Workload Equity in Vehicle Routing Problems: A Survey and Analysis

Over the past two decades, equity aspects have been considered in a growing number of models and methods for vehicle routing problems (VRPs). Equity concerns most often relate to fairly allocating workloads and to balancing the utilization of resources, and many practical applications have been reported in the literature. However, there has been only limited discussion about how workload equity should be modeled in VRPs, and various measures for optimizing such objectives have been proposed and implemented without a critical evaluation of their respective merits and consequences. This article addresses this gap with an analysis of classical and alternative equity functions for biobjective VRP models. In our survey, we review and categorize the existing literature on equitable VRPs. In the analysis, we identify a set of axiomatic properties that an ideal equity measure should satisfy, collect six common measures, and point out important connections between their properties and those of the resulting Pareto-optimal solutions. To gauge the extent of these implications, we also conduct a numerical study on small biobjective VRP instances solvable to optimality. Our study reveals two undesirable consequences when optimizing equity with nonmonotonic functions: Pareto-optimal solutions can consist of non-TSP-optimal tours, and even if all tours are TSP optimal, Pareto-optimal solutions can be workload inconsistent, i.e. composed of tours whose workloads are all equal to or longer than those of other Pareto-optimal solutions. We show that the extent of these phenomena should not be underestimated. The results of our biobjective analysis are valid also for weighted sum, constraint-based, or single-objective models. Based on this analysis, we conclude that monotonic equity functions are more appropriate for certain types of VRP models, and suggest promising avenues for further research.Comment: Accepted Manuscrip

Market risk management in a post-Basel II regulatory environment

We propose a novel method of Mean-Capital Requirement portfolio optimization. The optimization is performed using a parallel framework for optimization based on the Nondominated Sorting Genetic Algorithm II. Capital requirements for market risk include an additional stress component introduced by the recent Basel 2.5 regulation. Our optimization with the Basel 2.5 formula in the objective function produces superior results to those of the old (Basel II) formula in stress scenarios in which the correlations of asset returns change considerably. These improvements are achieved at the expense of reduced cardinality of Pareto-optimal portfolios. This reduced cardinality (and thus portfolio diversification) in periods of relatively low market volatility may have unintended consequences for banks’ risk exposure