Multi-Criteria Capital Budgeting with Incomplete Preference Information

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Multi-Criteria Capital Budgeting with Incomplete
Preference Information

• Pekka Mild, Juuso Liesiö and Ahti Salo
• Systems Analysis Laboratory
• Helsinki University of Technology
• P.O. Box 1100, 02150 HUT, Finland
• http//www.sal.hut.fi

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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)
• RD project portfolios (Stummer Heidenberger,
  2003)
• Nature conservation (Memtsas, 2003)

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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)

• 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)

• 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)

• 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 m20 takes one second, then m40 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

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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 j2,,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)

• 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)

• 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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RPM for project portfolio selection (2/4)

• 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)

• 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

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RPM for project portfolio selection (4/4)
Decision rules, e.g. minimax regret
Selected
Large numberof projects. Evaluated w.r.t.
multiple criteria.
Core projects Robust zone ? Choose

• Border line projectsuncertain zone
• Focus

Core
Wide intervals Loose weight statements
Narrower intervals Stricter weights
Border
Not selected
Exterior
Exterior projectsRobust zone ? Discard
Negotiation. Manual iteration. Heuristic rules.
Approach to promote robustness through incomplete
information (integrated sensitivity
analysis). Account for group statements
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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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Application to road pavement projects (2/6)

• 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

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Application to road pavement projects (3/6)

• 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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Application to road pavement projects (4/6)

• No information,
• 542 portfolios
• 103 core projects
• 16 exterior projects
• Augmentationsome 60 out of 104

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Application to road pavement projects (5/6)

• Rank ordering,
• 109 portfolios
• 127 core projects
• 32 exterior projects
• Augmentationsome 30 out of 64

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Application to road pavement projects (6/6)

• Rank order centroid ? variation,
• 4 portfolios
• 152 core projects
• 60 exterior projects
• Augmentationsome 5 out of 11
• 4 projects from the optimal portfolio at wroc are
  sensitive to the variation

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Recent 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)

• 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)

• 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

• 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
  RD 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 RPM
Decision rules, e.g. minimax regret
Selected
Large numberof projects. Evaluated w.r.t.
multiple criteria.
Core projects Robust zone ? Choose

• Border line projectsuncertain zone
• Focus

Narrowerintervals Stricter weights
Wide intervals Loose weight statements
Not selected
Exterior projectsRobust zone ? Discard
Negotiation. Manual iteration.
Model robustness through incomplete information
(cf. integrated sensitivity analysis). Account
for group statements
Gradual selection gt transparency w.r.t.
individual projects