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Accounts Receivable Challenge

Our August 2002 Challenge is inspired by a recent posting on the MS boards. Thanks to Harlan, Tim, and Tushar for a lively discussion around this problem. Here is the specific problem. An accounts receivable department receives a check from a customer for $4,556.92. Upon looking in the accounting system, there are 54 unpaid invoices, ranging from $77.74 to $5,465.45. The payment must be for some exact combination of entire invoices, but we don't know which invoices are being paid. The complete list of invoices for this specific problem is shown below.

On first examination, there could be 2^54 possible combinations - about 18 quadrillion - which would tend to take forever via a brute force method.

The August challenge is to document a method for determining which invoices the customer is paying. If there is more than one combination, note that. We are looking for the best general purpose algorithm that could be used every day by accounts receivable departments across the world for similar problems.

Whichever method is judged to work the best for the average accounts receivable problem such as this will win our soon-to-be-released Message Board CD.

Deadline for entries is August 31, 2002.

Here is the list of 54 open invoice amounts, which you can copy and paste to Excel: 

895.39
83.06
280.71
1021.7
219.1
1587.52
507.08
628.89
222.52
192.65
194.58
764.18
680.23
244.22
89.4
862.12
1842.59
329.97
444.98
630.92
440.93
324.84
978.53
144.77
230.72
456.68
126.69
2487.85
515.11
911.45
983.98
329.17
673.47
409.17
228.31
796.76
116.14
858.97
718.32
346.35
542.12
589.18
789.77
185.58
538.64
441.43
925.39
698.27
5465.45
160.62
722.73
691.83
77.74
365.43

If your local version of Excel requires the decimal place to be a comma, then copy and paste this set instead: 

895,39
83,6
280,71
1021,70
219,10
1587,52
507,8
628,89
222,52
192,65
194,58
764,18
680,23
244,22
89,40
862,12
1842,59
329,97
444,98
630,92
440,93
324,84
978,53
144,77
230,72
456,68
126,69
2487,85
515,11
911,45
983,98
329,17
673,47
409,17
228,31
796,76
116,14
858,97
718,32
346,35
542,12
589,18
789,77
185,58
538,64
441,43
925,39
698,27
5465,45
160,62
722,73
691,83
77,74
365,43

Results

First off, I never realized that I had posed a question which had 3,514 possible solutions.

We had a lively discussion with many possible directions. Ioannis popped in during the middle of the month and kept reporting success, but kept his winning macro a secret until the last minute. Simply for keeping us all baited so well, he caused a lot of anticipation for his macro, shown below.

IOANNIS's winning macro: 


Dim INV() As Long
Dim CHECK As Long
Dim MAX_CHECK_INVS_No As Integer
Dim Sol As Long
Dim RESUME_No() As Integer
Dim RES_No As Integer
Dim RESUME_CALC As Integer
Dim MAX_RESUME_No As Integer
Dim AA As Long
Dim MAX_INVS As Integer
Dim MAX_DEPTH As Integer

Sub Challenge()

' SORTING
Columns("B:B").Select
    Selection.Sort Key1:=Range("B1"), Order1:=xlAscending, Header:=xlNo, _
    OrderCustom:=1, MatchCase:=False, Orientation:=xlTopToBottom

' CLEAR SOLUTION AREA

Columns("H:H").Select
Selection.ClearContents

Columns("M:P").Select
Selection.ClearContents

Cells(11, 4).Select
Selection.Activate

Cells(18, 6) = 0
Cells(7, 6) = 0
CHECK = Cells(2, 6) * 100

TOT_INV = Cells(1, 6)
ReDim INV(TOT_INV + 1)

MaxSum = 0

For i = 1 To TOT_INV
   INV(i) = Cells(i, 2) * 100
   MaxSum = MaxSum + INV(i)
Next

SUM_INV = 0

MAX_INVS = 0

For i = 1 To TOT_INV
   SUM_INV = SUM_INV + INV(i)
   If SUM_INV = CHECK Then MAX_INVS = i: Exit For
   If SUM_INV > CHECK Then MAX_INVS = i - 1: Exit For
Next

Cells(3, 6) = MAX_INVS

MAX_CHECK_INVS_No = 1
i = 1

For i = TOT_INV To 2 Step -1
    SUM_INV = INV(i)
    If SUM_INV = CHECK Then GoTo Exit_for
    For k = 1 To i
        SUM_INV = SUM_INV + INV(k)
        If SUM_INV <= CHECK Then
           GoTo Exit_for
        End If
    Next
Next

Exit_for:
MAX_CHECK_INVS_No = i

INV(MAX_CHECK_INVS_No + 1) = MaxSum

Cells(4, 6) = MAX_CHECK_INVS_No

Sol = 0
AA = 0

Cells(14, 6) = Time

'Application.ScreenUpdating = False

    Find_Sol 0, "", 0
    
    Cells(10, 5) = Str(MAX_CHECK_INVS_No)
    Cells(15, 6) = Time
    
Application.ScreenUpdating = True

End Sub
    
Sub Find_Sol(No_01 As Integer, NN_01 As String, SINVS_01 As Long)
 
For No_02% = No_01 + 1 To MAX_CHECK_INVS_No
    
    NN_02$ = NN_01 + Str(No_02%)
    SINVS_02& = SINVS_01 + INV(No_02%)
        
    If SINVS_02& > CHECK Then Exit For
    If SINVS_02& = CHECK Then
       Sol = Sol + 1
       If Sol > 65536 Then Sol = 1: COPY_SOLUTIONS
       
       Cells(Sol, 8) = NN_02$
       'Cells(10, 5) = NN_02$
       Cells(15, 6) = Time
                     
    End If
    
    If (SINVS_02& + INV(No_02% + 1)) > CHECK Then
          
       If (INV(No_02%) = INV(No_02% + 1)) Then GoTo END_LOOP
       
       MAX_No_01% = MAX_CHECK_INVS_No + 1
       No_02% = No_02% + 1
       
START_LOOP:
      
       CH_No% = MAX_No_01% - No_02%
       
       If CH_No% > 1 Then
          CH_No_m% = CH_No% / 2 + No_02%
      
          If (SINVS_01 + INV(CH_No_m%)) > CHECK Then
             MAX_No_01% = CH_No_m%
             GoTo START_LOOP
          End If
          If (SINVS_01 + INV(CH_No_m%)) = CHECK Then
             Sol = Sol + 1
             If Sol > 65536 Then Sol = 1: COPY_SOLUTIONS
             Cells(Sol, 8) = NN_01 + Str(CH_No_m%)
             Exit For
          End If
          If (SINVS_01 + INV(CH_No_m%)) < CHECK Then
             No_02% = CH_No_m%
             GoTo START_LOOP
          End If
        Else
          If CH_No% = 1 Then
             No_02% = MAX_No_01% - 1
             If (SINVS_01 + INV(No_02%)) = CHECK Then
                Sol = Sol + 1
                If Sol > 65536 Then Sol = 1: COPY_SOLUTIONS
                   Cells(Sol, 8) = NN_01 + Str(No_02%)
                  Exit For
                End If
             End If
             Exit For
          End If
       End If
END_LOOP:
    
    Find_Sol No_02%, NN_02$, SINVS_02&
   
Next_No_02:
Next No_02

End Sub


Sub RESUME_LAST_SOLUTION()

TOT_INV = Cells(1, 6)

ReDim RESUME_No(TOT_INV)

LAST_SOLUTION_No = Cells(5, 6)
If LAST_SOLUTION_No = 0 Then Exit Sub
LAST_SOLUTION = Cells(LAST_SOLUTION_No, 8)

Range("D:D").Select
Selection.ClearContents

If LAST_SOLUTION <> "" Then

LAST_SOLUTION = Trim(LAST_SOLUTION) + " "

SOL_LEN = Len(LAST_SOLUTION)
START_LEN = 1

AA = 1
For i = START_LEN To SOL_LEN

No = InStr(i, LAST_SOLUTION, " ")

Cells(AA, 4) = Mid(LAST_SOLUTION, i, No - i)
RESUME_No(AA) = Cells(AA, 4)
i = No
AA = AA + 1

Next

End If


End Sub
Sub COPY_SOLUTIONS()

AC_NAME = ActiveSheet.Name

N = 0
SOL_NAME = Cells(2, 6)

Do
  N = N + 1
  SOL_NAME_01 = Trim(Str(SOL_NAME)) + "_" + Trim(Str(N))
Loop Until Exist_SHEET(SOL_NAME_01) = 0

Cells(7, 6) = N

Create_SOLUTIONS_PAGE (SOL_NAME_01)

Sheets(AC_NAME).Select
Range("H1:I65536").Select
Selection.Copy

Sheets(SOL_NAME_01).Select
Range("B1").Select
ActiveSheet.Paste
Columns("B:C").AutoFit
Range("B1").Select

Sheets(AC_NAME).Select
Range("H1:I65536").Select
Selection.ClearContents
Range("E10").Activate

End Sub
Function Exist_SHEET(SH_NAME)

Exist_SHEET = 0

For Each SH In Sheets
    If SH.Name = SH_NAME Then Exist_SHEET = 1: Exit For
Next SH
 
End Function

Sub Create_SOLUTIONS_PAGE(SH_NAME)

If Exist_SHEET(SH_NAME) Then
   Else
       Set NewSheet = Worksheets.Add
       NewSheet.Name = SH_NAME
End If

End Sub

Sub RESUME_Challenge()

If Cells(5, 6) = 0 Then Exit Sub

' SORTING
Columns("B:B").Select
    Selection.Sort Key1:=Range("B1"), Order1:=xlAscending, Header:=xlNo, _
    OrderCustom:=1, MatchCase:=False, Orientation:=xlTopToBottom

RESUME_LAST_SOLUTION

RESUME_CALC = 1

Cells(11, 4).Select
Selection.Activate

MAX_RESUME_No = Cells(8, 6)

CHECK = Cells(2, 6) * 100

TOT_INV = Cells(1, 6)
ReDim INV(TOT_INV + 1)

MaxSum = 0
For i = 1 To TOT_INV
    INV(i) = Cells(i, 2) * 100
    MaxSum = MaxSum + INV(i)
Next

SUM_INV = 0

MAX_INVS = 0

For i = 1 To TOT_INV
   SUM_INV = SUM_INV + INV(i)
   If SUM_INV = CHECK Then MAX_INVS = i: Exit For
   If SUM_INV > CHECK Then MAX_INVS = i - 1: Exit For
Next

Cells(3, 6) = MAX_INVS

MAX_CHECK_INVS_No = 1

For i = TOT_INV To 2 Step -1
    SUM_INV = INV(i)
    If SUM_INV = CHECK Then GoTo Exit_for
    For k = 1 To i
        SUM_INV = SUM_INV + INV(k)
        If SUM_INV <= CHECK Then
           GoTo Exit_for
        End If
    Next
Next_i:
Next
Exit_for:
MAX_CHECK_INVS_No = i
INV(MAX_CHECK_INVS_No + 1) = MaxSum

Cells(4, 6) = MAX_CHECK_INVS_No

CHECK = MaxSum * 2
RES_No = 0
Sol = Cells(5, 6)

Cells(18, 6).Value = Cells(19, 6).Value

Cells(14, 6) = Time
No_01% = RESUME_No(1)

'Application.ScreenUpdating = False

    RESUME_Find_Sol No_01%, "", 0

    Cells(10, 5) = Str(MAX_CHECK_INVS_No)
    Cells(15, 6) = Time
    
Application.ScreenUpdating = True

End Sub
    
Sub RESUME_Find_Sol(No_01 As Integer, NN_01 As String, SINVS_01 As Long)
    
For No_02% = No_01 + 1 To MAX_CHECK_INVS_No
    
    If RESUME_CALC = 1 Then
       RES_No = RES_No + 1
       If RES_No > MAX_RESUME_No Then
             RESUME_CALC = 2
             CHECK = Cells(2, 6) * 100
             No_02% = No_02% - 1
             Exit For
         Else
            No_02% = RESUME_No(RES_No)
       End If
    End If
    

    NN_02$ = NN_01 + Str(No_02%)
    SINVS_02& = SINVS_01 + INV(No_02%)
    
    If SINVS_02& > CHECK Then Exit For
    If SINVS_02& = CHECK Then
       Sol = Sol + 1
       If Sol > 65536 Then Sol = 1: COPY_SOLUTIONS
       
       Cells(Sol, 8) = NN_02$
       'Cells(10, 5) = NN_02$
       Cells(15, 6) = Time
                 
    End If
        
    If (SINVS_02& + INV(No_02% + 1)) > CHECK Then
           
       If (INV(No_02%) = INV(No_02% + 1)) Then GoTo END_LOOP
       
       MAX_No_01% = MAX_CHECK_INVS_No + 1
       No_02% = No_02% + 1
       
START_LOOP:
      
       CH_No% = MAX_No_01% - No_02%
       
       If CH_No% > 1 Then
          CH_No_m% = CH_No% / 2 + No_02%
      
          If (SINVS_01 + INV(CH_No_m%)) > CHECK Then
             MAX_No_01% = CH_No_m%
             GoTo START_LOOP
          End If
          If (SINVS_01 + INV(CH_No_m%)) = CHECK Then
             Sol = Sol + 1
             If Sol > 65536 Then Sol = 1: COPY_SOLUTIONS
             Cells(Sol, 8) = NN_01 + Str(CH_No_m%)
             Exit For
          End If
          If (SINVS_01 + INV(CH_No_m%)) < CHECK Then
             No_02% = CH_No_m%
             GoTo START_LOOP
          End If
        Else
          If CH_No% = 1 Then
             No_02% = MAX_No_01% - 1
             If (SINVS_01 + INV(No_02%)) = CHECK Then
                Sol = Sol + 1
                If Sol > 65536 Then Sol = 1: COPY_SOLUTIONS
                   Cells(Sol, 8) = NN_01 + Str(No_02%)
                  Exit For
                End If
             End If
             Exit For
          End If
       End If
END_LOOP:
         
    RESUME_Find_Sol No_02%, NN_02$, SINVS_02&

Next_No_02:
Next No_02

End Sub

Module 2

Dim Comp_No()
Function Print_No(Combination As String, Base As Integer) As Variant

Application.ScreenUpdating = False

Dim COMP As Integer
Dim Max_COMP As Integer
Dim ss, dd, i, k As Integer
Dim Co_02, Co_01 As Integer

Max_COMP = Val(SEPARATE_COMP(Combination, 0))
ReDim Comp_No(Max_COMP)

Dim Comp_SER(10, 2)

For COMP = 1 To Max_COMP
    Comp_No(COMP) = Val(SEPARATE_COMP(Combination, COMP))
Next

ss = 0: dd = 0

Select Case Max_COMP
 Case 1
      Print_No = CDec(1)
      For COMP = 1 To Comp_No(1) - 1
          Print_No = CDec(Print_No + Count_Combinations(COMP, Base))
      Next
 Case Base: Print_No = CDec(Max_COMP)
 Case Else
      Co_01 = Comp_No(1)
      For i = 2 To Max_COMP
          Co_02 = Comp_No(i)
          If Co_02 - Co_01 = 1 Then
                Co_01 = Co_02
                dd = dd + 1
            Else
                ss = ss + 1
                Comp_SER(ss, 1) = Co_01
                Comp_SER(ss, 2) = Co_02
                Co_01 = Co_02
          End If
      Next
      If ss = 0 Then
            Print_No = CDec(Max_COMP)
            For COMP = 1 To Comp_No(1) - 1
                Print_No = CDec(Print_No + Count_Combinations(COMP, Base))
            Next
        Else
            Print_No = CDec(Print_No + Max_COMP)
            For i = 1 To ss
                For k = Comp_SER(i, 1) + 1 To Comp_SER(i, 2) - 1
                   Print_No = CDec(Print_No + Count_Combinations(k, Base))
                Next
            Next
            For COMP = 1 To Comp_No(1) - 1
                Print_No = CDec(Print_No + Count_Combinations(COMP, Base))
            Next
            
      End If
End Select

Application.ScreenUpdating = True

End Function

Function Count_Combinations(No As Integer, Base As Integer) As Variant

Count_Combinations = CDec(2 ^ (Base - No))

End Function

Function SEPARATE_COMP(CELL_TEXT As String, No As Integer) As String
Application.ScreenUpdating = False

If CELL_TEXT = "" Then SEPARATE_COMP = "": Exit Function

' COUNT WORDS

CELL_TEXT = Trim(CELL_TEXT) + " "
TEXT_LEN% = Len(CELL_TEXT)
START_LEN% = 1
COUNTER_No% = 1
For i% = START_LEN% To TEXT_LEN%
    FOUNT_POSITION_No% = InStr(i%, CELL_TEXT, " ")
    WORD_FOUND = Mid(CELL_TEXT, i%, FOUNT_POSITION_No% - i%)
    i% = FOUNT_POSITION_No%
    If Trim(WORD_FOUND) <> "" Then
       COUNTER_No% = COUNTER_No% + 1
    End If
Next

MAX_WORDS% = COUNTER_No% - 1

If No = 0 Then
   SEPARATE_COMP = MAX_WORDS%
   Application.ScreenUpdating = True
   Exit Function
End If

' PUT WORDS IN ARRAY
ReDim WORDS_FOUND(MAX_WORDS%)

START_LEN% = 1
COUNTER_No% = 1

For i% = START_LEN% To TEXT_LEN%
   FOUNT_POSITION_No% = InStr(i%, CELL_TEXT, " ")
   WORD_FOUND = Mid(CELL_TEXT, i%, FOUNT_POSITION_No% - i%)
   i% = FOUNT_POSITION_No%
   If Trim(WORD_FOUND) <> "" Then
      WORDS_FOUND(COUNTER_No%) = WORD_FOUND
      COUNTER_No% = COUNTER_No% + 1
   End If
Next

If No > MAX_WORDS% Then No = MAX_WORDS%
SEPARATE_COMP = WORDS_FOUND(No)
    
Application.ScreenUpdating = True

End Function

Congratulations to IOANNIS and to everyone who participated in this month's challenge!