MaaS Platform Equilibrium Model

MaaS Platform Equilibrium Model is a tool designed to model the decisions of travelers and operators in a Mobility-as-a-Service (MaaS) platform, allowing platform subsidy plans to to achieve a desirable equilibrium. The model considers different types of services providers: Mobility-on-Demand (MoD) operators and traditional fixed-route transit operators. It facilitates efficient management and coordination between users, operators, and the platform within a mobility ecosystem.

The model takes the network structure of the operators, travelers' and operators' costs, traveler demand, and a system objective, and outputs the assignment of traveler demand, operators' operation decisions, and subsidy plans that optimizes the system objective. Potential system objectives includes minimizing system total costs, maximizing equity indices, minimizing GHG emissions, etc. The current tool consideres minimizing system total costs.

The tool is coded in Python 3.8.5.

Model inputs:

• Demand:

  • Number of travelers of each origin-destination (OD) pair for each traveler class considered.
    • Note: Travelers can be divided into different classes according to income level, age, disability, etc.
  • Trip utility of each OD pair for each traveler class considered.
    • Note: Trip utility represent the "gain" upon making this trip, needs to be calibrated beforehand. It can be related to trip purpose, income level, etc.

• Supply:

  • For each fixed-route transit operator: network link they potentially operate on, service frequency options, link operation costs for each link and service frequency, link travel costs.
    • Note: link travel costs is an representation of all measurable non-monetary costs on average, including travel time, average wait time, discomfort, etc.
    • Note: if system objectives other than cost are considered other information would be needed for the links, such as GHG emission per user for each link, etc.
  • For each MoD operator: service zones that they potentially cover, fleet size options, service cost per user in between each pair of service region, installation cost, access cost of each service zones as functions of number of travelers accessing the service through the zone and fleet size of the zone.
    • Note: all costs are trasferred into a common unit, e.g. $.
    • Note: the access cost function in this tool is defined as follows: t = t_coef*x^2/h, the coefficient t_coef is an input.

• Convergence parameters:

  • Tolerance ϵ of subgradient optimization (default: tolerance = 0.05).
  • Tolerance ε of Frank-Wolfe algorithm (default: FW_tol = 10**-2).
  • Required consecutive number of iterations meeting the tolerance of Frank-Wolfe algorithm (default: FW_conver_times = 5).

Model outputs:

• Equilibrium traveler flows of each travelers class on each operated link;
• Number of travelers of each origin-destination (OD) pair that choose not to use the MaaS platform;
• Operation decisions of operators:
  • For each fixed-transit operator: the links they decide to operate and corresponding service frequencies;
  • For each MoD operator: service regions that they decide to cover and corresponding fleet sizes;
• Subsidies per traveler on each link.

Input format:

There are 5 tables required as inputs, which are described as follows. Please name the input tables and columns exactly the same as instructed.

1. Network information table: "network.csv"

Columns:

• "link number": ID of the link.
• "from_node": ID of the start node of the link.
• "to_node": ID of the end node of the link.
• "travel cost": travel cost of the link ($).
• "operating cost": operating cost of the link ($). Only applies if it is a fixed-route link. For other link types, 0 should be put in.
• "capacity": capacity of the link (travelers/unit time). Only applies if it is a fixed-route link. For other link types, a number that is large enough (e.g. 10^8) should be put in.
• "operator": ID of the operator that owns the link.
• "link type": "fixed", "MOD", or "transfer". "fixed" means fixed-route transit link, "MOD" means MOD link, "transfer" means walking link.

2. Demand information table: "demand.csv"

Columns:

• "user_number": ID of the traveler group.
• "from_node": ID of the node from which the traveler group departs.
• "to_node": ID of the node to which the traveler group heads.
• "demand_amount": number of travelers in the traveler group.
• "utility": trip utility of each travelers in the traveler group ($).

3. MOD node renumbering table: "MODnodes.csv"

Columns:

• "fixed_node": MOD zones are represented by their centroids which are included in the network constructed by fixed-route transit links and transfer links (fixed network). This column is the ID of the node in the fixed network.
• "MOD_node": ID of the MOD node that represent the same location as the node in the fixed network in the same row.
• "direction": "bi" or not. Bidirection or not. "bi" as default.
• "operator": ID of the MOD operator that owns the node.
• "capacity": capacity of the MOD node when it is operated (number of users per time unit).
• "cost": cost of operating the MOD node ($). a one-time fixed cost representing the infrastructure cost.

4. MOD operation coeffcients table: "MODoper_coef.csv"

The access/wait cost functions of all the MoD operators on all MoD access links l is represented as follows.

$\tau(x_l;h_l)=0.5h_l^{-2}x_l$

where

$\tau$ is the travel cost of the access link $l$;

$x_l$ is the total flow on the access link $l$;

$h_l$ is the fleet size of the operator that covers the node accessed by access link $l$;

$t_coef$ is a parameter input by this table. The form is hard-coded in this tool.

The MoD operating cost per user on MOD link $l$ of all the MoD operators at all MoD nodes is represented as follows.

$m_l=h_l^2$

where

$m_l$ is the operation cost per person of the MOD link $l$;

$c_coef$is a parameter input by this table. The form is hard-coded in this tool.

Columns:

• "Operator": ID of the MOD operator.
• "t_coef": value of $t_coef$ for the MOD operator.
• "c_coef": value of $c_coef$ for the MOD operator.

5. Fleet size options table: "fleet_size_options.csv"

Columns:

• "operator": ID of MOD operator.
• "fleet_size_options": fleet size options of the MOD operator as a string separated by commas, e.g. "1,2,3".

Output format:

1. Optimal link flows per OD: "link_flows_od.csv"

Columns:

• "origin": node ID of the origin of the OD pair.
• "destination": node ID of the destination of the OD pair.
• "start": node ID of the start of the link.
• "end": node ID of the end of the link.
• "operator": operator ID of the operator that controls the link (0 if no one owns).
• "link_flow_of_the_od": solved link flow of the OD pair on the link (number of people per time unit).

2. Optimal path flows and subsidy needed per user each path: "path_flows_od.csv"

Columns:

• "origin": node ID of the origin of the OD pair.
• "destination": node ID of the destination of the OD pair.
• "path": path consists of a list of node IDs.
• "path_flow": solved flow on the path (number of people per time unit).
• "subsidy_per_person": solved subsidy per person on the path ($).

4. Operation decisions of fixed-route links: "fixed_route_operation_decisions.csv"

Columns:

• "start": node ID of the start of the link.
• "end": node ID of the end of the link.
• "operator": operator ID of the operator that controls the link.
• "operated_or_not": 1 if link operated, 0 otherwise.

6. Operation decisions of MOD nodes: "MOD_operation_decisions.csv"

Columns:

• "node": MOD node ID.
• "operator": operator ID of the MOD operator that controls the MOD node.
• "fleet_size": fleet size options of the MOD node.
• "chosen_or_not": 1 if the combination of MOD node and fleet size is choosen by the MOD operator, 0 otherwise.

8. Optimal fares: "fares.csv"

Columns:

• "start": node ID of the start of the link.
• "end": node ID of the end of the link.
• "operator": operator ID of the operator that controls the link.
• "buyer_optimal_fare": solved link fare for buyer optimal ($).
• "seller_optimal_fare": solved link fare for seller optimal ($).

10. Optimal user surplus: "user_surplus.csv"

Columns:

• "origin": node ID of the origin of the OD pair.
• "destination": node ID of the destination of the OD pair.
• "user_surplus_per_person_buyer_optimal": user surplus per person of the OD pair at buyer optimal ($).
• "user_surplus_per_person_seller_optimal": user surplus per person of the OD pair at seller optimal ($).

Example

We use a toy network shown in Fig. 1 to illustrate how the method works. All costs are in dollars ($). The solid links represent the fixed-route services, links with the same color are operated by the same operator. Link (21,22) and (21,23) are transfer links between lines, which are without capacity and operating cost with no owners. There are 3 MoD operators (blue, green, brown). The circles represent the service zones that MOD operators can choose from to operate (blue: A,B,C; green: B,C; brown: B,D). Zone A covers transit station node 1. Zone B covers transit station nodes 21,22,23. Zone C covers transit station node 3. Zone D covers transit station node 4.

The network is expanded into Fig. 2 by creating complete subgraphs for each MoD operator and adding MoD access links and egress links. Travel cost, operating cost, and capacities are labelled as shown in the legend.

Fleet size choices of all MoD operators are 1, 2, and 3. Installation cost of MoD nodes 7, 8, 9, 10, 11, 12, and 13 are 3, 3, 2, 2, 1, 1, and 3, respectively for all fleet size options. Travel demand is 1,000 from node 1 to 3, and 500 from node 1 to 4. Trip utility U_s is $9.50 for both OD pairs. The tolerance ϵ for subgradient optimization is 0.05. The tolerance ε of Frank-Wolfe is 0.01 and the required consecutive number of iterations meeting the tolerance is 5. No optimality gap control is applied, the algorithm is terminated when all branches are pruned.

Figure 1. Toy network.

Figure 2. Expended toy network.

The model inputs are as follows.

1. Network information table: "network.csv"

2. Demand information table: "demand.csv"

3. MOD node renumbering table: "MODnodes.csv"

4. MOD operation coeffcients table: "MODoper_coef.csv"

5. Fleet size options table: "fleet_size_options.csv"

After running the tool, the outputs are as follows.

1. Optimal link flows per OD: "link_flows_od.csv"

2. Optimal path flows and subsidy needed per user each path: "path_flows_od.csv"

3. Operation decisions of fixed-route links: "fixed_route_operation_decisions.csv"

4. Operation decisions of MOD nodes: "MOD_operation_decisions.csv"

5. Optimal fares: "fares.csv"

6.Optimal user surplus: "user_surplus.csv"

---

Please contact [email protected] for questions or more information.