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Implement plant storage

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Alinson S. Xavier
2025-09-19 14:14:50 -05:00
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@@ -49,33 +49,35 @@ The mathematical model employed by RELOG is based on three main components:
## Constants
| Symbol | Description | Unit |
| :-------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :------------- |
| $K^{\text{dist}}_{uv}$ | Distance between plants/centers $u$ and $v$ | km |
| $K^\text{cap-init}_p$ | 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 $c$ at time $t$ | tonne |
| $K^\text{out-var-len}_{cm}$ | Length of the $K^\text{out-var}_{c,m,*}$ vector. | -- |
| $K^\text{out-var}_{cmi}$ | 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 |
| $R^\text{expand}_{pt}$ | 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 |
| Symbol | Description | Unit |
| :---------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :------------- |
| $K^{\text{dist}}_{uv}$ | Distance between plants/centers $u$ and $v$ | km |
| $K^\text{cap-init}_p$ | 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 $c$ at time $t$ | tonne |
| $K^\text{out-var-len}_{cm}$ | Length of the $K^\text{out-var}_{c,m,*}$ vector. | -- |
| $K^\text{out-var}_{cmi}$ | 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 |
| $K^\text{storage-limit}_{pm}$ | Maximum amount of material $m$ that can be stored at plant $p$ at any time | 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 |
| $R^\text{expand}_{pt}$ | Cost to increase capacity of plant $p$ at time $t$ | \$/tonne |
| $R^\text{fix-exp}_{pt}$ | Increase in fixed operational cost for plant $p$ at time $t$ for every additional tonne of capacity | \$/tonne |
| $R^\text{fix-min}_{pt}$ | Fixed operating cost for plant $p$ at time $t$ at minimum capacity | \$ |
| $R^\text{fix}_{ct}$ | Fixed operating cost for center $c$ at time $t$ | \$ |
| $R^\text{open}_{pt}$ | 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 |
| $R^\text{storage}_{pmt}$ | Cost to store one tonne of material $m$ at plant $p$ at time $t$ for one year | \$/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 |
## Decision variables
@@ -88,8 +90,10 @@ The mathematical model employed by RELOG is based on three main components:
| $z^{\text{disp}}_{umt}$ | `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 |
| $z^{\text{input}}_{ut}$ | `z_input[u.name, t]` | Total amount received by plant/center $u$ 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 |
| $z^{\text{storage}}_{pmt}$ | `z_storage[p.name, m.name, t]` | Amount of input material $m$ stored at plant $p$ at the end of time $t$ | tonne |
| $z^{\text{process}}_{pt}$ | `z_process[p.name, t]` | Total amount of input material processed by plant $p$ at time $t$ | tonne |
## Objective function
@@ -171,6 +175,12 @@ The goal is to minimize a linear objective function with the following terms:
\sum_{p \in P} \sum_{(u,m) \in E^-(p)} \sum_{t \in T} R^\text{var}_{pt} y_{upmt}
```
- Plant storage cost, incurred for each tonne of material stored at the plant:
```math
\sum_{p \in P} \sum_{m \in M^-_p} \sum_{t \in T} R^\text{storage}_{pmt} z^{\text{storage}}_{pmt}
```
- Emissions penalty cost, incurred for each tonne of greenhouse gas emitted:
```math
@@ -190,13 +200,22 @@ The goal is to minimize a linear objective function with the following terms:
\end{align*}
```
- Plant input mix must have correct proportion
(`eq_input_mix[p.name, m.name, t]`):
- Definition of plant processing (`eq_z_process[p.name, t]`):
```math
\begin{align*}
& \sum_{u : (u,m) \in E^-(p)} y_{upmt}
= K^\text{mix}_{pmt} z^{\text{input}}_{pt}
& z^{\text{process}}_{pt} = z^{\text{input}}_{pt} + \sum_{m \in M^-_p} \left(z^{\text{storage}}_{p,m,t-1} - z^{\text{storage}}_{pmt}\right)
& \forall p \in P, t \in T
\end{align*}
```
- Plant processing mix must have correct proportion
(`eq_process_mix[p.name, m.name, t]`):
```math
\begin{align*}
& \sum_{u : (u,m) \in E^-(p)} y_{upmt} + z^{\text{storage}}_{p,m,t-1} - z^{\text{storage}}_{pmt}
= K^\text{mix}_{pmt} z^{\text{process}}_{pt}
& \forall p \in P, m \in M^-_p, t \in T
\end{align*}
```
@@ -205,7 +224,7 @@ The goal is to minimize a linear objective function with the following terms:
```math
\begin{align*}
& z^\text{prod}_{pmt} = K^\text{output}_{pmt} z^\text{input}_{pt}
& z^\text{prod}_{pmt} = K^\text{output}_{pmt} z^{\text{process}}_{pt}
& \forall p \in P, m \in M^+_p, t \in T
\end{align*}
```
@@ -220,7 +239,8 @@ The goal is to minimize a linear objective function with the following terms:
\end{align*}
```
- Plant can only be expanded if the plant is open, and up to a certain amount (`eq_exp_ub[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*}
@@ -251,15 +271,24 @@ The goal is to minimize a linear objective function with the following terms:
```
- Plants cannot process more than their current capacity
(`eq_input_limit[p.name,t]`)
(`eq_process_limit[p.name,t]`)
```math
\begin{align*}
& z^\text{input}_{pt} \leq K^\text{cap-min}_p x_{pt} + z^\text{exp}_{pt}
& z^\text{process}_{pt} \leq K^\text{cap-min}_p x_{pt} + z^\text{exp}_{pt}
& \forall p \in P, t \in T
\end{align*}
```
- Storage limit at the plants (`eq_storage_limit[p.name, m.name, t]`):
```math
\begin{align*}
& z^{\text{storage}}_{pmt} \leq K^\text{storage-limit}_{pm}
& \forall p \in P, m \in M^-_p, t \in T
\end{align*}
```
- Disposal limit at the plants (`eq_disposal_limit[p.name, m.name, t]`):
```math
@@ -343,7 +372,7 @@ The goal is to minimize a linear objective function with the following terms:
```math
\begin{align*}
& z^{\text{em-plant}}_{gpt} = K^\text{em-plant}_{gpt} z^{\text{input}}_{pt}
& z^{\text{em-plant}}_{gpt} = K^\text{em-plant}_{gpt} z^{\text{process}}_{pt}
& \forall g \in G, p \in P, t \in T
\end{align*}
```
@@ -356,3 +385,22 @@ The goal is to minimize a linear objective function with the following terms:
& \forall g \in G, t \in T
\end{align*}
```
- All stored materials must be processed by the end of the time horizon
(`eq_storage_final[p.name, m.name]`):
```math
\begin{align*}
& z^{\text{storage}}_{p,m,t^{max}} = 0
& \forall p \in P, m \in M^-_p
\end{align*}
```
- Initial storage is zero (`eq_storage_initial[p.name, m.name]`):
```math
\begin{align*}
& z^{\text{storage}}_{p,m,0} = 0
& \forall p \in P, m \in M^-_p
\end{align*}
```