Abstract
Cutting and Packing problems are hard Combinatorial Optimization problems that naturally arise in all industries and services where raw-materials or space must be divided into smaller non-overlapping items, so that waste is minimized. All the Cutting and Packing problems have in common the existence of a geometric sub-problem, originated by the natural item non-overlapping constraints. An important class of Cutting and Packing problems are the Irregular Packing problems that occur when raw materials have to be cut into items with irregular shapes. Irregular Packing problems, also known as Nesting problems, naturally arises in the garment, footwear, tools manufacturing and shipbuilding industries, among others. Each industrial application has its owns particular issues mainly related to the raw material's specific characteristics. Several challenges remain open in the Irregular Packing problems field. Some are due to the combinatorial nature of these problems. Others are of geometric nature, due to the non-convex and non-regular geometry of the items involved. Moreover these geometric challenges do not allow the combinatorial ones being properly tackled. This paper is mainly focused on presenting and discussing efficient tools and representations to tackle the geometric layer of nesting algorithms that capture the needs of the real-world applications of Irregular Packing problems. © 2013 IFAC.

Abstract

Abstract
The nesting problem, also known as irregular packing problem, belongs to the generic class of cutting and packing (C&P) problems. It differs from other 2-D C&P problems in the irregular shape of the pieces. This paper proposes a new mixed-integer model in which binary decision variables are associated with each discrete point of the board (a dot) and with each piece type. It is much more flexible than previously proposed formulations and solves to optimality larger instances of the nesting problem, at the cost of having its precision dependent on board discretization. To date no results have been published concerning optimal solutions for nesting problems with more than 7 pieces. We ran computational experiments on 45 problem instances with the new model, solving to optimality 34 instances with a total number of pieces ranging from 16 to 56, depending on the number of piece types, grid resolution and the size of the board. A strong advantage of the model is its insensitivity to piece and board geometry, making it easy to extend to more complex problems such as non-convex boards, possibly with defects. Additionally, the number of binary variables does not depend on the total number of pieces but on the number of piece types, making the model particularly suitable for problems with few piece types. The discrete nature of the model requires a trade-off between grid resolution and problem size, as the number of binary variables grows with the square of the selected grid resolution and with board size.

Abstract

Abstract
This paper presents an analytical approach for the evaluation of multi-user safety critical systems presenting a failure delayed behavior pattern. As a consequence of a failure event, the performance of these systems worsens progressively due to the internal fault tolerance mechanisms or the complacency of the users regarding the temporary unavailability of the services. A distinctive feature of the approach is the ability to handle stochastic models containing multiple processes with generalized distributions. The approach is based on the determination of analytical expressions to measure reliability, for instance, frequency and probability of failure states, which may be evaluated using general purpose mathematical tools. The paper first reviews other well-established techniques employed in the assessment of non-Markovian systems, particularly those based on stochastic Petri nets. The rationale of the new approach and its fundamental algorithms are presented together with a set of illustrative examples which highlight the strengths of the approach, as well as its limitations. Copyright (c) 2012 John Wiley & Sons, Ltd.

Abstract