George MAVROTAS Olena PECHAK

Laboratory of Industrial & Energy EconomicsSchool of Chemical EngineeringNational Technical University of Athens

George MAVROTASOlena PECHAK

8th Multi-criteria Meeting (HELORS)8-10 December 2011, Eretria, Greece

Dealing with uncertainty in project portfolio selection: Combining MCDA, Mathematical Programming and

Monte Carlo simulation: a trichotomic approach

8th MCDA meeting HELORS, Eretria, 8-10 December 2011

Contents

The project portfolio selection problem Methodology

MCDA & Mathematical Programming Incorporating uncertainty

Monte Carlo simulation Trichotomic approach

Case study Results and discussion Conclusions & Future research


Methodology


MCDA problematiques = What kind of problems we can address

4 problematiques (B. Roy) Description – Understanding of the MCDA problem Selection of the most preferred alternative Ranking of the alternatives Sorting of the alternatives in categories

Belton – Stewart (2002) Selection of a subset under constraints (Portfolio)


Description of the simple case of project selection (without constraints)

MCDA

Ranking

Selection of the first n-projects

Ranking

1η

2η

3η

4η

…

20η

…

55η

Top 20


If there are constraints...

ExampleSegmentation constrains

GeographicalSectoral

Logical constraintsPrecedenceMutually exclusive projects

Budget constraints . . .

The alternatives are no longer independent The top-n can only by chance fulfill the constraints Combinatorial problem


Remedy

Examine all possible, feasible combinations of projects and try to find the “most preferred”

Mathematical Programming

InputDecision variables (binary) projects

Xj=1 (j-th project selected), Xj=0 (j-th project not selected)Constraints feasible regionObjective function Sum of multicriteria scores

Output the best combination of projects that satisfy the constraints= optimal portfolio


Combination of MCDA and Math Prog

MCDAScoring of the projects

Mathematical ProgrammingInteger ProgrammingMost preferred portfolio

Typical example: Promethee V


Dealing with uncertainty

Project uncertaintyProject attributes (costs, performance…)

Environmental uncertaintyWeights of criteriaTotal budget…

We assume stochastic nature of the uncertainty (probabilities, distributions)

Monte Carlo simulation


Monte Carlo simulation & Optimization

A B

M

CA B

param1

param2

paramN

value1(i)

value2(i)

value3(i)

Solution of MP model

i =1…n

Project portfolio 1

Project portfolio 2

Project portfolio 3

Project portfolio n

. . .


Project allocation in sets

Trichotomic allocation of projects

green set

Projects selected in all optimal portfolios

red set

Projects not selected in none optimal portfolio

grey set

Projects selected in some optimal

portfolios

In each iteration we obtain an optimal portfolioEach project can be present (Xj=1) or not (Xj=0) in the optimal portfolio


Project allocation in sets

x1 x2 x3 x4 … xn

1 1 0 0 1 … 1

2 0 0 1 1 … 1

3 0 0 0 1 … 0

… … … … … … …

1000 1 0 0 1 … 1

Itera

tion

s

Projects


Remarks from phase 1

Usually there is no dominating portfolio1000 iterations about 1000 different portfolios

Trichotomic approach provides useful informationGreen set they are in under any circumstancesRed set they are out under any circumstancesGrey set we are not sure, we need more info

Exploit information from phase 1 and go to phase 2Only the grey set


The model of phase 2

Fix the values of green and red projects

Use as objective function coefficients the frequencies (fj) of the projects from the 1st phase

j

j

X 1 ( )

X 0 ( )

j gs green set

j rs red set

j= max Xn

jj

Z f


Results from phase 2

Two cases: No stochastic parameters in the constraints

One single run Final selection: the unique optimal portfolio

Stochastic parameters in the constraints Monte Carlo simulation – Optimization (1000 runs) Final selection: the dominating optimal portfolio

The portfolio with the highest frequency in 2nd phase If there is no clear winner comparison among the first two project-

wise comparison total budget adjustments


Illustration of the method

Set of projects

Multiple criteria

Multiple constraints

Uncertainty

red set

greyset

green set

Notselected

selected

1st phase 2nd phase

MCDAMC simulation

MathProg

1st phase infoMC simulation

MathProg


Case study

Input dataShakhsi-Niaei, M., Torabi, S.A., Iranmanesh, S.H. (2011) A comprehensive framework

for project selection problem under uncertainty and real-world constraints Computers and Industrial Engineering 61, 226-237.

40 projects for telecommunication company, classified in 3 groups: Basic - 2,7,9,12,13,14,17,22,23,26,28,37,38,39 Applied - 1,3,4,6,10,11,16,18,20,21,24,25,27,30,31,32,33,35,36 Developing - 5,8,15,19,29,34,40

Constraints Available total budget (we allow a 15% excess) Limits by project type (at most 20% Basic, 70% Applied, 40% Developing)


Case studyThe projects are evaluated by 5 criteria: Cost: Total project cost including all expenses required for project completion. Proposed methodology: Degree of being step-by-step, well planned, scientifically-proven,

disciplined, and proper for organization current status in the proposed methodology. The abilities of personnel: Work experience of project team related to concerned project. Scientific and actual capability: Scientific degree and educational certificates of project’s

team. Technical capability: Ability of providing technical facilities and infrastructures.

Uncertainties: Criteria weights Budget Costs Methodology Personnel qualification Scientific Technological

Environmental

Internal


Incorporating uncertainty with distributions

Weights triangular distribution

Total budget normal distribution max(6000; normal(6000, 300))

Scores uniform distribution

min mid maxcost 0.17 0.21 0.23meth 0.12 0.14 0.14pers 0.12 0.14 0.16sci 0.11 0.13 0.15tech 0.36 0.40 0.43


Project’s dataPro

ject

s’ d

ata


Results of the model – phase 1


Results

Phase 1 No clear dominating portfolio We may introduce threshold for green and red projects

1% projects are green if freq > 99% and red if freq < 1% Results

Green set - 7 projects (5, 8, 15, 19, 29, 34, 40) Red set - 3 projects (2, 9, 17) Grey set - 30 projects

Phase 2 Still no clearly dominating portfolio, but 2 combinations are most

preferred The difference between 2 most frequent portfolios --> 1 project.


Results

Results for 1000 iterations of seed 1513.

(we test 15 different seeds 15 MC experiments no significant difference)

Frequency 1 and 2 are the frequencies of top 2 portfolios.

Phase 1 Phase 2

ThresholdGREE

N RED GREYFrequency

1Frequency

2Time, sec

0 6 0 34 226 190 352

0.5 % 7 1 32 226 190 355

1% 7 3 30 226 190 367

2% 7 3 30 226 190 371

5% 8 3 29 226 190 399

10% 10 5 25 229 191 364

The most frequent portfolios differ only by 1 project 16 or 24 (only one of two may be in): Both are in the group of “applied” projects Have similar characteristics (in some criteria 24 performs weaker)

Final decision still to be made by a person according to the main goals.

min max min max min max min max min maxProject 16 374 486 2 6 4 8 1 3 2 6Project 24 385 416 1 4 1 4 1 5 3 5


The naïve approach: Consider the expected values

Expected values no uncertainty

Although the result may be almost the same…… we have more fruitful information than considering just the expected values We know which are the sure projects (green

and red) We can identify the “borderline projects” We know the probability of the preferred

combination from phase 2

Expected values1 12 03 14 05 16 17 08 19 0

10 011 012 013 114 015 116 017 018 019 120 121 122 023 024 125 026 027 128 129 130 131 132 133 034 135 136 137 038 139 140 1

Stochastic - 2 phase1010110100001011001110000011111101110111


Conclusions

The trichotomic approach is a structured method that can deal with uncertainties in project selection

Combination of MCDA, MP and MC Reduces information burden

Identification of the sure projects The DM can focus only on the grey projects

Flexible, not black box Can be adapted to a specific decision situation and DM

More fruitful information than the naïve “expected value” approach


Future work

To address separately the environmental (criteria weights, budget etc.) and internal project uncertainties

Iterative approach: to decrease the uncertainties on each new iteration Ask for more information only for the grey projects

To apply the trichotomic approach in group decision making Phase 1 unanimity principle Phase 2 majority principle


Thank you!

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George MAVROTAS Olena PECHAK

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