High-quality processes and rare nonconformities

Motivation

High-quality processes are processes in which nonconformities are rare. In that setting, attribute data are highly discrete, bounded below by zero, and often strongly asymmetric. Classical normal-approximation control limits may therefore be poorly calibrated, even when the usual chart is familiar and easy to apply.

IQCC addresses this problem with corrected p-chart limits, exact binomial false-alarm calculations, and a complete double-sampling np workflow including performance evaluation, automatic limit search, chart construction, and plotting.

Cornish-Fisher corrected p charts

IQCC distinguishes two corrected methods:

• type = "cf1" uses the first skewness correction;
• type = "cf2" uses the two-adjustment operational limits documented by Joekes and Barbosa (2013).

The following chunk is executed and compares all three limit methods.

methods <- c("normal", "cf1", "cf2")
p_results <- do.call(
  rbind,
  lapply(methods, function(method) {
    lim <- pchart_limits(p = 0.015, n = 20, type = method)
    risk <- pchart_alpha_risk(
      p = 0.015,
      n = 20,
      lcl = lim$lcl,
      ucl = lim$ucl
    )
    data.frame(
      method = method,
      lcl = lim$lcl,
      center = lim$center,
      ucl = lim$ucl,
      actual_alpha = risk,
      arl0 = ifelse(risk == 0, Inf, 1 / risk)
    )
  })
)
p_results
#>   method lcl center        ucl actual_alpha       arl0
#> 1 normal   0  0.015 0.09653924 0.0357458712   27.97526
#> 2    cf1   0  0.015 0.16120479 0.0002023458 4942.03542
#> 3    cf2   0  0.015 0.13031923 0.0031780828  314.65511

data(binomdata)
cchart.p(
  x1 = binomdata$Di[1:12],
  n1 = binomdata$ni[1:12],
  type = "cf2",
  x2 = binomdata$Di[13:25],
  n2 = binomdata$ni[13:25]
)

Double-sampling np charts

A double-sampling np chart uses two possible sampling stages. A first sample is inspected. If the count of nonconforming items is clearly acceptable, the process is not signaled. If it is clearly large, the chart signals immediately. If the count lies in an intermediate region, a second sample is inspected and the combined count is used for the final decision.

IQCC provides the complete DS-np workflow:

• dsnp_prob_accept() computes the total acceptance probability;
• dsnp_arl() computes average run length;
• dsnp_ass() computes average sample size;
• dsnp_limits() searches and ranks feasible limits;
• dsnp_design() searches sample sizes and limits under explicit ARL and ASS constraints;
• cchart.DSnp() classifies observations and constructs the chart.

Inspecting a published plan

n1 <- 34
n2 <- 162
wl <- 1.5
ucl1 <- 2.5
ucl2 <- 4.5
p0 <- 0.005
p1 <- 0.0075

published_plan <- data.frame(
  metric = c("P(accept | p0)", "ARL0", "ARL1", "ASS0"),
  value = c(
    dsnp_prob_accept(p0, n1, n2, wl, ucl1, ucl2)$pt,
    dsnp_arl(p0, n1, n2, wl, ucl1, ucl2)$arl,
    dsnp_arl(p1, n1, n2, wl, ucl1, ucl2)$arl,
    dsnp_ass(p0, n1, n2, wl, ucl1)$ass
  )
)
published_plan
#>           metric       value
#> 1 P(accept | p0)   0.9987553
#> 2           ARL0 803.4114304
#> 3           ARL1 193.2228555
#> 4           ASS0  35.9353364

The approximate published performance values are ARL0 = 803.41, ARL1 = 193.22, and ASS0 = 35.94.

Searching for limits

A reduced search space is used here to exercise the full algorithm without making documentation builds unnecessarily slow.

lim <- dsnp_limits(
  p0 = 0.05,
  n1 = 5,
  n2 = 10,
  alpha = 0.05,
  p1 = 0.10,
  max_results = 5
)
lim$best[, c("wl", "ucl1", "ucl2", "p_signal0",
             "arl0", "arl1", "ass0")]
#>    wl ucl1 ucl2  p_signal0     arl0     arl1     ass0
#> 1 0.5  1.5  2.5 0.04013256 24.91743 5.951221 7.036266

Searching a complete design

Joekes, Smrekar and Barbosa (2015) formulate the DS-np design problem as minimizing out-of-control ARL1, subject to a minimum ARL0 and a maximum in-control average sample size ASS0. The shift is written in the paper as p1 = gamma * p0. In IQCC, the corresponding bound is ass0_max:

design <- dsnp_design(
  p0 = 0.05,
  p1 = 0.10,
  n1_range = 5:6,
  n2_range = 8:10,
  arl0_min = 50,
  ass0_max = 6,
  objective = "arl1",
  max_results = 5
)
design$best[, c("n1", "n2", "wl", "ucl1", "ucl2",
                "ass0", "arl0", "arl1")]
#>   n1 n2  wl ucl1 ucl2     ass0     arl0     arl1
#> 1  5 10 1.5  2.5  2.5 5.214344 102.4701 17.84397

The published tables report selected plans but not the complete bounds used for the searches over n1 and n2 or every tie-breaking rule. IQCC tests recover a published Table 5 plan over an explicitly recorded local grid and validate the exhaustive algorithm independently on a small grid; this evidence should not be described as a reproduction of undocumented global search settings.

Setting ass0_max = NULL preserves the unconstrained search. The ASS used by dsnp_ass() and dsnp_design() is equation (15) of the paper: whenever the first-stage count enters the warning region, all n2 units are counted. The paper also describes stopping second-stage inspection as soon as rejection is certain, but it does not provide the corresponding expected-inspection formula; IQCC does not label that curtailed quantity as the current ASS.

Interpreting fractional limits

For a DS-np plan with first-stage count D1 and second-stage count D2, IQCC uses the following rule:

• accept at the first stage when D1 <= floor(wl);
• signal at the first stage when D1 >= floor(ucl1) + 1;
• otherwise inspect the second sample;
• after the second sample, accept when D1 + D2 <= floor(ucl2);
• signal after the second stage otherwise.

Constructing the chart

x1 <- c(0, 1, 2, 3, 1, 0, 2, 4, 1, 0)
x2 <- c(NA, NA, 2, NA, NA, NA, 3, NA, NA, NA)

chart <- cchart.DSnp(
  x1,
  n1 = 10,
  n2 = 20,
  p0 = 0.05,
  x2 = x2,
  wl = 1.5,
  ucl1 = 2.5,
  ucl2 = 4.5,
  p1 = 0.10,
  plot = TRUE
)

chart$limits
#> $wl
#> [1] 1.5
#> 
#> $ucl1
#> [1] 2.5
#> 
#> $ucl2
#> [1] 4.5
#> 
#> $wl_accept
#> [1] 1
#> 
#> $ucl1_reject
#> [1] 3
#> 
#> $ucl2_accept
#> [1] 4
chart$performance
#> $arl0
#> [1] 58.35236
#> 
#> $ass0
#> [1] 11.4927
#> 
#> $p_signal0
#> [1] 0.01713727
#> 
#> $arl1
#> [1] 7.531628
#> 
#> $ass1
#> [1] 13.8742
#> 
#> $p_signal1
#> [1] 0.1327734
chart$data
#>    index x1 x2 total         stage signal
#> 1      1  0 NA    NA  accept_first  FALSE
#> 2      2  1 NA    NA  accept_first  FALSE
#> 3      3  2  2     4 accept_second  FALSE
#> 4      4  3 NA    NA  signal_first   TRUE
#> 5      5  1 NA    NA  accept_first  FALSE
#> 6      6  0 NA    NA  accept_first  FALSE
#> 7      7  2  3     5 signal_second   TRUE
#> 8      8  4 NA    NA  signal_first   TRUE
#> 9      9  1 NA    NA  accept_first  FALSE
#> 10    10  0 NA    NA  accept_first  FALSE

cat("<!-- IQCC_EXECUTED_HIGH_QUALITY -->\n")

Practical workflow

A practical high-quality-process workflow in IQCC is:

1. Compare normal, CF1, and CF2 limits with pchart_limits().
2. Evaluate actual binomial risk with pchart_alpha_risk().
3. Use cchart.p() for routine p-chart monitoring.
4. Use dsnp_limits() when inspection effort motivates a double-sampling plan.
5. Compare candidates using false-alarm probability, out-of-control ARL, and ASS.
6. Supply the selected design to cchart.DSnp() for operational monitoring.