An algorithm for a cutting stock problem on a strip

A cutting stock problem is formulated as follows: a set of rectangular pieces must be cut from a set of sheets, so as to minimize total waste. In our problem the pieces are requested in large quantities and the set of sheets are long rolls of material. For this class of problems we have developed a fast heuristic based on partial enumeration of all feasible patterns. We then tested the effectiveness on a set of test problems ranging from practical to random instances. Finally, the algorithm has been applied to check the asymptotic behaviour of the solution when a continuous stream of pieces is requested and cutting decisions are to be made while orders are still arriving.     

Multi-job Cutting Stock Problem with Due Dates and Release Dates

The common feature of cutting stock problems is to cut some form of stock materials to produce smaller pieces of materials in quantities matching orders received. Most research on cutting stock problems focuses on either generating cutting patterns to minimize wastage or determining the required number of stock materials to meet orders. In this paper, we examine a variation of cutting stock problems that arises in some industries where meeting orders' due dates is more important than minimizing wastage of materials. We develop two two-dimensional cutting stock models with due date and release date constraints. Since adding due dates and release dates makes the traditional cutting stock problem even more difficult to solve, we develop both LP-based and non-LP-based heuristics to obtain good solutions. The computational results show that the solution procedures are easy to implement and work very well.     

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.     

Column generation and sequential heuristic procedure for solving an irregular shape cutting stock problem

The research addressing two-dimensional (2D) irregular shape packing has largely focused on the strip packing variant of the problem. However, it can be argued that this is a simplification. The materials from which pieces are required to be cut will ultimately have a fixed length either due to the physical dimensions of the material or through constraints on the cutting machinery. Hence, in order to cut all the pieces, multiple sheets may be required. From this scenario arises the 2D irregular shape cutting stock problem. In this paper, we will present implementations of cutting stock approaches adapted to handle irregular shapes, including two approaches based on column generation (CG) and a sequential heuristic procedure. In many applications, setup costs can be reduced if the same pattern layout is cut from multiple sheets; hence there is a trade-off between material waste and number of patterns. Therefore, we describe the formulation and implementation of an adaptation of the CG method to control the number of different patterns. CG is a common method for the cutting stock problem; however, when the pieces are irregular the sub-problem cannot be solved optimally. Hence we implement CG and solve the sub-problem using the beam search heuristic. Further, we introduce a version of CG for instances where the number of rows is less than the number of columns.     

Integrating two-dimensional cutting stock and lot-sizing problems

The two-dimensional cutting stock problem (2DCSP) consists in the minimization of the number of plates used to cut a set of items. In industry, typically, an instance of this problem is considered at the beginning of each planning time period, what may result in solutions of poor quality, that is, excessive waste, when a set of planning periods is considered. To deal with this issue, we consider an integrated problem, in which the 2DCSP is extended from the solution in only a single production planning period to a solution in a set of production planning periods. The main difference of the approach in this work and the ones in the literature is to allow sufficiently large residual plates (leftovers) to be stored and cut in a subsequent period of the planning horizon, which may further help in the minimization of the waste. We propose two integrated integer programming models to optimize the combined two-dimensional cutting stock and lot-sizing problems, minimizing the total cost, which includes material, waste and storage costs. Two heuristics based on the industrial practice to solve the problem were also presented. Computational results for the proposed models and for the heuristics are presented and discussed.     

Alternative configurations for cutting machines in a tube cutting mill

In a steel tube mill where an endless stream of steel tube is supplied from a manufacturing facility, trim waste is never made regardless of cutting patterns used and the standard cutting stock problem seems meaningless. Therefore, the continuous stock cutting problem with setup is introduced to minimize the sum of cutting time and pattern changing time to meet the given demand. We propose a new configuration of cutting machines to achieve higher production efficiency, namely the open-ended configuration as opposed to the traditional closed-ended configuration, thereby two variants of the problem are defined. We propose linear formulations for both problems using binary expansion of the number of pieces of different types in a pattern. Furthermore, we define the time for pattern change as a linear function of the number of knives used in the pattern to be more realistic. Computational studies suggest that the open-ended cutting machine may improve the production time by up to 44% and that our linear formulations are more efficient than the existing ones.     

New model and heuristic solution approach for one-dimensional cutting stock problem with usable leftovers

In the one-dimensional cutting stock problem with usable leftovers (1DCSPUL), items of the current order are cut from stock bars to minimize material cost. Here, stock bars include both standard ones bought commercially and old leftovers generated in processing previous orders, and cutting patterns often include new leftovers that are usable in processing subsequent orders. Leftovers of the same length are considered to be of the same type. The number of types of leftovers should be limited to simplify the cutting process and reduce the storage area. This paper presents an integer programming model for the 1DCSPUL with limited leftover types and describes a heuristic algorithm based on a column-generation procedure to solve it. Computational results show that the proposed approach is more effective than several published algorithms in reducing trim loss, especially when the number of types of leftovers is limited.     

A two-objective mathematical model without cutting patterns for one-dimensional assortment problems

This paper considers a one-dimensional cutting stock and assortment problem. One of the main difficulties in formulating and solving these kinds of problems is the use of the set of cutting patterns as a parameter set in the mathematical model. Since the total number of cutting patterns to be generated may be very huge, both the generation and the use of such a set lead to computational difficulties in solution process. The purpose of this paper is therefore to develop a mathematical model without the use of cutting patterns as model parameters. We propose a new, two-objective linear integer programming model in the form of simultaneous minimization of two contradicting objectives related to the total trim loss amount and the total number of different lengths of stock rolls to be maintained as inventory, in order to fulfill a given set of cutting orders. The model does not require pre-specification of cutting patterns. We suggest a special heuristic algorithm for solving the presented model. The superiority of both the mathematical model and the solution approach is demonstrated on test problems.     

A sequential heuristic procedure for one-dimensional cutting

The article examines the Sequential Heuristic Procedure (SHP) for optimising one-dimensional stock cutting when all stock lengths are different. In order to solve a bicriterial multidimensional knapsack problem with side constraints a lexicographic approach is applied. An item-oriented solution was found through a combination of approximations and heuristics that minimize the influence of ending conditions leading to almost optimal solutions. The computer program CUT was developed, based on the proposed algorithm. Two sample problems are presented and solved. A statistical analysis of parameters that affect material utilisation was also made.     

Heuristic and exact methods for the cutting sequencing problem

Cutting stock problems deal with the generation of a set of cutting patterns that minimizes waste. Sometimes it is also important to find the processing sequence of this set of patterns to minimize the maximum queue of partially cut orders. In such instances a cutting sequencing problem has to be solved. This paper presents a new mathematical model and a three-phase approach for the cutting sequencing problem. In the first phase, a greedy algorithm produces a good starting solution that is improved in the second phase by a tabu search, or a generalized local search procedure, while, in the last phase, the problem is optimally solved by an implicit enumeration procedure that uses the best solution previously found as an upper bound. Computing experience, based on 300 randomly generated problems, shows the good performance of the heuristic methods presented.     

Integrated lot-sizing and one-dimensional cutting stock problem with usable leftovers

This paper addresses the integration of the lot-sizing problem and the one-dimensional cutting stock problem with usable leftovers (LSP-CSPUL). This integration aims to minimize the cost of cutting items from objects available in stock, allowing the bringing forward production of items that have known demands in a future planning horizon. The generation of leftovers, that will be used to cut future items, is also allowed and these leftovers are not considered waste in the current period. Inventory costs for items and leftovers are also considered. A mathematical model for the LSP-CSPUL is proposed to represent this problem and an approach, using the simplex method with column generation, is proposed to solve the linear relaxation of this model. A heuristic procedure, based on a relax-and-fix strategy, was also proposed to find integer solutions. Computational tests were performed and the results show the contributions of the proposed mathematical model, as well as, the quality of the solutions obtained using the proposed method.

     

A solution procedure for a pattern sequencing problem as part of a one-dimensional cutting stock problem in the steel industry

To cut reinforcing bars for concrete buildings, machines are used which have compartments to store the cut orders until the requirement is met. Number and size of these compartments restrict kind and processing sequence of possible cutting patterns. In this paper we present the so-called “Sequencing algorithm” that tackles the problem of finding a processing sequence for the cutting patterns starting from an integer solution of the cutting stock problem and using an interpretation of relations between orders in patterns as a graph. Computational results are reported.     

Solving real-world cutting stock-problems in the paper industry: Mathematical approaches,experience and challenges

We discuss cutting stock problems (CSPs) from the perspective of the paper industry and the financial impact they make. Exact solution approaches and heuristics have been used for decades to support cutting stock decisions in that industry. We have developed polylithic solution techniques integrated in our ERP system to solve a variety of cutting stock problems occurring in real world problems. Among them is the simultaneous minimization of the number of rolls and the number of patterns while not allowing any overproduction. For two cases, CSPs minimizing underproduction and CSPs with master rolls of different widths and availability, we have developed new column generation approaches. The methods are numerically tested using real world data instances. An assembly of current solved and unsolved standard and non-standard CSPs at the forefront of research are put in perspective.     

One-dimensional cutting stock optimization in consecutive time periods

The paper deals with the general one-dimensional cutting stock problem (G1D-CSP), where optimization is not limited to a single order. Stock cutting is treated as a permanent business process in a company in which consecutive order sets need to be fulfilled either for production needs or for its customers. Exact demand for future orders is not known in advance. The unutilized and partly utilized stock lengths left after fulfilling current order sets are stored and used later. The goal is the reduction of trim loss and costs over a broader time-span. A new approach is suggested where previously developed method for G1D-CSP is modified. Several practical examples of the cutting process for several consecutive order sets are presented. An extension to a currently used typology for cutting stock problems is proposed.     

A branch-and-price algorithm for the two-stage guillotine cutting stock problem

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their widths and others down their heights. To solve this model, we propose an exact branch-and-price algorithm. To the best of our knowledge, this is the first contribution with regard to obtaining integer optimal solutions to Gilmore and Gomory model. Extensive results, on a set of real-world problems, indicate that the proposed algorithm delivers optimal solutions for instances with up to 809 items and that the hybrid cutting strategy often yields improved solutions. Furthermore, our computational study reveals that the proposed modelling and algorithmic strategy outperforms a recently proposed arc-flow model-based solution strategy.     

The combined cutting stock and lot-sizing problem in industrial processes

Despite its great applicability in several industries, the combined cutting stock and lot-sizing problem has not been sufficiently studied because of its great complexity. This paper analyses the trade-off that arises when we solve the cutting stock problem by taking into account the production planning for various periods. An optimal solution for the combined problem probably contains non-optimal solutions for the cutting stock and lot-sizing problems considered separately. The goal here is to minimize the trim loss, the storage and setup costs. With a view to this, we formulate a mathematical model of the combined cutting stock and lot-sizing problem and propose a solution method based on an analogy with the network shortest path problem. Some computational results comparing the combined problem solutions with those obtained by the method generally used in industry—first solve the lot-sizing problem and then solve the cutting stock problem—are presented. These results demonstrate that by combining the problems it is possible to obtain benefits of up to 28% profit. Finally, for small instances we analyze the quality of the solutions obtained by the network shortest path approach compared to the optimal solutions obtained by the commercial package AMPL.     

考虑生产计划的多周期切割问题优化研究

切割生产广泛存在于工业企业,是原材料加工的重要环节。已有文献主要关注单周期切割问题,但是切割计划也是生产计划的一部分,切割计划和生产计划应该协调优化,达到全局最优。本文研究考虑生产计划的多周期切割问题,目标是最小化运营成本,包括准备成本、切割成本、库存成本以及母材消耗成本。首先建立混合整数规划模型;提出动态规划启发式算法;最后对算例在多种情境下测试,分析成本因子变化对最优结果的影响。算法结果与CPLEX最优结果比较,平均误差为1.85%,表明算法是有效的。     

A Practical Approach for Determining Rectangular Stock Sizes

In this paper we address the problem of determining what rectangular sizes should be stocked in order to satisfy a bill of materials composed of smaller rectangles. As is common in many manufacturing operations, the stock material will be cut using two-staged guillotine cutting patterns. We first generate a large number of stock sizes ideally suited to the bill of materials. Then we use a facility location algorithm to consolidate the stock sizes down to an acceptable number. Given the solution of what rectangular sizes to stock, a second program is used to map bills of materials into the stock rectangles. The effectiveness of our approach is demonstrated by generating stock sizes for a real-world bill of materials with 392 distinct order sizes and over 7700 order pieces.     

An Improved Algorithm for the Non-Guillotine-Constrained Cutting-Stock Problem

A personal-computer-based algorithm to solve the non-guillotine-constrained two-dimensional cutting-stock problem is developed. The problem is constrained to single-sized rectangles placed orthogonally on a larger containing rectangle. The algorithm uses the linear combination of box lengths and widths that minimizes waste along the cutting stock's length and width to determine an optimal layout. The algorithm's performance is evaluated using two sets of test cases and compared to the results of other algorithms.     

A hierarchical approach for one-dimensional cutting stock problems in the steel industry that maximizes yield and minimizes overgrading

In the steel industry, as hot steel products exit the producing facility, they are cut at primary saws (hotsaws) into shorter pieces. After these pieces cool, they are inspected for defects and either applied directly to customer orders or are further cut to ordered lengths at secondary saws (cold saws). In this case study, we will describe a hierarchical algorithm, DYNACUT_CS, that efficiently and effectively generates cutting patterns for material that is to be cut at cold saws. DYNACUT_CS strives to maximize yield over all the material cut and simultaneously tries to minimize overgrading (applying higher quality material than specified by the customer). An example will be given to illustrate how the algorithm works. This approach has been implemented for a variety of products at several different Bethlehem Steel Corporation facilities.