Description

The problem is to find a sampling plan that minimizes cost
and yields variances of the population less than an upper bound.
Table 10.1 in the book is not consistent with the equation
statements and the objective function value must be a misprint.

Small Model of Type : NLP

Category : GAMS Model library

Main file : sample.gms

$title Stratified Sample Design (SAMPLE,SEQ=27)

$onText
The problem is to find a sampling plan that minimizes cost
and yields variances of the population less than an upper bound.
Table 10.1 in the book is not consistent with the equation
statements and the objective function value must be a misprint.


Bracken, J, and McCormick, G P, Chapter 10.2. In Selected Applications
of Nonlinear Programming. John Wiley and Sons, New York, 1968.

Keywords: nonlinear programming, sample design, statistics
$offText

Set
   h 'stratum' / 1*4  /
   j 'variate' / a, b /;

Table data(h,*)
          pop   a   b  cost
   1   400000  25   1     1
   2   300000  25   4     1
   3   200000  25  16     1
   4   100000  25  64     1;

Parameter
   vmax(j) 'maximum variance'  / a .04, b .01 /
   w(h)    'stratum weight'
   tpop    'total population'
   k1(h,j) 'constant one'
   k2(j)   'constant two';

tpop    = sum(h, data(h,"pop"));
w(h)    = data(h,"pop")/tpop;
k1(h,j) = sqr(w(h))*data(h,j);
k2(j)   = sum(h, w(h)*data(h,j)/data(h,"pop"));

display tpop, w, k1,k2;

Variable
   n(h)     'sample - original formulation'
   nr(h)    'reciprocal of sample size'
   c        'cost';

Equation
   vbal(j)  'variance balance - original formulation'
   vbalr(j) 'reciprocal variance balance'
   cbal     'cost balance - original formulation'
   cbalr    'reciprocal formulation of cost balance';

vbal(j)..  sum(h, k1(h,j)/n(h)) - k2(j)  =l= vmax(j);

vbalr(j).. sum(h, k1(h,j)*nr(h)) - k2(j) =l= vmax(j);

cbal..     c =e= sum(h, data(h,"cost")*n(h));

cbalr..    c =e= sum(h, data(h,"cost")/nr(h));

Model
   sample  'original formulation'   / vbal,  cbal  /
   sampler 'reciprocal formulation' / vbalr, cbalr /;

n.lo(h) = 100;
n.up(h) = data(h,"pop");
n.l(h)  = 200;
c.l     = sum(h, data(h,"cost")*n.l(h));

solve sample minimizing c using nlp;

nr.lo(h) = 1/data(h,"pop");
nr.up(h) = 1/100;
nr.l(h)  = 1/n.l(h);

solve sampler minimizing c using nlp;