A few weeks ago Gewilson asked on the Chandoo.org Forums, “Can I simplify my formula? “
=176 – (SUMIF($B10:$AF10,”PD”,$B11:$AF11) +SUMIF($B10:$AF10,”FA”,$B11:$AF11) +SUMIF($B10:$AF10,”PS”,$B11:$AF11) +SUMIF($B10:$AF10,”PN”,$B11:$AF11) +SUMIF($B10:$AF10,”F1″,$B11:$AF11) +SUMIF($B10:$AF10,”P1″,$B11:$AF11) +SUMIF($B10:$AF10,”F7″,$B11:$AF11))
SirJB7 responded with a nice Sumproduct solution:
=176 – SUMPRODUCT(((B$10:AF$10)=({“PD”;”FA”;”PS”;”PN”;”F1″;”F7″})) *(B$11:AF$11))
So Today we will pull this apart to see what inside, I think what we find may surprise you.
SirJB7’s Formula
=176 – SUMPRODUCT(((B$10:AF$10)=({“PD”;”FA”;”PS”;”PN”;”F1″;”F7″})) *(B$11:AF$11))
To Simplify things I am going to use a Truncated set of data and adjust the formula accordingly
We will examine:
=176 – SUMPRODUCT(((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”})) *(B$11:I$11))
This problem has a smaller Range B10:I10 instead of B10:AF10
as well as 2 less possible solutions {“PD”;”FA”;”PS”;”PN”} instead of {“PD”;”FA”;”PS”;”PN”;”F1″;”F7″}
The reason for this will soon become evident.
As usual you can download a Sample File to follow along with. Download Here.
Lets go:
=176 – SUMPRODUCT(((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”})) *(B$11:I$11))
We can see above that the formula is subtracting the result of a Sumproduct from a Fixed Number 176. So we really only need to focus on the Sumproduct part of the formula.
As we saw In Formula Forensics 007 – Sumproduct, Sumproduct adds up the products of the constituent arrays.
In this case
SUMPRODUCT(((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”;})) *(B$11:I$11))
Has only 1 constituent array. The array does consist of 2 components
SUMPRODUCT(((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”})) *(B$11:I$11))
These is a Logic component ((B$10:AF$10)=({“PD”;”FA”;”PS”;”PN”}))
and a Numerical Component (B$11:I$11)
Which are then multiplied together.
Looking at the Logical Component first
((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”}))
The formula is checking the Range B10:I10 against an Array of possible solutions {“PD”;”FA”;”PS”;”PN”}
That is, it is checking each value in our list {“PD”;”FA”;”PS”;”PN”}, against each cell in the range B10:I10.
If we type the above equation=((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”})) into a spare cell C14, and press F9
Excel returns ={FALSE,FALSE,FALSE,FALSE,TRUE,FALSE,FALSE,FALSE;FALSE,TRUE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE}
What the …
If we look closely at the above array we will see that it contains a lot of True/Falses separated by ,’s and a few ;’s
={FALSE,FALSE,FALSE,FALSE,TRUE,FALSE,FALSE,FALSE;FALSE,TRUE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE}
Specifically there are 4 blocks of 8 True/Falses separated by ,’s, each block is separated by a ;
In Total 4 x 8 = 32 Values
What this is, is an array representing the multiplication of the 8 cells in the range B10:I10 with each element of the possible solution array
Each row of the Array is separated from the next by a ;
Each element in each row is separated by a ,
This is best displayed like:
You can see why I simplified the size of the original problem.
So we have an Array of True/Falses ={FALSE,FALSE,FALSE,FALSE,TRUE,FALSE,FALSE,FALSE;FALSE,TRUE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE}
Which is now multiplied by the next component of the Sumproduct
({FALSE,FALSE,FALSE,FALSE,TRUE,FALSE,FALSE,FALSE;FALSE,TRUE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE}) *(B$11:I$11)
In a spare cell, say C23 enter
=({FALSE,FALSE,FALSE,FALSE,TRUE,FALSE,FALSE,FALSE;FALSE,TRUE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE;FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE,FALSE}) *(B$11:I$11) and press F9
Excel returns
={0,0,0,0,10,0,0,0;0,10,0,0,0,0,0,0;0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0}
Note that this array has the same 8 column x 4 Row layout as above, except that all the True have been replaced by the values in the Score cells B11:I11
Sumproduct now kicks in and adds these up
=Sumproduct({0,0,0,0,10,0,0,0;0,10,0,0,0,0,0,0;0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0})
To get 20
Which is subtracted from our original number
=176 – SUMPRODUCT(((B$10:I$10)=({“PD”;”FA”;”PS”;”PN”})) *(B$11:I$11))
= 176 – Sumproduct({0,0,0,0,10,0,0,0;0,10,0,0,0,0,0,0;0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0})
= 176 – 20
= 156
Download
You can download a copy of the above file and follow along, Download Here.
Other Posts In This Series
You can learn more about how to pull Excel Formulas apart in the following posts
We Need Your Help
I have received a few more ideas since last week and these will feature in coming weeks.
I do need more ideas though and so I need your help.
If you have a neat formula that you would like to share and explain, try putting pen to paper and draft up a Post like above or;
If you have a formula that you would like explained but don’t want to write a post also send it in to Chandoo or Hui.
19 Responses to “How to Distribute Players Between Teams – Evenly”
An excellent solution, especially for large data sets.
Another solution without using solver would be to assign the player with the highest score to Team 1, the 2nd to team 2, 3rd to team 3, 4th to team 3, 5th to team 2, 6th to team 1, 7th to team 1 and it continues. This method would end up with a Std Dev of 0.001247219. This works best with a distribution with lower Std Dev for the dataset.
Full Disclosure: this is not my idea, remember reading something a few years ago. Think it may have been Ozgrid
thinking back I now remember why I read about it. About 10 years back I had to distribute around 300 team members into 25-30 odd teams. Used this method based on their performance scores. I used the method I described to do this and the distribution was pretty fair.
Solver would have saved me a ton of time though 🙂
I think the issue with you first Solver approach was that you took the absolute value of the sum of team deviations (which should always be zero except for rounding) instead of the sum of the absolute values (which is a reasonable measure of how unbalanced the teams are).
Here's another simple algorithm you could use: you start from the top (with players sorted from high to low), and at each step allocate the next player to whichever team has the smallest total so far. You can implement it dynamically with some formulas so it will update automatically when the data changes.
If the scores were more widely distributed (so that this might end up with not all teams the same size), you could add a constraint to only pick among the teams which currently have fewest players at each step, or just stop adding to any team when it hits its quota.
When I tried it on the sample, I got the three teams below, with a STDEV of 0.000942809 (i.e. about half of what Solver got to).
Team 1: John, Hugo, Tom, Josh, Eric, Zane, Charles, Andrew
Team 2: Barry, Michael, Kenny, Joe, Xavier, Patrick, Oliver, William
Team 3: Henry, Steven, Ben, Frank, Kyle, Edward, Cameron, Lachlan
Thanks for sharing!
Hi,
I was looking at all the solutions and this is closest to what I intended to do. I am dividing a bunch of players into 3 soccer teams. Players availability is also a factor while deciding the teams.
So the steps the excel needs to do is as follows:
1) In availability column if "yes" go to next
2) Equally divide 'Goalkeepers', 'Strikers', 'Defenders' basis their quality
So the end result gives each 3 teams a balance of players playing at different positions.
Can this be done on Google spreadsheet with only availability as an input from the user and rest calculates by itself.
Sorry for asking such a pointed question, but I have been struggling to find a solution for it for sometime now!
Hi Ishaan,
I am working on a similar problem at the moment, so I am wondering if you ever found a solution and if you are willing to share what you did.
Hi everyone, this is a variation of the famous Knapsack Problem https://en.wikipedia.org/wiki/Knapsack_problem.
I had to use a VBA implementation recently as part of a problem, where we ar trying to allocate teams of an organization into different locations (we are a large company with many different team). The goal was to optimally allocate teams to individual buildings without putting too many teams into one building and not splitting teams apart.
As we had around 400 teams of different sizes, solver couldn't handle it anymore. Luckily there is a Knapsack algorithm implementation in VBA readily available on the internet :).
I also went with a heuristic approach first!
An interesting mathematical solution but what if Eric and Xavier can't stand each other or Patrick is best friends with Steven - the real life problems that effect "even" teams.
@Joe
You can add more criteria like
If Eric and Xavier can't stand each other
=OR(AND(E15=1,E16=1),AND(F15=1,F16=1),AND(G15=1,G16=1))
It must be False
If Patrick is best friends with Steven
=OR(AND(E5=1,E17=1),AND(F5=1,F17=1),AND(G5=1,G17=1))
It must be True
Note that the 2 formulas above are exactly the same
except for the ranges
One must be True = Friends
One must be False = Not Friends
Nice Post!
Just one question What if number of players are not even or equally divisible.
Nice post Hui!
I download your workbook and just try to change in options the Precision Restriction from 10E-6 to 10-8 and the Convergence from 10E-4 to 10E-10. The process take almost the same time, but the results was great.
The standard deviation I got was 0,000471.
Team 1: John, Tom, Kenny, Frank, Eric, Xavier, Edward, Zane
Team 2: Steven, Hugo, Ben, Joe, Josh, Oliver, Cameron, William
Team 3: Barry, Henry, Michael, Kyle, Patrick, Charles, Andrew, Lachlan
Great application of Solver! Thanks for the link!
Great explanation. Well done... However, I tried with 6 teams of 4 players and solver never did finish.
How about vba code for the same data set.
I have 3 column A B C wherein A has text and B has number Wherein C is blank. And in C1 been the header C2 where I want the name to come evenly distributed the number which is in Column B.
My Lastcolumn is 1000.
Sorry if I'm being slow here, but how is 'Team Score' calculated? I've gone through the explanation several times but it seems to just appear.
@Hrmft
This process uses the Solver Excel addin
Solver is effectively taking the model and trying different solutions until it gets a solution that meets all the criteria
Then solver puts the solution into the cell and moves to the next cell
So yes it appears to "just appear"
Hi ! Thank you so much ! Works great 🙂
I cannot get the fourth Equation to work in my excel spreadsheet
You have =($E$2:$G$25=0)+($E$2:$G$25=1)=1 as a SUMIF solution, I have, =($F$2:$H$13=0)+($F$2:$H$13=1)=1 as my solution but it does not work. The only thing I changed is the ranges. Any suggestions?
Thank you.
Jim
I cannot get the fourth Equation of TURE or FALSE statements to work in my excel spreadsheet You have =($E$2:$G$25=0)+($E$2:$G$25=1)=1 as a SUMIF solution, I have, =($F$2:$H$13=0)+($F$2:$H$13=1)=1 as my solution but it does not work. The only thing I changed is the ranges. Any suggestions?
Sorry I left some of it out in the previous question,
Thank you. Jim