Stock cutting to minimize cutting length

In this paper we investigate the following problem: Given two convex Pin and Pout, where Pin is completely contained in Pout, we wish to find a sequence of ‘guillotine cuts’ to cut out Pin, from Pout such that the total length of the cutting sequence is minimized. This problem has applications in stock cutting where a particular shape or design (in this case the polygon Pin) needs to be cut out of a given piece of parent material (the polygon Pout) using only guillotine cuts and where it is desired to minimize the cutting sequence length to improve the cutting time required per piece. We first prove some properties of the optimal solution to the problem and then give an approximation scheme for the problem that, given an error range &#x03B4, produces a cutting sequence whose total length is at most &#x03B4 more than that of the optimal cutting sequence. Then it is shown that this problem has optimal solutions that lie in the algebraic extension of the field that the input data belongs to — hence due to this algebraic nature of the problem, an approximation scheme is the best that can be achieved. Extensions of these results are also studied in the case where the polygons Pin and Pout are non-convex

Filling a rough-mill cutting order using a fuzzy logic controller

Because of their high cost, reducing the number of boards used to fill an order of dimension parts is an important goal in a lumber rough mill. FLCO (Fuzzy Logic Cutting Order) is a computer program that achieves this goal. Using the concepts of fuzzy logic control, FLCO provides a heuristic approach to this problem. FLCO incorporates a version of the CORY lumber cut-up software, and provides a model of a rough-mill system that is able to reduce the number of boards needed to fill a cutting order. FLCO allows different control methods to be used, including fuzzy logic control, dropping sizes from the cutting order as their demands are met, and no control. Because of the modularity of its design the FLCO code can easily incorporate other control methods. CORY lumber cut-up software provides sawing solutions for boards based on the values assigned to the sizes in a cutting order. After each board is sawn, the fuzzy logic controller adjusts the value of each size to achieve the objective of filling the demand for each size in a cutting order at about the same time. These new values are subsequently used by CORY when finding a sawing solution for the next board to be processed. Upon completion, FLCO reports the number of boards required to fill the cutting order, the average percent area yield, and the number of each piece recovered. Three cutting orders, three lumber grades and two types of control provide a means of determining the effects of fuzzy logic control on reducing the number of boards used to fill a cutting order. Across all lumber grades and cutting orders, fuzzy logic control greatly reduces the number of boards as compared with dropping sizes. Fuzzy logic control slightly reduces the number of boards as compared with a more complicated method of control across all lumber grades and most cutting orders, but the difference is not statistically significant. In addition, the effect of fuzzy logic control on individual sizes is also examined

Minimizing waste in the 2-dimensional cutting stock problem

The 2-dimensional cutting stock problem is an important problem in the garment manufacturing industry. The problem is to arrange a given set of 2-dimensional patterns onto a rectangular bolt of cloth such that the efficiency is maximised. This arrangement is called a marker. Efficiency is measured by pattern area I marker area. Efficiency varies depending on the shape and number of patterns being cut, but an improvement in efficiency can result in significant savings. Markers are usually created by humans with the aid of CAD software. Many researchers have attempted to create automatic marker making software but have failed to produce marker efficiencies as high as human generated ones. This thesis presents a mathematical model which optimally solves the 2-dimensional cutting stock problem. However, the model can only be solved in a practical amount of time for small markers. Subsequently, two compaction algorithms based on mathematical modelling have been developed to improve the efficiency of human generated markers. The models developed in this thesis make use of a geometrical calculation known as the no-fit polygon. The no-fit polygon is a tool for determining whether polygons A and B overlap. It also gives all feasible positions for polygons B with respect to polygon A, such that the two polygons do not overlap. For the case when both polygons A and B are non-convex, current calculation methods are either time consuming or unreliable. This thesis presents a method which is both computationally efficient and robust for calculating the no-fit polygon when polygons A and B are non-convex. When tested on a set of industrial markers, the compaction algorithms improved the marker efficiencies by over 1.5% on average

Modeling and analysis of three-dimensional robotic palletizing systems for mixed carton sizes

Currently, robot applications in warehousing are limited to very simple pallet loading where only one type of carton is palletized at any one time. This research investigates the situation where many cartons of various sizes must be placed on one pallet. This arises in retail businesses such as grocery or merchandise distributions;A mixed 0/1 integer programming model has been developed to solve for three-dimensional optimal pallet patterns. Since the pallet packing problem has been classified as NP-hard, a heuristic dynamic programming algorithm has also been developed. Only good solution may be obtained; however, less computation time is required when compared with the mixed 0/1 model. The determined pallet pattern which maximizes the utilization of the pallet cube is then used as the input data to a robot control program for automatic palletizing;A rhino XR-2 robot is employed to investigate automatic palletizing operations. A coordinate transformation program which allows the conversion of the Cartesian coordinate to the robot\u27s joint coordinate has been completed. Therefore, only the x, y, z coordinate values of box\u27s placement location are required to the robot control program. Two efficient palletizing methods are developed. One is dynamic pallet pattern, which will dynamically select a best match pattern according to incoming box sizes. The second is multi-pallet packing with turntables, which allows robot to simultaneously load two or more pallets. Two major simulation results are presented in this research. One is the simulation statistic for multi-pallet packing. Simultaneous loading of 1, 2, 3 and 4 pallets is investigated. The other simulation is to determine the length of look-ahead box queue on the conveyor in order to dynamically select a best match pallet pattern. A miniature robotic palletizing cell is employed to collect the palletizing statistics for evaluating the performance of the developed system and determining design alternatives