Your company sells a product whose demands over the next four months are $100,140,210$

and $180$ units, respectively. You can stock just enough supply to meet the demand each

month, or you can overstock to meet the demand for two or more consecutive months.

In the latter case, a holding cost of $$1\mathrm{.}2$ is charged per overstocked unit per month.

You estimate the unit purchase prices for the next $4$ months are $$15,$$$12,$$$10$ and $$14,$

respectively. A set$-$up cost of $$200$ is incurred each time a purchase order is placed. For

example, if your company wants to purchase enough products for next two months at the

beginning of the first month, it needs to pay

$$200+$$$15**(100+140)+$$$1\mathrm{.}2**140=$$$3968$;

if your company wants to purchase enough products for next three months at the beginning

of the first month, then it needs to pay

$$200+$$$15**(100+140+210)+$$$1\mathrm{.}2**140+$$$1\mathrm{.}2**210**2=$$$7622.$

Your company wants to develop a purchasing plan that will minimize the total costs of

ordering, purchasing and holding an item in stock.

$($a$)$ Find the optimal solution by Dijkstra's algorithm. Write down the steps clearly.

$($b$)$ Find the optimal solution by Dynamic Programming $($with backward recursion$).$

Write down the steps clearly.