In most classical scheduling models, it is assumed that a job is dispatched to a customer immediately after its processing completes. In many practical situations, however, a set of delivery dates may be fixed before any jobs are processed. This is particularly relevant where delivery is an expensive or complicated operation, for example, as with heavy machinery. A similar situation arises where customers find deliveries disruptive and thus require them to be made within a limited time interval that repeats periodically. A third possibility is that a periodic business function, for example, the supplier's billing cycle, effectively defines a delivery date, and includes all jobs that have been completed since the previous billing cycle. These situations are not adequately represented by classical scheduling models. We consider a variety of deterministic scheduling problems in which a job is dispatched to a customer at the earliest fixed delivery date that is no earlier than the completion time of its processing. Problems where the number of delivery dates is constant, and others where it is specified as part of data input, are studied. For almost all problems considered, we either provide an efficient algorithm or establish that such an algorithm is unlikely to exist. By doing so, we permit comparisons between the solvability of these fixed delivery date problems and of the corresponding classical scheduling problems.

South Africa is experiencing an increasing burden of noncommunicable diseases (NCDs). There is evidence of co-morbidity of several NCDs at small geographical areas in the country. However, the extent to which this applies to joint spatial autocorrections of NCDs is not known. The objective of this study was to derive and quantify multivariate spatial autocorrections for NCDrelated mortality in South Africa. The study used mortality attributable to cerebrovascular, ischaemic heart failure and hypertension captured by the country’s Department of Home Affairs for the years 2001, 2007 and 2011. Both univariate and pairwise spatial clustering measures were derived using observed, empirical Bayes smoothed and age-adjusted standardised mortality rates. Cerebrovascular and ischaemic heart co-clustering was significant for the years 2001 and 2011. Cerebrovascular and hypertension co-clustering was significant for the years 2007 and 2011, while hypertension and ischaemic heart co-clustering was significant for the year 2011. Co-clusters of cerebrovascular-ischaemic heart disease are the most profound and located in the south-western part of the country. It was successfully demonstrated that bivariate spatial autocorrelations can be derived for spatially dependent mortality rates as exemplified by mortality rates attributed to three cardiovascular conditions. The identified co-clusters of spatially dependent health outcomes may be targeted for an integrated intervention and monitoring programme.

In this article we fit a time-dependent generalised extreme value (GEV) distribution to annual maximum flood heights at three sites: Chokwe, Sicacate and Combomune in the lower Limpopo River basin of Mozambique. A GEV distribution is fitted to six annual maximum time series models at each site, namely: annual daily maximum (AM1), annual 2-day maximum (AM2), annual 5-day maximum (AM5), annual 7-day maximum (AM7), annual 10-day maximum (AM10) and annual 30-day maximum (AM30). Non-stationary time-dependent GEV models with a linear trend in location and scale parameters are considered in this study. The results show lack of sufficient evidence to indicate a linear trend in the location parameter at all three sites. On the other hand, the findings in this study reveal strong evidence of the existence of a linear trend in the scale parameter at Combomune and Sicacate, whilst the scale parameter had no significant linear trend at Chokwe. Further investigation in this study also reveals that the location parameter at Sicacate can be modelled by a nonlinear quadratic trend; however, the complexity of the overall model is not worthwhile in fit over a time-homogeneous model. This study shows the importance of extending the time-homogeneous GEV model to incorporate climate change factors such as trend in the lower Limpopo River basin, particularly in this era of global warming and a changing climate.

Abstract. In this paper we discuss a comparative analysis of the maximum likelihood (ML) and Bayesian parameter estimates of the generalised extreme value (GEV) distribution. We use a Markov Chain Monte Carlo (MCMC) Bayesian method to estimate the parameters of the GEV distribution in order to estimate extreme flood heights and their return periods in the lower Limpopo River basin of Mozambique. The return periods of extreme flood heights based on the Bayesian approach show an improvement over the frequentist approach based on the maximum likelihood estimation (MLE) method. However, both approaches indicate that the 13 m extreme flood height that occurred at Chokwe in the year 2000 due to cyclone Eline and Gloria had a return period in excess of 200 years, which implies that this event has a very small likelihood of being equalled or exceeded at least once in 200 years.

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