np Chart Analysis

Introduction

This report analyzes defect data from a semiconductor fabrication facility using an np control chart. The goal is to evaluate process stability and establish control limits for monitoring defective wafers.

Background

Problem Context

Silicon wafers undergo a series of highly precise fabrication steps including photolithography, etching, deposition, and chemical mechanical planarization (CMP). At each stage, wafers are inspected for defects such as particles, scratches, or pattern misalignments.

At the end of the photolithography stage, engineers inspect n = 200 wafers per lot and record the number of defective wafers. Data were collected for k = 50 consecutive lots during a period believed to represent a stable process.

The objective of this analysis is to construct an np control chart to determine whether the process appears to be in statistical control and to establish control limits for future monitoring. # Data

defects <- c(8, 7, 11, 3, 6, 11, 4, 4, 5, 8, 8, 6, 7, 6, 10, 7, 9, 5, 6, 8,
             8, 10, 4, 11, 4, 11, 6, 6, 9, 5, 4, 7, 5, 7, 8, 5, 10, 9, 7, 9,
             4, 9, 4, 7, 11, 8, 10, 8, 7, 5)

n <- 200
k <- length(defects)
sizes <- rep(n, k)

mean(defects)

## [1] 7.14

np Chart Formulas

For an np chart with constant sample size and estimated fraction nonconforming \(\bar{p}\):

\[CL = n\bar{p}\]

\[UCL = n\bar{p} + 3\sqrt{n\bar{p}(1-\bar{p})}\]

\[LCL = n\bar{p} - 3\sqrt{n\bar{p}(1-\bar{p})}\] # Calculations

pbar <- sum(defects) / (k * n)

CL <- n * pbar
UCL <- CL + 3 * sqrt(n * pbar * (1 - pbar))
LCL <- CL - 3 * sqrt(n * pbar * (1 - pbar))

LCL <- max(0, LCL)

pbar

## [1] 0.0357

CL

## [1] 7.14

UCL

## [1] 15.01184

LCL

## [1] 0

np Control Chart

np_chart <- qcc(defects,
                sizes = sizes,
                type = "np",
                title = "np Chart for Defective Wafers",
                xlab = "Lot Number",
                ylab = "Number of Defective Wafers")

---

2) Interpretation

# Interpretation

The np control chart shows that all points fall within the calculated control limits. 
Additionally, there are no runs rule violations or unusual patterns observed.

This indicates that the semiconductor fabrication process appears to be in statistical control. 
The variation observed in the number of defective wafers per lot is consistent with common-cause variation.

Therefore, the current process is stable and the calculated control limits can be used for future monitoring.
# Complete Code Appendix


``` r
library(qcc)

defects <- c(8, 7, 11, 3, 6, 11, 4, 4, 5, 8, 8, 6, 7, 6, 10, 7, 9, 5, 6, 8,
             8, 10, 4, 11, 4, 11, 6, 6, 9, 5, 4, 7, 5, 7, 8, 5, 10, 9, 7, 9,
             4, 9, 4, 7, 11, 8, 10, 8, 7, 5)

n <- 200
k <- length(defects)
sizes <- rep(n, k)

pbar <- sum(defects) / (k * n)

CL <- n * pbar
UCL <- CL + 3 * sqrt(n * pbar * (1 - pbar))
LCL <- CL - 3 * sqrt(n * pbar * (1 - pbar))
LCL <- max(0, LCL)

pbar
CL
UCL
LCL

np_chart <- qcc(defects,
                sizes = sizes,
                type = "np",
                title = "np Chart for Defective Wafers",
                xlab = "Lot Number",
                ylab = "Number of Defective Wafers")