# Interpretable Optimization: Why and How We Should Explain Optimization Models
## 3 Problem Reduction and Surrogate Model Building
### Experimental Design and Factor Reduction  
Factor reduction methods are crucial in experimental design for simplifying complex models while retaining the most significant factors. We will interpret factors as any relevant element of the problem (i.e., variables, constraints, or parameters) in the context of optimization problems. These techniques have been applied to models in many contexts to improve computational efficiency, enhance interpretability, and support decision-making, as they identify and remove factors with minimal impact on system performance. Various approaches exist, each suited to different problem structures and objectives. One common approach is Global Sensitivity Analysis (GSA), where methods like the [PERSON] method systematically perturb input factors to assess their influence on model outputs [56]. [PERSON] distinguishes between the important, interactive, and insignificant variables by computing the elementary effects. This distinction is the main takeaway from its application, rather than the specific result value (this is why some authors refer to it as a qualitative method). Important variables are those with a significant impact on the model's output, interactive variables that exhibit non-linear interactions, while insignificant variables contribute minimally to system performance and can be candidates for removal. In addition, the fact that the method is built on calculating a sample of one-at-a-time perturbations can cause the results to depend on calculation order, making them unstable if there is nonlinearity [56; 57]. Several improvements have been proposed to refine the [PERSON] method. [PERSON] et al. [58] introduced using the median instead of the mean for ranking parameters, making sensitivity assessments more robust against outliers. [PERSON] and [PERSON] [59] adapted the method for models with dependent inputs, resulting in more accurate sensitivity esti \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Benders’ Decomposition** & **Dantzig-Wolfe** &
\begin{tabular}{c} **Lagrangian** \\ **Relaxation** \\ \end{tabular} & **ADMM** \\ \hline \multirow{2}{*}{**Problem Structure**} & Complicating variables & Complicating constraints & Complex constraints & Separable objective \\  & LP/MIP problems & & & \\  & with complicating variables, continuous & LP problems with complicating constraints & Problems with nonlinear constraints & Large-scale convex problems \\  & non-complicating ones & & & \\
**Potential for** & & & & \\
**Interpretability** & Medium & Low & Medium & Medium \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of decomposition methods. mates for correlated systems. [PERSON] et al. [60] combined the Morris method with a sequential sampling strategy to improve the coverage of the input space and enhance sensitivity estimation. [PERSON] et al. [61] incorporated copula-based dependency structures for interactions between model parameters, making the method more suitable for complex systems with interrelated variables. Another extension, the controlled Morris method by [62; 63], introduced a sequential testing procedure that enhances factor screening without requiring predefined distributions. GSA has arguably not found extensive application due to its very intensive computational requirements. However, recent advances in computational capacity have motivated a renaissance in applications [64]. Design of Experiments (DOE) techniques offer another approach to factor reduction by structuring experiments efficiently. One avenue to reduce the number of necessary experiments is to perform only a fraction of factor combinations (Fractional Factorial Design). Fractional Factorial Design includes the Taguchi method [65], which uses orthogonal arrays to systematically vary input factors while minimizing the number of required experiments. This can efficiently increase the signal-to-noise ratio of experiments; however, this comes as a cost. Some of the interactions are ignored or aliased, so they will not be identified by this technique.