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To understand optimal inventory policies, we first need to delve into **inventory costs**. These costs include two key components: _ordering costs_ and _carrying costs_.
Ordering costs are incurred each time an order is placed. This could involve costs related to processing the order, shipping, and handling, represented by the variable \( h \), which is the cost per order. On the other hand, carrying costs reflect the expenses of holding inventory over time, such as storage, insurance, and opportunity costs. This is represented by \( s \), which is the carrying cost per item per year.
Thus, a crucial part of minimizing total inventory costs involves understanding and balancing these two components.

The **order quantity** determines how many items should be ordered each time to balance out the costs. We denote the optimal order quantity as \( q \). The goal is to find the value of \( q \) that minimizes total inventory costs.
Let's first consider the frequency of ordering. If \( Q \) represents the total sales per year, and \( N \) is the number of orders placed each year, then:

• \( N \) can be calculated as \( N = \frac{Q}{q} \), which shows how frequently we place orders based on our annual sales \( Q \) and our order quantity \( q \).

Balancing order frequency with quantity per order is essential for cost minimization.

Achieving **cost minimization** implies combining both ordering and carrying costs and finding a balance that minimizes them. The total inventory cost function is given by combining these individual costs:
When we combine the ordering cost \( h \textbackslash times N \) and the carrying cost \( s \textbackslash times \textbackslash \frac{q}{2} \), we obtain:

• \(\text{Total inventory cost} = h \textbackslash times \textbackslash \frac{Q}{q} + s \textbackslash \times \textbackslash \frac{q}{2}\)

The goal is to find the quantity \( q \) that minimizes this total cost function.

We utilize **calculus** to find the value of \( q \) that minimizes our total inventory cost. This involves taking the derivative of the total cost function with respect to \( q \) and setting it to zero:

• \(\frac{d}{dq}\bigg(h \textbackslash times \textbackslash \frac{Q}{q} + s \textbackslash times \textbackslash \frac{q}{2}\bigg) = 0\)

After taking the derivative, we simplify and solve for \( q \):

• \[-h \textbackslash \frac{Q}{q^2} + \textbackslash \frac{s}{2} = 0\]
• Rearranging and solving, we get: \( q^2 = \textbackslash \frac{2hQ}{s} \)
• Thus, \( q = \textbackslash sqrt{ \textbackslash \frac{2hQ}{s} } \)

This gives us the optimal order quantity for minimizing total inventory costs.