We consider a bilevel model where the leader wants to maximize revenues from a taxation scheme, while the follower rationally reacts to those tax levels. We focus our attention on the special case of a toll-setting problem defined on a multicommodity transportation network. We show that the general problem is NP-complete, while particular instances are polynomially solvable. Numerical examples are given.Pricing, Networks, Bilevel

The p-center problem consists of choosing p facilities among a set of M possible locations and assigning N clients to them in order to minimize the maximum distance between a client and the facility to which it is allocated. We present a new integer linear programming formulation for this min-max problem with a polynomial number of variables and constraints, and show that its LP relaxation provides a lower bound tighter than the classical one. Moreover, we show that an even better lower bound LB*, obtained by keeping the integrality restrictions on a subset of the variables, can be computed in polynomial time by solving at most O(log2(NM)) linear programs, each having N rows and M columns. We also show that, when the distances satisfy triangle inequalities, LB* is at least one third of the optimal value. Finally, we use LB* in an exact solution method and report extensive computational results on test problems from the literature. For instances where the triangle inequalities are satisfied, our method outperforms the running time of other recent exact methods by an order of magnitude. Moreover, it is the first one to solve large instances of size up to N = M = 1,817.

This study confirmed that radiotherapy, as delivered until the mid-1980s, increased the long-term risk of dying of cardiovascular diseases. The long-term risk of dying of cardiac disease is a particular concern for women treated for a left-sided breast cancer with contemporary tangential breast or chest wall radiotherapy. This risk may increase with a longer follow-up, even after 20 years following radiotherapy.

High doses of radiation strongly increase the risk of bone and soft tissue sarcoma.

The hub location problem with single assignment is the problem of locating hubs and assigning the terminal nodes to hubs in order to minimize the cost of hub installation and the cost of routing the traffic in the network. There may also be capacity restrictions on the amount of traffic that can transit by hubs. The aim of this paper is to investigate polyhedral properties of these problems and to develop a branch and cut algorithm based on these results. © Springer-Verlag 2004

The p‐Center problem consists of locating p facilities and assigning clients to them in order to minimize the maximum distance between a client and the facility to which he or she is allocated. In this paper, we present a basic Variable Neighborhood Search and two Tabu Search heuristics for the p‐Center problem without the triangle inequality. Both proposed methods use the 1‐interchange (or vertex substitution) neighborhood structure. We show how this neighborhood can be used even more efficiently than for solving the p‐Median problem. Multistart 1‐interchange, Variable Neighborhood Search, Tabu Search, and a few early heuristics are compared on small‐ and large‐scale test problems from the literature. © 2003 Wiley Periodicals, Inc.

T = (V, E) is a tree with nonnegative weights associated with each of its vertices. A fleet of vehicles of capacity Q is located at the depot represented by vertex v1. The Capacitated Vehicle Routing Problem on Trees (TCVRP) consists of determining vehicle collection routes starting and ending at the depot such that: the weight associated with any given vertex is collected by exactly one vehicle; the sum of all weights collected by a vehicle does not exceed Q; a linear combination of the number of vehicles and of the total distance traveled by these vehicles is minimized. The TCVRP is shown to be NP-hard. This paper presents lower bounds for the TCVRP based on the solutions of associated bin packing problems. We develop a linear time heuristic (upper bound) procedure and present a bound on its worst case performance. These lower and upper bounds are then embedded in an enumerative algorithm. Numerical results follow.

We consider the problem of determining a set of optimal tolls on the arcs of a multicommodity transportation network. The problem is formulated as a bilevel mathematical program where the upper level consists in a rm that raises revenues from tolls set on arcs of the network, while the lower level is represented by a group of users travelling on shortest paths with respect to a generalized travel cost.