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53 changes: 31 additions & 22 deletions docs/source/models/san.rst
Original file line number Diff line number Diff line change
Expand Up @@ -108,70 +108,79 @@ Optimal Objective Function Value
Unknown

Optimization Problem: Minimize Longest Path Plus Penalty with Stochastic Constraints (SAN-2)
============================================================================================
--------------------------------------------------------------------------------------------

Decision Variables
-------------------
^^^^^^^^^^^^^^^^^^

* **arc_means**

Objectives
-----------
Suppose that we can select :math:`\theta_i > 0` for each :math:`i`, but there is an associated cost.
^^^^^^^^^^

Suppose that we can select :math:`\theta_i > 0` for each :math:`i`, but there is an associated cost.
In particular, we want to minimize:

.. math::

\mathbb{E}[T(\theta)] + f(\theta),

where :math:`T(\theta)` is the (random) duration of the longest path from node :math:`a` to node :math:`i`, and
where :math:`T(\theta)` is the (random) duration of the longest path from node :math:`a` to node :math:`i`, and

.. math::

f(\theta) = \sum_{i=1}^{n} \theta_i^{-1},

where :math:`n` is the number of arcs.

The objective function is convex in :math:`\theta`.
The objective function is convex in :math:`\theta`.

Constraints
------------
We require that :math:`\theta_i > 0` for each :math:`i`.
^^^^^^^^^^^

We require that :math:`\theta_i > 0` for each :math:`i`.
Additionally, we allow :math:`n` stochastic constraints that restrict the expected time to reach node :math:`i`, of the form:

.. math::

\mathbb{E}[T_i(\theta)] \leq a_i.

Problem Factors
----------------
* **budget**: Maximum number of replications the solver is allowed to take.
*Default:* ``10000``
^^^^^^^^^^^^^^^

* **arc_costs**: Cost associated with each arc.
*Default:* ``(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)``
* **budget**: Maximum number of replications the solver is allowed to take.
*Default:* ``10000``

* **constraint_nodes**: Nodes with corresponding stochastic constraints.
*Default:* ``[6, 8]``
* **arc_costs**: Cost associated with each arc.
*Default:* ``(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)``

* **length_to_node_constraint**: Maximum expected length to corresponding constraint nodes.
*Default:* ``[5, 5]``
* **constraint_nodes**: Nodes with corresponding stochastic constraints.
*Default:* ``[6, 8]``

* **length_to_node_constraint**: Maximum expected length to corresponding constraint nodes.
*Default:* ``[5, 5]``

Fixed Model Factors
--------------------
^^^^^^^^^^^^^^^^^^^

* **N/A**

Starting Solution
------------------
^^^^^^^^^^^^^^^^^

* **initial_solution**: ``(8,) * 13``

Random Solutions
-----------------
^^^^^^^^^^^^^^^^

Each arc mean is sampled independently from a lognormal distribution with 2.5th and 97.5th percentiles equal to 0.1 and 10, respectively.

Optimal Solution
-----------------
^^^^^^^^^^^^^^^^

* **Unknown**

Optimal Objective Function Value
---------------------------------
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

* **Unknown**
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