Flu-related pandemics are not new phenomena and they have taken the valuable lives of millions over centuries. COVID-19, like other flu viruses, causes mild-to-severe respiratory illnesses that can lead to death in severe cases. Among various medical supplies needed to effectively treat flu-infected patients, and COVID-19, in particular, mechanical ventilators have proven to be a piece of essential medical equipment, because they minimize the conceivable health damage to patients and maximize their survival rates. For COVID-19-infected patients in critical conditions, denied ventilator access can result in a life-or-death struggle. Given the two uncertainties involved in ventilator stockpiling decisions, i.e., the exact timing of the next pandemic and its magnitude (demand for ventilators), governments face the following two risks (i) stockpiling too much (purchasing and holding) or (ii) stockpiling too little (potentially losing lives). Clearly, finding the right solution for such a trade-off is challenging. One politician pointed out that Operations Management (OM) techniques can be used as a remedy to help governments make a stockpiling decision that simultaneously balances the two competing risks. To illustrate the relevance and usefulness of OM techniques in this strategic decision-making, we use the ventilator demand data from the ongoing COVID-19 pandemic. As stated earlier, if ventilator access is denied to critical flu-infected patients, their lives are likely to perish. We assign a cost equal to $9.3 million for each patient's life lost (due to the unavailability of ventilators). The estimated cost of purchasing a high-end ventilator suitable for critical flu-infected patients is $65,000. This cost is not the only cost that governments need to consider; holding cost and salvage value must also be taken into account. From the past pandemics, we can find the average inter-arrival times between two successive major influenza pandemics is 15 years and governments thus need to factor the associated holding cost alongside the purchasing cost of a ventilator. Considering 15% of ventilators' purchasing cost as the annual holding cost, the average holding cost for 15 years will be $146,250 (= $65,00015% 15 years on average). The holding cost plus the purchasing cost is therefore $211,250 for each ventilator kept in the national stockpile. It is estimated that each patient requires to be connected to a ventilator for an average of two weeks. Assuming that the next pandemic will last 4 months (for simplicity, assuming each month has 4 weeks), an inventoried ventilator can serve up to 8 patients in a pandemic. Assume that ventilators will be discarded at the end of the next pandemic with their salvage value equivalent to 10% of the purchase price as they may have sat long enough in inventory that has lost some of their functionalities. Assume the demand distribution of ventilators for the next pandemic in Canada is normally distributed with an average of 1,000 and a standard deviation of 300. Imagine you are providing advice to the Prime Minister of Canada, Justin Trudeau, on this critical matter. Specifically, you are tasked to find the answers to the following questions.

1. What is the cost of underage?

2. What is the cost of overage?

3. Calculate the Critical Fractile. Use four decimal places.

4. What is the z-value associated with the critical fractile? Use four decimal places.

5. Calculate the optimal order quantity.

6. Calculate the expected number of ventilators shortage after the pandemic is done. Use six decimal places for the calculation of L(z).

7. Calculate the expected number of ventilators leftover. Use two decimal places.

8. What percentage of the order quantity is constituted by the safety capacity? (e.g. 0.6827). Do not include the % sign in your answer

Please explain steps in answer for me to understand the concepts.