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- (^) Prof. Ph.D. D.Sc. Mirosław Dytczak, Ph.D. Grzegorz Ginda, Multi-Criteria Methods Application
Lab, Department of Management in Power Engineering, Faculty of Management, AGH University of Science and Technology.
MIROSŁAW DYTCZAK, GRZEGORZ GINDA*
CONSTRUCTION SCHEDULE OF WORKS
AS COMBINATORIAL OPTIMIZATION PROBLEM
HARMONOGRAMOWANIE PRZEDSIĘWZIĘCIA JAKO
ZAGADNIENIE KOMBINATORYCZNE
A b s t r a c t
Construction schedule optimization generally deals with the identification of the feasible sequence of activities and allocation of modes that provide the most efficient construction performance according to the assumed evaluation criteria. The specific technological order of activities results in the numerous feasible sequences of activities and availability of alternative modes results in the numerous mode combinations. Construction scheduling becomes, therefore, a difficult combinatorial problem that is usually underestimated by the planners. This is the main reason for obtaining schedules that result in construction implementation lasting too long and costing too much. It is, nevertheless, possible to identify the construction schedules that provide excellent results. A simple, simulation- based approach is presented in this paper. Its originality results from nature of applied model and a way the calculations are made. The effectiveness of the approach is illustrated by means of a sample analysis based on data provided by [1]. The approach also proved useful for solving scheduling problems in engineering practice.
Keywords: construction, schedule, optimization, decision, support, activity sequence, mode, allocation, combinatorial problem
S t r e s z c z e n i e
Optymalizacja harmonogramu przedsięwzięcia polega na doborze takiej dopuszczalnej kolejności wykonywa- nia prac oraz przydzieleniu takich sposobów wykonania operacjom, które zapewnią najlepsze możliwe, zgodnie z przyjętymi kryteriami oceny harmonogramu, wykonanie przedsięwzięcia. Z technologicznego porządku prac wynika zwykle duża, a często nawet astronomiczna liczba ich dopuszczalnych uporządkowań. Złożoność procesu harmonogramowania dodatkowo podwyższa dostępność alternatywnych sposobów wykonania prac. Harmono- gramowanie przedsięwzięć stanowi więc w istocie trudne, a przy tym niedoceniane przez planistów, zagadnienie kombinatoryczne. W rezultacie otrzymujemy harmonogramy odpowiadające nadmiernie czaso- i kosztochłonnej realizacji przedsięwzięcia. Przy odrobinie wysiłku można jednak uzyskać harmonogramy, które odpowiadają krót- kiej i taniej realizacji przedsięwzięcia. Temu celowi służy również autorskie narzędzie symulacyjne przedstawione w pracy. Stanowi ono oryginalne podejście zarówno w zakresie modelu, jak i sposobu wykonywania obliczeń, do których dane zaczerpnięto z pracy [1]. Wyniki pracy zostały także zastosowane w praktyce inżynierskiej.
Słowa kluczowe: przedsięwzięcie budowlane, harmonogram, optymalizacja, decyzja, wspomaganie, kolejność prac, sposób wykonania, przydział, zagadnienie kombinatoryczne
TECHNICAL TRANSACTIONS
CIVIL ENGINEERING
2-B/
CZASOPISMO TECHNICZNE
BUDOWNICTWO
1. Introduction
Construction projects deal with the execution of many different activities associated
with the various building work being carried out. Sound project implementation requires
the careful preparation. Construction schedules are applied in this regard. Construction
scheduling deals with three primary decisions related to the choice of an appropriate
feasible sequence of activities, the project start date and the allocation of available modes
of construction activities. A specific schedule for a construction project results from a set of
such decisions. The effects of these decisions are evaluated by means of schedule evaluation
criteria. Project makespan T and total cost C are usually applied in this regard.
Note, that the technological order of activities is applied in numerous feasible
sequences of activities, however, the availability of alternative modes for activities cause
such scheduling to be problematic. Construction scheduling can then become a severe
combinatorial optimization problem. The problem is hard to solve in acceptable time even in
the case of construction projects that consist of average or small numbers of activities. Due
to this, we are usually forced to rely solely on approximation schemes while scheduling real
construction projects. Note, that mainly heuristic and metaheuristic approaches are applied
in this regard [2]. The application of such approaches brings advantages in the case of the
complex projects. This is because efficient “evolutionary” mechanisms allows for feasible
solutions to the scheduling problem. This approach also deals with several drawbacks. At
first, they comprise a kind of a black box. This fact discourages the conscious planners
from using such approaches which risk losing control over the accuracy of analysis results.
Secondly, the successful application of this approach relies on conducting time-consuming
introductory numerical experiments. These experiments provide appropriate values vital to
those parameters which influence the performance of the approaches.
The simple, yet powerful approach that addresses the drawbacks of the more advanced
optimization methods, is presented in this paper. The approach is based on simple numerical
simulation experiments [3] and provides near Pareto-efficient schedule for a construction
project in terms of simultaneous minimization of make span and total cost.
2. Schedule optimization model
The following assumptions are made while structuring the model. A construction project
consist of m activities, denoted by o(1), o(2)...o( m ). The activities represent the consecutive
construction works. There are oi alternative modes available for the activity o( i ), where i = 1,
2... m. It is assumed that each activity is executed by means of a single selected mode only. Each
alternative mode requires the application of a specific resource – manpower, equipment and
building materials. Building materials are considered to be a non-renewable resource because
they undergo continuous exhaustion in the course of the construction process. Note, that a given
set of structural and material solutions is considered. The same building materials are then
applied while executing a given construction activity, regardless of the selected mode. It is also
assumed that the necessary building materials are always available when required.
Manpower and equipment are the renewable resources because they become available
again as soon as an activity ends. It is assumed that the renewable resources are available
{ } k ∈ 1, 2∀ n Tk^ ≥ Tk^ −^1 ,^ (4)
C G xij ij j
o
i
m (^) i
( , x κ , ) =
= =
1 1
( ) (^) ( )
( ) ( , ) k
k G ij^ l k i j
x L ∈Ξ ∈ζ
The goal function presented by Eqn. (1) uses the simultaneous minimization of T and C.
Note, that to make T and C commensurable, they are divided by the reference values (^) T and
C , respectively. Normalized weights^ w 1 and^ w 2 express the relative importance of^ T^ and^ C.
Note that the goal function provides the appropriate project structure G out of the set of all
feasible project structures G. The corresponding allocation of available modes to activities
is also identified. The selected modes are represented by the binary decision variables xij ,
where: i = 1, 2… m and j = 1, 2… oi. The variables comprise the matrix of decision values x.
The makespan is expressed by means of the difference between time of terminal project
event occurrence Tn and the assumed time of starting project event occurrence T 0. It depends
on the assumed project structure G , the selected modes x and the regular duration of
activities corresponding with the available modes. Note, that the application of the j -th mode
applicable in the case of the activity o( i ) results in the regular activity duration τ ij. Regular
activity durations corresponding with all modes comprise the matrix τ. The total project
cost C depends on the applied project structure G , and the chosen modes x. Regular cost for
available modes is denoted by κij. Let us observe that matrices x , τ and κ have the same sizes.
Eqn. (2) assures that a single application mode is applied in the case of each activity and
Eqn. (3) enforces the natural occurrence of the consecutive project events: Tk , where k = 1,
2… n. Time of the occurrence of project events depends on the assumed project structure G
selected modes x , the regular duration of the activities κ , and the assumed time of occurrence
of the starting project event T 0. Note, that (^) k
expresses the set of activities terminated by the
k-th project event, where: k = 1, 2... n ., and t ( ) i s is the time of the occurrence of the starting
event for the activity o( i ).
The assumed sequence of the activities G is enforced by Eqn. (4). The following sum:
1
o i
ij ij j
x
∑ κdenotes the actual cost of the execution of the activity o( i ), where:^ i^ = 1, 2... m. The
total project cost is given in Eqn. (5). The actual duration of the activity o( i ) is equal to: 1
.
o i ij ij j
x
The Eqn. (6) deals with the possible competition for renewable resources between
different activities. It is assumed that the k -th conflict deals with a specific TMS. The number
of available items of that TMS is denoted by Ll ( k ). Note, that the number of possible conflicts
is denoted by Ξ and depends on the assumed project structure G. The set of modes invoλved
in the κ-th conflict, where: k = 1, 2... Ξ is denoted by ζ( k ). The involved modes are described
by the pairs of indices ( i , j ), where i and j express the number for the activity and the mode,
respectively ( i = 1, 2... m ; j = 1, 2... oi ).
Note, that the considered scheduling problem is a kind of the Multi-mode Resource
Constrained Project Scheduling Problem (RCPSP) [5] and is formulated as the mixed integer
linear programme (MILP). Both the exact and the approximate [2] approaches can be applied
to solve it. For example, the following exact approaches are applicable in this regard: Mixed
Integer programming, Dynamic Programming Constraint Programming, Branch and Bound,
and Benders Decomposition. The approaches mentioned provide exact optimization results but
become less efficient in the case of the construction projects consisting of the numerous activities.
3. The applied approach
The applied approach addresses the drawbacks of the exact and approximate methods. It
is based on the decomposition of the original problem into 2 levels:
- The upper level deals which provides the appropriate feasible project structure G*^ and
the corresponding selection of modes x *^ – note, that they define the near Pareto-efficient schedule.
- The lower level deals with the (lower level) tasks – the MILP problems, obtained for the
representative project structures G^ ∈ G^ and the following goal function which replaces the goal function given in Eqn. (1):
min
( , , , ) ,^ , x
x τ x τ F w
T G T T T
w
C G
C
= n −^ + (^ ) 1
0 0 2.^ (7)
Note, that the solution of the upper level problem is identical with the solution of the
original problem given in Eqns. (1–6). The simple ranking of the locally Pareto-efficient
solutions corresponding with the lower level tasks is enough in this regard. The ranking
corresponds with the decreasing values of the goal function given in Eqn. (7).
The simulations are applied to generate the representative project structures G. The
redundant representation of all feasible structures for a project is applied in this regard [3].
The representation is expressed by the acyclic, asymmetric and joined digraph G V E ( , ). It
consists of the vertices mapping all possible project events from the starting event labelled 0
to the latest possible terminating project event labelled m. The digraph arcs correspond with
all alternative locations of the activities in feasible project structures. Note, that the choice of
a single arc for each activity is enough to generate a feasible project structure.
Two different proposed approaches are finally applied [5]. They differ in the methods applied
to solve a lower level problem. The first detailed approach is called MC-LP and combines random
generation of representative project structures with linear programming to solve the lower level
tasks. The second detailed approach, called MC-MC, applies the random generation for both the
representative project structures G and the selection of modes x. The both approaches complement
each other. MC-LP is capable of providing more accurate results, while MC-MC is capable of
performing better in the case of projects with a considerable number of activities.
Note, that the definition of the number of the generated feasible structures N ' is required
in order to apply the MC-LP and MC-MC approaches. The number of generated allocations
of modes to activities ( N ") should be defined to make the MC-MC application possible. The
simple introductory simulation experiments are utilised to provide the required values for N '
and N ". The introductory experiment for estimating N ' deals with a number of lower level
tasks solved by the means of the MC-LP approach while the introductory experiment for the
MC-MC deals with a lower level task obtained for a single project structure. The following
Formulae are applied in this regard:
A project structure corresponding with the obtained near Pareto-efficient schedule is
presented in Fig. 2. The schedule corresponds with the goal function value F = 0.802 (+0.9%
compared with the Pareto-efficient schedule), T = 199 days (+2.4%) and C = 13.228,000 PLN.
The results are obtained in less than 33 seconds of mediocre CPU time.
N ‘ N ”
Fig. 3. Introductory analysis results for a sample project
5. Conclusions
The results obtained for the sample construction project confirm usability of the applied
approach for the rapid identification of near Pareto-efficient schedules. The schedules provided
by the approach are at most slightly worse than the Pareto-efficient schedule. Application of linear
programming techniques and Monte Carlo simulations makes the approach reliable both in the
case of projects consisting of smaller number of activities and in the case of the more complex
projects with a considerable number of activities. MC-MC proves also useful while solving non-
linear scheduling problems dealing with the influence of the surrounding environment.
R e f e r e n c e s
[1] Wojtkiewicz T., Wielokryterialna ocena harmonogramu w budownictwie , Politechnika
Opolska, Opole 2010 [Ph.D. dissertation, the supervisor: M. Dytczak].
[2] Zhou J., Love P.E.D., Wang X., Teo K.L., Irani Z., A review of methods and algorithms
for optimizing construction scheduling , Journal of the Operational Research Society, Vol. 64 (8), 2013, 1091-1105.
[3] Dytczak M., Ginda G., Lower-Level Decision Task Solution While Optimising
a Construction Project Schedule , Procedia Engineering, Vol. 57, 2013, 254-263.
[4] Węglarz J., Józefowska J., Mika M., Waligóra G., Project scheduling with finite or infinite
number of activity processing modes – a survey , European Journal of Operational Research, Vol. 208, 2011, 177-205.
[5] Dytczak M., Generation of project schedule structure along with solving low level
optimisation problems using different methods , Zeszyty Naukowe WSB we Wrocławiu 34 (2), 2013, 115-129.
