Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function

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1 American Journal of Operations Researc, 04, 4, Publised Online May 04 in SciRes. ttp://.scirp.org/ournal/aor ttp://dx.doi.org/0.436/aor Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response it Quadratic Cost Function Safiulla, Moammad Faisal Kan, Irfan Ali Department of Statistics & Operations Researc, Aligar Muslim University, Aligar, India Department of Computing and Informatics Saudi Electronics University, Riyad, Saudi Arabia Received 9 Marc 04; revised 9 April 04; accepted 6 April 04 Copyrigt 04 by autors and Scientific Researc Publising Inc. Tis ork is licensed under te Creative Commons Attribution International icense (CC BY). ttp://creativecommons.org/licenses/by/4.0/ Abstract In tis paper, te problem of non-response it significant travel costs in multivariate stratified sample surveys as been formulated of as a Multi-Obective Geometric Programming Problem (MOGPP). Te fuzzy programming approac as been described for solving te formulated MOGPP. Te formulated MOGPP as been solved it te elp of INGO Softare and te dual solution is obtained. Te optimum allocations of sample sizes of respondents and non respondents are obtained it te elp of dual solutions and primal-dual relationsip teorem. A numerical example is given to illustrate te procedure. Keyords Geometric Programming, Fuzzy Programming, Non-Response it Travel Cost, Optimum Allocations, Multivariate Stratified Sample Surveys. Introduction In sampling te precision of an estimator of te population parameters depends on te size of te sample and variability among te units of te population. If te population is eterogeneous and size of te sample depends on te cost of te survey, ten it is likely to be impossible to get a sufficiently precise estimate it te elp of simple random sampling from te entire population. In order to estimate te population mean or total it greater precision, te eterogeneous population is divided into mutually-exclusive, exaustive and non-overlap- Ho to cite tis paper: Safiulla, Kan, M.F. and Ali, I. (04) Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response it Quadratic Cost Function. American Journal of Operations Researc, 4, ttp://dx.doi.org/0.436/aor

2 Safiulla et al. ping strata ic ill be more omogeneous tan te entire population. Te entire population is called Stratified Random Sampling. Te problem of optimum allocation in stratified random sampling for univariate population is ell knon in sampling literature; see for example Cocran [] and Sukatme et al. []. In multivariate stratified sample surveys problems te non-response can appear en te required data are not obtained. Te problem of non-response may occur due to te refusal by respondents or teir not being at ome, making te information of sample inaccessible. Te problem of non-response occurs in almost all surveys. Te extent of nonresponse depends on various factors suc as type of te target population, type of te survey and te time of survey. For te problem of non-response in stratified sampling it may be assumed tat every stratum is divided into to mutually exclusive and exaustive groups of respondents and non respondents. Hansen and Huritz [3] presented a classical non-response teory ic as first developed for surveys in ic te first attempt as made by mailing te questionnaires and a second attempt as made by personal intervies to a sub sample of te non respondents. Tey constructed te estimator for te population mean and derived te expression for its variance and also orked out te optimum sampling fraction among te non respondents. El-Badry [4] furter extended te Hansen and Huritz s tecnique by sending aves of questionnaires to te non respondent units to increase te response rate. Te generalized El-Badry s approac for different sampling design as given by Foradori [5]. Srinat [6] suggested te selection of sub samples by making several attempts. Kare [7] investigated te problem of optimum allocation in stratified sampling in presence of nonresponse for fixed cost as ell as for fixed precision of te estimate. Kan et al. [8] suggested a tecnique for te problem of determining te optimum allocation and te optimum sizes of subsamples to various strata in multivariate stratified sampling in presence of non-response ic is formulated as a Nonlinear Programming Problem (NPP). Varsney et al. [9] formulated te multivariate stratified random sampling in te presence of non-response as a Multi-obective Integer Nonlinear Programming problem and a solution procedure is developed using lexicograpic goal programming tecnique to determine te compromise allocation. Fatima and Asan [0] addressed te problem of optimum allocation in stratified sampling in te presence of non-response and formulated as an All Integer Nonlinear Programming Problem (AINPP). Varsney et al. [] ave considered te multivariate stratified population it unknon strata eigts and an optimum sampling design is proposed in te presence of non-response to estimate te unknon population means using DSS strategy and developed a solution procedure using Goal Programming tecnique and obtained an integer solution directly by te optimization softare INGO. Ragav et al. [] as discussed te various multi-obective optimization tecniques in te multivariate stratified sample surveys in case of non-response. Geometric Programming (GP) is a smoot, systematic and an effective non-linear programming metod used for solving te problems of sample surveys, management, transportation, engineering design etc. tat takes te form of convex programming. Te convex programming problems occurring in GP are generally represented by an exponential or poer function. GP as certain advantages over te oter optimization metods because it is usually muc simpler to ork it te dual tan te primal one. Te degree of difficulty (DD) plays a significant role for solving a non-linear programming problem by GP metod. Geometric Programming (GP) as been knon as an optimization tool for solving te problems in various fields from 960 s. Duffin, Peterson and Zener [3] and also Zener [4] ave discussed te basic concepts and teories of GP it application in engineering in teir books. Beigtler, C.S., and Pililps, D.T. [5], ave also publised a famous book on GP and its application. Davis and Rudolp [6] applied GP to optimal allocation of integrated samples in quality control. Amed and Carles [7] applied geometric programming to obtain te optimum allocation problems in multivariate double sampling. Oa, A.K. and Das, A.K. [8] ave taken te Multi- Obective Geometric Programming Problem being cost coefficient as continuous function it eigted mean and used te geometric programming tecnique for te solutions. Maqbool et al. [9] and Safiulla et al. [0] ave discussed te geometric programming approac to find te optimum allocations in multivariate to-stage sampling and tree-stage sample surveys respectively. In many real-orld decision-making problems of sample surveys, environmental, social, economical and tecnical areas are of multiple-obectives. It is significant to realize tat multiple obectives are often non-commensurable and in conflict it eac oter in optimization problems. Te multi-obective models it fuzzy obectives are more realistic tan deterministic of it. Te concept of fuzzy set teory as firstly given by Zade []. ater on, Bellman and Zade [] used te fuzzy set teory for te decision-making problem. Tanaka et al. [3] introduces te obective as fuzzy goal over te α-cut of a fuzzy constraint set and Zimmermann [4] gave te concept to solve multi-obective linear-programming problem. Bisal [5] and Verma [6] developed fuzzy 74

3 Safiulla et al. geometric programming tecnique to solve Multi-Obective Geometric Programming (MOGP) problem. Islam [7] [8] as discussed modified geometric programming problem and its applications and also anoter fuzzy geometric programming tecnique to solve MOGPP and its applications. Fuzzy matematical programming as been applied to several fields. In tis paper, e ave formulated te problem of non-response it significant travel cost ere te cost is quadratic in n in multivariate stratified sample surveys as a Multi-Obective Geometric Programming problem (MOGPP). Te fuzzy programming approac as been described for solving te formulated MOGPP and te optimum allocations of sample sizes of respondents and non respondents are obtained. A numerical example is given to illustrate te procedure.. Formulation of te Problem In stratified sampling te population of N units is first divided into non-overlapping subpopulation called strata, of sizes N, N,, N,, N it N = N and te respective sample sizes itin strata are dran = it independent simple random sampling denoted by n, n,, n,, n it n. = n = t et for te stratum: N : Stratum size. Y : Stratum mean. S : Stratum variance. Wˆ ˆ = N N : te estimated stratum eigt among respondents. Wˆ ˆ = N N : te estimated stratum eigt among non-respondents. N : te sizes of te respondents. N = N N : te sizes of non respondents groups. t n : Units are dran from te stratum. Furter let out of n, n units belong to te respondents group. n = n n : Units belong to te non respondents group. n = n = : Te total sample size. A more careful second attempt is made to obtain information on a random subsample of size r out of n non respondents for te representation from te non respondents group of te sample. r = n k; =,,, : Subsamples of sizes at te second attempt to be dran from n non-respondent group of te t stratum. Were k and k denote te sampling fraction among non respondents. Since N and N are random variables ence teir unbiased estimates are given as: Nˆ = n N n : Te unbiased estimates of te respondents group. Nˆ = n N n : Te unbiased estimate of te non respondents group. y ;,, = p: denotes te sample means of t caracteristic measured on te n respondents at te first attempt. y ;,, r = p: denotes te r sub sampled units from non respondents at te second attempt. t Using te estimator of Hansen and Huritz [3], te stratum mean Y for caracteristic in te t stratum may be estimated by n y ny( r ) y = () n t t It can be seen tat y is an unbiased estimate of te stratum mean Y of te stratum for te caracteristic it a variance. W S W S v( y ) = S () n N r n t t ere S is te stratum variance of caracteristic in te stratum; =,,, p and =,,, given as: N S = yi Y N i= 75

4 Safiulla et al. ere y denote te value of te i N i= i t i unit of te Y = N y : is te stratum mean of y. i S is te stratum variance of te t stratum for t caracteristic in te Nˆ = ˆ i N i= t caracteristic. t stratum among non respondents, given by: Sˆ y Y Nˆ Y ˆ = N y i= i is te stratum mean of y among non respondents. i ˆ ˆ t W = N N is stratum eigt of non respondents in stratum. If te true values of S and S are not knon tey can be estimated troug a preliminary sample or te value of some previous occasion, if available, may be used. Furtermore, te variance of y =, Wy (ignoring fpc) is given as: = ( ˆ ˆ ) W S W S WWˆ Sˆ V y = W v y = = f nr ( ) ( ) = = n = r 0 (, ) (3) ere y is an unbiased estimate of te overall population mean Y of te t caracteristic and V ( y ) is as given in Equation (). Assuming a linear cost function te total cost C of te sample survey may be given as: C = c n c n c n 0 = = = ere c 0 = te per unit cost of making te first attempt, c = c = is te per unit cost for processing te t results of all te p caracteristics on te n selected units from respondents group in te stratum in te first p attempt and c = c : te per unit cost for measuring and processing te results of all te p caracteristics = t on te r units selected from te non respondents group in te stratum in te second attempt. Also, c t and c are per unit costs of measuring te caracteristic in first and second attempts respectively. As n is not knon until te first attempt as been made, te quantity W n may be used as its expected value. Te total expected cost Ĉ of te survey may be given as: ( 0 ) 0 (4) Cˆ = c c W n c r t n t r = = = = Te problem terefore reduces to find te optimal values of sample sizes of respondents respondents r ic are expressed as: ( ) W S W S WW S Min ( f0 ) = = n = r Subect to, =,,, p ( c0 c W) n cr t0 n t r C0 = = = = n, r 0 and =,,, 3. Geometric Programming Formulation n and non- Te folloing Multi-obective Nonlinear Programming Problem (MNPP) it te cost function quadratic in n and significant travel cost is defined in Equations (6) as follos: (5) 76

5 Safiulla et al. Min V Subect to ( ˆ π s ) a = = n c c W n c r t n t r C n,r 0 and =,,, = = = = =,,, P Similarly, te expression (6) can be expressed in te standard Primal GPP it cost function quadratic in n ere te travel cost is significant is given as follos: ( nr) Max f0, Subect fq nr, =,, p n, r 0, =,,, { } p [ ] i p ere,, 0,,,, i pi p i p i pi fq nr = di n n n r r r q= k i q p i p i or fq( nr, ) = di n r, di > 0, n > 0, q= 0,,,, k, i q [ ] = = p i : arbitrary real numbers, d i : positive and fq ( n ) : posinomials et for simplicity f0 = a and di = a = ( c c W ) C = c C = t C = t C ( ) c c W c t t ere f nr, n r n r r 0 0 q = = C0 = C0 = C0 = C0 Te dual form of te Primal GPP ic is stated in (7) can be given as: i i k k d [ ] i q i Max v0 = i ( i) q 0 i q [ ] = i q= i q [ ] Subect to i = ( ii) i [ 0] k pii ( iii) q i q [ ] i 0, q,,, k and i =,,, mk ( iv) =,, p. Te above formulated dual GPP (8) can be solved in te folloing to steps: Step : For te Optimum value of te obective function, te obective function alays takes te form: C 0 ( x ) ( s in te last constraint s) 0 0 Coeffi. of first term Coeffi. of Second term = Coeffi. of last term k 0 0 k ( s in te first constraint s) s in te last constraint s Te Multi-Obective obective function for our problem is: k i k d i q [ ] i i [ ] i i q [ ] q i q q= i s in te first constraint s Step : Te equations tat can be used for GPP for te eigts are given belo: i in te obective function = (Normality condition ) and for eac primal variable n & i & o [ ] r aving m terms. (6) (7) (8) n and 77

6 Safiulla et al. m k i= ( i ) ( n n r r ) for eac term exponent on & and & in tat term (Ortogonality condition) and i 0 (Positivity condition). Te above problem (8) as been solved it te elp of steps (-) discussed in Section (3) and te corresponding solutions 0i is te unique solution to te dual constraints; it ill also maximize te obective function for te dual problem. Next, te solution of te primal problem ill be obtained using primal-dual relationsip teorem ic is given belo. 4. Primal-Dual Relationsip Teorem If 0i primal problem (7) satisfies te system of equations: is a maximizing point for dual problem (8), eac minimizing points (,,, and,,, ) f 0 ( n) [ ] 0 iv, i J 0, = i, i J, v( 0 i) ere ranges over all positive integers for ic v( 0i) > 0. Te optimal values of sample sizes of te respondents ( n ) and non-respondents ( ) it te elp of te primal-dual relationsip teorem (9). 5. Fuzzy Geometric Programming Approac [ ] n n n n r r r r for (9) r can be calculated Te solution procedure to solve te problem (5) consists of te folloing steps: Step-: Solve te MOGPP as a single obective problem using only one obective at a time and ignoring te oters. Tese solutions are knon as ideal solution. Step-: From te results of step-, determine te corresponding values for every obective at eac solution ( ) ( ) p p derived. et n, r, n, r,, n, r,, n, r are te ideal solutions of te obective functions ( ) ( ) ( ) ( ) ( ) ( ) ( p) ( p) (, ), (, ),, (, ),, p(, ). ( ) ( ) ( ) ( ) ( p) ( p) U Max { f ( n, r ), f ( n, r ),, f (, )} p n r f n r f n r f n r f n r So = and ( ) ( ) 0 = f n, r, =,,, p. t U and be te upper and loer bonds of te obective function f0 nr,, =,,, p. Step 3: Te membersip function for te given problem can be define as: µ ( ) 0, if f0 nr, U U ( nr, ) f0 ( nr, ) f nr, =, if f ( nr, ) U, =,,, p (0) (, ) (, ), if f0 ( nr, ) 0 0 U nr nr Here U ( nr, ) is a strictly monotonic decreasing function it respect to Te membersip functions in Equation () i.e., µ f0 ( nr, ), =,,, p Terefore te general aggregation function can be defined as { p } ( nr, ) = ( f ( nr, )), ( f ( nr, )),, f ( nr, ) µ µ µ µ µ D D p Te fuzzy multi-obective formulation of te problem can be defined as: ( nr) 0 0 n r n r = C0 = C0 = C0 = C0 f nr. 0, Max µ, D Subect to ( c c W ) c t t ; n, r 0 and =,,, p. () 78

7 Te problem to find te optimal values of (, ) Safiulla et al. n r for tis convex-fuzzy decision based on addition operator (like Tiari et. al. [9]). Terefore te problem () is reduced according to max-addition operator as: Te above problem () reduces to Te problem (3) maximizes if te function F ( n) ( 0 (, )) p p U f nr Max µ D( n, r ) = µ ( f0 ( nr, )) = = = U Subect to fq ( nr, ) ; 0 µ ( f0 ( nr, )) n, r 0 and =,,, p. ( f0 ( nr, )) p U Max µ D ( n, r ) = = U U Subect to fq ( nr, ) ; n, r 0 and =,,, p. ( f0 ( nr, )) o = U problem (3) reduces into te primal problem (4) define as: p Min Fo ( nr, ) = 0 Subect to fq ( nr, ) ; n, r 0 and =,,, Te dual form of te Primal GPP ic is stated in (6) can be given as: i i k k d i q [ ] i Max v = i ( i) q 0 i q [ ] = i q= i q [ ] Subect i = ( ii) i [ 0] k pii iii q i q [ ] i 0, q,,, k and i =,,, mk ( iv ) () (3) attain te minimum values. Terefore te Te optimal values of sample sizes of te respondents ( n ) and non-respondents ( ) it te elp of te primal-dual relationsip teorem (9). 6. Numerical (4) (5) r can be calculated In Table, te stratum sizes, stratum eigts, stratum standard deviations, measurement costs and te travel costs itin te stratum are given for to caracteristics under study in a population stratified in four strata. Te data are mainly from Kan et al. [8]. Te travelling costs t 0 and t are assumed. Te total budget available for te survey is taken as C 0 = Te relative values of te variances of te non-respondents and respondents, tat is S S is assumed to be constant and equal to 0.5 for =, and =,,3,4. Hoever, tese ratios may vary from stratum to stratum and from caracteristic to caracteristic and can be andled accordingly. For solving MOGPP by using fuzzy programming, e sall first solve te to sub-problems: 79

8 Safiulla et al. Table. Data for four Strata and to caracteristics. N S S c c c t t Sub problem : On substituting te table values in sub-problem, e ave obtained te expressions given belo: Min f0 = n n n3 n r r r3 r 4 Subect to n n n n r r 0.00r3 0.00r n n n n r r r r4 n 0, r 0, n, r are integers; =,,, Te dual of te above problem (6) is obtained as: v( 0 i ) = (( 0 ) ) (( 0 ) ) (( 03 ) ) (( 04 ) ) (( ) ) (( ) ) (( ) ) (( ) ) Max ^ (( ) ( )); ( i) Subect to = ; ( normality condition) ( ii) ( ortogonality condition) 05 5 ( ) ( ) 0 ( iii) = 0, 0, 03, 04, 05, 06, 07, 08 > 0; ( positivity condition) ( iv),, 3, 4, 5, 6, 7, 8, 9, 0,,, 3, 4, 5, 6 0 (6) (7) 80

9 Safiulla et al. For ortogonality condition defined in expression 7(iii) are evaluated it te elp of te payoff matrix ic is defined belo ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Solving te above formulated dual problem (7) it te elp of ingo softare, e ave te corresponding dual solutions as follos: , , , , , , , , and v = Using te primal dual-relationsip teorem (9), e ave te optimal solution of primal problem: i.e., te optimal sample sizes of respondents and non respondents are computed as follos: (, ) = f nr v 0 0i 0i In expression (6), e first keep te r constant and calculate te values of n as: 8

10 Safiulla et al. ( i) ( i) ( i) ( i) f0 n, r v 0 n 474 f0 n, r v 0 n 30 f n, r = v n 49 f n, r = v n No, from te expression (6), e keep te n constant and calculate te values of r as: f0 ( nr, ) v( 0 i) r 65 f0 ( nr, ) v( 0 i) r 8 f nr, = v r 8 f nr, = v r 3 ( i) ( i) Te optimal values and te obective function value are given belo: n = 474, n = 30, n3 = 49 and n4 = 44; r = 65, r = 8, r = 8 and r = 3 and te obective value of te primal problem is Sub problem : On substituting te table values in sub-problem, e ave obtained te expressions given belo: Min f0 = n n n3 n ; r r r3 r4 Subect to n n n n r r 0.00r3 0.00r n n n n r r r r4 n 0, r 0, n, r are integers; =,,, Te dual of te above problem (8) is obtained as follos: v( 0 i ) = (( 0 ) ) (( 0 ) ) (( 03 ) ) (( 04 ) ) (( / 05 ) ) (( / 06 ) ) (( / 07 ) ) (( /08 ) ) Max ^ (( ) ( )); ( i) Subect to = ; ( normality condition) ( ii) ( ) ( ) 0 ( ortogonality condition) ( iii) = 0, 0, 03, 04, 05, 06, 07, 08 > 0; ( positivity condition) ( iv),, 3, 4, 5, 6, 7, 8, 9, 0,,, 3, 4, 5, 6 0 (8) (9) 8

11 Safiulla et al. For ortogonality condition defined in expression 9(iii) are evaluated it te elp of te payoff matrix ic is defined belo: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Solving te above formulated dual problems, e ave te corresponding solution as: ,.87303,.35806,.94348, , , , , and v = Te optimal values of sample sizes of respondents and non-respondents (, ) n r can be calculated it te 83

12 Safiulla et al. elp of te primal-dual relationsip teorem (9) as e ave calculated in te sub-problem are given as follos: n = 587, n = 37, n3 = 63 and n4 = 46; r = 8, r = 9, r = 8 and r = 3 and te obective value of te primal problem is No te pay-off matrix of te above problems is given belo: (, ) (, ) f nr f nr 0 0 ( ) ( ) ( n, r ) ( ) ( ) ( n, r ) f Te loer and upper bond of (, ) and (, ) f nr f nr can be obtained from te pay-off matrix f nr, and f nr, et µ ( nr, ) and µ ( nr, ) be te fuzzy membersip function of te obective function f0 (, ) ( nr ) respectively and tey are defined as: 0, ( nr), if f0, f0 ( nr, ) µ ( nr, ) =, if f0 ( nr, ) , if f0 ( nr, ) ( nr), if Z, Z ( nr, ) µ ( nr, ) =, if Z( nr, ) , if Z ( nr, ) nr and On applying te max-addition operator, te MOGPP, te standard primal problem reduces to te problem as: f0 n,r f0 n,r Maximize Subect to n n n n r r 0.00r 0.00r n n n n r r r r4 n 0, r 0, n, r are integers; =,,, (0) In order to maximize te above problem, e ave to minimize f0 nr f0 nr, subect to te con straints as described belo: (, ) (, ) 84

13 Safiulla et al n n n3 n 4 Min r r r3 r 4 Subect to n n n n r r 0.00r3 0.00r n n n n r r r r4 n 0, r 0, n, r are integers; =,,, () Degree of Difficulty of te problem () is = (4 (8 ) =5. Hence te dual problem of te above final formulated problem () is given as: v( 0 i ) = (( 0 ) ) (( 0 ) ) (( 03 ) ) (( 04 ) ) (( ) ) (( ) ) (( ) ) (( ) ) Max (( ) ( )); ( i) Subect to = ; ( normality condition) ( ii) ( ) 0 ( ortogonality condition) ( iii) = 0, 0, 03, 04, 05, 06, 07, 08 > 0;,, 3, 4, 5, 6, 7, 8, 9, 0,,, 3, 4, 5, 6 0 ( positivity condition) ( iv) For ortogonality condition defined in expression (iii) are evaluated it te elp of te payoff matrix ic is defined belo: ^ () 85

14 Safiulla et al ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) After solving te formulated dual problem () using lingo softare e obtain te folloing values of te dual variables ic are given as:.60739,.7708, ,.7033, , , , , and v( ) = Te optimal values of sample sizes of respondents and non-respondents (, ) n r can be calculated it te elp of te primal-dual relationsip teorem (9) as e ave calculated in te sub-problem are given as follos: 86

15 Safiulla et al. n = 537, n = 309, n = 04 and n = 44; r = 74, r = 8, r = 3 and r = and te obective value of te primal problem is Conclusion Tis paper provides an insigtful study of fuzzy programming for solving te multi-obective geometric programming problem (MOGPP). Te problem of non-response it significant travel costs ere te cost is quadratic in n in multivariate stratified sample surveys as been formulated of as a Multi-Obective Geometric Programming Problem (MOGPP). Te fuzzy programming approac as described for solving te formulated MOGPP. Te formulated MOGPP as been solved it te elp of INGO Softare [30] and te dual solution is obtained. Te optimum allocations of sample sizes of respondents and non respondents are obtained it te elp of dual solutions and primal-dual relationsip teorem. To ascertain te practical utility of te proposed metod in sample surveys problem in presence of non-response it significant travel cost ere te cost is quadratic in n, a numerical example is also given to illustrate te procedure. References [] Cocran, W.G. (977) Sampling Tecniques. 3rd Edition, Jon Wiley and Sons, Ne York. [] Sukatme, P.V., Sukatme, B.V., Sukatme, S. and Asok, C. (984) Sampling Teory of Surveys it Applications. Ioa State University Press, Ames and Indian Society of Agricultural Statistics, Ne Deli. [3] Hansen, M.H. and Huritz, W.N. (946) Te Problem of Non-Response in Sample Surveys. Journal of te American Statistical Association, 4, ttp://dx.doi.org/0.080/ [4] El-Badry, M.A. (956) A Sampling Procedure for Mailed Questionnaires. Journal of te American Statistical Association, 5, ttp://dx.doi.org/0.080/ [5] Fordori, G.T. (96) Some Non-Response Sampling Teory for To Stage Designs. Mimeograp Series No. 97, Nort Carolina State University, Raleig. [6] Srinat, K.P. (97) Multiple Sampling in Non-Response Problems. Journal of te American Statistical Association, 66, ttp://dx.doi.org/0.080/ [7] Kare, B.B. (987) Allocation in Stratified Sampling in Presence of Non-Response. Metron, 45, 3-. [8] Kan, M.G.M., Kan, E.A. and Asan, M.J. (008) Optimum Allocation in Multivariate Stratified Sampling in Presence of Non-Response. Journal of te Indian Society of Agricultural Statistics, 6, [9] Varsney, R., Asan, M.J. and Kan, M.G.M. (0) An Optimum Multivariate Stratified Sampling Design it Non- Response: A exicograpic Goal Programming Approac. Journal of Matematical Modeling and Algoritms, 65, [0] Fatima, U. and Asan, M.J. (0) Nonresponse in Stratified Sampling: A Matematical Programming Approac. Te Sout Pacific Journal of Natural and Applied Sciences, 9, [] Varsney, R., Namussear and Asan, M.J. (0) An Optimum Multivariate Stratified Double Sampling Design in Presence of Non-Response. Optimization etters, 6, [] Ragav, Y.S., Ali, I. and Bari, A. (04) Multi-Obective Nonlinear Programming Problem Approac in Multivariate Stratified Sample Surveys in Case of Non-Response. Journal of Statistical Computation and Simulation, 84, -36. ttp://dx.doi.org/0.080/ [3] Duffin, R.J., Peterson, E.. and Zener, C. (967) Geometric Programming: Teory & Applications. Jon Wiley & Sons, Ne York. [4] Zener, C. (97) Engineering Design by Geometric Programming. Jon Wiley & Sons, Ne York. [5] Beigtler, C.S. and Pilips, D.T. (976) Applied Geometric Programming. Wiley, Ne York. [6] Davis, M. and Scartz, R.S. (987) Geometric Programming for Optimal Allocation of Integrated Samples in Quality Control. Communications in Statistics Teory and Metods, 6, ttp://dx.doi.org/0.080/ [7] Amed, J. and Bonam, C.D. (987) Application of Geometric Programming to Optimum Allocation Problems in Multivariate Double Sampling. Applied Matematics and Computation,, ttp://dx.doi.org/0.06/ (87) [8] Oa, A.K. and Das, A.K. (00) Multi-Obective Geometric Programming Problem Being Cost Coefficients as Con- 87

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