Gerhard Wäscher

An improved typology of cutting and packing problems

The number of publications in the area of Cutting and Packing (C&P) has increased considerably over the last two decades. The typology of C&P problems introduced by Dyckhoff [Dyckhoff, H., 1990. A typology of cutting and packing problems. European Journal of Operational Research 44, 145–159] initially provided an excellent instrument for the organisation and categorisation of existing and new literature. However, over the years also some deficiencies of this typology became evident, which created problems in dealing with recent developments and prevented it from being accepted more generally. In this paper, the authors present an improved typology, which is partially based on Dyckhoff’s original ideas, but introduces new categorisation criteria, which define problem categories different from those of Dyckhoff. Furthermore, a new, consistent system of names is suggested for these problem categories. Finally, the practicability of the new scheme is demonstrated by using it as a basis for a categorisation of the C&P literature from the years between 1995 and 2004. � 2006 Elsevier B.V. All rights reserved.

Cutting and packing

Over a decade ago, the European Journal of Operational Research published its first special issue on Cutting and Packing [1] in conjunction with the founding of SICUP, the Special Interest Group on Cutting and Packing. The majority of the papers in the first special issue were focused on cutting problems in the aluminium, paper, and canvas industry, as well as container loading problems. In 1995, a second special issue appeared [2]. As before, the issue included a number of papers on traditional oneand two-dimensional cutting stock problems and container/pallet loading problems, but it also reported new research in nesting problems and introduced the use of metaheuristics such as simulated annealing and genetic algorithms for solving packing problems. Interest in solving these problems continues to grow: an increasing number of papers are being published in the literature each year on a range of cutting, packing, loading, and layout problems. Following the trend, this third special issue of EJOR reports current research that addresses a variety of cutting, packing and related problems where classical, as well heuristic approaches are applied. A first cluster of papers is dedicated to one-dimensional cutting and packing. De Carvalho reviews several models for problems of the cutting stock and bin packing type and analyses the relationship between them as grounds for the development of branch-and-price algorithms. In the two succeeding papers, new methods for the generation of integer solutions to the one-dimensional cutting stock problem with different standard lengths are presented. Belov and Scheithauer introduced an exact approach which uses Chvatal– Gomory cutting planes to tighten up the continuous relaxation of the classic model formulation. Despite the impressive computational results given, the suggested method may still not be applicable to all practical problems due to its computational requirements. In such cases, the heuristic solution method developed and evaluated by Holthaus may represent a feasible alternative. Important aspects of many real-world problems are considered in another two papers. Zak looks at cutting processes which extend over several stages. He proposes a solution method which is based on the classic column-generation procedure by Gilmore and Gomory and dynamically generates both rows (intermediate sizes) and columns (cutting patterns). However, the method does not guarantee that an optimal solution is found. In practice, the question of how to cut down orders from stock lengths is often interconnected with the problem of sequencing the cutting patterns in the most economical way. Armbruster describes such a problem at a steel service centre and suggests a European Journal of Operational Research 141 (2002) 239–240

Heuristics for the integer one-dimensional cutting stock problem: A computational study

In this paper the problem of generating integer solutions to the standard one-dimensional cutting stock problem is treated. In particular, we study a specific class of heuristic approaches that have been proposed in the literature, and some straightforward variants. These methods are compared with respect to solution quality and computing time. Our evaluation is based on having solved 4,000 randomly generated test problems. Not only will it be shown that two methods are clearly superior to the others but also that they solve almost any instance of the standard one-dimensional cutting stock problem to an optimum.ZusammenfassungIn der vorliegenden Arbeit betrachten wir das Problem der Bestimmung ganzzahliger Lösungen für das Standardproblem der eindimensionalen Zuschnittplanung. Insbesondere werden eine spezielle Klasse heuristischer Lösungsverfahren, die in der Literatur beschrieben sind, sowie einige naheliegende Varianten dieser Verfahren vorgestellt. Auf der Grundlage eines numerischen Experiments, bei dem 4.000 Probleme zufällig erzeugt und gelöst wurden, werden die Verfahren miteinander verglichen und im Hinblick auf die Kriterien „Lösungsqualität“ und „Rechenzeitbedarf“ beurteilt. Dabei zeigt sich nicht nur, daß zwei Verfahren deutlich besser als die übrigen einzustufen sind, sondern auch, daß mit ihrer Hilfe nahezu jede Problemausprägung des klassischen eindimensionalen Zuschneideproblems optimal gelöst werden kann.

Constraints in container loading – A state-of-the-art review

Container loading is a pivotal function for operating supply chains efficiently. Underperformance results in unnecessary costs (e.g. cost of additional containers to be shipped) and in an unsatisfactory customer service (e.g. violation of deadlines agreed to or set by clients). Thus, it is not surprising that container loading problems have been dealt with frequently in the operations research literature. It has been claimed though that the proposed approaches are of limited practical value since they do not pay enough attention to constraints encountered in practice.

CUTGEN1: a problem generator for the standard one-dimensional cutting stock problem

In this paper a problem generator for the Standard One-dimensional Cutting Stock Problem (1D-CSP) is developed. The problem is defined and its parameters are identified. Then it is shown what features have been included in the program in order to allow for the generation of easily reproducible random problem instances. Finally, by applying the generator a set of benchmark problems is identified.

Order Picking: A Survey of Planning Problems and Methods

Order picking — as a warehouse function — is often considered to be critical for the public manifestation of a supply chain. Underperformance can result both in unsatisfactory customer service and in high cost. In this paper the author will give an overview of planning problems and corresponding methods which have been suggested in the literature for the reduction of cost and the improvement of customer service in picker-to-product systems. In particular, the following central issues will be addressed: item location, order batching, and picker routing.

Pattern reduction in one-dimensional cutting stock problems

In industrial cutting operations the number of cutting patterns needed to satisfy a given set of orders may be crucial for the capacity load which can be achieved for the cutting equipment, since switching between different patterns often necessitates time-consuming set-ups. Therefore, in planning the cutting operations, one does not necessarily search for input-minimal cutting plans (i.e. cutting plans minimizing material costs) only, but also for plans with a small (or even minimal) number of cutting patterns. Such plans are usually generated in a two-step approach: in a first step, regardless of the number of patterns needed, an inputminimal cutting plan is generated while in a second step the number of patterns is reduced. In this paper a new method for the second step is presented, which can be considered as a generalization of other methods previously suggested for this purpose. The performance of the new method is evaluated on the basis of 1800 randomly generated problem instances. The solution quality of the new method turns out to be clearly superior to the existing methods, however, an increase in computing times occurs for some problem classes. In this case it is possible to speed up the proposed method considerably at the expense of a slight decrease in solution quality.

Tabu Search Heuristics for the Order Batching Problem in Manual Order Picking Systems

In a manual order picking system, order pickers walk or ride through a distribution warehouse in order to collect items requested by (internal or external) customers. In order to perform these operations efficiently, it is usually required that customer orders be combined into (more substantial) picking orders that are limited in size. The order batching problem considered in this paper deals with the question of how a given set of customer orders should be combined into picking orders such that the total length of all picker tours necessary for all of the requested items to be collected is minimized. For the solution of this problem the authors suggest two approaches based on the tabu search principle. The first is a (classic) tabu search (TS), and the second is the attribute-based hill climber (ABHC). In a series of extensive numerical experiments, these approaches are benchmarked against other solution methods put forward in the current literature. It is demonstrated that the proposed methods are superior to the existing methods and provide solutions which may allow distribution warehouses to operate more efficiently.

Simulated annealing for order spread minimization in sequencing cutting patterns

The Order Spread Minimization Problem (OSMP) is a sequencing problem that arises in the process of planning industrial cutting operations. As it can be looked upon as a generalization of the Travelling-Salesman Problem (TSP), it has to be classified as NP-complete. Thus heuristic algorithms are required in order to solve large problem instances. In this paper the authors suggest to apply Simulated Annealing (SA) to the OSMP. A specific version of SA is developed and compared to both an approach previously introduced into the literature by Madsen and a traditional 3-opt-procedure. The performance of these methods is compared on a set of 2400 randomly generated problem instances. SA appears to provide solutions which - in terms of solution quality - are equivalent to those generated by the 3-opt-procedure. However, computing times of the latter for solving large instances are prohibitive. In relation to Madsens approach SA provides significantly improved solutions at the expense of a moderate increase in computing times.

Order Batching in Order Picking Warehouses: A Survey of Solution Approaches

Order picking is a warehouse function dealing with the retrieval of articles from their storage location in order to satisfy a given demand specified by customer orders. Of all warehouse operations, order picking is considered to include the most cost-intensive ones. Even though there have been different attempts to automate the picking process, manual order picking systems are still prevalent in practice. This article will focus on order batching, one of the main planning issues in order picking systems. Order Batching has been proven to be pivotal for the efficiency of order picking operations. With respect to the availability of information about the customer orders, order batching can be distinguished into static batching and dynamic batching. Improved order batching reduces the total picking time required to collect the requested articles. According to experience from practice, this can result in significant savings of labor cost and into a reduction of the customer orders delivery lead time. The aim of this contribution is to provide comprehensive insights into order batching by giving a detailed state-of-the-art overview of the different solution approaches which have been suggested in the literature. Corresponding to the available publications, the emphasis will be on static order batching. In addition to this, the paper will also review the existing literature for variants and extensions of static order batching (e.g. due dates, alternative objective functions). Furthermore, solution approaches for dynamic order batching problems (like time window batching) will be presented.