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Implement plant expansion

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Alinson S. Xavier
2025-09-19 13:05:36 -05:00
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6
docs/src/format.md
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@@ -152,6 +152,7 @@ detail.
| `disposal cost ($/tonne)` | Dictionary mapping the name of each output product to the cost of disposing it at the plant. |
| `disposal limit (tonne)` | Dictionary mapping the name of each output product to the maximum amount allowed to be disposed of at the plant. May be `null` if unlimited. |
| `capacities` | List describing what plant sizes are allowed, and their characteristics. |
| `initial capacity (tonne)` | Capacity already available. If the plant has not been built yet, this should be `0`. |
The entries in the `capacities` list should be dictionaries with the following
keys:
@@ -160,9 +161,8 @@ keys:
| :---------------------------------- | :-------------------------------------------------------------------------------------------------- |
| `size (tonne)` | The size of the plant. |
| `opening cost ($)` | The cost to open a plant of this size. |
| `fixed operating cost ($)` | The cost to keep the plant open, even if the plant doesn't process anything. Must be a time series. |
| `variable operating cost ($/tonne)` | The cost that the plant incurs to process each tonne of input. Must be a time series. |
| `initial capacity (tonne)` | Capacity already available. If the plant has not been built yet, this should be `0`. |
| `fixed operating cost ($)` | The cost to keep the plant open, even if the plant doesn't process anything. |
| `variable operating cost ($/tonne)` | The cost that the plant incurs to process each tonne of input. Must be the same for all capacities. |
```json
{

84
docs/src/problem.md
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@@ -5,7 +5,7 @@
The mathematical model employed by RELOG is based on three main components:
1. **Products and Materials:** Inputs and outputs for both manufacturing and
recycling plants. This include raw materials, whether virgin or recovered,
recycling plants. This includes raw materials, whether virgin or recovered,
and final products, whether new or at their end-of-life. Each product has
associated transportation parameters, such as costs, energy and emissions.
@@ -52,22 +52,27 @@ The mathematical model employed by RELOG is based on three main components:
| Symbol | Description | Unit |
| :-------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :------------- |
| $K^{\text{dist}}_{uv}$ | Distance between plants/centers $u$ and $v$ | km |
| $K^\text{cap}_{p}$ | Capacity of plant $p$, if the plant is open | tonne |
| $K^\text{cap-init}_p$ | Initial capacity of plant $p$ | tonne |
| $K^\text{cap-max}_p$ | Maximum capacity of plant $p$ | tonne |
| $K^\text{cap-min}_p$ | Minimum capacity of plant $p$ | tonne |
| $K^\text{disp-limit}_{mt}$ | Maximum amount of material $m$ that can be disposed of (globally) at time $t$ | tonne |
| $K^\text{disp-limit}_{mut}$ | Maximum amount of material $m$ that can be disposed of at plant/center $u$ at time $t$ | tonne |
| $K^\text{em-limit}_{gt}$ | Maximum amount of greenhouse gas $g$ allowed to be emitted (globally) at time $t$ | tonne |
| $K^\text{em-plant}_{gpt}$ | Amount of greenhouse gas $g$ released by plant $p$ at time $t$ for each tonne of input material processed | tonne/tonne |
| $K^\text{em-tr}_{gmt}$ | Amount of greenhouse gas $g$ released by transporting 1 tonne of material $m$ over one km at time $t$ | tonne/km-tonne |
| $K^\text{mix}_{pmt}$ | If plant $p$ receives one tonne of input material at time $t$, then $K^\text{mix}_{pmt}$ is the amount of product $m$ in this mix. Must be between zero and one, and the sum of these amounts must equal to one. | tonne |
| $K^\text{out-fix}_{cmt}$ | Fixed amount of material $m$ collected at center $m$ at time $t$ | \$/tonne |
| $K^\text{out-fix}_{cmt}$ | Fixed amount of material $m$ collected at center $c$ at time $t$ | tonne |
| $K^\text{out-var-len}_{cm}$ | Length of the $K^\text{out-var}_{c,m,*}$ vector. | -- |
| $K^\text{out-var}_{c,m,i}$ | Factor used to calculate variable amount of material $m$ collected at center $m$. See `eq_z_collected` for more details. | -- |
| $K^\text{out-var}_{cmi}$ | Factor used to calculate variable amount of material $m$ collected at center $c$. See `eq_z_collected` for more details. | -- |
| $K^\text{output}_{pmt}$ | Amount of material $m$ produced by plant $p$ at time $t$ for each tonne of input material processed | tonne |
| $R^\text{collect}_{cmt}$ | Cost of collecting material $m$ at center $c$ at time $t$ | \$/tonne |
| $R^\text{disp}_{umt}$ | Cost to dispose of material at plant/center $u$ at time $t$ | \$/tonne |
| $R^\text{em}_{gt}$ | Penalty cost per tonne of greenhouse gas $g$ emitted at time $t$ | \$/tonne |
| $R^\text{fix}_{ut}$ | Fixed operating cost for plant/center $u$ at time $t$ | \$ |
| $R^\text{open}_{pt}$ | Cost to open plant $p$ at time $t$ | \$ |
| $R^\text{expand}_{pt}$ | Cost to increase capacity of plant $p$ at time $t$ | \$/tonne |
| $R^\text{fix}_{ct}$ | Fixed operating cost for center $c$ at time $t$ | \$ |
| $R^\text{fix-min}_{pt}$ | Fixed operating cost for plant $p$ at time $t$ at minimum capacity | \$ |
| $R^\text{fix-exp}_{pt}$ | Increase in fixed operational cost for plant $p$ at time $t$ for every additional tonne of capacity | \$/tonne |
| $R^\text{open}_{pt}$ | Cost to open plant $p$ at time $t$, at minimum capacity | \$ |
| $R^\text{rev}_{ct}$ | Revenue for selling the input product of center $c$ at this center at time $t$ | \$/tonne |
| $R^\text{tr}_{mt}$ | Cost to send material $m$ at time $t$ | \$/km-tonne |
| $R^\text{var}_{pt}$ | Cost to process one tonne of input material at plant $p$ at time $t$ | \$/tonne |
@@ -78,16 +83,17 @@ The mathematical model employed by RELOG is based on three main components:
| :--------------------------- | :------------------------------------------- | :------------------------------------------------------------------------------------------------------ | :----- |
| $x_{pt}$ | `x[p.name, t]` | One if plant $p$ is operational at time $t$ | binary |
| $y_{uvmt}$ | `y[u.name, v.name, m.name, t]` | Amount of product $m$ sent from plant/center $u$ to plant/center $v$ at time $t$ | tonne |
| $z^{\text{exp}}_{pt}$ | `z_exp[p.name, t]` | Extra capacity installed at plant $p$ at time $t$ above the minimum capacity | tonne |
| $z^{\text{collected}}_{cmt}$ | `z_collected[c.name, m.name, t]` | Amount of material $m$ collected by center $c$ at time $t$ | tonne |
| $z^{\text{disp}}_{umt}$ | `z_disp[u.name, m.name, t]` | Amount of product $m$ disposed of at plant/center $u$ at time $t$ | tonne |
| $z^{\text{em-plant}}_{gpt}$ | `z_em_plant[g.name, p.name, t]` | Amount of greenhouse gas $g$ released by plant $p$ at time $t$ | tonne |
| $z^{\text{em-tr}}_{guvmt}$ | `z_em_tr[g.name, u.name, v.name, m.name, t]` | Amount of greenhouse gas $g$ released at time $t$ due to transportation of material $m$ from $u$ to $v$ | tonne |
| $z^{\text{input}}_{ut}$ | `z_input[u.name, t]` | Total plant/center input at time $t$ | tonne |
| $z^{\text{prod}}_{umt}$ | `z_prod[u.name, m.name, t]` | Amount of product $m$ produced by plant/center $u$ at time $t$ | tonne |
| $z^{\text{em-tr}}_{guvmt}$ | `z_em_tr[g.name, u.name, v.name, m.name, t]` | Amount of greenhouse gas $g$ released at time $t$ due to transportation of material $m$ from $u$ to $v$ | tonne |
| $z^{\text{em-plant}}_{gpt}$ | `z_em_plant[g.name, p.name, t]` | Amount of greenhouse gas $g$ released by plant $p$ at time $t$ | tonne |
## Objective function
The goals is to minimize a linear objective function with the following terms:
The goal is to minimize a linear objective function with the following terms:
- Transportation costs, which depend on transportation distance
$K^{\text{dist}}_{uv}$ and product-specific factor $R^\text{tr}_{mt}$:
@@ -131,7 +137,9 @@ The goals is to minimize a linear objective function with the following terms:
\sum_{p \in P} \sum_{m \in M^+_p} \sum_{t \in T} R^\text{disp}_{pmt} z^\text{disp}_{pmt}
```
- Plant opening cost:
- Plant opening cost, incurred when the plant goes from non-operational at time
$t-1$ to operational at time $t$. Never incurred if the plant is initially
open:
```math
\sum_{p \in P} \sum_{t \in T} R^\text{open}_{pt} \left(
@@ -140,10 +148,20 @@ The goals is to minimize a linear objective function with the following terms:
```
- Plant fixed operating cost, incurred for every time period, regardless of
input or output amounts, as long as the plant is operational:
input or output amounts, as long as the plant is operational. Depends on the
size of the plant:
```math
\sum_{p \in P} \sum_{t \in T} R^\text{fix}_{pt} x_{pt}
\sum_{p \in P} \sum_{t \in T} \left(
R^\text{fix-min}_{pt} x_{pt} +
R^\text{fix-exp}_{pt} z^\text{exp}_{pt}
\right)
```
- Plant expansion cost, incurred whenever plant capacity increases:
```math
\sum_{p \in P} \sum_{t \in T} R^\text{expand}_{pt} \left(z^\text{exp}_{pt} - z^\text{exp}_{p,t-1} \right)
```
- Plant variable operating cost, incurred for each tonne of input material
@@ -160,6 +178,7 @@ The goals is to minimize a linear objective function with the following terms:
\sum_{p \in P} z^{\text{em-plant}}_{gpt} + \sum_{(u,v,m) \in E} z^{\text{em-tr}}_{guvmt}
\right)
```
## Constraints
- Definition of plant input (`eq_z_input[p.name, t]`):
@@ -201,12 +220,42 @@ The goals is to minimize a linear objective function with the following terms:
\end{align*}
```
- Plants have a maximum capacity; furthermore, if the plant is not open, its
capacity is zero (`eq_capacity[p.name,t]`)
- Plant can only be expanded if the plant is open, and up to a certain amount (`eq_exp_ub[p.name, t]`):
```math
\begin{align*}
& z^\text{input}_{pt} \leq K^\text{cap}_p x_{pt}
& z^\text{exp}_{pt} \leq \left(K^\text{cap-max}_p - K^\text{cap-min}_p) x_{pt}
& \forall p \in P, t \in T
\end{align*}
```
- Plant is initially open if initial capacity is positive:
```math
\begin{align*}
& x_{p,0} = \begin{cases}
0 & \text{ if } K^\text{cap-init}_p = 0 \\
1 & \text{otherwise}
\end{cases}
& \forall p \in P
\end{align*}
```
- Calculation of initial plant expansion:
```math
\begin{align*}
& z^\text{exp}_{p,0} = K^\text{cap-init}_p - K^\text{cap-min}_p
& \forall p \in P
\end{align*}
```
- Plants cannot process more than their current capacity
(`eq_input_limit[p.name,t]`)
```math
\begin{align*}
& z^\text{input}_{pt} \leq K^\text{cap-min}_p x_{pt} + z^\text{exp}_{pt}
& \forall p \in P, t \in T
\end{align*}
```
@@ -280,7 +329,8 @@ The goals is to minimize a linear objective function with the following terms:
\end{align*}
```
- Computation of transportation emissions (`eq_emission_tr[g.name, u.name, v.name, m.name, t`):
- Computation of transportation emissions
(`eq_emission_tr[g.name, u.name, v.name, m.name, t]`):
```math
\begin{align*}
@@ -293,7 +343,7 @@ The goals is to minimize a linear objective function with the following terms:
```math
\begin{align*}
& z^{\text{em-plant}}_{gpt} = \sum_{(u,m) \in E^-(p)} K^\text{em-plant}_{gpt} y_{upmt}
& z^{\text{em-plant}}_{gpt} = K^\text{em-plant}_{gpt} z^{\text{input}}_{pt}
& \forall g \in G, p \in P, t \in T
\end{align*}
```