Statistic Assignment: Specialty Toys Inc Sells A Variety Of
Statistic Assignment 2specialty Toys Inc Sells A Variety Of New And
Develop a managerial report addressing the following issues related to order quantities, demand distribution, stock-out probabilities, profit estimates, and recommendations for the Weather Teddy product, utilizing normal distribution assumptions, probabilities, and scenario analysis based on predicted demand and sales data.
Paper For Above instruction
Introduction
Specialty Toys Inc. faces a critical decision in determining the optimal order quantity for their new product, Weather Teddy, to maximize profitability while mitigating the risk of stock-outs or excess inventory. This managerial report explores the demand distribution, probability of stock-outs, profit projections under various demand scenarios, and formulates a recommendation based on quantitative analysis.
1. Demand Distribution and Probability Range
The forecasted demand for Weather Teddy is modeled as a normal distribution with a mean (μ) of 20,000 units and a standard deviation (σ) of 5,100 units, based on historical sales data. To analyze the variability in demand, we examine the probability that actual demand will fall between the two most extreme suggested order quantities: 15,000 units and 28,000 units.
Using Excel's NORM.DIST function, we can calculate:
- P(D ≤ 15,000) = NORM.DIST(15,000, 20,000, 5,100, TRUE) ≈ 0.1224
- P(D ≤ 28,000) = NORM.DIST(28,000, 20,000, 5,100, TRUE) ≈ 0.9545
Thus, the probability that demand falls between 15,000 and 28,000 units is:
P(15,000 ≤ D ≤ 28,000) = P(D ≤ 28,000) – P(D ≤ 15,000) ≈ 0.9545 – 0.1224 = 0.8321 or approximately 83.21%.
2. Stock-Out Probabilities for Selected Order Quantities
The likelihood of stock-outs signifies the probability that demand exceeds the ordered quantity. This is calculated as:
- For Q = 15,000 units:
- P(D > 15,000) = 1 – P(D ≤ 15,000) ≈ 1 – 0.1224 = 0.8776 or 87.76%
- For Q = 18,000 units:
- P(D > 18,000) = 1 – P(D ≤ 18,000) = 1 – NORM.DIST(18,000, 20,000, 5,100, TRUE) ≈ 1 – 0.5656 = 0.4344 or 43.44%
- For Q = 24,000 units:
- P(D > 24,000) = 1 – NORM.DIST(24,000, 20,000, 5,100, TRUE) ≈ 1 – 0.9512 = 0.0488 or 4.88%
- For Q = 28,000 units:
- P(D > 28,000) ≈ 1 – 0.9545 = 0.0455 or 4.55%
3. Profit Projections under Different Demand Scenarios
The profit calculation considers selling price ($24), cost ($16), and surplus value ($5), with demand scenarios at 10,000 units (worst case), 20,000 units (most likely case), and 30,000 units (best case). The profit formula employed is:
Profit = (Units sold × $24) + (Surplus units × $5) – (Order quantity × $16)
Calculations for each suggested order quantity:
| Order Quantity | Demand Scenario | Units Sold | Remaining Inventory | Profit Calculation | Projected Profit |
|---|---|---|---|---|---|
| 15,000 | 10,000 | 10,000 | 5,000 | 10,000×24 + 5,000×5 – 15,000×16 = | $25,000 |
| 15,000 | 20,000 | 15,000 | 0 | 15,000×24 + 0×5 – 15,000×16 = | $144,000 |
| 15,000 | 30,000 | 15,000 | 15,000 | 15,000×24 + 15,000×5 – 15,000×16 = | $75,000 |
| 18,000 | 10,000 | 10,000 | 8,000 | 10,000×24 + 8,000×5 – 18,000×16 = | $34,000 |
| 18,000 | 20,000 | 18,000 | 0 | 18,000×24 + 0×5 – 18,000×16 = | $144,000 |
| 18,000 | 30,000 | 18,000 | 12,000 | 18,000×24 + 12,000×5 – 18,000×16 = | $114,000 |
| 24,000 | 10,000 | 10,000 | 14,000 | 10,000×24 + 14,000×5 – 24,000×16 = | $54,000 |
| 24,000 | 20,000 | 20,000 | 4,000 | 20,000×24 + 4,000×5 – 24,000×16 = | $144,000 |
| 24,000 | 30,000 | 24,000 | 6,000 | 24,000×24 + 6,000×5 – 24,000×16 = | $84,000 |
| 28,000 | 10,000 | 10,000 | 18,000 | 10,000×24 + 18,000×5 – 28,000×16 = | $34,000 |
| 28,000 | 20,000 | 20,000 | 8,000 | 20,000×24 + 8,000×5 – 28,000×16 = | $144,000 |
| 28,000 | 30,000 | 28,000 | 0 | 28,000×24 + 0×5 – 28,000×16 = | $224,000 |
4. Optimal Order Quantity with 60% Service Level
To determine the order quantity Q corresponding to a 60% chance of satisfying demand (and thus a 40% chance of stock-out), we use the inverse normal distribution function:
Q = NORM.INV(0.6, 20,000, 5,100) ≈ 20,523 units (approximate)
Using this Q, the projected profits are calculated similarly under the three scenarios:
- At demand = 10,000 units: Profit ≈ 10,000×24 + (Q – 10,000)×5 – Q×16 ≈ $76,531
- At demand = 20,000 units: Profit ≈ 20,000×24 + 0×5 – Q×16 ≈ $152,235
- At demand = 30,000 units: Profit ≈ 30,000×24 + 0×5 – Q×16 ≈ $102,735
5. Recommended Order Quantity and Rationale
Based on the analysis, the recommended order quantity balances the probability of stock-out and profitability. The Q of approximately 20,500 units ensures a 60% service level, aligning with the management's risk appetite. The profit projections suggest significant gains under expected conditions, with manageable risk of shortage.
Expected profit under demand scenarios:
- Worst case (10,000): $76,531
- Most likely (20,000): $152,235
- Best case (30,000): $102,735
This approach prioritizes a higher probability of demand fulfillment while safeguarding profitability, making it a robust choice for the holiday season planning.
Conclusion
This analysis provides a comprehensive quantitative foundation for Specialty Toys Inc. to make an informed decision regarding order quantity for Weather Teddy. The recommended order quantity of approximately 20,500 units offers a strategic balance between risk and return, supported by demand distribution modeling, probability assessments, and scenario-based profit estimates.
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