A Car Dealer Believes That Demand For A 2014 Car Model Will

A Car Dealer Believes That Demand For A 2014 Car Model Will Be Normall

A car dealer believes that demand for a 2014 car model will be normally distributed with a mean of 200 and a standard deviation of 30. His cost of receiving a car model is $25,000, and he sells it for $40,000. Half of all leftover cars can be sold for $30,000. He is considering ordering 200, 220, 240, 260, 280, or 300 models of that car. How many should he order? Use Excel functions like Rand(), VLOOKUP(), MIN(), COUNTIF(), and relevant analysis to determine the optimal order quantity.

Paper For Above instruction

The problem presented involves determining the optimal inventory order quantity for a car dealership based on demand uncertainty modeled by a normal distribution. This scenario epitomizes the classic Newsvendor Model in operations management, which aims to balance overstocking and understocking costs to maximize profit. The demand for the 2014 car model is assumed to be normally distributed with a mean of 200 units and a standard deviation of 30 units, suggesting variability in consumer demand that must be managed through optimal ordering decisions.

Introduction

Inventory management, especially in the context of perishable or seasonal goods like cars, requires careful analysis of demand forecasts and associated costs. The dealership's problem aligns with the Newsvendor Model, which helps determine the order quantity that maximizes expected profit or minimizes expected costs considering demand variability. In this case, the dealer's revenues, costs, and salvage value influence the decision, making the analysis a suitable application of probability and statistical tools in operations research.

Demand Distribution and Cost Structure

The demand for the cars is normally distributed with parameters:

- Mean (\(\mu\)) = 200 units

- Standard deviation (\(\sigma\)) = 30 units

Costs and revenues include:

- Cost per car received = $25,000

- Selling price = $40,000

- Salvage value (for leftover cars) = $30,000 (50% of unsold cars can be sold at this price)

- The profit per unit sold is thus $15,000 ($40,000 - $25,000), and the salvage recouped on leftovers is $5,000 less than the original cost.

Optimal Order Quantity Using the Newsvendor Model

The critical ratio (CR) is a key component in determining the optimal order quantity, calculated as:

\[

CR = \frac{\text{Selling Price} - \text{Cost}}{\text{Selling Price} - \text{Salvage Price}}

\]

Applying the values:

\[

CR = \frac{40,000 - 25,000}{40,000 - 30,000} = \frac{15,000}{10,000} = 1.5

\]

Since CR exceeds 1, it indicates that the cost structure heavily favors overordering, but as probabilities cannot exceed 1, we interpret this as the dealership should order enough units to meet the high demand, considering the demand distribution and the costs involved.

Using Excel for Quantitative Analysis

To model this problem in Excel, the dealer can simulate demand scenarios and evaluate the expected profit at different order levels (200, 220, 240, 260, 280, 300). The steps can include:

- Generate a large number of demand samples using RAND() combined with the inverse normal distribution function (NORM.INV) to reflect demand variability.

- Calculate the profit or loss at each order quantity for each demand scenario, considering the salvage and selling prices.

- Use COUNTIF() to determine the frequency of profitable outcomes at each order quantity.

- Apply VLOOKUP() to identify the order quantity that maximizes expected profit based on simulation results.

Decision Making and Recommendations

Based on the simulation and probabilistic analysis, the optimal order quantity is the one where the expected marginal profit matches the expected marginal cost, generally corresponding to the demand level at or near the service level associated with the critical ratio.

In practice, the dealer might find that ordering slightly above the mean demand (e.g., 220 or 240 units) minimizes expected costs due to demand variability and salvage options. Excessive ordering beyond this range risks higher holding costs, while ordering less than the mean risks lost sales and lost revenue opportunity.

Conclusion

The application of the Newsvendor Model, supported by Excel simulation techniques, provides the dealership with an informed basis for decision-making. By accounting for demand variability, costs, and salvage options, the dealer can select an order quantity that optimizes profitability. Adapting this model regularly with updated demand forecasts and cost analyses will help sustain optimal inventory levels in a dynamic market.

References

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