We define formally the Knife Change Minimization Problem, we prove some properties which reduce the search space, and then describe some heuristsics. At one of the last stages of the paper construction process customer widths have to be cut out of jumbo reels. For example, the widths 50,40,60,40, 30,50,50,50 and 60,40,40,40 may have to be cut out of three jumbo reels of width 200. The collections of indivdidual widths (e.g. 50-40-60-40) are called patterns. The order in which to consider the patterns (i.e. the route) can be arbitrary, and the order in which to cut each pattern is arbitrary as well. Each different solution involves a different number of knife changes, e.g. the solution from above involves 12 knife changes, whereas the solution 50-40-40-60, 5-4-4-4 and 50-50-50-30 involves only 7 knife changes. The objective is to find the solution with the minimal number of knife changes, or, because the search space is immense, to approximate such a solution. We first give some auxiliary definitions describing operations on sequences, bags and sets. We then define formally the problem, the solution space and the cost function in terms of the above. We prove some properties which reduce the search space, and then we describe heuristics
