One-dimensional Cutting Stock Problem Research Articles

Hybrid heuristic for the one-dimensional cutting stock problem with usable leftovers and additional operating constraints

The One-Dimensional Cutting Stock Problem consists in cutting long bars into smaller ones, to satisfy customers’ demand, minimizing waste and cost. In this paper the standard problem is extended with the inclusion of additional constraints that are generally neglected in scientific literature, although relevant in many industrial applications. We also modified the standard objective function, by assuming that bars may have a different economical value and a different processing or shipping priority. Moreover, in line with business requirements, among solutions that generate the same cutting waste, we prefer the ones that generate a low number of leftovers, especially if leftovers are long, so that the likelihood of their reuse is high. To solve the problem, we propose a Simulated Annealing based heuristic, which exploits a specific neighbor search. The heuristic is implemented in a parametric way that allows the user to set the priorities of the bars and to choose the specific sub-set of constraints he or she wants to consider. The heuristic is finally tested on many problem instances, and it is compared to three benchmarks and to one commercial software. The outcomes of this comparative analysis demonstrate both its quality and effectiveness.

Solution Approach to Cutting Stock Problems Using Iterative Trim Loss Algorithm and Monte-Carlo Simulation

Cutting Stock Problems are the most studied NP-Hard combinatorial problems among optimization problems. An One-dimensional Cutting Stock Problem (1-CSP), which aims to create cutting patterns to minimize trim loss, is one of the best known optimization problems. The difficulty of the solution stages and the lack of a definite solution method that can be applied to all problems have caused these problems to attract a lot of attention by researchers. This study includes a hybrid solution algorithm combined with iterative trim loss algorithm and Monte Carlo simulations, and a comparative study of the method with the solution methods in the literature, for the solution of orders to be obtained with minimum cutting loss from the same type of stocks.

Minimizing the total waste in the one-dimensional cutting stock problem with the African buffalo optimization algorithm.

The one-dimensional cutting-stock problem (1D-CSP) consists of obtaining a set of items of different lengths from stocks of one or different lengths, where the minimization of waste is one of the main objectives to be achieved. This problem arises in several industries like wood, glass, and paper, among others similar. Different approaches have been designed to deal with this problem ranging from exact algorithms to hybrid methods of heuristics or metaheuristics. The African Buffalo Optimization (ABO) algorithm is used in this work to address the 1D-CSP. This algorithm has been recently introduced to solve combinatorial problems such as travel salesman and bin packing problems. A procedure was designed to improve the search by taking advantage of the location of the buffaloes just before it is needed to restart the herd, with the aim of not to losing the advance reached in the search. Different instances from the literature were used to test the algorithm. The results show that the developed method is competitive in waste minimization against other heuristics, metaheuristics, and hybrid approaches.

A Tree-Based Heuristic for the One-Dimensional Cutting Stock Problem Optimization Using Leftovers.

Cutting problems consist of cutting a set of objects available in stock in order to produce the desired items in specified quantities and sizes. The cutting process can generate leftovers (which can be reused in the case of new demand) or losses (which are discarded). This paper presents a tree-based heuristic method for minimizing the number of cut bars in the one-dimensional cutting process, satisfying the item demand in an unlimited bar quantity of just one type. The results of simulations are compared with the RGRL1 algorithm and with the limiting values for this considered type of problem. The results show that the proposed heuristic reduces processing time and the number of bars needed in the cutting process, while it provides a larger leftover (by grouping losses) for the one-dimensional cutting stock problem. The heuristic contributes to reduction in raw materials or manufacturing costs in industrial processes.

Solving One-Dimensional Cutting Stock Problems with the Deep Reinforcement Learning

It is well known that the one-dimensional cutting stock problem (1DCSP) is a combinatorial optimization problem with nondeterministic polynomial (NP-hard) characteristics. Heuristic and genetic algorithms are the two main algorithms used to solve the cutting stock problem (CSP), which has problems of small scale and low-efficiency solutions. To better improve the stability and versatility of the solution, a mathematical model is established, with the optimization objective of the minimum raw material consumption and the maximum remaining material length. Meanwhile, a novel algorithm based on deep reinforcement learning (DRL) is proposed in this paper. The algorithm consists of two modules, each designed for different functions. Firstly, the pointer network with encoder and decoder structure is used as the policy network to utilize the underlying mode shared by the 1DCSP. Secondly, the model-free reinforcement learning algorithm is used to train network parameters and optimize the cutting sequence. The experimental data show that the one-dimensional cutting stock algorithm model based on deep reinforcement learning (DRL-CSP) can obtain the approximate satisfactory solution on 82 instances of 3 data sets in a very short time, and shows good generalization performance and practical application potential.

Arc-flow formulations for the one-dimensional cutting stock problem with multiple manufacturing modes

In this paper, an integration of the one-dimensional cutting stock problem with an operational problem that arises in the manufacture of concrete poles is studied. Seeing that poles have a steel structure, different thicknesses of steel bars can be used in their manufacture. This variety in combining the materials to produce the structure of the poles is known as alternative production modes or multiple manufacturing modes. The problem considered here has the objective of minimizing the total cost to meet the demand for poles using the different available configurations. This problem has already been introduced in the literature and it has been formulated as an integer programming problem. To solve it, the column generation procedure was used. The contribution of this paper is to reformulate the cutting stock problem with multiple manufacturing modes using arc-flow formulations. Arc-flow formulations are promising tools to model and solve complex combinatorial problems. Computational tests are performed comparing the formulations using instances from the literature, which were generated based on the data from a civil construction plant. The arc-flow formulations increased the number of instances solved to proven optimality and also reduced solution time. Lower and upper bounds are also improved when compared with the solution proposed in the literature.

A method for demand-accurate one-dimensional cutting problems with pattern reduction.

The main objective in the one-dimensional cutting stock problem (1D-CSP) is to minimize material costs. In practice, it is useful to focus on auxiliary objectives, one of which is to reduce the number of different cutting patterns. This paper discusses the classical integer IDCSP, where only one type of stock object is included. Meanwhile, the demands of various items must be precisely satisfied in the constraints. In other words, no overproduction or underproduction is allowed. Therefore, to solve this issue, a variable-to-constant method based on a new mathematical model is proposed. In addition, we integrate the approach with two other representative methods to demonstrate its effectiveness. Both benchmark instances and real instances are used in the experiments, and the results show that the methodology is effective in reducing patterns. In particular, in terms of the solutions to the real-life instances, the proposed approach presents a 31.93 to 37.6% pattern reduction compared to other similar methods (including commercial software).

Contribution to solving a one-dimensional cutting stock problem with two objectives based on the generation of cutting patterns

Contribution to solving a one-dimensional cutting stock problem with two objectives based on the generation of cutting patterns

Computational Performance Evaluation of Column Generation and Generate-and-Solve Techniques for the One-Dimensional Cutting Stock Problem

The Cutting Stock Problem (CSP) is an optimisation problem that roughly consists of cutting large objects in order to produce small items. The computational effort for solving this problem is largely affected by the number of cutting patterns. In this article, in order to cope with large instances of the One-Dimensional Cutting Stock Problem (1D-CSP), we resort to a pattern generating procedure and propose a strategy to restrict the number of patterns generated. Integer Linear Programming (ILP) models, an implementation of the Column Generation (CG) technique, and an application of the Generate-and-Solve (G&S) framework were used to obtain solutions for benchmark instances from the literature. The exact method was capable of solving small and medium sized instances of the problem. For large sized instances, the exact method was not applicable, while the effectiveness of the other methods depended on the characteristics of the instances. In general, the G&S method presented successful results, obtaining quasi-optimal solutions for the majority of the instances, by employing the strategy of artificially reducing the number of cutting patterns and by exploiting them in a heuristic framework.

Pattern-based ILP models for the one-dimensional cutting stock problem with setup cost

Pattern-based ILP models for the one-dimensional cutting stock problem with setup cost

A BI-OBJECTIVE MULTIPERIOD ONE-DIMENSIONAL CUTTING STOCK PROBLEM

In this paper, we investigate the bi-objective multiperiod one-dimensional cutting stock problem that seeks to minimize the cost of production associated with the total length of cut objects (waste) and the inventory costs related to objects and items. A mathematical model is presented and heuristically solved by a column generation method. Computational tests were performed using the Weighted Sum method, the ε-Constraint method and a variation of the Benson method. The Pearson correlation coefficient was calculated in order to investigate the trade-off between the conflicting objectives of the problem. The results confirmed a strong negative correlation between the objective functions of the problem. All the applied scalar methods were able to find multiple efficient solutions for the problem in a reasonable computational time; however, the ε-Constraint and the modified Benson methods performed better.

An exact algorithm for an identical parallel additive machine scheduling problem with multiple processing alternatives

This paper develops an exact algorithm for the identical parallel additive machine scheduling problem by considering multiple processing alternatives to minimise the makespan. This research is motivated from an idea of elevating flexibility of a manufacturing system by using additive machines, such as 3D printers. It becomes possible to produce a job in a different form; a job can be printed in a complete form or in separate parts. This problem is defined as a bi-level optimisation model in which its upper level problem is to determine a proper processing alternative for each product, and its lower level problem is to assign the parts that should be produced to the additive machines. An exact algorithm, which consists of the linear programming relaxation of a one-dimensional cutting stock problem, a branch-and-price algorithm, and a rescheduling algorithm, is proposed to find an optimal solution of the problem. The experimental results show that the computational time of the algorithm outperforms a commercial solver (CPLEX). By examining how the parts are comprised when the processing alternatives are optimally selected, some useful insights are derived for designing processing alternatives of products.

A Bi-Integrated Model for coupling lot-sizing and cutting-stock problems

The lot-sizing and cutting-stock problems arise, in general, in manufacturing industries as subsequent processes. Many authors have addressed these problems separately; however, some studies have considered integrated approaches to these problems in recent practical applications. This paper aims to introduce a new model for integrated lot-sizing, one-dimensional cutting-stock and two-dimensional cutting-stock problems. This integrated model can be used to address the papermaking process as a three-phase subsequent process, which starts from the production of jumbos and ends with the paper sheets used in daily life. The first phase of the process is modeled according to a lot-sizing problem, where the quantities of jumbos are determined to meet the demand of the entire planning horizon. The second phase is modeled as a one-dimensional cutting-stock problem, where the jumbos are cut into smaller reels. The reels are used as raw material for the third phase of the process, where they are cut again into sheets employing a two-dimensional cutting-stock problem. A heuristic algorithm that uses the column generation and relax-and-fix techniques is applied to evaluate the proposed model from a series of experiments. Gains of up 16.5% were observed compared to the solution obtained with no integration among the production phases. Additional analyses of the model behavior under multiple scenarios resulted in significant findings, as the increasing costs’ reduction as more product types are considered and the use of more efficient cutting patterns resulting in reduced material waste.

A residual recombination heuristic for one-dimensional cutting stock problems

A residual recombination heuristic for one-dimensional cutting stock problems

An efficient mathematical model for solving one-dimensional cutting stock problem using sustainable trim

The cutting process is an important stage of the industries which are dealing with cutting of small pieces from large items in such a way so that the wastage should be minimum. In this study, we present an effective model for solving one-dimensional cutting stock problem (1D-CSP) using sustainable trim based on Cesàro means of order λ (λ is real >−1), with the provision of cutting at most two order lengths at a time, which is acceptable in many practical cases. Additionally, we present the comparison of the model with Residual Greedy Rounding (RGR) and CUT. It is shown that increased sustainable trim decreases the total trim loss by providing greater variety of stock lengths, which can be effectively used in future orders.

Modified Greedy Heuristic for the one-dimensional cutting stock problem

One can get an integer solution for the one-dimensional cutting stock problem by means of the constructive or residual heuristic. In this work we propose a change in the Constructive Greedy Heuristic that consists in building a cutting pattern by sorting in descending order the items of pair or odd length, priority being given to those which appear more frequently in the problem, cut from objects in stock with pair or odd length respectively, with the aim of minimizing the quantity of cut objects. Computing tests that were carried out showed the efficiency of the proposed heuristic when compared with other methods in the literature that generate an integer solution for the problem and will be presented at the end of this work.

Solving the one-dimensional cutting stock problem under discrete, uncertain, time-varying demands using a hybrid of special-purpose Benders' decomposition and column generation

Solving the one-dimensional cutting stock problem under discrete, uncertain, time-varying demands using a hybrid of special-purpose Benders' decomposition and column generation

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.

인공신경망을 활용한 선형재료절단문제의 휴리스틱 기법 선택

인공신경망을 활용한 선형재료절단문제의 휴리스틱 기법 선택

Meta-heuristics for the one-dimensional cutting stock problem with usable leftover

This work considers the one-dimensional cutting stock problem in which the non-used material in the cutting patterns may be used in the future, if large enough. We show that a multiobjective criteria to classify the solutions could be more accurate than previous classifications attempts, also we give a heuristic algorithm and two meta-heuristic approaches to the problem and we use them to solve practical and randomly generated instances from the literature. The results obtained by the computational experiments are quite good for all the tested instances.