Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm - for staged patterns - also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms. (c) 2007 Elsevier B.V. All rights reserved.

A linear optimization approach to the combined production planning model

Two fundamental processes usually arise in the production planning of many industries. The first one consists of deciding how many final products of each type have to be produced in each period of a planning horizon, the well-known lot sizing problem. The other process consists of cutting raw materials in stock in order to produce smaller parts used in the assembly of final products, the well-studied cutting stock problem. In this paper the decision variables of these two problems are dependent of each other in order to obtain a global optimum solution. Setups that are typically present in lot sizing problems are relaxed together with integer frequencies of cutting patterns in the cutting problem. Therefore, a large scale linear optimizations problem arises, which is exactly solved by a column generated technique. It is worth noting that this new combined problem still takes the trade-off between storage costs (for final products and the parts) and trim losses (in the cutting process). We present some sets of computational tests, analyzed over three different scenarios. These results show that, by combining the problems and using an exact method, it is possible to obtain significant gains when compared to the usual industrial practice, which solve them in sequence. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Orthogonal packing of identical rectangles within isotropic convex regions

A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations of the items, preserving the orthogonality constraint, are allowed. The solution method is based on a combination of branch and bound and active-set strategies for bound-constrained minimization of smooth functions. Numerical results show the reliability of the presented approach. (C) 2010 Elsevier Ltd. All rights reserved.

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50 000 instances). It is also effective for solving the instances of problem set Cover III (almost 100 000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes. Journal of the Operational Research Society (2010) 61, 306-320. doi: 10.1057/jors.2008.141 Published online 4 February 2009

Trophallaxis in Dehydrated Leaf Cutting Colonies of Atta sexdens rubropilosa (Hymenoptera: Formicidae)

Trophallaxis, the transfer of liquid among individuals by oral regurgitation or anal deposition, occurs in many insect groups including ants. The first indication that trophallaxis could occur in leaf cutting ants (Atta sexdens rubropilosa) was made by Autuori in 1942. He reported water collection by this ant species, and highlighted what in those days was an undescribed behavior for this species. In 2005, Da-Silva and Ribeiro presented preliminary results suggesting the existence of trophallaxis in A. sexdens rubropilosa. Here we report on a formal test of the hypothesis of trophallaxis in that species. Our approach was to test ant pairs in which only one individual (Group I) had access to blue-dyed water and the other individual (Group II), a nest-mate, came from a colony dehydrated by offering dry crushed corn for fungal growth. Positive results for trophallaxis were obtained in ants from four colonies and accounted for 33%-46% of all tests in which ants from Group I drank dyed water. These results indicate that trophallaxis occurs in this species.

Colony dehydration and water collection by specialized caste in the leaf-cutting ant Atta sexdens rubropilosa

Social organization enables leaf-cutting ants to keep appropriate micro-ecological nest conditions for the fungus garden (their main food), eggs, larvae and adults. To maintain stability while facing changing conditions, individual ants must perceive destabilising factors and produce a proper behavioral response. We investigated behavioral responses to experimental dehydration in leaf-cutting ants to verify if task specialization exists, and to quantify the ability of ant sub-colonies for water management. Our setup consisted of fourteen sub-colonies, ten of which were randomly assigned to different levels of experimental dehydration with silica gel, whereas the remaining four were controls. The ten experimental sub-colonies were split into two groups, so that five of them had access to water. Diverse ant morphs searched for water in dehydrated colonies, but mainly a caste of small ants collected water after sources had been discovered. Size specialization for water collection was replicable in shorter experiments with three additional colonies. Ants of dehydrated colonies accumulated leaf-fragments on the nest entrance, and covering the fungus garden. Behaviors that may enhance humidity within the nests were common to all dehydration treatments. Water availability increased the life span of dehydrated colonies.

Three-dimensional Reconstruction of Ovaries of Leaf-cutting Ant (Atta sexdens rubropilosa) Queens (Hymenoptera: Formicidae)

In this study, the ovary morphology of newly emerged ant queens of Atta sexdens rubropilosa was studied in whole mount preparations by confocal microscopy. The ovaries are composed of approximately 40 ovarioles, showing non-synchronic oocyte maturation. The terminal filament with clusters of undifferentiated cells was found at the distal end of the ovarioles. Next to this region is the germarium, composed of several elongated cystocytes interconnected by cytoplasmic bridges. The nurse cells (23-28 cells) result from asymmetric mitosis. Cytoskeleton analysis showed F-actin concentrated at the muscle cells of the external tunica and in fusomes inside the ovarioles. Microtubules were concentrated around the nuclei of the nurse and follicular cells. In contrast, the oocytes and the external tunica showed faint staining for tubulin.

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem`s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution-VSS-and the expected value of perfect information-EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.

Heuristics for the one-dimensional cutting stock problem with limited multiple stock lengths

This paper deals with the classical one-dimensional integer cutting stock problem, which consists of cutting a set of available stock lengths in order to produce smaller ordered items. This process is carried out in order to optimize a given objective function (e.g., minimizing waste). Our study deals with a case in which there are several stock lengths available in limited quantities. Moreover, we have focused on problems of low demand. Some heuristic methods are proposed in order to obtain an integer solution and compared with others. The heuristic methods are empirically analyzed by solving a set of randomly generated instances and a set of instances from the literature. Concerning the latter. most of the optimal solutions of these instances are known, therefore it was possible to compare the solutions. The proposed methods presented very small objective function value gaps. (C) 2008 Elsevier Ltd. All rights reserved.

A Lagrangian relaxation approach to a coupled lot-sizing and cutting stock problem

Industrial production processes involving both lot-sizing and cutting stock problems are common in many industrial settings. However, they are usually treated in a separate way, which could lead to costly production plans. In this paper, a coupled mathematical model is formulated and a heuristic method based on Lagrangian relaxation is proposed. Computational results prove its effectiveness. (C) 2009 Elsevier B.V. All rights reserved.

A coupling cutting stock-lot sizing problem in the paper industry

An important production programming problem arises in paper industries coupling multiple machine scheduling with cutting stocks. Concerning machine scheduling: how can the production of the quantity of large rolls of paper of different types be determined. These rolls are cut to meet demand of items. Scheduling that minimizes setups and production costs may produce rolls which may increase waste in the cutting process. On the other hand, the best number of rolls in the point of view of minimizing waste may lead to high setup costs. In this paper, coupled modeling and heuristic methods are proposed. Computational experiments are presented.