A model to determine the optimal sampling schedule of diet components

Various recommendations have been issued regarding the sampling schedule of diet components, especially forages. Their basis is unclear and none are justified from an economic standpoint. The objective of this research was to derive a general method for determining the optimal sampling design for forages. The process of forage removal from storage can be conceptualized as a quality control issue that can be monitored using a Shewhart X- bar chart. This procedure requires 3 control parameters: number of samples (n), sampling interval (h), and control limits (L). A quality cost function made of 4 parts is proposed: cost per cycle while the process is in- control (I); cost per cycle while the process is out- of- control (O); cost per cycle for sampling and analyses (A); and expected duration of a cycle (D). Thirteen inputs enter the cost function: the mean time that process is in control, the number of animals in the herd, the unit price of milk, the milk production loss due to white noise, the milk production loss from an abrupt change in forage composition, the time to sample and analyze one item, the expected time to discover the assignable cause, the expected time to fix the diet, the cost per false alarm, the cost to fix the diet, the fixed cost of sampling at each sampling time, the cost for each unit sampled, and the number of standard deviation slips when forage changes. The total quality cost per day C = (I + O + A)/D. The C function can be optimized with respect to n, h, and L to yield an optimal sampling schedule. Because n and h are discrete variables in a highly nonlinear function, parametric optimization algorithms cannot be used to optimize the function. A genetic algorithm was used for the minimization of C. Results showed that the optimal sampling designs are close to current practices in small herds of 50 cows, but very different in large herds of 1,000 cows, resulting in reduced total quality costs of $ 250/d. Total sensitivity of C was greatest for the number of cows in the herd, the shift in milk production when forage changes, the mean time that the process is in control, the price of milk, and the time to sample and analyze one item. Total sensitivities of n, h, and L were greatest for the mean time that the process is in control, the extent of the change in composition when there is a change in forage composition, the number of cows in the herd, the shift in milk production when forage changes, and the cost per unit sampled.