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.

The modified branch and bound algorithm and dotted board model for triangular shape items

Cutting Stock Problem (CSP) is a problem of cutting stocks into items with certain cutting rules. This study used the data where the rectangular stocks were cut into triangular shape items with various order sizes. This study used the Modified Branch and Bound Algorithm (MBBA) to determine the optimal cutting pattern then formulated it into a Dotted Board model. Based on the results, it showed that the MBBA produced three optimal cutting patterns, which used 6 times, 8 times, and 4 times respectively to fulfill the consumer demand. Then the cutting patterns were formulated into the Dotted Board model. Minimum trim loss of the three models are 1,774 cm 2, 1,720 cm 2dan 980 cm 2.

The cutting stock problem with multiple manufacturing modes applied to a construction industry

This paper addresses the problem of multiple manufacturing modes integrated into the cutting stock problem, based on a real-life application of the concrete pole manufacturing. The main aim is to propose, formulate and test this integrated problem, which can be applied to the construction industry and other contexts of multiple manufacturing modes with cutting processes. The motivation for this proposal is the construction industry, in which reinforced concrete structures can be reinforced by various combinations of one-dimensional steel bars of varying thicknesses and lengths. An integer programming mathematical formulation is proposed aiming to minimise the total cost and to meet a demand of final products with different possible configurations. A column generation procedure is used as the solution method together with a heuristic procedure to find an integer solution. Computational results were performed with practical instances in order to assess the value of the approach and with a set of random generated instances in order to explore the influence of parameters on the results. Some managerial insights are presented.

Integrated lot-sizing and cutting stock problem applied to the mattress industry

In many productive processes, two important problems arise in the production planning: the lot-sizing problem and the cutting stock problem. Generally, companies deal with these problems separately but, by considering them in an integrated way, better results can be obtained. In this paper, the integrated lot-sizing and three-dimensional cutting stock problem applied to the mattress industry is investigated, aiming at reducing costs and waste. A mathematical model of mixed integer programming was proposed and solved with an optimisation package. Computational tests based on data collected at a mattress factory were carried out, allowing the comparison of the solutions proposed by the model and the solutions adopted by the factory. Additional tests were performed with random data in order to evaluate the behaviour of the model for different cases. The results indicate that the model performs well, reducing the objective function costs for different data sets. Based on the results, some interesting options can be explored by the industry; for example, by increasing the number of cutting patterns up to a certain level, the number of possible combinations for cutting is increased, resulting in better use of the material and a consequent reduction in costs.

Linear Integer Optimization Model for Two-Stage Guillotine Cutting Stock Problem Using Branch and Bound Method in the Garment Industry

This paper discusses the Two-Stage Guillotine Cutting Stock Problem (2GCSP) in the garment industry, namely how to determine the two-stage guillotine pattern that is used to cut fabric stocks into several certain size t-shirt materials that are produced based on the demand for each size of the shirt. 2GCSP is modeled in the form of Linear Integer Optimization and finding solutions using the Branch and Bound method. In this paper also presented a Graphical User Interface with Maple software as an interactive tool to find the best fabric stock cutting patterns. The results show that the optimal solution can be determined by solving numerically using the Branch and Bound method and Maple optimization packages. The solution is shown with an illustration of the pattern and the amount of fabric cut based on the pattern.

3-Phase Matheuristic Model in Two-Dimensional Cutting Stock Problem of Triangular Shape Items

Cutting Stock Problem (CSP) is a problem of cutting stocks with certain cutting rules. This study used the data of rectangular stocks, which cut into triangular shape items with various order sizes. The Modified Branch and Bound Algorithm (MBBA) was used to determine the optimum cutting pattern then formulated it into the 3-Phase Matheuristic model which consisted of constructive phase, improvement phase, and compaction phase. Based on the results, it showed that the MBBA produces three optimum cutting patterns, which was used six times, eight times, and four times respectively to fulfill the consumer demand. Then the cutting patterns were formulated into the 3-Phase Matheuristic model whereas the optimum solution was the minimum trim loss for the first, second and third patterns.

A column generation on two-dimensional cutting stock problem with fixed-size usable leftover and multiple stock sizes

We investigate a two-dimensional two-stage cutting stock problem (2D-2CP) with multiple stock sizes when fixed-size usable leftovers are considered. A fixed-size usable leftover is an object with the predefined size that could be used later. Fixed-size usable leftovers can reduce the waste area and therefore can help construct better cutting patterns. A mathematical model is proposed in a set-partitioning form where the sub-problems corresponding to two-dimensional knapsack problem (2DKP) with fixed-size usable leftovers are generated for optimality testing. A column generation technique is applied in an attempt to find a solution that minimises a total waste. Moreover, two different practical integer solution finding strategies are proposed. The algorithms are tested on a real-world dataset using various initial solution patterns and analysed. Additionally, the effect of the usable leftovers is investigated when compared with the prior literature that used the same dataset on the 2D-2CP problem without considering usable leftovers.

Automating production planning and control in pallet manufacturing – A case study

In this work, we investigate and formalize the production planning for a highly automated pallet production and provide a solution approach. According to the final pallet to be produced, in pallet production boards have to be cut and stacked in order to fulfil the demand at the assembly line. The overall production problem primarily consists of a cutting stock problem with a constraining open stack problem. They are mathematically modelled and solved using a heuristic approach that aims at minimizing the waste of material. Various different approaches to generate the possible cutting patterns are introduced based on the chronology of orders. In a next step, a feasible production sequence of the cutting patterns is computed. Furthermore, the number of generated patterns for the cutting stock problem and its impact on the feasibility of the production sequence as well as on the solution quality is investigated.

A column generation on 2-dimensional cutting stock problem with fixed size usable leftover and multiple stock sizes

We investigate a two-dimensional two-stage cutting stock problem (2D-2CP) with multiple stock sizes when fixed-size usable leftovers are considered. A fixed-size usable leftover is an object with the predefined size that could be used later. Fixed-size usable leftovers can reduce the waste area and therefore can help construct better cutting patterns. A mathematical model is proposed in a set-partitioning form where the sub-problems corresponding to two-dimensional knapsack problem (2DKP) with fixed-size usable leftovers are generated for optimality testing. A column generation technique is applied in an attempt to find a solution that minimises a total waste. Moreover, two different practical integer solution finding strategies are proposed. The algorithms are tested on a real-world dataset using various initial solution patterns and analysed. Additionally, the effect of the usable leftovers is investigated when compared with the prior literature that used the same dataset on the 2D-2CP problem without considering usable leftovers.

Column-Randomized Linear Programs: Performance Guarantees and Applications

We propose a randomized method for solving linear programs with a large number of columns but a relatively small number of constraints. Since enumerating all the columns is usually unrealistic, such linear programs are commonly solved by column generation, which is often still computationally challenging due to the intractability of the subproblem in many applications. Instead of iteratively introducing one column at a time as in column generation, our proposed method involves sampling a collection of columns according to a user-specified randomization scheme and solving the linear program consisting of the sampled columns. While similar methods for solving large-scale linear programs by sampling columns (or, equivalently, sampling constraints in the dual) have been proposed in the literature, in this paper we derive an upper bound on the optimality gap that holds with high probability and converges with rate $1/\sqrt{K}$, where $K$ is the number of sampled columns, to the value of a linear program related to the sampling distribution. To the best of our knowledge, this is the first paper addressing the convergence of the optimality gap for sampling columns/constraints in generic linear programs without additional assumptions on the problem structure and sampling distribution. We further apply the proposed method to various applications, such as linear programs with totally unimodular constraints, Markov decision processes, covering problems and packing problems, and derive problem-specific performance guarantees. We also generalize the method to the case that the sampled columns may not be statistically independent. Finally, we numerically demonstrate the effectiveness of the proposed method in the cutting-stock problem and in nonparametric choice model estimation.

Constructing an Instance of the Cutting Stock Problem of Minimum Size Which Does Not Possess the Integer Round-Up Property

Constructing an Instance of the Cutting Stock Problem of Minimum Size Which Does Not Possess the Integer Round-Up Property

Trim Loss Optimization in Paper Production Using Reinforcement Artificial Bee Colony

In paper production, a jumbo reel is cut into multiple intermediate rolls, and each intermediate roll is then sheeted as finished goods. This problem is called a cutting stock problem and is proven to be NP-hard. The objective is to minimize material waste or trim loss from all the cuttings. In the case that any intermediate roll is not entirely used for its associated order, the intermediate roll itself could turn to be a dead stock. We use the concept of universal sizes of intermediate rolls to eliminate the dead stock. A pre-defined number of universal sizes of intermediate rolls is to be used to serve all the orders. The problem is solved using Reinforcement Artificial Bee Colony algorithm with Integer Linear Programming subroutine. This proposed approach is then tested with a set of 1,055 orders and 127 different sizes of sheet papers from a paper manufacturer. The results reveal that our method outperforms other algorithms. Our method offers the total trim loss of 3.51%, compared to the trim loss reported by the industry of at least 5%. This approach not only reduces the number of partially cut rolls, but also decreases the number of the jumbo reels needed to serve all the orders. Therefore, both the inventory cost and material cost can be saved.

Assessment of Stock Length for One Dimensional Cutting Stock Problem (1d-Csp) with Sustainable Trim Loss

Assessment of Stock Length for One Dimensional Cutting Stock Problem (1d-Csp) with Sustainable Trim Loss

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 Controlled Stability Genetic Algorithm With the New BLF2G Guillotine Placement Heuristic for the Orthogonal Cutting-Stock Problem

The orthogonal cutting-stock problem tries to place a given set of items in a minimum number of identically sized bins. Combining the new BLF2G heuristic with an advanced genetic algorithm can help solve this problem with the guillotine constraint. According to the item order, the BLF2G heuristic creates a direct placement of items in bins to give a cutting format. The genetic algorithm exploits the search space to find the supposed optimal item order. Other methods try to guide the evolutionary process. A new enhancement guides the evolutionary process, enriching the population via qualified individuals, without disturbing the genetic phase. The evolution of the GA process is controlled, and when no improvements after some number of iterations are observed, a qualified individual is injected to the population to avoid premature convergence to a local optimum. A generated set of order-based individuals enriches the evolutionary process with qualified chromosomes. The proposed method is compared with other heuristics and metaheuristics found in the literature on existing data sets.

A matheuristic algorithm for the one-dimensional cutting stock and scheduling problem with heterogeneous orders

Cutting stock and bin packing problems are common in several industrial sectors, such as paper, glass, furniture, steel industry, construction, transportation, among others. The classical cutting stock problem (CSP) ignores the production planning and scheduling of multiple customer orders. Nevertheless, in real industrial settings customer orders have to be planned and scheduled over time so as to meet demand and required due dates. We propose an integer linear programming model for the one-dimensional cutting stock and scheduling problem with heterogeneous orders. As this problem is a generalization of the classical single-period one-dimensional CSP, which is known to be NP-hard, it is difficult to solve real-sized instances to optimality. Thus, we propose a novel matheuristic algorithm based on a fix-and-optimize strategy hybridized with a random local search. The proposed matheuristic was tested on a set of 160 synthetic problem instances based on a real-world problem and compared with CPLEX solver. In larger instances, the proposed matheuristic performed better than CPLEX, with average relative percentage deviation (RPD) regarding objective values as high as 72%. On the other hand, in small instances CPLEX showed a marginal advantage, with best average RPD of 18% with relation to the matheuristic. We also performed a paired t-test with significance level 0.05 and null hypothesis: no difference between the proposed matheuristic and CPLEX. In small test instances, the performance of the proposed matheuristic was statistically indistinguishable from CPLEX, while in larger instances the matheuristic outperformed CPLEX in most cases with $$p\,\text {value} < 0.05$$.

An exact scalarization method with multiple reference points for bi-objective integer linear optimization problems

This paper presents an exact scalarization method to solve bi-objective integer linear optimization problems. This method uses diverse reference points in the iterations, and it is free from any kind of a priori chosen weighting factors. In addition, two new adapted scalarization methods from literature and the modified Tchebycheff method are studied. Each one of them results in different ways to obtain the Pareto frontier. Computational experiments were performed with random real size instances of two special problems related to the manufacturing industry, which involve lot sizing and cutting stock problems. Extensive tests confirmed the very good performance of the new scalarization method with respect to the computational effort, the number of achieved solutions, the ability to achieve different solutions, and the spreading and spacing of solutions at the Pareto frontier.

Implementation of branch and cut method on n-sheet model in solving two dimensional cutting stock problem

The problem of cutting raw materials to fulfill the size of demand using certain cutting pattern is called Cutting Stock Problem (CSP). CSP aims to find the optimal cutting pattern with the minimum trim loss. In this research, all possible cutting patterns were generated by pattern generation algorithm where the materials were cut based on the length on the first stage and the width on the rest stage. All patterns were modeled to N-Sheet model and solved using Branch and Cut method. Based on the results, it showed that the N-Sheet model ensured as minimum a possible trim loss to fulfill the demands.

Gilmore and gomory model on two dimensional multiple stock size cutting stock problem

Two-dimensional Multiple Stock Size Cutting Stock Problem (MSS2DCSP) is an optimization problem in determining the optimal cutting pattern of materials with length and width variables. The optimal cutting pattern is needed to minimize the number of materials used. This research implemented the pattern generation algorithm to generate Gilmore and Gomory models on every variation of material size. The first stage generated the pattern based on the length, followed by a pattern generated based on the width on the next stage. Then, Gilmore and Gomory models were generated according to the size of the material, where the Gilmore and Gomory model constraints ensured that strips cut in the first stage cutting were used in the second stage cutting and met all product requirements. The Branch and Bound method were used to determine the minimum size of the material sheets needed.

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.