Discrete modelling of uncertain continuous systems having an interval structure using higher-order integrators

In this paper, a higher-order integrator approach is proposed to obtain an approximate discrete-time transfer function for uncertain continuous systems having interval uncertainties. Because of the simple algebraic operations of this approach, the resulting discrete model is a rational function of the uncertain parameters. The problem of non-linearly coupled coefficients with exponential nature occurring in the exact discretetime transfer function is therefore circumvented. Furthermore, the interval structure of the uncertain continuous-time system is preserved in the resulting discrete model by using this approach. Formulae to obtain the lower and upper bounds for the coefficients of the discrete interval system are derived, so that digital simulation and design for the uncertain continuous systems can be performed by using the available robustness results in the discrete-time domain.     

A unified scheme of some Nonhomogenous Poisson process models for software reliability estimation

In this paper, we describe how several existing software reliability growth models based on Nonhomogeneous Poisson processes (NHPPs) can be comprehensively derived by applying the concept of weighted arithmetic, weighted geometric, or weighted harmonic mean. Furthermore, based on these three weighted means, we thus propose a more general NHPP model from the quasi arithmetic viewpoint. In addition to the above three means, we formulate a more general transformation that includes a parametric family of power transformations. Under this general framework, we verify the existing NHPP models and derive several new NHPP models. We show that these approaches cover a number of well-known models under different conditions.     

Economic production quantity (EPQ) models under an imperfect production process with shortages backordered

In this paper, we develop economic production quantity (EPQ) models to determine the optimal production lot size and backorder quantity for a manufacturer under an imperfect production process. The imperfect production process is characterised by the fraction of defective items at the time of production γ. The paper considers different cases of the EPQ model depending on (1) whether γ is known with certainty or is a random variable, and (2) whether imperfect items are drawn from inventory (a) as they are detected, (b) at the end of each production period or (c) at the end of each production cycle. Straightforward convexity results are shown and closed-form solutions are provided for the optimal order and backorder quantities for each of the cases we considered. We provide two numerical examples: one in which the defective probability follows a uniform distribution and the second which we assume follows a beta distribution, to illustrate the effects of yield variability and timing of the withdrawal of defectives on the optimal solutions. We obtain similar results for both numerical examples, which show that both the yield variability and the withdrawal timing are not critical factors.     

星载软件可靠性模型应用研究及实践

为定量评估软件的可靠性指标,介绍了利用软件可靠性模型评估软件可靠性的过程和方法;针对某星载嵌入式软件的失效趋势,根据模型的选择原则和方法,以及模型的预测质量的对比,最终选择了指数模型作为可靠性评估模型。对该软件在轨运行情况进行了可靠性评估,开展了基于该软件可靠性测试数据的可靠性评估,评估结果给出了该软件的可靠性水平。     

Optimal ordering and production policy for a recoverable item inventory system with learning effect

This article presents two models for determining an optimal integrated economic order quantity and economic production quantity policy in a recoverable manufacturing environment. The models assume that the unit production time of the recovery process decreases with the increase in total units produced as a result of learning. A fixed proportion of used products are collected from customers and then recovered for reuse. The recovered products are assumed to be in good condition and acceptable to customers. Constant demand can be satisfied by utilising both newly purchased products and recovered products. The aim of this article is to show how to minimise total inventory-related cost. The total cost functions of the two models are derived and two simple search procedures are proposed to determine optimal policy parameters. Numerical examples are provided to illustrate the proposed models. In addition, sensitivity analyses have also been performed and are discussed.     

The EPQ model under conditions of two levels of trade credit and limited storage capacity in supply chain management

An inventory problem involves a lot of factors influencing inventory decisions. To understand it, the traditional economic production quantity (EPQ) model plays rather important role for inventory analysis. Although the traditional EPQ models are still widely used in industry, practitioners frequently question validities of assumptions of these models such that their use encounters challenges and difficulties. So, this article tries to present a new inventory model by considering two levels of trade credit, finite replenishment rate and limited storage capacity together to relax the basic assumptions of the traditional EPQ model to improve the environment of the use of it. Keeping in mind cost-minimisation strategy, four easy-to-use theorems are developed to characterise the optimal solution. Finally, the sensitivity analyses are executed to investigate the effects of the various parameters on ordering policies and the annual total relevant costs of the inventory system.     

How to make the replenishment and payment strategy under flexible two-part trade credit

In the existing inventory models concerning the two-part trade credit, a common assumption is that the retailer either pays for all the ordered items within a short permissible delay period and receives a cash discount or pays for all the ordered items within a long permissible delay period at the regular price. In this paper, this unrealistic assumption is relaxed. We assume that the retailer may pay any fraction of the purchase cost within the short permissible delay period and receives a cash discount and then the rest is paid within the long permissible delay period. A decision model is proposed for a retailer to determine the optimal ordering policy and payment plan. The closed-form optimal solution to the model is developed and analyzed. Numerical studies show that a retailer can obtain more benefits from the proposed payment plan than from the extreme payment plan in the existing literature.     

Effective confidence interval estimation of fault-detection process of software reliability growth models

The quantitative evaluation of software reliability growth model is frequently accompanied by its confidence interval of fault detection. It provides helpful information to software developers and testers when undertaking software development and software quality control. However, the explanation of the variance estimation of software fault detection is not transparent in previous studies, and it influences the deduction of confidence interval about the mean value function that the current study addresses. Software engineers in such a case cannot evaluate the potential hazard based on the stochasticity of mean value function, and this might reduce the practicability of the estimation. Hence, stochastic differential equations are utilised for confidence interval estimation of the software fault-detection process. The proposed model is estimated and validated using real data-sets to show its flexibility.     

A fuzzy framework for coordinating pricing and inventory policies for deteriorating items under retailer partial trade credit financing

In this paper, we proposed a generalized economic order quantity (EOQ) – based inventory model using a trade credit policy in a fuzzy sense. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer, in turn, partially provides customers a permissible delay period N. This study considers fuzzy EOQ model to allow for: (1) selling price dependent demand rate which is imprecise in nature, (2) a profit maximization objective and (3) an imprecise holding cost, ordering cost, purchasing cost, interest earned and interest charged rate. Besides, the cases N ? M and N ? M are explored thoroughly. The objective function for the retailer in fuzzy sense is defuzzified using Modified Graded Mean Integration Representation Method. For the defuzzified objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.     

Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period

A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the trade credit scenario such as permissible delay in payments. In real life business, we observe that the demand is a function of both the selling price and credit period rather than the constant demand. Incorporating this demand function to the retailer of a supply chain, we develop an EPQ – based model for perishable items under two-echelon trade financing. The purpose of this paper is to maximize the profit by determining the optimal selling price, credit period and replenishment time. It is shown that the model developed by Jaggi et al. [Jaggi, J. K., Goyal, S. K., & Goel, S. K., 2008. Retailer’s optimal replenishment decisions with creditlinked demand under permissible delay in payments. European Journal of Operational Research, 190, 130–135] can be treated as a special case of this paper. Finally, through numerical examples, sensitivity analysis shows the influence of key model parameters.     

基于Web的动态几何软件领域模型及其应用

动态几何软件以其动态、直观的特点广泛应用于几何约束作图。针对数据结构缺乏对动态几何领域内可复用的抽象描述的问题,提出一种动态几何软件领域模型的设计方法。首先经过领域分析来识别并划分出最基本的上下文边界,然后通过领域模型设计得到动态几何软件核心领域模型,最后在体系结构建模过程中,在纵向与横向两个维度对动态几何软件进行解耦。实验结果表明,利用该领域模型设计方法研发的动态几何软件能正确地处理图形在临界位置退化的情形。该模型表达的领域知识同时适用于二维及三维的动态几何软件,并支持对不同设备分别设计布局与交互,实现了领域知识的高层次复用。     

The effect of inflation‐induced demand and trade credit on ordering policy of exponentially deteriorating and imperfect quality items

This paper develops an inventory model for exponentially deteriorating and imperfect quality items when a trade credit is offered by the supplier. As it has been seen that the changing rates of inflation do not just have an effect on the ordered quantity but they also stimulate the demand for products, so this paper considers the combined effect of inflation, deterioration, imperfect quality and ‘’ trade credit policy on the economic ordering policy using a discounted cash flow (DCF) approach. In the study, mathematical models are derived under two different circumstances, i.e., Case I: When the discount offered by the supplier with a condition of making the payment by a specified period, M₁ is accepted; and Case II: Discount is not taken and instead the payment is made by the end of the full credit period M (M > M₁). Finally, a numerical example demonstrating the applicability of the proposed model has been included.     

Determination of production run time and warranty length under system maintenance and trade credits

Manufacturers offer a warranty period within which they will fix failed products at no cost to customers. Manufacturers also perform system maintenance when a system is in an out-of-control state. Suppliers provide a credit period to settle the payment to manufacturers. This study considers manufacturer's production and warranty decisions for an imperfect production system under system maintenance and trade credit. Specifically, this study uses the economic production quantity to model the decisions under system maintenance and trade credit. These decisions involve how long the production run time and warranty length should be to maximise total profit. This study provides lemmas for the conditions of optimality and develops a theorem and an algorithm for solving the problems described. Numerical examples illustrate the solution procedures and provide a variety of managerial implications. Results show that simultaneously determining production and warranty decisions is superior to only determining production. This study also discusses the effects of the related parameters on manufacturer's decisions and profits. The results of this study are a useful reference for managerial decision-making and administration.     

基于PSO的软件可靠性模型参数估计方法

软件可靠性建模是一个重要的研究领域,现有的软件可靠性模型基本上是非线性函数模型,估计这些模型的参数比较困难。粒子群优化是一类适合求解非线性优化问题的随机优化方法,提出一种基于粒子群优化的软件可靠性模型估计参数方法,该方法的关键是构造合适的适应函数。用该方法分别估计了5个实际软件系统的指数软件可靠性模型以及对数泊松执行时间模型,实验结果表明：该方法参数估计的精度高,对模型的适应性强。     

A comparison of time domains for software reliability models

A comparison of time domains (i.e., execution time vs. calendar time) is made for software reliability models, with the purpose of reaching some general conclusions about their relative desirability. The comparison is made by using a generic failure intensity function that represents a large majority of the principal models. The comparison is based on how well the function fits the estimated failure intensity, where the failure intensity is estimated with respect to both kinds of time. The failure intensity in each time domain is examined for trends. Failure intensity estimates are calculated from carefully collected data. The execution time domain is found to be highly superior to the calendar time domain.     

基于结构的软件可靠性建模研究进展

软件体系结构作为提高软件系统质量、支持复杂软件开发和复用的重要手段,已经成为软件工程的一个重要研究领域,软件结构的研究对软件可靠性度量也带来的新的问题。分析和总结近年来基于软件体系结构的可靠性建模方法,并对其今后若干研究方向进行了探讨。     

Optimal manufacturer's pricing and lot-sizing policies under trade credit financing

In this paper, we extend Goyal's economic order quantity (EOQ) model to allow for the following four important facts: (1) the manufacturer's selling price per unit is necessarily higher than its unit cost, (2) the interest rate charged by a bank is not necessarily higher than the manufacturer's investment return rate, (3) the demand rate is a downward‐sloping function of the price, and (4) an economic production quantity (EPQ) model is a generalized EOQ model. We then establish an appropriate EPQ model accordingly, in which the manufacturer receives the supplier trade credit and provides the customer trade credit simultaneously. As a result, the proposed model is in a general framework that includes numerous previous models as special cases. Furthermore, we provide an easy‐to‐use closed‐form optimal solution to the problem for any given price. Finally, we develop an algorithm for the manufacturer to determine its optimal price and lot size simultaneously.     

Inventory policy of a deteriorating item with variable demand under trade credit period

In this paper, an inventory model of a deteriorating item with stock and selling price dependent demand under two-level credit period has been developed. Here, the retailer enjoys a price discount if he pays normal purchase cost on or before the first level of credit period, or an interest is charged for the delay of payments. In return, retailer also offers a fixed credit period to his customers to boost the demand. In this regard, the authors develop an EOQ model incorporating the effect of inflation and time value of money over all the costs. Keeping the business of seasonal products in mind, it is assumed that planning horizon of business is random and follows a normal distribution with a known mean and standard deviation. The model is formulated as retailer’s profit maximization problem for both crisp and fuzzy inventory costs and solved using a modified Genetic Algorithm (MGA). This algorithm is developed following fuzzy age based selection process for crossover and gradually reducing mutation parameter. For different values of MGA parameters, optimum results are obtained. Numerical experiments are performed to illustrate the model.     

An optimal production lot-sizing problem for an imperfect process with imperfect maintenance and inspection time length

This article develops an integrated model in considering the situations of an imperfect process with imperfect maintenance and inspection time for the joint determination of both economic production quantity (EPQ) and preventive maintenance (PM). This imperfect process has a general deterioration distribution with increasing hazard rate. Even with periodic PM, such a production system cannot be recovered as good as new. This means that the system condition depends on how long it runs. Also, the PM level can be distinct due to the maintenance cost. For convenience, it is assumed the age of system is reduced in proportional to the PM level. Further, during a production cycle, we need an inspection to see if the process is in control. This inspection might demand a considerable amount of time. In this article, we take PM level and inspection time into consideration to optimise EPQ with two types of out-of-control states. To see how the method works, we use a Weibull shock model to show the optimal solutions for the least costs.     

基于蚁群算法的软件可靠性模型参数估计方法

由于软件可靠性模型大多是非线性模型,导致其参数难于估计。总结了常用的软件可靠性模型的参数估计方法,提出一种基于蚁群算法的可靠性模型参数估计方法。通过对Musa软件可靠性模型分类方案中三个不同类型模型（G-O模型、Weibull模型以及M-O模型）的实验,发现本算法对不同模型具有很好的适应性,解决了应用传统数值计算方法时的无法收敛问题;与粒子群算法相比,本算法的收敛速度比粒子群算法快一倍以上,且对于部分实验对象的拟合结果精度比粒子群算法高一个数量级以上。