A hierarchy of bounds for stochastic mixed-integer programs

We consider general two-stage SMIPs with recourse, in which integer variables are allowed in both stages of the problem and randomness is allowed in the objective function, the constraint matrices (i.e., the technology matrix and the recourse matrix), and the right-hand side. We develop a hierarchy of lower and upper bounds for the optimal objective value of an SMIP by generalizing the wait-and-see solution and the expected result of using the expected value solution. These bounds become progressively stronger but generally more difficult to compute. Our numerical study indicates the bounds we develop in this paper can be strong relative to those provided by linear relaxations. Hence this new bounding approach is a complementary tool to the current bounding techniques used in solving SMIPs, particularly for large-scale and poorly formulated problems.     

Stochastic dynamics and Boltzmann hierarchy. III

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. II

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. I

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Bounding derivation lengths with functions from the slow growing hierarchy

Let be a (finite) rewrite system over a (finite) signature. Let be a strict well-founded termination ordering on the set of terms in question so that the rules of are reducing under . Then is terminating. In this article it is proved for a certain class of far reaching termination orderings (of order type reaching up to the first subrecursively inaccessible ordinal, i.e. the proof-theoretic ordinal of ) that – under some reasonable assumptions which are met in current applications – the derivation lengths function for is bounded by a function from the slow growing hierarchy of level determined by the order type of the underlying termination ordering. This result is a (correction of the proof of and a) strong generalization of theorem 8.1 in Cichon's article Termination orderings and complexity characterisations. Leeds, Proof Theory 1990, (Aczel, Simmons, and Wainer, editors), Cambridge University Press 1992, 171-193. Received: 2 February 1996     

A new approach for weight derivation using data envelopment analysis in the analytic hierarchy process

Recently, some researches have been carried out in the context of using data envelopment analysis (DEA) models to generate local weights of alternatives from pairwise comparison matrices used in the analytic hierarchy process (AHP). One of these models is the DEAHP. The main drawback of the DEAHP is that it generates counter-intuitive priority vectors for inconsistent pairwise comparison matrices. To overcome the drawbacks of the DEAHP, this paper proposes a new procedure entitled Revised DEAHP, and it will be shown that this procedure generates logical weights that are consistent with the decision maker's judgements and is sensitive to changes in data of the pairwise comparison matrices. Through a numerical example, it will be shown that the Revised DEAHP not only produces correct weights for inconsistent matrices but also does not suffer from rank reversal when an irrelevant alternative is added or removed.     

The analytic hierarchy process with stochastic judgements

The analytic hierarchy process (AHP) is a widely-used method for multicriteria decision support based on the hierarchical decomposition of objectives, evaluation of preferences through pairwise comparisons, and a subsequent aggregation into global evaluations. The current paper integrates the AHP with stochastic multicriteria acceptability analysis (SMAA), an inverse-preference method, to allow the pairwise comparisons to be uncertain. A simulation experiment is used to assess how the consistency of judgements and the ability of the SMAA-AHP model to discern the best alternative deteriorates as uncertainty increases. Across a range of simulated problems results indicate that, according to conventional benchmarks, judgements are likely to remain consistent unless uncertainty is severe, but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear.     

A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Using the main ideas of Tanaka, the measure-solution of a -dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to of solutions of approximating Boltzmann equations with cutoff is proved. Then, a result of Graham-Méléard is used and allows us to approximate with the empirical measure of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

     

Methods for scaling to doubly stochastic form

New methods for scaling square, nonnegative matrices to doubly stochastic form are described. A generalized version of the convergence theorem of Sinkhorn and Knopp (1967) is proved and applied to show convergence for these new methods. Tests indicate that one of the new methods has significantly better average and worst-case behavior than the Sinkhorn-Knopp method; for one of the 3 × 3 examples of Marshall and Olkin (1968), SK requires 130 times as many operations as the new algorithm to achieve row and column sums 1±10^-5.     

Application of stochastic analytic hierarchy process within a domestic appliance manufacturer

The stochastic analytic hierarchy process (SAHP) provides a mechanism for achieving more effective selection of alternatives in the form of considering multi and conflicting criteria using quantitative and qualitative information under uncertainty. In contrast to the traditional analytic hierarchy process, the SAHP uses probabilistic distributions to incorporate uncertainty that people have in converging their judgements of preferences into a Likert scale. The vector of priorities is calculated using Monte Carlo simulation, the final rankings are analysed for rank reversal using statistical analysis, and managerial aspects are introduced systematically. The present paper demonstrates an application of the SAHP in a world-class domestic appliance manufacturer. The case study was carried out by strictly following a disciplined and organized methodology for applying the SAHP developed by the authors. The results of this study were encouraging to key personnel within the company, establishing a greater opportunity to explore the applications of the SAHP in other core business processes.     

Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation

We construct weighted modifications of statistical modeling of an ensemble of interacting particles which is connected with approximate solution of a nonlinear Boltzmann equation.     

Methods of uniform optimal stochastic control

Translated from Upravlenie Nelineinymi Sistemami, Vsesoyuznyi Nauchno-Issledovatel'skii Institut Sistemnykh Issledovanii, Sbornik Trudov, No. 4, pp. 107–116, 1991.     

Methods of the stochastic approximation type for solving linear incorrectly posed problems

Siberian Mathematical Journal -     

On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes

Summary We consider systems of moderately interacting particles, which are divided into a finite number of different subpopulations, and show that in the limit as the population size tends to infinity the empirical processes of the subpopulations converge to the solution of a system of reaction-diffusion equations.This work has been supported by the Deutsche Forschungsgemeinschaft     

Methods based on data envelopment analysis for deriving group priorities in analytic hierarchy process

In recent years, group decision making has become one of the important issues in multiple criteria decision making, and analytic hierarchy process (AHP) is considered an appropriate method when dealing with this kind of problems. Many different approaches for attaining a group valuation in AHP have been developed. The applications most commonly employ the weighted geometric mean method. In the paper, we focus on the group AHP methods, which are based on the data envelopment analysis (DEA). First we discuss two methods for deriving a group priority vector: Wang and Chin’s DEA group method and Hosseinian et al.’s DEA-WDGD. Further, we propose a new WGMDEA method and compare all three methods with the WGMM on theoretical examples and on a real case study. The objective of the case study is to examine the current state of forest owners’ cooperatives. An analysis of the influence of forest owners’ cooperatives on private forest management in Slovenia was put forward. The A’WOT analysis, which is a combined method of AHP and SWOT analysis, an approach for identifying the strengths, weaknesses, opportunities and threats of the object under consideration, was performed.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.     

Corrections to the Article “Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation”

Corrections are given to the above-mentioned article.