This paper develops allocation methods for stratified sample surveys in which small area estimation is a priority. We assume stratified sampling with small areas as the strata. Similar to Longford (2006), we seek efficient allocation that minimizes a linear combination of the mean squared errors of composite small area estimators and of an estimator of the overall mean. Unlike Longford, we define mean-squared error in a model-assisted framework, allowing a more natural interpretation of results using an intra-class correlation parameter. This allocation has an analytical form for a special case, and has the unappealing property that some strata may be allocated no sample. We derive a Taylor approximation to the stratum sample sizes for small area estimation using composite estimation giving priority to both small area and national estimation.

A Markov model describes a randomly varying system that satisfies the Markov property. This means that future and past states at any time are independent of the current state. The most commonly used Markov models are Markov chains and higher-order Markov chains. Therefore, three types of Markov models are proposed in this chapter of the book: (i) Supply chain management, (ii) Markov queue waiting time monitoring, and (iii) Markov fuzzy time series forecasting. The introduction introduces the Markov chain (MC) and summarizes the most important aspects of Markov chain analysis. Using the classical (0, Q) policy, the first model explores a Markov queue coupled to a storage system. The second model focuses on the M/M/1/N service mode and develops a control chart for an Ms/Ms/1/N type simulated queue to monitor customer waiting times. The third is a higher-order Markov model (HOMM), which uses fuzzy sets to predict future states based on given hypothetical time series data. Numerical calculations are designed to find optimal order quantities, monitor customer wait times, and predict future HOMM conditions

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This paper develops optimal designs when it is not feasible for every cluster to be represented in a sample as in stratified design, by assuming equal probability two-stage sampling where clusters are small areas. The paper develops allocation methods for two-stage sample surveys where small-area estimates are a priority. We seek efficient allocations where the aim is to minimize the linear combination of the mean squared errors of composite small area estimators and of an estimator of the overall mean. We suggest some alternative allocations with a view to minimizing the same objective. Several alternatives, including the area-only stratified design, are found to perform nearly as well as the optimal allocation but with better practical properties. Designs are evaluated numerically using Switzerland canton data as well as Botswana administrative districts data.