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Equity in Stochastic Healthcare Facility Location

We consider issues of equity in stochastic facility location models for healthcare applications. We explore how uncertainty exacerbates inequity and examine several equity measures that can be used for stochastic healthcare location modeling. We analyze the limited literature on this subject and highlight areas of opportunity for developing tractable, reliable, and data-driven approaches that might be applicable within and outside healthcare operations. Our primary focus is on exploring various ways to model uncertainty, equity, and facility location, including modeling aspects (e.g., tractability and accuracy) and outcomes (e.g., equity/fairness/access performance metrics vs. traditional metrics like cost and service levels).


Introduction
Equity concerns naturally arise in many real-life applications (e.g., healthcare scheduling, facility location, disaster response operations, air traffic control, etc.), and it is crucial to address these concerns for the proposed solutions to be applicable, equitable, and acceptable. However, accounting for equity is a complex task primarily because there is no unique notion of equity that is generally accepted; the definition of equity often depends on the application. Moreover, uncertainty, an intrinsic property of real-life applications, interacts with equity in complicated and poorly understood ways. In particular, uncertainty increases the complexity of quantifying the loss in efficiency (e.g., minimizing the operational cost of vaccination clinics) incurred in the pursuit of equity (e.g., equitably allocating mobile vaccination clinics across geographical areas). In addition, deterministic models of most real-world problems are often challenging. Thus, incorporating both uncertainty and equity metrics or constraints may increase the complexity of these problems.
In this chapter, we consider issues of equity in stochastic facility location models for healthcare applications. The main motivation behind the attempt to establish equity in healthcare in generaland the need for inequity-averse models for healthcare facility (HCF) location in particular-is, of course, an ethical one: humans have equal rights, and therefore, nobody should be discriminated against by inequitable access to healthcare services or distribution of healthcare services. Another more pragmatic motivation for striving for equitable HCF location-allocation solutions is the need to avoid adverse health outcomes of vulnerable populations who often do not have proper or equal access to HCF (Gutjahr and Fischer, 2018). Finally, extending classical models with an emphasis on equity and equity-uncertainty interaction is practically relevant and technically interesting.
We first analyze different aspects and measures of equity in the literature. Then, we analyze recent static and mobile HCF location models to explore various ways to model uncertainty, equity, and facility location, including modeling aspects (e.g., tractability and accuracy) and outcomes (e.g., equity-, fairness-, or access-based performance metrics vs. traditional metrics like cost and service levels). Our goal is not to provide a comprehensive survey; instead, our goal is to highlight the issue of equity and access and the need for data-driven and tractable models to address emerging stochastic HCF problems.
The remainder of this chapter is organized as follows: In Section 2, we analyze existing definitions and metrics of equity as well as common methods to model these metrics. In Section 3, we briefly discuss the challenges of incorporating uncertainty and equity and demonstrate through a simple example that uncertainty and equity interact in ways that should not be ignored. In Section 4, we provide a high-level analysis of recent stochastic HCF location literature with a particular focus on studies that proposed and analyzed inequity-averse approaches. Maybe not surprisingly, and sadly, this analysis reveals that there is limited literature considering equity and equity-uncertainty interaction. Nevertheless, there are many opportunities to use the powerful tools of operations research to address equity concerns in emerging HCF location problems and derive inequity-averse stochastic HCF location approaches. In Section 5, we present some future research opportunities and open questions.

What is Equity, Anyway?
To account for equity (often synonymously called fairness) in facility location or any other decisions, one first needs to provide an exact meaning of equity. Despite the importance of the subject, there is no unique notion (definition) of equity that is generally accepted. Instead, there is a wide variety of notions of equity and fairness in the economics and decision theory literature that depend on the context. The primary concern for equity in resource allocation is treating entities fairly, such that everyone receives the same level of service and no one is at a disadvantage. The allocated resource(s) can be a particular good, or bad, or a chance of good or bad. The entities can be a population, group of people at some location or belonging to some social classes or organizations, etc. In general, most equity literature aims at equal distribution of benefits or disutilities between entities (Mostajabdaveh et al., 2019).
Although there is no single equity concept that we can use to design inequity-averse HCF location models, there are four key areas of equity research for health systems (Cardoso et al., 2015(Cardoso et al., , 2016. These are: • Equity of access: Informally, accessibility is the relative ease by which patients can reach a healthcare facility from a given location (Hawthorne and Kwan, 2013;Jin et al., 2015;Wang, 2012). Thus, patients should receive the care they need as close as possible to their place of residence or employment. In the case of emergency services, accessibility is the ability of a healthcare provider to reach the patients. Accessibility measures include both spatial and non-spatial factors (Wang, 2012). Spatial factors include the spatial separation between supply (e.g., surgical centers) and demand (e.g., patient population needing surgical care) and how they are connected in space. Thus it is a classical aspect in location analysis. Non-spatial factors include demographic (e.g., age, gender, sex, etc.) and socioeconomic (e.g., income, poverty, female-headed households, etc.) variables, which also vary across geographical areas.
• Equity of utilization: Utilization refers to the satisfied demand for different services. Ensuring equity of utilization means providing roughly equal service levels across services. Note that this diverges from the concept of "deliver the cheapest service" often observed in location models that seek cost minimization.
• Socioeconomic equity: Socioeconomic equity stipulates that the unsatisfied demand for population groups with lower income should not be greater than that of groups with higher income.
Decision-makers may also want to ensure that unmet demand for lower-income or vulnerable population groups is sufficiently low.
• Geographical equity: Geographical equity refers to the ability of the system to provide relatively equal levels of unsatisfied (satisfied) demand across geographical areas. Decision-makers often want to ensure that unsatisfied demand is not vastly different across geographical areas, or that some geographic areas do not lack healthcare service entirely.
Remark 1. Like any other metrics in any optimization problem, these four metrics can be targeted either by putting them in the objective function or in the constraints. For example, we might minimize the disparity in unsatisfied demand across income levels (socioeconomic equity) and/or add a constraint requiring the unsatisfied demand in any geographic area to be no more than a fixed maximum level (geographical equity).
Equity research in other application domains additionally considers social equity and diversity.
The social equity concept quantifies equity based on how any good received is proportional to the need (Levinson, 2010). For example, the volume of the demand for a particular health service may differ among demand nodes in rural and urban areas. If only a fraction of the demand can be satisfied, measures such as the proportion of the satisfied demand can be used to measure equity and service quality (Karsu and Morton, 2015). Diversity is another concept that is indirectly related to equity. Suppose, for example, that we want to select a set of locations to open vaccination centers.
The decision-maker may have concerns about diversity because they want certain population groups to have a certain degree of coverage or access to vaccination by the chosen location. One way to achieve this is to use quotas, i.e., ensuring that a certain proportion of the vaccination centers will be located to cover the groups of concern (Karsu and Morton, 2015). This approach treats people with different characteristics differently, such that the selected locations are diversified enough in the sense that they cover diverse groups of concerns.
In contrast to most of the HCF literature (see Section 4 and Ahmadi-Javid et al. (2017)), inequity measurement has found explicit and extensive consideration in the economic and decision theory literature and a few discussions in the humanitarian logistics literature. The commonly accepted theme is that there is no one-size-fits-all solution to ensure equity, and customized methods are needed to measure and handle application-specific equity concerns. Using transparent and explicit criteria that determine what is equitable and what is not is useful in ensuring that the decisions are acceptable, equitable, and implementable in practice.
There are also different operations research (OR) methods and metrics for incorporating equity in the decision process. The precise interpretation of each depends on both the structure of the problem at hand and the decision maker's understanding of equity (Karsu and Morton, 2015). Karsu and Morton (2015) give a comprehensive and deep survey on the use of equity concepts connected with optimization models. The authors distinguish between two types of inequity-averse optimization approaches, addressing "equitability" and "balance." Equitability is a concern whenever one deals with the allocation of goods across indistinguishable entities. In contrast, balance is a concern whenever goods are distributed across entities with different needs, claims, or preferences.
In the following subsections, we provide a high-level overview and analysis of the most common equity measures detailed in Karsu and Morton (2015).

The Rawlsian approach
The Rawlsian approach (Rawls, 1999) is one of the oldest, most common, and simplistic approaches used in OR to incorporate fairness or equity in optimization models. This approach represents equity preference by focusing on the worst-off entity, i.e., the minimum outcome level in a distribution. One can enforce a constraint that ensures that the minimum outcome is larger than a predefined level or seek to maximize the minimum outcome. For example, in the p-center problem, we seek to locate p facilities to minimize the maximum distance between any demand point and its nearest facility. As such, each demand point is assigned to the nearest facility, ensuring that each demand node is covered. Although the notion of coverage is well established in healthcare applications, in many HCF applications, we are often interested in the average distance that a patient has to travel to receive health service or the average distance that healthcare providers must travel to reach their patients (Daskin and Dean, 2005). Some studies extend the Rawlsian approach to a maximum lexicographic approach. That is, the welfare of the worst-off is first maximized subject to resource and other constraints, then the second worst-off, then the third worst-off, and so on (Kostreva et al., 2004). As pointed out by Karsu and Morton (2015), the lexicographic maximin approach is a regularization of the Rawlsian maximin approach and is inequity averse.

Approaches based on inequity indices
Various studies that involve equity incorporate an inequity index into the model, which often assigns a scalar value to any given distribution showing the degree of inequity. Inequity indices are often used to assess the disparity in distribution, and so they are related to several mathematical concepts of dispersion and variance. Inequity indices respect the anonymity property (Chakravarty, 1999) and often equal 0 when perfect equity occurs. Anonymity property indicates that an inequity measure does not depend on the labeling of individuals. As pointed out by Panzera and Postiglione (2020), the anonymity property implies that an inequity measure is permutation invariant, which means that very different spatial patterns can give rise to the same measure. Suppose we have a set of i ∈ I individuals (or demand points). Let x i donate the outcome value at node i. Below we use x to briefly summarize the most commonly used inequity indices in the literature (adapted from Karsu and Morton (2015)): • The deviation from the mean ( i∈I (x i −x), wherex is the mean value). This index measures the total deviation from the mean. In some applications, the mean of the outcome distribution is often unknown, especially at the time when the decisions leading to the outcome x are being made. Thus, the meanx is often approximated based on expert knowledge or derived endogenously in the model. Some studies use the total absolute deviation from the mean (i.e., i∈I |x i −x|). Note that the mean absolute deviation (MAD) disregards how these deviations are distributed. Thus, MAD does not provide an incentive to minimize the gap among high values of the outcome (i.e., above or equal to the average) and among low values of the outcome (i.e., below or equal to the average). For example, deviations 0.25 and 0.25 are equivalent to deviations 0.5 and 0. Other studies use the mean squared deviation, the maximum componentwise deviation from mean, or only the positive or negative deviation from the mean as a measure of inequity. Mathematically, if the mean is (assumed) known, then using MAD in the objective function or a constraint may yield a linear optimization problem. For example, min{ i∈I |x i −x| : where X is a set of (potentially) linear constraints on x.
• The range, or difference between the minimum and maximum levels of outcomes (max i x i − min i x i ). Some studies also minimize this range normalized by the minimum outcome or enforce a constraint that ensures that min i x i max i x i ≥ β, where β is an equity or fairness parameter Chang et al. (2006). This index is used in many applications owing to its being simple and easy to understand. However, as pointed out by Karsu and Morton (2015), by considering the two extremes (e.g., most and least deprived), this index is rather crude as it fails to distinguish allocations with the same values of extremes but different levels of other values.
Thus, the range is sensitive to extreme values and ignores the interior of the distribution.
Note that the normalized range may lead to a non-linear and complex formulation.
• The variance or standard deviation. A small variance means a low dispersion of the outcome.
Both variance and standard deviation typically result in non-linear optimization problems.
• The Gini Coefficient ( i∈I j∈I |x i − x j |/2|I| i∈I x i ). The Gini coefficient (and indices derived based on it) is one of the most widely used measures of income inequity in an economy that satisfies the Pigou-Dalton principle of transfers (PD), which states that any transfer from a poorer person to a richer person, other things remaining the same, should always lead to a less equitable allocation. Note that the Gini index is often a dimensionless quantity, and thus, it cannot often be incorporated in a natural way with other terms in a multi-criteria problem (Gutjahr and Fischer, 2018). Moreover, this measure has the disadvantage of being highly non-linear, possibly making the resulting optimization problem extremely complex. The Gini index will always assume a value between 0 (indicating total equity) and 1 (indicating total inequity) • Sum of pairwise (absolute) differences ( i∈I j∈I |x i − x j |) and sum of squared deviations between all pairs. In contrast to MAD, the sum of pairwise absolute differences (SAD) does consider the spread of the outcome. Like MAD, the SAD term can be linearized (e.g., • The deviation from a predefined target: If the predefined target is the best possible output, for example, then satisfying it indicates perfect equity. Thus the larger the deviation, the larger the inequity. Minimizing this deviation is often referred to as minimizing regret. Related measures include minimizing maximum regret, minimizing absolute regret, etc. Remark 2. Selecting one of the above (or other) indices implies a particular assumption on the decision maker's or optimizer's attitude to equity.

Approaches based on inequity-averse aggregation functions
Karsu and Morton (2015) propose approaches based on inequity-averse aggregation functions, which use the aggregation function of the distribution vector in the model that would encourage equitable distributions. Unlike an inequity index, which only focuses on the inequity in a distribution, an inequity-averse aggregation function reflects concerns for both equity and efficiency. There are several ways to capture equity in this approach. For example, some studies use aggregation functions that have convenient mathematical properties such as convexity. Marín et al. (2010) use ordered median functions as objective functions of flexible discrete location problems. Ordered median functions are weighted total cost functions, in which the weights are rank-dependent. Ordered median functions with appropriate weights are inequity-averse in the sense that they are strictly concave. We refer to Karsu and Morton (2015) for more details on inequity-averse aggregation functions.

Equity versus Uncertainty
Uncertainty is intrinsic to many HCF location problems since various key input parameters such as demand, costs, and travel times are often unpredictable. While inequity-averse optimization in a deterministic context is conceptually relatively simple, though often computationally nontrivial and demanding, location decisions under uncertainty represented by suitable stochastic models introduce additional challenges for the following primary reasons. First, because of uncertainty, the value estimates are often not perfectly accurate, whereby the ex-post realized values of the impact (e.g., access to care, percentage of satisfied demand across geographical locations) and alternative decisions rarely coincide with their ex-ante estimated values.
Second, in most real-world applications such as HCF location, it is unlikely that we can accurately infer the actual distributions of random parameters and thus quantify the impacts of decisions on equity, especially with limited data or no information during the planning stage. Even when historical data is available, the quality of such data may not be sufficient to estimate the distribution of uncertain factors accurately, and future uncertainty is often not distributed as the past. Various studies show that different distributions can typically explain raw data of uncertain parameters, indicating distributional ambiguity (i.e., uncertainty in distribution type (Esfahani and Kuhn, 2018;Vilkkumaa and Liesiö, 2021)).
Third, incorporating equity measures and addressing both uncertainty and distributional ambiguity may increase the overall complexity of HCF location problems. However, ignoring uncertainty and equity-uncertainty interaction may lead to devastating costs and health outcomes. Adverse outcomes associated with poor HCF location decisions include increased costs, disparities in service, and increased illness or death. For example, a hard-to-access healthcare facility is likely to be associated with increased morbidity (disease) and mortality (death).
In this section, we demonstrate through a simple example that uncertainty and equity interact in ways that should not be ignored-the decision one should make in the presence of both considerations is often different from the decision under either consideration in isolation. As detailed Ahmadi-Javid et al. (2017) and Daskin and Dean (2005), and in Section 4, most of the existing HCF location models are extensions of basic discrete location problems (in which facilities can be established only at candidate locations), including p-median, p-center, covering-based, and fixed charge models. For brevity and illustrative purposes, herein, we next analyze extensions of the p-median and p-center using some of the linear equity metrics discussed in the previous section.
Specifically, we define different equity objectives based on the sum or maximum of pairwise (absolute) differences in distances or demand-weighted distances. shortly, when we discuss stochastic models. We assume that all demand nodes are also potential facility locations, denoted by I, but this assumption is straightforward to relax if necessary. Using y i,j 1, if demand point i is assigned to a facility at candidate location j, 0, otherwise.
the notation in the table 1, we define the following common feasible region among all formulations: Constraint (C1) specifies the total number of facilities to be established. Constraints (C2) ensure that each demand point is assigned to only one facility, and constraints (C3) limit assignments to open facilities. Constraints (C4) are integrality constraints. Table 2 presents several deterministic formulations for locating p facilities, including both classical facility location problems such as the p-median and p-center, as well as equity-based formulations. We do not claim that the equity-based approaches listed here are the best models for considering equity. Rather, we chose these measures because they are used frequently and because they are useful for illustrating the interaction between uncertainty and equity.
In the next section, we compare the optimal solutions obtained using the deterministic formulations in Table 2 and their stochastic programming (SP) counterparts. In the SP, travel time and demand are modeled as random variables that follow fully known probability distributions. The objective is to minimize the expectation of the objective function, where the expectation is taken with respect to an assumed known distribution. We approximate solutions to the SP models using their sample average approximation (SAA). That is, we generate a sample of N scenarios (each scenario consists of a vector of realizations of demand and travel time which are drawn independently from the distributions corresponding to each node and pair of nodes, respectively), and then optimize the sample average of the objective. (The technical details of SAA are out of the scope of this chapter, and we refer the reader to Kim et al. (2015); Kleywegt et al. (2002); Mak et al. (1999); Shapiro et al. (2021), and references therein, for a thorough discussion.). For example, the Here, the d n i,j values are the travel times (distances) under the nth sample, and z n is the optimal objective function value of the (deterministic) p-center problem under the nth sample.

Example: Where to locate a new hospital in Lehigh County?
In this subsection, we consider locating a single hospital (i.e., p = 1) in a service region based on Lehigh County, which is located in the Lehigh Valley region of the U.S. state of Pennsylvania. We consider a dataset consisting of 21 nodes: the 20 largest communities in Lehigh County according to the 2010 census, plus the city of Easton. 1 We calculated the distance and travel time between each pair of nodes using the Google API. We use these travel times as the d i,j values for the deterministic models. For the stochastic models we set both the average travel time (µ d i,j ) and the standard deviation of travel times (σ d i,j ) between each pair of nodes (i, j) equal to the calculated travel time, for all (i, j).
We use the population estimate in each county based on the 2010 U.S. census (see Table 3) to construct the following demand structure. We use the population percentage (weight) at each node to generate the mean (average) demand at each node i ∈ I as µ w i = population% × 1000 (i.e., total demand of 1000). To a certain extent, this structure reflects what may be observed in real life, i.e., locations with more population typically create greater demand. We set the standard deviation as σ w i = 0.5µ w i , for all i ∈ I. We generate the following two sets of N data samples for the parameters w and d.
• Set 2: w i ∼ LogN with mean µ w i and standard deviation σ w i , and d i,j ∼ LogN with mean µ d i,j and standard deviation σ d i,j .
We solve the SAA counterparts of the deterministic models in Table 2 with the generated data samples. In addition, we solve the deterministic formulations with one scenario (N = 1).
Tables 4 and 5 present the optimal solutions under Sets 1 and 2. We make the following observation from these Tables. First, the optimal location can change when we consider uncertainty

Is the Stochastic HCFL Literature Inequity-averse?
In this section, we provide a high-level analysis of recent stochastic approaches for HCF location, focusing on studies proposing inequity-averse approaches published between 2004 and 2017.
By inequity-averse, we mean any approach that considers one or more of the considerations mentioned above or other equity-related objectives or constraints. Our goal is to bring attention to a fundamental and timely question: Is the Stochastic HCF Location Literature Inequity-averse?.
We next analyze the limited literature considering equity and uncertainty, highlighting existing equity-related objectives or constraints and the challenges of incorporating these.

Non-emergency HCF location
Non-emergency medical (health) services include medical treatment, observation, prevention, testing, and other healthcare services provided to patients whose conditions are not urgent or considered an emergency. Non-emergency facilities include primary care facilities (hospitals, outpatient and primary care clinics, etc.), blood banks, specialized centers (e.g., organ transplant centers), medical laboratories, mobile healthcare units, drugstores, etc. Next, we review some of the literature on major non-emergency HCFs.  (2015) proposed a mixed-integer linear programming model for a robust and reliable medical service center location network design problem, which simultaneously takes uncertainty in demand, transfer cost, and system disruptions into account. However, the model did not include any explicit equity measure.

Blood Banks
A blood bank is a free-standing HCF or part of a larger HCF (e.g., hospital) that collects blood samples from donors, then stores and prepares them for transfusion to recipients. Timely access to different blood banks (e.g., blood transfusion providers, blood centers, mobile blood banks units) is crucial in health systems. One of the main challenges to blood bank location is that human blood is scarce, perishable, and often in high demand (e.g., for cancer patients, surgical patients, patients presenting to emergency rooms, etc.). Furthermore, both demand and supply of blood are stochastic and subject to various disruptions.

Jabbarzadeh et al. (2014) presents a robust location-allocation model for dynamic supply chain
network design for the supply of blood in disasters. Uncertainty considered includes the capacity of a temporary blood facility, capacity of a permanent blood facility, and maximum blood supply of each donor group. The objective is to minimize the total cost of the network (the combined costs of locating blood facilities, transportation, and holding), while ensuring that the network is robust to major disasters. Fahimnia et al. (2017) also proposed a stochastic bi-objective supply chain design model for the efficient (cost-minimizing) and effective (delivery time minimizing) blood supply in disasters. Uncertainty considered includes the' cost of moving a mobile blood facility, unit operational cost at each mobile blood facility, local blood center, and regional blood center, unit transportation cost between blood facilities, and blood demand at each hospital. Although both studies considered uncertainty, they did not incorporate any equity objective or constraints.

Organ transplant centers
Organ transplant centers (OTC) are the main components of organ transplantation programs in in most healthcare systems. The demand for organs is a major random factor that is often larger than the supply, which is also random. As a result, organ transplants suffer from long waiting

Medical laboratories
A medical laboratory is a HCF where tests are carried out on clinical specimens from the patient to aid in diagnosis, treatment, and disease prevention. Although medical laboratories are critical for public health, there is a lack of research on the location of these HCFs. Notably, Saveh-Shemshaki et al. (2012) propose a p-median-based model for designing a network of tuberculosis testing laboratories to reduce transportation times and thereby decrease overall test turnaround time. They use the travel time from any region to any laboratory as a measure of equity. Accordingly, they included a constraint that allows decision-makers to specify an upper bound for origin-destination transportation time. Their results suggest that the optimal locations and capacities are not sensitive to this additional equity constraint. Modeling uncertainty was suggested as a future direction.

Long-term care centers
Long-term (nursing) care is an HCF that provides rehabilitative, restorative, and ongoing nursing care to patients or residents who need assistance with their health or daily living activities. The location of long-term care facilities is crucial to provide the best and most equitable possible services to aged people who represent the major demand group needing social and medical services. Given that this type of HCF provides medical care and social services to inpatients, the simultaneous determination of location, optimal capacity levels (e.g., number of beds), inventory levels, and locations are essential. Similarly, the location of rehabilitation HCF (i.e., HCF devoted to the physical rehabilitation of patients with, e.g., neurological, orthopedic other medical conditions), doctors' offices, and drugstores are not studied as other types of HCFs. An analysis of equity concerns related to, for example, the distribution of these HCFs and access to them under uncertainty is an important future research area.

Emergency HCF location
Life-threatening emergencies, such as a severe injury, stroke, or heart attack, require the services of emergency HCFs. In addition, emergency HCFs provide service for patients with an injury or illness that does not appear to be life-threatening, but the treatment of such patients cannot wait until the next day or for a primary care doctor to see them. Ahmadi-Javid et al. (2017) classify emergency HCFs according to whether they perform under permanent or temporary emergencies.
Permanent emergency HCFs provide service regularly, including emergency off-site public access devices, trauma centers, and ambulance stations. In contrast, temporary emergency HCFs are constructed to respond to unexpected health (e.g., infectious disease outbreak) and other situations (e.g., disaster). Next, we analyze the literature on emergency and trauma centers, ambulance stations, and temporary medical centers.

Emergency and trauma centers
Emergency departments or emergency centers are permanent emergency facilities that provide medical and surgical care to both patients arriving in need of immediate care or walk-ins (unscheduled patients). They can be part of a hospital or free-standing. Locating these facilities has not received as much attention as other emergency facilities such as ambulance stations. Therefore, there is a need for inequity-averse stochastic models for the location of emergency centers, particularly those associated with hospitals and clinics or ambulance stations. Silva and Serra (2008) is one of the early studies that presented a priority queuing-based covering location problem for locating emergency services considering different service priority levels. They implemented a greedy randomized adaptive search procedure to solve randomly-generated problem instances. Silva and Serra (2008) did not focus on equity.
Trauma centers are hospitals that provide specialized medical and nursing care to patients suffering from major traumatic injuries (e.g., falls, motor vehicle collisions and accidents, gunshot wounds, etc.). Patients are typically transported to trauma centers via helicopters or ambulances.
To uncertainty. In addition, equity and equity-uncertainty interaction has not been considered.

Ambulance stations
Emergency Medical Services, more commonly known as EMS, is a system that provides out-ofhospital acute medical care and transfers patients to emergency centers/departments within or outside hospitals and trauma centers for definitive care. EMS aims to reduce the elapsed time to respond to an emergency. Ambulance stations and ambulances are major resources to achieve this aim. Ambulance stations are responsible for dispatching ambulances that provide EMSs on the scene or during transport to an emergency center/department, a trauma center, etc. The ambulance then returns to a predetermined station to await another call. For an efficient and timely response, ambulance stations must be located at appropriate points to provide adequate coverage and minimize the response time. The OR community has paid significant attention to the location of ambulance stations, the deployment (location, relocation, fleet sizing) of ambulances in the stations, and the dispatch of ambulances to the demand points or emergency sites.
Most of the existing models for this type of HCF are extensions of the basic maximum or set covering location, maximum expected coverage location Daskin (1982Daskin ( , 1983, and p-center location models. Note that by focusing on maximizing the demand that can be covered, traditional covering models favor locating ambulances in more densely populated areas, resulting in longer response times for patients in more rural areas. That is, traditional covering models may lead to solutions in which the coverage pattern is quite good for those nodes counted as covered but extremely poor for those not covered, highlighting the need for equity-based models.
Most existing models for EMS are stochastic due to the stochastic nature of EMS operations.
Noyan (2010)  Specifically, to deal with the issue of equity, they assigned an envy function (a function of the distance from a demand zone to its closest EMS station and the distance from a demand zone to its backup EMS stations weighted by priority of the serving stations and weighted by the proportion of demand) to each pair of demand nodes, for each level of priority. This value indicates the dissatisfaction level of a demand node with its serving station in comparison with other demand nodes that have the same level of priority.
To address the issue of fairness in semi-rural/semi-urban communities, Chanta et al. (2014) propose a bi-objective covering location model for locating EMS ambulances at preexisting rescue stations that balances efficiency (i.e., maximizing expected coverage) and equity. Specifically, they propose the following alternative objective functions for improving fairness in rural areas: minimize the maximum distance between uncovered demand zones and their closest opened station, minimize the number of uncovered rural demand zones, and minimize the number of uncovered demand zones. Chanta et al. (2014) use the ǫ-constraint method to solve their multi-objective model. Khodaparasti et al. (2016) studied balancing efficiency and equity in a location-allocation EMS model under uncertainty using data envelopment analysis. Enayati et al. (2019) proposed a mul-ticriteria optimization approach to study the trade-offs in equity and efficiency for simultaneously optimizing location and multipriority dispatch of ambulances.

Temporary medical centers
Temporary medical centers (TMCs) provide healthcare services to victims of large-scale and catastrophic disasters. Examples of TMCs include Red Cross medical tents, casualty collection points, and any temporary HCF established before the disaster to play a short term role in the immediate aftermath. TMC location has some stochastic characteristics similar to those we see in humanitarian logistics that should be considered. These include the possibility of TMCs being completely destroyed during the disaster, changes in the capacities and functionality of roads between TMCs and disaster sites, uncertainty in the fraction of usable medical supply in the aftermath, and uncertainty in the size and location of demand. In addition, helicopters are often used to transport medical supplies and people to the TMC and transport victims needing further care from TMCs to other HCFs. Thus, one should also consider helicopter-feasible locations. Despite the importance of TMC, as pointed out by Ahmadi-Javid et al. (2017), only a few papers addressed TMC location problems, and surprisingly under deterministic conditions. None of these studies incorporated equity objectives or constraints.

Future Directions
While great research efforts have been reported to improve stochastic HCF location theory, much work is still needed to incorporate equity and derive inequity-averse stochastic HCF location approaches and insights. We discuss a few critical research opportunities for the future in the following subsections.

Analyzing Equity Measures
The World Health Organization defines health equity as the absence of unfair and avoidable or remediable differences in health among population groups defined socially, economically, demographically, or geographically (WHO 2021). Accordingly, when making HCF location-allocation decisions, one should ensure equal distributions of HCF, equal access to HCF, and equal utilization of healthcare services/HCFs across geographical areas, social groups, demographics, socioeconomic groups, and identified/non-identified groups under normal and abnormal (e.g., disaster, conflict, etc.) conditions. However, to ensure this, one should first find the right measure of equity. But, is the obvious objective the right one? Are the obvious and classical constraints the right ones? Is the impact of decisions obtained from classical models with the obvious objectives equitable when compared across various people or groups of people?
To date, we do not have a formal analysis of equity objectives and constraints under uncertainty.
Thus, the first step toward developing tractable and realistic stochastic inequity-averse HCF location approaches is to rigorously analyze the mathematical similarities and differences between different mathematical representation of a measure of equity under uncertainty. This is to identify subsets of mathematically similar equity metrics and accordingly limit the scope of modeling and comparison (and later optimization) to the subset of mathematically different metrics. Such analysis will facilitate studying equity-equity trade-offs (based on distinct notions of equity arising from a choice of population grouping or equity metric) in addition to studying equity-efficiency trade-offs.
Then, one can analyze the subset of different measures to identify their mathematical complexity (e.g., linearity, convexity, etc). This will, in turn, determine the complexity of optimization models.
For example, one might resort to a linear metric that yield solutions that are near-optimal rather than using a nonlinear and complex metric that yields optimal solutions (if the problem can even be solved).

Capturing Uncertainty: Optimizer's Curse and Trade-offs
Acknowledging the inevitable uncertainty and the uncertainty-equity interaction in HCF location settings is crucial to devise inequity-averse models for emerging real-life HCF location problems.
One possibility for hedging against uncertainty is to capture it in the parameters underlying optimization models to support the decision-making process.
There are three main stochastic optimization (SO) frameworks: stochastic programming (SP), robust optimization (RO), and distributionally robust optimization (DRO). The main distinguishing feature between these frameworks concerns the knowledge of the probability distribution of the underlying random vector. Hence, adopting one of these modeling frameworks depends mainly on the available information regarding uncertainty and its distribution. If we know the uncertainty distribution or have sufficient and high-quality data to represent it, we resort to SP. In SP, we typically consider two-or multi-stage models in which the first-stage (here-and-now ) decisions represent decisions to implement before any uncertainty is revealed. Then, in the subsequent stages, we often use a discrete number of scenarios (from the assumed known distribution) to capture the uncertainty and make recourse (wait-and-see) decisions that adapt to these scenarios.
The SP decision problem evaluates the objective and optimizes the decisions only for the given training sample (which may come from a biased distribution). Thus the resulting SP decisions may be overfitted or optimistically biased. In reality, possible future uncertainty realizations may differ from all training samples used in the optimization. Thus, SP solutions may demonstrate disappointing out-of-sample performance under the true distribution. Disappointing consequences in healthcare include, but are not limited to, disparities in healthcare service and distribution of services, poor access to care, increased mortality and morbidity of the vulnerable population, and increased costs. Mathematically, solutions to SP decision problems display an optimistic in-sample risk/outcome that cannot be realized out of sample. This phenomenon is known as the optimizer's curse (i.e., optimization based on imperfect estimates of distribution leads to biased decisions) and is reminiscent of overfitting effects in statistics (Esfahani and Kuhn, 2018;Smith and Winkler, 2006). Unfortunately, as shown in Section 4, most existing studies employ SP or other "samplebased" approaches.
In contrast to SP, RO assumes that the decision-maker has no distributional knowledge about the underlying uncertainty, except for its support. RO seeks a particular measure of robustness against the uncertainty represented as deterministic variability in the value of the problem's parameters. Specifically, in RO, we assume that uncertain parameters belong to an uncertainty set of possible outcomes with some structure (e.g., ellipsoid or polyhedron). Then, optimization is based on the worst-case scenario occurring within this set, which inevitably leads to over-conservatism and suboptimal decisions for other more likely scenarios.
DRO is an alternative ambiguity-averse approach to model uncertainty that offers a middle ground instead of the black-or-white view of the SP and RO approaches about knowing the distribution of uncertain parameters. In DRO, we model the distribution of uncertainty as a decision variable that belongs to an ambiguity set, i.e., a family of all possible distribution of uncertainty that share some characteristics (e.g., mean, range, co-variance, etc). Optimization is then based on the distribution within this ambiguity set.
DRO has recently become a promising approach for addressing real-life stochastic optimization problems due to the following striking benefits. First, DRO models are more "honest" than their SP and RO counterparts as they acknowledge the presence of distributional uncertainty. Various theoretical results show that DRO solutions faithfully mitigate the optimizer's curse and the associated disappointing out-of-sample consequences. Second, by allowing uncertainty to follow any distribution within the ambiguity set, DRO alleviates the unrealistic assumption of the decisionmaker's complete knowledge of distributions. Third, one can use intuitive and easy-to-approximate statistics (e.g., based on expert knowledge) to construct the ambiguity set. For example, we could estimate the average demand for primary care in each city of the Lehigh Valley in Pennsylvania (U.S.). Then, we can construct a mean-range ambiguity set of demand distributions, where the range is used to represent the error margin in the estimates that we seek protection against. Accordingly, a DRO model could optimize primary care clinic location decisions against all possible distributions in the ambiguity set that share these mean and range values. Some ambiguity sets also allow for incorporating the possibly small available data in the optimization while hedging against the estimation error of the associated empirical distribution (e.g., DRO using the Wasserstein metric Esfahani and Kuhn (2018)).
Fifth, DRO may reflect decision-makers' ambiguity aversion and preference to err on the side of caution and seek robust location decisions to safeguard performance in adverse scenarios and mitigate direct and indirect costs of care and inequity (e.g., poor population health outcomes). Finally, even when complete distributional information is available, DRO models often perform well. This suggests that it is worthwhile to consider DRO as an alternative approach for modeling uncertainty in HCF location. Yet, despite the potential advantages, there is no inequity-averse DRO approach with equity and efficiency concerns for facility location within or outside healthcare applications.
Indeed, there is a trade-off to using different approaches to model uncertainty that must also be considered when deriving inequity-averse HCF location models. SP may provide an excellent basis if we unrealistically assume that we know the distributions of uncertain parameters. However, as we mentioned above, SP solutions often underdeliver and disappoint. In contrast, by focusing on hedging against ambiguity, DRO solutions may seem conservative and pessimistic. Thus, models that minimize the trade-off between considering distributional ambiguity (DRO pessimism) and following distributional belief (SP optimism) may offer a middle ground.
Unfortunately, this trade-off has not been studied within the context of facility location (except in Shehadeh and Tucker (2020)) or in inequity-averse HCF location in particular.
These gaps and challenges call for more efforts toward rigorous analyses of equity under different scenarios of information availability and ambiguity. Formal analyses are also needed to answer questions about when to use each type of modeling approach (SP, DRO, or a trade-off between them) and about what is the value of adopting each for inequity-averse HCF location. Extending classical models to derive distribution-free and tractable inequity-averse HCF location models is both theoretically and practically relevant. It will have worthwhile contributions to location science, optimization under uncertainity, and computational operations research.

Dynamic Mapping and Databases
Location is the central element in all spatial analyses and varies in size and shape depending on the context. For many decades, locations or places played an important role in understanding health patterns, disease patterns, and healthcare service distributions. Historically, maps have been the primary source for storing and communicating spatial information. In the past two decades, the advancement in computer power and the emergence of Geographic Information Systems (GIS; a system that captures, creates, stores, manages, analyzes, and maps all types of data) has allowed for better mapping and more widespread, complex, and comprehensive analyses than previously.
GIS connects data to a map, integrating location data (so it is possible to know where things are) with all types of descriptive information (so it is possible to understand what things are like in a specific location). Such advances have made it possible for governments, researchers, and others to seek answers to previously overly complex and unfeasible questions.
Data accuracy, correctness, and completeness are crucial elements affecting our ability to use GIS to analyze health and equity issues effectively. Unfortunately, despite the significant improvement in technologies to obtain geographic data, we often need to geocode specific data such as patient data (volume, disease, etc) and HCF utilization by different groups to undertake a particular geographical analysis. As pointed out by Lyseen et al. (2014), in the absence of standardized data collection methods or databases, this process imposes a significant challenge for health information systems in collecting data with adequate granularity, ensuring reliability and validity of the relevant health data, and maintaining appropriate privacy and security for the collected data. Moreover, even in a perfect situation when the needed data has been collected, it is often unavailable to researchers due to privacy laws regarding patients' medical information Lyseen et al. (2014).
Several developed countries and research groups create and publish interactive GIS maps of HCF locations with basic information. However, these are individual efforts; researchers and governments often work individually, and there are no integrated or standardized databases even within the same country. Therefore, policies and strategies for developing standards-compliant and reliable databases on HCF and the ability to integrate these in GIS dynamically as information unfolds are important research directions and a prerequisite to developing data-driven inequity-averse HCF location models. Such databases may include but are not limited to real-time data on the location, type, and characteristics of existing HCFs (location, scope of services, quality of service, types of insurance accepted, etc.), utilization of existing HCFs (by, e.g., social and racial groups), characteristics (socioeconomic, education level, demographic, etc.) of people living in each geographical area (country, state, county, city, block, etc.), and demand and access for healthcare and health services across geographical areas and (social, economic, religious, racial, etc.). This information is essential for developing inequity-averse HCF location approaches that better mimic reality.
A key challenge in collecting and using data for HCF location problems is the inevitability of bias in the data. A number of types of bias can arise, including selection bias (when groups are overor under-represented in the data compared to their true populations), historical bias (bias caused by events that occurred before the data collection or analysis), confirmation bias (researchers' tendencies to choose or trust data that supports their existing hypotheses), and availability bias (bias due to the use of data that are available, even if those are not the most relevant data). Bias in healthcare data have already been shown to produce negative outcomes (for example, an algorithm that interprets imaging results and that is trained on gender-biased data will produce worse outcomes for under-represented genders Larrazabal et al. (2020)), and it stands to reason that biased data used for HCF location studies will similarly lead to harms for some groups. See, e.g., Healthcare Information and Management Systems Society (HIMSS) (2021); Catalog of Bias (2021) for more on data bias.

Multicriteria Approaches
The specific approach or criteria to capture inequity are context-dependent and depend on the decision maker's perception of equity. One can, of course, consider a multicriteria objective and a method to optimize multiple equity criteria (equity of access, socioeconomic equity, geographical equity) and efficiency criteria (e.g., transportation cost, operational costs, etc). However, to the best of our knowledge, and according to a recent survey on HCF location Ahmadi-Javid et al.
(2017) and our analysis in Section 4, within the limited literature considering equity in stochastic HCF location, most studies only consider one equity-related policy objective, and only a few multiobjective models exist for the joint attainment of different equity objectives under uncertainty.
When we seek both equity (e.g., equitable allocation of vaccination clinics across urban and rural areas) and efficiency (e.g., minimum operational cost), there is the additional challenge of mathematically integrating them in a tractable model. As we mentioned earlier, considering equity may degrade efficiency. Thus, two fundamental questions arise in integrating equity and efficiency: how to regulate the trade-off between the two under uncertainty and potential distributional ambiguity?
and what is the price of equity (i.e., the efficiency difference between selecting an inequity-averse approach and not using an inequity-averse approach)?
In theory, multi-criteria optimization problems such as HCF location with equity and efficiency concerns can be modeled and solved using multi-objective optimization (MOO) approaches such as interactive approach, hierarchical approach, or a weighted objective function (see, e,g., Ehrgott (2005) for a detailed discussion of MOO). However, multicriteria location models assume that decision-makers can articulate their objectives or have a well-defined metric for each objective. In most real-world applications such as HCF location, the decision maker's priorities, goals, and equity concerns are not easy to articulate and could vary over time and among different decision-makers.
Moreover, we often need to incorporate the (potentially conflicting) perspective of many stakeholders. For example, locating hospitals is a process that must take into consideration many different stakeholders Burkey et al. (2012), including patients who need access to the hospital, clinical staff who want an attractive and easy-to-reach workplace, taxpayers who want value for their dollars, politicians who want to demonstrate their ability to deliver a better quality of life and healthcare services, and more.
Moreover, equity measures generally compare the impacts of actions (e.g., location decisions) on different groups and possibly weigh such effects based on the characteristics of a group (e.g., needs for a particular health service). Unfortunately, decision-makers cannot always quantify or predict HCF location decisions' direct/indirect impact on patient outcomes and access to care, among others. Thus, the cost associated with HCF location decisions is often subject to ambiguity and cannot be explicitly measured. Even in a perfect situation when we have a precise definition of equity and efficiency metrics, these metrics are often conflicting, and it is often challenging to arbitrate the conflict between them. In addition, the incorporation of uncertainty increases the complexity of multi-criteria HCF location models.
These challenges call for more efforts toward bridging the gap between MOO (and other operations research) approaches, decision-makers' perspectives of equity and efficiency, and the actual cost or criteria used in practice. The ability to quantify the trade-off between efficiency and equity would help decision-makers make informed and better location decisions (Karsu and Morton, 2015).
In particular, to make decisions, a decision-maker needs to understand (a) what the efficiency loss might be and (b) what the equity loss might be for a specific solution. Although various studies analyze the price of fairness or price of equity (see, e.g., Bertsimas et al. (2012Bertsimas et al. ( , 2013) this price has not been adequately analyzed in HCF location contexts under uncertainty.
Analyzing the robustness of solutions obtained from different inequity-averse HCF location approaches under uncertainty is another venue for future research. Incorporating multiple equity measures and an analysis of the impact of uncertainty on each and the trade-off among them could shed some light on the question of how similar or different equity measures are.
There are some attempts to analyze the commonality and differences of different equity measures in general facility location models (Batta et al., 2014;Karsu and Morton, 2015;Mulligan, 1991;López-De-Los-Mozos and Mesa, 2003). However, such analyses have not been conducted in the HCF location literature.

Mobile HCF
A mobile healthcare facility (MHCF) is a facility-like-vehicle that can serve patients in a way similar to a static HCF when stationary but can also move from one place to another to provide health services to communities (Attipoe-Dorcoo et al., 2020;Malone et al., 2020;McGowan et al., 2020;Shehadeh, 2020;Santa González et al., 2020). They offer a wide variety of (prevention, testing, diagnostic) health services and are often staffed by a combination of physicians, nurses, community health workers, and other health professionals. As longstanding community-based service delivery models, mobile facilities have the potential to help underserved communities overcome common barriers to accessing healthcare, including availability, time, geography, and trust (Clark et al., 2011;Guruge et al., 2010;Sommers, 2015), and have demonstrated improvements in health outcomes and reductions in cost (Brown-Connolly et al., 2014;Oriol et al., 2009;Song et al., 2013).
MHCF also offer an alternative healthcare delivery option when a disaster, conflict, or other event causes stationary HCFs (e.g., hospitals) to close or stop operations (Blackwell and Bosse, 2007;Du Mortier and Coninx, 2007;Fox-Rushby and Foord, 1996;Gibson et al., 2011 is practically important to support decisions in all applications of mobile HCF and technically interesting.

Robust and Resilient HCF locations
The supply and demand for healthcare are often not balanced. In addition, multiple disruptive events can shock the healthcare network and exaggerate the imbalance between supply and demand. Long-term disruptions including disasters, infectious disease outbreaks, and economic and political conflicts can cause significant shocks to HCF operations over a long period. Unfortunately, vulnerable populations often suffer the most during such a disruption. Thus, we need strategies for building a robust and resilient inequity-averse HCF location framework that can adjust to changes caused by long-term disruptions. Resilience considers the healthcare network's vulnerability to disruption, i.e., the system's ability to function even if one or more nodes (HCFs) are removed from the network or disabled (e.g., a temporary or permanent closure of a hospital or a halt in operations).
Furthermore, developing models for inequity-averse HCF location incorporating the dynamic unfolding of disruptive events is practically relevant and theoretically interesting. The emerging complex healthcare challenges also demand new models that account for equity when pursuing actions such as reorganizing existing HCF networks, which may involve relocation or closure of existing facilities, redistribution of capacity, and mergers of facilities.

Conclusion
Healthcare and HCF location problems have become noticeably more essential to society in the past decade due to the global decrease in birth rates, higher longevity and associated growth in the elderly population, outspread of chronic and infectious diseases, increase in environmental and economic problems, conflicts, and disasters, and the associated impact on health and healthcare supply and demand. These globally pervasive trends continue to create new challenges and equity concerns that classical HCF location models cannot address.
In this chapter, we focused on the issue of equity in stochastic HCF locations. We recognize that uncertainty is an intrinsic property of HCF location and explore how uncertainty and equity interact in ways that should not be ignored. The primary goal is to bring attention to equity issues and provide an understanding of the existing recent effort to incorporate equity measures in stochastic HCF locations. Our key findings are • There is limited literature considering equity in HCF location and equity-uncertainty interaction.
• Within the limited literature, most studies use the distance that a patient must travel or the travel time to HCF to measure equity and accessibility, with minimizing the maximum distance to the nearest facility as the most frequent measure. Some other studies account for factors such as spatial accessibility, multi-modal travel, temporal service availability, competition, multiple and hierarchical services, socio-economic and personal factors, etc. in order to more realistically reproduce users' behaviors (Bruno et al., 2020;Higgs et al., 2019;Jin et al., 2019;Mathon et al., 2018;Mayaud et al., 2019;Lin et al., 2018;Shin and Lee, 2018;Yin et al., 2018).
• Most studies, especially in EMS, use covering-based models to ensure equity. However, covering-based models favor locating HCFs in more densely populated areas, potentially resulting in longer travel or response times for patients in more rural areas. In addition, traditional covering models may lead to solutions in which the coverage pattern is quite good for those nodes counted as covered but extremely poor for those not covered.
• Some studies investigate the trade-off between one or more equity objectives and multiple efficiency objectives. However, no study has analyzed the equity-equity trade-off (i.e., the trade-off between equity metrics) or the equity-efficiency trade-off with multiple equity metrics.
• Stochastic optimization (SO) models with recourse have been adopted widely to hedge against uncertainty, particularly in two-stage settings. Recourse models result when (first-stage) decisions must be fixed before information relevant to uncertainty is available, and some (secondstage) recourse decisions are delayed until this information is available (Birge and Louveaux, 2011;Higle, 2005). Thus, recourse decisions offer an opportunity to adjust to the received information about the uncertain data. The two-stage structure of stochastic HCF location problems (e.g., choose number and locations of facilities before we know the future demand and take actions once demand uncertainty has been revealed, for example, by assigning customers to facilities) has made them attractive to be modeled using recourse models (Ahmadi-Javid et al., 2017;Snyder, 2006).
• Existing SO models for HCF location assumes that the distribution of uncertainty is fully known or there is sufficient data to model it. To date, there are no data-driven and distributionfree inequity-averse HCF location models.
• While there is extensive literature on static facility location problems, there is limited literature on stochastic mobile facility locations. To date, there are no inequity-averse mobile healthcare facility deployment, routing, and scheduling models.
• No studies rigorously analyze the mathematical similarities, differences, and complexity of equity objectives and constraints. In addition, there are no standardized and open-access databases for HCF locations that incorporate all the geographical, socioeconomic, and other relevant data that one needs to derive and test inequity-averse models.
We discuss and provide new directions for future research opportunities by recognizing these challenges and gaps in the literature. Our main recommendations for future research include: • Conducting formal analyses of equity metrics and their mathematical complexity under uncertainty and distributional ambiguity • Developing data-driven, distribution-free, and tractable multi-criteria inequity-averse static and mobile HCF location approaches • Developing standardized, granular, and dynamic HCF databases and integrate these with recent GIS technology • Developing robust and resilient inequity-averse stochastic HCF location approaches