Network Pricing Problem Research Articles

A Catalog of Formulations for the Network Pricing Problem

We study the network pricing problem where the leader maximizes revenue by determining the optimal amounts of tolls to charge on a set of arcs, under the assumption that the followers will react rationally and choose the shortest paths to travel. Many distinct single-level reformulations of this bilevel optimization program have been proposed; however, their relationship has not been established. In this paper, we aim to build a connection between those reformulations and explore the combination of the path representation with various modeling options, allowing us to generate 12 different reformulations of the problem. Moreover, we propose a new path enumeration scheme, path-based preprocessing, and hybrid framework to further improve performance and robustness when solving the final model. We provide numerical results, comparing all the derived reformulations and confirming the efficiency of the novel dimensionality reduction procedures.

Competitive location and pricing on a line with metric transportation costs

Consider a three-level non-capacitated location/pricing problem: a firm first decides which facilities to open, out of a finite set of candidate sites, and sets service prices with the aim of revenue maximization; then a second firm makes the same decisions after checking competing offers; finally, customers make individual decisions trying to minimize costs that include both purchase and transportation. A restricted two-level problem can be defined to model an optimal reaction of the second firm to known decision of the first.For non-metric costs, the two-level problem corresponds to Envy-free Pricing or to a special Network Pricing problem, and is APX-complete even if facilities can be opened at no fixed cost. Our focus is on the metric 1-dimensional case, a model where customers are distributed on a main communication road and transportation cost is proportional to distance. We describe polynomial-time algorithms that solve two- and three-level problems with opening costs and single 1st level facility. Quite surprisingly, however, even the two-level problem with no opening costs becomes NP-hard when two 1st level facilities are considered.

Network pricing problem with unit toll

In the so‐called network pricing problem an authority owns some arcs of the network and tolls them, while users travel between their origin and destination choosing their minimum cost path. In this article, we consider a unit toll scheme, and in particular the cases where the authority is imposing either the same toll on all of its arcs, or a toll proportional to a given parameter particular to each arc (for instance a per kilometer toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo‐polynomial, proposing ad‐hoc solution algorithms and relating these problems to the parametric shortest path problem. We then address a robust approach using an interval representation to take into consideration uncertainty on parameters. We show how to modify the algorithms for the deterministic case to solve the robust counterparts, maintaining their complexity class. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(1), 83–93 2017

The envy-free pricing problem, unit-demand markets and connections with the network pricing problem

A common problem faced in economics is to decide the pricing of products of a company, since poorly chosen prices might lead to low profit. One important model for this is the unit-demand envy-free pricing problem, in which one considers that every consumer buys an item that maximizes his own profit, and the goal is to find a pricing of the items that maximizes the expected profit of the seller. This is a practical and interesting problem which is, unfortunately, not in APX unless P=NP. We present two new MIP formulations for this problem and experimentally compare them to previous ones from the literature. We describe three models to generate different random instances for general unit-demand auctions, that we designed for the computational experiments. Each model has a nice economic interpretation. Our results show that one of our MIP formulations can sometimes lead to better results than the previous ones from the literature. We also consider a variant of the network pricing problem in which one has to price toll arcs in a highway, and prove that it is as hard to approximate as the envy-free pricing problem.

Mixed-logit network pricing

This paper addresses a network pricing problem where users are assigned to the paths of a transportation network according to a mixed logit model, i.e., price sensitivity varies across the user population. For its solution, we propose algorithms based on combinatorial approximations, and show that the smoothing effect induced by both the discrete choice and price sensitivity features of the model help in determining near-global solutions. This stands in contrast with simpler formulations where the main difficulty is due to the combinatorial nature of the problem. From an economic point of view, we provide an estimate of the proportion of revenue raised from the various population segments, an information that can be used for policy purposes.

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

This paper considers a pricing problem on a network with connected toll arcs and proposes a Dantzig-Wolfe reformulation for it. The model is solved with column generation and the gap between the optimal integer value and the linear relaxation optimal value is shown to be at least as good as the one from the mixed-integer formulation proposed in the literature. Numerical results on different sets of instances are reported, showing that in many cases the proposed model performs strictly better.

A Network Pricing Formulation for the revenue maximization of European Air Navigation Service Providers

In Europe, all Air Navigation Service Providers (ANSPs) finance their activities by charging airlines to use their airspace. These ‘en route charges’ usually account for a significant fraction of the cost of a flight, and they can therefore influence the route choice: airlines may decide to fly longer routes to avoid countries with higher charges. Then ANSPs’ traffic and revenue do depend on the charge they impose on their own airspace. It follows that if ANSPs look for the maximization of their revenues, they must choose an optimal charge to impose on their airspace. We show that this optimal charge can be identified through a Network Pricing Problem formulation in the form of Bilevel Programming where the leader (i.e. the ANSP) owns a set of arcs (the airways in its national airspace) and charges the commodities (i.e. the flights) passing through them. As the en route charges are proportional to a Unit Rate value fixed by the ANSP, by exploiting the structure of the problem we propose an exact algorithm to compute the optimal Unit Rate and apply it to a case study.

A Tabu search algorithm for the network pricing problem

In this paper, we propose an efficient Tabu Search procedure for solving the NP-hard network pricing problem. By exploiting the problem's features, the algorithm allows the near-optimal solution of problem instances that are out of reach of exact combinatorial methods.

An exact algorithm for the network pricing problem

This work focuses on an improved exact algorithm for addressing an NP-hard network pricing problem. The method involves an efficient and partial generation of candidate solutions, a recursive scheme for generating improved upper bounds, and a column generation procedure for solving the network-structured subproblems. Its efficiency is assessed against both randomly generated instances involving three distinct topologies as well as instances based on real life situations in telecommunication and freight transportation.

A polyhedral study of the network pricing problem with connected toll arcs

AbstractWe consider the problem of setting revenue‐maximizing tolls on a subset of arcs of a transportation network, assuming that the users of the network are assigned to shortest paths with respect to the sum of tolls and initial costs. Our main results are concerned with a polyhedral study of the problem, i.e., the design of valid inequalities and facets for this pricing problem and some of its variants. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

A smoothing heuristic for a bilevel pricing problem

In this paper, we provide a heuristic procedure, that performs well from a global optimality point of view, for an important and difficult class of bilevel programs. The algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques. Although the algorithm cannot guarantee global optimality, very good solutions can be obtained through the use of a suitable set of parameters. The algorithm has been tested on large-scale instances of a network pricing problem, an application that fits our modeling framework. Preliminary results show that on hard instances, our approach constitutes an alternative to solvers based on mixed 0–1 programming formulations.

Closure to discussion of "integration of price cap and yardstick competition schemes in electrical distribution regulation"

In the UK, allocation of costs to the network users has, largely, been left in the hands of the distributors with the regulator having no direct influence over the prices charged to individual customer. Distribution companies largely use conventional pricing models that attempts to identify costs attributable to different customer groups domestic, industrial and commercial. On the other hand, one of the basic objectives of the ongoing reforms of electricity supply industries around the world is to promote competition in the generation and retail segments of the industry. Both generators and retailers require access to network services and therefore, a necessary pre-condition for competition to develop is open and nondiscriminatory access to transmission and distribution networks by all. Electricity prices at wholesale electricity market in UK, average at about 2‐2.5 p/kWh while the retail price of electricity is currently about 6‐7 p/kWh. Transmission and distribution networks, together with the supply business are responsible for the difference between retail and wholesale prices. With energy provided much closer to the load, the value of dispersed kWh may be higher since the costs of “bought-in” network services to facilitate product delivery from point of purchase to point of sale, is likely to be considerably lower. What is important to recognize is that the issue of competitiveness of embedded generation is a network pricing problem. At present however, distribution network pricing, conceptualized for passive distribution networks, does not recognize this impact and is not able to facilitate competition in generation. Clearly, location specific impact of dispersed generation cannot be captured by the generic cost allocation scheme currently used and this considerably disadvantages the competitiveness of dispersed sources.