A GRAPH-SEARCH APPROACH ON RESOURCE-CONSTRAINED

SCHEDULING PROBLEMS AND ITS APPLICATION TO

ADVANCED DRIVER ASSISTANCE SYSTEMS

Christoph Endres and Christian M

uller

German Research Center for Artiﬁcial Intelligence, DFKI GmbH, Saarbruecken, Germany

Keywords:

Automotive, Presentation scheduling, Conﬂict resolution.

Abstract:

In this paper we present a problem which is a variation of the resource-constrained project scheduling problem

and a graph-based approach to solve it. The problem is deﬁned as resource-constrained scheduling problem

(RCSP). Particularly, we apply the approach to the problem of scheduling a large number of driver warnings

based on car-to-car communication (also known as cooperative vehicles). Data is presented from the project

SIM

, a large-scale ﬁeld test in the area of the Hessian city of Frankfurt, where 120 cars participate in a

number of controlled tests in three main scenarios: the rural road scenario (basic complexity), the motorway

scenario (intermediate complexity), and the urban road scenario (high complexity). We argue that, due to its

run-time behaviour, our graph-based approach is suitable for the particular application domain at hand. Results

are presented in terms of quality of the solution (conﬂict resolution), runtime behavior and pruning effects to

the size of the search tree. In addition to the scenarios derived from the actual ﬁeld test, a hyper-real stress test

is presented to demonstrate the performance of our solution.

1 INTRODUCTION

The resource-constrained project scheduling problem

(RCPSP) is a combinatorial optimization problem

(Blazewicz et al., 1983). Due to its high practical

importance, it has been analyzed for several decades

now. RCPSP is an extension of the job shop schedul-

ing problem (JSP). (Garey and Johnson, 1979) pro-

vides the following deﬁnition, which is used as a basis

here.

Deﬁnition 1 (JSP). The job shop scheduling problem

(JSP) consists of a set A of activities each with a pos-

itive integer-valued duration, d

. A is partitioned into

projects (jobs), and with each project is associated a

total ordering on that set of activities. Each activity

speciﬁes a resource on which it must execute without

interruption. No activities that require the same re-

source can overlap in their execution.

Finding a solution to this problem with minimal

makespan (difference between minimum start time

and maximum end time) is NP-hard. RCPSP is more

complicated than JSP, since each task requires not

only a processor but also additional scarce resources.

Since it can be reduced to the simpler JSP, it is NP-

hard as well (Garey and Johnson, 1979; Blazewicz

et al., 1983).

In this paper, we introduce the resource-constrained

scheduling problem (RCSP), which is similar but adds

additional constraints to start- and endtime of activ-

ities. Unlike the original RCPSP, it has a dynamic

component from disruption management. (Clausen

et al., 2001) deﬁnes a disruption as ”a state during

the execution of the current operation, where the de-

viation from plan is sufﬁciently large that the plan

has to be changed substantially.” In a similar manner,

(Kuster et al., 2007) deﬁnes disruption management

(DM) as ”the process of responding to an unforseen

disturbance occuring during the execution of planned

and scheduled operations.”. According to the author,

DM aims at the selection of appropriate repair actions

to minimize the negative impact typically associated

with disruption.

In the automotive domain, particularly in the ﬁeld

of Advanced Driver Assistance Systems (ADAS), we

face the challenge of scheduling a growing number of

assistance systems trying to communicate simultane-

ously with the driver over a limited amount of com-

munication channels. This differs from RCPSP in a

way that scheduling priority lies not on the order of

activities but on preserving the requested presentation

times as good as possible. Moreover, following (Baier

334

Endres C. and Müller C..

A GRAPH-SEARCH APPROACH ON RESOURCE-CONSTRAINED SCHEDULING PROBLEMS AND ITS APPLICATION TO ADVANCED DRIVER

ASSISTANCE SYSTEMS.

DOI: 10.5220/0003743303340339

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 334-339

ISBN: 978-989-8425-95-9

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

rural road scenario

motorway scenario

urban road scenario

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Figure 1: The SIM

test ﬁeld around Frankfurt/Main, Ger-

many, is comprised of three test scenarios that imply differ-

ent degrees of resource utilization from low (rural road sce-

nario) to high (urban road scenario). 120 cars are involved

in the ﬁeld test.

et al., 2007), we do not only need to be able to distin-

guish between plans that satisfy goals and those that

do not without providing further means of discrimina-

tion between successful plans. For the application at

hand, we need to have information about how ”good”

a plan is, thus ”enabling the planner to distinguish be-

tween successful plans of differing quality” (ibid.).

Furthermore, the nature of the application demands

for an anytime behaviour. The desired algorithm to

solve the problem should be able to output a solution

at any time, which should be as close as possible to

the optimal solution.

2 BACKGROUND

RCSP consists of a set of tasks (activities) with ﬁxed

start times and durations, which should be executed

without interruption on a given (limited) resource.

Thus, conﬂicts occur when two or more tasks attempt

to use the same resource simultaneously. The objec-

tive is to resolve all conﬂicts while modifying the set

of tasks as little as possible.

For RCPSP, several algorithms have been pro-

posed, reaching back to the 1960’s with a Branch-and-

Bound approach by (Johnson, 1967), adopted later

by (Stinson et al., 1978; Christoﬁdes et al., 1987;

Brucker et al., 1998). (Pritsker et al., 1969; Patterson

and Roth, 1976) proposed Zero-One programming

solutions, while more recent approaches use Linear

(Brucker and Knust, 2000), Constraint (Demassey

et al., 2005) and Genetic Programming (Kuster et al.,

2007).

The Branch-and-Bound approach is closest to our

solution presented here. It is based on graph search

and effective bounding (e.g. pruning). Linear, Integer

and Zero-One-Programming are variations of a stan-

dard optimization problem. Genetic programming is

a search heuristic which mimics natural selection be-

haviour.

It is important to keep in mind that our problem

presented here differs signiﬁcantly from the original

RCPSP and thus, the approaches mentioned here can-

not simply be adopted.

For solving RCSP in the given highly dynamic do-

main, we need an anytime algorithm which ﬁnds a

good (not necessary perfect) solution very fast. Both

tree search and genetic algorithms fulﬁll these re-

quirements. In this paper, we present a tree search

approach with effective pruning.

3 APPLICATION DOMAIN

We apply the approach to the problem of scheduling

a large number of driver warnings based on car-to-car

communication (also known as cooperative vehicles).

Particularly, the system is developed for the project

SIM

The SIM

test ﬁeld is located in the Frankfurt-

Rhine-Main area of Hessen, Germany. With up to 120

vehicles and more than 100 roadside stations installed

by the Hessian trafﬁc centre (VZH) and the Integrated

Trafﬁc Management Center Frankfurt (IGLZ), car-

to-x communication is tested under real conditions.

The Frankfurt-Rhine-Main area is an important Ger-

man trafﬁc hub with major trafﬁc generators such as

Frankfurt Airport, Frankfurt Trade Fair and the sta-

dium. Large parts of the area are characterised by

high trafﬁc density and therefore allow experiments

on all road safety and trafﬁc efﬁciency functions un-

der normal as well as high load conditions. Figure 1

shows a map of the test ﬁeld. It is comprised of three

scenarios: I Rural road, II / III Motorway, and IV Ur-

ban road. The order of the scenarios represents their

increasing use of the limited HMI resources. These

assumptions are based on the number of applications

active in the respective area, the trafﬁc density as well

A GRAPH-SEARCH APPROACH ON RESOURCE-CONSTRAINED SCHEDULING PROBLEMS AND ITS

APPLICATION TO ADVANCED DRIVER ASSISTANCE SYSTEMS

335

as the density of road site stations (stationary wireless

communication units) used in the ﬁeld test.

Figure 2 gives an overview on the complexity

of the scenarios and speciﬁes the role of RCSP in

the given application context. A large number of

presentations try to access (potentially) simultane-

ously a limited resource (the Human-Machine Inte-

face, HMI). Not all requests are equally important.

Output channels are distributed over different modal-

ities: multiple visual channels (symbols and main

screen), auditory outputs, and text-to-speech (TTS).

Hence, unlike traditional presentation planning, we

look at a highly dynamic scenario in which presen-

tation requests come in at any time during runtime.

As indicated by the dashed triangle, a second re-

source limitation has to be taken into account: the

limited cognitive resource of the driver. Here ques-

tions should be addressed like for example, how long

the minimal display time for an information should

be, or wether or not overlapping presentations on dif-

ferent channels (TTS for one message and simultane-

ous display of another message) is beneﬁcial. How-

ever, this is beyond the scope of this paper. We refer

the interested reader to (Cao et al., 2009; Cao et al.,

2010; Mahr et al., 2010).

So called presentation requests are issued by the

”functions” listed on the right side of Figure 2. Func-

tions act as individual subsystems that have no knowl-

edge on the available resources and activities of other

functions. Presentation requests are issued (”an-

nounced”) as soon as the need for a warning becomes

apparent. Hence, the scheduler can allocate the HMI

resources beforehand (planning time does not delay

crucial warning messages). Nevertheless, the scenario

involves a multiplicity of presentations and is highly

dynamic.

4 PROPOSED GRAPH-BASED

SOLUTION

In short, our problem is a dynamic set of tasks (pre-

sentation requests) simultaneously trying to access a

limted resource. Tasks have a start time, duration, a

hard minimal duration, a priority, and a desired re-

source (presentation strategy). We look for a solution

which resolves resource conﬂicts while modifying the

tasks as little as possible. A more formal defnition is

given below.

We propose a two-step graph-based solution to the

problem. Step one is transforming dynamic planning

to static planning at runtime. Dynamic planning thus

is by construction equivalent to having a series of

static planning problems. The main problem here is

to identify subsets of the set of presentation tasks and

the appropriate order of resolving multiple different

conﬂict sets. Identifying overlapping presentations is

a straightforward task at ﬁrst sight. However, since

the solution to that conﬂict could again cause other

conﬂicts, the appropriate set of tasks to be included in

the conﬂict set is not always obvious. Here we have to

solve the trade-off between scope of the solution and

runtime, which is heavily depending on the amount of

tasks in the conﬂict set.

Step two of our approach is transforming our

problem into a tree search problem for which well

known algorithms can be applied.

Let T be a set of presentation tasks T =

,...,t

}. Let each presentation task t

consist

of a start time, an end time, a priority and additional

information about the presentation such as rendering

information, distance to event, etc. which are not of

immediate relevance for the scheduling.

Deﬁnition 2 (Scheduling Problem RCSP). The RCSP

P is deﬁned as:

P = < T

,M > where

• T

is a set of conﬂicting presentation tasks, T

= {t

, t

,..., t

• M is a set of modifying actions (short: modi-

ﬁcations) as conﬂict resolving strategies, M =

,...,m

}

In our implementation, modiﬁcations are postpon-

ing, preponing, shortening (beginning or end of task),

switching resource (not considered in this paper) and

canceling a task.

Deﬁnition 3 (Search Tree). We deﬁne a search tree

S(P ) as follows:

• The root R of the tree contains the set of conﬂict-

ing tasks T

• Each edge e is a pair of a presentation task and a

modiﬁcation applied to it, e = < t

• A cost function p(E), which deﬁnes a positive in-

teger penalty for each edge, based on the severity

of the modiﬁcation (see Deﬁnition 5).

• Each node is the result of the modiﬁcation of the

previous edge to the parent node

• The penalty p(n) of a node n is the sum of the

penalties of all edges on the branch leading to it

• A node n containing a presentation task set T

without conﬂicts is a leaf.

By design, the leaf with the lowest cost is the best so-

lution to our problem. Using the tree S (P ) described

in Deﬁnition 3, we can now search the solution with

the Breadth First Search (BFS) Algorithm 1.

ICAART 2012 - International Conference on Agents and Artificial Intelligence

336

IV Urban road scenario

II"/"III"Motorway"Scenario"

I Rural road scenario

Traffic flow information and navigation

1 Foresighted road/traffic information

2 Road works information system

3 Advanced route guidance and navigation

Local danger alert

5 Obstacle warning

6 Congestion warning

7 Road weather warning

8 Emergency vehicle warning

Driving assistance

9 In-vehicle signage/traffic rule violation warning

10 Traffic light phase assistant/ violation warning

11 Extended electronic brake light

12 Intersection and cross traffic assistance

Additional services

13 Internet-based usage of services

14 Location-dependent services

limited resource

attention, decision

making**

Traffic management

4 Alternative route management

limited resource

Human-Machine

Interface

(HMI)

Resource-

constraint

scheduling

problem*

*addressed here **not addressed here but part of our research

Figure 2: Complexity of the scenarios I – IV and the role of RCSP in the given application context.

Algorithm 1: Scheduling using BFS tree search.

1: solution ← nil

2: minPenalty ← ∞

3: i ← 0

4: N

← {R}

5: while |N i| > 0 do

6: for all n ∈ N

7: for all m ∈ M do

8: if applicable(m,n) then

9: n’ ← apply(m,n)

10: N

i+1

← N

i+1

∪ {n’}

11: if isSolution(n’) and p(n’)<minPenalty

then

12: minPenalty ← p(n’)

13: solution ← n’

14: end if

15: end if

16: end for

17: end for

18: i ← i+1

19: end while

Pruning. In order to reduce complexity and run-

time, several pruning mechanisms are applied in the

algorithm: A branch with a total penalty bigger than

the best solution found so far minus the minimum ac-

tion penalty will be pruned; a branch that encodes a

solution already encoded in another branch (maybe

with permutated actions) will be pruned. Additional

implicit pruning is achieved using the function appli-

cable(a,n) (see line 8 of Algorithm 1):

Deﬁnition 4 (applicable(m,n)). A modiﬁcation m is

applicable to a node n if and only if:

• n is not a leaf

• applying m to n will not create another conﬂict

• m will not shorten any task below a given minimal

display time

• p(n’) < minPenalty

• n’ /∈ N

, 0 ≤ j ≤ i + 1

• m will not move any task completely out of its

original scope

Anytime Behavior. Our algorithm shows an ”any-

time” behavior, i.e. it incrementally ﬁnds better plans

(Baier et al., 2007). Once a solution is found, its met-

ric value can be used as a bound for future solutions.

The metric we use here is a penalty of the actions

needed in order to modify the conﬂict set to a conﬂict-

free set. It is deﬁned in more detail below.

5 EMPIRICAL EVALUATION

In preparation of the SIM

ﬁeld test which will be-

gin in 2012, applications are tested on the basis of

pre-recorded traces containing GPS positions, car-to-

car communication messages, and vehicle data (ve-

locity, heading, gas pedal status, break pedal status,

index lights, etc.). The scenarios lined-up at the X-

axis in Figure 3 correspond to what we have seen ear-

lier in Figures 1 and 2. Additionally, a scenario with

hyper-realistic complexity was added (here denoted

Scenario V). Each scenario was further speciﬁed with

three concrete evaluation cases (I.1, I.2, I.3, .... V.3).

The dashed black line corresponds to the actual com-

plexity of the evaluation cases:

Deﬁnition 5 (Complexity). The complexity of a con-

ﬂict set T

cx(T

) =

∑

ol(t

)

· |T

| · |ol(t

)| · alloc(ms),

A GRAPH-SEARCH APPROACH ON RESOURCE-CONSTRAINED SCHEDULING PROBLEMS AND ITS

APPLICATION TO ADVANCED DRIVER ASSISTANCE SYSTEMS

337

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I Rural road

scenario

IV Urban road

scenario

V Hyper-realistic

scenario (I, II/III, IV)

II / III Motorway(

Scenario(

complexity

(see definition in the text)

runtime (s)

treedepth

I.1 I.2 I.3 II.2 II.1 III.1 IV.1 IV.2 V.1 V.3 IV.3

V.2

Figure 3: For each type of scenario, several examples of

varying complexity based on ﬁeld data are used to test the

algorithm presented in this paper. Hyperrealistic in the

sense of unrealistic high) scenarios are used for stress test-

ing. Runtime and tree depth match the tendency of growing

complexity from left to right in each scenario type.

where ol is the temporal overlap of two conﬂicting

tasks in milliseconds and alloc(ms) is the percentage

of the makespan allocated by the unmodiﬁed tasks.

Figure 3 shows that the complexity of the test

cases has a positive trend from left to right(see solid

black trend line), which appeals to our intuition. Nev-

ertheless, there is a signiﬁcant case-speciﬁc complex-

ity within each of the scenarios. We see, for exam-

ple, that the variance of complexity in Motorway sce-

nario is higher than both Rural road scenario and Ur-

ban road scenario. Note that the complexity of V.1 is

lower than the ones of I.3 and III.1. This is because

V.1 is a hyper-real Rural road test case, V.2 is hyper-

real Motorway test case, and V.3 is a hyper-real Urban

road test case.

Both run-time as well as tree-depth differed sig-

niﬁcantly, which can be seen by regarding the grey

(square) solid lines respectively the dashed (triangle)

lines. Runtime relates to the time needed to ﬁnd the

optimal solution. It was measured on a regular state-

of-the-art desktop PC.

For each of the evaluation cases, a solution was

found by the scheduling algorithm. In order to be able

to distinguish between successful solutions of differ-

ent quality, we deﬁne quality via a penalty for the

modifying actions it involves: the fewer the modiﬁ-

cations, the better the solution.

Deﬁnition 6 (Penalty of a RCSP Solution). Be M a

set of modiﬁcations, M = {m

,...,m

} and t a task

with given positive priority prio(t). The penalty p for

m applied to t is p(m,t) = d(m) · prio(t) · ∆, where

d(m) deﬁnes the general desirablility of an action as

positive interger value and ∆ specifying the temporal

aspect of the modiﬁcation (e.g. postponing or short-

ening).

The penalty of a solution is then deﬁned as

penalty (log)

(see definition in the text)

time in ms (log)

50#

500#

5.000#

50.000#

500.000#

5# 50# 500#

I.1

I.2

I.3

II.1

II.2

III.1

IV.1

IV.2

IV.3

V.1 (I)

V.2 (III)

V.3 (IV)

Figure 4: Quality of a solution annotated as penalty (see

Deﬁnition 6) as a function of runtime (ms). Solutions with

qualities close to the respective optimal solutions are found

after a very short time. Note that presentation tasks are is-

sued at least a few seconds before start time. Hence, the

planning time does not delay the warning.

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I.1

I.2

I.3

II.1

II.2

III.1

IV.1

IV.2

IV.3

V.1 (I)

V.2 (III)

V.3 (IV)

number of nodes

tree depth

Figure 5: Number of nodes as a function of tree depth show-

ing the effect of pruning. The symbols correspond to the

point when the optimal solution was found.

∑

p(m,t) of all (m,t) contained in it.

The order of steps taken towards a certain solu-

tion is in our case not important. This is respected by

pruning duplicate and redundant solutions. As a side

effect, the penalty for a solution is well deﬁned and

independent of the way it was constructed.

Figure 4 shows the correlation between runtime

and solution quality in terms of the penalty we de-

ﬁned. Our algorithm provides a ﬁrst solution typi-

cally in the ﬁrst few milliseconds and then ﬁnds in-

creasingly better solutions fast. After 50-100 ms, we

are very close to, or in some cases already at, the op-

timal solution. Note that the optimal solution is not

at penalty zero, so the asymptotic behaviour of the

graph is not aimed towards the X-axis. Interrupting

ICAART 2012 - International Conference on Agents and Artificial Intelligence

338

the scheduling process at this point leads to a satis-

factory result.

In Figure 5, the effects of pruning are depicted.

The tree depth is correlated to the number of nodes

calculated on that level using our BFS tree search. If

no pruning was applied, the graph would be a straight

line on the logarithmic scale. Using the penalty of the

best solution found so far (which drops rather rapidly

as we saw in Figure 4) as an upper bound enables us

to severly prune the tree fast. The symbols correspond

to the point when the optimal solution was found.

6 CONCLUSIONS AND

OUTLOOK

In this paper, we presented the Resource-constrained

Scheduling Problem (RCSP), which is like the well

known Resource-constrained Project Scheduling

Problem (RCPSP) a combinatorial optimization

problem. The automotive domain, in which we

encountered this problem, is highly dynamic and

requires fast responses, e.g. an anytime behaviour of

the algorithm tackling this problem. We presented

a tree-search based solution for the RCSP, analyzed

its runtime behaviour, solution quality increase and

space consumption. Although the algorithm is space-

consuming when running until termination, very

good results are produced fast and the calculation can

be stopped then. We argue that the solution presented

is suitable for the scenario at hand. As a next step,

we will alternatively implement a genetic algorithm

for the problem, analyze it and compare it to the

performance of the algorithm presented here. This

will, based on a categorization of problems according

to its complexity, lead to a hybrid approach using the

most suitable mechanism for each problem.

This work was funded within the project SIM

by the German Federal Ministries of Economics

and Technology as well as Education and Research,

and supported by the Federal Ministry of Transport,

Building and Urban Development.

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