1 Helsinki University of Technology Systems Analysis Laboratory Multi-Criteria Capital Budgeting with Incomplete Preference Information Pekka Mild, Juuso

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Helsinki University of Technology Systems Analysis Laboratory

Multi-Criteria Capital Budgeting with Multi-Criteria Capital Budgeting with

Incomplete Preference InformationIncomplete Preference Information

Pekka Mild, Juuso Liesiö and Ahti SaloSystems Analysis Laboratory

Helsinki University of TechnologyP.O. Box 1100, 02150 HUT, Finland

http://www.sal.hut.fi


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Multi-criteria capital budgeting (1/2)Multi-criteria capital budgeting (1/2)

Choose a subset of projects, a project portfolio, from a large

set of proposals (e.g. 50) subject to scarce resources Each project evaluated w.r.t. multiple criteria Project value as a weighted sum of criterion-specific scores Portfolio value as sum its constituent projects’ values Several application areas, e.g.

– Healthcare systems (Kleinmuntz & Kleinmuntz, 1999)

– R&D project portfolios (Stummer & Heidenberger, 2003)

– Nature conservation (Memtsas, 2003)


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Multi-criteria capital budgeting (2/2)Multi-criteria capital budgeting (2/2)

Find a feasible portfolio which maximizes the overall value– Large number of projects

– Criteria, i = 1,…,n scores , weights

– Project value

– Portfolio , overall value

– Resources k = 1,…,q resource consumption

– Budget vector , the set of feasible portfolios

With precise weights and scores the optimal portfolio is obtained as a solution

to the binary LP-problem

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Incomplete preference information (1/2)Incomplete preference information (1/2)

Set of feasible weights– Linear constraints– Several weight vectors are consistent with the given preference statements

– E.g. criterion 1 is the most important of three criteria

– Interval sensitivity analysis (cf. Lindstedt et al., 2001)

Interval scores– Lower and upper bounds for the criterion-specific scores of each project

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Incomplete preference information (2/2)Incomplete preference information (2/2)

Portfolio p dominates p’ ( ) iff

– The value of projects included in both portfolios is canceled

pairwise dominance check is an LP-problem

The set of non-dominated portfolios

– With precise scores and no a priori weight information (i.e. ), the set of non-dominated portfolios corresponds to the set of Pareto-optimal solutions

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Computation of non-dominated portfolios (1/2)Computation of non-dominated portfolios (1/2)

Dominance checks require pairwise comparisons Number of possible portfolios is high

– m projects lead to 2m possible portfolios, i.e.

– Typically high number of feasible portfolios as well

– Brute force enumeration of all possibilities not computationally attractive

» If m=20 takes one second, then m=40 takes 13 days

Combinatorial problem– Corresponds to an n-objective q-dimensional knapsack problem

– Score intervals and weight information are handled with a specific algorithm

based on dynamic programming

| | 2mP


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Computation of non-dominated portfolios (2/2)Computation of non-dominated portfolios (2/2)

Outline of the algorithm– Portfolios that use resources efficiently are stored in

– Projects are added one by one,

1) Let

2) For j=2,…,m do

3) Obtain

Effective implementation– If is sorted by portfolio cost, fewer pairwise comparisons are needed in 2b)

– The size of can be reduced by discarding portfolios that cannot end-up non-

dominated by adding projects

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Robust Portfolio Modeling (RPM)Robust Portfolio Modeling (RPM)

Incomplete information in multi-criteria capital budgeting– Non-dominated portfolios are of interest

– Computational challenges in large problems

– Portfolio features open new opportunities for decision support» Portfolio is an m-tuple of project-specific yes/no decision

Robust portfolio selection– Accounts for the lack of complete information

– Consideration of all non-dominated portfolios

– Reasonable performance across the full range of permissible parameter values

– “What portfolios/projects can be defended - knowing that we have only

incomplete information?”


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RPM for project portfolio selection (1/4)RPM for project portfolio selection (1/4)

Portfolio-oriented selection– Consider non-dominated portfolios as decision alternatives– Decision rules: Maximax, Maximin, Central values, Minimax regret– Methods based on exploring the “solution space” for a compromize

» E.g. aspiration levels (c.f. Stummer and Heidenberger, 2003)

Project-oriented selection– Portfolio is a set of project-specific yes/no decisions– Project compositions of non-dominated portfolios typically overlap– Which projects are incontestably included in a non-dominated portfolio?– Robust decisions on individual projects in the light of incomplete information


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Core index of a project– Share of non-dominated portfolios in which a project is included

– Project-specific performance measure derived in the portfolio context» Accounts for competing projects, scarce resources and other portfolio constraints

Core and exterior– Core projects are included in all non-dominated portfolios,

– Exterior projects are not included in any of the nd-portfolios,

– Border line projects are included in some of the nd-portfolios,

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RPM for project portfolio selection (3/4)RPM for project portfolio selection (3/4) Gradual process

– Select the core projects» Robust choices w.r.t. incomplete information

– Discard the exterior projects» Despite the lack of complete information, these can be safely discarded

– Focus attention to the borderline projects» Specify information, i.e. narrower score intervals and/or stricter weight statements» Narrower score intervals for core and exterior projects do not affect the core indexes» Negotiation, manual iteration

Core and exterior expand with more complete information– Additional information (s.t. ) can reduce the set

– No new portfolio can become non-dominated

– Unique portfolio has no borderline projects

,w w v vS S S S NP


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Approach to promote robustness through incomplete information (integrated sensitivity analysis).Account for group statements


Decision rules, e.g. minimax regret

•Narrower intervals•Stricter weights

•Wide intervals•Loose weight

statements

Large number

of projects.

Evaluated w.r.t.

multiple criteria.

Border line projects

“uncertain zone” Focus

Exterior projects

“Robust zone” Discard

Core projects“Robust zone”

Choose

Core

Border

Exterior

Negotiation.Manual iteration.Heuristic rules.

Se

lecte

dN

ot se

lecte

d

Gradual selection: Transparency w.r.t. individual projectsTentative conclusions at any stage of the process


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Application to road pavement projects (1/6)Application to road pavement projects (1/6)

Real-life data from Finnish Road Administration– Selection of the annual pavement programme in one major road district

Large set of m = 223 project proposals – Generated by a specific road condition follow-up system– Coherent road segments proposals are considered independent

Criteria (n = 3) derived from technical measurements– Damage sum in the proposed site– Annual cost savings attained by road users (if repaired)– Durability life of the repair

Budget of 16.3 M€ allowing some 160 projects Prevailing praxis based mainly on one criterion

– Benefit to cost analysis and manual iteration w.r.t. the damage coverage


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Illustrative data analysis with RPM tools Three pre-set incomplete weight specifications

– No information

– Rank-ordering

– Rank order centroid wroc = (0.61, 0.28, 0.11) and 10% relative interval on each

criterion

– Set inclusion

Rank-ordering set by experts at Finnish Road Administration Complete score information

0 | 0, 1w i iS w w w 1 2 3| , 1rank

w iS w w w w w

| 0.9 1.1 , 1roc roc rocw i i i iS w w w w w

0 rank rocw w wS S S


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Evolution of the core index w.r.t. completeness of information Approximate core indexes

– Computed from the set of potentially optimal (supported efficient) portfolios

Prior decision as a reference– Dominating solutions found

– Similar performance w.r.t. all criteria can be reached at 1.3M€ lower cost

Positive feedback– Transparent and simple model

– Use of incomplete preference information

– Downsizing the manual iteration task


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No information, 542 portfolios 103 core projects 16 exterior projects Augmentation:

some 60 out of 104

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Rank ordering, 109 portfolios 127 core projects 32 exterior projects Augmentation:

some 30 out of 64

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Rank order centroid

variation, 4 portfolios 152 core projects 60 exterior projects Augmentation:

some 5 out of 11 4 projects from the

optimal portfolio at

wroc are sensitive to

the variation

rocwS


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Recent applications of RPMRecent applications of RPM Road pavement project selection Strategic product portfolio selection

– A telecommunications company setting a product strategy– Some 50 products for which a yes/no decision had to be made– A group decision, score intervals to capture the opinions of all stakeholders– Core indexes were used to describe the attractiveness of projects

Ex post evaluation of an innovation programme– Scoring model derived from ex post evaluation data – Incomplete criterion weights– Comparative analysis between the sets of core and exterior projects– Identifying success factors from ex ante data

Paper machine efficiency analysis– Paper quality modeled through multicriteria overall value– Selecting the sets best and worst production periods– Comparative analysis between the sets of core and exterior projects


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Conclusions (1/2) Conclusions (1/2)

Systematic and structured process– Each project proposal treated equally

– Gradual selection tentative conclusions at any stage

– Helps focus attention to critical projects (the borderline projects)

Transparency– Simple and transparent model

– Intuitive performance measures on different units of analysis

Effect of uncertainty on individual projects– Gradual selection: at which step a project is included in the core

– Gradual “what if” analysis: which projects are jeopardized by which variation

Robustness through integrated sensitivity analysis


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Conclusions (2/2) Conclusions (2/2)

Groups statements through the use of intervals – Negotiation over the borderline projects

– Select a portfolio that best satisfies all views

Project interdependencies– Synergies, mutually exclusive projects or strategic balance requirements can be

modeled with linear constraints

– Knapsack formulation becomes a general multi-objective integer linear

programming problem

– Need for new algorithms that handle score intervals


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References References

» Kleinmuntz, C.E, Kleinmuntz, D.N., (1999). Strategic approach to allocating capital in healthcare organizations, Healthcare Financial Management, Vol. 53, pp. 52-58.

» Lindstedt, M., Hämäläinen, R.P., Mustajoki, J., (2001). Using Intervals for Global Sensitivity Analysis in Multiattribute Value Trees, in M. Köksalan and S. Zionts (eds), Lecture Notes in Economics and Mathematical Systems 507, pp. 177 - 186.

» Memtsas, D., (2003). Multiobjective Programming Methods in the Reserve Selection Problem, European Journal of Operational Research, Vol. 150, pp. 640-652.

» Stummer, C., Heidenberg, K., (2003). Interactive R&D Portfolio Analysis with Project Interdependencies and Time Profiles of Multiple Objectives, IEEE Trans. on Engineering Management, Vol. 50, pp. 175 - 183.


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Gradual selection in RPMGradual selection in RPMDecision rules, e.g. minimax regret

No

t sele

cted

•Narrowerintervals•Stricter weights

•Wide intervals•Loose weight

statements

Model robustness through incomplete information (cf. integrated sensitivity analysis).Account for group statements

Large number

of projects.

Evaluated w.r.t.

multiple criteria.

Border line projects

“uncertain zone” Focus

Exterior projects

“Robust zone” Discard

Core projects“Robust zone”

Choose

Negotiation.Manual iteration.

Gradual selection => transparency w.r.t. individual projects

Se

lecte

d

Core

Border

Exterior

Documents

1 Helsinki University of Technology Systems Analysis Laboratory Multi-Criteria Capital Budgeting with Incomplete Preference Information Pekka Mild, Juuso