Application of Geometric Programming to Optimum Allocation Problems in Multivariate Doubling Sampling *

Javed Ahmed and Charles D. Bonham

Hrrnge Science Depurtment Colorc~lo St&e University

Fort Collins, Colorado 80523

M3STRACT

Optimunr allocation in double sampling or two-phase sampling is easy when tlealing with one variable. However, a simple technique has not been available when an investigator is interested in estimating more than one variable using the double- sampling technique. This paper describes application of geometric programming to the solution of optimum-allocation problems in multivariate double sampling. The solution described here is much simpler than complex analytical techniques described in statistical literature.

I. INTRODUCTION

Optimum allocation is an attempt to attain specified levels of precision at the minimum cost. The problem of optimum allocation in double sampling with a single factor is described in books on sampling techniques. However, a simple technique for optimal allocation in multivariate double sampling has not been available. Kokan [l] proposed a nonlinear-programming solution for optimal allocation in multivariate surveys, but did not discuss its applicability to double sampling. Kokan and Khan [2] described an analytical solution of an allocation problem in multivariate surveys and also discussed its application to double sampling.

Double sampling is extensively employed in range surveys for appraisal of the weight of forage available. Optimum allocation of sampling effort for estimating total forage was described by Wilm et al. [3], Schumacher and

* C~~XXI~ State University Science Series No. 3024

APPLIED MATHEMATICS AND COMPUTATION 21:157-169 (1987) 157

c Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 10017 00983003/87/$03.Fio

1 Fj8 J.4vED AHLIED AND (:H.\RL.ES D. HONII \I1

(~~1apn1~111 [-&I. amI Francis et al. IS]. IIowever, a siniple dt~s~riptioti of ;I tt~chlrique for optimum allocatioli it] dolll~lc~ saillplirig to tlcttwiiiirc the

c~oirtrilnitioii of individual species to total forage productioii has iiot prcviollsly

I~etr available. A techniqlle which simplifies this complex problellr and \~hic% rltilize\ simple arithmetic a11c1 a relatively new technique of geometric pro- graiiiiiiiiig is clesuil~ed here for optiiilrun-allo~~itioii prol~lenia iii Itlrllti\-ariatc clo~ilk sampling.

OIIW the allocatioii prol~leni which iiiiiiimizes co5t for fixed precisiolr Ie\cl

is ~olvecl, then a series of optimal plans of \.aryiiig co5ts ;urtl precisioli le\cl\

CYUI IW calclllated. The sampler can then choose the plar~ \vhic,h Ilest fit\ specific requirements. The techniql~e, althollgh cle~~eloped for sptxics I)iortl;thh

e\tirlutiotr ill this paper. is generally applical~lc ill all sitrlations of tlot~ldc~

s;iiiipIiiIg.

11. OI’TI~IUM .4LLOC:,4TION IN MULTIV.4KIATP: DOITRLE SAMPLING:

The notation r~secl here is that of Cockran [6]. It is assi~riieJ that tlo~~l~le sariipling iiivolves I) factors of interest. It is also assii~irecl that it i\ iiot po\5il)le to keep a separate account of the time or cost to iiic;is~ire or giiess c2ch variate at every sampling point. Then the total cost of d011lk uiiipliirg i\

c: = llC1( + n’c,,,, (ll.l.a)

where c,, is the cost to obtain one direct estimate (measurement of 2, factors) and c,,> is the cost to obtain one indirect estimate (guessing the values of p factors). The n and n’ are the sample sizes of direct and indirect sampling. respectively. Usually c,, is large in relation to c,:, and fixed costs of sampling do not enter into optimization problems [l].

The optimum allocation problem is

Minimize C: = rw,, + rl’c,,, (11.1.1))

srll)ject to

v,, v,,, 1 + 2 < vi’ (j=l,...,p).

n n’ (11.2)

where VI” is the specified variance of the jth variate, \;, ar~cl L’,,, are the

sample variances of jth variate in direct and indirect sa;nples. ‘the other notation is the same as already described.

III. PREPROCESSING OF THE PROBLEM

The usual analytical approach for optimization of nonlinear problems is to use calculus and/or Lagrange multipliers. The geometric-programming ap- proach of Duffin et al. [7] is a relatively new technique to solve algebraic nonlinear problems; in combination with simple arithmetic, it is used here to solve the allocation problem. The allocation problem can be reduced to the following mathematical form: Find the vector

x = (XI, q,) (xl=n and x2= n’) (1II.l.a)

sl~h that

xi > 0, j = 1,2, (111.2.a)

and function g(x), corresponding to the cost function, is minimized. In matrix notation: Find

x=(x,J2.‘...J,,) (111.1.1))

which satisfies

x > 0,

and minimize f(x), where

A(mxn)= [vii] and vO= [D:,v~,...,v~~].

(111.2.b)

(111.3)

Given that there are only two unknowns, xl and x2, and the number of constraints is greater than two, the chances that more than two constraints will be binding at optimality are low. Therefore, the first step is to determine which constraints, if any, are irrelevant. Divide all constraints by VP. This process, sometimes called rwrmdizing the constraints, allows for easy identi- fication of the lower bounds for x1 and x2 and also puts the constraints in the

160 JAVED ;\HhlED .\ND (:II.\RLES D. HONII;\\I

c,orru,t fornl for geometric progralntiiing:

k,,=$ !

(III.l)

(111.5)

where u 1 and (I,, are the constants of the two terms in the ith constraint, and k, and k, are the normalized constants in the ith constraint.

The second step is to check for the dominated constraints. If k ,, > k I, and k,, > k,,, then eliminate constraint j for all i and j.

The third step is to find the lower bound for x, and s,. Let

)11 l = lllax( k,,). (111.6)

In2 = max( k?,). (111.7)

Then x1 is lower bounded by ~1, a11d s, 1,~ tt~, and the cost is lower-lx~unded

by cIull + c2rr12, where uI = c,, and c2 = c,,,.

If c1111y one constraint remains after processiilg, the ~illocatioil prolhl~ is aiitoiliatically solved. If two constraints are I~inding. then optinrizatiolr ill-

valves the solutiolr of two equatiorls and two lmknowns.

I\!. (:~OhlETKI(:-PKO(:RAhlMIN<: APPKO:\t:H

Follo\z,ing the georrletl-ic-prog~~llrillg 1uleh for polynollliuls [ 81, the dloc-

atioii prolhii is solved iii four steps as follows:

Stq, ( i ). The optiimiin valrie of the ohjectivc furrctioir is Avays of tlrc.

fc )I'111

i

coeff. of 1st term ” ’ g,,(s)* =

I(‘, I

i

coeff. of second term I’ ’ X

?l‘? I

coeff. of last term 1’ I,\, x ‘.. x

1 ‘c‘ I.,,1 I

Chmdric Progrmming in Double Smnipling 161

From step (i) and the optimum-allocation problem,

(IV.1)

S!C~, (ii). The equations generated for a geometric program are

c W’S in the objective function = 1, (IV.2.a)

and for each primal variable x1, given n variables and 1~ terms,

i = ,I,

c (IX, for each term) ,=I

x (exponent on xi in that term) = 0. (IV.3.a)

In this case,

W, + 1oa = 1,

(l)w,+(0)Wg+( -l)W:~+(O)W,=o,

(o)w,+(l)W,+(o)W,+( -l)ti,=O.

(IV.2.b)

(1V.n.b)

(IV.3.c)

Now, combining (IV.2.b), (IV.S.b), and (IV.3.c)

w,+wo = 1, (IV.4.a)

Wl -w:3 =o, (IV.5.a)

w2 _ w4 = 0, (IV.6.a)

which is a set of three simultaneous linear equations in four unknowns. Therefore, the degree of difficulty is 1. The degree ofdifficulty is the number of weights (W’S) minus the number of terms, plus one. The problems with degree of difficulty only 0 can be solved immediately. The above set of equations may be solved in terms of one w, say wl:

w,=l- WI, (IV.4.b)

w:J = WI> (IV.5.b)

w,=w,=l-IC,. (1V.S.b)

162 JAVED AHMED AND (:HARLES D. HONII.\\I

St(,)) ( iii ). The contriblltion of terms ill the constraints to optimal solrltioll i\ al\vays proportional to their weights. 111 thi5 c’asc

k, 11‘ 1

~ - Wl. (Iv.;) s , 11‘; + fl‘,

Frorr~ (I\‘.T) and (IV.8) \ve have

(1V.X)

(lV.9.l))

S/cl~ ( it.). The primal variables may be fonnd 1))

1st term in objective function &,( ,r)* = ~--

2nd term in 0l)jective flmctiotl

1c‘ , 11‘)

last term in objective frmction

‘c’ I.,rc (I\;. 1O.a)

111 thi\ case,

(1V.lO.b)

Since w1 and x$ are already known (IV.7 and IV.S.b), Equation (IV.10.1))

can be solved for xf in terms of the constants C and k. Theu

k,.r,

clx; cp

x; - k, k,=

1-5 (1V.ll.a)

XI x;

The eqlution (IV. I 1.a) implies that

(1V.ll.b)

Ckmctric. Progrcmniing in lh4ble Sampling 163

and from (IV.9.b)

x* - k?.

’ - I- k,,‘x, ’ (IV.9.c)

,4 FORTRAN computer program for interactive use for the solution of the allocation problem is given in the appendix. The program can handle up to 10 constraints, but can be easily modified to handle more constraints. The program does not give integer solutions for the sample sizes, but the mmrbers may be rounded off.

V. NUMERICAL EXAMPLE

Data from a sample survey, using a double-sampling technique, were collected from northeastern Colorado in 1979. The variance of double sam- pling for each species on the left side of the constraint was calculated using Equation 12.49 of [6]. The standard error was calculated for each species, and its squared value was used as a precision specification on the right-hand side of the constraint. The costs to obtain one direct and one indirect sample were estimated at $5.00 and $0.50, respectively. The optimum allocation problem was then of the form

Minimize cost = 5r, +0.5x,

subject to

6 100 -+-t-_-o.5 Xl X2

40 175 -+----_l.O Xl x2

5 + 3 < 0.2 XI x2

2 17 - + - < 0.1 XI x2

3 21 - + - < 0.1 Xl x2

(constraint for blue gramma),

(constraint for sand dropseed)

(constraint for three-awn),

(constraint for globe mallow),

(constraint for goosefoot),

(V.1.)

(V.2.a)

(V.2.b)

(V.2.c)

(V.2.d)

(V.2.e)

164 JAVED AHMED .\ND (:H,\RLES D. HONH.\\l

.After elitrtitiatittg irrelevant constraitit~. (V.2.a) atd (V.2.c) Lverr forttld to

Iw Ititttling at optitnality. The nortdized cotlstrainta (\‘.?a) md (\‘.2.l)) arc‘

12 Boo - + ~ $ 1.0, (V.3.a s , .I. 1

By \ttl)atittttitlg the values of k,. k. c, ant1 (‘? iii (IV. 11 .I)). \vca liaw

0.3 X 40 X 17.5 S] = 30+ --

Ti . (\-.4.;1)

= 66.3,57.5, or say 66, (\‘.4.I))

;urtl I)y sttltstittttittg the values of k,, k,. ant1 s, itt (IV.1 l.c.), \ve IMVC

(V3.a)

= 3:39.,57, or say 440. (\‘.5.1))

‘I’lrc cost of satiiplittg for the specified variatlces is

cost = $5 X 66 + $0.5 X 340 (\‘.C.a)

= $5.50.00. (\‘.A.l))

If the fitraticial resources are limited, optional sampling plats c’att Iw

calcttlated for some fixed cost. If the satiiplitig cost is C” atid the fihetl cmt i\

(: ‘. theta the fixetl-cost sample with

c C’ = ” ( r is a constant)

r (17.7)

_ and “1 St

r > 0. (\‘A) I r ’

which satisfies

Vi’ < TV,?

165

(v.9)

VI. C:oNCLUSIO~

The optimum allocation problem for multivariate double sampling, which miirimizes the cost for specified precision levels, can be solved easily using simple arithmetic and a geometric-programming approach rather than by iising complex analytical methods. The solution of the allocation problem also allows the investigator to choose a series of optional plans for varying costs and degrees of precision. The most binding constraints may not be important practically. The investigator can then choose the plan which best fits in terms of the budget and precision requirements. The procedure described here will be foimd useful in many fields where double sampling is applicable.

;\PI’ENDIX. FORTRAN COMPUTER PROGRAM

The variables used in this program are:

COST = cost values for the objective function.

DOM = the array of dominance vector. Tells which constraint dominates

the constraints under analysis. FACTOR = the factor by which the cost of initial plan is reduced to get the

optional plans. KR = normalized variance coefficients (the right hand sides of all the

constraints are set equal to 1). L F E AS = a logical variable. The system is feasible if . t rue .

N = the number of constraints in the optimization problem. NUMI = the direct sample size of the optional plans. NUM2 = indirect sample size of the optional plans. OPT1 = the option to reenter the data. OPT2 = the option to get additional optional plans. PNUM = the number of the optional plans which the sampler desires. PLAN = the cost for the optional plans. RHS = the right-hand-side vector of the constraints. S I GMA = the estimated variances of the means for the initial plan, “plan 0.” S I GMA2 = the estimated variances of the means for the optional plans. T COST = total cost of sampling. VAR = the elements which are being sampled.

JAVED AHhlED AND (:H:ZRLES D. HONH:\\l 166

Xl x2

100

602

604

606

10

610

612

15

614

616

c

618

620

C

C

20

622

624

626

628

c

200

630

C

= the tirunl)er of direct samples. = the numl)er of the indirect satnph.

DIMENSION COST(Z), VARClO,Z), RHSCIO), SIGMACIO), SIGMA2(10,10),

+P COSTCIO), NUMlClO), NUMZClO)

REAL TCOST,Xl,X2,COST,VAR,RHS,NUM1,NUM2,PCOST,SIGMA,SIGMA2

INTEGER N, PNUM

(WRITE (6,602)

FORMAT (*HOW MANY CONSTRAINTS*)

READ (5,*)N

WRITE (6,604)

FORMAT (*COST FUNCTION*)

READ (5,*)CCOSTCI),I=1,2)

WRITE (6,606)

FORMATC*INPUT CONSTRAINT COEFFICIENTS AND RIGHT-HANDSIDE* + /*(SEPARATED BY A COMMA OR A SPACE)*)

DO IO I=l,N

WRITE (6,608)1

FORMAT (*CONSTRAINT*, 12, *:*)

READ (5,*) VARCI,l), VARCI,Z), RHSCI)

CONTINUE

WRITE (6,610)

FORMAT (///25X, *OPTIMUM ALLOCATION PROBLEM*)

WRITE (6,612) CCOSTCI),I=1,2)

FORMAT C//, 15X,*MINIMIZE COST=*,F6_2,*Nl + *,F6_2,*NZ*/

+15x, *SUBJECT TO*)

DO 15 I=l,N

WRITE (6,614) VARCI,l),VARCI,2),RHS(I)

FORMAT (/20X,F6.2,*/Nl+*,F6.2,*/NZ .LE. *,F6.3)

WRITEC6,616)

FORMAT (/28X),*N2 .GT. Nl .GT. O*)

FIND OUT IF THE DATA IS TO BE RE-ENTERED.

WRITE (6,618)

FORMAT (/*DO YOU WANT TO CHANGE THE DATA*/ + *CENTER 0 FOR ~0, 1 FOR YES)*)

READ (5,*) OPT1

IFCOPTI .EP.l) GO TO 100

WRITEC6,620)

FORMAT (/////,25X,*INITIAL OPTIMUM ALLOCATION PLAN*)

CALL THE SUBROUTINE NONLIN TO FIND THE OPTIMUM ALLOCATION

CALL NONLIN CN,TCOST,Xl,X2,COST,VAR,RHS)

WRITE THE INITIAL OPTIMUM ALLOCATION PLAN.

DO 20 I=l,N

SIGMACI)=VARCI,l)/Xl+VAR(I,Z)/X2

WRITE (6,622) Xl,XZ,TCOST,N

FORMATC//,5X,*THE NUMBER OF DIRECT SAMPLES*,BX,*=*,FlO.l,//

+5X,*THE NUMBER OF INDIRECT SAMPLES*,6X,*=*,FlO.l,/

+/5X,*THE COST OF SAMPLING WITH THIS PLAN=$*,F9.2

+///25X,*VARIANCES OF THE MEANS*/lZX,=C*--------*))

WRITE (6,624) N,CI,I=l,N),N FORMAT (15X,=(*VAR*,I2,3X)/l2X,=(B(*-*)))

WRITE (6,626) N, CRHS(I),I=~,N)

FORMATClX,*SPECIFIED*,3X,=(F6.2,2X))

WRITE (6,628) N, CSIGMACI),I=l,N)

FORMATClX,*ESTIMATED*,4X,=(F6.2,2X))

ADDITIONAL SAMPLING PLANS.

WRITE (6,630) FORMATC///*HOW MANY SAMPLING PLANS DO YOU WANT*)

READ (5,*) PNUM

IFCPNUM.EQ.0) GO TO 900

c d’(. 167

C

632

150

634

636

638

640

642

644

646

16

ia 17

19

648

650

2

C

10

WRITE (6,632)

FORMATc*ENTER THE COST OF EACH PLAN* + /*(SEPARATED BY A COMMA OR A SPACE)*)

READ(~,*) cPCOSTcI),I=1,PNUM)

WRITEc6,634)

FORMATc///25X,*OPTIONAL OPTIMUM ALLOCATION PLANS*)

WRITEc6,636) N

FORMATc//l1X,*COST*,2X,*SAMPLES*,l2X,*VARIANCES OF THE*

+*MEANS*/,11X,8c*-*I,2X,llc*-*),2X,=c8c*-*~~~

WRITEc6,638) N,cI,I=l,N)

FORMAT(llX,*DOLLARS*,5X,*Nl*,4X,*N2*,3X,=(*VAR*,I2,3X))

WRITEc6,640) N

FORMAT~11X.8~*-*~.2X.ll~*-*~.2X_=~8(*-*~~~

WRITEc6,64;) N, cRHSiI),I=l,N) . FORMAT (*SPECIFIED VARIANCES*, 13~,=(~6.2,2~))

WRITEc6,644)

FORMATc*OPTIONAL PLANS*)

WRITEc6.646) TCOST,X1,X2,N,cSIGMAO,I=l,N~

FORMATc/3X,*PLAN O*,F8.2,2X,F6.1,1X,F6.l,=cF6.2,2X))

DO 16 I=l,PNUM

FACTOR=TCOST/PCOSTcI)

NUMlcI)=Xl/FACTOR

NUM2cI)=X2/FACTOR

CONTINUE

CALCULATE THE VARIANCES OF THE OPTIONAL PLANS

DO 17 M=l,PNUM

DO 18 I=l,N

SIGMA2cM,I)=VARcI,1)/NUMlcM)+VAR(I,2)/NUM2cM~

CONTINUE

CONTINUE

DO 19 N=l,PNUM

WRITE~6,648~M,PCOST~M~,NUMl~M~,NUM2~M~,N,~SIGMA2~M,I~,I=l,N~

FORMAT~/3X,*PLAN*,I2,F~.2,2X,F6.l,lX,F6.l,=~F6.2,2X~~

ARE ADDITIONAL PLANS NEEDED?

WRITEc6,650)

FORMATc/*DO YOU WANT ADDITIONAL OPTIONAL PLANS* + /*ENTER 0 FOR NO, 1 FOR YES*)

READCS,*) OPT2

IFcOPT2.EQ.l) GO TO 200

STOP

END

SUBROUTINE NONLIN cN,TCOST,X,X2,COST,VAR,RHS)

DIMENSION COSTc2),VARC10,2),RHS(10),DOM(10),KR(10,2~

REAL TCOST,Xl,XZ,COST,VAR,RHS,KR

INTEGER N,DOM

LOGICAL LFEAS

INITIALIZE ALL OF THE WORKING ARRAYS TO ZERO

DO 2 I=l,N DOM(I)=O LFEAS = -FALSE.

NORMALIZE ALL THE CONSTRAINTS TO OBTAIN THE GEOMETRIC

PROGRAMMING STANDARD FORM.

DO 10 I=l,N

DO 10 J=1,2

KRcI,J)=VARcI,J)/RHS(I)

CONTINUE

CHECK FOR THE DOMINATED CONSTRAINTS.

IFcN.LE.1) GO TO 31

DO 30 I=l,N

DO 20 J=l,N

IF CI.EQ.J) GO TO 20

IF cKRcI,1).LE.KRcJ,1).AND.KRcI,2).LE.KR(J,2)) GO TO 25

168 JAVED AHMED AND (:HARL,ES D. RONH:\?rl

20

25

30

31

40

C

60

50

C

65

70

99

100

CONTINUE

GO TO 30

THE CONSTRAINT I IS DOMINATED BY CONSTRAINT J.

DOMCI)=J

CONTINUE

CONTINUE

ASSUME ONLY ONE CONSTRAINT IS TIGHT. APPLY GEOMETRIC

PROGRAMMING OPTIMIZATION BY CLOSED FORM SOLUTION.

DO 40 I=l,N

IF CDOMCI).NE.O) GO TO 40

X1=KRCI,1)+SQRTCCOST(2)*KRo*KR(I,2)/COSTCl~~

XZ=KR(I,2)/(1-(KR(I,1)/X1))

CHECK FOR THE FEASIBILITY OF THE SOLUTION.

CALL FEAS CLFEAS,N,VAR,RHS,TCOST,Xl,X2,COST)

IF CLFEAS) RETURN

CONTINUE

SYSTEM HAS MORE THAN ONE TIGHT CONSTRAINTS.

CHECK UNDOMINATED CONSTRAINTS, TWO AT A TIME.

IF CN.EQ.1) GO TO 65

M=N-1

DO 50 1=1,M

IF CDOMCII.NE.0) GO TO 50

K=I+1

DO 60 J=K,N

IF CDOMCJ).NE.O) GO TO 60

X2=~KR~I,2~*KR~J,1~-KR~J,2~*KR~I,l~~/~KR~J,1~-KR~I,l~~

Xl=KR(I,l)/(l-(KR(I,Z)/XZ))

CHECK FOR FEASIBILITY OF SOLUTION.

CALL FEAS CLFEAS,N,VAR,RHS,TCOST,Xl,X2,COST)

IF C.NOT.LFEAS) TO TO 60

RETURN

CONTINUE

CONTINUE

NO TWO EQUATIONS SATISFY THE SYSTEM.

X2 WANTS TO BE LESS THAN X1. SET X1=X2.

FIND BINDING CONSTRAINT IF X1=X2. SUM=0

DO 70 I=l,N

IF CSUM_GE_KRCI,1)+KRCI,2)I GO TO 70 SUM=KRCI,l)+KRCI,2)

CONTINUE

Xl =SUM

x2=x1

CHECK FEASIBILITY OF SOLUTION

CALL FEAS CLFEAS,N,VAR,RHS,TCOST,Xl,X2,COST)

IF C.NOT.LFEAS) GO TO 99

RETURN

SYSTEM IS STILL INFEASIBLE. SOMETHING IS WRONG.

IO=6

WRITE (6,100)

FORMAT (*FATAL ERROR IN NONLIN. SYSTEM OF EQUATIONS IS

INFEASIBLE*.

STOP

END

SUBROUTINE FEAS CLFEAS,N,VAR,RHS,TCOST,Xl,X2,COST)

SUBROUTINE FEAS CLFEAS,N,VAR,RHS,TCOST,Xl,XZ,COST)

DIMENSION COSTC2),VARC10,2),RHS(10)

REAL COST,VAR,RHS,KR,Xl,X2,LHSC~o)

INTEGER N,DOM

LOGICAL LFEAS

CHECK FEASIBILITY FOR GIVEN X1 AND X2.

IF (Xl .GT.XZ) RETURN

DO 10 I=l,N

Gonwtric Programming in Double Sampling

LHS(I)=VAR(I,l)/X1+VAR(I,2)IXZ

IF (LHS(I).GT.RHS(I)) RETURN

10 CONTINUE

C SYSTEM IS FEASIBLE

LFEAS=.TRUE.

C CALCULATE TOTAL COST.

TCOST=COST(l)*Xl+COST(2)*X2

RETURN

END

REFERENCES

169

1

2

3

4

5

6

7

8

A. R. Kokan, Optimum allocation in multivariate surveys, J. Roy. Stntist. Sot.

126:X37-565 (1963).

A. R. Kokan and S. Khan, Optimum allocation in multivariate surveys: An analytical solution. J. Roy. Statist. Sot. l:llFj-125 (1967). H. G. Wilm, D. F. Costello, and 6. E. Klipple, Estimating forage yield by the

tlodde sampling method, Amer. SM. Agron. J. 36:194-203 (1944).

F. X. Schumacker and R. A. Chapman, Sampling methods in forestry and range uranagement, Duke Univ. School of Forestry Bull. No. 7, 1948.

R. C. Francis, G. M. Van Dyne, and B. K. Williams, An evaluation of weight estimation double sampling as a method of botanical analysis, J. Environ. MczncIgc.

X:5%72 (1979).

W. G. Co&ran, Sampling Techniques, Wiley, New York, 1977, p. 628. R. J. Duffin, E. L. Peterson, and C. Zener, Geolnetric Programming. Wiley, New York, 1967, p. 278. R. E. D. Woolsey and H. S. Swanson, Operations Research for Immediatr

,Applicntion: A Quick and Dirty h4anua1, Harper and Row, New York. 1975, p. 204.