Scheduling can be formalized as a Constraint Satisfaction Problem (CSP). Within this framework activities belonging to a plan are interconnected via temporal constraints that account for slack among them. Temporal representation must include methods for constraints propagation and provide a logic for symbolic and numerical deductions. In this paper we describe a support framework for opportunistic reasoning in constraint directed scheduling. In order to focus the attention of an incremental scheduler on critical problem aspects, some discrete temporal indexes are presented. They are also useful for the prediction of the degree of resources contention. The predictive method expressed through our indexes can be seen as a Knowledge Source for an opportunistic scheduler with a blackboard architecture

Gini, G.

Paolucci, E.

Patriarca, E.

Sem, M.

English

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Predit: a Temporal Predictive Framework for Scheduling Systems
E.Paolucci, E.Patriarca, M.Sem, G.Gini
Politeenico di Milano
Piazza Leonardo da Vinci 32
20133 Milano - Italy
Paolucci@ipmel 1.polimi.it
Abstract
Scheduling can be formalized as a Constraint Satisfac-
t/on Problem (CSP). Within this framework activities
belonging to a plan are interconnected via temporal
conswaJnts that account for slack among them. Tempo-
ral representation must include methods for constraints
propagation and provide a logic for symbolic and nu-
merical deductions.
In this paper we describe a support framework for
opportunistic reasoningin constraintdirectedschedul-
ing. In order to focus the attention of an incremental
scheduler on critical problem aspects, some discrete
temporal indexes are presented. They are also useful for
the prediction of the degree of resources contention.
The predictive method expressed through our indexes
can be seen as a Knowledge Source foran opporumi_c
scheduler with a blackboard architecture.
1. Formalization of scheduling problem and
strategies for its solution
Schedulin_g can be formalized as a Constraint Satisfac-
tion Problem (CSP)[Keng and Yun, 1989]. This ap-
proach is concerned with the assignment of values to
variables subject to a set of _ts. In scheduling
variables are constituted by activities start limes and
from resources allocation; for this reason we have w
deal explicitely with two types of constraints: temporal
relationsamong tasks and resources capacity [Fox, 86].
In our approach we assume that we have a set of plans
to be scheduled, where a plan is defined as a partial
ordering of activities. Each activity may require one or
more resources and for each of them there can be
alternative choices. Beside resources capacity can he
used _ntemporady by different tasks; for the sake of
simplicity we will assume an resources with unary
capacity.
Scheduling is an NP-hard problem and methods re-
quired for its solution must face this complexity. In our
research we decided to focus our attention on contribu-
tion in scheduling coming from AI, and particularly on
oppommistic reasoning [Hayes-Roth, 79].
We me concerned with the issue of how it's possible to
focus the attention of an incremental scheduler on the
most critical scheduling choices in order to evaluate
which are the most critical points, which decisions seem
to be the most promising in reducing search complexity
and improving quality of resulting schedule.
Our strategy is to identify the most "solvable" aspects
of the problem through the evaluation of the degree of
interaction existing among activities belonging to dif-
ferem orders. The aim is to reduce the number of steps
required to obtain a solutiorL
The necessity to overcome the limits of partial decom-
position approach, such as order-based and resource-
based decompositions, has led us towards an
event-based perspective whit chronologically-grouped
information.
This basic search strategy is realized through most-con-
strained and least-impact policies. Every step is divided
into two pans: first the most-constrained policy selects
dynamically on which agent must be focused schedul-
ing attention; then, the least-impact policy chooses for
that agent a value whose impact on the rest of the
non-scheduledagents is as _nall as possibile. The goal
is the identification of critical activities lhat heavily rely
on the possession of highly contended temporal inter-
yah or resources because of inn-a-order and inter-order
interactions Oook-ahead strategy).
This two policies need numeric indexes which, analyz-
ing the particular _ of a problem, are ableto
measure the in_raction among a_vities and resources
in terms of variable looseness and value goodness
[Sadeh end Fox, 88].
Variable looseness is the measure of how constrained
is a resource or an acevity; value goodness measmes
which variable value, among an the feasible ones, gives
the least impact (i.e. a sort of maximum sack) on
availability of feasible (and good) values for non-sche-
deled variables.
We identified some numeric indexes that contain infor-
marion required to realize an event-based policy: these
indexes are useful for different reasons:
2
w
1. This work has been part_ly supported by CNR PRF
Project and Mauro grants
https://ntrs.nasa.gov/search.jsp?R=19930009502 2020-03-17T07:50:38+00:00Z
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[] they makepossible to point out critical resources
armactiviues;
[] they identify "island of certainty" that Will be a
part of problem solution;
[] they give information about activities start times
that flare the least impact on non-scheduled acti-
vities.
[] an internal bound (INT), which represems the
mihimun time interval which must separate the
ep_ of _ first !apse from the begin of the second
of two related lapses;
[] an external bound (EXT), which represents the
maximum time interval from the begin of the first
lapse to the end of the second.
This behaviour is a sort of "oppommistic reasoning"
[Hayes-Roth, 79]i this term has been used to charac-
terize a problem-solving process where reasoning is
consistently directed towards those actions that appear
most promising for solving a problem.
Our predictive approach, used together with an oppor-
tunistic reasoning, is also useful to detect un_tisfiable
CSPs as s_n as possible, simply by analyzing the
indexes we defined. In this sense the system can be
viewed as a Knowledge Source in a blackboard archi-
tecture, which assumes responsibility for preventive
analysis of activities interactions and for the detection
of prospective bottlenecks.
2. The predictive approach: basic assumptions
The main goal of our research was to provide a simple
but complete inference mechanism to supportschedul-
ing, working in a discrete time domain. This mechanism
is based on some indexes and is designed to perform an
a-priori guidance for search in scheduling domain. We
kept a particular attention on the efficiency and on the
speed of such a mechanism, because we realized that
such properties are necessary in scheduling systems for
real applicative environments like, for instance, manu-
facturing ones. For this reason we decide to consider a
discrete representation of time instead of a continous
one.
Our indexes are based on the constraints analysis (and
on the propagation of the temporal ones) and on a
particular representation of existing time relations.
In terms of constraints analysis we differentiate be-
tween._str.i.'cU'om _ preferences [Fox, 86]. Temporal
preferences are-represented through utility functions
defined__pn__fivity Starttimes that maps possible values
onto utility levels ranging from 0 to 1. Moreover in our
analysis we consider the existence of intra-order
(among activities belonging to an order) and inter-order
_n_ (among activities belonging to different or-
Through these two parameters it's possible to model
any temporal relation in a scheduling problem. They are
simpler than thirteen ADen's primitive relations; more-
over, INT and EXT improve greatly the efficiency of
numeric temporal reasoning, that is instead a limit in
Allen's primitive.
3. The Predit indexes
Temporal relation constraints are used to describe par-
tial orderings among activities as provided by the pro-
cess planning step.
We will refer to the graph defined by these constraints,
for a given CSP, as the CSP's Temporal Constraint
Graphs (TCG).
We have to schedule a set of activities (A1,A2 ..... AN).
Let Ik be the time interval associated with AL Activities
are connected by a set of temporal relation constraints,
thereby forming a TCG. We view TCGs as undirected
graphs. An Arc in a TCG indicates the presence of a
temporal relation between two intervals represented by
the couple INT-EXT (Fig. 1).
Ai
Additionally there are capacity constraints limiting the
use of each resource to only one activity at a time. The
next example presents a simple case of a TCG com-
posed of two orders.
ders).
The model a_lopmted _:represehfi_g dtfie_::_f_r_.al
sorting about temporal relation is based on the concept
of lapse, that is defined as the period of dine associated
with an __activity.In a temporal axis a lapse is represented
by two temporal parameters, namely st/n-time and end
time. Relations between different lapses are expressed
by two parameters [Paolucci, 90]:
In order to provide a predictive support for opportunis-
................... ...... := = ::=tic schedulers operating in a discrete time domain we
have considered interactions among activities caused
by temporal relations.
The first issue we faced was to detect as soon aspossible
during the scheduling the possible arising of conflicts
due to interactions among activities.
151
Forthisissuewedefinedanindexcalled
D.ggggg(CD),whichmeasuresthehowfight is the link
existing between two generic activities Ai and Aj con-
nected by a temporal relation constraint (represented by
INT-EX'r) in a contraint graph.
Temporal relations between intervals may simply be
expressed by using potential inequalities associated
with the bounds of intervals such as :
[21 Di + Dj + IN'I" $ EX'T
The first inequality verifies that time interval composed
of a_vities durations and Internal Bound is included in
maximum temporal window allowed by Aj latest end
time and Ai earliest start time.
The second inequality controls that the same time inter-
val doesn't violate External Bound temporal constraint.
These inequalities lead to define the CD formula
through a mu/fiplication of their members:
CD_= ( oi . D/+ INT ) z
EXT. ( e-_- S?_)
0 <CDi/< 1
Vk = Ak duration _ = imemal bound be-
tween Ai and Aj
STk = Ak start time EXT = external bound be-
twee_1 Ai and Aj
ETk = Ak end time
The Constrxim Degree is calculated on the notion of
slack between two activities fled by temporal links.
u CDij = I means that AI allows no slack to Aj
(most constrained)
u CDij = 0 means thatAi allows maximumclackto
Aj.
The C'D computational alg,ritran considers an con-
nected activities from the beginning to the end of the
gra_ Therefore, for ending activities we set CD index
to zero (ending activities are not constrained, with tem-
poral relation, with any other activity in the graph).
The validity of CD index is preserved by a pgevious
optimization procedure in order to adjust activities tem-
poral windows cutting out start time values that can
never be involved in CD computaticm (the same is
made for other indexes).
152
To sum up, the CD index detects (following the most-
constrained policy) the most critical activities with II:-
spec_ w inwa-ordertemporal relations (exp.,seal by
IRT and EXT) and to temporal windows (expn_,sed by
activity start and end time).
The second index, called Preferential Start Time
(PST), is a local measure of value goodness and glo-
bally, ameasure of variable looseness for activities start
times. It helps in choosing among all admissible start
times the one that minimizes future conflicts. It is
calculated between each pair of activities connected in
the TCG (i.e. Ai and Aj) and it depends on the start time
of the first activity (i.e. sti).
The main goal of PST index was to introduce some
es_mat/on rule for activity start times in order to ident-
ify the least impact values arising from intra-order
interactions.
PST index is computed for every activity start time
evaluated between earliest start time (STi), or value
allowed by INT-EXT, and latest start time (ETi-Di), or
value allowed by INT-EXT, increasing stiwith thefixed
time unit.
PST is expressed by the ratio:
[4] PSTo( st_) = ._# ( st_)
EXT- Di- D/- INT
0 _ PST_-Oti) S 1
where:
u i/lfij(b'[i ) ----reLl_ive irlg_rtHl_ boLll_
u EXT-D_-Dj-INT = maximum slack between the
activities
The numerator is calculatedfor _ values from Earliest
Start Time to the maximum allowed by temporal con-
straints,increasingeachtime_ with achosentime unit.
It may be also considered as"actual" slack betweenthe
two mivites corresponding m s_ value.
Therefore, the denomtrmtormay be viewed as the max/-
mum slack between the two activities. The closer is
PSTij valt_ tO{Xte,the greater is the slack between Ai
and Aj. "therefore, for a generic activity PST measures
for each admissible start time its goodness and likel/-
hood to minimize future _hedu]/ng conflicts.
To computeactivitylndlvldual Demand forresources,
we have combined the value goodness of every start
time (expressedby PST) with the activities durations.
Mo_,over, as assumed in [Sadeh-Fox. 88], an activity
=E_
g_
Ai can use a resotlrce Rj if Ai i$ active at time t and Ai
uses Rj at time t to fulfill its resource requiremmt.
From each PST graph we achieve an Individual De-
mand graph (whose values are expressed by ID index)
for each activity, expanding PST values with a lapse
equal to the activity duration and adding all values in
function of time. We obtain a histogram representing
activity resource demand in function of time.
of time. AD shows when resource competition is par-
ticularly high and which are activities that heavily rely
in the possession of these resources. AD values must he
tightly evaluated in function of lime because temporal
constraint propagation doesn't atlow for any resour_
preference (as explained before, we assume all resour-
ces with unary capacity). Therefore, AD index can
estimate the amount of contention for each resource
over temporal axis but only as a function of start time.
Moreoyer, it's easy to improve tiffs approach repre-
senting, for example, resources preferences with utility
functions and propagating these resources reservations
through the TC_ graph.
Figure 2 shows a simple example in order to illustrate
our graphic results concerning temporal discrete in-
dexes presented above.
Individual Demand va!.u_ arc combined to measure
reso_ A t,_sie Demand (AD), +always in function .......
0rderl
AD Resource R3
10 - l0
...... II
.4
.|
....... @
Tk_e
F_,_ 3
0.II,.
O,lll
II, I.
Oil.
@.
Pst a7 Order 2
ll.ll I.ll
I0 II t4 II tl m II I+ II II
YLme
110
Figure4
Next results are concerned with the reasonable steps
that an Opportunistic Scheduler should achieve in order
to produce the final Gann chart.
Order2
P 'IP
Hgure 2
The temporal constraint a_ciated at _ linker is
equal for all couple of activity and it is expressed by
INTri0 and EXT=40. However, these values may he
optimized as described before.
Start Tune and End Time are expressed by numbers
above each activity and the same is made for requested
resources. For the sake of simplicity, in this example we
AI A2 A3
O.25 0.2 02
Table 1: CD mluts for order I activities
A6 A7
0.25 0.25
Table 2: CD values for order 2 activities
A4 A5
o,_82 ........ o :
A8
0
Among the agg_gate demands, the most highly con-
tended resource is 1t3 (fig. 3), required by A2, A3, A4,
A7; the next activities we will focus our attention on are
A4, ,4,3 and A7 (because A2 has an alternative in R1).
havenotintroducedpreferendalstartdmes (soactivity TakingaIcokattheCD indexesoforderIand order 2,
start times are equally preferred). A7 appears to he the most constrained activity because
of its highest CD value. Now, A7 PSI" graph (fig. 4)
153
presents a maximum for:t=20 and scheduling A? with
st=20 we can assign the resource R1 to the activity A2
at the same start time.
The same considerations based on temporal indexes
evaluation allow the identification of other activity pref-
erential start times leading to the Gann chart presented
below in fig. 5.
GANYr CHART order 1
GAN'/'r CHART order 2
41 II tO 16 tO fill, _I0 _q+L 40 411 liO 16
410 16 7O
I_ 16 70 -
5
The quality of a schedule is based on the capability of
the scheduler to satisfy a set of performance measures.
Moreover, a satisfiable schedule is always a com-
promise between the attempt to meet performance re-
quired and the necessity to respect all its constraints:
schedule quality mirrors this trade-off. Each set of
organizational constraints has its effects on final pro-
duction schedules and, following the CSP formulation,
if we change the constraints the solution will change
tOO.
In order to improve schedule quality, our research is
focusingon the evaluation of whichimpactmighthave
an unexpected event on the resulting solution. PRED1T
approachthroughtheevaluationof discretetemporal
indexesproducesrelativelyaccuratearlypredictions
of activities behaviour as soon as PREDrr receives their
changes and as long as constraints remain constant
during indexes computation. The ability to react to
changes that occm"in dynamic environments pn3viding
a feasible solution in a sufficiently short time is very
important expociaIly in manufacturing scheduling do-
main.
predictive supportinanoppommisticschedulingsys-
tem.
We implemented this model in a MS-DOS environment
with a particular attention towards speed performances.
Our experiments indicate that our approach is success-
ful in supportingopportunistic scheduling. This system
is very effident (it takes few seconds to calculate in-
dexes in non-l_ivial real problems).
Ourmodel seems to be highly appropriate for problems
where the costs of backtracldng is high because it's able
to point out scheduling decisions that will minimize
intra-order and inter-order conflicts.Itincreases signi-
ficantly the performances of an opportunistic scheduler,
making it possible to introduce such a tools in real
applications.
Moreover thepoliciesusedby Preditto conu-olthe
solutionsearch(mustconstrainedandleastimpact)can
beusedalsoindynamicmanufacturingenvironments.
We aredevelopingourresearchinthisense,alsou'ying
to _pport reactive scheduling and to manage multia-
gent production control systems.
References
[Fox, 1896], Fox M., Observation on the role of con-
straints in pmhlem solving, Proceedings Sixth Cana-
dian Conlerence on,41, Montreal, 1986.
[Hayes-Roth, 79], Hayes-Roth B. et al., Modeling plan-
ning as an incrementalopportunisticproblem,Proceed-
ings 3rd IJCAI, Tokio, 1979.
[Keng and Yun, 89], Keng. N. and Yun D., A planning
scheduling methodology for the constrained resource
problem', Proceedings IJCA11989, pp. 998-1002.
[Paolucci, 90], Paolucci et. al., "CRONOS-III: require-
merits for a knowledge-based scheduling tool covering
a broad class of production environments", in Expert
systems in engineering, G.Goulob and W.Nejdl (eds.),
Springer-Vedag, 1990.
[SadehandFox, 88], Sadeh N. and Fox M., Preference
propagation in temporal capacity con.qraint graphs,
Techmcal report CMU-CS-88-19.L
4. Conducting remarks
The approach we presented in this paper constitutes the
basis for integrating an event-based mechanism and a
154


Predit: A temporal predictive framework for scheduling systems

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