  Mathematical optimization offers highly-effective tools for finding solutions
for problems with well-defined goals, notably scheduling. However, optimization
solvers are often unexplainable black boxes whose solutions are inaccessible to
users and which users cannot interact with. We define a novel paradigm using
argumentation to empower the interaction between optimization solvers and
users, supported by tractable explanations which certify or refute solutions. A
solution can be from a solver or of interest to a user (in the context of
'what-if' scenarios). Specifically, we define argumentative and natural
language explanations for why a schedule is (not) feasible, (not) efficient or
(not) satisfying fixed user decisions, based on models of the fundamental
makespan scheduling problem in terms of abstract argumentation frameworks
(AFs). We define three types of AFs, whose stable extensions are in one-to-one
correspondence with schedules that are feasible, efficient and satisfying fixed
decisions, respectively. We extract the argumentative explanations from these
AFs and the natural language explanations from the argumentative ones.
