Marcos Nereu Arenales

Lot sizing and furnace scheduling in small foundries

A lot sizing and scheduling problem prevalent in small market-driven foundries is studied. There are two related decision levels: (1) the furnace scheduling of metal alloy production, and (2) moulding machine planning which specifies the type and size of production lots. A mixed integer programming (MIP) formulation of the problem is proposed, but is impractical to solve in reasonable computing time for non-small instances. As a result, a faster relax-and-fix (RF) approach is developed that can also be used on a rolling horizon basis where only immediate-term schedules are implemented. As well as a MIP method to solve the basic RF approach, three variants of a local search method are also developed and tested using instances based on the literature. Finally, foundry-based tests with a real-order book resulted in a very substantial reduction of delivery delays and finished inventory, better use of capacity, and much faster schedule definition compared to the foundrys own practice.

Staged and constrained two-dimensional guillotine cutting problems: An AND/OR-graph approach

Abstract An AND/OR-graph approach has been proposed by Morabito et al. to solve non-staged and unconstrained two-dimensional guillotine cutting problems. Basically, this approach consists of representing cutting patterns as complete paths in an AND/OR graph (where nodes and arcs correspond to rectangles and cuts, respectively) and choosing a search strategy to traverse or enumerate the nodes of the graph. In this present paper, the AND/OR-graph approach is extended to solve staged and constrained problems. Computational experiments with examples from the literature are performed and indicate that this approach generates good and fast solutions using a microcomputer. In addition to the fact that the AND/OR-graph approach can handle important constraints, it can be easily extended to solve cutting and packing problems with multiple dimensions.

Joint rolling-horizon scheduling of materials processing and lot-sizing with sequence-dependent setups

Abstract A lot sizing and scheduling problem from a foundry is considered in which key materials are produced and then transformed into many products on a single machine. A mixed integer programming (MIP) model is developed, taking into account sequence-dependent setup costs and times, and then adapted for rolling horizon use. A relax-and-fix (RF) solution heuristic is proposed and computationally tested against a high-performance MIP solver. Three variants of local search are also developed to improve the RF method and tested. Finally the solutions are compared with those currently practiced at the foundry.

An AND/OR-graph approach to the solution of two-dimensional non-guillotine cutting problems

This work is concerned with unconstrained two-dimensional non-guillotine cutting problems, where a rectangular plate is cut into a number of rectangular pieces in such a way as to optimize an objective function. We reduce the state-space of the problem by considering non-guillotine cutting patterns which are combinations of guillotine and simple non-guillotine cuts. These cutting patterns are represented as complete paths in an AND/OR-graph and a branch and bound method is described. We provide rules and heuristics that reduce the graph search and present computational results from randomly generated examples as well as from the literature.

An and--or-graph approach for two-dimensional cutting problems

Abstract The problem of generating guillotine cutting patterns for a rectangular plate is studied and a type of structure is proposed for representing the solution of the problem, called and-or graph , which is commonly used in the Artificial Intelligence environment. To search the graph we combined two classical strategies: depth-first and hill-climbing . Further, some heuristics are considered and computational results are presented, relating their performance on examples from both literature as well as randomly generated.

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.

Optimizing the cutting of stock plates in a furniture company

In this study we analysed practical aspects of the application of a cutting stock model to a Brazilian company that manufactures furniture on a large scale with a high degree of standardization. The model is based on the classical approach of Gilmore and Gomory (1965, Operations Research, 14, 94-120), which combines a linear program and a column generation procedure. Besides the two-stage and three-stage guillotine cutting patterns, we also considered one-group guillotine patterns that improve the productivity of the cutting equipment. Examples derived from the furniture company are used to illustrate some of the trade-offs involved, in particular the trade-off between cutting simpler patterns and patterns that yield less waste material, but reduce the productivity of the cutting machine.

The one-dimensional cutting stock problem with usable leftover – A heuristic approach

In this work we consider a one-dimensional cutting stock problem in which the non-used material in the cutting patterns may be used in the future, if large enough. This feature introduces difficulties in comparing solutions of the cutting problem, for example, up to what extent a minimum leftover solution is the most interesting one when the leftover may be used. Some desirable characteristics of good solutions are defined and classical heuristic methods are modified, so that cutting patterns with undesirable leftover (not large enough to be used, nor too small to be acceptable waste) are redesigned. The performance of the modified heuristics is observed by solving instances from the literature, practical instances and randomly generated instances.

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

Abstract 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.

A lot-sizing problem in an automated foundry

Abstract This work consists of the study of a foundry which has only one furnace and several moulding machines producing a known demand of different types of items which can be made of different alloys. There are two important and linked decision levels in this foundry: (1) what alloys should be produced in the furnace in each period, and (2) the quantity of items to be produced in each moulding machine. Two different cases are highlighted here: a single alloy can produce all the items, and different alloys are needed to produce the items. Assuming that the production bottleneck is the furnace, efficient problem-specific solution methods are proposed.