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

Symmetric and Asymmetric Rounding

If rounded data are used in estimating moments and regression coefficients, the estimates are typically more or less biased. The purpose of the paper is to study the bias inducing effect of rounding, which is also seen when population moments instead of their estimates are considered. Under appropriate conditions this effect can be approximately specified by versions of Sheppard's correction formula. We discuss the conditions under which these approximations are valid. We also investigate the efficiency loss that comes along with rounding. The rounding error, which corresponds to the measurement error of a measurement error model, has a marginal distribution which can be approximated by the uniform distribution. We generalize the concept of simple rounding to that of asymmetric rounding and study its effect on the mean and variance of a distribution under similar circumstances as with simple rounding

AFPTAS results for common variants of bin packing: A new method to handle the small items

We consider two well-known natural variants of bin packing, and show that these packing problems admit asymptotic fully polynomial time approximation schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins). It has been known for a while that the most basic problem admits an AFPTAS. In this paper, we develop methods that allow to extend this result to other variants of bin packing. Specifically, the problems which we study in this paper, for which we design asymptotic fully polynomial time approximation schemes, are the following. The first problem is "Bin packing with cardinality constraints", where a parameter k is given, such that a bin may contain up to k items. The goal is to minimize the number of bins used. The second problem is "Bin packing with rejection", where every item has a rejection penalty associated with it. An item needs to be either packed to a bin or rejected, and the goal is to minimize the number of used bins plus the total rejection penalty of unpacked items. This resolves the complexity of two important variants of the bin packing problem. Our approximation schemes use a novel method for packing the small items. This new method is the core of the improved running times of our schemes over the running times of the previous results, which are only asymptotic polynomial time approximation schemes (APTAS)

Planning drinking water for airplanes

The management of the Dutch national airline company KLM intends to bring a sufficient amount of water on board of all flights to fulfill customer’s demand. On the other hand, the surplus of water after a flight should be kept to a minimum to reduce fuel costs. The service to passengers is measured with a service level. The objective of this research is to develop models, which can be used to minimize the amount of water on board of flights such that a predefined service level is met. The difficulty that has to be overcome is the fact that most of the available data of water consumption on flights are rounded off to the nearest eighth of the water tank. For wide-body aircrafts this rounding may correspond to about two hundred litres of water. Part of the problem was also to define a good service level. The use of a service level as a model parameter would give KLM a better control of the water surplus. The available data have been analyzed to examine which aspects we had to take into consideration. Next, a general framework has been developed in which the service level has been defined as a Quality of Service for each flight: The probability that a sufficient amount of water is available on a given flight leg. Three approaches will be proposed to find a probability distribution function for the total water consumption on a flight. The first approach tries to fit a distribution for the water consumption based on the available data, without any assumptions on the underlying shape of the distribution. The second approach assumes normality for the total water consumption on a flight and the third approach uses a binomial distribution. All methods are validated and numerically illustrated. We recommend KLM to use the second approach, where the first approach can be used to determine an upper bound on the water level

Probabilistic Rounding and Sheppard's Correction

When rounded data are used in place of the true values to compute the variance of a variable or a regression line, the results will be distorted. Under suitable smoothness conditions on the distribution of the variable(s) involved, this bias, however, can be corrected with very high precision by using the well-known Sheppard’s correction. In this paper, Sheppard’s correction is generalized to cover more general forms of rounding procedures than just simple rounding, viz., probabilistic rounding, which includes asymmetric rounding and mixture rounding