I am writing a Macro using the Solver tool to solve a number of problems in the same manner.

After it has solved each case the pop up box entitled "Solver Results" appears.

What is the code to Keep Solver Solution and automatically select the OK button to avoid having to click the OK button for every case it has solved?

That is very much easy.
You just need to copy below code to replace the final row "SolverSolve"

SolverSolve UserFinish:=True
SolverFinish KeepFinal:=1


Regards,
nkyung