Decision Optimization

Dear community team,

I'm trying to solve an optimization problem by using the epsilon-constraint method. As I cannot find any related document about, how we can do that by OPL (just a simple template might be found in StackOverflow) I have worked with the tiny example that was illustrated by Alex Fleischer to do that. The problem is as follows:

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");
  writeln("===============================================================");
};

 main {
  var epsilon=0.0001; 
  thisOplModel.generate();
  cplex.setObjCoef(thisOplModel.co2emission,0);
  cplex.solve();
  thisOplModel.postProcess();
  var cost=thisOplModel.cost.solutionValue;
  thisOplModel.cost.LB=cost;
  thisOplModel.cost.UB=cost+epsilon;
  cplex.setObjCoef(thisOplModel.co2emission,1);
  cplex.solve();
  thisOplModel.postProcess();
};​

After solving the problem the results are:

The minimum cost is 2500
CO2 emission is 6
We will use 5 40 seats buses 0 30 seats buses and 0 buses 50 seats
===============================================================
The minimum cost is 2500
CO2 emission is 4
We will use 0 40 seats buses 0 30 seats buses and 4 buses 50 seats
===============================================================

As far as I know, the e-constraints method can produce a non-dominance solution, in contrast to the lexicographic method.
What I am interested to know is, how the above solution is non-dominance?


Thanks in advance
Abbas

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Abbas Omidi
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#DecisionOptimization