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Abstract Industrial software often has many parameters that critically impact performance. Frequently, these are left in a sub-optimal configuration for a given application because searching over possible configurations is costly and, except for developer instinct, the relationships between parameters and performance are often unclear and complex. While there have been significant advances in automated parameter tuning approaches recently, they are typically black-box. The high-quality solutions produced are returned to the user without explanation. The nature of optimisation means that, often, these solutions are far outside the well-established settings for the software, making it difficult to accept and use them. To address the above issue, a systematic approach to software parameter optimization is presented. Several well-established techniques are followed in sequence, each underpinning the next, with rigorous analysis of the search space. This allows the results to be explainable to both end users and developers, improving confidence in the optimal solutions, particularly where they are counter-intuitive. The process comprises statistical analysis of the parameters; single-objective optimization for each target objective; functional ANOVA to explain trends and inter-parameter interactions; and a multi-objective optimization seeded with the results from the single-objective stage. A case study demonstrates application to business-critical software developed by the international airline Air France-KLM for measuring flight schedule robustness. A configuration is found with a run-time of 80% that of the tried-and-tested configuration, with no loss in predictive accuracy. The configuration is supplemented with detailed analysis explaining the importance of each parameter, how they interact with each other, how they influence run-time and accuracy, and how the final configuration was reached. In particular, this explains why the configuration included some parameter settings that were outwith the usually recommended range, greatly increasing developer confidence and encouraging adoption of the new configuration.