Two-dimensional Cutting Stock Problem Research Articles

Heuristics for the two-dimensional cutting stock problem with usable leftover

Utilization of residue is a challenge in engineering practice, because unreasonable cutting causes excess materials wasted and increases the production cost. This work considers the residual two-dimensional cutting stock problem with usable leftover in which unused parts of cutting patterns can be used for future orders. We propose an algorithm that combines the iterative sequential value correction heuristic with the beam search heuristic, considering both the accumulation and the reusability of leftovers to reduce the material consumption. Cutting plans are constructed iteratively and the best one are chosen as the solution. Cutting patterns in the cutting plan are generated sequentially by recursive techniques, and potentially usable leftover are accumulated by beam search heuristic. Item values are corrected after each pattern to diversify cutting plans. Three sets of simulations under different number of periods, over medium and large instances from the literature, are used to demonstrate the effectiveness of the heuristics. Computational results show that the algorithm provides better solutions, which can save a considerable amount of plate in a long-term production period. The utilization of wastages can save a considerable amount of stock plate and contract the production cost of enterprises in the long-term production period.

Two-dimensional cutting stock problem with multiple stock sizes

Two-dimensional cutting stock problem with multiple stock sizes

The two-dimensional cutting stock problem with usable leftovers and uncertainty in demand

When dealing with cutting problems, the generation of usable leftovers proved to be a good strategy for decreasing material waste. Focusing on practical applications, the main challenge in the implementation of this strategy is planning the cutting process to produce leftovers with a high probability of future use without complete information about the demand for any ordered items. We addressed the two-dimensional cutting stock with usable leftovers and uncertainty in demand, a complex and relevant problem recurring in companies due to the unpredictable occurrence of customer orders. To deal with this problem, a two-stage formulation that approximates the uncertain demand by a finite set of possible scenarios was proposed. Also, we proposed a matheuristic to support decision-makers by providing good-quality solutions in reduced time. The results obtained from the computational experiments using instances from the literature allowed us to verify the matheuristic performance, demonstrating that it can be an efficient tool if applied to real-life situations.

Two dimensional guillotine cutting stock and scheduling problem in printing industry

We address a two-dimensional cutting stock problem combined with production scheduling that arises in paper printing industry where printing orders arrive daily, together with their specifications and desired delivery dates The planner has to decide the printing patterns, their quantities to be printed, and their production dates. The printing pattern should comply with guillotine cuts having three stages. The objective is to minimize the total printing cost composed of paper and plate costs. For plates containing orders with print on both sides, two printing options exist. With the printing options and the option of placing copies of orders in only one pattern, the problem differs from previous studies and becomes challenging to be solved by a mathematical model. We develop a nonlinear integer programming (NIP) model to obtain exact solutions for small-sized instances and an efficient Genetic Algorithm (GA) to solve real-sized problems. We design packing routines that generate balanced printing patterns by putting multiple copies of items in one pattern. The GA also takes advantage of complex heuristics for production scheduling and pattern reduction. We evaluate the performance of the GA against the NIP model. Computational experiments on large-sized real-world data demonstrate the efficiency of the GA.

The two-dimensional cutting stock problem with usable leftovers: mathematical modelling and heuristic approaches

The two-dimensional cutting stock problem with usable leftovers: mathematical modelling and heuristic approaches

Contribution to Solving a Two-Dimensional Cutting Stock Problem with Two Objectives

Contribution to Solving a Two-Dimensional Cutting Stock Problem with Two Objectives

Models for two- and three-stage two-dimensional cutting stock problems with a limited number of open stacks

We address three variants of the two-dimensional cutting stock problem in which the guillotine cutting of large objects produces a set of demanded items. The characteristics of the variants are the rectangular shape of the objects and items; the number of two or three orthogonal guillotine stages; and a sequencing constraint that limits the number of open stacks to a scalar associated with the number of automatic compartments or available space near the cutting machine. These problems arise in manufacturing environments that seek minimum waste solutions with limited levels of work-in-process. Despite their practical relevance, we are not aware of mathematical models for them. In this paper, we propose an integer linear programming (ILP) formulation for each of these variants based on modelling strategies for the two-dimensional guillotine cutting stock problem and the minimisation of the open stacks problem. The first two variants deal with exact and non-exact 2-stage patterns, and the third with a specific type of 3-stage patterns. Using a general-purpose ILP solver, we performed computational experiments to evaluate these approaches with benchmark instances. The results show that several equivalent solutions of the cutting problem allow obtaining satisfactory solutions with a reduced number of open stacks.

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 Study of Ising Formulations for Minimizing Setup Cost in the Two-Dimensional Cutting Stock Problem

We proposed the method that translates the two-dimensional CSP for minimizing the number of cuts to the Ising model. After that, we conducted computer experiments of the proposed model using the benchmark problem. From the above, the following results are obtained. (1) The proposed Ising model adequately represents the target problem. (2) Acceptance rates were as low as 0.2% to 9.8% and from 21.8% to 49.4%. (3) Error rates from optimal solution were as broad as 0% to 25.9%. For future work, we propose the following changes: (1) Improve the Hamiltonian for constraints. (2) Improve the proposed model to adjust more complex two-dimensional CSP and reduce the number of spins when it deals with large materials and components. (3) Conduct experiments using a quantum annealer.

Implementation of arc flow model incapacitated multi-period cutting stock problem with the pattern set up cost to minimize the trim loss

Two-dimensional Cutting Stock Problem (CSP) is a problem in cutting raw materials where the trim loss is on two sides, namely the width and length sides. This research implemented the Arc Flow model incapacitated multi-period with set up cost to minimize the trim loss in the cutting of paper. The cutting patterns were generated by the Pattern Generation (PG) algorithm. Furthermore, it was formulated to a linear Arc Flow model where the constraints indicated the number of demands per item. The solution of the model was completed using the LINGO 13.0 application. The optimal solution of the Arc Flow model showed that the quantity demands for the second and third types of items were fulfilled. The maximum amount of inventory contained in the second type of item for the second period was 132,517 sheets. Excess inventory will become a surplus. Based on the arc flow model solution, it turned out that no trim loss was produced, or in other words, trim loss is equal to zero.

Applying Material Flow Cost Accounting and Two-Dimensional, Irregularly Shaped Cutting Stock Problems in the Lingerie Manufacturing Industry

The textiles and garment industry plays an important role in Thailand’s economic growth, despite facing competition in product quality and rising production costs. Meeting diverse consumer needs and satisfaction has become increasingly difficult, as environmental issues become a major concern for firms internationally. Entrepreneurs require sophisticated strategic management techniques to maintain organizational productivity. Growing industries generate material losses, while negatively impacting the environment. Companies may account for their waste, but in reality, actual productivity is much lower, since hidden wastes are mostly unaccounted for and unquantified. A key barrier to reducing waste is that potential cost savings by revising waste management processes are not calculated. To solve this problem, material flow cost accounting (MFCA) was introduced to reduce negative product costs in a ladies’ lingerie company by identifying and evaluating the quantity and cost of concealed material waste. An effective meta-heuristic called the Two-Dimensional Cutting Stock Problem—Tabu Search algorithm (2DCSP-TS) was then proposed based on the idea of finding a layout that minimized a bin length. The multi-phase arrangement strategy embedded in it can obtain near-optimal conditions for solving realistic-sized problems. To illustrate the effectiveness of the proposed methods, numerical experimental results were compared with those of the current practice. From the numerical experiments, it was found that the proposed technique is an efficient method for reducing negative product costs.

Column Generation Model in Capacitated Multi-Periods Cutting Stock Problem with Pattern Set-Up Cost

Cutting Stock Problem (CSP) determines the cutting of stocks with standard length and width to meet the item’s demand. The optimal cutting pattern will minimize the usage of stocks and trim loss. This research implemented the pattern generation algorithm to form the Gilmore-Gomory and Column Generation model in two-dimensional CSP. The CSP in this research had three periods of cutting with different capacities in each period. The Column Generation model added the pattern set-up cost as the constraint. The Gilmore-Gomory model ensured that the first stage’s strips were used in the second stage and met the item’s demand. Based on the Column Generation model’s solution, the 1st period used the 2nd, 4th, and 5th patterns, the 2nd period used 4th and 5th patterns, and the 3rd period did not use any patterns. The first and second periods fulfilled all of the demands.

Solving two-dimensional cutting stock problem via a DNA computing algorithm

Two-dimensional cutting stock problem (TDCSP) is a well-known combinatorial optimization problem in which a given set of two-dimensional small pieces with different shapes should be cut from a given main board so that the demand of each small piece is satisfied and the total waste is minimized. Since TDCSP is an NP-complete problem, it is unsolvable in polynomial time on electronic computers. However, using the structure of DNA molecules, DNA computing algorithms are capable to solve NP-complete problems in polynomial time. In this paper, a DNA computing algorithm based on the sticker model is presented to find the optimal solution to TDCSP. It is proved that the time complexity of this algorithm on DNA computers is polynomial considering the number of small pieces and the length and width of the main board.

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.

An Innovative Genetic Algorithm for a Multi-Objective Optimization of Two-Dimensional Cutting-Stock Problem

ABSTRACTThis paper addressed an important variant of two-dimensional cutting stock problem. The objective was not only to minimize trim loss, as in traditional cutting stock problems, but rather to minimize the number of machine setups. This additional objective is crucial for the life of the machines and affects both the time and the cost of cutting operations. Since cutting stock problems are well known to be NP-hard, we proposed an approximate method to solve this problem in a reasonable time. This approach differs from the previous works by generating a front with many interesting solutions. By this way, the decision maker or production manager can choose the best one from the set based on other additional constraints. This approach combined a genetic algorithm with a linear programming model to estimate the optimal Pareto front of these two objectives. The effectiveness of this approach was evaluated through a set of instances collected from the literature. The experimental results for different-size problems show that this algorithm provides Pareto fronts very near to the optimal ones.

Two-dimensional cutting stock problem with sequence dependent setup times

Motivated by a firm in the technical textile industry, we study a two-dimensional cutting stock problem with sequence dependent setup times and permissible tolerances. We provide a sequential heuristic with feedback loop based on the approach of Gilmore and Gomory and formulate the sequencing problem as a mixed integer program. We derive a lower bound algorithm and demonstrate the near-optimal performance of our heuristic. Finally, we use real data to test our heuristic and illustrate its applicability to a problem of realistic size.

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.

An Application Traditional and Nontraditional Approach on Lock Industry for Optimization of Cutting Layout

Engineering becomes more advanced and the business in the industrial world becomes more competitive. The method of implementing optimization technique is essential. The nesting of two dimensional shapes for the press tool design is general optimization problem. In mass production industries; the small inefficiencies will have the huge wastage. This is also known as two-dimensional Cutting Stock Problem (CSP). The aim is to minimize wastage there by increasing the profit by means of arranging the parts into the master sheet. The parts are to be arranged in the sheet in a acceptable position. It needs a various algorithm for placing the parts into the sheets like Heuristic, Meta heuristic, Bottom left heuristics etc.

Modeling Two-Dimensional Guillotine Cutting Problems via Integer Programming

We propose a framework to model general guillotine restrictions in two-dimensional cutting problems formulated as mixed-integer linear programs (MIPs). The modeling framework requires a pseudopolynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state-of-the-art MIP solver, can tackle instances of challenging size. We mainly concentrate our analysis on the guillotine two-dimensional knapsack problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. We also show how the modeling of general guillotine cuts can be extended to other relevant problems such as the guillotine two-dimensional cutting stock problem and the guillotine strip packing problem (GSPP). Finally, we conclude the paper discussing an extensive set of computational experiments on G2KP and GSPP benchmark instances from the literature.