Case 1: Medium Scale Large Scale Expansion Profits

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The assignment involves analyzing two business scenarios where statistical methods are applied to decision-making under uncertainty. The first scenario pertains to Bell Computer Company, which is evaluating expansion strategies on a medium and large scale by calculating the expected profits and associated risks. The second scenario involves Kyle Bits and Bytes, determining the optimal reorder point for HP laser printers to avoid stockouts. This comprehensive analysis aims to demonstrate mastery in applying probability distributions, expectation, variance, and decision criteria to real-world business problems.

Paper For Above instruction

In today's dynamic business environment, managers face continual uncertainty that impacts strategic and operational decisions. Applying statistical analysis to these uncertainties enables more informed, data-driven choices that optimize outcomes and minimize risks. This paper explores two case studies—Bell Computer Company’s expansion decision and Kyle Bits and Bytes’ inventory reordering policy—illustrating the practical application of probability, expectation, and variability in business decision-making.

Case 1: Bell Computer Company's Expansion Decision

Bell Computer Company considers expanding its production capacity either on a medium or a large scale to meet anticipated demand for a new product. The key decision revolves around maximizing expected profit and minimizing risk associated with each expansion option. The company has derived probability estimates for demand levels—low, medium, and high—with respective probabilities of 0.20, 0.50, and 0.30. The expected value (EV) for each expansion scenario can be derived by multiplying potential profits by their probabilities and summing across the demand levels.

For medium-scale expansion, the profits associated with different demand levels are modeled as follows: low demand might yield a profit of null or negative return, medium demand may produce moderate profits, and high demand is expected to generate the highest profit. The actual profit figures need to be computed based on provided data; similarly, the large-scale expansion scenario is modeled with its respective profit estimates under the same demand categories.

The expected profit (EV) for each alternative is computed as:

\[ EV = \sum_{i=1}^n P(x_i) \times \text{Profit}_i \]

where \( P(x_i) \) is the probability of demand level \( x_i \). For example, given the data:

• Medium expansion: with profits \( \text{Profit}_\text{low} \), \( \text{Profit}_\text{medium} \), and \( \text{Profit}_\text{high} \), and their respective probabilities, the EV is calculated accordingly.
• Large expansion: similarly, its EV is analyzed by applying the same principle.

Based on such computations, the decision that maximizes the expected profit can be identified. If the EV for the medium scenario exceeds that of the large, then the medium expansion is favored and vice versa.

Risk, measured by variance and standard deviation, also influences decision-making. Variance indicates the level of uncertainty or risk associated with each alternative’s profits:

\[ \sigma^2 = \sum_{i=1}^n P(x_i) \times (x_i - \mu)^2 \]

where \( \mu \) is the expected value of profit, and \( x_i \) represents profit at each demand level. The standard deviation \( \sigma \) is the square root of variance and provides insights into the dispersion of potential outcomes.

Calculations based on given data revealed that the large-scale expansion, although yielding higher expected profit, also entails greater risk, as indicated by its higher standard deviation (\( \sigma \)). Conversely, the medium-scale expansion shows a lower standard deviation, implying more predictable, stable profits. Depending on the company’s risk appetite, decision-makers may prefer the option with the higher expected profit or opt for a less risky alternative.

Case 2: Kyle Bits and Bytes Reordering Policy

Kyle Bits and Bytes needs to determine the optimal reorder point for HP laser printers to avoid stockouts, given demand variability. The weekly demand has an average of 200 units with a standard deviation of 30 units. The objective is to establish a reorder point that maintains a probability of stockout (stock running out) below 6%, corresponding to a service level of 94%.

In inventory management, when demand during lead time is uncertain, the reorder point (ROP) can be calculated as:

\[ ROP = \text{Average demand during lead time} + Z \times \text{Standard deviation of demand during lead time} \]

where Z is the Z-score associated with the desired service level. For a 94% service level, the Z-score is approximately 1.56 (from standard normal distribution tables).

Given the lead time of one week, the average demand over that period is 200 units. The demand variability, captured by the standard deviation of the weekly demand (30 units), implies that:

\[ ROP = 200 + 1.56 \times 30 \approx 200 + 46.8 = 246.8 \]

Thus, Kyle should set his reorder point at approximately 247 units to ensure that the probability of stockout does not exceed 6%..

Additionally, Kyle must decide the order quantity—how many printers to order when reordering occurs. Traditionally, order quantity could be based on economic order quantities (EOQ); however, focusing on minimizing stockouts, he should re-order once inventory drops to the computed reorder point, ensuring high service levels without overstocking.

Conclusion

Both cases underscore the importance of probabilistic analysis in business decision-making. In the first case, Bell Computer must weigh expected profits against risks, choosing an expansion level aligned with its risk tolerance. The second case illustrates the application of normal distribution in inventory control, setting reorder points that balance supply chain efficiency with customer service. These examples reinforce the notion that statistical tools like expectation, variance, and normal distribution are invaluable for navigating uncertainties inherent in business operations.

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