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. 89–112, 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.     

A dependent project evaluation and review technique: A Bayesian network approach

The Program Evaluation and Review Technique (PERT) dates back to 1959. This method evaluates the uncertainty distribution of a project’s completion time given the uncertain completion times of the activities/tasks comprised within it. Each activity’s uncertainty was defined originally by a unique two parameter beta PERT distribution satisfying what is known to be the PERT mean and PERT variance. In this paper, a three-parameter PERT family of bounded distributions is introduced satisfying that same mean and variance, generalizing the beta PERT distribution. Their additional flexibility allows for the modeling of statistical dependence in a continuous Bayesian network, generalizing in turn the traditional PERT procedure where statistical independence is assumed among beta PERT activity durations. Through currently available Bayesian network software and the construction of that PERT family herein, the coherent monitoring of remaining project completion time uncertainty given partial completion of a project may become more accessible to PERT analysts. An illustrative example demonstrating the benefit of monitoring of remaining project completion time uncertainty as activities complete in that Bayesian fashion shall be presented, including expressions and algorithms for the specification of the three prior parameters for each activity in the project network to adhere to classical the PERT mean and PERT variance and a degree of statistical dependence between them.     

航空复杂产品项目活动时间混合分布估算模型的改进

航空复杂产品开发项目具有复杂性、随机性、多目标性的特点,并且产品本身小批量、多品种的生产模式使得无法大量积累历史数据。因此,航空复杂产品项目活动时间存在高度的不确定性。如何能够更加合理、准确的描述其项目活动时间,对优化航空复杂产品的研制过程、缩短研制周期以及降低研制成本具有重大的意义。本文在Hahn基于PERT所建立的Beta Rectangular混合分布模型的基础上,保留其期望值的表达式,对其方差表达式进行改进。按照Malcolm、José等人的思想,将最可能值m与Beta分布的众数相对应,并且考虑其对方差的影响。仅保留PERT的一个假设,使方差的推导更多地依照于Beta分布而不是较多的近似。仿真结果表明,改进后的混合分布模型不但更具柔性,而且可以更加科学准确地描述航空复杂产品调度中高度不确定的任务持续时间状况。     

An alternative for robust estimation in Project Management

Recently, Hahn (2008) has proposed the mixture between the uniform and the beta distributions as an alternative to the beta distribution in PERT methodology which allows for varying amounts of dispersion and a greater likelihood of more extreme tail-area events. However, this mixture lacks a closed cumulative distribution function expression and its parameters remain a difficult interpretation. In addition, the kurtosis limit of the beta distribution is 3. Due to their higher kurtosis and easier elicitation we consider the Two-Sided Power and the Generalized Biparabolic distributions good candidates for applying Hahn’s idea (2008) and for building the Uniform-Two Sided Power (U-TSP) and the Uniform-Generalized Biparabolic (U-GBP) distributions. Using the same example from Hahn (2008) we are going to demonstrate that we can obtain more accuracy and a greater likelihood of more extreme tail area events. These distributions could be applied in other heavy-tailed phenomena which are usual in business contexts.     

Security improvement on an anonymous key agreement protocol based on chaotic maps

In 2009, Tseng et al. proposed a password sharing and chaotic map based key agreement protocol (Tseng et al.’s protocol). They claimed that the protocol provided mutual authentication between a server and a user, and allowed the user to anonymously interact with the server to establish a shared session key. However, in 2011, Niu et al. have proved that Tseng et al.’s protocol cannot guarantee user anonymity and protocol security when there is an internal adversary who is a legitimate user. Also it cannot provide perfect forward secrecy. Then Niu et al. introduced a trust third party (TTP) into their protocol designing (Niu et al.’s protocol). But according to our research, Niu et al.’s protocol is found to have several unsatisfactory drawbacks. Based on reconsidering Tseng et al.’s protocol without introducing TTP, we give some improvements to meet the original security and performance requirements. Meanwhile our proposed protocol overcomes the security flaws of Tseng et al.’s protocol.     

On the Distribution of Activity Time in PERT

In PERT analysis the activity-time distribution is assumed to be a beta distribution, and the mean and variance of the activity time are estimated on the basis of the ‘pessimistic’, ‘most likely’ and ‘optimistic’ completion times, which are subjectively determined by an analyst. In this paper, on the basis of the study of the PERT assumptions, we present an improvement of these estimates. It is also shown that, by means of additional reasonable assumptions, the activity-time distribution in PERT analysis may be essentially simplified.     

The robustness of mean and variance approximations in risk analysis

This paper examines further the problem of estimating the mean and variance of a continuous random variable from estimates of three points within the distribution, typically the median or mode and two extreme fractiles. The problem arises most commonly in PERT and risk analysis where it can usually be assumed that the distribution in question is bell-shaped and positively skewed, often typified by a Beta distribution. Over the years, a number of alternative approximations have been proposed, usually as modifications to the original PERT formulae. The accuracy of a number of these approximations is investigated based not only on a Beta distribution, but also for three other commonly used bell-shaped, positively skewed distributions, namely the Gamma, Lognormal and F distributions. It is shown that a balanced weighted average of the median and the 4% fractiles provides a consistent estimator of the distribution mean across all four distributions. Furthermore, reasonably accurate estimates of the variance can also be obtained by treating the three fractiles as defining an equivalent discrete distribution with the same probability weight as in the formula for the mean.     

计划评审技术的改进

由于计划评审技术存在的不足,导致项目管理存在一定的偏差.对计划评审技术中的活动期望时间和方差的公式做了改进,从而提高了项目管理的精确性.还指出计划评审技术中活动期望时间和方差的计算公式,实际上是改进后公式的特殊情形.     

The Completion Time of PERT Networks

This paper proposes an approximation for the completion time mean and variance of PERT networks in which the durations of individual activities are Normally and independently distributed. The proposed approximation involves the manipulation of distribution parameters only and hence can be computed manually, even for large networks.     

Quantifying the bullwhip effect in a supply chain with stochastic lead time

In a recent paper, Dejonckheere, Disney, Lambrecht, and Towill [European Journal of Operational Research 147 (2003) 567] used control systems engineering (transfer functions, frequency response, spectral analysis) to quantify the bullwhip effect. In the present paper, we, like Chen, Ryan, Drezner, and Simchi-Levi [Management Science 46 (2000) 436], use the statistical method. But our method extends Dejonckheere et al. and Chen et al. in that we include stochastic lead time and provide expressions for quantifying the bullwhip effect, both with information sharing and without information sharing. We use iid demands in a k-stage supply chain for both. By contrast, Chen et al. provide lower bounds using autoregressive demand for information sharing and for information not sharing (with zero safety factor for stocks). Dejonckheere et al. validate Chen et al.’s results for a 2-stage supply chain without information sharing, using both autoregressive and iid normally distributed demands. We estimate the mean and variance of lead-time demand (LTD) from historical LTD data, rather than from the component period demands and lead time. Nevertheless, we also calculate the variance amplification like Chen et al., but with gamma lead times. With constant lead times, which Chen et al. used, our method yields lower variance amplification. As for the effect of information, we find that the variance increases nearly linearly in echelon stage with information sharing but exponentially in echelon stage without information sharing.     

The simplified solution procedure for deteriorating items under stock-dependent demand and two-level trade credit in the supply chain management

Min et al. [1] (J. Min, Y.W. Zhou, J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Appl. Math. Model. 34 (2010) 3273–3285.) develop an inventory model for deteriorating items under stock-dependent demand and two-level trade credit. They provide the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions that could maximize the retailer’s average profit per unit time. Basically, their paper is correct and interesting. Recently, several researchers have been showing a huge interest in developing simple and easy to implement solution procedures in management science. Therefore this paper indicates that Min et al.’s solution procedure can be further improved and simplified. So, the main purpose of this paper is to present simple and easy to understand solution procedures to locate the optimal solutions of an inventory model that considers deteriorating items under stock-dependent demand and two-level trade credit.     

Portfolio selection and duality under mean variance preferences

This paper uses duality to analyze an investor’s behavior in a n-asset portfolio selection problem when the investor has mean variance preferences. The indirect utility and wealth requirement functions are used to derive Roy’s identity, Shephard’s lemma and the Slutsky equation. In our simple Slutsky equation the income effect is characterized by decreasing absolute risk aversion (DARA) and the substitution effect is always positive [negative] with respect to an asset’s holding if the asset’s mean return [risk] increases. Substitution effect and income effect work in the same direction presupposed mean variance preferences display DARA.     

A simplified expression of the Shapley function for fuzzy game

In this paper, a simplified expression of the Shapley function for games with fuzzy coalition is proposed, which can be regarded as the generalization of Shapley functions defined in some particular games with fuzzy coalition. The simplified expression of the Shapley function is compared with two definitions established by Butnariu, Tsurumi et al. A conclusion is drawn that the simplified expression of the Shapley function is equivalent to Butnariu’s definition when characteristic function is a game with proportional values, and is equivalent to Tsurumi’s definition when characteristic function is a game with Choquet integral forms. Furthermore, from an angle of interaction between two participation levels, the properties of the two games defined by Butnariu and Tsurumi are respectively studied.     

A dexterous randomized response model for estimating a rare sensitive attribute using Poisson distribution

Taking the clue from the pioneer work of Land et al. (2011), we have suggested an unbiased estimator of the mean number of persons possessing the rare sensitive attribute in presence of the known proportion of persons possessing a rare unrelated attribute. The variance expression is derived. The empirical studies performed and graphical representation show that the new model is more efficient than the one recently considered by Land et al. (2011).     

The simplified solution algorithm for an integrated supplier-buyer inventory model with two-part trade credit in a supply chain system

Ho et al. [Ho, C.H., Ouyang, L.Y., Su, C.H., 2008. Optimal pricing, shipment and payment policy for an integrated supplier-buyer inventory model with two-part trade credit, European Journal of Operational Research 187, 496-510] discussed the integrated inventory model with two-part trade credit and presented an algorithm to solve it. Basically, Ho et al.’s inventory model is correct and interesting. However, this paper indicates that the solution algorithm described in Ho et al. (2008) can be simplified further. So, this paper can not only derive the optimally closed-form formulations for the optimal numbers of shipments but also develop different algorithms to improve those in Ho et al. (2008). Numerical examples illustrate that the algorithm to locate the optimal solution is rather accurate and rapid.     

混合von Mises 模型的参数估计

有限混合von Mises模型在天文学、生物学、地理和医药等许多领域都有重要的应用．可是,不论样本量有多大,此模型的似然函数都是无界的．因此,参数的最大似然估计(MLE)是不相合的．我们发现,与混合正态模型一样,上述困难可以通过引入关于分布浓度参数的一个惩罚函数或对参数空间添加适当的约束来克服．在此文中,我们从理论上证明了这两种方法是可行的,相应的参数估计是强相合的,且是渐近有效的．我们还通过计算机模拟来探讨这些新方法在有限样本情况下的统计性质,并与现有的矩估计作了比较．结果发现,惩罚极大似然估计在均方误差方面表现最佳．最后我们还分析了一组实际数据,以进一步介绍新的估计方法．     

Comparison of three semiparametric methods for estimating dependence parameters in copula models

Three semiparametric methods for estimating dependence parameters in copula models are compared, namely maximum pseudo-likelihood estimation and the two method-of-moment approaches based on the inversion of Spearman’s rho and Kendall’s tau. For each of these three asymptotically normal estimators, an estimator of their asymptotic (co)variance is stated in three different situations, namely the bivariate one-parameter case, the multivariate one-parameter case and the multivariate multiparameter case. An extensive Monte Carlo study is carried out to compare the finite-sample performance of the three estimators under consideration in these three situations. In the one-parameter case, it involves up to six bivariate and four-variate copula families, and up to five levels of dependence. In the multiparameter case, attention is restricted to trivariate and four-variate normal and t copulas. The maximum pseudo-likelihood estimator appears as the best choice in terms of mean square error in all situations except for small and weakly dependent samples. It is followed by the method-of-moment estimator based on Kendall’s tau, which overall appears to be significantly better than its analogue based on Spearman’s rho. The simulation results are complemented by asymptotic relative efficiency calculations. The numerical computation of Spearman’s rho, Kendall’s tau and their derivatives in the case of copula families for which explicit expressions are not available is also investigated.     

Speeding up continuous GRASP

Continuous GRASP (C-GRASP) is a stochastic local search metaheuristic for finding cost-efficient solutions to continuous global optimization problems subject to box constraints (Hirsch et al., 2007). Like a greedy randomized adaptive search procedure (GRASP), a C-GRASP is a multi-start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion. In this paper, we describe several improvements that speed up the original C-GRASP and make it more robust. We compare the new C-GRASP with the original version as well as with other algorithms from the recent literature on a set of benchmark multimodal test functions whose global minima are known. Hart’s sequential stopping rule (1998) is implemented and C-GRASP is shown to converge on all test problems.     

A simple cost minimization procedure for the (Q, r) inventory system with a specified fixed cost per stockout occasion

For fast-moving A items, Silver et al. [E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Production Planning and Scheduling, third ed., John Wiley & Sons, New York, NY, 1998] explore the (Q, r) inventory system with a specified cost per stockout occasion. However, a number of difficulties have impeded the implement of their solution procedure. That is, the total relevant cost function is not convex in general, so the convergence of the Silver et al.’s solution procedure to the optimal solution of the total relevant cost function is not necessarily true. An easier and more accurate solution procedure is proposed to overcome the shortcoming of the Silver et al.’s solution procedure.