Matroids are Immune to Braess Paradox

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.Comment: 21 page

Equilibrium Computation in Resource Allocation Games

We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either games and, thus, leads -- via our first algorithm -- to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd [2016].Comment: This version contains some typo corrections onl

Location Problems in Supply Chain Design: Concave Costs, Probabilistic Service Levels, and Omnichannel Distribution

Location of facilities such as plants, distribution centers in a supply chain plays critical role in efficient management of logistics activities. Real-life supply chains are generally large in size with multiple echelons, prone to disruptions and uncertainties, and constantly evolving to meet customer demands in a fast and reliable way. Therefore, it is quite challenging to identify these locations while balancing the trade-off between costs and service levels. In this thesis, we investigate three supply chain design problems addressing various issues that complicate the location of facilities in a supply chain. The first paper investigates a multilevel capacitated facility location problem. Such problems commonly arise in large scale production-distribution supply chains with plants at one echelon, and distribution centers / warehouse at another, and there is hierarchy of flow between facilities and to the end customers such as retail stores. The operating costs at facilities and transportation costs on arcs are assumed to be concave. The concave functions model economies of scale in operations (such as production, handling, transportation) performed at large scale and emission of green house gases from transportation activities. The mathematical model for our problem is nonlinear (concave) for which we present two formulations. The first formulation is a prevalent mixed-integer nonlinear program, and second is a purely nonlinear programming problem. Extensive computations are performed to measure the efficiency of two formulations, and managerial insights are provided to understand the behavior of the model under different scenarios of concavities. The second work focuses on e-commerce supply chains that have a common objective of providing fast and reliable deliveries of customers’ orders. The order delivery time primarily depends on the time taken to process the order at the facilities and travel time from facilities to customers. These two times are uncertain in practice, therefore, to capture the combined effect of both uncertainties, we introduce a mathematical model with a requirement that all customer orders should be delivered within a committed time with some probabilistic guarantee. The problem is formulated as a dynamic (multiperiod) capacitated facility location problem with modular capacities. The probabilistic service level constraints make the problem nonconvex. We present two linear binary programming reformulations, and develop an exact branch-and-cut algorithm utilizing the reformulations to solve large size instances. We also include sensitivity analysis to study the change in network configuration under various modeling parameters. An increase in online sales every year is driving many brick-and-mortar retailers to follow an omni-channel retailing approach that would integrate their online sales channel with store sales. Omnichannel retailing requires a considerable change in current practices. For instance, a retailer generally decides if there is a need of new distribution facilities, which stores should be used as fulfillment centers as well, where to keep safety stocks, from where to serve online demand, among others. To study these aspect, in the third paper, we propose a novel mathematical model for the design of omnichannel distribution network along with allocation of safety stock to the facilities. The original problem is nonlinear which can be reformulated as conic quadratic mixed integer programming problem. The problem is solved using a branch-and-cut solution algorithm. Further, we present several managerial insights related to fulfillment and safety stock decisions using a small example

Drone-Delivery Network for Opioid Overdose -- Nonlinear Integer Queueing-Optimization Models and Methods

We propose a new stochastic emergency network design model that uses a fleet of drones to quickly deliver naxolone in response to opioid overdoses. The network is represented as a collection of M/G/K queuing systems in which the capacity K of each system is a decision variable and the service time is modelled as a decision-dependent random variable. The model is an optimization-based queuing problem which locates fixed (drone bases) and mobile (drones) servers and determines the drone dispatching decisions, and takes the form of a nonlinear integer problem, which is intractable in its original form. We develop an efficient reformulation and algorithmic framework. Our approach reformulates the multiple nonlinearities (fractional, polynomial, exponential, factorial terms) to give a mixed-integer linear programming (MILP) formulation. We demonstrate its generalizablity and show that the problem of minimizing the average response time of a network of M/G/K queuing systems with unknown capacity K is always MILP-representable. We design two algorithms and demonstrate that the outer approximation branch-and-cut method is the most efficient and scales well. The analysis based on real-life overdose data reveals that drones can in Virginia Beach: 1) decrease the response time by 78%, 2) increase the survival chance by 432%, 3) save up to 34 additional lives per year, and 4) provide annually up to 287 additional quality-adjusted life years