ENBIS/1 © Chris Hicks University of Newcastle upon Tyne An analysis of the use of the Beta...

ENBIS/1

© Chris HicksUniversity 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

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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.

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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.

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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

1

11

))(()(

)())((

ab

xbaxxf 0,, bxa

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Figure 1 Beta function ),,( xf for α = β

Symmetric

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Figure 2 Beta function ),,( xf for α < β

Left skewed

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Figure 2 Beta function ),,( xf for α > β

Right skewed

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Planning estimates: optimistic (to), pessimistic (tp) and most likely (tm)

6

4 pmo ttt 6

op tt

ss

ss

2

2)1(

s

s

)1(

Mean Standard deviation

Alpha Beta

op

os tt

t

ops tt

Where μs and σs 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)

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RelationshipsLet us assume that to =X * tm and tp = Y * tm and substitute into previous equations

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For any values of X and Y we can calculate α and β and find the PDF

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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.

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(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

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Experimental Design

Experiment X Y α β

1 0.2 2 3.68 4.27

2 0.8 10 0.79 3.55

3 0.2 10 1.15 4.05

4 0.8 2 1.73 4.50

(Full factorial design with 1000 replicates)

Note: Tm is assumed to be the Company’s estimated operation time

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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

Beta LxLy HxHy LxHy HxLy

Mean (days) 275.8 873.4 777.3 351.1

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Conclusions• The paper has established the relationships between the

planning parameters to, tm and tp.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.

ENBIS/1 © Chris Hicks University of Newcastle upon Tyne An analysis of the use of the Beta...

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