Interval scheduling to maximize bandwidth provision

We study an interval scheduling problem in which each job j is associated with a time interval Ij, a minimum aj and maximum bj required bandwidths, and a weight wj. We are given W colors. We need to assign to each job j between aj and bj colors, such that each color is assigned at each time to at most one interval. The weighted bandwidth allocated to job j is the number of colors allocated to it times |Ij|, and the weighted bandwidth of a coloring is the sum of the weighted bandwidth of all jobs. The objective is to find a coloring with maximum weighted bandwidth. A coloring is (circularly) contiguous if the set of colors for each job forms an (circular) interval.