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

Outline 1.Context and Background Material 2.Fuzzy Ant System 3.Bee System 4.Q&A and Discussion Transportation Modeling: An Artificial Life Approach * Presented by Matthew Nizol CSE 914, Spring 2014 Michigan State University February 6 th , 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.

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) Context • Using Artificial Life methods • Fuzzy Ant System • Bee System • To solve combinatorial optimization problems

Example: Traveling Salesman Problem 56 Lansing Flint 91 65 70 54 Ann Detroit Arbor 44 Some Transportation-Related COPs • Traveling Salesman Problem • Vehicle Routing Problem • Highway Alignment Optimization • Transit Line Schedule Synchronization

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 �� Example: Traveling Salesman Problem 56 Lansing Flint 91 65 70 54 Ann Detroit Arbor 44

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 Big-Oh and Polynomial Time g(n) f(n) Time Input Size

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 Strategies Inspired by Nature • Genetic Algorithms • Simulated Annealing • Neural Nets • Swarm Intelligence

Artificial Ants and the TSP An artificial ant : 1.Has memory 2.Makes non-deterministic choices 3.Communicates via pheromones 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

Ant System Example 5; 1 Ant 1 A B Start 9; 1 9;1 7; 1 5; 1 D C 4; 1 Ant System Example 5; 1 A B 9; 1 7; 1 6;1 5; 1 D C 4; 1

Artificial Pheromone • After each iteration, pheromone on edge 𝑗, 𝑘 is updated: 𝑢 �� = 𝑓𝑤𝑏𝑞𝑝𝑠𝑏𝑢𝑗𝑝𝑜 × 𝑢 �� + (𝑏𝑜𝑢 ¡𝑣𝑞𝑒𝑏𝑢𝑓𝑡) ¡ Ant System Example 5; 1 A B 9; 1 7; 1 9;1 5; 1 Ant 2 D C Start 4; 1

Artificial Pheromone • After each iteration, pheromone on edge 𝑗, 𝑘 is updated: 𝒏 𝑢 �� = �𝜍 × 𝑢 �� � + �(𝑱 𝒍𝒋𝒌 × 𝚬 𝒍 ) 𝒍�𝟐 • Where: • 𝜍 is the evaporation rate • 𝒍 represents the 𝒍 𝒖𝒊 ant • 𝒏 is the number of ants • 𝑱 𝒍𝒋𝒌 indicates (1 or 0) whether ant 𝒍 visited (𝒋, 𝒌) • 𝜠 𝒍 is the change in pheromone due to the 𝒍 𝒖𝒊 ant Artificial Pheromone • After each iteration, pheromone on edge 𝑗, 𝑘 is updated: 𝑢 �� = �𝝇 × 𝑢 �� � + 𝑏𝑜𝑢 ¡𝑣𝑞𝑒𝑏𝑢𝑓𝑡 • Where: • 𝝇 is the evaporation rate

Ant System Example • Set 𝜍 = 𝟏. 𝟔 • Set 𝑅 = 100 • Set 𝑀 � = 25 • Set 𝑀 � = 30 � ��� • ∆ � = ¡ � � ¡ = ¡ �� = 𝟓 � ��� • ∆ � = ¡ � � = ¡ �� = 𝟒. 𝟒 Artificial Pheromone • After each iteration, pheromone on edge 𝑗, 𝑘 is updated: � �(𝐽 ��� × 𝑹 𝑢 �� = �𝜍 × 𝑢 �� � + ) 𝑴 𝒍 ��� • 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 • 𝑴 𝒍 is the length of the tour taken by the 𝒍 𝒖𝒊 ant

Ant System Example 5; 4.5 Ant 1 Start, A B Iteration 2 9; 3.8 7; 7.8 9; 7.8 5; 3.8 D C 4; 4.5 Ant System Example 5; (0.5*1 + 4)= 4.5 A B 9; (0.5*1 + 3.3) = 3.8 9; 7.8 7; (0.5*1 + 3.3 + 4)= 7.8 5; 3.8 D C 4; 4.5

Probability of Node Selection • An ant at node 𝑗 ¡ travels to node 𝑘 ¡ with probability: � × (𝑤𝑗𝑡𝑗𝑐𝑗𝑚𝑗𝑢𝑧 ¡𝑝𝑔 ¡(𝑗, 𝑘)) 𝒖 𝒋𝒌 𝑄𝑠𝑝𝑐 𝑘 = ¡ 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠 • Where: • 𝒖 𝒋𝒌 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 = ¡ 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: � × (𝑜 �� ) � 𝑢 �� 𝑄𝑠𝑝𝑐 𝑘 = ¡ (𝒖 𝒋𝒍 ) 𝜷 (𝒐 𝒋𝒍 ) 𝜸 ∑ 𝒍∈𝑽 • Where: • 𝑢 �� is the quantity of pheromone on (i, j) • 𝛽 is a tuning parameter • 𝑜 �� is the inverse of the distance from i to j • 𝛾 is a tuning parameter • U is the set of unvisited nodes Probability of Node Selection • An ant at node 𝑗 ¡ travels to node 𝑘 ¡ with probability: � × (𝒐 𝒋𝒌 ) 𝜸 𝑢 �� 𝑄𝑠𝑝𝑐 𝑘 = ¡ 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑏𝑢𝑗𝑝𝑜 ¡𝑔𝑏𝑑𝑢𝑝𝑠 • Where: • 𝑢 �� is the quantity of pheromone on (i, j) • 𝛽 is a tuning parameter • 𝒐 𝒋𝒌 is the inverse of the distance from i to j • 𝜸 is a tuning parameter

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 optimizing terminal departure times Schedule Synchronization Transit Line Transfer Station

Fuzzy Set Theory • Set membership can be partial and can overlap: Tall Short Medium 1 Height 0 • Each element has membership degree • E.g., Bob is Short with truth degree 0.4 • Truth degree ≠ Probability 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 optimizing terminal departure times

FAS for Traveling Salesman Medium Small Long Distance Distance Distance 1 0 d ij Strong Medium Weak Pheromone Pheromone Pheromone 1 t ij 0 Fuzzy Ant System (FAS) • Modifies Ant System • Replaces exact values with fuzzy sets • Distance • Pheromone intensity • Replaces transition probability with Fuzzy Logic

FAS for Schedule Synchronization Model passengers transferring from line 𝑗 to line 𝑘 at time 𝑣 as: P iju 1 Number of 0 Passengers Uncertainty 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 D Line 3 Line 2 Line 1 “Ride” ¡line, ¡computing ¡arrival, ¡ Ant 1 departure, and wait times O FAS for Schedule Synchronization Possible Terminal D Departure Times Line 3 Line 2 Line 1 O

FAS for Schedule Synchronization D Line 3 Line 2 Line 1 Ant 2 Ant 1 O FAS for Schedule Synchronization D Line 3 Line 2 Line 1 Ant 1 O

FAS Results for 50 Lines 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

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 • Inspired by foraging behavior of bees in nature • Similar to ideas found in the Ant System • Used to solve Traveling Salesman Problem

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

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

Bee System: Time Hierarchy 1. 𝑡 nodes visited during a stage � �� 2. ¡ stages during an iteration � 3. 𝑁 iterations Bee System Example Stage 2, Iteration 1 H Start of Stage 2 Bee 3 Bee 2 Bee 1 S = 2 (two nodes visited per stage)