In his pioneering papers, Igor Zaslavsky started an algorithmic (constructivist) analysis of fuzzy logic. In this paper, we extend this analysis to fuzzy mathematics and fuzzy data processing. Specifically, we show that the two mathematically equivalent representations of a fuzzy number – by a membership function and by α-cuts – are not algorithmically equivalent, and only the α-cut representat...

This paper introduces a class of cuts, called reachability cuts, for the Vehicle Routing Problem with Time Windows (VRPTW). Reachability cuts are closely related to cuts derived from precedence constraints in the Asymmetric Traveling Salesman Problem with Time Windows and to k-path cuts for the VRPTW. In particular, any reachability cut dominates one or more k-path cuts. The paper presents sepa...

This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these...

A market basket survey for beef retail cut composition at the retail level (four stores each from two chains in each city) was conducted in 11 US cities from January to March 2006. Beef cuts (n=17,495) were measured for external fat thickness with cuts from the chuck (0.05cm), round (0.05cm), and miscellaneous (0.04cm) having less (P<0.05) fat than cuts from the loin (0.11cm) and rib (0.11cm). ...

We present α-expansion β-shrink moves, a simple generalization of the widely-used αβswap and α-expansion algorithms for approximate energy minimization. We show that in a certain sense, these moves dominate both αβ-swap and α-expansion moves, but unlike previous generalizations the new moves require no additional assumptions and are still solvable in polynomial-time. We show promising experimen...

In technology mapping, enumeration of subcircuits or cuts to be replaced by a standard cell is an important step that decides both the quality of the solution and execution speed. In this work, we view cuts as set of edges instead of as set of nodes and based on it, provide a classification of cuts. It is shown that if enumeration is restricted to a subclass of cuts called unidirectional cuts, ...

Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use...

The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. “Nearly flat” means that every outer face normal forms a sufficiently small angle φ < Φ with the ẑ-axis orthogonal to the halfspace bounding pl...

Cutting planes for a mixed-integer program are linear inequalities which are satisfied by all feasible solutions of the latter. These are fundamental objects in mixed-integer programming that are critical for solving large-scale problems in practice. One of the main challenge in employing them is that there are limitless possibilities for generating cutting planes; the selection of the stronges...