Data Analysis Mathematics Linear Algebra Statistics
New Version 7.0!  QuickStart Samples

# Mixed Integer Programming QuickStart Sample (Visual Basic)

Illustrates how to solve mixed integer programming by solving Sudoku puzzles using the linear programming solver in Visual Basic.

```Option Infer On

' The linear programming classes reside in their own namespace.
Imports Extreme.Mathematics.Optimization
' Vectors and matrices are in the Extreme.Mathematics namespace
Imports Extreme.Mathematics
Imports Extreme.Mathematics.LinearAlgebra

Namespace Extreme.Numerics.QuickStart.VB

Module MixedIntegerProgramming

' Illustrates solving mixed integer programming problems
' using the classes in the Extreme.Mathematics.Optimization
' namespace of the Extreme Optimization Numerical Libraries for .NET.
Sub Main()
' In this QuickStart sample, we'll use the Mixed Integer
' programming capabilities to solve Sudoku puzzles.
' The rules of Sudoku will be4 expressed in terms of
' linear constraints on binary variables.

' First, create an empty linear program.
Dim lp As New LinearProgram()

' Create an array of binary variables that indicate whether
' the cell at a specific row and column contain a specific digit.
' - The first index corresponds to the row.
' - The second index corresponds to the column.
' - The third index corresponds to the digit.
Dim variables(8, 8, 8) As LinearProgramVariable

' Create a binary variable for each digit in each row and column.
' The AddBinaryVariable method creates a variable that can have values of 0 or 1.
For row As Integer = 0 To 8
For column As Integer = 0 To 8
For digit As Integer = 0 To 8
Dim name As String = String.Format("x{0}{1}{2}", row, column, digit)
variables(row, column, digit) = lp.AddBinaryVariable(name, 0.0)
Next
Next
Next

' Now add constraints that represent the rules of Sudoku.

' There are 4 rules in Sudoku. They are all of the kind
' where only one of a certain set of combinations
' of (row, column, digit) can occur at the same time.
' We can express this by stating that the sum of the corresponding
' binary variables must be one.

' AddConstraints is a helper function defined below.
' For each combination of the first two arguments,
' it builds a constraint by iterating over the third argument.

' Rule 1: each posiion contains exactly one digit
AddConstraints(lp, Function(row, column, digit) variables(row, column, digit))
' Rule 2: each digit appears once in each row
AddConstraints(lp, Function(row, digit, column) variables(row, column, digit))
' Rule 3: each digit appears once in each column
AddConstraints(lp, Function(column, digit, row) variables(row, column, digit))
' Rule 4: each digit appears exactly once in each block
variables(3 * (block Mod 3) + (index Mod 3), 3 * (block \ 3) + (index \ 3), digit))

' We represent the board with a 9x9 sparse matrix.
' The nonzero entries correspond to the numbers

' Let's see if we can solve "the world's hardest Sudoku" puzzle:
' http:'www.mirror.co.uk/fun-games/sudoku/2010/08/19/world-s-hardest-sudoku-can-you-solve-dr-arto-inkala-s-puzzle-115875-22496946/
Dim rows() = {0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8}
Dim columns() = {2, 3, 0, 7, 1, 4, 6, 0, 5, 6, 1, 4, 8, 2, 3, 7, 1, 3, 8, 2, 7, 5, 6}
Dim digits() As Double = {5, 3, 8, 2, 7, 1, 5, 4, 5, 3, 1, 7, 6, 3, 2, 8, 6, 5, 9, 4, 3, 9, 7}
Dim board = Matrix.CreateSparse(9, 9, rows, columns, digits)

' Now fix the variables for the for the digits that are already on the board.
' We do this by setting the lower bound equal to the upper bound:
For Each triplet In board.NonzeroComponents
variables(triplet.Row, triplet.Column, CInt(triplet.Value) - 1).LowerBound = 1.0
Next

' Solve the linear program.
Dim solution = lp.Solve()

' Scan the variables and print the digit if the value is 1.
For row As Integer = 0 To 8
For column As Integer = 0 To 8
Dim theDigit As Integer = 0
For digit As Integer = 0 To 8
If (variables(row, column, digit).Value = 1.0) Then
theDigit = digit + 1
Exit For
End If
Next
Console.Write(IIf(theDigit > 0, theDigit.ToString(), "."))
Next
Console.WriteLine()
Next

Console.Write("Press Enter key to exit...")

End Sub

' Helper function that creates a constraint:
Dim coefficients() As Double = {1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0}
Sub AddConstraints(ByVal lp As LinearProgram, ByVal variable As Func(Of Integer, Integer, Integer, LinearProgramVariable))
For i As Integer = 0 To 8
For j As Integer = 0 To 8
Dim variables(8) As LinearProgramVariable
For k As Integer = 0 To 8
variables(k) = variable(i, j, k)
Next