Dynamic resource allocation and coordination for high-load crisis volunteer management Open Access

Humanitarian emergencies or crises result from natural or man-made events that can be threatening to the well-being of a large number of people, especially to those who are vulnerable. According to its website, the Red Cross responds to over 65,000 natural and man-made disasters in the USA alone every year. According to the 2015 World Disasters Report by the International Federation of Red Cross and Red Crescent Societies (IFRC, 2015), the number of total disasters between 2005 and 2014 not including wars, diseases or epidemics is 6,311 (an average of 631 disasters per year worldwide). This amounted to 839,342 deaths, 1,935,585 affected people and over $1.6 trillion in estimated damages. Querying the Emergency Events Database (http://emdat.be/emdat_db) for natural disasters between years 2000 and 2017 shows 7,451 natural disasters, which resulted in 1.3m deaths and $2.31 trillion in damages.

Government and non-profit emergency response organizations usually respond to such crises, and these organizations rely not only on their staff but also on hundreds or thousands of volunteers who offer their services. The Corporation for National and Community Service (Corporation for National and Community Service (CNCS), 2007) reported that volunteers in 2014 contributed 7.7bn h equaling an estimated $173bn for their services. The Urban Institute reported that four in five non-profit organizations (with budgets over $25,000) use volunteers and more than nine in ten organizations can take on more volunteers (Hager and Brudney, 2011). The Red Cross website shows that 90 percent of the humanitarian work is carried out by volunteers and 95 percent of disaster relief workers are volunteers. In other words, volunteers are the lifeblood of many non-profit organizations, and many of the services provided by nonprofits would cease to exist without volunteers (Handy et al., 2008).

The study of spontaneous volunteers in 2015/2016 flooding by Harris et al. (2017) in the UK reveals that volunteer management at disasters should be aware of the possibility that there is a need for “surge capacity” which cannot be met by official responders. As such, spontaneous volunteers should not be excluded but rather: “have a system for greeting and noting contact details; avoid immediate rejection” (p. 369), and consider reassigning tasks to them that have a low risk, requiring fewer skills and oversight.

Although volunteers play a pivotal role in disaster response, they also bring a unique set of management challenges to organizations, which often find themselves in the midst of a complex workforce optimization problem when a crisis hits. There are profound differences between managing paid employees and volunteers, and the two very often require completely different approaches (Sampson, 2006). Managing volunteer labor can in fact be significantly more difficult because of the non-compulsory nature of the work. For example, volunteers cannot be held to do regularly scheduled shifts or even be available when and where organizations require them. Furthermore, volunteers, if dissatisfied, can leave with little cost to themselves, which can be costly for the organization. In times of disasters, moreover, volunteer supply can be unpredictable in both number and skill requiring careful attention on part of the organization – to recruit them, assign them to tasks according to their preferences and capabilities, and motivate and retain them, all the while advancing the mission of providing disaster relief. In addressing these concerns, organizations also must cope with sudden and irregular volunteer arrivals, short lead time for planning, the uniqueness of each case, uncertainty, complexity of coordination, limited infrastructure and scarcity of resources on the ground (Mayorga et al., 2017; Balcik and Beamon, 2008; Sheu, 2007). Organizations may also need to provide housing and food to volunteers in larger-scale scenarios (Simo and Bies, 2007). In addition, the phenomenon of material convergence is also known to occur, whereby the flood of resources arriving in the aftermath of a disaster often hinders the efficiency of high-priority resource allocation and utilization (Holguín-Veras et al., 2012a). As a result, the logistical problem in matching the volunteer supply to the demand for services in emergency situations can be very complex.

We focus on the development and application of models that would best optimize the available labor supply and related resources in times of crisis. In particular, in high-load crisis scenarios volunteer resources need to be allocated in rapidly changing conditions where there are many unmet demands. In such scenarios, relying on an ad hoc approach generally leads to suboptimal outcomes, and a more formalized and automated approach is necessary. Our paper takes this approach as a point of departure and addresses volunteer resource allocation in particularly high-load crisis scenarios, an area which has received little research attention in the past.

The main contributions of this paper are as follows. First, we conduct interviews with managers who have worked in large and well-known volunteer organizations to identify major distinguishing features of crisis volunteer resource allocation problems. We provide a detailed characterization firmly rooted in reality and highlight the unique needs of this problem class in contrast to more well-studied types of scheduling and resource allocation problems. Second, we provide a flexible and extensible mixed integer programming (MIP) framework for modeling crisis resource allocation problems which enables optimal assignment of volunteers, renewable and non-renewable resources to tasks in highly volatile crisis scenarios without requiring probabilistic resource or demand characterizations. This framework provides a formulation of the distinguishing features identified for this problem class using a very small set of constructs to enable flexible and practical applications across a broad range of problems.

Importantly, the framework proposed is designed to work within the constantly evolving scenarios characterizing this problem class of resource allocation in times of humanitarian crisis. While stochastic methods are often successfully applied to handle uncertainty, they often rely on assumptions that cannot be guaranteed in crisis scenarios. Thus, the proposed framework employs a dynamic resource reallocation approach. We also show through several examples that this framework may readily be extended to incorporate scenario-specific concerns. Our approach fills a gap in the literature and presents a new direction for solving humanitarian logistics problems. We demonstrate the viability of this approach on problems of realistic size and scale through computational experiments on a large set of benchmark problems spanning a broad set of problem characteristics.

In their review of the literature on humanitarian logistics, Kara and Savaser (2017) classified OR problems in disaster management literature into several categories dominated by location and routing problems. The category of “allocation” received only 6 percent of the papers reviewed, and even scarcer were studies address the allocation of human resources to specific tasks. In their extensive literature review of optimization models in emergency logistics, Caunhye et al. (2012) stated, “Research on manpower management during large-scale emergencies is lacking. Most of the uncertainties that affect post-disaster operations also affect the planning and scheduling of manpower” (p. 11). Even less is published on planning and scheduling of volunteer labor, with a few exceptions (Mayorga et al., 2017; Lassiter et al., 2015; Falasca and Zobel, 2012; Garcia and Rabadi, 2011; Sampson, 2006; Gordon and Erkut, 2004). Volunteer labor assignment problems are different from traditional labor assignment problems as methods that work with paid labor cannot be easily adapted to unpaid labor especially in emergency scenarios by non-profit organizations (Sampson, 2006). This requires a greater understanding of the organization’s objectives and challenges as well as the non-trivial costs of recruiting and managing volunteers which include matching their skills to their preferences for particular tasks and schedules (Handy and Srinivasan, 2005, 2004).

Altay and Green (2006) published a review article on operations research/management science (OR/MS) about disaster operations management in which they reviewed the literature up to year 2004. Galindo and Batta (2013) followed up with another review paper with a similar framework for the literature between 2005 and 2010. Both papers concluded that despite the significant growth in the OR/MS literature for disaster operations management, research gaps still existed. While mathematical programming was the most used method in the papers reviewed, virtually none of the reviewed papers addressed the issue of human resource allocation and scheduling including the planning and scheduling of volunteers, tasks and other resources. A similar conclusion is evident in a recent literature review on post-disaster response and recovery planning by Ozdamara and Ertem (2015); the literature examined issues related to relief delivery, vehicle routing, casualty transportation, traffic management and mass evacuation; however, it did not tackle the issues of scheduling volunteers and other resources.

Simpson and Hancock (2009), who reviewed evolution of OR in emergency management (for all emergencies, not only for disasters), identified volunteer management as one of the research areas in need of further development. The authors cited Wright et al. (2006) and Buck et al. (2006), who recognized the dearth of research incorporating volunteers and emergent groups despite their playing a pivotal role they play in humanitarian crises in the early phases. In the recent years, there have been more OR/MS publications on disaster operations management and humanitarian logistics, but they are mostly focused on facility location, demand planning, managing inventory, relief distribution, routing and transportation (e.g. Moreno et al., 2016; Lei et al., 2016; Tofighi, et al., 2016; Van der Laan et al., 2016; Zokaee et al., 2016; Huang et al., 2015; Battini et al., 2014; Das and Hanaoka, 2014; Safeer et al., 2014; Wex et al., 2014).

Although there are a number of papers on resource allocation, their scope is mostly limited to the allocation of equipment, vehicles, commodities, etc., and rarely address human resource allocation and scheduling explicitly, with the exception of the few papers discussed below. Although a vast body of the OR literature covers workforce planning and scheduling, it is widely accepted that differences between commercial and humanitarian logistics are quite distinct (e.g. see Beamon, 2004; Beamon and Kotleba, 2006; Sampson, 2006; Van Wassenhove, 2006; Holguín-Veras et al., 2012b), hence not easily adaptable from one to a another. Caunhye et al. (2012) noted in their review paper on the use of optimization models in emergency logistics that research on human resource management during large-scale emergencies is lacking, and post-disaster operations affect the planning and scheduling of manpower.

Almost all OR/MS review papers emphasized the need to consider uncertainty for humanitarian logistics problems, including Hoyos et al. (2015) in their review of the OR literature related to disaster operations management with emphasis on stochastic components. The vast majority of stochastic models make certain assumptions about the statistical and probabilistic distribution used, which, in reality, may be unknown or invalid, although robust scheduling can deal with uncertainty in which some slack (or idle) time is optimally inserted in the schedule to account for future schedule changes. This approach may produce good results in repetitive and non-emergency scenarios; however, it may not work well or be valid in high-load crises, as it is unlikely that idle time will be inserted into a schedule for disaster management. Furthermore, the robust scheduling approach typically focuses on the start time of the tasks without much consideration to human resource shifting, which may arise in crises situations and must therefore be considered. In our paper, we take a different and more practical approach to handle uncertainty by using a deterministic but dynamic MIP scheduling model that gets updated and re-optimized as circumstances on the ground change. Caunhye et al. (2012) concluded that volunteers, survivors and people unaffected by the disaster may join and leave relief workforces in a dynamic fashion, and researchers such as Holguín-Veras et al. (2012b) identified optimal dynamic allocation of resources as one of the areas in need of further research.

While researchers have previously considered volunteer scheduling, to our knowledge, no papers have addressed the planning and scheduling of human resources (or volunteers) in conjunction with other renewable and non-renewable resources. Gordon and Erkut (2004) developed a spreadsheet-based support tool specifically for the annual Edmonton Folk Music Festival and produced master and individual volunteer schedules using a combination of integer and goal programming. However, it was tailored to the festival activities making it less applicable to other events and organizations. Garcia and Rabadi (2011) developed constraint and goal programming models to enable a church schedule its weekly operations utilizing volunteer teams. Each team has its own scheduling objective: either to maximize participation or the amount of time each volunteer has “off.” Conflicts between jobs and volunteers are prevented in addition to respecting several hard constraints. The previous papers were limited in their models and methods to non-emergency events; humanitarian emergencies are inherently different and require modeling with a different set of objectives and constraints.

Falasca and Zobel (2012) introduced a bi-criterion mathematical model assigning volunteers to tasks in order to balance the unmet demand against the number of instances of volunteers being scheduled to undesirable tasks, over the scheduling horizon. They employed both efficiency frontier and fuzzy logic as solution methods to enable the decision maker to understand the trade-offs between these two objectives. Their approach, however, has several limitations: first, they assumed volunteers are interchangeable in terms of skills; second, they do not consider other (renewable and non-renewable) resources that may have to be simultaneously allocated to tasks; and, finally, their model finds solutions to a static problem, which implies the model is unable to reschedule volunteers nor reassign tasks when things on the ground change.

A recent paper by Mayorga et al. (2017) considered uncertainty in the number of volunteers who may spontaneously show up to help in the recovery effort of a disaster (e.g. debris cleaning), and modeled the problem as a multi-server queueing system. In addition to assuming that the inter-arrival and service times are exponential, the authors also treated all volunteers as a homogenous group with respect to skills and assumed that other necessary resources are readily available.

Lassiter et al. (2015) considered a robust optimization approach for allocating and training a limited number of volunteers to a variety of tasks in order to minimize the total unmet demand, when the demand for each task group is uncertain. They did task matching by introducing skill levels for tasks and volunteers, and took volunteer preferences into account by presenting the trade-offs between volunteers’ preferences and the unmet demand. This required managers to examine Pareto optimality curves which maybe unpractical in humanitarian crises. Furthermore, they do not make a distinction between the different resource types (renewable vs non-renewable) that must be allocated simultaneously for a task to be completed. Such practical constraints are addressed in our model, as discussed below.

Our approach, which is based on extensive discussions with humanitarian organizations including the Red Cross, the Corporation for National and Community Service and others, differs from existing models and contributes to the literature in several important ways. First, we consider allocation and scheduling of volunteers based on their skill and preferences in conjunction with the presence of other renewable and non-renewable resources. We make no assumptions about the distributions of the task demand, availability of resources and the presence of any other restrictions; rather we take the inputs and update the plan as the circumstances on the ground change over a rolling scheduling horizon. For instance, as task and/or resources levels change over time, the decision makers need only to update the model with the new information to obtain a new schedule. To avoid unnecessarily reshuffling resources, our modeling framework strikes a balance between the schedule quality and stability. This framework models and addresses the core concerns that underlie many types of volunteer resource allocation problems while providing extensibility to be adapted to specific scenarios.

In order to gain a thorough understanding of crisis volunteer management, we supplemented an extensive literature review by interviewing volunteer managers who have worked in major organizations including the Red Cross, Points of Light, AmeriCorps and the Corporation for National and Community Service (Clay, 2016; Nankervis, 2016; Payne, 2016). Some of these organizations focus on providing a specific type of support, while others focus largely on coordinating the efforts of different organizations. In general, we found a strong agreement between the literature and the interviews. However, the latter provided a great deal of additional information and a much more detailed understanding of crisis volunteer management. In this section, we outline crisis volunteer management practices as they occurred in some of the major organizations and identify a set of recurrent characteristics found across the spectrum of volunteer resource allocation problems.

Responses to crises depend on the specific nature of the crisis. However, there are very general response structures for different types of events, such as hurricanes or earthquakes, and normally a significant amount of improvisation is required to adapt to the specific crisis. The nature of the event, will no doubt dictate the number of volunteers and the different types of work needed. Smaller responses typically involve 50–100 volunteers, while large responses may involve several hundreds or thousands (Clay, 2016; Payne, 2016). After an event unfolds, there is typically an assessment of the situation followed by a determination of the organizations immediate objectives. Volunteer organizations tend to prioritize two types of needs above all others: first, those related to safety and preservation of lives, and second, the specific needs of children, the elderly, the uninsured and other groups who have no other means of meeting them (Nankervis, 2016; Clay, 2016).

When disasters occur, it is typical for one organization to take on a role as the local lead and focus on coordinating efforts. Different volunteer organizations tend to specialize in different types of work, and so the division of labor often (but not always) falls along organizational lines (Nankervis, 2016; Clay, 2016). Accordingly, in many cases, individual organizations are given general ownership and management of specific responsibilities. However, some cases require higher degrees of coordination across multiple organizations, particularly when resources need to be shared or when tasks require multiple kinds of skilled workers (Payne, 2016). As a result, the process of allocating volunteer resources varies from one crisis to another. One widely used approach is to have volunteers sign themselves up for tasks rather than be allocated (Payne, 2016). In high-demand scenarios where there is a shortage or significant excess of volunteer resources, or when there are rapidly changing requirements, this approach can lead to suboptimal results; a more coordinated approach is needed. In some cases, this coordinated approach may involve volunteer allocation primarily by and within individual organizations, while other cases may require a more centralized approach for proper coordination. Thus, methods aimed at enhancing volunteer resource allocation and coordination must be able to work at both levels, at higher, more centralized levels and lower, more organization-centric levels.

While there is considerable variation in crisis volunteer management scenarios and practices, we identified six recurrent characteristics of crisis volunteer resource allocation problems which are summarized below.

Crises are inherently uncertain and evolving, and one of the distinguishing characteristics of crisis volunteer management is the need to be able to cope with changing circumstances and rapidly adapt. This capability is absolutely essential to success (Clay, 2016; Nankervis, 2016). As a crisis unfolds, plans must often be changed on short notice and, often, there is a lack of information on the specific needs far in advance. Meeting the stated objectives of safety and care for vulnerable groups generally carries the highest priority in the decision-making process. Other more diverse circumstances creating different needs do not pose a threat to the organizations because they allow for broader volunteer engagement and better utilization of the diverse skills of the volunteers. At the same time, these diverse needs may require changes in plans and resource allocation, and, at some level, this may impose undesirable logistical costs and may even adversely impact volunteer morale. As a result, it is important to try to minimize disruption where possible, particularly in the reallocation of certain resources. An additional consequence of rapidly changing conditions is that planning horizons are typically shorter in crisis scenarios than in other contexts; indeed planning horizons of only a few days are typical (Payne, 2016).

As experience has shown, there is almost always a huge response from ordinary citizens who want to help and who unexpectedly turn up at the disaster site (Helsloot and Ruitenberg, 2004). Indeed, it is common to see hundreds and thousands of volunteers spontaneously arrive at the scene (Barsky et al., 2007). This can create congestion, interfere with ongoing response activities, and those volunteers will themselves require resources, thereby creating further pressures on emergency response systems (McLennan et al., 2016). Therefore, to utilize volunteers effectively, schedules will need to be adjusted and reassignments made as new volunteers arrive (Sauer et al., 2014). This, however, should not be at the expense of the schedule stability, which can be maintained by minimizing the number of changes to the current schedule.

The need to incorporate different types of volunteers with different skills is required in certain volunteer scheduling contexts (Lassiter et al., 2015). From our interviews, we found this to be an essential feature of crisis volunteer management as well. From a planning perspective, volunteers are viewed as renewable resources to be allocated along with other resources such as equipment, food etc. Volunteers with different skills are seen as different types of resources, although it is entirely possible that some volunteers have multiple skills and may belong to multiple resource categories (Payne, 2016). While the specific manner of dividing volunteers into classes varies based on the context, there is a pervasive distinction between permanent or regular volunteers and event-based (sometimes called scenario-based) volunteers which exists within all forms of crisis volunteer management. Permanent or regular volunteers are those who have committed to long-term service with an organization and have received appropriate training. By contrast, event-based volunteers often show up to offer help in the midst of a crisis and typically have little or no formal volunteer training and are given a training session before being deployed.

One of the ways crisis volunteer allocation and scheduling differs from many other types of scheduling and allocation lies in the interdependencies of the resources rather than tasks at hand. In workforce and project scheduling, there are often task many interdependencies that require the optimization of task sequences. In contrast, in crisis volunteer management, tasks generally have fixed times and the interdependencies occur between resource types. An ubiquitous example of this involves ensuring that there are adequate ratios of regular to event-based volunteers on each task to provide supervision and guidance. A soft form of this constraint has to do with the fact that many volunteers often come in groups (such as church-groups or school-groups) and prefer to work alongside others in their group. Clearly, such preferences, while important, are not of the highest priority but should be accommodated whenever possible (Nankervis, 2016).

It is well-known in the volunteer management literature that it is essential to work within volunteer availabilities (Balcik and Beamon, 2008; Sheu, 2007). This is no different in crisis volunteer management, and volunteers must be scheduled to tasks only according to their availabilities.

Crisis scenarios can range from excesses to shortages of volunteers, and in the case of excesses, it is a highly frustrating experience for volunteers to offer their time and not be utilized (Mook et al., 2007; Liao-Troth, 2008), and this can introduce significant planning challenges (Clay, 2016; Nankervis, 2016; Simo and Bies, 2007). Furthermore, it is also important for volunteer skills and preferences to be considered (Handy and Srinivasan, 2004, 2005), and this holds true in crisis volunteer scenarios as well. Volunteers want to be used and feel useful, and utilizing them to the fullest extent possible is desirable for many reasons. Volunteer labor provides both significant cost savings and fosters the goodwill of volunteers, making them more likely to continue offering their services in the future. Good management of volunteers further fosters positive public perception and garners ongoing support for the major volunteer organizations.

Ensuring workers are not overworked is basic to all forms of workforce management. In crisis scenarios, volunteers arrive as they are able and are often present on the scene for a relatively short duration (several days to a week), and it is important in crisis volunteer management to ensure that volunteers have adequate time off in between their tasks. In cases that involve longer-term volunteer utilization, more traditional labor constraints may also apply (such as a maximum of 40 h per week).

While each crisis is unique, these six characteristics appear to occur across the broad spectrum of crisis volunteer management scenarios. In some cases, the practice of having volunteers select their desired tasks may lead to satisfactory achievement of objectives; in others, assignment is the route for optimizing the use of volunteers. In situations that involve frequent and significantly changing resource needs, or where there is a shortage or surplus of volunteers, a more coordinated and flexible approach is required to best meet objectives and balance all concerns.

In this section, we provide a problem formulation based on the key needs of crisis volunteer management and resource allocation identified above. We then provide a core MIP model for the basic form of the problem which addresses the key concerns that underlie all the types of crisis volunteer resource allocation reviewed. Finally, we provide additional constraints and extensions likely to be present in common variations of the problem.

We assume a set R of renewable resources (primarily volunteers, but can include equipment and other types of resources as well) and a set J of distinct resource types. There is a set T of tasks, where each task k∈T requires at least D_jk units of resource type j. In cases where task k is allocated less than D_jk units of resource type j, there is a shortage penalty S_jk incurred for each unit short. We also assume an upper limit L_jk on the units of resource type j that may be allocated to task k. For each resource i∈R and resource type j∈J, there is an indicator r_ij=1 if resource i is of type j and 0 otherwise. Each resource i also has an availability indicator A_ik=1 if resource i is available to work on task k and 0 otherwise. In order to prevent resources from being assigned to conflicting tasks, a conflict indicator u_ikk' is employed, where u_ikk'=1 if resource i cannot be assigned to both tasks k and k' and 0 otherwise. This facilitates both ensuring appropriate time off for volunteers between assignments as well as prevention of simultaneous assignments. Additionally, there is the need to incorporate resource interdependencies. The most common type is to ensure there are adequate numbers of permanent or regular volunteers paired with event-based volunteers on each task. Other types of resource dependencies may also exist (e.g. every bulldozer must have a qualified driver). To represent such dependencies, we utilize a dependency ratio d_ijk which represents the minimum ratio (expressed as a fraction) of units of resource type i required per resource type j on task k. Finally, to address the need to ensure that each volunteer is maximally used, there is a satisfaction benefit factor B_ik which is incurred when resource i is assigned to task k. This may also be used to allow individual volunteers to give higher or lower preferences to certain tasks.

In order to meet the rapidly-fluctuating needs of crisis volunteer management, our approach assumes the frequent reallocation of resources, aiming to best meet changing objectives while minimizing unwanted reallocation and schedule disruptions where possible. In some cases, tasks may need to be interrupted and modified in progress, while, in other cases, resources deposited at the site of certain future tasks may need to be reallocated to the site of other more pressing tasks as more information is learned. Accordingly, we generalize the view of resource allocation decisions to be both updatable and to possess a prior history in an earlier plan revision. This is represented by a prior task allocation indicator y_ik for each resource i, where y_ik=1 if resource i was previously allocated to task k and 0 otherwise. Deviations in allocation from one planning iteration to the next can involve the moving of certain resources away from some tasks and onto others. Though necessary, such deviations are often undesirable and consequently we associate penalties with these occurrences: $mik+=$ penalty for assigning resource i to task k in a new schedule revision when not previously allocated (i.e. positive deviation cost), and $mik−=$ penalty for not assigning resource i to task k in the new schedule revision when previously allocated (i.e. negative deviation cost). We note that in this framework, time is assumed to be connected to tasks whenever it needs to be considered, rather than represented as an explicit modeling construct. This allows for more flexible modeling capabilities.

The basic objective is to allocate resources to tasks to maximize the total allocation benefit attained minus the sum of the total shortage and deviation costs, while respecting the task conflicts as well as resource availability and dependency constraints. This use of a weighted sum-based approach was chosen over others because it provides a great deal of flexibility to optimize the desired outcomes in relative order while doing so in a conceptually simple manner. It neither requires separate solution for different objectives nor does it require complicated or highly precise setting of weights to be effective. Different overarching concerns, such as shortages of resources vs changes in resource allocation, may be given weights of differing magnitudes to ensure the more important concerns are always optimized ahead of the less important ones. Within a specific concern (such as maximizing the total satisfaction benefit attained), differing weights of similar magnitude may then be used to optimize that particular outcome.

We first define the following decision variables:

• x_ik=1 if resource i is allocated to task k in the updated schedule, 0 otherwise;

• v_jk= shortage (in units) of resource type j on task k;

• $wik+=1$ if resource i is assigned to task k in the new schedule revision but not in the previous one, 0 otherwise (i.e. positive deviation); and

• $wik−=1$ if resource i is assigned to task k in the previous schedule revision but not in the new one, 0 otherwise (i.e. negative deviation).

The MIP model is formulated as follows:

s.t.:

In this model, the objective function (1) is to maximize the total allocation benefit attained minus the sum of the total shortage and deviation costs. Constraints in (2) ensure no resource scheduled outside its availability. Constraints in (3) ensure no conflicting assignments are made. Constraints in (4) ensure the minimum required resource dependency ratios for each task. This can be used to make sure event-based volunteers are paired with regular volunteers, or to ensure no bulldozer is assigned to a task without someone to drive it for example. Constraints in (5) together with (11) ensure v_jk is the shortage in units of resource j on task k, or 0 if there is no shortage. Constraints in (6) and (7) ensure that the positive and negative deviation indicators for each resource and task (⁠$wik+$ and $wik−$⁠) are properly set. Constraints in (8) ensure that no task is allocated more than its upper limit for any resource type. Constraints in (9) ensure that only the needed types of resources are allocated to each task. Finally, constraints in (10) and (11) ensure the appropriate variable domains.

We illustrate the nature of this problem by a simplified example. In this example, we assume there are three resource classes: regular volunteers (8), bulldozers (1) and bulldozer drivers (1). Resources 1–8 are regular volunteers, resource 9 is the bulldozer driver and 10 is the bulldozer. In addition, there are four simultaneous tasks with initial demands as follows: Task 1 requires two regular volunteers; Task 2 requires three regular volunteers, one bulldozer and one bulldozer driver; Task 3 requires two regular volunteers; and Task 4 also requires two regular volunteers. It is assumed that a driver must be allocated wherever a bulldozer is allocated.

In this scenario, the most important concern is assumed to be resource shortages for the tasks (particularly the bulldozer and driver, which are mission-critical) followed by volunteer satisfaction. To model the concerns in this order, we assume that Tasks 1 and 2 have an initial shortage cost of 2 units per regular volunteer, and Tasks 3 and 4 have costs of 1 unit per volunteer (reflecting an initial priority for Tasks 1 and 2). The bulldozer has an initial shortage cost of 50 units for Task 2, reflecting its criticality. We additionally assume that each volunteer receives 1 unit of benefit for each task assigned. A solution for this initial scenario is shown in the first row of Table I.

Now assume that just as these tasks are beginning, the volunteers report back new information which changes the scenarios. In particular, Task 4 now requires four regular volunteers plus a bulldozer and driver. In addition, the task priorities have changed and are reflected by updated shortage costs: Tasks 1 and 2 now have shortage costs of 1 unit per regular volunteer, and Tasks 3 and 4 now cost 2 units per regular volunteer short. Additionally, Task 4 has a bulldozer shortage cost of 80 units, reflecting its priority over Task 2. Reallocation costs are 1 unit for each volunteer moved away from their initially assigned task and 20 units for reassigning the bulldozer away from its initial task. A solution for this example is shown in the second row of Table I.

The objective function of the initial schedule is 8 [=9 (total benefit received for task allocation of volunteers) – 1 (regular volunteer shortage in Task 4)]. The objective function for the updated schedule is −70 [= 9 (total benefit received) – 5 (Task 1 and 2 shortages) – 50 (Task 3 bulldozer shortage) – 20 (bulldozer reallocation cost) – 4 (total volunteer reallocation costs)]. It is important to note that the cost and benefit weights are chosen to reflect the relative importance of priorities, and these allow the schedule to be constructed and reconstructed in a manner that addresses the different concerns in priority order and leads to the most beneficial result. In this example, the most pressing tasks received the resources first, and when the conditions changed, the updated schedule minimized the weighted movement, reassigning only five out of the ten total resources and resulting in 50 percent schedule stability. Had reallocations been deemed costlier than the shortages, there would have been less reallocation resulting in a higher stability.

In addition to the core formulation above, we also show how the model may be extended to accommodate additional types of requirements. We show three extensions addressing concerns frequently found in volunteer scheduling. Because of the core model’s structural simplicity, it should be noted that many other types of extensions may be likewise formulated and incorporated with relative ease, and depending on the crisis at hand, such constraints can be activated or deactivated.

In some cases, it may be necessary to specify a limit on time worked per time period (e.g. no more than 40 h per week). This is complementary to ensuring that volunteers get enough time off between assignments (e.g. ensuring that a volunteer does not work consecutive shifts may be accomplished by using constraint 3 above). To ensure an overall limit over a given time period, we may first define the following:

• H=set of time periods;

• h_l=set of tasks in time period l∈H, where h_l ⊆ T;

• t_k=time duration for task k; and

• E_il=maximum total time resource i can be allocated during period l.

We can then add the following constraint:

In addition to renewable resources, there may be limited consumable resources that likewise must be allocated. In order to accomplish this, we first define the following:

• J′=set of consumable resource classes;

• V_j=available units of consumable type j;

• $Djk'$=demand (in units) for consumable resource type j by task k;

• $Sjk'$=shortage penalty per unit of consumable resource type j for task k;

• $xjk'$=units of consumable resource j to allocate task k;

• $m'ik−,m'ik+=$penalty for reassigning consumable resource i away from or to task k, respectively; and

• $w'ik−,w'ik+=1$ if consumable resource i is reassigned away from or to task k, respectively, 0 otherwise.

Next we subtract the following expression in the objective function to minimize the total consumable resource shortage and reallocation penalties:

We note that the desire to use all volunteers whenever possible has no analogue with consumable resources. Consequently, there is no reason to allocate more of these resources than needed to a task, so we add (14) to accomplish this and (15) to ensure the total amount of any consumable resource is not exceeded:

In many cases, it may be desirable for certain groups of individuals to work together. This is particularly important when volunteer groups train and travel together. To incorporate this extension, we first define the following: G=set of resource groups, where each g∈G is a group of resources that must be allocated together.

Here, g ⊆ R for each g∈G. We then add the constraints in (16) to ensure the resources in each group are allocated together:

In some cases, a softer form of this requirement may be desired such as, for instance, requiring that anyone in a particular group is always allocated to tasks with at least one other member of their group. Here we may assume a number q_g ⩽ |g| for group g such that anyone from g must always have at least q_g − 1 members of their group allocated with them on any task. To accomplish this, we may first define binary variable X_gk=1 iff any member of g is allocated to task k. We then add the following constraints:

Constraints in (17) ensure X_gk=1 if any member of g is allocated to task k while those in (18) ensure at least q_g members of g are allocated to each task.

Another concern which may occasionally arise is the need to integrate transportation costs into the decision making (Clay, 2016). To incorporate this concern, we assume each task has an implicit location. We define a sequence of work shifts E={e₁, e₂,…,e_|E|}, where each e_i ⊆ T and e_i+1 denotes the shift immediately following e_i. In order to account for transportation activities, we force all resources to a task in each shift. Accordingly, we also include dummy OFF tasks in each shift (and perhaps multiple ones representing different locations) with zero demand, infinite upper limit and zero resource dependencies in order to model time off. Let c_jk denote the transportation cost of going from task j to task k. Noting that costs only apply when going from one task to the next, c_jk=0 for all tasks j and k not in consecutive shifts. We then define binary variable b_ijk=1 if resource i is assigned to both tasks j and k. To integrate transportation costs into the formulation, we first subtract these costs as shown in the following equation from the objective function:

Then we add the following constraints:

Constraints in (20) ensure that b_ijk=1 whenever both x_ij and x_ik are 1. Constraints in (21) force each resource into exactly one task assignment during each shift (if only a dummy OFF shift) in order to account for all transportation activities. Finally, to avoid conflicting constraints, we let T' designate all non-OFF tasks and rewrite the indexing over tasks in constraints (9) as k∈T'.

Volunteer managers cannot be expected to have expertise in integer programming. Furthermore, it is highly impractical to deal with mathematical models, solvers, data, weight determination and results interpretation in the midst of a crisis scenario. Consequently, the modeling framework proposed is envisioned to be embedded within a resource allocation software tool. Such a tool would conceivably connect to appropriate volunteer and resource databases, allow users to define tasks and update their resource requirements in real time, set resource allocation and reallocation priorities and modify them in real time, and finally build and automatically update the schedule or resource allocation plan in response to these updates. The tool would likely also provide users the ability to make manual adjustments to the generated plans.

A related consideration has to do with how the benefit and penalty weights would be determined. Although there are different approaches possible, one straightforward approach would be for users to prioritize the importance of allocating different resource types to different tasks, ensuring each volunteer gets utilized according to their preferences, minimizing disruptions to the allocation plan when updates occur, and other objectives. In the context of a software tool, these priorities could readily be converted to weights of differing magnitude to ensure that the different objectives are optimized in order of priority.

To illustrate this with a simple example, consider a scenario where all assignment benefits and deviation costs are equal (i.e. $Bik=mik−=mik+$⁠, for all i, k) and where resource shortages should be prioritized over assignment preferences and reassignments. Let U designate an upper bound on the number of assignments and reassignments possible for the given problem (the sum of the number of assignment and reassignment variables suffices). Then we can ensure that shortages are always optimized above assignment preferences and reassignments by choosing each S_jk such that $Sjk>U(Bik+mik−+mik+)$⁠.

Although the design and implementation of a software tool is beyond the scope of this paper, it is readily conceivable for the proposed integer programming framework to be incorporated into such a tool, thereby enhancing the decision-making by optimally allocating and reallocating resources in real time.

A key objective of this research is to provide an approach to volunteer resource allocation that can realistically be incorporated into the decision-making process to improve high-load crisis volunteer management, specifically in scenarios with changing requirements and significant mismatches in the supply and demand of resources. Computational tractability is a crucial requirement for this, and such an approach must be capable of producing solution for problems of varying conditions and realistic sizes. In order to validate our MIP approach as computationally viable, we performed extensive experimentation on a large set of benchmark problems generated to span a wide variety of problem characteristics. While our benchmark problems do contain certain realistic aspects, the primary intent of this experimentation is to validate the computational tractability of our approach across a broad spectrum of problem characteristics that may be encountered. In this section, we describe the major aspects of the computational experimentation.

The experiments we conducted involved generating and solving many benchmark problem sets with differing characteristics. For these experiments, we implemented two separate models: the core MIP model from Section 4.2, and the core model with the transportation constraints from Section 4.3.4 These models were implemented in IBM ILOG OPL and solved using the CPLEX solver, and we developed a Python program to both generate the benchmark data and run the experiments. Parallel experiments were carried out for each model. Within each experiment problems are organized into sets, where each set contains problems generated from a specific distribution (discussed further below). Each problem corresponds to a single experimental trial.

As noted in the problem formulation, the model is intended to be run at repeated intervals to respond to changing circumstances and thus presupposes a prior resource allocation. Accordingly, each trial entails a sequence of four distinct steps:

1. Generation of a new initial problem.

2. Solution of the initial problem.

3. Altering of the initial problem to simulate changing conditions while also incorporating the resource allocation prescribed by the previous solution. This produces an updated problem version.

4. Solution of the updated problem version.

In these experiments, both the generation of initial problems and the altering to produce updated problems are based on distributions specified by different parameters. Each initial problem has no resources previously allocated to any tasks (i.e. all y_ik=0), and in each updated problem, the allocations resulting from the initial solution are incorporated (i.e. each x_ik from the initial solution becomes y_ik in the updated problem). After all problem solutions were obtained, the solution times were analyzed to assess the computational tractability of this approach. All benchmark problems and corresponding solutions may be obtained by contacting the authors.

The benchmark problems are designed to generally resemble the structure of volunteer allocation problems in crisis scenarios while providing the ability to be altered along multiple dimensions. As stated above, the main objective of these experiments is to validate computational tractability rather than to simulate scenarios with perfect realism. Consequently, while our benchmark problems contain realistic aspects, certain problem features (such as the number of resources or resource requirements) may be deliberately exaggerated in many cases.

While problems differ widely, there are a few features that each problem shares. One common feature encountered in all types of crisis volunteer management is the need to require minimum ratios of permanent or regular volunteers to event-based volunteers in each task assignment. Each problem is thus conceived to have a designated resource class of permanent volunteers as well as multiple types of event-based volunteer resources. Additionally, because planning horizons are generally only two to three days in length, all problems utilize a three-day time horizon and assume two 12 h shifts per day (Clay, 2016). It is further assumed that no volunteer can be allocated to two or more consecutive shifts to prevent being overworked. Beyond these basic fixed features, a set of parameters are used to generate problems with different characteristics.

The parameters shown in Table II are used to generate the specific model inputs for each problem instance. In every trial, each problem has both initial and updated versions. This allows the investigation of performance both in circumstances where there is no prior allocation and where there is prior resource allocation as well as where rescheduling and reallocation must occur in response to condition changes. Accordingly, some of the parameters concern the initial problem generation while others control how the problem will be altered in its updated form. In realistic scenarios, a problem may be altered from one planning iteration to the next by any combination of the following: addition or subtraction of tasks; addition or subtraction of resources; changes in resource availabilities; or changes in resource demands. Computationally, it is possible to simulate the addition of tasks by zeroing out all resource requirements for certain tasks in the initial version and then increasing these requirements in the updated version. A similar approach may be used for representing the subtraction of tasks and for addition and subtraction of resources. Each of these changes has the net effect of shifting demand for specific resources. Accordingly, the primary control of problem alteration degree used in these experiments is the TD parameter. The manner in which the parameters above are used to generate the specific model inputs in both initial and updated versions is shown in Table III.

Finally, we note our choice of using uniform random distributions rather than normal distributions. This type of distribution is frequently used in the generation of scheduling benchmark problems because it induces higher levels of variance and tends to result in harder problems (Rabadi et al., 2006; Weng et al., 2001). This is consistent with our stated objective of evaluating computational tractability.

In order to generate a sufficiently wide range of problems, we employ a 2^k factorial experiment design to facilitate the manipulation of four problem characteristic factors in a combinatorial manner. This design is applied to experiments on each of the two models. These factors are as follows: problem size (PS), resource demand (RD), variability (V) and emphasis (E). Problem size is concerned with the number of resources and the number of tasks involved, where larger problem sizes involve more resources and tasks. Resource demand is concerned with the amounts of resources needed by each task relative to the amounts available. Higher resource demands involve tasks that require more resources. Variability is concerned with how much change occurs in resource demand between the initial and updated version of a problem. Higher levels of variability involve greater changes in resource demands. Finally, the emphasis involves the different orders of priority reflected in the objective function. There are two such emphases: quality and stability. Quality relates to how well a solution meets the task demands, while stability relates to how minimally resource allocations change from one planning iteration to the next. An emphasis on quality has higher penalties for resource shortages and lower penalties for resource reallocations, while an emphasis on stability does the opposite.

In our experimental design, we utilize two distinct levels for each design factor, low (−) and high (+), and investigate every combination of factors leading to 2⁴=16 distinct problem classes. For each problem class, ten replicates were generated for both initial and updated problem versions, leading to a total of 320 benchmark problems for each model (160 initial and 160 updated), or 640 problems in all. While factors 1–3 are primarily quantitative in nature, the emphasis (E) factor is categorical. For this factor, we let the low-level designate quality and the high-level designate stability. For all other factors, the low and high levels indicate low or high degrees of the factor present. In Table IV, the problem generation parameter settings are shown for each different factor level.

As a final note, we point out that the problem size levels above translate to a range of 370 volunteers with 36 tasks to 1,450 volunteers with 72 tasks.

All computational experiments were executed on a computer having a 2.3 GHz quad-core processor with 32 GB RAM running Windows Server 2012. The model was implemented in OPL and solved with CPLEX using IBM ILOG Optimization Studio 12.4, and the program for generating the data and running the experiments was written in Anaconda Python.

A summary of results from the computational experiments for the base model are shown in Table V. The results summarize the times (in seconds) required to obtain optimal solutions for each of the problem sets. The problem sets are additionally separated by initial and updated problem versions to better understand the expected solution times in each case.

Perhaps the most immediate observation to be made is that optimal solutions were able to be obtained in less than 2 min for all problems, with approximately 75 percent being obtained in less than 10 s. This includes very large problems having 1,450 volunteers and 72 tasks with high resource demands and variability. Because we are primarily interested in computational tractability rather than understanding how the different factors affect solution times, we omit a full Analysis of Variance (ANOVA) with respect to solution time. However, it can be clearly seen that larger problem sizes (P+) consistently require longer solution times in the initial problem versions, as do those with higher resource demands (RD+). In particular, the combination of P+ and RD+ produced the hardest problems and required the longest solution times by far.

Another interesting insight is that the updated problem versions were solved in less than 10 s, with the majority being solved in less than one second. This holds true even for problem-classes with high levels of variability (V+). This indicates that the most computationally difficult tasks generally lie in obtaining the initial solutions rather than in reallocating resources. Thus, optimal reallocation of resources can be done very quickly even in cases where there is a large change in requirements. One of the most critical aspects of crisis volunteer management is the need to respond quickly and accurately to changes when they occur, and this result demonstrates the potential of our approach to support this kind of decision making.

Table VI shows a summary of results for the transportation-based model. In this case, all problems are solved within 5 s. This indicates that rather than increasing complexity, these added transportation constraints, in general, narrow the search space and allow optimal solutions to be found quickly.

As has been discussed above, the changing nature of crisis scenarios often requires a reallocation of resources. Depending on the nature of the conditions and the relative importance of concerns, some conditions will require greater amounts of resource reallocation than others. We define the resource allocation constancy (RAC) measure in order to quantify resource allocation stability between an initial and updated problem version. RAC expresses the percentage of resource allocation that remains constant in an initial and updated problem solution. Letting c_ik=1 iff y_ik=x_ik, RAC is defined as follows:

In these experiments, the four conditions and concerns which are commonly encountered were altered systematically and in combinations. The effects of these factor combinations on the amount of reallocation are summarized in Table VII, expressed as mean RAC. In addition, an ANOVA determined statistically significant relationships between factor combinations and the resource allocation stability. These results are shown in Tables VIII and IX.

In Table VII, a number of interesting relationships between the problem design factors and the resource allocation stability across model types can be seen. The emphasis factor (E) clearly affects the stability, with an emphasis on quality (−) generally resulting in lower stability levels than an emphasis on stability (+). The significance of this relationship is clearly seen in both the core and transportation models as shown in Tables VIII and IX.

The variability factor (V) likewise has a strong relationship to stability with the high level (+) resulting in significantly lower stability scores overall, particularly in the cases where the emphasis is on quality. In Tables VIII and IX, this is seen to hold true in both the core and transportation models, with the V main effect and the V:E interaction effect both showing significance at α=0.001 in both models. The resource demand factor (R) also has a significant effect, with the low level (−) resulting in lower stability scores overall. These differences are most pronounced when the emphasis is on quality, and as can be seen in Tables VIII and IX, this relationship holds true in both the core and transportation models. While this may seem counterintuitive at first glance, it appears to be due to the fact that generally, in lower resource demand scenarios, the resources are more interchangeable. This relationship was reflected in the manner in which the problems were generated as shown in Table IV. In cases where changes in resource allocation are lightly penalized, we may expect to see more reallocation of resources.

In summary, these results indicate that our model has the potential to find solutions in short amounts of time for problems with a broad range of characteristics. This includes cases that go far beyond what is typical in terms of number of resources and tasks, resource demands and variability. Accordingly, our approach shows promise to improve decision making in crisis volunteer management. In addition, we have found that resource demand and variability levels as well as emphasis have a strong relationship to resource reallocation stability across the two different model types.

In this paper, we studied the problem of crisis volunteer resource allocation, particularly under high-load conditions where there is a mismatch between available volunteer resources and demands or where there are extreme and frequent changes in requirements. Through a combination of literature reviews and interviews with managers from major volunteer organizations, we identified six key characteristics of crisis volunteer resource allocation problems. Based on these six characteristics, we developed an MIP-based framework for modeling crisis volunteer resource allocation problems. This framework is designed to be utilized within the constantly changing conditions that are inherent to this problem-class present in crisis management. Furthermore, it provides adaptability via a dynamic resource reallocation approach that minimizes the impact of changes while meeting the desired objectives. This framework also utilizes a small set of constructs to support modeling a broad range of problems, and its simplicity further supports adding scenario-specific extensions with ease. It may be applied in either organization-centric allocation scenarios or scenarios that require coordination of resources between multiple organizations. We further demonstrated the viability of this approach for solving problems of realistic size and scale through a set of computational experiments run on a large set of benchmark problems. These problems were generated to contain a very wide range of conditions that may be encountered in real life. In even the most demanding problems, we were able to obtain initial optimal solutions in under 2 min, and in all cases optimal reallocation was accomplished in under 10 s. Finally, we showed that the objective function emphases as well as the resource demand and variability levels have a strong impact on the amount of reallocation stability, both individually and in combination.

While this research shows potential to enhance decision making in high-load crisis volunteer resource allocation, there is more work needed in this area to begin impacting practice. One of the major limitations of the proposed methodology is its reliance upon integer programming, a technique which is complex and unfamiliar to a typical volunteer manager, and which cannot be utilized in crisis scenarios without first being embedded within a user-friendly software system. Further work in designing an appropriate software system is thus needed to enhance utilization. This research does not address how volunteer and resource optimization should best be integrated into the decision-making process, and there are likely to be significant challenges surrounding this concern, particularly given the diversity of organizations and the different approaches used in resource allocation in crisis scenarios. Moreover, in cases where there is a good match between available volunteers and demand and where conditions are more static, ad hoc approaches may provide satisfactory results and provide a more personal touch to volunteer management. It may thus be preferable to operate in an ad hoc mode when possible and switch into an optimization mode when the complexity of the situation becomes hard to manage. Further research can provide answers in how to incorporate resource and volunteer optimization into the decision-making process as well as how best to switch into and out of this optimization performance mode.