The correct answer is C. 81 . In DOE, the total number of runs starts with the number of treatment combinations in the design, then is multiplied by the number of replicates. For a 3-level design with 4 factors , a full factorial design would require 3⁴ = 81 treatment combinations. Since this question specifies a 1/3 fractional factorial , the base design is reduced to 81 ÷ 3 = 27 treatment combinations. With 3 replicates , the total required number of runs becomes 27 × 3 = 81 .

This is consistent with CSSBB DOE principles, where the number of runs depends on the number of factors, levels, fraction, and replicates. The CSSBB materials also emphasize that replicates are used to reduce experimental error and increase the power of the experiment, and that the design choice must balance information needs with budget and practicality. The source further discusses screening and multi-level designs, noting that 3-level designs are used when non-linear effects are expected. Therefore, after applying both the fractional reduction and the 3 replicates, the required total is 81 runs .