In this article, the Fuzzy technique is used to identify the optimal distribution of sample sizes in Conversed Tertiary-objective fractional goal programming. LINGO software provides the answer to formulated bi-goal fractional programming. For practical resolution, if the value of the problem is non - integer then the solution of the problem is obtained through Branch and Bound method

?ZER, ?zen

A. Lone, Mushtaq.

A. Mir, S.

Fayaz Khan, Omar

Khalifa, M.

Lone, Rafiq

Arab Journals Platform

 *Corresponding author e-mail: mushtastat11@gmail.com   © 2022 NSP    Natural Sciences Publishing Cor. J. Stat. Appl. Pro. 11, No. 3, 811-818 (2022) 811   Journal of Statistics Applications & Probability An International Journal   http://dx.doi.org/10.18576/jsap/110305   Conversed Fractional Goal Programming: A Mathematical Programming Approach  Mushtaq. A. Lone1,* , S. A. Mir1, M. Khalifa2, Omar Fayaz Khan1, Özen ÖZER3, Rafiq Lone4 and Tabasum Mushtaq5  1Department of Statistics, FOH, SKUAST –K, P.O. Box 190025, J&K India 2Political Science Department, College of Administrative Sciences, Applied Science University, Kingdom of Bahrain 3Department of Mathematics, Faculty of Science and Arts, Kirklareli University, P.O. Box 39100 Kayal, Turkey 4Department of Commerce, Faculty of Management, ICSC, P.O. Box 190025, J&K India 5Department of Statistics, GDC Shopian P.O. Box 30004, J&K India  Received: 3 Feb. 2021, Revised: 5 Aug. 2021, Accepted: 7 Aug. 2021. Published online: 1 Sep. 2022.    Abstract: In this article, the Fuzzy technique is used to identify the optimal distribution of sample sizes in Conversed Tertiary-objective fractional goal programming. LINGO software provides the answer to formulated bi-goal fractional programming. For practical resolution, if the value of the problem is non - integer then the solution of the problem is obtained through Branch and Bound method.   Keywords: Optimal solution, Stratification, Fuzzy programming, LINGO and Fractional goal Programming. 1 Introduction  Statistics is widely used in all fields’ of scientific inspection and one of the important sections of statistics is stratified sampling Design. Reducing the heterogeneity of the population units is appropriate to increase the estimate's precision. It is accomplished using stratification technique, where total population comprising of items is first distributed among L sub-units  of sizes one-to-one are termed as strata and together make the entire population i.e. . The strata/blocks are within homogeneous & between they are heterogeneous. Once the blocks created, a sample from each stratum is taken independently. In every block if the procedure of SRS (Simple Random Sampling) is taken, the entire practice is termed as stratified random sampling. should be known in stratified design, the consideration of allocation of sample sizes is most important in every block/stratum and problem of optimally choosing the sample sizes either to minimize cost subject to variance or vice versa is known as the optimal allocation problem. In stratified sampling designs the problem of optimal allocation conferred by numerous authors (see, for example, [1-19] etc).  It is found that a number of optimization problems from different sector of sciences like engineering, agriculture, health, planning, etc and economics have a need of  optimizing the ratio between  two tasks physical &/or economic. Multi-task  problems, which cares about decision-making issues, especially in the public sector where there are several conflicting purposes, was developed as a result of the optimization programming technique having more than one objective function. The multi-objective methodology is a popular alternative that can be applied together in public & private segments. Our daily routines are full of decision-making situations with several objectives that are incommensurable. In Refs. [20, 21] the obtained integer optimal allocation in formulated Linear /Non -linear Plus fraction programming problem and in different situations. In mathematical modeling an expanding economy through a generalized fractional programming approach used [22]. Various algorithms found in ([23,24,25,26]) used to solve linear fractional programming problems. [27] presented two alternative algorithms—the fuzzy weighting average algorithm and the fuzzy critical point algorithm—to determine the ideal location for the facility centre. This article discusses a fuzzy programming technique in which [28] converts a tertiary goal function into a bi-goal  and solution of the programming problem is achieved by using Mathematical programming software [29].  For practical NLNNN ,...,, 21NNLi i=å =1iN812                                                                                 M. A. Lone et al: Allocation problem using fuzzy method…   © 2022 NSP Natural Sciences Publishing Cor.  scenario, in rare cases, changing non-integer sample sizes to the adjacent integral value may yield an impossible and suboptimal outcome if the acquired result is not an integer. In that scenario, the Branch and Bound approach would be used instead of non-integer values to the neighbouring integer value.  2 Problem Formulations Suppose size N of finite population is distributed among M strata of different sizes  in such a way that. It is also assumed that the drawing of the sample sizes from each stratum is made independently. Let = studying characteristic and  = impartial estimation of the population's average ( ).   Let = impartial estimation of the stratum’s average  s.t    and 𝑦" = !"∑ 𝑁#$#%! 𝑦"# =  ∑ 𝑊#$#%! 𝑦"# is an impartial estimation of population mean . This estimate's accuracy is restrained by variance of the sample-based estimation of the population's attributes. , where,  ,  and . Subsequently  keeps going and it is adequate to min. .  Furthermore, the CV is   The optimum allocation problem comprises of determining the sample sizes in such a way which minimizes the overall variance tested by the cost function.  The cost function is given as                                                                                                         (i) where,  charge per sample in  block,  is  overhead charges and C is the survey's entire available budget. Let. Now, optimal allocation of different sample sizes is determined from the below formulated tertiary objective allocation problem:    MNNN !,2,NMi iN =å =1 inYy Yiy Y i( )1 1,2,3,...1nii Ly yi ihni h= " =å=Y2 21MaW Si i ii=å=NiWi N=1 1xi n Ni i= -1M aiNi iå=( )1M aiV yni i= å=1/21M aiCV Yni ié ù= ê úåê ú=ë û1MoC c c ni ii= + å=cithi co*oC Cc- =( )1MV y a xiii= å= J. Stat. Appl. Pro. 11, No. 3, 811-818 (2022)/ http://www.naturalspublishing.com/Journals.asp                                                        813              © 2022 NSP            Natural Sciences Publishing Cor.  Min           Min                                                                                                       (ii) Min  Sub. to   Where   is prefaced variance of the population mean estimate? 3 Fuzzy Method for Conversed Fractional Goal Programming   The outlined proposed algorithm for (MONLPP) is under: I : Identify the highest value of say it .  II. Perform division to each individual objective by and for left over . III: we obtain fractional programming . IV: Construct the membership function for pth objective. V: Use Taylor’s first order theorem to transform the membership function. VI: Frame the Fuzzy mathematical goal programming problem. VII: Determine later evaluating converted mathematical programme in Step 6. VIII: Terminate. 4 Numerical Example The below mention information has been barrow from [10],in which  population comprises 64 units, the stratum weights & variance of a population that has been distributed among 3 divisions with  studying one feature is  given as under in tabular form 3.1. Table 3.1: Information about strata with one feature.    **1Mc c ni ii= å=*1Mn nii= å=1/2( )1M aiCV Yni ié ùê ú= åê ú=ë û*( )1M aiV y vnii= £å=LiniN ini ,...,3,2,1integer,2 =££v*( )K np*( )K np*( )K np ( )1K np-( )t nz*n      1 16 0.2500 540.0625 62.9375 33.7539 2 20 0.3125 14.6737 27.6000 1.4330 3 28 0.4375 7.2540 14.0714 1.3885 iiN iiNWN=2iS2iY ia814                                                                                 M. A. Lone et al: Allocation problem using fuzzy method…   © 2022 NSP Natural Sciences Publishing Cor.   It is Assumed the available budget denoted by C =100 items with  and = 30 items (above cost). The total given budget aimed at the field study is = 70 items. It is further assumed that the charge t of dimension  in different units are  aimed at the below cost relationship   In the direction of  and to fulfill the subsequent fuzzy objectives and vague aspirations are 0.3170 and 0.0002997 s.t. .Therefore 0.56 & 0.00067 are tolerance limits to the above mentioned objectives respectively. Thu we describe the membership functions as: co co*C iC15.1,4 321 === candcc1MoC c c ni ii= + å=n0002997.0)(3170.0)( 21 andnandn ££ xx111 111 ( ) 0.31700.56 ( )( ) 0.3170 ( ) 0.560.56 0.31700 ( ) 0.56if nnn if nif nzzµ zz£ìï -ï= £ £í -ï³ïî J. Stat. Appl. Pro. 11, No. 3, 811-818 (2022)/ http://www.naturalspublishing.com/Journals.asp                                                        815              © 2022 NSP            Natural Sciences Publishing Cor.   Thus, first order series about points by Taylor  = (16, 3.85, 3.32) and  = (16, 20, 28) for                        Consequently, FGP can be defined as Min  S. t                                                                     (v)   The above-described relationship with 1 a highest value, and the following is the aspiration level of the above-described relationship with unity:  , Where  is the over deviational variable. FGP is described as    222 221 ( ) 0.00029970.00067 ( )( ) 0.0002997 ( ) 64.0.00067 0.00029970 ( ) 0.00067if nnn if nif nzzµ zz£ìï -ï= £ £í -ï³ïî*1n*2n)()( 21 nandn µµ633.00090.0013.006.0)()(9965.00385.002965.0.00150.0)()(321*212321*111+-+==+--==nnnnnnnnnnµµµµ)(ntµ*( )1M aiV y Vnii= £å=LiniN ini ,...,3,2,1integer,2 =££1)( =+ +tt n dµ+td816                                                                                 M. A. Lone et al: Allocation problem using fuzzy method…   © 2022 NSP Natural Sciences Publishing Cor.   Min  S. t                                 (vii)                           After evaluating (vii) using LINGO s, we have Thus optimal allocation is  Lingo Software's Branch and Bound approach produced the following results:    5 Conclusions It is determined that a fuzzy technique is used in developed tertiary objective stratified sampling design, which is translated into bi-objective, to determine the best distribution of sample sizes. Turning non-integer values to neighbouring integral values can occasionally produce an impractical result in real-world settings. Branch and Bound Method has been applied to determine integer optimum sample size allocation.  Conflicts of Interest: The authors affirm that the publishing of this paper does not involve any conflicts of interest.   ++ + 21 dd282,202,16290.23885.14330.17539.331633.00090.0013.006.019965.00385.002965.0.00150.012132123211321£££££££++=+-++=++--++nnnnnnnnnnnndd.3,19.4,16 321 === nandnn 1 2 316, 5, 3.n n and n= = = J. Stat. Appl. Pro. 11, No. 3, 811-818 (2022)/ http://www.naturalspublishing.com/Journals.asp                                                        817              © 2022 NSP            Natural Sciences Publishing Cor.  References   [1] Ghosh, S.P, A note on stratified random sampling with multiple characters. Calcutta Statistical Bulletin,8, 81-89, 1958. [2] Yates, F, Sampling Methods for Censuses and Surveys. 3rd ed. London: Charles Griffin  (1960). [3] Aoyama. H, Stratified Random Sampling with Optimum Allocation for Multivariate Populations, Annals of the institute of statistical Mathematic,14, 251–258 (1963). [4] Folks, J.K., Antle, C.E, Optimum allocation of sampling units to the strata when there are R responses of interest. Journal of American Statistical Association,60, 225-233 (1965).  [5] Hartley H.O, Multiple purpose optimum allocation in stratified sampling, in Proc. of the  American Statistical Association, Social Statistics Section. 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Conversed Fractional Programming Approach Goal Programming: A Mathematical Programming Approach

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