Operations Research Course ISC612  Project material 
Chanceconstrained spare parts optimization problem
To do
 We start from your
problem description
and formulation.

Here
is an interesting work on probabilistic programming (also known as
chanceconstrained programming).
 First, prove that your problem, as
formulated here, can be
decomposed in two separate problems that have the same formulation
and can be solved separately (consider critical and noncritical
items). From now on we only deal with one of the two problems (call
it P), as the other can be solved in exactly the same
way.
 Secondly, the current notation is not very clear. I would like you
to use the following notation: set of items A, unit
costs c, demands d, probabilities associated to each
item p, decision variables x (all these symbols are
indexed by i in A); global
probability q. Rewrite the formulation of P using
these symbols.
 Third, reformulate P such that no variables ever appear on
the sum limits. This can be done by introducing binary
variables. You have up to 10th january to do all this. If you can't
do it, contact me.