FUZZIFICATION OF THE RESOURCE-CONSTRAINED
PROJECT SCHEDULING PROBLEM
A Fight against Nature
Anikó Csébfalvi, György Csébfalvi and Sándor Danka
University of Pécs, Pécs, Hungary
Keywords: Project scheduling, Stochastic scheduling, Fuzzy scheduling, Resource-constrained project, Heuristic
algorithm, Simulation.
Abstract: In a recent article (Bhaskar et al., 2011) the authors proposed a heuristic method for the resource-
constrained project scheduling problem (RCPSP) with fuzzy activity times. The apropos of this state-of-the-
art work, we try identify and illuminate a popular misconception about fuzzification of RCPSP. The main
statement of their approach, similarly to the other fuzzy approaches, is simple: the project completion time
can be represented by a "good" fuzzy number. This statement is naturally true: in a practically axiomatic
fuzzy thinking and model building environment, using only fuzzy operators and rules, we get a fuzzy output
from the fuzzy inputs. But the real problem is deeper. The possibilistic (fuzzy) approach, traditionally,
defines itself against the probabilistic approach, so in the "orthodox" fuzzy community everything is
prohibited which is connected to somehow to the probability theory. For example, the Central Limit
Theorem (CLT) is in the taboo list of this community. We have to emphasize, CLT is a humanized
description of a miracle of nature. When we fight against CLT, we fight against nature. The situation in the
"neologist" fuzzy community is not better, because they try to redefine somehow the probability theory
within the fuzzy approach without using "forbidden" statistical terms. In this paper, we will show that the
nature is working totally independently from our "magic" abstractions. According to the robustness of CLT,
the distribution function of the completion time of real-size projects remains nearly normal, which is a
manager friendly, natural and usable result. An abstraction and its "natural" operators are unable to modify
the order of nature. When we want to add a practical scheduling method to the project managers we have to
destroy the borders between the probabilistic and possibilistic approaches and have to define a "unified"
approach to decrease the gap between scientific beliefs and reality. In this paper we present a unified
(probabilistic/possibilistic) model for RCPSP with uncertain activity durations and a concept of a heuristic
approach connected to the theoretical model. It will be shown, that the uncertainty management can be built
into any heuristic algorithm developed to solve RCPSP with deterministic activity durations. The essence
and viability of our unified model will be illustrated by a fuzzy example presented in the recent fuzzy
RCPSP literature.
1 INTRODUCTION
In a recent article (Bhaskar et al., 2011) the authors
proposed a heuristic method for the resource-
constrained project scheduling problem (RCPSP)
with fuzzy activity durations. The apropos of this
state-of-the-art work, we try identify and illuminate
a popular misconception about fuzzification of
RCPSP. The main statement of their approach,
similarly to the other fuzzy approaches, is simple:
the project completion time can be represented by a
"good" fuzzy number. This statement is naturally
true: in a practically axiomatic fuzzy thinking and
model building environment, using only fuzzy
operators and rules, we get a fuzzy output from the
fuzzy inputs. But the real problem is deeper. The
possibilistic (fuzzy) approach, traditionally, defines
itself against the probabilistic approach, so in the
"orthodox" fuzzy community everything is
prohibited which is connected to somehow to the
probability theory. For example, the Central Limit
Theorem (CLT) is in the taboo list of this
community. We have to emphasize, CLT is a
humanized description of a miracle of nature. When
286
Csébfalvi A., Csébfalvi G. and Danka S..
FUZZIFICATION OF THE RESOURCE-CONSTRAINED PROJECT SCHEDULING PROBLEM - A Fight against Nature.
DOI: 10.5220/0003644402860291
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 286-291
ISBN: 978-989-8425-83-6
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
we fight against CLT, we fight against nature. The
situation in the "neologist" fuzzy community is not
better, because they try to redefine somehow the
probability theory within the fuzzy approach without
using "forbidden" statistical terms. In this paper, we
will show that the nature is working totally
independently from our "magic" abstractions.
According to the robustness of CLT, the distribution
function of the completion time of real-size projects
remains nearly normal, which is a manager friendly,
natural and usable result. An abstraction and its
"natural" operators are unable to modify the order of
nature. When we want to add a practical scheduling
method to the project managers we have to destroy
the borders between the probabilistic and
possibilistic approaches and have to define a
"unified" approach to decrease the gap between
scientific beliefs and reality. In this paper we present
a new unified (probabilistic/possibilistic) model and
a conception of a heuristic connected to the unified
model for RCPSP with uncertain activity durations.
In Section 2 we present a unified theoretical model.
In Section 3 we describe the conception of the
uncertainty management according to the theoretical
model. The essence and viability of our unified
model will be illustrated by a fuzzy example in
Section 4. Finally, Section 5 draws conclusions from
this study.
2 THEORETICAL MODEL
In this section we describe the theoretical model for
RCPSP with uncertain activity durations. The
approach produces “robust” schedules which are
immune against uncertainties in the activity
durations. The optimality criterion is defined as a
linear combination (weighted sum) of resource-
feasible makespans connected to the key terms of
the applied uncertainty formulation. Theoretically
the optimal robust schedule searching process is
formulated as a multi-objective mixed integer linear
programming problem (MOMILP) where the
number of objectives corresponds to the number of
key terms (parameters) of uncertainty formulation.
In this paper, we replaced the MOMILP with a
MILP by scalarization. The resulting MILP can be
solved directly in the case of small-scale projects
within reasonable time. The proposed model is
based on the so-called “forbidden set” concept. The
output of the model is the set of the optimal conflict
repairing relations. Obviously, the solution of the
problem depends on the choice of the weights for the
objective functions.
In order to model uncertain activity durations in
projects, we consider the following resource-
constrained project-scheduling problem: A single
project consists of
N real activities
{}
Ni 2 1 ,...,, .
In this paper, without loss of generality, we
assume that each activity duration can be described
by three parameters:
{
}
221 iii
DDD ,, ,
{}
Ni 2 1 ,...,, ,
where triplet
{
}
221 iii
DDD ,, may define a triangular
membership function in the possibilistic approach,
or a density function from beta distribution in the
probabilistic approach. We have to note, that in the
probabilistic approach the triplet is estimated from a
sample using standard statistical tools, assuming that
the future can be described from the past, but in the
possibilistic approach it is only an abstraction which
describe the future according to knowledge of the
project managers.
The fuzzy community, under the spell of the
challenging but manageable nature of the
membership function (it is non-smooth composite of
linear segments) tries to recreate everything from the
beginning. For example, "normalization" is a "coded
message" that the triangle is not a density function,
and the horizontal line corresponding to "α-cut" is a
theoretically questionable replacement of the two
vertical lines, which define the confidence interval
in the probabilistic approach. Changing the position
of α we change our risk-taking habit, but, at the
same time, we omit/add duration segments with
totally different left/right tail probability (Figure 1).
Our opinion about the uncertainty management
in project scheduling is very simple: we have to
replace the triangular membership function with the
equivalent triangular density function, have to let the
CLP to work. Formalisms which in the uncertainty
dimension, try to redefine statistical terms without
statistical terms, are meaningless and misleading.
The activities are interrelated by precedence
constraints: Precedence constraints force an activity
not to be started before all its predecessors are
finished. These are given by network-relations
ji , where ji means that activity j cannot
start before activity
i is completed. Furthermore,
activity
(
)
1 0
+
=
=
Nii is defined to be the unique
dummy source (sink). Let
NR be the set of the
network relations.
FUZZIFICATION OF THE RESOURCE-CONSTRAINED PROJECT SCHEDULING PROBLEM - A Fight against
Nature
287
1 i
D
2 i
D
3 i
D
1 i
D
2 i
D
3 i
D
α
1
1 3
2
ii
DD
Possibilistic Approach
Probabilistic Approach
50
.
=
π
50
.
=
π
50.=
π
01.=
π
Figure 1: Possibilistic and probabilistic approaches.
Let
R
denote the number of renewable
resources required for carrying out the project. Each
resource
{}
Rr ,...,1 has a constant per period
availability
r
R . In order to be processed, each real
activity
{}
Ni ...,2, 1, requires 0
i r
R units of
resource
{}
R , ... 1,r over its duration.
A schedule is network-feasible if satisfies the
predecessor-successor relations:
jii
SDS + , for each
NRji
(1)
Let
denote the set of network-feasible
schedules. For a network feasible schedule
S ,
let
{
}
{}
TDStSiA
iiit
,...,1 t +<= , denote the
set of active (working) activities in period
t
and let
=
t
Ai
it
rU
r r
,
{}
Tt ,...,1 ,
{}
Rr ,...,1
(2)
be the amount of resource
r
used in period
t
, where
T is an upper bound of the resource-feasible
makespan.
A network-feasible schedule
S is resource-
feasible if satisfies the resource constraints:
rt
RU
r
,
{}
Tt ,...,1 ,
{}
Rr ,...,1
(3)
Let
denote the set of resource-feasible
schedules. The presented unified MILP formulation
is based on the forbidden set concept.
In MILP model the total number of zero-one
variables is
RR , and the formulation is based on
well-known "big-M" constraints. The presented
MILP model is a modified and simplified version of
the original forbidden set oriented model developed
by Alvarez-Valdés and Tamarit for the deterministic
case.
A forbidden activity set is identified such that:
(1) all activities in the set may be executed
concurrently, (2) the usage of some resource by
these activities exceeds the resource availability, and
(3) the set does not contain another forbidden set as
a proper subset. A resource conflict can be repaired
explicitly by inserting a network feasible precedence
relation between two forbidden set members, which
will guarantee that not all members of the forbidden
set can be executed concurrently. We note, that an
inserted explicit conflict repairing relation (as its
side effect) may be able to repair one or more other
conflicts implicitly, at the same time.
Let
=
=
N
i
i
DT
1
3
, which is an “extremely weak”
resource-feasible upper bound and fix the position of
the unique dummy sink in period
1+
T
. Naturally,
this “weak” upper bound can be replaced by any
“stronger” one.
Let
1ii
DD
=
,
{
}
Ni ...,2, 1,
, and let
F
denote
the number of forbidden sets and let
f
RR denote the
set of explicit repairing relations for forbidden set
f
F
f
F ,
{
}
Ff ,...,,21
according to the "optimistic"
durations and resource-feasible upper-bound
T
.
Let
{}
=
f
f
FfRRRR ,...,,21
(4)
denote the set of all the possible repairing relations.
In the forbidden set oriented model, a resource-
feasible schedule is represented by the set of the
inserted resource conflict repairing relations
(Alvarez-Valdés and Tamarit, 1993). According to
the implicit resource constraint handling, in this
model the resource-feasibility is not affected by the
feasible activity shifts (movements).
In the time oriented model, a resource-feasible
schedule is represented by the activity starting times.
In this model, according to the explicit resource
constraint handling, an activity movement may be
able to destroy the resource-feasibility.
It is very important to note, that after inserting an
appropriate conflict repairing set, the "immunised"
schedule will invariant to the duration change. In
other words, the schedule will be resource-feasible
on the set of the possible (and allowed) activity
durations because we immunised it according to the
optimistic (shorter) durations:
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
288
[]
31 iii
DDD ,= ,
{}
Ni ...,2, 1,
(5)
Let
pi
S denote the starting time of activity i ,
where
{}
1 2 1 0
+
Ni ,...,,, and
{}
321 ,,p . By
definition, in the optimistic, most likely, and
pessimistic schedules the durations are optimistic,
most likely, and pessimistic durations:
pii
DD =
,
{}
Ni 2 1 ,...,, ,
{}
321 ,,p
(6)
Defining the binary decision variables:
RRj i
j i
Y
ij
= ,
otherwise0
inserted if1
(7)
the following MILP model arises:
1
3
1
min
+
=
pN
p
p
SW
(8)
1
,
f
RRji
ij
Y
{}
1 Ff ,...,
(9)
(
)
()
ijip
j
i
jpipip
YDSSSDS +++ 1
RRji
,
{}
321 ,,p
(10)
jpipip
SDS + , NRji
,
{}
321 ,,p
(11)
The objective function (8) minimizes the linear
combination of the resource-constrained makespans,
where the weights characterize risk-taking habit of
the project manager (for example: "best pessimistic"
may be a good scheduling policy, when the project
manager is a risk-avoider).
Constraint set (9) assures the resource feasibility
(we have to repair each resource conflict explicitly
or implicitly, therefore from each conflict repairing
set we have to choose at least one element).
Constraint sets (10) take into consideration the
precedence relations between activities in the
function of the inserted repairing relations.
Constraint sets (11) take into consideration the
original precedence (network) relations between
activities.
In the "big-M" formulation
(
)
i
i
SS define the
earliest (latest) starting time of activity
i , in the
optimistic schedule according to upper-bound
T .
We have to note again, that the optimal solution
is a function of
p
W ,
{
}
321 ,,p weights. According
to the model construction in the optimal schedule
every possible activity movement is resource
feasible, and schedule is “robust” because it is
invariant to the variability of activity durations. In
other words, a non-critical activity movement (a
non-critical delay) or longer (but possible) activity
duration is unable to destroy the resource feasibility
of the schedule.
3 HEURISTIC ALGORITHM
In this section, describe the conception of a heuristic
algorithm connected to the presented theoretical
model, Without loss of generality, we assume that
we have a deterministic list scheduling algorithm
width forward-backward improvement (FBI) to
produce resource-feasible schedules in a arbitrary
metaheuristic frame. According to the essence of the
algorithm, we generate the resource-feasible
schedules by taking the selected activities one by
one in the given activity order and scheduling them
at the earliest (latest) feasible start time using the
optimistic activity durations. After that, using FBI
we try to improve the quality of the generated
schedule.
When the algorithm, in the forward-backward
list scheduling process, inserts a precedence relation
between an already scheduled activity and the
currently scheduled activity whenever they are
connected without lag, than we get a schedule
without "visible" resource-conflicts in which,
according to applied "thumb rule", the number of
"hidden" conflicts is drastically decreased.
The importance of the "thumb rule" may be
explained by the fact, that in this way we are able to
resolve resource conflicts, without explicit forbidden
set computation. After that, the algorithm is able (in
exactly one step) to repair all of the hidden
(invisible) conflicts, inserting always the “best”
conflict repairing relation for each forbidden set.
In this context “best” means a relation between
two forbidden set members for which the lag is
maximal. Naturally, the algorithm memorizes the
best schedule found so far by computing the
durations of the schedule according to the key point
durations. In the search process, according to the
"from optimistic to pessimistic" strategy, the
algorithm resolves the visible (hidden) resource
conflicts using the optimistic durations, after that
replaces the optimistic durations with the most
likely, and pessimistic ones. The algorithm exploits
the fact, that we can not destroy the resource-
feasibility, replacing the optimistic durations in a
conflict free optimistic solution with longer
durations.
After the "best conflict repairing combination"
searching phase, the makespan distribution function
is generated by simulation. In the simulation phase
we have to replace the membership functions with
FUZZIFICATION OF THE RESOURCE-CONSTRAINED PROJECT SCHEDULING PROBLEM - A Fight against
Nature
289
the appropriate density functions (for example: we
replace a triangular membership function with a
triangular density function and use a triangular
random number generator to get duration instances).
4 EXAMPLE
The algorithm of the proposed approach has been
programmed in Compaq Visual Fortran 6.5. To
solve the presented problem to optimality the
callable version of Cplex 12.2 was used. The
computational results were obtained by running the
algorithm on a 1.8 GHz Pentium IV IBM PC under
Microsoft Windows XP
®
operating system. The
conception of Section 3 was inserted to the "Sounds
of Silence" harmony search metaheuristic frame
(Csébfalvi et al., 2008).
The fuzzy example was borrowed from Bhaskar,
Pal, and Pal. The project is shown in Table 1 and
Figure 2. The project has only one renewable
resource type. The "weak" upper-bound is
397
=
T .
Table 1: A fuzzy RCPSP.
Activity Duration
Resource
requirement
1 {42, 50, 61} 8
2 {36, 40, 42} 17
3 {35, 50, 79} 12
4 {39, 50, 59} 3
5 {16, 25, 30} 13
6 {43, 51, 57} 17
7 {52, 58, 69} 16
Resource availability 30
According to the presented fuzzy RCPSP
algorithm, which is based on a "distance base
ranking of fuzzy numbers" method, the "good"
schedule obtained by the heuristic is:
{}
278 249 212 ,, .
1
2
3
4
5
6
7
30
R
1
50
100
150
176
50 100 150 176
Figure 2: A fuzzy RCPSP with optimistic durations.
Because the project is extremely small, we can
prove by explicit enumeration, that this result is
wrong. According to
397=T setting and using the
optimistic duration estimations, the problem has
only two forbidden sets (see Table 2). The implicit
enumeration tree is presented in Figure 3.
Table 2: Forbidden sets and repairing relations.
Forbidden sets
Explicit
repairs
Implicit
repairs
1 {2,6}
2
6
6
2
2
3
2
4
2 {2,3,4}
2
3
3
2
2
4
4
2
3
4
4
3
6
2
{ , , }
{208, , }
{208, 249, }
{208, 249, 308}
{208, , 308}
{208, 249, 308}
{ , 249, }
{208, 249, }
{208, 249, 308}
{ , 249, 288}
{212, 249, 288}
{ , , 288}
{212, , 288}
{212, 249, 288}
{ , 249, 288}
{212, 249, 288}
Figure 3: Explicit enumeration tree.
The problem has two non-dominated solutions:
{
}
308 249 208 ,, and
{
}
288 249 212 ,, , which illustrate
the fact, that a good optimistic schedule not
necessarily will be a good pessimistic one and vice
versa. The presented "good" solution from (Bhaskar
et al., 2011) is better then a non-dominated solution,
which is impossible.
When we apply the model of Section 2 to the
presented fuzzy problem with unit weights, we get
{
}
288 249 212 ,, as optimal solution within 0.05 sec.
In this case, the optimal resource conflict repairing
relations are:
62 and 24 . The optimistic
optimal solution is presented in Figure 4.
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
290
1
2
3
4
5
6
7
30
R
1
50
100
150
200
212
50 100 150 200 212
Figure 4: Optimal solution with optimistic durations.
The problem is really simple. The applied
harmony search metaheuristic reached the optimal
solution in the random repertoire uploading phase
setting the repertoire size to ten.
After the "best conflict repairing combination"
searching phase, the makespan distribution function
is generated by simulation. In the simulation phase
we replaced the membership functions with density
functions (in this case we replaced the triangular
membership functions with a triangular density
functions and used a triangular random number
generator to get duration instances). We have to
mention it, that simulation is a cheap operation, so
the sample size may be large enough. In the
presented example we set the sample size to ten
thousand. Using the Kolmogorov-Smirnov test, we
can not reject a null hypothesis that the sample
comes from a normal distribution with the following
parameters:
10.0 256.4, 0.158, ===
σ
μ
π
(12)
where
1580
.
=
π
is the probability of the largest
difference (in absolute value) between the observed
and theoretical distribution functions when the null
hypothesis is true with mean
4256.=
μ
and standard
deviation
010
.
=
σ
.
The histogram in Figure 5 reviles the fact, that
the nature knows nothing about the fuzzification and
does its best according to the CLP.
212 249 288256
Figure 5: Makespan estimation by simulation.
5 CONCLUSIONS
In this paper, a new unified theoretical model and a
concept of the corresponding heuristic approach to
solve RCPSP with uncertain activity durations were
presented. In the proposed heuristic approach, the
uncertainty management is invariant to the applied
heuristic frame; therefore it can be built into any
other heuristic developed to solve RCPSPs. The
essence and viability of our unified approach was
illustrated by a fuzzy example presented in the
recent fuzzy RCPSP literature. A fast and effective
metaheuristic algorithm for large problems is under
development and will be presented in a forthcoming
paper.
REFERENCES
Alvarez-Valdés, R., Tamarit, J. M., 1993. The project
scheduling polyhedron: Dimension, facets and lifting
theorems, Journal of Operational Research, 96, 204-
220.
Bhaskar, T., Pal, M. N., Pal, A. K., 2011. A heuristic
method for RCPSP with fuzzy activity times,
European Journal of Operational Research, 208, 57-
66.
Csébfalvi, G., Csébfalvi, A., Szendrői, E., 2008. A
harmony search metaheuristic for the resource-
constrained project scheduling problem and its multi-
mode version, In Proceedings of the Eleventh
International Workshop on Project Management and
Scheduling, Istanbul, 56–59.
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Nature
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