$4.$ Dynamic Lot Sizing $($reasoning$)$

Suppose that the demand over the five periods is $(20,30,20,30,30),$ and a practitioner make the

following inventory management plan: the amount of products produced at the beginning of each

period is $(30,90,0,0,10).$ The inventory holding cost per period per product is $$1.$

$(1)$ What is the inventory at the end of each period?

$(2)$ You may find that the inventory level at the end of fourth period is not zero, yet the production

is nonzero in period $5.$ Clearly, it is not an optimal inventory management schedule. What is the

maximum cost that can be saved, through decreasing the last production before period $4$ and

increase the production after period $4$ at the same amount, so that the production at period $5$ begins

with zero inventory?

$5.$ Dynamic Lot sizing $($computing$)$

Solve the following dynamic lot sizing problem: determine the order$/$production quantity in each

period and the level of inventory at the end of each period, using the following policies

$-$ EOQ

$-$ Silver$-$meal heuristic

$-$ Part$-$period balancing

$-$ Wagner$-$Whitin procedure

The demand over the six$-$month horizon is $(20,40,50,30,10,30).$ The order setup cost is $$100,$ if

the order is placed in the first three months, and the order setup cost is $$150$ if the order is placed