A market requires a program to manage product inventory. The demand for the

product follows a normal distribution with a mean of $20$ and a variance of $7.$ The

company aims to maintain a safety stock of at least $5$ products, and the maximum

number of products available for ordering in each period is $22.$ If the demand cannot

be met, it will be backordered to be fulfilled in subsequent periods. However, the

policy is to avoid shortages if possible. The supplier offers different ordering costs per

item based on the quantity ordered. For orders less than $8,$ the cost is $$3$ per product.

For orders between $8$ and $18,$ the cost is $$2$ per item, and for orders greater than $18,$

the cost is $$1$ per product.

Write a program for inventory management. Test your program by generating

demand data for $10$ periods and provide the amount of orders in each period, the

inventory at each period, and the total ordering cost. Determine if any shortages

occurred and, if so$,$ in which periods. Provide the code, data, and results.