$1)$ A company$$s distribution network has a single plant and $10$ regional warehouses for the U$.$S$.$ market. The annual demand of the company is D$,$ which is assumed to be known. The customer zones are assigned to the warehouses in such a way that their demands are equal. Each warehouse incurs a fixed cost of K for each order from the plant, and the annual holding cost per unit is h$.$ The warehouses use the EOQ policy to order from the plant. The company decides to close four warehouses and assign their customer demand equally to the remaining six.

a$)$ What will be the impact of this risk pooling strategy on the total inventory carried in the supply chain?

b$)$ Suppose the company wants to reduce the inventory by $50\%,$ how many warehouses should it close?