## Wednesday, March 25, 2009

### rminlp’s

Hi all,
I am working in pharma and here I need you help for a problem related to a pharmacokinetic equation.
I would like to have the (X,Y,S, T, U, V, W) possible values that will lead to a value of Z=5.7.
Z= 1/X*1/(Y/(1+(U×V)/(W×(U+S)))+(1-Y)) )
knwing that:
0.57<X<0.79
0.00016<W<0.00032
0.89<Y<0.94
0.0936<U<0.967
V =0.05 ou 0.072
S = 29 ou 5.3

This GAMS model calculates some solutions:

variables z,x,y,u,v,w,s;
equation e;

e.. z =e= (1/X)*1/(Y/(1+(U*V)/(W*(U+S)))+(1-Y));

x.lo = 0.57;    x.up = 0.79;
w.lo = 0.00016; w.up = 0.00032;
y.lo = 0.89;    y.up = 0.94;
u.lo = 0.0936;  u.up = 0.0967;

binary variable b1,b2;
equation e1,e2;
e1.. v =e= 0.05*b1 + 0.072*(1-b1);
e2.. s =e= 29*b2 + 5.3*(1-b2);

z.fx = 5.7;

model m /all/;

b1.fx=0; b2.fx=0; solve m minimizing z using rminlp;
b1.fx=0; b2.fx=1; solve m minimizing z using rminlp;
b1.fx=1; b2.fx=0; solve m minimizing z using rminlp;
b1.fx=1; b2.fx=1; solve m minimizing z using rminlp;

I could only find feasible solutions for b2=0 (i.e. s=5.3). This was confirmed by the global solver Baron. Here they are:

 LOWER          LEVEL          UPPER         MARGINAL ---- VAR z                  5.7000         5.7000         5.7000         1.0000      ---- VAR x                  0.5700         0.6019         0.7900          .          ---- VAR y                  0.8900         0.8900         0.9400         EPS         ---- VAR u                  0.0936         0.0936         0.0967         EPS         ---- VAR v                 -INF            0.0720        +INF             .          ---- VAR w                  0.0002         0.0003         0.0003         EPS         ---- VAR s                 -INF            5.3000        +INF             .          ---- VAR b1                  .              .              .             EPS         ---- VAR b2                  .              .              .             EPS LOWER          LEVEL          UPPER         MARGINAL ---- VAR z                  5.7000         5.7000         5.7000         1.0000      ---- VAR x                  0.5700         0.5700         0.7900         EPS         ---- VAR y                  0.8900         0.9400         0.9400         EPS         ---- VAR u                  0.0936         0.0965         0.0967          .          ---- VAR v                 -INF            0.0500        +INF             .          ---- VAR w                  0.0002         0.0003         0.0003         EPS         ---- VAR s                 -INF            5.3000        +INF             .          ---- VAR b1                 1.0000         1.0000         1.0000         EPS         ---- VAR b2                  .              .              .             EPS