Allocating slacks in stochastic PERT network

The SPERT problem was defined, in a game theory framework, as the fair allocation of the slack or float among the activities in a PERT network previous to the execution of the project. Previous approaches tackle with this problem imposing that the durations of the activities are deterministic. In this paper, we extend the SPERT problem into a stochastic framework defining a new solution that tries also to maintain the good performance of some other approaches that have been defined for the deterministic case. Afterward, we present a polynomial algorithm for this new solution that also could be used for the calculation of other approaches founded in the deterministic SPERT literature.     

Local path fitting: A new approach to variational integrators

A new approach for constructing variational integrators is presented. In the general case, the estimation of the action integral in a time interval [_tk,_tk+1]$[t_k,t_{k+1}]$ is used to construct a symplectic map (_qk,_qk+1)→(_qk+1,_qk+2)$(q_k,q_{k+1})\to (q_{k+1},q_{k+2})$. The basic idea, is that only the partial derivatives of the estimated action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, gives rise to a new integral that depends on the Euler–Lagrange vector itself (which in the continuous and exact case vanishes) and not on the Lagrangian. Since this new integral can only be computed through a numerical method based on some internal grid points, we can locally fit the exact curve by demanding the Euler–Lagrange vector to vanish at these grid points. Thus, the integral vanishes, and the process dramatically simplifies the calculation of high order approximations. The new technique is tested in high order solutions of the two-body problem with high eccentricity (up to 0.99) and of the outer planets of the solar system.     

A probabilistic approach to antigen-antibody recognition

Expressions for the expected value and the variance of the longest matched segment on the surfaces of two random shapes are obtained. Applications of these estimates to antibody-antigen pattern recognition during immune response are considered. Methods of estimation of the involved probabilities are presented with relevant computational results.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

A probabilistic approach to polynomial inequalities

We define a probability measure on the space of polynomials over ? ⁿ in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities.Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of ? ₁ ⁿ tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.     

A probabilistic approach to modelling uncertain logical arguments

Argumentation can be modelled at an abstract level using a directed graph where each node denotes an argument and each arc denotes an attack by one argument on another. Since arguments are often uncertain, it can be useful to quantify the uncertainty associated with each argument. Recently, there have been proposals to extend abstract argumentation to take this uncertainty into account. This assigns a probability value for each argument that represents the degree to which the argument is believed to hold, and this is then used to generate a probability distribution over the full subgraphs of the argument graph, which in turn can be used to determine the probability that a set of arguments is admissible or an extension. In order to more fully understand uncertainty in argumentation, in this paper, we extend this idea by considering logic-based argumentation with uncertain arguments. This is based on a probability distribution over models of the language, which can then be used to give a probability distribution over arguments that are constructed using classical logic. We show how this formalization of uncertainty of logical arguments relates to uncertainty of abstract arguments, and we consider a number of interesting classes of probability assignments.     

A stochastic approach to the flood control problem

We propose a stochastic model in conjunction with reliability analysis concepts to improve estimates for the protection volume that should be allocated in a reservoir to control a flood wave. In this approach, the inflow that reaches the reservoir during a flood is considered to be a load, and the reservoir capacity to control this flood is considered to be the resistance that the reservoir offers against the propagation of the flood. Here, the load and the resistance are modeled as a diffusion stochastic process, and the protection volume is determined via Itô's formula. In this scenario, an explicit formula for the failure risk is derived. The parameter inference is carried out by a Bayesian approach for a time discrete version of the load, and the estimates are obtained by using Monte Carlo Markov Chain Algorithms (MCMC). The maximum likelihood estimators are used in the comparison. The record utilized comprises nine years of daily inflow rates during flood periods that come to the Chavantes hydroelectric power plant (CHPP) in Southeast Brazil. The protection volumes estimated through the proposed model are compared to the volumes obtained by other existing methods.     

A Simulation Solution to the PERT Problem

The completion times of a set of paths through a PERT networkare expressed in terms of the multivariate normal distribution.The inverse transformation of the multivariate normal distribution.The inverse transformation of the multivariate normal distributionto the multivariate independent normal distribution is thenused to obtain a random sample of the completion times for theset of paths. Additional paths are introduced progressivelyuntil their contribution becomes insignificant. Economy, interms of computational load, is achieved by the need to generatethe completion times of entire paths only, and by the use ofsimulation variance reduction techniques.     

A stochastic frontier approach to running performance

^** Email: e.sterken{at}eco.rug.nl Long-distance running performance depends on age of the runnerand race distance. In this paper we apply a stochastic frontierapproach to estimate optimal running performance across distancesfrom 5000 m up to and including the marathon. We present agecorrection factors for 5000, 10000 and 15000 m, half-marathonand marathon distances for women and men. Two conclusions emerge:(1) official age grading tables are too optimistic for olderrunners and (2) running performance–age curves differacross distances.     

A path probability approach to the solute transport problem

An alternative solute transport algorithm, termed the path probability approach, is presented. The method may be more effective than typical finite methods in some numerical solution problems. The method differs from predominant numerical techniques in that the conventional conservation differential is replaced by a series of equations defining solute migration in terms of a finite number of representative particle position histories and corresponding probabilities. The governing physical assumptions applied in model development are conditionally consistent with those of conventional Lagrangian and Eulerian conceptualizations, but mathematical expression and development of the problem differ from traditional approaches. The method represents dispersion orthogonal to the primary axis of advective transport explicitly, according to arbitrary user-defined probability functions; longitudinal dispersion effects result from the existence of velocity shear effects along the primary axis of movement. It is possible to incorporate Fickian dispersive processes, and thereby reproduce results obtainable with traditional stream tube models; but non-Fickian alternatives can also be explored. Simulation results of the path probability model are compared to analogous results from a representative finite difference model for a hypothetical test flow channel. The comparison demonstrated the ability of the new model to effectively generate simulation results with a computational effeciency considerably higher than conventional techniques under the conditions tested.     

Mixture densities for project management activity times: A robust approach to PERT

PERT is a widely utilized framework for project management. However, as a result of underlying assumptions about the activity times, the PERT formulas prescribe a light-tailed distribution with a constant variance conditional on the range. Given the pervasiveness of heavy-tailed phenomena in business contexts as well as inherently differing levels of uncertainty about different activities, there is a need for a more flexible distribution which allows for varying amounts of dispersion and greater likelihoods of more extreme tail-area events. In particular, we argue that the tail-area decay of an activity time distribution is a key factor which has been insufficiently considered previously. We provide a distribution which permits varying amounts of dispersion and greater likelihoods of more extreme tail-area events that is straightforward to implement with expert judgments. Moreover, the distribution can be integrated into the PERT framework such that the classic PERT results represent an important special case of the method presented here.     

Advertising for a new product introduction: A stochastic approach

We formulate a stochastic extension of the Nerlove and Arrow’s advertising model in order to analyze the problem of a new product introduction. The main idea is to introduce some uncertainty aspects in connection both with the advertising action and the goodwill decay, in order to represent the random consequences of the advertising messages and of the word-of-mouth publicity, respectively. The model is stated in terms of the stochastic optimal control theory and a general study is attempted using the stochastic Maximum Principle. Closed form solutions are obtained under linear quadratic assumptions for the cost and the reward functions. Such optimal policies suggest that the decision-maker considers both the above mentioned phenomena as opportunities to increase her/his final reward. After stating some general features of the optimal solutions, we analyze in detail three extreme cases, namely the deterministic model and the stochastic models with either the word-of-mouth effect only, or the lure/repulsion effect only. The optimal policies provide us with some insight on the general effects of the advertising action. Supported by MIUR and University of Padua.     

A probabilistic approach to heat diffusion on symmetric spaces

We give an interpretation of heat diffusion phenomena on symmetric spaces as described by Anker and Setti⁽²⁾ in terms of the asymptotic behavior of Brownian motion.     

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.     

A New Approach to the Activity-time Distribution in PERT

We consider the problem of computing the density function of the completion time of PERT stochastic networks. An activity-time distribution is suggested which is stable to maximization and close to the beta distribution. Thus classical network-reducing methods, e.g. Martin's algorithm, can be essentially simplified.     

A stochastic approach to inverse scattering in geophysical layers

This paper considers wave propagation in a stratified geophysical medium to determine seismic properties of stratigraphic layers by agreement between observed data and calculated response. Discrete and continuous problems are discussed. If the medium has a stochastic nature, we employ random reflection coefficients and can, in some cases, relate to imbedding ideas of Bellman, Vasudevan, and Soong [11]. An interesting new approach is suggested also, based on the work of Adomian, in new methods for the solution of linear and nonlinear stochastic operator equations representing the successive strata or a single propagation medium as stochastic systems since density and other properties are random functions of position and successive layers can be represented by products of stochastic operators. The solution of the resulting equations can be made for strong nonlinearities and strongly fluctuating variables by the recently developed methods (see particularly [15]).