Bin packing problem

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In bin packaging issues , objects with different volumes must be packaged into a limited amount of garbage or containers per volume V in a way that minimizes the amount of waste used. In the theory of computational complexity, this is a matter of NP-hard combination. The decision issue (deciding whether the object will match the specified amount of waste) is NP-complete.

There are many variations of this problem, such as 2D packing, linear packaging, heavy packing, packing based on cost, and so on. They have many applications, such as filling containers, loading trucks with heavy capacity constraints, backing up media files and technology mapping in the design of programmable array semiconductor chips.

Bin packaging issues can also be seen as a special case of stock-cutting issues. When the amount of waste is limited to 1 and each item is characterized by volume and value, the problem of maximizing the value of items that can be accommodated in the trash is known as a backpack problem.

Despite the fact that bin packaging problems have the complexity of NP-hard computing, the optimal solution to the enormous problem of problems can be produced with sophisticated algorithms. In addition, many heuristics have been developed: for example, first fit algorithms provide quick but often non-optimal solutions, involving placing each item onto the first tray that will match. It takes ? ( n Ã, n ), where n is the number of elements to pack. The algorithm can be made much more effective by first sorting the list of elements into descending order (sometimes known as the first-fit drop-down algorithm), although this still does not guarantee the optimal solution, and for longer lists can increase the running time of the algorithm. However, it is known that there is always at least one ordering of goods that allows the first fit to produce the optimal solution.

The bin packaging variant that occurs in practice is when an item can share space when packed into a tray. In particular, a set of items can occupy less space when packed together rather than the number of sizes each. This variant is known as packing a VM because when virtual machines (VMs) are packaged in the server, their total memory requirements can be reduced because the pages shared by the VM only need to be stored once. If an item can share space in an arbitrary way, bin packaging problems are difficult to estimate. However, if the space division matches the hierarchy, as with sharing memory in virtual machines, bin packaging issues can be approached efficiently. Another variant of the bin packaging that is interesting in practice is the so-called online bin packing. Here the objects with different volumes should arrive in a sequence and the decision maker must decide whether to select and package the items currently observed, or let them pass. Any decision without withdrawal.

Video Bin packing problem

Pernyataan formal

di mana ${\ displaystyle y_ {i} = 1}  ÂÂ$ jika bin ${\ displaystyle i}  ÂÂ$ digunakan dan ${\ displaystyle x_ {ij} = 1}  ÂÂ$ jika item ${\ displaystyle j}  ÂÂ$ dimasukkan ke dalam bin ${\ displaystyle i}  ÂÂ$ .

Maps Bin packing problem

Algoritme cocok pertama

This is a direct greedy approximation algorithm. Algorithm processes items in random order. For each item, try to place an item on the first bin that can accommodate the item. If no bin is found, it will open a new bin and place the item in the new bin.

This algorithm reaches the approximation factor 2; the amount of waste used by this algorithm is no more than twice the optimum amount of waste. In other words, it is impossible for two trash cans to be at most half full because such a possibility implies that at any point, exactly one bin is at most half full and newly opened to accommodate the size of the goods at most V/2 But since the first has at least one space V Ã,/Ã,2, the algorithm will not open a new bin for any item of the greatest size V Ã,/Ã, 2. Only after the bin is filled with more than V Ã,/Ã,2 or if an item of size greater than V Ã,/Ã,2 arrives, the algorithm may open a new bin.

Jadi jika kita memiliki B sampah, setidaknya B Â-1 sampah lebih dari setengah penuh. Oleh karena itu, ${\ displaystyle \ sum _ {i = 1} ^ {n} a_ {i} & gt; {\ tfrac {B-1} {2}} V}  ÂÂ$ . Karena ${\ displaystyle {\ tfrac {\ sum _ {i = 1} ^ {n} a_ {i}} {V}}}  ÂÂ$ adalah batas bawah dari nilai optimal OPT , kita mendapatkan B Â-1Â & lt; Â 2 OPT dan karena itu B Â <= Â 2 OPT . Lihat analisis di bawah ini untuk hasil perkiraan yang lebih baik.

Modified the first fit decreases (MFFD) increases FFD for items larger than half bin by grouping items by size into four large, medium, small, and small size classes, corresponding to items of size & gt; 1/2 bin, & gt; 1/3 bin, & gt; 1/6 bin, and smaller items respectively. Then the result goes through five phases:

1. Give a box for each big item, the biggest message to the smallest.
2. Continue forward through the trash can. In each: If the smallest remaining items are not matching, skip this bin. If not, place the largest appropriate remaining medium item.
3. Continue backwards through the trash that does not contain any moderate items. In each: If the two smallest items remaining do not fit, skip this bin. If not, place the smallest remaining items and the rest of the largest small item accordingly.
4. Continue forward through all the trash. If the smallest item left from any size class does not match, skip this bin. If not, place the largest item that fits and stay on this tray.
5. Use FFD to pack the rest of the stuff into a new basket.

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Analysis of approximate algorithms

The best matching and first strategy is one of the simplest heuristic algorithms to solve bin packaging problems. They have been shown to use no more than 11/9 OPT Ã, Ã, 1 garbage (where OPT is the amount of waste provided by the optimal solution). The simpler this, the First Fit Decreasing (FFD) strategy, operates by first sorting items to be inserted in descending order by size, and then inserting each item into the first bin in the list with enough space left. Sometimes, however, one does not have an option to sort input, for example, when faced with an online bin packaging problem. In 2007, it was proven that bound 11/9Ã, OPT Ã, Ã,6/9 for FFD was very strict. MFFD uses no more than 71/60 OPT Ã, 1 garbage (ie limited by about 1.18 OPT , compared to about 1.22 OPT for FFD). In 2013, DÃÆ'³sa and Sgall provide a tight upper limit for the first-fit strategy (FF), indicating that it never takes more than 17/10 OPT garbage for any input.

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The exact algorithm

Martello and Toth developed an appropriate algorithm for the problem of 1-D bin packaging, called MTP. A faster alternative was the Bin Solution algorithm proposed by Korf in 2002 and later improved; This second paper reports the average time to complete one million samples with 80 items on the 450 MHz Sun Ultra 10 workstation is 31 ms.

Further improvements are presented by Schreiber and Korf in 2013. The new Upgrade Bin Upgrade algorithm is shown up to five times faster than Solution Bin on non-trivial issues with 100 items, and exceeds BCP (branch-and-cut-and-price) algorithm by Belov and Scheithauer on the problem of having less than 20 rubbish as the optimal solution. Which algorithm performs best depends on problem properties such as item count, optimal garbage count, unused space in optimal solution, and accuracy of value.

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See also

• If the amount of garbage has to be fixed or restricted, and the size of the trash should be minimized, it is a different problem that is equivalent to the Multiprocessor scheduling problem
• Guillotine Problems
• Packaging problem
• Partition problem
• Problems with part number

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References

Source of the article : Wikipedia