Math Is Fun Forum

MORE COMPLEX REAL LIFE RULES:
1) Each saleswoman has 3 months to qualify. [There are also 2 month qualifications, 
   but by figuring out the 3 month algorithm, a 2 month qualification calculation should be able to use that same algorithm,
   or a slight modification of it.
   There are only 2 and 3 month qualifications]
2) Each saleswoman can have unlimited "Promote-Outs" in each 3 months to qualify, however having more than 3 is quite rare.
3) The maximum number of points a saleswoman is able to use over the 3 months FOR EACH Promote-Out is 128000 [THIS IS ALWAYS A CONSTANT]
4) For month1, this saleswoman needs to make at least the following over all her Promote-Out for month1: 110360  [THIS VARIES PER SITUATION, DEPENDING ON OTHER SALES NUMBERS THE SALESWOMAN HAS]
   For month2, this saleswoman needs to make at least the following over all her Promote-Out for month2:      0  [THIS VARIES PER SITUATION, DEPENDING ON OTHER SALES NUMBERS THE SALESWOMAN HAS]
   For month3, this saleswoman needs to make at least the following over all her Promote-Out for month3: 113320  [THIS VARIES PER SITUATION, DEPENDING ON OTHER SALES NUMBERS THE SALESWOMAN HAS]
NOTE: This saleswoman happens to have 2 Promote-Outs


Again: We are looking BACK over time, from the third month back to the first to see how we can get her to qualify for promotion to the next level.
So, for instance, if this analysis was for the current day, since today is in May 2011: 
month1 = March 2011
month2 = April 2011
month3 = May 2011

Here are her points for her 2 Promote-Outs over the past 3 months.

m1p1 = 105234   |   m1p2 = 50306.50   //She'll need to use the following number of points from this, to promote to the next level, 
(it is wisest to just use the minimum needed): 110360 [Rule 4 (for month1) = 110360]
___________________________________
m2p1 =  66940   |   m2p2 = 37778      //She really should not use any of this, since it'll be using the "slots" that could be used if she makes the points next month [Rule 4 (for month2) = 0]
___________________________________
m3p1 =  99591   |   m3p2 = 45985      //She happens to need the following number of points this month: 113320 [Rule 4 (for month3) = 113320]



By just adding the numbers up, not trying to optimize the saleswoman's dispersion of points, the saleswoman would NOT advance to the next level:

Here is why:

So, for month1:
~~~~~~~~~~~~~~~ 
m1p1 + m1p2 = 105234 + 50306.50 = 155540.5 (by the way, remember, for month1 only 110360 is needed) [Rule 4 (for month1) = 110360]
Now, p1 has the following available to be used for the next 2 months: p1_slot_left = 128000 - 105234    = 22766   [Rule 3 = 128000] [m1p1 = 105234]
Now, p2 has the following available to be used for the next 2 months: p2_slot_left = 128000 -  50306.50 = 77693.5 [Rule 3 = 128000] [m1p2 =  50306.50]

Then if without optimization, all of the points are used in month2.

So, for month2:
~~~~~~~~~~~~~~~
Since p1_slot_left = 22766,   though m2p1 = 66940 , since p1_slot_left < m2p1 , only p1_slot_left can be used, so: m2p1_used = p1_slot_left = 22766
Since p2_slot_left = 77693.5, though m2p2 = 37778,  since p2_slot_left > m2p2 , only m2p2 can be used, so m2p2_used = m2p2 = 37778

m2p1_used + m2p2_used = 22766 + 37778 = 60544 (by the way, remember, for month2 0 is needed however) [Rule 4 (for month2) = 0]


So, for month3:
~~~~~~~~~~~~~~~
Now, p1_slot_left (for month3) = p1_slot_left (for month2) - m2p1_used = 22766 - 22766 = 0
Now, p2_slot_left (for month3) = p2_slot_left (for month2) - m2p2_used = 77693.5 - 37778 = 39915

Since p1_slot_left = 0, though m3p1 = 99591 , since p1_slot_left < m3p1 , only p1_slot_left can be used, so: m3p1_used = p1_slot_left = 0
since p2_slot_left = 39915, though m3p2 = 45985,  since p2_slot_left < m3p2 , only p2_slot_left can be used, so: m3p2_used = p2_slot_left = 39915

BUT: We need 113320 [Rule 4 (for month3) = 113320], so the saleswoman does NOT qualify to promote to the next level.

This is how her points were used:

m1p1_used = 105234   |  m1p2_used = 50306.50      
____________________________________________
m2p1_used =  22766   |  m2p2_used = 37778      
____________________________________________
m3p1_used =      0   |  m3p2_used = 39915


===============================

HOWEVER: If we optimize, this is how she COULD use the points to show she SHOULD promote:


m1p1_used = 60053.50 |  m1p2_used =  50306.50      60053.50 + 50306.50 = 110360 (which is how much she needs [Rule 4 (for month1) = 110360] -- so she has saved some p1 "slots" from m1p1 to use in m3p1)   
_____________________________________________
m2p1_used =     0    |  m2p2_used =      0         she needs nothing here (so she is saving the p1 "slots" and the p2 "slots" from m2p1 and m2p2 to use in m3p1 and mp3p2) [Rule 4 (for month2) = 0]
_____________________________________________
m3p1_used = 67946.50 |  m3p2_used =  45985         67946.50 + 45985 = 113931.5 (all she needed was 113320). [Rule 4 (for month2) = 113320]


This is how my mind figured out this example, though more complex examples (for instance more than one "maxed-out" Promote-Out) would need a more complex formula:

NOTE: I use the term "maxed-out" Promote-Out to indicate that the points in the Promote-Out over all qualification months equal more than the maximum amount allowed.
For instance, in the above example: m1p1 + m2p1 + m3p1 > [Rule 3 = 128000] [BUT YOU WILL LEARN LATER IN THIS POST, THAT I THINK I NOW HAVE AN ALGORITHM TO FIGURE THAT OUT TOO :) ]

Since we don't need m2p1 and m2p2, just know we will use 0 of each (and not include them in the "slot" calculations below).
Then add up what we have in m1p1 + m3p1 (p1 "slot") --> 105234 + 99591  = 204825 (this is over 128000 [Rule 3 = 128000], so we know we will NOT be able to use all of it.)
Then add up what we have in m1p2 + m3p2 (p2 "slot") --> 50306.50 + 45985 = 96291.5 (this is under 12800 [Rule 3 = 128000], so we know we can use all of this.)

We will need m1p1 + m1p2 to equal 110360 [Rule 4 (for month1) = 110360] We can use all of m1p2, so --> 110360 - 50306.50 = m1p1_used = 60053.5
So we have the following left for p1 "slots" [Rule 3 = 128000] 128000 - m1p1_used  =  128000 - 60053.50 =  67946.5 [p1_slot_left]
(We will call p1_slot_left that occurs after m1p1_used calculations are made: m1p1_slot_left .)
Since nothing is needed for month2 qualifications m2p1_slot_left = m1p1_slot_left
Since m3p1 has 99591, we can not use all of it, but can only use  67946.5 [m2p1_slot_left] 
m3p1_used = m2p1_slot_left = 67946.5 (now m3p1_slot_left = 0)
For m3p2_used: m3p2 = 45985 ; m2p2_slot_left = 77693.5 ( [Rule 3 = 128000] - m1p2_used - m2p2used = 128000 - 50306.50 - 0 = 77693.5 )
m3p2 <  m2p2_slot_left, so: m3p2_used = m3p2.
m3p2_used = 45985

NOTE: My writing this post has helped me complete an pseudo-code for this example. (Writing does clarify thinking :), at least mine :) )

1) More than one Promote-Out has more points than can be used

if (RULE4_Month2 = 0) then   //NOTE: RULE4_Month2 turns out to very often be 0.

    if ((m1p1 + m3p1) > RULE3) then
        p1_maxout = true;
    else
        p1_maxout = false;
    end if

    if ((m1p2 + m3p2) > RULE3) then
        p2_maxout = true;
    else
        p2_maxout = false;
    end if

else 

    if ((m1p1 + m2p1 + m3p1) > RULE3) then
        p1_maxout = true;
    else
        p1_maxout = false;
    end if

    if ((m1p2 + m2p1 + m3p2) > RULE3) then
        p2_maxout = true;
    else
        p2_maxout = false;
    end if

end if


if ( (p1_maxout = true) && (p2_maxout = false) ) then

   m1p2_used = m1p2;
   m1p1_used = RULE4_Month1 - m1p2;

elseif ((p1_maxout = false) && (p2_maxout = false) ) then

   m1p1_used = m1p1; 
   m1p2_used = m1p2;

elseif ( (p1_maxout = false) && (WillBeAbleToUseAllPoints2 == false) ) then

   m1p1_used = m1p1;
   m2p2_used = RULE4_Month1 - m1p1;

else

  //NOTE: I have figured out another algorith for the case where more than one group Promote-Out Point is "maxed-out"
  //Please see that algorithm later in this post.

end if

m1p1_slot_left = RULE3 - m1p1_used;
m1p2_slot_left = RULE3 - m1p2_used;


if (RULE4_Month2 = 0) then   //NOTE: RULE4_Month2 turns out to very often be 0.

    if (m3p1 >= p1_slot_left) then
        m3p1_used = p1_slot_left
    else
        m3p1_used = m3p1
    end if

    if (m3p2 >= p2_slot_left) then
        m3p2_used = p2_slot_left
    else
        m3p2_used = m3p2
    end if

else

    if (m2p1 >= p1_slot_left) then
        m2p1_used = p1_slot_left
    else
        m2p1_used = m2p1
    end if

    if (m2p2 >= p2_slot_left) then
        m2p2_used = p2_slot_left
    else
        m2p2_used = m3p2
    end if

end if

I have created a group of Constants for data that we know in advance:

P_ALLOWED // the number in Rule 3
M1P1 // the promote-out points for the first  promote-out for month1
M1P2 // the promote-out points for the second promote-out for month1 [Note, since there can be an unlimited number of promote-outs, I will be using an array of M1P's in my code
M2P1 // the promote-out points for the first  promote-out for month2
M2P2 // the promote-out points for the second promote-out for month2 [Note, since there can be an unlimited number of promote-outs, I will be using an array of M2P's in my code
M3P1 // the promote-out points for the first  promote-out for month3
M3P2 // the promote-out points for the second promote-out for month3 [Note, since there can be an unlimited number of promote-outs, I will be using an array of M3P's in my code
M1_NEEDED // the number of points needed for month1 (Rule 4)
M2_NEEDED
M3_NEEDED

The following are derived values:
m1p1_perc // the percentage of m1p1 that is used in comparison with m1p2_perc to get the M1_NEEDED. m1p1_perc + m1p2_perc = 100% (when there are 2 groups of "Promote-Out Points")
m1p2_perc
m2p1_perc
m2p2_perc
m3p1_perc
m3p2_perc

m1p1_used // the number of points for month1 that the first Promote-Out group uses to meet the goal.
m1p2_used
m2p1_used
m2p2_used
m3p1_used
m3p2_used

m1p1_slot_left // the number of points "slots" left for the first Promote-Out group after the m1p1_used have been subtracted from P_ALLOWED.


I have found that the following is ALWAYS true:

(M1P1 / (M1P1 + M1P2))  + (M1P2 / (M1P1 + M1P2))  = 1

m1p1_perc = M1P1 / (M1P1 + M1P2)
m1p2_perc = M1P2 / (M1P1 + M1P2)

m1p1_used = M1_NEEDED * m1p1_perc
m1p2_used = M1_NEEDED - m1p1_used

m1p1_slot_left = P_ALLOWED - m1p1_used
m1p2_slot_left = P_ALLOWED - m1p2_used

m2p1_perc = M2P1 / (M2P1 + M2P2)
m2p2_perc = M2P2 / (M2P1 + M2P2)

m2p1_used = M2_NEEDED * m2p1_perc
m2p2_used = M2_NEEDED - m2p1_used

if (m2p1_used > m1p1_slot_left) then
    m2p1_used = m1p1_slot_left
end if

if (m2p2_used > m1p2_slot_left) then
    m2p2_used = m1p2_slot_left
end if

m2p1_slot_left = m1p1_slot_left - m1p2_used
m2p2_slot_left = m1p2_slot_left - m2p2_used


m3p1_perc = M3P1 / (M3P1 + M3P2)
m3p2_perc = M3P2 / (M3P1 + M3P2)

m3p1_used = M3_NEEDED * m3p1_perc
m3p2_used = M3_NEEDED - m3p1_used

if (m3p1_used > m2p1_slot_left) then
    m3p1_used = m2p1_slot_left
end if

if (m3p2_used > m2p2_slot_left) then
    m3p2_used = m2p2_slot_left
end if


So: Now I have the algorithms to get the amount to use for each Promote-Out group in each month, to give the saleswoman the greatest chance of promoting to the next level  :)