We've discussed improving weekday service quite a bit on here, but I was also interested in improving weekend service as well. Right now there are 7 revenue round trips and I believe that schedule can be operated with 4 sets. The equipment for trains 70, 72, 74, and 76 should be on it's first run of the day as there's no revenue moves that arrive at Port Jervis before 76 departs.
Many of the current trips are local, meaning that it takes way too long to reach Secaucus and Hoboken, likely deterring much potential ridership. I see train 74 as the most important trip to make express, given how well it's timed for people going into the city on weekends for leisure purposes. I always take train 72 when I go in on weekends because it's the only express in the morning. If what I'm doing in the city doesn't fit with that departure, I either change it so it does, or don't take the train at all. I see no reason for 82 or 69 to become express, and the rest would benefit, but aren't quite as important as 74 IMO. BTW, train 77 is usually pretty busy, so the demand for increased service is definitely there.
There's no obvious place for the 71's equipment to go, so I'm guessing it just finishes at Port Jervis at around 11:30am. That equipment could be used for another revenue run that departs around 2 hours before train 78 does, but timed to meet any opposing moves at the sidings to fill in that 4+ hour gap between 76 and 78 arriving at Hoboken. The set could return to Port Jervis on a train that was between 79 and 81, with 81 being pushed back about 30-40 minutes and being made express. Adding one extra trip and having some or most of the existing trips start running express would probably increase ridership quite a bit.
Another thing I noticed with the weekend schedule is that many of the 2 hour gaps in local service are caused by a Port Jervis train making local stops in the other direction, so a trip made to fill in a 2 hour gap wouldn't have a revenue return trip. This is the case with the gap between train 1719 and 1723. If there was a train between them, it would depart Hoboken at about 2:25pm, arrive Suffern at about 3:40, and return to Hoboken on a trip departing Suffern at 4:08. However, train 78 is covering most of the stops from Suffern to Ridgewood so there's no return trip really necessary. This means that making a Port Jervis local into an express and adding a local trip to cover the skipped stops would likely eliminate a 2 hour gap in local service on it's run in the other direction.