  We prove a strong law of large numbers for directed last passage times in an
independent but inhomogeneous exponential environment. Rates for the
exponential random variables are obtained from a discretisation of a speed
function that may be discontinuous on a locally finite set of discontinuity
curves. The limiting shape is cast as a variational formula that maximises a
certain functional over a set of weakly increasing curves.
  Using this result, we present two examples that allow for partial analytical
tractability and show that the shape function may not be strictly concave, and
it may exhibit points of non-differentiability, flat segments, and
non-uniqueness of the optimisers of the variational formula. Finally, in a
specific example, we analyse further the macroscopic optimisers and uncover a
phase transition for their behaviour.
