Multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost

In stratified sampling, usually the cost function is taken as a linear function of sample sizes n _h . In many practical situations, the linear cost function does not approximate the actual cost incurred adequately. For example, when the cost of travelling between the units selected in the sample within a stratum is significant, instead of a linear cost function, a cost function that is quadratic in √n _h will be a more close approximation to the actual cost. In this paper, the problem is formulated as multi-objective nonlinear integer programming problem with quadratic cost under three different situations, i.e. complete, partial or null information about the population. A numerical example is also presented to illustrate the computational details.     

Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function

In multivariate stratified sample survey with L strata, let p-characteristics are defined on each unit of the population. To estimate the unknown p-population means of each characteristic, a random sample is taken out from the population. In multivariate stratified sample survey, the optimum allocation of any characteristic may not be optimum for others. Thus the problem arises to find out an allocation which may be optimum for all characteristics in some sense. Therefore a compromise criterion is needed to workout such allocation. In this paper, the procedure of estimation of p-population means is discussed in the presence of nonresponse when the use of linear cost function is not advisable. A solution procedure is suggested by using lexicographic goal programming problem. The numerical illustrations are given for its practical utility.     

On the Problem of Compromise Allocation in Multi-Response Stratified Sample Surveys

In stratified sample surveys, the problem of determining the optimum allocation is well known due to articles published in 1923 by Tschuprow and in 1934 by Neyman. The articles suggest the optimum sample sizes to be selected from each stratum for which sampling variance of the estimator is minimum for fixed total cost of the survey or the cost is minimum for a fixed precision of the estimator. If in a sample survey more than one characteristic is to be measured on each selected unit of the sample, that is, the survey is a multi-response survey, then the problem of determining the optimum sample sizes to various strata becomes more complex because of the non-availability of a single optimality criterion that suits all the characteristics. Many authors discussed compromise criterion that provides a compromise allocation, which is optimum for all characteristics, at least in some sense. Almost all of these authors worked out the compromise allocation by minimizing some function of the sampling variances of the estimators under a single cost constraint. A serious objection to this approach is that the variances are not unit free so that minimizing any function of variances may not be an appropriate objective to obtain a compromise allocation. This fact suggests the use of coefficient of variations instead of variances. In the present article, the problem of compromise allocation is formulated as a multi-objective non-linear programming problem. By linearizing the non-linear objective functions at their individual optima, the problem is approximated to an integer linear programming problem. Goal programming technique is then used to obtain a solution to the approximated problem.     

Fuzzy multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost and parabolic fuzzy numbers

This article deals with the uncertainties in a multivariate stratified sampling problem. The uncertain parameters of the problem, such as stratum standard deviations, measurement costs, travel costs and total budget of the survey, are considered as parabolic fuzzy numbers and the problem is formulated as a fuzzy multi-objective nonlinear programming problem with quadratic cost function. Using α-cut, parabolic fuzzy numbers are defuzzified and then the compromise allocations of the problem are obtained by fuzzy programming for a prescribed value of α. To demonstrate the utility of the proposed problem a numerical example is solved with the help of [LINGO User?s Guid. Lindo Systems Inc., 1415 North Dayton Street, Chicago,Illinois-60622, (USA), 2013] software and the derived compromise optimum allocation is compared with deterministic and proportional allocations.     

Multiobjective Stochastic Multivariate Stratified Sampling in Presence of Nonresponse

The case of nonresponse in multivariate stratified sampling survey was first introduced by Hansen and Hurwitz in 1946 considering the sampling variances and costs to be deterministic. However, in real life situations sampling variance and cost are often random (stochastic) and have probability distributions. In this article, we have formulated the multivariate stratified sampling in the presence of nonresponse with random sampling variances and costs as a multiobjective stochastic programming problem. Here, the sampling variance and costs are considered random and converted into a deterministic NLPP by using chance constraint and modified E-model. A solution procedure using three different approaches are adopted viz. goal programming, fuzzy programming, and D₁ distance method to obtain the compromise allocation for the formulated problem. An empirical study has also been provided to illustrate the computational details.     

An Optimal Multivariate Stratified Sampling Design Using Dynamic Programming

Numerous optimization problems arise in survey designs. The problem of obtaining an optimal (or near optimal) sampling design can be formulated and solved as a mathematical programming problem. In multivariate stratified sample surveys usually it is not possible to use the individual optimum allocations for sample sizes to various strata for one reason or another. In such situations some criterion is needed to work out an allocation which is optimum for all characteristics in some sense. Such an allocation may be called an optimum compromise allocation. This paper examines the problem of determining an optimum compromise allocation in multivariate stratified random sampling, when the population means of several characteristics are to be estimated. Formulating the problem of allocation as an all integer nonlinear programming problem, the paper develops a solution procedure using a dynamic programming technique. The compromise allocation discussed is optimal in the sense that it minimizes a weighted sum of the sampling variances of the estimates of the population means of various characteristics under study. A numerical example illustrates the solution procedure and shows how it compares with Cochran's average allocation and proportional allocation.     

Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the case of non-response

In multivariate cases, usually the minimization of sampling variances is considered as an objective under a cost constraint. Since the variances are not unit free, it is more logical to consider the minimization of the squared coefficients of variation as an objective. In this paper, the problem of optimum compromise allocation in multivariate stratified sampling in the case of non-response as a multi-objective all-integer nonlinear programming problem is described. A solution procedure using four different approaches is considered, namely the value function, goal programming,∈-constraint and distance based, to obtain the compromise allocation for non-response. A numerical example is also presented to illustrate the computational details.     

Designing of a mixed-chain sampling plan based on the process capability index with chain sampling as the attributes plan

In this article, we propose a new mixed chain sampling plan based on the process capability index C_pk, where the quality characteristic of interest having double specification limits and follows the normal distribution with unknown mean and variance. In the proposed mixed plan, the chain sampling inspection plan is used for the inspection of attribute quality characteristics. The advantages of this proposed mixed sampling plan are also discussed. Tables are constructed to determine the optimal parameters for practical applications by formulating the problem as a non linear programming in which the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level and limiting quality level under the operating characteristic curve. The practical application of the proposed mixed sampling plan is explained with an illustrative example. Comparison of the proposed sampling plan is also made with other existing sampling plans.     

Optimum multivariate stratified double sampling design: Chebyshev's Goal Programming approach

In stratified sampling when strata weights are unknown a double sampling technique may be used to estimate them. A large simple random sample from the unstratified population is drawn and units falling in each stratum are recorded. A stratified random sample is then selected and simple random subsamples are obtained out of the previously selected units of the strata. This procedure is called double sampling for stratification. If the problem of non-response is there, then subsamples are divided into classes of respondents and non-respondents. A second subsample is then obtained out of the non-respondents and an attempt is made to obtain the information by increasing efforts, persuasion and call backs. In this paper, the problem of obtaining a compromise allocation in multivariate stratified random sampling is discussed when strata weights are unknown and non-response is present. The problem turns out to be a multiobjective non-linear integer programming problem. An approximation of the problem to an integer linear programming problem by linearizing the non-linear objective functions at their individual optima is worked out. Chebyshev's goal programming technique is then used to solve the approximated problem. A numerical example is also presented to exhibit the practical application of the developed procedure.     

Comparison of computer programs for simple linear L 1 regression

A number of efficient computer codes are available for the simple linear L ₁ regression problem. However, a number of these codes can be made more efficient by utilizing the least squares solution. In fact, a couple of available computer programs already do so.

We report the results of a computational study comparing several openly available computer programs for solving the simple linear L ₁ regression problem with and without computing and utilizing a least squares solution.     

Trade-off between cost and variance for a multi-objective compromise allocation in stratified random sampling

In the present paper, a multi-objective goal optimization mechanism is developed by trading off between cost and variance. Both are adversaries to each other while allocating a sample size even in stratified sampling design. Discussion section shows how these adversaries put their influence on optimal selection. This is a dual optimization procedure in which variance or mean square error is optimized in the first step and then considering some compromise on variance, cost is optimized. The process is applied to both individual and multi-objective programming models.     

Estimation of Population Means in Multivariate Stratified Random Sampling

In multivariate surveys where p (> 1) characteristics are defined on each unit of the population, the problem of allocation becomes complicated. In the present article, we propose a method to work out the compromise allocation in a multivariate stratified surveys. The problem is formulated as a Multiobjective Integer Nonlinear Programming Problem. Using the value function technique, the problem is converted into a single objective problem. A formula for continuous sample sizes is obtained using Lagrange Multipliers Technique (LMT) that can provide a near optimum solution in some cases. It may give an initial point for any integer nonlinear programing technique.     

Probability generating function of GPED<Subscript>2</Subscript>

The Probability generating function of a random variable which has Generalized Polya Eggenberger Distribution of the second kind (GPED ₂) is obtained. The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable from GPED ₂. The results of Bazargan-Lari (2004) follow as special cases.     

Unbiased L1 and L∞ estimation

Sielken and Heartely 1973 have shown that the L₁ and L_∞ estimation problems may be formulated in such a way as to yield unbiased estimators of in the standard linear model y = Xβ + ε In this paper we will show that the L₁ estimation problem is closely related to the dual of the L_∞ estimation problem and vice versa. We will use this resu;t to obtain four fistiner lineat programming problems which yield unbiased L₁ and L_∞ estimators of β.     

Distribution of the range when sample size has a GPED<Subscript>1</Subscript>

The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable having the Generalized Polya Eggenberger Distribution of the first kind (GPED ₁). The results of Raghunandanan and Patil (1972) and Bazargan-lari (1999) follow as special cases. The p.d.f of rangeR is obtained, when the distribution of the sample sizeN belongs to Katz family of distributions, as a special case. An erratum to this article is available at .     

Using a Truncated C p Statistic for Variable Selection in Multiple Linear Regression

In multiple linear regression analysis each lower-dimensional subspace L of a known linear subspace M of ? ⁿ corresponds to a non empty subset of the columns of the regressor matrix. For a fixed subspace L, the C _p statistic is an unbiased estimator of the mean square error if the projection of the response vector onto L is used to estimate the expected response. In this article, we consider two truncated versions of the C _p statistic that can also be used to estimate this mean square error. The C _p statistic and its truncated versions are compared in two example data sets, illustrating that use of the truncated versions may result in models different from those selected by standard C _p .     

Selective Maintenance in System Reliability with Random Costs of Repairing and Replacing the Components

In this paper we consider the problem of determining the optimum number of repairable and replaceable components to maximize a system's reliability when both, the cost of repairing the components and the cost of replacement of components by new ones, are random. We formulate it as a problem of non-linear stochastic programming. The solution is obtained through Chance Constrained programming. We also consider the problem of finding the optimal maintenance cost for a given reliability requirement of the system. The solution is then obtained by using Modified E-model. A numerical example is solved for both the formulations.     

Methodology and analysis for comparing discrete linear l1 approximation codes

This is the first of a projected series of papers dealing with computational experimentation in mathematical programming. This paper provides early results of a test case using four discrete linear L₁ approximation codes. Variables influencing code behavior are identified and measures of performance are specified. More importantly, an experimental design is developed for assessing code performance and is illustrated using the variable “problem size”.     

Determining the optimal allocation of testing resource for modular software system using dynamic programming

Abstract

Reliability is a major concern in the process of software development because unreliable software can cause failure in the computer system that can be hazardous. A way to enhance the reliability of software is to detect and remove the faults during the testing phase, which begins with module testing wherein modules are tested independently to remove a substantial number of faults within a limited resource. Therefore, the available resource must be allocated among the modules in such a way that the number of faults is removed as much as possible from each of the modules to achieve higher software reliability. In this article, we discuss the problem of optimal resource allocation of the testing resource for a modular software system, which maximizes the number of faults removed subject to the conditions that the amount of testing-effort is fixed, a certain percentage of faults is to be removed and a desired level of reliability is to be achieved. The problem is formulated as a non linear programming problem (NLPP), which is modeled by the inflection S-shaped software reliability growth models (SRGM) based on a non homogeneous Poisson process (NHPP) which incorporates the exponentiated Weibull (EW) testing-effort functions. A solution procedure is then developed using a dynamic programming technique to solve the NLPP. Furthermore, three special cases of optimum resource allocations are also discussed. Finally, numerical examples using three sets of software failure data are presented to illustrate the procedure developed and to validate the performance of the strategies proposed in this article. Experimental results indicate that the proposed strategies may be helpful to software project managers for making the best decisions in allocating the testing resource. In addition, the results are compared with those of Kapur et al. (2004), Huang and Lyu (2005), and Jha et al. (2010) that are available in the literature to deal the similar problems addressed in this article. It reveals that the proposed dynamic programming method for the testing-resource allocation problem yields a gain in efficiency over other methods.     

Statistical analysis of process capability indices with measurement errors: The case ofC p

Process capability indices (PCIs) have been widely used in manufacturing industries to previde a quantitative measure of process potential and performance. While some efforts have been dedicated in the literature to the statistical properties of PCIs estimators, scarce attention has been given to the evaluation of these properties when sample data are affected by measurement errors. In this work we deal with the problem of measurement errors effects on the performance of PCIs. The analysis is illustrated with reference toC _p , i.e. the simplest and most common measure suggested to evaluate process capability. The authors would like to thank two anonymous referees for their comments and suggestion that were useful in the preparation and improvement of this paper. This work was partially supported by a MURST research grant.