## 2 Literature Review
### Summary and Research Gap  
Table 1 summarizes the key characteristics of the reviewed papers, organized according to several core principles. The first criterion, product flow, distinguishes between studies considering a single-product production environment and those that address multi-product settings. The second, sampling, identifies whether the inspection strategy includes a defined sampling rate--meaning only a portion of the products is inspected--or assumes full inspection of all units. Detection error refers to whether the model accounts for the possibility of misclassification during inspection; if this column is left blank, it implies that the study assumes perfect detection accuracy. The inspection methodologies criterion indicates whether multiple approaches to inspection, such as human-based and technology-driven methods, are considered. Lastly, the solution approach describes the methods used to address the inspection optimization problem in each study. Over the years, research on inspection allocation has progressively evolved from abstract, simplified models toward frameworks that better reflect the complexity of real-world manufacturing systems. Early contributions often assumed idealized conditions--such as perfect detection, single-product flows, or unlimited inspection resources--that limited their practical relevance. More recent studies have moved toward integrating typical industrial features and constraints, including multiple product flows, resource limitations (e.g., availability of inspection stations or personnel), detection errors, production variability, and even preventive maintenance scheduling. This shift reflects a broader trend in the literature toward enhancing the industrial applicability of inspection models, driven by the need for robust, cost-effective solutions in diverse and dynamic production environments. Despite these advancements, as highlighted in Table 1, many existing approaches still fall short of fully capturing the complexity of modern systems--particularly in their limited consideration of multiple inspection technologies, lack of sampling rate optimization, or focus on single-product contexts. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Modeling Structure**} \\ \hline \multirow{3}{*}{**Reference**} & \multirow{3}{*}{\begin{tabular}{c} **Product** \\ **Flow** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Sampling** \\ **Rate** \\ **Identification** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Detection** \\ **Error** \\ **Consideration** \\ \end{tabular} } & \multicolumn{2}{c}{\
\begin{tabular}{c} **Inspection** \\ **Technology** \\ **Selection** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **Solution** \\ **Approach** \\ \end{tabular} } \\ \hline
[33] &
\begin{tabular}{c} Multi \\ Single \\ \end{tabular} & & & & & Meta-heuristic \\
[38] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & & MILP \\
[39] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & & MILP \\
[40] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & & MILP \\
[35] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & & Heuristic \\
[34] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & Meta-heuristic \\
[36] &
\begin{tabular}{c} Multi \\ \end{tabular} & & & & Meta-heuristic \\
[37] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & Simulation- \\
[42] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & & based \\  & & & & & optimization \\
[41] &
\begin{tabular}{c} Single \\ \end{tabular} & & & & & Dynamic \\  & & & & programming \\
**This paper** & **Multi** & & & & & **MILP** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Literature review summary. Motivated by existing gaps in the literature, this paper presents a novel MILP model that introduces several new and previously unaddressed capabilities in the context of inspection allocation.