We present an extension of vendor-managed inventory (VMI) problems by considering advertising and pricing policies. Unlike the results available in the literature, the demand is supposed to depend on the retail price and advertising investment policies of the manufacturer and retailers, and is a random variable. Thus, the constructed optimization model for VMI supply chain management is a stochastic bi-level programming problem, where the manufacturer is the upper level decision-maker and the retailers are the lower-level ones. By the expectation method, we first convert the stochastic model into a deterministic mathematical program with complementarity constraints (MPCC). Then, using the partially smoothing technique, the MPCC is transformed into a series of standard smooth optimization subproblems. An algorithm based on gradient information is developed to solve the original model. A sensitivity analysis has been employed to reveal the managerial implications of the constructed model and algorithm: (1) the market parameters of the model generate significant effects on the decision-making of the manufacturer and the retailers, (2) in the VMI mode, much attention should be paid to the holding and shortage costs in the decision-making.

Simulation models of nutrient utilisation ignore that variation in pig system components can influence the predicted mean and variance of the performance of a group of pigs. The objective of this study was to develop a methodology to investigate how variation in feed composition would (a) affect the outputs of a nutrient utilisation model and (b) interact with variation that arises from the traits of individual pigs. We used a P intake and utilisation model to address these characteristics. Introduction of stochasticity gave rise to a number of methodological challenges – for example, how to generate variation in both feed composition and pigs and account for correlations between ingredients when modelling variation associated with feed mixing efficiency. Introducing variation in feed composition and pig phenotype resulted in moderate decreases in mean digested, retained and excreted P predicted for a population of pigs and an increase in their associated CV. A lower percentage of pigs in the population were predicted to meet their requirements during the feeding period considered, by comparison with the no-variation scenario. Variation in feed ingredient composition contributed more to performance variation than variation due to mixing efficiency. When variations in both feed composition and pig traits were considered, it was the former rather than the latter that had the dominant influence on variability in pig performance. The developed framework emphasises the consequences of random variability on the predictions of nutrient utilisation models. Such consequences will have a significant impact on decisions about management strategies such as feeding that are subject to variation.

We develop and test an intuitively simple dynamic network model to describe the type of time-varying connectivity structure present in many technological settings. The model assumes that nodes have an inherent hierarchy governing the emergence of new connections. This idea draws on newly established concepts in online human behaviour concerning the existence of discussion catalysts, who initiate long threads, and online leaders, who trigger feedback. We show that the model captures an important property found in e-mail and voice call data – ‘dynamic communicators’ with sufficient foresight or impact to generate effective links and having an influence that is grossly underestimated by static measures based on snaphots or aggregated data.

The development of stochastic investment models for actuarial and investment applications has become an important area of interest to actuaries. This paper reports the application of some techniques of modern time series and econometric analysis to Australian inflation, share market and interest rate data. It considers unit roots, cointegration and state space models. Some of the results from this analysis are not reflected in the published stochastic investment models.

There has recently been some debate about the usefulness of financial economics to the actuarial profession. The consensus appears to be that the techniques are potentially valuable, but the published material is not quite ‘oven ready’. Some development work is required before the material can be applied.

The author has carried out much of the necessary development work on behalf of various clients over the last few years. On some occasions the results have conflicted with more conventional actuarial methods. The main aim of the paper is to bring the new techniques into the public domain so that they can be properly discussed by the profession, and adopted more widely if appropriate.

The paper contains a number of worked examples using techniques developed by financial economists. The author has listed the computer code which generated the examples and deposited copies on the Internet, so that others can explore the issues with a minimal development overhead.

The paper examines the suitability of the stable family of distributions with the Maturity Guarantees Working Party's stochastic investment model (Ford et al, 1980). It then examines the effect of replacing the Gaussian assumption made by the working party with a more general stable distribution. It also explains how the appropriate stable distribution can be fitted.

The paper compares and contrasts the theories and methodologies of financial economics with generalisations drawn from practical experience of institutional investment management. A Dynamic Equilibrium Model is put forward as a much more realistic theoretical framework than the Sharpe Diagonal Model, a downside theory of risk is developed from sports analogies, and a detailed model of human behaviour, incorporating benchmark levels of ‘unintelligent’, ‘intelligent’, ‘optimal’ and ‘rational’ behaviour, is suggested as a replacement for the very restrictive simplifying assumptions of existing theories. On the basis of a classification system derived from parallels with the history of astronomy, it is concluded that, as a scientific framework, the current theories of financial economics are seriously incomplete, the two crucial areas of incompleteness being the absence of a downside theory of risk and the absence of a realistic model of human behaviour under conditions of risk and uncertainty.

The impact that asset allocation has on the economic capital and the risk adjusted performance of financial services firms is considered in this article. A stochastic modelling approach is used in conjunction with a life insurance annuity firm illustrative example. It is shown that traditional solvency driven deterministic approaches to financial services firm asset allocation can yield sub optimal results in terms of minimising economic capital or maximising risk adjusted performance. Our results challenge the conventional wisdom that the assets backing life insurance annuities and financial services firm capital should be invested in low risk, bond type, assets. Implications for firms, customers, capital providers and regulators are discussed.

The impact that capital structure and capital asset allocation have on financial services firm economic capital and risk adjusted performance is considered. A stochastic modelling approach is used in conjunction with banking and insurance examples. It is demonstrated that gearing up Tier 1 capital with Tier 2 capital can be in the interests of bank Tier 1 capital providers, but may not always be so for insurance Tier 1 capital providers. It is also shown that, by allocating a bank or insurance firm’s Tier 1 and Tier 2 capital to higher yielding, more risky assets, risk adjusted performance can be enhanced. These results are particularly pertinent with the advent of the new Basel 2 and Solvency 2 risk based capital initiatives, for banks and insurers respectively.

Real-time and near real-time precise GPS positioning requires shorter GPS solution convergence time. Residual tropospheric delay, which exists as a result of the limitations of existing tropospheric correction models, is a limiting factor for quick GPS solution convergence. This paper proposes a new approach to tropospheric delay modelling, which overcomes the limitations of existing models. In this approach, the bulk of the tropospheric delay is accounted for using the NOAA-generated tropospheric correction model, while the residual tropospheric delay component is accounted for stochastically. First, the NOAA tropospheric correction model is used to generate daily time series of zenith total tropospheric delays (ZTDs) at ten IGS reference stations spanning North America for many days in 2006. The NOAA ZTDs are then compared with the new highly-accurate IGS tropospheric delay product to obtain daily residual time series at 5 minute intervals. Finally, the auto-covariance functions of the daily residual tropospheric delay series are estimated at each of the ten reference stations and then used to find the best empirical covariance function in the least squares sense. Of the three potential covariance functions examined, it is shown that the exponential cosine function gives the best fit most of the time, while the second-order Gauss-Markov model gives the worst fit. The first-order Gauss-Markov fits are close to those of the exponential cosine. Additionally, the model coefficients seem to be season independent, but change with geographical location.

A stochastic simulation was used to investigate the value of crossbreeding information in a two line crossbreeding system in poultry under different genetic scenarios. Populations consisted of 25 sires and 250 dams, which were mated to produce both purebred and crossbred progeny. The next generation parents were selected either based on purebred information (PLS) (sib means, and own performance for females) or additional crossbred sib means were included (CCPS). The trait under selection was controlled by 20 loci with varying degree of dominance. Pure lines differed in initial allele frequencies. If the trait was controlled by loci with partial dominance, little or no extra benefit was obtained from including crossbred information over the pure line information. Under complete dominance and overdominance CCPS outperformed PLS. As a practical rule, CCPS is better than PLS if the ratio between dominance variance and total genetic variance is around 0·3 or higher. In this case the most probable cause of the dominance variation is loci with full or overdominance.

A model of a birthdate distribution for a herd of beef cows is constructed using the probability distributions of the variables that affect reproduction in the cow — anoestrous interval, oestrous cycle length, conception to each oestrus, gestation length, period of mating and the prior calving frequency distribution. The model is general and can be reparamaterized to deal with issues such as intervention to synchronize oestrous cycles among cows in the herd by changing the form of the relevant probability distributions.

The model is applied to the question of what time to begin mating in a herd of beef cows. The average calf live weight at day 200, herd conception rate and proportion of cows calving before the planned start of calving were calculated from the model output. The model parameters given by the anoestrous period, conception rate to each oestrus and the regression between prior calving date and anoestrous period, were varied in a factorial design to investigate a range of circumstances found on a farm. Prior calving distributions were generated by random sampling from eight actual calving frequency distributions.

Generally starling mating earlier produced an advantage in terms of extra calf live weight and herd conception rate. However, the proportion of the herd calving earlier than expected increased with early mating. Thus, the feasibility of early mating depends on the cost to the farmer of dealing with early calving cows as well as the advantage of heavier older calves.

Altering the fixed parameters in the model (variances and covariances, prior calving distributions, mating period) to accommodate the circumstances of herds run under different conditions may produce different results. Model structure allows easy alteration of these parameters and also the introduction of different probability distributions for some variables. This might be necessary to model oestrous synchronization and artificial insemination, issues not considered in this paper.

Stochastic simulation is used to compare different systems of mating to reduce rates of inbreeding in selection programmes with phenotypic or animal model best linear unbiased prediction (BLUP) evaluation. Compensatory mating (the mating between individuals from the largest selected families to individuals from the smallest) turns out to be proportionately about 0-30 more effective than minimum coancestry matings for situations with low rates of inbreeding, such as phenotypic selection or high population size, although the advantage is less apparent if common environmental effects are important. A modification of this system of mating is proposed which can be applied for overlapping generations, and this is shown to reduce rates of inbreeding proportionately by about 0-50 more than for discrete generations. Under high inbreeding, however, such as for BLUP selection and small population size, minimum coancestry matings, or even avoidance of sib matings are more effective. A procedure combining compensatory and minimum coancestry matings is also simulated and gives the largest reductions in the rate of inbreeding. The effects of these and other systems of mating on the rate of inbreeding are shown to occur through a reduction in the cumulative effect of selection and a deviation from Hardy-Weinberg proportions.

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992).

In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

In an earlier paper (Bartholomew (1977)) the problem of maintaining a grade (or age) structure by controlling the recruitment flows was discussed. Here we carry out a parallel analysis for the case where control is by promotion. We show how to calculate the probability that a structure can be maintained and investigate the performance of several strategies for keeping a structure as near as possible to that desired. As before, the results demonstrate the limitations of a deterministic analysis of the problem.

Grade and age structures in manpower systems are often far from ideal. This fact raises the question of how the flows of people — and particularly the recruitment flow — should be controlled in order to attain and maintain a more desirable structure. The problem has received considerable attention from a deterministic point of view. This paper adopts a stochastic approach to the study of maintainability and shows, among other things, that the problem is more subtle than the deterministic analysis suggests.

A stochastic model for inter-plant competition in a single-species population is presented, in which the probability that a given plant will be eliminated is defined in terms of the sizes and relative spatial locations of rival plants. The properties of the model are then investigated and illustrations of particular cases given, with particular emphasis on the distribution of the sizes of surviving plants. Empirical comparisons are made between the properties of the model and some experimental results due to Ford (1975b). Some possible extensions to the basic model are suggested.