Objective

In mathematical terms, the goal is to maximize the following objective function

\[f(\mathbf{x}) = \sum_{i = 1}^{n_{\max}} p_{\mathrm{com}}(i) c_{i} + \sum_{i = 1}^{n_{\max}} \sum_{j = 1}^{j_{\max}(i)} \sum_{\sigma = 1}^{\sigma_{\max}} p_{\mathrm{res}}(\sigma) r_{i, j, \sigma}\]

under the constraints that will be discussed in later sections. Here, \(\mathbf{x}\) is the tuple of all variables where

\[\mathbf{x} := (\dots, c_{i}, \dots r_{i, j, \sigma}, \dots)\]

where \(1 \leq i \leq n_{\max}\), \(1 \leq j \leq j_{\max}(i)\), and \(1 \leq \sigma \leq \sigma_{\max}\).

Note

There are many hidden variables not apearing in \(\mathbf{x}\) above, such as \(e_{i, j, \sigma}, s_{i, j, \sigma}, h_{i, j, \sigma}, l_{i, j, \xi},\) etc. These are used in constraints in relation to other variables.