  Electromagnetic follow-up of gravitational wave detections is very resource
intensive, taking up hours of limited observation time on dozens of telescopes.
Creating more efficient schedules for follow-up will lead to a commensurate
increase in counterpart location efficiency without using more telescope time.
Widely used in operations research and telescope scheduling, mixed integer
linear programming (MILP) is a strong candidate to produce these
higher-efficiency schedules, as it can make use of powerful commercial solvers
that find globally optimal solutions to provided problems . We detail a new
target of opportunity scheduling algorithm designed with Zwicky Transient
Facility in mind that uses mixed integer linear programming. We compare its
performance to \texttt{gwemopt}, the tuned heuristic scheduler used by the
Zwicky Transient Facility and other facilities during the third LIGO-Virgo
gravitational wave observing run. This new algorithm uses variable-length
observing blocks to enforce cadence requirements and ensure field
observability, along with having a secondary optimization step to minimize slew
time. \blue{We show that by employing a hybrid method utilizing both this
scheduler and \texttt{gwemopt}, the previous scheduler used, in concert, we can
achieve an average improvement in detection efficiency of 3\%-11\% over
\texttt{gwemopt} alone} for a simulated binary neutron star merger data set
consistent with LIGO-Virgo's third observing run, highlighting the potential of
mixed integer target of opportunity schedulers for future multimessenger
follow-up surveys.
