Exploring Graphs with Time Constraints by Unreliable Collections of Mobile Robots

A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored. We concentrate on line graphs and rings, for which we give positive results. We start with the case of the collections involving only reliable robots. We give algorithms finding optimal times needed for exploration when the robots are assigned to fixed initial positions as well as when such starting positions may be determined by the algorithm. We extend our consideration to the case when some number of robots may be unreliable. Our most surprising result is that solving the line exploration problem with robots at given positions, which may involve crash-faulty ones, is NP-hard. The same problem has polynomial solutions for a ring and for the case when the initial robots' positions on the line are arbitrary. The exploration problem is shown to be NP-hard for star graphs, even when the team consists of only two reliable robots

The Complexity of Flow Expansion and Electrical Flow Expansion

FlowExpansion is a network design problem, in which the input consists of a flow network and a set of candidate edges, which may be added to the network. Adding a candidate incurs given costs. The goal is to determine the cheapest set of candidate edges that, if added, allow the demands to be satisfied. FlowExpansion is a variant of the Minimum-Cost Flow problem with non-linear edge costs. We study FlowExpansion for both graph-theoretical and electrical flow networks. In the latter case this problem is also known as the Transmission Network Expansion Planning problem. We give a structured view over the complexity of the variants of FlowExpansion that arise from restricting, e.g., the graph classes, the capacities, or the number of sources and sinks. Our goal is to determine which restrictions have a crucial impact on the computational complexity. The results in this paper range from polynomial-time algorithms for the more restricted variants over NP-hardness proofs to proofs that certain variants are NP-hard to approximate even within a logarithmic factor of the optimal solution

Fault-Tolerant Hotelling Games

The $n$-player Hotelling game calls for each player to choose a point on the line segment, so as to maximize the size of his Voronoi cell. This paper studies fault-tolerant versions of the Hotelling game. Two fault models are studied: line faults and player faults. The first model assumes that the environment is prone to failure: with some probability, a disconnection occurs at a random point on the line, splitting it into two separate segments and modifying each player's Voronoi cell accordingly. A complete characterization of the Nash equilibria of this variant is provided for every $n$. Additionally, a one to one correspondence is shown between equilibria of this variant and of the Hotelling game with no faults. The second fault model assumes the players are prone to failure: each player is removed from the game with i.i.d. probability, changing the payoffs of the remaining players accordingly. It is shown that for $n \geq 3$ this variant of the game has no Nash equilibria

Waste processing facility location problem by stochastic programming: Models and solutions

The paper deals with the so-called waste processing facility location problem (FLP), which asks for establishing a set of operational waste processing units, optimal against the total expected cost. We minimize the waste management (WM) expenditure of the waste producers, which is derived from the related waste processing, transportation, and investment costs. We use a stochastic programming approach in recognition of the inherent uncertainties in this area. Two relevant models are presented and discussed in the paper. Initially, we extend the common transportation network flow model with on-and-off waste-processing capacities in selected nodes, representing the facility location. Subsequently, we model the randomly-varying production of waste by a scenario-based two-stage stochastic integer linear program. Finally, we employ selected pricing ideas from revenue management to model the behavior of the waste producers, who we assume to be environmentally friendly. The modeling ideas are illustrated on an example of limited size solved in GAMS. Computations on larger instances were realized with traditional and heuristic algorithms, implemented within MATLAB. © Springer Nature Switzerland AG 2019

Measuring Incineration Plants' Performance using Combined Data Envelopment Analysis, Goal Programming and Mixed Integer Linear Programming

Incineration plants produce heat and power from waste, reduce waste disposal to landfills, and discharge harmful emissions and bottom ash. The objective of the incineration plant is to maximize desirable outputs (heat and power) and minimize undesirable outputs (emissions and bottom ash). Therefore, studying the overall impact of incineration plants in a region so as to maximize the benefits and minimize the environmental impact is significant. Majority of prior works focus on plant specific decision making issues including performance analysis. This study proposes a hybrid Data Envelopment Analysis (DEA), Goal Programming (GP) and Mixed Integer Linear Programming (MILP) model to assess the performance of incineration plants, in a specific region, to enhance overall power production, consumption of waste and reduction of emissions. This model not only helps the plant operators to evaluate the effectiveness of incineration but also facilitates the policy makers to plan for overall waste management of the region through decision-making on adding and closing plants on the basis of their efficiency. Majority of prior studies on incineration plants emphasize on how to improve their performance on heat and power production and neglect the waste management aspects. Additionally, optimizing benefits and minimizing negative outputs through fixing targets in order to make decision on shutting down the suboptimal plants has not been modeled in prior research. This research combines both the aspects and addresses the overall performance enhancement of incineration plants within a region from both policy makers and plant operators’ perspectives. The proposed combined DEA, GP and MILP model enables to optimize incineration plants performance within a region by deriving efficiency of each plant and identifying plants to close down on the basis of their performance. The proposed model has been applied to a group of 22 incineration plants in the UK using secondary data in order to demonstrate the effectiveness of the model.

Managing facility disruption in hub-and-spoke networks: formulations and efficient solution methods

Hub disruption result in substantially higher transportation cost and customer dissatisfaction. In this study, first a mathematical model to design hub-and-spoke networks under hub failure is presented. For a fast and inexpensive recovery, the proposed model constructs networks in which every single demand point will have a backup hub to be served from in case of disruption. The problem is formulated as a mixed integer quadratic program in a way that could be linearized without significantly increasing the number of variables. To further ease the model’ computational burden, indicator constraints are employed in the linearized model. The resulting formulation produced optimal solutions for small and some medium size instances. To tackle large problems, three efficient particle swarm optimisation-based metaheuristics which incorporate efficient solution representation, short-term memory and special crossover operator are proposed. We present the results for two scenarios relating to high and low probabilities of hub failures and provide managerial insight. The computational results, using problem instances with various sizes taken from CAB and TR datasets, confirm the effectiveness and efficiency of the proposed problem formulation and our new solution techniques

Neighbourhood Reduction in Global and Combinatorial Optimization: The Case of the p-Centre Problem

Neighbourhood reductions for a class of location problems known as the vertex (or discrete) and planar (or continuous) p-centre problems are presented. A brief review of these two forms of the p-centre problem is first provided followed by those respective reduction schemes that have shown to be promising. These reduction schemes have the power of transforming optimal or near optimal methods such as metaheuristics or relaxation-based procedures, which were considered relatively slow, into efficient and exciting ones that are now able to find optimal solutions or tight lower/upper bounds for larger instances. Research highlights of neighbourhood reduction for global and combinatorial optimisation problems in general and for related location problems in particular are also given