$1.1.$ State clearly and explain the difference between chance$-$constrained programming and two stage stochastic with recourse, using the following dimensions:

$($i$)$ Concept of the formulation.

$($ii$)$ Difficulty of solving.

$($iii$)$ Interpretation of the solutions.

For each dimension provide clear justifications for your claims.

$2.$ An inventory manager has two products to supply for his customers. Each month he must make the order for the two products. The products are not useable after the month. If during the month he realized the quantity he orders is finished he can replenish with a higher cost. Assume the cost at the beginning of the month are c$1$ and c$2$ and the cost of both products are b$1$ and b$2.$ Formulate the problem as a two$-$stage stochastic programming and derive the solution if you know the cumulative distribution of the demand for products $1$ and $2$ are H$($x$)$ and L$($x$)$respectively$.$