Suppose The Demand For A Company's Product In Weeks 1 And 2

Suppose That The Demand For A Companys Product In Weeks 1 2 And 3 A

Suppose that the demand for a company’s product in weeks 1, 2, and 3 are each normally distributed, with mean demands of 50, 45, and 65 units respectively, and standard deviations of 10, 5, and 15 units respectively. Assuming these demands are independent, the total demand over three weeks is also normally distributed. The mean total demand is calculated by summing the individual weekly means: 50 + 45 + 65 = 160 units. The variance of the total demand is the sum of the individual variances: 10^2 + 5^2 + 15^2 = 100 + 25 + 225 = 350. The standard deviation of the total demand is the square root of the total variance: √350 ≈ 18.71 units. It might be tempting to sum the standard deviations directly, but this approach is incorrect because variances (not standard deviations) are additive for independent variables. The standard deviation of the total demand reflects the spread of the combined distribution, which in this case is approximately 18.71 units.

Given the current stock of 180 units and no future shipments, the company’s total demand over three weeks is normally distributed with mean 160 and standard deviation approximately 18.71. To determine the probability of running out of stock, we calculate the probability that demand exceeds 180 units: P(Demand > 180). This requires standardizing the demand to a Z-score: Z = (180 - 160) / 18.71 ≈ 20 / 18.71 ≈ 1.07. Referencing standard normal distribution tables or using statistical software, the probability that demand exceeds this value is P(Z > 1.07) ≈ 0.1421, or roughly 14.21%. Therefore, there is about a 14.2% chance that the company will run out of units over the three-week period if no additional stock is received, highlighting the importance of inventory planning and risk management in supply chain operations.

Paper For Above instruction

Effective inventory management is crucial for companies to meet customer demand while minimizing costs associated with stockouts and excess inventory. In this context, understanding demand variability over time and calculating the probability of inventory depletion are essential. The problem at hand involves assessing the likelihood that a company's current stock of 180 units will be insufficient over a three-week period, given the probabilistic nature of demand in each week modeled as normally distributed variables.

Initially, the demands for weeks 1, 2, and 3 are characterized by their means and standard deviations: 50, 45, and 65 units with standard deviations of 10, 5, and 15 units, respectively. Assuming these demands are independent, the aggregate demand over the three-week period also follows a normal distribution, with its mean and variance derived from those of the individual weeks. The total mean demand is straightforwardly calculated as the sum of weekly means: 50 + 45 + 65, resulting in 160 units. For the variance, the sum of the squares of the weekly standard deviations is computed: 10^2 + 5^2 + 15^2, which equals 350. Taking the square root provides the overall standard deviation: √350 ≈ 18.71 units.

It's critical to recognize that the standard deviation of the total demand cannot simply be the sum of individual standard deviations because variances add for independent variables, not standard deviations. This understanding ensures correct calculation of the combined distribution's spread. Once the total demand distribution parameters are established, we evaluate the probability that demand exceeds the available stock. Standardizing the threshold (180 units) converts the problem into finding the probability that a standard normal variable exceeds a certain Z-score. This Z-score is calculated as (180 - 160) / 18.71 ≈ 1.07.

Using standard normal distribution tables or computational tools, the probability of demand surpassing 180 units, given a mean of 160 and standard deviation of approximately 18.71, is about 14.21%. This indicates a moderate risk — roughly a 14% chance — that the company will run out of stock within the three-week period if no replenishments occur.

This analysis demonstrates how probabilistic modeling informs inventory decisions. Companies can use such calculations to establish safety stock levels, balancing the costs of excess inventory against the risks of stockouts. Incorporating demand variability into planning strategies enhances operational resilience and customer satisfaction, especially in supply chain environments characterized by uncertainty and fluctuating demands.

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