Assignment 7 Due Wed Oct 17, 2018 - Stevens Fall 2015

Assignment 7 Due Wed Oct 17 2018 Stevens Fall 20151 Of 1cee 4200

In this assignment, you are asked to analyze data from a previous study involving ten pump companies, each providing an initial bid for pump operation based on anticipated annual cost savings attributed to increased efficiency. The core tasks include conducting an incremental analysis to determine the most cost-effective pump based on the rate of return (ROR), and evaluating how uncertainties in pump costs and interest rates (assumed at 10%) could influence decision-making. Additionally, a simplified explanation of the uncertainty propagation methodology, including first-order error analysis, is required to communicate findings clearly to a non-technical stakeholder.

Paper For Above instruction

Introduction

The process of selecting the optimal pump from a set of alternatives involves understanding the economic trade-offs and uncertainties associated with each option. The primary metric in such evaluations is frequently the rate of return (ROR), which reflects the economic efficiency of each pump considering costs, savings, and salvage values. Beyond calculating RORs for each pump, it is crucial to perform an incremental analysis to identify which pump offers the best economic advantage when compared directly to alternatives. This approach involves systematic comparisons to ascertain the most cost-effective choice, considering uncertainties in key parameters such as initial costs, savings, salvage values, and interest rates.

Data Overview and Preliminaries

The data provided include costs, projected savings, salvage values, and the annual lifespan of ten different pumps (Pumps 1 through 10). The specific data points are summarized as follows:

  • Initial costs range from \$6.65k to \$8.49k.
  • Annual cost savings vary from \$0.627k to \$1.487k.
  • Salvage values at end of operational life range from \$0.527k to \$1.074k.
  • Projected pump lifespans are between approximately 1.22 and 1.487 years.

Using these data, we will determine the ROR for each pump, compare pumps incrementally, and analyze how uncertainty affects our choice, especially between specific pairs like Pump 8 and Pump 6.

Calculation of Rate of Return (ROR) for Each Pump

The ROR signifies the discount rate that equates the present worth of costs and benefits. To compute RORs, we use the formula that balances initial investment, annual savings, salvage value, and operational life, typically derived from the net present value (NPV) equation:

NPV = -Initial Cost + Σ (Annual Savings / (1+ROR)^t) + Salvage / (1+ROR)^n = 0

Given the data, nonlinear iterative methods or approximation techniques (such as trial-and-error or financial calculator functions) are employed to estimate each pump's ROR. This process reveals the economic efficiency of each pump, providing a basis for comparison.

Incremental Analysis for Pump Selection

The incremental analysis compares pumps sequentially based on their RORs. First, rank the pumps from highest to lowest ROR. Next, compare each pump's ROR to that of the next most efficient alternative, calculating the incremental net benefit and cost. The decision rule is to select the pump with the highest incremental benefit that exceeds a predetermined threshold, considering both economic advantage and uncertainty implications.

For example, when comparing Pump 8 and Pump 6, the incremental analysis involves calculating the difference in their costs, savings, and salvage values, and then determining the incremental ROR. The pump with the higher incremental ROR and net benefit is preferred, provided the difference exceeds the margin of error owing to uncertainties.

Uncertainty Propagation and First-Order Error Analysis

Decision-making under uncertainty requires understanding how variations in key parameters—initial costs, savings, salvage values, and interest rates—affect the ROR calculations. First-order error analysis provides a simplified way to estimate the impact of such uncertainties using Taylor series approximations.

The method involves computing the total differential of the ROR with respect to each uncertain parameter. If the parameters are assumed independent, their individual variances contribute linearly to the overall uncertainty in ROR estimation. Mathematically, this is expressed as:

V(f) = Σ (∂f/∂x_i)^2 * V(x_i)

where V(f) is the variance of the function of interest (e.g., ROR), ∂f/∂x_i are partial derivatives of ROR with respect to each parameter, and V(x_i) are the variances of the parameters. This analysis highlights which parameters significantly influence the decision and guides the assessment of whether observed differences are statistically meaningful or within the margin of error.

Case Study: Comparing Pump 8 vs. Pump 6

Focusing on a specific comparison, consider Pump 8 and Pump 6. Using their respective data, we first compute their RORs. Suppose Pump 8 has a higher ROR than Pump 6, but both parameters have associated uncertainties—for instance, costs and interest rates. We then calculate the sensitivity of the ROR to these parameters, determining the potential range of variation due to a 10% interest rate fluctuation and cost uncertainties.

Applying the first-order error propagation, we estimate how these uncertainties could potentially bridge or widen the gap between the two pumps’ RORs. If, within the bounds of uncertainty, Pump 6 could outperform Pump 8, the decision becomes less clear-cut. Conversely, if the confidence interval around Pump 8’s ROR remains well above Pump 6’s, we can confidently select Pump 8 despite uncertainties.

Implications for Management

It is crucial for decision-makers to understand not only the numerical comparison but also the confidence level associated with each choice. Incorporating the uncertainty analysis ensures that investments are made with awareness of risks and potential variability in outcomes. Clear communication with non-technical stakeholders involves emphasizing which parameters most influence the decision and quantifying the associated risks.

Conclusion

In conclusion, the systematic comparison of pumps using ROR, coupled with incremental analysis, provides a robust framework for selecting the most cost-effective pump. Incorporating uncertainty propagation through first-order error analysis adds a layer of confidence, helping to avoid costly misjudgments due to parameter variations. This integrated approach ensures a data-driven, transparent, and risk-aware decision-making process that aligns with managerial and financial objectives.

References

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