Consider the problem 1|p_j=1|L_max. Show how to solve this problem by formulating an assignment problem, the solution of which gives a solution to your scheduling problem. That is, given an instance of 1|p_j=1|L_max, you should give an algorithm to convert it into an assignment problem (minimum weight bipartite matching). Then you should show that an optimal solution to the resulting assignment problem can be used to construct an optimal solution to the scheduling problem.