Improved method for ALT plan optimization

Peter Arrowsmith

This paper describes a simplified method for obtaining the optimum stress levels and allocation for a candidate ALT plan, to minimize the variance of the estimated time to fractional failure, extrapolated to the use condition. Why another test plan optimization method? Fixed allocations of test units, such as nL:nM:nH = 4:2:1 for the low, middle and high levels, respectively, of a three stress level test plan, or a fixed fractional allocation, e.g. nM/n = 0.2, may not be optimal, particularly over a wide range of total sample size (n), or when there are more than three stress levels. Ma and Meeker, “Strategy for Planning ALTs with Small Sample Sizes”, IEEE Trans. Rel. vol 59(4), 610-619 (2010) imposed two constraints and suggested using a three step method to select a candidate test plan. For a three level plan the constraints were equally were spaced standardized stresses, or ξM = (ξL+ξH)/2, and equal number of expected failures for the low and middle stress levels. The steps involve 1) finding candidate test plans in the region (ξL, nL and n) for which the probability of zero failures at one or more stress levels Pr{ZFP1} meets a target value, e.g. Pr{ZFP1} ≤1%. Step 2) is to find a candidate test plan than minimizes the large sample approximation variance of the parameter of interest, typically the log time to failure for the pth percentile, or Avar[ln(tp)], extrapolated to the normal use condition. This requires calculating the Fisher information matrix. Finally, step 3) uses Monte Carlo simulation to fine tune the candidate test plan based on the estimated variance of the time to p% failure for the actual number of samples. This paper proposes an optimization method that is simpler, with a single step instead of the above steps 1 & 2 to obtain candidate test plans using a spreadsheet calculator (e.g. Excel), without the need to calculate the Avar. In addition, the new method requires only the constraint of equally spaced standardized stresses. The basis of the method is to find the minimum lower stress level that achieves a target Pr{ZFP1} value. Evaluation of candidate test plans was performed using Monte Carlo simulation and the results support the assumption that minimum lowest stress corresponds to lower variance of the time to p% failure at the use condition, for a given sample allocation. The results obtained with the new method will be discussed in detail in the paper, and compared to test plans made using traditional methods, including fixed allocation and the Ma and Meeker approach.