1. MATHEMATICS 3 Operational Analysis Štefan Berežný Applied informatics Košice - 2010

2. Štefan Berežný Lecture For Applied Informatics 2 PERT PERT = Project (or Program) Evaluation and Review Technique: PERT is a model for project management designed to analyze and represent the tasks involved in completing a given project.

3. Štefan Berežný Lecture For Applied Informatics 3 PERT PERT – especially, the time needed to complete each task, and identifying the minimum time needed to complete the total project. PERT – is applied to very large-scale, one- time, complex, non-routine infrastructure and Research and Development projects.

4. Štefan Berežný Lecture For Applied Informatics 4 PERT The PERT method is similar to CPM method, but it is applied to stochastic models. Most is used in projects which are not adequately tested and there are enough high-quality estimates of each activity. For each activity it is necessary to determine the next 3 estimates:

5. Štefan Berežný Lecture For Applied Informatics 5 PERT Optimistic time (a): the minimum possible time required to accomplish a task, assuming everything proceeds better than is normally expected. This estimate reflects the shortest possible duration of the action in the implementation of activities expected ideal running.

6. Štefan Berežný Lecture For Applied Informatics 6 PERT Pessimistic time (b): the maximum possible time required to accomplish a task, assuming everything goes wrong. Pessimistic estimate of the duration of the activity. This rating reflects the longest possible time for completion of the action in the implementation of this activity is calculated with all logically possible obstacles.

7. Štefan Berežný Lecture For Applied Informatics 7 PERT Normal time (m): the best estimate of the time required to accomplish a task, assuming everything proceeds as normal. The estimate of normal activity (Let us denote by m) that corresponds to the most likely estimate of the duration of the activity.

8. Štefan Berežný Lecture For Applied Informatics 8 PERT Expected time ( ): Suppose that the duration of action t ij is a random variable that can be described by a known statistical distribution, which is best known. distribution of β, which can be approximated by a normal Gaussian distribution.

9. Štefan Berežný Lecture For Applied Informatics 9 PERT The numerical characteristics of this distribution is: s - standard deviation s 2 - variance - average duration of activity t ij

10. Štefan Berežný Lecture For Applied Informatics 10 PERT Example: Using the PERT method, find the critical path in the project, which is shown in the chart below. (The figures are in weeks in order: estimate: optimistic, normal and pessimistic). Also see: (1) the likelihood of the project, if extend the duration of two weeks over a time corresponding to the critical path method using CPM? (2) what should be the duration of the project that this time the project executed with a probability of 90%?

11. Štefan Berežný Lecture For Applied Informatics 11 PERT 2, 3, 10 12, 15, 30 5, 7, 15 4, 5, 6 0, 0, 0 8, 10, 18 2, 2, 2 4, 6, 8 4, 6, 14 3, 4, 5 7, 10, 25 2, 3, 4 4, 7, 14 5, 6, 7 8, 9, 16 V1V1 V8V8 V2V2 V9V9 V7V7 V6V6 V3V3 V4V4 V5V5

12. Štefan Berežný Lecture For Applied Informatics 12 PERT Solution: First, we verify whether the graph is acyclic by ATN (ATN = Algorithm for Topological Numbering of the vertecies). If it is, so we can find some of its topological number.

13. Štefan Berežný Lecture For Applied Informatics 13 PERT Solution: In the table is calculated the average duration of each activity and their variances. Then we get a new network, which will have a topological numbering of the verticies and the average duration of each activity and to the network we apply the algorithm to find the critical path. The algorithm is shown in the graph with special vertices, but the chart table for the network.

14. Štefan Berežný Lecture For Applied Informatics 14 PERT 4 17 8 5 0 11 2 6 7 4 12 3 8 6 10 1 3 2 4 5 8 6 7 9

15. Štefan Berežný Lecture For Applied Informatics 15 PERT Solution: T = 38 weeks. The project will be completed with a probability of 50% for 38 weeks. The critical path is 1 - 4 - 6 - 8 - 9 Hence we get the total variance is equal to the sum of the variances of sub-activities which form the critical path, thus we get that:

16. Štefan Berežný Lecture For Applied Informatics 16 PERT Solution: 

17. Štefan Berežný Lecture For Applied Informatics 17 PERT Solution: If we to prolong the project for 2 weeks, so we get that x = 40, m = T = 38 and  = 4. Hence we get that value

18. Štefan Berežný Lecture For Applied Informatics 18 PERT Solution: The value of the standardized normal distribution function is F (0,5) = 0.69146. Based on the calculated values, we found that the likelihood of the project for 40 weeks is approximately 70%.

19. Štefan Berežný Lecture For Applied Informatics 19 PERT Solution: If we put the  (u) = 0,9, both tables we see that this equality is satisfied for the value u = 1.29 and receive equal:

20. Štefan Berežný Lecture For Applied Informatics 20 PERT Solution: From there to express x = 43.16. The project will be implemented in approximately 90% probability for 43 weeks.

21. Štefan Berežný Lecture For Applied Informatics 21 Thank you for your attention Štefan Berežný Department Of Mathematics FEI TU Košice B. Němcovej 32 040 02 Košice