Problem 78 - University Calculus: Early Transcendentals

Question

One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is $$A(q)=\frac{k m}{q}+c m+\frac{h q}{2}$$ where \(q\) is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be): \(k\) is the cost of placing an order (the same, no matter how often you order); \(c\) is the cost of one item (a constant); \(m\) is the number of items sold each week (a constant); and \(h\) is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). Find \(d A / d q\) and \(d^{2} A / d q^{2}\)

Step-by-Step Solution

Answer

First derivative: \(-\frac{k m}{q^2} + \frac{h}{2}\); second derivative: \(\frac{2 k m}{q^3}\).

Step 1: Understand the problem

You need to find the first and second derivatives of the function \( A(q) = \frac{k m}{q} + c m + \frac{h q}{2} \) with respect to \( q \). These derivatives will help understand how the average weekly cost changes as the order quantity \( q \) changes.

Step 2: Compute the first derivative

Differentiate each term in the function \( A(q) \) with respect to \( q \). For \( \frac{k m}{q} \), use the power rule: it becomes \(-\frac{k m}{q^2} \). The second term \( c m \) is a constant with respect to \( q \), so its derivative is 0. For \( \frac{h q}{2} \), differentiate to get \( \frac{h}{2} \). Therefore, the first derivative is \( \frac{dA}{dq} = - \frac{k m}{q^2} + \frac{h}{2} \).

Step 3: Compute the second derivative

Differentiate the first derivative \( \frac{dA}{dq} = -\frac{k m}{q^2} + \frac{h}{2} \) with respect to \( q \). The derivative of \(-\frac{k m}{q^2} \) is \( \frac{2 k m}{q^3} \) using the power rule again, and the second term \( \frac{h}{2} \) is a constant, thus its derivative is 0. So \( \frac{d^2 A}{dq^2} = \frac{2 k m}{q^3} \).

Key Concepts

Derivatives

Derivatives are a foundational concept in differential calculus. They are used to measure how a function changes as its input changes. In the context of this exercise, we are dealing with the cost function \( A(q) = \frac{k m}{q} + c m + \frac{h q}{2} \), where \( q \) represents order quantity. To find the derivative of \( A(q) \) with respect to \( q \), we need to apply differentiation rules.

• The derivative \( \frac{dA}{dq} \) gives us the rate at which the average weekly cost \( A(q) \) changes as the order quantity \( q \) changes.
• The first derivative \( \frac{dA}{dq} = -\frac{k m}{q^2} + \frac{h}{2} \) is derived by differentiating each term individually. Constant terms like \( c m \) disappear as their rate of change with respect to \( q \) is zero.
• The second derivative \( \frac{d^2 A}{dq^2} = \frac{2 k m}{q^3} \) informs us about the curvature of \( A(q) \) – essentially how the rate of change \( \frac{dA}{dq} \) itself is changing.

Understanding these derivatives helps us analyze behaviors such as minimizing the cost function with respect to order quantity.

Inventory Management

Inventory management involves efficiently overseeing the constant flow of units into and out of an existing inventory. The primary aim is to ensure that the right amount of stock is available at the right time, thus minimizing costs while maximizing profits. A critical aspect of inventory management is determining the optimal order quantity, often guided by balancing ordering and holding costs.Using the formula \( A(q) = \frac{k m}{q} + c m + \frac{h q}{2} \), businesses can manage costs through:

• Ordering Costs: These are the expenses related to placing and receiving orders. \( k \) represents the fixed cost of ordering, which remains constant regardless of order size.
• Holding Costs: These costs associate with storing unsold goods. The term \( \frac{h q}{2} \) reflects the linear relationship between order quantity and holding costs.
• Balancing Act: Finding the right \( q \) is crucial. Ordering too much increases holding costs, while ordering too little can increase ordering frequency and thus increase costs.

A mastery of these aspects allows a company to minimize costs and optimize supply chains effectively.

Cost Function Analysis

Cost function analysis is an essential tool that helps businesses understand their cost behavior concerning different variables like production levels or, in this case, order quantities. Analyzing and deriving cost functions provide valuable insights into which areas might benefit from optimization.In the given function \( A(q) = \frac{k m}{q} + c m + \frac{h q}{2} \) we analyze:

• Variable Costs: These are costs that vary depending on the number of goods produced or sold. The function helps us identify how variable costs influence total cost incurrence.
• Optimization Potential: By understanding \( \frac{dA}{dq} \) and \( \frac{d^2A}{dq^2} \), businesses can determine the influence of changing order quantities on costs, allowing identification of optimal cost strategies.
• Strategic Planning: Understanding cost functions helps businesses forecast budgets, make strategic decisions on production, and set pricing models that ensure competitiveness in the market.

Cost function analysis thus not only aids in understanding current cost structures but also in planning and strategy formulation for future operations.