1. ENBIS/1 © Chris Hicks University of Newcastle upon Tyne An analysis of the use of the Beta distribution for planning large complex projects Chris Hicks, Business School Fouzi Hossen, Mechanical & Systems Engineering

2. ENBIS/2 © Chris Hicks University of Newcastle upon Tyne Introduction Large complex projects are often planned using project management systems based upon the Project Evaluation and Review Technique (PERT). PERT models uncertainties using the Beta distribution based upon estimates of optimistic, pessimistic and most likely activity durations. The Probability Density Function for a Beta distribution can be uniform, symmetric or skewed.

3. ENBIS/3 © Chris Hicks University of Newcastle upon Tyne Objectives To explore the relationship between the planning values used, the Beta distribution parameters and shape. A case study then establishes the cumulative impact of uncertainties using data obtained from a company that produces complex capital goods.

4. ENBIS/4 © Chris Hicks University of Newcastle upon Tyne The General Beta distribution Г represents the Gamma function α and β are the shape parameters and a and b are the lower and upper bounds

5. ENBIS/5 © Chris Hicks University of Newcastle upon Tyne Figure 1 Beta function for α = β Symmetric

6. ENBIS/6 © Chris Hicks University of Newcastle upon Tyne Figure 2 Beta function for α < β Left skewed

7. ENBIS/7 © Chris Hicks University of Newcastle upon Tyne Figure 2 Beta function for α > β Right skewed

8. ENBIS/8 © Chris Hicks University of Newcastle upon Tyne Planning estimates: optimistic (to), pessimistic (tp) and most likely (tm) Mean Standard deviation AlphaBeta Where μ s and σ s refer to the standard Beta distribution, which has a lower bound of 0 and an upper bound of 1 PERT parameters and the Beta distribution (Source: Moitra, 1990)

9. ENBIS/9 © Chris Hicks University of Newcastle upon Tyne Relationships Let us assume that to =X * t m and tp = Y * t m and substitute into previous equations

10. ENBIS/10 © Chris Hicks University of Newcastle upon Tyne

11. ENBIS/11 © Chris Hicks University of Newcastle upon Tyne

12. ENBIS/12 © Chris Hicks University of Newcastle upon Tyne

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14. ENBIS/14 © Chris Hicks University of Newcastle upon Tyne For any values of X and Y we can calculate α and β and find the PDF

15. ENBIS/15 © Chris Hicks University of Newcastle upon Tyne Case Study Objective is to establish the cumulative effect of uncertainty through a series of simulation experiments. Data obtained from a collaborating capital goods company.

16. ENBIS/16 © Chris Hicks University of Newcastle upon Tyne (No. 15) (No. 11) (No. 17) Typical product, considered in simulation Uncertainties are cumulative because an assembly cannot start until all the necessary components and sub assemblies are available

17. ENBIS/17 © Chris Hicks University of Newcastle upon Tyne Experimental Design ExperimentXYαβ 10.223.684.27 20.8100.793.55 30.2101.154.05 40.821.734.50 (Full factorial design with 1000 replicates) Note: T m is assumed to be the Company’s estimated operation time

18. ENBIS/18 © Chris Hicks University of Newcastle upon Tyne Results Histograms of lead-time for a typical component, assembly and the product are provided in the paper. The effective of cumulative uncertainty is to move the distributions to the right. Probability of meeting a due date produced by a deterministic planning system is very low. Product Lead Time BetaLxLyHxHyLxHyHxLy Mean (days)275.8873.4777.3351.1

19. ENBIS/19 © Chris Hicks University of Newcastle upon Tyne Conclusions The paper has established the relationships between the planning parameters t o, t m and t p. and the Beta parameters and the PDF. A case study has investigated the cumulative effect of uncertainties at assembly and product level. The results showed that lead-time was sensitive to the planning assumptions used. The lead time was 3 times longer in the worst case than the best case, despite the fact that the same most likely times were used. The probability of meeting due dates established by deterministic planning systems was very small. Managers should: i) minimise uncertainty; ii) take into account uncertainty in planning.