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I am trying to figure out all possible combinations of packaging items together. Basically it's an integer programming problem where you have i number of items, j number of boxes and binary variable x[i,j].

There are many constraints, but I built a simple problem with one volume constraint first, which is the sum of volume of items assigned to box j cannot exceed the volume of box j.

I need all feasible solutions for this constraint. I used AMPL with cplex but there is no cplex option for finding all feasible points.

I wonder if it is possible to get all feasible solutions using lpSolveAPI package using R. I attach my AMPL codes below for better understanding. Thank you.

set items;
set containers;

param items_vol {i in items};
param containers_vol {j in containers};

var x{i in items, j in containers} binary;
var y{j in containers} binary;

minimize containers_volume: sum{i in items, j in containers} 
containers_vol[j] * x[i,j];

subject to volumes {j in containers}:
sum {i in items} x[i,j] * items_vol[i] <= containers_vol[j];

subject to usage {i in items}:
sum {j in containers} x[i,j] = 1;

subject to usage2 {j in containers}:
y[j] - sum{i in items} x[i,j]  <= 1
Ray
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1 Answers1

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There are two different approaches to enumerate all feasible integer solutions.

The first is to add a cut to the constraints that forbids the previously found solution. I.e.

 1. solve problem
 2. if infeasible: done
 3. record optimal solution x[i] into a[i]
 4. add cut of the form
      sum(i, (2*a[i]-1)*x[i]) <= sum(i,a[i])-1
 5. go to step 1.

I assumed here the binary variables are x[i].

The second approach is to use the Cplex solution pool. This is much faster if there are many solutions.

PS: LpSolveApi documents things like get.solutioncount and select.solution. I don't think that actually works.

Erwin Kalvelagen
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