bins

The Algorithms Heuristic Algorithms NF – Next Fit FF – First Fit WF – Worst Fit In last 2, bins stay open until they are completely full. Decreasing Versions NFD – Next Fit D FFD – First Fit D WFD – Worst Fit D With these first sort wts into decreasing order, then pack.

Next Fit Packing Algortihm Only one bin open at a time! Pack item in current bin. If item will not fit in current bin: close this bin & open a new bin. Continue until all items packed.

Example using Next Fit Packing Wts 6, 6, 5, 5, 5, 4, 4, 4, 4, 2, 2, 2, 2, 3, 3, 7, 7, 5, 5, 8, 8, 4, 4, 5

Next Fit Packing Algortihm Advantages – Do not need to know all the weights in advance; only need to know the remaining space in the bin. Disadvantages – The bin packed early on may have had room for small items that come later in the list. Best method for assembly-line packing.

First Fit & Worst Fit Packing Keep all bins open until full. Then close. To pack an item, look at open bins. FF Place item in first bin in which it fits. WF Place item in bin with the most space (and into which item fits). Only open new bin if item will not fit in any of the already open bins.

16 Bin Packing Methods Next Fit & First Fit & Worst Fit NF A new bin is opened if the weight to be packed next will not fit in the bin that is currently being filled; the current bin is then closed. FF The next weight to be packed is placed in the lowest-numbered bin already opened into which it will fit. If it does not fit in any open bins, a new bin is opened. WF The next weight to be packed is placed into the open bin with the largest amount of room remaining. If it does not fit in any bins, open a new bin.

17 Bin Packing Methods Next Fit Decreasing (NFD) First Fit Decreasing(FFD) Worse Fit Decreasing(WFD) Decreasing-Time Heuristics NFD, FFD, WFD – Create the priority list by listing the weights in order of decreasing size before applying selected bin-packing method.

Example: Next Fit A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9 ' lengths(!). How should you cut the boards so as to minimize the cost of material? Use (1) NF and (2) NFD. Next Fit:Next Fit:6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 46, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4,

Example: Next Fit A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9' lengths(!). How should you cut the boards so as to minimize the cost of material? Use (1) NF and (2) NFD. Next Fit Decreasing:Next Fit Decreasing:8, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 28, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2,

Example Pack the following weights: 6, 5, 7, 3, 4, 2 in bins of size 10 using (1) NF and (2) NFD. Next Fit:NFD: 7, 6, 5, 4, 3, 2 3

Example: First Fit A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9' lengths(!). How should you cut the boards so as to minimize the cost of material? Use (1) FF and (2) FFD. First Fit:First Fit:6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 46, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4,

Example: First Fit A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9 1 lengths(!). How should you cut the boards so as to minimize the cost of material? Use (1) FF and (2) FFD. First Fit Decreasing:First Fit Decreasing:8, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 28, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2,

Example 4 Pack the following weights: 6, 5, 7, 3, 4, 2 in bins of size 10 using (1) FF and (2) FFD. First Fit:FFD: 7, 6, 5, 4, 3,

Example: Worst Fit A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9' lengths(!). How should you cut the boards so as to minimize the cost of material? Use (1) WF and (2) WFD. Worst Fit:Worst Fit:6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 46, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4,

Example: Worst Fit A wall unit requires boards of lengths: 6, 5, 4, 5, 3, 2, 2, 2, 7, 5, 5, 8, 4, 4 Stock boards come in 9 1 lengths(!). How should you cut the boards so as to minimize the cost of material? Use (1) WF and (2) WFD. Worst Fit Decreasing:Worst Fit Decreasing:8, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 28, 7, 6, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2,

Example 4 Pack the following weights: 6, 5, 7, 3, 4, 2 in bins of size 10 using (1) WF and (2) WFD. Worst Fit:WFD: 7, 6, 5, 4, 3,

Exercise Find a question of the form: Use indicated bin-packing algorithm to place given weight in the supplied, already partially filled, bins. Place wts in the bins and select the next wt to be added in such a way that each of the 3 algorithms (NF, FF, WF) will place the wt in a di ff erent bin.