Input-Output Multiobjective Optimization Approach for
Food-Energy-Water Nexus
Isaac Okola
School of Technology, KCA University, Thika Highway, Nairobi, Kenya
Keywords: Input-Output Theory, Food-Energy-Water Nexus, Multiobjective Optimization.
Abstract: Food, energy and water are essential for human survival. These resources consume each other thus enhancing
security in one resource can reduce security in another resource. Multiobjective optimization approaches have
been used to understand the complexity associated with the Food, Energy Water (FEW) Nexus. However
most of these approaches focus on either maximizing resource production or minimizing resource
consumption in the FEW Nexus but not addressing the two simultaneously. To achieve sustainability of the
FEW Nexus sustainable consumption and production of the resources need to be emphasized. In this paper,
the Input-Output theory is used to develop a multiobjective optimization approach that minimises resource
intensities. Minimising resource intensities results into minimised consumption and maximised production of
resources in the nexus. Using the developed approach simulations are carried out to demonstrate its
applicability in FEW Nexus. The results show that the approach can be used to explore alternative ways of
minimizing consumption and maximizing production simultaneously based on the concept of non-dominated
solutions.
1 INTRODUCTION
Sustainable consumption and production in Food,
Energy and Water(FEW) nexus has a direct or
indirect link in achieving many if not all of the
Sustainable Development Goals developed by United
Nations. Many countries are concerned about
addressing Sustainable Development Goals in the
midst of pressures emanating from rapid population
growth and climate change. Climate change and the
ever increasing population has led to increase in
competition and trade-offs between food, energy and
water (Bian & Liu, 2021). The FEW nexus approach
is one that can be used to address the challenges of
ever growing demand for food, energy and water
(Miralles-Wilhelm & Muñoz, 2017). The nexus has
become very important in addressing sustainability
issues (Dalla Fontana et al., 2021).The nexus depicts
some complex interactions with hidden feedback
connections among food, energy and water resources.
The production of a specific resource requires the
consumption of one or the two other resources thus
playing a big role in determining the demand, supply
and availability of the resources in the FEW Nexus
(White et al., 2018). In relation to attaining global
sustainability, managing the FEW Nexus has become
a big challenge (Taniguchi et al., 2017). This is due
to the fact that increasing security of one resource
may have a negative consequence on another resource
(Abdi et al., 2020).
To achieve sustainability in the FEW Nexus,
resource consumption and production need to be
optimized simultaneously and this may conflict each
other (Okola et al., 2019). Multi-objective
optimization approaches can be used to address
conflicts in the FEW Nexus because they are known
to deal with multiple conflicting objectives in real
world problems. These approaches provide non-
dominated solutions that identify trade-offs and
synergies in FEW Nexus.
In a minimization problem a solution x
i
is non-
dominated as compared to x
j
when each objective
value of x
j
is not less than that of x
i
and at least one
objective value of x
j
is greater than x
i
(Srinivas &
Deb, 1994). Evolutionary algorithms have a great
potential in solving multi-objective optimization
problems. They evolve solutions in each generation
thus being able to produce non-dominated solutions
which are closer to the pareto-front. However, to the
best of our knowledge there is no evidence of studies
that have looked at how resource intensities in FEW
Okola, I.
Input-Output Multiobjective Optimization Approach for Food-Energy-Water Nexus.
DOI: 10.5220/0011271500003271
In Proceedings of the 19th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2022), pages 155-160
ISBN: 978-989-758-585-2; ISSN: 2184-2809
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
155

Nexus can be minimised simultaneously using a
multiobective optimisation approach
2 INPUT-OUTPUT THEORY IN
FEW NEXUS
2.1 Input-Output Theory
The concept of a resource being consumed to produce
another can be formulated using an Input-Output
model that was designed and developed by Professor
Wassily Leontief (Dietzenbacher & Lahr, 2004).
Based on this theory, a resource sector is consumed
by another sector and the final demand. For instance,
the output of a resource sector is used to produce itself,
other resources and for the domestic and industrial
consumption. This is indicated by equation (1).
𝑥

= 𝑥



+ 𝑦



(1)
where x
i
is the output of resource sector i, x
ij
is the
consumption of resource sector i to produce a
resource sector j, and y
ij
is the consumption of
resource sector i to fulfil the final demand j.
The amount of resource sector i required to
produce one unit of resource sector j is given as a
fraction after dividing the amount consumed to
produce a resource sector x
ij
by the total output of a
resource sector x
j
. This fraction is expressed by
equation (2).
𝑎

=




or 𝑥

=𝑎

𝑥

(2)
Where a
ij
is considered to be technological
coefficient describing the amount of resource sector i
consumed to produce a single unit of resource sector j.
Equation (1) and equation (2) can be combined in
a matrix and a vector form using equation (3).
𝐴𝑋 + 𝑌 = 𝑋 (3)
where A is a matrix of intensity or technological
coefficients, Y is a vector of final demands, and X is
a vector of outputs.
2.2 Resource Consumption and
Production in FEW Nexus
The use of Input-Output theory in Food-Energy-
Water Nexus has been demonstrated in (Karnib,
2016, 2017a, 2017b, 2018). In this nexus, there exists
complex interactions where water is used to produce
energy and food, energy is used to produce water and
food and food can be used to produce energy.
Therefore the consequences occurring in one sector
affect the other sectors (Mahlknecht et al., 2020).
Consumption of resources in the FEW Nexus can
be represented using variables as indicated in Table 1
below. The number of food resources can be denoted
by q, energy resources by m and water resources by n
(Karnib, 2018). The number of final demands can be
denoted by h.
Table 1: Consumption of resources in the FEW Nexus.
Resource Consumption Variable
The consumption of water i (w
i
)
to produce energy j (e
j
)
w
i
e
j
The consumption of water i (w
i
)
to produce food j (f
j
)
w
i
f
j
The consumption of water i (w
i
)
by demand j (d
j
)
w
i
d
j
The consumption of energy i (e
i
)
to produce water j (w
j
)
e
i
w
j
The consumption of energy i (e
i
)
to produce food j (f
j
)
e
i
f
j
The consumption of energy i (e
i
)
to produce energy j (e
j
)
e
i
e
j
The consumption of energy i (e
i
)
by demand j (d
j
)
e
i
d
j
The consumption of food i (f
i
) to
produce energy j (e
j
)
f
i
e
j
The consumption of food i (f
i
) to
produce food j (f
j
)
f
i
f
j
The consumption of food i (f
i
) by
demand j (d
j
)
f
i
d
j
Using equation (1), resource consumption and
production in FEW Nexus can be formulated as
equations (4),(5) and (6).
𝑤

𝑒



+𝑤

𝑓



+𝑤

𝑑



= 𝑤

where i=1, 2, …..,n
(4)
𝑒

𝑤



+𝑒

𝑒



+𝑒

𝑓



+𝑒

𝑑



= 𝑒

where i=1, 2, …..,m
(5)
𝑓

𝑒



+𝑓

𝑓



+𝑓

𝑑



= 𝑓

where i=1, 2, …..,q
(6)
ICINCO 2022 - 19th International Conference on Informatics in Control, Automation and Robotics
156

The total resource consumptions can be
represented using equation 7, 8 and 9.
∑
𝑤



= 𝑤 (7)
∑
𝑒



= 𝑒 (8)
∑
𝑓



= 𝑓 (9)
Equations 10 to 14 represent the resource
consumption in production of other resources.
∑


𝑤

𝑒

=𝑤𝑒


(10)
Where we is the amount of water used to produce
energy.
∑


𝑤

𝑓



=𝑤𝑓 (11)
Where wf is the amount of water used to produce
food.
∑


𝑒

𝑤



=𝑒𝑤 (12)
Where ew is the amount of energy used to
produce water.
∑


𝑒

𝑓



=𝑒𝑓 (13)
Where ef is the amount of energy used to
produce food.
∑


𝑓

𝑒

=𝑓𝑒


(14)
Where fe is the amount of food used to produce
energy.
2.3 Formulation of Objective Functions
The main objective is to minimise the amount of a
resource used to produce another resource and at the
same time maximising the production of the other
resource. This is achieved by using technological
coefficient specified by equation 2. Therefore,
minimisation of the intensities implies minimising
consumption as well as maximising production
simultaneously. By combining equations 7 to 14,
objective functions are formulated using equations 15
to 19.
𝑀𝐼𝑁 𝑤𝑒/𝑒 (15)
𝑀𝐼𝑁 𝑤𝑓/𝑓 (16)
𝑀𝐼𝑁 𝑒𝑤/𝑤 (17)
𝑀𝐼𝑁 𝑒𝑓/𝑓 (18)
𝑀𝐼𝑁 𝑓𝑒/𝑒 (19)
3 SIMULATIONS
Simulations were performed using gamultiobj which
is a NSGA-II (Deb et al., 2002) based Multiobjective
Genetic Algorithm function implemented in
MATLAB. This function is a controlled elitist
algorithm that prefers solutions with better fitness
values and those that have low fitness values but they
increase diversity of the population. Two simulations
were performed using Business As Usual(BAU)
resource consumption data obtained from the work of
Karnib (Karnib, 2018) to demonstrate the feasibility
of Multiobjective Optimization Algorithms in FEW
Nexus.
A fitness function that takes a row vector of a
given number of decision variables was specified.
The objective functions formulated in section 2.3
were incorporated in this fitness function that returns
a vector of objective function values. The fitness
function was executed using gamultiobj with the
specified lower and upper bounds of the given
problem.
At first, based on BAU consumption data,
resource intensities were calculated using equations
15 to 19. Then simulations are done to provide results
of optimisation. The obtained results are used as a
basis of comparison of resource intensities based on
BAU values and the resource intensities obtained
after optimisations. The comparisons were done to
establish whether the resource intensities are reducing
despite the consumption values increasing. Reduction
in intensities is an indication of two scenarios. The
first one is when there is simultaneous reduction in a
resource used to produce another resource and an
increase of the resource being produced. The second
one is when the amount of change in consumption is
small while the amount of change in production of a
resource is large.
In the first simulation, the low bound vector of our
fitness function is set to the BAU consumption values
for both intersectoral and final demand values while
the upper bound vector is set such that the
intersectoral values do not have any upper limit while
the final demand values are set to BAU values thus
making the final demand to be fixed. This is to make
the demand values constant. The purpose of these
settings is to demonstrate the behaviour of the FEW
Nexus when the demand is fixed but the intersectoral
consumption changes while minimising the resource
consumption intensities.
In the second simulation, the aim was to find out
the behaviour of the FEW Nexus when both
intersectoral consumption and the demand of the
resources are changing while minimising resource
Input-Output Multiobjective Optimization Approach for Food-Energy-Water Nexus
157

intensities. In this case the lower bound and upper
bound vectors for intersectoral consumption values
are set to BAU and infinity values respectively.
Similarly, the lower bound for final demand is set to
BAU values while the upper bound is set to infinity
values.
The Multiobjective Genetic Algorithm function
was executed with the above settings and multiple
non dominating solutions were obtained. From these
many solutions we selected separately only those that
indicated the minimum intensity values for
consumption related to water for energy, water for
food, energy for water, energy for food and food for
energy.
4 RESULTS
The simulations carried out highlighted the potential
of using Multiobective Optimisation Algorithms in
understanding resource consumption and production
in FEW Nexus. The algorithm generated many non-
dominated solutions depicting various alternatives
that can be taken by the decision maker. Only five
solutions were selected for demonstration purposes.
Each selected solution represented a scenario where a
resource intensity was having the minimum value as
compared to the same resource intensity values
appearing in other solutions generated by the
algorithm. Therefore, we selected five solutions such
that the water-energy, water-food, energy-water,
energy-food and food-energy intensities were the
lowest respectively.
The values entered in Table 2 and Table 3 are
obtained by subtracting the BAU intensity values
from the ones obtained after the optimisation process.
A negative value indicates a downward tendency of a
resource intensity while a positive value indicates an
upward tendency of a resource intensity. Based on
these results, we can argue that although the amount
of some resources consumed can be above the BAU
values, it is still possible to achieve a reduction of
minimised resource intensities.
Table 1 summarises the results from the first
simulation. It is important to note that the water-
energy resource intensities in all the solutions have
higher values than the ones obtained from the BAU
values. The first row shows the intensities for the
solution where the water-energy intensity has the
lowest value.
In this row there is an upward tendency for the
water-energy, water-food and energy-food intensities
while energy-water and food-energy intensities show
a downward tendency. The second row is where the
water-food intensity is the lowest. In this case water
–energy and food-energy intensities have an upward
tendency while water –food, energy –water and
energy –food intensities have a downward tendency.
In the third row, water-energy, water-food and
energy-food intensities have an upward tendency
while energy-water and food-energy intensities have
a downward tendency. This is a row showing
intensities for a solution that has the lowest value for
energy-water intensity. The fourth row is the solution
where there is the lowest energy-food intensity. In
this row there is upward tendencies in water-energy
and food-energy intensities while downward
tendencies are observed in water-food, energy-water
and energy-food intensities. In the last row where the
food-energy intensity is the lowest, the water-energy,
water-food, energy-water, energy-food have upward
tendencies while food-energy having a downward
tendency.
Table 2: The differences between BAU and Optimisation values from the 1
st
simulation.
Min. Intensity
Water-Energy Water-Food Energy-Water Energy-Food Food-Energy
Water-Ener
gy
0.001 4.5E-05 -0.0003 1.35E-05 -2E-06
Water-Food 0.001 -8E-07 -0.00043 -4.4E-07 2E-06
Ener
gy
-Water 0.001 0.00012 -0.00053 2.08E-16 -4E-17
Energy-Food 0.001 -8E-07 -0.00043 -4.4E-07 2E-06
Food-Energy 0.012 0.03659 0.08569 0.037289 -0.002
Table 3: The differences between BAU and Optimisation values from the 2
nd
simulation.
Min. Intensit
y
Water-Ener
gy
Water-Food Ener
gy
-Water Ener
gy
-Food Food-Ener
gy
Water-Ener
gy
0.000968 0.000724 0.000131 0.00043 9.05E-06
Water-Food 0.001013 -0.00013 -0.00046 -5.8E-05 8.83E-06
Energy-Water 0.003376 0.006689 -0.00372 0.002054 3.91E-06
Energy-Food 0.001071 -1.9E-05 -3E-05 -8.5E-05 -5E-06
Food-Ener
gy
0.011439 0.020998 0.071247 0.028517 -0.00226
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Table 2 summarises the results from the second
simulation. The first row indicates the solution where
the water-energy intensity is the lowest. This row
indicates an upward tendency for all the resource
intensities. In the second row, the solution is where
the water-food intensity is the lowest. In this case just
like in the first simulation, the water –energy and
food-energy intensities have an upward tendency
while water –food, energy –water and energy –food
intensities have a downward tendency. The third row
shows intensities for a solution with the lowest value
for energy-water intensity. In this row, it is only the
energy-water intensity that has a downward tendency
as the others have upward tendencies. The solution
presented in the fourth row is the one with the lowest
energy-food intensity. In this row the upward
tendency is indicated only in water-energy intensity
while other resource intensities show downward
trends. The fifth row represents intensities for a
solution where the food-energy intensity has the
lowest value. In this row only the food-energy
intensity has a downward tendency.
5 DISCUSSION
The findings from the simulations indicate that after
the intensity minimisation process, there are those
resource intensities that will have upward tendencies
while others will have downward tendencies from the
BAU values. It is also noted that water-energy
intensity always has an upward tendency.
The Input-Output theory has the assumption that
the total amount of a resource produced is the amount
consumed by other resources. Therefore, the upward
tendency of a resource intensity means a more
increase in a resource consumption to produce
another resource as compared to the resource
produced.
The increase in water-energy intensity as
compared to BAU in all cases is an indication that
water is heavily consumed. This implies more water
is used to produce energy and food as well meeting
the final demand. The intensity value has increased
because the rate of change of water consumption is
more than that of energy production.
An increase in water-food intensity implies more
water is available for food. The water-food intensity
increases because there is an increase of water
consumption rate as compared to food production
rate. Also energy-water intensity has reduced because
the water production rate has increased as compared
to the rate of energy consumption.
Also it is noted that reduction in energy-water
intensity implies water-energy, water-food and
energy food intensities can increase while food-
energy intensity can reduce. Water-energy and food-
energy intensities can increase while water-food and
energy-food intensities can reduce. There is also a
case where water-energy, water-food and energy-
food intensities can increase while food-energy
intensity can reduce.
6 CONCLUSION AND FURTHER
WORK
The proposed approach can support various
alternatives of optimization of resource consumption
and production in the FEW Nexus. The results show
that when the resource intensities are minimized
simultaneously, the consumption of water to produce
energy will always be high. However, the
consumption of energy and food to produce other
resources maybe increase or reduce. Most existing
approaches are not able to demonstrate ways of how
to minimize the resource intensities simultaneously
therefore making this approach a novel one. The
design and development of a novel Many-Objective
Optimization algorithm that is suitable to handle five
or more objectives is considered as future work.
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