Abstract
Continuous network design problem (CNDP) is searching for a transportation network configuration to minimize the sum of the total system travel time and the investment cost of link capacity expansions by considering that the travellers follow a traditional Wardrop user equilibrium (UE) to choose their routes. In this paper, the CNDP model can be formulated as mathematical programs with complementarity constraints (MPCC) by describing UE as a non-linear complementarity problem (NCP). To address the difficulty resulting from complementarity constraints in MPCC, they are substituted by the Fischer-Burmeister (FB) function, which can be smoothed by the introduction of the smoothing parameter. Therefore, the MPCC can be transformed into a well-behaved non-linear program (NLP) by replacing the complementarity constraints with a smooth equation. Consequently, the solver such as LINDOGLOBAL in GAMS can be used to solve the smooth approximate NLP to obtain the solution to MPCC for modelling CNDP. The numerical experiments on the example from the literature demonstrate that the proposed algorithm is feasible.
Highlights
The network design problem (NDP) is seeking of a transportation network configuration that minimizes some objective functions, subject to a traditional Wardrop user equilibrium (UE) as the constraints
The objective of the Continuous network design problem (CNDP) is to minimize the sum of the total system travel time and the investment cost to expand the link capacity, while route choice behaviour of travellers follows UE described by non-linear complementarity problem (NCP)
The CNDP model can be formulated as the following mathematical programs with complementarity constraints (MPCC): min f^v, xw, y, twh =
Summary
The network design problem (NDP) is seeking of a transportation network configuration that minimizes some objective functions, subject to a traditional Wardrop user equilibrium (UE) as the constraints. The CNDP’s objective is to minimize the sum of the total system travel time and the investment cost by expanding the link capacity, while route choice behaviour of travellers follows UE, which is described by non-linear complementarity problem (NCP). Solving MPCC is a hard task because the Mangasarian Fromovitz constraint qualification (MFCQ) does not hold at any feasible point [29, 30] To circumvent these problems, some algorithmic approaches have focused on avoiding this formulation. Some assumptions used in this paper are presented as follows [6]: 1) The link travel time function tij(vij,yi),(i,j)!A is strictly increasing and continuously differentiable with respect to the link flow vij,(i,j)!A, for any fixed link capacity expansion yij,(i,j)!A. are all continuous with respect to (vij,yij). Are all continuous with respect to (vij,yij). 3) The capacity expansion cost function gij(yij),(i,j)!A is continuously differentiable with respect to yij
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