Description

Example from Chapter 1.5.3 page 45-48 and 276-277 (The Wall Design)

Conejo A J, Castillo E, Minguez R, and Garcia-Bertrand R, Decomposition
Techniques in Mathematical Programming, Springer, Berlin, 2006.

Contributor: Jan-H. Jagla, January 2009

Small Model of Type : BP

Category : GAMS EMP library

Main file : ccmg153.gms

$title Bilevel Programs in Engineering Example 1.5.3 (CCMG153,9)

$onText
Example from Chapter 1.5.3 page 45-48 and 276-277 (The Wall Design)

Conejo A J, Castillo E, Minguez R, and Garcia-Bertrand R, Decomposition
Techniques in Mathematical Programming, Springer, Berlin, 2006.

Contributor: Jan-H. Jagla, January 2009

$offText

variables cost
          a              width
          b              hight
          f0             safety
          t              force
          h              soil offset
          beta           associated reliability index
          z1             random variable for t
          z2             tandom variable for h
          relindex       reliability index;

equations defobj,safety,minsafety,geometry,
          reliability,defrelindex,overturning,
          defbeta2,defbeta3,defz1,defz2;

scalars
   gamma         'unit weight [kN/m3]'      / 23   /
   muh           'mean of h [m]'            /  3   /
   mut           'mean of t [kN]'           / 50   /
   sigmah        'std of h [m]'             /  0.2 /
   sigmat        'std of t [kN]'            / 15   /
   f00           overturning failure safety /  1.5 /
   beta0         reliabilty bound           /  3   /
   b0            minimum height             /  4   /;

* classical design
defobj.. cost =e= a*b;
safety.. f0 =e= (sqr(a)*b*gamma)/(2*muh*mut);
minsafety.. f0 =g= f00;
geometry.. b =e= 2*a;
b.lo = b0;

* additional equation for modern design
reliability.. beta =g= beta0;
defbeta2.. beta =e= relindex;
defrelindex.. relindex =e= (sqr(a)*b*gamma)/(2*h*t);
overturning.. relindex =e= 1;

* additional equations for mixed design
defbeta3.. beta =e= sqrt(sqr(z1) + sqr(z2));
defz1.. z1 =e= (t-mut)/sigmat;
defz2.. z2 =e= (h-muh)/sigmah;

model classic classic design / defobj,safety,minsafety,geometry                                                                   /
      modern  modern design  / defobj,       minsafety,geometry,reliability,defbeta2,defrelindex,overturning                      /
      mixed   mixed design   / defobj,safety,minsafety,geometry,reliability,         defrelindex,overturning,defbeta3,defz1,defz2 /;

* need to avoid 0*0
a.l = 0.1;
b.l = 0.1;
* need to avoid divison by zero
h.l  = muh;
t.l  = mut;
z1.l = 1;
z2.l = 1;

* Now we solve the mixed problem with the proposed relaxation method
* We will increase the safty factor (f00) until we reach the desired
* failure mode probability (beta0).
scalar  epsilon / 1e-6 /;

model subproblem / defbeta3,defrelindex,overturning,defz1,defz2 /;
option nlp=conopt,solvelink=%solveLink.callModule%,solprint=off,limrow=0,limcol=0;

set v iteration counter / v1*v10 /;
parameter report(v,*) iteration report;
beta.l=0;
loop(v$((beta0-beta.l) > epsilon),
   solve classic us nlp min cost;
   report(v,'cost') = cost.l;
   report(v,'a')    = a.l;
   report(v,'b')    = b.l;
   report(v,'f00')  = f00;
   a.fx = a.l;
   b.fx = b.l;
   solve subproblem min beta us nlp;
   report(v,'beta') = beta.l;
* Relax
   a.lo = 0;
   a.up = inf;
   b.lo = 0;
   b.up = inf;
* Update failure safety
   f00 = f00 + 0.3*max(beta0-beta.l,0);
);

display report;

* Store the solution of the proposed relaxation method
scalars a_l
        b_l
        cost_l
        f0_l
        beta_l;
a_l    = a.l;
b_l    = b.l;
cost_l = cost.l;
f0_l   = f0.l;
beta_l = beta.l;

*Now we use EMP to solve the modern design and the mixed design problem
option solvelink=%solveLink.callScript%,solprint=on;

File emp extended MP info file handle / '%emp.info%' / ;
putclose emp '* modern design' / 'bilevel a b min beta f0 t h relindex defbeta2 defrelindex overturning';
solve modern us emp min cost;

putclose emp '* mixed design' / 'bilevel a b min beta f0 t h z1 z2 relindex defrelindex overturning defbeta3 defz1 defz2';
solve mixed us emp min cost;

abort$(   (cost.l - cost_l > epsilon)
       or (f0.l   - f0_l   > epsilon)
       or (beta.l - beta_l > epsilon)) 'The solution obtained for the mixed design problem is worse than the obtained by the relaxation method'