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
| \(W_{ij}\) |
Waste transported from city \(i\) to disposal \(j\) (Mg/day) |
| \(R_{kj}\) |
Residual waste transported from disposal \(k\) to disposal \(j\) (Mg/day) |
| \(Y_j\) |
Operational status (on/off) of disposal \(j\) (binary) |
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
| \(a_{ij}\) |
Cost of transporting waste from source \(i\) to disposal \(j\) ($/Mg-km) |
| \(l_{ij}\) |
Distance between source \(i\) and disposal \(j\) (km) |
\[
\begin{align}
&\text{Transportation Costs} = \\[0.5em]
&\qquad \sum_{i \in I, j \in J} a_{ij}\,l_{ij}\,W_{ij}
\end{align}
\]
Objective: Disposal Costs
| \(c_j\) |
Fixed costs of operating disposal \(j\) ($/day) |
| \(b_{j}\) |
Variable cost of disposing waste at disposal \(j\) ($/Mg) |
\[
\begin{align}
&\text{Disposal Costs} = \\[0.5em]
&\qquad \sum_{j \in J} \left[c_j + b_j \sum_{i \in I} W_{ij}\right]
\end{align}
\]
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 \(j\) means we need new indicator variables.
\[Y_j = \begin{cases}0 & \text{if} \sum_{i \in I} W_{ij} = 0 \\[0.5em] 1 & \text{if} \sum_{i \in I} W_{ij} > 0\end{cases}\]
Waste Mass-Balance Constraints
- Need to dispose of all waste from each source \(i\): \[\sum_{j \in J} W_{ij} = S_i\]
- Capacity limit at each disposal site \(j\): \[\sum_{i \in I} W_{ij} \leq K_j\]
Facility Costs
| 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): \[2466 Y_1 + 60(W_{11} + W_{21} + R_{21})\]
MRF Costs
Recycling: 40%, $35/Mg recycled
Fixed Costs: $400,000/yr
Tipping Cost: $5/Mg handled
Total MRF Cost ($/day): \[
\begin{align}
&1096 Y_2 + 5(W_{12} + W_{22}) +
\\ &\qquad 0.4(35)(W_{12} + W_{22})
\end{align}
\]
LF Costs
Fixed Costs: $700,000/yr
Tipping Cost: $40/Mg handled
Total LF Cost ($/day): \[1918 Y_2 + 40(W_{13} + W_{23} + R_{13} + R_{23})\]
Transportation Costs
Transportation Cost: $1.50/Mg-km
Total Tranportation Cost ($/day): \[
\begin{align}
1.5&{\Large[}15W_{11} + 5W_{12} + 30W_{13} \\
& \quad 10W_{21} + 15W_{22} + 25W_{23} \\
& \quad 18R_{13} + 15R_{21} + 32R_{23}{\Large]}
\end{align}
\]
Final Objective
Combining the transportation and disposal costs and simplifying: \[
\begin{align}
\min_{W, R, Y} \qquad & 82.5 W_{11} + 26.5W_{12} + 85W_{13} + 75W_{21} + \\[0.25em]
& \quad 41.5W_{22} + 77.5W_{23} + 67R_{13} + 82.5R_{21} + \\[0.75em]
& \quad 88R_{23} + 2466Y_1 + 1096Y_2 + 1918Y_3
\end{align}
\]
City Mass-Balance Constraints
City 1: \(W_{11} + W_{12} + W_{13} = 100\)
City 2: \(W_{21} + W_{22} + W_{23} = 170\)
Residual Mass-Balance Constraints
Recycling Rate: 40%
WTE Residual Ash: 20%
\(R_{13} = 0.2(W_{11} + W_{21} + R_{21})\)
\(R_{21} + R_{23} = 0.6(W_{12} + W_{22})\)
Disposal Limit Constraints
WTE: \(W_{11} + W_{21} + R_{21} \leq 150\)
MRF: \(W_{12} + W_{22} \leq 130\)
LF: \(W_{13} + W_{23} + R_{23} + R_{13} \leq 200\)
Commitment and Non-Negativity Constraints
\[
\begin{align}
Y_1 &= \begin{cases}0 &\quad \text{if } W_{11} + W_{21} + R_{21} = 0 \\ 1 & \quad \text{else} \end{cases} \\
Y_2 &= \begin{cases}0 &\quad \text{if } W_{21} + W_{22} = 0 \\ 1 & \quad \text{else} \end{cases} \\
Y_3 &= 1 \\[0.5em]
W_{ij}, R_{ij} &\geq 0
\end{align}\]
How To Incorporate Indicator Constraints in JuMP
Some solvers allow you to directly translate indicator constraints into JuMP. For example,
@constraint(waste, commit2, !Y[2] => {W[1,2] + W[2,2] == 0})
implements \[Y_2 = \begin{cases}0 &\quad \text{if } W_{21} + W_{22} = 0 \\ 1 & \quad \text{else} \end{cases}
\]
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:
\[MY_1 \geq W_{21} + W_{22}\]
where \(M\) is so large that it is greater than any feasible value of the RHS.
Example Solution
After putting this all into JuMP (you know how!), optimal objective is $26,879/day.
\(Y_1 = 1\)
\(Y_2 = 1\)
\(Y_3 = 1\)
\(W_{12} = 100\) Mg/day
\(W_{23} = 170\) Mg/day
\(R_{13} = 7.5\) Mg/day
\(R_{21} = 37.5\) Mg/day
\(R_{23} = 22.5\) Mg/day
