The purpose of this article is to
obtain all subsets of a set by using of Classical or Fuzzy set theory and a VB
code written by Myrna Larson. The power set has many applications in the fields
of engineering such as control valves in piping and process engineering,
control keys in electrical engineering, field data analysis in geophysics (does
not different among the arrangements of Wenner, Schlumberger or Polygon - Net in resistivity
test or Geoelectric investigation) and data analysis of NDT tests.
And so we can find the
applications of power set in financial management, strategic management and
risk management. For instance, I would like to refer you to my article of “EMFPS: Efficient Portfolio of Assets (The Optimization for Risk,
Return and Probability)” posted on link http://emfps.blogspot.com/2011/10/emfps-efficient-portfolio-of-assets.html
where I stated: “....To change Rp(i) and probabilities simultaneously into above limited range, we should
obtain all Permutations without Repetition by using of VB codes in excel...”
The concept of a set is the
collection of the numbers or elements where we consider capital letters for
them such as A, B, C,...
For example, set A has 4 elements as
follows:
A = {1,2,3,4}
If all elements of set B are
included in set A, we can say that set B is a subset of set A.
If B = {2,4}, set B is a subset of
set A.
The power set of set A is a set such
as C which is contained all subsets of set A even empty set and set A as
follows:
C = {{}, {1}, {2}, {3}, {4}, {1,2},
{1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4},
{1,2,3,4}}
As we see, the total elements of set
C is equal to (2^n) where “n” is all elements of set A.
How can we get the power set of a
set by using of excel?
Methodology
Assume that set A is a subset of
Universe set U. In the reference with Characteristic function (membership
function) in set theory which is the base of Fuzzy set theory, if each element
of set A is a member of set U, we will assign 1 as membership function
otherwise 0 (zero) as follows:
Where: x is member of set U.
Example:
If we have: A = {1,,3,7,9} and U = {1,2,3,4,5,6,7,8,9}
By using of membership function, we
can write set A as follows:
A = {(1,1), (2,0), (3,1), (4,0),
(5,0), (6,0), (7,1),(8,0), (9,1)}
According to above mentioned, there
is very important rule that we should consider when we want to use from excel.
This note is below cited:
Rule: If we
have not set U, we have to consider 1 or 0 for each element of set A.
Let me start step by step this
method by using of excel:
Ø Type “c” into cell A1 in your spreadsheet
Ø Type into cell A2 the number of elements in your set
Ø Regarding to above rule, for
each element of your set, you should type 0 and 1 respectively into cell A3,
A4, A5, A6 and so on. For example, if your set has 5 elements, you should fill
cells as follows:
A1 = c
A2 = 5
A3 = 0
A4 = 1
A5 = 0
A6 = 1
A7 = 0
A8 = 1
A9 = 0
A10 = 1
A11 = 0
A12 = 1
Ø Now, we can run macro from VB code written by Myrna Larson which is
as follows:
Option Explicit
'Written by Myrna
Larson - Microsoft Excel MVP
Dim vAllItems As Variant
Dim Buffer() As String
Dim BufferPtr As Long
Dim Results As Worksheet
Sub ListPermutations()
Dim Rng As Range
Dim PopSize As Integer
Dim SetSize As Integer
Dim Which As String
Dim N As Double
Const BufferSize As Long = 4096
Set Rng = Range(Range("A1"),
Cells(Rows.Count, "A").End(xlUp))
PopSize = Rng.Cells.Count - 1
If PopSize < 1 Then Goto DataError
SetSize = Rng.Cells(2).Value
If SetSize > PopSize Then Goto DataError
Which = UCaseS(Rng.Cells(1).Value)
Select Case Which
Case "C"
N =
Application.WorksheetFunction.Combin(PopSize, SetSize)
Case "P"
N = Application.WorksheetFunction.Permut(PopSize,
SetSize)
Case Else
Goto DataError
End Select
If N > Cells.CountLarge Then Goto DataError
Application.ScreenUpdating = False
Set Results = Worksheets.Add
vAllItems = Rng.Offset(2, 0).Resize(PopSize).Value
Redim Buffer(1 To BufferSize) As String
BufferPtr = 0
If Which = "C" Then
AddCombination PopSize, SetSize
Else
AddPermutation PopSize, SetSize
End If
vAllItems = 0
Application.ScreenUpdating = True
Exit Sub
DataError:
If N = 0 Then
Which = "Enter your data in a
vertical range of at least 4 cells. " _
& String$(2, 10) _
& "Top cell must contain the letter
C or P, 2nd cell is the number " _
& "of items in a subset, the
cells below are the values from which " _
& "the subset is to be
chosen."
Else
Which = "This requires "
& Format$(N, "#,##0") & _
" cells, more than are available
on the worksheet!"
End If
MsgBox Which, vbOKOnly, "DATA ERROR"
Exit Sub
End Sub
Private Sub AddPermutation(Optional PopSize As Integer = 0, _
Optional SetSize As Integer = 0, _
Optional NextMember As Integer = 0)
Static iPopSize As Integer
Static iSetSize As Integer
Static SetMembers() As Integer
Static Used() As Integer
Dim i As Integer
If PopSize <> 0 Then
iPopSize = PopSize
iSetSize = SetSize
Redim SetMembers(1 To iSetSize) As Integer
Redim Used(1 To iPopSize) As Integer
NextMember = 1
End If
For i = 1 To iPopSize
If Used(i) = 0 Then
SetMembers(NextMember) = i
If NextMember <> iSetSize Then
Used(i) = True
AddPermutation , , NextMember +
1
Used(i) = False
Else
SavePermutation SetMembers()
End If
End If
Next i
If NextMember = 1 Then
SavePermutation SetMembers(), True
Erase SetMembers
Erase Used
End If
End Sub 'AddPermutation
Private Sub AddCombination(Optional PopSize As Integer = 0, _
Optional SetSize As Integer = 0, _
Optional NextMember As Integer = 0, _
Optional NextItem As Integer = 0)
Static iPopSize As Integer
Static iSetSize As Integer
Static SetMembers() As Integer
Dim i As Integer
If PopSize <> 0 Then
iPopSize = PopSize
iSetSize = SetSize
Redim SetMembers(1 To iSetSize) As Integer
NextMember = 1
NextItem = 1
End If
For i = NextItem To iPopSize
SetMembers(NextMember) = i
If NextMember <> iSetSize Then
AddCombination , , NextMember + 1,
i + 1
Else
SavePermutation SetMembers()
End If
Next i
If NextMember = 1 Then
SavePermutation SetMembers(), True
Erase SetMembers
End If
End Sub 'AddCombination
Private Sub SavePermutation(ItemsChosen() As Integer, _
Optional FlushBuffer As Boolean = False)
Dim i As Integer, sValue As String
Static RowNum As Long, ColNum As Long
If RowNum = 0 Then RowNum = 1
If ColNum = 0 Then ColNum = 1
If FlushBuffer = True Or BufferPtr = UBound(Buffer()) Then
If BufferPtr > 0 Then
If (RowNum + BufferPtr - 1) >
Rows.Count Then
RowNum = 0
ColNum = ColNum + 1
If ColNum > 256 Then Exit Sub
End If
Results.Cells(RowNum,
ColNum).Resize(BufferPtr, 1).Value _
= Application.WorksheetFunction.Transpose(Buffer())
RowNum = RowNum + BufferPtr
End If
BufferPtr = 0
If FlushBuffer = True Then
Erase Buffer
RowNum = 1
ColNum = 0
Exit Sub
Else
Redim Buffer(1 To UBound(Buffer))
End If
End If
'construct the next set
For i = 1 To UBound(ItemsChosen)
sValue = sValue & ", "
& vAllItems(ItemsChosen(i), 1)
Next i
'and save it in the buffer
BufferPtr = BufferPtr + 1
Buffer(BufferPtr) = Mid$(sValue, 3)
End Sub 'SavePermutation
Ø Since we have some repeated cells, we should use Data Tab – Data
Tools – Remove Duplicates
Ø Due to each subsets is into one cell, to split the elements of
subsets into other cells, we should utilize Data Tab – Data Tools – Text to
Columns
Ø If the elements of your set are the numbers, you can multiply all
subsets of 0 and 1 to numbers to return codes to numbers.
Ø If the elements of your set are letters (no numbers), you can use
from above step plus Vlookup formula in excel.
Here, I have brought an example as
follows:
If we have set A = {a,b,c,d,e,f,},
the power set will generated as follows:
c
|
6
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
As we can see, the power set of set
A has 64 sets (2^6).
Each row (subset) can be considered
as a scenario to analyze data.
If you want to analyze your data by
using What – If – Analysis Tool in excel, Scenario analysis is limited to 32
rows. Therefore, the best tool is Solver – add.
To be continued….
Note:
“All spreadsheets and calculation notes are available. The people, who are
interested in having my spreadsheets of this method as a template for further
practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call
me on my cellphone: +989109250225. Please be informed these spreadsheets are
not free of charge.”
