Determine the value-weight ratio of every item. The unbounded knapsack problem is fairly easy to solve: The knapsack problem has a long history, dating back to at least 1897 and possibly much earlier.įor a single knapsack, there are three basic versions of the problem: The knapsack problem aims to maximize the combined value of items placed into a knapsack of limited capacity. This article presents a more efficient way of handling the bounded knapsack problem. IntroductionĪ common solution to the bounded knapsack problem is to refactor the inputs to the 0/1 knapsack algorithm. # Example 1 p K <- read.table ( textConnection ( "item weight value pieces map 9 150 1 compass 13 35 1 water 153 200 2 sandwich 50 160 2 glucose 15 60 2 tin 68 45 3 banana 27 60 3 apple 39 40 3 cheese 23 30 1 beer 52 10 3 suntan_cream 11 70 1 camera 32 30 1 T-shirt 24 15 2 trousers 48 10 2 umbrella 73 40 1 waterproof_trousers 42 70 1 waterproof_overclothes 43 75 1 note-case 22 80 1 sunglasses 7 20 1 towel 18 12 2 socks 4 50 1 book 30 10 2" ), header = TRUE, row.names = "item" ) ks <- knapsack ( K $ value, K $ weight, 400 ) # profit: 1030, capacity: 396 is <- sort ( ks $ indices ) row.names ( K ) # "map" "compass" "water" # "sandwich" "glucose" "banana" # "suntan_cream" "waterproof_trousers" "waterproof_overclothes" # "note-case" "sunglasses" "socks" # Example 5: Bounded knapsack problem ks <- knapsack ( K $ value, K $ weight, 400, K $ pieces ) # profit: 1200, capacity: 385 for ( i in 1 : length ( ks $ indices )) cat ( row.The attached file is an HttpHandler written in C#. Other packages implementing knapsack routines. “Knapsack Problems: Algorithms andĬomputer Implementations”. In the sequence profit/weight are equal instead of decreasing. This routines do not work correctly if too many or all of the elements 
Vector of indices, with components only 0 or 1. Profits/weights, j=1.(n-1) must be monotonically Sum of all weights is greater or equal to the capacity Maximum weight is smaller than the capacity The input problem description must satisfy the following conditions: Select a subset of the items, described by the 0/1 vector x, so as to Problem formulation: Given a set of n items and a knapsack, with Inf with integers in one vector as bounds. It implements the enumerative algorithm described in section 3.6.3 of theĪ mixed bounded/unbounded knapsack problem can be formulated by mixing The components x_j can get as large as possible. With bounds=Inf it solves the unbounded knapsack problem, that is It implements the transformation method (that is, transformed into anĮquivalent 0-1 knapsack problem) described in section 3.2 of Martello If bounds is a single integer, it will be expanded to the length of With bounds an integer vector it solves the bounded knapsack problemįor integer profits and weights, i.e. Of Martello and Toth's book “Knapsack Problems”. It implements the branch-and-bound algorithm described in section 2.5.2 Solves the 0-1 single knapsack problem for integer profits and weights Logical Fortran check routine enabled cannot be turned off.
Integer vector of weights, of the same length.īounds on the multiplicity of items can be NULL,
Integer vector of profits, with length greater or equal 2. Knapsack ( profits, weights, capacity, bounds = NULL, check = TRUE )
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