static void Main()
{
// We are looking for the optimal mix of two materials that have different properties.
// Each material is made up of particles of different sizes, think two types of gravel
// that have different size rocks in them.
// We measure the properties of the material by measuring the volume of material that
// can pass through two different sieves:
// Sieve 1, for example, has 10mm holes in it.
// Sieve 2, for example, has 2mm holes in it.
// For each material we measure the volume (in %) of the material that can pass through each sieve.
// Material 1:
// - 70% passes through sieve 1
// - 30% also passes through sieve 2
// So that means:
// - 30% of the volume of material 1 are particles > 10 mm
// - 40% of the volume of material 1 are particles <= 10 mm and > 2 mm
// - 30% of the volume of material 1 are particles < 2 mm
// Material 2 has different properties:
// - 90% passes through sieve 1
// - 75% also passes through sieve 2
// Our goal is to find a mix of both materials that comes as close as possible to
// our target particle distribution.
// For this example, let's say we want:
// Target mix:
// - 80% passes through sieve 1
// - 50% also passes through sieve 2
// Se would like to find out how to mix both materials (what % of each in the mix) so that
// the mix has the desired properties?
// Creating the model.
// [START model]
CpModel model = new CpModel();
// [END model]
// [START variables]
// These are the variables we are actually interested in.
// The volume percentages of each material in the mix. How much (in %) of each material should we put in the mix?
IntVar volPctMaterial1 = model.NewIntVar(0, 10000, "v%1");
IntVar volPctMaterial2 = model.NewIntVar(0, 10000, "v%2");
// The rest of the variables are intermediate variables.
// First we have the percentages of the mix that pass through both sieves.
// These should be as close as possible to our target (80/50 in our example).
IntVar sieve1PassPctForMix = model.NewIntVar(0, 100000000, "Y1");
IntVar sieve2PassPctForMix = model.NewIntVar(0, 100000000, "Y2");
// These variables will contain the difference between the actual and target pass percentages for the mix.
// For example: if the target is 80% through sieve 1, but our solution finds a mix where 75% passes through sieve 1,
// then the passPctDeltaForSieve1 is 5% (80% - 75%).
IntVar passPctDeltaSieve1 = model.NewIntVar(-100000000, 100000000, "passPctDeltaSieve1");
IntVar passPctDeltaSieve2 = model.NewIntVar(-100000000, 100000000, "passPctDeltaSieve2");
// These variables are the ABS() (positive) values of our delta variables.
// We will optimize for the lowest sum of these ABS() delta variables, so that our mix comes as close as
// possible to our desired pass percentages.
IntVar passPctDeltaAbsSieve1 = model.NewIntVar(0, 100000000, "passPctDeltaAbsSieve1");
IntVar passPctDeltaAbsSieve2 = model.NewIntVar(0, 100000000, "passPctDeltaAbsSieve2");
// [END variables]
// [START constraints]
// Add the mixture pass percentage constraints (Y1 = v%1*Y11 + v%2*Y12 + ... + v%n*Y1n).
// The formula for the amount of the mix that passes through each sieve is the weighted sum of the pass
// percentages of the materials in the mix.
// For example: if the mix is 50% material 1, 50% material 2, and 70% of material 1 passes through sieve 1,
// while 90% of material 2 passes through sieve 1,
// then 80% of the mix would pass through sieve 1 (= 50%*70% + 50%*90%).
model.Add(sieve1PassPctForMix == 7000 * volPctMaterial1 + 9000 * volPctMaterial2);
model.Add(sieve2PassPctForMix == 3000 * volPctMaterial1 + 7500 * volPctMaterial2);
// The volume percentages should add up to 100% constraint.
// The mix can for example be 40% material 1, 60% material 2: 40 + 60 = 100
model.Add(volPctMaterial1 + volPctMaterial2 == 10000);
// We calculate the difference between our target pass percentages and our target
model.Add(passPctDeltaSieve1 == sieve1PassPctForMix - 80000000);
model.Add(passPctDeltaSieve2 == sieve2PassPctForMix - 50000000);
// We now calculate the absolute values of these differences, so we can minimize their sum.
model.AddAbsEquality(passPctDeltaAbsSieve1, passPctDeltaSieve1);
model.AddAbsEquality(passPctDeltaAbsSieve2, passPctDeltaSieve2);
// Uncommenting the line below makes the solution INFEASIBLE. Why?
model.Add(volPctMaterial2 == 3000); // 30% of the mix needs to be material 2
// [END constraints]
// We minimize the sum of the absolute difference between the actual and target pass percentages for the mix.
model.Minimize(new SumArray(new IntVar[] {
passPctDeltaAbsSieve1,
passPctDeltaAbsSieve2,
}));
// [START solve]
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.Solve(model);
// [END solve]
if (status == CpSolverStatus.Feasible || status == CpSolverStatus.Optimal)
{
Console.WriteLine("ObjectiveValue: " + solver.ObjectiveValue);
Console.WriteLine("volPctMaterial1 = " + solver.Value(volPctMaterial1));
Console.WriteLine("volPctMaterial2 = " + solver.Value(volPctMaterial2));
Console.WriteLine("sieve1PassPctForMix (Y1) = " + solver.Value(sieve1PassPctForMix));
Console.WriteLine("sieve2PassPctForMix (Y2) = " + solver.Value(sieve2PassPctForMix));
Console.WriteLine("passPctDeltaSieve1 = " + solver.Value(passPctDeltaSieve1));
Console.WriteLine("passPctDeltaSieve2 = " + solver.Value(passPctDeltaSieve2));
Console.WriteLine("passPctDeltaAbsSieve1 = " + solver.Value(passPctDeltaAbsSieve1));
Console.WriteLine("passPctDeltaAbsSieve2 = " + solver.Value(passPctDeltaAbsSieve2));
}
else
{
Console.WriteLine("Status was " + status + ", which is not feasible. Response stats: " + solver.ResponseStats());
}
Console.ReadLine();
}