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# Input and Output Data Formats
In this page, we describe the input and output JSON formats used by RELOG. In addition to these, RELOG can also produce [simplified reports](reports.md) in tabular data format.
## Input Data Format (JSON)
RELOG accepts as input a JSON file with three sections: `parameters`, `products` and `plants`. Below, we describe each section in more detail.
### Parameters
The **parameters** section describes details about the simulation itself.
| Key | Description
|:--------------------------|:---------------|
|`time horizon (years)` | Number of years in the simulation.
|`building period (years)` | List of years in which we are allowed to open new plants. For example, if this parameter is set to `[1,2,3]`, we can only open plants during the first three years. By default, this equals `[1]`; that is, plants can only be opened during the first year. |
|`distance metric` | Metric used to compute distances between pairs of locations. Valid options are: `"Euclidean"`, for the straight-line distance between points; or `"driving"` for an approximated driving distance. If not specified, defaults to `"Euclidean"`.
#### Example
```json
{
"parameters": {
"time horizon (years)": 2,
"building period (years)": [1],
"distance metric": "driving",
}
}
```
### Products
The **products** section describes all products and subproducts in the simulation. The field `instance["Products"]` is a dictionary mapping the name of the product to a dictionary which describes its characteristics. Each product description contains the following keys:
| Key | Description
|:--------------------------------------|:---------------|
|`transportation cost ($/km/tonne)` | The cost to transport this product. Must be a time series.
|`transportation energy (J/km/tonne)` | The energy required to transport this product. Must be a time series. Optional.
|`transportation emissions (tonne/km/tonne)` | A dictionary mapping the name of each greenhouse gas, produced to transport one tonne of this product along one kilometer, to the amount of gas produced (in tonnes). Must be a time series. Optional.
|`initial amounts` | A dictionary mapping the name of each location to its description (see below). If this product is not initially available, this key may be omitted. Must be a time series.
| `disposal limit (tonne)` | Total amount of product that can be disposed of across all collection centers. If omitted, all product must be processed. This parameter has no effect on product disposal at plants.
| `disposal cost ($/tonne)` | Cost of disposing one tonne of this product at a collection center. If omitted, defaults to zero. This parameter has no effect on product disposal costs at plants.
Each product may have some amount available at the beginning of each time period. In this case, the key `initial amounts` maps to a dictionary with the following keys:
| Key | Description
|:------------------------|:---------------|
| `latitude (deg)` | The latitude of the location.
| `longitude (deg)` | The longitude of the location.
| `amount (tonne)` | The amount of the product initially available at the location. Must be a time series.
#### Example
```json
{
"products": {
"P1": {
"initial amounts": {
"C1": {
"latitude (deg)": 7.0,
"longitude (deg)": 7.0,
"amount (tonne)": [934.56, 934.56]
},
"C2": {
"latitude (deg)": 7.0,
"longitude (deg)": 19.0,
"amount (tonne)": [198.95, 198.95]
},
"C3": {
"latitude (deg)": 84.0,
"longitude (deg)": 76.0,
"amount (tonne)": [212.97, 212.97]
}
},
"transportation cost ($/km/tonne)": [0.015, 0.015],
"transportation energy (J/km/tonne)": [0.12, 0.11],
"transportation emissions (tonne/km/tonne)": {
"CO2": [0.052, 0.050],
"CH4": [0.003, 0.002]
},
"disposal cost ($/tonne)": [-10.0, -12.0],
"disposal limit (tonne)": [1.0, 1.0],
},
"P2": {
"transportation cost ($/km/tonne)": [0.022, 0.020]
},
"P3": {
"transportation cost ($/km/tonne)": [0.0125, 0.0125]
},
"P4": {
"transportation cost ($/km/tonne)": [0.0175, 0.0175]
}
}
}
```
### Processing plants
The **plants** section describes the available types of reverse manufacturing plants, their potential locations and associated costs, as well as their inputs and outputs. The field `instance["Plants"]` is a dictionary mapping the name of the plant to a dictionary with the following keys:
| Key | Description
|:------------------------|:---------------|
| `input` | The name of the product that this plant takes as input. Only one input is accepted per plant.
| `outputs (tonne/tonne)` | A dictionary specifying how many tonnes of each product is produced for each tonnes of input. For example, if the plant outputs 0.5 tonnes of P2 and 0.25 tonnes of P3 for each tonnes of P1 provided, then this entry should be `{"P2": 0.5, "P3": 0.25}`. If the plant does not output anything, this key may be omitted.
|`energy (GJ/tonne)` | The energy required to process 1 tonne of the input. Must be a time series. Optional.
|`emissions (tonne/tonne)` | A dictionary mapping the name of each greenhouse gas, produced to process each tonne of input, to the amount of gas produced (in tonne). Must be a time series. Optional.
| `locations` | A dictionary mapping the name of the location to a dictionary which describes the site characteristics (see below).
Each type of plant is associated with a set of potential locations where it can be built. Each location is represented by a dictionary with the following keys:
| Key | Description
|:------------------------------|---------------|
| `latitude (deg)` | The latitude of the location, in degrees.
| `longitude (deg)` | The longitude of the location, in degrees.
| `disposal` | A dictionary describing what products can be disposed locally at the plant.
| `storage` | A dictionary describing the plant's storage.
| `capacities (tonne)` | A dictionary describing what plant sizes are allowed, and their characteristics.
The `storage` dictionary should contain the following keys:
| Key | Description
|:------------------------|:---------------|
| `cost ($/tonne)` | The cost to store a tonne of input product for one time period. Must be a time series.
| `limit (tonne)` | The maximum amount of input product this plant can have in storage at any given time.
The keys in the `disposal` dictionary should be the names of the products. The values are dictionaries with the following keys:
| Key | Description
|:------------------------|:---------------|
| `cost ($/tonne)` | The cost to dispose of the product. Must be a time series.
| `limit (tonne)` | The maximum amount that can be disposed of. If an unlimited amount can be disposed, this key may be omitted. Must be a time series.
The keys in the `capacities (tonne)` dictionary should be the amounts (in tonnes). The values are dictionaries with the following keys:
| Key | Description
|:--------------------------------------|:---------------|
| `opening cost ($)` | 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.
#### Example
```json
{
"plants": {
"F1": {
"input": "P1",
"outputs (tonne/tonne)": {
"P2": 0.2,
"P3": 0.5
},
"energy (GJ/tonne)": [0.12, 0.11],
"emissions (tonne/tonne)": {
"CO2": [0.052, 0.050],
"CH4": [0.003, 0.002]
},
"locations": {
"L1": {
"latitude (deg)": 0.0,
"longitude (deg)": 0.0,
"disposal": {
"P2": {
"cost ($/tonne)": [-10.0, -12.0],
"limit (tonne)": [1.0, 1.0]
}
},
"storage": {
"cost ($/tonne)": [5.0, 5.3],
"limit (tonne)": 100.0,
},
"capacities (tonne)": {
"100": {
"opening cost ($)": [500, 530],
"fixed operating cost ($)": [300.0, 310.0],
"variable operating cost ($/tonne)": [5.0, 5.2],
},
"500": {
"opening cost ($)": [750, 760],
"fixed operating cost ($)": [400.0, 450.0],
"variable operating cost ($/tonne)": [5.0, 5.2]
}
}
}
}
}
}
}
```
### Geographic database
Instead of specifying locations using latitudes and longitudes, it is also possible to specify them using unique identifiers, such as the name of a US state, or the county FIPS code. This works anywhere `latitude (deg)` and `longitude (deg)` are expected. For example, instead of:
```json
{
"initial amounts": {
"C1": {
"latitude (deg)": 37.27182,
"longitude (deg)": -119.2704,
"amount (tonne)": [934.56, 934.56]
},
}
}
```
is is possible to write:
```json
{
"initial amounts": {
"C1": {
"location": "us-state:CA",
"amount (tonne)": [934.56, 934.56]
},
}
}
```
Location names follow the format `db:id`, where `db` is the name of the database and `id` is the identifier for a specific location. RELOG currently includes the following databases:
Database | Description | Examples
:--------|:------------|:---------
`us-state`| List of states of the United States. | `us-state:IL` (State of Illinois)
`2018-us-county` | List of United States counties, as of 2018. IDs are 5-digit FIPS codes. | `2018-us-county:17043` (DuPage county in Illinois)
### Current limitations
* Each plant can only be opened exactly once. After open, the plant remains open until the end of the simulation.
* Plants can be expanded at any time, even long after they are open.
* All material available at the beginning of a time period must be entirely processed by the end of that time period. It is not possible to store unprocessed materials from one time period to the next.
* Up to two plant sizes are currently supported. Variable operating costs must be the same for all plant sizes.
* Accurate driving distances are only available for the continental United States.
## Output Data Format (JSON)
To be documented.

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# RELOG: Reverse Logistics Optimization
**RELOG** is an open-source supply chain optimization package focusing on reverse logistics and reverse manufacturing. The package uses Mixed-Integer Linear Programming to determine where to build recycling plants, what size should these plants have and which customers should be served by which plants. The package supports custom reverse logistics pipelines, with multiple types of plants, multiple types of product and multiple time periods.
```@raw html
<center>
<img src="assets/ex_transportation.png" width="1000px"/>
</center>
```
### Table of Contents
```@contents
Pages = ["usage.md", "format.md", "reports.md", "model.md"]
Depth = 3
```
### Source Code
* [https://github.com/ANL-CEEESA/RELOG](https://github.com/ANL-CEEESA/RELOG)
### Authors
* **Alinson S. Xavier,** Argonne National Laboratory <axavier@anl.gov>
* **Nwike Iloeje,** Argonne National Laboratory <ciloeje@anl.gov>
### License
```text
RELOG: Reverse Logistics Optimization
Copyright © 2020, UChicago Argonne, LLC. All Rights Reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted
provided that the following conditions are met:
1. Redistributions of source code must retain the above copyright notice, this list of
conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice, this list of
conditions and the following disclaimer in the documentation and/or other materials provided
with the distribution.
3. Neither the name of the copyright holder nor the names of its contributors may be used to
endorse or promote products derived from this software without specific prior written
permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR
IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY
AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR
CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
```

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# Optimization Model
In this page, we describe the precise mathematical optimization model used by RELOG to find the optimal logistics plan. This model is a variation of the classical Facility Location Problem, which has been widely studied in the operations research literature. To simplify the exposition, we present the simplified case where there is only one type of plant.
## Mathematical Description
### Sets
Symbol | Description
:-------|:------------
$L$ | Set of locations holding the original material to be recycled
$M$ | Set of materials recovered during the reverse manufacturing process
$P$ | Set of potential plants to open
$T = \{ 1, \ldots, t^{max} \}$ | Set of time periods
### Constants
#### Plants
Symbol | Description | Unit
:-------|:------------|:---
$c^\text{disp}_{pmt}$ | Cost of disposing one tonne of material $m$ at plant $p$ during time $t$ | \$/tonne/km
$c^\text{exp}_{pt}$ | Cost of adding one tonne of capacity to plant $p$ at time $t$ | \$/tonne
$c^\text{open}_{pt}$ | Cost of opening plant $p$ at time $t$, at minimum capacity | $
$c^\text{f-base}_{pt}$ | Fixed cost of keeping plant $p$ open during time period $t$ | $
$c^\text{f-exp}_{pt}$ | Increase in fixed cost for each additional tonne of capacity | \$/tonne
$c^\text{var}_{pt}$ | Variable cost of processing one tonne of input at plant $p$ at time $t$ | \$/tonne
$c^\text{store}_{pt}$ | Cost of storing one tonne of original material at plant $p$ at time $t$ | \$/tonne
$m^\text{min}_p$ | Minimum capacity of plant $p$ | tonne
$m^\text{max}_p$ | Maximum capacity of plant $p$ | tonne
$m^\text{disp}_{pmt}$ | Maximum amount of material $m$ that plant $p$ can dispose of during time $t$ | tonne
$m^\text{store}_p$ | Maximum amount of original material that plant $p$ can store for later processing. | tonne
#### Products
Symbol | Description | Unit
:-------|:------------|:---
$\alpha_{pm}$ | Amount of material $m$ recovered by plant $t$ for each tonne of original material | tonne/tonne
$m^\text{initial}_{lt}$ | Amount of original material to be recycled at location $l$ during time $t$ | tonne
#### Transportation
Symbol | Description | Unit
:-------|:------------|:---
$c^\text{tr}_{t}$ | Transportation cost during time $t$ | \$/tonne/km
$d_{lp}$ | Distance between plant $p$ and location $l$ | km
### Decision variables
Symbol | Description | Unit
:-------|:------------|:---
$q_{mpt}$ | Amount of material $m$ recovered by plant $p$ during time $t$ | tonne
$u_{pt}$ | Binary variable that equals 1 if plant $p$ starts operating at time $t$ | Boolean
$w_{pt}$ | Extra capacity (amount above the minimum) added to plant $p$ during time $t$ | tonne
$x_{pt}$ | Binary variable that equals 1 if plant $p$ is operational at time $t$ | Boolean
$y_{lpt}$ | Amount of product sent from location $l$ to plant $p$ during time $t$ | tonne
$z^{\text{disp}}_{mpt}$ | Amount of material $m$ disposed of by plant $p$ during time $t$ | tonne
$z^{\text{store}}_{pt}$ | Amount of original material in storage at plant $p$ by the end of time period $t$ | tonne
$z^{\text{proc}}_{mpt}$ | Amount of original material processed by plant $p$ during time period $t$ | tonne
### Objective function
RELOG minimizes the overall capital, production and transportation costs:
```math
\begin{align*}
\text{minimize} \;\; &
\sum_{t \in T} \sum_{p \in P} \left[
c^\text{open}_{pt} u_{pt} +
c^\text{f-base}_{pt} x_{pt} +
\sum_{i=1}^t c^\text{f-exp}_{pt} w_{pi} +
c^{\text{exp}}_{pt} w_{pt}
\right] + \\
&
\sum_{t \in T} \sum_{p \in P} \left[
c^{\text{store}}_{pt} z^{\text{store}}_{pt} +
c^{\text{proc}}_{pt} z^{\text{proc}}_{pt}
\right] + \\
&
\sum_{t \in T} \sum_{l \in L} \sum_{p \in P}
c^{\text{tr}}_t d_{lp} y_{lpt}
\\
&
\sum_{t \in T} \sum_{p \in P} \sum_{m \in M} c^{\text{disp}}_{pmt} z_{pmt}
\end{align*}
```
In the first line, we have (i) opening costs, if plant starts operating at time $t$, (ii) fixed operating costs, if plant is operational, (iii) additional fixed operating costs coming from expansion performed in all previous time periods up to the current one, and finally (iv) the expansion costs during the current time period.
In the second line, we have storage and variable processing costs.
In the third line, we have transportation costs.
In the fourth line, we have the disposal costs.
### Constraints
* All original materials must be sent to a plant:
```math
\begin{align*}
& \sum_{p \in P} y_{lpt} = m^\text{initial}_{lt}
& \forall l \in L, t \in T
\end{align*}
```
* Amount received equals amount processed plus stored. Furthermore, all original material should be processed by the end of the simulation.
```math
\begin{align*}
& \sum_{l \in L} y_{lpt} + z^{\text{store}}_{p,t-1}
= z^{\text{proc}}_{pt} + z^{\text{store}}_{p,t}
& \forall p \in P, t \in T \\
& z^{\text{store}}_{p,0} = 0
& \forall p \in P \\
& z^{\text{store}}_{p,t^{\max}} = 0
& \forall p \in P
\end{align*}
```
* Plants have a limited processing capacity. Furthermore, if a plant is closed, it has zero processing capacity:
```math
\begin{align*}
& z^{\text{proc}}_{pt} \leq m^\text{min}_p x_p + \sum_{i=1}^t w_p
& \forall p \in P, t \in T
\end{align*}
```
* Plants have limited storage capacity. Furthermore, if a plant is closed, is has zero storage capacity:
```math
\begin{align*}
& z^{\text{store}}_{pt} \leq m^\text{store}_p x_p
& \forall p \in P, t \in T
\end{align*}
```
* Plants can only be expanded up to their maximum capacity. Furthermore, if a plant is closed, it cannot be expanded:
```math
\begin{align*}
& \sum_{i=1}^t w_p \leq m^\text{max}_p x_p
& \forall p \in P, t \in T
\end{align*}
```
* Amount of recovered material is proportional to amount processed:
```math
\begin{align*}
& q_{mpt} = \alpha_{pm} z^{\text{proc}}_{pt}
& \forall m \in M, p \in P, t \in T
\end{align*}
```
* Because we only consider a single type of plant, all recovered material must be immediately disposed of. In RELOG's full model, recovered materials may be sent to another plant for further processing.
```math
\begin{align*}
& q_{mpt} = z_{mpt}
& \forall m \in M, p \in P, t \in T
\end{align*}
```
* A plant is operational at time $t$ if it was operational at time $t-1$ or it was built at time $t$. This constraint also prevents a plant from being built multiple times.
```math
\begin{align*}
& x_{pt} = x_{p,t-1} + u_{pt}
& \forall p \in P, t \in T \setminus \{1\} \\
& x_{p,1} = u_{p,1}
& \forall p \in P
\end{align*}
```
* Variable bounds:
```math
\begin{align*}
& q_{mpt} \geq 0
& \forall m \in M, p \in P, t \in T \\
& u_{pt} \in \{0,1\}
& \forall p \in P, t \in T \\
& w_{pt} \geq 0
& \forall p \in P, t \in T \\
& x_{pt} \in \{0,1\}
& \forall p \in P, t \in T \\
& y_{lpt} \geq 0
& \forall l \in L, p \in P, t \in T \\
& z^{\text{store}}_{pt} \geq 0
& p \in P, t \in T \\
& z^{\text{disp}}_{mpt}, z^{\text{proc}}_{mpt} \geq 0
& \forall m \in M, p \in P, t \in T
\end{align*}
```

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# Simplified Solution Reports
In addition to the full output format described in [data formats](format.md), RELOG can also generate a number of simplified reports in tabular data format (CSV), which can be more easily processed by spreadsheet software (such as Microsoft Excel), by data analysis libraries (such as Pandas) or by relational databases (such as SQLite).
In this page, we also illustrate what types of charts and visualizations can be produced from these tabular data files. The sample charts have been produced using Python, matplotlib, seaborn and geopandas.
## Plants report
Report showing plant costs, capacities, energy expenditure and utilization factors. Generated by `RELOG.write_plants_report(solution, filename)`.
| Column | Description
|:--------------------------------------|:---------------|
| `plant type` | Plant type.
| `location name` | Location name.
| `year` | What year this row corresponds to. This reports includes one row for each year.
| `latitude (deg)` | Latitude of the plant.
| `longitude (deg)` | Longitude of the plant.
| `capacity (tonne)` | Capacity of the plant at this point in time.
| `amount received (tonne)` | Amount of input material received by the plant this year.
| `amount processed (tonne)` | Amount of input material processed by the plant this year.
| `amount in storage (tonne)` | Amount of input material in storage at the end of the year.
| `utilization factor (%)` | Amount processed by the plant this year divided by current plant capacity.
| `energy (GJ)` | Amount of energy expended by the plant this year.
| `opening cost ($)` | Amount spent opening the plant. This value is only positive if the plant became operational this year.
| `expansion cost ($)` | Amount spent this year expanding the plant capacity.
| `fixed operating cost ($)` | Amount spent for keeping the plant operational this year.
| `variable operating cost ($)` | Amount spent this year to process the input material.
| `storage cost ($)` | Amount spent this year on storage.
| `total cost ($)` | Sum of all previous plant costs.
### Sample charts
* Bar plot with total plant costs per year, grouped by plant type (in Python):
```python
import pandas as pd
import seaborn as sns; sns.set()
data = pd.read_csv("plants_report.csv")
sns.barplot(x="year",
y="total cost ($)",
hue="plant type",
data=data.groupby(["plant type", "year"])
.sum()
.reset_index());
```
```@raw html
<img src="../assets/ex_plant_cost_per_year.png" width="500px"/>
```
* Map showing plant locations (in Python):
```python
import pandas as pd
import geopandas as gp
# Plot base map
world = gp.read_file(gp.datasets.get_path('naturalearth_lowres'))
ax = world.plot(color='white', edgecolor='50', figsize=(13,6))
ax.set_ylim([23, 50])
ax.set_xlim([-128, -65])
# Plot plant locations
data = pd.read_csv("nimh_plants.csv")
points = gp.points_from_xy(data["longitude (deg)"],
data["latitude (deg)"])
gp.GeoDataFrame(data, geometry=points).plot(ax=ax);
```
```@raw html
<img src="../assets/ex_plant_locations.png" width="1000px"/>
```
## Plant outputs report
Report showing amount of products produced, sent and disposed of by each plant, as well as disposal costs. Generated by `RELOG.write_plant_outputs_report(solution, filename)`.
| Column | Description
|:--------------------------------------|:---------------|
| `plant type` | Plant type.
| `location name` | Location name.
| `year` | What year this row corresponds to. This reports includes one row for each year.
| `product name` | Product being produced.
| `amount produced (tonne)` | Amount of product produced this year.
| `amount sent (tonne)` | Amount of product produced by this plant and sent to another plant for further processing this year.
| `amount disposed (tonne)` | Amount produced produced by this plant and immediately disposed of locally this year.
| `disposal cost ($)` | Disposal cost for this year.
### Sample charts
* Bar plot showing total amount produced for each product, grouped by year (in Python):
```python
import pandas as pd
import seaborn as sns; sns.set()
data = pd.read_csv("plant_outputs_report.csv")
sns.barplot(x="amount produced (tonne)",
y="product name",
hue="year",
data=data.groupby(["product name", "year"])
.sum()
.reset_index());
```
```@raw html
<img src="../assets/ex_amount_produced.png" width="500px"/>
```
## Plant emissions report
Report showing amount of emissions produced by each plant. Generated by `RELOG.write_plant_emissions_report(solution, filename)`.
| Column | Description
|:--------------------------------------|:---------------|
| `plant type` | Plant type.
| `location name` | Location name.
| `year` | Year.
| `emission type` | Type of emission.
| `amount (tonne)` | Amount of emission produced by the plant this year.
### Sample charts
* Bar plot showing total emission by plant type, grouped type of emissions (in Python):
```python
import pandas as pd
import seaborn as sns; sns.set()
data = pd.read_csv("plant_emissions_report.csv")
sns.barplot(x="plant type",
y="emission amount (tonne)",
hue="emission type",
data=data.groupby(["plant type", "emission type"])
.sum()
.reset_index());
```
```@raw html
<img src="../assets/ex_emissions.png" width="500px"/>
```
## Products report
Report showing primary product amounts, locations and marginal costs. Generated by `RELOG.write_products_report(solution, filename)`.
| Column | Description
|:--------------------------------------|:---------------|
| `product name` | Product name.
| `location name` | Name of the collection center.
| `latitude (deg)` | Latitude of the collection center.
| `longitude (deg)` | Longitude of the collection center.
| `year` | What year this row corresponds to. This reports includes one row for each year.
| `amount (tonne)` | Amount of product available at this collection center.
| `amount disposed (tonne)` | Amount of product disposed of at this collection center.
| `marginal cost ($/tonne)` | Cost to process one additional tonne of this product coming from this collection center.
## Transportation report
Report showing amount of product sent from initial locations to plants, and from one plant to another. Includes the distance between each pair of locations, amount-distance shipped, transportation costs and energy expenditure. Generated by `RELOG.write_transportation_report(solution, filename)`.
| Column | Description
|:--------------------------------------|:---------------|
| `source type` | If product is being shipped from an initial location, equals `Origin`. If product is being shipped from a plant, equals plant type.
| `source location name` | Name of the location where the product is being shipped from.
| `source latitude (deg)` | Latitude of the source location.
| `source longitude (deg)` | Longitude of the source location.
| `destination type`| Type of plant the product is being shipped to.
| `destination location name`| Name of the location where the product is being shipped to.
| `destination latitude (deg)` | Latitude of the destination location.
| `destination longitude (deg)` | Longitude of the destination location.
| `product`| Product being shipped.
| `year`| Year.
| `distance (km)`| Distance between source and destination.
| `amount (tonne)`| Total amount of product being shipped between the two locations this year.
| `amount-distance (tonne-km)`| Total amount being shipped this year times distance.
| `transportation cost ($)`| Cost to transport this amount of product between the two locations for this year.
| `transportation energy (GJ)`| Energy expended transporting this amount of product between the two locations.
### Sample charts
* Bar plot showing total amount-distance for each product type, grouped by year (in Python):
```python
import pandas as pd
import seaborn as sns; sns.set()
data = pd.read_csv("transportation_report.csv")
sns.barplot(x="product",
y="amount-distance (tonne-km)",
hue="year",
data=data.groupby(["product", "year"])
.sum()
.reset_index());
```
```@raw html
<img src="../assets/ex_transportation_amount_distance.png" width="500px"/>
```
* Map of transportation lines (in Python):
```python
import pandas as pd
import geopandas as gp
from shapely.geometry import Point, LineString
import matplotlib.pyplot as plt
from matplotlib import collections
# Plot base map
world = gp.read_file(gp.datasets.get_path('naturalearth_lowres'))
ax = world.plot(color='white', edgecolor='50', figsize=(14,7))
ax.set_ylim([23, 50])
ax.set_xlim([-128, -65])
# Draw transportation lines
data = pd.read_csv("transportation_report.csv")
lines = [[(row["source longitude (deg)"], row["source latitude (deg)"]),
(row["destination longitude (deg)"], row["destination latitude (deg)"])
] for (index, row) in data.iterrows()]
ax.add_collection(collections.LineCollection(lines,
linewidths=0.25,
zorder=1,
alpha=0.5,
color="50"))
# Draw source points
points = gp.points_from_xy(data["source longitude (deg)"],
data["source latitude (deg)"])
gp.GeoDataFrame(data, geometry=points).plot(ax=ax,
color="0.5",
markersize=1);
# Draw destination points
points = gp.points_from_xy(data["destination longitude (deg)"],
data["destination latitude (deg)"])
gp.GeoDataFrame(data, geometry=points).plot(ax=ax,
color="red",
markersize=50);
```
```@raw html
<img src="../assets/ex_transportation.png" width="1000px"/>
```
## Transportation emissions report
Report showing emissions for each trip between initial locations and plants, and between pairs of plants. Generated by `RELOG.write_transportation_emissions_report(solution, filename)`.
| Column | Description
|:--------------------------------------|:---------------|
| `source type` | If product is being shipped from an initial location, equals `Origin`. If product is being shipped from a plant, equals plant type.
| `source location name` | Name of the location where the product is being shipped from.
| `source latitude (deg)` | Latitude of the source location.
| `source longitude (deg)` | Longitude of the source location.
| `destination type`| Type of plant the product is being shipped to.
| `destination location name`| Name of the location where the product is being shipped to.
| `destination latitude (deg)` | Latitude of the destination location.
| `destination longitude (deg)` | Longitude of the destination location.
| `product`| Product being shipped.
| `year`| Year.
| `distance (km)`| Distance between source and destination.
| `shipped amount (tonne)`| Total amount of product being shipped between the two locations this year.
| `shipped amount-distance (tonne-km)`| Total amount being shipped this year times distance.
| `emission type` | Type of emission.
| `emission amount (tonne)` | Amount of emission produced by transportation segment this year.
### Sample charts
* Bar plot showing total emission amount by emission type, grouped by type of product being transported (in Python):
```python
import pandas as pd
import seaborn as sns; sns.set()
data = pd.read_csv("transportation_emissions_report.csv")
sns.barplot(x="emission type",
y="emission amount (tonne)",
hue="product",
data=data.groupby(["product", "emission type"])
.sum()
.reset_index());
```
```@raw html
<img src="../assets/ex_transportation_emissions.png" width="500px"/>
```

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Usage
=====
## 1. Installation
To use RELOG, the first step is to install the [Julia programming language](https://julialang.org/) on your machine. Note that RELOG was developed and tested with Julia 1.8 and may not be compatible with newer versions. After Julia is installed, launch the Julia console, then run:
```julia
using Pkg
Pkg.add(name="RELOG", version="0.6")
```
After the package and all its dependencies have been installed, please run the RELOG test suite, as shown below, to make sure that the package has been correctly installed:
```julia
Pkg.test("RELOG")
```
## 2. Modeling the problem
The two main model components in RELOG are **products** and **plants**.
A **product** is any material that needs to be recycled, any intermediary product produced during the recycling process, or any product recovered at the end of the process. For example, in a NiMH battery recycling study case, products could include (i) the original batteries to be recycled; (ii) the cathode and anode parts of the battery; (iii) rare-earth elements and (iv) scrap metals.
* The model assumes that some products are initially available at user-specified locations (described by their latitude, longitude and the amount available), while other products only become available during the recycling process.
* Products that are initially available must be sent to a plant for processing during the same time period they became available.
* Transporting products from one location to another incurs a transportation cost (`$/km/tonne`), spends some amount of energy (`J/km/tonne`) and may generate multiple types of emissions (`tonne/tonne`). All these parameters are user-specified and may be product- and time-specific.
A **plant** is a facility that converts one type of product to another. RELOG assumes that each plant receives a single type of product as input and converts this input into multiple types of products. Multiple types of plants, with different inputs, outputs and performance characteristics, may be specified. In the NiMH battery recycling study case, for example, one type of plant could be a *disassembly plant*, which converts *batteries* into *cathode* and *anode*. Another type of plant could be *anode recycling plant*, which converts *anode* into *rare-earth elements* and *scrap metals*.
* To process each tonne of input material, plants incur a variable operating cost (`$/tonne`), spend some amount of energy (`GJ/tonne`), and produce multiple types of emissions (`tonne/tonne`). Plants also incur a fixed operating cost (`$`) regardless of the amount of material they process. All these parameters are user-specified and may be region- and time-specific.
* Plants can be built at user-specified potential locations. Opening a plant incurs a one-time opening cost (`$`) which may be region- and time-specific. Plants also have a limited capacity (in `tonne`), which indicates the maximum amount of input material they are able to process per year. When specifying potential locations for each type of plant, it is also possible to specify the minimum and maximum capacity of the plants that can be built at that particular location. Different plants sizes may have different opening costs and fixed operating costs. After a plant is built, it can be further expanded in the following years, up to its maximum capacity.
* Products received by a plant can be either processed immediately or stored for later processing. Plants have a maximum storage capacity (`tonne`). Storage costs (`$/tonne`) can also be specified.
* All products generated by a plant can either be sent to another plant for further processing, or disposed of locally for either a profit or a loss (`$/tonne`). To model environmental regulations, it is also possible to specify the maximum amount of each product that can be disposed of at each location.
All user parameters specified above must be provided to RELOG as a JSON file, which is fully described in the [data format page](format.md).
## 3. Running the optimization
After creating a JSON file describing the reverse manufacturing process and the input data, the following example illustrates how to use the package to find the optimal set of decisions:
```julia
# Import package
using RELOG
# Solve optimization problem
solution = RELOG.solve("/home/user/instance.json")
# Write full solution in JSON format
RELOG.write(solution, "solution.json")
# Write simplified reports in CSV format
RELOG.write_plants_report(solution, "plants.csv")
RELOG.write_transportation_report(solution, "transportation.csv")
```
For a complete description of the file formats above, and for a complete list of available reports, see the [data format page](format.md).
## 4. What-If Analysis
Fundamentally, RELOG decides when and where to build plants based on a deterministic optimization problem that minimizes costs for a particular input file provided by the user. In practical situations, it may not be possible to perfectly estimate some (or most) entries in this input file in advance, such as costs, demands and emissions. In this situation, it may be interesting to evaluate how well does the facility location plan produced by RELOG work if costs, demands and emissions turn out to be different.
To simplify this what-if analysis, RELOG provides the `resolve` method, which updates a previous solution based on a new scenario, but keeps some of the previous decisions fixed. More precisely, given an optimal solution produced by RELOG and a new input file describing the new scenario, the `resolve` method reoptimizes the supply chain and produces a new solution which still builds the same set of plants as before, in exactly the same locations and with the same capacities, but that may now utilize the plants differently, based on the new data. For example, in the new solution, plants that were previously used at full capacity may now be utilized at half-capacity instead. As another example, regions that were previously served by a certain plant may now be served by a different one.
The following snippet shows how to use the method:
```julia
# Import package
using RELOG
# Optimize for the average scenario
solution_avg, model_avg = RELOG.solve("input_avg.json", return_model=true)
# Write reports for the average scenario
RELOG.write_plants_report(solution_avg, "plants_avg.csv")
RELOG.write_transportation_report(solution_avg, "transportation_avg.csv")
# Re-optimize for the high-demand scenario, keeping plants fixed
solution_high = RELOG.resolve(model_avg, "input_high.json")
# Write reports for the high-demand scenario
RELOG.write_plants_report(solution_high, "plants_high.csv")
RELOG.write_transportation_report(solution_high, "transportation_high.csv")
```
To use the `resolve` method, the new input file should be very similar to the original one. Only the following entries are allowed to change:
- **Products:** Transportation costs, energy, emissions and initial amounts (latitude, longitude and amount).
- **Plants:** Energy and emissions.
- **Plant's location:** Latitude and longitude.
- **Plant's storage:** Cost.
- **Plant's capacity:** Opening cost, fixed operating cost and variable operating cost.
## 5. Advanced options
### 5.1 Changing the solver
By default, RELOG internally uses [Cbc](https://github.com/coin-or/Cbc), an open-source and freely-available Mixed-Integer Linear Programming solver developed by the [COIN-OR Project](https://www.coin-or.org/). For larger-scale test cases, a commercial solver such as Gurobi, CPLEX or XPRESS is recommended. The following snippet shows how to switch from Cbc to Gurobi, for example:
```julia
using RELOG, Gurobi, JuMP
gurobi = optimizer_with_attributes(
Gurobi.Optimizer,
"TimeLimit" => 3600,
"MIPGap" => 0.001,
)
RELOG.solve(
"instance.json",
output="solution.json",
optimizer=gurobi,
)
```
### 5.2 Multi-period heuristics
For large-scale instances, it may be too time-consuming to find an exact optimal solution to the multi-period version of the problem. For these situations, RELOG includes a heuristic solution method, which proceeds as follows:
1. First, RELOG creates a single-period version of the problem, in which most values are replaced by their averages. This single-period problem is typically much easier to solve.
2. After solving the simplified problem, RELOG resolves the multi-period version of the problem, but considering only candidate plant locations that were selected by the optimal solution to the single-period version of the problem. All remaining candidate plant locations are removed.
To solve an instance using this heuristic, use the option `heuristic=true`, as shown below.
```julia
using RELOG
solution = RELOG.solve(
"/home/user/instance.json",
heuristic=true,
)
```