EMFPS: How Can We Get the Power Set of a Set by Using of Excel? (The Case Analysis of Big Data)

Following to my previous article posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html, let me tell you one of the applications of this method in Risk Management.  In this article, I am willing to solve a case in which the template is referred 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.

The Case: To Find Minimum Risk for Portfolio of Assets in Variety of the Outcomes

If we have had a range of probability for our outcomes where there is not the same probability assigned to the outcomes, how can we calculate minimum risk for the portfolio of assets? What are the probability distributions assigned to the expected portfolio return in which we will have minimum risk?

Let me start the solution of this case by using of the example stated in 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 as follows:

Referring to step (3), I define the range of probability for six outcomes equal to 10% to 21%. 

10% ≤ Pr ≤ 21%

The solution is step by step below cited:

Ø  On the spreadsheet of excel, we should type:

A1 = p

A2 = 6

A3 = 0.1, A4, = 0.11, A5 = 0.12, A6 = 0.13, A7 = 0.14, A8 = 0.15, A9 = 0.16, A10 = 0.17

A11 = 0.18, A12 = 0.19, A13 = 0.2, A14 = 0.21

Ø  Using from VB code written by Myrna Larson mentioned in article of  “EMFPS: How Can We Get the Power Set of a Set by Using of Excel?” posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html

Ø  We will have 665280 cells filled with different permutations (please check it by formula = PERMUT (12,6) on excel)

               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  Using from formula = =IF(SUM(A2:F2)=1,A2,"") on excel to find all combinations of probabilities which are equal to 1or 100%. Copy and paste this formula for all 665280 cells.

Ø  Using from formula: Rv= SUM [Rp(i) * P(i)] to find the expected value of portfolio returns. Copy and paste this formula for all 665280 cells.

Ø  Using from formula: Qr = {SUM [(Rp(i) – Rv)^2) * P(i)]}^0.5 to find Standard Deviation of the expected portfolio returns. Copy and paste this formula for all 665280 cells.

Ø   Using from formula: CV = Qr / Rv to find Coefficient of Variation. Copy and paste this formula for all 665280 cells.

Now, we will be able to find the minimum Qr and CV for the range of 10% ≤ Pr ≤ 21% as follows:

Qr(min)	0.1993867
CV(min)	0.0144341

There are 12 combinations of the probability distributions (in the range of 10% to 21%) assigned to the outcomes which show us the minimum risk as follows:

	C1	C2	C3	C4	C5	C6
Outcome (1)	0.21	0.21	0.21	0.21	0.21	0.21
Outcome (2)	0.18	0.18	0.18	0.18	0.19	0.19
Outcome (3)	0.19	0.19	0.2	0.2	0.18	0.18
Outcome (4)	0.2	0.2	0.19	0.19	0.2	0.2
Outcome (5)	0.1	0.12	0.1	0.12	0.1	0.12
Outcome (6)	0.12	0.1	0.12	0.1	0.12	0.1

C7     C8    C9         C10      C11     C12 Outcome (1)                   0.21 0.21 0.21 0.21 0.21 0.21 Outcome (2)                0.19 0.19 0.2 0.2 0.2 0.2 Outcome (3)                    0.2 0.2 0.18 0.18 0.19 0.19 Outcome (4)                0.18 0.18 0.19 0.19 0.18 0.18 Outcome (5)                  0.1 0.12 0.1 0.12 0.1 0.12 Outcome (6)   0.12 0.1 0.12 0.1 0.12 0.1

(C1 to C12 are all 12 combinations)

What do you think about using of my previous simulation model mentioned in article of “Application of Pascal’s Triangular plus Monte Carlo Analysis to Find the Least Squares Fitting for a Limited Area” posted on link: http://emfps.blogspot.com/2012/05/application-of-pascals-triangular-plus_23.html?

I have already calculated the minimum risk by using of this simulation model. The final results are as follows:

CV(min)	0.014946785
Qr(min)	0.206693106

As you can see, the error is approximately equal to 3.6%.

How can we decrease the error by using of this simulation model?

Definitely we will be able to decrease the error by increasing the calculation iteration.

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.”

 

How Can We Get the Power Set of a Set by Using of Excel?

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.”