Fina412buyu Manufacturing Has Been Contracted To Provide Sal

Fina412buyu Manufacturing Has Beencontracted To Provide Sael Electron

Fina412buyu Manufacturing Has Beencontracted To Provide Sael Electron

Fina412buyu Manufacturing has been contracted to provide SAEL Electronics with printed circuit and motherboards (PC) boards under specific terms: manufacturing 100,000 PC boards within one month and potentially an additional 100,000 boards in three months if SAEL exercises its option. The payment per board is $5, with manufacturing costs comprising a fixed setup cost of $250,000 per batch regardless of size and a marginal cost of $2 per board. The decision for BUYU is whether to produce all 200,000 boards now or produce 100,000 now and the remaining only if SAEL exercises its option, considering a 50% chance of the option being exercised. The analysis includes assessing potential profits, decision tree construction, expected value analysis, probability ranges for optimal decisions, and the impact of risk aversion.

Paper For Above instruction

The manufacturing decision faced by BUYU involves a nuanced analysis of costs, revenues, probabilities, and risk preferences. This paper examines the potential profit involved in manufacturing all 200,000 PC boards now versus a staged approach, constructs a decision tree to visualize choices and uncertainties, evaluates expected profits to identify the optimal decision, analyzes the probability range where the decision is optimal, and considers how risk aversion alters the decision-making process.

1. Potential Profit of Manufacturing All 200,000 Boards Now

Manufacturing all 200,000 boards immediately involves significant upfront fixed costs and variable costs. The fixed setup cost per batch is $250,000, and since all 200,000 boards would be produced in one batch, the total setup cost remains $250,000. The variable manufacturing cost for 200,000 boards totals $400,000 (200,000 boards * $2 per board). Therefore, total manufacturing costs are:

  • Fixed setup cost: $250,000
  • Marginal costs for 200,000 boards: $400,000

Total manufacturing cost = $250,000 + $400,000 = $650,000.

The total revenue depends on the number of boards SAEL purchases. With 100,000 boards sold at $5 each, revenue is $500,000. If SAEL exercises its option to buy the additional 100,000 boards, revenue increases to $1,000,000; otherwise, it remains at $500,000.

The profit calculation assumes all manufactured boards are sold, whether in part or total:

  • If SAEL does not exercise the option: Profit = Revenue ($500,000) – Manufacturing costs ($650,000) = -$150,000 (a loss).
  • If SAEL exercises the option: Profit = Revenue ($1,000,000) – Manufacturing costs ($650,000) = $350,000.

Given the 50% probability that SAEL will exercise the option, the expected profit if all 200,000 boards are manufactured now is:

Expected Profit = (0.5 $350,000) + (0.5 -$150,000) = $175,000 - $75,000 = $100,000.

2. Decision Tree for the BUYU Decision

The decision tree begins with BUYU choosing between manufacturing all 200,000 boards immediately or manufacturing 100,000 now and the remaining 100,000 later if SAEL exercises its option. The subsequent chance node reflects SAEL's decision to exercise its option (with a 50% chance) or not.

  • Decision Node 1: Manufacture all now OR manufacture 100,000 now and wait.
  • Chance Node: SAEL exercises the option (50%) OR does not (50%).

In the staged approach:

  • If SAEL exercises, the remaining 100,000 boards are produced with the same costs, and revenue is double the initial amount.
  • If SAEL does not exercise, no additional production occurs, and revenue remains from the initial 100,000 boards.

3. Expected Profit and Preferred Course of Action

Calculating the expected profit for each scenario:

Manufacture all 200,000 now:

  • Expected profit = (Probability SAEL exercises profit if exercised) + (Probability SAEL does not exercise profit no exercise)

As previously computed, the expected profit is $100,000.

Manufacture 100,000 now and wait:

This approach requires analyzing the expected profit of waiting:

  • If SAEL exercises, BUYU produces remaining 100,000 boards at a cost of $250,000 fixed + $2 per board = $250,000 + $200,000 = $450,000, with revenue of $500,000, thus profit = $50,000.
  • If SAEL does not exercise, only initial 100,000 boards are sold, profit = (100,000 $5) – (setup cost $250,000 + 100,000 $2) = $500,000 – ($250,000 + $200,000) = $50,000.

Expected profit of staged production, assuming a 50% chance of exercise:

Expected profit = 0.5 $50,000 + 0.5 $50,000 = $50,000.

Therefore, manufacturing all now yields an expected profit of $100,000, higher than staging at $50,000, suggesting that manufacturing all now is the preferred option based on expected values.

4. Range of Probabilities for SAEL Exercise and Value of Perfect Information

To determine the probability range where manufacturing all now remains optimal, set the expected profit of manufacturing now equal to staged production:

Expected profit manufacturing now = Expected profit staged = $50,000.

Expected profit of manufacturing now depends on the probability p that SAEL will exercise:

  • Expected profit = p $350,000 + (1 – p) (–$150,000) = 350,000p – 150,000(1 – p) = 350,000p – 150,000 + 150,000p = (500,000p) – 150,000

Setting this equal to the staged expected profit ($50,000):

500,000p – 150,000 = 50,000

= 500,000p = 200,000

= p = 0.4 (or 40%)

Thus, if the probability that SAEL exercises exceeds 40%, manufacturing all now is the optimal decision; if less, staging is preferable.

The expected value of perfect information (EVPI) quantifies the maximum amount BUYU should be willing to pay for perfect knowledge of SAEL’s decision. Since perfect information would ensure a decision yielding the higher profit, EVPI is computed as the difference between the expected profit with perfect information and the expected profit without perfect information:

Expected profit with perfect info:

  • If SAEL always exercises: profit = $350,000
  • If SAEL never exercises: profit = –$150,000

Expected profit with perfect information:

= 0.5 $350,000 + 0.5 (–$150,000) = $100,000.

EVPI = $100,000 – expected profit without perfect info ($100,000) = $0, indicating that in this case, perfect information does not increase expected profit beyond the current expected value.

5. Impact of Risk Aversion with a $100,000 Tolerance

Considering buyu's risk aversion changes the decision-making process. Instead of maximizing expected profit, BUYU now seeks to maximize expected utility, given a risk tolerance of $100,000. The utility function could reflect diminishing marginal returns or risk aversion; a common approach is to model utility as a concave function of profit, such as a logarithmic or quadratic utility.

Assuming a quadratic utility function: U(profit) = profit – (risk tolerance)^2 / 2. If we compute the expected utility for both options:

Manufacture all now:

  • Expected profit = $100,000
  • Variance is driven by the uncertainty in SAEL's decision; assuming the profits are $350,000 with a 50% chance and –$150,000 with 50% chance, variance is:
  • Variance = 0.5 ($350,000 – $100,000)^2 + 0.5 (–$150,000 – $100,000)^2
  • = 0.5 ($250,000)^2 + 0.5 (–$250,000)^2 = 0.5 62,500,000,000 + 0.5 62,500,000,000 = 62,500,000,000

Standard deviation ≈ $250,000.

Expected utility for manufacturing all now:

U = $100,000 – (62,500,000,000)/(2*(100,000)^2) = $100,000 – 62,500,000,000 / 20,000,000 = $100,000 – 3,125 ≈ $96,875.

Manufacture 100,000 now and wait:

  • Expected profit = $50,000 with lower variance because profits are less extreme
  • Calculate variance similarly; the profits are either $50,000 if exercise, or $50,000 if not, hence zero variance, utility maximizes at the mean profit of $50,000.

Thus, considering risk aversion, manufacturing all now remains preferable because the expected utility is higher ($96,875) compared to staged production ($50,000 utility, as variance is zero). The risk-averse decision aligns with the expected profit decision when risk is sufficiently accounted for.

Conclusion

The analysis indicates that manufacturing all 200,000 boards immediately is the optimal decision based on expected profit, probability thresholds, and risk preferences. Decision trees help visualize uncertainties, and considering risk aversion adjusts the decision but ultimately affirms manufacturing now as the preferable strategy under the assumptions provided.

References

  • Birge, J. R., & Louveaux, F. (2011). Introduction to Stochastic Programming. Springer.
  • Ferguson, T. S. (2013). A Course in Sequential and Decision Processes. Springer.
  • Huang, Y., & Guo, P. (2022). Manufacturing Cost Management and Decision Support. Journal of Manufacturing Systems, 61, 1-15.
  • Keeney, R. L., & Raiffa, H. (1993). Decisions with Multiple Objectives: Preferences and Value Trade-offs. Cambridge University Press.
  • Magnanti, T. L., & Birge, J. R. (2014). Optimization in Manufacturing. Wiley.
  • Pierskalla, W. P., & Tarn, T. J. (2013). Cost and Cost-Effectiveness in Manufacturing. Operations Research, 62(2), 408-422.
  • Sharpe, W. F., & Alexander, G. J. (1990). Investments. Prentice Hall.
  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury.
  • Zhao, R., & Zhang, H. (2019). Strategic Manufacturing Decisions Under Uncertainty. IEEE Transactions on Engineering Management, 66(3), 374-386.
  • Zou, H., Liu, Z., & Chen, Y. (2021). Risk Management in Manufacturing: A Statistical Approach. Manufacturing & Service Operations Management, 23(2), 242-257.

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