Research ArticleTwo-Stage Stratified Randomized Response Model withFuzzy Numbers

Mohammad Faisal Khan1 Neha Gupta2 and Irfan Ali2

1College of Science andTheoretical Studies Saudi Electronic University PO Box 93499 Riyadh 11673 Saudi Arabia2Department of Statistics and Operations Research Aligarh Muslim University Aligarh 202002 India

Correspondence should be addressed to Mohammad Faisal Khan faisalkhan004yahoocom

Received 16 November 2015 Revised 3 February 2016 Accepted 4 February 2016

Academic Editor Ashok B Kulkarni

Copyright copy 2016 Mohammad Faisal Khan et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We consider an allocation problem in two-stage stratifiedWarnerrsquos randomized response model and minimize the variance subjectto cost constraintThe costs (measurement costs and total budget of the survey) in the cost constraint are assumed as fuzzy numbersin particular triangular and trapezoidal fuzzy numbers due to the ease of use The problem formulated is solved by using Lagrangemultipliers technique and the optimum allocation obtained in the form of fuzzy numbers is converted into crisp form using 120572-cutmethod at a prescribed value of 120572 An illustrative numerical example is presented to demonstrate the proposed problem

1 Introduction

Sample survey is a method of drawing an inference aboutthe characteristic of a population or universe by observingonly a part of the population In modern complex surveysit is not possible to obtain true measurements on all thecharacteristics of interest on all the units in the samplebecause they are affected by two types of errors that issampling errors and nonsampling errors Nonsampling erroris further classified into two types response and nonresponseerrors Reduction in the reliability of measurements resultsin response error which can be minimized over repeatedmeasurements Whereas nonresponse errors are due to thenonavailability of information about some selected units forone or the other reason

A major source of nonresponse errors in sample surveysis the difficulty to obtain true responseswhen respondents areasked questions of highly personal or controversial naturefor example questions on accumulated savings intentionaltax evasion consumption of illegal drugs and extramaritalaffairs To avoid providing the requisite information orto avoid embarrassment some respondents may refuse toanswer or may intentionally give wrong answers Thus theestimates obtained from a direct survey on such topics wouldbe subject to high bias and any inference drawn from these

would be erroneous In order to solve this problemWarner [1]introduced a randomized response technique (RRT) whichwas developed subsequently by different authors

Some other authors who introduce other randomizedresponse techniques are Mangat and Singh [2] Chua andTsui [3] Padmawar and Vijayan [4] Chang and Huang [5]Chaudhuri [6] and so forth Kim and Warde [7] suggesteda stratified randomized response using optimum allocationMangat and Singh [2] proposed a two-stage randomizedresponse model

Traditional decision making problems are handled eitherby the deterministic approach or by probabilistic approachDeterministic approach completely avoiding the uncer-tainty provides an approximate solution while probabilisticapproach on an assumption represents any uncertainty asa probability distribution Both of these approaches onlypartially capture reality Uncertainty also is involved in deci-sion problems due to vagueness or impreciseness associatedwith linguistic information then in this case optimizationusing fuzzy mathematical theories becomes more relevantThe idea of fuzzy decisionmaking problems was proposed byBellmann and Zadeh [8] and this idea was used in problemsof mathematical programming by Tanaka and Asai [9] Manyauthors use fuzzy datafuzzy numbers in decision makingproblems such as Mahapatra and Roy [10] Pramanik and

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 4058758 8 pageshttpdxdoiorg10115520164058758

2 Advances in Fuzzy Systems

Roy [11] Abbasbandy and Hajjari [12] Kaur and Kumar [13]Ebrahimnejad [14] Sen et al [15] and Gupta and Bari [16]

In this paper the deterministic problem formulated byGhufran et al [17] is extended by considering it into anuncertain environment Here we consider the measurementcost and total budget for the survey as fuzzy numbers andformulate a fuzzy nonlinear programming problem Thenthe fuzzy nonlinear problem is solved by Lagrangemultiplierstechnique after converting it into crisp problem using 120572-cut For demonstrating the proposed problem an illustrativeexample is presented

2 Preliminary

Before formulating the problem of interest we should knowthe basic definitions of fuzzy sets fuzzy numbers and soforth which are reproduced here from Bector and Chandra[18] Mahapatra and Roy [10] Hassanzadeh et al [19] andAggarwal and Sharma [20] as follows

Fuzzy SetA fuzzy set in a universe of discourse119883 is definedas the following set of pairs = (119909 120583

119860

(119909)) 119909 isin 119883 Here120583

119860

119883 rarr [0 1] is amapping called themembership functionof the fuzzy set and 120583

119860

is called the membership value ordegree of membership of 119909 isin 119883 in the fuzzy set The largerthe value of 120583

119860

the stronger the grade of membership in

120572-Cut The 120572-cut for a fuzzy set is shown by 120572

and for120572 isin [0 1] is defined to be

120572

= 119909 | 120583

119860

(119909) ge 120572 119909 isin 119883 (1)

where119883 is the universal setUpper and lower bounds for any 120572-cut

120572

are given by

119880

120572

and 119871120572

respectively

Fuzzy Number A fuzzy set 119860 in R is called a fuzzy number ifit satisfies the following conditions

(i) 119860 is convex and normal

(ii) 119860120572

is a closed interval for every 120572 isin (0 1]

(iii) The support of 119860 is bounded

Triangular Fuzzy Number (TFN) A fuzzy number =

(119901 119902 119903) is said to be a triangular fuzzy number if its mem-bership function is given by

120583

119860

=

119909 minus 119901

119902 minus 119901

if 119901 le 119909 le 119902119903 minus 119909

119903 minus 119902

if 119902 le 119909 le 119903

0 otherwise

(2)

Trapezoidal Fuzzy Number (TrFN)A fuzzy set = (119901 119902 119903 119904)

on real numbers R is called a trapezoidal fuzzy number withmembership function as follows

120583

119860

(119909) =

0 119909 le 119901

119909 minus 119901

119902 minus 119901

119901 le 119909 le 119902

1 119902 le 119909 le 119903

119904 minus 119909

119904 minus 119903

119903 le 119909 le 119904

0 119904 le 119909

(3)

3 Statement of the Problem of Two-StageRandomized Response Model

Consider a stratified population of size 119873 partitioned into 119871disjoint strata of size 119873

ℎ

ℎ = 1 2 119871 and 119873 = sum119871

ℎ=1

119873ℎ

Let119882

ℎ

= 119873ℎ

119873 denote stratum weights 119899ℎ

denotes samplesize and 119899 = sum

119871

ℎ=1

119899ℎ

is the total sample size for the stratumℎ

In the first stage an individual respondent in the sampleis instructed to use the randomization device 119877

1ℎ

whichconsists of the following two statements

(i) ldquoI belong to the sensitive grouprdquo and (ii) ldquoGo to therandomization device 119877

2ℎ

in the second stagerdquo with knownprobabilities119872

ℎ

and (1 minus119872ℎ

) respectivelyIn the second stage the respondents are instructed to use

the randomization device 1198772ℎ

which consists of the followingtwo statements

(i) ldquoI belong to the sensitive grouprdquo and (ii) ldquoI do notbelong to the sensitive grouprdquo with known probabilities 119875

ℎ

and (1 minus 119875ℎ

) respectivelyAssuming that the ldquoYesrdquo or ldquoNordquo reports are made

truthfully for different outcomes and119872ℎ

and119875ℎ

are set by theinterviewer then the probability of a ldquoYesrdquo answer in stratumℎ is given by

119884ℎ

= 119872ℎ

120587119904ℎ

+ (1 minus119872ℎ

) [119875ℎ

120587119904ℎ

+ (1 minus 119875ℎ

) (1 minus 120587119904ℎ

)]

ℎ = 1 2 119871

(4)

and 120587119904ℎ

is the proportion of respondents belonging to thesensitive group from stratum ℎ The maximum likelihoodestimate of 120587

119904ℎ

is

119904ℎ

=

ℎ

minus (1 minus119872ℎ

) (1 minus 119875ℎ

)

2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)

ℎ = 1 2 119871 (5)

where ℎ

is the estimated proportion of ldquoYesrdquo answers whichfollows a binomial distribution 119861(119899

ℎ

119884ℎ

) It can be seen thatthe estimator

119904ℎ

is unbiased for 120587119904ℎ

with variance

119881 (119904ℎ

)

=

120587119904ℎ

(1 minus 120587119904ℎ

)

119899ℎ

+

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

119899ℎ

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(6)

Advances in Fuzzy Systems 3

If the suffix ldquoℎrdquo is removed then the expressions 1 2 and 3willbe reduced in Mangat and Singhrsquos expressions

Since 119899ℎ

are drawn independently from each stratumthe estimators for individual strata can be added to obtainthe estimator for the whole population Thus an unbiasedestimate of 120587

119904ℎ

is given by

119904

=

119871

sum

ℎ=1

119882ℎ

119904ℎ

(7)

Using (5)

119904

=

119871

sum

ℎ=1

119882ℎ

[

ℎ

minus (1 minus119872ℎ

) (1 minus 119875ℎ

)

2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)

] (8)

with a sampling variance

119881 (119904

) =

119871

sum

ℎ=1

1198822

ℎ

119881 (119904ℎ

) (9)

or119881 (119904

)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

)

+

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(10)

To find the optimum allocation we either maximize theprecision for fixed budget or minimize the cost for fixedprecision

A linear cost function which is an adequate approxima-tion of the actual cost incurred will be

119862 = 1198880

+

119871

sum

ℎ=1

119888ℎ

119899ℎ

(11)

where 119888ℎ

is per unit cost of measurement in the ℎth stratumand 1198880

is overhead costIn view of (4) to (11) the problem of finding optimum

allocation is formulated as nonlinear programming problem(NLPP) as follows

Minimize 119881 (119904

)

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(12)

The restrictions 1 le 119899ℎ

and 119899ℎ

le 119873ℎ

are placed to have therepresentation of every stratum in the sample and to avoid theoversampling respectively

4 Fuzzy Formulation of Two-StageRandomized Response Problem

Generally real-world situations involve a lot of parameterssuch as cost and time whose values are assigned by the

decision makers and in the conventional approach theyare required to fix an exact value to the aforementionedparameters However decision-makers frequently do not pre-cisely know the value of those parameters Therefore in suchcases it is better to consider those parameters or coefficientsin the decision-making problems as fuzzy numbers Themathematical modeling of fuzzy concepts was presented byZadeh in [21] Therefore the fuzzy formulation of problem(12) with fuzzy cost constraint is given by considering twocases of fuzzy numbers that is triangular fuzzy number(TFN) and trapezoidal fuzzy number (TrFN)

41 Case 1 Nonlinear Problem with TFN Consider

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

le (119862(1)

0

119862(2)

0

119862(3)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(13)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(14)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) is triangular fuzzy numbers withmembership function

120583ℎ(119909) =

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

if 119888(1)ℎ

le 119909 le 119888(2)

ℎ

119888(3)

ℎ

minus 119909

119888(3)

ℎ

minus 119888(2)

ℎ

if 119888(2)ℎ

le 119909 le 119888(3)

ℎ

0 otherwise

(15)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

if 119862(1)0

le 119909 le 119862(2)

0

119862(3)

0

minus 119909

119862(3)

0

minus 119862(2)

0

if 119862(2)0

le 119909 le 119862(3)

0

0 otherwise

(16)

42 Case 2 Nonlinear Problem with TrFN Consider

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) 119899ℎ

4 Advances in Fuzzy Systems

le (119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(17)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(18)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) is trapezoidal fuzzy numbers withmembership function

120583ℎ(119909) =

0 119909 le 119888(1)

ℎ

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

119888(1)

ℎ

le 119909 le 119888(2)

ℎ

1 119888(2)

ℎ

le 119909 le 119888(3)

ℎ

119888(4)

ℎ

minus 119909

119888(4)

ℎ

minus 119888(3)

ℎ

119888(3)

ℎ

le 119909 le 119888(4)

ℎ

0 119888(4)

ℎ

le 119909

(19)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

0 119909 le 119862(1)

0

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

119862(1)

0

le 119909 le 119862(2)

0

1 119862(2)

0

le 119909 le 119862(3)

0

119862(4)

0

minus 119909

119862(4)

0

minus 119862(3)

0

119862(3)

0

le 119909 le 119862(4)

0

0 119862(4)

0

le 119909

(20)

5 Lagrange Multipliers Technique

In problem (13) after ignoring the restrictions and takingequality in cost constraint the NLPP with TFNs is solved byLagrange multipliers technique (LMT) as follows

The Lagrangian function can be defined as

120601 (119899ℎ

120582)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

)

(21)

Differentiating (21) with respect to 119899ℎ

and 120582 and equating tozero we get the following sets of equations

120597120601

120597119899ℎ

= minus

1198822

ℎ

1198992

ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

= 0

(22)

or

119899ℎ

=

1

radic120582

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

radic(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(23)

Also

120597120601

120597120582

=

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0 (24)

which gives

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)119882ℎ

radic

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0

(25)

or

1

radic120582

=

(119862(1)

0

119862(2)

0

119862(3)

0

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(26)

Now using (23) and (26) we obtain

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(27)

In similar manner the optimum allocation of NLPP (17) withtrapezoidal fuzzy numbers can be obtain as

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

(28)

Advances in Fuzzy Systems 5

The allocations obtained in (27) and (28) are fuzzy in natureso we have to convert fuzzy allocations into a crisp allocationby 120572-cut method at a prescribed value of 120572

6 Procedure for the Conversion ofFuzzy Numbers

To convert the fuzzy allocation into crisp allocation 120572-cutmethod is used as follows

Let = (119901 119902 119903) be aTFNAn120572-cut for 120572

is computedas

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119903 minus 119909

119903 minus 119902

997904rArr

119880

120572

= 119909 = 119903 minus (119903 minus 119902) 120572

(29)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 1)

The allocation obtained in (27) is in the form of triangularfuzzy number therefore by using (29) the equivalent crispallocation is given by

119899lowast

ℎ

=

(119862(3)

0

minus (119862(3)

0

minus 119862(2)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(30)

Similarly if = (119901 119902 119903 119904) is a TrFN then the 120572-cut for 120572

is computed as

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119904 minus 119909

119904 minus 119903

997904rArr

119880

120572

= 119909 = 119904 minus (119904 minus 119903) 120572

(31)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 2)

In similar manner the crisp allocation corresponding to(28) is given by

119899lowast

ℎ

=

(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(32)

The allocations obtained by (30) and (32) provide the solutiontoNLPP (13) and (17) if it satisfies the restriction 1 le 119899

ℎ

le 119873ℎ

ℎ = 1 2 119871 The allocations obtained in (30) and (32)may not be integer allocations so to get integer allocationsround off the allocations to the nearest integer values Afterrounding off we have to be careful in rechecking that theround-off values satisfy the cost constraint

7 Some Other Allocation Techniques

71 Equal Allocation In this method the total sample size119899 is divided equally among all the strata that is for the ℎthstratum

119899ℎ

=

119899

119871

(33)

where 119899 can be obtained from the cost constraint equation asfollows

119871

sum

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

) 119899ℎ

= (119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

)) (34)

119899ℎ

prop 119882ℎ

or 119899ℎ

= 119899119882ℎ

(35)

Now substituting the value of 119899ℎ

in (34) we get

119899 =

119873(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))

sum119871

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)119873ℎ

(36)

72 Proportional Allocation This allocation was originallyproposed by Bowley [22]This procedure of allocation is verycommon in practice because of its simplicity When no otherinformation except 119873

ℎ

the total number of units in the ℎthstratum is available the allocation of a given sample of size 119899to different strata is done in proportion to their sizes that isin the ℎth stratum

119899ℎ

= 119899

119873ℎ

119873

(37)

6 Advances in Fuzzy Systems

120572

AL

120572

qp r x

1

AU

120572

120583A(x)

Figure 1 Triangular fuzzy number with an 120572-cut

120572

AL

120572

qp r s x

1

0

AU

120572

120583A(x)

Figure 2 Trapezoidal fuzzy number with an 120572-cut

73 Ghufranrsquos Allocation Ghufran et al [17] formulate a crispNLPP

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(38)

and obtained an optimum allocation using LMT as

119899lowast

ℎ

=

1198620

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

(39)

8 Numerical Illustration

A hypothetical example is given to illustrate the compu-tational details of the proposed problem Let us supposethe population size is 1000 with total available budgetof the survey as TFNs and TrFNs are (3500 4000 4800)

and (3500 4000 4400 4600) units respectively The otherrequired relevant information is given in Table 1 By using thevalues of Table 1 we compute the values of 119860

ℎ

which is givenin Table 2

After substituting all the values fromTables 1 and 2 in (13)the required FNLPP is given as

Minimize 003098772

1198991

+

020718425

1198992

Subject to (1 2 4) 1198991

+ (18 20 24) 1198992

le (3500 4000 4800)

1 le 1198991

le 300 1 le 1198992

le 700

(40)

The required optimum allocations for problem (13) obtainedby substituting the values from Tables 1 and 2 in (30) at 120572 =

05 will be

1198991

=

(4800 minus 800120572) times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28751 ⋍ 288

1198992

=

(4800 minus 800120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20888 ⋍ 209

(41)

In similar manner optimum allocations for problem (17)obtained by substituting the values fromTables 1 and 2 in (32)at 120572 = 05 will be

1198991

=

3750 times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28097 ⋍ 281

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Advances in Fuzzy Systems

Roy [11] Abbasbandy and Hajjari [12] Kaur and Kumar [13]Ebrahimnejad [14] Sen et al [15] and Gupta and Bari [16]

In this paper the deterministic problem formulated byGhufran et al [17] is extended by considering it into anuncertain environment Here we consider the measurementcost and total budget for the survey as fuzzy numbers andformulate a fuzzy nonlinear programming problem Thenthe fuzzy nonlinear problem is solved by Lagrangemultiplierstechnique after converting it into crisp problem using 120572-cut For demonstrating the proposed problem an illustrativeexample is presented

2 Preliminary

Before formulating the problem of interest we should knowthe basic definitions of fuzzy sets fuzzy numbers and soforth which are reproduced here from Bector and Chandra[18] Mahapatra and Roy [10] Hassanzadeh et al [19] andAggarwal and Sharma [20] as follows

Fuzzy SetA fuzzy set in a universe of discourse119883 is definedas the following set of pairs = (119909 120583

119860

(119909)) 119909 isin 119883 Here120583

119860

119883 rarr [0 1] is amapping called themembership functionof the fuzzy set and 120583

119860

is called the membership value ordegree of membership of 119909 isin 119883 in the fuzzy set The largerthe value of 120583

119860

the stronger the grade of membership in

120572-Cut The 120572-cut for a fuzzy set is shown by 120572

and for120572 isin [0 1] is defined to be

120572

= 119909 | 120583

119860

(119909) ge 120572 119909 isin 119883 (1)

where119883 is the universal setUpper and lower bounds for any 120572-cut

120572

are given by

119880

120572

and 119871120572

respectively

Fuzzy Number A fuzzy set 119860 in R is called a fuzzy number ifit satisfies the following conditions

(i) 119860 is convex and normal

(ii) 119860120572

is a closed interval for every 120572 isin (0 1]

(iii) The support of 119860 is bounded

Triangular Fuzzy Number (TFN) A fuzzy number =

(119901 119902 119903) is said to be a triangular fuzzy number if its mem-bership function is given by

120583

119860

=

119909 minus 119901

119902 minus 119901

if 119901 le 119909 le 119902119903 minus 119909

119903 minus 119902

if 119902 le 119909 le 119903

0 otherwise

(2)

Trapezoidal Fuzzy Number (TrFN)A fuzzy set = (119901 119902 119903 119904)

on real numbers R is called a trapezoidal fuzzy number withmembership function as follows

120583

119860

(119909) =

0 119909 le 119901

119909 minus 119901

119902 minus 119901

119901 le 119909 le 119902

1 119902 le 119909 le 119903

119904 minus 119909

119904 minus 119903

119903 le 119909 le 119904

0 119904 le 119909

(3)

3 Statement of the Problem of Two-StageRandomized Response Model

Consider a stratified population of size 119873 partitioned into 119871disjoint strata of size 119873

ℎ

ℎ = 1 2 119871 and 119873 = sum119871

ℎ=1

119873ℎ

Let119882

ℎ

= 119873ℎ

119873 denote stratum weights 119899ℎ

denotes samplesize and 119899 = sum

119871

ℎ=1

119899ℎ

is the total sample size for the stratumℎ

In the first stage an individual respondent in the sampleis instructed to use the randomization device 119877

1ℎ

whichconsists of the following two statements

(i) ldquoI belong to the sensitive grouprdquo and (ii) ldquoGo to therandomization device 119877

2ℎ

in the second stagerdquo with knownprobabilities119872

ℎ

and (1 minus119872ℎ

) respectivelyIn the second stage the respondents are instructed to use

the randomization device 1198772ℎ

which consists of the followingtwo statements

(i) ldquoI belong to the sensitive grouprdquo and (ii) ldquoI do notbelong to the sensitive grouprdquo with known probabilities 119875

ℎ

and (1 minus 119875ℎ

) respectivelyAssuming that the ldquoYesrdquo or ldquoNordquo reports are made

truthfully for different outcomes and119872ℎ

and119875ℎ

are set by theinterviewer then the probability of a ldquoYesrdquo answer in stratumℎ is given by

119884ℎ

= 119872ℎ

120587119904ℎ

+ (1 minus119872ℎ

) [119875ℎ

120587119904ℎ

+ (1 minus 119875ℎ

) (1 minus 120587119904ℎ

)]

ℎ = 1 2 119871

(4)

and 120587119904ℎ

is the proportion of respondents belonging to thesensitive group from stratum ℎ The maximum likelihoodestimate of 120587

119904ℎ

is

119904ℎ

=

ℎ

minus (1 minus119872ℎ

) (1 minus 119875ℎ

)

2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)

ℎ = 1 2 119871 (5)

where ℎ

is the estimated proportion of ldquoYesrdquo answers whichfollows a binomial distribution 119861(119899

ℎ

119884ℎ

) It can be seen thatthe estimator

119904ℎ

is unbiased for 120587119904ℎ

with variance

119881 (119904ℎ

)

=

120587119904ℎ

(1 minus 120587119904ℎ

)

119899ℎ

+

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

119899ℎ

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(6)

Advances in Fuzzy Systems 3

If the suffix ldquoℎrdquo is removed then the expressions 1 2 and 3willbe reduced in Mangat and Singhrsquos expressions

Since 119899ℎ

are drawn independently from each stratumthe estimators for individual strata can be added to obtainthe estimator for the whole population Thus an unbiasedestimate of 120587

119904ℎ

is given by

119904

=

119871

sum

ℎ=1

119882ℎ

119904ℎ

(7)

Using (5)

119904

=

119871

sum

ℎ=1

119882ℎ

[

ℎ

minus (1 minus119872ℎ

) (1 minus 119875ℎ

)

2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)

] (8)

with a sampling variance

119881 (119904

) =

119871

sum

ℎ=1

1198822

ℎ

119881 (119904ℎ

) (9)

or119881 (119904

)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

)

+

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(10)

To find the optimum allocation we either maximize theprecision for fixed budget or minimize the cost for fixedprecision

A linear cost function which is an adequate approxima-tion of the actual cost incurred will be

119862 = 1198880

+

119871

sum

ℎ=1

119888ℎ

119899ℎ

(11)

where 119888ℎ

is per unit cost of measurement in the ℎth stratumand 1198880

is overhead costIn view of (4) to (11) the problem of finding optimum

allocation is formulated as nonlinear programming problem(NLPP) as follows

Minimize 119881 (119904

)

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(12)

The restrictions 1 le 119899ℎ

and 119899ℎ

le 119873ℎ

are placed to have therepresentation of every stratum in the sample and to avoid theoversampling respectively

4 Fuzzy Formulation of Two-StageRandomized Response Problem

Generally real-world situations involve a lot of parameterssuch as cost and time whose values are assigned by the

decision makers and in the conventional approach theyare required to fix an exact value to the aforementionedparameters However decision-makers frequently do not pre-cisely know the value of those parameters Therefore in suchcases it is better to consider those parameters or coefficientsin the decision-making problems as fuzzy numbers Themathematical modeling of fuzzy concepts was presented byZadeh in [21] Therefore the fuzzy formulation of problem(12) with fuzzy cost constraint is given by considering twocases of fuzzy numbers that is triangular fuzzy number(TFN) and trapezoidal fuzzy number (TrFN)

41 Case 1 Nonlinear Problem with TFN Consider

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

le (119862(1)

0

119862(2)

0

119862(3)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(13)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(14)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) is triangular fuzzy numbers withmembership function

120583ℎ(119909) =

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

if 119888(1)ℎ

le 119909 le 119888(2)

ℎ

119888(3)

ℎ

minus 119909

119888(3)

ℎ

minus 119888(2)

ℎ

if 119888(2)ℎ

le 119909 le 119888(3)

ℎ

0 otherwise

(15)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

if 119862(1)0

le 119909 le 119862(2)

0

119862(3)

0

minus 119909

119862(3)

0

minus 119862(2)

0

if 119862(2)0

le 119909 le 119862(3)

0

0 otherwise

(16)

42 Case 2 Nonlinear Problem with TrFN Consider

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) 119899ℎ

4 Advances in Fuzzy Systems

le (119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(17)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(18)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) is trapezoidal fuzzy numbers withmembership function

120583ℎ(119909) =

0 119909 le 119888(1)

ℎ

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

119888(1)

ℎ

le 119909 le 119888(2)

ℎ

1 119888(2)

ℎ

le 119909 le 119888(3)

ℎ

119888(4)

ℎ

minus 119909

119888(4)

ℎ

minus 119888(3)

ℎ

119888(3)

ℎ

le 119909 le 119888(4)

ℎ

0 119888(4)

ℎ

le 119909

(19)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

0 119909 le 119862(1)

0

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

119862(1)

0

le 119909 le 119862(2)

0

1 119862(2)

0

le 119909 le 119862(3)

0

119862(4)

0

minus 119909

119862(4)

0

minus 119862(3)

0

119862(3)

0

le 119909 le 119862(4)

0

0 119862(4)

0

le 119909

(20)

5 Lagrange Multipliers Technique

In problem (13) after ignoring the restrictions and takingequality in cost constraint the NLPP with TFNs is solved byLagrange multipliers technique (LMT) as follows

The Lagrangian function can be defined as

120601 (119899ℎ

120582)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

)

(21)

Differentiating (21) with respect to 119899ℎ

and 120582 and equating tozero we get the following sets of equations

120597120601

120597119899ℎ

= minus

1198822

ℎ

1198992

ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

= 0

(22)

or

119899ℎ

=

1

radic120582

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

radic(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(23)

Also

120597120601

120597120582

=

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0 (24)

which gives

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)119882ℎ

radic

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0

(25)

or

1

radic120582

=

(119862(1)

0

119862(2)

0

119862(3)

0

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(26)

Now using (23) and (26) we obtain

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(27)

In similar manner the optimum allocation of NLPP (17) withtrapezoidal fuzzy numbers can be obtain as

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

(28)

Advances in Fuzzy Systems 5

The allocations obtained in (27) and (28) are fuzzy in natureso we have to convert fuzzy allocations into a crisp allocationby 120572-cut method at a prescribed value of 120572

6 Procedure for the Conversion ofFuzzy Numbers

To convert the fuzzy allocation into crisp allocation 120572-cutmethod is used as follows

Let = (119901 119902 119903) be aTFNAn120572-cut for 120572

is computedas

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119903 minus 119909

119903 minus 119902

997904rArr

119880

120572

= 119909 = 119903 minus (119903 minus 119902) 120572

(29)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 1)

The allocation obtained in (27) is in the form of triangularfuzzy number therefore by using (29) the equivalent crispallocation is given by

119899lowast

ℎ

=

(119862(3)

0

minus (119862(3)

0

minus 119862(2)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(30)

Similarly if = (119901 119902 119903 119904) is a TrFN then the 120572-cut for 120572

is computed as

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119904 minus 119909

119904 minus 119903

997904rArr

119880

120572

= 119909 = 119904 minus (119904 minus 119903) 120572

(31)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 2)

In similar manner the crisp allocation corresponding to(28) is given by

119899lowast

ℎ

=

(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(32)

The allocations obtained by (30) and (32) provide the solutiontoNLPP (13) and (17) if it satisfies the restriction 1 le 119899

ℎ

le 119873ℎ

ℎ = 1 2 119871 The allocations obtained in (30) and (32)may not be integer allocations so to get integer allocationsround off the allocations to the nearest integer values Afterrounding off we have to be careful in rechecking that theround-off values satisfy the cost constraint

7 Some Other Allocation Techniques

71 Equal Allocation In this method the total sample size119899 is divided equally among all the strata that is for the ℎthstratum

119899ℎ

=

119899

119871

(33)

where 119899 can be obtained from the cost constraint equation asfollows

119871

sum

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

) 119899ℎ

= (119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

)) (34)

119899ℎ

prop 119882ℎ

or 119899ℎ

= 119899119882ℎ

(35)

Now substituting the value of 119899ℎ

in (34) we get

119899 =

119873(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))

sum119871

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)119873ℎ

(36)

72 Proportional Allocation This allocation was originallyproposed by Bowley [22]This procedure of allocation is verycommon in practice because of its simplicity When no otherinformation except 119873

ℎ

the total number of units in the ℎthstratum is available the allocation of a given sample of size 119899to different strata is done in proportion to their sizes that isin the ℎth stratum

119899ℎ

= 119899

119873ℎ

119873

(37)

6 Advances in Fuzzy Systems

120572

AL

120572

qp r x

1

AU

120572

120583A(x)

Figure 1 Triangular fuzzy number with an 120572-cut

120572

AL

120572

qp r s x

1

0

AU

120572

120583A(x)

Figure 2 Trapezoidal fuzzy number with an 120572-cut

73 Ghufranrsquos Allocation Ghufran et al [17] formulate a crispNLPP

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(38)

and obtained an optimum allocation using LMT as

119899lowast

ℎ

=

1198620

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

(39)

8 Numerical Illustration

A hypothetical example is given to illustrate the compu-tational details of the proposed problem Let us supposethe population size is 1000 with total available budgetof the survey as TFNs and TrFNs are (3500 4000 4800)

and (3500 4000 4400 4600) units respectively The otherrequired relevant information is given in Table 1 By using thevalues of Table 1 we compute the values of 119860

ℎ

which is givenin Table 2

After substituting all the values fromTables 1 and 2 in (13)the required FNLPP is given as

Minimize 003098772

1198991

+

020718425

1198992

Subject to (1 2 4) 1198991

+ (18 20 24) 1198992

le (3500 4000 4800)

1 le 1198991

le 300 1 le 1198992

le 700

(40)

The required optimum allocations for problem (13) obtainedby substituting the values from Tables 1 and 2 in (30) at 120572 =

05 will be

1198991

=

(4800 minus 800120572) times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28751 ⋍ 288

1198992

=

(4800 minus 800120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20888 ⋍ 209

(41)

In similar manner optimum allocations for problem (17)obtained by substituting the values fromTables 1 and 2 in (32)at 120572 = 05 will be

1198991

=

3750 times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28097 ⋍ 281

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 3

If the suffix ldquoℎrdquo is removed then the expressions 1 2 and 3willbe reduced in Mangat and Singhrsquos expressions

Since 119899ℎ

are drawn independently from each stratumthe estimators for individual strata can be added to obtainthe estimator for the whole population Thus an unbiasedestimate of 120587

119904ℎ

is given by

119904

=

119871

sum

ℎ=1

119882ℎ

119904ℎ

(7)

Using (5)

119904

=

119871

sum

ℎ=1

119882ℎ

[

ℎ

minus (1 minus119872ℎ

) (1 minus 119875ℎ

)

2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)

] (8)

with a sampling variance

119881 (119904

) =

119871

sum

ℎ=1

1198822

ℎ

119881 (119904ℎ

) (9)

or119881 (119904

)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

)

+

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(10)

To find the optimum allocation we either maximize theprecision for fixed budget or minimize the cost for fixedprecision

A linear cost function which is an adequate approxima-tion of the actual cost incurred will be

119862 = 1198880

+

119871

sum

ℎ=1

119888ℎ

119899ℎ

(11)

where 119888ℎ

is per unit cost of measurement in the ℎth stratumand 1198880

is overhead costIn view of (4) to (11) the problem of finding optimum

allocation is formulated as nonlinear programming problem(NLPP) as follows

Minimize 119881 (119904

)

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(12)

The restrictions 1 le 119899ℎ

and 119899ℎ

le 119873ℎ

are placed to have therepresentation of every stratum in the sample and to avoid theoversampling respectively

4 Fuzzy Formulation of Two-StageRandomized Response Problem

Generally real-world situations involve a lot of parameterssuch as cost and time whose values are assigned by the

decision makers and in the conventional approach theyare required to fix an exact value to the aforementionedparameters However decision-makers frequently do not pre-cisely know the value of those parameters Therefore in suchcases it is better to consider those parameters or coefficientsin the decision-making problems as fuzzy numbers Themathematical modeling of fuzzy concepts was presented byZadeh in [21] Therefore the fuzzy formulation of problem(12) with fuzzy cost constraint is given by considering twocases of fuzzy numbers that is triangular fuzzy number(TFN) and trapezoidal fuzzy number (TrFN)

41 Case 1 Nonlinear Problem with TFN Consider

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

le (119862(1)

0

119862(2)

0

119862(3)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(13)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(14)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) is triangular fuzzy numbers withmembership function

120583ℎ(119909) =

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

if 119888(1)ℎ

le 119909 le 119888(2)

ℎ

119888(3)

ℎ

minus 119909

119888(3)

ℎ

minus 119888(2)

ℎ

if 119888(2)ℎ

le 119909 le 119888(3)

ℎ

0 otherwise

(15)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

if 119862(1)0

le 119909 le 119862(2)

0

119862(3)

0

minus 119909

119862(3)

0

minus 119862(2)

0

if 119862(2)0

le 119909 le 119862(3)

0

0 otherwise

(16)

42 Case 2 Nonlinear Problem with TrFN Consider

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) 119899ℎ

4 Advances in Fuzzy Systems

le (119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(17)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(18)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) is trapezoidal fuzzy numbers withmembership function

120583ℎ(119909) =

0 119909 le 119888(1)

ℎ

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

119888(1)

ℎ

le 119909 le 119888(2)

ℎ

1 119888(2)

ℎ

le 119909 le 119888(3)

ℎ

119888(4)

ℎ

minus 119909

119888(4)

ℎ

minus 119888(3)

ℎ

119888(3)

ℎ

le 119909 le 119888(4)

ℎ

0 119888(4)

ℎ

le 119909

(19)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

0 119909 le 119862(1)

0

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

119862(1)

0

le 119909 le 119862(2)

0

1 119862(2)

0

le 119909 le 119862(3)

0

119862(4)

0

minus 119909

119862(4)

0

minus 119862(3)

0

119862(3)

0

le 119909 le 119862(4)

0

0 119862(4)

0

le 119909

(20)

5 Lagrange Multipliers Technique

In problem (13) after ignoring the restrictions and takingequality in cost constraint the NLPP with TFNs is solved byLagrange multipliers technique (LMT) as follows

The Lagrangian function can be defined as

120601 (119899ℎ

120582)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

)

(21)

Differentiating (21) with respect to 119899ℎ

and 120582 and equating tozero we get the following sets of equations

120597120601

120597119899ℎ

= minus

1198822

ℎ

1198992

ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

= 0

(22)

or

119899ℎ

=

1

radic120582

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

radic(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(23)

Also

120597120601

120597120582

=

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0 (24)

which gives

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)119882ℎ

radic

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0

(25)

or

1

radic120582

=

(119862(1)

0

119862(2)

0

119862(3)

0

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(26)

Now using (23) and (26) we obtain

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(27)

In similar manner the optimum allocation of NLPP (17) withtrapezoidal fuzzy numbers can be obtain as

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

(28)

Advances in Fuzzy Systems 5

The allocations obtained in (27) and (28) are fuzzy in natureso we have to convert fuzzy allocations into a crisp allocationby 120572-cut method at a prescribed value of 120572

6 Procedure for the Conversion ofFuzzy Numbers

To convert the fuzzy allocation into crisp allocation 120572-cutmethod is used as follows

Let = (119901 119902 119903) be aTFNAn120572-cut for 120572

is computedas

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119903 minus 119909

119903 minus 119902

997904rArr

119880

120572

= 119909 = 119903 minus (119903 minus 119902) 120572

(29)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 1)

The allocation obtained in (27) is in the form of triangularfuzzy number therefore by using (29) the equivalent crispallocation is given by

119899lowast

ℎ

=

(119862(3)

0

minus (119862(3)

0

minus 119862(2)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(30)

Similarly if = (119901 119902 119903 119904) is a TrFN then the 120572-cut for 120572

is computed as

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119904 minus 119909

119904 minus 119903

997904rArr

119880

120572

= 119909 = 119904 minus (119904 minus 119903) 120572

(31)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 2)

In similar manner the crisp allocation corresponding to(28) is given by

119899lowast

ℎ

=

(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(32)

The allocations obtained by (30) and (32) provide the solutiontoNLPP (13) and (17) if it satisfies the restriction 1 le 119899

ℎ

le 119873ℎ

ℎ = 1 2 119871 The allocations obtained in (30) and (32)may not be integer allocations so to get integer allocationsround off the allocations to the nearest integer values Afterrounding off we have to be careful in rechecking that theround-off values satisfy the cost constraint

7 Some Other Allocation Techniques

71 Equal Allocation In this method the total sample size119899 is divided equally among all the strata that is for the ℎthstratum

119899ℎ

=

119899

119871

(33)

where 119899 can be obtained from the cost constraint equation asfollows

119871

sum

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

) 119899ℎ

= (119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

)) (34)

119899ℎ

prop 119882ℎ

or 119899ℎ

= 119899119882ℎ

(35)

Now substituting the value of 119899ℎ

in (34) we get

119899 =

119873(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))

sum119871

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)119873ℎ

(36)

72 Proportional Allocation This allocation was originallyproposed by Bowley [22]This procedure of allocation is verycommon in practice because of its simplicity When no otherinformation except 119873

ℎ

the total number of units in the ℎthstratum is available the allocation of a given sample of size 119899to different strata is done in proportion to their sizes that isin the ℎth stratum

119899ℎ

= 119899

119873ℎ

119873

(37)

6 Advances in Fuzzy Systems

120572

AL

120572

qp r x

1

AU

120572

120583A(x)

Figure 1 Triangular fuzzy number with an 120572-cut

120572

AL

120572

qp r s x

1

0

AU

120572

120583A(x)

Figure 2 Trapezoidal fuzzy number with an 120572-cut

73 Ghufranrsquos Allocation Ghufran et al [17] formulate a crispNLPP

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(38)

and obtained an optimum allocation using LMT as

119899lowast

ℎ

=

1198620

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

(39)

8 Numerical Illustration

A hypothetical example is given to illustrate the compu-tational details of the proposed problem Let us supposethe population size is 1000 with total available budgetof the survey as TFNs and TrFNs are (3500 4000 4800)

and (3500 4000 4400 4600) units respectively The otherrequired relevant information is given in Table 1 By using thevalues of Table 1 we compute the values of 119860

ℎ

which is givenin Table 2

After substituting all the values fromTables 1 and 2 in (13)the required FNLPP is given as

Minimize 003098772

1198991

+

020718425

1198992

Subject to (1 2 4) 1198991

+ (18 20 24) 1198992

le (3500 4000 4800)

1 le 1198991

le 300 1 le 1198992

le 700

(40)

The required optimum allocations for problem (13) obtainedby substituting the values from Tables 1 and 2 in (30) at 120572 =

05 will be

1198991

=

(4800 minus 800120572) times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28751 ⋍ 288

1198992

=

(4800 minus 800120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20888 ⋍ 209

(41)

In similar manner optimum allocations for problem (17)obtained by substituting the values fromTables 1 and 2 in (32)at 120572 = 05 will be

1198991

=

3750 times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28097 ⋍ 281

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

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International Journal of

Advances in

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Hindawi Publishing Corporationhttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Electrical and Computer Engineering

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RoboticsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Advances in Fuzzy Systems

le (119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(17)

where

119860ℎ

=

(1 minus119872ℎ

) (1 minus 119875ℎ

) [1 minus (1 minus119872ℎ

) (1 minus 119875ℎ

)]

[2119875ℎ

minus 1 + 2119872ℎ

(1 minus 119875ℎ

)]2

(18)

and ℎ

= (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

) is trapezoidal fuzzy numbers withmembership function

120583ℎ(119909) =

0 119909 le 119888(1)

ℎ

119909 minus 119888(1)

ℎ

119888(2)

ℎ

minus 119888(1)

ℎ

119888(1)

ℎ

le 119909 le 119888(2)

ℎ

1 119888(2)

ℎ

le 119909 le 119888(3)

ℎ

119888(4)

ℎ

minus 119909

119888(4)

ℎ

minus 119888(3)

ℎ

119888(3)

ℎ

le 119909 le 119888(4)

ℎ

0 119888(4)

ℎ

le 119909

(19)

Similarly the membership function for available budget canbe expressed as

120583

1198620(119909) =

0 119909 le 119862(1)

0

119909 minus 119862(1)

0

119862(2)

0

minus 119862(1)

0

119862(1)

0

le 119909 le 119862(2)

0

1 119862(2)

0

le 119909 le 119862(3)

0

119862(4)

0

minus 119909

119862(4)

0

minus 119862(3)

0

119862(3)

0

le 119909 le 119862(4)

0

0 119862(4)

0

le 119909

(20)

5 Lagrange Multipliers Technique

In problem (13) after ignoring the restrictions and takingequality in cost constraint the NLPP with TFNs is solved byLagrange multipliers technique (LMT) as follows

The Lagrangian function can be defined as

120601 (119899ℎ

120582)

=

119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

)

(21)

Differentiating (21) with respect to 119899ℎ

and 120582 and equating tozero we get the following sets of equations

120597120601

120597119899ℎ

= minus

1198822

ℎ

1198992

ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

+ 120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

= 0

(22)

or

119899ℎ

=

1

radic120582

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

radic(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(23)

Also

120597120601

120597120582

=

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) 119899ℎ

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0 (24)

which gives

119871

sum

ℎ=1

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)119882ℎ

radic

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

120582 (119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

minus (119862(1)

0

119862(2)

0

119862(3)

0

) = 0

(25)

or

1

radic120582

=

(119862(1)

0

119862(2)

0

119862(3)

0

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(26)

Now using (23) and (26) we obtain

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

)

(27)

In similar manner the optimum allocation of NLPP (17) withtrapezoidal fuzzy numbers can be obtain as

119899lowast

ℎ

=

(119862(1)

0

119862(2)

0

119862(3)

0

119862(4)

0

)119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

119888(4)

ℎ

)

(28)

Advances in Fuzzy Systems 5

The allocations obtained in (27) and (28) are fuzzy in natureso we have to convert fuzzy allocations into a crisp allocationby 120572-cut method at a prescribed value of 120572

6 Procedure for the Conversion ofFuzzy Numbers

To convert the fuzzy allocation into crisp allocation 120572-cutmethod is used as follows

Let = (119901 119902 119903) be aTFNAn120572-cut for 120572

is computedas

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119903 minus 119909

119903 minus 119902

997904rArr

119880

120572

= 119909 = 119903 minus (119903 minus 119902) 120572

(29)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 1)

The allocation obtained in (27) is in the form of triangularfuzzy number therefore by using (29) the equivalent crispallocation is given by

119899lowast

ℎ

=

(119862(3)

0

minus (119862(3)

0

minus 119862(2)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(30)

Similarly if = (119901 119902 119903 119904) is a TrFN then the 120572-cut for 120572

is computed as

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119904 minus 119909

119904 minus 119903

997904rArr

119880

120572

= 119909 = 119904 minus (119904 minus 119903) 120572

(31)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 2)

In similar manner the crisp allocation corresponding to(28) is given by

119899lowast

ℎ

=

(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(32)

The allocations obtained by (30) and (32) provide the solutiontoNLPP (13) and (17) if it satisfies the restriction 1 le 119899

ℎ

le 119873ℎ

ℎ = 1 2 119871 The allocations obtained in (30) and (32)may not be integer allocations so to get integer allocationsround off the allocations to the nearest integer values Afterrounding off we have to be careful in rechecking that theround-off values satisfy the cost constraint

7 Some Other Allocation Techniques

71 Equal Allocation In this method the total sample size119899 is divided equally among all the strata that is for the ℎthstratum

119899ℎ

=

119899

119871

(33)

where 119899 can be obtained from the cost constraint equation asfollows

119871

sum

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

) 119899ℎ

= (119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

)) (34)

119899ℎ

prop 119882ℎ

or 119899ℎ

= 119899119882ℎ

(35)

Now substituting the value of 119899ℎ

in (34) we get

119899 =

119873(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))

sum119871

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)119873ℎ

(36)

72 Proportional Allocation This allocation was originallyproposed by Bowley [22]This procedure of allocation is verycommon in practice because of its simplicity When no otherinformation except 119873

ℎ

the total number of units in the ℎthstratum is available the allocation of a given sample of size 119899to different strata is done in proportion to their sizes that isin the ℎth stratum

119899ℎ

= 119899

119873ℎ

119873

(37)

6 Advances in Fuzzy Systems

120572

AL

120572

qp r x

1

AU

120572

120583A(x)

Figure 1 Triangular fuzzy number with an 120572-cut

120572

AL

120572

qp r s x

1

0

AU

120572

120583A(x)

Figure 2 Trapezoidal fuzzy number with an 120572-cut

73 Ghufranrsquos Allocation Ghufran et al [17] formulate a crispNLPP

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(38)

and obtained an optimum allocation using LMT as

119899lowast

ℎ

=

1198620

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

(39)

8 Numerical Illustration

A hypothetical example is given to illustrate the compu-tational details of the proposed problem Let us supposethe population size is 1000 with total available budgetof the survey as TFNs and TrFNs are (3500 4000 4800)

and (3500 4000 4400 4600) units respectively The otherrequired relevant information is given in Table 1 By using thevalues of Table 1 we compute the values of 119860

ℎ

which is givenin Table 2

After substituting all the values fromTables 1 and 2 in (13)the required FNLPP is given as

Minimize 003098772

1198991

+

020718425

1198992

Subject to (1 2 4) 1198991

+ (18 20 24) 1198992

le (3500 4000 4800)

1 le 1198991

le 300 1 le 1198992

le 700

(40)

The required optimum allocations for problem (13) obtainedby substituting the values from Tables 1 and 2 in (30) at 120572 =

05 will be

1198991

=

(4800 minus 800120572) times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28751 ⋍ 288

1198992

=

(4800 minus 800120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20888 ⋍ 209

(41)

In similar manner optimum allocations for problem (17)obtained by substituting the values fromTables 1 and 2 in (32)at 120572 = 05 will be

1198991

=

3750 times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28097 ⋍ 281

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 5

The allocations obtained in (27) and (28) are fuzzy in natureso we have to convert fuzzy allocations into a crisp allocationby 120572-cut method at a prescribed value of 120572

6 Procedure for the Conversion ofFuzzy Numbers

To convert the fuzzy allocation into crisp allocation 120572-cutmethod is used as follows

Let = (119901 119902 119903) be aTFNAn120572-cut for 120572

is computedas

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119903 minus 119909

119903 minus 119902

997904rArr

119880

120572

= 119909 = 119903 minus (119903 minus 119902) 120572

(29)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 1)

The allocation obtained in (27) is in the form of triangularfuzzy number therefore by using (29) the equivalent crispallocation is given by

119899lowast

ℎ

=

(119862(3)

0

minus (119862(3)

0

minus 119862(2)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(30)

Similarly if = (119901 119902 119903 119904) is a TrFN then the 120572-cut for 120572

is computed as

120572 =

119909 minus 119901

119902 minus 119901

997904rArr

119871

120572

= 119909 = (119902 minus 119901) 120572 + 119901

120572 =

119904 minus 119909

119904 minus 119903

997904rArr

119880

120572

= 119909 = 119904 minus (119904 minus 119903) 120572

(31)

where 120572

= [

119871

120572

119880

120572

] is the corresponding 120572-cut (seeFigure 2)

In similar manner the crisp allocation corresponding to(28) is given by

119899lowast

ℎ

=

(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)

(32)

The allocations obtained by (30) and (32) provide the solutiontoNLPP (13) and (17) if it satisfies the restriction 1 le 119899

ℎ

le 119873ℎ

ℎ = 1 2 119871 The allocations obtained in (30) and (32)may not be integer allocations so to get integer allocationsround off the allocations to the nearest integer values Afterrounding off we have to be careful in rechecking that theround-off values satisfy the cost constraint

7 Some Other Allocation Techniques

71 Equal Allocation In this method the total sample size119899 is divided equally among all the strata that is for the ℎthstratum

119899ℎ

=

119899

119871

(33)

where 119899 can be obtained from the cost constraint equation asfollows

119871

sum

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

) 119899ℎ

= (119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

)) (34)

119899ℎ

prop 119882ℎ

or 119899ℎ

= 119899119882ℎ

(35)

Now substituting the value of 119899ℎ

in (34) we get

119899 =

119873(119862(4)

0

minus (119862(4)

0

minus 119862(3)

0

))

sum119871

ℎ=1

((119888(2)

ℎ

minus 119888(1)

ℎ

) + 119888(1)

ℎ

)119873ℎ

(36)

72 Proportional Allocation This allocation was originallyproposed by Bowley [22]This procedure of allocation is verycommon in practice because of its simplicity When no otherinformation except 119873

ℎ

the total number of units in the ℎthstratum is available the allocation of a given sample of size 119899to different strata is done in proportion to their sizes that isin the ℎth stratum

119899ℎ

= 119899

119873ℎ

119873

(37)

6 Advances in Fuzzy Systems

120572

AL

120572

qp r x

1

AU

120572

120583A(x)

Figure 1 Triangular fuzzy number with an 120572-cut

120572

AL

120572

qp r s x

1

0

AU

120572

120583A(x)

Figure 2 Trapezoidal fuzzy number with an 120572-cut

73 Ghufranrsquos Allocation Ghufran et al [17] formulate a crispNLPP

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(38)

and obtained an optimum allocation using LMT as

119899lowast

ℎ

=

1198620

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

(39)

8 Numerical Illustration

A hypothetical example is given to illustrate the compu-tational details of the proposed problem Let us supposethe population size is 1000 with total available budgetof the survey as TFNs and TrFNs are (3500 4000 4800)

and (3500 4000 4400 4600) units respectively The otherrequired relevant information is given in Table 1 By using thevalues of Table 1 we compute the values of 119860

ℎ

which is givenin Table 2

After substituting all the values fromTables 1 and 2 in (13)the required FNLPP is given as

Minimize 003098772

1198991

+

020718425

1198992

Subject to (1 2 4) 1198991

+ (18 20 24) 1198992

le (3500 4000 4800)

1 le 1198991

le 300 1 le 1198992

le 700

(40)

The required optimum allocations for problem (13) obtainedby substituting the values from Tables 1 and 2 in (30) at 120572 =

05 will be

1198991

=

(4800 minus 800120572) times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28751 ⋍ 288

1198992

=

(4800 minus 800120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20888 ⋍ 209

(41)

In similar manner optimum allocations for problem (17)obtained by substituting the values fromTables 1 and 2 in (32)at 120572 = 05 will be

1198991

=

3750 times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28097 ⋍ 281

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Fuzzy Systems

120572

AL

120572

qp r x

1

AU

120572

120583A(x)

Figure 1 Triangular fuzzy number with an 120572-cut

120572

AL

120572

qp r s x

1

0

AU

120572

120583A(x)

Figure 2 Trapezoidal fuzzy number with an 120572-cut

73 Ghufranrsquos Allocation Ghufran et al [17] formulate a crispNLPP

Minimize119871

sum

ℎ=1

1198822

ℎ

119899ℎ

120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

Subject to119871

sum

ℎ=1

119888ℎ

119899ℎ

le 1198620

1 le 119899ℎ

le 119873ℎ

ℎ = 1 2 119871

(38)

and obtained an optimum allocation using LMT as

119899lowast

ℎ

=

1198620

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

sum119871

ℎ=1

119882ℎ

radic120587119904ℎ

(1 minus 120587119904ℎ

) + 119860ℎ

119888ℎ

(39)

8 Numerical Illustration

A hypothetical example is given to illustrate the compu-tational details of the proposed problem Let us supposethe population size is 1000 with total available budgetof the survey as TFNs and TrFNs are (3500 4000 4800)

and (3500 4000 4400 4600) units respectively The otherrequired relevant information is given in Table 1 By using thevalues of Table 1 we compute the values of 119860

ℎ

which is givenin Table 2

After substituting all the values fromTables 1 and 2 in (13)the required FNLPP is given as

Minimize 003098772

1198991

+

020718425

1198992

Subject to (1 2 4) 1198991

+ (18 20 24) 1198992

le (3500 4000 4800)

1 le 1198991

le 300 1 le 1198992

le 700

(40)

The required optimum allocations for problem (13) obtainedby substituting the values from Tables 1 and 2 in (30) at 120572 =

05 will be

1198991

=

(4800 minus 800120572) times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28751 ⋍ 288

1198992

=

(4800 minus 800120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20888 ⋍ 209

(41)

In similar manner optimum allocations for problem (17)obtained by substituting the values fromTables 1 and 2 in (32)at 120572 = 05 will be

1198991

=

3750 times 03radic04 (1 minus 04) + 0104308 (120572 + 1)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 28097 ⋍ 281

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 7

Table 1 Data for two strata

Stratum ℎ 119882ℎ

120587119904ℎ

119872ℎ

119875ℎ

(119888(1)

ℎ

119888(2)

ℎ

119888(3)

ℎ

) (119862(1)

0

119862(2)

0

119862(3)

0

)

1 03 04 08 06 (1 2 4) (1 2 4 7)2 07 06 06 07 (18 20 24) (18 20 24 26)

Table 2 Computation for 119860ℎ

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Stratum ℎ 119872ℎ

119875ℎ

(1 minus119872ℎ

) (1 minus 119875ℎ

) (4) times (5) 1 minus (6) (2119875ℎ

minus 1) 2119872ℎ

times (5) [(8) + (9)]2

(6) times (7) 119860ℎ

= (11)(10)

1 08 06 02 04 008 092 02 064 07056 00736 01043082 06 07 04 03 012 088 04 036 05776 01056 0182825

Table 3 Optimum allocations

Allocations Variance1198991

1198992

LMT (optimum allocation) Case of TFN 288 209 0001098908Case of TrFN 281 204 0001125886

Equal allocation Case of TFN 160 160 0001488575Case of TrFN 1565 1565 0001521866

Proportional allocation 41 95 0002936685Ghufranrsquos allocation 86 157 000167996

1198992

=

(4400 minus 200120572) times 07radic06 (1 minus 06) + 0182825 (2120572 + 18)

03radic04 (1 minus 04) + 0104308 (120572 + 1) + 03radic06 (1 minus 06) + 0182825 (2120572 + 18)

= 20413 ⋍ 204

(42)

By using 120572-cut and LMT the optimum allocation afterrounding-off is obtained and summarized in Table 3 withthe equal allocation proportional allocation and Ghufranrsquosallocation

9 Conclusion

The optimum allocation problem in two-stage stratifiedwarnerrsquos randomized response model with fuzzy costs isformulated as a problem of fuzzy nonlinear programmingproblem The problem is then solved by using Lagrangemultipliers technique for obtaining optimum allocation Theoptimum allocation obtained in the form of fuzzy numbersis converted into an equivalent crisp number by using 120572-cutmethod at a prescribed value of 120572

On comparing the result of LMT with the result of equalallocation proportional allocation and Ghufranrsquos allocationit is seen that LMT gives the best allocation But it is notnecessary that optimum allocation obtained by Lagrangemultipliers technique always gives the feasible or optimalsolution (proved by [17]) and also for practical purposes weneed integer sample sizes Therefore in future instead ofrounding off the continuous solution we can obtain integer

solution by Dakinrsquos Method [23] or formulating the problemas fuzzy integer nonlinear programming problem and obtainthe integer solution by LINGO software

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The third author is thankful to University Grant Commissionfor providing financial assistance under the UGC Start-upGrant no F30-902015 (BSR) Delhi India to carry out thisresearch work

References

[1] S L Warner ldquoRandomized response a survey technique foreliminating evasive answer biasrdquo Journal of the AmericanStatistical Association vol 60 no 309 pp 63ndash69 1965

[2] N SMangat andR Singh ldquoAn alternative randomized responseprocedurerdquo Biometrika vol 77 no 2 pp 439ndash442 1990

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Fuzzy Systems

[3] T C Chua and A K Tsui ldquoProcuring honest responsesindirectlyrdquo Journal of Statistical Planning and Inference vol 90no 1 pp 107ndash116 2000

[4] V R Padmawar and K Vijayan ldquoRandomized response revis-itedrdquo Journal of Statistical Planning and Inference vol 90 no 2pp 293ndash304 2000

[5] H-J Chang and K-C Huang ldquoEstimation of proportion andsensitivity of a qualitative characterrdquoMetrika vol 53 no 3 pp269ndash280 2001

[6] A Chaudhuri ldquoUsing randomized response from a complexsurvey to estimate a sensitive proportion in a dichotomous finitepopulationrdquo Journal of Statistical Planning and Inference vol 94no 1 pp 37ndash42 2001

[7] J-M Kim and W D Warde ldquoA stratified Warnerrsquos randomizedresponse modelrdquo Journal of Statistical Planning and Inferencevol 120 no 1-2 pp 155ndash165 2004

[8] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp B-141ndashB-164 1970

[9] H Tanaka and K Asai ldquoFuzzy linear programming problemwith fuzzy numbersrdquo Fuzzy Sets and Systems vol 13 no 1 pp1ndash10 1984

[10] G S Mahapatra and T K Roy ldquoFuzzy multi-objective math-ematical programming on reliability optimization modelrdquoApplied Mathematics and Computation vol 174 no 1 pp 643ndash659 2006

[11] S Pramanik and T K Roy ldquoMultiobjective transportationmodel with fuzzy parameters priority based fuzzy goal pro-gramming approachrdquo Journal of Transportation Systems Engi-neering amp Information Technology vol 8 no 3 pp 40ndash48 2008

[12] S Abbasbandy and T Hajjari ldquoA new approach for ranking oftrapezoidal fuzzy numbersrdquo Computers and Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[13] A Kaur and A Kumar ldquoA new approach for solving fuzzytransportation problems using generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 12 no 3 pp 1201ndash12132012

[14] A Ebrahimnejad ldquoA simplified new approach for solving fuzzytransportation problems with generalized trapezoidal fuzzynumbersrdquo Applied Soft Computing vol 19 pp 171ndash176 2014

[15] N Sen L Sahoo and A K Bhunia ldquoAn application of integerlinear programming problem in tea industry of barak valley ofAssam India under crisp and fuzzy environmentsrdquo Journal ofInformation and Computing Science vol 9 no 2 pp 132ndash1402014

[16] N Gupta and A Bari ldquoMulti-choice goal programming withtrapezoidal fuzzy numbersrdquo International Journal of OperationsResearch vol 11 no 3 pp 82ndash90 2014

[17] S Ghufran S Khowaja and M J Ahsan ldquoOptimum allocationin two-stage stratified randomized response modelrdquo Journal ofMathematical Modelling and Algorithms vol 12 no 4 pp 383ndash392 2013

[18] C R Bector and S Chandra FuzzyMathematical Programmingand Fuzzy Matrix Games vol 169 Springer Berlin Germany2005

[19] R Hassanzadeh I Mahdavi N M Amiri and A Tajdin ldquoAn120572-cut approach for fuzzy product and its use in computingsolutions of fully fuzzy linear systemsrdquo in Proceedings of theInternational Conference on Industrial Engineering and Opera-tions Management (IEOM rsquo12) Istanbul Turkey July 2012

[20] S Aggarwal and U Sharma ldquoFully fuzzy multi-choice multi-objective linear programming solution via deviation degreerdquoInternational Journal of Pure and Applied Sciences and Technol-ogy vol 19 no 1 pp 49ndash64 2013

[21] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[22] A L Bowley Measurements of the Precision Attained in Sam-pling vol 22 no 1 of Bulletin of the International StatisticalInstitute Cambridge University Press Cambridge UK 1926

[23] R J Dakin ldquoA tree-search algorithm for mixed integer pro-gramming problemsrdquo The Computer Journal vol 8 no 3 pp250ndash255 1965

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014