\$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;