Shapley effect application for variance-based sensitivity analysis of the few-group cross-sections

Analyzing the variance of complex computer models is an essential practice to assess and improve that model by identifying the influential parameters that cause the output variance. Variance-based sensitivity analysis is the process of decomposing the output variance into components associated with each input parameter. In this study, we applied a new concept of variance-based sensitivity analysis inspired by the game theory proposed by Shapley. The technique is called the Shapley effect, and it investigates the contribution of each input parameter as well as its interactions with every other parameter in the system by exploring all possible permutations between them. The Shapley effect is compared to the common Sobol indices technique (first order and total effects) to investigate their performance under correlated and uncorrelated parameters. The Shapley effect demonstrated superior performance when compared to the Sobol indices for correlated input parameters. Shapley effect captured the correlation between the input parameters, expressing the variance contribution in a single index instead of two indices, and normalization of the fractional indices is preserved without over or underestimation. On the other hand, the two algorithms we selected to calculate Sobol indices under correlated inputs experienced different issues including: over/underestimating the output variance, first order effect could be larger than the total effect, possibility of negative indices, unnormalized fractional indices, and difficulty to interpret the results. However, Sobol showed satisfactory performance when the inputs are uncorrelated as the numerical values and input ranking were in good agreement with Shapley effect. The main disadvantage of Shapley effect is its large computational cost especially for high dimensional problems where the number of possible input subsets becomes very large. The results of our tests showed that the thermal fission cross-section carried most of the uncertainty at BOL, and its contribution declines after fuel burnup, which is replaced by the uncertainty contribution of the fast cross-section parameters.