Question 2 of 6: SPC Problem-Solving Tools, Deteriorating Processes, and CUSUM

Question 2: SPC Problem-Solving Tools, Deteriorating Processes, and CUSUM (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Three of the seven basic SPC tools

The seven basic tools are the cause-and-effect (Ishikawa/fishbone) diagram, check sheet, Pareto chart, flow chart, scatter diagram, control chart, and histogram; any three are described here. The cause-and-effect diagram organizes potential causes of a quality problem into major branches (commonly the "6 M's": Man, Machine, Material, Method, Measurement, Mother Nature/Environment) feeding into the effect; it is used in a team brainstorming session to structure root-cause analysis, e.g. mapping every plausible cause of excess weld porosity before collecting data. The Pareto chart is a descending bar chart of defect types (or cost/frequency categories) with a cumulative-percentage line overlaid, used to apply the 80/20 rule and focus improvement effort on the "vital few" defect categories that account for most of the loss, e.g. showing that 3 of 12 defect codes account for 78% of scrap cost. The scatter diagram plots one variable against another to reveal (or rule out) a relationship between a suspected cause and an effect, e.g. plotting furnace temperature against tensile strength to check whether temperature is actually driving strength variation before spending resources controlling it.

(b) Monitoring a deteriorating (tool-wear) process, and why EWMA is unsuitable for trend processes

Yes — traditional Shewhart control charts can still be used on a deteriorating process, but not with fixed, symmetric $3\sigma$ limits computed once and left in place, since a genuinely trending mean will simply walk out through the upper (or lower) limit on a predictable schedule rather than signal a discrete assignable cause. The chart is adapted by fitting the deterministic trend itself (e.g. modeling tool wear as a regression of the characteristic on time or on parts produced) and plotting the residuals from that trend on a conventional chart with fixed limits — an unexpected point still signals an assignable cause distinct from the modeled wear. Alternatively, the center line and limits can be updated on a moving schedule (a chart with a systematically shifting target, matching the expected wear rate between tool changes) so the chart tracks the expected drift and only flags a departure from the expected trend, not the trend itself.

EWMA (and CUSUM) charts are built specifically to be highly sensitive to small, sustained step or ramp shifts away from a constant, otherwise stationary in-control mean — their memory of past observations is what gives them that sensitivity. A trend process, by contrast, does not have a constant mean at all: the mean is deterministically moving as a normal, expected feature of the process (tool wear, catalyst depletion, fouling). Applying an EWMA with limits calibrated for a stationary mean to a genuinely trending process means the statistic will chase the trend and drift outside its limits continuously, generating a permanent false alarm that carries no diagnostic information — the chart cannot distinguish "the trend is proceeding as expected" from "something new has gone wrong," which is precisely the distinction a control chart exists to make. The chart must instead be built against a moving target that already incorporates the expected trend (as in part (b)'s first paragraph), not against a flat EWMA baseline.

(c) Why CUSUM beats $\bar X$ for small shifts, V-mask vs. tabular CUSUM, and V-mask limitations

An $\bar X$ chart bases its decision on the current sample alone; its ability to detect a shift of size $\delta\sigma$ depends only on how far that one sample's average falls from the center line, so a small shift ($\delta\lesssim1\sigma$) produces only a small, easily-missed departure and the chart's average run length to detection is long. A CUSUM chart instead accumulates the deviations of every sample from the target, $C_i=\sum_{j\le i}(\bar x_j-\mu_0)$: a small but persistent shift adds up sample after sample, so the cumulative sum's drift eventually becomes large and unmistakable even though no single sample would have triggered an $\bar X$ signal. This makes CUSUM (and EWMA) markedly faster than a Shewhart chart specifically for small, sustained shifts, at some cost of sensitivity to detecting one single large, isolated shift.

The V-mask procedure plots the raw cumulative sum $C_i$ and overlays a V-shaped mask (apex a distance $d$ ahead of the most recent point, arms of half-angle $\theta$) at each new point; the process is judged out of control the first time a historical point on the CUSUM path falls outside (above or below) the V's arms. The tabular (algorithmic) CUSUM instead maintains two one-sided running sums, $C_i^+=\max[0,\,C_{i-1}^++(\bar x_i-\mu_0-K)]$ and $C_i^-=\max[0,\,C_{i-1}^--(\bar x_i-\mu_0+K)]$ (with reference value $K$, typically $\tfrac12\delta\sigma$), and signals when either exceeds a decision interval $H$ (i.e. $C_i^+\gt H$ or $C_i^-\gt H$). The two are mathematically equivalent (the V-mask's $d$ and $\theta$ map directly onto the tabular form's $H$ and $K$) but the tabular form is far easier to implement in software, gives an unambiguous numerical signal instead of a geometric judgment call, and — critically — separates the upward and downward cumulative sums, so it directly reports both the fact of an out-of-control signal and its estimated time of onset.

The limitations of the V-mask specifically are: it is a graphical/geometric procedure, so the out-of-control decision (does any historical point fall outside the current mask?) is comparatively awkward to automate and easy to apply inconsistently by eye; the choice of mask parameters $d$ and $\theta$ is not intuitive to a practitioner in the way a numerical decision interval is; the mask is normally applied by inspecting the entire historical path retrospectively each time a new point is added, which is cumbersome for real-time monitoring; and, unlike the tabular form, it does not naturally separate the upward- and downward-drift evidence, making it harder to diagnose the direction and starting point of a detected shift.