This document summarizes a study that uses harmony search optimization and Monte Carlo simulation to optimize the time-cost tradeoff for construction projects with uncertain activity durations. Markov chains are used to model crew performance variability over time. The harmony search algorithm evaluates solutions by running Monte Carlo simulations to obtain probabilistic time and cost distributions, which are compared using a Kolmogorov-Smirnov test to determine statistical dominance between solutions. The approach is demonstrated on a sample project network problem.

1. Time Cost Trade off Optimization Using
Harmony Search and Monte-Carlo Method
Authors:
Mohammad Lemar ZALMAI
Osman Hürol Türkakın
Ekrem MANİSALI
Istanbul, Turkey
October, 21-25, 2014

2. • Introduction
• Existing Solutions
• Harmony Search Method
• Pareto set
• A test problem
• Conclusion
Overview

3. • Project cost and project duration are main
factors in construction management.
• Both the trade-off between the project cost and
the project completion time are considerable
aspects for schedulers.
• In the real life, uncertainty of the environment
increases hardness of the problem
Introduction

4. • The main objectives are minimizing both
project duration and project cost.
• Every activity have several options that ends
different durations with different costs.
• In meta-heuristics optimization projects are
optimized by using Genetic Algorithms
Particle Swarms
Problem Description

5. • In this study, the durations of activites are
considered as uncertain.
• This uncertainity modelled by using Monte-
Carlo Method and Markov Chains
• In the modelling the crew show different
performance day by day.
Problem Description

6. • In the most of early studies the duration and
the total cost of the project considered as
constant.
• In early solutions analytical solutions are used
• After (Feng et al,1997) metaheuristics are
mainly used for this problem.
Existing Solutions

7. • There are some studies model uncertainity by
using Stochastic methods.
• (Feng et al,2000) use simulation techniques,
and optimized by using Genetic Algorithms.
• Harmony Search Algorithm is firstly used by
(Geem Z.W. ,2010) for time-cost trade off
problems in construction projects.
Existing Solutions

8. • 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 musician’s
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.
Harmony Search

9. Taken from A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice
By: Lee, Kang Seok, and Zong Woo Geem. Computer Methods in Applied Mechanics and Engineering
Harmony Search

10. HM =
𝑦1
1
𝑦2
1
… 𝑦 𝑛
1
𝑍1
𝑦1
2
𝑦2
2
… 𝑦 𝑛
2 𝑍2
⋯ … ⋮ ⋯ ⋯
𝑦1
𝐻𝑀𝑆
𝑦2
𝐻𝑀𝑆
… 𝑦 𝑛
𝐻𝑀𝑆
𝑍 𝐻𝑀𝑆
(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
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.
Harmony Search

12. • In multi-objective optimization problems if a
candidate 1 is better than the another
candidate. It calles candidate 1 strictly
dominates candidate 2.
• There are two kinds of domination alternatives
between two candidates
Pareto Domination

13. 𝐶𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒 2 𝑖𝑠 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑑 𝑏𝑦 𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒 2 if c1 ≤ c2 and t1 < t2
cost
time
1 2
1t 2t
2c1c

14. cost
time
1
2
1t
2t
2c
1c
𝐶𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒 2 𝑖𝑠 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑑 𝑏𝑦 𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒 2 if c1 < c2 and t1 ≤ t2

15. Pareto Set
cost
time
1
4
2
3
5
6
7

16. • In this study, a modified approach is used. In
this case candidate results are statistical
distributions instead of single value.
• Two sample Kolmogorov Smirnov test is a non
parametric statistical test that compares two
statistical distributions.
• Two sample K-S test is used for determination
of domination of one candidate to another.
Statistical Domination

17. • There are two samples that compared each
other.
𝐻 𝑜: 𝑋1 = 𝑋2
𝐻1: 𝑋1 ≠ 𝑋2
𝐻1: 𝑋1 > 𝑋2
𝐻1: 𝑋1 < 𝑋2
Kolmogorov-Smirnov Test

18. X and Y values of both
distributions are tested
with Kolmogrov-Smirnov
Test with a significant
level of 0.05
Statistical Domination

19. • Generation of random activity durations is
main objections of Monte Carlo method.
• Three performance levels are defined for
crews.
• And the performance level of the crew affects
to the production rate and the probability of
performance levels of the next time period
Monte-Carlo implemention

20. perform=[.4 .3 .3; .2 .6 .2 ; .3 .3 .4];
HIGH PERFORMANCE
LOW PERFORMANCE
MEDIUM PERFORMANCE
0.6
0.2
0.2
0.4
0.4
0.3
0.3
0.3
0.3
Markov Chain of the Performance
Levels

21. 0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10 11
Performance
Days
Sample Results of Markov Chains

22. • In this study, the harmony search is performed
to optimization 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 next slide.
A Test Problem

23. 1 5
A
10
15
B
C
2520 30
D
E
d1 F
G
Project network

24. • The activity quantity and average work rate for each work
mode is shown.
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
Activities

25. 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
Costs of Activity modes

26. Results

27. • This study solves the indeterminable situations
by using both Monte Carlo method and
Markov chains.
• With implemention of both methods, the
harmony search optimization can be more
usable for indeterminable situations.
Conclusion

28. Authors:
Mohammad Lemar ZALMAI (lemar_zalmai07@hotmail.com)
Osman Hürol Türkakın (turkakin@istanbul.edu.tr)
Ekrem MANİSALI (ekmanisa@istanbul.edu.tr)
Istanbul, Turkey
October, 21-25, 2014
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