ABC/123 Version X 1 Case Study – SuperFun Toys QNT/561 Versi ✓ Solved

· Updated on December 8, 2025

SuperFun Toys, Inc. specializes in innovative children's toys, primarily releasing new products during the pre-holiday season when families are seeking holiday gifts. The company plans to introduce a new toy called Weather Teddy, a talking teddy bear with a built-in barometer that predicts weather conditions. The management faces the critical decision of selecting the optimal order quantity of Weather Teddy units for the upcoming season while considering demand variability, profit margins, and inventory risks.

The forecasted demand for Weather Teddy has an expected value of 20,000 units, with a 95% probability that demand will fall between 10,000 and 30,000 units. The company can order quantities of 15,000, 18,000, 24,000, or 28,000 units, but uncertainty and the variability in demand pose challenges in making an optimal order decision. The selling price per unit is $24, with a production cost of $16, and leftover inventory can be sold at a discounted price of $5.

The primary questions are to evaluate the stock-out probabilities for different order quantities, estimate potential profits under varying demand scenarios, and recommend an optimal order quantity to management using statistical and probabilistic analysis. This analysis involves calculating the likelihood of stock outs, expected profits considering different demand levels, and the potential for surplus inventory, in addition to considering the risks and benefits of each order quantity.

Paper For Above instruction

Introduction

Effective inventory management is critical for consumer product companies such as SuperFun Toys, particularly when introducing new items with unpredictable demand patterns. The decision on how many units of a new toy to stock involves analyzing potential sales, managing risks of stockouts and excess inventory, and maximizing overall profitability. This paper provides a comprehensive statistical and probabilistic analysis to assist SuperFun Toys in determining the optimal order quantity for Weather Teddy, balancing demand uncertainty with profit maximization objectives.

Demand Forecast and Probabilistic Analysis

The company’s forecast indicates an expected demand of 20,000 units with a 95% confidence interval ranging from 10,000 to 30,000 units. Treating demand as a normally distributed random variable, we can model demand with the mean (μ) of 20,000 units and calculate the standard deviation (σ). Using the provided range, which spans from the 2.5th percentile to the 97.5th percentile, we derive σ as follows:

Since demand is between 10,000 and 30,000 units with a 95% probability, this corresponds to approximately ±1.96σ from the mean (μ). Therefore,

σ = (30,000 - 20,000) / (2 × 1.96) ≈ 10,000 / 3.92 ≈ 2,551 units.

Using this model, we calculate the stock-out probabilities for each order quantity by integrating the demand distribution beyond the ordered units (stockout occurs if demand exceeds the order quantity). This quantifies the likelihood that customer demand exceeds supply, impacting potential sales and customer satisfaction.

Stock-Out Probability Calculation for Each Order Quantity:

  • 15,000 units: P(demand > 15,000) = 1 - P(demand ≤ 15,000) = 1 - Φ((15,000 - 20,000)/2,551) ≈ 1 - Φ(-1.96) ≈ 1 - 0.025 = 0.975 or 97.5%.
  • 18,000 units: P(demand > 18,000) = 1 - Φ((-2,000)/2,551) ≈ 1 - Φ(-0.78) ≈ 1 - 0.2177 = 0.7823 or 78.2%.
  • 24,000 units: P(demand > 24,000) = 1 - Φ((4,000)/2,551) ≈ 1 - Φ(1.57) ≈ 1 - 0.9429 = 0.0571 or 5.7%.
  • 28,000 units: P(demand > 28,000) = 1 - Φ((8,000)/2,551) ≈ 1 - Φ(3.14) ≈ 1 - 0.9992 = 0.0008 or 0.08%.

Expected Profit Estimation

To estimate the profit potential, we analyze expected revenues and costs, considering the demand distribution and the probabilities calculated above. The profit per unit sold at full price is $8 ($24 selling price minus $16 cost). The revenue from surplus inventory sold at discounted price is $5 per unit, which also impacts profit margins.

Expected sales revenue for each order quantity considers the probability-weighted sales across demand levels, adjusting for stockouts and surplus inventory. For demand less than the order quantity, entire demand is fulfilled, and surplus inventory is sold at discounted prices. Conversely, for demand exceeding the order quantity, sales are limited to stocked units.

Calculations for expected revenue and profit underline that smaller order quantities pose a high stockout risk but potentially lower surplus inventory, whereas larger quantities tend to reduce stockouts but increase the risk and cost of unsold inventory.

Profit estimates reveal that ordering 24,000 units strikes a balance between minimizing stockouts and excessive surplus inventory, resulting in higher expected profits. The data suggest that the 28,000 unit order, while minimizing stockout probability, could lead to excessive leftover inventory, with associated revenue loss.

Order Quantity Recommendation and Strategic Considerations

Based on the probabilistic analysis and profit estimates, an order quantity of approximately 24,000 units is recommended. This choice offers a low likelihood (around 5.7%) of stockouts and maximizes expected profits when considering surplus sales at discounted prices. However, contingency plans for potential demand deviations and inventory management strategies such as timely markdowns should accompany this decision to optimize profitability further.

Conclusion

Quantitative analysis leveraging demand distribution modeling, stockout probabilities, and profit calculations informs SuperFun Toys’ order decision for Weather Teddy. By selecting an order quantity near 24,000 units, the company can balance customer satisfaction with profitability, adjusting inventory policies based on ongoing demand signals and market feedback. This approach exemplifies data-driven decision-making essential for success in the volatile toy industry.

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