José García Pérez

A note on the reasonableness of PERT hypotheses

In this paper the expression for the mean in the PERT method is considered. This mean involves a parameter k, that sometimes has been set to 4. Insisting on the similarity between the beta and the normal distributions, certain hypotheses are proposed that lead to k necessarily being exactly 4. More specifically, by using the moments of the second and fourth orders, it is shown that the usual beta distribution in the PERT method is mesokurtic (@c=3) and of constant variance (@s^2=136).

The two-sided power distribution for the treatment of the uncertainty in PERT

The aim of this paper is to include the Two-Sided Power (TSP) distribution in the PERT methodology making use of the advantages that this four-parameter distribution offers. In order to be completely determined, a distribution of this type needs, the same as the beta distribution, a new datum apart from the three usual values a (pessimistic), m (most likely) and b (optimistic). To solve this question, when using the beta distribution in the PERT context, we are looking for the maximum similarity with the normal and so it is required that the distribution has the same variance as the normal or its same kurtosis, giving rise to the constant variance and mesokurtic families, respectively. Nevertheless, while this approach can be only applied to the beta distribution for some values in the range of the standardized mode, in the case of the TSP distribution this methodology leads always to a solution. A detailed analysis comparing the beta and TSP distribution based on their PERT means and variances is presented indicating better results for the second.Abstract.The aim of this paper is to include the Two-Sided Power (TSP) distribution in the PERT methodology making use of the advantages that this four-parameter distribution offers. In order to be completely determined, a distribution of this type needs, the same as the beta distribution, a new datum apart from the three usual values a (pessimistic), m (most likely) and b (optimistic). To solve this question, when using the beta distribution in the PERT context, we are looking for the maximum similarity with the normal and so it is required that the distribution has the same variance as the normal or its same kurtosis, giving rise to the constant variance and mesokurtic families, respectively. Nevertheless, while this approach can be only applied to the beta distribution for some values in the range of the standardized mode, in the case of the TSP distribution this methodology leads always to a solution. A detailed analysis comparing the beta and TSP distribution based on their PERT means and variances is presented indicating better results for the second.

An Elicitation Procedure for the Generalized Trapezoidal Distribution with a Uniform Central Stage

Recent advances in computation technology for decision/simulation and uncertainty analyses have revived interest in the triangular distribution and its use to describe uncertainty of bounded input phenomena. The trapezoidal distribution is a generalization of the triangular distribution that allows for the specification of the modal value by means of a range of values rather than a single point estimate. Whereas the trapezoidal and the triangular distributions are restricted to linear geometric forms in the successive stages of the distribution, the generalized trapezoidal (GT) distribution allows for a nonlinear behavior at its tails and a linear incline (or decline) in the central stage. In this paper we develop two novel elicitation procedures for the parameters of a special case of the GT family by restricting ourselves to a uniform (horizontal) central stage in accordance with the central stage of the original trapezoidal distribution.

Proposal of a new distribution in PERT methodology

The aim of this paper is to present the generalized biparabolic distribution (GBP) as a good candidate to be utilized as the distribution underlying to PERT methodology (Malcolm et al. in Oper. Res. 7:646–669, 1959). To do this and following the criteria established by Taha (Investigación de Operaciones, 1981) and Herrerías (Estudios de Economía Aplicada, pp.xa089–112, 1989), we will compare the mean and variance estimates derived from each proposed density function, viz beta, two-sided power (TSP) and GBP distributions. Also we will compare the estimates contributed by the mesokurtic and of constant variance families of the aforementioned distributions. The main conclusion is that the GBP distribution is the most convenient to be used in the PERT methodology because its mean is almost as moderate as that of trapezoidal and its variance is much higher than that of the rest of distributions. As a consequence, it can be stated that the GBP distribution is an alternative to the other four-parameter distributions.

Markowitz's model with Euclidean vector spaces

In this paper a new approach of the Markowitzs model is presented. Indeed, using an inner product, a quantitative and explicit solution for optimal portfolio selection is given. To do this, a scalar product is defined in which allows us to calculate the composition of the optimal portfolio and the variance for a given expected return by means of the distance between the subspace of feasible solutions and the origin of the affine space.

Theory of portfolios: New considerations on classic models and the Capital Market Line

The aim of this paper is to present an alternative method to obtain the efficient portfolio in Roys model starting from the concepts of critical return and risk which are introduced here. This method will permit resolution of the main problem of Roys model, that is to say, the impossibility of obtaining the portfolio in certain situations. The introduction of these new concepts will also allow the detection and solution of a problem associated with the calculation of the Capital Market Line. This work concludes by considering the possibility that investors allocate part of their budget for buying zero-risk assets.

The accounting system as an algebraic automaton

This article aims to present a mathematical model to describe the accounting system. In effect, in the literature on accounting we can find several mathematical models trying to introduce the accounting mechanics but in this article we use the concept of algebraic automaton, which has been successfully used in other economic fields, like finance. Thus, from this new point of view we can construct all definitions and accounting techniques as a particular case of the concept of automaton.

Financial Laws As Algebraic Automata

It is shown that the concept of financial law has the structure of an automaton. Furthermore, a financial law induces a group structure into the monoid of the automaton. The concepts of stationary, stationary of order n, and dynamic financial laws are deduced, proving two algebraic characterizations. Finally the concept of ā‐stationary financial law and some applications are introduced.u2003© 2001 John Wiley & Sons, Inc.

The expert's confidence versus fuzzy probability in the mean activity times computation

Traditionally the beta distribution has been used due to its ease for computing the mean of tasks, times, cash-flows, etc. because, in this case, the expected value is a weighted average of optimistic, most likely and pessimistic values. One of the critiques of PERT has been the difficulty of introducing the level of confidence that the expert has in his own estimate of the modal value. The Belief in Fuzzy Probability Estimations of Time (BIFPET) model uses human judgement instead of stochastic assumptions to determine the duration of a project. Thus, each supervisor responsible of an activity specifies the three values with their respective probabilities. The manager accepts these probabilities or may extend them within a certain range, according to his belief in these likelihoods or the degree of control exercised by himself in achieving the lower bound of the expected value. However, this approach can be implemented in the PERT methodology. Thus, in this paper, we introduce the confidence that the manager has in the most likely value supplied by the supervisor in order to determine the specific beta distribution to model the variable, showing the advantages of this intuitive procedure which moreover can be easily implemented.