In production industries, parameter identification, sensitivity analysis and multi-dimensional visualization are vital steps in the planning process for achieving optimal designs and gaining valuable information. Sensitivity analysis and visualization can help in identifying the most-influential parameters and quantify their contribution to the model output, reduce the model complexity, and enhance the understanding of the model behavior. Typically, this requires a large number of simulations, which can be both very expensive and time consuming when the simulation models are numerically complex and the number of parameter inputs increases. There are three main constituent parts in this work. The first part is to substitute the numerical, physical model by an accurate surrogate model, the so-called metamodel. The second part includes a multi-dimensional visualization approach for the visual exploration of metamodels. In the third part, the metamodel is used to provide the two global sensitivity measures: i) the Elementary Effect for screening the parameters, and ii) the variance decomposition method for calculating the Sobol indices that quantify both the main and interaction effects. The application of the proposed approach is illustrated with an industrial application with the goal of optimizing a drilling process using a Gaussian laser beam.