Suppose Your Manager Presents You With The Following Informa

Suppose Your Manager Presents You With The Following Information About

Suppose your manager presents you with information about machines that could be used for a job, including their costs and process variability. The specific width of the specification is 0.48 mm. The decision involves selecting the most appropriate machine based on cost and process consistency, but additional information may be required to make a definitive recommendation. Answers should be rounded to three decimal places.

Paper For Above instruction

When evaluating machinery for a specific process, especially in manufacturing and quality control, understanding the trade-offs between cost and process variability is crucial. The primary goal is to select the machine that not only meets the specification width requirement but also aligns with cost-efficiency and quality standards. This paper discusses the process of analyzing the provided data, applying statistical quality control principles, and making an informed decision based on the given parameters.

The data presented includes four different machines (A, B, C, D), each with associated costs per unit and their process standard deviations. The costs are as follows: Machine A and B at $12 each, Machine C at $19, and D at $10. The process standard deviations are 0.071 mm for A, 0.060 mm for B, 0.054 mm for C, and 0.062 mm for D. The narrower the process standard deviation, the more consistent the machine is in quality control, which is essential when meeting strict specification limits.

Given the specification width of 0.48 mm, the goal is to determine which machine provides the best balance of cost and process stability. The evaluation involves calculating process capability indices (Cp), which measure how well a process meets specifications, considering its variability (standard deviation). The formula for Cp is:

Cp = (USL - LSL) / 6σ

where USL (Upper Specification Limit) and LSL (Lower Specification Limit) define the acceptable process bounds, and σ represents the process standard deviation. Assuming the process is centered, calculating Cp allows us to estimate the process's potential capability to meet the specifications.

Although specific USL and LSL values are not provided directly, using the process variability data, we can estimate the capability assuming the width of 0.48 mm is the specification limit. For simplicity, we consider that the process mean is centered within the specification limits, and thus the process capability index can be approximated by substituting the process standard deviation and the half-width of the specification limit (which is 0.24 mm, i.e., half of 0.48 mm) into the calculation:

Cp ≈ 0.24 / (3σ)

Calculating this for each machine provides insight into their respective process capabilities. Higher Cp values indicate processes capable of consistently producing within the specification limits with less variability.

For Machine A:

Cp_A = 0.24 / (3 × 0.071) ≈ 0.24 / 0.213 ≈ 1.127

For Machine B:

Cp_B = 0.24 / (3 × 0.060) ≈ 0.24 / 0.180 ≈ 1.333

For Machine C:

Cp_C = 0.24 / (3 × 0.054) ≈ 0.24 / 0.162 ≈ 1.481

For Machine D:

Cp_D = 0.24 / (3 × 0.062) ≈ 0.24 / 0.186 ≈ 1.290

Based on these calculations, Machine C exhibits the highest process capability index, suggesting it is the most capable machine to produce within the specification limits reliably, despite its higher cost. Machine B also demonstrates a strong capability with a slightly lower cost, making it a viable candidate. Machine D, while less capable than B and C, could still be suitable considering its lower cost, depending on the criticality of process precision and production volume.

However, selecting the optimal machine requires considering additional factors, including the total cost of ownership, the frequency of process adjustments, maintenance costs, and potential impact on product quality. The choice may also be influenced by whether process centering adjustments are needed to reduce the process mean and improve capability further. Moreover, if the process is not centered, the process capability indices must be recalculated considering process mean shifts, which could alter the decision dynamics.

In conclusion, based on the process capability analysis, Machine C appears to offer the best balance of low variability and high capability, justifying its higher cost in scenarios where quality and consistency are paramount. Machine B remains a strong candidate bridging performance and cost efficiency, while Machines A and D, with higher variability, might be less suitable unless their costs are significantly advantageous or if further process control measures are implemented.

References

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