Simultaneous Network Line Planning and Traffic Assignment

Simultaneous Network Line Planning and Traffic Assignment Karl Nachtigall and Karl Jerosch Technical University Dresden, Faculty of Traffic Sciences, Institute for Logistics and Aviation Abstract. One of the basic problems in strategic planning of public and rail transport is the line planning problem to find a system of lines and its associated frequencies. The objectives of this planning process are usually manifold and often contradicting. The transport operator wants to minimize cost, whereas passengers want to have travel time shortest routes without any or only few changings between different lines. The travel quality of a passenger route depends on the travel time and on the number of necessary changings between lines and is usually measured by a disutility or impedance function. In practice the disutility strongly depends on the line plan, which is not known, but should be calculated. The presented model combines line planning models and traffic assignment model to overcome this dilemma. Results with data of Berlin’s city public transportion network are reported. Key words: line planning problem, integer programming 1 Related literature In the last years a lot of mathematical integer programming models have been proposed (see e.g. [1–3, 6, 10]). The line plan model presented by Borndörfer, Grötschel and Pfetsch [1, 2] minimizes the combination of line costs and system travel time disregarding transfers between lines and waiting times. To make the model feasible for real instances of local traffic systems, the line frequency variables are not forced to be integer, however knowing that the fractual solution can be quite far from the integer optimum. Because in general the minimum system travel time (called Beckmann-User-Equality) do not really reflect the passenger’s ”selfish” behavior, the use of the system travel time seems to be disadvantageous. Both facts are respected in the model of Schöbel and Scholl [11]. This line plan model minimizes the passengers inconvenience under the restriction of a fixed budget for all line costs, whereas the passengers inconvenience is the sum of the travel time and the time needed for transfers concerning one origin-destination relation. The model in [11] assumes that the given passenger travel demand must be satisfied by the line plan. For all origin-destination pairs the passengers will ATMOS 2008 8th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems http://drops.dagstuhl.de/opus/volltexte/2008/1589 2 Simultaneous Network Line Planning and Traffic Assignment travel on a time shortest path. In the model presented here not all passenger demand must be covered. This is controlled by a limited budget for the operational cost of the line plan. For larger networks the model of Schöbel and Scholl suffers from large memory requirements for solving the shortest path problem in the change& go network, which contains for each line and each pair of consecutive served stations a special travel edge. Large number of lines and stations requires a big amount of memory. We try to overcome this problem by the use of a column generation scheme. The resulting pricing problem for those passenger flow variables is a shortest path problem. 2 The Basic Combined Model 2.1 General notations The proposed model is restricted to the case to find a line plan for one homogeneous transport carrier. Each of the edges e ∈ E of the underlying network (V, E) is assigned with a homogeneous travel time te . The nodes or vertices n ∈ V of the network represent stations. Terminal stations are nodes, at which a line is allowed to start or to terminate. By a potential line L we will understand a path between two terminal nodes, which running time is in maximum the product of the detour factor ρ and the shortest travel time between the two terminal nodes. For simplicity we use the notation e ∈ L to indicate, that the line L uses edge e. By L|(A,B) we denote the set of all edges, which are used by line L running from station A to station B. This set might be empty, if A and B are not served by L. 2.2 Traffic Assignment The traffic assignment problem is modelled by some kind of multi-commodity flow problem, where we consider for each travel demand pair (O, D) and lines L ∈ L partial passenger routes pO,D (A,B),L . The variables ϕO,D (A,B),L define the traffic flow of all OD-passengers using line L between station A and station B. A total passenger route p from the origin node O to the destination node D is simply the concatenation of partial routes (see figure 1). The disutility of impedance of a passenger route is a measure for the inconvenience for the trip by this route. In traffic assignment theory this impedance is modelled in dependence of the travel time, the number of changings and the frequency of the service. In practice, there are used rather complex and nonlinear models for this function (see e.g. [4, 7]). Here, we simplify the model by using a linear approximation: For a path p denote the travel time by t(p) and define the number of line changings by c(p), which are penalized by the parameter β. Then imp(p) := t(p) + βc(p) is a reasonable measure for the disutility of using the trip route p. Simultaneous Network Line Planning and Traffic Assignment ϕO,D (A,B),L2 A 3 D B ϕO,D (O,A),L1 ϕO,D (B,D),L3 O Fig. 1. Partial passenger route flow variables In order to keep the resulting model linear, we do not use the common 1 approach ([4]) to measure quality or utility by a function f (p) ∼ imp(p) , but define the travel quality by ω(p) = ρ · t∗OD − t(p) − βc(p) (1) t∗OD denotes the minimum possible travel time from origin O to destination D. The detour factor ρ ≥ 1 should be defined in such a way, that passengers will accept all those routes for which the travel time added with the change penalty is at most a ρ− multiple of the minimum travel time. The quality measure ω(p) can be interpreted as that time, what a passenger can save by using a connection p compared to the maximal accepted travel time for that OD-relation. The best quality is given by t∗OD ·(ρ − 1). If the travel time or/and the number of necessary changings becomes too large, the quality turns out to be negative. If there exists no alternative, more favourable path with positive quality, we will assume to ’loose’ those passengers. Figure 2 illustrates this approach. In order to split the total travel time onto the line parts, we define tL;(A,B) to be the travel time of line L from station A to station B. The quality weights  −tL;(A,B),    −t L;(A,B) − β, O,D ω(A,B),L := ∗  ρtOD − tL;(A,B) − β,    ∗ ρtOD − tL;(A,B) , if A = O and B 6= D if A 6= O and B 6= D if A 6= O and B = D if A = O and B = D obviously have the property, that its sum on each total path from origin node O to destination node D equals with (1) (see Figure 3). Combining flow conservation laws and travel demand leads to the traffic assignment model: (...truncated)