# An Adaptive Linear Programming Algorithm with Parameter Learning
## 3 Parameter Learning to Dynamically Change the RMC
### Step 2--Developing the Evaluation Indices (EIs)  
To manage different design preferences for multiple goals, we obtain solutions using multiple design scenarios. As the design scenarios are discrete and cannot enumerate all situations, we use the limited, discrete solutions to predict a satisficing solution space. By implementing multiple design scenarios, we acquire the results of the weight-sum deviations, active constraints, accumulated constraints, etc., using which, we develop statistical-based evaluation indices (EIs) as shown in Table 6. Statistical-based quality control has been used to monitor model profiles [34], so we leverage the statistics to obtain EIs. Our development of the EIs is based on the index for the robust concept exploration method, error margin index (EMI) [31], and design capability index (DCI) [35]. This is based on the central limit theorem that the results as samplings of each criterion follow Gaussian distributions, and the sampling statistics represent their characteristics. Therefore, we use the mean (\(\mu\)) and the standard deviation (\(\sigma\)) as EIs and tune the RMC by minimizing the \(\mu\) and \(\sigma\) of each EI. Index for evaluating the weight-sum deviation--\(\mu_{Z}\) and \(\sigma_{Z}\). To maintain the goal fulfillment (one minus weight-sum deviation) relatively insensitive to design scenario changes, we control the center and spread of the sample solutions by minimizing \(\mu_{Z}\) and \(\sigma_{Z}\). Index for evaluating the robustness--\(\mu_{\text{Nab}}\), \(\sigma_{\text{Nab}}\), \(\mu_{\text{Naoc}}\), and \(\sigma_{\text{Naoc}}\). Using the simplex algorithm, we obtain the vertex solution to the approximated linear problem. At a vertex solution, there is at least one active constraint (AOC) or active bound (AB) of the approximated linear problem, but we prefer fewer AOCs and ABs so the chance of losing a solution due to potential errors or variations is relatively small, so we minimize the \(\mu_{\text{Nab}}\), \(\sigma_{\text{Nab}}\), \(\mu_{\text{Naoc}}\), and \(\sigma_{\text{Naoc}}\). Index for evaluating the computational complexity--\(\mu_{\text{Nac}}\), \(\sigma_{\text{Nac}}\), \(\mu_{\text{Nit}}\), and \(\sigma_{\text{Nit}}\). As the approximation accuracy is not necessarily improved by increasing the number of accumulated constraints (Nacc) or the number of approximation iterations (Nit), we need to find the ranges of Nacc and Nit associated with good weight-sum deviations and robustness. We desire Nacc and Nit to be acceptable under all design scenarios and be insensitive to scenarios changing, so we measure \(\mu_{\text{Nacc}}\), \(\sigma_{\text{Nacc}}\), \(\mu_{\text{Nit}}\), and \(\sigma_{\text{Nit}}\) and identify their appropriate ranges. In summary, we tune the RMC by satisfying the EIs--\(\mu_{\text{Nab}}\), \(\sigma_{\text{Nab}}\), \(\mu_{\text{Naoc}}\), \(\sigma_{\text{Naoc}}\), \(\mu_{\text{Nacc}}\), \(\sigma_{\text{Nacc}}\), \(\mu_{\text{Nict}}\), and \(\sigma_{\text{Nit}}\)-- to obtain desired ranges (DEI). Then, the RMC tuning becomes: \"given\" the parameters and variables, \"find\" the value of decision variables that can \"satisfy\" constraints and bounds and \"minimize\" an objective. Here, we rank the EIs in the iTH iteration among all iterations and choose the first \(\kappa\) items with minimum values. These are the most important \(\kappa\) where EIs attain their best and most stable performance. Algorithm 3 is a summary of the RMC parameter learning algorithm. \begin{table}
\begin{tabular}{c c} \hline
**Criteria** & **Meaning and Representation** \\ \hline Weight-sum deviations & The unfulfilled percentage of the goals compared with their targets \\ \hline Robustness & Whether a solution is away from the boundary of the system as defined by the nonlinear model \\ \hline Approximation accuracy & Whether the nonlinear constraints are approximated well by a set of linear constraints in the sub-region that contains the solutions that fulfill the goals to an extent \\ \hline \end{tabular}
\end{table}