Assoc. Prof. Dr. Niyazi Onur Bakır
onur.bakir@altinbas.edu.tr Altınbas University Ümmügülsüm Erdağı
ummugulsumerdagi21@gmail.com Industrial Engineer
Altınbas University
Assistant Professor Dr. Doğu Cağdaş ATİLLA
cagdas.atilla@altinbas.edu.tr Electrical and Electronics Engineering Altınbas University
Dr.Engin Sansarcı engin.sansarci@altinbas.edu.tr Altınbas University
MAKALE
This study is carried out by Ümmügülsüm Erdağı, who is the captain of EVA Optimization & Project Tracking Team and recently graduated from Altınbas University, Department of Industrial Engineering, and EVA Team advisors, Assoc. Prof. Dr. Niyazi Onur Bakır, Dr. Engin Sansarcı and Asst. Prof. Dr.
Doğu Cağdaş Atilla.
ARGE Dergisi 31 ABSTRACT
Monte Carlo simulation is a useful tool for doing risk assessment under uncertainty and is widely used in physical and mathematical problems of different application areas such as finance, marketing, engineering, project management, energy, manufacturing, R&D etc. In this study, we apply the Monte Carlo simulation method to analyze the potential risks on the project completion time for EVA (Electric Vehicle of Altınbas Team) Project. The study is carried out in MATLAB environment and the results are evaluated by different graphs, outputs, etc.
Keywords: Monte Carlo simulation, Risk analysis, Project time management.
1. INTRODUCTION
Eva is a recurring and university-funded R&D Project that seeks to produce highly efficient electric vehicles each academic year. A notable accomplishment was made when Eva team ranked 3rd among 82 universities in Teknofest-TUBITAK Efficiency Challenge 2019. The deadline of each recurring project is the annual Teknofest-TUBITAK Efficiency Challenge, hence there is limited time for completion of tasks. We perform a risk analysis of the project completion time using Monte Carlo simulation in order to predict possible scenarios and make changes on the resource sharing of the tasks, if required.
Eva Project consists of tasks which mostly have uncertain duration. Critical Path Method (CPM) is not appropriate for our Project Time Management (PTM) analysis since it is a deterministic model and not used under uncertainty. For this reason, our team has decided to use Monte Carlo simulation to produce different scenarios depending on randomly selected durations of related tasks [1]. Monte Carlo simulation uses a set of random numbers generated from probability distributions of inputs and gives an output for each repeated simulation run. The number of simulation runs can be in thousands, or more. Higher number of runs result in a more precise analysis since the more stochastic permutations of uncertainty are considered.
2. DATA SET
Except for a few number of tasks, task completion times of the Eva Project are mostly uncertain. To represent this uncertainty, we use the triangular probability distributions of the task times. Accordingly, information required in our risk analysis consists of task names, task numbers, task dependencies, minimum task time, most likely task time and maximum task time. This information is collected from the project subunits including Vehicle Control System Unit, Motor Unit, Motor Driver Unit, Embedded Recharging Unit, Battery Management System Unit, Telemetry Unit and Mechanic Unit. Then, all work packages are merged, and one table is created according to task dependencies between the subunits. This table consists of 120 tasks that have certain or uncertain completion time and their data but only first eight tasks of the table are shared in Table 1. Task names are not shared due to project privacy and representative letters are used instead.
Table 1. Work package
3. METHODS
3.1. Identification of Probability Distributions for Inputs
One of the most important steps in Monte Carlo simulation is assignment of the probability distributions for inputs. There are many distributions that can be used in Monte Carlo simulation such as normal, lognormal, uniform, triangular, beta pert etc. Selected distribution depends on the application area. Since triangular and beta pert distributions are commonly used in project time management, we decide to assign triangular probability distributions for the tasks that have uncertain time. The triangular distribution is a continuous distribution defined on the interval x∈[a,b], where a<b and a≤c≤b. Letter a,b and c are the parameters of this distribution that correspond to the minimum time, maximum time, and the most likely time, respectively.
The density function (P(x)), and the cumulative distribution function (F(x)) of the triangular distribution are:
The probability density and the cumulative distribution functions are graphed and illustrated in Figures 1 and 2, respectively.
After a probability distribution is selected for task times, a time matrix is created to serve as an input for the MATLAB algorithm, this matrix is given below.
3.2. Random Variable Generation
Function rand () is used in MATLAB to generate uniformly distributed random numbers in the interval (0,1). Our MATLAB algorithm uses these random numbers to generate random times from the triangular distributions of the task times. Generated random numbers in the interval (0,1) are assumed to be probability values by the algorithm and the algorithm uses them as input in inverse cumulative distribution functions of the task times.
Thus, it generates task times from these ICDFs. Formula for generating variate from ICDF of the triangular distribution by random numbers is given below.
u: random number generated by rand() function in the interval (0,1).
F(c): Cumulative probability of c (most likely value) F(c)=((c-a))/((b-a))
The algorithm creates task times by using random number generation for the tasks that have uncertain time while it takes the known durations as fixed times for other tasks. After these steps, a set of random task times
Figure 1: Probability density function of the triangular distribution (pdf)
Figure 2: Cumulative distribution function of the triangular distribution (cdf)
Tmx3 = Time matrix of the tasks for 1 ≤ i ≤ m and 1 ≤j ≤ 3
where i: task number, ti1: minimum completion time for task i, ti2: most likely completion time for task i, ti3: maximum completion time for task i. Eva Project consists of 120 tasks this year, so m is assumed to be equal to 120. Also, ti1, ti2 and ti3 are assumed equal for the tasks whose durations are known.
After tasks are linked, the start and finish times of tasks and the project completion time are produced with a set of random task times by one simulation run.
4. RESULTS
Different start and finish times are produced for the tasks by Monte Carlo simulation to investigate how critical path changes as the result of each simulation run, generated times for the first eight tasks given in Table 1 can be seen in Table 2.
Then 20000 simulation runs are executed for a more precise analysis. The MATLAB algorithm generates a different set of task times and project completion time in each simulation run. At the end, a total of 20000 project completion times are produced as output values. After this step, a frequency histogram of output values is created in MATLAB environment (as seen in Figure 3) to evaluate the occurrence of different project completion times.
Project completion times take values between 322 and 422 days. Additionally, duration times in the interval from 359 to 369 days have higher frequency as seen in the Figure 3.
After creating the histogram, simulation outputs are fitted to the normal distribution. Normal distribution is selected for this process because histogram and some statistical values of outputs are appropriate for it. The plot of the fitted normal distribution can be found in Figure 4.
Dmxm: Dependency matrix of the tasks for 1≤i≤m and 1≤j≤m.
where dij shows the dependency between task i and task j. If task j is the precondition of task i, dij is assumed to be equal to 1, otherwise dij is assumed to be equal to 0 and m is assumed to be equal to 120 again.
Table 2. Generated start and finish times for the tasks
Figure 3: Frequency histogram of project completion times
ARGE Dergisi 33
5. CONCLUSIONS
Each year, Eva project must be completed approximately in 360 days before the start of the Efficiency Challenge organized by TUBITAK. It consists of many tasks with uncertain duration. Because of this uncertainty and limited time until the deadline, our team decided to perform a risk analysis of project completion time with Monte Carlo simulation. The idea was to inform the project team members about possible scenarios, and to stimulate discussion for potentially new arrangements in the project plan, resource sharing, etc. Monte Carlo simulation is run 20000 times and the results are illustrated in this manuscript. A project duration of 361 days has the highest frequency and potential durations in the 359-369 days interval have highest number of occurrences as seen in the Figure 3. Mean and the standard deviation of simulation output values are equal to 364.5427 and 13.1551, respectively. The probability of completing project within 360 days is calculated as 0.3649. The results suggest that completing project within 360 days is subject to a significant risk. Therefore, instead of continuing with the current project and resource sharing plan, execution of tasks must be streamlined and accelerated.
6. REFERENCES
[1] Wolfgang Tysiak (2011). - Risk Management in Projects: The Monte Carlo Approach versus PERT. - The 6th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems:
Technology and Applications 15-17 September 2011, Prague, Czech Republic. DOI: 10.1109/
Figure 4: The frequency histogram, and the pdf of the project completion times
Figure 5: Cumulative distribution function of project completion times
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