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MIE376 Lec8 Column Generation

MIE376 Lec8 Column Generation - MIE376 Mathematical...

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MIE376 Mathematical Programming Lecture Notes Daniel Frances 2011 1 Lecture 3: Cutting Stock Problem with Column Generation This is a classical problem that leads to a totally new expansion of the capabilities of linear programming. Suppose you sell boards of lumber to the retail market in lengths of 3 m., 5 m. and 9 m. to meet a demand of 25 units of 3 m. boards 20 units of 5 m. boards 15 units of 9 m. Length Suppose also that you can only buy boards in the wholesale market in 17 m lengths. How many 17 m lengths do we need and how should we cut them to meet the demand at minimum waste. Unfortunately we cannot simply start to define decision variables to formulate this problem into a mathematical model. We need some preliminary analysis. It turns out that one of the issues we have to deal with, is to define in advance all the possible ways that we can cut a 17 m board into one or more of the target lengths. This is best done methodically by using a table of all possible cuts: Cut Type 3 m 5 m 9 m waste 1 5 0 0 2 2 4 1 0 0 3 2 2 0 1 4 2 0 1 2 5 1 1 1 0 6 0 3 0 2 Note that we have omitted any cut which results in a waste ≥ 3 m. Now the problem can be formulated as follows: Let x i be the number of boards cut according to cut type i. and z the overall waste. Then we could formulate the following LP problem: Min z = 2x 1 + x 3 + 2x 4 + 2x 6 subject to 5x 1 + 4x 2 + 2x 3 + 2x 4 + x 5 ≥ 25 x 2 +2x 3 +x 5 + 3x 6 ≥ 20
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