Reducing the number of cuts in generating three-staged cutting patterns

Three-staged guillotine patterns are widely used in the manufacturing industry to cut stock plates into rectangular items. The cutting cost often increases with the number of cuts required. This paper focuses on the rectangular two-dimensional cutting stock problem, where three-staged guillotine patterns are used, and the objective is to minimize the sum of plate and cutting costs. The column generation framework is used to solve the problem. It uses a pattern-generation procedure to obtain the patterns. The cutting cost is considered in both the pattern-generation procedure and the objective of the linear programming formulation. The computational results indicate that the approach can reduce the number of cuts, without increasing the plate cost.     

Combined primary and secondary log breakdown optimisation

At sawmills logs are converted into boards by a series of cutting operations. Primary cuts reduce logs into slabs of wood, secondary cuts produce boards. Boards incorporating natural defects such as knots (branch sections) are inferior to clear boards. The aim of the sawmill is to cut logs to produce boards of greatest value. However, when logs are pruned, knots are only exposed after primary cutting. This complicates the conversion problem.To effectively convert logs into boards the interrelated effects of the cutting phases must be recognized. In this paper linked dynamic programming formulations are developed. The inner (secondary) formulation determines the optimal sequence for cutting a slab into boards. The result is passed to the master (primary) formulation that determines the optimal sequence for cutting the log into slabs. The objective functions can be modified to maximise either total value of boards (incorporating quality) or total volume. Results from simulations indicate that significant increases in value are possible when quality is considered.     

An integrated approach to the one-dimensional cutting stock problem in coronary stent manufacturing

This paper presents a two-stage approach for pattern generation and cutting plan determination of the one-dimensional cutting stock problem. Calculation of the total number of patterns that will be cut and generation of the cutting patterns are performed in the first stage. On the other hand, the second stage determines the cutting plan. The proposed approach makes use of two separate integer linear programming models. One of these models is employed by the first stage to generate the cutting patterns through a heuristic procedure with the objective of minimizing trim loss. The cutting patterns obtained from Stage 1 are then fed into the second stage. In this stage, another integer linear programming model is solved to form a cutting plan. The objective of this model is to minimize a generalized total cost function consisting of material inputs, number of setups, labor hours and overdue time; subject to demand requirements, material availability, regular and overtime availability, and due date constraints. The study also demonstrates an implementation of the proposed approach in a coronary stent manufacturer. The case study focuses on the cutting phase of the manufacturing process followed by manual cleaning and quality control activities. The experiments show that the proposed approach is suitable to the conditions and requirements of the company.     

桉树扦插育苗正交试验设计分析

根据扦插育苗试验表明，影响泰国赤桉苗高、根长及发根数的主要因素各不相同。综合分析表明：其主要因素是ＢＡ浓度，插条浸泡药液的时间和蔗糖浓度，其较优的插条处理方法是将插条在０．６％蔗糖＋３００ＰＰｍＩＢＡ溶液中浸泡２小时     

A hybrid approach based on genetic algorithms to solve the problem of cutting structural beams in a metalwork company

This work presents a hybrid approach based on the use of genetic algorithms to solve efficiently the problem of cutting structural beams arising in a local metalwork company. The problem belongs to the class of one-dimensional multiple stock sizes cutting stock problem, namely 1-dimensional multiple stock sizes cutting stock problem. The proposed approach handles overproduction and underproduction of beams and embodies the reusability of remnants in the optimization process. Along with genetic algorithms, the approach incorporates other novel refinement algorithms that are based on different search and clustering strategies. Moreover, a new encoding with a variable number of genes is developed for cutting patterns in order to make possible the application of genetic operators. The approach is experimentally tested on a set of instances similar to those of the local metalwork company. In particular, comparative results show that the proposed approach substantially improves the performance of previous heuristics.     

New upper bounds for the two-dimensional orthogonal non-guillotine cutting stock problem

The two-dimensional orthogonal non-guillotine cutting stockproblem (NGCP) appears in many industries (e.g. the wood andsteel industries) and consists of cutting a rectangular mastersurface into a number of rectangular pieces, each with a givensize and value. The pieces must be cut with their edges alwaysparallel to the edges of the master surface (orthogonal cuts).The objective is to maximize the total value of the pieces cut. New upper bounds on the optimal solution to the NGCP are described.The new bounding procedures are obtained by different relaxationsof a new mathematical formulation of the NGCP. Various proceduresfor strengthening the resulting upper bounds and reducing thesize of the original problem are discussed. The proposed newupper bounds have been experimentally evaluated on test problemsderived from the literature. Comparisons with previous boundingprocedures from the literature are given. The computationalresults indicate that these bounds are significantly betterthan the bounds proposed in the literature.     

Heuristic for the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns

Two-staged patterns are often used in manufacturing industries to divide stock plates into rectangular items. A heuristic algorithm is presented to solve the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns. It uses the column-generation method to solve the residual problems repeatedly, until the demands of all items are satisfied. Each pattern is generated using a procedure for the constrained single large object placement problem to guarantee the convergence of the algorithm. The computational results of benchmark and practical instances indicate the following: (1) the algorithm can solve most instances to optimality, with the gap to optimality being at most one plate for those solutions whose optimality is not proven and (2) for the instances tested, the algorithm is more efficient (on average) in reducing the number of plates used than a published algorithm and a commercial stock cutting software package.     

An effective solution for a real cutting stock problem in manufacturing plastic rolls

We confront a practical cutting stock problem from a production plant of plastic rolls. The problem is a variant of the well-known one dimensional cutting stock, with particular constraints and optimization criteria defined by the experts of the company. We start by giving a problem formulation in which optimization criteria have been considered in linear hierarchy according to expert preferences, and then propose a heuristic solution based on a GRASP algorithm. The generation phase of this algorithm solves a simplified version which is rather similar to the conventional one dimensional cutting stock. To do that, we propose a Sequential Heuristic Randomized Procedure (SHRP). Then in the repairing phase, the solution of the simplified problem is transformed into a solution to the real problem. For experimental study we have chosen a set of problem instances of com-mon use to compare SHRP with another recent approach. Also, we show by means of examples, how our approach works over instances taken from the real production process. All authors are supported by MEC-FEDER Grant TIN2007-67466-C02-01 and by contract CN-05-127 of the University of Oviedo and the company ERVISA, and by FICYT under grant BP04-021.     

Application of Heuristic Techniques to the Cutting-Stock Problem for Worktops

This paper addresses a cutting-stock problem which arises in the manufacture of furniture. An heuristic has been developed to schedule the cutting of worktops of varying shapes and sizes from available raw material.The work was carried out for a local manufacturing company. Interface software written at that company allows the program to be integrated into their computer system in such a way that it can easily be run by the user.Details of the heuristic are given, together with results of some numerical experiments designed to test the robustness of the algorithm and to give some guidance as to the optimal batch size and dimensions of the raw material.Performance of the algorithm in situ supports the results of numerical experiments, the computer solution giving some improvements over the previously obtained manual solution with respect to trim loss without incurring an unacceptable build-up of offcuts.     

An iterative sequential heuristic procedure to a real-life 1.5-dimensional cutting stock problem

This paper addresses a real-life 1.5D cutting stock problem, which arises in a make-to-order plastic company. The problem is to choose a subset from the set of stock rectangles to be used for cutting into a number of smaller rectangular pieces so as to minimize total production cost and meet orders. The total production cost includes not only material wastage, as in traditional cutting stock problems, but also production time. A variety of factors are taken into account, like cutter knife changes, machine restrictions, due dates and other work in progress limitations. These restrictions make the combinatorial structure of the problem more complex. As a result, existing algorithms and mathematical models are no longer appropriate. Thus we developed a new 1.5D cutting stock model with multiple objectives and multi-constraints and solve this problem in an incomplete enumerative way. The computational results show that the solution procedure is easy to implement and works very well.     

On a pattern sequencing problem to minimize the maximum number of open stacks

Consider a wood cutting setting where different panels have to be cut from large boards. Each cut panel size is put into a stack which remains opened until the last panel of that size is cut. The problem considered here deals with the sequencing of the patterns in order to minimize the maximum number of open stacks during the production run. A branch and bound method for solving this problem is presented which uses a greedy type scheme to select the next node to branch.     

A Partitioned Cutting-stock Problem Applied in the Meat Industry

The paper reports on a linear programming application in the meat industry. The problem is formulated as a variant of the cutting-stock or trim problem, where the objective is to maximize the return from selling products yielded from cutting patterns applied to animal carcasses. One feature of the formulation is the partitioning of cutting patterns among carcass sections. Since the sections are relatively independent, this partitioning vastly reduces the number of cutting patterns in the formulation. Implementation is on a personal computer, and the system is used by a meat company for market planning. The system uses a commercial database to handle data entry and solution reporting, and has been found to be extremely user-friendly.     

Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints

The paper examines a new problem in the irregular packing literature that has many applications in industry: two-dimensional irregular (convex) bin packing with guillotine constraints. Due to the cutting process of certain materials, cuts are restricted to extend from one edge of the stock-sheet to another, called guillotine cutting. This constraint is common place in glass cutting and is an important constraint in two-dimensional cutting and packing problems. In the literature, various exact and approximate algorithms exist for finding the two dimensional cutting patterns that satisfy the guillotine cutting constraint. However, to the best of our knowledge, all of the algorithms are designed for solving rectangular cutting where cuts are orthogonal with the edges of the stock-sheet. In order to satisfy the guillotine cutting constraint using these approaches, when the pieces are non-rectangular, practitioners implement a two stage approach. First, pieces are enclosed within rectangle shapes and then the rectangles are packed. Clearly, imposing this condition is likely to lead to additional waste. This paper aims to generate guillotine-cutting layouts of irregular shapes using a number of strategies. The investigation compares three two-stage approaches: one approximates pieces by rectangles, the other two approximate pairs of pieces by rectangles using a cluster heuristic or phi-functions for optimal clustering. All three approaches use a competitive algorithm for rectangle bin packing with guillotine constraints. Further, we design and implement a one-stage approach using an adaptive forest search algorithm. Experimental results show the one-stage strategy produces good solutions in less time over the two-stage approach.     

The Cutting Stock Problem in the Flat Glass Industry

The cutting stock problem occurs where large rectangles of some material require cutting into smaller rectangles, in the most appropriate way, to satisfy an order book. A linear programming approach to the problem has been suggested by P. C. Gilmore and R. E. Gomory. An application of this approach in the glass industry is described which is shown to be inadequate since it only satisfies a wastage criterion. In practice, multiple criteria must be satisfied and two alternative approaches using linear programming and heuristic scheduling are proposed.     

Cutting Down Trim-loss for Competitive Advantage: a Singaporean Company's Experience

This paper concerns a study of the cutting-stock problem faced by a small company in the building components industry in Singapore. It involves the development of a simple heuristic procedure which is able to recommend cutting patterns tailored to the specific needs of the company. To help determine the appropriate stock lengths to order, a simulation-cum-sampling approach is proposed. The pattern-generating heuristic has been tested using real-life project data. The results obtained are very encouraging in that the goals set by the management of the firm are met well within the numerous constraints faced.     

Two-dimensional Cutting Stock with Multiple Stock Sizes

In this paper an algorithm for a cutting stock problem in the wood industry is presented. Cuts are of guillotine type and requirements have to be met exactly, i.e. no over- or under-production is allowed. There are several different board sizes from which panels can be cut and the problem is to find the best mix of boards and respective cutting patterns that satisfies the demand for panels with minimum wastage. The heuristic algorithm uses a pattern-building procedure combined with an enumeration scheme for the mix of boards.     

Cutting optimization of structural tubes to build agricultural light aircrafts

In this study we deal with the one-dimensional cutting of metallic structural tubes used in the manufacturing of agricultural light aircrafts. The problem is modeled by mixed integer linear formulations aiming to minimize material trim losses and considering the possibility of generating remainders (leftovers) with enough size to reuse. To validate the application of the models in practice, we carried out experiments with real data of order lists from Ipanema, an agricultural airplane produced by a Brazilian aeronautical company. The models were solved using a modeling language and an optimization software. The computational results show that the models are useful in supporting decisions in this cutting process.     

The one-dimensional cutting stock problem with usable leftovers – A survey

In this article, we review published studies that consider the solution of the one-dimensional cutting stock problem (1DCSP) with the possibility of using leftovers to meet future demands, if long enough. The one-dimensional cutting stock problem with usable leftovers (1DCSPUL) is a problem frequently encountered in practical settings but often, it is not dealt with in an explicit manner. For each work reviewed, we present the application, the mathematical model if one is proposed and comments on the computational results obtained. The approaches are organized into three classes: heuristics, item-oriented, or cutting pattern-oriented.     

MANAGEMENT VARIATION AND PRICE EXPECTATIONS IN AN INTERTEMPORAL FOREST SECTOR MODEL

The assessment of wood supply in long-run forest sector analysis by harvest scheduling approaches entails an assumption of rational behavior among forest owners. The implications of the assumption are investigated by projecting wood supply in southern Sweden under alternate price expectations and with different requirements on the variation in forest management practices. The results suggest that the formation of price expectations is of prime importance in modeling wood supply and influences the effect of variation.