A Minimum-Risk Dynamic Assignment Mechanism Along with an Approximation, Heuristics, and Extension from Single to Batch Assignments

07/02/2020 ∙ by Kirk Bansak, et al. ∙ University of California, San Diego 0

In the classic linear assignment problem, items must be assigned to agents in a manner that minimizes the sum of the costs for each item-agent assignment, where the costs of all possible item-agent pairings are observed in advance. This is a well-known and well-characterized problem, and algorithms exist to attain the solution. In contrast, less attention has been given to the dynamic version of this problem where each item must be assigned to an agent sequentially upon arrival without knowledge of the future items to arrive. This study proposes an assignment mechanism that combines linear assignment programming solutions with stochastic programming methods to minimize the expected loss when assignments must be made in this dynamic sequential fashion, and offers an algorithm for implementing the mechanism. The study also presents an approximate version of the mechanism and accompanying algorithm that is more computationally efficient, along with even more efficient heuristic alternatives. In addition, the study provides an extension to dynamic batch assignment, where items arrive and must be assigned sequentially in groups. An application on assigning refugees to geographic areas in the United States is presented to illustrate the methods.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 14

page 16

page 25

page 26

page 27

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

In the classic linear assignment problem, items must be assigned to agents in a manner that minimizes the sum of the costs for each item-agent assignment, where the costs of all possible item-agent pairings are observed in advance. This is a well-known and well-characterized problem, and many algorithms exist to attain the solution. In contrast, less attention has been given to the dynamic version of this problem where each item must be assigned to an agent sequentially upon arrival without knowledge of the future items to arrive.

This study proposes an assignment mechanism that combines linear assignment programming solutions with stochastic programming methods to minimize the expected loss when assignments must be made in this dynamic sequential fashion, and offers an algorithm for implementing the mechanism. The study also presents an approximate version of the mechanism and accompanying algorithm that is more computationally efficient, as well as an extension to the dynamic assignment of batches of items. An application on refugee assignment is presented to illustrate the methods.

Section II introduces the classic static formulation of the linear assignment problem. Section III delineates the properties of the dynamic context of interest in this study. Section IV then develops a minimum-risk assignment rule for this dynamic context—i.e., a rule that minimizes the expected loss incurred from needing to assign items sequentially rather than via the static optimal solution that is employed when all items are observed simultaneously. Section V shows how this assignment rule can be implemented via simulation methods, presenting a formal mechanism along with an implementing algorithm. Section VI then derives an approximate version of the mechanism, along with an accompanying algorithm, that improves upon computational efficiency. Section VII describes how the mechanisms can easily accommodate situations in which the number of agents does not equal the number of items. Section VIII presents alternative heuristic mechanisms that further improve upon computational efficiency. Using real-world data from one of the largest refugee resettlement agencies in the United States, Section IX then illustrates the mechanisms via a simulated application on assigning refugees to geographic areas in the United States. Section X provides an extension to dynamic batch assignment, showing how the proposed mechanisms and accompanying algorithms for one-by-one dynamic assignment can be adapted to the context where items arrive and must be assigned sequentially in groups rather than individually. Finally, Section XI concludes.

Ii Classic Linear Assignment Problem

Let there be items that must be assigned to agents.111This is easily generalized to capacity units among agents for . For item , let denote the agent to which item is assigned. Let denote the cost incurred from the assignment of item to agent , and let denote the x cost matrix containing for all and .

Let denote a full assignment of items to agents such that each item is assigned to exactly one agent and each agent is assigned to exactly one item (i.e. a bijection). The classic linear assignment problem is to determine the optimal , or , such that the sum of the costs is minimized:

Equivalently, the problem can also be formulated as follows:

subject to the constraints that

where X is a binary x matrix, where for entry

This is a linear programming problem that can be solved via well-known methods, such as the Hungarian algorithm

(Kuhn, 1955; Munkres, 1957) and related methods (e.g. see Bertsekas, 1991). This standard setup will be referred to as the “static” version of the linear assignment problem.

Iii Dynamic Assignment

A Background

In many real-world applications, however, assignment of items to agents must proceed by (logistical, physical, financial, or other) necessity according to additional constraints or specifications that are not present in the static formulation of the assignment problem. Solutions to the static version of the problem do not always easily generalize to such cases.

Possible constraints and alternative specifications include inter alia the need to assign items dynamically (i.e. sequentially), imperfectly or unobserved assignment costs for certain item-agent pairings, unknown future item arrivals, agent availability that changes over time, and the ability to change assignments given new information. This study is interested in a version of the “dynamic” assignment problem, as described below.

B Dynamic Context of Interest

The dynamic version of the problem considered in this study shares the following similarities with the static case: there are agents and items, one item must be assigned to each of the agents, each of the agents is available for the assignment of any item (provided an item has not already been assigned to it), and the goal is to minimize the sum of the assignment costs. Note that this is easily generalized to capacity units among agents for , which will be discussed later in Section VII. In addition, the dynamic version considered here is also defined by the following additional features:

  1. Blind Sequentiality: The items are assigned in an order that is exogenously determined and unknown in advance, and each item must be assigned before item is assigned.

  2. Non-anticipativity: Each item is assigned with knowledge of , the

    -dimensional vector with elements that denote the costs of assigning item

    to each of the agents, but without knowledge of for all .222See Shapiro et al. (2009) for usage of non-anticipativity in stochastic programming contexts.

  3. Permanence: Assignments cannot be changed once they are made.

Note that this setup does not assume or require that there are a finite number of item types, nor does it assume or require a known fixed pool of items that the items will be drawn from.

In contrast to the static case, these additional features have important implications for the information that is available at each stage in the assignment process and hence for optimal decision-making procedures. First, instead of being able to determine the optimal assignment for all items simultaneously, the fact that each item must be assigned sequentially implies that an optimal decision rule must be developed for an individual item . Furthermore, the optimal decision rule for item can utilize but cannot utilize for all , which are unobserved. In addition, assuming items and their cost vectors are independent of one another, there is also no value from utilizing for all , since earlier assignments cannot be changed due to permanence.

Real-world use cases include assigning refugees or asylum-seekers to different geographic areas with predetermined capacity, assigning jobs/loads to a predetermined set of workers, and other problems characterized by the following features:

  • There is known, limited capacity (agents or agent bandwidth) to serve/accept items.

  • Service is first-come, first-served.

  • Service is required immediately due to high costs or lack of capacity for delaying/holding items.

  • Items cannot be reallocated after initial assignment.

  • It is possible to know (or estimate) the cost of assigning a new item to any given agent.

C Related Literature

Past scholarship on dynamic assignment mechanisms differ from the present study in terms of their characterizations of the dynamic context and/or their objectives.

Dynamic assignment problems have been considered, in particular, in transportation and shipping applications (Spivey and Powell, 2004; Powell et al., 2002; Godfrey and Powell, 2002a, b; Pillac et al., 2013), though with problem features that are distinct from the dynamic context considered in this study. For instance, Spivey and Powell (2004) present a dynamic assignment mechanism that seeks to minimize the expected costs over the future. However, in their context, the agents available at any given time are unknown and potentially changing, there is a finite and fixed set of item types, at the discrete arrival times any number of agents and items may be available (or not available) for assignment, and items need not be assigned immediately (though the cost could increase for a delayed assignment). Given these complexities, their mechanism provides approximate solutions, and finding optimal solutions is computationally intractable.

Other research has introduced different types of complexity and wrinkles to the classic assignment problem. For instance, Ünver (2010) presents a dynamic matching mechanism meant for the kidney exchange market, focused on minimizing waiting time, a fundamentally different objective than that in this study. Korsah et al. (2007) present what they call a dynamic version of the Hungarian algorithm, where “dynamic” refers to the ability of the algorithm to efficiently repair an initial solution if any of the costs change, rather than re-implementing the Hungarian algorithm on the full problem.

Finally, there has also been work on dynamic matching in the economics literature on market design, with researchers investigating the extent to which market design mechanisms can be adapted from the usual static setting to the dynamic setting (Andersson et al., 2018; Kurino, 2009; Kennes et al., 2014; Kadam and Kotowski, 2018b, a; Doval, 2014). Unlike the objective of this study, market design research on dynamic matching has not focused on optimizing the final costs/payoffs of the assignment process. Instead, the focus is placed on preserving key mechanism properties that are desirable in matching theory, often as a function of the preferences of the units involved. Such properties include matching stability, pareto efficiency, strategyproofness, and envy-freeness.

Iv Dynamic Risk Minimization Objective

Let there be items that must be assigned to agents. Let

be a random vector generated by an unknown probability distribution

, let denote a realization of the random vector for item , and let denote the element of (i.e. the cost of assignment of item to agent ). Let denote the order of item assignment. Let denote an assignment for items such that each item is assigned to one agent and no two items are assigned to the same agent. That is subject to . Thus, as previously defined: an assignment of items to agents such that each item is assigned to exactly one agent and each agent is assigned to exactly one item.

Now define the following:

That is, denotes the minimum cost of any assignment of all remaining items () to all remaining agents (all agents to which items were not assigned) while satisfying the requirement that no two items among are assigned to the same agent as defined by .

The following marginal loss function for a single assignment of item

to a particular agent under can be defined as follows:

In this function, denotes the sum of costs for a conditional optimal assignment where the assignment for item is fixed with agent and the remaining items after item are assigned by the optimal static process to the remaining available agents. The final term, , denotes the sum of the costs for the optimal static assignment of all items starting with item to the remaining available agents. Thus, given all realizations of for , this definition of loss (or regret) measures the increased cost for assigning all remaining items except for by the optimal static process and fixing item ’s assignment with agent , relative to assigning all remaining items including by the optimal static process.

Now, let the following denote the expected loss (risk) of assigning item to under the probability distribution , given that only is observed (i.e. the cost vectors for items are not observed):

The following minimum-risk dynamic assignment rule for item , , can thus be formulated:

This assignment rule exhibits a similar structure as multistage stochastic programming problems (Shapiro et al., 2009), where an optimal decision must be made in a particular stage that takes into account what is expected to occur (but not yet observed) over the full sequence of later stages that will ensue. In this case, the decision stage is the assignment of item , and the assignment of all items encompasses the sequence of later stages.

Furthermore, the following two assumptions are made:

  1. Stagewise independence: Each cost vector is independent of all others.333See Shapiro et al. (2009) for usage of stagewise independence in stochastic programming contexts.

  2. Stationarity of stochastic process: The sequence of new cost vectors is produced by the same probability distribution, , that produced past cost vectors.

Let denote the expected loss of applying the minimum-risk assignment rule for item :

Under the stagewise independence and stationarity assumptions, the expected sum of the expected loss of applying the minimum-risk assignment rule for all items (that is, ) equals the expected difference between the overall sum of costs resulting from the minimum-risk dynamic assignment and the sum of costs that would have resulted from a static optimal assignment (i.e. ) if all items were observed simultaneously. A proof of this property can be found below.

V Simulation-Based Implementation

In the dynamic context, at the point when item must be assigned, the only element of the loss function itself that is observed is . However, the expectation of the loss function can be estimated. As in other stochastic programming contexts, an optimal decision rule for a single stage (i.e. an assignment rule for any item ) can be formulated via Monte Carlo simulation applied to the sequence of later stages (i.e. the assignment of all items ). That is, given the assumptions of stagewise independence and stationarity, the expectation of the loss can be estimated via simulation, specifically by taking random draws of to create simulated complementary sets of vectors to consider alongside . In many applications, there should be historical data on past items and their cost vectors, and thus vectors for can be randomly drawn from the historical data. If historical data do not exist, the random draws must be generated by explicitly modeling .

Let a randomly drawn vector set be denoted by , where denotes the set drawn, , and is large. Then, for each the simulated loss for assigning item to each of the possible agents can be computed, and then finally, the average loss across all draws can be computed for each of the possible agent assignments. The agent assignment resulting in the lowest average loss is thus the minimum-risk assignment for item . This mechanism can be represented as follows.

Mechanism 1

Minimum-Risk Dynamic Linear Assignment Mechanism

where corresponds to as applied to the randomly drawn items (cost vectors) in the set . For , and hence effectively drops out of the expression.

Note that the final expression in the loss function () does not depend upon so it can be dropped from the mechanism. Practical implementation of the mechanism is delineated in the algorithm below.

Algorithm 1

Minimum-Risk Dynamic Linear Assignment Algorithm

  • Consider items numbered , and begin with a known set of agents numbered . Let denote the set of available agents and initialize . Pre-specify a large number of simulation iterations to perform.

  • for each item in :

    • Determine and fix (i.e. observed cost vector for item upon the item’s arrival).

    • for in :

      • Simulate (randomly draw) vectors, .

      • for each :

        • Provisionally assign item to agent .

        • Solve static linear assignment problem for simulated items/vectors under the assumption that item has been assigned to agent .

        • Store resulting sum of costs for the assignment of items . Denote this sum by .

    • For each , compute the average sum of costs, .

    • Assign item to the agent with the lowest average sum of costs: .

    • Remove from .

Vi Approximate Implementation

The process above requires the solution of many static linear assignment problems. Specifically, when assigning item , for each of the draws, the algorithm must solve linear assignment problems (or a number corresponding to the number of available agents, if there are fewer than agents each with capacity for a certain number of items, as discussed in a later section). Thus, this is a computationally expensive procedure. To improve efficiency, an approximate solution that requires solving only one static linear assignment problem for each of the draws can be formulated as follows.

First, consider that the expected loss for assigning item to agent can be re-expressed as follows:

where denotes the optimal assignment for item under an optimal static assignment. In other words, the expected loss for assigning item to agent is the conditional loss given that is not optimal scaled by the probability that is not optimal. Hence, under reasonable conditions,444For the location with the highest probability of being optimal, in the subset of the sample space where it is not optimal, it must not incur a loss that is “too much higher” than that of other locations. the expected loss can be approximately minimized by minimizing the probability that is not optimal. That is, the approximate minimum-risk decision rule proceeds by choosing the that minimizes . The mechanism is as follows.

Mechanism 2

Approximate Minimum-Risk Dynamic Linear Assignment Mechanism

where corresponds to as applied to the randomly drawn items (cost vectors) in the set . For , and hence effectively drops out of the expression.

In other words, for each random vector set , this mechanism finds the agent that comprises part of the solution to the static linear assignment problem with respect to , and then finds the modal value of across all random vector sets. This comprises the agent that is the highest probability of being the optimal and hence the approximate solution, . The algorithm is delineated below.

Algorithm 2

Approximate Minimum-Risk Dynamic Linear Assignment Algorithm

  • Consider items numbered , and begin with a known set of agents numbered . Let denote the set of available agents and initialize . Pre-specify a large number of simulation iterations to perform.

  • for each item in :

    • Determine and fix .

    • for in :

      • Simulate (randomly draw) vectors, .

      • Solve static linear assignment problem for item/vector and simulated items/vectors with all agents in .

      • Store resulting agent to which item is assigned, denoted by .

    • Assign item to the agent that is the modal assigned agent in : .

    • Remove from .

Vii Extension to Agents

The mechanisms presented above can be easily extended to the case where there are items but agents, each with a specified number of capacity units denoting the number of items that can be assigned to agent , provided that . To do so, the assignments must simply be reconceptualized as assignments of items to agent capacity units. In other words, denotes the agent capacity unit to which item is assigned, it continues to be the case that (each agent capacity unit can only have one item assigned to it), and if and correspond to the same agent. This is equivalent to augmenting a x cost matrix such that each column is duplicated according to the number of capacity units belonging to the respective agent.

The implementing algorithms would be modified accordingly as follows. The set would continue to correspond to the set of available agents, now initialized as , and for any agent , . At the end of the outer loop corresponding to the assignment of item , instead of simply removing agent from , would be decremented by , and agent would be removed from if and only if after being decremented.

Viii Alternative Heuristic Mechanisms

By randomly drawing vectors , the simulation-based assignment mechanisms described above utilize the full set of available information on the probability distribution giving rise to . However, the trade-off is computational expense. In situations where the computational expense of these approaches exceeds the available resources, more efficient approaches may be necessary even if they result in increased loss.

Alternative heuristic mechanisms are proposed here that make more limited, efficient use of the available information to perform dynamic assignment of items to agents. Specifically, rather than using information about the full distribution of , these approaches utilize marginal distributional information on the individual elements of (i.e. for a given agent , the information on the distribution of costs across the underlying population of items).

The approaches proceed by first using the historical data (or a modeling process) to construct the empirical (or modeled) quantile function for each agent, which characterizes the distribution of costs for assigning a random item to the agent in question. Let

denote the quantile function for agent , let denote the quantile of the cost of assigning a specific item to agent , and let denote the vector of quantiles for a specific item where the th element corresponds to the quantile of the cost of assigning item to agent for .

Mechanism 3

Weighted Cost-Quantile Assignment

For item , let be a standardized version of , and construct the weighted average . Let denote the th element of . Assign item to an agent as follows:

Mechanism 4

Sequential Cost-Quantile Assignment

For item , identify the lowest cost elements in corresponding to agents to which previous items have not yet been assigned. Of those elements, choose the lowest quantile element in . Let this element be the agent assignment for item .

For each of these mechanisms, and may be viewed as tuning parameters whose values can be selected through backtesting on historical data.

Ix Application: Assigning Refugees to Geographic Destinations

To illustrate the performance of the minimum-risk and approximate minimum-risk dynamic assignment mechanisms, the mechanisms are applied to real-world data on refugees resettled in the United States.

These data include (de-identified) information on refugees of working age (ages 18 to 64; N = 33,782) who were resettled in 2011-2016 in the United States by one of the largest U.S. refugee resettlement agencies. During this period, the placement officers at the agency centrally assigned each refugee family to one of approximately 40 resettlement locations in the United States. The data contain details on the refugee characteristics (such as age, gender, origin, and education), their assigned resettlement locations, and whether each refugee was employed at 90 days after arrival. According to the agency’s actual assignment procedures, refugees are assigned periodically in cohorts, rather than dynamically. However, the general structure of the assignment process and the nature of the available data in the United States are similar to those of other countries where dynamic assignment is actually implemented. Hence, testing the dynamic assignment mechanisms on U.S. data is still informative for understanding the relative performance of these mechanisms in general.

Family-location outcome scores are constructed for each refugee family who arrived in the third quarter (Q3) of 2016, specifically focusing on refugees who were free to be assigned to different resettlement locations (561 families), in contrast to refugees who were predestined to specific locations on the basis of existing family or other ties. For each family, a vector of outcome scores is constructed, where each element corresponds to the mean probability within that family of finding employment if assigned to a particular location. These outcome score vectors are used in place of cost vectors, with the analogous objective of maximizing expected employment. In other words, in this application, minimizing costs is defined as maximizing expected employment.

To generate each family’s outcome score vector across each of the locations, the same methodology is employed as in Bansak et al. (2018), using the data for the refugees who arrived from 2011 up to (but not including) 2016 Q3 to generate models that predict the expected employment success of a family (i.e. the mean probability of finding employment among working-age members of the family) at any of the locations, as a function of their background characteristics. These models were then applied to the families who arrived in 2016 Q3 to generate their predicted employment success at each location, which comprise their outcome score vectors. See Bansak et al. (2018) for details.

Free-case refugee families who were resettled in the final two weeks of September 2016 (the last two weeks of data available in the data set) are employed as the cohort to be assigned dynamically in this backtest. That is, the mechanisms explored in this study are applied to this particular cohort, which is comprised of 68 families, in the specific order in which they are logged as having actually arrived into the United States. To further mimic the real-world process by which these families would be assigned dynamically to locations, real-world capacity constraints are also employed, such that each location can only receive the same number of cases that were actually received in this two-week period.

Figure 1: Results from U.S. Refugee Assignment Application

All free-case refugee families assigned in 2016 Q3 but prior to the final two weeks of September (493 families) serve as the pool of historical data referenced by the various dynamic assignment mechanisms. That is, this is the pool from which outcome score vectors are randomly drawn for the minimum-risk and approximate minimum-risk mechanisms, and the pool from which quantile information is extracted for the weighted cost-quantile and the sequential cost-quantile mechanisms.

Figure 1 shows the results, with the -axis denoting each particular mechanism, and the -axis measuring the mean outcome score realized when applying each mechanism.555Similar results for the 2016 Q2, 2016 Q1, and 2015 Q4 refugee cohorts are presented in the Supplementary Materials. The first two mechanisms included serve as baseline results. The first represents the ideal scenario whereby an optimal static assignment is made. This optimal assignment cannot actually be performed in a dynamic context, but its results set the upper bound of what is achievable by any mechanism. The second mechanism displayed is a myopic greedy assignment, whereby each family is assigned sequentially to the location with the highest expected outcome score for that family, out of the locations with remaining capacity. This greedy procedure is a common approach to dynamic optimization problems given its simplicity and efficiency. However, its simplicity also drastically limits its ability to achieve outcomes near the globally optimal solution. The difference between the optimal assignment results and greedy assignment results represent a window for potential improvement via the dynamic mechanisms presented in this study.

The next two mechanisms displayed are the minimum-risk and approximate minimum-risk mechanisms, followed by the results of the weighted cost-quantile mechanism at various values of () and the sequential cost-quantile mechanism at various values of . As can be seen, the minimum-risk and approximate minimum-risk mechanisms exhibit the best performance of all the dynamic mechanisms, achieving mean outcome scores that are about 80% of the way toward the static global optimum from the greedy baseline. In addition, the minimum-risk mechanism just barely beats out the approximate minimum-risk mechanism, suggesting that the approximation method is relatively efficient, at least in this context.

As can also be seen, the weighted cost-quantile and sequential cost-quantile mechanisms also improve upon the myopic greedy assignment. At their optimal values of and , these mechanisms achieve mean outcome scores that make it about halfway from the greedy results to the optimal results. Notably, this occurs with virtually no loss of computational efficiency relative to the greedy mechanism.

Figure 2: Minimum-Risk Assignment + Expected Loss

In implementing the (exact) minimum-risk mechanism, the expected loss associated with the assignment of each family is estimated and stored. The sum of the expected loss can then be computed over all of the families. In theory, the sum of the expected loss should equal, in expectation, the discrepancy between the minimum-risk assignment sum of costs and the optimal assignment sum of costs. As shown in Figure 2, this appears to be the case for these U.S. refugee data: adding the estimated mean of the expected loss to the minimum-risk mean outcome score yields a value almost exactly equal to the mean outcome score under the static optimal assignment.666Note, the results displayed correspond to mean outcome scores, whereas the mechanism is defined in terms of the sum of costs (sum of the complement of the outcome scores). Thus, for comparison, the sum of the expected loss is also scaled accordingly to the mean of the expected loss. The mean outcome score under the optimal assignment is , whereas the mean expected loss added to the mean outcome score under the minimum-risk assignment yields a value of .

X Extension to Batch Assignments

In the dynamic formulation of the assignment problem presented above, each item is observed and must be assigned one by one. At the opposite end of the spectrum is, of course, the classic static formulation of the assignment problem, where all items are observed and assigned simultaneously. There also exists a middle ground between the dynamic and static formulations, whereby items are observed and must be assigned in groups or batches. For instance, in the refugee assignment realm, there may be periodic (e.g. weekly, monthly) cohorts of refugee arrivals that can be assigned in batches rather than on a purely one-by-one basis. This section extends the dynamic assignment mechanisms presented above to this “dynamic-batch” context.

As before, assume that are sequentially indexed items, . will continue to denote the minimum cost of any assignment of all remaining items beginning with () to all remaining agents (all agents to which items were not assigned) while satisfying the requirement that no two items among are assigned to the same agent as defined by . That is:

Now, however, to take batch assignment into account, also assume that the sequence of items is split along cut points to form sequential batches of items. For the th batch, let denote the sequentially indexed set of items in the batch, where . Hence, given that the items in the batch are sequentially indexed, . Further, given that the batches are also sequentially ordered, ; that is, the first item in a particular batch immediately follows (in the overall sequence of ) the last item in the immediately preceding batch. As an example, imagine a full set of items, indexed , that are split into two batches (), the first with items and the second with items. Then and . In this case, for the second batch, , , , , and .

Further let denote a vector of agents to which the sequence of items in () are assigned, where the elements of the vector are ordered to match the sequence of items:

Recall that previously, the marginal loss function for a single assignment of item to a particular agent under was defined as . To take batch assignment into account, now define the marginal loss function for the assignment of all items in batch to agents as follows:

Here, denotes the cost of the assignment of item to a particular agent . Hence, denotes the sum of costs for a conditional optimal assignment where the assignment for each item is fixed with agent and the remaining items after item (i.e. the rest of the items not belonging to batch or any batch where ) are assigned by the optimal static process to the remaining available agents.

Now, let the following denote the expected loss (risk) of assigning the items in batch to under the probability distribution , given that only for all are observed (i.e. the cost vectors for items are not observed):

The following minimum-risk dynamic-batch assignment rule for batch , , can thus be formulated:

subject to

Extending the simulation-based methods presented earlier to implement this rule, where again denotes the number of draws and indexes a particular draw, leads to the following mechanism.

Mechanism 5

Minimum-Risk Dynamic-Batch Linear Assignment Mechanism

subject to

where corresponds to as applied to randomly drawn items (cost vectors) in the set . For , and hence effectively drops out of the expression.

In practice, however, implementing this minimum-risk dynamic-batch assignment rule is not necessarily feasible given the combinatorial complexity of the vector . As an example, consider a full sequence of items , and denote the sequence in the first batch as (i.e. ). With items and agents, the vector has unique permutations that must be assessed. Compare this to applying the dynamic assignment mechanism presented earlier to each item in the batch one-by-one (Mechanism 1), which would require assessing possibilities for the first item, for the second item, and so on, for a total of assessments.

Instead, to achieve an approximate implementation of the minimum-risk dynamic-batch assignment rule, Mechanisms 1 and 2 can be directly extended to batches through a simple modification that leads to no increase in computational complexity: by incorporating the set of cost vectors of all items in the batch as additional information in the optimization process, but not altering the one-by-one assignment procedure. Under this modification, for any item , Mechanisms 1 and 2 to proceed as before except that would be computed/solved with respect to modified cost vector sets that include the observed cost vectors of all items for , where denotes the batch to which item belongs. Specifically, let , which is the set of observed cost vectors of the items in subsequent to item . Further, let , where as before this is a randomly drawn cost vector set and denotes the draw. To extend Mechanisms 1 and 2 to batches, let , and simply employ in place of .

Xi Conclusion

This study has proposed mechanisms for minimizing the costs of assigning items to agents when those assignments must be made dynamically (i.e. the assignment of each item must occur prior to the observance of the following items). By employing stochastic programming methods that simulate the distribution of costs across future items that have not yet arrived, for each new item assignment, these mechanisms minimize (or approximately minimize) the final expected cost of this assignment relative to the ideal theoretical scenario whereby all items could be optimally assigned simultaneously. In doing so, these mechanisms can attain a final sum of assignment costs (for all items) that is substantially lower than the myopic greedy assignment heuristic that simply assigns each item to its own lowest-cost agent that is still available. An extension to dynamic batch assignment has also been provided, showing how the proposed mechanisms and accompanying algorithms for one-by-one dynamic assignment can be adapted to the context where items arrive and must be assigned sequentially in groups rather than individually.

To illustrate the potential of these mechanisms, they are applied to real-world refugee data in the United States. The assignment of refugees and asylum seekers to geographic areas represents one immediately relevant dynamic assignment use case, as facilitating self-sufficiency or other integration outcomes is often a core goal of refugee and asylum agencies across the world, but the institutional, logistical, and financial realities these agencies face in certain countries often requires that they allocate refugees or asylum seekers to geographic locations within the country on a prompt, case-by-case basis. As such, these mechanisms help build upon recent research on outcome-based refugee assignment and could be integrated into existing refugee assignment mechanisms (Bansak et al., 2018; Gölz and Procaccia, 2019; Trapp et al., 2018; Acharya et al., 2019; Andersson et al., 2018).

References

  • Acharya et al. (2019) Acharya, A., Bansak, K., and Hainmueller, J. (2019). Combining outcome-based and preference-based matching: The g-constrained priority mechanism. IPL Working Paper Series, Working Paper No. 19-03 (arXiv preprint arXiv:1902.07355).
  • Andersson et al. (2018) Andersson, T., Ehlers, L., and Martinello, A. (2018). Dynamic refugee matching. Working Paper 2018:7, Lund University, Department of Economics.
  • Bansak et al. (2018) Bansak, K., Ferwerda, J., Hainmueller, J., Dillon, A., Hangartner, D., Lawrence, D., and Weinstein, J. (2018). Improving refugee integration through data-driven algorithmic assignment. Science, 359(6373):325–329.
  • Bertsekas (1991) Bertsekas, D. P. (1991). Linear network optimization: algorithms and codes. MIT press.
  • Doval (2014) Doval, L. (2014). A theory of stability in dynamic matching markets. Technical report.
  • Godfrey and Powell (2002a) Godfrey, G. A. and Powell, W. B. (2002a). An adaptive dynamic programming algorithm for dynamic fleet management, i: Single period travel times. Transportation Science, 36(1):21–39.
  • Godfrey and Powell (2002b) Godfrey, G. A. and Powell, W. B. (2002b). An adaptive dynamic programming algorithm for dynamic fleet management, ii: Multiperiod travel times. Transportation Science, 36(1):40–54.
  • Gölz and Procaccia (2019) Gölz, P. and Procaccia, A. D. (2019). Migration as submodular optimization. In

    Proceedings of the AAAI Conference on Artificial Intelligence

    , volume 33, pages 549–556.
  • Kadam and Kotowski (2018a) Kadam, S. V. and Kotowski, M. H. (2018a). Multiperiod matching. International Economic Review, 59(4):1927–1947.
  • Kadam and Kotowski (2018b) Kadam, S. V. and Kotowski, M. H. (2018b). Time horizons, lattice structures, and welfare in multi-period matching markets. Games and Economic Behavior, 112:1–20.
  • Kennes et al. (2014) Kennes, J., Monte, D., and Tumennasan, N. (2014). The day care assignment: A dynamic matching problem. American Economic Journal: Microeconomics, 6(4):362–406.
  • Korsah et al. (2007) Korsah, G. A., Stentz, A. T., and Dias, M. B. (2007). The dynamic Hungarian algorithm for the assignment problem with changing costs. Technical Report CMU-RI-TR-07-27, Carnegie Mellon University, Pittsburgh, PA.
  • Kuhn (1955) Kuhn, H. W. (1955). The hungarian method for the assignment problem. Naval Research Logistics (NRL), 2(1-2):83–97.
  • Kurino (2009) Kurino, M. (2009). House allocation with overlapping agents: A dynamic mechanism design approach. Technical report, Jena Economic Research Papers.
  • Munkres (1957) Munkres, J. (1957). Algorithms for the assignment and transportation problems. Journal of the Society for Industrial and Applied Mathematics, 5(1):32–38.
  • Pillac et al. (2013) Pillac, V., Gendreau, M., Guéret, C., and Medaglia, A. L. (2013). A review of dynamic vehicle routing problems. European Journal of Operational Research, 225(1):1–11.
  • Powell et al. (2002) Powell, W. B., Shapiro, J. A., and Simão, H. P. (2002). An adaptive dynamic programming algorithm for the heterogeneous resource allocation problem. Transportation Science, 36(2):231–249.
  • Shapiro et al. (2009) Shapiro, A., Dentcheva, D., and Ruszczyński, A. (2009). Lectures on Stochastic Programming: Modeling and Theory. SIAM.
  • Spivey and Powell (2004) Spivey, M. Z. and Powell, W. B. (2004). The dynamic assignment problem. Transportation Science, 38(4):399–419.
  • Trapp et al. (2018) Trapp, A. C., Teytelboym, A., Martinello, A., Andersson, T., and Ahani, N. (2018). Placement optimization in refugee resettlement. Working Paper 2018:23, Lund University, Department of Economics.
  • Ünver (2010) Ünver, M. U. (2010). Dynamic kidney exchange. The Review of Economic Studies, 77(1):372–414.

Proof of Relationship between Expected Sum of Costs under Minimum-Risk Dynamic Assignment and under Static Optimal Assignment

Under the minimum-risk dynamic assignment rule, the expected loss for the assignment of any item , conditional upon observing , is:

Since does not depend upon , it can be pulled out of the minimization operator:

Let denote the cost of assigning item via the minimum-risk dynamic assignment rule to location , and let denote the expected sum of costs of the optimal assignment for all other items, given that has been assigned to . Hence, given the assignment of item via the minimum-risk dynamic assignment rule, the expected loss conditional on observing is:

Rearranged, and showing the same result for item , yields:

Now note that, by the assumptions of stationarity and stagewise independence:

This is because both the left-hand and right-hand side are expected sums of the costs from an optimal static assignment of items given the same set of available agents (all agents to which items have not been assigned) under the same (stationary) probability distribution that (independently across stages) gives rise to the items’ cost vectors. In addition note that, trivially, .

Therefore, the expected sum of the costs for assigning each item via the minimum-risk dynamic assignment rule is:

In other words, the expected sum of the costs of applying the minimum-risk dynamic assignment rule to all items equals the expected sum of costs of applying the optimal static linear sum solution (if all items could be observed and assigned simultaneously) plus the expected sum of the expected loss of employing the minimum-risk dynamic assignment rule at each stage.

Supplementary Materials (SM)

Mechanism Mean Outcome Score
Optimal (Non-Dynamic) 0.439
Greedy 0.376
Minimum-Risk 0.427
Approximate Minimum-Risk 0.424
Weighted Cost-Quantile
0.8 0.378
0.6 0.386
0.5 0.391
0.4 0.399
0.2 0.407
0.0 0.397
Sequential Cost-Quantile
8 0.398
7 0.402
6 0.405
5 0.401
4 0.397
3 0.390
2 0.378
Table S1: U.S. Refugee Assignment Application, 2016 Q3
Figure S1: Results from U.S. Refugee Assignment Application, 2016 Q2
Figure S2: Results from U.S. Refugee Assignment Application, 2016 Q1
Figure S3: Results from U.S. Refugee Assignment Application, 2015 Q4