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Dynamic Programming Knapsack Problem
Dynamic Programming Knapsack Problem
Cutting stock problem
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The cutting stock problem is an optimization problem, or more specifically, an integer linear programming problem. It arises from many applications in industry. Imagine that you work in a paper mill and you have a number of rolls of paper of fixed width waiting to be cut, yet different customers want different numbers of rolls of various-sized widths. How are you going to cut the rolls so that you minimize the waste (amount of left-overs)?
Solving this problem to optimality can be economically significant: a difference of 1% for a modern paper machine can be worth more than one million USD per year.
Formulation and solution approaches
The standard formulation for the cutting stock problem (but not the only one) starts with a list of m orders, each requiring qj, j = 1,…,m pieces. We then construct a list of all possible combinations of cuts (often called “patterns”), associating with each pattern a positive integer variable xi representing how many times each pattern is to be used. The linear integer program is then:
minimise
subject to and
, integer
where aij is the number of times order j appears in pattern i and ci is the cost (often the waste) of pattern i. The precise nature of the quantity constraints can lead to subtly different mathematical characteristics. The above formulation’s quantity constraints are minimum constraints (at least the given amount of each order must be produced, but possibly more). In this case waste minimisation is equivalent to minimising the number of utilised master rolls. The most general formulation has two-sided constraints (for which minimising waste is no longer equivalent to minimising the number of master rolls):
This formulation applies not just to one-dimensional problems. Many variations are possible, including one where the objective is not to minimise the waste, but to maximise the total value of the produced items, allowing each order to have a different value.
In general, the number of possible patterns grows exponentially as a function of m, the number of orders. As the number of orders increases, it may therefore become impractical to enumerate the possible cutting patterns.
An alternative is to use a Delayed Column Generation approach. This method solves the cutting stock problem by starting with just a few patterns. It generates additional patterns when they are needed. For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program. The knapsack problem has well-known methods to solve it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be much more efficient than the original approach, particularly as the size of the problem grows. The column generation approach was pioneered by Gilmore and Gomory in a series of papers published in the 1960′s. . Gilmore and Gomory showed that this approach is guaranteed to converge to the (fractional) optimal solution, without needing to enumerate all the possible patterns in advance.
A limitation of the original Gilmore and Gomory method is that it does not handle integrality, so the solution may contain fractions, e.g. a particular pattern should be produced 3.67 times. Rounding to the nearest integer often does not work, in the sense that it may lead to a sub-optimal solution and/or under- or over-production of some of the orders (and possible infeasibility in the presence of two-sided demand constraints). This limitation is overcome in modern algorithms, which can solve to optimality (in the sense of finding solutions with minimum waste) very large instances of the problem (generally larger than encountered in practice ).
The cutting stock problem is often highly degenerate, in that multiple solutions with the same waste are possible. This degeneracy arises because it is possible to move items around, creating new patterns, without affecting the waste. This give arise to a whole collection of related problems which are concerned with some other criterion, such as the following:
The minimum pattern count problem: to find a minimum-pattern-count solution amongst the minimum-waste solutions. This is a very hard problem, even when the waste is known. There is a conjecture that any equality-constrained one-dimensional instance with n orders has at least one minimum waste solution with no more than n + 1 patterns. No upper bound to the number of patterns is known either, examples with n + 5 are known.
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Knapsack Problems $153.72 This book provides a full-scale presentation of all methods and techniques available for the solution of the Knapsack problem. This most basic combinatorial optimization problem appears explicitly or as a subproblem in a wide range of optimization models with backgrounds such diverse as cutting and packing, finance, logistics or general integer programming. This monograph spans the range from a co… |
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Dynamic-Programming Approaches to Single- and Multi-Stage Stochastic Knapsack Problems for Portfolio Optimization $27.95 This is a NAVAL POSTGRADUATE SCHOOL MONTEREY CA report procured by the Pentagon and made available for public release. It has been reproduced in the best form available to the Pentagon. It is not spiral-bound, but rather assembled with Velobinding in a soft, white linen cover. The Storming Media report number is A500263. The abstract provided by the Pentagon follows: This thesis proposes new metho… |
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Where are the hard knapsack problems? [An article from: Computers and Operations Research] $5.95 This digital document is a journal article from Computers and Operations Research, published by Elsevier in . The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.Description: The knapsack problem is believed to be one of the ”easier” NP-hard problems. Not only can it be solved in pseudo-polynom… |
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