Transportation Modeling: An Artificial Life Approach * Presented by - - PDF document

Transportation Modeling: An Artificial Life Approach*

Presented by Matthew Nizol CSE 914, Spring 2014 Michigan State University February 6th, 2014

* Panta Lučić and Dušan Teodorović. "Transportation modeling: an artificial life approach," in Proceedings of the 14th IEEE International Conference on Tools with Artificial Intelligence, pp. 216-223, 2002.

Outline

1.Context and Background Material 2.Fuzzy Ant System 3.Bee System 4.Q&A and Discussion

Context

• Using Artificial Life methods
• Fuzzy Ant System
• Bee System
• To solve combinatorial
• ptimization problems

Combinatorial Optimization Problems (COPs)

• Finite set 𝑇 of feasible solutions
• Objective function 𝑔: 𝑇 ¡ → ℝ
• Optimal solution: s ∈ 𝑇 that

minimizes or maximizes 𝑔(𝑡)

Some Transportation-Related COPs

• Traveling Salesman Problem
• Vehicle Routing Problem
• Highway Alignment Optimization
• Transit Line Schedule

Synchronization

Example: Traveling Salesman Problem

Detroit Flint Lansing Ann Arbor

70 56 54 65 44 91

Example: Traveling Salesman Problem

Detroit Flint Lansing Ann Arbor

70 56 54 65 44 91

Why is TSP Hard?

• Previous example:
• 3 choices for first node
• 2 choices for second node
• 1 choice for third node
• 3 x 2 x 1 = 6 possible solutions
• In general, O(n!) possible solutions
• n! grows very quickly: 20! > 2 × 10

Big-Oh and Polynomial Time

Input Size f(n) g(n) Time

Hardness

• Class P: Can solve in polynomial time
• Class NP: Can verify solution in

polynomial time

• NP-hard: At least as hard as any NP

problem

• Strategy: Seek near-optimal solutions

Strategies Inspired by Nature

• Genetic Algorithms
• Simulated Annealing
• Neural Nets
• Swarm Intelligence

Swarm Intelligence

• Inspired by behavior of social insects
• Individuals follow simple rules
• Communication between individuals
• Ants leave pheromone trail
• Bees dance to advertise food sources
• Intelligent behavior emerges from these

interactions

Ant System

• Proposed by Colorni, et al in 1991
• Inspired by behavior of ants in nature
• Used to solve Traveling Salesman

Problem

• Primary inspiration for Bee System

and Fuzzy Ant System

Artificial Ants and the TSP

An artificial ant : 1.Has memory 2.Makes non-deterministic choices 3.Communicates via pheromones

Ant System Example

D B A C

7; 1 5; 1 5; 1 6;1 4; 1 9; 1

Ant System Example

D B A C

7; 1 5; 1 5; 1 9;1 4; 1 9; 1

Ant 1 Start

Ant System Example

D B A C

7; 1 5; 1 5; 1 9;1 4; 1 9; 1

Ant 2 Start

Artificial Pheromone

• After each iteration, pheromone on edge 𝑗, 𝑘 is updated:

𝑢 = 𝑓𝑤𝑏𝑞𝑝𝑠𝑏𝑢𝑗𝑝𝑜 × 𝑢 + (𝑏𝑜𝑢 ¡𝑣𝑞𝑒𝑏𝑢𝑓𝑡) ¡

Artificial Pheromone

• After each iteration, pheromone on edge 𝑗, 𝑘 is updated:

𝑢 = 𝝇 × 𝑢 + 𝑏𝑜𝑢 ¡𝑣𝑞𝑒𝑏𝑢𝑓𝑡

• Where:
• 𝝇 is the evaporation rate

Artificial Pheromone

• After each iteration, pheromone on edge 𝑗, 𝑘 is updated:

𝑢 = 𝜍 × 𝑢 + (𝑱𝒍𝒋𝒌× 𝚬𝒍)

𝒏 𝒍𝟐

• Where:
• 𝜍 is the evaporation rate
• 𝒍 represents the 𝒍𝒖𝒊 ant
• 𝒏 is the number of ants
• 𝑱𝒍𝒋𝒌 indicates (1 or 0) whether ant 𝒍 visited (𝒋, 𝒌)
• 𝜠𝒍 is the change in pheromone due to the 𝒍𝒖𝒊 ant

Artificial Pheromone

• After each iteration, pheromone on edge 𝑗, 𝑘 is updated:

𝑢 = 𝜍 × 𝑢 + (𝐽× 𝑹 𝑴𝒍

• )
• Where:
• 𝜍 is the evaporation rate
• 𝑙 represents the 𝑙 ant
• 𝑛 is the number of ants
• 𝐽 indicates (1 or 0) whether ant 𝑙 visited (𝑗, 𝑘)
• 𝑹 is an arbitrary constant
• 𝑴𝒍 is the length of the tour taken by the 𝒍𝒖𝒊 ant

Ant System Example

• Set 𝜍 = 𝟏. 𝟔
• Set 𝑅 = 100
• Set 𝑀 = 25
• Set 𝑀 = 30
• ∆= ¡
• ¡ = ¡
• = 𝟓
• ∆= ¡
• = ¡
• = 𝟒. 𝟒

Ant System Example

D B A C

5; (0.5*1 + 4)=4.5 5; 3.8 4; 4.5 7; (0.5*1 + 3.3 + 4)=7.8 9; 7.8 9; (0.5*1 + 3.3) = 3.8

Ant System Example

D B A C

5; 4.5 5; 3.8 4; 4.5 7; 7.8 9; 7.8 9; 3.8

Ant 1 Start, Iteration 2

Probability of Node Selection

• An ant at node 𝑗 ¡travels to node 𝑘 ¡with probability:

𝑄𝑠𝑝𝑐 𝑘 = ¡ 𝑞ℎ𝑓𝑠𝑝𝑛𝑝𝑜𝑓 ¡𝑟𝑢𝑧 ¡𝑝𝑜 ¡ 𝑗, 𝑘 × (𝑤𝑗𝑡𝑗𝑐𝑗𝑚𝑗𝑢𝑧 ¡𝑝𝑔 ¡(𝑗, 𝑘)) 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠

Probability of Node Selection

• An ant at node 𝑗 ¡travels to node 𝑘 ¡with probability:

𝑄𝑠𝑝𝑐 𝑘 = ¡ 𝒖𝒋𝒌

× (𝑤𝑗𝑡𝑗𝑐𝑗𝑚𝑗𝑢𝑧 ¡𝑝𝑔 ¡(𝑗, 𝑘))

𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠

• Where:
• 𝒖𝒋𝒌 is the quantity of pheromone on (i, j)
• 𝜷 is a tuning parameter

Probability of Node Selection

• An ant at node 𝑗 ¡travels to node 𝑘 ¡with probability:

𝑄𝑠𝑝𝑐 𝑘 = ¡ 𝑢

× (𝒐𝒋𝒌)𝜸

𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠

• Where:
• 𝑢 is the quantity of pheromone on (i, j)
• 𝛽 is a tuning parameter
• 𝒐𝒋𝒌is the inverse of the distance from i to j
• 𝜸 is a tuning parameter

Probability of Node Selection

• An ant at node 𝑗 ¡travels to node 𝑘 ¡with probability:

𝑄𝑠𝑝𝑐 𝑘 = ¡ 𝑢

× (𝑜)

∑ (𝒖𝒋𝒍)𝜷(𝒐𝒋𝒍)𝜸

𝒍∈𝑽

• Where:
• 𝑢 is the quantity of pheromone on (i, j)
• 𝛽 is a tuning parameter
• 𝑜is the inverse of the distance from i to j
• 𝛾 is a tuning parameter
• U is the set of unvisited nodes

Schedule Synchronization

Transit Line Transfer Station

Schedule Synchronization

• Given:
• A transit system with transfer stations
• Line headways
• Stop time at stations
• Travel time between stations
• Number of transferring passengers
• Goal: Minimize waiting time by
• ptimizing terminal departure times

Schedule Synchronization

• Given:
• A transit system with transfer stations
• Line headways
• Stop time at stations
• Travel time between stations
• Number of transferring passengers
• Goal: Minimize waiting time by
• ptimizing terminal departure times
• Set membership can be partial and can overlap:
• Each element has membership degree
• E.g., Bob is Short with truth degree 0.4
• Truth degree ≠ Probability

Fuzzy Set Theory

Short Medium Tall

Height

1

Fuzzy Ant System (FAS)

• Modifies Ant System
• Replaces exact values with fuzzy sets
• Distance
• Pheromone intensity
• Replaces transition probability with

Fuzzy Logic

FAS for Traveling Salesman

Small Distance Medium Distance Long Distance Weak Pheromone Medium Pheromone Strong Pheromone

dij tij

1 1

FAS for Traveling Salesman

• Fuzzy transition function:

If distance is SHORT and trail intensity is STRONG Then utility is VERY HIGH

• Probability calculated for each target

node based on fuzzy utility

• Otherwise the same as Ant System

FAS for Schedule Synchronization

Model passengers transferring from line 𝑗 to line 𝑘 at time 𝑣 as:

Piju Number of Passengers

1

Uncertainty

FAS for Schedule Synchronization

O Line 1 Line 2 Line 3 D

Possible Terminal Departure Times

FAS for Schedule Synchronization

O Line 1 Line 2 Line 3 D Ant 1 “Ride” ¡line, ¡computing ¡arrival, ¡ departure, and wait times

FAS for Schedule Synchronization

O Line 1 Line 2 Line 3 D Ant 1

FAS for Schedule Synchronization

O Line 1 Line 2 Line 3 D Ant 1 Ant 2

FAS for Schedule Synchronization

• On next iteration, ants pick next node

using fuzzy rules that consider:

• Estimated wait time
• Pheromone levels
• Node selection based on rules such as

If wait time is SMALL and trail intensity is STRONG Then utility is VERY HIGH

FAS Results for 50 Lines

Bee System

• Inspired by foraging behavior
• f bees in nature
• Similar to ideas found in the

Ant System

• Used to solve Traveling

Salesman Problem

Behavior of Bees in Nature

• 1. Find Food Source
• 2. Return Nectar to Hive
• 3. Choose one of the following:
• a. Return to food source alone
• b. Perform ¡“waggle ¡dance” ¡
• c. Abandon food source

Bee System Example

H S = 2 (two nodes visited per stage) Stage 1, Iteration 1

Bee System Example

H Bee 1 Bee 2 Bee 3 S = 2 (two nodes visited per stage) Stage 1, Iteration 1

Bee System Example

H Bee 1 Bee 2 Bee 3 S = 2 (two nodes visited per stage) Stage 1, Iteration 1

Bee System Example

H Bee 1 Bee 2 Bee 3 S = 2 (two nodes visited per stage) Stage 1, Iteration 1

Bee System Example

H Bee 1 Bee 2 Bee 3 S = 2 (two nodes visited per stage) Stage 2, Iteration 1 Start of Stage 2

Bee System: Time Hierarchy

• 1. 𝑡 nodes visited during a stage

2.

• ¡stages during an iteration
• 3. 𝑁 iterations

Probability of Choosing a Node

Probability that bee 𝑙 at node 𝑗 during stage 𝑣 + 1 ¡and iteration 𝑨 chooses node 𝑘:

𝑞

𝑣 + 1, 𝑨 = ¡

exp ¡ (−1 × 𝑒𝑗𝑡𝑢𝑏𝑜𝑑𝑓 × 𝑗𝑢𝑓𝑠𝑏𝑢𝑗𝑝𝑜 × 1 𝑞𝑝𝑞𝑣𝑚𝑏𝑠𝑗𝑢𝑧) 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠

Probability of Choosing a Node

Probability that bee 𝑙 at node 𝑗 during stage 𝑣 + 1 ¡and iteration 𝑨 chooses node 𝑘:

𝑞

𝑣 + 1, 𝑨 = ¡

exp ¡ (−𝒃𝒆𝒋𝒌 𝒜 𝑞𝑝𝑞𝑣𝑚𝑏𝑠𝑗𝑢𝑧) 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠 Where:

• 𝒃 is a tuning parameter
• 𝒆𝒋𝒌 is the distance from node 𝒋 to node 𝒌
• 𝒜 is the iteration index

Probability of Choosing a Node

Probability that bee 𝑙 at node 𝑗 during stage 𝑣 + 1 ¡and iteration 𝑨 chooses node 𝑘:

𝑞

𝑣 + 1, 𝑨 = ¡

exp ¡ (−𝑏𝑒 𝑨 ∑ 𝒐𝒋𝒌(𝒔)

𝒜𝟐 𝒔𝐧𝐛𝐲 ¡ (𝒜𝒄,𝟐)

) 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠 Where:

• 𝑏 is a tuning parameter
• 𝑒 is the distance from node 𝑗 to node 𝑘
• 𝑨 is the iteration index
• 𝒄 is the memory length
• 𝒐𝒋𝒌(𝒔) counts the bees that visited link (𝒋, 𝒌) during iteration 𝒔

Probability of Retaining Tour

Probability that bee 𝑙 does not abandon their partial tour after stage 𝑣 during iteration 𝑨:

𝑞 𝑣 + 1, 𝑨 = exp ¡ (

,

((,))

×

)

Where:

• 𝑀 𝑣, 𝑨 is the length of bee 𝑙’s ¡tour ¡during ¡stage ¡𝑣 and

iteration 𝑨

Waggle Dancing

• Bees usually advertise tours they retain
• Communication is via waggle dancing
• Bees that abandon their tour watch

dancing bees to select a new tour

• The probability of following a dancer:
• Increases with shorter tour lengths
• Increases with more bees advertising the

same tour

Benchmark Results

Benchmark Optimal Value Best Bee System Value %-Difference Run Time (seconds) Eil51 428.87 428.87 37 Berlin52 7544.37 7544.37 1 St70 677.11 677.11 22 Pr76 108159 108159 11 Kroa100 21285.4 21285.4 10 Eil101 640.21 640.21 1741 Tsp225 3859 3876.05 0.44 5153 A280 2586.77 2600.34 0.53 13465

Time Complexity

1 10 100 1000 10000 100000 50 100 150 200 250 300 Run Time (seconds) Graph Order (Number of Cities)

Bee System: Run Time vs Graph Order on TSP Benchmarks

Uncertainty: Algorithm

• Source: Behavior of Algorithm
• Mitigation: N/A
• Leveraging controlled uncertainty

Uncertainty: State Space

• Source: State space size
• Solution space: Is solution optimal?
• Input space: Is algorithm validated?
• Mitigation:
• Non-deterministic search
• Benchmark Comparisons

Uncertainty: Environment

• Sources:
• Trains:
• Arrival / travel times
• System malfunctions
• Number of transfer

passengers

• Other:
• Cultural expectations
• Geography
• Mitigation?
• Buses and trucks:
• Weather
• Road conditions
• Traffic
• Driver behavior
• Policy changes

Uncertainty: Environment

• Sources:
• Trains:
• Arrival / travel times
• System malfunctions
• Number of transfer

passengers

• Other:
• Cultural expectations
• Geography
• Mitigation: Fuzzy Logic
• Buses and trucks:
• Weather
• Road conditions
• Traffic
• Driver behavior
• Policy Changes

Limitations

• Tuning parameter guidance
• Validation
• Time complexity
• Justification for probability functions
• Justification for complexity
• Many types of uncertainty not

considered

Research Challenges

• How do we validate these systems?
• Can we improve the time complexity?
• How can we handle other types of

uncertainty?

• Might different mathematical models

improve results or running time?

• How might these systems be applied to

dynamically adaptive systems?

References

• 1. Colorni, ¡et ¡al. ¡ ¡“Heuristics ¡from ¡Nature ¡for ¡Hard ¡

Combinatorial ¡Optimization ¡Problems,” ¡International ¡ Transactions in Operational Research, vol. 3, no. 1, pp. 1- 21, 1996.

• 2. Cormen, et al. Introduction to Algorithms. 2nd Edition.

MIT Press, 2003.

• 3. W. Domschke. ¡ ¡“Schedule ¡Synchronization ¡for ¡Public ¡

Transit Networks,” ¡Operations ¡Research ¡Spektrum, vol. 11, issue 1, pp. 17-24, 1989.

• 4. Laporte, Gilbert. ¡ ¡“The ¡Vehicle ¡Routing ¡Problem: ¡An ¡
• verview of exact ¡and ¡approximate ¡algorithms,” ¡European ¡

Journal of Operational Research, vol. 59, pp.345-358, 1992.

References

5. Panta Lučić and Dušan Teodorović. "Transportation modeling: an artificial life approach," in Proceedings of the 14th IEEE International Conference on Tools with Artificial Intelligence,

• pp. 216-223, 2002.

6. Panta Lučić. ¡ ¡“Modeling ¡Transportation ¡Problems ¡Using ¡ Concepts ¡of ¡Swarm ¡Intelligence ¡and ¡Soft ¡Computing,” ¡PhD ¡ Dissertation, Virginia Polytechnic Institute and State University, 2002. 7. Stuart Russell and Peter Norvig. Artificial Intelligence: A Modern Approach. 2nd Edition. Prentice Hall, 2003. 8. Dušan Teodorović, et al. ¡ ¡“Bee ¡Colony ¡Optimization: ¡Principles ¡ and ¡Applications,” ¡8th Seminar on Neural Network Applications in Electrical Engineering, pp. 151-156, 2006.

Questions and Discussion