One-dimensional Cutting Stock Problem Research Articles

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.

Solving the One-dimensional Cutting Stock Problem under Discrete, Uncertain, Time-varying Demands using a Hybrid of Special-purpose Benders' Decomposition and Column Generation

This study shows the integration of Benders' decomposition, column generation, and a special-purpose algorithm for solving the one-dimensional cutting stock problem under discrete, uncertain, time-varying demands. The results show that there is a linear relationship between the processing time and the number of scenarios involved in the raw material cutting patterns. Moreover, the algorithm results are not different from using column generation. During the search for production patterns, the quality of the solutions relies on the convergence tolerance. As the tolerance approaches zero, the solutions become near-optimal, but the computational time increases substantially. In addition, the experimental results indicate that the proposed method is suitable for large-scale problems.

A Least-Loss Algorithm for a Bi-Objective One-Dimensional Cutting-Stock Problem

This article presents a new model and an efficient solution algorithm for a bi-objective one-dimensional cutting-stock problem. In the cutting-stock—or trim-loss—problem, customer orders of different smaller item sizes are satisfied by cutting a number of larger standard-size objects. After cutting larger objects to satisfy orders for smaller items, the remaining parts are considered as useless or wasted material, which is called “trim-loss.” The two objectives of the proposed model, in the order of priority, are to minimize the total trim loss, and the number of partially cut large objects. To produce near-optimum solutions, a two-stage least-loss algorithm (LLA) is used to determine the combinations of small item sizes that minimize the trim loss quantity. Solving a real-life industrial problem as well as several benchmark problems from the literature, the algorithm demonstrated considerable effectiveness in terms of both objectives, in addition to high computational efficiency.

The Effect of Welding on the One-Dimensional Cutting-Stock Problem: The Case of Fixed Firefighting Systems in the Construction Industry

Safety is paramount in the construction industry and the fixed sprinkler and water spray systems used in firefighting involve networks of pipes of various lengths. Manufacturers of such fixed firefighting systems need to either cut the existing stocks to length—a (one-dimensional) cutting-stock problem—or lengthen the existing stocks or leftover segments through welding, a (one-dimensional) cutting-stock problem with welding. Best industry practice safety requirements allow only one weld per length of pipe. The case of a Hungarian manufacturer of fixed firefighting systems motivates this article, which argues that the cutting-stock problem with welding (with single- or multiple-size stocks) may be converted to an equivalent cutting-stock problem (with multiple-size stocks). Readily available algorithms and software may then be used to generate an optimal cutting plan for the equivalent cutting-stock problem. Subject to certain restrictions, the optimal cutting plan for the equivalent cutting-stock problem may then be converted to cutting patterns for the original cutting-stock problem with welding.

Simulation of the stochastic one-dimensional cutting stock problem to minimize the total inventory cost

This paper utilizes Monte Carlo simulation to identify the distribution of the optimal values of the one-dimensional cutting stock problem to help decide the stock order quantity before the demand mix becomes known. Due to the lead time when ordering the raw material, it is not possible to wait and see what the demand mix is. Thus, a “here and now” approach of stochastic programming is used to hedge against future demand uncertainty. The LINGO software is used as a subroutine of a MATLAB code to set up and solve the model in each iteration by drawing random demand values. Once the raw material order is placed and later the demand mix becomes known, decisions concerning the cutting patterns and shortage or overage amounts are determined with a second mathematical model. This paper presents a novel solution approach (using the optimizer software LINGO as a subroutine in a MATLAB simulation code) to a classic problem encountered in production planning by proposing a new method to account for demand randomness. Two raw material ordering rules (mean plus ½ standard deviation and mean plus 1 standard deviation) were considered and implemented using three random cut demand cases (exponential, uniform, and normal). For the exponentially distributed cut length demands, both rules resulted in ordering an excessive number of rails suggesting a need for a different ordering rule. For the normally distributed cut length demands, in some cases shortages may result for either of the two rules. For uniformly distributed cut length demands both ordering rules were effective in meeting cut length demands.

Modification of Haessler’s sequential heuristic procedure for the one-dimensional cutting stock problem with setup cost

Abstract Paper aims We propose a modified Sequential Heuristic Procedure (MSHP) to reduce the cutting waste and number of setups for the One-Dimensional Cutting Stock Problem with Setup Cost. Originality This heuristic modifies Haessler’s sequential heuristic procedure (1975) by adapting the Integer Bounded Knapsack Problem to generate cutting patterns, instead of the original lexicographic search employed. The solution strategy is to generate different cutting plans using MSHP, and then to use an integer programming model to seek even better results. Research method It is a axiomatic research, ordinary in studies of Operational Research. Main findings In the computational experiments, we demonstrate the effectiveness of the algorithm with two sets of benchmark instances by comparing it with other approaches, and obtaining better solutions for some scenarios. Implications for theory and practice The approach is suitable for practitioners from different industrial settings due to its easily coding and possible adaptation for problem extensions.

Large proper gaps in bin packing and dual bin packing problems

We consider the one-dimensional skiving stock problem, also known as the dual bin packing problem, with the aim of maximizing the best known dual and proper dual gaps. We apply the methods of computational search of large gaps initially developed for the one-dimensional cutting stock problem, which is related to the bin packing problem. The best known dual gap is raised from 1.0476 to 1.1795. The proper dual gap is improved to 1.1319. We also apply a number of new heuristics developed for the skiving stock problem back to the cutting stock problem, raising the largest known proper gap from 1.0625 to 1.1.

Cutting stock problems with nondeterministic item lengths: a new approach to server consolidation

Based on an application in the field of server consolidation, we consider the one-dimensional cutting stock problem with nondeterministic item lengths. After a short introduction to the general topic we investigate the case of normally distributed item lengths in more detail. Within this framework, we present two lower bounds as well as two heuristics to obtain upper bounds, where the latter are either based on a related (ordinary) cutting stock problem or an adaptation of the first fit decreasing heuristic to the given stochastical context. For these approximation techniques, dominance relations are discussed, and theoretical performance results are stated. As a main contribution, we develop a characterization of feasible patterns by means of one linear and one quadratic inequality. Based on this, we derive two exact modeling approaches for the nondeterministic cutting stock problem, and provide results of numerical simulations.

Algorithms for the one-dimensional two-stage cutting stock problem

In this paper, we consider a two-stage extension of one-dimensional cutting stock problem which arises when technical requirements inhibit cutting large stock rolls to demanded widths of finished rolls directly. Therefore, demands on finished rolls are fulfilled through two subsequent cutting processes, in which rolls produced in the former are used as input for the latter, while the number of stock rolls used is minimized. We tackle the pattern-based formulation of this problem which typically has a very large number of columns and constraints. The special structure of this formulation induces both a column-wise and a row-wise increase when solved by column generation. We design an exact simultaneous column-and-row generation algorithm whose novel element is a row-generating subproblem that generates a set of columns and rows. For this subproblem, which is modeled as an unbounded knapsack problem, we propose three algorithms: implicit enumeration, column generation which renders the overall methodology nested column generation, and a hybrid algorithm. The latter two are integrated in a well-known knapsack algorithm which forges a novel branch-and-price algorithm for the row-generating subproblem. Extensive computational experiments are conducted, and performances of the three algorithms are compared.

Evaluation of procurement scenarios in one-dimensional cutting stock problem with a random demand mix

The one-dimensional cutting stock problem describes the problem of cutting standard length stock material into various specified sizes while minimizing the material wasted (the remnant or drop as manufacturing terms). This computationally complex optimization problem has many manufacturing applications. One-dimensional cutting stock problems arise in many domains such as metal, paper, textile, and wood. To solve it, the problem is formulated as an integer linear model first, and then solved using a common optimizer software. This paper revisits the stochastic version of the problem and proposes a priority-based goal programming approach. Monte Carlo simulation is used to simulate several likely inventory order policies to minimize the total number of shortages, overages, and the number of stocks carried in inventory.

A comparative study of exact methods for the bi-objective integer one-dimensional cutting stock problem

This article addresses the bi-objective integer cutting stock problem in one dimension. This problem has great importance and use in various industries, including steel mills. The bi-objective model considered aims to minimize the frequency of cutting patterns to meet the minimum demand for each item requested and the number of different cutting patterns to be used, being these conflicting objectives. In this study, we apply three classic methods of scalarization: weighted sum, Chebyshev metric and ε-Constraint. This last method is developed to obtain all of the efficient solutions. Also, we propose and test a fourth method, modifying the Chebyshev metric, without the insertion of additional variables in the formulation of the sub-problems. The computational experiments with randomly generated real size instances illustrate and attest the suitability of the bi-objective model for this problem, as well as the applicability of all the proposed exact algorithms, thus showing that they are useful tools for decision makers in this area. Moreover, the modified metric method improved with respect to the performance of the classical version in the tests.

Effect of demand variations on steel bars cutting loss

ABSTRACTThe stock cutting process of construction steel bars causes trim loss. The demand assortment can vary considerably from job to job depending on the designs of building components and the progress of the projects. This research employs one-dimensional cutting stock problem model and conducts three sets of experiment on the variations of three parameters in order to discover the relationships between the demand variation and the cutting loss. The combination of the demanded lengths and numbers are altered to simulate many different decent jobs. Then, statistical techniques are applied to analyse the minimum cutting loss resulted from the tests. The results show that the less uniformity of the demand assortment produces more trim loss of the job. A highly unbalanced proportion of long demanded lengths also increases loss. The outcomes of this research are strategies to arrange effective cutting jobs, which will reduce the trim loss along the project timeline and consequently save the project cost.

A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems

We consider the one-dimensional cutting stock problem which consists in determining the minimum number of given large stock rolls that has to be cut to satisfy the demands of certain smaller item lengths. Besides the standard pattern-based approach of Gilmore and Gomory, containing an exponential number of variables, several pseudo-polynomial formulations were proposed in the last decades. Much research has dealt with arcflow models, their relationship to the standard model, and possible reduction methods, whereas the one-cut approach has not attracted that much scientific interest yet.In this paper, we aim to compare both alternative formulations from a theoretical and numerical point of view. As a theoretical main contribution, we constructively prove the equivalence of the continuous relaxations of the one-cut model, the arcflow model, and the pattern-based model. In particular, the relationship between the one-cut model and the pattern-based model has remained an open question since the one-cut approach was proposed.Moreover, in order to make a computational comparison of the arcflow model and the one-cut model, we present how reduction methods, partly originating from arcflow considerations, can successfully be transferred to the one-cut context. Furthermore, we derive relations between the numbers of variables and constraints in both models, and investigate their influences in numerical simulations.

Decomposition methods for the lot-sizing and cutting-stock problems in paper industries

• We propose new models for cutting-stock problem integrated with lot-sizing problem. • We combine column generation method and an adaptive large neighborhood search. • We analyze real-world data instances. We investigate the one-dimensional cutting-stock problem integrated with the lot-sizing problem in the context of paper industries. The production process in paper mill industries consists of producing raw materials characterized by rolls of paper and cutting them into smaller rolls according to customer requirements. Typically, both problems are dealt with in sequence, but if the decisions concerning the cutting patterns and the production of rolls are made together, it can result in better resource management. We investigate Dantzig–Wolfe decompositions and develop column generation techniques to obtain upper and lower bounds for the integrated problem. First, we analyze the classical column generation method for the cutting-stock problem embedded in the integrated problem. Second, we propose the machine decomposition that is compared with the classical period decomposition for the lot-sizing problem. The machine decomposition model and the period decomposition model provide the same lower bound, which is recognized as being better than the linear relaxation of the classical lot-sizing model. To obtain feasible solutions, a rounding heuristic is applied after the column generation method. In addition, we propose a method that combines an adaptive large neighborhood search and column generation method, which is performed on the machine decomposition model. We carried out computational experiments on instances from the literature and on instances adapted from real-world data. The rounding heuristic applied to the first column generation method and the adaptive large neighborhood search combined with the column generation method are efficient and competitive.

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.

One-Dimensional Cutting Stock Problem with Cartesian Coordinate Points

The cutting stock problem is used in many industrial processes and recently has been considered as one of the most important research topics. It is basically describes in two ways, One –dimensional and Two-dimensional Cutting Stock Problems (CSP). An optimum cutting stock problem can be defined as cutting a main sheet into smaller pieces while minimizing the total wastage of the raw material or maximizing overall profit obtained by cutting smaller pieces from the main sheet. In this study, One-dimensional CSP is discussed. Modified Brach and Bound algorithm for One-dimensional cutting stock problem is coded and programmed in the Matlab programming environment to generate feasible cutting patterns. At the same time, Cartesian coordinate points are derived from the developed algorithm. In our approach, the case study is pertained to the real data used at L.H. Chandrasekara & Brothers (Pvt Ltd) in Sri Lanka for its production.

Spreadsheet solutions of the one-dimensional cutting stock problem

The cutting stock problem encompasses cutting parts available in stock, which are called objects, to produce in specified quantities smaller pieces which are called items and optimizing an objective function. The cutting stock problem is common for many industrial processes – steel, paper tube and construction industries. One of the most important phases in solution of the one-dimensional cutting stock problem is a pattern generating procedure which can be performed in a spreadsheet.

A vulcanising decision planning as a particular one-dimensional cutting stock problem with limited part-related tooling in make-to-order industrial environments

A specialised case of the classical one-dimensional cutting stock problem (1D-CSP) with six main additional features is adapted in this paper to model and solve planning unit operations with limited resources in the make-to-order industrial environment, generating a new decision-making problem. The objective is to satisfy demand using the minimum number of manufacturing cycles. Although there is a large number of applications, this paper proposes a decision model directed at the vulcanising operation during the manufacturing of rubber curved hoses in the automotive industry. Because each vulcanising cycle (VC) uses a considerable amount of resources, the need to minimise the total number of VCs is of importance due to its direct relationship with lead time and productivity. An integer-programming model based on a network optimisation formulation is proposed to solve the problem to optimality and allows for construction of the mathematical model. In addition, due to the limited capacity of computers to solve large instances, a heuristic is developed to obtain near-optimal solutions. Scheduling issues are found in the optimal solution of the integer-programming model, but the heuristic overcomes such obstacles. Numerical experiments demonstrate the efficiency of the heuristic.

Mathematical models and a heuristic method for the multiperiod one-dimensional cutting stock problem

The multiperiod cutting stock problem arises in the production planning and programming of many industries that have the cutting process as an important stage. Ordered items are required in different periods of a finite planning horizon. It is possible to bring forward or not the production of items. Unused inventory in a certain period becomes available for the next period, all together with new inventory which may come to be acquired in the market. Based on mixed integer optimization models from the literature, extensions are proposed to deal with the multiperiod case and a residual heuristic is used. Computational experiments showed that effective gains can be obtained when comparing multiperiod models with the lot for lot solution, which is typically used in practice. Most of the instances are solved satisfactorily with a high performance optimization package and the heuristic method is used for solving the hard instances.

Minimal proper non-IRUP instances of the one-dimensional cutting stock problem

We consider the well-known one dimensional cutting stock problem (1CSP). Based on the pattern structure of the classical ILP formulation of Gilmore and Gomory, we can decompose the infinite set of 1CSP instances, with a fixed number n of demanded pieces, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property (IRUP) we use the proper LP relaxation of the Gilmore and Gomory model that allows us to consider the 1CSP as the bin packing problem (BPP). We computationally show that all 1CSP instances with n≤9 have the proper IRUP, while we give examples of proper non-IRUP instances with n=10 and proper gap 1. Proper gaps larger than 1 occur for n≥11. The largest known proper gap is raised from 1.003 to 1.0625. The used algorithmic approaches are based on exhaustive enumeration and integer linear programming. Additionally we give some theoretical bounds showing that all 1CSP instances with some specific parameters have the proper IRUP.