Waste Management and Network Models
Lecture 20
November 11, 2024
Review and Questions
Mixed Integer Linear Programming
- Problems involving integer variables break assumptions of LP.
- Can solve using branch and bound or related algorithms.
- Iteratively solve a sequence of restricted LPs until integer solution is found.
Example: Unit Commitment
- Power systems problem involving which generators to operate.
- Binary variables reflect operational status.
- Much more complicated than economic dispatch!
Questions
Text: VSRIKRISH to 22333
Solid Waste Management
Waste Load Allocation
Some relevant questions:
- Where do we send waste?
- What types of facilities do we build/operate?
Source: EPA
Waste Allocation As A Network
Want to model flows between waste sources and sinks (disposal sites).
Network Parameterization
| Variable | Definition |
|---|---|
| Waste produced at source |
|
| Capacity of disposal |
|
| Waste transported from source |
What Is Missing?
This network representation takes into account the graph structure of the system, but nothing else.
What else might we need?
System-Specific Information Needs
To complete the model, need:
- System dynamics (fate & transport);
- Objectives
- Costs
- Management goals and/or regulatory constraints
Waste Management Example
Example System
Three disposal options:
- Waste-to-Energy (WTE)
- Materials Recovery Facility (MRF)
- Landfill (LF)
Waste Management Decision Variables
What are our decision variables?
Waste Management Decision Variables
| Variable | Definition |
|---|---|
| Waste transported from city |
|
| Residual waste transported from disposal |
|
| Operational status (on/off) of disposal |
Objective: Minimize Total Costs
What are some components of total system cost?
Cost Components
- Transportation of waste
- Disposal: fixed costs and variable costs
Objective: Transportation Costs
| Variable | Definition |
|---|---|
| Cost of transporting waste from source |
|
| Distance between source |
Objective: Disposal Costs
| Variable | Definition |
|---|---|
| Fixed costs of operating disposal |
|
| Variable cost of disposing waste at disposal |
Is this expression for disposal costs right?
Objective: Disposal Costs
The prior expression is only correct if all disposal facilities are operating.
The option to not operate disposal facility
Waste Mass-Balance Constraints
- Need to dispose of all waste from each source
- Capacity limit at each disposal site
Problem Formulation
Facility Costs
| Facility | Fixed Cost ($/yr) | Tipping Cost ($/Mg) | Recycling Cost ($/Mg recycled) |
|---|---|---|---|
| WTE | 900,000 | 60 | – |
| MRF | 400,000 | 5 | 35 |
| LF | 700,000 | 40 | – |
MRF Recycling Rate: 40%
WTE Costs
Fixed Costs: $900,000/yr
Tipping Cost: $60/Mg handled
Total WTE Cost ($/day):
MRF Costs
Recycling: 40%, $35/Mg recycled
Fixed Costs: $400,000/yr
Tipping Cost: $5/Mg handled
Total MRF Cost ($/day):
LF Costs
Fixed Costs: $700,000/yr
Tipping Cost: $40/Mg handled
Total LF Cost ($/day):
Transportation Costs
Transportation Cost: $1.50/Mg-km
Total Tranportation Cost ($/day):
Final Objective
Combining the transportation and disposal costs and simplifying:
City Mass-Balance Constraints
City 1:
City 2:
Residual Mass-Balance Constraints
Recycling Rate: 40%
WTE Residual Ash: 20%
Disposal Limit Constraints
WTE:
MRF:
LF:
Commitment and Non-Negativity Constraints
How To Incorporate Indicator Constraints in JuMP
Some solvers allow you to directly translate indicator constraints into JuMP. For example,
implements
How To Incorporate Indicator Constraints in Other Solvers
Not all frameworks support this type of syntax. So what can we do?
Use a “big-M” reformulation:
where
Example Solution
After putting this all into JuMP (you know how!), optimal objective is $26,879/day.
Example Solution
Key Takeaways
Network Problems
- Explicitly model flows between network components;
- Often are limits on flow or node capacities;
- Fate & transport dynamics can result in residuals and extra flows.
- Often are non-obvious results for how to route relevant stocks through network.
Another example: power flow modeling through a transmission network.
Solid Waste Management
- Network induced by flows between sources and sinks
- Different types of facilities can produce different residual waste products.
- Extra residual waste can mean facilities have to operate.
Network Models
- Can have more advanced network models with highly-specified routes
- Transportation networks
- Power transmission networks
- Pipeline networks
- Can also add flow constraints rather than just mass-balance.
- Flows could also be bidirectional.
Upcoming Schedule
Next Lectures
- Wednesday: Prelim 2
- Next Week: Stochastic Optimization and Scenario Trees
