Expert answer:Resource Leveling Paraphrasing and Summarizing

Expert answer:summaries the methodology, result and issues with the proposed methods from the attached papers talking about resource leveling in construction projects.I need around 300 words about each paper (total 1200 for 5 papers).
1_s2.0_s0926580512001677_main.pdf

1_s2.0_s0926580512000787_main.pdf

a_novel_resource_leveling_approach_for_construction_project_based_on_differential_evolution___2014___tran_et_al.pdf

a_new_resource_leveling_model_for_the_construction_project_management_of_building_structures_the_genetic_algorithm___2014___prof__yesug.pdf

yang_nature_book_part.pdf

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Automation in Construction 29 (2013) 161–172
Contents lists available at SciVerse ScienceDirect
Automation in Construction
journal homepage: www.elsevier.com/locate/autcon
The Resource Leveling Problem with multiple resources using an adaptive
genetic algorithm
Jose Luis Ponz-Tienda a, Víctor Yepes b,⁎, Eugenio Pellicer c, Joaquin Moreno-Flores a
a
b
c
School of Building Eng., Universitat Politècnica de València, Camino de Vera sn, 46022, Valencia, Spain
ICITECH, Universitat Politècnica de València, Camino de Vera sn, 46022, Valencia, Spain
School of Civil Eng., Universitat Politècnica de València, Camino de Vera sn, 46022, Valencia, Spain
a r t i c l e
i n f o
Article history:
Accepted 3 October 2012
Available online 29 October 2012
Keywords:
Project scheduling
Resource leveling
Genetic algorithms
Benchmarking
a b s t r a c t
Resource management ensures that a project is completed on time and at cost, and that its quality is as
previously defined; nevertheless, resources are scarce and their use in the activities of the project leads to
conflicts in the schedule. Resource leveling problems consider how to make the resource consumption as
efficient as possible. This paper presents an Adaptive Genetic Algorithm for the Resource Leveling Problem,
and its novelty lies in using the Weibull distribution to establish an estimation of the global optimum as a termination condition. The extension of the project deadline with a penalty is allowed, avoiding the increase in
the project criticality. The algorithm is tested with the Project Scheduling Problem Library PSPLIB. The proposed algorithm is implemented using VBA for Excel 2010 to provide a flexible and powerful decision support
system that enables practitioners to choose between different feasible solutions to a problem in realistic
environments.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Project management is the process of the coordination and integration of activities in an efficient and effective manner using limited
resources. It consists of linking resources to their respective deliverables and assembling them into the whole project [1]. Resource
management is an intrinsic element of project management [2–4];
resource management ensures that the project is completed on
time and at cost and that the quality is as previously defined [5–7].
This is even more necessary for project-based companies such as
contractors [3,8,9]. In fact, project scheduling problems are one of
the most important problems that practitioners deal with in scheduling, especially when they need to achieve the most efficient resource
consumption without increasing the prescribed makespan of the
project.
However, because resources are scarce, the use of resources in the
activities of the project leads to conflicts in the schedule [10]. Project
scheduling problems comprise not only resource-constrained problems but also Resource Leveling Problems, among others [11]. These
two kinds of problem consider resource consumption in two different
ways: in the former it is seen as a constraint, and in the latter the
problem is to make it as efficient as possible. Even though these two
⁎ Corresponding author.
E-mail addresses: jopontie@csa.upv.es (J.L. Ponz-Tienda), vyepesp@cst.upv.es
(V. Yepes), pellicer@upv.es (E. Pellicer), jmflores@mat.upv.es (J. Moreno-Flores).
0926-5805/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.autcon.2012.10.003
approaches may seem similar, they are conceptually different. Both
have been widely studied by researchers and applied by practitioners,
although these two groups are unaware of the differences between
the approaches and the serious limitations imposed by the heuristics
used in the commercial software.
These two problems are defined as non-deterministic polynomialtime hard (NP-hard) problems [12]. The first approach is a regular problem known as the Resource Constrained Project Scheduling Problem; its
objective is to reduce the makespan without exceeding the constraints
of resource availability [13,12]. The second, known as the Resource
Leveling Problem (from now on, RLP) is a non-regular problem; its objective is to achieve the most efficient resource consumption without
increasing the prescribed makespan of the project [14,12]. The two
problems can be combined together as a multi-objective optimization problem, but there is always one main objective (usually the
makespan); the other objective (usually the efficient resource consumption) is secondary.
Nevertheless, conventional analytical and heuristic methods are neither flexible nor productive when solving the RLP [15]. Some reasons for
this inefficiency are, on the one hand, that exact procedures simplify the
real problems so are not useful at offering optimal solutions with acceptable computational effort [16] and, on the other hand, that heuristics offer solutions which are far from optimal, so that it is necessary
to apply metaheuristic algorithms to complex and realistic projects
[17]. Recently, important approaches have been made by researchers
to improve the efficiency of resource consumption, proposing different
heuristics which are applicable to small projects; simple examples
try to show the merits of a particular algorithm, without establishing
162
J.L. Ponz-Tienda et al. / Automation in Construction 29 (2013) 161–172
clear criteria for a performance comparison between the different
algorithms [18].
Following this line of work, Liao et al. [11] proposed some ideas to
advance the RLP in realistic environments; these authors made several proposals for the development or the improvement of the RLP. Regarding resource allocation, these authors proposed the use of a
decision support system to assist project managers, as well as the development of benchmarking tests for performance assessment and
comparison [11]. Concerning resource leveling, they suggested the
use of multiple resources allowing the extension of the project deadline with a penalty [11]. We take these proposals as challenges to be
overcome in this paper, contributing a little to the corpus of knowledge in this field.
Therefore, in this paper we present an Adaptive Genetic Algorithm
(AGA) for the RLP with multiple resources allowing the extension of
the project deadline with a penalty; for this purpose, we use the Weibull
distribution as a termination condition, establishing an estimation of
the global optimum. The proposed algorithm is tested with the standard
“project scheduling problem library” (PSPLIB) [18], presenting a complete set of benchmarking tests. A decision support system is also
used in order to implement this algorithm. Without loss of generality,
we consider the classical resource leveling objective function: the
total squared utilization cost for a given schedule.
The remainder of this paper is organized as follows. Section 2 provides the classification and formulation of the RLP. Section 3 details
the different solving procedures: exact, heuristic, and metaheuristic algorithms with the new use of the Weibull distribution as a termination
condition. Section 4 describes the algorithm proposed for the RLP with
multiple resources. Computational results and the benchmarking test
are explained in Section 5. Finally, conclusions are drawn.
2. Classification and formulation of the Resource Leveling Problem
The general formulation of the RLP requires us to consider the following elements:
1. The set of activities, N:
N ¼ fj1 ; j2 ; ⋯; jn g
ð1Þ
6. The set of costs, C:
C ¼ fc1 ; c2 ; ⋯; ck g:
7. The set SS: to distribute the performance of the activities along the
elements of the set T one needs to allocate a starting time for each
activity, given by the ordered set, SS:
SS ¼ fSS1 ; SS2 ; ⋯; SSn g:
ð7Þ
SSi, 1 ≤ i ≤ n, is the starting time of the activity ji. T can be considered as the starting time of a finish dummy activity SSfinish, and
then SS becomes:
n
o
SS ¼ SS1 ; SS2 ; ⋯; SSn ; SSfinish :
ð8Þ
Obviously, the schedule SS is not unique; on the contrary, there are
a large number of different possibilities, according to the logic and
restrictions of the project to be performed. Each of these schedules
has significant differences in the efficiency of resource consumption, and this is the reason for finding the values of SS which optimize this efficiency.
8. The functions ri(S,t), 1 ≤i ≤ k: given a schedule SS, the function ri(S,t)
is defined as the consumption of the resource ri in the period of time
t, belonging to the set T, in such a way that the consumption of the
resource ri throughout the project is given by:

ui1 ¼ r i ðS; t 1 Þ; ui2 ¼ r i ðS; t 2 Þ; ⋯; uip ¼ r i S; t p :
ð9Þ
9. The function f: Given a schedule SS, the efficiency of resource consumption depends on the layout of its use. Therefore, it becomes
fundamental to establish an optimal criterion for the distribution
of the resources. This is the role we want f to play in the development of the problem. Hence, the function f will be different for
each optimization criterion to be considered.
Once we have the elements that compose the problem, a general
formulation could be:
Minimize
n being the total number of activities.
2. The set of durations, D:
ð6Þ
k
X
ci f ½r i ðS; t Þ
ð10Þ
i¼1
subject to:
D ¼ fd1 ; d2 ; ⋯; dn g
ð2Þ
where di, 1 ≤ i ≤ n is the assigned duration for each activity.
3. The set of periods of time in which these activities have to be distributed:
n
o
T ¼ t 1 ; t 2 ; ⋯; t p
ð3Þ
tp being the deadline of the project, from now on denoted T .
4. The set of resources, R:
R ¼ fr 1 ; r 2 ; ⋯; r k g
ð4Þ
SSfinish ≤T
ð11Þ
SSi þ di þ γ ij ≤SSj ; for all i which are successors to j
ð12Þ
uij ≤aij
ð13Þ
where γij is the lead/lag between i and j.
Having done this, the choice of the function f, which defines the
criterion for the optimization of the resource consumption, provides
different ways of solving the problem. In the case of the RLP, the optimization criterion focuses on getting the resource consumption as
level as possible. Consequently, a suitable choice of f could be:
f ½ri ðS; t Þ ¼
k being the total number of resources.
5. The set of availabilities of the resources, A:
T
X
ðuit −ait Þ2
:
T
t¼1
ð14Þ
And Eq. (10) turns into:
A ¼ fait ; 1≤i≤k; 1≤t≤pg
where ait is the availability of the resource ri in the period t.
ð5Þ
k
k X
T̄¯
X
X
ðu −ait Þ2
ci ⋅f ½ri ðS; t Þ ¼
ci it
:
Minimize
T̄¯
i¼1
i¼1 t¼1
ð15Þ
J.L. Ponz-Tienda et al. / Automation in Construction 29 (2013) 161–172
Next, we can simplify the problem by taking into account the
following:
1. ci = 1, for all 1 ≤ i ≤ k
2. ait = a, for all 1 ≤ i ≤ k and for all 1 ≤ t ≤ p
And then Eq. (15) becomes:
2
k X
T̄¯
X
ðuit −aÞ
Minimize
:
T̄¯
i¼1 t¼1
ð16Þ
As the minimum depends neither on how far the variable is
shifted nor on the constant values, we can take a = 0 and T ¼ 1:
Minimize
k X
T̄¯
X
2
urt :
ð17Þ
163
Eq. (10), into a linear programming formulation. Other linear programming formulations have to be used in order to be able to specify
the resource constraints in a correct and solvable form.
The most efficient formulations are based on integer and binary
programming, and can be of two kinds, depending on whether the
decision variables establish the period of execution or the finishing
time of the activities.
The first formulation for the Resource Leveling Problem in the
multi-mode case is based on a set of binary decision variables xjst
[22] that establish the period in which the activities are finished:

if job j is finished in mode s at the end of period t
;
otherwise
h
i
for j∈J; s∈Mj ; and t∈ ESj þ dj ; ⋯; LF j :
xjst ¼
1;
0;
ð19Þ
r¼1 t¼1
The objective function expressed in Eq. (17) is known as a minimum squares optimization, and was introduced by Burgess and
Killebrew [19] in a heuristic algorithm in which the near-optimality
is determined by the schedule with the minimum total sum of the
squares of resource consumption for each period, as illustrated in
Fig. 1. Other measures for the objective function are the Minimum
Moment (MOM) proposed by Harris [20], and more recently the
entropy-maximization proposed by Christodoulou et al. [21], using
the maximality and sub-additivity properties of the entropy function.
To compare the leveling effectiveness between different projects,
we use the Resource Improvement Coefficient (RIC) developed by
Robert Harris [20], a measure that is independent of the total resource
demand and is given by Eq. (18). The RIC relates the variation of a
selected resource histogram to an ideal resource histogram which is
a rectangle-shaped resource histogram (the ideal leveled schedule
corresponds to a RIC value of one):
T
X
T ⋅ ut 2
RIC ¼
t¼1
T
X
ut
The variables xjst can only be executed in one mode, and are defined over the interval between the earliest and latest finishing
times (the delimiting periods) of the activities of the project. These
limits are established using the traditional forward and backward
pass calculations for the unconstrained problem.
The vector SS of Starting Scheduled period for each activity is defined
by:
SSj ¼
Pj
X
0
@
LF j
X
1
t⋅xjst −djs A:
The objective function to minimize the minimum total sum of the
squares of resource consumption for each period is:
Minimize
0
12
Pj
LF j
R X
T̄¯
X
X
X
ck ⋅@ ∑
urst ⋅
xjst A :
j∈Eðt Þ s¼1
k¼1 t¼1
!2 :
ð18Þ
A different kind of non-regular objective function for Eq. (14) can
be considered if the objective function to be optimized represents the
Net Present Value Problem. In this case, the objective function represents the net present value of the project, which is to be maximized,
and this is used in practice when expensive resources have to be
purchased.
ð21Þ
q¼EF j
Subject to:
P j LF j
X
X
t¼1
ð20Þ
t¼EF j
s¼1
xjst ¼ 1
ð22Þ
for j∈J
s¼1 t¼EF j
Pj
LF j
X
X
P j LF j

X
X
t⋅xist ≥0 for j∈J and i∈P j
t−djs ⋅xjst −γij −
s¼1 t¼EF j
ð23Þ
s¼1 t¼EF j
h
i
h
i
xjst ∈f0; 1g for j∈J; s∈ 1; ⋯; P j and t∈ ESj þ dj ; ⋯; LF j :
ð24Þ
3. Solving procedures
The previous conceptual linear programming cannot be solved directly, because there is no easy way to translate the set S, used in
Fig. 1. Initial
The objective function modeled in Eq. (21) minimizes the total
sum of the squares of resource consumption for each period.
Eq. (22) specifies that only one mode and one completion time are
T
T
X
X
ukt 2 ¼ 10:669; leveled
ukt 2 ¼ 6:477.
t¼1
t¼1
164
J.L. Ponz-Tienda et al. / Automation in Construction 29 (2013) 161–172
allowed for every activity. The precedence constraints are given in
Eq. (23). Finally Eq. (24) specifies that the decision variables are binary.
Another possible formulation for the RLP is also based on binary
programming, but the decision variables xjst establish the period in
which the activities are executed [23]:

if job j is processed in mode s in period t
;
otherwise
h
i
for j∈J; s∈Mj ; and t∈ ESj þ 1; ⋯; LF j :
xjst ¼
1;
0;
ð25Þ
The vector SS of Starting Scheduled period for each activity is defined by:
SSj ¼
Pj
X
0
1
LF j

X
@
t⋅xjst ⋅ xjst−1 −xjst A:
s¼1
ð26Þ
ESj
The objective function will be:
0
12
Pj
R X
T
X
X
@
Minimize
ck ⋅ ∑
urst ⋅xjst A
j∈Eðt Þ s¼1
k¼1 t¼1
ð27Þ
subject to:
Pj
LF j
X
X
xjst ¼ dj
ð28Þ
for j∈J
s¼1 t¼ESj þ1
t

X
djs ⋅ xjst −xj;tþ1 −
xjsq ≤0
h
i
for j∈J; s∈Mj ; and t∈ ESj þ 1; ⋯; LF j −1 ð29Þ
q¼ESj þ1
di ⋅xjst −γij −
t−1
X
xist ≤0
h
i
for j∈J; i∈P j ; s∈Mj ; and t∈ ESj þ 1; ⋯; LF i ð30Þ
q¼ESj þ1
h
i
h
i
xjst ∈f0; 1g for j∈J; s∈ 1; ⋯; P j and t∈ ESj þ 1; ⋯; LF j :
ð31Þ
The objective function in Eq. (27) minimizes the total sum of the
squares of resource consumption for each period. Eq. (28) ensures
that each activity is executed for dj time units. The execution of the
activities without pre-emption is modeled in Eq. (29). The precedence constraints are given in Eq. (30). Finally, Eq. (31) specifies
that the decision variables are binary. Easa [24] proposed a different
binary integer formulation, preserving the precedence constraints
by limiting the shifting of the activities to the free float, which must
be equal to or greater than zero.
The Resource Leveling Problem is NP-Hard even if only one resource
is considered [24,12]. The time complexity function is of the order of
O(q n) with q being a positive constant. The universe of schedules for
an instance is:

J Mj
∏ ∏ Ht js þ T −mkp þ 1
ð32Þ
j¼1 s¼1
with Htjs being the total float of the activity j processed in mode s, T the
prescribed makespan,mkp the makespan for the resource unconstrained
problem (RUPSP or resource relaxation of the RCPSP) and Mj the execution modes of activity j.
If the prescribed makespan is established to be equal to the
makespan for the RUPSP and only one execution mode is considered,
Eq. (32) can be simplified to:

J
∏ Ht js þ 1 :
j¼1
ð33Þ
3.1. Exact algorithms
Exact algorithms based upon implicit enumeration, integer programming, and dynamic programming techniques, have been proposed
to solve the RLP. Easa [24] used a mixed binary-integer programming
technique that guarantees the optimum leveling. Exhaustive enumeration procedures were presented by Ahuja [25]. Bandelloni et al. [26]
developed an optimal technique based on dynamic programming.
Recently, Rieck et al. [27] proposed a new mixed-integer linear model
formulation and domain-reducing pre-processing techniques for the
RLP based on smart discrete-time formulations.
The branch-and-bound technique [28] is probably the most widely
used exact solution technique for solving project scheduling problems,
as it is the only technique which allows for the generation of optimal solutions with acceptable computational effort. Neumann and Zimmermann [29] describe a branch-and-bound procedure that reduces the
set of all feasible solutions by successively scheduling activities for approximately solving the problem. Recently, a lower bound improved
method (Maximum Allowable Daily Resources Method) to branch the
nodes has been developed by Mutlu [30], which may form a basis for
the performance evaluation of heuristic and metaheuristic procedures
for the RLP. Recently, Gather et al. [31] presented a new enumeration
scheme embedded into a branch-and-bound framework using a constructive lower bound as well as pre-processing techniques and Hariga
and El-Sayegh [32] a new mixed integer binary linear optimization
model that allow activity splitting and minimizes its associated costs.
The RLP as an NP-Hard problem has a phenomenon of “combinatorial
explosion” [15], especially for large-scale projects. For this reason, exact
algorithms are only efficient for small projects.
3.2. Heuristics algorithms
To avoid the explosion problem, heuristic rules are mostly used to
solve the RLP. These are simple rules or sets of rules which aim to obtain
a “good” solution (locally optimal) for a difficult problem but do not
guarantee the best solution (globally optimal). Heuristic algorithms
can be of two kinds, construction and improvement procedures. Construction procedures are used to establish a feasible solution to the
problem, and the other procedures are used to improve it.
The first heuristic procedure for the RLP was proposed by Burgess
and Killebrew [19], establishing the minimum squares as the performance measure. The Burgess–Killebrew algorithm is an improvement
procedure that readjusts the starting time of each activity and reduces
the variability to a near optimum (local optimum). The first step of
the algorithm con …
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