Single machine bicriteria scheduling with equal-length jobs to minimize total weighted completion time and maximum cost

Bicriteria scheduling problems are of significance in both theoretical and applied aspects. It is known that the single machine bicriteria scheduling problem of minimizing total weighted completion time and maximum cost simultaneously is strongly NP-hard. In this paper we consider a special case where the jobs have equal length and present an $O(n^{3}\log n)$ algorithm for finding all Pareto optimal solutions of this bicriteria scheduling problem.