Bilevel Programming: Appli- Cations, Blp:applications

Bilevel programming (see Bilevel programming: formulation) is ideally suited to model situations where the decision maker does not have full control over all decision variables. Five such situations are described in this article. The rst example involves the improvement of a road network through either capacity expansion , traac signals synchronization, vehicle guidance systems, etc. While management may be assumed to control the design variables, it can only aaect indirectly the travel choices of the network users. Let x denote the design vector , y the ow vector, X the set of feasible design variables and c i (x; y) the travel delay along link i. One wishes to minimize over the set X the system travel cost P i y i c i (x; y), where the vector y is required to be an equilibrium traac assignment corresponding to the design vector x. Neglecting the latter equilibrium requirement could lead to suboptimal policies. However, as shown in 4] for a continuous variant of the network design problem, eecient heuristic procedures can generate near-optimal solutions at a low computational cost. Indeed it is in the interest of both the management and the network users to minimize travel delays, although the former is interested in minimizing total travel time, while the users optimize their own travel time. Next consider the maximization of revenues raised from tolls set on a transportation network. If tolls are set too high, traac on the corresponding arcs will drop and revenues will be aaected negatively. Conversely, low toll values will generate low revenues. One could strike the right balance by maximizing total revenue, subject to the network users y achieving an equilibrium with respect to the toll vector x. In the case where the network is uncongested, users are assigned to shortest paths linking their respective origin and destination. This yields the bilevel program with bilinear objectives max x;y P i2I 1 x i y i s.t. y 2 arg min y 0 2Y P i2I 1 (c i + x i)y 0 i + P i2I 2 c i y 0 i ; where I 1 represents the set of toll arcs, I 2 the set of toll-free arcs, and Y the polyhedron of demand-feasible ow vectors. In 3] it has been shown that this problem is reducible to a linear bilevel program with an economic interpretation in terms of`second-best' choices, and can also be …