I feel silly enough being 75K based in WAS with dwindling trips to the west coast but it's pretty ridiculous to be based in SIN with an annual trip to BWI... nonetheless the easiest way to get segments is through 10 segments on the west coast bookended with a flight to/from WAS. There's a new LAX-IAD red-eye on the schedule so you should be able to do this in 2 days plus a red-eye.

Cheapest segments are often intra-california (LAX-SJC/SMF or SMF-SAN, rarely LAX-SFO), fastest would be SEA-PDX and back again, but flight times to SEA aren't as useful as those to California for a segment run and I'm not sure you could do a flight to/from SEA with 5 round trips to Portland and back any quicker than doing trips just in California.

Staring on 8/25 you could take the 6:15am IAD-LAX getting into LAX at 9:15am and have time for 3 intra CA round trips that day with time for 3 more the next day then the red-eye back - time for 14 segments when you only need 12!

There might also be a creative way to fly something like IAD-LAX-SFO-LAX-SFO-SEA-JNU-ANC-SEA-PDX-GEG-SEA-WAS taking advantage of the ANC-SEA red-eye and getting back to DC on an afternoon flight but that would be more expensive and miserable than just shuttling around CA.