bpp_approx | R Documentation |

Solves the Bin Packing problem approximately.

```
bpp_approx(S, cap, method = c("firstfit", "bestfit", "worstfit"))
```

`S` |
vector of weights (or sizes) of items. |

`cap` |
same capacity for all the bins. |

`method` |
which approximate method to use. |

Solves approximately the Bin Packing problem for numeric weights and bins, all having the same volume.

Possible methods are "firstfit", "bestfit", and "worstfit". "firstfit" tries to place each item as early as possible, "bestfit" such that the remaining space in the bin is as small as possible, and "worstfit" such that the remaining space is as big as possible.

Best results are achieved with the "bestfit" method. "firstfit" may be a reasonable alternative. For smaller and medium-sized data the approximate results will come quite close to the exact solution, see the examples.

In general, the results are much better if the items in `S`

are sorted
decreasingly. If they are not, an immediate warning is issued.

A list of the following components:

`nbins` |
minimum number of bins. |

`xbins` |
index of the bin each item is assigned to. |

`sbins` |
sum of item sizes in each bin. |

`filled` |
total volume filled in the bins (as percentage). |

The Bin Packing problem can be solved as a Linear Program. The formulation is a bit tricky, and it turned out 'lpSolve' does not solve medium-sized problems in acceptable time. (Tests with 'Rglpk' will follow.)

Hans W. Borchers

Silvano Martello. "Bin packing problems". In: 23rd Belgian Mathematical Optimization Workshop, La-Roche-en-Ardennes 2019.

Function `binpacking`

in package 'knapsack' (on R-Forge).

```
## (1)
S <- c(50, 3, 48, 53, 53, 4, 3, 41, 23, 20, 52, 49)
cap <- 100
bpp_approx(S, cap, method = "bestfit")
## exact -- $nbins 4, filled 99.75 %
## firstfit -- $nbins 6, filled 66.5 %
## bestfit -- $nbins 5, filled 79.8 %
## ! when decreasingly sorted, 'bestfit' with nbins = 4
## (2)
S <- c(100,99,89,88,87,75,67,65,65,57,57,49,47,31,27,18,13,9,8,1)
cap <- 100
bpp_approx(S, cap, method = "firstfit")
# firstfit: 12 bins; exact: 12 bins
## Not run:
## (3)
S <- c(99,99,96,96,92,92,91,88,87,86,
85,76,74,72,69,67,67,62,61,56,
52,51,49,46,44,42,40,40,33,33,
30,30,29,28,28,27,25,24,23,22,
21,20,17,14,13,11,10, 7, 7, 3)
cap <- 100
bpp_approx(S, cap)
# exact: 25; firstfit: 25; bestfit: 25 nbins
## (4)
# 20 no.s in 1..100, capacity 100
set.seed(7013)
S <- sample(1:100, 20, replace = TRUE)
cap <- 100
bpp_approx(sort(S, decreasing = TRUE), cap, method = "bestfit")
# exact: 12 bins; firstfit and bestfit: 13; worstfit: 14 bins
## End(Not run)
```

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