A geometric modelling of nonlinear RLC networks

A new approach to nonlinear RLC circuits, which is based on the geometric Birkhoffian formalism, is described in this note. The configuration space and a special Pfaffian form, called Birkhoffian, are obtained from the constitutive relations of the involved resistors, inductors and capacitors and from Kirchhoff's laws. No assumptions are placed upon the topology of the network. Properties of the corresponding Birkhoffian such as its regularity, or its dissipativeness, are discussed in this context. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On zero-degree stochastic geometric programs

The Mellin transform is used to encode randomness in the constraint and objective function coefficients using the substituted dual function. This enables one to obtain statistical moments and the probability distribution of the optimal objective valueZ*. Advantage is taken of the form of the dual function and the limiting property of the lognormal distribution to prove that the probability distribution ofZ* approximates the lognormal distribution, independent of the distribution of the parameters. This is of importance because those probability distributions are seldom known; even if they are, a derivation of the distribution ofZ* is apt to be elusive. Further, the larger the number of stochastic parameters in the geometric program, the more closely, in general, does the distribution ofZ* approximate the lognormal distribution. Illustrative examples are provided.Credit is due to Keith R. Weiss who developed the examples. The Office of Naval Research supported the work under Contract No. N000-14-75-C-0254.     

Characterizations of the exponential distribution by stochastic ordering properties of the geometric compound

Under the reliability NBU/NWU conditions, the exponential distribution is characterized by stochastic ordering properties which link the geometric compound with minimum order statistics or spacings of order statistics. This somewhat answers a question posed by Kakosyan, Klebanov and Melamed (1984,Characterization of Distributions by the Method of intensively Monotone Operators, Springer, New York). We also show the related results based on the residual life in a renewal process and on record values. Finally, some fundamental properties of the NBUC/NWUC classes of life distributions are investigated.     

Panel discussion on geometric modelling

Panel discussion on geometric modelling     

A structure-conveying modelling language for mathematical and stochastic programming

We present a structure-conveying algebraic modelling language for mathematical programming. The proposed language extends AMPL with object-oriented features that allows the user to construct models from sub-models, and is implemented as a combination of pre- and post-processing phases for AMPL. Unlike traditional modelling languages, the new approach does not scramble the block structure of the problem, and thus it enables the passing of this structure on to the solver. Interior point solvers that exploit block linear algebra and decomposition-based solvers can therefore directly take advantage of the problem’s structure. The language contains features to conveniently model stochastic programming problems, although it is designed with a much broader application spectrum.     

Analytical and stochastic modelling techniques

In mining supply chains, large combinatorial optimization problems arise. These are NP-hard and typically require a large number of computing resources to solve them. In particular, the run-time overheads can become increasingly prohibitive with increasing problem sizes. Parallel methods provide a way to manage such run-time issues by utilising several processors in independent or shared memory architectures. However it is not obvious how to adapt serial optimisation algorithms to perform best in a parallel environment. Here, we consider a resource constrained scheduling problem which is motivated in mining supply chains and present two popular meta-heuristics, ant colony optimization (ACO) and simulated annealing and investigate how best to parallelize these methods on a shared memory architecture consisting of several cores. ACO’s solution construction framework is inherently parallel allowing a relatively straightforward parallel implementation. However, for best performance, ACO needs an element of local search. This significantly complicates the paralellization. Several alternative schemes for parallel ACO with elements of local search are considered and evaluated empirically. We find that ACO with local search is the most effective single-threaded algorithm. The best parallel implementation can obtain similar quality results to the serial method in significantly less elapsed time.     

Judgemental modelling based on geometric least square

The Analytic Hierarchy Process developed by T.L. Saaty has received widespread attention in the literature. However this has not been without some criticisms, including the problems of meaning of consistency and large data requirements. This paper addresses the last two problems and presents a new method of calculating preference vectors. To the user the procedure is essentially the same, but the data requirements can be made less onerous and at the same time feedback is provided permitting a greater understanding of the data inputs. Preliminary results indicate the acceptability of the proposed methodology.     

A stochastic integral arising in discounting continuous cash flows and certain transformed characteristic functions

Transformed characteristic functions are universally recognized as the most powerful tools for investigating distribution functions of complicated stochastic models. The paper is mainly devoted to the establishment of properties and applications of a particular convolution model. More precisely, the paper derives the characteristic function of a convolution model based on a stochastic integral and provides applications of this model in discounting continuous cash flows.     

A survey of stochastic modelling approaches for liberalised electricity markets

Liberalisation of energy markets, climate policy and the promotion of renewable energy have changed the framework conditions of the formerly strictly regulated energy markets. Generating companies are mainly affected by these changing framework conditions as they are exposed to the different risks from liberalised energy markets in combination with huge and largely irreversible investments. Uncertainties facing generating companies include: the development of product prices for electricity as well as for primary energy carriers; technological developments; availability of power plants; the development of regulation and political context, as well as the behaviour of competitors.     

Dynamic programming for stochastic target problems and geometric flows

Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications in this direction are also discussed. Received October 24, 2000 / final version received July 24, 2001?Published online November 27, 2001     

A geometric programming approach to the Van der Waerden conjecture on doubly stochastic matrices

LetD _p be the set of all doubly stochastic square matrices of orderp i.e. the set of allp × p matrices with non-negative entries with row and column sums equal to unity. The permanent of ap × p matrixA = (a _ij ) is defined byP(A) = _Sp II _i=1 ^p a _i(i) whereS _p is the symmetric group of orderp. Van der Waerden conjectured thatP(A) p !/p ^p for all A AD _p with equality occurring if and only ifA = J _p , whereJ _p is the matrix all of whose entries are equal to 1/p.The validity of this conjecture has been shown for a few values ofp and for generalp under certain assumptions. In this paper the problem of finding the minimum of the permanent of a doubly stochastic matrix has been formulated as a reversed geometric program with a single constraint and an equivalent dual program is given. A related problem of reversed homogeneous posynomial programming problem is also studied.     

Parameter estimation in geometric process with Weibull distribution

We consider geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Weibull. Explicit estimators of the parameters in GP are derived by using the method of modified maximum likelihood (MML) proposed by Tiku [24]. Asymptotic distributions and consistency properties of these estimators are obtained. We show that our estimators are more efficient than the widely used modified moment (MM) estimators via Monte Carlo simulation study. Further, two real life examples are given at the end of the paper.     

A stochastic permutation-flowshop scheduling problem minimizing in distribution the schedule length

Consider n jobs (J₁,J₂,…,J_n) and m machines (M₁,M₂…,M_m). Upon completion of processing of J_i, 1 ? i ? n, on M_j 1 ? j ? m ? 1, it departs with probability p_i or moves to M_j+1 with the complementary probability, 1?p_i. A job completing service on M_m departs. The processing time of j_i on M_j possesses a distribution function F_j. It is proved that sequencing the jobs in a nondecreasing order of p_i minimizes in distribution the schedule length.     

Distributed modelling of deterministic and stochastic population dynamics

In many problems of practical purpose, one is interested not only in the study of the total population of the overall system, but also in the distribution of the population depending upon some characteristic parameters. This necessity is patent in biological systems defined on a wide range of distributed features, but it is also in order when one tries to analyze social systems by means of population models. As a matter of fact, with this objective in mind, we deal with infinite species population models.This study discusses this question, proposes a distributed version for the logistic equation, and examines the existence of stationary solutions. The problem of the influence of environmental noise on the dynamics of the distributed population is considered, and a model is proposed. A detailed analysis is carried out around the stationary solution via a linearization technique, and the covariance equation is derived.     

Relief inventory modelling with stochastic lead-time and demand

The irregular demand and communication network disruption that are characteristics of situations demanding humanitarian logistics, particularly after large-scale earthquakes, present a unique challenge for relief inventory modelling. However, there are few quantitative inventory models in humanitarian logistics, and assumptions inherent in commercial logistics naturally have little applicability to humanitarian logistics. This paper develops a humanitarian disaster relief inventory model that assumes a uniformly distributed function in both lead-time and demand parameters, which is appropriate considering the limited historical data on relief operation. Furthermore, this paper presents different combinations of lead-time and demand scenarios to demonstrate the variability of the model. This is followed by the discussion of a case study wherein the decision variables are evaluated and sensitivity analysis is performed. The results reveal the presence of a unique reorder level in the inventory wherever the order quantity is insensitive to some lead-time demand values, providing valuable direction for humanitarian relief planning efforts and future research.     

On modelling and forecasting of vector stochastic processes

Some deterministic and multivariate stochastic model identification techniques for setting up a suitable dynamic model for an observed multivariate process are described. Results of a case study involving the problem of multinodal load forecasting are presented in order to illustrate the practical utility of the methods discussed.     

Comparative analysis of asynchronous cellular automata in stochastic pharmaceutical modelling

In pharmaceutical modelling, cellular automata have been used as an established tool to represent molecular changes through discrete structural interactions. The data quality provided by such modelling is found suitable for the early drug design phase where flexibility is paramount. While both synchronous (CA) and asynchronous (ACA) types of automata have been used, analysis of their nature and comparative influence on model outputs is lacking. In this paper, we outline a representative probabilistic CA for modelling complex controlled drug formulations and investigate its transition from synchronous to asynchronous update algorithms. The key investigation points include quantification of model dynamics through three distinct scenarios, parallelisation performance and the ability to describe different release phenomena, namely erosion, diffusion and swelling. The choice of the appropriate update mechanism impacts the perceived realism of the simulation as well as the applicability of large-scale simulations.     

Estimating completion-time distribution in stochastic activity networks

This paper deals with simulation-based estimation of the probability distribution for completion time in stochastic activity networks. These distribution functions may be valuable in many applications. A simulation method, using importance-sampling techniques, is presented for estimation of the probability distribution function. Separating the state space into two sets, one which must be sampled and another which need not be, is suggested. The sampling plan of the simulation can then be decided after the probabilities of the two sets are adjusted. A formula for the adjustment of the probabilities is presented. It is demonstrated that the estimator is unbiased and the upper bound of variance minimized. Adaptive sampling, utilizing the importance sampling techniques, is discussed to solve problems where there is no information or more than one way to separate the state space. Examples are used to illustrate the sampling plan.