| I want to introduct something about M. G. Sandwich / | | | | For the one-dimensional case, the new patterns are |
| Cap Paper. 17/21gsm, 22x28cm, 25x35cm, or as your | | | | introduced by solving an auxiliary optimization problem |
| requirement. M. G. Sandwich / Cap Pap | | | | called the knapsack problem, using dual variable |
| The cutting stock problem is an optimization problem, | | | | information from the linear program. The knapsack |
| or more specifically, an integer linear programming | | | | problem has well-known methods to solve it, such as |
| problem. It arises from many applications in industry. | | | | branch and bound and dynamic programming. The |
| Imagine that you work in a paper mill and you have a | | | | Delayed Column Generation method can be much |
| number of rolls of paper of fixed width waiting to be | | | | more efficient than the original approach, particularly |
| cut, yet different customers want different numbers | | | | as the size of the problem grows. The column |
| of rolls of various-sized widths. How are you going to | | | | generation approach was pioneered by Gilmore and |
| cut the rolls so that you minimize the waste (amount | | | | Gomory in a series of papers published in the 1960's. . |
| of left-overs)? | | | | Gilmore and Gomory showed that this approach is |
| Solving this problem to optimality can be economically | | | | guaranteed to converge to the (fractional) optimal |
| significant: a difference of 1% for a modern paper | | | | solution, without needing to enumerate all the |
| machine can be worth more than one million USD per | | | | possible patterns in advance. |
| year. | | | | A limitation of the original Gilmore and Gomory |
| Formulation and solution approaches | | | | method is that it does not handle integrality, so the |
| The standard formulation for the cutting stock | | | | solution may contain fractions, e.g. a particular pattern |
| problem (but not the only one) starts with a list of m | | | | should be produced 3.67 times. Rounding to the |
| orders, each requiring qj, j = 1,...,m pieces. We then | | | | nearest integer often does not work, in the sense |
| construct a list of all possible combinations of cuts | | | | that it may lead to a sub-optimal solution and/or |
| (often called "patterns"), associating with each | | | | under- or over-production of some of the orders |
| pattern a positive integer variable xi representing how | | | | (and possible infeasibility in the presence of two-sided |
| many times each pattern is to be used. The linear | | | | demand constraints). This limitation is overcome in |
| integer program is then:minimisesubject to and | | | | modern algorithms, which can solve to optimality (in |
| , integerwhere aij is the number of times order j | | | | the sense of finding solutions with minimum waste) |
| appears in pattern i and ci is the cost (often the | | | | very large instances of the problem (generally larger |
| waste) of pattern i. The precise nature of the | | | | than encountered in practice ). |
| quantity constraints can lead to subtly different | | | | The cutting stock problem is often highly degenerate, |
| mathematical characteristics. The above formulation's | | | | in that multiple solutions with the same waste are |
| quantity constraints are minimum constraints (at least | | | | possible. This degeneracy arises because it is possible |
| the given amount of each order must be produced, | | | | to move items around, creating new patterns, |
| but possibly more). In this case waste minimisation is | | | | without affecting the waste. This give arise to a |
| equivalent to minimising the number of utilised master | | | | whole collection of related problems which are |
| rolls. The most general formulation has two-sided | | | | concerned with some other criterion, such as the |
| constraints (for which minimising waste is no longer | | | | following: |
| equivalent to minimising the number of master rolls): | | | | The minimum pattern count problem: to find a |
| This formulation applies not just to one-dimensional | | | | minimum-pattern-count solution amongst the |
| problems. Many variations are possible, including one | | | | minimum-waste solutions. This is a very hard problem, |
| where the objective is not to minimise the waste, | | | | even when the waste is known. There is a |
| but to maximise the total value of the produced | | | | conjecture that any equality-constrained |
| items, allowing each order to have a different value. | | | | one-dimensional instance with n orders has at least |
| In general, the number of possible patterns grows | | | | one minimum waste solution with no more than n + 1 |
| exponentially as a function of m, the number of | | | | patterns. No upper bound to the number of patterns |
| orders. As the number of orders increases, it may | | | | is known either, examples with n + 5 are known. |
| therefore become impractical to enumerate the | | | | The minimum stack problem: this is...(and so on) To |
| possible cutting patterns. | | | | get More information , you can visit some products |
| An alternative is to use a Delayed Column Generation | | | | about designer tote bag, advance ballast, . The M. G. |
| approach. This method solves the cutting stock | | | | Sandwich / Cap Paper products should be show |
| problem by starting with just a few patterns. It | | | | more here! |
| generates additional patterns when they are needed. | | | | |