How to code Epsilon constraint method in CPLEX

Question

How to solve a bi-/multi-objective problem by the Epsilon constraint method in CPLEX using OPL?

Thanks!

Answer 1

In OPL CPLEX you can directly use multiobjective

int nbKids=200;
float costBus40=500;
float costBus30=400;
float costBus50=625;
     
dvar int+ nbBus40;
dvar int+ nbBus30;
dvar int+ nbBus50;

dvar float cost;
dvar float co2emission;
     
minimize
  staticLex(cost,co2emission);
     
subject to
{
 cost==costBus40*nbBus40  +nbBus30*costBus30+nbBus50*costBus50;
 co2emission==nbBus50+nbBus40*1.1+nbBus30*1.2;

  40*nbBus40+nbBus30*30+nbBus50*50>=nbKids;
}

execute DISPLAY_After_SOLVE
{
  writeln("The minimum cost is ",cost);
  writeln("CO2 emission is ",co2emission);
  writeln("We will use ",nbBus40," 40 seats buses ",nbBus30,
  " 30 seats buses and ", nbBus50," buses 50 seats");
}

but if you prefer to use flow control you can write

int nbKids=200;
float costBus40=500;
float costBus30=400;
float costBus50=625;

dvar int+ nbBus40;
dvar int+ nbBus30;
dvar int+ nbBus50;

dvar float cost;
dvar float co2emission;

minimize
  cost+co2emission;

subject to
{
 cost==costBus40*nbBus40  +nbBus30*costBus30+nbBus50*costBus50;
 co2emission==nbBus50+nbBus40*1.1+nbBus30*1.2;

  40*nbBus40+nbBus30*30+nbBus50*50>=nbKids;
}

execute DISPLAY_After_SOLVE
{
  writeln("The minimum cost is ",cost);
  writeln("CO2 emission is ",co2emission);
  writeln("We will use ",nbBus40," 40 seats buses ",nbBus30,
  " 30 seats buses and ", nbBus50," buses 50 seats");
}

main
 {
   var epsilon=0.01;
   thisOplModel.generate();
   // remove co2emission from the objective by setting coef 0
   cplex.setObjCoef(thisOplModel.co2emission,0);
   cplex.solve();
   thisOplModel.postProcess();
   var cost=thisOplModel.cost.solutionValue;
  
   // add a new constraint that cost should not move up too much
   thisOplModel.cost.UB=cost*(1+epsilon);

   // keep only CO2 emission in the objective
   cplex.setObjCoef(thisOplModel.co2emission,1);
   cplex.setObjCoef(thisOplModel.cost,0);
   cplex.solve();
   thisOplModel.postProcess();
    
 }

or if you want to update bounds on a constraint

int nbKids=200;
float costBus40=500;
float costBus30=400;
float costBus50=625;

dvar int+ nbBus40;
dvar int+ nbBus30;
dvar int+ nbBus50;

dvar float cost;
dvar float co2emission;

minimize
  cost+co2emission;

subject to
{
 cost==costBus40*nbBus40  +nbBus30*costBus30+nbBus50*costBus50;
 co2emission==nbBus50+nbBus40*1.1+nbBus30*1.2;

  40*nbBus40+nbBus30*30+nbBus50*50>=nbKids;
  
  ctMaxCost:cost<=maxint;
}

execute DISPLAY_After_SOLVE
{
  writeln("The minimum cost is ",cost);
  writeln("CO2 emission is ",co2emission);
  writeln("We will use ",nbBus40," 40 seats buses ",nbBus30,
  " 30 seats buses and ", nbBus50," buses 50 seats");
}

main
 {
   var epsilon=0.01;
   thisOplModel.generate();
   // remove co2emission from the objective by setting coef 0
   cplex.setObjCoef(thisOplModel.co2emission,0);
   cplex.solve();
   thisOplModel.postProcess();
   var cost=thisOplModel.cost.solutionValue;
  
   // add a new constraint that cost should not move up too much
   thisOplModel.ctMaxCost.UB=cost*(1+epsilon);

   // keep only CO2 emission in the objective
   cplex.setObjCoef(thisOplModel.co2emission,1);
   cplex.setObjCoef(thisOplModel.cost,0);
   cplex.solve();
   thisOplModel.postProcess();
    
 }