COMPETITION AND COOPERATION BETWEEN SUPPLIERSIN MULTIMODAL NETWORK DESIGN PROBLEMS
C. Bandieraa*, R.D. Connorsb, F.ViticaFaculty of Science, Technology and Communication, University of Luxembourg,
Esch-sur-Alzette, Luxembourg
claudia.bandiera@uni.lubFaculty of Science, Technology and Communication, University of Luxembourg,
Esch-sur-Alzette, Luxembourg
richard.connors@uni.lu
And
Institute for Transport Studies, University of Leeds, Leeds, UK
r.d.connors@its.leeds.ac.uk
cUniversity of Luxembourg, Esch-sur-Alzette, Luxembourg
francesco.vit@uni.lu* Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

March 31, 2022
Keywords: EPEC, Multimodal Network, Profit Maximization, Supernetwork, VariationalInequality

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INTRODUCTION

The introduction of new mobility services in the transport market (e.g., carsharing, bike sharing, escooters, etc.) increases both the multimodal alternatives offered to travellers, and the competition
among service providers for attracting customers. Sometimes cooperation between different
Mobility Service Providers (MSPs) is observed, for instance when services are pooled via integrated
fare systems or via the Mobility-as-a-Service paradigm, or thanks to strategic partnerships designed
to offer a service with adequate resources and capture sufficient demand to be economicallysustainable.
Recent works (Nair and Miller-Hooks, 2014, Nguyen et al., 2022) try to capture the interactions
between classical transport services and shared services, but it remains unclear how such systems
respond under various scenarios: suppliers entering or exiting the network; increasing heterogeneityof the users; changes in user preferences and external incentives/regulation (such as governmental
policies or subsidies). In order to capture such phenomena, and represent the interactions between
multiple MSPs and travellers of the transport network, this paper formulates an Equilibrium Problem
with Equilibrium Constraints (EPEC). At the upper level, each MSP seeks to maximize profits, andtheir objective functions contain the specific types of costs and revenues faced by each of them. At
the lower level, users are divided into classes that capture their heterogeneity in terms of daily trip
chains and personal characteristics. The users’ multimodal network is non-separable and therefore
the equilibrium assignment is written as Variational Inequality (VI).

2METHODOLOGY

In this paper the behaviours of, and relationships between MSPs and users are modelled as asupernetwork (or hypernetwork (Sheffi and Deganzo, 1978)). We do not explicitly represent the real
underlying transport infrastructure; instead we divide the transport network into several unimodal
networks (colored links and layers in Figure 1) each representing one type of mobility service beingoffered.
Travellers are divided into K classes based on their personal attributes and daily trip chains. For eachclass, their sequence of activities is modelled as a graph (black layer in Figure 1), where nodes
indicate activity locations and links indicate trips to move from one activity location to the next.

Figure 1: Supernetwork trip chain (adaptation from Carlier et al., 2003)
Each traveller must choose a path through the multimodal network in order to access their sequenceof activity locations. A path (trip chain) can comprise three different types of links. Access links(black dashed lines) allow users to access a mode of transport from their origin (Home), and egressfrom a mode of transport to reach their final destination (Home). Mode-specific links (horizontallink) indicate trips made from one activity location to another using a specific mode of transport(designated by colour). Interchange links (vertical black link) allow users to move from one modeof transport to another.

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We assume that each MSP seeks to maximize the profit arising from their mobility service,represented here as a uniquely defined layer of the supernetwork. They collect revenues based on
how many travellers use the service, and accrue costs primarily depending on the size of their vehicle
fleet. This is the decision variable for each MSP: the capacity or number of vehicles they provide.
Moreover, at equilibrium, MSPs decide how to strategically distribute these vehicles amongst thelinks of their network layer. One innovative aspect of our formulation is the flexibility to represent
costs and revenues for different mobility services, such as car-sharing, bike-sharing, bus, train, escooter and taxi, and hence calculate their profits. In the special case of only one service provider,this problem collapses into the more conventional Network Design Problem.
Each user class is assigned to the multimodal network following fixed demand traffic equilibrium,using a path-based adaptation of the multi-class and multicriteria network equilibrium model(Nagurney, 2000). Explicit enumeration of paths is used for now. The lower level equilibrium
decision variables are the vector of path flows.
One aspect that increases the complexity of the problem is the interdependency between flows on
parallel links of the supernetwork that represent the same real transport link of the underlying
infrastructure network. Consequently, the link costs are non-separable. The users’ equilibrium istherefore formulated as a VI, and the algorithm used to calculate it is the Extragradient method (ormodified projection method) (Nagurney, 1999).

3EXAMPLE AND DISCUSSION

In this section, we show the application of the described methodology solving the users’ equilibrium
at the lower level for one class of travellers. The upper level equilibrium for different MSPs definedin our formulation it is illustrated below, but not explicitly solved it in this paper.
We first illustrate a simple scenario in which a user class with a Home-Work-Home tour has twomodes available: private car and a bus service (Figure 2 left). The costs perceived by users are flow
dependent. More specifically, the private car’s travel time will increase with congestion, while theFigure 2: Example with two modes of transport and one MSPTRISTAN XI Symposium

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attractiveness of the bus service will decrease based on the waiting time at the bus stop. Here the bus
company is the only MSP in the network, and they seek to maximize their profit by varying thecapacity of their service (“Profit Variation” in Figure 2).
The equilibrium between a bus provider and a single user class is shown on the right side of Figure
2. The abscissa of all plots shows increasing the capacity (frequency) of the bus. As expected, thisresults in users swapping from private car to bus (“Path Flow Variation”). The lower right graph ofFigure 2 (“Path Cost Variation”) shows the change in path costs, and that they coincide.
Convergence of the equilibrium algorithm was assured using a standard path cost gap function.

Figure 3: Example with Three modes of transport and two MSPs
Building on this base scenario, we introduce a car sharing service. Bus service capacity is fixed
corresponding to maximum profit from above, the number of vehicles of the shared service isincreased (abscissa of all plots in Fig 3). The system equilibrium changes drastically (Figure 3).
While the bus service has a dedicated lane, the same roads are used by private car and car sharing
who therefore both contribute to the congestion experienced. Initially the car sharing service isunused, due to the absence of vehicles in the network. As the number of car sharing vehicles
increases, users are less attracted from private car and bus; the bus MSP’s profit naturally decreases.
From this example, it is clear that the bus provider needs to respond (in terms of their service
frequency) to the car sharing MSP’s fleet size, to mitigate against loss of profits. In Figure 4 weshow the variation in profits for car share MSP and bus MSP at different combinations of theirdecision variables, illustrating the upper level MSP equilibrium problem.
Despite this simple illustrative example, the methodology accommodates multiple MSPs at the upper
level adopting different strategies. Typically they compete in pursuit of profit, and to become market
leader. In other contexts that we are currently investigating, cooperation may occur with the aim ofMSPs collectively increasing their share of demand and hence revenues.
The lower level equilibrium representing users’ choices depends on the MSPs’ behaviour: the modesof transport available, type of mobility subscriptions, prices and fleet sizes. At the same time, theuser equilibrium influences MSPs’ decisions at the upper level via revenues accrued. A poorly usedmobility service will push the MSP to change strategy, or it will disappear from the market because
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it is unprofitable. Additionally, we are now applying our model to study how a new MSP entering
the market affects users’ equilibrium at the lower level, and competition at the upper level whereMSPs will change their strategies in order to protect their profits.

Figure 4: Profit Variation with capacity

Acknowledgment
This research is part of the project MaaS4All funded by Fonds National de la Recherche
Luxembourg (GRANT_NUMBER: 13769009).

References
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