IBM Optimization Studio OPL, why constraint is not

Question

Here is a description of the optimization problem I need to solve, but with a small twist. I need to add two constraints:

• The first constraint: From each group we want to choose only one product, which means that we can't allow two products from the same group to be in the same basket(i.e. Product11 and Product12 should never be in the same basket )
• The second constraint: In the user basket we only want products from the categories that the user is interested in. i.e if user is interested in the category 'Protein' he should never find in his basket a product from the category 'Carbs' nor 'Fat'.

Accordingly I have changed the OPL code products.mod:

{string} categories=...;

{string} groups[categories]=...;

{string} allGroups=union (c in categories) groups[c];

{string} products[allGroups]=...;
{string} allProducts=union (g in allGroups) products[g];

float prices[allProducts]=...;

int Uc[categories]=...;
float Ug[allGroups]=...;

float budget=...;



dvar boolean z[allProducts]; // product out or in ?


dexpr int xg[g in allGroups]=(sum(p in products[g]) z[p]);
dexpr int xc[c in categories]=(1<=sum(g in groups[c]) xg[g]);




maximize
sum(c in categories) Uc[c]*xc[c]+
sum(c in categories) sum(g in groups[c]) Uc[c]*Ug[g]*xg[g];
subject to
{
ctBudget:// first constraint
    sum(p in allProducts) z[p]*prices[p]<=budget;
ctGroups: // second constraint 
    forall( g in allGroups )
        xg[g]==1;
ctCategories: // third constraint 
    forall( c in categories )
        Uc[c]==xc[c];
}
{string} solution={p | p in allProducts : z[p]==1};
 execute
 {
   writeln("xg=",xc);
   writeln("xg=",xg);
   writeln("Solution=",solution);

  }

Here the code for products.data

categories={"Carbs","Protein","Fat"};
groups=[{"Meat","Milk"},{"Pasta","Bread"},{"Oil","Butter"}];
products=[
{"Product11","Product12"},{"Product21","Product22","Product23"},
{"Product31","Product32"},{"Product41","Product42"},
{"Product51"},{"Product61","Product62"}];

prices=[1,1,3,3,2,1,2,1,3,1,2,1];


Uc=[1,0,0];
Ug=[0.8,0.2,0.1,1,0.01,0.6];
budget=2;

The results given by IBM Studio are the following: {Product12,Product31}; while the result I want is either {Product11} or {Product12}.

I have also noticed this in the conflicts tab:

And this in the relaxation tab:

So I have five questions:

1. I don't see any conflicts between the constraints, because if we choose the product "Product12" (or Product11") we respect all the constraint and the budget would be <= 2 because price["Product12"]==1.
2. I don't understand why does the optimizer chose to not respect the last constraint, and to maximize the objective function instead.
3. If the optimizer did not use any relaxation, would this lead to an infeasible model (no solution to the problem)? I don't understand why? For me choosing only "Product12" (or "Product11") is a perfect solution with no need to any relaxation.

4. How to oblige the optimizer to not relax the last constraint? (Note that changing the settings file, products.ops to relax only labeled constraints as in the documentation did not help, as I want to relax only one constraint )

5. In the documentation about relaxing infeasible models I found this:

Be aware, however, that infeasibility may be the consequence of an error in the modeling of another constraint.

Is this my case?

Thank you in advance for the help

Answer 1

• on no.1+2 = you have some things which are not defined in the model... can you say if AllGroups and groups exist separately or the 2 are the same then what is the data for those? Also you use "products" and "AllProducts", the same Q as for "groups". Would you paste here a full .mod and .dat which you have run and produced the relaxed result you showed...? Once I can at least reproduce the problem you show, I can start looking at the "why's" :-)
• on no.3 = yes, it supposed to
• on no.4 = the way you could arrive to a non-relaxed model is that you remove the naming for the constraints. I.e. EVERY constraint which is named is considered to be relaxed if w/o relaxation there could be no solution. Every non-named constraint is "hard", i.e. it HAS to be respected, can not be relaxed. Simply remove or comment out these lines: ctBudget:// first constraint, ctGroups: // second constraint, ctCategories: // third constraintif you want all constraint to be respected as they are with the given data...