# The Convoy Movement Problem

Τεχνίτη Νοημοσύνη και Ρομποτική

23 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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The Convoy Movement Problem

Dr Andy Verity
-
Harrison

17
th

April 2012

Overview

2

A quick introduction to optimisation

Routing military convoys

Formulation as an integer programme

Branch
-
and
-
bound

Genetic algorithms

Lagrangian relaxation

Reformulation and random sampling

CMP

Computational Complexity

3

Polynomial time

NP, P and NP
-
complete

Worst
-
case measure

CMP

NP

NP
-
complete

P

Linear Programming

4

A linear programme

maximise

z

= 4
x

+ 3
y

subject to

-
¼x + y ≤ 4

x + y ≤ 9

3x
-

y ≤ 15

x ≥ 0

y ≥ 0

Simplex

Interior point

Integer Programming

2

3

4

5

1

6

7

1

2

3

4

5

6

7

Optimal

solution

CMP

The Convoy Movement
Problem

5

Given

A route network,

A set of convoys, each of which has

a length,

an origin node and a destination node; and

an earliest start time and a deadline

Can we route each convoy

from its origin to its destination

satisfying the convoy’s earliest start time and deadline;

so that no pair of convoys occupies the same part of the
route network simultaneously; and

the maximum (or cumulative) completion time is minimised?

CMP

Integer programming
formulation

6

minimise

i

cost(

i
,
k
i
)

+
e
i

+
d
i

+
w
i

subject to

v
i

=

t
i

0 ≤ i ≤ m

1
,

cost(

i
,
k
i
)

+
e
i

+
d
i

+
w
i

f
i

0 ≤ i ≤ m

1
,

Q
ip

Q
jp

=

a
ip

+
w
i

<

a
jq

{

if
a
ip
-
1

<
a
jq
-
1

when
a
ip

=
a
jq

,
a
ip
-
1

=
a
jq
-
1

a
ip

+
w
i

<

a
jq

{

if
a
ip
-
1

<
a
jq
-
1

when
a
ip

=
a
jq
-
1
,
a
ip
-
1

=
a
jq

CMP

Lee, Y.N., McKeown, G.P., Rayward
-
Smith, V.J., 1996. The convoy movement problem with initial delays. In: Rayward
-
Smith, V.J., Oman, I., Reeves, C.R., Smith, G.D. (Eds.), Modern Heuristic Search Methods. Wiley, pp. 213

234.

route for convoy
i

type of convoy
i

earliest start time of convoy
i

delay of convoy
i

time window of convoy
i

i

destination of convoy
i

Each convoy
finishes before its

occupancy of node
p

by convoy
i

Prevents more
than one convoy
occupying the
same vertex at the
same time

arrival time of convoy
i

at
p
th

node on route

Prevents
overtaking

Prevents passing in
opposite direction

Early work

7

Branch
-
and
-
bound

Hybrid Genetic Algorithm and Branch
-
and
-
Bound

Genetic algorithm selects delays

Branch
-
and
-
bound generates paths

Genetic Algorithm only

Genetic algorithm selects delays and from
r
-
shortest paths

CMP

Lee, Y.N., McKeown, G.P., Rayward
-
Smith, V.J., 1996. The convoy movement problem with initial delays. In: Rayward
-
Smith, V.J., Oman, I., Reeves, C.R., Smith, G.D. (Eds.), Modern Heuristic Search Methods. Wiley, pp. 213

234.

Number
of vertices

Number
of edges

Number of
convoys

Problem 1

160

212

17

Problem

2

530

724

25

Results

8

CMP

Lee, Y.N., McKeown, G.P., Rayward
-
Smith, V.J., 1996. The convoy movement problem with initial delays. In: Rayward
-
Smith, V.J., Oman, I., Reeves, C.R., Smith, G.D. (Eds.), Modern Heuristic Search Methods. Wiley, pp. 213

234.

Branch
-
and
-
bound

Hybrid

Genetic algorithm

Best
solution

Run
time

Best
solution

Run
time

Best
solution

Run
time

Problem 1

15,792

242.00s

(279.09s
*
)

15,793

172.79

15,792

254.28

Problem

2

35,644

454.78s

(579.43s
*
)

35,631

14,509.81

*

terminated

Route Graphs

9

CMP

time

finish

start

Earliest
start time

Convoy length

Convoy speed

Deconfliction

Lagrangian relaxation

10

Primal problem

Dual problem

Minimising the dual problem obtains a lower bound

For convoy movement problem:

The complicating constraints are those associated with
deconfliction

CMP

minimise

cx

subject to

Ax

=

b

Dx

e

x

0

minimise

cx +

(b
-
Ax
)

subject to

Dx

e

x

0

Time
-
space network

11

CMP

Results

12

CMP

Chardaire, P.C., McKeown, G.P., Verity
-
Harrison, S.A., Richardson, S.B. Solving a Time
-
Space Network Formulation of
the Convoy Movement Problem.
Operations Research
53
(2): 219

230, 2005

Branch
-
and
-
bound

Lagrangian

relaxation

Best
solution

Run time

Upper

bound

Lower
bound

Run time

Problem 1

15,792

242.00s

(279.09s
*
)

15,792

15,792

3.0s

Problem

2

35,644

454.78s

(579.43s
*
)

34,701

34,206

42.3s

Reformulation

13

Applied Dijkstra
-
based heuristic at heart of time
-
space
network approach

Randomly sampled convoy permutations

CMP

Tuson, A. and Verity
-
Harrison, A. Problem Difficulty of Real Instances of Convoy Planning.
Journal of the Operational
Research Society

56
(7):763

775, 2005

Lagrangian

relaxation

Sampled

Upper

bound

Lower
bound

Run
time

Expected
evaluations

Problem 1

15,792

15,792

3.0s

16.7

Problem

2

34,701

34,206

42.3s

31.2

Any questions?