Suppose that I have a table A with each of its n columns corresponding to an "individual" and each of its m rows corresponding to a "task". The element of the jth row and ith column, s(ji), is a real number, which can be interpreted as a score measuring the quality of match between individual i and task j.
A second table, B has n columns and two rows. The first row of column i specifies the maximum number of tasks that individual i can be assigned to. The second row specifies the minimum number of tasks that individual i can be assigned to (the two values might be equal).
A third table, C has m columns and two rows. The first row of column j specifies the maximum number of individuals that task j can be assigned to. The second row specifies the minimum number of individuals that task j can be assigned to (the two values might be equal).
The elements of B and C are natural numbers.
I would like to output an n column, m row table with element I(ji) equal to 1 if individual i is assigned to task j and zero otherwise, such that the constraints given in the second and third tables are satisfied and the output maximises the total score,
$$\sum_{i,j}I_{ij}s_{ji}.$$
As an example, suppose we have the following input:
A={{5,3},{1,2}}B={{1,1},{1,0}}C={{1,1},{1,1}}Table A says, for instance, that individual 1 has a score of 5 for task 1 and 1 for task 2. Table B says that individual 1 should be assigned exactly 1 task, while individual 2 should be assigned between 0 and 1 tasks. Table C says each task should have no less than one individual and no more than one individual assigned. The solution (and desired output) is
{{1,0},{0,1}}i.e. we assign individual 1 to task 1 and individual 2 to task 2 because that maximises the score (5+2) subject to the constraints. While we could get a higher score by, for example, giving task 1 to both individuals (total score is then 5+3), this would violate the constraint in C.
This is for practical use, so a numerical approach that finds an approximately correct solution would also be fine. If there is a more convenient way to supply the scores/constraints then that is also fine.
I have no idea how to even start programming this in Mathematica. Is anyone able to offer any direction?
