07 Prescriptive Modeling

.title[Prescriptive Modeling and Waste Load Allocation] <br> .subtitle[BEE 4750/5750] <br> .subtitle[Environmental Systems Analysis, Fall 2022] <hr> .author[Vivek Srikrishnan] <br> .date[September 19, 2022]

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# Outline

<hr>

1. Questions?
2. Prescriptive Modeling
3. Waste Load Allocation Problems

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# Poll

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URL: [https://pollev.com/vsrikrish](https://pollev.com/vsrikrish)

Text: **VSRIKRISH** to 22333, then message]

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***Any questions?***

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# Last Class

<hr>

* Fate and Transport Modeling
  * Dissolved Oxygen
  * Simulation Workflow

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# CRUD, Revisited

<hr>

Recall the CRUD wastewater model from [Lecture 2](https://viveks.me/environmental-systems-analysis/assets/lecture-notes/02-intro-modeling/index.html#13).

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# CRUD, Revisited

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Environmental authorities have sampled water from the river and determined that concentrations exceed the legal standard (1 mg/L).

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# CRUD, Revisited

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We want to design a CRUD removal plan to get back in compliance.

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# Prescriptive Modeling

<hr>

If we want to design a treatment strategy, we are now in the world of *prescriptive modeling*.

**Recall**: Precriptive modeling is intended to specify an action, policy, or decision.

* Descriptive modeling question: "What happens if I do something?"
  * Prescriptive modeling question: "What should I do?"

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# Decision Model Components

<hr>

This means that we need a *decision model*.

What information do we need to specify a decision?

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# Components of a Decision Model

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**Objective**: What are we trying to *do*?

* *e.g* minimizing cost or impact

**Constraints**: What limits do we have to observe?

* *e.g.* budgetary or regulatory constraints

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# Decision Model for CRUD Treatment

<hr>

Let's apply this framework to the CRUD release problem.

* What might our objective be?
  * What constraints do we have?

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# Formulating An Objective

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Let's say that our objective is to minimize costs, and relevant constraints include the regulatory standard.

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# Formulating An Objective

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Treating CRUD costs $ \$50 E^2 \text{ per } 1000 \  m^3,$ where $E$ is the treatment efficiency.

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# Formulating An Objective

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This makes the treatment cost

$$
C(E_1, E_2) = 50(100)E_1^2 + 50(60)E_2^2 = 5000E_1^2 + 3000E_2^2.
$$

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# Formulating An Objective

<hr>

Treatment Cost:

(E*1, E*2) = 50(100)E*1^2 + 50(60)E*2^2 = 5000E*1^2 + 3000E*2^2.:$

Then the objective is to minimize the cost, or

$$
\min_{E_1, E_2} 5000E_1^2 + 3000E_2^2.
$$

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# Developing Constraints

<hr>

But we can't choose just any $E_1$ and $E_2$ to minimize the cost, or we would just choose $E_1=E_2=0$.

For example, we need to comply with the regulatory standard:

CRUD concentration $< 1$ mg/l.

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# Developing Constraints

<hr>

**What information can we bring to bear?**

Since we know that the concentrations are highest at the points of discharge, we can check whether each of those points is in compliance with the standard.

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# Mass Balance at Point 1

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Total CRUD after factory 1 release: $\color{blue}\text{100} + \color{red} 1000(1-E_1) \color{black} \ \text{kg/d}$

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# Mass Balance at Point 1

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Our standard is $1 \ \text{mg/L} = 10^{-3} \ \text{kg/m}^3$, and the volume of the inflow is $\color{blue}500,000 + \color{red}100,000 \color{black}= 600,000 \ \text{m}^3\text{/d}$.

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# Constraint at Point 1

<hr>

So:

$$
\begin{aligned}
100 + 1000(1-E_1) &\leq 600 \\\\
\color{blue}1000E_1 &\color{blue}\geq 500
\end{aligned}
$$

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# Mass Balance at Release 2

<hr>

We [had derived](https://viveks.me/environmental-systems-analysis/assets/lecture-notes/02-intro-modeling/index.html#31) that the CRUD concentration at release 2 is:

$$
(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) \ \text{kg/d}.
$$

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# Mass Balance At Release 2

<hr>

The volume at release 2 is $660,000 \ \text{m}^3\text{/d}$, so the constraint is:

$$
(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) \leq 660.
$$

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# Constraint At Release 2

<hr>

Simplifying:

$$
\begin{aligned}
(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) &\leq 660  \\\\
(1100 - 1000E_1) 0.835 + 1200(1 - E_2) &\leq 660  \\\\
2119 - 835E_1 - 1200E_2 &\leq 660\\\\
\color{blue} 835E_1 + 1200E_2 &\color{blue}\geq 1459
\end{aligned}
$$

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# Model Formulation

<hr>

Combining our objective and our regulatory constraints:

$$
\begin{alignat}{3}
& \min_{E_1, E_2} &\color{green}\quad  5000E_1^2 + 3000E_2^2 & \notag \\\\
& \text{subject to:} & \color{blue}1000 E_1 &\color{blue}\geq 500 \notag \\\\
& & \color{blue}835E_1 + 1200E_2 &\color{blue}\geq 1459 \notag \\\\
\end{alignat}
$$

**Is this complete?**

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# Model Formulation

<hr>

$$
\begin{alignat}{3}
& \min_{E_1, E_2} &\color{green}\quad  5000E_1^2 + 3000E_2^2 &  \\\\
& \text{subject to:} & \color{blue}1000 E_1 &\color{blue}\geq 500 \\\\
& & \color{blue}835E_1 + 1200E_2 &\color{blue}\geq 1459 \\\\
& & \color{purple}E_1, E_2 &\color{purple}\geq 0 \\\\
& & \color{purple}E_1, E_2 &\color{purple}\leq 1
\end{alignat}
$$

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# Ok...so how do we solve this?

<hr>

More general discussion next time, but this example is relatively straightforward to solve graphically.

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# Plot the Decision Space

<hr>

```julia
using Plots
using LaTeXStrings
using Measures

# define objective function
a = range(0, 1, step=0.05)
b = range(0, 1, step=0.05)
f(a, b) = 5000 * a.^2 + 3000 * b.^2
# plotting contours
contour(a,b,(a,b)->f(a,b), nlevels=15,
  c=:heat, linewidth=5, colorbar = false,
  contour_labels = true, grid = false,
  right_margin=8mm)
xaxis!(L"E_1", ticks=0:0.1:1,
  limits=(0, 1))
yaxis!(L"E_2", ticks=0:0.1:1,
  limits=(0, 1))
```

```
"/private/var/folders/24/8k48jl6d249_n_qfxwsl6xvm0000gn/T/jl_DW9KSa/build/crud-base-plot.svg"
```

]

---

# Plot the Feasible Region

<hr>

```julia
# plot constraints
vline!([0.5], color=:green, linewidth=3,
  label=false) # Equation 2
plot!(a, (1459 .- 835 .* a) ./ 1200,
  color=:green, linewidth=3,
  label=false) # Equation 3
# plot feasible region
fa = a[a .>= 0.5]
fb = (1459 .- 835 .* a[a .>= 0.5])./1200
plot!(fa, fb, fillrange=1,
  label="Feasible Region", opacity=0.4,
  color=:green, legend=:bottomleft)
scatter!([0.5], [(1459 - 835 * 0.5) / 1200],
  markershape=:star, color=:yellow,
  markersize=20, label="Optimum")
```

```
"/private/var/folders/24/8k48jl6d249_n_qfxwsl6xvm0000gn/T/jl_DW9KSa/build/crud-feasible-plot.svg"
```

]

---

# The Solution!

<hr>

$$
E_1 = 0.5, E_2 = 0.85
$$

and the cost of this treatment plan is

$$
C(0.5, 0.85) = \$ 3417.
$$

**Does this solution make sense**?

] .right-column[![CRUD Problem Feasible Region](figures/crud-feasible-plot.svg) ]

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# Waste Load Allocation Problems

<hr>

This is an example of a *waste load allocation* problem.

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# Waste Load Allocation Problems

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**Key question**: How do we restrict waste discharges to control or limit environmental impact?

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center: left

# Waste Load Allocation Problems

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Each source is allocated a "load" they can discharge based on waste fate and transport.

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# Waste Load Allocation Problem

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Waste loads affect quality $Q$ based on F&T model:

$$
Q=f(W_1, W_2, \ldots, W_n)
$$

So the general form for a prescriptive waste load allocation model:

$$
\begin{aligned}
\text{determine} & \quad  W_1, W_2, \ldots, W_n \notag \\\\
\text{subject to:} & \quad f(W_1, W_2, \ldots, W_n) \geq Q^* \notag
\end{aligned}
$$

How do we choose certain variables to **optimize** a target outcome, possibly subject to some constraints.

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<hr>

# Next Class

<hr>

* Optimization Models
  * Solution Methods
  * Solving the CRUD Example