An overview of Stackelberg pricing in networks

The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bi-linear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This manuscript contains a compilation of theoretical and algorithmic results on the Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then, we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and the development of algorithms: establishing NP-hardness, approximability, and polynomially solvable cases, and an efficient exact branch-and-bound algorithm.mathematical applications;

An overview of Stackelberg pricing in networks

The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bilinear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This paper contains a compilation of theoretical and algorithmic results on the network Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and for the development of algorithms: establishing NP-hardness, approximability, special cases solvable in polynomial time, and an efficient exact branch-and-bound algorithm.Economics ;

The Hub Location and Pricing Problem

This paper introduces the joint problem of locating hubs on a network and determining transportation prices between the hubs. Two levels of decision makers are present in the problem acting non-cooperatively: hub transportation provider and customers. The objective of the hub transportation provider is to locate hubs and to set the prices (per unit of commodity) of crossing the hub arcs maximizing its prot, whereas the customers aim is to send their commodities, in the cheapest way, having the possibility of using the hub arcs at the price set by the hub transportation provider or using the existing network at a predefinedtariff. The problem is modeled as a nonlinear bilevel programming formulation, which is in turn linearized, and strengthened through variable reductions as well as valid inequalities. The case in which the price of each hub arc is determined by applying a common discount factor to the predefined tariff in the existing network is also studied. Computational results of mixed integer programming models and a metaheuristic on instances adapted from the literature are presented

Bilevel Modelling of Energy Pricing Problem

International audienceCost minimization problem of a smart grid operator is integrated into the revenue optimization problem of an energy provider. Bilevel programming approach is applied to model the problem. The results of a classical exact method and two heuristic methods are compared

On the sensitivity of local flexibility markets to forecast error : A bi-level optimization approach

The large-scale integration of intermittent distributed energy resources has led to increased uncertainty in the planning and operation of distribution networks. The optimal flexibility dispatch is a recently introduced, power flow-based method that a distribution system operator can use to effectively determine the amount of flexibility it needs to procure from the controllable resources available on the demand side. However, the drawback of this method is that the optimal flexibility dispatch is inexact due to the relaxation error inherent in the second-order cone formulation. In this paper we propose a novel bi-level optimization problem, where the upper level problem seeks to minimize the relaxation error and the lower level solves the earlier introduced convex second-order cone optimal flexibility dispatch (SOC-OFD) problem. To make the problem tractable, we introduce an innovative reformulation to recast the bi-level problem as a non-linear, single level optimization problem which results in no loss of accuracy. We subsequently investigate the sensitivity of the optimal flexibility schedules and the locational flexibility prices with respect to uncertainty in load forecast and flexibility ranges of the demand response providers which are input parameters to the problem. The sensitivity analysis is performed based on the perturbed Karush-Kuhn-Tucker (KKT) conditions. We investigate the feasibility and scalability of the proposed method in three case studies of standardized 9-bus, 30-bus, and 300-bus test systems. Simulation results in terms of local flexibility prices are interpreted in economic terms and show the effectiveness of the proposed approach.</p

Integrated service selection, pricing and fullfillment planning for express parcel carriers - Enriching service network design with customer choice and endogenous delivery time restrictions

Express parcel carriers offer a wide range of guaranteed delivery times in order to separate customers who value quick delivery from those that are less time but more price sensitive. Such segmentation, however, adds a whole new layer of complexity to the task of optimizing the logistics operations. While many sophisticated models have been developed to assist network planners in minimizing costs, few approaches account for the interplay between service pricing, customer decisions and the associated restrictions in the distribution process. This paper attempts to fill this research gap by introducing a heuristic solution approach that simultaneously determines the ideal set of services, the associated pricing and the fulfillment plan in order to maximize profit. By integrating revenue management techniques into vehicle routing and eet planning, we derive a new type of formulation called service selection, pricing and fulfillment problem (SSPFP). It combines a multi-product pricing problem with a cycle-based service network design formulation. In order derive good-quality solutions for realistically-sized instances we use an asynchronous parallel genetic algorithm and follow the intuition that small changes to prices and customer assignments cause minor changes in the distribution process. We thus base every new solution on the most similar already evaluated fulfillment plan. This adapted initial solution is then iteratively improved by a newly-developed route-pattern exchange heuristic. The performance of the developed algorithm is demonstrated on a number of randomly created test instances and is compared to the solutions of a commercial MIP-solver.Series: Schriftenreihe des Instituts für Transportwirtschaft und Logistik - Supply Chain Managemen

Exact solution of the evasive flow capturing problem

The Evasive Flow Capturing Problem is defined as the problem of locating a set of law enforcement facilities on the arcs of a road network to intercept unlawful vehicle flows traveling between origin-destination pairs, who in turn deviate from their route to avoid any encounter with such facilities. Such deviations are bounded by a given tolerance. We first propose a bilevel program that, in contrast to previous studies, does not require a priori route generation. We then transform this bilevel model into a single-stage equivalent model using duality theory to yield a compact formulation. We finally reformulate the problem by describing the extreme rays of the polyhedral cone of the compact formulation and by projecting out the auxiliary variables, which leads to facet-defining inequalities and a cut formulation with an exponential number of constraints. We develop a branch-and-cut algorithm for the resulting model, as well as two separation algorithms to solve the cut formulation. Through extensive experiments on real and randomly generated networks, we demonstrate that our best model and algorithm accelerate the solution process by at least two orders of magnitude compared with the best published algorithm. Furthermore, our best model significantly increases the size of the instances that can be solved optimally

Exact solution of the evasive flow capturing problem

The Evasive Flow Capturing Problem is defined as the problem of locating a set of law enforcement facilities on the arcs of a road network to intercept unlawful vehicle flows traveling between origin-destination pairs, who in turn deviate from their route to avoid any encounter with such facilities. Such deviations are bounded by a given tolerance. We first propose a bilevel program that, in contrast to previous studies, does not require a priori route generation. We then transform this bilevel model into a single-stage equivalent model using duality theory to yield a compact formulation. We finally reformulate the problem by describing the extreme rays of the polyhedral cone of the compact formulation and by projecting out the auxiliary variables, which leads to facet-defining inequalities and a cut formulation with an exponential number of constraints. We develop a branch-and-cut algorithm for the resulting model, as well as two separation algorithms to solve the cut formulation. Through extensive experiments on real and randomly generated networks, we demonstrate that our best model and algorithm accelerate the solution process by at least two orders of magnitude compared with the best published algorithm. Furthermore, our best model significantly increases the size of the instances that can be solved optimally.</p

A Choice-Driven Service Network Design and Pricing Including Heterogeneous Behaviors

The design and pricing of services are two of the most important decisions faced by any intermodal transport operator. The key success factor lies in the ability of meeting the needs of the shippers. Therefore, making full use of the available information about the demand helps to come up with good design and pricing decisions. With this in mind, we propose a Choice-Driven approach, incorporating advanced choice models directly into a Service Network Design and Pricing problem. We evaluate this approach considering three different mode choice models: one deterministic with 4 attributes (cost, time, frequency and accessibility); and two stochastic also accounting for unobserved attributes and shippers heterogeneity respectively. To reduce the computational time for the stochastic instances, we propose a predetermination heuristic. These models are compared to a benchmark, where shippers are solely cost-minimizers. Results show that the operator profits can be significantly improved, even with the deterministic version. The two stochastic versions further increase the realized profits, but considering heterogeneity allows a better estimation of the demand