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Implement transportation emissions
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@@ -39,6 +39,7 @@ The mathematical model employed by RELOG is based on three main components:
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| $C$ | Set of collection and distribution centers |
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| $P$ | Set of manufacturing and recycling plants |
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| $M$ | Set of products and materials |
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| $G$ | Set of greenhouse gases |
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| $M^+_u$ | Set of output products of plant/center $u$. |
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| $M^-_u$ | Set of input products of plant/center $u$. |
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| $T$ | Set of time periods in the planning horizon. We assume $T=\{1,\ldots,t^{max}\}.$ |
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@@ -48,35 +49,37 @@ The mathematical model employed by RELOG is based on three main components:
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## Constants
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| Symbol | Description | Unit |
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| :-------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :---------- |
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| $K^{\text{dist}}_{uv}$ | Distance between plants/centers $u$ and $v$ | km |
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| $K^\text{cap}_{p}$ | Capacity of plant $p$, if the plant is open | tonne |
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| $K^\text{disp-limit}_{mt}$ | Maximum amount of material $m$ that can be disposed of (globally) at time $t$ | tonne |
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| $K^\text{disp-limit}_{mut}$ | Maximum amount of material $m$ that can be disposed of at plant/center $u$ at time $t$ | tonne |
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| $K^\text{mix}_{pmt}$ | 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 |
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| $K^\text{output}_{pmt}$ | Amount of material $m$ produced by plant $p$ at time $t$ for each tonne of input material processed | tonne |
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| $R^\text{tr}_{mt}$ | Cost to send material $m$ at time $t$ | \$/km-tonne |
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| $R^\text{collect}_{cmt}$ | Cost of collecting material $m$ at center $c$ at time $t$ | \$/tonne |
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| $R^\text{disp}_{umt}$ | Cost to dispose of material at plant/center $u$ at time $t$ | \$/tonne |
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| $R^\text{fix}_{ut}$ | Fixed operating cost for plant/center $u$ at time $t$ | \$ |
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| $R^\text{open}_{pt}$ | Cost to open plant $p$ at time $t$ | \$ |
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| $R^\text{rev}_{ct}$ | Revenue for selling the input product of center $c$ at this center at time $t$ | \$/tonne |
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| $R^\text{var}_{pt}$ | Cost to process one tonne of input material at plant $p$ at time $t$ | \$/tonne |
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| $K^\text{out-fix}_{cmt}$ | Fixed amount of material $m$ collected at center $m$ at time $t$ | \$/tonne |
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| $K^\text{out-var}_{c,m,i}$ | Factor used to calculate variable amount of material $m$ collected at center $m$. See `eq_z_collected` for more details. | -- |
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| $K^\text{out-var-len}_{cm}$ | Length of the $K^\text{out-var}_{c,m,*}$ vector. | -- |
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| Symbol | Description | Unit |
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| :-------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :------------- |
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| $K^{\text{dist}}_{uv}$ | Distance between plants/centers $u$ and $v$ | km |
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| $K^\text{cap}_{p}$ | Capacity of plant $p$, if the plant is open | tonne |
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| $K^\text{disp-limit}_{mt}$ | Maximum amount of material $m$ that can be disposed of (globally) at time $t$ | tonne |
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| $K^\text{disp-limit}_{mut}$ | Maximum amount of material $m$ that can be disposed of at plant/center $u$ at time $t$ | tonne |
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| $K^\text{mix}_{pmt}$ | 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 |
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| $K^\text{output}_{pmt}$ | Amount of material $m$ produced by plant $p$ at time $t$ for each tonne of input material processed | tonne |
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| $K^\text{tr-em}_{gmt}$ | Amount of greenhouse gas $g$ released by transporting 1 tonne of material $m$ over one km at time $t$ | tonne/km-tonne |
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| $R^\text{tr}_{mt}$ | Cost to send material $m$ at time $t$ | \$/km-tonne |
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| $R^\text{collect}_{cmt}$ | Cost of collecting material $m$ at center $c$ at time $t$ | \$/tonne |
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| $R^\text{disp}_{umt}$ | Cost to dispose of material at plant/center $u$ at time $t$ | \$/tonne |
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| $R^\text{fix}_{ut}$ | Fixed operating cost for plant/center $u$ at time $t$ | \$ |
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| $R^\text{open}_{pt}$ | Cost to open plant $p$ at time $t$ | \$ |
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| $R^\text{rev}_{ct}$ | Revenue for selling the input product of center $c$ at this center at time $t$ | \$/tonne |
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| $R^\text{var}_{pt}$ | Cost to process one tonne of input material at plant $p$ at time $t$ | \$/tonne |
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| $K^\text{out-fix}_{cmt}$ | Fixed amount of material $m$ collected at center $m$ at time $t$ | \$/tonne |
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| $K^\text{out-var}_{c,m,i}$ | Factor used to calculate variable amount of material $m$ collected at center $m$. See `eq_z_collected` for more details. | -- |
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| $K^\text{out-var-len}_{cm}$ | Length of the $K^\text{out-var}_{c,m,*}$ vector. | -- |
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## Decision variables
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| Symbol | JuMP name | Description | Unit |
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| :--------------------------- | :------------------------------- | :------------------------------------------------------------------------------- | :----- |
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| $x_{pt}$ | `x[p.name, t]` | One if plant $p$ is operational at time $t$ | binary |
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| $y_{uvmt}$ | `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 |
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| $z^{\text{collected}}_{cmt}$ | `z_collected[c.name, m.name, t]` | Amount of material $m$ collected by center $c$ at time $t$ | tonne |
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| $z^{\text{disp}}_{umt}$ | `z_disp[u.name, m.name, t]` | Amount of product $m$ disposed of at plant/center $u$ at time $t$ | tonne |
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| $z^{\text{input}}_{ut}$ | `z_input[u.name, t]` | Total plant/center input at time $t$ | tonne |
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| $z^{\text{prod}}_{umt}$ | `z_prod[u.name, m.name, t]` | Amount of product $m$ produced by plant/center $u$ at time $t$ | tonne |
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| Symbol | JuMP name | Description | Unit |
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| :--------------------------- | :------------------------------------------- | :------------------------------------------------------------------------------------------------------ | :----- |
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| $x_{pt}$ | `x[p.name, t]` | One if plant $p$ is operational at time $t$ | binary |
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| $y_{uvmt}$ | `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 |
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| $z^{\text{collected}}_{cmt}$ | `z_collected[c.name, m.name, t]` | Amount of material $m$ collected by center $c$ at time $t$ | tonne |
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| $z^{\text{disp}}_{umt}$ | `z_disp[u.name, m.name, t]` | Amount of product $m$ disposed of at plant/center $u$ at time $t$ | tonne |
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| $z^{\text{input}}_{ut}$ | `z_input[u.name, t]` | Total plant/center input at time $t$ | tonne |
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| $z^{\text{prod}}_{umt}$ | `z_prod[u.name, m.name, t]` | Amount of product $m$ produced by plant/center $u$ at time $t$ | tonne |
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| $z^{\text{tr-em}}_{guvmt}$ | `z_tr_em[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 |
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## Objective function
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@@ -264,4 +267,13 @@ The goals is to minimize a linear objective function with the following terms:
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& \sum_{p \in P} z^\text{disp}_{pmt} + \sum_{c \in C} z^\text{disp}_{cmt} \leq K^\text{disp-limit}_{mt}
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& \forall m \in M, t \in T
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\end{align*}
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```
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```
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- Computation of transportation emissions (`eq_tr_em[g.name, u.name, v.name, m.name, t`)
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```math
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\begin{align*}
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& z^{\text{tr-em}}_{guvmt} = K^{\text{dist}}_{uv} K^\text{tr-em}_{gmt} y_{uvmt}
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& \forall g \in G, (u, v, m) \in E, t \in T
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\end{align*}
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```
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