Comparison of Approaches for Calculating the Probability of a Project Completion

Project management is nowadays a widely used and discussed discipline. This fact is substantiated by numerous scientific articles, books and publications dealing with these problems (Berganitiños and Vidal-Puga, 2009; Černá, 2008). This discipline is also included in the courses of numerous faculties focusing on economy both in the Czech Republic (International project management association, 2011) and abroad. Experts are also associated in various professional organizations or associations (Korecký and Trkovský, 2011; Společnost pro projektové řízení Česká republika, 2011).


Introduction
Project management is nowadays a widely used and discussed discipline. This fact is substantiated by numerous scientific articles, books and publications dealing with these problems (Berganitiños and Vidal-Puga, 2009;Černá, 2008). This discipline is also included in the courses of numerous faculties focusing on economy both in the Czech Republic (International project management association, 2011) and abroad. Experts are also associated in various professional organizations or associations (Korecký and Trkovský, 2011; Společnost pro projektové řízení Česká republika, 2011).
Project managers and other members of the project team use different approaches to solve PERT method (Doskočil and Doubravský, 2012;Seal, 2001). These approaches are based on various methods (Hajdu,

Abstract
The paper deals with a comparison of two different approaches (PERT method and Monte Carlo method) for calculation of the probability of a project completion. The PERT method is commonly used in a project management; the Monte Carlo is used less. The base assumption of authors can be expressed: The difference between the results obtained by the Monte Carlo method and PERT method is not significant with increasing number of simulations (iterations). For this reason, the hypothesis was formulated: There is no statistically significant difference between the calculated probabilities, i.e. both approaches are identical from application's point of view. The case study describes a model of a project, which is shown by a network chart. This chart contains 18 nodes and 18 real activities and 6 fiction's activities. Each activity is expressed by three time estimates, i.e. pessimistic, most likely and optimistic time. The planned date of completion of the project was selected at 200 time units and it was calculating the probability of completion of the project by PERT method and Monte Carlo method. Time duration of each project activity by the Monte Carlo method is successively obtained for 10, 100, 300, 500, 1000, 5000 and 10000 simulations. The calculated probabilities of project completion were compared using statistical hypothesis testing. The hypothesis was rejected for all simulations. It follows that there is difference between the approaches from application's point of view. probabilistic analysis of the project (Doskočil and Doubravský, 2013). Some papers show another approach to analyze the critical paths in a project network with fuzzy activity times. It is a used method of fuzzy numbers (Relich, 2013), where the activity time is represented by a triangular fuzzy number. This method is compared with classical CPM and PERT methods.
The paper deals with a PERT method. This method is signed as stochastic method. Its aim is an identification of critical path in a chart. The chart represents a model of a project. The implementation of the PERT algorithm is based on the critical path method-CPM (Trietsch and Baker, 2012). The paper focuses on the comparison of two different approaches (deterministic approach, Monte Carlo method), calculation of probability analysis and their influence on the calculation of the planning time of the project and their probabilities.

PERT Method
Three estimates of activity duration were provided: optimistic, most likely and pessimistic. Subsequently, activity duration mean times (1) were computed according to the following formula (Relich, 2010; Plevný and Žižka, 2005): (1) Where: tij -activity duration, aij -optimistic estimate of activity duration, mij -most likely estimate of activity duration, bij -pessimistic estimate of activity duration.
Variances (2) and standard deviations (3) of activity duration were also calculated. The following formulas were used for their calculation: ( For the purposes of the time analysis, basic characteristic times were calculated in accordance with traditional approaches. For more detailed information see related publications (Černá, 2008;Wisniewski, 1996).
Using incidence matrix, the earliest times for each node (4) were calculated as follows: Where: ETNj -Earliest Time of Node, EFTij -Earliest Finish Time of Activity, ESTij -Earliest Start Time of Activity, tij -Activity Duration.
The latest times for each node (5) were calculated as follows: The total float of activity (6) from ith node to jth node was calculated as follows: Where: ETNi -Earliest Time of Node, LTNj -Latest Time of Node, tij -Activity Duration.

Monte Carlo Method
Monte Carlo method is a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo method is often used in simulations of mathematical and economical systems (it is used to model phenomena with significant uncertainty in inputs, such as the calculation of risk in business). This method is most suited to calculation by a computer (Salling and Leleur, 2011). It can be used when it is infeasible to compute an exact result with a deterministic algorithm (Vose, 2008).   34 5 10 5 4 14 5 16 5 20 4 6 1 6 6 The graphical representation of predecessor and relationships of project (the network chart) is shown in Fig. 1. Network chart consists of 18 nodes, 18 real activities and 6 fictions activities. All calculations were computed using own programs creating in MS Excel.  and their variances are in column 5. Column 6 to 10 represents information about the times characteristics of activities.   Table 3 presents the solving of the same case study which is calculation using Monte Carlo methods. For each simulation there is time duration of the project in the last row.

Conclusion
The paper deals with a time and probability analysis in stochastic chart PERT. The paper focuses on the comparison of two different approaches (PERT method and Monte Carlo method) calculation for probability analysis. Because the Monte Carlo method belongs to simulation methods, the results were successively obtained for 10, 100, 300, 500, 1000, 5000 and 10000 simulations (iterations). Since the calculation of the probability of project completion to 200 time units is based on using the normal distribution, values were generated using the normal distribution in the Monte Carlo method. The calculated probabilities of project completion were compared using statistical hypothesis testing. The assumption was expressed: The difference between the results obtained by the Monte Carlo method and PERT method is not significant with increasing number of simulations (iterations). For this reason the hypothesis was formulated: The results (probabilities) are not statistically significant. For testing the significance of difference between the calculated probabilities, the test of parameter p for alternative distribution was used. The null hypothesis was formulated: There is no statistically significant difference between the calculated probabilities, i.e. both approaches are identical from an application point of view. Opposite the null hypothesis, the alternative hypothesis was formulated: there is a statistically significant difference between the calculated probabilities. This null hypothesis was rejected for all testing. It follows that there is a difference between the approaches (it depends whether we use a PERT method or Monte Carlo method).