I and another three colleagues have an ongoing paper about the application of sociometry to multiple human resource allocation to multiple projects. The problem is mathematically quite challenging but we are on a dead end concerning one of the outcomes we are dealing with.
Particularly, we have been applying meta-heuristics and evolutionary algorithms to solve the problem of allocating groups of people so as to maximize cohesion among them. The research is quite interesting and we think it is going to open multiple and very interesting research and industry application in the near future.
However, we are stuck in one part. We are trying to calculate the number of viable combinations of people who can work either full-time, part-time or not work at all in several simultaneous projects and we need a person with advanced knowledge in combinatorics to give us a hand. We are willing to pay or to put his/her name as co-author on the paper.
Getting straight to the point, the problem statement is as follows:
There are N people (i=1...N) who can be selected to work in P simultaneous projects (j=1...P). Each person can have a dedication of work full-time (1), half-time (0.5) or not work (0), that is, three possible allocations (0, 0.5, 1).
Now, we know that each project j requires Rj people. How many different and viable combinations are there?
Numerical Example:
People available Project j=1 Project j=2 Project j=3=P
i=1 0 or 0.5 or 1 0 or 0.5 or 1 0 or 0.5 or 1 Row sum =
Thanks Peter, I am sure your answer points in the right way, at least concerning the 2^N denominator. There are, however, many things I don't understand. Maybe we can work with an example to illustrate my concerns. Let's think about a problem with 4 people and 2 projects which require 2 people each.
Does the division of labor in your work combinations mean that each person can only work one job or do you mean to insist that only the "man-hours" add up to R_j = 2, 1, and 1, respectively? If so, does that mean that you could have all available workers working if some elect to work two half-time jobs? I think I need clarity on why one worker necessarily has to be left out.
Thanks Peter. I don't know why the rest of the answer I wrote just disappeared! but yes, your answer is totally right!
To be honest, I am not an expert in statistics so I don't undertstand much of the terminology you are using (nodes, edges,non-isomorphic and the like) but I can see that you know what you are saying since your answers are correct.
Would you like to be a co-autor then or to be paid? There are currently four of us (five authors in case you accepted). We now need the generic expressions to calculate the number of viable (different) combinations for sets of people of any size allocated to sets of projects of any size. Indeed, the case you have solved is just for one area of expertise, but I guess the rest will require only multiplying these combinations by the combinations obtained for the other areas of expertise (with other people and other Project requirements) to reach the final amount of combinations for the complete problem.
Do you mind continuing the conversationby email? [email protected]? Thanks again.
Concerning Mr. Duane Lee, the problema statements is just as Mr. Peter Breuer has stated it. We intend to keep dividing people time into more divisions, but we are first to try to deal with the situation of full-times, half-times and no-times. In case you had other useful insights do not hesitate to state them. Thanks to you too.
Thanks Peter. We really appreciate the time you have spend solving this problem. Your contribution will be duly ackowledged as we agreed.
Thanks too for all you laborious explanations trying to make that someone like me understood the mathematical and combinatorics vision of this problem.
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