$title 'GUROBI test suite - multi objective' (GUROBI04,SEQ=712)
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Contributor: Michael Bussieck
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Sets
     i   canning plants   / seattle, san-diego /
     j   markets          / new-york, chicago, topeka / ;

Parameters

     a(i)  capacity of plant i in cases
       /    seattle     350
            san-diego   600  /

     b(j)  demand at market j in cases
       /    new-york    325
            chicago     300
            topeka      275  / ;

Table d(i,j)  distance in thousands of miles
                  new-york       chicago      topeka
    seattle          2.5           1.7          1.8
    san-diego        2.5           1.8          1.4  ;

Scalar f  freight in dollars per case per thousand miles  /90/ ;

Parameter c(i,j)  transport cost in thousands of dollars per case ;

          c(i,j) = f * d(i,j) / 1000 ;

Variables
     x(i,j)   shipment quantities in cases
     tcost    total transportation costs in thousands of dollars
     pSeattle total production in Seattle
     z        combined objective function;

Positive Variable x ; x.up(i,j) = 1e5;

Equations
     defobj      define objective function
     defcost     define objective function
     defpSeattle define total production in Seattle
     supply(i)   observe supply limit at plant i
     demand(j)   satisfy demand at market j ;

Scalar psDirection optimization direction for production in Seattle;

defobj ..      z  =e=  tcost + psDirection*0.1*pSeattle;

defcost ..     tcost  =e=  sum((i,j), c(i,j)*x(i,j)) ;

defpSeattle .. pSeattle =e=  sum(j, x('Seattle',j)) ;

supply(i) ..   sum(j, x(i,j))  =l=  a(i) ;

demand(j) ..   sum(i, x(i,j))  =g=  b(j) ;

Model transport /all/ ;

$ echo multobj 1 > gurobi.opt
option solver=gurobi;
transport.optfile = 1;

* Maximize production in Seattle
psDirection = -1;
Solve transport using mip minimizing z ;
abort$(transport.modelstat <> 1) 'expect optimal solution';
abort$(abs(pSeattle.l-350)>1e-6) 'expect max production of 350 in Seattle', pSeattle.l;

* Minimize production in Seattle
psDirection = 1;
Solve transport using mip minimizing z ;
abort$(transport.modelstat <> 1) 'expect optimal solution';
abort$(abs(pSeattle.l-300)>1e-6) 'expect min production of 300 in Seattle', pSeattle.l;