Methods and Applications of (max,+) Linear Algebra

Projet META2Exotic semirings such as the ``$(\max,+)$ semiring'' $(\mathbb{R}\cup\{-\infty\},\max,+)$, or the ``tropical semiring'' $(\mathbb{N}\cup\{+\infty\},\min,+)$, have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities). Despite this apparent profusion, there is a small set of common, non-naive, basic results and problems, in general not known outside the $(\max,+)$ community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of $(\max,+)$ results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of discrete event systems, analysis of Markov decision processes with average cost). Basic techniques include: solving all kinds of systems of linear equations, sometimes with exotic symmetrization and determinant techniques; using the $(\max,+)$ Perron-Frobenius theory to study the dynamics of $(\max,+)$ linear maps. We point out some open problems and current developments

Bad Data Injection Attack and Defense in Electricity Market using Game Theory Study

Applications of cyber technologies improve the quality of monitoring and decision making in smart grid. These cyber technologies are vulnerable to malicious attacks, and compromising them can have serious technical and economical problems. This paper specifies the effect of compromising each measurement on the price of electricity, so that the attacker is able to change the prices in the desired direction (increasing or decreasing). Attacking and defending all measurements are impossible for the attacker and defender, respectively. This situation is modeled as a zero sum game between the attacker and defender. The game defines the proportion of times that the attacker and defender like to attack and defend different measurements, respectively. From the simulation results based on the PJM 5 Bus test system, we can show the effectiveness and properties of the studied game.Comment: To appear in IEEE Transactions on Smart Grid, Special Issue on Cyber, Physical, and System Security for Smart Gri

Creating Low-carbon Communities: Evaluating the Role of Individual Agency and Systemic Inequality in San Jose, CA

Following a scholarly need to test compelling community level sociodemographic representations of environmental behaviors and outcomes, a sequential mixed method approach was used to evaluate the connections of human agency and systemic inequalities with carbon footprints. Statistical analyses of the 2016 SDG San Jose Dashboard data of city blocks and 2009 - 2013 ACS survey data were supplemented with interviews with eight climate action-oriented community engagement professionals in the South Bay. Boundary limiting socioeconomic conditions for systemic inequalities and human agency, dimensions of Gidden’s Structuration model, were specified. Partially supporting structural inequality theories, socioeconomic resources, primarily, and to a lesser extent dominant race concentration, were associated with larger carbon footprints, particularly when wealth was concentrated. Both human (time driven alone) and demographic (senior and mid-aged blocks) agencies were also in part at play in shrinking or even enlarging carbon footprints, in wealthier communities. These findings not only contributed to the literature on climate action, but also highlighted the need for targeted interventions in communities of different socioeconomic standing

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