GEOMETRIC PROGRAMMING APPROACH TO OPTIMUM ALLOCATION IN MULTIVARIATE TWO- STAGE SAMPLING DESIGN

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1 Electroic Joural of Applied Statistical Aalysis EJASA, Electro. J. App. Stat. Aal. (0), Vol. 4, Issue, 7 8 e-issn , DOI 0.85/i v4p7 008 Uiversità del Saleto GEOMETRIC PROGRAMMING APPROACH TO OPTIMUM ALLOCATION IN MULTIVARIATE TWO- STAGE SAMPLING DESIGN Shoat Maqbool, Abdul H. Mir, Shaeel A. Mir Divisio of Agricultural Statistics, SK Uiversity of Agricultural Scieces & Techology of Kashmir, Idia. Received April 00; Accepted 4 November 00 Available olie 6 April 0 Abstract: Geometric programmig provides a poerful tool for solvig oliear problems here o-liear relatios ca be ell preseted by a epoetial or poer fuctio. I real life situatios applicatios of geometric programmig are soud i egieerig desig, samplig desig etc. I is paper, e problem of allocatio i first stage ad secod stage uits i multivariate to stage samplig is cosidered. The problem is formulated as a cove programmig problem i liear obective fuctio. A solutio procedure is developed to solve e resultig maematical programmig problem by usig geometric programmig techique. The computatioal details of e procedure are illustrated rough a umerical eample. Keyords: To Stage samplig, No-liear programmig, cove programmig, geometric programmig.. Itroductio I may surveys e use of to stage samplig desigs ofte specifies to stages of selectio: clusters or primary samplig uits (PSUs) at e first stage, ad subsamples from PSUs at secod stage as a secodary samplig uits (SSUs). For e large-scale surveys, stratificatio may precede selectio of e sample at ay stage. Aalysis of to-stage desigs are ell documeted he a sigle variable is measured ad e meods to obtai e optimum allocatios of samplig uits to each stage are readily available The problem of optimum allocatio i tostage samplig i a sigle character is described i stadard tets o samplig (see Cochra smaqbool007@yahoo.co.i 7

2 Geometric programmig approach to optimum allocatio i multivariate to- stage samplig desig [] ). Hoever he more a oe characteristic are uder study e procedures for determiig optimum allocatios are ot ell defied. The traditioal approach is to estimate optimal sample size for each characteristic idividually ad e choose e fial samplig desig from amog e idividual solutios. I practice it is ot possible to use is approach of idividual optimum allocatios because a allocatio, hich is optimum for oe characteristic, may ot be optimum for oer characteristic. Moreover, i e absece of strog positive correlatio betee e characteristics uder study e idividual optimum allocatios may differ a lot ad ere may be o obvious compromise. I certai situatios some criterio is eeded to or out a acceptable samplig desig hich is optimum i some sese for all e characteristics. Geometric programmig (GP), a systematic meod for solvig e class of maematical programmig problems at ted to appear maily i egieerig desig, as first developed by Duffi ad Zeer i e early 960s, ad furer eteded by Duffi et al. [4]. Davis ad Rudolph [3] use geometric programmig to optimal allocatio of itegrated samples i quality cotrol. Shiag [6] ad Shaoia et.al [7] used G.P for egieerig desig problems. The paper is preseted as follos: First a allocatio problem is formulated i a to-stage samplig desig i sectio ad geometric programmig approach is used to solve it sectio 3. A umerical illustratio is e preseted i sectio 4 ad e fial commets ad coclusio is give i sectio 5.. Formulatio of e Problem Let us assume at e populatio cosists of NM elemets grouped ito N first-stage uits of M secod-stage uits each. Let ad m be e correspodig sample sizes selected i equal probability ad iout replacemet at each stage. Let y be e value of e populatio at r secodary stage uit i e h,,..., N, r,,..., M,,..., p. r character: hr h primary stage uit for We defie for m yhr yh = Sample mea per sub uit at e r m yh y = Overall sample mea per sub uit (elemet). h M yhr Yh = Mea per elemet at e h first stage uit. M Y S b N Yh = Mea per elemet i e populatio. N h N Y h Y h N h primary stage uit. = True variace betee first stage uit meas. character, 7

3 Maqbool, S. et al., Electro. J. App. Stat. Aal. (0), Vol 4, Issue, 7 8. S N M y hr h r N M Y h = True variace ii first stage uits. I case of equal first-stage uits a ubiased estimate of Y is V y Sb S,,..., p N m NM (see proof i Appedi) The total cost fuctio of a to stage samplig procedure may be give by: y i its samplig variace as, () C C Cm () Where: C The cost of e survey i approachig a sigle primary stage uit. C C The cost of eumeratig e p C character per elemet. = The cost of eumeratig all e p characters per SSu. Suppose at it is required to fid e values of ad m so at e cost C is miimized, subect to e upper limits o e variaces. If N ad M are large, e from (), e limits o e variaces may be epressed as: Sb S v,,..., p. (3) m Where v is e upper limits o e variaces of various characters. Here S b is e variace amog primary stage uits meas ad S is e variace amog subuits ii primary uits for characteristic respectively. The problem erefore reduces to fid ad m hich: Miimize C C Cm (4) Sb S Subect to v,,..., p (5) m, m (6) (I each primary stage uit at least oe secodary stage uit has to be eumerated as egative values of PSUs ad SSUs are of o practical use). 3. Geometric programmig approach 73

4 Geometric programmig approach to optimum allocatio i multivariate to- stage samplig desig Geometric programmig (GP) is a techique for miimizig a fuctio called a posyomials subect to several costraits. A posyomial is a polyomial i several variables i positive coefficiets i all terms ad e poer to hich e variables are raised ca be ay real umbers. Bo e cost fuctio ad e variace costrait fuctios are posyomials. G.P trasforms e primal problem of miimizig a posyomial subect to posyomial costraits to a dual problem of maimizig a fuctio of e eights o each costrait. Usually ere are feer costraits a strata, so e trasformatio simplifies e procedure. The problem (4) - (6) as such taes e folloig maematical form: Fid e vector =, ( ad m hich miimizes C( ) = subect to g( )= i a iq q i v, q C C C i i q,..., p ) ad i 0, i, (9) m We have substituted i e above equatios:, m, Sbq a q, Sq aq for q,..., p It may be oted at e obective fuctio (7) is liear ad e costraits (8) are oliear ad e stadard GP (Primal) problem stated i to subscripts is reduced to: Miimize f 0 Subect to fq, q,... p 0,,... Where posyomial q is: (0) (7) (8) f ( ) di iq q pi, d i 0, 0, q 0,,..., p, () here deotes e umber of posyomial terms i e fuctio, is e umber of variables ad e epoets p are real costats. For our allocatio problem, e obective fuctio C() i give i (7) ad (8) has,, p p, p p 0, di Ci, i,, ad e q costrait has,, p p, p p 0 ad di aiq, i,. (see Maqbool & Pirzada[5]). The dual of GP problem stated i (0) is give by: 74

5 Maqbool, S. et al., Electro. J. App. Stat. Aal. (0), Vol 4, Issue, 7 8. p i p i d i[ q ] i Maimize i q0 i[ q] i q i [ q] () subect to i (3) p q0 i[ q] i i[0] P 0 (4) i i 0, q 0,..., p i,..., p (5) Folloig Woolsey ad Saso [9] ad Duffi et al [4], e allocatio problem (7) & (8) ill be solved i four steps as follos: Step : The Optimum value of e obective fuctio is alays of e form Coeff. of first term Coeff. of Secod term C0( ) Coeff. of last term... K ' s i e first costra its ' s i e last costra its K ' s i e first costraits ' s i e last costrait s (6) For our problem e obective fuctio is: C a v C Cost = 3 4 Where, a v (7) Step : The equatios geerated for geometric program for e eights are 75

6 Geometric programmig approach to optimum allocatio i multivariate to- stage samplig desig ' s i e obective fuctio (8) ad for each primal variable give variables ad terms m for each term ep oet o i at term 0 i i (9) I our case: (Normalizatio coditio, see (3)) (0) ( 3 4 ) (0) ( ) (0) 0 () ( 3 4 0) () (0) ( ) 0 () Equatios () & () are Orogoality coditios, see (4). Collectively, ese coditios are referred to as dual costraits. For more details see Duffi et al. [4]. No combiig (0), () ad (), e get:, 0, hich is a set of ree liear equatios i four uos. The above set of equatios may be solved i terms of oe, say., 3, 4 Step 3: The cotributio of terms i e costraits to optimal solutio is alays proportioal to eir eights. I is case: (3) 4 (4) 3 4 From e above equatios (3) & (4), e get: hich implies: (5) 76

7 Maqbool, S. et al., Electro. J. App. Stat. Aal. (0), Vol 4, Issue, 7 8. Step 4: The primal variables may be foud by: C ( ) 0 first term i obective last term i obective fuctio... K fuctio sec od term i obective fuctio I is case: C C, here ad (6) Sice ad are already o from (3) ad (5), e above equatio ca be solved for i terms of e costats C ad, e: C C (7) The above equatio implies at: C (8) C From equatios (5) ad (8), e ca easily calculate e optimum values for are ad m. 4. Numerical illustratio We cosider Charavary [] for umerical illustratio, here dispersio matri for characters i a sample of 0 PSUs ad 8 SSUs i a situatio he each PSU as dra i equal probability at each stage is give belo; e cost of eumeratig a PSU is estimated as ad at of SSU as: 77

8 Geometric programmig approach to optimum allocatio i multivariate to- stage samplig desig Table. Dispersio Matri. Dispersio due to Degree of Freedom S.P. Matri Covariace Matri Betee PSUs Wii PSUs Betee SSUs Total I is case e values of N ad M are ot o, ey may be assumed to be ifiite. Also e data may be tae as derived from a pilot survey ad a similar survey is to be plaed for hich e require e best values of ad m hich miimizes e total cost. Therefore from e above dispersio matri, e have: S =0.0078, Sb =0.0037, S =0.043 Sb = These are sample estimates ad are subect to e samplig fluctuatios. No our problem is to miimize: Miimize C Cm (9) Sb S Subect to v (30) Sb S (3) v, The upper boud of v is calculated usig e loer 5 percet poit of =0.7, e have: distributio i 9 d.f. v = , v = (3) 78

9 Maqbool, S. et al., Electro. J. App. Stat. Aal. (0), Vol 4, Issue, 7 8. The upper cofidece bouds of S at 95 percet cofidece level are: Upper boud of S Upper boud of S = (33) The upper cofidece bouds of Sb at 95 percet cofidece level are: Upper boud of Sb Upper boud of Sb =0.080 (34) Usig e values (3), (33) ad (34), our problem becomes: Miimize C (35) Subect to (36) (37) The ormalized costraits are:, (38) (39) Which gives: X X X X (40) (4) Let us tae e costrait (4) as active (if bo costraits ere active, e oe ould ot be able to fid a optimal dual solutio or a optimal solutio to e origial solutio, see Shiag[8] ad Maqbool & pirzada [5]. The K 6. 4 ad K Substitutig e values of K, K, C ad C i equatio (8) ad (5), e get: 79

10 Geometric programmig approach to optimum allocatio i multivariate to- stage samplig desig ad compute m ad roudig yields: 3 m 33 93, usig e values of ad C , e get e total cost as: Therefore e optimum values are 3 ad m 93, i.e m 3. This shos at e solutio is feasible. Thus, e require a sample of 3 primary stage uits ad 3 secodary stage uits i each primary stage uit givig us a total of m 93 elemetary uits for e sample. The above results ca easily be verified rough GP optimizatio algorims available o iteret (see GPGLP [0] & XGP []. 4. Commets ad Coclusio Optimum allocatio i to stage samplig is easy he dealig i oe variable. Hoever a simple techique has ot bee available he oe is iterested i estimatig more a oe variable. This paper is a attempt to utilize geometric programmig approach to e solutio of optimum allocatio problems i multivariate to-stage samplig. The solutio described here is much simpler a comple aalytical techiques described i statistical literature. Geometric programmig has already sho its poer i practice i e past. I real orld applicatios, e parameters i e geometric program may ot be o precisely due to isufficiet iformatio. The umerical result illustrates e feasibility ad effectiveess of e preset approach. Wi e availability of GP optimizatio softare, e ider applicatios of e proposed approach ca be utilized i double samplig desig havig multiple characters ad i case of respose error ( itervieer variability), various agricultural surveys here to stage samplig desigs are frequetly employed for differet research studies. 80

11 Maqbool, S. et al., Electro. J. App. Stat. Aal. (0), Vol 4, Issue, 7 8. Appedi: Proof of equatio () V y S S, N b m NM Wi simple radom samplig at bo e stages: = E Yi y E E y E V y V E y E V y V ˆ V E ˆ E ˆ V Sice E y N = Yi Y., because: yi, e first term o e right is e variace of e mea per subuit for a oe stage simple radom sample of uits, hece by usig e basic eorems of SRS ( see Cochra []): V N (4) N E y Sb Furermore, i yi y ad simple radom samplig used at e secod stage: M m V y Sr m M / (43) Where S r yr yr /( M ) is e variace amog subuits for e h primary uit. Whe e average over e first stage samples: h S r averages to N h r S S N, 8

12 Geometric programmig approach to optimum allocatio i multivariate to- stage samplig desig Hece: M m EV y S / m (44) M Addig (4) ad (44) gives: V y S S, N b m NM If e igore e terms idepedet of ad m, e get e variace: Sb S V y m Refereces []. Cochra, W.G. (977). Samplig techiques. Ne Yor: Joh Wiley & Sos. []. Charavary, I. M. (955). O e problem of plaig a multistage survey for multiple correlated characters. Shahya, 4,-6. [3]. Davis, M., Rudolf, E.S. (987). Geometric programmig for optimal allocatio of itegrated samples i quality cotrol. Comm.Stat.Theo.Me.,6 (), [4]. Duffi, R.J., Peterso, E.L., Zeer, C. (967). Geometric programmig: Theory & applicatios. Ne Yor: Joh Wiley & Sos. [5]. Maqool, S., Pirzada, S. (007). Optimum allocatio i multivariate to-stage samplig: A aalytical solutio. Jour.of aalysis & Computatio, Vol 3, No, [6]. Shaoia, Qu, Kecu, Z., Fusheg, W. (008). A global optimizatio usig liear relaatio for geeralized geometric programmig. Europ. Jour.of Oper. Res.,90, [7]. Shiag-Tai Liu (008). Posyomial geometric programmig i iterval epoets ad coefficiets. Europ. Jour.of Oper. Res.,86,7-7. [8]. Suhatme, P.V., Suhatme, B.V. Suhatme, S. (954). Samplig eory of surveys i applicatios. Ioa State Uiversity Press.,USA. [9]. Woolsey, R.E., Saso, H.S. (975). Operatios Research for immediate applicatios: A quic ad dirty maual. Ne Yor: Harper & Ro. [0]. GPGLP: ftp://ftp.pitt.edu/dept/ie/gp/. []. XGP: ftp://col.biz.uioa.edu/dist/xu/doc/softare.html. 8