Interesting Algebra Problem: Predicting Qualification

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Hi,

I work for a multi-level marketing company and am creating an application that will help Sellers to predict whether they will promote to the next level.

I am stuck today on the following problem, and am hoping some algebra-wizzes out there can assist me.

Here is a real example of what has happened to a Seller in the past.

BUSINESS RULE: A Seller can only use a total of 128000 maximum points from each DownLine over the 3 month qualification period.

Note: This Seller has 2 Downlines, so 12800 x 2 = 24800 points total can be used over 3 months. There are enough points available, the trick is to determine how many points from each Downline to use each month so that the maximum points rule (above) for each Downline is not broken. Even if glancing at this, you can tell me the answer, what I need is an algebraic formula that I can use with different numbers for different Sellers to derive the answer. For instance: I know intuitively, the following (which is the second bold group near the bottom of this email), but I want to be able to derive it via an algorithm:
Month2TotalDownline1PointsUsedToReachNEEDED: 0
Month2TotalDownline2PointsUsedToReachNEEDED: 0

MaxPointsOver3MonthsUsableFromADowline:  128000

Month1PointsNEEDED:                      110360              
Month2PointsNEEDED:                           0
Month3PointsNEEDED:                      113320

Month1TotalDownLine1Points:              105234
Month1TotalDownLine2Points:               50306.50   
Month1TotalAllLegsPoints:                155540.50

Month2TotalDownLine1Points:               99591
Month2TotalDownLine2Points:               45985
Month2TotalAllLegsPoints:                145567

Month3TotalDownLine1Points:               99591
Month3TotalDownLine2Points:               45985
Month3TotalAllLegsPoints:                145567


Month1TotalDownline1PointsUsedToReachNEEDED:  ????  // total Month1 = 110360
Month1TotalDownline2PointsUsedToReachNEEDED:  ????  // total Month1 = 110360

Month2TotalDownline1PointsUsedToReachNEEDED:  ????  // total Month1 = 0
Month2TotalDownline2PointsUsedToReachNEEDED:  ????  // total Month1 = 0

Month3TotalDownline1PointsUsedToReachNEEDED:  ????  // total Month1 = 113320
Month3TotalDownline2PointsUsedToReachNEEDED:  ????  // total Month1 = 113320

Thank you so much for whatever help any of you are able to give me to assist me in solving this problem and improving my algebra so I am better able to solve problems like this and assist others with them in the future!

:) Anne

Posted 14-May-11 17:15pm

AnneThorne

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YvesDaoust · Answer 1 · 2011-07-27T21:43:00

Because of the specific terminology in use, I cannot understand the details. More specifically, I have no idea what a Downline is nor what AllLegs can mean.

Anyway, it seems to me that this problem can be formulated as a set of linear inequations of the form 0 <= x <= M, x + y + z <= N, or similar, and you need to either find an admissible solution or maximize some criterion.

You will need to resort to the techniques of linear programming (http://en.wikipedia.org/wiki/Linear_programming[^]). The set of admissible solutions corresponds to a polyedron in space (actually a 6-polytope in 6-hyperspace), and possibly use the Simplex algorithm (http://en.wikipedia.org/wiki/Simplex_algorithm[^]).

This may look scary at first glance, but is manageable. You first step in this search is to establish the inequations between all your parameters.

Last minute: good news for you, Excel has an add-in module called the Solver that is able to handle such linear programming problems (http://office.microsoft.com/en-us/excel-help/about-solver-HP005198368.aspx?CTT=3[^]).