I am trying to distribute 15 tasks to two people. Each task can only be assigned to one person and each person has a time budget.

I want to express this problem as a linear program (ultimately in the standard linear form) so that I can write a computer program to solve it. $$\begin{array}{|c|c|c|c|c|} \hline Workers& Task 1 & Task2 & Task3 & ... & Time budget \\ \hline Jack& 0.5& 0.25& 0.25 & & 9\\ \hline John& 0.75& 1.0 &0.75 & & 8\\ \hline \end{array}$$

I can find plenty of examples how to do it without a budget constraint. Without it, it is an Assignment Problem that can be solved with The Hungarian Method.

• or.stackexchange.com – Rodrigo de Azevedo Nov 6 '19 at 8:50
• @RodrigodeAzevedo : Do you think or.stackexchange.com should handle all questions with the "operations-research" tag ? – Kuifje Nov 6 '19 at 10:22
• @Kuifje I merely think that the OP should be aware of the existence of OR SE. Naturally, people over there are much more enthusiastic about it than people here. – Rodrigo de Azevedo Nov 6 '19 at 10:25
• The question is tagged "linear-programming", but I think it will require integer linear programming. – prubin Nov 6 '19 at 23:19
• If you want help have to give a reply to the comments/answers. – callculus Nov 7 '19 at 16:14

Hints: Let binary decision variable $$x_{i,j}$$ indicate whether person $$i$$ is assigned task $$j$$. Now you need a constraint that each task gets assigned to exactly one person, as well as a time budget constraint for each person.