Multi-Criteria Capital Budgeting with Incomplete Preference Information

1. 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

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)

3. 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

4. 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

5. 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

6. Computation of non-dominated portfolios (1/2) • Dominance checks require pairwise comparisons • Number of possible portfolios is high • m projects lead to 2mpossible 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

7. 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

8. 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?”

9. 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

10. 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,

11. 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

12. Transparency w.r.t. individual projectsTentative conclusions at any stage of the process Gradual selection: RPM for project portfolio selection (4/4) Decision rules, e.g. minimax regret Selected Core projects “Robust zone”  Choose Large numberof projects. Evaluated w.r.t. multiple criteria. • Border line projects“uncertain zone” • Focus Core •Wide intervals •Loose weight statements •Narrower intervals •Stricter weights Border Not selected Exterior Exterior projects“Robust zone”  Discard Negotiation. Manual iteration. Heuristic rules. Approach to promote robustness through incomplete information (integrated sensitivity analysis). Account for group statements

13. 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

14. 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

15. 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

16. Application to road pavement projects (4/6) • No information, • 542 portfolios • 103 core projects • 16 exterior projects • Augmentation:some 60 out of 104

17. Application to road pavement projects (5/6) • Rank ordering, • 109 portfolios • 127 core projects • 32 exterior projects • Augmentation:some 30 out of 64

18. Application to road pavement projects (6/6) • 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

19. 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

20. 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

21. 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

22. 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 Systems507, 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.

23. Core Border Exterior Gradual selection in RPM Decision rules, e.g. minimax regret Selected Core projects “Robust zone”  Choose Large numberof projects. Evaluated w.r.t. multiple criteria. • Border line projects“uncertain zone” • Focus •Wide intervals •Loose weight statements •Narrowerintervals •Stricter weights Not selected Exterior projects“Robust zone”  Discard Negotiation. Manual iteration. Model robustness through incomplete information (cf. integrated sensitivity analysis). Account for group statements Gradual selection => transparency w.r.t. individual projects