### Risk Analysis of Stochastic PERT Graph

#### Abstract

Purpose of the article: The paper deals with a time and probability analysis of stochastic graph PERT. The paper focuses on the comparison of two different approaches calculation of probability analysis. Concretely the planning time of the project was calculated. A sample PERT network graph was examined, which comprised 18 nodes and 18 real activities and 6 fictions activities. For the purpose of the analysis, the basic characteristic times were calculated in accordance with traditional approaches related to the PERT method.

Methodology/methods: The implementation of the PERT algorithm is based on the critical path method (CPM). It was calculated the basic time charactetistics of the project and identificed the critical path. For probability analysis was also calculated expected value, variance and standard deviance of the activities. For calculation of the planning time was used distribution function of standardized normal distribution. The PERT algorithm is realized by using spreadsheet in the MS Excel.

Scientific aim: of the paper is comparison of two different approaches calculation of the probability analysis and their influence on the calculation of the planning time of the project.

Findings: Two different approaches calculation of the probability analysis shows on different result of values of project planning time. Approach II better reflects the difference between the values of variances of project activities. The value of variance depends on the input values of three time durations s activity estimates (pessimistic, most likely, optimistic). For higher values of probability there is a bigger difference between the values of planned times that are calculated by two described approaches.

Conclusions: The problem was solved using the example project whose model (network graph) contained 18 nodes and 24 activities. For each activity have been known three time estimates (pesimitic, most likely, optimistic). Based on these estimates were calculated expected values of the duration activities and their variances. Expected values of the duration activities were used as input values to calculate the time characteristics. Variances of the activities were used as input values to calculate the variance at the nodes. For these calculations two approaches was used. The expected value of project duration (value of earliest time in last node) was the same for both approaches. For the approach I is a value of the variance in the last node less than for the approach II. These values were used as input data for calculation of planning time of the project at various levels of probability according to the standardized normal distribution. From obtained results dependence between the probability and size of the differences in planned times were observed. This difference increases with a probability going to one. Based on the analysis a recommendation shows to use the approach II under conditions when there are large variations between optimistic (pessimistic) estimates of activity durations and the most likely estimate of activity duration. It causes great differences in values of the variances of the activities. The approach II better reflects this dissimilarity in the variances of the activities. This approach provides longer planning times of the project opposite the approach I.

Methodology/methods: The implementation of the PERT algorithm is based on the critical path method (CPM). It was calculated the basic time charactetistics of the project and identificed the critical path. For probability analysis was also calculated expected value, variance and standard deviance of the activities. For calculation of the planning time was used distribution function of standardized normal distribution. The PERT algorithm is realized by using spreadsheet in the MS Excel.

Scientific aim: of the paper is comparison of two different approaches calculation of the probability analysis and their influence on the calculation of the planning time of the project.

Findings: Two different approaches calculation of the probability analysis shows on different result of values of project planning time. Approach II better reflects the difference between the values of variances of project activities. The value of variance depends on the input values of three time durations s activity estimates (pessimistic, most likely, optimistic). For higher values of probability there is a bigger difference between the values of planned times that are calculated by two described approaches.

Conclusions: The problem was solved using the example project whose model (network graph) contained 18 nodes and 24 activities. For each activity have been known three time estimates (pesimitic, most likely, optimistic). Based on these estimates were calculated expected values of the duration activities and their variances. Expected values of the duration activities were used as input values to calculate the time characteristics. Variances of the activities were used as input values to calculate the variance at the nodes. For these calculations two approaches was used. The expected value of project duration (value of earliest time in last node) was the same for both approaches. For the approach I is a value of the variance in the last node less than for the approach II. These values were used as input data for calculation of planning time of the project at various levels of probability according to the standardized normal distribution. From obtained results dependence between the probability and size of the differences in planned times were observed. This difference increases with a probability going to one. Based on the analysis a recommendation shows to use the approach II under conditions when there are large variations between optimistic (pessimistic) estimates of activity durations and the most likely estimate of activity duration. It causes great differences in values of the variances of the activities. The approach II better reflects this dissimilarity in the variances of the activities. This approach provides longer planning times of the project opposite the approach I.

#### Keywords

network graph; PERT method; node criticality; mean value; probability analysis

#### JEL Classification

C10; C44

#### Full Text:

PDF (Czech)*Online: 2014-07-31*