A Branch and Bound Approach to Optimal Allocation in Stratified Sampling

For practical applications of any allocations, integer values of the sample sizes are required. This could be done by simply rounding off the non-integer sample sizes to the nearest integral values. When the sample sizes are large enough or the measurement cost in various strata are not too high, the rounded off sample allocation may work well. However for small samples in some situations the rounding off allocations may become infeasible and non-optimal. This means that rounded off values may violate some of the constraints of the problem or there may exist other sets of integer sample allocations with a lesser value of the objective function. In such situations we have to use some integer programming technique to obtain an optimum integer solution. Keywords:  Stratified sampling, Non-linear Integer Programming, Allocation Problem,  Langrangian Multiplier, Branch & Bound Techniqu

Estimation of Finite Population Mean in Multivariate Stratified Sampling under Cost Function Using Goal Programming

In practical utilization of stratified random sampling scheme, the investigator meets a problem to select a sample that maximizes the precision of a finite population mean under cost constraint. An allocation of sample size becomes complicated when more than one characteristic is observed from each selected unit in a sample. In many real life situations, a linear cost function of a sample size nh is not a good approximation to actual cost of sample survey when traveling cost between selected units in a stratum is significant. In this paper, sample allocation problem in multivariate stratified random sampling with proposed cost function is formulated in integer nonlinear multiobjective mathematical programming. A solution procedure is proposed using extended lexicographic goal programming approach. A numerical example is presented to illustrate the computational details and to compare the efficiency of proposed compromise allocation

NEAR OPTIMUM MULTIVARIATE STRATIFIED SAMPLING DESIGN WITH RANDOM MEASUREMENT COSTS

Usually in sample surveys information on more than one characteristic are collected and the data obtained are analyzed to get the required estimates for the multivariate population under study. If stratified sampling design is to be applied on such a population the individual optimum allocations don’t help much unless the characteristics are highly correlated. Therefore, in multivariate stratified sampling we need to work out an allocation that is optimum for all characteristics in some sense, that is, near optimum for all characteristics. Such an allocation is called a compromise allocation. Furthermore, in surveys usually the per unit measurement costs are taken as deterministic, that is, they remain constant throughout the survey. In practice the costs of measurement of different characteristics in various strata may change during the course of survey for reasons beyond the control of the sampler. Thus in some practical situations the measurement costs may become a random variable and the  problem of obtaining a compromise allocation becomes a Stochastic Integer Nonlinear Programming Problem (SINLPP). The present paper addresses the problem of obtaining an integer compromise allocation for multivariate stratified sampling with random cost of measurements. A solution procedure has been developed for the formulated problem. A practical application of the procedure is also given through a numerical example to illustrate the computational details

On The Efficiency of Some Techniques For Optimum Allocation In Multivariate Stratified Survey.

In multivariate stratified sampling, the major concern is on the problem of estimation of more than one population characteristics which often make conflicting demands on sampling technique. In this type of survey, an allocation which is optimum for one characteristic may not be optimum for other characteristics. In such situations a compromise criterion is needed to work out a usable allocation which is optimum for all characteristics in some sense. This study is focuses on the efficiency of some techniques for optimum sample allocation which are Yates/Chatterjee, Booth and Sedransk and Vector maximum criterion (VMC) on the set of real life data stratified into six strata and two variates with desired variances using: (i) method of minimum variance with fixed sample size and (ii) an arbitrary fixing of variances. The stratum sample sizes  among the classes were obtained to examine the criterion that will produce the smallest n. In this paper, it was discovered that VMC and Booth and Sedransk are superior to Yates/Chatterjee. Even though, no universal conclusion can be drawn, the work clearly brings out the fact that the best allocation is not always obvious and that sufficient care is necessary in the choice of allocation of the sample sizes to different strata with several items. Keywords:- Stratified Survey; Optimum Allocation; Vector Maximum Criterion (VMC); Yates/Chatterjee;Booth and Sedrans

Stochastic-optimization of equipment productivity in multi-seam formations

Short and long range planning and execution for multi-seam coal formations (MSFs) are challenging with complex extraction mechanisms. Stripping equipment selection and scheduling are functions of the physical dynamics of the mine and the operational mechanisms of its components, thus its productivity is dependent on these parameters. Previous research studies did not incorporate quantitative relationships between equipment productivities and extraction dynamics in MSFs. The intrinsic variability of excavation and spoiling dynamics must also form part of existing models. This research formulates quantitative relationships of equipment productivities using Branch-and-Bound algorithms and Lagrange Parameterization approaches. The stochastic processes are resolved via Monte Carlo/Latin Hypercube simulation techniques within @RISK framework. The model was presented with a bituminous coal mining case in the Appalachian field. The simulated results showed a 3.51% improvement in mining cost and 0.19% increment in net present value. A 76.95yd³ drop in productivity per unit change in cycle time was recorded for sub-optimal equipment schedules. The geologic variability and equipment operational parameters restricted any possible change in the cost function. A 50.3% chance of the mining cost increasing above its current value was driven by the volume of material re-handled with 0.52 regression coefficient. The study advances the optimization process in mine planning and scheduling algorithms, to efficiently capture future uncertainties surrounding multivariate random functions. The main novelty includes the application of stochastic-optimization procedures to improve equipment productivity in MSFs --Abstract, page iii

Improving the representativeness of a simple random sample: an optimization model and its application to the continuous sample of working lives

This paper proposes an optimization model for selecting a larger subsample that improves the representativeness of a simple random sample previously obtained from a population larger than the population of interest. The problem formulation involves convex mixed-integer nonlinear programming (convex MINLP) and is, therefore, NP-hard. However, the solution is found by maximizing the size of the subsample taken from a stratified random sample with proportional allocation and restricting it to a p-value large enough to achieve a good fit to the population of interest using Pearson’s chi-square goodness-of-fit test. The paper also applies the model to the Continuous Sample of Working Lives (CSWL), which is a set of anonymized microdata containing information on individuals from Spanish Social Security records and the results prove that it is possible to obtain a larger subsample from the CSWL that (far) better represents the pensioner population for each of the waves analyzed

Estimation of origin-destination matrices from traffic counts: theoretical and operational development

This thesis deals with the o-d estimation problem from indirect measures, addressing two main aspects of the problem: the identification of the set of indirect measures that provide the maximum information with a resulting reduction of the uncertainty on the estimate; once defined the set of measures, the choice of an estimator to identify univocally and as much reliable as possible the estimate. As regards the former aspect, an innovative and theoretically founded methodology is illustrated, explicitly accounting for the reliability of the o-d matrix estimate. The proposed approach is based on a specific measure, named Synthetic Dispersion Measure (SDM), related to the trace of the dispersion matrix of the posterior demand estimate conditioned to a given set of sensors locations. Under the mild assumption of multivariate normal distribution for the prior demand estimate, the proposed SDM does not depend on the specific values of the counted flows – unknown in the planning stage – but just on the locations of such sensors. The proposed approach is applied to real contexts, leading to results outperforming the other methods currently available in the literature. In addition, the proposed methodology allows setting a formal budget allocation problem between surveys and counts in the planning stage, in order to maximize the overall quality of the demand estimation process. As regard the latter aspect, a “quasi-dynamic” framework is proposed, under the assumption that o-d shares are constant across a reference period, whilst total flows leaving each origin vary for each sub-period within the reference period. The advantage of this approach over conventional within-day dynamic estimators is that of reducing drastically the number of unknowns given the same set of observed time-varying traffic counts. The quasi-dynamic assumption is checked by means of empirical and statistical tests and the performances of the quasi-dynamic estimator - whose formulation is also given – are compared with other dynamic estimators

Application of stochastic programming to management of cash flows with FX exposure

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 23/06/2006.In this thesis we formulate a model for foreign exchange (FX) exposure management and multi-currency cash management taking into consideration random fluctuations of exchange rates and net revenues of a multinational firm (MNF). The central decision model used in this thesis is a scenario-based stochastic programming (SP) recourse model. A critical review of alternative scenario generation methods is given followed by analysis of some desirable properties of the scenario tree. The application of matching statistical moments of a probability distribution to generate a multiperiod scenario tree for our problem is described in detail. A four-stage SP decision model is formulated using the random parameter values. This model evaluates currency / cash flows hedging strategies, which provide rolling decisions on the size and timing of the forward positions. We compute an efficient frontier from which an investor can choose an optimal strategy according to his risk and return preferences. The flexibility of the SP model allows an investor to analyse alternative risk-return trading strategies. The model decisions are investigated by making comparisons with decisions based purely on the expected value problem. The investigation shows that there is a considerable improvement to the "spot only" strategy and provides insight into how these decisions are made. The contributions of the thesis are summarised below. (i) The FX forward scenario trees are derived using an arbitrage-free pricing strategy and is in line with modem principles of finance. (ii) Use of the SP model and forward contracts as a tool for hedging decisions is novel. (iii) In particular smoothing of the effects in exchange rates and the smoothing of account receivables are examples of innovative modelling approaches for FX management

Matching on-the-fly in Sequential Experiments for Higher Power and Efficiency

We propose a dynamic allocation procedure that increases power and efficiency when measuring an average treatment effect in sequential randomized trials. Subjects arrive iteratively and are either randomized or paired via a matching criterion to a previously randomized subject and administered the alternate treatment. We develop estimators for the average treatment effect that combine information from both the matched pairs and unmatched subjects as well as an exact test. Simulations illustrate the method's higher efficiency and power over competing allocation procedures in both controlled scenarios and historical experimental data.Comment: 20 pages, 1 algorithm, 2 figures, 8 table