Access to this full-text is provided by MDPI.
Content available from Mathematics
This content is subject to copyright.
Citation: 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
Academic Editors: Kin Keung Lai,
Lean Yu and Jian Chai
Received: 21 June 2024
Revised: 27 August 2024
Accepted: 2 September 2024
Published: 5 September 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
mathematics
Article
Mathematical Model for Optimal Agri-Food Industry Residual
Streams Flow Management: A Valorization Decision
Support Tool
Íñigo Barasoain-Echepare 1,*, Marta Zárraga-Rodríguez 1, Adam Podhorski 1, Fernando M. Villar-Rosety 1,
Leire Besga-Oyanarte 2, Sofía Jaray-Valdehierro 2, Tamara Fernández-Arévalo 2, Luis Sancho 2,
Eduardo Ayesa 2, Jesús Gutiérrez-Gutiérrez 1and Xabier Insausti 1
1Tecnun School of Engineering, University of Navarra, Manuel de Lardizábal 13, 20018 San Sebastián, Spain;
mzarraga@unav.es (M.Z.-R.); apodhorski@unav.es (A.P.); fmdvillar@unav.es (F.M.V.-R.);
jgutierrez@unav.es (J.G.-G.); xinsausti@unav.es (X.I.)
2
CEIT Basque Research and Technology Alliance (BRTA), Manuel de Lardizábal 15, 20018 San Sebastián, Spain;
lbesga@ceit.es (L.B.-O.); sjarayv@ceit.es (S.J.-V.); tfernandez@ceit.es (T.F.-A.); lsancho@ceit.es (L.S.);
eayesa@ceit.es (E.A.)
*Correspondence: ibarasoaine@unav.es; Tel.: +34-943-219-877
Abstract: We present a mathematical model for agri-food industry residual streams flow management,
which serves as a decision support tool for optimizing their valorization. The aim is to determine,
under a cost-benefit analysis approach, the best strategy at a global level. The proposed mathematical
model provides the optimal valorization scenario, namely the set of routes followed by agri-food
industry residual streams that maximizes the total profit obtained. The model takes into account the
complete stoichiometry of the residual stream at each step of the valorization route. Furthermore, the
model allows for the calculations of different scenarios to support decision-making. The proposed
approach is illustrated through a case study using a real-case network of a region. The case study
bears evidence that the use of the model can lead to significant profit increases compared to those
obtained with current practices. Moreover, notable profit improvements are obtained in the case
study if the selling price of all the value-added products considered increases or if the processing
cost of the animal feed producer decreases. Therefore, our model enables the detection of key factors
that influence the optimal strategy, making it a powerful decision-support tool for optimizing the
valorization of agri-food industry residual streams.
Keywords: agri-food residual streams; valorization; optimization; mathematical modeling; value chain
MSC: 90-10; 90B06; 90B50; 90B90
1. Introduction
Optimal management of agri-food industry residual streams is linked to sustainable
food processing, and the reutilization (valorization) of such organic waste is a relevant
issue of global concern under a circular economy approach.
Agri-food industry organic waste deposited in landfills represents a cost to the agri-
food processor and is responsible for greenhouse emissions and air pollution as well
as groundwater contamination. The social and environmental negative impact would
be mitigated if the organic waste of one food processing sector is used as a resource
(feedstock) for another. Hence, the use of agri-food industry residual streams as raw
material for obtaining new value-added products is of great interest and makes valorization
a revenue source.
In the literature there exist many references regarding agri-food waste valorization,
mainly under technical and economic aspects, but they are focused on one valorization
Mathematics 2024,12, 2753. https://doi.org/10.3390/math12172753 https://www.mdpi.com/journal/mathematics
Mathematics 2024,12, 2753 2 of 15
process or on one particular agri-food waste. However, valorization requires collaboration
across the entire industry chain, and improving the logistics system is a challenge to be
addressed [
1
]. Therefore, a broader approach is needed in order to determine the best
strategy at a global level. Such an approach must consider the logistics of multi-step
valorization routes in big networks. It should also include environmental aspects for
management prioritization. Moreover, for the optimization of the valorization processes
considered in those routes, the complete stoichiometry of the residual streams should be
taken into account.
To address these challenges, in this paper we present a mathematical model for opti-
mal agri-food industry residual streams flow management. This model allows to find the
optimal valorization scenario handling together different agri-food industries, with their
corresponding residual streams, multiple bio-based industries, and all the available routes
that residual streams can follow through bio-based industry processes (i.e., valorization
processes). Our model includes the complete stoichiometry of the residual streams for
accurate process modeling and supports the inclusion of economic, social, and environ-
mental factors. Furthermore, variation of the model’s parameters enables the detection of
key factors that influence the optimal strategy, making the mathematical model a powerful
decision support tool for optimizing the valorization of agri-food industry residual streams.
The rest of the paper is structured as follows: In Section 2we present a literature review.
In Section 3, we present the description of the mathematical model and the optimization
problem to be solved. In Section 4, the use of the proposed model is illustrated through a
case study. Finally, in Section 5, we present the conclusions of this work.
2. Literature Review
Although transport and treatment of agri-food industry residual streams can some-
times be very expensive, those costs may be covered and eventually overcome by higher
benefits due to valorization’s economic, social, and environmental impact. In fact, bio-
based industries have gained special attention under the European Circular Economy
strategy since they allow to recover valuable compounds and to obtain new value-added
products from residual streams, reducing both waste and environmental footprint. How-
ever, according to [
2
], literature evidencing the application of waste valorization models is
still limited.
Commonly adopted strategies for the management and valorization of agri-food in-
dustry residual streams are to process them in order to obtain biofuels, bioproducts, animal
feed, or high value-added compounds (see, e.g., [
3
–
6
]). Therefore, there will be multiple
possible valorization scenarios available for agri-food industry residual streams flow man-
agement, and the aim of this work is to find the optimal one. In this study, a valorization
scenario is understood as a set of agri-food industries, with their corresponding residual
streams, a set of bio-based industries, and a set of available routes that residual streams can
follow from agri-food industries through different bio-based industries processes (i.e., val-
orization processes). It should be mentioned that valorization processes, in addition to
obtaining value-added products, may generate new residual streams, which in turn can be
re-introduced in another valorization process as feedstock.
In the literature, different methodological approaches have been employed to evaluate
several potential waste valorization strategies: life cycle assessment (LCA), life cycle
sustainability assessment (LCSA), life cycle costing (LCC), cost-benefit analysis (CBA), full
cost accounting (FCA), and variations on multi-criteria decision analysis (MCDA). Those
methodologies assess different valorization scenarios according to their economic viability,
environmental and social sustainability, or technological maturity [
7
]. Following one of
these approaches, a company can determine what is likely to be its optimal valorization
option (see, e.g., [
8
,
9
]). In this work, we follow a CBA approach where profits of multiple
potential strategies are compared. However, the aim is to determine the best strategy at a
global level and not necessarily help a specific company in its decision-making.
Mathematics 2024,12, 2753 3 of 15
For the optimization of agri-food industry residual streams flow management, it is nec-
essary to consider logistics but also the mass and energy flows that allow the optimization
of industrial processes.
Currently, there are commercial programs for the simulation of industrial processes
such as Aspen Plus
®
, Chemcad
®
, Pro/II
®
, Prosim
®
, SuperPro
®
, or COCO
®
among many
others. These programs, based on mass and energy balances, allow predicting the response
of the system to scenarios or disturbances in a wide range of sectors (chemical, petroleum,
pharmaceutical, bio-technological, wastewater treatment, etc.). However, the model library
for food sector processes has not followed the same development, and there are still missing
models of numerous main processes (see [
10
]). The existing commercial products focus
on providing tools for the description of the bio-industry and allow the residual stream
generated in the different sectors to be taken as raw material. The biggest limitation of
those commercial products is that they are mainly based on the analysis of specific facilities
and not on broader analyses of big networks with various industries, in which logistics
plays a key role.
Parallel to the development and updating of these commercial programs, the scientific
community is making considerable progress in the development of models to simulate and
predict the value-added products that can be obtained through fermentation (see, e.g., [
11
–
14
])
and extraction [15,16].
Furthermore, the aforementioned commercial programs base optimizations or deci-
sions on technical and economic factors, although there are more and more attempts to
include environmental aspects. An example of this is the Aspen Plus “Environmental &
Safety Analysis” module. Numerous works are collected in the literature that combine
LCA and industrial process modeling (see, e.g., [17–19]).
In the literature, there also exist many references focused on the productivity and
sustainability of just one valorization process with one input and one output (see [
20
–
24
]
among others). Their aim is to model and optimize the performance of a single operation.
Our mathematical model has a more global approach, and as many valorization processes
as wanted can be considered. It should be mentioned that in order to find the optimal
valorization scenario, the complete stoichiometry of the residual stream may be essential.
For the logistic analysis, there are specific programs such as Arena
®
, Flexsim
®
, or
TRUX Haul-it, among others, that allow the analysis of commercial or logistic operations.
There are programs for optimising the time of decision-making but not for the optimization
of mass flows such as industrial simulators.
In the literature, waste exploitation is not a very common issue. There exist references
dealing with the design of biomass supply chains for the production of bio-based products
(see [
25
–
30
] and references therein). The supply chain involves raw material suppliers,
processing plants, and demand. Unlike here, in many of those references, biomass is the
raw material considered instead of agri-food industry residual streams, and it usually
comes from dedicated crops. Moreover, in many of those references, only biorefineries
are regarded as bio-based industries, while in this work as many management options as
wanted can be considered. Furthermore, the design of a supply chain is linked to meeting
customers’ demands, while in our case such an approach is not entirely appropriate. Our
work is focused on the management and valorization of residual streams, and hence, talking
about meeting customers’ demands makes no sense.
In the literature, there exist also many references dealing with the problem of where to
locate a new biorefinery or the problem of establishing the number, size, and location of all
the potential biorefineries in the network (see, e.g., [
31
–
37
]). In this work, we assume that
the number and location of the bio-based industries are known. Nevertheless, our model
allows us to study different layouts of the network in order to determine the optimal one.
Mathematics 2024,12, 2753 4 of 15
3. Methodology
In this section, we first present the description of the model considered and the
optimization problem to be solved. Secondly, we introduce the mathematical formulation
of the optimization problem.
3.1. Model Description and Assumptions
The model considered represents a valorization scenario and consists of a set of
agri-food industries (AFI), a set of bio-based industries (BBI), a set of available valoriza-
tion routes, a set of residual streams generated, and a residual stream distribution. The
mathematical model is developed according to the following criteria:
•
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 applica-
tions 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 valoriza-
tion 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 trans-
port 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 inclusionof economic, social, and environmental costs/benefits.
•
BBIs are assumed to have enough capacity for processing all the residual streams that
they receive.
The aim of this work is to determine the optimal valorization scenario, that is, the resid-
ual streams distribution that maximizes the sum of profits generated at BBIs throughout the
different available steps of the valorization routes from the AFIs through one or more BBIs.
3.2. Mathematical Formulation of the Optimization Problem
In this subsection, we present the mathematical formulation of the optimization
problem to be solved. First we introduce some notation.
Mathematics 2024,12, 2753 5 of 15
Assume, without loss of generality, a valorization scenario with
N
AFIs (AFI
1
, AFI
2
, ...,
AFI
N
) and
N
BBIs (BBI
1
, BBI
2
, ..., BBI
N
). Figure 1shows an
S
-step model of the valorization
scenario considered, with
s∈ {
1,
. . .
,
S}
being the step number of the valorization route.
The
S
-th step is the final step where all the valorization routes of the residual streams
have ended. That is, either the residual stream has been deposited in a landfill (or in an
incineration plant, in a waste water treatment plant, etc.) or the residual stream generated
cannot be used as feedstock for another valorization process. A connection between BBIs
(or between AFIs and BBIs) in Figure 1represents a possible path of the valorization route.
Figure 1. Simplified diagram of an S-step model.
Let
Q
be the number of by-products and chemical compounds that the residual streams
considered in the scenario might contain. Then, a residual stream is modeled as a
Q×
1 real
column vector. The residual stream at the output of the
BBIi
at step
s
,
s∈ {
1,
. . .
,
S−
1
}
, is
given by
xi(s) = Ai
N
∑
j=1
α(s)
j,ixj(s−1),i∈ {1, . . . , N}, (1)
with
xi(
0
)
being a
Q×
1 real column vector that models the residual stream generated at
AFIi,Aibeing the Q×Qreal matrix that models the relation between the residual stream
at the input and the residual stream at the output of
BBIi
, and
α(s)
j,i≥
0 being the proportion
of residual stream that is received at BBIifrom BBIjat step s(or from AFIjif s=1).
According to the assumptions,
α(s)
j,i=
0 if
i=j
, for all
s∈ {
2,
. . .
,
S}
. Moreover,
∑N
i=1α(s)
j,i=1 for all j∈ {1, . . . , N}and s∈ {1, . . . , S}.
Let
Mi
be the number of different value-added products at the output of
BBIi
. The rev-
enue from the sale of those value-added products obtained at step
s
,
s∈ {
1,
. . .
,
S}
,
is given by
ri(s) = p⊤
iBi
N
∑
j=1
α(s)
j,ixj(s−1),i∈ {1, . . . , N}, (2)
with
⊤
denoting transpose,
pi
being a
Mi×
1 real column vector with the selling prices of
the value-added products obtained at
BBIi
,
Bi
being the
Mi×Q
real matrix that models
the relation between the residual stream at the input and the value-added products at the
output of BBIi, and xj(s)given in (1).
Mathematics 2024,12, 2753 6 of 15
The cost derived from the valorization of the residual stream received at
BBIi
at step
s
,
s∈ {1, . . . , S}, is given by
ci(s) =
N
∑
j=1cj,i(s)⊤α(s)
j,ixj(s−1),i∈ {1, . . . , N}, (3)
with
cj,i(s)
being a
Q×
1 real column vector that models the costs at
BBIi
derived from
the valorization of the residual stream received from
BBIj
(or from
AFIj
if
s=
1) and
xj(s)
given in (1).
Therefore, from
(2)
and
(3)
, the profit obtained at
BBIi
at step
s
,
s∈ {
1,
. . .
,
S}
,
is given by
bi(s) =
N
∑
j=1bj,i(s)⊤α(s)
j,ixj(s−1),i∈ {1, . . . , N},
with
bj,i(s) = B⊤
ipi−cj,i(s)
. Observe that
bj,i(s)
models the profit obtained at
BBIi
when
the residual stream received from
BBIj
(or from
AFIj
if
s=
1) is processed. It should be
mentioned that the profit vector,
bj,i(s)
, supports the inclusion of revenues from the sales of
value-added products obtained, costs derived from the valorization of the residual stream
(transport, energy, labour, etc.), as well as social and environmental costs/benefits.
Table 1shows a summary of the different model parameters and decision variables.
Table 1. Summary of the different parameters and decision variables in the proposed mathemati-
cal model.
Parameters Description
xi(s)Residual stream at the output of BBIiat step s
AiRelation between the residual stream at the input and the output of BBIi
ri(s)Revenue from the sale of value-added products obtained at BBIiat step s
piSelling prices of the value-added products obtained at BBIi
BiRelation between the residual stream at the input and the value-added products at the output of BBIi
ci(s)Cost derived from the valorization of the residual stream received at BBIiat step s
cj,i(s)Costs at BBIiderived from the valorization of the residual stream received from BBIj
bi(s)Profit obtained at BBIiat step s
bj,i(s)
Profit obtained at
BBIi
at step
s
from the sale of value-added products obtained from the residual stream coming
from BBIj
Variables Description
α(s)
j,iProportion of residual stream received at BBIifrom BBIjat step s
Now we present the mathematical formulation of the optimization problem to be
solved, which is expressed as the maximization of an objective function subject to a number
of constraints. We aim to obtain the residual stream distribution that maximizes the sum of
profits generated throughout the different available steps of the valorization routes. Hence,
we aim to solve the following maximization problem:
Maximize S
∑
s=1
N
∑
i=1
N
∑
j=1bj,i(s)⊤α(s)
j,ixj(s−1)
Subject to
α(s)
i,j≥0i,j∈ {1, . . . , N},s∈ {1, . . . , S}
n
∑
j=1
α(s)
i,j=1i∈ {1, . . . , N},s∈ {1, . . . , S}
α(s)
i,j=0i=j,s∈ {2, . . . , S}
(4)
Mathematics 2024,12, 2753 7 of 15
The maximization is with respect to
α(s)
i,j
, with
i
,
j∈ {
1,
. . .
,
N}
and
s∈ {
1,
. . .
,
S}
,
and the objective function of the optimization problem
(4)
is a non-linear function. There
are numerical methods that are able to solve the aforementioned optimization problem in a
reasonable amount of time.
4. Results of a Case Study
In this section, the proposed approach is illustrated through a case study. Using a
real-case network of a region, we compare, under a CBA approach, the current agri-food
industry residual streams management strategy with the optimal management strategy
obtained by solving the optimization problem presented in Section 3.2. Our mathemat-
ical model allows for the calculations of different scenarios to support decision-making.
Different factors affect the revenue and/or the cost of the case under study. A sensitivity
analysis that takes these factors into account has been carried out to show the potential of
the mathematical model.
4.1. Case Study Description
Consider the region in Figure 2a. The set of agri-food industries and the set of BBIs
that can be found in the considered region are shown in Figure 2b,c, respectively.
(a)
(b) (c)
Figure 2. Considered region, agri-food industries and BBIs. (a) A 100
×
100 kilometer region of
northern Spain considered for optimization. (b) Plot of the different agri-food industries found in the
considered region. Each point corresponds to one agri-food industry. (c) Plot of the different BBIs
found in the considered region. Each point corresponds to a different BBI.
Although the mathematical model allows us to consider as many agri-food industries
and BBIs as desired, for the sake of simplicity, we will select for the example a valorization
scenario with two agri-food industries (
AFI1
and
AFI2
) and five BBIs: two anaerobic
digesters operating at mesophilic temperature (
BBI1
and
BBI2
), two composting plants (
BBI3
and
BBI4
), and one animal feed producer (
BBI5
). Detailed descriptions of the treatments
Mathematics 2024,12, 2753 8 of 15
carried out in the anaerobic digester, the composting plant, and the animal feed producer
are depicted in Figures 3–5respectively.
Figure 3. Diagram representing the treatments carried out in the anaerobic digester.
Figure 4. Diagram representing the treatments carried out in the composting plant.
Figure 5. Diagram representing the treatments carried out in the animal feed producer.
As it can be seen in Figure 6, the residual streams may go through different valorization
routes. At first, both residual streams may be sent to any of the anaerobic digesters, to any
of the composting plants (ending the valorization route), or to the animal feed producer
(ending the valorization route as well). In the case that the agri-food industry residual
stream is sent to an anaerobic digester, the resulting residual stream is then sent to any of
the composting plants, ending the valorization route.
The residual streams in the scenario are modeled considering two different by-products
(potato peels, which is the processing waste from
AFI1
, and artichoke bracts, which is the
processing waste from
AFI2
) and 68 chemical compounds. Hence, a residual stream is
modeled as a 70
×
1 real column vector where each entry reflects the amount in tons of
by-product or chemical compound that the residual stream contains. The complete list
of the considered chemical compounds can be found in [dataset] [
38
]. For this example,
the considered amount of potato peels is 2 tons and the considered amount of artichoke
bracts is 5 tons.
Residual streams at the output of
AFI1
and
AFI2
,
x1(
0
)
and
x2(
0
)
, can be found in
[dataset] [
38
]. According to
(1)
residual streams at the output of
BBI1
and
BBI2
at
s=
1 are
given by
x1(1) = A1α(1)
1,1 x1(0) + α(1)
2,1 x2(0),
x2(1) = A2α(1)
1,2 x1(0) + α(1)
2,2 x2(0),
Mathematics 2024,12, 2753 9 of 15
with
A1
and
A2
being the 70
×
70 real matrices that model the relation between the residual
stream at the input and the residual stream at the output of
BBI1
and
BBI2
, respectively.
Matrices A1and A2can be found in [dataset] [38].
Figure 6. Diagram representing the possible valorization routes of the proposed example.
Three value-added products can be obtained: biogas, which is used to produce elec-
tricity and heat (from
BBI1
and
BBI2
), compost (from
BBI3
and
BBI4
), and animal feed
(from
BBI5
). Hence, the matrices that model the relation between the residual stream at
the input and the value-added products at the output of
BBIi
,
Bi
with
i∈ {
1,
. . .
, 5
}
, turn
out to be 1
×
70 real row vectors (which can be found in [dataset] [
38
]). Moreover,
pi
with
i∈ {
1,
. . .
, 5
}
, have only one entry,
pi
, which corresponds to the selling price of the
value-added product obtained at
BBIi
,
i∈ {
1,
. . .
, 5
}
. Revenue from the sale depends on
the amount of value-added products obtained and the selling price of such products.
In this example, we classify the costs derived from the valorization of a residual stream
into two categories: processing cost (
cp
) and transport cost (
ct
). We assume that processing
cost depends on the total mass processed at the BBI and that transport cost additionally
depends on the BBI from which the waste stream processed comes (distance).
We now compute the profit vector at each BBI in this example, which depends on
the tons of residual stream processed. A summary of the information provided by the
companies for computing the profit vector can be found in Table 2. Distances between each
industry (di,j), computed using openrouteservice [39], are presented in Table 3.
Table 2. Value of the different model parameters.
Parameters Value Unit Notation
Processing
cost
BBI1
BBI2
9.3 €/ton cp1,cp2
BBI3
BBI4
18.77 €/ton cp3,cp4
BBI5436.51 €/ton cp5
Transport cost 0.0468 €/km ton ct
Selling
price
biogas 490 €/ton p1,p2
compost 34 €/ton p3,p4
animal feed 316.83 €/ton p5
Mathematics 2024,12, 2753 10 of 15
Table 3. Distance matrix of the considered industries, in kilometres.
AFI1AFI2BBI1BBI2BBI3BBI4BBI5
AFI10 13.01 34.74 39.53 54.8 35.17 17.59
AFI20 16.88 50.92 41.89 31.37 16.57
BBI10 34.61 38.75 55.93 38.91
BBI20 50.62 78.45 60.1
BBI30 72.13 57.33
BBI40 18.41
BBI50
Using the information in Tables 2and 3, we compute the profit obtained at
BBIi
at step
s, with s∈ {1, 2}, when residual stream from BBIj(or from AFIjif s=1) is processed as
bj,i(s) = pi·B⊤
i−cpi +dj,ict170 i,j∈ {1, . . . , 5},
where 1His the H×1 vector of ones.
Hence, according to (4), we aim to solve the following maximization problem:
Maximize 5
∑
i=1
2
∑
j=1bj,i(1)⊤α(1)
j,ixj(0) +
4
∑
i=3
2
∑
j=1bj,i(2)⊤α(2)
j,ixj(1),
Subject to
α(1)
i,j≥0i∈ {1, 2},j∈ {1, . . . , 5},
α(2)
i,j≥0i∈ {1, 2},j∈ {3, 4},
∑5
j=1α(1)
i,j=1i∈ {1, 2},
∑4
j=3α(2)
i,j=1i∈ {1, 2}
(5)
The objective function of the optimization problem is a non-linear function. Non-linear
optimization problems are considered to be harder than linear problems, and optimizing
the non-linear objective function analytically is not an easy task. There are numerical
methods that are able to solve the aforementioned optimization problem in a reasonable
amount of time and, among them, we use the well-known gradient descent or steepest
descent method (see, e.g., [40]).
4.2. Optimization Results and Sensitivity Analysis
In the considered example, the current agri-food industry strategy for the management
of processing waste is to send all the residual streams to produce animal feed. Therefore,
α(1)
1,5 =
1,
α(1)
2,5 =
1, and the profit obtained for the considered amount of by-products
generated by AFI
1
and AFI
2
would be
−
161€. Solving the maximization problem in
(5)
, we
obtain that the optimal strategy for the management of the processing waste is to use all the
residual streams to generate biogas in the first step and to produce compost in the second
step. That is,
α(1)
1,1 =
1,
α(1)
2,1 =
1,
α(2)
1,3 =
1, and the profit obtained for the considered amount
of by-products generated by AFI
1
and AFI
2
would be 592€. This result bears evidence that
the use of the model can lead to significant profit increases compared to those obtained
with current industry practices. Figure 7compares the current and the optimal residual
stream distribution.
Mathematics 2024,12, 2753 11 of 15
(a) (b)
Figure 7. Diagram representing residual stream distribution. (a) Current distribution. (b) Optimal
distribution obtained by solving the maximization problem in (5).
As it has been mentioned, different factors may affect the profit and the optimal
residual stream distribution of the case under study. Just to show the potential of the
model, a brief sensitivity analysis has been carried out to study the impact of variations in
parameters related with the revenues or with the costs.
As an example, we have proposed two groups of scenarios. In the first group of
scenarios, we have changed the value of parameters related to the revenue obtained from
the sales. Specifically, the selling prices of the biogas, the compost, and the animal feed have
been changed by
±
35%. These adjustments were made separately as well as simultaneously,
resulting in eight different scenarios. Table 4shows the variation in the solution of
(5)
after
adjusting the prices, and Figure 8also depicts the optimal residual stream distributions in
this group of scenarios. As it is shown, the optimal residual stream distribution changes
only in two scenarios: when the selling price of animal feed increases or when the selling
prices of all the value-added products increase. If the selling price of animal feed increases,
the optimal management strategy would be to send all the residual streams to produce
animal feed. If the selling prices of all the value-added products increase, the optimal
management strategy would be to use the artichoke bracts to produce animal feed and to
use the potato peels to produce first biogas and secondly compost. Interestingly, a decrease
in the selling price of the animal feed does not affect either the residual stream distribution
or the profit obtained.
Table 4. Summary of the impact variations in selling prices (an up arrow represents an increase and a
down arrow represents a decrease).
Value-Added
Product
Selling
Price Profit Optimal
Strategy
↑35% ↑35% Figure 8c
biogas ↓35% ↓35% Figure 8c
↑35% ↑11.5% Figure 8c
compost ↓35% ↓11.5% Figure 8c
↑35% ↑44.1% Figure 8a
animal feed ↓35% 0% Figure 8c
↑35% ↑48.6% Figure 8b
all ↓35% ↓46.4% Figure 8c
Mathematics 2024,12, 2753 12 of 15
(a) (b)
(c)
Figure 8. Results obtained when modifying selling prices. (a) Optimal distribution when the animal
feed price increases 35%. (b) Optimal distribution when all selling prices increase 35%. (c) Optimal
distribution for the rest of resulting scenarios
In the second group of scenarios, we have changed the value of parameters related
to the processing costs of the valorization processes. Specifically, the processing cost of
the anaerobic digestion plant, the composting plant, and the animal feed producer have
changed by
±
35%. As in the first group, these adjustments were made both separately and
simultaneously, resulting in eight different scenarios. Table 5shows the variation in the
solution of
(5)
after adjusting the processing costs, and Figure 9depicts the optimal residual
stream distributions in the second group of scenarios. As it is shown, the optimal residual
stream distribution remains unchanged in every scenario, except when the processing cost
of BBI
5
(that is, the animal feed producer) decreases. In that case, the optimal management
strategy would be to send all the residual streams to produce animal feed.
The result bears evidence that our model enables us to study the impact of different
factors on the profit and on the optimal residual stream distribution. This brief sensitivity
analysis shows the potential of the mathematical model as a powerful decision-support
tool for optimizing the valorization of agri-food industry residual streams.
Mathematics 2024,12, 2753 13 of 15
Table 5. Summary of the impact of variations in costs derived from the valorization of the resid-
ual streams (an up arrow represents an increase and a down arrow represents a decrease).
BBI Processing
Cost Profit Optimal
Strategy
↑35% ↓3.8% Figure 9b
BBI1, BBI2↓35% ↑3.8% Figure 9b
↑35% ↓6.4% Figure 9b
BBI3, BBI4↓35% ↑6.4% Figure 9b
↑35% 0% Figure 9b
BBI5↓35% ↑53.2% Figure 9a
↑35% ↓10.4% Figure 9b
all ↓35% ↑10.4% Figure 9b
(a) (b)
Figure 9. Results obtained when modifying the production costs. (a) Optimal distribution when the
processing cost of animal feed production decreases 35%. (b) Optimal distribution for the rest of
resulting scenarios.
5. Conclusions
In this paper, we have presented a mathematical model for optimal agri-food industry
residual streams flow management. Our model includes the complete stoichiometry of the
residual streams for accurate process modeling and supports the inclusion of economic,
social, and environmental factors. Furthermore, variation of the model’s parameters
enables the detection of key factors that influence the best recommended way for the
valorization of agri-food industry residual streams from a holistic perspective. This makes
the mathematical model a powerful decision-support tool.
To show the potential of our model, the proposed approach is illustrated through
a case study using a real-case network of a region. Although it is a simplified example,
the results obtained bear evidence that the use of our model can lead to significant profit
increases compared to those obtained with current practices. Moreover, the brief sensitive
analysis conducted uncovers several useful management insights and shows that the model
has a wide range of possible applications, such as:
•
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.
Mathematics 2024,12, 2753 14 of 15
•
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 val-
orization 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.
Interesting future lines would be to explore strategies for addressing possible data
limitations in real-world applications, to study how to evaluate in the long run the possible
effects of the optimal strategy on the environment and the economy, and to extend the
model in order to relax the linearity assumption of the valorization processes.
Author Contributions: Conceptualization, T.F.-A., E.A., J.G.-G. and X.I.; Methodology, Í.B.-E., A.P.
and F.M.V.-R.; Software, Í.B.-E. and A.P.; Resources, L.B.-O., S.J.-V., T.F.-A. and L.S.; Writing—original
draft, Í.B.-E. and M.Z.-R.; Writing—review and editing, X.I. All authors have read and agreed to the
published version of the manuscript
Funding: This work was developed in the context of the European Model2Bio project (H2020-BBI-JTI-
2019. No. 887191). This project has received funding from the Bio Based Industries Joint Undertaking
(JU) under grant agreement No. 887191. The JU receives support from the European Union’s Horizon
2020 research and innovation programme and the Bio Based Industries Consortium.
Data Availability Statement: The dataset used for the case study is available at https://data.
mendeley.com/datasets/3s8sfwpf5v (accessed on 26 March 2024).
Conflicts of Interest: The authors declare no conflicts of interest.
References
1.
Zhang, Q.; Zhang, H. Assessing Agri-Food Waste Valorization Challenges and Solutions Considering Smart Technologies: An
Integrated Fermatean Fuzzy Multi-Criteria Decision-Making Approach. Sustainability 2024,16, 6169. [CrossRef]
2.
Leder, N.; Kumar, M.; Rodrigues, V.S. Influential factors for value creation within the Circular Economy: Framework for Waste
Valorisation. Resour. Conserv. Recycl. 2020,158, 104804. [CrossRef]
3.
Berenguer, C.V.; Andrade, C.; Pereira, J.A.M.; Perestrelo, R.; Câmara, J.S. Current Challenges in the Sustainable Valorisation of
Agri-Food Wastes: A Review. Processes 2023,11, 20. [CrossRef]
4.
Girotto, F.; Alibardi, L.; Cossu, R. Food waste generation and industrial uses: A review. Waste Manag. 2015,45, 32–41. [CrossRef]
5.
Malenica, D.; Kass, M.; Bhat, R. Sustainable Management and Valorization of Agri-Food Industrial Wastes and By-Products as
Animal Feed: For Ruminants, Non-Ruminants and as Poultry Feed. Sustainability 2023,15, 117. [CrossRef]
6.
Mirabella, N.; Castellani, V.; Sala, S. Current options for the valorization of food manufacturing waste: A review. J. Clean. Prod.
2014,65, 28–41. [CrossRef]
7.
Stone, J.; Garcia-Garcia, G.; Rahimifard, S. Development of a pragmatic framework to help food and drink manufacturers select
the most sustainable food waste valorisation strategy. J. Environ. Manag. 2019,247, 425–438. [CrossRef]
8.
Lin, Z.; Ooi, J.K.; Woon, K.S. An integrated life cycle multi-objective optimization model for health-environment-economic nexus
in food waste management sector. Sci. Total Environ. 2022,816, 151541. [CrossRef]
9.
Maier, J.M.; Sowlati, T.; Salazar, J. Life cycle assessment of forest-based biomass for bioenergy: A case study in British Columbia,
Canada. Resour. Conserv. Recycl. 2019,146, 598–609. [CrossRef]
10.
Madoumier, M.; Azzaro-Pantel, C.; Tanguy, G.; Gésan-Guiziou, G. Modelling the properties of liquid foods for use of process
flowsheeting simulators: Application to milk concentration. J. Food Eng. 2015,164, 70–89. [CrossRef]
11.
Alexandropoulou, M.; Antonopoulou, G.; Lyberatos, G. A novel approach of modeling continuous dark hydrogen fermentation.
Bioresour. Technol. 2018,250, 784–792. [CrossRef] [PubMed]
12.
Montecchio, D.; Yuan, Y.; Malpei, F. Hydrogen production dynamic during cheese whey Dark Fermentation: New insights from
modelization. Int. J. Hydrogen Energy 2018,43, 17588–17601. [CrossRef]
13.
Van Ngo, A.; Nguyen, H.T.; Van Le, C.; Goel, R.; Terashima, M.; Yasui, H. A dynamic simulation of methane fermentation process
receiving heterogeneous food wastes and modelling acidic failure. J. Mater. Cycles Waste Manag. 2016,18, 239–247. [CrossRef]
14.
Xiong, Z.; Hussain, A.; Lee, J.; Lee, H.S. Food waste fermentation in a leach bed reactor: Reactor performance, and microbial
ecology and dynamics. Bioresour. Technol. 2019,274, 153–161. [CrossRef] [PubMed]
Mathematics 2024,12, 2753 15 of 15
15.
Muhammad, A.; Younas, M.; Rezakazemi, M. Quasi-dynamic modeling of dispersion-free extraction of aroma compounds using
hollow fiber membrane contactor. Chem. Eng. Res. Des. 2017,127, 52–61. [CrossRef]
16.
Neviani, M.; Aliakbarian, B.; Perego, P.; Paladino, O. Extraction of polyphenols from olive pomace: Mathematical modeling and
technological feasibility in a high temperature and high pressure stirred reactor. Chem. Eng. Res. Des. 2019,141, 32–46. [CrossRef]
17.
Kalakul, S.; Malakul, P.; Siemanond, K.; Gani, R. Integration of life cycle assessment software with tools for economic and
sustainability analyses and process simulation for sustainable process design. J. Clean. Prod. 2014,71, 98–109. [CrossRef]
18.
Rathnayake, M.; Chaireongsirikul, T.; Svangariyaskul, A.; Lawtrakul, L.; Toochinda, P. Process simulation based life cycle
assessment for bioethanol production from cassava, cane molasses, and rice straw. J. Clean. Prod. 2018,190, 24–35. [CrossRef]
19.
Righi, S.; Baioli, F.; Dal Pozzo, A.; Tugnoli, A. Integrating Life Cycle Inventory and Process Design Techniques for the Early
Estimate of Energy and Material Consumption Data. Energies 2018,11, 970. [CrossRef]
20.
Cheng, L.; Anderson, C. Too conservative to hedge: How much does a corn ethanol facility lose? Int. J. Prod. Econ. 2017,
193, 654–662. [CrossRef]
21.
García-Velásquez, C.; Defryn, C.; van der Meer, Y. Life cycle optimization of the supply chain for biobased chemicals with local
biomass resources. Sustain. Prod. Consum. 2023,36, 540–551. [CrossRef]
22.
Wang, G. Multiobjective Sustainable Operation Dynamics and Optimization. IEEE Trans. Eng. Manag. 2021,68, 1126–1138.
[CrossRef]
23.
Cooney, R.; de Sousa, D.B.; Fernández-Ríos, A.; Mellett, S.; Rowan, N.; Morse, A.P.; Hayes, M.; Laso, J.; Regueiro, L.;
Wan, A.H.; et al.
A circular economy framework for seafood waste valorisation to meet challenges and opportunities for
intensive production and sustainability. J. Clean. Prod. 2023,392, 136283. [CrossRef]
24.
García-Velásquez, C.; Leduc, S.; van der Meer, Y. Design of biobased supply chains on a life cycle basis: A bi-objective optimization
model and a case study of biobased polyethylene terephthalate (PET). Sustain. Prod. Consum. 2022,30, 706–719. [CrossRef]
25.
Atashbar, N.Z.; Labadie, N.; Prins, C. Modeling and optimization of biomass supply chains: A review and a critical look.
IFAC-PapersOnLine 2016,49, 604–615. [CrossRef]
26.
Sun, O.; Fan, N. A Review on Optimization Methods for Biomass Supply Chain: Models and Algorithms, Sustainable Issues, and
Challenges and Opportunities. Process Integr. Optim. Sustain. 2020,4, 203–226. [CrossRef]
27.
Tapia, J.F.D.; Samsatli, S.; Doliente, S.S.; Martinez-Hernandez, E.; Ghani, W.A.B.W.A.K.; Lim, K.L.; Shafri, H.Z.M.; Shaharum,
N.S.N.B. Design of biomass value chains that are synergistic with the food–energy–water nexus: Strategies and opportunities.
Food Bioprod. Process. 2019,116, 170–185. [CrossRef]
28.
Zhang, Y.; Hu, G.; Brown, R.C. Integrated supply chain design for commodity chemicals production via woody biomass fast
pyrolysis and upgrading. Bioresour. Technol. 2014,157, 28–36. [CrossRef]
29.
Lo, S.L.Y.; How, B.S.; Leong, W.D.; Teng, S.Y.; Rhamdhani, M.A.; Sunarso, J. Techno-economic analysis for biomass supply chain:
A state-of-the-art review. Renew. Sustain. Energy Rev. 2021,135, 110164. [CrossRef]
30.
Stellingwerf, H.M.; Guo, X.; Annevelink, E.; Behdani, B. Logistics and Supply Chain Modelling for the Biobased Economy: A
Systematic Literature Review and Research Agenda. Front. Chem. Eng. 2022,4, 778315. [CrossRef]
31.
Aboytes-Ojeda, M.; Castillo-Villar, K.K.; Eksioglu, S.D. Modeling and optimization of biomass quality variability for decision
support systems in biomass supply chains. Ann. Oper. Res. 2022,314, 319–346. [CrossRef]
32.
Galanopoulos, C.; Giuliano, A.; Barletta, D.; Zondervan, E. An integrated methodology for the economic and environmental
assessment of a biorefinery supply chain. Chem. Eng. Res. Des. 2020,160, 199–215. [CrossRef]
33.
Jayarathna, L.; Kent, G.; O’Hara, I.; Hobson, P. A Geographical Information System based framework to identify optimal location
and size of biomass energy plants using single or multiple biomass types. Appl. Energy 2020,275, 115398. [CrossRef]
34.
López-Díaz, D.C.; Lira-Barragán, L.F.; Rubio-Castro, E.; Ponce-Ortega, J.M.; El-Halwagi, M.M. Optimal location of biorefineries
considering sustainable integration with the environment. Renew. Energy 2017,100, 65–77. [CrossRef]
35.
Panteli, A.; Giarola, S.; Shah, N. Lignocellulosic supply chain MILP model: A Hungarian case study. Comput. Aided Chem. Eng.
2016,38, 253–258. [CrossRef]
36.
Serrano-Hernandez, A.; Faulin, J. Locating a Biorefinery in Northern Spain: Decision Making and Economic Consequences.
Socio-Econ. Plan. Sci. 2019,66, 82–91. [CrossRef]
37.
Xie, Y.; Zhao, K.; Hemingway, S. Optimally Locating Biorefineries: A GIS-Based Mixed Integer Linear Programming Approach.
In Proceedings of the 51st Annual Transportation Research Forum, 2010-03, Arlington, VA, USA, 11–13 March 2010; Number
1430-2016-118685, p. 15. [CrossRef]
38.
Barasoain-Echepare, Í. Mathematical Model for Optimal Agri-Food Industry Residual Streams Flow Management Dataset; Tecnun School
of Engineering, University of Navarra: San Sebastián, Spain, 2024. [CrossRef]
39. HeiGIT gGmbH. Openrouteservice. 2022. Available online: https://openrouteservice.org/ (accessed on 12 March 2024).
40. Dahlquist, G.; Björck, Å. Numerical Methods; Courier Corporation: North Chelmsford, MA, USA, 2003.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
Available via license: CC BY 4.0
Content may be subject to copyright.
