enbis/1 © chris hicks university of newcastle upon tyne an analysis of the use of the beta...
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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
ENBIS/2
© Chris HicksUniversity 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.
ENBIS/3
© Chris HicksUniversity 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.
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© Chris HicksUniversity 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
1
11
))(()(
)())((
ab
xbaxxf 0,, bxa
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© Chris HicksUniversity of Newcastle upon Tyne
Figure 1 Beta function ),,( xf for α = β
Symmetric
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© Chris HicksUniversity of Newcastle upon Tyne
Figure 2 Beta function ),,( xf for α < β
Left skewed
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© Chris HicksUniversity of Newcastle upon Tyne
Figure 2 Beta function ),,( xf for α > β
Right skewed
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© Chris HicksUniversity of Newcastle upon Tyne
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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© Chris HicksUniversity of Newcastle upon Tyne
RelationshipsLet us assume that to =X * tm and tp = Y * tm and substitute into previous equations
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© Chris HicksUniversity of Newcastle upon Tyne
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© Chris HicksUniversity of Newcastle upon Tyne
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© Chris HicksUniversity of Newcastle upon Tyne
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© Chris HicksUniversity of Newcastle upon Tyne
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© Chris HicksUniversity of Newcastle upon Tyne
For any values of X and Y we can calculate α and β and find the PDF
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© Chris HicksUniversity 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.
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© Chris HicksUniversity 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
ENBIS/17
© Chris HicksUniversity of Newcastle upon Tyne
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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© Chris HicksUniversity 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
Beta LxLy HxHy LxHy HxLy
Mean (days) 275.8 873.4 777.3 351.1
ENBIS/19
© Chris HicksUniversity of Newcastle upon Tyne
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.
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