Certainly! Let$$s break down the problem step by step:

a$)$ Overage Cost and Underage Cost per Unit:

Overage cost per unit: The cost of holding excess inventory beyond demand. It is the difference between the cost of the product and its salvage value.

Overage cost $=$ Cost per unit $-$ Salvage value

Overage cost $=$ $$6-$ $$2=$ $$4$ per unit

Underage cost per unit: The cost of not meeting demand. It is the difference between the selling price and the cost of the product.

Underage cost $=$ Selling price $-$ Cost per unit

Underage cost $=$ $$13-$ $$6=$ $$7$ per unit

b$)$ Optimal Order Quantity $($EOQ$)$ with Normal Distribution: Given:

Mean demand $=350$ units

Standard deviation of demand $=110$ units

We want to maximize expected profit. The critical ratio $($ratio of underage cost to total cost$)$ is given by:

$[\backslash$text$\{$Critical Ratio$\}=\backslash$frac$\{\backslash$text$\{$Underage Cost$\}\}\{\backslash$text$\{$Underage Cost$\}+\backslash$text$\{$Overage Cost$\}\}]$

Using the inverse of the cumulative distribution function $($CDF$)$ for the normal distribution, we find the optimal order quantity:

$[\backslash$text$\{$Optimal Order Quantity $($EOQ$)\}=\backslash$text$\{$Inverse CDF$\}(\backslash$text$\{$Critical Ratio$\},\backslash$text$\{$Mean Demand$\},\backslash$text$\{$Standard Deviation of Demand$\})]$

Calculating the EOQ:

$[\backslash$text$\{$Optimal Order Quantity $($EOQ$)\}\backslash$approx $388\mathrm{.}36]$

c$)$ Optimal Ordering Quantity with Discrete Distribution for Demand: Given the discrete demand distribution:

Table

Quantity Probability

$1000\mathrm{.}05$

$1500\mathrm{.}05$

$2000\mathrm{.}15$

$2500\mathrm{.}15$

$3000\mathrm{.}20$

$3500\mathrm{.}15$

$4000\mathrm{.}15$

$4500\mathrm{.}05$

$5000\mathrm{.}05$