$2)$ Consider a dynamic lot$-$sizing problem with two products $1$ and $2$ under a big time$-$bucket setting. For each product i$,$ the demand at time t is Di$,$t$.$ For each product i and in each period t$,$ the setup cost is Ai$,$t$,$ the production cost is ci$,$t and the holding cost for carrying a unit of inventory from period t to t $+1$ is hi$,$t$.$ All demand must be met in each period, with no backorders. There is no initial inventory for either product. There are no capacity constraints, and the production facility can produce both products in each period if needed. Assume that the demand for product $1$ can be satisfied using product $2,$ but only product $2$ can satisfy the demand for product $2.($Think of product $2$ as being of better quality.$)$ In each period t and for i in $\{1,2\},$ let Xi$,$t denote the quantity of product i that is used to meet the demand for product $1.$

$($a$)(2$ points$)$ What equation relates X$1,$t and X$2,$t for each t$?$

$($b$)(3$ points$)$ State the flow balance equations for the inventory of each product

$($c$)(3$ points$)$ Formulate this lot$-$sizing problem as an MILP. State all the decision variables, the objective and all the constraints that the decision variables must satisfy.