We show that given two vertices of a polytope one cannot in general nd a hyperplane containing the vertices, that has two or more facets of the polytope in one closed half-space. Our result refutes a long-standing conjecture. We prove the result by constructing a 4-dimensional polytope that provides the counterexample. Also, we show that such a cutting hyperplane can be found for each pair of v...

We consider the problem of fairly dividing a two dimensional heterogeneous good among multiple players. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols primarily consider a one-dimensional resource, or allocate each player multiple infinitesimally small “pieces”. In practice, however, the two dimensional shape of the allo...

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Combining the use of our data structure for characterizing feasible packings with our new classes of lower bounds, and other heuristics, we develop a two-level tree search algorithm for solving higher-dimensional packing problems to optimality. Computational re...

Consider the feasibility problem in higher-dimensional orthogonal packing. Given a set I of d-dimensional rectangles, we need to decide whether a feasible packing in a d-dimensional rectangular container is possible. No item rotation is allowed and item edges are parallel to the coordinate axes. Typically, solution methods employ some bounds to facilitate the decision. Various bounds are known,...

We consider a one-dimensional cutting stock problem (CSP) in which the stock widths are not used to fulfill the order but kept for use in the future for the industrial-use paper production. We present a new model based on the flexible stock allocation and trim loss control to determine the production quantity. We evaluate our approach using a real data and show that we are able to solve industr...

The mechanics of the cutting process on the microscopic level differ fundamentally from the conventional macro cutting. For example, the tool edge radius influences the cutting mechanism in micro machining significantly with regard to the effective rake angle, the minimum chip thickness, the dominance of ploughing, and the related elasto-plastic deformation of the workpiece material. These phen...

This paper proposes a method to vibration analysis in order to on-line monitoring of milling process quality. Adapting envelope analysis to characterize the milling tool materials is an important contribution to the qualitative and quantitative characterization of milling capacity and a step by modeling the three-dimensional cutting process. An experimental protocol was designed and developed f...

In this work we consider the 3-staged 2-dimensional cutting stock problem, which appears in many real-world applications such as glass and wood cutting and various scheduling tasks. We suggest a variable neighborhood search (VNS) employing “ruin-and-recreate”based very large neighborhood searches (VLNS). We further present a polynomial-sized integer linear programming model (ILP) for solving th...