Problem 29 - Applied Calculus

Question

A business sells an item at a constant rate of \(r\) units per month. It reorders in batches of \(q\) units, at a cost of \(a+b q\) dollars per order. Storage costs are \(k\) dollars per item per month, and, on average, \(q / 2\) items are in storage, waiting to be sold. [Assume \(r, a, b, k\) are positive constants.] (a) How often does the business reorder? (b) What is the average monthly cost of reordering? (c) What is the total monthly cost, \(C\) of ordering and storage? (d) Obtain Wilson's lot size formula, the optimal batch size which minimizes cost.

Step-by-Step Solution

Answer

(a) Reorders every \(\frac{q}{r}\) months. (b) Average cost is \(\frac{ra}{q} + rb\). (c) Total cost \(C = \frac{ra}{q} + rb + \frac{q}{2} k\). (d) Optimal \(q = \sqrt{\frac{2ra}{k}}\).

Step 1: Calculate Reorder Frequency

To find how often the business reorders, we divide the reorder quantity by the rate at which items are sold. The reorder frequency is given by the formula: \[ T = \frac{q}{r} \] where \(T\) is the time in months between reorders. Thus, the business reorders every \(\frac{q}{r}\) months.

Step 2: Calculate Reordering Cost

The cost to reorder is \(a + bq\) per order. To find the average monthly cost of reordering, calculate the number of orders per month: \(\frac{r}{q}\). Then multiply by the reorder cost: \[ \text{Average monthly reordering cost} = \left(\frac{r}{q}\right) (a + bq) = \frac{ra}{q} + rb \]

Step 3: Calculate Average Storage Cost

The average storage cost per month is based on the average quantity held \(\frac{q}{2}\) and the storage cost \(k\). Hence, the average monthly storage cost is: \[ \text{Average monthly storage cost} = \frac{q}{2} \, k \]

Step 4: Determine Total Monthly Cost

The total monthly cost, \(C\), combines reordering and storage costs: \[ C = \frac{ra}{q} + rb + \frac{q}{2} \, k \]

Step 5: Derive Wilson's Lot Size Formula

To find the order quantity that minimizes the total monthly cost, we take the derivative of \(C\) with respect to \(q\) and set it to zero: \[ C' = -\frac{ra}{q^2} + \frac{k}{2} = 0 \] Solving for \(q\) yields: \[ q = \sqrt{\frac{2ra}{k}} \] which is Wilson's lot size formula.

Key Concepts

Reorder Frequency

Reorder frequency tells us how often a business needs to purchase more inventory to keep up with its sales rate. It's a fundamental part of inventory management because it ensures that the company has enough stock to meet customer demand without overstocking, which can lead to higher storage costs.

In our exercise, the reorder frequency is calculated using the formula:

• Reorder Frequency (\(T\)) = \(\frac{q}{r}\)

Here, \(q\) represents the quantity of units ordered at once, and \(r\) is the rate at which these units are sold per month. This formula essentially tells us the number of months between each reorder.
For instance, if a business orders 100 units and sells at a rate of 25 units per month, it will reorder every 4 months.

Reordering Cost

Reordering cost refers to the expenses incurred every time a new order is placed to replenish inventory. It's crucial for businesses to minimize these costs as they can quickly add up, especially in high-turnover industries.

In the exercise, the reordering cost for each order is given by the formula:

• Reordering Cost per Order = \(a + bq\)

Where \(a\) is a fixed cost per order, and \(bq\) is a variable cost dependent on the order size. To find out the average reordering cost per month, we calculate the number of orders per month (\(\frac{r}{q}\)) and multiply it by the per-order cost:

• Average Monthly Reordering Cost = \(\left(\frac{r}{q}\right)(a + bq)\)
• Simplified to = \(\frac{ra}{q} + rb\)

This breakdown helps businesses see exactly where their reordering costs are coming from, allowing for more strategic ordering plans.

Storage Cost

Storage cost is another essential factor in inventory management, representing the cost to hold inventory over a period. Effective management of these costs is vital to maintain profitability.

In our case, the average number of items held in storage is \(\frac{q}{2}\), since, on average, half of each order is still unsold. The formula for calculating the average monthly storage cost is straightforward:

• Average Monthly Storage Cost = \(\frac{q}{2} \, k\)

Where \(k\) is the storage cost per item per month. By knowing this cost, businesses can evaluate how much it costs them to hold inventory and look for ways to optimize their storage efficiency.

Wilson's Lot Size Formula

Wilson's lot size formula, also known as the Economic Order Quantity (EOQ) model, helps determine the optimal order quantity that minimizes the total costs associated with ordering and holding inventory.

The total monthly cost (\(C\)) is calculated by adding the reordering and storage costs:

• Total Monthly Cost = \(\frac{ra}{q} + rb + \frac{q}{2} \, k\)

By taking the derivative of \(C\) with respect to \(q\) and setting it to zero, we derive Wilson's lot size formula:

• \(q = \sqrt{\frac{2ra}{k}}\)

This formula helps businesses pinpoint the order quantity that achieves a balance between ordering too frequently (which increases reordering costs) and ordering too much at once (which spikes storage costs). Understanding and implementing Wilson's lot size formula ensures efficiency in inventory management, helping businesses consistently meet demand while keeping costs in check.