On a Reduction for a Class of Resource Allocation Problems

In the resource allocation problem (RAP), the goal is to divide a given amount of a resource over a set of activities while minimizing the cost of this allocation and possibly satisfying constraints on allocations to subsets of the activities. Most solution approaches for the RAP and its extensions allow each activity to have its own cost function. However, in many applications, often the structure of the objective function is the same for each activity, and the difference between the cost functions lies in different parameter choices, such as, for example, the multiplicative factors. In this article, we introduce a new class of objective functions that captures a significant number of the objectives occurring in studied applications. These objectives are characterized by a shared structure of the cost function depending on two input parameters. We show that, given the two input parameters, there exists a solution to the RAP that is optimal for any choice of the shared structure. As a consequence, this problem reduces to the quadratic RAP, making available the vast amount of solution approaches and algorithms for the latter problem. We show the impact of our reduction result on several applications, and in particular, we improve the best-known worst-case complexity bound of two problems in vessel routing and processor scheduling from [Formula: see text] to [Formula: see text]. Summary of Contribution: The resource allocation problem (RAP) with submodular constraints and its special cases are classic problems in operations research. Because these problems are studied in many different scientific disciplines, many conceptual insights, structural properties, and solution approaches have been reinvented and rediscovered many times. The goal of this article is to reduce the amount of future reinventions and rediscoveries by bringing together these different perspectives on RAPs in a way that is accessible to researchers with different backgrounds. The article serves as an exposition on RAPs and on their wide applicability in many areas, including telecommunications, energy, and logistics. In particular, we provide tools and examples that can be used to formulate and solve problems in these areas as RAPs. To accomplish this, we make three concrete contributions. First, we provide a survey on algorithms and complexity results for RAPs and discuss several recent advances in these areas. Second, we show that many objectives for RAPs can be reduced to a (simpler) quadratic objective function, which makes available the extensive collection of fast and efficient algorithms for quadratic RAPs to solve these problems. Third, we discuss the impact that RAPs and the aforementioned reduction result can make in several application areas.

Stochastic Knapsack Revisited: The Service Level Perspective

A key challenge in the resource allocation problem is to find near-optimal policies to serve different customers with random demands/revenues, using a fixed pool of capacity (properly configured). Three classes of allocation policies, responsive (with perfect hindsight), adaptive (with information updates), and anticipative (with forecast information) policies, are widely used in practice. We analyze and compare the performances of these policies for both capacity minimization and revenue maximization models. In both models, the performance gaps between optimal anticipative policies and adaptive policies are shown to be bounded when the demand and revenue of each item are independently generated. In contrast, the gaps between the optimal adaptive policies and responsive policies can be arbitrarily large. More importantly, we show that the techniques developed, and the persistency values obtained from the optimal responsive policies can be used to design good adaptive and anticipative policies for the other two variants of resource allocation problems.

Epidemic Control and Resource Allocation: Approaches and Implications for the Management of COVID-19

The experience with COVID-19 underscores a classic public policy choice problem: how should policymakers determine how to allocate constrained budgets, limited equipment, under-resourced hospitals and stretched personnel to limit the spread of the virus. This article presents an overview of the general literature on resource allocation in epidemics and assess how it informs our understanding of COVID-19. We highlight the peculiarities of the pandemic that call for a rethinking of existing approaches to resource allocation. In particular, we analyse how the experience of COVID-19 informs our understanding and modelling of the optimal resource allocation problem in epidemics. Our delineation of the literature focuses on resource constraint as the key variable. A qualitative appraisal indicates that the current suit of models for understanding the resource allocation problem requires adaptations to advance our management of COVID-19 or similar future epidemics. Particularly under-studied areas include issues of uncertainty, potential for co-epidemics, the role of global connectivity, and resource constrained problems arising from depressed economic activity. Incorporating various global dimensions of COVID-19 into resource allocation modelling such a centralized versus decentralized resource control and the role of geostrategic interests could yield crucial insights. This will require multi-disciplinary approaches to the resource allocation problem. JEL Classifications: I14, I18, E61, D60, H4, H12

Is Equality Always Desirable? Analyzing the Trade-Off Between Fairness and Attractiveness in Crew Rostering

Millions of employees around the world work in irregular rosters. The quality of these rosters is of utmost importance. High-quality rosters should be attractive on an individual level, but also divide the work fairly over the employees. We develop novel methodology to compute the trade-off between fairness and attractiveness in crew rostering. First, we propose an intuitive fairness scheme for crew rostering and analyze its theoretical performance. To do so, we introduce the approximate resource-allocation problem. This extension of the resource-allocation problem provides a framework for analyzing decision making in contexts where one relies on approximations of the utility functions. Fairness is a typical example of such a context due to its inherently subjective nature. We show that the scheme has “optimal” properties for a large class of approximate utility functions. Furthermore, we provide a tight bound on the utility loss for this scheme. We then present a unified approach to crew rostering. This approach integrates our proposed fairness scheme with a novel mathematical formulation for crew rostering. We call the resulting problem the Fairness-Oriented Crew Rostering Problem and develop a dedicated exact Branch-Price-and-Cut solution method. We conclude by applying our solution approach to practical instances from Netherlands Railways, the largest passenger railway operator in the Netherlands. Our computational results confirm the importance of taking the fairness–attractiveness trade-off into account. This paper was accepted by Yinyu Ye, optimization.

Dominant Resource Fairness with Meta-Types

Inspired by the recent COVID-19 pandemic, we study a generalization of the multi-resource allocation problem with heterogeneous demands and Leontief utilities. Unlike existing settings, we allow each agent to specify requirements to only accept allocations from a subset of the total supply for each resource. These requirements can take form in location constraints (e.g. A hospital can only accept volunteers who live nearby due to commute limitations). This can also model a type of substitution effect where some agents need 1 unit of resource A \emph{or} B, both belonging to the same meta-type. But some agents specifically want A, and others specifically want B. We propose a new mechanism called Dominant Resource Fairness with Meta Types which determines the allocations by solving a small number of linear programs. The proposed method satisfies Pareto optimality, envy-freeness, strategy-proofness, and a notion of sharing incentive for our setting. To the best of our knowledge, we are the first to study this problem formulation, which improved upon existing work by capturing more constraints that often arise in real life situations. Finally, we show numerically that our method scales better to large problems than alternative approaches.

Improving Streaming Capacity In Tree & Meshed Based P2P Live Streaming Systems

The streaming capacity for a channel is defined as the maximum streaming rate that can be achieved by every user in the channel. In the thesis, we investigated the streaming capacity problem in both tree-based and mesh-based Peer-to-Peer (P2P) live streaming systems, respectively. In tree-based multi-channel P2P live streaming systems, we propose a crosschannel resource sharing approach to improve the streaming capacity. We use cross-channel helpers to establish the cross-channel overlay links, with which the unused upload bandwidths in a channel can be utilized to help the bandwidth-deficient peers in another channel, thus improving the streaming capacity. In meshed-based P2P live streaming systems, we propose a resource sharing approach to improve the streaming capacity. In mesh-based P2P streaming systems, each peer exchanges video chunks with a set of its neighbors. We formulate the streaming capacity problem into an optimal resource allocation problem. By solving the optimization problem, we can optimally allocate the link rates for each peer, thus improve the streaming capacity.

Improving Streaming Capacity In Tree & Meshed Based P2P Live Streaming Systems

The streaming capacity for a channel is defined as the maximum streaming rate that can be achieved by every user in the channel. In the thesis, we investigated the streaming capacity problem in both tree-based and mesh-based Peer-to-Peer (P2P) live streaming systems, respectively. In tree-based multi-channel P2P live streaming systems, we propose a crosschannel resource sharing approach to improve the streaming capacity. We use cross-channel helpers to establish the cross-channel overlay links, with which the unused upload bandwidths in a channel can be utilized to help the bandwidth-deficient peers in another channel, thus improving the streaming capacity. In meshed-based P2P live streaming systems, we propose a resource sharing approach to improve the streaming capacity. In mesh-based P2P streaming systems, each peer exchanges video chunks with a set of its neighbors. We formulate the streaming capacity problem into an optimal resource allocation problem. By solving the optimization problem, we can optimally allocate the link rates for each peer, thus improve the streaming capacity.