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

Abstract

Monitoring the nutritional composition of forage can be conceptualized as a quality control process that can be accomplished using control charts such as the Shewhart X-bar chart. A sampling schedule for an X-bar chart is defined by 3 parameters: the number of samples n taken at each sampling time, the sampling interval h, and the control limits L. All 3 parameters affect the performance of the chart, and thus, the total quality cost (TQC). A TQC function consists of cost per cycle while the process is in-control, cost per cycle while the process is out-of-control, cost per cycle for sampling and analyses, and the expected duration of a cycle, with a cycle defined as the time between the start of successive in-control periods. A general TQC function was derived for a renewal reward process. Optimization of this TQC function allows for the determination of the optimal n, h, and L values that minimize the total daily quality costs. The model assumes an abrupt change in composition of the forage when the process goes out-of-control. It also assumes a normal distribution of the measurements when the process is in-control and an absence of outlier measurements. The objective of this research was to evaluate model robustness to departure from these 3 basic assumptions. A series of Monte Carlo simulations was performed while varying the average time that the process is in control from 5 to 90 d using 1) errors of measurements that follow a standard normal distribution (SN); 2) SN with ± 3.5 SD outliers inserted with a frequency of 1, 5, and 10%; 3) log normal error of measurements with SD = 1; and finally 4) SN with a gradual shift of the mean from 0 to 1.5 SD over 7, 14, and 28 d. The model was very robust to the presence of outliers; the average change in TQC was less than 1%, even with a frequency of outliers of 10%. The model is also very robust to asymmetry in the distribution of the measurements (i.e., probability distribution function with a long right tail): the log normal distribution, as opposed to the assumed normal, resulted in an increase in TQC of less than 1.4%. Finally, the gradual shift in mean composition did not result in an increased TQC but in a 17.3% decrease compared with an abrupt change. The model appears very robust to departure from normality, presence of outliers, and a skewed distribution of measurements. Gradual changes in the process are readily detected by the optimum X-bar chart with the conventional decision criterion, and monitoring performance is not markedly improved by augmenting the number of decision criteria in the X-bar chart or by the addition of a cumulative sum chart. Because of its robustness, the model can be applied to optimize forage sampling on dairy farms, with expected savings ranging between $80 and $100/cow per yr.