diff --git a/docs/source/models/san.rst b/docs/source/models/san.rst index b2970d47..b12ff9ea 100644 --- a/docs/source/models/san.rst +++ b/docs/source/models/san.rst @@ -108,22 +108,24 @@ 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:: @@ -131,11 +133,12 @@ where :math:`T(\theta)` is the (random) duration of the longest path from node : 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:: @@ -143,35 +146,41 @@ Additionally, we allow :math:`n` stochastic constraints that restrict the expect \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**