# Constraint Programming OR-Tools finding Matrix Combinations

As a similar post of mine Find all Combinations of a Matrix I am trying to find matrix combinations with entries $$>0$$ meaning for a matrix

$$\begin{bmatrix} 0 & 1 & 3 \\ 5 & 2 & 1 \\ 0 & 0 & 10 \end{bmatrix}$$

The following combinations: $$(5,1,3); (5,2,1); (5,1,10);...$$

I have decided to use constraint-programming with OR-Tools for this case to find all the solutions.

Furthermore, I want to use added constraints so that at most X rows should get used for a combination, e.g., max 2 rows $$(5,1,3)$$ but not (5,1,10).

At the moment I have the following model:

1. $$x_{i,j}$$ is a binary decision variable with $$i$$ being a row and $$j$$ being a column
2. $$y_i$$ a binary variable stating whether row $$i$$ is getting used for the combinations
3. I need exactly one row per column:

$$\sum_ix_{i,j} = 1 \ \forall j \in J$$

1. Choose only a matrix entity if it is $$>0$$ with $$p_{i,j}$$ stating the matrix elements: $$x_{i,j} \leq p_{i,j} \ \forall i,j$$

2. Help constraint to see whether a row is used for a combination by setting $$y_i$$ to $$0$$ or $$1$$ $$\sum_jx_{i,j} \leq y_i \cdot |J| \ \forall i \in I$$

3. Limit the use of rows for the combinations by a number of $$q$$. $$\sum_iy_i = q$$

1) (Solved) As of now, I am getting weird results when I define the variable $$y_i$$. By that, I mean that I am getting the same result multiple times.

2) Also, how to find which row is used in the combinations? As of now, the constraint 5 is inadequate, since y can still get the value 1 although the whole row is not chosen.

y = {}
for i in range(matrixRows): # rows
y[i] = model.NewBoolVar('vendor_%d' % (i))


If I comment it out the results are fine.

Here you can find the complete code with constraints 5 and 6 commented out (definition of $$y_i$$ included):

    from __future__ import print_function
from ortools.sat.python import cp_model

class SolutionPrinter(cp_model.CpSolverSolutionCallback):

# def __init__(self,x,y,matrixRows,matrixColumns,matrix):
def __init__(self,x,matrixRows,matrixColumns,matrix):
cp_model.CpSolverSolutionCallback.__init__(self)
self._x = x
# self._y = y
self._matrixRows = matrixRows
self._matrixColumns = matrixColumns
self._matrix = matrix
self._solution_count = 0

def OnSolutionCallback(self):
self._solution_count += 1
for j in range(self._matrixColumns):
for i in range(self._matrixRows):
if self.Value(self._x[(i,j)]):
print('x: %d for x(%d,%d) and matrix value:%d' %(self.Value(self._x[(i,j)]),i,j,matrix[i][j]))
# for v in self._x:
# print('%s = %i' % (v, self.Value(v)), end=' ')
print()

def SolutionCount(self):
return self._solution_count

def main(matrix):
model = cp_model.CpModel()
matrixRows = len(matrix)
matrixColumns = len(matrix)
# define variables
x = {}
for i in range(matrixRows): # rows
for j in range(matrixColumns):
x[(i,j)] = model.NewBoolVar('company_%d,%d' % (i,j))

y = {}
for i in range(matrixRows): # rows
y[i] = model.NewBoolVar('vendor_%d' % (i))

# choose exactly one vendor per attribute
for j in range(matrixColumns):
model.Add(sum(x[(i,j)] for i in range(matrixRows)) == 1)

# to choose value x(i,j), the parameter of the matrix has to be >0
for i in range(matrixRows):
for j in range(matrixColumns):

# # help constraint to count whether a vendor is in the mix
# for i in range(matrixRows):
#         model.Add(sum(x[i, j] for j in range(matrixRows)) <= y[i] *matrixColumns)
#
# # set the max vendors
# model.Add(sum(y[i] for i in range(matrixRows)) <= 2)

#call the solver and display the results
solver = cp_model.CpSolver()
# solution_printer = SolutionPrinter(x,y,matrixRows,matrixColumns,matrix)
# the following is an object of the class SolutionPrinter
solution_printer = SolutionPrinter(x,matrixRows,matrixColumns,matrix)
status = solver.SearchForAllSolutions(model, solution_printer)
print()
print('Solutions found: %i' % solution_printer.SolutionCount())

if __name__ == '__main__':
matrix = [[0,1,3],[5,0,1],[0,0,10]]
main(matrix)


You get the same result $$2^{\operatorname{len}(y)}$$ times because $$y_i$$ is not constrained.

print("y:", [self.Value(y) for y in self._y.values()])

x: 1 for x(1,0) and matrix value:5
x: 1 for x(0,1) and matrix value:1
x: 1 for x(1,2) and matrix value:1
y: [0, 0, 0]

x: 1 for x(1,0) and matrix value:5
x: 1 for x(0,1) and matrix value:1
x: 1 for x(1,2) and matrix value:1
y: [1, 0, 0]

...
y: [1, 0, 1]

...
y: [0, 0, 1]

...
y: [0, 1, 1]

...
y: [0, 1, 0]

...
y: [1, 1, 0]

...
y: [1, 1, 1]


PS: I would check matrix[i][j] != 0 before creating the boolean.

Edit:

model.Add(sum(x[i, j] for j in range(matrixRows)) > 0).OnlyEnforceIf(y[i])
model.Add(sum(x[i, j] for j in range(matrixRows)) == 0).OnlyEnforceIf(y[i].Not())

• This is great! Do you mean I should implement the following matrix[i][j] != 0 as a constraint? It is just a parameter and not a variable so I do not understand how it would help me. Furthermore, what does $\operatorname{len}(y)$ mean? – Georgios Dec 3 '19 at 20:29
• What I mean is that you don't have to create the boolean if it is always going to be $0$, the code will be uglier though. And $\operatorname{len}(y)$ = number of $y_i$. – Stradivari Dec 3 '19 at 21:42
• Thanks for the $\operatorname{len}(y)$ explanation. Now I get that you used the python len method. I use the $y$ boolean variable since I do not always want to choose the same amount of rows for the combinations. Or did you mean with boolean something else? – Georgios Dec 4 '19 at 15:21
• Or how should I know when a row gets chosen? The current implementation of the following constraint is inadequate $\sum_ix_{i,j}\leq y_i\cdot |J|\ \forall i \in I$ – Georgios Dec 4 '19 at 15:32