Mathematical Model for Optimal Agri-Food Industry Residual Streams Flow Management: A Valorization Decision Support Tool
Abstract
:1. Introduction
2. Literature Review
3. Methodology
3.1. Model Description and Assumptions
- A residual stream from a given AFI (by-products) is reintroduced as raw material in a valorization route. There may be different routes available. Without loss of generality, we assume that each AFI generates only one by-product (if more than one by-product were generated, as many instances of the AFI as needed could be used).
- Available valorization routes for a given residual stream are made up of different steps. Each step corresponds to the set of BBIs that can use such a residual stream as raw material.
- The relations between the input and the outputs of a BBI are modeled as linear applications and therefore can be modeled by using a matrix. Mathematical models used for describing mass transformations in the bio-industries processes are out of the scope of this paper. However, since many of those transformations are usually modeled by using differential equations, it is not unusual to assume that some valorization processes conform to a linear transformation and therefore can be modeled by using a transformation matrix.
- BBIs use a residual stream as raw material to produce value-added products and can also generate a new residual stream. The relation between the residual stream at the input and at the output of a BBI and the relation between the residual stream at the input and the value-added products at the output of a BBI are modeled separately.
- The residual stream generated in one BBI must be sent to the next step of the valorization route.
- If a BBI does not produce a residual stream (e.g., landfills, incineration plants, waste water treatment plants, etc.) or the residual stream generated cannot be used as feedstock for another valorization process, then the valorization route ends.
- The residual stream might be modeled considering the by-products that it contains as well as considering its chemical composition.
- It is assumed that the composition of the residual stream does not change with transport or storage. This assumption may not be true in real conditions, but it could be easily overcome by including in the model a transformation matrix that represents those changes.
- A residual stream from a given BBI is never reintroduced in the same BBI.
- The profit obtained in a BBI at each step is the difference between the revenue from the sale of the value-added products obtained and the cost derived from the valorization of the residual stream used as feedstock in such step.
- The model supports the inclusion of economic, social, and environmental costs/benefits.
- BBIs are assumed to have enough capacity for processing all the residual streams that they receive.
3.2. Mathematical Formulation of the Optimization Problem
4. Results of a Case Study
4.1. Case Study Description
4.2. Optimization Results and Sensitivity Analysis
5. Conclusions
- Given a valorization network of industries in a region, the model allows to improve the profits compared to those achieved through existing industry practices.
- Enables to test different scenarios to make informed decisions. For instance, it enables us to study the impact of variations in the selling prizes and/or the valorization costs (individually and simultaneously) in the profit and in the optimal residual stream distribution. This knowledge helps to recognize potential challenges or opportunities in advance.
- Helps to detect key factors that influence the optimal strategy and to identify optimal conditions for achieving the best strategy at a global level.
- Allows us to deal with the problem of establishing the number, size, and location of all the potential BBIs of a valorization network and to study different layouts of the network in order to determine the optimal one.
- Given a complex valorization network of industries, with many different multi-step valorization routes available, the model enables to avoid unprofitable valorization options.
- Helps to anticipate the impact of potential risks by simulating different scenarios. This will improve the resilience of the valorization network, leading to more strategic decision-making.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Parameters | Description |
---|---|
Residual stream at the output of at step s | |
Relation between the residual stream at the input and the output of | |
Revenue from the sale of value-added products obtained at at step s | |
Selling prices of the value-added products obtained at | |
Relation between the residual stream at the input and the value-added products at the output of | |
Cost derived from the valorization of the residual stream received at at step s | |
Costs at derived from the valorization of the residual stream received from | |
Profit obtained at at step s | |
Profit obtained at at step s from the sale of value-added products obtained from the residual stream coming from | |
Variables | Description |
Proportion of residual stream received at from at step s |
Parameters | Value | Unit | Notation | |
---|---|---|---|---|
Processing cost | BBI1 BBI2 | 9.3 | €/ton | |
BBI3 BBI4 | 18.77 | €/ton | ||
BBI5 | 436.51 | €/ton | ||
Transport cost | 0.0468 | €/km ton | ||
Selling price | biogas | 490 | €/ton | |
compost | 34 | €/ton | ||
animal feed | 316.83 | €/ton |
0 | 13.01 | 34.74 | 39.53 | 54.8 | 35.17 | 17.59 | |
0 | 16.88 | 50.92 | 41.89 | 31.37 | 16.57 | ||
0 | 34.61 | 38.75 | 55.93 | 38.91 | |||
0 | 50.62 | 78.45 | 60.1 | ||||
0 | 72.13 | 57.33 | |||||
0 | 18.41 | ||||||
0 |
Value-Added Product | Selling Price | Profit | Optimal Strategy |
---|---|---|---|
biogas | ↑35% | ↑35% | Figure 8c |
↓35% | ↓35% | Figure 8c | |
compost | ↑35% | ↑11.5% | Figure 8c |
↓35% | ↓11.5% | Figure 8c | |
animal feed | ↑35% | ↑44.1% | Figure 8a |
↓35% | 0% | Figure 8c | |
all | ↑35% | ↑48.6% | Figure 8b |
↓35% | ↓46.4% | Figure 8c |
BBI | Processing Cost | Profit | Optimal Strategy |
---|---|---|---|
BBI1, BBI2 | ↑35% | ↓3.8% | Figure 9b |
↓35% | ↑3.8% | Figure 9b | |
BBI3, BBI4 | ↑35% | ↓6.4% | Figure 9b |
↓35% | ↑6.4% | Figure 9b | |
BBI5 | ↑35% | 0% | Figure 9b |
↓35% | ↑53.2% | Figure 9a | |
all | ↑35% | ↓10.4% | Figure 9b |
↓35% | ↑10.4% | Figure 9b |
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Barasoain-Echepare, Í.; Zárraga-Rodríguez, M.; Podhorski, A.; Villar-Rosety, F.M.; Besga-Oyanarte, L.; Jaray-Valdehierro, S.; Fernández-Arévalo, T.; Sancho, L.; Ayesa, E.; Gutiérrez-Gutiérrez, J.; et al. Mathematical Model for Optimal Agri-Food Industry Residual Streams Flow Management: A Valorization Decision Support Tool. Mathematics 2024, 12, 2753. https://doi.org/10.3390/math12172753
Barasoain-Echepare Í, Zárraga-Rodríguez M, Podhorski A, Villar-Rosety FM, Besga-Oyanarte L, Jaray-Valdehierro S, Fernández-Arévalo T, Sancho L, Ayesa E, Gutiérrez-Gutiérrez J, et al. Mathematical Model for Optimal Agri-Food Industry Residual Streams Flow Management: A Valorization Decision Support Tool. Mathematics. 2024; 12(17):2753. https://doi.org/10.3390/math12172753
Chicago/Turabian StyleBarasoain-Echepare, Íñigo, Marta Zárraga-Rodríguez, Adam Podhorski, Fernando M. Villar-Rosety, Leire Besga-Oyanarte, Sofía Jaray-Valdehierro, Tamara Fernández-Arévalo, Luis Sancho, Eduardo Ayesa, Jesús Gutiérrez-Gutiérrez, and et al. 2024. "Mathematical Model for Optimal Agri-Food Industry Residual Streams Flow Management: A Valorization Decision Support Tool" Mathematics 12, no. 17: 2753. https://doi.org/10.3390/math12172753