The present status of lattice calculations of fD, fB and some mass splittings are discussed. When one includes the uncertainties due to discretization errors, the results do not yet have a suucient precision to be relevant to phenomenological applications. There are, however, good prospects of cutting down the uncertainties by a factor of 2 or more soon.

This paper deals with the classical two-dimensional cutting stock problem, which consists of generating cutting patterns to produce bi rectangular items of sizes (li,wi), i=1,..,m, from rectangular objects of sizes (L,W), so that an objective is optimized, for example, the cost or trim loss. However, to cut the patterns they should be sequenced in such a way as to optimize another objective, fo...

In the production process for wooden furniture, raw material costs account more than 50% of furniture costs, and utilization rate materials depends mainly on layout scheme. Therefore, a reasonable is an important measure to reduce costs. This paper investigates solid wood board cutting stock problem (CSP) establishes optimization model, with goal highest possible original boards. An ant colony-...

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting — dividing a heterogeneous resource among agents with different preferences. Resourceand population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all o...

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...

Cutting stock problems and bin packing problems are basically the same problems. They differ essentially on the variability of the input items. In the first, we have a set of items, each item with a given multiplicity; in the second, we have simply a list of items (each of which we may assume to have multiplicity 1). Many approximation algorithms have been designed for packing problems; a natur...

The modi ed integer round-up property (MIRUP) for a linear integer minimization problem means that the optimal value of this problem is not greater than the optimal value of the corresponding LP relaxation rounded up plus one. In earlier papers the MIRUP was shown to hold for the so-called divisible case and some other subproblems of the one-dimensional cutting stock problem. In this paper we e...

The G12 project is developing a software environment for stating and solving combinatorial problems by mapping a high-level model of the problem to an efficient combination of solving methods. Model annotations are used to control this process. In this paper we explain the mapping to branch-and-price solving. G12 supports the selection of specialised sub-problem solvers, the aggregation of iden...

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm —for staged pat...

The modified integer round-up property (MIRUP) for a linear integer minimization problem means that the optimal value of this problem is not greater than the optimal value of the corresponding LP relaxation rounded up plus one. In earlier papers the MIRUP was shown to hold for the so-called divisible case and some other subproblems of the one-dimensional cutting stock problem. In this paper we ...