Prescriptive Modeling
Lecture 13
September 25, 2023
Review and Questions
Last Classes
- Simulation modeling as a means for data generation.
- Uncertainty and Monte Carlo
Questions?
Text: VSRIKRISH to 22333
Decision Problems
Wastewater Model Revisited
Diagram of CRUD release from two factories
Consider the wastewater model from Lecture 5.
Wastewater Model Revisited
Environmental authorities have sampled water from the river and determined that concentrations exceed the legal standard (1 mg/L).
We want to design a CRUD removal plan to get back in compliance.
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.
Constraints
Constraints often come from engineering or societal restrictions.
- Regulatory limits on pollution;
- Supply constraints;
- Budgetary constraints;
- Engineering constraints.
Decision Model for the Wastewater Model
Let’s apply this framework to the CRUD release problem.
- What might our objective be?
- What constraints do we have?
Formulating an Objective
Let’s say that our objective is to minimize costs, and relevant constraints include the regulatory standard.
Treating CRUD costs \(\$50 E^2 \text{ per } 1000 \ \text{m}^3,\) where \(E\) is the treatment efficiency.
Formulating an Objective
Diagram of CRUD release from two factories
This makes the daily treatment cost \[C(E_1, E_2) = 50(100)E_1^2 + 50(60)E_2^2 = 5000E_1^2 + 3000E_2^2.\]
Formulating an Objective
Daily Treatment Cost (the metric):
\[C(E_1, E_2) = 50(100)E_1^2 + 50(60)E_2^2 = 5000E_1^2 + 3000E_2^2.\]
Then if the objective is to minimize the cost, we write:
\[\min_{E_1, E_2} 5000E_1^2 + 3000E_2^2.\]
Developing Constraints
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.
Developing Constraints
What information can we bring to bear?
Recall (from a while ago) our assumption that CRUD decays as it moves downriver.
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.
Mass Balance at Release 1
Total CRUD after factory 1 release: \(\color{blue}\text{100} + \color{red} 1000(1-E_1) \color{black} \ \text{kg/d}\)
Mass Balance at Release 1
- Total CRUD after factory 1 release: \(\color{blue}\text{100} + \color{red} 1000(1-E_1) \color{black} \ \text{kg/d}\)
- Our standard is \(1 \ \text{mg/L} = 10^{-3} \ \text{kg/m}^3\)
- The volume of the inflow is \(600,000 \ \text{m}^3\text{/d}\).
So the constraint until we hit Release 2 is: \[100 + 1000(1-E_1) \leq 600 \Rightarrow \boxed{1000E_1 \geq 500} \]
Mass Balance at Release 2
We had derived that the CRUD concentration at release 2 is:
\[(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) \ \text{kg/d}.\]
Constraint at Release 2
The volume at release 2 is \(660,000 \ \text{m}^3\text{/d}\), so the constraint is:
\[(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) \leq 660.\]
Constraint at Release 2
Simplifying:
\[\begin{aligned} (1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) &\leq 660 \\\\ (1100 - 1000E_1) 0.835 + 1200(1 - E_2) &\leq 660 \\\\ 2119 - 835E_1 - 1200E_2 &\leq 660 \end{aligned}\]
\[\Rightarrow \boxed{835E_1 + 1200E_2 \geq 1459}\]
Model Formulation
Combining our objective and our regulatory constraints:
\[\begin{alignat}{2} & \min_{E_1, E_2} & 5000E_1^2 + 3000E_2^2 \notag \\\\ & \text{subject to:} & 1000 E_1 &\geq 500 \notag \\ & & 35E_1 + 1200E_2 &\geq 1459 \notag \end{alignat}\]
Is this complete?
Model Formulation
\[\begin{alignat}{3} & \min_{E_1, E_2} &\quad 5000E_1^2 + 3000E_2^2 & \\\\ & \text{subject to:} & 1000 E_1 &\geq 500 \\ & & 835E_1 + 1200E_2 &\geq 1459 \\ & & \color{purple}E_1, E_2 &\;\color{purple}\geq 0 \\ & & \color{purple}E_1, E_2 &\;\color{purple}\leq 1 \end{alignat}\]
OK…now what?
We’ve formulated our model, but now we need to solve it to find \(E_1\) and \(E_2\). This is the realm of optimization.
In this case, the problem is straightforward as it is so low-dimensional.
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:
\[\begin{aligned} \text{determine} & \quad W_1, W_2, \ldots, W_n \notag \\\\ \text{subject to:} & \quad f(W_1, W_2, \ldots, W_n) \geq Q^* \notag \end{aligned}\]
Key Takeaways
Key Takeaways
- Prescriptive modeling involves decision models.
- Key components of a decision model:
- Objectives;
- Constraints.
- Need to bring to bear relevant engineering, economic, and environmental information to formulate these.
Upcoming Schedule
Next Classes
Wednesday: How do we solve this decision problem (and many others)?
Friday: Introduction to linear programming.
Assessments
- HW 3 assigned today, due 10/6.