C$.$ Consider the following production planning problem over a finite planning horizon of T periods. The objective is to find the production plan that minimizes total cost over the planning horizon. The demand in period t is known and is Dt units. You do not have to satisfy all demand, unsatis$\_$ed demand is lost. The unit cost for not satisfying demand is $ b$.$ There is a yield loss in production process. When X $>0$ units are produced, only aX can be used to satisfy the demand in that period $(0<$ a $<1).$ Notice that a is a known constant. The products that cannot be used to satisfy the demand are discarded immediately. The production lead$-$time is negligible, that is products produced successfully in a certain period can be used to satisfy the demand in that period. There is no production capacity constraint.

There are both fixed and variable costs of production. When X $>0$ units are produced, a production cost of K $+$ cX is incurred. Notice that, production cost is incurred even if the item produced cannot be used to satisfy the demand. It is possible to carry inventory as long as$,$ ending inventory level does not exceed Imax units. Inventory holding cost is $ h per unit per period. Model the problem using Dynamic Programming using backward recursion. Clearly provide the following with an appropriate notation and verbal explanations:

$1.$ Stage

$2.$ Decision$($s$)$ to make

$3.$ State

$4.$ Stage transformation $($Express the next stage's state in terms of the

current stage's state and the decisions made$)$

$5.$ Action space $($allowable actions$)$ at each stage

$6.$ State space for a stage

$7.$ A mathematical expression for the backward recursive function for any

stage other than the last stage

$8.$ What does your recursive function represent? Provide a verbal explanation

for the backward recursive function for any stage other than the last stage.

$9.$ Backward recursive function for the last stage

D$.$ Consider the following version of the problem in Question $3.$ There is still a yield loss in production process. When X $>0$ units are produced, only aX can be used to satisfy the demand in that period. But, now a is a random variable that takes the value of ai with probability pi for i $=1,.......$ n$.0<$ ai $<1,$ i $=1,.......$ n and $$Pi from i to n$.$

The rest of the problem situation is exactly the same as in Question $3.$ Now, your objective is to minimize expected cost over the planning horizon.

What modifications are required in your answer to Question $3?$ Clearly state required modifications on the components of DP that you provide for Question $3.$