Finding the UE flow pattern over a
transportation network
Methods:
Heuristic equilibration techniques
Applying the convex combinations methods
Shortest paths over a network
1.HEURISTIC EQUILIBRATION
TECHNIQUES
Network loading (All- or -nothing):
Assigning the OD entries to the network for
constant travel time.
All paths connecting 𝑟 − 𝑠 are fixed travel time
(Example: 𝑡𝑎 = 𝑡𝑎 0 ∀ 𝑎 )
Each OD trip rates 𝑞𝑟𝑠 assign to link that is on
minimum travel time path.
CAPACITY RESTRAINT
Idea: perform all – or – nothing process based on
previous travel times.
Algorithm:
Step 𝟎: Initialization
𝑡𝑎 = 𝑡𝑎 (0)
Perform all – or – nothing → {𝑥𝑎0 }
𝑛 ≔ 1
Step 1: Update
𝑡𝑎 = 𝑡𝑎 𝑥𝑎𝑛−1 ∀ 𝑎
Step 2: Network loading
Perform all – or – nothing based on 𝑡𝑎𝑛 → 𝑥𝑎𝑛
Step 3: Convergence test
If max 𝑥𝑎𝑛 − 𝑥𝑎𝑛−1 ≤ 𝜖 → stop
𝑎
Otherwise, 𝑛 = 𝑛 + 1 → Step 1
Improve: Using combination of the last two
travel times. (Smoothing)
Algorithm:
Step 𝟎: Initialization
𝑡𝑎 = 𝑡𝑎 (0)
Perform all – or – nothing → {𝑥𝑎0 }
𝑛 ≔ 1
Step 1: Update
𝜏𝑎 = 𝑡𝑎 𝑥𝑎𝑛−1 ∀𝑎
Step 2: Smoothing
𝑡𝑎𝑛 = 0.75𝑡𝑎𝑛−1 + 0.25𝜏𝑎𝑛 ∀𝑎
Step 3: Network loading
Perform all – or - nothing based on 𝑡𝑎𝑛 → {𝑥𝑎𝑛 }
Step 4: Stopping rule:
n=N→ Step 5
 otherwise: 𝑛 = 𝑛 + 1, → Step 1
Step 5: Averaging
𝑥𝑎∗
=
1
4
3
𝑛−𝑘
𝑥
𝑘=0 𝑎
and stop.
INCRMENTAL ASSIGNMENT
Assign a portion entry to network at each
interaction.
Algorithm:
Step 𝟎: Initialization
𝑛
𝑞𝑟𝑠
=
𝑞𝑟𝑠
;
𝑁
𝑛 ≔ 1; 𝑥𝑎0 = 0 ∀𝑎
Step 1: Update
𝑡𝑎𝑛 = 𝑡𝑎 𝑥𝑎𝑛−1 ∀𝑎
Step 2: Incremental loading
Perform all – or – nothing based on 𝑡𝑎𝑛 and trip
𝑛 → 𝑤𝑛
rate 𝑞𝑟𝑠
𝑎
Step 3: Flow summation
𝑥𝑎𝑛 = 𝑥𝑎𝑛−1 + 𝑤𝑎𝑛 ∀ 𝑎
Step 4: Stopping rule
If 𝑛 = 𝑁: stop
Otherwise: 𝑛 = 𝑛 + 1 and go to step 1
2.APPLYING THE CONVEX
COMBINATIONS METHOD
UE problem:
𝑥𝑎
𝑎 0 𝑡𝑎 (𝜔) 𝑑𝜔
Min 𝑧 𝒙 =
Subject to
𝑓𝑘𝑟𝑠 = 𝑞𝑟𝑠
𝑓𝑘𝑟𝑠 ≥ 0
𝑥𝑎 =
2.1𝑎
∀𝑟, 𝑠
(2.1b)
∀𝑘, 𝑟, 𝑠
(2.1𝑐)
𝑟𝑠 𝑟𝑠
𝑓
𝑟,𝑠,𝑘 𝑘 𝛿𝑎,𝑘
∀𝑎
(2.1c)
Applying the convex combination algorithm: at
each interaction find 𝐲 𝒏 :
Min 𝑧 𝑛 𝒚 = 𝑎 𝑡𝑎𝑛 𝑦𝑎
Subject to
𝑟𝑠
𝑔
∀𝑟, 𝑠
𝑘 𝑘 = 𝑞𝑟𝑠
𝑔𝑘𝑟𝑠
≥0
∀𝑘, 𝑟, 𝑠
𝑦𝑎
= 𝑘,𝑟,𝑠 𝑔𝑘𝑟𝑠 𝛿𝑎,𝑘
Sub problem: with each OD pair r-s:
Min 𝑧 𝑛 𝒈𝑟𝑠 = 𝑘 𝑐𝑘𝑟𝑠 𝑔𝑘𝑟𝑠
Subject to
𝑟𝑠
𝑔
𝑘 𝑘 = 𝑞𝑟𝑠
𝑔𝑘𝑟𝑠
≥0
∀𝑘
⇒ finding the path m
𝑟𝑠
𝑐𝑚
Obtain
= min 𝑐𝑘𝑟𝑠
𝑘
𝑟𝑠
𝑔𝑚
= 𝑞𝑟𝑠 𝑎𝑛𝑑 𝑔𝑘𝑟𝑠 = 0 ∀ 𝑘 ≠ 𝑚
ALGORITHM
Step 𝟎: Initialization
𝑡𝑎 = 𝑡𝑎 0 , 𝑛 = 1
Perform all – or – nothing → {𝑥𝑎1 }
Step 1: Update
𝑡𝑎𝑛 = 𝑡𝑎 𝑥𝑎𝑛 ∀𝑎
Step 2: Direction finding
Perform all – or –nothing based on {𝑡𝑎𝑛 } → 𝑦𝑎𝑛
Step 3: Line search
Find 𝛼𝑛 :
min
0≤𝛼≤1
𝑛 +𝛼 (𝑦 𝑛 −𝑥 𝑛 )
𝑥𝑎
𝑛 𝑎
𝑎
𝑡𝑎
𝑎 0
𝜔 𝑑𝜔
Step 4: Move
𝑥𝑎𝑛+1 = 𝑥𝑎𝑛 + 𝛼𝑛 𝑦𝑎𝑛 − 𝑥𝑎𝑛 ∀𝑎
Step 5: Convergence test
If
𝑛+1 −𝑥 𝑛
𝑥
𝑎 𝑎
𝑎
𝑛
𝑎 𝑥𝑎
2
≤ 𝜖 → stop
Otherwise: 𝑛 = 𝑛 + 1 and go to step 1
COMMENT
Number of interactions is affected by the
congestion level
Actual application: four to six interactions.
3.SHORTEST PATHS
Using label – correcting methods to find
shortest path from a root node to all other node
Idea
Notation
Sequence list S
𝑡𝑖𝑗 : travel time on link 𝑖 → 𝑗
With each node 𝑖:
𝑙𝑖 : current label
𝑝𝑖 : predecessor node
Algorithm
Input: root node =0, network
Step 0: Initialization
𝑙𝑖 = ∞, 𝑝𝑖 = 0, l0 = 0
𝑆 = {0}
Step 1:
Select a node 𝑖 from 𝑆
All node 𝑗 adjoin 𝑖:
If 𝑙𝑖 + 𝑡𝑖𝑗 < 𝑙𝑗 then 𝑙𝑗 = 𝑙𝑖 + 𝑡𝑖𝑗 , 𝑝𝑗 = 𝑖,
𝑆 ∪ {𝑗}
𝑆 = 𝑆\{𝑖}
𝑆=
Shortest path from root- node: trace back from
node to root by 𝑝𝑖
Step 2: Stopping condition
If 𝑆 = ∅ then Stop
Else go to step 1