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So a company has n available projects and k employees on the bench. Each project has a "number of hours" associated with it. Each employee has an hourly rate that the parent company gets paid gets paid if he is on a project. Not all employees can be assigned to any project i.e. each employee has a subset of the n projects he can work on. I want to assign the employees to projects so that I can maximize what the company makes from the assignment. Each project can be assigned to only one employee and each employee can work on only one project.

I am thinking of using dynamic programming but am unable to reach a recursion which I can use to fill a table. I for a m x n matrix, where m is the projects and n are the employees. Matrix[i][j]= the amount earned by the company if project i is assigned to employee j. I am stuck on how to maximize this.

Any help will be appreciated!

user2995244
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  • I think the problem is not well defined. It seems that each worker can work on arbitrary many projects at the same time (but only one worker per project). This way, you would just assign each projekt to the worker which would induce the biggest payoff. – Martin Jan 27 '16 at 09:37
  • I am sorry. I will edit the question. Each employee can work on only one project. – user2995244 Jan 27 '16 at 16:32

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This looks like an a 'assignment problem' which can formulated as a linear programming problem. Here is an example formulation: enter image description here

The first part is just inventing some random data. The variable x is a binary variable which usually means we need to solve the model as a mixed integer programming model. But this assignment problem has a property in which variables are integer automatically, so we can actually solve as an LP. (I solved here as an 'RMIP' which means: drop the integrality conditions on x).

The formulation is a little bit complicated as I don't want to include x(i,k)'s in the model that are not allowed.

This is an easy LP so you should be able to solve large problems quickly.

Erwin Kalvelagen
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