# Fairness in maximal covering location problems

Víctor Blanco† and Ricardo Gázquez†
†Institute of Mathematics (IMAG), Universidad de Granada
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper provides a general mathematical optimization based framework to
incorporate fairness measures from the facilities’ perspective to Discrete and
Continuous Maximal Covering Location Problems. The main ingredients to
construct a function measuring fairness in this problem are the use of: (1)
ordered weighted averaging operators, a family of aggregation criteria very
popular to solve multiobjective combinatorial optimization problems; and (2)
$\alpha$-fairness operators which allow to generalize most of the equity
measures. A general mathematical optimization model is derived which captures
the notion of fairness in maximal covering location problems. The models are
firstly formulated as mixed integer non-linear optimization problems for both
the discrete and the continuous location spaces. Suitable mixed integer second
order cone optimization reformulations are derived using geometric properties
of the problem. Finally, the paper concludes with the results obtained on an
extensive battery of computational experiments on real datasets. The obtained
results support the convenience of the proposed approach.

###### Key words and phrases:

Maximal covering location; Continuous Location; Fair resource allocation;
Ordered Weighted Averaging Problem; Mixed Integer Non Linear Programming.

##  1\. Introduction

The term fairness is defined as “the quality of treating people equally or in
a way that is right or reasonable” (Cambridge Dictionary). It is an abstract
but widely studied concept in Decision Sciences in which some type of
indivisible resources are to be shared among different agents. The importance
of fairness issues in resource allocation problems has been recognized and
well studied in a variety of settings with tons of applications in different
fields (see e.g., Kelly et al., 1998; Luss, 1999; Li and Li, 2006; Jiang et
al., 2021). Fair allocations should imply impartiality, justice and equity in
the allocation patterns, which are usually quantified by means of inequality
measures that are minimized. Several measures have been proposed in the
literature to this end, although the most popular one is the max-min (or min-
max) approach which assures that the most harmed agent in the share is as less
damaged as possible (see e.g., Megiddo, 1974; Hayden, 1981; Jaffe, 1981;
Bertsekas et al., 1992). Other proposals of fairness measures are the minimum
envy (Lipton et al., 2004; Caragiannis et al., 2009; Espejo et al., 2009;
Netzer et al., 2016) or certain families of ordered weighted averaging
criteria (Ogryczak and Trzaskalik, 2006; Hurkała and Hurkała, 2013; Ogryczak
et al., 2014), among others.

In this paper, we analyze the notion of fairness in the context of Location
Science. Facility location problems aim to optimally determine the position of
one or more facilities in order to satisfy the demand of a set of users. There
is a vast variety of location problems, which are commonly classified by the
nature of users and facilities, the optimization criteria, the type of demand
to be satisfied, among many others (Hamacher and Nickel, 1998). The interested
reader is referred to the recent book Laporte et al. (2019) for a fresh view
on the developments on both theoretical an applied aspects of location
problems.

We analyze here a particular family of location problems, the so-called
Covering Location (CL) problems, that arise whenever a decision-maker aims to
cover a given demand in case the facilities have a limited coverage area.
These types of problems appear in many practical situations in which the
services to be located are not able to satisfy the demand outside their
coverage area. One of the most interesting CL problems is the Maximal Covering
Location Problem (MCLP) whose goal is to find the positions of a given number
of services, $p$, each of them with a coverage area, maximizing the covered
demand of the users. This problem was firstly introduced by Church and ReVelle
(1974), and since then the MCLP has attracted significant attention from both
researchers and practitioners (Wei and Murray, 2015), both by its technical
merit and practical interest. As in other Location Science problems, it is
usual to analyze two different types of location spaces for the MCLP, in terms
of the nature of the facilities that are to be located. In the discrete MCLP,
the services are selected from a given finite set of potential facilities,
whereas in the continuous MCLP, the services are allowed to be located at any
position of the decision space (usually, the plane).

This paper provides a general mathematical optimization based framework to
incorporate fairness measures from the facilities’ perspective to discrete and
continuous MCLPs. We assume that a given number of services has to be located
to maximize the demand covered by the different agents. In a fairly ideal
solution, one would desire to “independently” maximize the covered demand of
each of the services, not affecting negatively to the demand coverage of the
others. However, since the demands are usually indivisible, in most cases, an
advantageous solution for one service (one covering a large amount of the
demand) harms others. As already occurs in other resource allocation problems,
one may prefer to slightly sacrifice the overall covered demand in order to
balance the different covered demands among the open services. This might be
applicable to the case of the location of public schools, in which it is
preferable to find an homogeneous distribution of kids among the schools, or
the location of routers with high capacities, where a “good” location for them
would be the one in which the performance of all the routers can be better
used instead of saturating some and leaving others covering a small amount of
users.

###  1.1. Related work

In many resource reallocation problems there has been special attention to the
notion of fairness. For this reason, it has been widely recognized in the
literature and, due also to the subjective implications the term fairness has,
it has been studied in a wide variety of ways. The applications of fair
resource allocation range from social or humanitarian contexts to engineering
applications or location problems.

The social context is an important field where the fairness has been used.
Gross (2008) highlights the importance of studying the notions of equity and
justice and its incorporation into decision problems such as the fair
allocation of water in irrigation farms. Huang and Rafiei (2019) analyze the
notion of fairness in the field of humanitarian logistics.

In engineering applications, as in telecommunication networks, where limited
resources must be allocated among the different competing entities, the notion
of fairness is also crucial (see e.g., Kleinberg et al., 1999; Luss, 1999;
Bonald and Massoulié, 2001; Ogryczak et al., 2014).

The allocation of public resources is one of the most studied applications
when analyzing equity and fairness. For instance, in the allocation of beds
and ambulances in emergency medical services. In Leclerc et al. (2012, Chapter
4), the authors review different measures previously proposed in the
literature and study their incorporation to the allocation of ambulances.
Another example of the importance of equity allocation comes from healthcare
scheduling. Specifically, in the allocation of beds and other resources to
patients where fairness allows to reduce mortality (see e.g, Zhou et al.,
2020).

In Location Science some equity measures have been also analyzed (Espejo et
al., 2009; Chanta et al., 2014). A review of measures for equitable facility
location problems is given in Marsh and Schilling (1994); Barbati and Piccolo
(2016).

Nevertheless, the notion of fairness can be interpreted in many different ways
and there is no universal principle that is accepted as “the most fair”
(Bertsimas et al., 2012). The most usual strategy to incorporate fairness in
resource allocation problems is to quantify the degree of fairness associated
with any feasible action through a fairness measure, a function that maps each
feasible solution into a real value indicating its degree of equity.
Therefore, different measures have been proposed in the literature in order to
capture the notion of fairness. The most popular one is the max-min (or min-
max) approach which assures that the most damaged agent in the share is as
less damaged as possible (see e.g., Jiang et al., 2021; Kelly et al., 1998; Li
and Li, 2006; Luss, 1999). The max-min ratio is given by the maximum pairwise
ratio between the resources allocated to the agents. Furthermore, some authors
have incorporated fairness in decision problem through _envy_ measures,
understood as the overall deviation in the gain (or cost) between all pairs of
agents (see e.g., Lipton et al., 2004; Caragiannis et al., 2009; Espejo et
al., 2009; Chanta et al., 2014; Netzer et al., 2016).

Other popular measure that has been widely investigated is the Jain’s index
(Jain et al., 1984) which is obtained by normalizing the square mean of an
entropy function. The Jain’s index measures the _equity_ of an allocation
pattern, being the Jain’s index 1 for the most fair allocation in which all
the agents receive (or pay) the same amount. This index was generalized by Lan
et al. (2010), where the authors propose a method to construct fairness
schemes based on different axioms. The authors propose fairness measures
generated by power functions generalizing the Jain’s index.

An alternative approach that is very used in recent mathematical optimization
based problems is to construct objective functions to guide the optimization
approach to obtain fair solutions. This is the case of the so-called
$\alpha$-fairness (Kelly et al., 1998; Mo and Walrand, 2000; Lan et al., 2010;
Bertsimas et al., 2011, 2012). The $\alpha$-fairness functions were introduced
by Atkinson (1970) and was defined as the one maximizing the constant
elasticity social welfare. This family of functions generalizes well-known
criteria in the the allocation of resources, as the utilitarian principle
($\alpha=0$), which is neutral toward inequalities; proportional fairness
($\alpha=1$), that was introduced by Nash (1950); and for
$\alpha\rightarrow\infty$ the measure converges to the max-min criterion. As
we will observe, other possibilities of $\alpha$ allow to find a trade-off
between efficiency and fairness, opening room for different and suitable
allocation patterns.

An ideal approach to find solutions to fully satisfy all the agents involved
in the allocation problem is to model it as a multiobjective problem. However,
solving this problem may be computationally difficult in practice (Ehrgott and
Gandibleux, 2000). Moreover, the solution of a multiobjective problem is, in
general, not a single solution but a solution set, the Pareto frontier, which
can be useless in many decision problems. Thus, it is widely accepted to use
aggregation operators of each of the agents utilities to find compromise
solutions. Several aggregation criteria have been proposed to construct
solutions satisfying all the different agents in the decision problem. The
most popular aggregation criteria are: the weighted sum aggregation, the
minimum of the values (resulting in max-min problems) and the lexicographic
order (Ogryczak et al., 2014).

Another general family of aggregation criteria that has been successfully
applied is Ordered Weighted Averaging (OWA) functions, introduced by Yager
(1988). These operators have been also applied to encourage fairness in the
obtained solutions (see e.g., Ogryczak and Trzaskalik, 2006; Hurkała and
Hurkała, 2013; Ogryczak et al., 2014). An OWA operator maps a vector to a
weighted sum of its coordinates in which the weights are assigned to their
sorted values. These operators allow to generalize most of the robust
statistical measures, as the mean, the minimum, the $k$-mean, the trimmed mean
values or the Gini index, and have been applied in different fields (see e.g.,
Argyris et al. (2022); Blanco et al. (2018, 2021); Marín et al. (2022), among
many others).

The use of OWA operators in Facility Location is not new and several authors
have studied the incorporation of these operators to the classical problems
through the so-called ordered median location problems (see Puerto and
Fernández, 1994). Furthermore, when the weights defining the OWA function are
nonnegative and monotone (non decreasing) the OWA operator is considered fair
aggregation criteria, allowing to find fair distribution of the demand.

In this paper, we analyze a novel version of one of the core family of
problems in Facility Location, Covering Location problems. The location of the
facilities in CL is characterized by the fact that the facilities are allowed
to give service to the users at a limited distance from them. This type of
problems has been applied in different fields, as in the location of emergency
services (Toregas et al., 1971), mail advertising (Dwyer and Evans, 1981),
archaeology (Bell and Church, 1985), among many others. The interested reader
is referred to Laporte et al. (2019, Chapter 5) for further details and recent
advances on CL problems. In particular, we answer the question of how to
incorporate fairness in the Maximal Covering Location problem (MCLP). In the
MCLP, introduced by Church and ReVelle (1974), it is assumed the existence of
a budget for opening facilities and the goal is to accommodate it to satisfy
as much demand as possible. As usual in location problems, one can consider
different frameworks based on the nature of the solution space for the
facilities: discrete or continuous spaces. While the discrete setting is more
adequate when locating physical services, (as ATMs, stores, hospitals, etc),
the continuous location space is known to be more adequate to determine the
positions of routers, alarms or sensors, that can be more flexibly positioned.
This type of space is also useful to determine the set of potential facilities
that serves as input for a discrete version of the problem. One of the main
difference between these two families of problems (from the mathematical
optimization viewpoint) is that in the discrete one the distances between the
facilities and the users are given as input data (or can be preprocessed
before solving the problem), while in the continuous case, the distances are
part of the decision and they must incorporated to the optimization problem.

As far as we know, the incorporation of fairness measures into CL problems has
been studied in Drezner and Drezner (2014), in which the max-min approach has
been applied to the gradual maximal covering location problem. There, the
authors incorporate the worst-case fairness criteria from the user’s
perspective, i.e. in order to enforce equity between the partial coverage of
all users. In networks, Rahmattalabi et al. (2020) consider the selection of a
subset of nodes to cover their adjacent nodes with fairness constraints with
applications to social networks. Asudeh et al. (2020) analyze a covering
location problem with fairness constraints minimizing the pairwise deviations
between the different covered sets. Korani and Sahraeian (2013) study a hub
covering problem with _equity_ allocation constraints.

Despite of these applications, the efficiency measure used in the MCLP is the
overall covered demand, that is, as much covered demand the better. However,
when one looks at the individual utilities of each of the constructed
facilities, one may obtain solutions with highly saturated facilities in
contrast to others that only cover a small amount of demand, which results in
unfair systems from the facilities’ perspective. Moreover, in many situations
this type of unfair solutions are also undesirable from the users’ viewpoint
which may see reduced the quality of the required service, as in the location
of telecommunication servers which have a higher probability to fail in case
of being saturated (being other capable to give service to these users) or in
the student assignment process to schools, in which a higher number of alumni
allocated to a school may deteriorate the education system. As far as we know,
this problem has not been previously investigated in the literature in the
context of Covering Location.

###  1.2. Contributions

In this paper, we provide a flexible mathematical optimization based framework
to incorporate fairness measures from the facilities’ perspective to Discrete
and Continuous MCLPs. This generalization of the fairness measure for the MCLP
is based on adequately combining the two main tools mentioned above: OWA
operators and $\alpha$-fairness operators.

Our specific contributions in this paper are:

  1. (1)

To define a novel fairness measure combining OWA and $\alpha$-fairness
operators that can be incorporated to the objective function of the MCLP.

  2. (2)

To describe a general mathematical optimization model which captures the
notion of fairness from the facilities’ perspective in MCLPs.

  3. (3)

To provide a mixed integer non-linear optimization formulation for the two
main location spaces in the facility location problems: discrete and
continuous.

  4. (4)

To derive MISOCO reformulations for the problem, suitable to be solved with
off-the-shelf optimization softwares.

  5. (5)

To test on a battery of computational experiments the computational
performance of the approaches and their managerial insights.

###  1.3. Paper structure

The remainder of the paper is organized as follows. Section 2 is devoted to
recall the used tools for our proposed method and states the notation for the
rest of the paper. In Section 3 we introduce the generalized fair maximal
covering problem. In Section 4 we present a mathematical optimization
formulation for the problem, both in the discrete and the continuous location
spaces. The results of our computational experience are reported in Section 5.
Finally, the paper ends with some conclusions and future research lines.

##  2\. Preliminaries

In this section we introduce the notation used for the rest of the sections as
well as the main results for analyzing the problem studied in this paper.

###  2.1. Maximal Covering Location Problem

Consider a finite set of demand points in a $d$-dimensional space,
$\mathcal{A}=\\{a_{1},\ldots,a_{n}\\}\subseteq\mathbb{R}^{d}$, indexed by the
set $N=\\{1,\ldots,n\\}$. Each demand point $a_{i}\in\mathcal{A}$ has
associated a non-negative demand weight $\omega_{i}$. Throughout the paper we
often call a demand point interchangeably by the node $a_{i}$ or by the index
$i$. Demand points may represent users or regions and the weights allow one to
give more importance to different users or take into account the
size/population of each of the regions. We are also given a potential set of
facilities, $\mathcal{X}\subseteq\mathbb{R}^{d}$, where the _services_ are to
be chosen. The set $\mathcal{X}$ is not necessarily finite.

Each potential position for the facilities, $X\in\mathcal{X}$ is endowed with
a coverage area. It is usual to define the coverage areas as Euclidean balls
with certain coverage radii. In this paper, we consider ball-shaped coverage
areas in the form:

$\mathbb{B}_{R}(X)=\\{z\in\mathbb{R}^{d}:\|z-X\|\leq R\\}$

where $R>0$ is the radius and $\|\cdot\|$ is a $\ell_{\tau}$-based norm
($\tau\geq 1$), that is:

$\|x\|_{\tau}=\left(\displaystyle\sum_{l=1}^{d}|x|^{\tau}\right)^{\frac{1}{\tau}},$

or a polyhedral norm with a symmetric polytope $B$ (with respect to the
origin) in $\mathbb{R}^{d}$:

$\|x\|_{B}=\min\\{\displaystyle\sum_{g=1}^{G}|\beta_{g}|:x=\displaystyle\sum_{g=1}^{G}\beta_{g}e_{g}\\},$

where $\\{\pm e_{1},\ldots,\pm e_{G}\\}$ are the extreme points of $B$.

A demand point $a\in\mathcal{A}$ is said to be covered by the facility
$X\in\mathcal{X}$ if it belongs to the coverage area of $X$, i.e.,
$a\in\mathbb{B}_{R}(X)$. Given $X\in\mathcal{X}$, we denote by
$\mathcal{C}(X)$, the index set of demand points covered by $X$, i.e.,

$\mathcal{C}(X)=\\{i\in N:a_{i}\in\mathbb{B}_{R}(X)\\}.$

The goal of the Maximal Covering Location Problem (MCLP), introduced by Church
and ReVelle (1974), is to determine the positions of $p$ new facilities in
$\mathcal{X}$, that is $\\{X_{1},\ldots,X_{p}\\}\subseteq\mathcal{X}$,
maximizing the ($\omega$-weighted) covered demand. Denoting by
$P=\\{1,\ldots,p\\}$ the index set for the facilities to be located, the MCLP
can be formally formulated as the following optimization problem:

$\max_{X_{1},\ldots,X_{p}\in\mathcal{X}}\displaystyle\sum_{i\in\bigcup_{j\in
P}\mathcal{C}(X_{j})}\omega_{i}.$

Two main families of MCLP arise based on the nature of the metric set from
where the facilities are to be located. In case $\mathcal{X}$ is a finite set,
one obtain the Discrete MCLP that was introduced by Church and ReVelle (1974).
On the other hand, in case $\mathcal{X}=\mathbb{R}^{d}$ one obtain the
Continuous MCLP that was firstly formulated by Church (1984) for the planar
case ($d=2$). In this paper, we will analyze an unified and particular version
of the MCLP for both type of location spaces.

###  2.2. Ordered Weighting Averaging operators

As in most decision problems in which several agents are involved, the MCLP
also exhibits a compromise in view of the users that require the services to
be located or for the decision makers that construct and locate the facilities
to give the different services to the users. In this paper, we analyze the
MCLP from the point of view of the different decision makers that invest on
the installation of the facilities to give service to the users. In an ideal
solution one would desire to determine the position of the services to
maximize, separately, the demand covered by each service. This approach will
result on modeling the problem as a multiobjective mathematical optimization
problem (with objectives the demands covered by each of the facilities) which
can be cumbersome in practice. Instead, we aggregate the $p$ objective
functions by means of Ordered Weighted Averaging (OWA) operators, introduced
by Yager (1988).

OWA operators are functions in the form
$\Phi_{\mathbf{\lambda}}:\mathbb{R}^{p}\rightarrow\mathbb{R}$ with associated
weighting vector $\mathbf{\lambda}=(\lambda_{1},\ldots,\lambda_{p})$, with
$\lambda_{j}\in[0,1],\;\forall j\in\\{1,\ldots,p\\}$ and
$\sum_{j=1}^{p}\lambda_{j}=1$. We denote as $P=\\{1,\ldots,p\\}$ the index set
for coordinates of vectors in $\mathbb{R}^{p}$. For
$\mathbf{W}=(W_{1},\ldots,W_{p})\in\mathbb{R}^{p}$, the OWA operator is
defined as:

$\Phi_{\mathbf{\lambda}}(W_{1},\ldots,W_{p})=\displaystyle\sum_{j=1}^{p}\lambda_{j}W_{(j)},$
(OWA)

where $W_{(j)}$ is the $j$th-largest value in the vector $\mathbf{W}$, i.e.,
$W_{(j)}\in\\{W_{1},\ldots,W_{p}\\}$ such that $W_{(1)}\leq\ldots\leq
W_{(p)}$. OWA operators are weighted sums of the different criteria, but where
weights are associated with the position of the criteria when they are sorted
in non decreasing order.

When aggregating different criteria by means of an OWA operator, a crucial
step is to provide the adequate weights $\lambda$ that are assigned to the
sorted sequence. The most popular OWA operators ($\min$ –
$\lambda=(1,0,\ldots,0)$ and $\max$– $\lambda=(0,\ldots,0,1)$) serve as
reference weights to define the notion of orness of a vector of
$\lambda$-weights defining an OWA operator:

${\rm orness}(\lambda)=\displaystyle\sum_{j=1}^{p}\frac{p-i}{p-1}\lambda_{i}$

The degree of orness emphasizes the higher (better) values or the lower
(worse) values in a set of attributes associated with the different
agents/services. Given a vector of $\lambda$ weights, as closer its orness to
$1$, closer to the $\min$-operator while as closer to $0$, closer to the
$\max$-operator. Assuming that all the criteria are to be minimized, the
$\min$-operator allows one to generate solutions protected under worst-case
scenario (pessimistic), while the $\max$-operator produces solution in which
the best situation for all the criteria is assumed (optimistic). In between,
one can find an equilibrium between those extreme choices. In particular, for
$\lambda=(\frac{1}{p},\ldots,\frac{1}{p})$– the mean operator, its orness
degree takes value $0.5$. In Table 1 we show a list with some of the most
popular OWA operators and their orness degree.

$OWA$ $\lambda$ Operator ${\rm orness}$ Average $\lambda_{j}=\frac{1}{p}$
$\frac{1}{p}\displaystyle\sum_{j\in P}W_{j}$ $\frac{1}{2}$ Minimum
$\lambda_{1}=1,\lambda_{j}=0$ ($j\geq 2$) $\min_{j\in P}W_{j}$ 1 $k$-Average
$\lambda_{j}=\frac{1}{k}$ ($j\leq k$), $\lambda_{j}=0$ ($j>k$)
$\frac{1}{k}\displaystyle\sum_{j=1}^{k}W_{(k)}$ $1-\frac{k-1}{2(p-1)}$
$\alpha$-Min-Average $\lambda_{1}=\frac{1}{1+(p-1)\alpha}$,
$\lambda_{j}=\frac{\alpha}{1+(p-1)\alpha}$ ($j\geq 2$)
$\frac{1-\alpha}{1+(p-1)\alpha}\min_{j\in
P}W_{j}+\frac{\alpha}{1+(p-1)\alpha}\displaystyle\sum_{j\in P}W_{j}$
$\frac{-p\alpha+p+2\alpha}{2p\alpha-2\alpha+2}$ Gini
$\lambda_{j}=\frac{2(p-j)+1}{p^{2}}$ for all $j$
$\frac{1}{p^{2}}\displaystyle\sum_{j\in
P}W_{j}+\frac{2}{p^{2}}\displaystyle\sum_{j\in P}(p-j)W_{(j)}$
$\frac{4p+1}{6p}$ Harmonic $\lambda_{j}=\frac{1}{p}\left(H(p)-H(j-1)\right)$
($H(k)=\displaystyle\sum_{\ell=1}^{k}\frac{1}{\ell}$)
$\frac{1}{p}\displaystyle\sum_{j\in P}\left(H(p)-H(j-1)\right)W_{(j)}$
$\frac{3}{4}$

Table 1. Some examples of fair OWA operators and its orness.

It is clear that an OWA operator (identified with a $\lambda$-weight) is not
uniquely determined by its orness degree (unless its orness degree is in
$\\{0,1\\}$ or $p=2$). Thus, several optimization-based methods have been
proposed in order to construct, with different paradigms, $\lambda$-weights
with a given orness degree $\beta\in(0,1)$ (see, e.g., Filev and Yager, 1995;
Fullér and Majlender, 2001; Liu and Chen, 2004). The main idea when searching
for $\lambda$-weights with a given orness degree is to solve problems in the
form:

$\displaystyle\min$ $\displaystyle\;\;\mathcal{L}(\lambda)$ s.t.
$\displaystyle{\rm orness}(\lambda)=\beta,$
$\displaystyle\lambda\in\mathbb{R}^{p}_{+},$

where $\mathcal{L}$ is a loss function measuring some properties of the
weights. For instance, if
$\mathcal{L}(\lambda)=-\sum_{j=1}^{p}\lambda_{j}\log\lambda_{j}$ one obtain
the maximal entropy monotone OWA (O’Hagan, 1988), or choosing
$\mathcal{L}(\lambda)=\sum_{j=1}^{p}(\lambda_{j}-\overline{\lambda})^{2}$ one
obtains the minimum variance weights (Fullér and Majlender, 2003), where
$\overline{\lambda}$ stands for the mean of the vector $\lambda$.

There are several ways to sort a list of (unknown) values and compute the OWA
operator. The most known are the OT proposed by Ogryczak and Tamir (2003) and
the BEP reformulation proposed by Blanco et al. (2014). In the former, the
authors provide a suitable linear optimization representation of the problem
of minimizing the sum of the $k$ largest (equivalently, smallest) linear
functions on a polyhedral set in $\mathbb{R}^{d}$. This representation is
extended to the minimization of monotone OWA functions by means of a
telescopic sum of $k$-sum functions.

###  2.3. Fairness

As already mentioned, fairness is a crucial requisite when making decisions in
many situations in which different agents are involved. Since the notion of
fairness is by itself weak and it is not clear, in general, a list of
axioms/desirable properties have been stated for a vector of values to be
considered fair. Lan et al. (2010) provides an axiomatic approach to fairness
measures which served at starting point to derive new fairness measures
fulfilling a list of minimum required properties. Here, we provide some useful
of desirable properties of fairness schemes. Let be
$\Psi:\mathbb{R}^{p}_{+}\rightarrow\mathbb{R}$ a fairness scheme, and
$\mathbf{W}=(W_{1},\ldots,W_{p})\in\mathbb{R}^{p}_{+}$ the allocation of
resources. Some of these properties were stated in (Lan et al., 2010; Jain et
al., 1984; Barbati and Piccolo, 2016).

  1. (1)

Continuity: $\Psi$ is a continuous function. This axiom assures that, locally,
small changes in the allocation do not significantly affect the measure.

  2. (2)

Population size independence: Equal resource allocations are, eventually,
independent of the number of users, i.e.,
$\lim_{p\to\infty}\frac{\Psi(\mathbf{1}_{p+1})}{\Psi(\mathbf{1}_{p})}=1$.

  3. (3)

Pareto optimality: If $W_{j}\leq\bar{W}_{j},\forall j\in\\{1,\ldots,p\\}$, and
$W_{j}<\bar{W}_{j}$ for at least some $j$, then
$\Psi(\mathbf{W})\leq\Psi(\mathbf{\bar{W}})$.

  4. (4)

Symmetry: $\Psi(W_{1},\ldots,W_{p})=\Psi(W_{\sigma(1)},\ldots,W_{\sigma(p)}),$
where $\sigma$ is an arbitrary permutation of the indices.

  5. (5)

Bounded: The value of allocation given by the scheme is bounded.

  6. (6)

Scale and metric independence: The measure is independent of scale, i.e, the
unit of measurement does not matter.

Here, we consider a particular measure of fairness which has gained attention
in the operations research community (Bertsimas et al., 2011, 2012), the
$\alpha$-fairness scheme, that was introduced by Atkinson (1970). According to
this scheme, the decision maker decides on the allocation of the resources by
maximizing the constant elasticity social welfare operator $\Psi_{\alpha}$
parameterized by $\alpha\geq 0$ and for
$\mathbf{W}=(W_{1},\ldots,W_{p})\in\mathbb{R}^{p}_{+}$, the amount of
resources allocated into $p$ agents,

$\Psi_{\alpha}(W_{1},\ldots,W_{p})=\left\\{\begin{array}[]{cl}\frac{1}{1-\alpha}\displaystyle\sum_{j=1}^{p}W_{j}^{1-\alpha}&\mbox{if
$\alpha\geq 0$, $\alpha\neq 1$,}\\\
\displaystyle\sum_{j=1}^{p}\log(W_{j})&\mbox{if
$\alpha=1$}.\end{array}\right.$

The parameter $\alpha$ is known as the _inequality aversion parameter_ since
it controls the difference between the allocation resources $W_{j}$. Consider
an agent $k$ with lower amount of resources $W_{k}$ than the agent
$k^{\prime}$ with amount of resources $W_{k^{\prime}}$. If we increase the
resources in $W_{k}$ then we would have a higher welfare than if we increase
in $W_{k^{\prime}}$. Thus, an increase in the resources of $W_{k}$ would be
more desirable to reduce the unfairness in the allocation of the resources.
This can be performed by tuning, adequately, the parameter $\alpha$ since when
it increases, the difference between the values of resources for the agents
decreases (for a constructive proof of this, see Lan et al., 2010). This
property, also known Principle of transfer or Pigou-Dalton (Erkut, 1993),
yielding then to fair solutions.

Therefore, moving the $\alpha$ value, one can represent classical fairness
measures. For $\alpha\rightarrow\infty$, the scheme converges to the max-min
fairness, whereas if $\alpha=0$ one obtain the sum, which is neutral toward
inequalities. For $\alpha=1$, one obtain the proportional fairness measure
that has been widely studied in the literature (Nash, 1950).

##  3\. The generalized Fair Maximal Covering Location Problem

As already mentioned, the Maximal Covering Location Problem (MCLP), in its
different versions, can be seen as a resource allocation problem, in which the
overall demand of the covered users is shared among the different services
that are located. Thus, a high coverage of the total demand (no matter which
service is providing the coverage) is appropriate from a global perspective,
but from an individual viewpoint, one can easily get unfair allocation
patterns. Furthermore, the MCLP usually exhibits multiple optimal solutions,
that is, different subsets of $p$ services covering the maximum possible
covered demand, and then, optimization solvers output an arbitrary one,
possibly not the fairest.

In the following example we illustrate this situation in a toy instance.

######  Example 1.

We consider a randomly generated set of $200$ demand points with coordinates
in $[0,20]\times[0,20]$. We assign to each of them a random integer weight in
$[0,100]$. We assume that three $(p=3)$ services are to be located chosen from
the same set of demand points. The coverage areas for all the potential
location of the services are disks of radius $5$. The solution obtained by
Gurobi for the classical MCLP is shown in Figure 1(a). In such a solution,
$65.7\%$ of the demand is covered, and the distribution of (weighted) users
among the services is $(1723,2365,2804)$, that is, there are two services
covering close to one thousand more clients than the other one. Other feasible
(non optimal for the MCLP) solution for the problem is also shown in Figure
1(b), in which $62.58\%$ of the demand is covered and whose distribution of
covered demand is $(2126,2162,2278)$. This solution, although covers $3\%$
less demand than the classical MCLP, is clearly much more balanced than the
MCLP solution, since all the users cover approximately the same demand, but
still efficient.

(a) Classical MCLP.

(b) Fairer solution for MCLP.

Figure 1. Optimal solutions obtained with the MCLP.

In what follows we introduce a new fairness measure that we incorporate into
the MCLP in order to provide fair coverage of the demands.

######  Definition 1.

Let $\alpha\geq 0$ and $\lambda\in\mathbb{R}_{+}^{p}$ with
$\sum_{j=1}^{p}\lambda_{j}=1$ and $\lambda_{1}\geq\cdots\geq\lambda_{p}$. The
$(\alpha,\mathbf{\lambda})$-fair operator is a function
$F_{\alpha,\lambda}:\mathbb{R}^{p}_{+}\rightarrow\mathbb{R}^{+}$ defined as:

$F_{\alpha,\mathbf{\lambda}}(W_{1},\ldots,W_{p}):=\left\\{\begin{array}[]{cl}\frac{1}{1-\alpha}\displaystyle\sum_{j=1}^{p}\lambda_{j}W_{(j)}^{1-\alpha}&\mbox{if
$\alpha\neq 1$,}\\\
\displaystyle\sum_{j=1}^{p}\lambda_{j}\log(W_{(j)})&\mbox{if
$\alpha=1$}.\end{array}\right.$ ($(\alpha,\mathbf{\lambda})$-Fair)

It is straightforward to check that $F_{\alpha,\mathbf{\lambda}}$ is a concave
function that verifies all of the desirable properties described in Section
2.3.

The ($(\alpha,\mathbf{\lambda})$-Fair) operator depends of $p+1$ parameters
($\alpha$ and the $\lambda$-weights) and combines the $\alpha$-fairness
measure introduced by Atkinson (1970) and the OWA operators introduced in
Yager (1988). In case $\alpha=0$, the operator turns into the
$\mathbf{\lambda}$-OWA operator, whereas if
$\mathbf{\lambda}=(\frac{1}{p},\ldots,\frac{1}{p})$ it becomes the
$\frac{1}{p}$-weighted $\alpha$-fairness function
$\frac{1}{p}\;\Psi_{\alpha}$. This combination of these two operators allows
us to derive a unified framework to deal with fairness in maximal covering
location problems, that we detail below.

We are given a set of demand points in a $d$-dimensional space,
$\mathcal{A}=\\{a_{1},\ldots,a_{n}\\}\subseteq\mathbb{R}^{d}$, indexed by the
set $N=\\{1,\ldots,n\\}$, and each of the points endowed with a demand weight
$\omega_{i}\geq 0$ for all $i\in N$. We are also given a metric space
$\mathcal{X}\subseteq\mathbb{R}^{d}$ endowed with a distance measure
$\|\cdot\|$ and a radius, $R$, for each $X\in\mathcal{X}$, which is assumed to
be the same for all the facilities to be located (although it is not a
limitation of the results provided in this paper). The goal of the Fair MCLP
is to find the position of the $p$ facilities to locate,
$X_{1},\ldots,X_{p}\in\mathcal{X}$, maximizing the $(\alpha,\lambda)$-fair
operator of the demands covered by these services. Formally, given
$\mathbf{\lambda}$ and $\alpha$, the $(\alpha,\mathbf{\lambda})$-Fair Maximal
Covering Location Problem ($(\alpha,\lambda)$-FMCLP, for short) can be model
as the following optimization problem:

$F^{*}_{\alpha,\mathbf{\lambda}}=\max_{X_{1},\ldots,X_{p}\in\mathcal{X}}F_{\alpha,\mathbf{\lambda}}(W(X_{1}),\ldots,W(X_{p}))$
(${\rm FMCLP}_{\alpha,\mathbf{\lambda}}$)

where $W(X_{j})$ is the covered demand of facility $X_{j}$, assumed that each
demand point is accounted as covered by at most one of the facilities. We
denote by $\mathcal{W}_{\alpha;\mathbf{\lambda}}=$ $(W(X_{1}^{*})$, $\ldots,$
$W(X_{p}^{*}))\in\mathbb{R}^{p}_{+}$ a coverage vector of each of the
facilities in the problem above. We also denote by $\mathcal{W}^{\rm
sum}_{\alpha;\mathbf{\lambda}}=\sum_{j=1}^{p}W(X_{p}^{*})$ the total covered
demand in the solution and by $\mathcal{W}^{\rm
min}_{\alpha;\mathbf{\lambda}}=\displaystyle\min_{j=1,\ldots,p}W(X_{p}^{*})$
the demand covered by the service with smallest coverage in the solution.

The problem that we introduced above allows one to determine the position of
the facilities that are fair from the individual viewpoint, but, how much is
one willing to lose when imposing fairness to the MCLP? _The price of
fairness_ in any allocation rule is a notion studied in Bertsimas et al.
(2011) in order to measure the efficiency loss under a fair allocation
compared to the one that maximizes the overall sum of the users utilities. In
our case, the solution of (${\rm FMCLP}_{\alpha,\mathbf{\lambda}}$) is
compared against the solution of the classical MCLP in order to know how far
is a fair solution to the best coverage of the give demand.

######  Definition 2.

The price of fairness measure for (${\rm FMCLP}_{\alpha,\mathbf{\lambda}}$) is
defined as the index:

${\rm PoF}\eqref{fmclp:0}=\dfrac{\mathcal{W}^{\rm
sum}_{0;(\frac{1}{p},\ldots,\frac{1}{p})}-\mathcal{W}^{\rm
sum}_{\alpha;\mathbf{\lambda}}}{\mathcal{W}^{\rm
sum}_{0;(\frac{1}{p},\ldots,\frac{1}{p})}},$ (PoF)

The price of fairness indicates the relative deviation of the covered demand
when solving the maximal $(\alpha,\lambda)$-FMCLP with respect to the solution
of the classical MCLP which attains the maximal possible coverage. Thus, the
price of fairness is a value between $0$ and $1$ measuring how close is the
effectiveness of the obtained fair solution with respect to the most effective
covering solution. A price of fairness equal to $0$ indicates that (${\rm
FMCLP}_{\alpha,\mathbf{\lambda}}$) is able to construct a fair allocation
without loss of efficiency (at the maximum possible coverage). In contrast, a
price of fairness with value $1$ indicates that (${\rm
FMCLP}_{\alpha,\mathbf{\lambda}}$) got the worst global coverage. Thus, as
closer this measure to $0$ the better. In general it provides the percent loss
of coverage with respect to the maximal possible coverage of an instance,
allowing one to quantify the price one has to pay when imposing
$(\alpha,\lambda)$-fairness.

On the other hand, one can also measure how far is a fair solution from the
fairest share, which is obtained when solving the max-min covering location
problem, i.e., comparing the demand covered (in our fair MCLP) by the service
covering the smallest demand in the solution with respect to the solution in
which the coverage of the service covering the least demand is maximized. This
measure was called in Bertsimas et al. (2012) the _price of efficiency_.

######  Definition 3.

The price of efficiency measure for (${\rm FMCLP}_{\alpha,\mathbf{\lambda}}$)
is defined as:

${\rm PoE}\eqref{fmclp:0}=\dfrac{\mathcal{W}^{\rm
min}_{0;(1,0,\ldots,0)}-\mathcal{W}^{\rm
min}_{\alpha;\mathbf{\lambda}}}{\mathcal{W}^{\rm min}_{0;(1,0,\ldots,0)}},$
(PoE)

The Price of Efficiency is interpreted as the percent loss in the minimum
demand coverage guarantee compared to the maximum minimum covered demand
guarantee. This index also takes value in $[0,1]$, in which a value of $0$
means that the $(\alpha,\lambda)$-FMCLP obtains the fairest solution, while a
value of $1$ indicates the least fair solution in which there is a service not
covering any demand.

Other widely used measure of fairness is the _envy_. In the MCLP the envy of
facility positioned in $X_{j}\in\mathcal{X}$, whose covered demand is
$W(X_{j})$, for a facility located at $X_{k}\in\mathcal{X}$, whose covered
demand is $W(X_{k})$, is defined as:

${\rm envy}(X_{j},X_{k})=\max\\{0,W(X_{k})-W(X_{j})\\}$

that is, facility $j$ suffers envy of value $W(X_{k})-W(X_{j})$ from facility
$k$ in case $k$ covers more demand than $j$. The overall envy of a set of $p$
facilities $X_{1},\ldots,X_{p}\in\mathcal{X}$ is the overall sum of all the
pairwise envies. From this, the _Gini index_ is defined as the ratio of this
total envy and the all the covered demand by the facilities multiplied by $2p$
(the overall number of pairwise comparisons):

######  Definition 4.

The Gini index is defined as:

${\rm Gini}(X_{1},\ldots,X_{p})=\frac{\displaystyle\sum_{j,k\in P}{\rm
envy}(X_{j},X_{k})}{2p\displaystyle\sum_{j\in P}W(X_{j})}$ (Gini)

We will see in our computational experience that the family of
$(\alpha,\mathbf{\lambda})$-FMCLP exhibits differences when varying the values
of $\alpha$ and $\lambda$ with respect to the three measures that we described
above (PoF, PoE and Gini). A trade-off solution between these measures will be
desirable from the point of view of efficiency and also of fairness.

##  4\. Mathematical optimization formulations for
$(\alpha,\mathbf{\lambda})$-FMCLP

In this section we derive a suitable mathematical optimization formulation to
model the $(\alpha,\mathbf{\lambda})$-FMCLP. We will present different
formulations for the problem for both the discrete case ($\mathcal{X}$ being a
finite pre-specified set) and the continuous case
($\mathcal{X}=\mathbb{R}^{d}$). The nature of the domain of this problem
directly affect the development of resolution strategies for it.

A general formulation for the problem considers the following decision
variables:

$x_{ik}=\begin{cases}1,&\text{if node $a_{i}$ is covered by the $k$-th
selected facility in $\mathcal{X}$},\\\
0,&\text{otherwise},\end{cases}\quad\text{ for all }i\in N,\>k\in P.$

and $X_{k}\in\mathbb{R}^{d}$: coordinates of the $k$-th selected facility in
$\mathcal{X}$.

(${\rm FMCLP}_{\alpha,\mathbf{\lambda}}$) can be formulated as follows:

$\displaystyle\max\;$ $\displaystyle
F_{\alpha,\mathbf{\lambda}}\Big{(}\displaystyle\sum_{i\in
N}\omega_{i}x_{i1},\ldots,\displaystyle\sum_{i\in N}\omega_{i}x_{ip}\Big{)}$
(1a) s.t. $\displaystyle\displaystyle\sum_{k\in P}x_{ik}\leq 1,\;\forall i\in
N,$ (1b) $\displaystyle a_{i}\in\mathbb{B}(X_{k})\mbox{ if
$x_{ik}=1$},\;\forall i\in N,k\in P,$ (1c) $\displaystyle
x_{ik}\in\\{0,1\\},\;\forall i\in N,k\in P,$ (1d) $\displaystyle
X_{k}\in\mathcal{X},\;\forall k\in P,$ (1e)

The objective function aims to maximize the
$(\alpha,\mathbf{\lambda})$-fairness of the demand coverage by each of the
facilities. Constraint (1b) assures that each covered demand point is counted
at most once as covered. (1c) ensures the adequate definition of the
$x$-variables.

The feasible set of the problem above will described then by a set of linear
and second-order cone representable inequalities on binary and continuous
variables. One of the main difficulties of the model above stems on the
representation of the objective function $F_{\alpha,\lambda}$ which consists
of the following two ingredients:

Sorting::

Representing the order given by the OWA operator into an optimization problem
is a difficult challenge. In Section 2.2 we describe the two most popular
formulations for this operator on the values $W_{k}^{1-\alpha}$ for $k\in P$.
A third representation is based on the $x$-variables that we consider in our
problem, by sorting the selected facilities in $\mathcal{X}$ in non-increasing
order of the demand coverage, i.e, enforcing the following constraints:

$\displaystyle\sum_{i\in N}\omega_{i}x_{ik}\leq\displaystyle\sum_{i\in
N}\omega_{i}x_{ik+1},\;\forall k\in P,$ (Sorting)

$(1-\alpha)$-powers::

Observe that the $(1-\alpha)$ powers of the coverage of each facility appear
in the objective function. Denoting by $W_{k}=\displaystyle\sum_{i\in
N}\omega_{i}x_{ik}$ and by $Z_{k}=W_{k}^{1-\alpha}$, for $k\in P$, assuming
that $W_{1}\leq\ldots\leq W_{p}$, the objective function above can be written
as the linear function:

$F_{\alpha,\mathbf{\lambda}}(W_{1},\ldots,W_{p})=\frac{1}{1-\alpha}\displaystyle\sum_{k\in
P}\lambda_{k}Z_{k}$

as long as it is fulfilled that $Z_{k}\leq W_{k}^{1-\alpha}$ (for $\alpha<1$)
or $Z_{k}\geq W_{k}^{1-\alpha}$ (for $\alpha>1$) for all $k\in P$. Assuming
that $\alpha$ is rational, we get that there exists $p,q\in\mathbb{Z}_{+}$
with $p\geq q\geq 1$ and $\gcd(p,q)=1$ such that:

$\frac{1}{1-\alpha}=\begin{cases}\frac{p}{q}&\mbox{if $\alpha<1$},\\\
-\frac{q}{p}&\mbox{if $\alpha>1$}\end{cases}$

Thus, the power-constraints can be rewritten as:

$Z_{k}^{p}\leq W_{k}^{q},\text{for all $k\in P$}.$ (Powers)

These constraints can be conveniently rewritten as a set of quadratic second-
order cone constraints following a simplification of the results in Blanco et
al. (2014).

In the rest of the section we describe how to represent constraints (1c) in a
suitable mathematical optimization formulation. This representation highly
depends on the nature of the set of potential coordinates for the facilities
$\mathcal{X}$. We analyze the cases in which $\mathcal{X}$ is a finite set and
the one where $\mathcal{X}=\mathbb{R}^{d}$.

###  4.1. Continuous facilities

We analyze here the case where the potential set from which the coordinates of
the services are chosen is the entire space, i.e.,
$\mathcal{X}=\mathbb{R}^{d}$. In this case, the norm-based covering contraints
(1c) can be rewritten as

$\|X_{k}-a_{i}\|\leq R+U_{i}(1-x_{ik}),\forall i\in N,k\in P.$ (Norms)

where $U_{i}$ is a big enough constant ($U_{i}>\|a_{i}-a_{i^{\prime}}\|$ for
all $i^{\prime}\in N$). It ensures that in case $i$ is allocated to the $k$th
selected facility ($X_{k}$), then $a_{i}$ must belong to
$\mathbb{B}_{R}(X_{k})$.

In case $\|\cdot\|$ is the Euclidean norm, it is well known that these
constraints can be re-written as the following set of linear and second-order
cone inequalities:

$\displaystyle v_{ikl}\geq a_{il}-X_{kl},$ $\displaystyle\>\forall i\in
N,\>\forall k\in P,$ (2a) $\displaystyle v_{ikl}\geq-a_{il}+X_{kl},$
$\displaystyle\>\forall i\in N,\>\forall k\in P,$ (2b) $\displaystyle
s_{ik}\geq\displaystyle\sum_{l=1}^{d}v_{ikl}^{2},$ $\displaystyle\>\forall
i\in N,\>\forall k\in P,$ (2c) $\displaystyle s_{ik}\leq R+U_{i}\,(1-x_{ik}),$
$\displaystyle\>\forall i\in N,\>\forall k\in P,$ (2d) $\displaystyle
v_{ikl}\geq 0,$ $\displaystyle\>\forall i\in N,\>\forall k\in P,$ (2e)
$\displaystyle s_{ik}\geq 0,$ $\displaystyle\>\forall i\in N,\>\forall k\in
P.$ (2f)

where $v_{ikl}$ represents (via (2a) and (2b)) the absolute value
$|a_{il}-X_{kl}|$ (here $a_{il}$ and $X^{t}_{kl}$ stand for the $l$-th
coordinate of demand point $a_{i}$ and the $k$-th facility, respectively) and
$s_{ik}$ determines the euclidean distance (by (2c)) as the squared sum of the
absolute differences between the coordinates of $a_{i}$ and $X_{k}$.
Consequently, the problem simplifies to a mixed-integer second-order cone
optimization (MISOCO) problem, which can be solved using any of the available
off-the-shelf solver.

Actually, by the results in Blanco et al. (2014), one can also consider block-
norm based distances (deriving linear optimization models) or
$\ell_{\tau}$-norms (with $\tau\geq 1$) inducing again mixed-integer second-
order cone optimization problems.

As already observed in other maximal covering location problems (see Blanco
and Gázquez (2021); Blanco et al. (2022)) the norm-based constraints (Norms)
can also be rewritten as linear constraints. This linearization is based on
projecting out the $X$-variables by ensuring that these can be constructed
from the $x$-variables. The observation that will allow this formulation is a
direct consequence of the following lemma whose proof is straightforward:

######  Lemma 1.

Let $N_{1},\dots,N_{p}\subseteq N$ be $p$ nonempty disjoint subsets of $N$.
Then, there exists $X_{1},\ldots,X_{p}\in\mathbb{R}^{d}$ such that the points
of $N_{k}$ are covered by $X_{k}$, for all $k\in P$, if and only if

$\bigcap_{i\in N_{k}}\mathbb{B}_{R}(a_{i})\neq\emptyset,\forall k\in P.$

From this result, (Norms) can be replaced by:

$\displaystyle\sum_{i\in Q}x_{ik}\leq|Q|-1,\forall k\in P,\>\forall Q\subseteq
N:\bigcap_{i\in Q}\mathbb{B}_{R}(a_{i})=\emptyset,\\\ $ (3)

These exponentially many constraints assure that those subsets of incompatible
demand points cannot be covered by the same facility. Thus, if a solution in
the $x$-variables, $\bar{x}$, fullfilling (3), we can construct the
coordinates of the services that must verify

$X_{k}\in\bigcap_{i\in N:\atop\bar{x}_{ik}=1}\mathbb{B}_{R}(a_{i}),\text{for
all $k\in P$. }$

Actually, these coordinates can be found (in poly-time) for a given feasible
solution $\bar{x}$ either solving a second order cone optimization problem or
by solving a one-center facility location problem. This strategy has been
successfully applied in the literature by Blanco and Gázquez (2021); Blanco et
al. (2022).

Nevertheless, the above formulation requires incorporating the exponentially
many constraints (3). However, they can be simplified reducing from
exponential to polynomial many constraints by means of Helly’s Theorem (Helly,
1923) (see also Danzer et al., 1963). Invoking that result, only intersections
of $(d+1)$-wise balls are needed to check, allowing to replace Constraints (3)
by:

$\displaystyle\sum_{i\in Q}x_{ik}\leq|Q|-1,\forall k\in P,\>\forall Q\subseteq
N:\bigcap_{i\in Q}\mathbb{B}_{R}(a_{i})=\emptyset,\text{ with }|Q|=d+1.\\\ $
(4)

Despite this simplification, the number of constraints is still $O(n^{d})$
making the problem difficult to solve. Then, we propose to consider an
incomplete formulation (removing (4)) and iteratively incorporating these
constraints on-the-fly, as needed.

The selection of the constraints to incorporate in each iteration is found
using the following separation strategy: After solving the problem with none
or part of the constraints (4) a solution, say $\bar{\mathbf{x}}$, is
obtained. Then, for each $k\in P$ the define set $Q_{k}=\\{i\in
N:\bar{x}_{ik}=1\\}$. One can check for the validity of the set $Q_{k}$ as a
feasible cluster of demand points for our problem by solving the $1$-center
problem for the points in such a set. In case the optimal coverage radius
obtained is less than or equal to $R$, one can conclude that $Q_{k}$ is a
valid subset of demand points that can be covered by the same server.
Otherwise, the solution violates the relaxed constraints, and thus we add the
cut

$\displaystyle\sum_{i\in Q_{k}}x_{ik^{\prime}}\leq|Q_{k}|-1,\forall
k^{\prime}\in P,$ (5)

to ensure that such a solution is no further deemed feasible and thus obtained
again.

The $1$-center problem with Euclidean distances on the plane is known to be
solvable in polynomial time (see e.g., Elzinga and Hearn, 1972). Extensions to
higher dimensional spaces and generalized covering shapes have been recently
proved to be also poly-time solvable (Blanco and Puerto, 2021).

###  4.2. Discrete facilities

Let us assume that the potential set of facilities if finite, that is,
$\mathcal{X}=\\{b_{1},\ldots,b_{m}\\}\subseteq\mathbb{R}^{d}$. We denote by
$M=\\{1,\ldots,m\\}$ its index set. The model in this case can be simplified
taking into account that the subset of potential facilities that are able to
cover each single demand point can be pre-computed. It allows also to avoid
the use of the $X$-variables, replacing them by the following decision
variables to decide which of the potential facilities from
$\\{b_{1},\ldots,b_{m}\\}$ are open and which is the position of the demand
covered by each facility in the ordered vector.

$y_{jk}=\begin{cases}1,&\text{if the covered demand of facility $j$ is the
$k$-th largest},\\\ 0,&\text{otherwise},\end{cases}\quad\text{ for all }j\in
M,\>k\in P.$

Then, (1) can be alternatively formulated as:

$\displaystyle\max\;$ $\displaystyle\frac{1}{1-\alpha}\displaystyle\sum_{k\in
P}\lambda_{k}Z_{k}$ (6a) s.t.
$\displaystyle\eqref{eq:fmclp2},\eqref{sorting},\eqref{powers}$ (6b)
$\displaystyle
x_{ik}\leq\displaystyle\sum_{j\in\mathcal{C}(i)}y_{jk},\;\forall i\in N,k\in
P,$ (6c) $\displaystyle\displaystyle\sum_{j\in M}y_{jk}=1,\;\forall k\in P,$
(6d) $\displaystyle\displaystyle\sum_{k\in P}y_{jk}\leq 1,\;\forall j\in M,$
(6e) $\displaystyle x_{ik}\in\\{0,1\\},\;\forall i\in N,\forall k\in P,$ (6f)
$\displaystyle y_{jk}\in\\{0,1\\},\;\forall j\in M,k\in P.$ (6g)

Apart from (6b), the covering constraints (1c) are rewritten using the
$y$-variables using (6c)–(6e). Constraints (6c) assure that the demand points
can be assigned to a facility if it is sorted in any position (equivalently,
if it is open). Constraints (6d) enforces that a single facility is assigned
to a position and (6e) that at most one position is assigned to a facility
(those facilities not assigned to a positions will be not open). Both
constraints together with (Sorting) assure that exactly $p$ facilities are
open, each of them in a different order in the coverage sequence.

The classical formulations for the MCLP use one-index binary variables to
determine the _open_ facilities. Note that in our case it is not enough, since
the positions of the activated services in the sorting coverage sequence are
needed to allow the allocation of the demand points to a facility only in case
it is open.

##  5\. Computational study

In this section, we report the results of the computational experience that we
have run in order to analyze the performance of the proposed methoddologies.

We consider the dataset provided in Orlandini (2019) where the coordinates
(latitude and longitude) of Residential Schools and student hostels operated
by the federal government in Canada are detailed. The coordinates of this
dataset have been normalized in the unit square.

From the whole dataset we have constructed different instances with sizes,
$n$, ranging in $\\{45,90,120,179\\}$ (the first $n$ demand nodes in the
complete instance). The demands have been randomly generated by a uniform
distribution in $(0,1)$. The set of generated instances are available in the
github repository github.com/vblancoOR/fmclp. The number of facilities to be
located, $p$, ranges in $\\{5,10,15,20\\}$, and we consider the radius for the
services, $R$, ranging in $\\{0.1,0.15\\}$.

We run our models by choosing the $\lambda$-parameters of the OWA operator in
$\\{\texttt{W},\texttt{C},\texttt{K},\texttt{D},\texttt{G},\texttt{H}\\}$ (see
Table 1, where C \- Minimum, D \- $\alpha-$Min-Average, G \- Gini, H \-
Harmonic, K \- $k-$Average, and W \- Average). The $\alpha$-parameter was
chosen in $\\{0,0.5,1,2\\}$ except for problems under the C-objective, where
the value of $\alpha$ does not affect the result, and we only solved the
instances for $\alpha=0$.

Finally, we have solved all these instances for the two types of domains:
discrete and continuous (we refer them in the rest of the currently section as
Disc and Cont). With that, a total of 672 instances for each type of location
space (a total of 1344 instances) have been solved in our computational study.
For the continuous MCLP, we observed that the nonlinear formulation (Norms)
have a worse performance than the linear formulations proposed in the
literature (see Blanco and Gázquez, 2021; Blanco et al., 2022), as expected.
Thus, we provide the results of using formulation (5) in our experiments.

All the experiments have been run on a virtual machine in a physical server
equipped with 8 threads from a processor AMD EPYC 7402P 24-Core Processor, 64
Gb of RAM and running a 64-bit Linux operating system. The models were coded
in Python 3.7 and we used Gurobi 9.1 as optimization solver. A time limit of 2
hour was fixed for all the instances.

We analyze in this section the computational performance of our approach as
well as the managerial insights of the obtained solutions in terms of fairness
and efficiency.

###  5.1. Computational performance

In order to analyze the computational difficulty of the proposed approach, in
Table 2 we summarize the obtained results in terms of consumed CPU time, MIP
Gap and unsolved intances. In column Time we show the average required
computational times for those instances solved up to optimality within the
time limit (2 hours). In column GAP0 we report the percent of instances that
were solved in the time limit. The rest of the columns show the information
for those instances not optimally solved. GAP1 shows the percentage of
unsolved instances but MIPGap less than $1\%$, whereas GAP5 is the average
percentage of instances that are with MIPGap in $(1\%,5\%]$. Finally in column
GAP unsolved, we show the average MIPGap of the instances that we are not able
to solve within the time limit. All the results are averaged for each fixed
value of $n$ (number of demand points), $p$ (number of facilities to locate)
and the type of location space (Disc and Cont).

The first observation is that the continuous problems are more computationally
challenging to solve than the discrete ones. Even being both problems non
linear in most of the cases because of the $\alpha$-fairness objective
function, the distance-based constraints needed for the continuous problems
make those problems much more difficult to solve (in terms of CPU time and
unsolved instances).

The table also highlights the percentage of solved instances. As expected, as
the number of facilities increases, the number of solved instances decreases.
The MIPGap, however, are relatively small, being around 75% the instances that
were solved within the time limit, with less than 5% of MIPGaps, 55% being
with less than 1% MIPGap. The unique exception are the instances with $n=45$
and $p=20$. In those instances, a $100\%$-coverage can be obtained in all
cases, but the number of feasible allocation is huge, being the underlying
combinatorial problem very hard to solve.

$n$ $p$ CPUTime (secs.) GAP0 GAP1 GAP5 GAP unsolved Disc Cont Disc Cont Disc
Cont Disc Cont Disc Cont 45 5 2.00 25.50 100% 100% 100% 100% 100% 100% 0% 0%
10 1672.71 1557.52 71.43% 42.86% 80.95% 71.43% 85.71% 83.33% 9.44% 4.70% 15
192.04 159.60 11.90% 11.90% 28.57% 38.10% 54.76% 78.57% 44.68% 6.09% 20 66.41
524.13 9.52% 9.52% 19.05% 23.81% 30.95% 57.14% 67.15% 60.52% 90 5 142.73
329.32 100% 80.95% 100% 100% 100% 100% 0% 0.03% 10 366.22 2003.43 61.90%
33.33% 97.62% 83.33% 100% 97.62% 0.24% 0.98% 15 1174.07 191.58 21.43% 11.90%
69.05% 35.71% 80.95% 92.86% 3.13% 2.10% 20 56.66 1487.77 9.52% 7.14% 21.43%
16.67% 40.48% 38.10% 14.85% 7.74% 120 5 41.27 850.91 85.71% 83.33% 100% 100%
100% 100% 0.09% 0.17% 10 1269.24 275.26 59.52% 9.52% 95.24% 50.00% 97.62%
88.1% 0.52% 2.36% 15 958.53 868.72 30.95% 14.29% 78.57% 21.43% 85.71% 80.95%
2.05% 3.44% 20 51.28 445.71 9.52% 9.52% 30.95% 11.90% 66.67% 64.29% 7.96%
4.47% 179 5 102.51 2306.03 78.57% 57.14% 100% 69.05% 100% 100% 0.03% 1.73% 10
1897.44 TL 38.10% 0% 78.57% 9.52% 100% 28.57% 0.79% 10.85% 15 1642.58 82.43
7.14% 2.38% 28.57% 9.52% 73.81% 45.24% 3.26% 23.16% 20 2224.03 17.19 7.14%
4.76% 14.29% 7.14% 45.24% 42.86% 6.45% 7.38%

Table 2. Average CPU time (CPUTime), percentage of instances solved within
time limit (GAP0), percentage of instances solved with less than 1% and 5% of
gap (GAP0 and GAP5), and average of MIP GAP of unsolved instances (GAP
unsolved) averaged by size $n$ and number of centers $p$ for the discrete
location problem.

In Figure 2 we show the results of columns GAP0, GAP1 and GAP5. Each bar
represents all the instances for each combination of $n$, $p$ and Disc/Cont
location problem. Each bar is partitioned into the four possible sets:
optimally solved within the time limit (GAP0), unsolved but with MIPGap less
than $1\%$ (GAP1), unsolved by with MIPGap in $(1\%,5\%]$ (GAP5) and unsolved
with MIPGap greater than $5\%$ (GAP+).

Although percentage of solved instances within the time limit decreases as $p$
increases, the percentage of instances that have less than a 5% gap is stable
for each value of $n$. In addition we observed for values of $p\leq 10$, we
were able to solve in most of the cases with less than one 5% for the two
types of location spaces.

Figure 2. Percentage of instances optimally solved within the time limit
(GAP0), unsolved but with MIPGap less than $1\%$ (GAP1), unsolved by with
MIPGap in $(1\%,5\%]$ (GAP5) and unsolved with MIPGap greater than $5\%$
(GAP+) averaged for each fixed value of $n$ (number of demand points), $p$
(number of facilities to locate) and the type of location space (Disc and
Cont).

In Figure 3 we show a similar table but averaging by the type of problem
($\lambda$) and by the values of $\alpha$. We observe that C and W were less
computationally demanding than the others, with close to $60\%$ of the
instances optimally solved. We also observed that around $70\%$ of the
instances are solved with less than a 5% gap.

However, for the value of $\alpha=1$ the figure shows a higher percentage of
instances with gap greater than 5%, caused by the piecewise linear
representation of Gurobi of the logarithmic function, which requires a new set
of variables and constraints in the problem.

(a) By type of problem $\lambda$.

(b) By value of $\alpha$.

Figure 3. Percentage of instances optimally solved within the time limit
(GAP0), unsolved but with MIPGap less than $1\%$ (GAP1), unsolved by with
MIPGap in $(1\%,5\%]$ (GAP5) and unsolved with MIPGap greater than $5\%$
(GAP+) averaged for parameters involved in the problem $\lambda$–problem type
and $\alpha$.

###  5.2. Managerial insights

In what follows we analyze the quality of the obtained solutions with the
different approaches in terms of fairness and efficiency. As noted above, the
MIPGaps for the instances unsolved within the time limit is small, being most
of the instances assumed to be optimally solved (with certain degree of
accuracy). In order to show the quality of the solutions, we consider those
instances with a MIPGap smaller than 5%.

Figures 4 and 5 show the values of the price of fairness (PoF) and the price
of efficiency (PoE) defined in Section 3 for the discrete and continuous
location problems, respectively. These values are shown for each type of
problem ($\lambda$) considered and for each value of $\alpha$. The highest
values of the PoF are obtained with C (the one for which PoE is the smallest)
whereas the highest values of the PoE are obtained with W (the one with
smallest PoF). However, when the parameter $\alpha$ increases, we obtain a
range of fair solutions between the classical MCLP
($\alpha=0,\;\lambda=\texttt{W}$) and the maximin approach (C), balancing
efficiency and fairness.

The performance of the solutions for each of the problem types $\lambda$ seems
to indicate a relationship between the PoF and the PoE, in the sense that the
largest PoF is related to the smallest PoE (or vice versa) as can be seen for
C (or W). Nevertheless, this is not true in general. For instance, in the
discrete case (Figure 4), we got that D have a value of PoF between those
obtained by G and K but its PoE is below K. That is, the solution for D is
better than K both in efficiency and fairness.

Finally, for the discrete location problem we can conclude that the trade-off
between fairness and efficiency is promising when considering as
$\lambda\in\\{\texttt{D},\texttt{G}\\}$, obtaining fairer solutions by
increasing the value of $\alpha$. If one is willing to lose more efficiency to
gain fairness the better operator seems to be the harmonic (H).

(a) Price of fairness (PoF).

(b) Price of efficiency (PoE).

Figure 4. PoF and PoE averaged for each value of $\alpha$ for Discrete
location problem.

For the continuous problems, similar conclusions are derived. Specifically,
the harmonic operator (H) seems to provide fair solutions with values of PoE
close to zero, but still with not so high values of PoF as C.

(a) Price of fairness (PoF).

(b) Price of efficiency (PoE).

Figure 5. PoF and PoE averaged for each value of $\alpha$ for Continuous
location problem.

These graphs show that for different values the decision maker has a wide
range of possibilities to decide in choosing between the fairest or the one
that covers the most. This is the first conclusion that we can draw from the
results and that verifies the starting hypothesis, a new fairness measurement
function for allocation problems.

More detailed graphs of the averaged results can be found in A.

One can draw similar conclusions to those presented above. Nevertheless, one
can draw other interesting observations from the averaged results. For
instance, in Figure 8, for $n=45$, a breakpoint can be observed for H in
$\alpha=1$. Observe that for small values of $n$ and large value of $p$, only
a few instances were solved with MIPGap less than 5%, and then, the maximum
averaged value of PoF is reached.

Another behavior that needs further explanation is for when we average by $p$
for both type of location spaces, and for both PoF and PoE. For $p=20$, we
observe that for $\lambda\in\\{\texttt{H},\texttt{K}\\}$ the graphs drop to 0
for $\alpha\in\\{1,2\\}$ in discrete case and for $\lambda=\texttt{G}$ in
continuous case. In this case, none of the instances were solved with MIPGap
under $\%5$. Moreover, for the continuous case with $p=20$, Figure 14 shows
the PoF obtained for H is 0 for all $\alpha$ values, i.e., it fails to improve
the classical MCLP solution, but the PoE shows in Figure 15 that when
$\alpha\in\\{0,0.5\\}$ a better solution in terms of efficiency was obtained,
even if the coverage is equal. This empirically proof that one can find
alternative maximal coverage solutions with different allocation of the users
of customers, one of them fairer than others.

We have also analyzed the behavior of the solutions obtained with our
approaches with respect to the Gini index (Section 3). The Gini index measures
the global pairwise envy of the different facilities (in this case the larger
the difference in covered demand between two facilities, the larger the Gini
index). In Figure 6 we show the overall performance of all the solved
instances in terms of the average Gini indices for each of the used
$\lambda$-weights and each of the values of $\alpha$. Observe that the
behavior is very similar to PoE. Again, the C models outperforms the rest of
models for all $\alpha$, while W exhibits the worse behavior, being the
maximum-envy solution. In the middle, D provides a trade off and H has also a
better performance than W. For the continuous location case, both K and H have
similar Gini indices than C, in contrast to the PoE. This is because the
difference between the facilities is the same as C and therefore minimal, but
does not reach the minimum coverage.

(a) Disc location problem.

(b) Cont location problem.

Figure 6. Gini averaged for each value of $\alpha$ for both types of location
spaces.

Finally, in Figure 7 we illustrate different optimal solutions obtained for a
selected instance for different values of ($\lambda$, $\alpha$) of fairness
measure. From left top to right bottom we draw: (C, any $\alpha$),
$(\texttt{K},1)$, $(\texttt{D},0.5)$, $(\texttt{D},0)$, $(\texttt{W},0.5)$ and
$(\texttt{W},0)$. The reader can observe that, geometrically, a greater
concentration of facilities is obtained for fairer solutions, being not only
the position of the facilities the factor that affect the fairness of a
solution, but also the allocation of the users to them, which can be more
easily when different facilities are able to cover the same demand points.

(a) C (62.46%)

(b) K and $\alpha=1$ (67.18%)

(c) D and $\alpha=0.5$ (67.79%)

(d) D and $\alpha=0$ (70.02%)

(e) W and $\alpha=0.5$ (73.13%)

(f) W and $\alpha=0$ (74.09%)

Figure 7. Example of fairness distribution for some results when $n=179$ and
$p=10$, and in parenthesis the weighted covered percentage.

##  6\. Conclusions and further research

We present in this paper a novel fairness measure for Maximal Covering
Location Problems, that combines the OWA operators, early introduced by Yager
(1988), and the $\alpha$-fairness scheme introduced by Atkinson (1970).

We develop suitable mathematical optimization models that allow to capture the
notion of fairness in the MCLP for the two main types of location spaces that
are studied in the literature: the discrete and continuous. The models are
then reformulated as MISOCO problem, using the geometrical insights of the
problem, and then, the programs that we propose are able to be solved with the
available off-the-shelf software.

We have tested our models using a real data set containing the locations of
residential schools and student hostels in Canada. Applying our new scheme to
this dataset, we empirically observed that our models provide different
solutions in terms of fairness than using the OWA operators and the
$\alpha$-fairness scheme separately. Therefore, we conclude that the
combination of both in the same allocation scheme provides the decision maker
with a wide range of options to find a trade-off between fairness and
efficiency.

Further research lines on fairness topic include the using of this combination
of schemes into different location problems, as the set covering problem,
location problems with capacities and also its incorporation to queue problems
where the facilities has to deal with the management of waiting users when the
facility is saturated, and fair solutions of the problem could be a
successfully too to deal with this problematic.

## Acknowledgments

This research has been partially supported by Spanish Ministerio de Ciencia e
Innovación, AEI/FEDER grant number PID2020-114594GBC21, Junta de Andalucía
projects P18-FR-1422/2369 and projects FEDERUS-1256951, B-FQM-322-UGR20,
CEI-3-FQM331 and NetmeetData (Fundación BBVA 2019). The first author was also
partially supported by the IMAG-Maria de Maeztu grant CEX2020-001105-M /AEI
/10.13039/501100011033. The second author was also partially supported by
Spanish Ministry of Education and Science grant number PEJ2018-002962-A, and
the PhD Program in Mathematics at the Universidad de Granada.

##  Appendix A Discussions

(a) $n=45$.

(b) $n=90$.

(c) $n=120$.

(d) $n=179$.

Figure 8. Price of fairness averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$n\in\\{45,90,120,179\\}$ Discrete location problem.

(a) $n=45$.

(b) $n=90$.

(c) $n=120$.

(d) $n=179$.

Figure 9. Price of efficiency averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$n\in\\{45,90,120,179\\}$ for Discrete location problem.

(a) $p=5$.

(b) $p=10$.

(c) $p=15$.

(d) $p=20$.

Figure 10. Price of fairness averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$p\in\\{5,10,15,20\\}$ Discrete location problem.

(a) $p=5$.

(b) $p=10$.

(c) $p=15$.

(d) $p=20$.

Figure 11. Price of efficiency averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$p\in\\{5,10,15,20\\}$ for Discrete location problem.

(a) $n=45$.

(b) $n=90$.

(c) $n=120$.

(d) $n=179$.

Figure 12. Price of fairness averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$n\in\\{45,90,120,179\\}$ Continuous location problem.

(a) $n=45$.

(b) $n=90$.

(c) $n=120$.

(d) $n=179$.

Figure 13. Price of efficiency averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$n\in\\{45,90,120,179\\}$ for Continuous location problem.

(a) $p=5$.

(b) $p=10$.

(c) $p=15$.

(d) $p=20$.

Figure 14. Price of fairness averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$p\in\\{5,10,15,20\\}$ Continuous location problem.

(a) $p=5$.

(b) $p=10$.

(c) $p=15$.

(d) $p=20$.

Figure 15. Price of efficiency averaged by $\alpha\in\\{0,0.5,1,2\\}$ and
$p\in\\{5,10,15,20\\}$ for Continuous location problem.

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