I want to create a dynamic array that returns me X values based on given probabilities. For instance:
Imagine this is a gift box and you can open the box N times. What I want is to have N random results. For example, I want to get randomly 5 of these two rarities but based on their chances.
I have this following formula for now:
=index(A2:A3,randarray(5,1,1,rows(A2:A3),1). And this is the output I get:
The problem here is that I have a dynamic array with the 5 results BUT NOT BASED ON THE PROBABILITIES.
How can I add probabilities to the array?
Here is how you could generate a random outcome with defined probabilities for the entries (Google Sheets solution, not sure about Excel):
=ARRAYFORMULA(
VLOOKUP(
RANDARRAY(H1, 1),
{
{0; OFFSET(C2:C,,, COUNTA(C2:C) - 1)},
OFFSET(A2:A,,, COUNTA(C2:C))
},
2
)
)
This whole subject of random selection was treated very thoroughly in Donald Knuth's series of books, The Art of Computer Programming, vol 2, "Semi-Numerical Algorithms". In that book he presents an algorithm for selecting exactly X out of N items in a list using pseudo-random numbers. What you may not have considered is that after you have chosen your first item the probability array has changed to (X-1)/(N-1) if your first outcome was "Normal" or X/(N-1) if your first outcome was "Rare". This means you'll want to keep track of some running totals based on your prior outcomes to ensure your probabilities are dynamically updated with each pick. You can do this with formulas, but I'm not certain how the back-reference will perform inside an array formula. Microsoft's dynamic array documentation indicates that such internal array references are considered "circular" and are prohibited.
In any case, trying to extend this to 3+ outcomes is very problematic. In order to implement that algorithm with 3 choices (X + Y + Z = N picks) you would need to break this up into one random number for an X or not X choice and then a second random number for a Y or not Y choice. This becomes a recursive algorithm, beyond Excel's ability to cope in formulas.
Related
I have a table on Excel with data as the following:
Meaning, I have different JPH based on the %SMALL unit and the number of active stations.
I need to create a matrix like the following (with %SMALL on horizontal and STATIONS on vertical axes):
And the formula for each cell should:
Take the input of Stations (column "B")
Check, for that specific Stations number, the amount of data on the other table (like make a filter on STATIONS for the specific number)
Perform an VLOOKUP for checking the JPH based on the %SMALL value on row 2
Interpolate for the exact JPH value, if not found on table
For now, I was able to create the last part (the VLOOKUP and the interpolation), with the following:
=IFERROR(VLOOKUP(C2;'EARLY-STATIONS'!$F:$H;3;FALSE);AVERAGE(OFFSET(INDEX('EARLY-STATIONS'!$H:$H;MATCH(C2;'EARLY-STATIONS'!$F:$F;1));0;0;2;1)))
The problem I'm facing is than with this, the calculation is not checking the number of stations, so the Iteration is not accurate.
Unfortunately I cannot use VBA macros to solve this.
Any clue?
This is an attempt because more clarity is needed in terms of all possible scenarios to consider, based on different input data and how to understand the "extrapolation" process. This approach understands as extrapolation the average of two values (lower and greater), but the idea can be customized to any other way to calculate it. Per tags listed in the question I assume there is no Excel version constraint. This is O365 solution:
=LET(sm, A2:A10, st, B2:B10, jph, C2:C10, smx, F1:J1, sty, E2:E4, NULL, "",
GETLk, LAMBDA(x,y,mode, FILTER(jph, (st=y)
* (sm = INDEX(sm, XMATCH(x, sm, mode))), NULL)),
GET, LAMBDA(x,y, LET(f, FILTER(jph, (jph=GETLk(x,y, 1))
+ (jph=GETLk(x,y, -1)), NULL), IF(#f=NULL, NULL, AVERAGE(f)))),
HREDUCE, LAMBDA(yi, DROP(REDUCE("", smx, LAMBDA(ac,x,
HSTACK(ac, GET(x, yi)))),,1)),
DROP(REDUCE("", sty, LAMBDA(ac,y, VSTACK(ac, HREDUCE(y)))),1))
The above formula spills the entire result, I don't think for this case you can use a LOOKUP-like function.
Here is the output:
The highlighted cells where the average is calculated.
Explanation
The main idea is to use DROP/REDUCE/HSTACK/VSTACK pattern to generate the grid. Check my answer to the following question: how to transform a table in Excel from vertical to horizontal but with different length on how to apply it.
We use two user LAMBDA functions to abstract some calculations:
GETLk(x,y,mode), filters jph name based on %SMALL and Stations columns values, based on input values x (x-axis value from the grid), y (y-axis value form the grid) respectively. The third input argument mode, is for doing the approximate search in XMATCH (1-next largest, -1 next smallest). In case the value exist in the input table, XMATCH returns the same value in both cases.
GET(x,y) has the logic to find the value or if the value doesn't exist to calculate the average. It uses the previous LAMBDA function GETLk. We filter for jph values that match the input values (x,y), but we use an OR condition in the FILTER (+), to select both lower or greater values. If the value exist, returns just one value otherwise two values are returned by FILTER (f). Finally if f is not empty we return the average, otherwise the value we setup as NULL.
HREDUCE: Concatenate the result by columns for a given row of the grid. Check the referred question for more information about it.
Given an NxM array of positive integers, how would one go about selecting integers so that the maximum sum of values is achieved where there is a maximum of x selections in each row and y selections in each column. This is an abstraction of a problem I am trying to face in making NCAA swimming lineups. Each swimmer has a time in every event that can be converted to an integer using the USA Swimming Power Points Calculator the higher the better. Once you convert those times, I want to assign no more than 3 swimmers per event, and no more than 3 races per swimmer such that the total sum of power scores is maximized. I think this is similar to the Weapon-targeting assignment problem but that problem allows a weapon type to attack the same target more than once (in my case allowing a single swimmer to race the same event twice) and that does not work for my use case. Does anybody know what this variation on the wta problem is called, and if so do you know of any solutions or resources I could look to?
Here is a mathematical model:
Data
Let a[i,j] be the data matrix
and
x: max number of selected cells in each row
y: max number of selected cells in each column
(Note: this is a bit unusual: we normally reserve the names x and y for variables. These conventions can help with readability).
Variables
δ[i,j] ∈ {0,1} are binary variables indicating if cell (i,j) is selected.
Optimization Model
max sum((i,j), a[i,j]*δ[i,j])
sum(j,δ[i,j]) ≤ x ∀i
sum(i,δ[i,j]) ≤ y ∀j
δ[i,j] ∈ {0,1}
This can be fed into any MIP solver.
Introduction
I have written code to give me a set of numbers in '36 by q' format ( 1<= q <= 36), subject to following conditions:
Each row must use numbers from 1 to 36.
No number must repeat itself in a column.
Method
The first row is generated randomly. Each number in the coming row is checked for the above conditions. If a number fails to satisfy one of the given conditions, it doesn't get picked again fot that specific place in that specific row. If it runs out of acceptable values, it starts over again.
Problem
Unlike for low q values (say 15 which takes less than a second to compute), the main objective is q=36. It has been more than 24hrs since it started to run for q=36 on my PC.
Questions
Can I predict the time required by it using the data I have from lower q values? How?
Is there any better algorithm to perform this in less time?
How can I calculate the average number of cycles it requires? (using combinatorics or otherwise).
Can I predict the time required by it using the data I have from lower q values? How?
Usually, you should be able to determine the running time of your algorithm in terms of input. Refer to big O notation.
If I understood your question correctly, you shouldn't spend hours computing a 36x36 matrix satisfying your conditions. Most probably you are stuck in the infinite loop or something. It would be more clear of you could share code snippet.
Is there any better algorithm to perform this in less time?
Well, I tried to do what you described and it works in O(q) (assuming that number of rows is constant).
import random
def rotate(arr):
return arr[-1:] + arr[:-1]
y = set([i for i in range(1, 37)])
n = 36
q = 36
res = []
i = 0
while i < n:
x = []
for j in range(q):
if y:
el = random.choice(list(y))
y.remove(el)
x.append(el)
res.append(x)
for j in range(q-1):
x = rotate(x)
res.append(x)
i += 1
i += 1
Basically, I choose random numbers from the set of {1..36} for the i+q th row, then rotate the row q times and assigned these rotated rows to the next q rows.
This guarantees both conditions you have mentioned.
How can I calculate the average number of cycles it requires?( Using combinatorics or otherwise).
I you cannot calculate the computation time in terms of input (code is too complex), then fitting to curve seems to be right.
Or you could create an ML model with iterations as data and time for each iteration as label and perform linear regression. But that seems to be overkill in your example.
Graph q vs time
Fit a curve,
Extrapolate to q = 36.
You might want to also graph q vs log(time) as that may give an easier fitted curve.
Excel
Need to find nearest float in a table, for each integer 0..99
https://www.excel-easy.com/examples/closest-match.html explains a great technique for finding the CLOSEST number from an array to a constant cell.
I need to perform this for many values (specifically, find nearest to a vertical list of integers 0..99 from within a list of floats).
Array formulas don't allow the compare-to value (integers) to change as we move down the list of integers, it treats it like a constant location.
I tried Tables, referring to the integers (works) but the formula from the above web site requires an Array operation (F2, control shift Enter), which are not permitted in Tables. Correction: You can enter the formula, control-enter the array function for one cell, copy the formulas, then insert table. Don't change the search cell reference!
Update:
I can still use array operations, but I manually have to copy the desired function into each 100 target cells. No biggie.
Fixed typo in formula. See end of question for details about "perfection".
Example code:
AI4=some integer
AJ4=MATCH(MIN(ABS(Table[float_column]-AI4)), ABS(Table[float_column]-AI4), 0)
repeat for subsequent integers in AI5...AI103
Example data:
0.1 <= matches 0
0.5
0.95 <= matches 1
1.51 <= matches 2
2.89
Consider the case where target=5, and 4.5, 5.5 exist in the list. One gives -0.5 and the other +0.5. Searching for ABS(-.5) will give the first one. Either one is decent, unless your data is non-monotonic.
This still needs a better solution.
Thanks in advance!
I had another problem, which pushed to a better solution.
Specifically, since the Y values for the X that I am interested in can be at varying distances in X, I will interpolate X between the X point before and after. Ie search for less than or equal, also greater than or equal, interpolate the desired X, then interpolate the Y values.
I could go a step further and interpolate N - 1 to N + 1, which will give cleaner results for noisy data.
I'm a total beginner with CPLEX and OPL, so maybe you can help me with the coding of a mixed integer programming model.
In my case: I have an optimization function including a parameter transportation cost which are specific for the starting point (Hubs h), the destination (DCs i), the transported good (Products k) and the mode of transportation (TransportOptions r) used.
I wrote it like this:
float transportC_Hub_DC[Hubs][DCs][Products][TransportOptions] = ...;
//transport cost of one unit of gook k vor starting point h to destination i using transportation option r
I would like to fill this array with its multiple dimensions from an excel spread sheet. At the moment my spreadsheet has the four indexes in separate columns and the specifice transportation cost in another column. It looks like this:
Excel Datasheet
My problem is that I do not know how to make the programme understand how the transportation cost data are ordered. How does the programme know that in the first cell of the column "transportation cost" is the cost for the specific combination of the different indexes? So how do I tell the programme that I used h=1, i=1, k=1, r=1 in the first cell and h=1, i=1, k=1, r=2 in the second cell and not h=1, i=1, k=2, r=1 in the second cell? What do I have to write in the model or the data file in CPLEX to make this clear?
See technote http://www-01.ibm.com/support/docview.wss?rs=0&context=SSCMS55&uid=swg21401340&loc=en_US&cs=utf-8&cc=us&lang=all
The idea is to read a tuple set and then turn your tuple set into a 4D array.