## 1 Introduction
Nevertheless, they are costly, non-value-added activities that must be judiciously integrated into production systems through targeted strategies [30]. An inspection-oriented quality-assurance strategy seeks to allocate an economically appropriate level of inspection activity by balancing the various costs associated with inspection, repair, replacement due to quality failure and, where applicable, warranty penalties incurred when a non-conforming product is delivered to a customer [31]. Inspections may be conducted following any operation within a Multi-Stage Manufacturing System (MSMS), and the decision can be made to inspect either all products or only a subset. The inspection allocation problem, concerning the decision-making process that defines which product characteristics to inspect and at which stage of the production system, has been extensively discussed in the literature since the second half of the 20 th century [30; 31; 32]. However, the scientific literature continues to exhibit important gaps, particularly in developing modeling approaches capable of jointly supporting the selection of suitable inspection technologies and the determination of the optimal sampling rate, defined as the proportion of produced items inspected within a batch, in order to minimize the overall quality-related costs, which encompass prevention, inspection, rework,and failure costs. While recent advancements in inspection technologies raise important questions--such as whether quality checks should be performed by human operators or automated systems--existing research offers limited guidance on how to make these decisions in an integrated and systematic way. To address this gap, this paper proposes a novel Mixed-Integer Linear Programming (MILP) model that simultaneously identifies the most suitable inspection technologies from a set of available alternatives based on process-specific requirements and optimizes the sampling rate by determining how many products should be inspected, depending on the selected technology and the production system's operating conditions. In this paper, the three inspection methodologies are considered alternative options, allowing parallel adoption across different stages of the assembly process. While it would theoretically be possible to deploy both a robot and a human operator for inspection tasks at the same station, such a configuration would require a detailed study of their interaction dynamics, which falls outside the scope of this research. The proposed MILP formulation was selected due to its ability to provide exact and interpretable solutions, which are particularly suitable for supporting strategic decision-making processes in quality inspection planning. Although the complexity of the model increases with the size of the problem instance, the computational times remain acceptable for medium-scale instances and consider the strategic scope of the decisions being addressed. It is also acknowledged that heuristic and metaheuristic approaches could be valuable alternatives for very-large-scale instances. In this regard, the MILP model proposed herein can serve as a benchmark for future studies exploring heuristic and metaheuristic approaches. The model was tested on synthetic data to assess its computational limitations and on data coming from a real-life case study in the automotive sector to verify its applicability in a real-world context and derive further insights. In particular, three options of inspection methodologies are considered in the real-world case study: visual inspections performed by human operators, visual inspections performed by human operators using hand-held devices, and visual inspections performed by CNNs, with the necessary images collected via automated solutions such as fixed or mobile cameras. To guide the study, the following research questions are addressed:  
* How does the structural complexity of the inspection optimization model evolve with respect to key input dimensions, and what are the implications for computational scalability?