Prescriptive Modeling
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?”
Decision Models
To make a decision, we need certain pieces of information which:
- define decision options (or alternatives);
- provide one or more objectives to assess performance;
- specify constraints to tell us what decisions are possible or acceptable.
Objectives
Typical objectives can include:
- Minimizing costs (or maximizing profits);
- Minimizing environmental impacts;
- Maximizing some other performance metric.
Decision Modeling for Wastewater Treatment
Goal: Identify treatment levels for each factory which ensure compliance with regulatory standard of \(1 \ \text{mg}/\text{L}\).
Decision Variables
What are the decision variables?
Objectives and Metrics
Objectives are goals, such as “minimize cost” or “maximize environmental quantity”.
Metrics are functions which measure some relevant quantity, in this case the specific cost function.
Many different metrics can be used to specify an objective function!
Objective Function
The objective function includes a goal and a metric:
\[\begin{align*}
\min_{E_1, E_2} \quad &50(100)E_1^2 + 50(60)E_2^2 \\
= \min_{E_1, E_2} \quad &5000E_1^2 + 3000E_2^2.
\end{align*}\]
Identifying Constraints
What are relevant constraints?
What information do we need?
Constraint for First Segment
Where does the maximum value of the first segment occur?
Constraint for First Segment
Where does the maximum value of the first segment occur?
\[100 + 1000(1-E_1) \leq 600 \]
\[\Rightarrow \boxed{1000E_1 \geq 500}\]
Constraint for Second Segment
What is the concentration at the second release with treatment level \(E_2\)?
Constraint for Second Segment
What is the concentration at the second release with treatment level \(E_2\)?
\[(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) \leq 660 \]
\[\begin{aligned}
(1100 - 1000E_1) 0.835 + 1200(1 - E_2) &\leq 660 \\[0.5em]
2119 - 835E_1 - 1200E_2 &\leq 660
\end{aligned}\]
\[\Rightarrow \boxed{835E_1 + 1200E_2 \geq 1459}\]
Additional Constraints?
We have two concentration compliance constraints:
\[\begin{align*}
1000 E_1 &\geq 500\\[0.5em]
835E_1 + 1200E_2 &\geq 1459
\end{align*}\]
Are these a complete set?
Boundary Constraints
We need to add boundary constraints for \(E_1\), \(E_2\) to avoid implausible treatment levels.
\[\begin{align*}
1000 E_1 &\geq 500\\[0.5em]
835E_1 + 1200E_2 &\geq 1459\\[0.5em]
\color{purple}E_1, E_2 &\;\color{purple}\geq 0 \\[0.5em]
\color{purple}E_1, E_2 &\;\color{purple}\leq 1
\end{align*}\]
Final Problem
\[\begin{alignat}{3}
& \min_{E_1, E_2} &\quad 5000E_1^2 + 3000E_2^2 & \\\\
& \text{subject to:} & 1000 E_1 &\geq 500 \\
& & 835E_1 + 1200E_2 &\geq 1459 \\
& & E_1, E_2 &\;\geq 0 \\
& & E_1, E_2 &\;\leq 1
\end{alignat}\]
Plotting the Decision Space
Plotting the Feasible Region
The Solution!
So the solution occurs at the intersection of the two constraints, where:
\[E_1 = 0.5, E_2 = 0.85\]
and the cost of this treatment plan is
\[C(0.5, 0.85) = \$ 3417.\]
Does this solution make sense?
Waste Load Allocation Problem
This is an example of a waste load allocation problem.
Each source is allocated a “load” they can discharge based on waste fate and transport.
Waste Load Allocation Problem
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:
\[\begin{aligned}
\text{determine} & \quad W_1, W_2, \ldots, W_n \notag \\\\
\text{subject to:} & \quad f(W_1, W_2, \ldots, W_n) \geq Q^* \notag
\end{aligned}\]
