PLANNING FOR THE CONVOY MOVEMENT PROBLEM

Anand Kumar, I. Murugeswari, Deepak Khemani and N. S. Narayanaswamy

Department of Computer Science and Engineering, Indian Institute of Technology - Madras, Chennai, India

Keywords:

Convoy routing, Convoy movement problem, Logistics, Transport.

Abstract:

Convoy movement problem has signiﬁcant practical applications. The problem has been attempted in many

styles with varying results. We address it as an AI Search and Planning and Scheduling problem. The work

focuses on modeling the convoy movement problem using PDDL and attempting a solution using existing

planning methods and planners. Initial results indicated problems of scalability. To address this we propose a

two stage planning process.

1 INTRODUCTION

The convoy movement problem (CMP) is a logisti-

cal problem and addresses the issue of planning and

scheduling a ﬂeet of convoys between their sources

and their respective destinations. Usually convoys are

many vehicles long and take days to reach their desti-

nation.

Convoys are often employed when a large num-

ber of men and or large amounts of material have to

be moved. In many situations movement by road is

the only option. Planning convoys is of strategic im-

portance and it is a time consuming job. This work

presents a study on modeling the convoy movement

problem from a planning and scheduling perspective.

Toward the end present results from our experiments

on moderately large problem instances using existing

domain independent planners.

2 CONVOY MOVEMENT

PROBLEM

The convoy movement problem has two parts to it.

First is ﬁnding out the route the convoy is going to

take to reach its destination and the second is ﬁx-

ing the time of the actions so that the resulting plan

satisﬁes all relevant constraints. Route selection and

scheduling of convoys can be seen as two interdepen-

dent activities. The route selected has to be schedule-

able within the deadlines goals and vice-versa there

should be a viable route that within bounds of the im-

posed deadlines. There are constraints in the prob-

lem that involve either or both of route selection and

scheduling.

Routing Constraints. Certain roads may not support

a convoy because the infrastructure may not be

able to support say movement of heavy vehicles.

Some roads may not allow bi-directional trafﬁc

ﬂow. Constraints under this category are most of-

ten static.

Scheduling Constraints. These are usually deadline

constraints. There are four common forms in

which they are imposed. They are “Earliest ar-

rival time”, “Latest arrival time”, “Earliest de-

parture time” and “Latest departure time”.

Mixed Constraints. Cities most often impose con-

straints on when convoys can enter city trafﬁc

zones. Some roads along a route might become

available for trafﬁc only during certain time of

day. Constraints of this type affects both route

planning and scheduling.

3 RELATED WORK

The convoy movement problem has been tried in dif-

ferent forms. Most often it has been formulated as an

optimization problem (Chardaire et al., 2005; Gold-

stein et al., 2010; Montana et al., 1999). Some of

the earlier work consider convoys as point objects and

do not model their length explicitly (Chardaire et al.,

2005; Montana et al., 1999).

In (Goldstein et al., 2010) the problem they con-

sider is most similar to the work discussed above.

495

Kumar A., Murugeswari I., Khemani D. and S. Narayanaswamy N..

PLANNING FOR THE CONVOY MOVEMENT PROBLEM.

DOI: 10.5220/0003739004950498

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 495-498

ISBN: 978-989-8425-95-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

They consider convoys in the case of disaster recov-

ery and management. Genetic algorithm is used for

ﬁnding the solution. The initial state consists of the

shortest path between the source and destination for

each convoy. The cost function in this case considers

a linear combination of penalties for vertex overlaps

(no. of common vertexes between paths) and edge

overlaps (no. of common edges between paths).

In (Chardaire et al., 2005) the cost function de-

ﬁned is focused on the time taken for the convoys to

reach their destination over a given path. Length of

the convoy is modeled indirectly as a guard time in-

terval. We start from a set of simple paths between

source and destination and optimize using integer pro-

gramming.

In (Montana et al., 1999) genetic algorithms are

used for route selection and convoy scheduling. In

this work the convoy schedules are optimized for min-

imum cargo weight time and minimum civilian trafﬁc

disruption. These are objectives that the current work

does not attempt to address.

4 PDDL MODEL

PDDL has chosen as the domain description language

because this allows use of existing planners for study.

The models were developed incrementally. The initial

model is a single domain incorporating operators that

together model all constraints listed in 4.1.

4.1 Constraints Modeled

The set of constraints that we have currently modeled

are

• Edges have direction and may not allow two way

trafﬁc.

• Nodes that allow halting have a speciﬁed capacity.

• Convoys have size which is proportional to their

length.

• Each convoy has its own length.

• Each convoy has its own speed.

• Each convoy has to maintain a minimum distance

from the convoy ahead of it.

4.2 Model Details

Roads in the networks are modeled as directed edges

in a graph. This allows trafﬁc directions to be mod-

eled. To impose this constraint, it is assumed that con-

voys always move from left to right along an edge.

This allows us to make the move operators require

the convoys to move from the left end node, marked

by asserting (left-vertex ?v - vertex ?e - edge), to the

right end node, marked by asserting (right-vertex ?v

- vertex ?e - edge). Nodes where convoys can halt

are marked as halting grounds using the predicate

(halting-ground ?v - vertex). Convoys have heads

and tails. The tail follows the head taking the same

path. This is achieved by asserting the (head-edge ?c

- convoy ?v - vertex ?e - edge) predicate whenever the

head enters a new edge and is de-asserted after the

tail passes through. Fluents are used to model the size

(convoy-size ?c - convoy), speed (convoy-speed ?c -

convoy) and capacity (free-space ?v - vertex) param-

eters. The minimum inter-convoy distance that is to

observed is a constant so we can model this constraint

by increasing each convoys length appropriately.

4.2.1 Operators

There are four classes of operators in this model.

These operators are enter, exit, unwind and move. As

we are handling convoy length in the model, a single

move, enter, exit or unwind operator does not sufﬁce.

The operators enter, exit and unwind each have three

cases and the move operator has nine cases. Thus the

model in total has 18 operators. The operators are dis-

cussed in detail below.

4.3 Observations

State space planners using planning graph heuristic

perform poorly because they spend most of the time

grounding actions. The domain has 18 operators in

PDDL and each of these have at least 3 parameters.

The parameters are of type convoy, vertex and edge

respectively. Now assume a 8x8 grid domain. This

has 81 vertexes with 89 edges. Suppose we have 8

convoys this amountsto 81x89x8 variationsof one op-

erator. So we have a total of 81x89x8x18 = 1038096

ground actions. This exponential blow up severely

affects scaling. Plan space planners also performed

poorly in our experiments. This can be attributed

to poor heuristics and the high branching factors in-

volved in the domain.

5 TWO STAGE PLANNING

5.1 Two Stage Process

To overcome the scaling issue introduced by sec-

tion 4.3, we introduce a two staged planning method.

The ﬁrst stage is broadly concerned with ﬁnding con-

voy routes and the second stage deals with the addi-

ICAART 2012 - International Conference on Agents and Artificial Intelligence

496

Simpliﬁed

domain and

problem

Stage 1

Planning

Process

path for

each convoy

Domain with

additional

constraints

Stage 2

Planning

Output

ﬁnal plan

Figure 1: Two stage planning.

tional constraints that arise due to length of the con-

voys. The ﬁrst stage uses a simpliﬁed domain model

as compared to the one detailed in section 4. This

model models point convoys and edge exclusivity for

convoys.

The second stage deals with all the constraints

dealt with in the monolithic model, in fact the same

PDDL model is augmented and used again. We al-

ter this model by removing the generic move opera-

tor with ground actions for each convoy. As noted

in section 4.2.1 the move operator has 9 variants and

contributes signiﬁcantly against scaling. This leads

to faster planning times. Figure 1 depicts the whole

process.

5.1.1 PDDL Details

The ﬁrst stage of the two stage model uses a simpli-

ﬁed domain model. This model is geared toward ﬁnd-

ing routes for all the convoys while making sure that

more than one convoy does not enter an edge simul-

taneously. It does not model convoy lengths. It has

three operators in all. They are enter, exit and move.

The second stage uses the monolithic model dis-

cussed in section 4 with the move operators replaced

by appropriately grounded actions for each convoy as

discussed in section 5.1

5.2 Experiment Results

The trial runs were conducted on two sets of problems

with the same object count. One is a set of random

graphs and the other a grid graph. All the graphs are

directed. All experiment runs were done using LPG-

td (Gerevini et al., 2006) as the planner for both stages

of planning. Results from the experiments conducted

are given in Tables 1 and 2. The planning process has

an overall timeout of 20 minutes. The experiments

Table 1: Result on random graph.

Nodes, Edges Convoys Stage 1 (s) Stage 2 (s)

25 , 80

10 0.1 0.02

20 0.09 0.02

30 0.09 0.02

100, 360

10 1.1 0.07

20 1.15 0.12

30 1.15 0.16

225, 840

10 5.4 0.38

20 5.37 0.13

30 5.42 0.13

400,1520

10 15.0 0.61

20 15.0 0.3

30 15.0 0.3

Table 2: Result on grid graph.

Nodes, Edges Convoys Stage 1 Stage 2

25, 80

10 0.11 0.45

20 0.09 0.08

30 0.1 0.07

100, 360

10 1.32 9.96

20 1.38 0.86

30 1.38 16.0

225, 840

10 7.62 4.65

20 8.29 50.0

30 8.26 474.0

400, 1520

10 - -

20 - -

30 - -

were conducted on a machine with Q9550 processor

and 4 GB of ram.

From the above tables we can see that the newpro-

cess has given results on domain which are much big-

ger than the domains we were able to deal with in the

single stage planning process. Another interesting re-

sult that we can observe from the above tables is that

the topology of the graph plays an important role in

the time consumed for the planning process. More ex-

periments have to be made for making strong claims

in this regard. At present we are speculating that tight

topologies like the grid lead to very few actions being

pruned, weakening of the heuristic used or both.

6 FUTURE WORK

6.1 Case for Local Search

In the real world convoys are used in logistics plan-

ning and execution. This translates to a decision sup-

port system in many cases. One characteristic of this

system that is of interest is that we might want to

be able to plan around hazards when execution fails.

PLANNING FOR THE CONVOY MOVEMENT PROBLEM

497

Figure 2: Convoy crossover.

This would also mean preservation of as much of the

existing plan as possible. Another characteristic of

interest is that of convoy interaction. This can lead to

unsatisﬁed deadlines and/or other constraints which

in turn leads to backtracking and a ripple effect. Both

these characteristics suggest the use of local search.

6.2 Case for a Knowledgeable Heuristic

In ﬁgure 2 if the small but faster convoy is arriv-

ing a little later than the larger and slower convoy

then we would like to have the larger convoy wait for

the smaller convoy to crossover. We would like to

consider a heuristic that considers the convoy length,

speed and the time constraints associated with each

convoy. We would like to investigate whether a

heuristic can be formed for a dynamic scenario like

this which can take into point temporal constraints.

6.3 Complete Search in Split Planning

When dealing with goals that have deadlines, which

are common in this domain, the split planning ap-

proaching discussed in section 5 can fail to ﬁnd a

suitable plan. This is because the second stage is al-

ready committed to certain routes, and a solution may

not exist for those routes. An interesting development

would be a backtracking feedback based planner. The

feedback can result in the plan from previous stage

begin altered according to the reason for failure.

7 CONCLUSIONS

In this paper we have looked at some of the prob-

lems that arise when we look at path ﬁnding and

scheduling of convoys when their length is signiﬁcant

and modeled explicitly. We have experimented with

Sapa (Do and Kambhampati, 2003), Crikey (Coles

et al., 2009), LPG (Gerevini and Serina, 2002) and

LPG-td (Gerevini et al., 2006), an extension of LPG,

at various stages of model development. LPG-td per-

formed better than the other planners overall. By

splitting the problem into two stages and planning

separately we were able to improve scale and reduce

planning time. We plan to improve the system by

attempting to make it complete and also explore if

this methodology of split planning can be generalized

over multiple stages.

REFERENCES

Chardaire, P., McKeown, G. P., Verity-Harrison, S. A., and

Richardson, S. B. (2005). Solving a time-space net-

work formulation for the convoy movement problem.

Operations Research, 53(2):219–230.

Coles, A., Fox, M., Halsey, K., Long, D., and Smith, A.

(2009). Managing concurrency in temporal planning

using planner-scheduler interaction. Artiﬁcial Intelli-

gence, 173(1):1 – 44.

Do, M. B. and Kambhampati, S. (2003). Sapa: A multi-

objective metric temporal planner. Journal of Artiﬁ-

cial Intelligence Research, 20:155–194.

Gerevini, A., Saetti, A., and Serina, I. (2006). An approach

to temporal planning and scheduling in domains with

predictable exogenous events. Journal of Artiﬁcial In-

telligence Research (JAIR), 25:187–231.

Gerevini, A. and Serina, I. (2002). Lpg: A planner based

on local search for planning graphs with action costs.

In International Conference on Automated Planning

and Scheduling/Artiﬁcial Intelligence Planning Sys-

tems, pages 13–22.

Goldstein, D., Shehab, T., Casse, J., and Lin, H.-C. (2010).

On the formulation and solution of the convoy routing

problem. Transportation Research Part E: Logistics

and Transportation Review, 46(4):520 – 533. Selected

papers from the Second National Urban Freight Con-

ference, Long Beach, California, December 2007.

Montana, D., Bidwell, G., Vidaver, G., and Herrero, J.

(1999). Scheduling and route selection for military

land moves using genetic algorithms. In Evolutionary

Computation, 1999. CEC 99. Proceedings of the 1999

Congress on, volume 2, pages 3 vol. (xxxvii+2348).

ICAART 2012 - International Conference on Agents and Artificial Intelligence

498