# Comparison of Approaches for Calculating the Probability of a Project Completion

##### Journal of Eastern Europe Research in Business and Economics

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Karel Doubravský and Radek Doskočil

Brno University of Technology (BUT), Faculty of Business and Management (FBM), Department of Informatics, Kolejní 2906/4, Brno 612 00, Czech Republic

Volume 2015 (2015), Article ID 638688, Journal of Eastern Europe Research in Business and Economics, 7 pages, DOI: 10.5171/2015.638688

Received date : 3 March 2014; Accepted date : 23 June 2014; Published date : 14 September 2015

**Cite this Article as:**
Karel Doubravský and Radek Doskočil (2015), “Comparison of Approaches for Calculating the Probability of a Project Completion”, Journal of Eastern Europe Research in Business & Economics, Vol. 2015 (2015), Article ID 638688,
DOI: 10.5171/2015.638688

Copyright © 2015. Karel Doubravský and Radek Doskočil. Distributed under Creative Commons CC-BY 4.0

**Abstract**

**Keywords:**
Monte Carlo method, PERT method, probability of project completion, statistical testing.

**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; ÄŒerná, 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 ÄŒeská 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, 2013; Hashemin, Ghomi and Modarres, 2012; Yaghoubi et al, 2011) for calculating the variation of nodes. This fact causes a difference in the results of 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.

**Materials and Methods**

**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):

Where:

t_{ij} — activity duration,

a_{ij} — optimistic estimate of activity duration,

m_{ij} – most likely estimate of activity duration,

b_{ij} — pessimistic estimate of activity duration.

Using incidence matrix, the earliest times for each node (4) were calculated as follows:

Where:

ETN_{j} – Earliest Time of Node,

EFT_{ij} – Earliest Finish Time of Activity,

EST_{ij} – Earliest Start Time of Activity,

t_{ij} – Activity Duration.

The latest times for each node (5) were calculated as follows:

Where:

LTN_{i} – Latest Time of Node,

LFT_{ij} – Latest Start Time of Activity,

LFT_{ij} – Latest Finish Time of Activity,

t_{ij} – Activity Duration.

The total float of activity (6) from ith node to jth node was calculated as follows:

Where:

ETN_{i} – Earliest Time of Node,

LTN_{j} – Latest Time of Node,

t_{ij} – Activity Duration.

**Monte Carlo Method**

**Results**

**Table 1: Representation of the project (Source: Edited by (Rais and Doskočil, 2011))**

**Figure 1: Network chart (Source: Own work)**

**Approach I: PERT Method**

**Table 2: Calculation of Activity Duration Mean Time and Activity Variance (Source: Own work)**

**Approach II: Monte Carlo Method**

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.

**Table 3: First 10 simulations of the Monte Carlo method (Source: Own work)**

**Discussion**

The values of probability for Monte Carlo method are seen in Tab. 4.

**Table 4: Calculated probabilities and result of the test (Source: Own work)**

**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. Concretely, the planning time of the project was calculated. A sample network chart was examined and contains 18 nodes and 18 real activities and 6 fictions activities. For the purpose of the analysis, the basic time characteristic were calculated in accordance with traditional approaches related to the PERT method.

**Acknowledgments**

This paper was supported by grant FP-S-13-2148 ‘The Application of ICT and Mathematical Methods in Business Management’ of the Internal Grant Agency at Brno University of Technology.

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