A SCHEDULING TECHNIQUE OF PLANS WITH PROBABILITY

AND TEMPORAL CONSTRAINTS

Bassam Baki and Maroua Bouzid

GREYC

University of Caen

Bd Marchal Juin

14032 Caen Cedex

Keywords:

Decision support systems, Temporal Constraint Reasoning, Planning, AND/OR Graph and probability.

Abstract:

In this paper, we address the problem of scheduling plans with probability and temporal constraints. We

illustrate our problem with an AND/OR graph, where we try to ﬁnd a plan of tasks that satisﬁes all temporal

constraints and precedence relations between tasks, has a high probability of execution, a minimal cost and a

reduced time. Each task has a set of temporal constraints, a set of probabilities and a set of constant costs. Our

planner uses the temporal constraint propagation technique to simplify the resolution of a given problem.

We describe one approach to deal with a problem that has paid a little attention of planing community. This

problem is to combine temporal and probabilistic planning.

1 INTRODUCTION

While today’s planners can handle large problems

with durative actions and time constraints, few re-

searches have considered the problem concerning un-

certainty and the different durations of actions. For

example, IxTeT ((Ghallab and Laruelle, 1994), (La-

borie and Ghallab, 1995)) is a time-map manager that

handles symbolic constraints (precedence), numeric

constraints (intervals) and sharable resources. This

planner provides an initial complete plan which is

then run by temporal executive following a cycle: in-

tegrate external messages, repair the plan if needed,

decide which actions to execute. Other planners,

like “C-Buridan” ((Draper et al., 1994), (Kushmer-

ick et al., 1994)) deal with probabilistic information-

producing actions and contingent execution. their al-

gorithm is sound and complete but they do not con-

sider any kind of constraints between tasks. Their so-

lutions are satisfying rather than optimal.

Many existing works have been developed on tem-

poral planning or probabilistic planning but little at-

tention has been paid to the combination of these two

planning techniques.

In this paper, we address the problem of scheduling

plans with probability and temporal constraints. We

illustrate our problem with an AND/OR graph, where

we try to ﬁnd a plan of tasks that satisﬁes all temporal

constraints and precedence relations between tasks,

has a high probability of execution, a minimal cost

and a reduced time.

This problem consists in ﬁnding a set of tasks to

be executed respecting all temporal constraints and

precedence relations between tasks. By temporal con-

straints we mean the start times, the end times and

possible execution durations of tasks. By precedence

relation we mean the order of execution of tasks and

the delays between tasks. Our planner uses the tem-

poral constraint propagation technique (Bresina and

Washington, 2000) to simplify the resolution of a

given problem. Practically, that can consist in with-

drawing the values which do not belong to any solu-

tion. This ﬁltering avoids many attempts at resolution

which are likely to fail.

In the following, we consider that an agent is con-

cerned with achieving a set of tasks connected to-

gether in a network that forms an acyclic directed

AND/OR graph where nodes are tasks and edges cor-

respond to precedence relations. Given a set of tasks,

a start time, an end time, a set of durations, a set of

probabilities and a set of constant costs for each task,

a time delay and a precedence relation between tasks,

our approach allows the agent to determine the set of

tasks to be executed. The goal will be achieved and

all temporal and precedence constraints will be satis-

ﬁed in order to guarantee a correct execution.

A plan leading to the achievement of the desired goals

and satisfying temporal constraints would be consid-

Baki B. and Bouzid M. (2005).

A SCHEDULING TECHNIQUE OF PLANS WITH PROBABILITY AND TEMPORAL CONSTRAINTS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 77-84

DOI: 10.5220/0001169300770084

 SciTePress

ered as admissible. From all admissible plans, we se-

lect the one with the minimal expected utility to be

executed.

The paper is organized as follows: in Section 2,

we describe some basic temporal concepts concern-

ing the knowledge of an agent, tasks and the repre-

sentation of a temporal precedence graph. In Sec-

tion 3 we formulate our problem; then in Section 4

we describe a temporal constraint propagation algo-

rithm. Section 5 shows how to calculate probabilities

of intervals of tasks, and in Section 6 we select the

admissible plan which has the minimal expected util-

ity. Finally, we present some analysis and experiment

results in section 8 and we conclude in Section 9.

2 FORMAL FRAMEWORK

In this section, we describe the notion of a task and

a temporal precedence graph of tasks that we use

throughout the rest of the paper.

2.1 Task Description

The goal of the agent is to choose a subset of exe-

cuting tasks to convert some initial states into some

desired goals.

Deﬁnition 1 We call a task, the action realized by

an agent throughout a duration of time. For a task

t, one associates the list < (I

−

, I

, ∆

, P r

, C

) >,

where :

• I

−

is the earliest start time, i.e., the earliest time at

which the execution of the task can start;

• I

is the latest end time, i.e., the time at which the

execution of the task must ﬁnish;

−

, I

] is the time window referring to absolute

time during which task t can be executed;

• ∆

= {d

, d

, ..., d

} is a set of possible durations

of time such that d

∈ ℜ (i = 1...m), is a period of

time necessary to accomplish the task t;

• P r

= {pr

, pr

, ..., p r

} is a set of probabilities

such that pr

∈ ℜ (0 ≤ pr

≤ 1 and i = 1...m) is

the probability to execute t during d

∈ ∆

;

• C

= {c

, c

, ..., c

} is a set of costs payed to ac-

complish the task t such that c

∈ ℜ (i = 1...m) is

the cost to execute t during d

∈ ∆

In this framework, we shall consider a task to be an

atomic activity which can be undertaken and accom-

plished in whole. No medium states are taken into

account - either the task is successfully accomplished

or it fails.

2.2 Precedence Constraints

In real world, The execution of task depends on cer-

tain conditions like the time, the resources and/or the

execution of other tasks. In this last case, we talk

about precedence constraints in the form of partial or-

der of a set of tasks. In this paper, We distinguish

between two kinds of precedence constraints :

1. Conjunctive Precedence Constraint: let

, t

, . . . , t

} be tasks. There is a conjunc-

tive precedence constraint between t and the set of

tasks {t

, t

, . . . , t

}, denoted [t

, t

, . . . , t

] → t,

if t can be executed only if all tasks t

, t

, . . . , t

have already been executed.

2. Disjunctive Precedence Constraint: let

, t

, . . . , t

} be tasks. There is a disjunc-

tive precedence constraint between t and the set of

tasks t

, t

, . . . , t

, noted t

| . . . |t

→ t, if t can

be executed if at least one of tasks t

, t

, . . . , t

executed.

A special case is when t has a unique predeces-

sor (t

′

→ t), in this situation we talk about simple

precedence constraint.

Note also that, it is often the case that after execut-

ing the preceding task one should wait for some time

before the execution of the following task becomes

possible. We shall call δ

the time delay between

tasks t

and t

2.3 Temporal Precedence Graph

The goal of the agent is to execute a subset of tasks

represented by an acyclic temporal graph where nodes

represent tasks (described in 2.1) and arcs repre-

sent precedence constraints between tasks (described

in 2.2). More formally :

Deﬁnition 2 Let T be a set of tasks as described

in 2.1, E be a set of conjunctive and disjunctive

precedence constraints such that E = {c|c =

, t

, . . . , t

] → t or c = t

| . . . |t

→ t where

, t

, . . . , t

∈ T } and D be a set of delay con-

straints such that D = {δ

|δ

∈ ℜ, t

, t

∈

T and ∃ an arc from t

to t

}, a temporal prece-

dence graph is an acyclic oriented graph noted G =

(T, E, D) where nodes are elements of T and arcs are

elements of E.

Note that, nodes of the set T of any temporal prece-

dence graph G can be divided into three disjoint sets

T = T

∪ T

of tasks having no preceding

tasks, intermediate tasks having preceding tasks and

being predecessors for other tasks and ﬁnal tasks be-

ing no predecessors.

Figure 1 represents a temporal precedence graph G =

(T, E, D), where T = {t

, ..., t

}, E represents the

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

Figure 1: A temporal precedence graph of tasks

precedence constraints between tasks and D repre-

sents delay between tasks. We represent conjunctive

precedence constraints by arcs.

3 STATING THE PROBLEM

Given a temporal precedence graph of tasks G =

(T, E, D) where T is a set of temporal tasks as de-

scribed in 2.1, E represents conjunctive and disjunc-

tive precedence constraints as described in 2.3 and D

is a set of delay between tasks, the problem to be

solved is to ﬁnd a single plan of executable tasks in

T so that some goals which are of interest are satis-

ﬁed. In fact a plan is speciﬁed by a partial order of

tasks. More formally :

Deﬁnition 3 A temporal plan generation problem is

a couple (G, T

′

) where G = (T, E, D) is a temporal

precedence graph of tasks and T

′

⊂ T

⊂ T is a

subset of ﬁnal tasks in T .

The plan should satisfy all the constraints i.e.

AND/OR graph, precedence and temporal con-

straints. We are interested in ﬁnding a plan that has a

high probability to be executed during a reduced time

and with a reduced cost. To do that, we proceed in

four stages:

1. Determining of Feasible Plans. A feasible plan is

a subgraph of the initial graph G where constraints

are conjunctive or simple. This subgraph represents

path in the graph starting by initial tasks and ending

by goal tasks. We use both depth-ﬁrst and back-

wards search methods to ﬁnd all feasible plans. T

being the set of all goal tasks and T

the set of all

initial tasks, we start with the goal tasks to be ac-

complished, and we ﬁnish when all tasks in T

are

accomplished.

For example, consider the problem given by T

} and T

= {t

} for the graph given in

Figure 1. Feasible plans, marked with a bold

line, are [{[t

, t

], [t

, t

]}, t

, t

] and

[{[t

, t

], [t

, t

]}, t

, t

2. Determining of Admissible Plans. We call admis-

sible plan, a feasible plan where all temporal con-

straints of tasks and constraints between tasks are

satisﬁed.

3. Selection of The Most Likely Temporal Plan.

This step selects from all admissible plans, the best

likely one to be executed. This last must have a

high probability of execution and reduced cost and

time.

4. Selection of The Most Likely Temporal Schedul-

ing. This step selects from all scheduling of the

selected admissible plan, the one with the minimal

total expected utility.

We denote by P

the set of all feasible plans in tem-

poral precedence graph G and by T (P) the subset of

T of all the tasks occurring in the feasible plan P.

In the next section we present the propagation al-

gorithm of intervals execution allowing to determine

from the feasible plan set, the set of admissible plans.

4 CONSTRAINT PROPAGATION

ALGORITHM

Given a temporal plan generation problem (G, T

′

)

where G is a temporal precedence graph and T

′

a subset of ﬁnal tasks, we determine for each task in

each feasible plan the set of temporal intervals during

which a task can be executed by propagating temporal

constraints through the graph. Recall that a feasible

plan is a subgraph where constraints are conjunctive

or simple.

This propagation organizes the graph into levels so

that: l

is the level containing initial tasks (T

), l

contains all nodes that are constrained only by ini-

tial tasks, l

contains all nodes whose predecessors in-

clude nodes at level l

i−1

. For each node in any given

level l

, we compute all its possible execution inter-

vals from its predecessors.

Given a task t, we call s

(respectively e

) a possi-

ble start time of t (respectively a possible end time of

t), S

(respectively E

) the set of possible start times

of t (respectively the set of possible end times of t)

and I

the set of all possible execution intervals of

the t. A possible execution interval is formed by a

possible start time and a possible end time. I

, I

, ..., I

} where I

= [s

, e

] is the possible ex-

ecution interval to task t taking its duration d

∈ ∆

In the next section, we describe the algorithm used to

calculate the execution intervals of each task in each

feasible plan P.

A SCHEDULING TECHNIQUE OF PLANS WITH PROBABILITY AND TEMPORAL CONSTRAINTS

4.1 Execution Intervals of Tasks

Given a task t belongs a feasible plan P (t ∈ T (P))

where ∆

= {d

, d

, ..., d

} is its set of durations,

−

is its earliest start time and I

is its latest end time,

the possible times for starting and ending execution of

t are calculated as follows :

• level l

: t

∈ T

is an initial task) : Suppose that

the agent can begin its execution at a given time

noted start

time.

– S

= {s

= max(I

−

, start

time)}

– For i = 1 to i = m where m is the number of

possible execution durations of task t

= {e

= s

+ d

}

– I

= {I

= [s

, e

] where i = 1..m}. I

is a

possible execution interval to task t.

• level l

: for each task in level l

, its possible start

times are computed as all the times at which the

predecessor tasks can ﬁnish. Thus we deﬁne the

set of possible start and end times of each task at

level l

as follows :

1. If t ∈ T

∪ T

where ∆

= {d

, d

, ..., d

}

is the set of execution durations of t and t has a

unique task predecessor t

′

→ t) where E

′

, ..., e

′

} then :

(a) Determine the set S

(initialized to ∅) like that :

– For i = 1 to i = p (p is the number of possible

execution end times of task t

′

)

= S

∪ {s

= max(I

−

, e

′

+ δ

′

)}

(b) Determine the set E

(initialized to ∅) like that :

– For i = 1 to i = p

– For j = 1 to j = m

= E

∪ {e

= s

+ d

}

We call I

= [s

, e

] a possible execution in-

terval of t where s

∈ S

and e

∈ E

2. If t ∈ T

∪ T

where ∆

= {d

, d

, ..., d

}

is the set of execution durations of t which has

a set of direct task predecessors {t

, t

, . . . , t

}

([t

, t

, . . . , t

] → t) and if for each task t

(i =

1..n), E

= {e

, e

, ..., e

} then :

(a) Determine the set S

(initialized to ∅) of possi-

ble execution start times of t like that :

– For each task t

– For k = 1 to k = j

= S

∪ {s

= max(I

−

, max(e

))}

(b) Determine the set E

(initialized ∅) of possible

execution end time of task t like that :

– For k = 1 to k = p (p represents the cardinal-

ity of the set S

calculated in 2a)

– For r = 1 to r = m (m is the number of

possible execution durations of task t)

= E

∪ {e

= s

+ d

}

We call I

= [s

, e

] a possible execution inter-

val of t where s

∈ S

and e

∈ E

4.2 Admissibility of Plans

A feasible plan P ∈ P

, in order to become an ad-

missible one, must satisfy all the temporal constraints.

More formally, for any task t ∈ T (P), for each

∈ E

, the following condition : e

≤ I

must hold.

If a possible end time of E

exceeds I

, we consider

that execution interval I

= [s

, e

] is not valid. In the

other hand, if all possible end times of E

exceed I

we consider that task t is not able to execute. Thus

we consider the plan which contains such task is not

admissible.

We denote by P

the set of all admissible plans P.

We have P

P, where P is an admissible plan.

5 PROBABILISTIC TEMPORAL

PROPAGATION

We describe in this section how we can weight each of

those intervals calculated as above, with a probability.

This probabilistic weight allows us to know the prob-

ability for a task to be executed during a given interval

of time. For that, a probability propagation algorithm

among the graph of tasks is described using for each

node its execution time probability and the end-time

probabilities of its predecessors.

The probability of a possible execution interval I

de-

pends on its start time (the end time of the previous

tasks) and the probability of execution time pr

. To

simplify, we consider in the rest of this paper, that no

time delay between tasks.

In the next section, we compute the probability that

the execution for a task t to occur during an interval

where s

is a possible start time and e

is a possible

end time of task t.

5.1 Probability Propagation

Algorithm

Before a task t can start its execution, all its direct

predecessors must be ﬁnished. The probability for the

execution of t start at s

is deﬁned by :

1. If t has an only direct predecessor t

′

→ t) in the

admissible plan P where E

′

= {e

′

, ..., e

′

} and

= {s

, s

, ..., s

} is the set of possible execu-

tion start time of task t then :

• The probability that task t starts its execution at

when its predecessor t

′

ﬁnishes its execution

at e

′

, noted pr

start

′

), is calculated as fol-

lowing :

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

– For i = 1 to i = m do

– For j = 1 to j = p do

– If s

< e

′

then pr

start

′

) = 0

– If s

≥ e

′

then pr

start

′

) = 1

2. If t has a set of direct predecessors {t

, t

, . . . , t

}

in the admissible plan P i.e. [t

, t

, . . . , t

] →

t then let t

∈ {t

, t

, . . . , t

} where E

, e

, . . . , e

} such that j

represents the num-

ber of possible end times of task t

and let S

, s

, ..., s

} be the set of possible start times of

t then :

• The probability that task t starts its execution at

(r = 1...m) when each direct predecessor

∈ {t

, t

, . . . , t

} of t ﬁnishes its execution

at e

(k = 1..j

) is calculated as following :

– If s

< max{e

} then pr

start

) = 0

– If s

≥ max{e

} then pr

start

) = 1

A special case has to be considered for the ﬁrst tasks.

Indeed, these initial tasks (∈ T

) have no predeces-

sors, the probability of starting the execution of an

initial task t at s

is given by : pr

start

) = 1.

5.2 Execution Intervals Probability

In this section, we describe the method used to calcu-

late the execution intervals probability which depends

on execution durations and execution start times.

Let pr

) be the probability that the execution of

task t takes d

units of time when it starts at s

The probability of an execution interval I

of task t

that has an only direct predecessor t

′

is the probability

P r

′

) that interval I

= [s

, e

] is the interval

during which a task t is executed, if the predecessor

task ends at e

′

. This probability measures the proba-

bility that a task t starts its execution at s

and ends at

. It is deﬁned as following :

P r

′

) = pr

start

′

) ∗ pr

)

where e

′

is the end time of the last executed task.

Indeed, a task t can start its execution only when its

predecessor t

′

ﬁnishes. That is why the probability

of execution interval depends on the predecessor end

time.

In the case where t, such that ∆

, d

, ..., d

}, has a set of direct predecessors

, t

, . . . , t

}, where t

∈ {t

, t

, . . . , t

}, ends its

execution at e

= 1..j

s.t. j

represents the

number of possible execution end times of task t

we have :

P r

) = pr

start

) ∗ pr

)

where r = 1..m.

A special case has to be considered for the ﬁrst task.

The probability that an initial task t executes in the

interval I

is equal to the probability that it starts its

execution at s

and takes d

units of time. More for-

mally :

P r

) = pr

start

) ∗ pr

)

For example, suppose 0.7 is the probability for the

execution of task t that has an only direct predeces-

sor t

′

, to take 2 units of time when it starts at 5

(P r

= 2|s

= 5) = 0.7). We assume that the end

execution of the predecessor task t

′

is 3. The proba-

bility of the interval [5, 7] is :

P r

= [5, 7]|e

′

= 3) = 1 × 0.7 = 0.7

If for example task t

′

ends its execution at 7, we ob-

tain :

P r

= [5, 7]|e

′

= 7) = 0 × 0.7 = 0

6 SELECTION OF THE MOST

LIKELY ADMISSIBLE PLAN

As the order of searching for plans is arbitrary, the

search may produce several admissible plans. In this

case, further criteria will be applied (Probability, cost

and time) to analyze and compare them so as to select

one to be executed. In this section we describe the

method we use to choose the most likely admissible

plan to be executed.

We are interested in ﬁnding a unique execution plan

that has a high probability and reduced cost and time.

To do that, we calculate expected utilities of cost and

time of all tasks. Then, we calculate the expected util-

ity of tasks of each admissible plan, ﬁnally we select

the one with the smallest expected utility to be the

most likely plan to be executed. We detail this as-

sumption in the next section.

6.1 Expected Utilities of Cost and

Time of Tasks

For each task in an admissible plan P, we calculate its

expected utility of cost and time. The expected utility

of cost is calculated in function of costs associated

of execution durations and of execution probabilities

associated of execution intervals (calculated in 5.2).

The expected utility of time is calculated in func-

tion of possible execution durations and of execution

probabilities associated of execution intervals (calcu-

lated in 5.2). More formally :

Let t, where ∆

= {d

, d

, ..., d

} is the set of

possible durations of t, C

= {c

, c

, ..., c

} is the

set of execution costs of t where c

is the cost to ac-

complish t taking duration d

(k = 1..m).

A SCHEDULING TECHNIQUE OF PLANS WITH PROBABILITY AND TEMPORAL CONSTRAINTS

1. If t has an only direct predecessor t

′

such that

′

= {e

′

, e

′

, . . . , e

′

} is its set of possible ex-

ecution end times, and if the probability of t to be

executed in duration d

where its predecessor t

′

has

ﬁnished its execution at e

′

is noted P r

′

)

(section 5.2), then :

• Expected Utility of cost of t is calculated as fol-

lowing :

µ(cost(t|t

′

)) =

j=1

k=1

P r

′

) ∗ c

To simplify, we denote µ(cost(t|t

′

)) by

µ(cost(t)).

• Expected Utility of time of t is calculated as fol-

lowing :

µ(time(t|t

′

)) =

j=1

k=1

P r

′

) ∗ d

To simplify, we denote µ(time(t|t

′

)) by

µ(time(t)).

2. If t has a set of direct predecessor t

, t

, . . . , t

then, let E

= {e

, e

, . . . , e

} be the

set of possible execution end times of t

∈

, t

, . . . , t

} then :

• Expected Utility of cost of t is calculated as fol-

lowing :

µ(cost(t|t

, . . . , t

)) =

i=1

µ(cost(t|t

))

To simplify, we denote µ(cost(t|t|t

, . . . , t

))

by µ(cost(t)).

• Expected Utility of time of t is calculated as fol-

lowing :

µ(time(t|t

, . . . , t

)) =

i=1

µ(time(t|t

))

To simplify, we denote µ(time(t|t

, . . . , t

)) by

µ(time(t)).

In the next section, we calculate the total expected

utility of each admissible plan P.

6.2 Expected Utility of Plans

The expected utility of a plan P is calculated in func-

tion of expected utilities of cost and time (calculated

above).

The total expected utility of cost of a plan P is cal-

culated by adding all expected utilities of cost of all

tasks in this plan. More formally :

µ(cost(P)) =

i=1

µ(cost(t

))

where n is the number of tasks in P.

The total expected utility of time of a plan P is calcu-

lated as following:

• µ(time(P)) is initialized to max(µ(time(t

)))

where t

∈ T

– If t has an only direct predecessor t

′

with

expected utility µ(time(t

′

)), then we add

µ(time(t

′

)) to the total expected utility of time

of the plan. More formally :

µ(time(P)) = µ(time(P)) + µ(time(t

′

))

– If t has a set of direct predecessors

, t

, . . . , t

} then, we add the maximum

of expected utilities of tasks t

, t

, . . . , t

to the

total expected utility of time of the plan. More

formally :

µ(time(P)) = µ(time(P))+max

i=1

µ(time(t

))

To calculate the total expected utility for each plan

P ∈ P

, we assign a value to each plan according to

the preferences of the user. We translate these pref-

erences into a function µ(P) : µ called function of

utility. This function is prone to the cost and the time

that are balanced by coefﬁcients α and β. This al-

lows us to adjust the relative importance of the vari-

ous utilities according to the preferences of the user.

This utility function is described by :

µ(P) = αµ(cost(P)) + βµ(time(P))

where α + β = 1 and P ∈ P

Among all admissible plans, we choose the one

with the smallest utility to be executed :

∗

= argmin

P∈P

µ(P)

This plan is the most likely one to be executed satis-

fying all constraints and that has a high probability to

be executed with reduced cost an time of execution.

In the next section, we determinate the most likely

scheduling to be executed.

7 SELECTION OF THE MOST

LIKELY SCHEDULING

We call scheduling of a plan P, the set of tasks of

P where each task is executed in a well deﬁned in-

terval. If there is a simple precedence constraint be-

tween two tasks t

and t

→ t

), then the execu-

tion interval of task t

must occur before the one of

task t

. More formally, let T (P) = {t

, t

, . . . , t

}

be the set of tasks of an admissible plan P, and

= {I

, I

, . . . , I

} be the set of execution in-

tervals of task t

in T (P), we have :

ord

= {(t

, I

), (t

, I

), ..., (t

, I

)}

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

is a possible scheduling in P where k

= 1..j

such

that if t

→ t

and t

∈ P

ord

) then e

≤ s

We have also :

P =

[

ord

In the worst case, the total number of scheduling of

a plan P is equal to the product of cardinalities of all

possible execution interval sets of all tasks in T (P).

More formally this number is equal to :

No(P

ord

) = j

∗ j

∗ . . . ∗ j

In the next section, we calculate the expected utility

of cost and the expected utility of time for each task

(t, I

) ∈ P

ord

7.1 Expected Utilities of Cost and

Time of Tasks of Scheduling

for each P

ord

{(t

, I

), (t

, I

), ..., (t

, I

)}, we calcu-

late the expected utility of cost and the expected

utility of time for each task t

to be executed in I

More formally :

1. If (t, I

) ∈ P

ord

has an only direct predecessor

′

, I

′

) ∈ P

ord

where (t, I

) → (t, I

) then :

• The expected utility of cost of (t, I

) is calcu-

lated as following :

µ(cost((t, I

)|(t

′

, I

′

))) = P r

′

)∗c

• The expected utility of time is calculated as fol-

lowing :

µ(time((t, I

)|(t

′

, I

′

))) = P r

′

)∗d

2. If (t, I

) ∈ P

ord

has a set of direct predeces-

sors {(t

, I

), (t

, I

), ..., (t

, I

)} then, let

, I

) ∈ {(t

, I

), (t

, I

), ..., (t

, I

)}

then :

• The expected utility of cost of (t, I

) is calcu-

lated as following :

µ(cost((t, I

)|(t

, I

), (t

, I

), ..., (t

, I

)))

i=1

µ(cost((t, I

)|(t

, I

)))

• The expected utility of time of (t, I

) is calcu-

lated as following :

µ(time((t, I

)|(t

, I

), (t

, I

), ..., (t

, I

)))

i=1

µ(time((t, I

)|(t

, I

)))

In the next section, we calculate the total expected

utility of a scheduling P

ord

7.2 Expected Utility of Scheduling

To calculate the total expected utility of a scheduling

ord

, we calculate ﬁrst the total expected utilities of

cost and time of P

ord

as following :

1. The total expected utility of cost of P

ord

is calcu-

lated by adding all expected utilities of all tasks in

ord

. More formally :

µ(cost(P

ord

)) =

i=1

µ(cost(t

, I

))

where n is the number of tasks of P.

2. The total expected utility of time of P

ord

is calcu-

lated as following :

• µ(time(P

ord

)) is initialized to

max(µ(time(t

, I

))) where t

∈ T

– If (t, I

) has an only direct predecessor

′

, I

′

) such that its expected utility is

µ(time(t

′

, I

′

)) then we add this expected

utility to the total expected utility of time of

plan. More formally :

µ(time(P

ord

)) =

µ(time(P

ord

)) + µ(time(t

′

, I

′

))

– If (t, I

) has a set of direct predecessors

{(t

, I

), (t

, I

), ..., (t

, I

)} then, we

add the maximum of expected utilities of these

tasks to the total expected utility of time of plan.

More formally :

µ(time(P

ord

)) =

µ(time(P

ord

)) + max

i=1

µ(time(t

, I

))

For each scheduling P

ord

{(t

, I

), (t

, I

), ..., (t

, I

)} of the ad-

missible plan P chosen to be executed, we calculate

its utility function, where α + β = 1 and, deﬁned by :

µ(P

ord

) = α ∗ µ(cost(P

ord

)) + β ∗ µ(time(P

ord

))

From all scheduling formed by P we choose the one

with the smallest expected utility to be executed :

∗

exec

= argmin

ord

∈P

µ(P

ord

)

This plan determines the expected intervals of execu-

tion in which tasks should be executed by the agent.

8 ANALYSIS AND EXPERIMENTS

The number of execution intervals for each task t de-

pends on the number of its direct precedence tasks, its

A SCHEDULING TECHNIQUE OF PLANS WITH PROBABILITY AND TEMPORAL CONSTRAINTS

time window [I

−

, I

] and the cardinality of its dura-

tion set |∆

|. When one of these constraints increases,

this number increases. The number of plans in a graph

depends on the number of execution intervals of tasks,

the number of tasks and the number of precedence

constraints. This number increases when the number

of execution intervals or the number of tasks increases

and it decreases when the number of precedence con-

straints increases.

In the worst case, the number of execution intervals

for an initial task is equal to the cardinality of its du-

ration set. The number of execution intervals for an

intermediate or ﬁnal task which has an only direct pre-

decessor is equal to the cardinality of its duration set

multiplied by the cardinality of the set of the ends of

the direct predecessor task. The number of execution

intervals for an intermediate or ﬁnal task which has

a set of direct predecessors is equal to the cardinality

of its duration set multiplied by all the cardinalities

of the sets of the ends of the direct predecessor tasks.

More formally, in the worst case :

• If t ∈ T

then |I

| = |∆

• If t ∈ T

∪ T

where t has an only direct prede-

cessor t

′

then |I

| = |∆

| ∗ |E

′

• If t ∈ T

∪T

and t has a set of direct predecessors

, t

, . . . , t

} then |I

| = |∆

| ∗

i=1

In our experimental tests, we used these parameters :

the total number of tasks, the number of plans we can

obtain if all temporal constraints are correct (worst

case) and the number of plans obtained without the

ones violating temporal constraints.

We remarked that theoretically, for 50 tasks with 5

execution intervals for each, we obtain 250000 plans

but with our approach, we generate only 20569 plans.

These ﬁrst experimental results consolidate our idea

of the founded good of the approach. However, ad-

ditional experimental tests on the other factors, as

the size of the temporal windows and the number of

precedence constraints, are to be analyzed.

9 CONCLUSION

In this paper we present an approach of temporal

probabilistic task planning where we construct a plan

of tasks that satisﬁes all constraints and executes with

a high probability during a reduced time and with a

reduced cost. Our approach allows the agent to deter-

mine the set of tasks to execute and when to execute

them by respecting all temporal and precedence con-

straints. This approach is one of the ﬁrst techniques

combining probability and time in planning.

Future work will be focused on other kind of exper-

imental factors and on comparing our approach with

other ones. Another issue consists in ﬁnding other

heuristics to choose the most likely plan to be exe-

cuted and reducing the search space then comparing

them with the heuristics presented in this paper.

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