Optimal Sampling Schedule of Diet Components: Model Robustness to Departure from Assumptions¹

Graphical representation of a quality cycle; 1/λ = mean time process is in-control, τ = expected time of occurrence of abrupt change after last sampling while in-control, T_c = expected time between the occurrence of abrupt change and the next sampling time, h = sampling interval, T_e = expected time until an out-of-control signal occurs, T₁ = expected time to investigate the cause of the change, T₂ = expected time to reformulate and implement new diets, and T_f = expected time to investigate the cause of the change, reformulate and implement new diets.

Probability distribution functions of the standard normal (solid) and log-normal (dashed) distributions with equal means (= 0) and standard deviations (= 1); x is a standardized random variable and P(x) is the value of the probability functions at x.

Optimal sampling schedule, total quality cost calculated analytically, and cost obtained by the simulation model as a function of the mean time process is in control (1/λ): —— = the sampling interval (h); —Δ— = number of standard deviations used as control limits of the X-bar chart (L); — ♢ — = number of samples taken (n); —■— = the total daily quality cost calculated using the analytical formula (C); and —○— = the total daily quality cost calculated through the simulation process.

Frequency of false alarms, average run length when out-of-control (ARL₁), and total quality cost per day as a function of the mean time process is in control (1/λ) when all assumptions are met: — ♢ — = average run length; —Δ— = frequency of false alarms; and — ○ — = total quality cost calculated through the simulation process.

Effect of the presence of 10% outliers (±3.5 SD) on the frequency of false alarms, average run length when out-of-control (ARL₁), and total quality cost per day as a function of the mean time process is in control (1/λ): —♢— = average run length; —— = frequency of false alarms; —○— = total quality cost calculated through the simulation process; and —Δ— = total quality cost calculated using the analytical formula.

Effect of asymmetry (log-normal distribution of samples) on the frequency of false alarms, average run length when out-of-control (ARL₁), and total quality cost per day as a function of the mean time process is in control (1/λ): ♢ = average run length; —— = frequency of false alarms; —○— = total quality cost calculated through the simulation process; and —Δ— = total quality cost calculated using the analytical formula.

Effect of a gradual shift in mean composition over a period of 28 d on the frequency of false alarms, average run length when out-of-control (ARL₁), and total quality cost per day as a function of the mean time process is in control (1/λ): ♢ = average run length; —— = frequency of false alarms; —○— = total quality cost calculated through the simulation process; and —Δ— = total quality cost calculated using the analytical formula.

Comparison of standard X-bar chart and augmented X-bar in combination with a cumulative sum (CUSUM) chart on the total quality cost per day over a range of mean time that process is in control (1/λ): —Δ— = total quality cost calculated through the simulation process using the standard X-bar chart; and —○— = total quality cost calculated through the simulation process using the X-bar chart with augmented X-bar chart rules in conjunction with a CUSUM chart.