Talk at

1
An Electromagnetism-like Mechanism for Solving
the Multiple Depot Vehicle Routing Problem

• Talk at
• Department of Industrial Engineering
  Management,I-Shou University.
• July. 21, 2006
• B.Y. Huang
• Committees Dr. Chin-Shiuh Shieh Dr. Nai-Chie
  Wei
• Research Advisers Dr. Peitsang Wu Dr. I-Ming
  Chao

2
Outline

• Introduction
• Routing Problems
• Motivation
• Research Objectives
• Literature Review
• Solution Methods
• Methodologies
• The Mathematical Formulation of the MDVRP
• The Electromagnetism-like Mechanism
• Illustrated Examples Analyses
• Conclusions Future Research

Introduction
3
Routing Problems
Node-Covering Problems
Arc-Covering Problems
Chinese Postman Problem
Traveling Salesman Problem
. . .
Vehicle Routing Problem (VRP)
Multiple Depot VRP
. . .
Introduction
4
Introduction
5
Traveling Salesman Problem
D
Introduction
6
Vehicle Routing Problem
D
Introduction
7
Multiple Depot VRP
D
D
D
Introduction
8
Motivation

• Exciting Problem
• Practical Applications
• Industrial Relevance
• Importance to Society

GDP
According toIndustrial Technology Research
Institute (2002)
Introduction
9
Research Objectives

• Objectives
• Solve the MDVRP
• Good performance be investigated
• Tool
• The EM algorithm
• Execution
• C program language

Introduction
10
Research Scope and Restrictions

• Network
• non-directional and symmetrical network
• corresponding to Euclidian Space
• Depot
• limitless volume of stock
• needless to consider the vehicle loading time
• Customer
• demand quantity, place coordinate, and
  merchandise categories, etc., are all already
  known and fixed
• Vehicle
• needless to consider driving speed, drivers state
• limited carries capacity

Introduction
11
Thesis Architecture
Introduction
12
Literature Review

• The MDVRP is NP-hard (Lenstra et al, 1981)
• Current Methods in VRP
• Exact Methods
• Dynamic Programming
• Langrangean relaxation
• Branch bound
• Approximate Algorithms and Heuristics
• Savings Algorithm (Clarke and Wright, 1964)
• Route first, cluster second Cluster first,
  route second
• Tabu search
• Genetic algorithm
• Simulated annealing
• Threshold accepting, etc.

Literature Review
13
Solution Methods for MDVRP

• Exact Procedure
• Branch and bound
• Laporte et al. (1984) customers ? 50 depots ? 8
  
• Laporte et al. (1988) customers ? 80 depots ? 3

Literature Review
14
Solution Methods for MDVRP

• Heuristic Algorithms
• Savings Algorithm
• Tillman (1971)
• Two-Phase-Approaches
• Wren and Holliday (1972) applied cluster first,
  route second way for two depots and up to 176
  cities
• Raft (1982) introduced 2-opt exchange procedure
• Chao et al. (1993) used the "record-to-record"
• Giosa et al. (1999) described the Assignment
  Algorithms
• Meta-Heuristic Algorithms
• Renaudl et al. (1994) introduced the tabu search
  heuristic

Literature Review
15
The Mathematical Formulation of the MDVRP
Methodologies
16
N set of all nodes, m number of depots NV
number of vehicles Kv capacity of vehicle v di
demand of customer i
ti service duration in node i yi the serial
number of node i be served on a route Tv
maximum duration of a route by vehicle v tij
transportation duration between node i and node j
cij transportation cost between node i and
node j
Methodologies
17
Electromagnetism-like Mechanism

• Birbil and Fang (2003) constructed a mechanism
  that likes the attraction-repulsion mechanism of
  the electromagnetism theory.
• Chiang (2005) used the EM to solve the traveling
  salesman problem (TSP) and the results
  corresponded to his expected
• Yu (2005) in his thesis described the EM could
  suitable for the Object Sequencing and
  Grouping Problems

Methodologies
18
Electromagnetism-like Mechanism

• General Scheme

(Birbil and Fang, 2003)
Methodologies
19
General Scheme

• Initialize

Methodologies
20
General Scheme

• Local search

Methodologies
21
General Scheme

• Total force calculation

F
Methodologies
22
General Scheme

• Move along the total force

Methodologies
23
The EM for MDVRP

• The Activity-List (AL)
• The Random-Key (RK)

Methodologies
24
An AL form of the EM algorithm
Methodologies
25
The three types of the EM algorithms

• The prototype EM Algorithm the original type of
  the EM algorithm
• The improved EM Algorithm add a swap mechanism
  (The 2-Opt method) to the EM algorithm
• The intensification EM Algorithm construct
  initial solutions for the improved EM algorithm.

Executed on an Intel Celeron 2.8GHz personal
computer with 512 Mb RAM
Methodologies
26
Characteristics of test problems
Described by Christofides and Eilon (1969)
Illustrated Examples Analyses
27
Parameters of the EM Algorithm
Wu and Chiangs researches (2005)
Illustrated Examples Analyses
28
The Prototype EM Algorithm
Illustrated Examples Analyses
29
Results in the Prototype EM algorithm
13.8
Illustrated Examples Analyses
30
Illustrated Examples Analyses
31
The Improved EM Algorithm
IMPROVED EM
5. 2-opt Method ()
6. 7. 8. 9.
Illustrated Examples Analyses
32
Results in the Improved EM Algorithm
11.3
Illustrated Examples Analyses
33
Illustrated Examples Analyses
34
The Intensification EM Algorithm


Illustrated Examples Analyses
35
Results in the Intensification EM Algorithm
6.3
Illustrated Examples Analyses
36
Illustrated Examples Analyses
37
Summary of computational results
Illustrated Examples Analyses
38
Summary of computational results
Type(1) the Prototype EM algorithm Type(2)
the Improvement EM algorithm Type(3) the
Intensification EM algorithm
Illustrated Examples Analyses
39
Summary of computational results
ANOVA for the error
Illustrated Examples Analyses
40
Conclusions

• In our researches, The improved EM algorithm is
  better than the original (prototype) EM
  algorithm. When the improved EM algorithm accedes
  to the initial solutions construction method, we
  can improve the results.
• The EM algorithm is possible to solve the MDVRP
  because the transportation cost is close to the
  best known cost.

Conclusions Future Research
41
Future Researches

• Combine other meta-heuristic algorithms with the
  EM algorithm, the performance of the new
  integration may be better.
• Apply other local search methods to improve the
  efficiency of the EM algorithm might produce
  better results and spend less time.

Conclusions Future Research
42
Future Researches

• We can apply this method for other problems,
  e.g., the VRP, and VRPTW, have not be solved by
  this new meta-heuristic algorithm.
• Combine other meta-heuristic algorithms with the
  EM algorithm, the performance of the new
  integration may be better.

Conclusions Future Research
43
Thank you for your attention!
QA