All the input variables are positive float (x > 0). We have $M$ agents with limited amount of time $t_1,\dots,t_M$, $N$ tasks $task_1,\dots,task_N$ associated with duration $d_1,\dots, d_N$. Cost for assigning agent $i$ to task $j$ is $c_{ij}$. Time assigning agent $i$ to task $j$ is $x_{ij}$.
We define task as satisfied if the sum of of time allocated to it, by the agents, equals to its duration (binary value): $$SAT(task_j) = 1 \iff d_j=\sum_{i=1}^Mx_{ij}$$
We want to find the $x_{ij}$ that get the maximum number of satisfied tasks while minimizing the cost: $$MIN\sum_{j=1}^N \sum_{i=1}^Mc_{ij}x_{ij}$$ While $$\sum_{j=1}^N \left(SAT(task_j)\right) = MAX-SATISFIED$$
The main idea is to find the solution with lowest cost in the set of solutions with maximal number of satisfied tasks (The satisfied tasks is the most important objective).
Is there a name for this variation of the assignment problem? Also is there an known efficient algorithm?