Explanation:

Part $1$:

Expected units short:

To find the expected units short, you can calculate the z$-$value for the desired service level and then use it to find the safety stock. The safety stock is the inventory maintained above the average demand to reduce the risk of running out of stock. The z$-$value can be looked up in a standard normal distribution table, corresponding to the desired service level. For example, for a service level of $95\%,$ the z$-$value is approximately $1\mathrm{.}65.$ To calculate the safety stock, you would multiply the z$-$value by the standard deviation and add it to the mean demand. Subtracting the mean from this value gives you the expected units short.

Expected profit:

To calculate the expected profit, you can subtract the cost of the vegetables from the revenue generated by selling them. This profit can be multiplied by the probability distribution of demand and summed to find the expected profit.

Part $2$:

Optimal order quantity:

To maximize daily profits, you can use the Economic Order Quantity $($EOQ$)$ model. You can use the EOQ formula, which takes into account the demand, ordering cost, and holding cost to determine the optimal order quantity. The formula for EOQ is the square root of $[(2*$ demand $*$ ordering cost$)/$ holding cost$].$

Expected units short:

You can calculate the expected units short using the safety stock approach described in Part $1.$ Once the optimal order quantity is determined, you can use the same method to find the expected units short.

Expected profit:

Similarly, once the optimal order quantity is known, you can calculate the expected profit by considering the revenue generated from selling the vegetables minus the cost of purchasing them and then accounting for the probability distribution of demand.

Step $2$

Explanation:

Part $3$:

Optimal order size:

Incorporating the penalty of stocking out changes the approach for determining the optimal order size. You would need to consider the total cost, including the penalty cost, of stockouts in addition to the ordering and holding costs. The new EOQ formula would take this penalty cost into account to identify the optimal order size.

Expected units short:

The expected units short can be calculated using the safety stock approach, considering the penalty cost for stockouts as well.

Expected profit:

The expected profit can be derived using the updated costs and incorporating the penalty for stockouts. This would involve calculating the revenue, the cost of purchasing the vegetables, and the penalty cost for stockouts, considering the probability distribution of demand.

Part $4$:

Optimal order size:

Given the empirical distribution of demand, the new optimal order size can be determined using the EOQ formula, considering the stockout penalty as in Part $3.$

Expected units short:

Using the safety stock approach with the updated optimal order size and the probability distribution of demand will provide the expected units short.

Expected profit:

Similar to the previous parts, the expected profit can be calculated by considering the revenue, cost, and penalty cost associated with stockouts, and accounting for the probability distribution of demand based on the new optimal order size.

Answer

It's important to note that these calculations and optimization methods would provide guidance for determining the best order quantity to maximize profits while considering the risk of stockouts and associated penalties, helping to fine$-$tune the inventory management strategy for $"8$ Asparagus and $9$ Beets."