Objective
=========

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

.. math::

   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, :math:`\mathbf{x}` is the tuple of all variables where

.. math::

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

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

.. note::

   There are many hidden variables not apearing in :math:`\mathbf{x}` above,
   such as
   :math:`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.