Project cost and project duration are main factors in construction management. In real-life projects, both the trade-off between the project cost and the Project completion time, and the uncertainty of the environment are considerable aspects for decision-makers. Moreover, in some projects, activity durations show their complexity with time-dependence as well as randomness. Time cost trade off problem have been solved by using vast of methods such as genetic algorithms, particle swarm optimization. Most of these studies considered deterministic project values and includes CPM calculations for fitness value calculation. However most of construction projects are stochastic processes. In this paper, a stochastic time–cost trade-off problem is introduced. This study takes into account that activity duration and cost are uncertain variables. this study differs from other studies that for each fitness value calculation. The proposed model is dealt with an intelligent algorithm combining stochastic simulation and Harmony search, where stochastic simulation technique is employed to estimate random functions and Harmony search is designed to search optimal schedules under different decision-making criteria. Beside this Monte Carlo simulation is made for fitness value calculation in order to make more realistic optimization. Finally, some numerical experiments are given to illustrate the algorithm effectiveness.

1. 1
Time Cost Trade off Optimization Using Harmony Search and
Monte-Carlo Method
Mohammad Lemar ZALMAİ
1
, Osman Hürol Türkakın 2
Ekrem MANİSALI3
1
Graduate StudentDepartment of Civil Engineering, Istanbul University, Istanbul, Turkey,
lemar_zalmai07@hotmail.com
2
Research Assistant Department of Civil Engineering, Istanbul University, Istanbul, Turkey,
turkakin@istanbul.edu.tr
3
Professor Department of Civil Engineering, Istanbul University, Istanbul, Turkey, ekmanisa@istanbul.edu.tr
Abstract
Project cost and project duration are main factors in construction management. In real-life projects, both the
trade-off between the project cost and the Project completion time, and the uncertainty of the environment are
considerable aspects for decision-makers. Moreover, in some projects, activity durations show their complexity
with time-dependence as well as randomness. Time cost trade off problem have been solved by using vast of
methods such as genetic algorithms, particle swarm optimization. Most of these studies considered deterministic
project values and includes CPM calculations for fitness value calculation. However most of construction
projects are stochastic processes. In this paper, a stochastic time–cost trade-off problem is introduced. This study
takes into account that activity duration and cost are uncertain variables. this study differs from other studies
that for each fitness value calculation. The proposed model is dealt with an intelligent algorithm combining
stochastic simulation and Harmony search, where stochastic simulation technique is employed to estimate
random functions and Harmony search is designed to search optimal schedules under different decision-making
criteria. Beside this Monte Carlo simulation is made for fitness value calculation in order to make more realistic
optimization. Finally, some numerical experiments are given to illustrate the algorithm effectiveness.
Keywords: Construction Project, Monte carlo simulation, Harmoni search method, time cost trade off problem ,
Critical path method
1 Introduction
Completing a project with minimal time as well as minimal cost is a critical factor for scheduling a project.
However, because completion speed tends to be correlated with cost (e.g., usually, time can be saved if more
workers are hired), the relationship between time and cost is called time cost trade off curve.
The Stochastic time-cost trade off problems refers to durations and total costs of the activities are stochastically
related each other instead of deterministically. For real-life project decision-makers, the analysis of the time–cost
trade-off is one of the most important aspects of project scheduling and control.
In early attemps time cost problem has been treated as deterministic approach such as linear programming
(Elmaghraby 1993) and linear programming and integer programming (LP/IP) hybrid (Liu et al. 1995).
Generally in literature time cost trade off problem solved as an deterministic problems and beside that many
uncertain factors are ignored.

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The project completion time may be variational due to many external factors, such as the change of weather, the
increase of productivity level, etc. Since Freeman introduced the probability theory into the project scheduling
problem (Freeman 1966), many authors have taken into account the nondeterministic factors for characterizing
the uncertainty in real projects. The reader may refer to [A. Charnes (1962), ] D. Golenko-Ginzburg (1997), H.
Ke, B. Liu (2005)] to see different types of project scheduling problem with stochastic activity duration times. In
1985, Wollmer [W.J. Gutjahr, C. Strauss (1985)] discussed a stochastic version of the deterministic linear time–
cost trade-off problem, in which some discrete random variables were used to depict the uncertainty in the
problem. In 2000, Gutjahr et al. [W.J. Gutjahr (2000)] designed a modified stochastic branch-and-bound
approach and applied it to a specific stochastic discrete time–cost trade-off problem. Ke et al. [H. Ke, W. Ma, Y.
Ni (2009)] built two models for stochastic time–cost trade-off problem with the philosophies of chance-
constrained programming and dependent-chance programming.
Traditionally, time-cost trade-off problem (TCTP) has been addressed by mathematical programming models
such as linear programming (Kelley 1961), dynamic programming (Elmaghraby 1993) and linear programming
and integer programming (LP/IP) hybrid (Liu et al. 1995). However, the linear programming model requires
specific assumptions for functions involved (Yang 2007) and the dynamic programming model requires
numerous trials (Xiong and Kuang 2008).
The paper is organized as follows: In the next section we briefly define the stocastic time-cost trade-off
problem, then relevant researchs belong to it and finally the new approach that combines simulation with
Harmony Search (HS).
2 Existing Techniques
Existing techniques for construction time-cost trade-off problems can be categorized into three areas: heuristics,
mathematical programming, and simulation. The following sections briefly describe the strengths and
weaknesses of the existing techniques.
2.1 Heuristic Methods
Heuristic methods are based on rules of thumb that generally lack mathematical rigor. They provide good
solutions but do not guarantee optimality. Examples include Fondahl’s method (1961), Prager’s structural model
(1963), Siemens’s effective cost slope model (1971), and Moselhi’s structural stiffness method (1993). In
resource-constrained scheduling, intensive research has been conducted under the consideration of renewable
resource constraints in terms of minimizing project duration. Heuristic methods have enjoyed the majority of the
research effort because they can find good solutions with far less computational effort than other methods.
Heuristics can be classified into two types: (1) Serial heuristics, in which processes are first prioritized and retain
their values throughout the scheduling procedure; and (2) parallel heuristics, in which process priorities are
updated each time a process is scheduled. Examples are Bell and Han (1991), Kahattab and Choobineh (1991),
and Boctor (1993). These heuristics provide good solutions; however, the performance is problem-dependent,
and good solutions are not guaranteed.
2.2 Mathematical Programming Models
Mathematical programming methods convert time-cost trade-off problems to mathematical models and utilize
linear programming, integer programming, or dynamic programming to solve them. Kelly (1961) formulated
time-cost trade-off problems by assuming linear time-cost relationships within processes. Other approaches, such
as those by Hendrickson and Au (1989) and Pagnoni (1990), also use linear programming to solve the time-cost
trade-off problem. Linear programming approaches are suitable for problems with linear time-cost
relationships but fail to solve those with discrete time-cost relationships. Meyer and Shaffer (1963) and Patterson
and Huber (1974) solved time-cost problems including both linear and discrete relationships by using mixed
integer programming. However, integer programming requires a prohibitive amount of computational effort once
the number of options to complete an activity becomes too large or the network becomes too complex. Burns et
al. (1996) took a hybrid approach that used (1) linear programming to find a lower bound of the
trade-off curve; and (2) integer programming to find the exact solution for any desired duration. Robinson
(1975), Elmagraby (1993), and De et al. (1995) used dynamic programming to solve time-cost trade-off
problems for a special class of networks that can be decomposed to pure series or parallel sub networks.

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2.3 Simulation
Finding the optimal solutions for construction time-cost optimization in a stochastic network has proven to be a
difficult problem to solve. For example, if the durations of the options were random variables (continuous or
discrete), the evaluation process would involve excessive numerical integration, which makes it practically
impossible. Simulation techniques have been used to enhance the study of stochastic project networks; however,
many studies focus only on estimating project duration duration or expenditures (Dobin 1985; Weiss 1986). Few
studies have been conducted to optimize the project in a stochastic network. Examples are Wan’s perturbation
analysis (1994) and Kidd (1987). They provide a good strategy to analyze the results of simulation; however, the
above analyses do not attempt to optimize the entire project time-cost trade-off curve. Heuristic methods,
mathematical models, and simulation techniques show both strengths and weaknesses. The heuristic approaches
select the processes to be shortened or expanded based on certain selection criteria, but they do not guarantee
optimal solutions. On the other hand, mathematical models require great computational effort, and some
approaches do not provide the optimal solution either. In addition, for largescale networks, neither heuristic
methods nor mathematical programming models can obtain optimal solutions efficiently. Simulation techniques
provide a good estimate for the optimal solutions; however, a guide to analyze the result of the simulation must
be provided in order to find the solutions efficiently. The state of research suggests the need to develop a more
efficient algorithm to conduct optimal time-cost tradeoff analysis under uncertainties.
2.4 Harmony Search Method
The harmony search algorithm is an optimization technique which took inspiration from music phenomenon.
The music instruments are played with certain discrete musical notes based on musicians’ experiences or
randomness. In same logic decision variables can be defined with certain discrete values based on computational
mentality or randomness in the optimization process.
Figure 3 shows the details of the analogy between music improvisation and engineering optimization. In music
improvisation, each player sounds any pitch within the possible range, together making one harmony vector. If
all the pitches make a good harmony, that experience is stored in each player’s memory, and the possibility to
make a good harmony is increased next time. Similarly in engineering optimization, each decision variable
initially chooses any value within the possible range, together making one solution vector. If all the values of
decision variables make a good solution, that experience is stored in each variable’s memory, and the possibility
to make a good solution is also increased next time.
Figure. 1. (Relationship between music improvement and engineering optimisation) (Lee ,
Geem, 2005).
2.4.1 HS Model
The algorithm of Harmony Search (HS) starts with a matrix named Harmony memory (HM) where random
solution vectors are put in:

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HM =
⎣
⎢
⎢
⎡
𝑦1
1
𝑦2
1
… 𝑦𝑛
1
𝑍1
𝑦1
2
𝑦2
2
… 𝑦𝑛
2
𝑍2
⋯ … ⋮ ⋯ ⋯
𝑦1
𝐻𝑀𝑆
𝑦2
𝐻𝑀𝑆
… 𝑦𝑛
𝐻𝑀𝑆
𝑍 𝐻𝑀𝑆
⎦
⎥
⎥
⎤
(1)
(y1 , y2 , . . . , yn) is the solution vector; and HMS : solution vector number in the Harmony memory. The
updating of the memory is showed below,
𝑦𝑖
𝑁𝑒𝑤
= �
𝑘, 𝑘 ∈ 𝐾𝑖
𝑦𝑖, 𝑦 ∈ {𝑦𝑖
1
, 𝑦𝑖
2
, … , 𝑦𝑖
𝐻𝑀𝑆
𝑃𝑅𝑎𝑛𝑑𝑜𝑚
𝑃 𝑀𝑒𝑚𝑜𝑟𝑦
(2)
A random number is generated and if the random number is appeared before 𝑃𝑟𝑎𝑛𝑑𝑜𝑚 the parameter of the new
solution candidate selected randomly. Otherwise the random number is after 𝑃𝑚𝑒𝑚𝑜𝑟𝑦 the new parameter is one
of existing parameters in the memory. And if the new parameter is better than one of the member in the memory
it replaces the existing member. This process is called domination. The flowchart of the calculation is showed in
figure 2.
Figure. 2. Flowchart for HS TCTP model

5. ML.Zalmai, OH Turkakin, E Manisalı
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3 Application
In this study, the harmony search is performed to optimisation of the project by its project duration and project
cost stochastically, the main aspect of this study all activities has a quantity such as surface area of a wall or the
volume of the concrete. And for each activities and each modes, the average work rate is defined. the Network of
the project is depicted in figure 3.
1 5
A
10
15
B
C
2520 30
D
E
d1 F
G
Figure. 3. Project network
And the activity quantity and average work rate for each work mode is shown in the table 1. For instance for
activity A there are 3 different modes defined. And three different average work rate occured due to changes
technological conditions or labor hour per day. As mode changes, the average work rate changes.
Table 1. Work modes for activities
Activities Activity Quantity Work Rate Mode 1 Work Rate Mode 2 Work Rate Mode 3
A 300 20 30 40
B 250 10 20 30
C 350 20 30 40
D 300 20 30 40
E 650 20 30 40
F 550 80 100 120
G 250 100 120 140
Harmony search technique changes the modes of activity, in order to find the optimum schedule. For each fitness
value calculation, Monte Carlo analysis is applied as a result there are two data set one of them is cost data of
the project and the other is project duration data. these data is produced by using Monte Carlo method is
described in detail respectively.
3.1 Monte Carlo analysis
First of all determining the activity durations is essential process in order to start Monte Carlo solution. Markov
Chain method is used to determine the activity duration. There are three performance levels are defined. And it
affects the work rate so that the activity duration of the activities are changeable. these performance levels are
high performance level medium performance level and low performance level. When the labors works in the
medium performance level the actual work rate equals to the work rate, as defined the table 1. When the
performance level increases the work rate increases respectively. Markov chain model is used for modelling the
fluctuations of the performance level day by day for the workers. When the performance level is high for a day,
in the next day the performance level is expected to be high again. For three levels the performance levels has a
high probability in order to stay constant for the next day. This phenomena is described in the figure 4.

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HIGH PERFORMANCE
LOW PERFORMANCE
MEDIUM PERFORMANCE
0.6
0.2
0.2
0.4
0.4
0.3
0.3
0.3
0.3
Figure. 4. Markov chain diagram
The diagram shows that, the probability of performance occurrence in the next day. For example when the labor
show low performance in the next day they show low performance again with the probability of 40 percent, and
medium performance with the probability of 30 percent and high performance with probability of 30 percent. A
sample output of daily performance workers is showed in figure 5.
Figure. 5. Daily performances
0,6
0,8
1
1,2
1,4
0 1 2 3 4 5 6 7 8 9 10 11
Performance
Days

7. ML.Zalmai, OH Turkakin, E Manisalı
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When the performance level is high, the work rate increases %20 percent and the when the performance level is
low the work rate decreases 20 percent. To find the activity duration the worker finishes a quantity of the job and
the work rate is changes by the performance day by day when the job’s quantity is finished, the activity duration
is determined. In the table below, the produced activity durations for one time is shown.
Table 2. Generated activity durations
Activities Activity Quantity Work Rate Mode Generated Activity
duration
A 300 20 16
B 250 10 25
C 350 20 18
D 300 20 17
E 650 20 34
F 550 80 8
G 250 100 3
Bu using these values, project duration is calculated by using critical path method. Beside the project duration
the project cost is have to be calculated. The direct and indirect cost is determined for activities and for the
modes which is shown in the table below. When the performance level is low the indirect costs are increased
due to delaying of activity ending. So that project costs are increased.
Table 3. Costs for each modes
Activities
Mode 1 Mode 2 Mode 3
Direct Indirect Direct Indirect Direct Indirect
A 2000 200 2500 250 3000 300
B 1000 100 1500 150 2000 200
C 3000 300 4000 400 5000 500
D 1100 110 1500 150 2000 200
E 2000 200 2500 250 3000 300
F 4100 410 5060 560 6600 660
G 400 40 505 50 600 60
3.2 Domination
In this study, a different domination technique is used which utilizes a hypothesis test that determines whether
the new candidate is better than the another one with a significance level. Two-sample Kolmogorov-Smirnov test
is used for domination test.
3 Conclusion
In the construction sector there are various conditions that affects the workers performance. However most of
optimization studies suppose the conditions of the construction site is determinable. This study solves the
indeterminable situations by using both Monte Carlo method and Markov chains. and these methods results
evaluated by using statistical hypothesis. With this method, the harmony search optimization can be used for
indeterminable situations.
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