On multivariate-multiobjective stratified sampling design under probabilistic environment: A fuzzy programming technique

1 Introduction

In many real-life practices, the populations may vary in their accessibility. Some parts of the population may be in remote locations, gated buildings or other inaccessible areas. For such situations, the choice of sampling design affects the results of the survey. Then the stratified sampling seems to be the best choice of the sampling technique. For obtaining detailed information about the characteristics of the population, a multivariate stratified sample survey is carried out by splitting the population into $L$ strata. It is assumed that all the $p$ characteristics are defined in each unit of the population. The estimation of unknown population means of $p$ characteristics is required and may be carried out using the Nonlinear Programming Problem (NLPP). Cochran (1977) has been shown that the individual optimum allocation of one character may not remain optimum for others. A compromise criterion may be required to obtain the best allocation, which helps obtain precise information about population parameters. Therefore the allocation based on some compromised criterion is called compromise allocation in multivariate stratified sampling design. Most of the authors (Neyman, 1934; Kokan and Khan, 1967; Chatterjee, 1968; Ahsan, 1975; Khan et al., 1997; Semiz, 2004; Kozak, 2006; Varshney et al., 2012, 2015; Fatima et al., 2014; Muhammad et al., 2015; Muhammad and Husain, 2017; Varshney and Mradula, 2019) discussed the problems of allocation and worked out compromise allocation in multivariate stratified sample surveys. A compromise allocation is obtained either by suggesting different compromise criteria or using the suggested criteria under different conditions, i.e. in the availability of auxiliary information, presence of nonresponse, etcetera. In many sampling designs, the stratum variances are not known in advance but maybe estimable. From the deterministic point of view, such problems may be formulated as NLPPs. However, if the nature of the estimated variances is also considered, it will be an additional restriction to the problem, and therefore the compromise allocation may not be obtained easily. For such situations, the Stochastic Nonlinear Programming Problem (SNLPP) may help to work out the required compromise allocation to obtain sufficient information about population parameters (See (Charnes and Cooper, 1963; Prékopa, 1978; Díaz-García and Garay Tapia, 2007; Kozak and Wang, 2010; Haseen et al., 2016)). Díaz-García and Ramos-Quiroga (2014) discussed and provided results by solving SNLPPs with a fixed linear cost function. The concept of fuzzy set theory has been discussed by Zadeh (1965) and then Bellman and Zadeh (1970). They utilized the fuzzy approach for dynamic issues. The idea of the fuzzy set was given by Zimmermann (1978) to convert the multiobjective linear programming problem into a single objective linear programming problem. Many authors solved sample allocation problems by using fuzzy programming techniques (See (Gupta et al., 2013; Ali and Hasan, 2013; Varshney et al., 2017; Haq et al., 2020). Fuzzy programming is one of many available optimization techniques that deal with optimization problems under uncertainty. The technique is flexible and thus helps decision-makers have a better understanding of their problems. Such techniques may be applied when situations are not clearly defined and also have uncertainty. The fuzzy programming technique is a more appropriate technique for solving the problem when the data has anykind of uncertainity. Fuzzy programming has been studied and applied recently by several authors in different areas (Elsisi, 2019a, 2019b, 2020; Fakhrzad and Goodarzian, 2019; Elsisi and Soliman, 2020; Fathollahi-Fard et al., 2020; Goodarzian and Hosseini-Nasab, 2021; Lu et al., 2020).

Under the probabilistic environment, the use of the nonlinear cost function is proposed by considering the labour cost as part of the survey's total cost. In this paper, the compromise criterion is suggested for determining the compromise allocation for a multiobjective-multivariate stochastic nonlinear programming problem for the fixed cost in the probabilistic situation. The solution procedure is also given to solve the formulated problem by using an appropriate nonlinear programming technique.

In Section 2, the notations and formulation of the problem are given. The formulated problems' solutions are suggested using two deterministic approaches: modified E-model and chance constraints in Section 3. In the modified E-model formulation, the solution strategy is recommended by utilizing a fuzzy goal programming technique. For the chance constraints model, the solutions are obtained using Lagrange multiplier and integer nonlinear programming techniques. In Section 4, the numerical illustration is discussed by considering the Iris data set, and the data set is obtained by simulation carried out by the software R. The formulated problem is solved through the modified E-model approach and chance constraints technique. The solution procedures are suggested by utilizing fuzzy goal programming problem, Lagrange multiplier method and integer nonlinear programming problem. MATLAB software is used to solve the formulated NLPPs. A comparative study is included by considering some other allocations, as discussed in Section 4.1.1, with the proposed allocation. Finally, the conclusion has been made for using the proposed technique in Section 5.

2 Framework of the problem

Assuming a population of $N$ units that is partitioned into strata of sizes $N_1,N_2,\dots ,N_L$ units such that ${\sum }_{h=1}^L{N}_h=N.$

For $h^{\mathit{th}}$ stratum, the following notations are introduced as follows:

$N_h:$ Stratum size

$W_h=\frac{N_h}{N}:$ Stratum weight

$n_h:$ Sample size

$y_{\mathit{hi}}$ : Observational value of $i^{\mathit{th}}$ stratum unit/stratum sample.

${\overline{Y}}_h=\frac{1}{N_h}{\sum }_{i=1}^{N_h}{y}_{\mathit{hi}}:$ Stratum mean

${\overline{y}}_h=\frac{1}{n_h}{\sum }_{i=1}^{n_h}{y}_{\mathit{hi}}:$ Sample mean

$S_h^2=\frac{1}{N_h-1}{\sum }_{i=1}^{N_h}{\left(y_{\mathit{hi}}-{\overline{Y}}_h\right)}^2:$ Stratum mean square

$s_h^2=\frac{1}{n_h-1}{\sum }_{i=1}^{n_h}{\left(y_{\mathit{hi}}-{\overline{y}}_h\right)}^2:$ Sample mean square.

Furthermore,

$\overline{Y}=\frac{1}{N}{\sum }_{h=1}^L{\sum }_{i=1}^{N_h}{y}_{\mathit{hi}}=\frac{1}{N}{\sum }_{h=1}^L{N}_h{\overline{Y}}_h={\sum }_{h=1}^L{W}_h{\overline{Y}}_h:$ describes the overall population mean.

If the estimated value of $\overline{Y}$ is needed, then the stratified sample mean $${\overline{y}}_{\mathit{st}}=\sum _{h=1}^L{W}_h{\overline{y}}_h,$$

gives an unbiased estimator for $\overline{Y}$ with the sampling variance $$V({\overline{y}}_{\mathit{st}})=\sum _{h=1}^L\frac{W_h^2{S}_h^2}{n_h}-\sum _{h=1}^L\frac{W_h{S}_h^2}{N}$$

In a multivariate stratified population where $p$ characteristics are given on each population element, then the $p$ population means ${\overline{Y}}_j;j=1,2,\dots ,p$ are to be estimated. Since the individual optimum allocation may not be optimum for other characteristics. Let $y_{\mathit{jhi}}$ denotes the value obtained from $i^{\mathit{th}}$ element in $h^{\mathit{th}}$ stratum having $j^{\mathit{th}}$ characteristic and ${\overline{Y}}_{\mathit{jh}}=\frac{1}{N_h}{\sum }_{i=1}^{N_h}{y}_{\mathit{jhi}}$ be the stratum mean of $y_{\mathit{jhi}}$ . Then the sample means for all characteristics in $h^{\mathit{th}}$ stratum are calculated by $${\overline{y}}_{\mathit{jh}}=\frac{1}{n_h}\sum _{i=1}^{n_h}{y}_{\mathit{jhi}};j=1,2,\dots ,p.$$

For $j^{\mathit{th}}$ characteristic, an unbiased estimate of the overall population mean ${\overline{Y}}_j$ is given by ${\overline{y}}_{\mathit{jst}}$ and is expressed by $${\overline{y}}_{\mathit{jst}}=\sum _{h=1}^L{W}_h{\overline{y}}_{\mathit{jh}}$$ with its sampling variance $$V({\overline{y}}_{jst})=\sum _{h=1}^L\frac{W_h^2{S}_{\mathit{jh}}^2}{n_h}-\sum _{h=1}^L\frac{W_h{S}_{\mathit{jh}}^2}{N}$$ where $S_{\mathit{jh}}^2$ is the stratum variance of $j^{\mathit{th}}$ characteristic in $h^{\mathit{th}}$ stratum for $j=1,2,\dots ,p$ , $h=1,2,\dots ,L$ and can be calculated by $$S_{\mathit{jh}}^2=\frac{1}{N_h-1}\sum _{i=1}^{N_h}{\left(y_{\mathit{jhi}}-{\overline{Y}}_{\mathit{jh}}\right)}^2.$$

For a multivariate stratified sample survey, the linear cost function may be considered for the overall budget of the survey (Cochran, 1977) and may be expressed as $$C=c_0+\sum _{h=1}^L{c}_h{n}_h,$$

or $$C-c_0=C_0=\sum _{h=1}^L{c}_h{n}_h,$$ where $C_0$ denotes the cost to measure all sampling units in all strata, $c_h$ is the per-unit measurement cost of measuring $p$ characteristics on the selected unit in $h^{\mathit{th}}$ stratum, $n_h$ is the $h^{\mathit{th}}$ stratum sample size and $c_0$ is the overhead cost to conduct the survey. If the travel cost within the stratum come into consideration, then the cost function may not have remained linear. Beardwood et al. (1959) suggested the nonlinear cost function for this case. They showed that the distance between $n$ arbitrarily dispersed points is proportional to $\sqrt{n}$ . The nonlinear cost function, which includes travel costs, may be expressed as $$C=c_0+\sum _{h=1}^L{c}_h{n}_h+\sum _{h=1}^L{t}_h\sqrt{n_h},$$

or

Practically, some other cost factors may be considered, like costs on the reward to respondents, labour costs, etc. Whenever the interviewers want to collect detailed information from the selected respondents, there will be a requirement for more human resources available for a specified time. For these prerequisites, the labour costs may be used for conducting the survey, and therefore the cost function may be expressed as

for various values of $\lambda$ (Muhammad and Husain, 2017).

The problem may be formulated in two ways by using a deterministic approach, and the solution is obtained either by minimizing the variance $V({\overline{y}}_{jst})$ for a fixed cost or by minimizing the cost of the survey with variances of specified limits. Therefore two optimization problems may be described as

respectively.

If the true values of $S_{\mathit{jh}}^2$ are unknown, then they may be computed through a starter test or the values of past events (Kozak, 2006).

3 Determination of identical probabilistic sampling variances and cost

If $S_{\mathit{jh}}^2$ are considered as random variables, then the problems defined in (3) and (4) become SNLPPs. These problems may be converted into their equivalent deterministic problems. Several techniques, like modified E-model, E-model, V-model, chance constraints, etc., are available to solve deterministic problems (Charnes and Cooper, 1963). In this manuscript, modified E-model and chance constraints methods are used to convert the problems into the deterministic form.

4 Application

A population of size $N=9500,$ with three strata and two characteristics are taken and obtained by simulating 150 observations of Iris data. Iris data set is available in the Software R domain (R Development Core Team, 2018). These observations are divided into three strata where two characteristics (that is, length and width of a leaf of a particular species of flower) are measured on each population unit. The population units for three strata of sizes 3000, 3000 and 3500 are generated by the simulation of Iris data using the software R and the values of $s_{\mathit{jh}}^2$ and $C_{\mathit{yjh}}^4$ are computed and reported in Table 1. The values of $c_h,{\tau }_h,\omega$ and $\lambda$ are assumed for numerical illustration accordingly and given in Table 1. The total cost for conducting the survey is taken as $1000$ units.