When Cp and Cpk valuers of a process are equal, the process average will be

Cp (Process Capability Index): Cp measures whether the process spread is narrower than the specification width. It is calculated as the ratio of the specification width (the difference between the Upper Spec Limit (USL) and Lower Spec Limit (LSL)) to the process width (the difference between the Upper Control Limit (UCL) and Lower Control Limit (LCL)). Mathematically, Cp = Spec Width / Process Width.

Cpk (Process Capability Index with Centering): Cpk not only considers the process spread but also accounts for the centering of the process relative to the specification width. It takes into account how well the process is centered over the nominal desired by the design engineer. Cpk is calculated as the minimum of two values: Cpk = min(Cp, (UCL - Mean) / 3σ, (Mean - LCL) / 3σ), where σ represents the process standard deviation.

Now, let’s address the scenario when Cp and Cpk are equal:

If Cp = Cpk, it means that the process spread (process width) is equal to the specification width, and the process is centered between the specification limits. In other words, the process is capable of consistently producing parts within specifications.

When Cp and Cpk are equal, the process average (mean) will be equal to the midpoint of the specification. This is because the process is centered, and the spread of values falls within the specified limits.

Therefore, the correct answer is B. equal to the midpoint of the specification1.

Remember that process capability analysis helps determine whether a manufacturing process can consistently produce parts that meet specifications. The goal is to have a narrow and well-centered process relative to the specification limits2.

References: 1: Simplilearn: Process Capability