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Problem: Given graph G = (A,N), edge capacity function C : E → Z+. There are k ≥ 1 commodities, each with its own source si, sink ti, and demand dk(i). The objective is to minimize total cost such that we can send dk(i) units of commodity k from si to ti for each i simultaneously, without violating the capacity constraint of any edge. Algorithm: Column generation procedure; Dantzig-Wolfe decomposition
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 4 Choosing the node with minimum cost as the current node, repeating the same procedure
1
9
8
Network assignment
link
(1,2) (2,3) …
Transportation Planning
Objectives of Network Assignment
•Predict the paths to be taken by each trip.
•Output shows the paths that all trips will take, and therefore the number of cars on each roadway and the number of passengers on each transit route.
Transportation Planning
links or arcs nodes source =origin sink =destination
Figure 1. Directed network of links and nodes
Байду номын сангаас
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Transportation Planning
Process of Network Assignment •construct a map representing the vehicle and transit networks •define intersections (nodes) and sections between intersection (links) •identify the length, type of facility, number of lanes, speed, travel time, etc. associated with each link •For transit, identifies fares, headway, route descriptions on a separate network •information allows a computer to determine the paths a traveler may take between any two points on the network and assign trips between zones to these paths.
Step 1 Initialization
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 2 Consider all its unvisited neighbors and calculate their distance, if less than the previous record, updating.
华中科技大学管理学院
谭志加 Office #616 zjatan@mail.hust.edu.cn
Transportation Planning
Land use and socioeconomic projection
Trip generation Trip distribution Transportation system specification The 4-Step Model
Problem: to find the cheapest way to ship prescribed amounts of a commodity from specified origins to specified destinations through a transportation network. Algorithm: Network Simplex Method
Transportation Planning
Overview of Network Flow Problems Minimum cost flow problem
Linear programming formulation:
min
subject to
j: i , j A
i , j A
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 4 Choosing the node with minimum cost as the current node, repeating the same procedure
cij xij
supply or demand
xij
j: j ,i A
iN i
x ji b i , for i N
b 0
0 xij uij , for all i, j A
Transportation Planning
Overview of Network Flow Problems Multi-commodity flow problem
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 4 Choosing the node with minimum cost as the current node, repeating the same procedure
n 1 for i s xij x ji for all i N s j: i , j A j: j ,i A 1
xij 0 for all i, j A
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Modal choice
Network assignment
Direct (user) impact
Transportation Planning
7 5 6 4 3 2
truck
(4,1) xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx
Transportation Planning
Transportation Planning
Network Representation
•A network (see Figure 1) consists of a set of nodes and a set of links connecting these nodes. •Each link is associated with a direction of flow, termed directed link, and the network is thus called a directed network. •Each link is associated with an impedance (or level of service) that affects the flow using it. The units of measurement of this impedance could be time or costs, utility etc. •A path is a sequence of directed links leading from one node to another. •A connected network is one in which it is possible to get from any node to any other node by following a path (or a route) through the network.
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 3 Choosing the node with minimum cost as the current node, repeating the same procedure.
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 5 No unvisited node? Stop.
Transportation Planning
Overview of Network Flow Problems Minimum cost flow problem
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Linear programming formulation:
min
subject to
i , j A
cij xij
n nodes in the network
Problem: to determine for every non-source node i a shortest length directed path from source node s to i. Alternatively, we might view the problem as sending 1 unit of flow as cheaply as possible (with arc costs as Cij) from node s to each of the nodes in the network. Variances: (1) to determine for every source node i a shortest length directed path from node i to sink t; (2) to determine a shortest length directed path for several sources and sinks. Algorithm: Dijkstra's algorithm; Reverse Dijkstra's algorithm; bidirectional Dijkstra's algorithm.
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 4 Choosing the node with minimum cost as the current node, repeating the same procedure
1
9
8
Network assignment
link
(1,2) (2,3) …
Transportation Planning
Objectives of Network Assignment
•Predict the paths to be taken by each trip.
•Output shows the paths that all trips will take, and therefore the number of cars on each roadway and the number of passengers on each transit route.
Transportation Planning
links or arcs nodes source =origin sink =destination
Figure 1. Directed network of links and nodes
Байду номын сангаас
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Transportation Planning
Process of Network Assignment •construct a map representing the vehicle and transit networks •define intersections (nodes) and sections between intersection (links) •identify the length, type of facility, number of lanes, speed, travel time, etc. associated with each link •For transit, identifies fares, headway, route descriptions on a separate network •information allows a computer to determine the paths a traveler may take between any two points on the network and assign trips between zones to these paths.
Step 1 Initialization
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 2 Consider all its unvisited neighbors and calculate their distance, if less than the previous record, updating.
华中科技大学管理学院
谭志加 Office #616 zjatan@mail.hust.edu.cn
Transportation Planning
Land use and socioeconomic projection
Trip generation Trip distribution Transportation system specification The 4-Step Model
Problem: to find the cheapest way to ship prescribed amounts of a commodity from specified origins to specified destinations through a transportation network. Algorithm: Network Simplex Method
Transportation Planning
Overview of Network Flow Problems Minimum cost flow problem
Linear programming formulation:
min
subject to
j: i , j A
i , j A
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 4 Choosing the node with minimum cost as the current node, repeating the same procedure
cij xij
supply or demand
xij
j: j ,i A
iN i
x ji b i , for i N
b 0
0 xij uij , for all i, j A
Transportation Planning
Overview of Network Flow Problems Multi-commodity flow problem
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 4 Choosing the node with minimum cost as the current node, repeating the same procedure
n 1 for i s xij x ji for all i N s j: i , j A j: j ,i A 1
xij 0 for all i, j A
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Modal choice
Network assignment
Direct (user) impact
Transportation Planning
7 5 6 4 3 2
truck
(4,1) xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx
Transportation Planning
Transportation Planning
Network Representation
•A network (see Figure 1) consists of a set of nodes and a set of links connecting these nodes. •Each link is associated with a direction of flow, termed directed link, and the network is thus called a directed network. •Each link is associated with an impedance (or level of service) that affects the flow using it. The units of measurement of this impedance could be time or costs, utility etc. •A path is a sequence of directed links leading from one node to another. •A connected network is one in which it is possible to get from any node to any other node by following a path (or a route) through the network.
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 3 Choosing the node with minimum cost as the current node, repeating the same procedure.
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Step 5 No unvisited node? Stop.
Transportation Planning
Overview of Network Flow Problems Minimum cost flow problem
Transportation Planning
Overview of Network Flow Problems Shortest path problem
Linear programming formulation:
min
subject to
i , j A
cij xij
n nodes in the network
Problem: to determine for every non-source node i a shortest length directed path from source node s to i. Alternatively, we might view the problem as sending 1 unit of flow as cheaply as possible (with arc costs as Cij) from node s to each of the nodes in the network. Variances: (1) to determine for every source node i a shortest length directed path from node i to sink t; (2) to determine a shortest length directed path for several sources and sinks. Algorithm: Dijkstra's algorithm; Reverse Dijkstra's algorithm; bidirectional Dijkstra's algorithm.