key: cord-0553582-1ui73vdb
authors: Kaya, Yaren Bilge; Maass, Kayse Lee; Dimas, Geri L.; Konrad, Renata; Trapp, Andrew C.; Dank, Meredith
title: Improving Access to Housing and Supportive Services for Runaway and Homeless Youth: Reducing Vulnerability to Human Trafficking in New York City
date: 2022-01-31
journal: nan
DOI: nan
sha: bcd14c0ce35bb34c5f26344841af2e8a8f01d8a3
doc_id: 553582
cord_uid: 1ui73vdb

Recent estimates indicate that there are over 1 million runaway and homeless youth and young adults (RHY) in the United States (US). Exposure to trauma, violence, and substance abuse, coupled with a lack of community support services, puts homeless youth at high risk of being exploited and trafficked. Although access to safe housing and supportive services such as physical and mental healthcare is an effective response to youth s vulnerability towards being trafficked, the number of youth experiencing homelessness exceeds the capacity of available housing resources in most US communities. We undertake an informed, systematic, and data-driven approach to project the collective capacity required by service providers to adequately meet the needs of homeless youth in New York City, including those most at risk of being trafficked. Our approach involves an integer linear programming model that extends the multiple multidimensional knapsack problem and is informed by partnerships with key stakeholders. The mathematical model allows for time-dependent allocation and capacity expansion, while incorporating stochastic youth arrivals and length of stays, services provided in periodic fashion and service delivery time windows. Our RHY and service provider-centered approach is an important step toward meeting the actual, rather than presumed, survival needs of vulnerable youth, particularly those at-risk of being trafficked.

Human trafficking (HT) is the criminal commercial exchange and exploitation of humans for monetary gain or benefit -a globally prevalent, violation of human rights (Gajic-Veljanoski and Stewart, 2007) . HT is rarely a one-time event, but rather a process that can be conceptualized as a series of event-related stages over time during which various risks and intervention opportunities may arise: recruitment; possible transit; exploitation; and re-integration (Zimmerman C, 2011) . A key and largely understudied means to disrupt trafficking networks is through reducing an individual's vulnerability during any of the four stages.

Runaway and homeless youth and young adults (RHY) are particularly vulnerable to exploitation and trafficking (Wright et al., 2021; Hogan and Roe-Sepowitz, 2020; Middleton et al., 2018) .

Several factors are known to raise the risk of trafficking for this population, such as a history of physical abuse, emotional neglect, and low self-esteem (Hannan et al., 2017) . Such circumstances, coupled with a lack of community support, put RHY at a high risk of HT. One study on child sexual exploitation found that most victims in the study sample had experienced homelessness or persistent housing instability (Curtis et al., 2008) . Labor traffickers are also known to prey on vulnerabilities by providing housing along with the employment opportunity, making it more difficult for youth to leave the trafficking situation because they have nowhere else to go (Bigelsen and Vuotto, 2013) . When youth leave a trafficking situation, they need to have near-immediate access to stable shelter and other support services, as well as financial resources to ensure they are not re-victimized and are able to recover (Duncan and DeHart, 2019; Okech et al., 2012) . In a recent study with individuals who were able to exit child sexual exploitation, the survivors interviewed confirmed the need for social services to provide ongoing safety and basic needs, such as shelter, to fully exit trafficking and exploitative experiences (Bruhns et al., 2018) . For all these reasons, access to appropriate housing can significantly decrease the likelihood of youth experiencing sexual or labor exploitation and trafficking (Davy, 2015; Potocky, 2010) .

Trafficking prevention and rehabilitative services have been shown to be effective in disrupting trafficking activity by decreasing vulnerability and associated risk factors. Of the varied prevention and rehabilitative services from which those at-risk of trafficking and HT survivors may benefit, access to safe housing is widely agreed to be the most pressing need (Murphy, 2016; Dank et al., 2015; Clawson et al., 2006) . Yet, while the exact numbers of available beds for RHY in the US is unknown, it is widely acknowledged that demand greatly exceeds supply (Clawson et al., 2009) .

As a trafficking intervention, shelter provision extends beyond the mere supplying of beds.

Shelters are linked to a dynamic and shifting landscape of networked support services such as medical treatment, psycho-social care, education, life-skills training, and legal advocacy that aids in a holistic approach to rehabilitation and trafficking network disruption (Ide and Mather, 2019; Clawson, 2007) . The ability to engage at-risk populations in shelter services is a critical component of service provision. Programs that provide at-risk youth and trafficking survivors with options and the ability to make their own choices further reduce vulnerability (Hopper et al., 2010) . Yet, shelters are unique in the varied services they offer, as well as inclusion and exclusion criteria that dictate who may receive services at the shelter (Clawson, 2007) . These restrictions may depend on a shelter's funding sources, state law, or federal policy. It is therefore important to place youth in shelters that are able to meet the unique profile of needs of at-risk youth and HT survivors. This study is motivated by the appreciation that shelter and associated services disrupt trafficking activity by decreasing vulnerability for those at-risk of trafficking, including HT survivors. We seek to alleviate current capacity limitations and improve access to housing and support services for homeless and unstably housed youth and young adults as a mechanism to reduce the supply of potential HT victims. The study population of this study are homeless, unstably housed and at-risk youth in New York City (NYC) ages 16-24, we note that NYC has the largest such population in the US (Morton et al., 2019) .

Informed by partnerships and interviews with RHY who have experienced homelessness, shelter service providers, and the NYC Mayor's Office, we develop a mixed-integer linear programming (MILP) model to project the cost-minimizing shelter and service capacity expansion that meets the collective needs of youth. Our RHY-centered approach allows us to provide a much clearer picture of the actual, rather than presumed, needs of homeless youth. To the best of our knowledge, our study is the first to (i) incorporate primary data collection to estimate the support services and resources available to RHY, and (ii) develop an optimization model to determine the cost-minimizing approach to expand capacity under stochastic demand patterns including RHY arrivals, stay durations, and varying service frequencies and intensities.

The remainder of the paper is structured as follows. We provide a review of the literature in Section 2; explain our data collection process in Section 3; propose a mixed-integer linear programming framework in Section 4; present our computational setup, and discuss results, analysis and insights from our computational experiments in Section 5. We conclude by summarizing our contributions, limitations and future research directions in Section 6.

Operations Research (OR) techniques are applicable to challenges in HT, and engineering systems analysis can produce important insights concerning HT operations (Konrad et al., 2017; Caulkins et al., 2019) . Analytical techniques employed in disaster preparedness may be adapted to allocate limited funds and resources for anti-trafficking operations. However, such techniques likely need to be extended to be effective as trafficking is an ongoing, rather than a single disaster event (He and Zhuang, 2016) . Contextual similarities exist between allocating funds for anti-trafficking services and models addressing fund allocation for hunger relief (Sengul Orgut et al., 2016) . Similarly, capacity allocation and location models such as Kilci et al. (2015) and Li et al. (2012) could be adapted to increase the efficiency and effectiveness of public services provided to HT victims and populations vulnerable to HT. However, simply adapting existing OR approaches to address HT is insufficient and irresponsible -the intricacies of individual agency must be incorporated for successful implementation of OR techniques . Our work draws upon, and contributes to, two primary areas: (i) OR for disrupting the supply side of HT networks and (ii) effective and appropriate allocation of scarce resources via capacity allocation and expansion.

Research related to OR and analytics efforts to reduce homelessness and the supply of sex and labor trafficking victims has only recently received attention and largely focus on three broad areas: the scope of the problem (Kosmas et al., 2020; Brelsford and Parakh, 2018) , frameworks for addressing the crisis (Taylor, 2018; Kővári and Pruyt, 2014) , and appropriate allocation of scarce resources to combat HT and homelessness (Chan et al., 2018; Petry et al., 2021) . A second body of OR literature focuses on detection of hidden HT victims (Keskin et al., 2021; Kapoor et al., 2017; Mensikova and Mattmann, 2017) , movement patterns of covert traffickers (McDonald et al., 2021; Yao et al., 2021; Amin, 2010) and long-term intervention approaches to prevent trafficking or retrafficking by increasing the efficiency and effectiveness of public services (Amadasun, 2020) , as well as improving access to supportive and rehabilitative services (Azizi et al., 2018) . However, little attention has been paid to using OR techniques to examine the problem of allocating scarce resources for anti-trafficking efforts, with the exceptions of Konrad (2019) and Maass et al. (2020) .

To increase trafficking awareness among at-risk populations, Konrad (2019) proposes a resource allocation model for trafficking prevention programs in Nepal, aiding decision makers in evaluating how to allocate limited funds in the context of trafficking awareness. To improve stabilization and the eventual reintegration of trafficked persons, Maass et al. (2020) present an optimization model that allocates a budget for locating residential shelters in a manner that maximizes a measure of societal impact. The model evaluates the trade-off in the cost of opening and operating shelters in each location with the health benefits, labor productivity gained, and reduction in criminal justice costs. Although both Maass et al. (2020) and Konrad (2019) investigate HT-related interventions, they emphasize the broader system and the optimal budget allocation rather than the effective capacity allocation of supportive and rehabilitative services provided to vulnerable populations.

Although not explicitly focused on reducing vulnerability to HT, other OR literature addresses the importance of reducing homelessness. Chan et al. (2018) propose an Artificial Intelligence-based decision maker to support the current long-term housing assignment process of RHY. They consider the resource-constrained assignment of various types of housing programs available to youth with different needs. In another study, Azizi et al. (2018) investigate how to prioritize heterogeneous homeless youth on a waiting list for scarce housing resources of different types. They use mixedinteger programming and machine learning to present a policy that trades off efficiency, fairness, and interpretability. Both Chan et al. (2018) and Azizi et al. (2018) focus on how to make the housing assignment process of homeless youth more efficient while working with scarce resources. Both sets of authors propose approaches that prioritize certain groups of youth, rather than expanding housing resources to meet a greater number of homeless youth's needs. Moreover, neither of these studies consider that youth may need particular resources at varied levels at different time periods, nor do they incorporate a time component in their models to reflect the dynamic demands seen in practice. Additionally, both studies assume that there is a centralized list of youth who are waiting to be placed into housing and are available to be assigned simultaneously. It has been well documented that for emergency and transitional independent living shelters, youth often tend to self-select the shelter they visit, possibly due to location, convenience, or practical necessity. The aforementioned assumption may therefore limit their study population to youth in search of long-term housing.

The problem of planning capacity expansion in facilities that provide different products and services has received considerable attention, and many mathematical programming formulations have been proposed. Earlier studies on this topic mainly focused on expanding electric utility capacity, modeling the problem as a linear program (Steiner, 1957; Williamson, 1966; Sherali et al., 1982) . A number of analytics-based methods have been applied to capacity expansion in healthcare settings. Lovejoy and Li (2002) investigate how a hospital can best invest in operating room capacity to provide high-quality service while protecting its profitability by considering the trade-off among three performance criteria: wait time, scheduled procedure start-time reliability, and hospital profits. In another study, Akcali et al. (2006) presents a network flow model that incorporates facility performance and budget constraints to determine optimal hospital bed capacity over a finite planning horizon. Similarly, Woodall et al. (2013) combine simulation and optimization to improve patient flow within the Duke Cancer Institute. These studies are representative of many healthcare planning and scheduling models while featuring an objective function centered on efficiency or patient satisfaction, such as cost minimization, revenue maximization, and waiting time minimization.

While such studies were instrumental in pioneering capacity expansion and allocation models within healthcare systems, they lack a number of critical elements such as the impact on individual and aggregate health outcomes. To fill this gap, Deo et al. (2013) developed an integrated capacity allocation model that incorporates clinical (disease progression) and operational (capacity constraint) aspects for chronic disease treatment, and investigates how operational decisions can improve population-level health outcomes. As our study examines the chronic problem of youth homelessness in the context of trafficking, we use a similar approach to form our objective function, integrating both system efficiency and youth preference.

Against the backdrop of capacity allocation literature, our decision to use capacity expansion models was motivated by an opportunity to consider service delivery time windows and client preferences in a novel manner. Some of the first research efforts to consider service delivery time windows for clients include Kolen et al. (1987) and Desrochers et al. (1992) . Both studies focused on routing limited capacity vehicles while minimizing their total distances to fulfill known demands. While these studies were an important step in the time window literature, they are unable to capture the stochastic nature of client demands and preferences. Demand stochasticity is a necessary component of the real-world modeling conditions and is included in more recent studies (Gocgun and Puterman, 2013; Patrick et al., 2008; Jalilvand et al., 2021) . The work of Gocgun and Puterman (2013) employ a Markov Decision Process perspective to assign randomly arriving chemotherapy patients to future appointment dates within clinically established time windows. While the methods of Gocgun and Puterman (2013) were effective to capture the randomness of demands, none of the existing studies consider varied service needs within different time windows. Furthermore, existing approaches do not consider preferences of the client. In contrast, we (i) introduce the time window concept into our capacity expansion model; (ii) consider the unique preferences of youth, and (iii) incorporate a variety of service needs within different time windows.

Building homeless shelter and service workforce capacity is important to strengthen the ability to deliver effective services to homeless and vulnerable populations (Mullen and Leginski, 2010) .

To the best of our knowledge, no OR-based work exists that focuses on building shelter and service capacity for the homeless as a means to decrease vulnerability to trafficking. To address this gap in literature we focus on projecting the cost-minimizing capacity to deploy to organizations that provide housing and supportive services to homeless youth while considering: (i) multiple organizations that provide multiple capacitated services, (ii) organizations that only serve certain demographics, (iii) stochastic youth arrivals and stay durations, (iv) varying service frequencies and intensities, (v) service delivery time windows, (vi) services provided in periodic and non-periodic fashion, and (vii) youth abandonment.

This section presents a high-level overview of the data acquisition process. New York City (NYC) has the highest rate of homelessness in the United States (The U.S. Department of Housing and Urban Development, 2021), with an estimated 14,946 homeless children and 18,370 single adults sleeping in the NYC municipal shelter system on any given night (Coalition For The Homeless, 2021). Our study population is at-risk runaway, homeless, and unstably housed youth and young adults ages 16-24 in NYC and we consider service providers as the non-profit organizations that provide shelter services to this population. For brevity, we will refer to these populations as RHY and RHY organizations. To better understand existing resources and capacities of RHY organizations, service needs and preferences of the youth, as well as the practicality and feasibility of implementation of any model results, we collected data and feedback from multiple stakeholders: RHY organizations, RHY youth, the New York City Mayor's Office, and the New York Coalition for Homeless Youth. Regular meetings were held with the New York City Mayor's Office and the New York Coalition for Homeless Youth to obtain feedback and suggestions for improvements throughout the data collection and model building phases. These data were used to create provision and need profiles of service providers and RHY in NYC, respectively.

To assess the existing capacities and resources, as well as the nature of these different services in NYC, we conducted structured interviews with five RHY organizations that fund, support, and provide services to RHY. These interviews revealed: (i) the particular demographics of the youth served, (ii) the types of services offered by each organization, (iii) the amount of resources available for different services, (vi) the average length of stay of youth and (v) the services outsourced through referrals to other RHY organizations. The RHY organizations we interviewed provide different types of programs to RHY such as crisis/emergency, transitional independent living (TIL), and long-term housing. The majority of these organizations provide TIL services to youth, accordingly the study focus is the TIL programs. The average length of stay (LOS) of RHY is highly dependent on the RHY organization, program, and youth themselves; in our modeling, the LOS of youth follows a normal distribution. Reasons such as safety issues, mental health problems, strict RHY organization rules, finding stable housing or reuniting with family might cause RHY to abandon the RHY organization earlier than expected. Thus, in our modeling, we assume that a portion of RHY have shorter average length of stay due to abandonment.

Covid-19 protocols and restrictions put immense operating pressure on RHY organizations limiting our data collection. To supplement the data we were able to gather, we also collected publicly available data regarding providers. We supplemented the interview data with public data to generate different RHY organization profiles. These RHY organization demographics and service profiles used in our model can be found in Table 8 and 10 of the Appendix. We also included an incompatibility set to represent a hypothetical RHY organization that could serve any individual regardless of their demographic. This RHY organization is the last RHY organization in our model and was designed to catch any youth who are otherwise unable to be served by the existing resources in NYC, a concept which is further explored in Section 4.1. Additionally, we defined an overflow shelter that will provide services to RHY when capacity within RHY organizations is insufficient to fulfill the needs of any youth.

To represent the demographics, needs and desires of RHY in NYC we generated a simulated dataset of youth profiles using a mix of primary and secondary data. Sources include data from NYC RHY programs, domain experts, and reports, which we used to generate proportions of varied youth demographics and needs in NYC. These demographics are available in Table 11 and Table 12 of the Appendix. Once the proportions were established, we assigned different youth demographics, needs and desires to create a simulated dataset of RHY profiles to be used. Granular data on demographics, needs and desires of RHY are not publicly available and therefore in our simulated dataset we assume all features in a youths profile are conditionally independent.

We formulate the operational challenge of matching homeless youth to RHY organizations that provide housing and support services. Using mathematical optimization, we project the costminimizing capacity to meet the collective needs of youth.

Let T be the number of days and S the set of RHY organizations. We consider two types of RHY organizations serving homeless youth in NYC: (i) RHY shelters that offer housing (bed, food and basic necessities) and a variety of services in-house, denoted by S b ⊆ S and (ii) Service providers that do not offer housing (beds) but provide other support services like medical assistance, mental health support and legal assistance (such as hospitals and mental health clinics) denoted by

Each RHY organization s ∈ S b provides various services in-house, such as housing, healthcare and mental health support to youth with varying intensities. Let P be the set of services provided to youth and E p be the set of intensity levels available for each service p ∈ P . For most services, we consider 3 different intensities shown as ∈ E p : low, medium and high intensity depending on the amount of resources required to provide a particular service to youth. However, certain services have fewer intensity levels, for example, the housing service provided through RHY organizations has a single level of resource intensity, whereas mental health support services have 3 levels of intensity (low intensity: group therapy, medium intensity: weekly individual therapy, high intensity: seeing a psychiatrist and receiving medication). For ease of notation, we use index i to represent every unique service-intensity pair (p, ), where i ∈ I = {(p, ) : p ∈ P , ∈ E p } . For brevity, we refer to each i ∈ I as a service, rather than its more precise "service-intensity" name. A list of all services with their corresponding intensity options can be found in Table 9 of the Appendix. Figure   1 illustrates the two types of RHY organizations providing different services to RHY in NYC.

Each RHY organization s ∈ S b provides a variety of services to a subset of RHY demographics.

For example, some organizations only provide services to females, some are specifically welcoming of LGBTQ+ people, some only serve youth under 21 years, and some do not accept families (RHY that have children of their own). These restrictions cause some youth to have particularly challenging time accessing services. To identify which demographics have reduced access, as discussed earlier in Section 3.1, we include an incompatibility set as an additional RHY organization where youth who are unable to be placed in an existing RHY organization are assigned.

Let Y be the set of homeless youth in the system. Youth y ∈ Y arrives independently to the system on day l y ∈ T with distinct need profile η y and demographics profile α y . Needs profile η y represents the needs that youth y seeks from an RHY organization such as bed, financial assistance, or medical assistance, as well as the intensities and frequencies of these services. Each service i in needs profile η y has a corresponding duration d y,i and frequency f y,i . Demographic profiles α y carry the age, gender, sexual orientation, child status and HT victim information of youth y and each attribute in the demographics profile is denoted as n ∈ N . For example α y [1] is a binary value that denotes whether youth y is a 16-year-old, and α y [|N |] is a binary value that denotes whether youth y has been a HT victim. An illustration of two distinct youth needs profiles are depicted in Figure 2 . We use these needs profiles to match homeless youth with RHY organizations that serve their demographic and have available capacity to fulfill their unique needs. A representation of the matching process is depicted in Figure 3 . Sometimes RHY organizations have insufficient capacity at their facility to provide services to youth and must use creative approaches to meet demand such as by providing hotel vouchers.

We model this capability through the concept of an overflow shelter, which enables youth referral outside of the shelter. The overflow shelter captures the overall additional capacity needed in NYC after the in-house capacity of RHY organizations are extended as far as possible. Therefore, after the matching process, if the capacity within the RHY organization is insufficient to fulfill the needs of each youth, we project the additional capacity required by each in-house service, as well as how many youth should be directed to the overflow shelter. The reason we allow the capacity expansion through the overflow shelter is because RHY organizations have limited capacity within their facilities for additional resources, a phenomenon observed in areas where real estate is at a premium, such as NYC. Such an approach acknowledges the challenge of expanding in-house capacity by building another facility or expanding into a neighboring facility.

We expand upon our explanation of the generalized model framework outlined in Section 4.1. The sets, parameters and decision variables used in our optimization model are summarized in Tables   1, 2 the cost of adding extra resources to in-house services, and (iii) the cost of directing youth to the overflow shelter. Assigning youth to RHY organizations that offer services in-house (S b ) has zero cost since we focus on the capacity expansion, and assignment to organizations that only offer support services such as hospitals and mental health clinics incur positive costs. These considered costs are sufficient to inform the capacity expansion required by each service provider to fulfill youth's needs (Elluru et al., 2019; Alumur et al., 2021) .

Our model is given by: (1)-(4). Objective function (1) 

The first system of constraints (2) are related to the assignment of youth while considering the existing capacities within RHY organizations. The first constraint set (2a) ensures that the number of youth assigned to a RHY organization to receive a service i at shelter s at time t does not exceed Duration of service i ∈ I for youth y ∈ Y f y,i

Number of times service i ∈ I should be provided to youth y ∈ Y ω y,i Time between appointments while providing service i ∈ I to youth y ∈ Y [a y,i , b y,i ] Earliest and latest possible start times of service i ∈ I to youth y ∈ Y k i Periodicity flexibility of service i ∈ I α y Demographics profile of youth y ∈ Y β s Demographics profile of RHY organization s ∈ S η y List of requested services by youth y ∈ Y σ s List of services offered by RHY organization s ∈ S c s,i,t Capacity of service i ∈ I in RHY organization s ∈ S at time t ∈ T µ s,i

Maximum number of in-house resources that RHY organization s ∈ S can have for service i ∈ I within their facility r y,s,i

Cost of assigning youth y ∈ Y to RHY organization s ∈ S for service i ∈ I γ s,i

Cost of allocating one unit of extra resource to RHY organization s ∈ S, for service i ∈ I λ s,i

Cost of directing youth to overflow shelter from RHY organization s ∈ S to receive service i ∈ I Table 3 : List of decision variables used in mathematical modeling.

U y,s,i Binary Youth to RHY organization assignment decision variables are equal to 1 if youth y ∈ Y is assigned to organization s ∈ S to receive service i ∈ I, 0 otherwise X t y,s,i

Time-dependent youth to RHY organization assignment decision variables are equal to 1 if youth y ∈ Y is assigned to RHY organization s ∈ S to receive service i ∈ I at time t ∈ T , 0 otherwise E t s,i Continuous Extra resource decision variables are equal to the amount of extra resources required to fulfill youths' collective demand at RHY organization s ∈ S, to provide service i ∈ I at time t ∈ T O t s,i Continuous Overflow shelter decision variables are equal to the number of youths that are directed to the overflow shelter through RHY organization s ∈ S, to receive service i ∈ I at time t ∈ T the existing capacity of the service (c s,i,t ), extra in-house resources added to that service (E t s,i ) and the number of youth directed to the overflow shelter (O t s,i ). Constraint set (2b) imposes an upper bound µ s,i on the number of extra resources that can be added to an in-house service.

Constraint set (2c) ensures that a youth y is receiving service i from a single RHY organization at time t. Constraint set (2d) ensures continuity of care; that is, it ensures that youth y is receiving a service from a single RHY organization throughout the duration of their stay in the system.

Finally, constraint set (2e) ensures that youth y is not matched with RHY organization s, if RHY organization s does not serve the demographic of youth y. Variable domains are stated in (2e)-(2i).

t∈T i∈I

X t y,s,i ∈ {0, 1}, ∀y ∈ Y, ∀s ∈ S, ∀i ∈ I, ∀t ∈ T,

E t s,i ∈ {0, 1}, ∀s ∈ S, ∀i ∈ I, ∀t ∈ T,

O t s,i ∈ {0, 1}, ∀s ∈ S, ∀i ∈ I, ∀t ∈ T.

(2i)

Interviews with RHY organizations revealed that the timing of service provision (e.g. immediately upon arrival, in a few weeks) is nearly as important as the list of services provided to youth.

All RHY organizations provide essential services that every youth must receive within the first 72 hours of arrival, such as: sexually transmitted disease testing, case management, and mental health assessment. We define the service delivery start time windows for each youth and service pair by [a y,i , b y,i ]; the distributions they follow can be seen in Table 13 of the Appendix, and the system of constraints corresponding to service delivery time windows can be seen in (3). Constraint set (3a)

requires that a youth is not assigned to a service before their earliest start time, or after their latest end time. Constraint set (3b) imposes that the onset of provision of service i to youth y occurs between the earliest start time a y,i and latest start time b y,i . s∈S a y,i −1 t=0 |T | t=b y,i +d y,i +1 X t y,s,i = 0, ∀y ∈ Y, ∀i ∈ I,

The periodicity of services is a necessary component to model real-world conditions. Therefore, the following system of constraints in (4) change depending on whether the service is likely to be provided to youth in a periodic fashion (i ∈ I ω ) or not (i ∈ I nω ). If the service is not periodically served, constraint set (4a) ensures that the summation over time of the time-dependent assignment variable X t y,s,i is equal to the number of times youth requested the service f y,i . In this case the time between the appointments is not significant. On the other hand, if the service is provided to youth in a periodic fashion, the time between the appointments should be equal to the periodicity ω y,i , which we define as follows:

Thus, the periodicity ω y,i is equal to the duration of service d y,i divided by the number of times the service is requested, f y,i . To reflect operational reality, we also introduce parameter k i so as to allow flexibility in assigning youth to appointments within a small window of tolerance around ω y,i .

For example, if service i is provided to youth y weekly every Monday, with flexibility k i we ensure that youth y can schedule their next appointment within [−k i , +k i ] days of every Monday. Thus, if the service is provided to youth in periodic fashion, constraint set (4b) assures that the summation of X t y,s,i over time is equal to the frequency f y,i while considering the periodicity parameter ω y,i and the flexibility parameter k i . Constraint set (4c) imposes that youth y is assigned to receive service i at most once within the same periodicity flexibility window

We now present the computational setup for our experiments, followed by results of solving the mathematical models with varying model parameters, as well as discussions regarding insights. All experiments were conducted using Gurobi Optimizer version 9.1 (2021) and Python 3.8.12, with up to 64 GB memory on an HPC cluster. Each instance was run with the Gurobi MIPGap optimality tolerance parameter set to 0.01. We conduct a variety of experiments on our simulated datasets using formulation (1)-(4).

We evaluate the optimal capacity expansion for 8 RHY organizations (|S| = 8) that provide various transitional independent living services (TIL) at different intensity levels (|I| = 40) in NYC over a 6-month period (|T | = 180 days of the current resources are idle for any of the incoming 500 youth to use.

In light of the uncertainty regarding RHY data, we performed extensive sensitivity analyses around several key model parameters to determine their effect on the optimal capacity expansion for NYC RHY organizations. Table 4 summarizes the ranges of values used in our sensitivity analysis. Results and insights of sensitivity analyses are presented in Section 5.3.

This section discusses the results of our base expansion model, which provides the optimal capacity expansion required by RHY organizations considering our base input parameters. The capacity expansion we present assumes that youth are matched with RHY organizations as efficiently as possible, thus, the capacity expansion we present is likely to represent a conservative estimate on the extra resources required by RHY organizations to fulfill the collective needs of youth.

We allow for four different bed types that a youth can receive: (i) existing in-house beds, (ii) extra in-house beds that expand the capacity within the facility, (iii) overflow shelter beds when bed space in the facility has been maximally expanded, and (iv) incompatibility set (Ψ) beds when none of the existing 8 RHY organizations are able to serve a particular youth's demographic. The demographics each RHY organization serves, services they provide and the number of beds they currently have are provided in Tables 8 and 10 of the Appendix. Figure 4 depicts the number of youth receiving these different types of beds from 8 of these RHY organizations in the optimal solution to our base model.

All of the RHY organizations in the system used their existing capacity and added the maximum amount of extra beds that their facility is able to handle, yet still needed to resort to creative options for beds, accommodating youth via the overflow shelter. Such results underscore the critical need for more RHY organizations in NYC. Out of 500 youth, a total of 67 youth received an extra inhouse bed, 81 youth had to receive a bed through the overflow shelter due to capacity restrictions within existing facilities, and 2 youth had to be placed in the incompatibility set (Ψ) as none of the 8 RHY organizations were able to serve their particular demographic. The model demonstrates that simply adding more capacity to existing shelters is insufficient as certain demographics particularly experience access challenges. Figure 5 represents the percentage of capacity expansion required by each RHY organization. The average total percentage increase in resources needed for extra in-house beds and overflow shelters, collectively, across the 8 organizations is approximately 68%. Figure 4 shows that the second RHY organization requires the greatest amount of additional capacity. This is because this organization has relatively fewer entry restrictions compared with other organizations, thus can more broadly serve different populations. Accordingly, the optimal solution increases the capacity of Organization 2 more so than any other shelter. Notably, this organization already has 80 TIL beds within their facility. Therefore, considering the youth who are directed to the incompatibility set Ψ, and the significant need for capacity expansion in Organization 2, our results illustrate the impact of collective entry requirements (such as age and gender restriction) on meeting the needs of youth in the city. 

Recall that RHY organizations provide support services to youth (i) with existing in-house resources, (ii) with extra in-house resources added to expand the capacity within the facility and (iii) through referrals (S o ). Figure 6 illustrates the number of youth who received these services by source (note that the intensities of services are combined). In our simulated data, medical assistance is required by almost every youth, which is currently provided primarily through referrals.

Additionally, a large portion of legal and financial assistance must be provided through referrals, while educational assistance requires additional in-house resources as it is less costly to provide.

When considering the addition of resources, referrals are preferred over in-house services as the latter is more costly. However, as Figure 6 illustrates, there is a constant need for referrals for specific services: medical, legal, and financial assistance. This underscores that adding extra in-house resources would increase convenience and access for youth, even if more costly. Figure 7 breaks down the number of additional appointments required by each RHY organization to provide the support services (note that the extra in-house resources and referrals are combined). These results demonstrate that the additional resources needed for medical, legal, financial and educational assistance are significant in reducing youth vulnerability to trafficking. It can be seen in Figure 7 that the number of youth requesting mental health support, substance abuse, childcare and practical assistance is lower than other support services. This reflects youth needs and mirrors insights gleaned from meetings with key stakeholders. RHY hesitate requesting mental health and substance abuse support as judgment, doubt, pride, fear, and misinformation may come into play. 

We now discuss sensitivity analysis insights gained by varying model parameters as per Table 4, including changes in the optimal capacity expansion plan and RHY organization types to expand.

The number of RHY seeking TIL opportunities from the NYC municipal shelter system is highly correlated with external variables such as weather conditions, safety concerns, and the political environment . Accordingly, we perform a sensitivity analysis on youth arrival rate, using information from our service provider interviews.

All considered RHY organizations show a similar capacity expansion behavior, which we illustrate with the largest shelter, RHY Organization 2. Figure 8 shows the number of youth directed to receive an overflow shelter bed. Up to 10 beds can be added to Organization 2 (E t 2,1 = 10). In the scenario with the fewest youth (|Y | = 300), only adding extra in-house beds appears sufficient, although 11 youth remain unable to receive services with existing resources. Unsurprisingly, the number of youth being directed to the overflow shelter reaches its maximum when |Y | = 700; even when 10 more beds are added in-house, 97 youth are unable to receive a bed within the facility. Figure 8 : The effect of total number of youth in the system, on the day-to-day trend in number of youth at the overflow shelter referred from Organization 2 over 6-month period. Overflow Cost Change -92% 0% +56%

From a system-wide perspective, our base expansion model assumes that 500 youth arrive to the system over a six-month period. Considering alternative arrival numbers, in Table 5 we see that the six-month average of the overflow shelter need increases by 279% when the number of youth increases by 40% (|Y | = 700); and decreases by 97% when the number of youth decreases by 40% (|Y | = 300). While these re-sults are unsurprising, quantifying the extent to which the arrival rate affects capacity is critical supporting data for capacity expansion decisions.

Interviews with multiple service providers revealed that the average length of stay of youth is around 60 days. Yet, as discussed in Section 3.1, the service duration (d y,i ) that youth y receives greatly depends on the individual youth, organization and service type. Thus, we vary the length of stay that youth y receives service i (d y,i ), which in our base expansion model follows a normal distribution with mean of 60 days and standard deviation of 15 days. Figure 9 : The effect of average length of stay on the day-to-day trend in number of youth at the overflow shelter referred from Organization 2 over 6-month period. Variation in duration d y,i also affects the number of times a youth needs each service (f y,i ), thereby influencing the costminimizing capacity expansion. As seen in Figure 9 , decreasing the average length of stay by 20% for 500 youth almost completely eliminates the need for the overflow shelter. However, reduced length of stays at a shelter might disrupt the rehabilitation process and increase future vulnerability and recidivism for those at risk of trafficking. On the other hand, a 20% increase in the duration results in 77 youth that are unable to access an existing bed, which also raises their vulnerability to trafficking.

A shorter service duration results in reduced overflow costs, slightly lower referral costs, and less capacity expansion as revealed in Table 6 . There is a tradeoff when considering voluntarily reducing the service duration for youth; while this may alleviate some current capacity limitations and temporarily improve access to housing and support services, it will likely disrupt the much needed efforts for youth to be able to successfully exit trafficking and exploitative experiences. Capacity expansion plans should consider the average service duration of youth in each organization, and should provide for the possibility of extending the stay of youth until a safe and stable living arrangement is identified.

A youth may stop receiving a particular service or abandon the system completely for various reasons such as feeling limited by organizational restrictions, avoiding conflict and abuse, relapse, health concerns, or the inability to find stable housing (Donley and Wright, 2012) . Uncertainty exists regarding the number of youth abandoning the system while receiving services. Accordingly, we performed sensitivity analysis regarding the percentage of youth abandoning the system (θ), which in our base model is set to 20%. Service provider interviews revealed that almost half of youth who abandon the system leave within the first three days of arrival. Therefore when modeling abandonment, we assume that half of the youth that abandon have a service duration (d y,i ) that follows a normal distribution with mean of 3 days and standard deviation of 0.5 days, whereas the other half has their duration decrease to 33.3% of the original duration. Figure 10 : The effect of abandonment on the day-to-day trend in the number of youth at the overflow shelter referred from Organization 2 over 6-month period. Table 7 shows that a 10% increase in the abandonment percentage decreases the average need for overflow by 86%. In comparison, a 10% decrease in abandonment increases the average overflow by 119%. When the percentage of youth abandoning the system is between 10-30%, Figure 10 shows that in Organization 2 the in-house capacity and extra in-house resources are exhausted, leading to the need for overflow around day 40. When the abandonment level is at 10%, the number of referred youth actively in the overflow shelter reaches a daily maximum of 54. In contrast, only 11 youth are directed when abandonment increases to 30%. 

Access to scarce healthcare resources is a common challenge during the COVID-19 pandemic, especially for vulnerable populations (Van Dorn et al., 2020) . We evaluate the healthcare needs of RHY as access to healthcare would reduce their vulnerability to HT (Duncan and DeHart, 2019) . Over the course of the COVID-19 pandemic, semi-regular meetings with NYC stakeholders revealed that the number of youth arriving to the RHY organizations had decreased due to concerns of infection. RHY organizations also experienced significant staff availability and attrition due to positive COVID-19 cases and various vaccine mandates. To evaluate the effect of COVID-19 on the need for capacity expansion, we decrease the existing capacity of in-house services to 50% of the base model and decrease the number of youth arriving to Y = 400.

While the halving of in-house capacity left more physical space for extra in-house resources, staffing beds during the pandemic remained a challenge. As can be seen in Figure 11a , the need to direct youth to the overflow shelter beds still exists, with 86 of the 400 youth unable to receive an existing bed. Figure 11b shows that in an optimal expansion, only 23% of youth could be accommodated with existing in-house medical resources; and while an additional 5% could be accommodated by adding extra in-house resources, a full 72% of the youth must have their medical needs met through referrals. However, during the pandemic referrals to larger healthcare institutions became less desirable due to health concerns (Li and Yu, 2020) and more difficult to access as healthcare resources were redirected and social distancing took effect. Such a situation highlights the need to additional in-house medical resources. Our model is able to evaluate the tradeoff between the increased costs of providing in-house services and access to these services in situations such as a pandemic.

Operations research and analytics research efforts to disrupt the supply of sex and labor trafficking victims are relatively new (Dimas et al., 2022) . We use optimization-based techniques to evaluate cost-minimizing capacity expansion options for shelters serving runaway and homeless youth and young adults (RHY), which we believe is a first such attempt to address the vulnerability of a population extremely susceptible to trafficking (Wright et al., 2021; Hogan and Roe-Sepowitz, 2020) .

We present an integer linear programming model that incorporates stochastic youth arrivals and length of stay; youth abandonment; service delivery time windows, as well as periodic and nonperiodic services. Through careful variable definition, we allow for three types of time-dependent capacity expansion: adding extra resources to in-house services, directing youth to an overflow shelter when there is no longer an ability to expand in-house capacity, and an incompatibility set that serves youth who are otherwise unable to receive housing and support services due to demographic mismatch. While we illustrate our approach with a case study aimed at expanding transitional independent living (TIL) service capacity for RHY organizations in New York City (NYC), the same (or a similar) model could be readily employed for other non-profit shelter organizations.

We discovered new insights that have the potential to substantially impact HT disruption efforts by increasing the accessibility of services aimed at reducing youth vulnerability of being trafficked and exploited. Our systematic and data-driven analysis addresses extremely resource-constrained contexts by providing a capacity expansion strategy for NYC with practical impact. Overall, our study represents an innovative use of mathematical modeling to address a capacity expansion problem with a broader societal impact. The aforementioned insights were informed through analysis of model output based on estimated RHY profiles using a mix of primary and secondary data. While a potential limitation, we believe our secondary sources have provided a reasonable representation of the current demographic and needs profiles of youth, and moreover a solid foundation for further studies. The results reported in this study can be used as a template for future analysis, including the analysis of primary data collection on RHY demographics, needs and desires.

While this study specifically focuses on capacity expansion of RHY organizations within NYC, there remains great potential for organizations in other locations provided sufficient data exists.

Further extensions include optimally locating the additional overflow shelter and extending the model to allow RHY organizations to share resources with one another. Additionally, as demand for housing and support services greatly exceeds the existing capacity, the optimal deployment will require more capacity than is feasible to add at one time. Therefore, identifying an actionable capacity expansion plan that details how to implement the capacity deployment over time would offer additional utility. Moreover, there remains potential to embed our optimization approach into a decision-support tool to further facilitate the decision-making process in NYC.

In conclusion, this study provides preliminary evidence of the value of incorporating the preferences and needs of vulnerable populations into humanitarian operations research problems. Our approach benefits government and nonprofit decision-makers by offering a means to effectively evaluate the allocation and expansion of scarce resources, while readily enabling sensitivity analyses to examine the effect of demand changes on the optimal expansion of housing and support services.

The authors confirm that the aggregated data parameters supporting the findings of this study are available within the article and its supplementary materials. As stated in the Institutional Review

Board, due to the nature of this research, participants of this study (such as RHY organizations) did not agree for their data to be shared publicly; thus detailed, identifiable supporting data is not available. 

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The RHY and RHY organization demographic, needs and service profiles are shown in Appendix. Age 21 below 1 1 1 1 1 1 1 1 21+ 1 0 1 1 0 0 1 1 Gender Cis-gender Male 0 1 1 0 1 1 1 1 Cis-gender Female 0 1 1 1 1 1 1 1 Transgender Male 1 1 1 0 1 1 1 1 Transgender Female 1 1 1 1 1 1 1 1 Non-binary 1 1 1 0 1 1 1 1 Genderqueer 1 1 1 0 1 1 1 1 Intersex 1 1 1 0 1 1 1 1

Heterosexual/straight 1 1 1 1 1 1 1 1 Gay 1 1 1 0 1 1 1 1 Lesbian 1 1 1 1 1 1 1 1 Bisexual 1 1 1 1 1 1 1 1 Queer 1 1 1 1 1 1 1 1 Questioning 1 1 1 1 1 1 1 1 Asexual 1 1 1 1 1 1 1 1 Pansexual 1 1 1 1 1 1 1 1 Other Children 0 1 1 1 1 1 0 0 Citizen 1 1 1 1 1 1 1 1 Immigrant 1 1 0 1 1 1 1 1 HT Victim 1 1 1 1 1 1 1 1 Bed l y,i a y,i + T RIA(1, 2, 4) Mental Health l y,i a y,i + T RIA(1, 2, 4) Medical l y,i a y,i + T RIA(1, 2, 4) Substance abuse l y,i a y,i + T RIA(2, 3, 7) Crisis 24 hour services l y,i a y,i + T RIA(2, 3, 7) Long term housing l y,i a y,i + T RIA(2, 3, 14) Legal l y,i a y,i + T RIA(2, 3, 14) Service coordination l y,i a y,i + T RIA(2, 3, 7) Practical l y,i a y,i + T RIA(2, 3, 7) Financial l y,i a y,i + T RIA(2, 3, 7) Life Skills l y,i a y,i + T RIA(2, 3, 7) Employment l y,i a y,i + T RIA(2, 3, 14) Education l y,i a y,i + T RIA(2, 3, 14) Childcare l y,i a y,i + T RIA(2, 3, 14)