instructions.jmd

---
title : Lake Problem Simulation-Optimization Example
author : BEE 4750/5750
date : Fall 2022
weave_options:
template: ../bee4750html.tpl
doctype: md2html
---

# Background

Due to ongoing economic activity, a town emits phosphorous into a shallow lake (with a concentration of $a_t$), which also receives non-point source runoff (concentration $y_t$) from the surrounding area. The concentration of the lake at time $t+1$ is given by
```math
X_{t+1} = X_t + a_t + y_t + \frac{X_t^q}{1+X_t^q} - bX_t,
```
where:

| Parameter | Value |
| :------: | :------ |
| $a_t$ | point-source phosphorous concentration from the town |
| $y_t$ | non-point-source phosphorous concentration |
| $q$ | rate at which phosphorous is recycled from sediment |
| $b$| rate at which phosphorous leaves the lake |

and $X_0 = 0$, $y_t \sim LogNormal(\log(0.02), 0.2)$, $q=2.5$, and $b=0.4$.

# Questions (we might not get to address all of these)

1. Try to find the best solution you can (maximizing the objective $1/T \sum_{t=1}^T a_t$) which does not allow the lake to eutrophy (exceeding its critical threshold value) with more than 20% probability.

    Some things to experiment with:
    * How many Monte Carlo samples are you using?
    * What bounds are you searching over for the releases?
    * How many function evaluations are you using for the search?
    * What is the impact of different [`Metaheuristics.jl` algorithms](https://docs.juliahub.com/Metaheuristics/aJ70z/3.2.12/algorithms/)?

2. What are the tradeoffs that you notice between these choices, the performance of any found feasible solutions, and speed? What might you recommend?

3. How well does your solution perform against slightly different of $q$, $b$, and/or the variance and mean of $y$? Is the solution reasonably "robust" (in other words, can it satisfy the constraints if your estimate of those parameters is wrong)? How might you decide this?