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This paper is an attempt to work out a compromise allocation to construct combined ratio estimates under multivariate double sampling design in presence of non-response when the population mean of the auxiliary variable is unknown. The problem has been formulated as a multi-objective integer non-linear programming problem. Two solution procedures are developed using goal programming and fuzzy programming techniques. A numerical example is also worked out to illustrate the computational details. A comparison of the two methods is also carried out.

Often in sample surveys the main variable is highly correlated to another variable called an auxiliary variable and the data on auxiliary variable are either available or can be easily obtained. In this situation to obtain the estimate of the parameters regarding the main variable the auxiliary information can be used to enhance the precision of the estimate. Ratio and Regression Methods and double sampling technique are some examples. When data are collected on the sampled units of the main variable due to one or the other reason, data for all the selected units cannot be obtained. This result is an incomplete and less informative sample. This phenomenon is termed as “non response”. [

In the present paper, we considered combined ratio estimators of the population means of a multivariate stratified population using double sampling in presence of non-response. Compromise allocations at first and second phase of double sampling are obtained by formulating the problems as multi-objective integer non-linear programming problems. Solution procedures are developed by using goal programming and fuzzy programming techniques. A numerical example is also worked out to illustrate the computational details. A comparison of the two methods is also carried out.

When auxiliary information is available, the use of Ratio method of estimation is well known in univariate stratified sampling. Formulae are also available to work out optimum allocations to various strata [

In Section 2 of the manuscript combined ratio estimates for the population means of the “p” characteristics in presence of non-response using double sampling are constructed. Section 3 formulates the problem of obtaining compromise allocations for phase-I and phase-II of the double sampling as an integer nonlinear programming problem (INLPP). Sections 4 and 5 show that how these INLPP’s can be transformed to apply the Goal Programming Technique (GPT) and the Fuzzy Programming Technique (FPT) to solve the transformed problems. Section 6 provides an application of the techniques through a numerical data. In the last Section 8 gives the conclusion and the future work trend for interested readers.

Consider a multivariate stratified population of size

known in advance then the strata weights

For the

In double sampling for stratification the combined ratio estimate of the population mean of the

where “CR” and “DS” stand for “combined ratio” and “double sampling” respectively.

Further,

The sampling variance of

where

stratum for

In the presence of non-response, let out of the

is drawn and interviewed with extra efforts. Where

An combined ratio estimate

where

Using the results presented in [

where,

The total cost of the survey may be given

where,

Since

In Phase I, we obtain the sample size

Expression (5) can be expressed as

where the terms independent of

The cost constraint (7) becomes

where

Thus the multi-objective formulation of the problem at Phase I becomes

(see [

Ignoring the term independent from

where

The cost constraint becomes

where

Then the multi-objective formulation of the problem at Phase II becomes

Let

Further let

denote the variance under the compromise allocation, where

Obviously

Let

We have

or

or

A suitable compromise criterion to work out a compromise allocation at phase-I will then be to minimize the sum of deviations

(See [

The goal is now to minimize the sum of deviations from the respective optimum variances.

Similarly, at phase II Goal Programming formulation of the problem (15) will be

To obtain Fuzzy solution we first compute maximum value

where

The difference of the maximum value

The Fuzzy Programming Problem (FPP) corresponding to the (11) at phase I is given by the following NLPP

where

Similarly, the Fuzzy Programming Problem corresponding to the (15) at phase II is given by the following NLPP

where

The NLPPs may be solved by using the optimization software [

The data in

It is assumed that

In the last column of

The total cost for the survey is taken as

1 | 0.32 | 0.4 | 0.5 | 1 | 2 | 3 | 784 | 242 | 341 | 1444 | 481 | 628 |

2 | 0.21 | 0.5 | 0.6 | 1 | 3 | 4 | 576 | 192 | 250 | 676 | 255 | 294 |

3 | 0.27 | 0.6 | 0.7 | 1 | 4 | 5 | 1024 | 341 | 445 | 1936 | 645 | 842 |

4 | 0.20 | 0.65 | 0.75 | 1 | 5 | 6 | 2916 | 972 | 1268 | 6084 | 2028 | 2645 |

Group | ||||||||
---|---|---|---|---|---|---|---|---|

1 | Respondent | 361.06 | 157.04 | 767.82 | 255.76 | 333.93 | ||

Non-respondent | 310.55 | 88.73 | 135.07 | 454.76 | 151.48 | 197.93 | ||

2 | Respondent | 373.79 | 124.60 | 162.24 | 449.92 | 169.72 | 195.67 | |

Non-respondent | 326.29 | 108.76 | 141.62 | 353.81 | 133.46 | 153.88 | ||

3 | Respondent | 930.15 | 309.75 | 404.22 | 1272.88 | 424.07 | 553.60 | |

Non-respondent | 560.28 | 186.85 | 243.48 | 1165.98 | 388.46 | 507.10 | ||

4 | Respondent | 2355.98 | 785.33 | 1024.48 | 2690.53 | 896.84 | 1169.70 | |

Non-respondent | 1013.08 | 337.69 | 440.53 | 2403.55 | 801.18 | 1044.93 |

Using estimated values of strata weights the values of

Using data from

For

Using optimization software LINGO we get the optimal solution as

For

Using optimization software LINGO we get the optimal solution as

Using data from

Using optimization software LINGO we get the optimal solution as

with

As in Section 6.1.1 for the given data the individual optimum allocations for each the two characteristics using NLPP (15) are:

For

For

For the given data, as in Section 6.1.2 Goal Programming Problem (20) gives the following optimal solution

with

To obtain fuzzy solution we first obtained the maximum value and minimum value as given in (21) for each characteristic by using individual optimum allocation worked out in Section 6.1.1

and

After computing the optimum allocation and optimum variances for two characteristics the compromise optimal solution for the above problem can be obtained by solving the given Fuzzy Programming Problem (FPP) of (23)

Using optimization software LINGO we get the optimal solution as

Similarly, using data from

In the following results obtained using Goal Programming Technique and Fuzzy Programming Technique are summarized.

The Goal Programming and Fuzzy Programming technique and some other techniques like Dynamic Programming and Separable Programming can be used to solve a wide variety of mathematical programming problems. These techniques may be of great help in solving multivariate sampling problem also. Like determining

Techniques | Allocations | Variances | Trace | Cost incurred | ||||
---|---|---|---|---|---|---|---|---|

GPT | 175 | 72 | 134 | 152 | 1900 | |||

FPT | 171 | 75 | 135 | 151 | 1900 |

Techniques | Allocations | Variances | Trace | Cost incurred | ||||
---|---|---|---|---|---|---|---|---|

GPT | 22 | 10 | 20 | 24 | 350 | |||

FPT | 22 | 10 | 20 | 24 | 350 |

the number of strata, strata boundaries and compromise allocations in multivariate stratified sampling. Little work has been done to solve the above mentioned optimization problems in real life situations. For example when the estimates of the population parameters used in formulating the problems are themselves treated as random variables with assumed or known distributions. In such cases the formulated problems becomes a multivariate stochastic programming. Further, apart from a linear cost function, nonlinear functions may be used that may include travel cost, labour cost, rewards to the respondent and incentives to the investigators etc. Interested researchers may expose these situations.

The authors are thankful to the Editor for his valuable remarks and suggestions that helped us a lot in improving the standard of the paper. This research work is partially supported by the UGC grant of Emeritus Fellowship to the author Mohammed Jameel Ahsan for which he is grateful to UGC.