A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size

We consider a two-dimensional cutting stock problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as its subproblem the solution of a two-dimensional knapsack problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. [3].     

Fast heuristic for constrained homogenous T-shape cutting patterns

Homogenous T-shape (HTS) cutting patterns are welcomed when the two-phase process is used to produce rectangular pieces from the stock plate, where the plate is cut into homogenous strips at the first phase, and the strips are divided into pieces at the second phase. A heuristic is presented for generating constrained HTS patterns, where the objective is to maximize the pattern value that is equal to the total value of the included pieces, observing the upper bound constraint on the frequency of each piece type. The heuristic is based on dynamic programming and branch-and-bound techniques. It can yield solutions close to optimal with short computation time. By providing good initial solutions, the heuristic can greatly improve the time efficiency of an existing exact branch-and-bound algorithm.     

T-shape homogenous block patterns for the two-dimensional cutting problem

This paper presents an algorithm for unconstrained T-shape homogenous block cutting patterns of rectangular pieces. A vertical cut divides the stock sheet into two segments. Each segment consists of sections that have the same length and direction. A section contains a row of homogenous blocks. A homogenous block consists of homogenous strips of the same piece type. Each cut on the block produces just one strip. The directions of two strips cut successively from a block are either parallel or orthogonal. The algorithm uses a dynamic programming recursion to generate optimal blocks, solves knapsack problems to obtain the block layouts on the sections and the section layout on segments of various lengths, and optimally selects two segments to compose the cutting pattern. The computational results indicate that the algorithm is efficient in improving material usage, and the computation time is reasonable.     

2DCPackGen: A problem generator for two-dimensional rectangular cutting and packing problems

Cutting and packing problems have been extensively studied in the literature in recent decades, mainly due to their numerous real-world applications while at the same time exhibiting intrinsic computational complexity. However, a major limitation has been the lack of problem generators that can be widely and commonly used by all researchers in their computational experiments. In this paper, a problem generator for every type of two-dimensional rectangular cutting and packing problems is proposed. The problems are defined according to the recent typology for cutting and packing problems proposed by Wäscher, Haußner, and Schumann (2007) and the relevant problem parameters are identified. The proposed problem generator can significantly contribute to the quality of the computational experiments run with cutting and packing problems and therefore will help improve the quality of the papers published in this field.     

Simplest optimal cutting patterns for equal rectangles

This paper presents branch-and-bound algorithms that can guarantee the simplest optimal cutting patterns of equal rectangles. An existing linear algorithm determines the global upper bound exactly. The branching process ends when a branch of a lower bound equal to the global upper bound is found.     

A heuristic for the one-dimensional cutting stock problem with usable leftover

A heuristic algorithm for the one-dimensional cutting stock problem with usable leftover (residual length) is presented. The algorithm consists of two procedures. The first is a linear programming procedure that fulfills the major portion of the item demand. The second is a sequential heuristic procedure that fulfills the remaining portion of the item demand. The algorithm can balance the cost of the consumed bars, the profit from leftovers and the profit from shorter stocks reduction. The computational results show that the algorithm performs better than a recently published algorithm.     

The two-dimensional cutting stock problem revisited

In the strip packing problem (a standard version of the two-dimensional cutting stock problem), the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed. The k-stage Guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing. We present a complete analysis of the quality of k-stage Guillotine strip packings versus globally optimal packings: k=2 stages cannot guarantee any bounded asymptotic performance ratio. k=3 stages lead to asymptotic performance ratios arbitrarily close to 1.69103; this bound is tight. Finally, k=4 stages yield asymptotic performance ratios arbitrarily close to 1.Steve Seiden died in a tragic accident on June 11, 2002. This paper resulted from a number of email discussions between the authors in spring 2002.     

New upper bounds for the two-dimensional orthogonal non-guillotine cutting stock problem

The two-dimensional orthogonal non-guillotine cutting stockproblem (NGCP) appears in many industries (e.g. the wood andsteel industries) and consists of cutting a rectangular mastersurface into a number of rectangular pieces, each with a givensize and value. The pieces must be cut with their edges alwaysparallel to the edges of the master surface (orthogonal cuts).The objective is to maximize the total value of the pieces cut. New upper bounds on the optimal solution to the NGCP are described.The new bounding procedures are obtained by different relaxationsof a new mathematical formulation of the NGCP. Various proceduresfor strengthening the resulting upper bounds and reducing thesize of the original problem are discussed. The proposed newupper bounds have been experimentally evaluated on test problemsderived from the literature. Comparisons with previous boundingprocedures from the literature are given. The computationalresults indicate that these bounds are significantly betterthan the bounds proposed in the literature.     

Exact algorithm for generating two-segment cutting patterns of punched strips

Metal plates are often divided into items in two stages. First a guillotine shear cuts the plate into strips at the shearing stage, and then a stamping press punches out the items from the strips at the punching stage. This paper presents an algorithm for generating optimal two-segment cutting patterns of strips at the shearing stage. An orthogonal cut divides the plate into two segments, each of which contains strips of the same direction and length. The algorithm uses dynamic programming techniques to determine the optimal strip layouts on segments of various lengths, and selects two segments to appear in the optimal pattern. The segments are considered in increasing order of their lengths, so that dominant properties can be used to shorten the computation time. The computational results indicate that the algorithm is efficient in both material utilization and computation time.     

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.     

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.     

The cutting stock problem for large sections in the iron and steel industries

The characteristics of a cutting stock problem for large sections in the iron and steel industries are as follows:(1) There is a variety of criterions such as maximizing yield and increasing effeciency of production lines. (2) A cutting stock problem is accompanied by an optimal stock selection problem. A two-phase algorithm is developed, using an heuristic method. This algorithm gives nearly optimal solutions in real time. It is applied to both batch-solving and on-line solving of one-dimensional cutting of large section. The new algorithm has played an important role in a large-section production system to increase the yield by approximately 2.5%.     

Reducing the number of cuts in generating three-staged cutting patterns

Three-staged guillotine patterns are widely used in the manufacturing industry to cut stock plates into rectangular items. The cutting cost often increases with the number of cuts required. This paper focuses on the rectangular two-dimensional cutting stock problem, where three-staged guillotine patterns are used, and the objective is to minimize the sum of plate and cutting costs. The column generation framework is used to solve the problem. It uses a pattern-generation procedure to obtain the patterns. The cutting cost is considered in both the pattern-generation procedure and the objective of the linear programming formulation. The computational results indicate that the approach can reduce the number of cuts, without increasing the plate cost.     

A comparative numerical analysis for the guillotine two-dimensional cutting problem

In this work, the behavior of four algorithms in the resolution of the two-dimensional constrained guillotine cutting problem is analyzed. This problem is concerned about the way a set of pieces should be cut from a plate of greater dimensions, considering guillotine cutting and a constrained number of times a piece can be cut from the plate. In this study three combinatorial and two heuristic methods are considered. In the combinatorial methods from the set of pieces, a minimum loss layout is constructively generated based on Wang's algorithm. In addition, an evolutionary and an annealing type approach are considered. All of these models have been implemented on a high performance Silicon Graphics machine. Performance of each algorithm is analyzed both in terms of percentage waste and running time. In order to do that, a set of 1000 instances are classified according to their combinatorial degree and subsequently evaluated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Development of some cutting stock algorithms

A multivariant scheme of the south-eastern angle method for a two dimensional mass cutting stock problem is discussed in the paper. The use of a flexible linear programming package LP-LGIT for the solution of such problems with various column generators is described. The paralleling of a computational scheme of the method of the generating structures is suggested.     

Solving binary cutting stock problems by column generation and branch-and-bound

We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.This research was supported by NSF and AFOSR grant DDM-9115768     

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.     

The one-dimensional cutting stock problem with due dates

The one-dimensional cutting stock problem is the problem of cutting stock material into shorter lengths, in order to meet demand for these shorter lengths while minimizing waste. In industrial cutting operations, it may also be necessary to fill the orders for these shorter lengths before a given due date. We propose new optimization models and solution procedures which solve the cutting stock problem when orders have due dates. We evaluate our approach using data from a large manufacturer of reinforcement steel and show that we are able to solve industrial-size problems, while also addressing common cutting considerations such as aggregation of orders, multiple stock lengths and cutting different types of material on the same machine. In addition, we evaluate operational performance in terms of resulting waste and tardiness of orders using our model in a rolling horizon framework.     

An integer programming model for two- and three-stage two-dimensional cutting stock problems

In this paper, an integer programming model for two-dimensional cutting stock problems is proposed. In the problems addressed, it is intended to cut a set of small rectangular items of given sizes from a set of larger rectangular plates in such a way that the total number of used plates is minimized.