$Title Ajax Paper Company Production Schedule  (AJAX,SEQ=60)

$Ontext

   This sample model is taken from the cybernet pds/apex sample
   library of models. A paper manufacturer can produce four different
   types of paper on three different machines. Given a fixed demand
   schedule the objective is to find a production plan that
   maximizes monthly profits.


CDC, PDS/APEX Sample Model Library, 1977. Control Data Corporation

24 May 2009: (Michael Saunders)
             For class MS&E 318 / CME 338, we have added two more models
             with the BP and BPDN objectives.

$Offtext

 Sets  m     machines at mills  / machine-1, machine-2, machine-3 /
       g     paper grades       / 20-bond-wt, 25-bond-wt, c-bond-ext, tissue-wrp /


 Table prate(g,m)  production rate (tons per hour)

            machine-1  machine-2  machine-3
 20-bond-wt    53         52         49
 25-bond-wt    51         49         44
 c-bond-ext    52         45         47
 tissue-wrp    42         44         40


 Table pcost(g,m)  production cost ($ per ton)

            machine-1  machine-2  machine-3
 20-bond-wt    76         75         73
 25-bond-wt    82         80         78
 c-bond-ext    96         95         92
 tissue-wrp    72         71         70


 Table dempr(g,*)     demand and prices

              demand  price
 20-bond-wt    30000    77
 25-bond-wt    20000    81
 c-bond-ext    12000    99
 tissue-wrp     8000   105


 Parameter  avail(m)     available machine time (hours per month)
                       / machine-1  672,
                         machine-2  600,
                         machine-3  480 /

 Parameter  lambda1      weight on sum of variables in BPDN objective
                       / 100.0 /
 Parameter  lambda2      weight on least-squares part of BPDN objective
                       / 1 /

 Variables  outp(g,m)    production  (tons per month)
            profit       profit      ($ per month)
            BPobj        Basis Pursuit objective
            BPDNobj      Basis Pursuit DeNoising objective
            rescap(m)    residuals on capacity constraints
            resdem(g)    residuals on demand constraints

 Positive Variable outp

 Equations  cap(m)       machine capacity  (hours per month)
            dem(g)       demand            (tons per month)
            pdef         profit definition ($ per month)
            capBPDN(m)   capacity with residuals
            demBPDN(g)   demand   with residuals
            BP           Basis Pursuit def (sum of positive variables)
            BPDN         BP Denoising  def (weighted sum of vars + least-squares)
            ;

 cap(m)..   sum(g, outp(g,m)/prate(g,m)) =l= avail(m);

 dem(g)..   sum(m, outp(g,m)) =e= dempr(g,"demand");

 pdef..     profit  =e= sum( g,    dempr(g,"demand")*dempr(g,"price"))
                      - sum((g,m), pcost(g,m)*outp(g,m));

 capBPDN(m)..  sum(g, outp(g,m)/prate(g,m)) + rescap(m) =l= avail(m);

 demBPDN(g)..  sum(m, outp(g,m))            + resdem(g) =e= dempr(g,"demand");

 BP..      BPobj   =e= lambda1*sum((g,m), outp(g,m)) - profit;

 BPDN..    BPDNobj =e= lambda1*sum((g,m), outp(g,m)) - profit
                     + 0.5*lambda2*sum(m, sqr(rescap(m)))
                     + 0.5*lambda2*sum(g, sqr(resdem(g)));

 Models    ajax     /cap, dem, pdef/
           ajaxBP   /cap, dem, pdef, BP  /
           ajaxBPDN /capBPDN, demBPDN, pdef, BPDN/;


*******************************************
* Run just the nonlinear model ajaxBPDN.
* We could say "option nlp    = minos;" to
* specify the solver here, but it is more
* flexible to say "gams ajaxBPDN nlp=minos"
* at runtime.
*******************************************
option sysout = on;
ajaxBPDN.optfile = 1;
Solve ajaxBPDN using nlp minimizing BPDNobj;


$Stitle BPDN report definitions

 Parameter mtrBPDN(m,*)  machine time report
           parBPDN(g,*)  production allocation report ;

 mtrBPDN(m,"avail-h")  = avail(m);
 mtrBPDN(m,"used-h")   = capBPDN.l(m);
 mtrBPDN(m,"unused-h") = avail(m) - capBPDN.l(m);
 mtrBPDN(m,"marginal") = - capBPDN.m(m);
 mtrBPDN(m,"residual") = rescap.l(m);

 parBPDN(g,"demand")   = dempr(g,"demand");
 parBPDN(g,"sold")     = demBPDN.l(g);
 parBPDN(g,"unsold")   = dempr(g,"demand") - demBPDN.l(g);
 parBPDN(g,"marginal") = demBPDN.m(g);
 parBPDN(g,"residual") = resdem.l(g);

 Display profit.l, mtrBPDN, parBPDN, outp.l, outp.m;