Problem 45 - Thomas Calculus

Question

Wilson lot size formula One of the formulas for inventory management says 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, radios, 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). a. Your job, as the inventory manager for your store, is to find the quantity that will minimize \(A(q) .\) What is it? (The formula you get for the answer is called the Wilson lot size formula. b. Shipping costs sometimes depend on order size. When they do, it is more realistic to replace \(k\) by \(k+b q,\) the sum of \(k\) and a constant multiple of \(q .\) What is the most economical quantity to order now?

Step-by-Step Solution

Answer

a. The formula for the quantity is \( q = \sqrt{\frac{2km}{h}} \). b. With shipping, the quantity is \( q = \sqrt{\frac{k m}{b + \frac{h}{2}}} \).

Step 1: Differentiate the Cost Function

To find the quantity \( q \) that minimizes \( A(q) \), we first find the derivative \( A'(q) \) with respect to \( q \). The given function is \( A(q) = \frac{k m}{q} + c m + \frac{h q}{2} \). Differentiating each term:\( \frac{d}{dq} \left( \frac{k m}{q} \right) = -\frac{k m}{q^2} \),\( \frac{d}{dq} (c m) = 0 \) (since \( c m \) is constant),\( \frac{d}{dq} \left( \frac{h q}{2} \right) = \frac{h}{2} \).Thus, \( A'(q) = -\frac{k m}{q^2} + \frac{h}{2} \).

Step 2: Set the Derivative to Zero

To find the minimum, set the derivative \( A'(q) \) equal to zero: \(-\frac{k m}{q^2} + \frac{h}{2} = 0\).Solving for \( q \), we have \(-\frac{k m}{q^2} = -\frac{h}{2}\).This implies \( \frac{k m}{q^2} = \frac{h}{2} \).

Step 3: Solve for q

From \( \frac{k m}{q^2} = \frac{h}{2} \), solve for \( q^2 \) by multiplying both sides by \( q^2 \):\( k m = \frac{h q^2}{2} \).Rearrange to get \( q^2 = \frac{2km}{h} \).Taking the square root of both sides gives \( q = \sqrt{\frac{2km}{h}} \). This is the Wilson lot size formula, the optimal order quantity.

Step 4: Modify Cost Function with Shipping Cost

Now consider the modified problem where \( k \) is replaced by \( k + b q \):\( A(q) = \frac{(k + b q)m}{q} + c m + \frac{h q}{2} \).Simplify to get:\( A(q) = \frac{k m}{q} + b m + c m + \frac{h q}{2} \).Differentiate with respect to \( q \).

Step 5: Differentiate the Modified Cost Function

Find the derivative of the modified \( A(q) \):\( A'(q) = -\frac{k m}{q^2} + \frac{h}{2} \).Again, differentiate the additional term \( b m \) to get \( b \), so the complete derivative is now:\( A'(q) = -\frac{k m}{q^2} + b + \frac{h}{2} \).

Step 6: Solve for Economic Order Quantity with Shipping Costs

Set the new derivative to zero for minimization:\(-\frac{k m}{q^2} + b + \frac{h}{2} = 0\).Solving gives \( -\frac{k m}{q^2} = -b - \frac{h}{2} \).Thus, \( \frac{k m}{q^2} = b + \frac{h}{2} \).Rearranging, \( q^2 = \frac{k m}{b + \frac{h}{2}} \).Therefore, \( q = \sqrt{\frac{k m}{b + \frac{h}{2}}} \). This is the revised lot size formula accounting for shipment cost.

Key Concepts

Inventory Management

Inventory management plays a crucial role in ensuring a business runs smoothly. The goal is to maintain a balance between supply and demand while minimizing costs. Effective inventory management helps a company to avoid running out of stock or holding excess inventory. It involves several activities such as ordering, storing, and using a company's inventory.
Optimizing inventory levels ensures that products are available when customers need them, which enhances customer satisfaction and reduces carrying costs.

• Ordering Costs: These include costs related to creating and processing an order.
• Holding Costs: Also known as carrying costs, these cover storage, utilities, and other expenses incurred while inventory is held.
• Shortage Costs: Costs that arise when inventory is insufficient to meet demand.

The Wilson lot size formula aids in determining an optimal order quantity, reducing overall costs associated with inventory.

Cost Function Differentiation

Differentiation is a mathematical technique used to find how a function changes at any given point. When applied to cost functions in inventory management, it helps determine the order quantity that minimizes costs. In the Wilson lot size problem, the cost function is given as:\[ A(q) = \frac{k m}{q} + c m + \frac{h q}{2} \] where:

• \( k \) is the fixed cost of an order,
• \( m \) is the number of units sold per week,
• \( c \) is the cost of one item, and
• \( h \) is the holding cost per unit.

To minimize \( A(q) \), we find the derivative \( A'(q) \) and set it to zero to solve for \( q \). This process identifies the optimal point where the costs involved in purchasing and holding inventory are minimized.

Economic Order Quantity

The Economic Order Quantity (EOQ) model determines the ideal order quantity that minimizes the sum of ordering, holding, and stockout costs. Derived as the Wilson lot size formula, it gives managers a mathematical basis for decision-making. The formula is expressed as:\[ q = \sqrt{\frac{2km}{h}} \]Here, the EOQ formula balances the trade-off between ordering frequency and carrying inventory. Ordering in larger quantities reduces the frequency of orders, thus lowering ordering costs. However, it increases holding costs. Conversely, ordering smaller quantities more frequently reduces holding costs but raises ordering costs.
The aim is to find a "sweet spot" where combined costs are minimized, ensuring efficient inventory management and contributing to improved cash flow and profitability.

Shipping Costs Adjustment

Shipping costs can alter the dynamics of inventory management. When shipping costs depend on order quantities, it adds another layer of complexity. This modification is made by incorporating a variable cost component into the standard inventory formula. In our problem, the fixed ordering cost \( k \) is updated to \( k + bq \), where \( b \) represents the shipping cost per unit order increment.Updating the formula, it now becomes:\[ A(q) = \frac{(k + bq)m}{q} + c m + \frac{h q}{2} \] To find the new optimal quantity, differentiate this revised cost function with respect to \( q \) and solve:\[ q = \sqrt{\frac{k m}{b + \frac{h}{2}}} \]This adjusted formula considers both constant and variable costs associated with ordering and shipping, allowing for more realistic business forecasting and planning.