1. 1 Helsinki University of Technology Systems Analysis Laboratory 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. Helsinki University of Technology Systems Analysis Laboratory 2 Multi-criteria capital budgeting (1/2) n Choose a subset of projects, a project portfolio, from a large set of proposals (e.g. 50) subject to scarce resources n Each project evaluated w.r.t. multiple criteria n Project value as a weighted sum of criterion-specific scores n Portfolio value as sum its constituent projects’ values n Several application areas, e.g. –Healthcare systems (Kleinmuntz & Kleinmuntz, 1999) –R&D project portfolios (Stummer & Heidenberger, 2003) –Nature conservation (Memtsas, 2003)

3. Helsinki University of Technology Systems Analysis Laboratory 3 Multi-criteria capital budgeting (2/2) n 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. Helsinki University of Technology Systems Analysis Laboratory 4 Incomplete preference information (1/2) n 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) n Interval scores –Lower and upper bounds for the criterion-specific scores of each project

5. Helsinki University of Technology Systems Analysis Laboratory 5 Incomplete preference information (2/2) n Portfolio p dominates p’ () iff –The value of projects included in both portfolios is canceled  pairwise dominance check is an LP-problem n 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. Helsinki University of Technology Systems Analysis Laboratory 6 Computation of non-dominated portfolios (1/2) n Dominance checks require pairwise comparisons n Number of possible portfolios is high –m projects lead to 2 m 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 n 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. Helsinki University of Technology Systems Analysis Laboratory 7 Computation of non-dominated portfolios (2/2) n 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 n 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. Helsinki University of Technology Systems Analysis Laboratory 8 Robust Portfolio Modeling (RPM) n 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 n 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. Helsinki University of Technology Systems Analysis Laboratory 9 RPM for project portfolio selection (1/4) n 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) n 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. Helsinki University of Technology Systems Analysis Laboratory 10 RPM for project portfolio selection (2/4) n 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 n 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. Helsinki University of Technology Systems Analysis Laboratory 11 RPM for project portfolio selection (3/4) n 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 n 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. Helsinki University of Technology Systems Analysis Laboratory 12 Approach to promote robustness through incomplete information (integrated sensitivity analysis). Account for group statements RPM for project portfolio selection (4/4) 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. Selected Not selected Gradual selection: Transparency w.r.t. individual projects Tentative conclusions at any stage of the process

13. Helsinki University of Technology Systems Analysis Laboratory 13 Application to road pavement projects (1/6) n Real-life data from Finnish Road Administration –Selection of the annual pavement programme in one major road district n Large set of m = 223 project proposals –Generated by a specific road condition follow-up system –Coherent road segments  proposals are considered independent n 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 n Budget of 16.3 M€ allowing some 160 projects n Prevailing praxis based mainly on one criterion –Benefit to cost analysis and manual iteration w.r.t. the damage coverage

14. Helsinki University of Technology Systems Analysis Laboratory 14 Application to road pavement projects (2/6) n Illustrative data analysis with RPM tools n Three pre-set incomplete weight specifications –No information –Rank-ordering –Rank order centroid w roc = (0.61, 0.28, 0.11) and  10% relative interval on each criterion –Set inclusion n Rank-ordering set by experts at Finnish Road Administration n Complete score information

15. Helsinki University of Technology Systems Analysis Laboratory 15 Application to road pavement projects (3/6) n Evolution of the core index w.r.t. completeness of information n Approximate core indexes –Computed from the set of potentially optimal (supported efficient) portfolios n Prior decision as a reference –Dominating solutions found –Similar performance w.r.t. all criteria can be reached at 1.3M€ lower cost n Positive feedback –Transparent and simple model –Use of incomplete preference information –Downsizing the manual iteration task

16. Helsinki University of Technology Systems Analysis Laboratory 16 Application to road pavement projects (4/6) n No information, n 542 portfolios n 103 core projects n 16 exterior projects n Augmentation: some 60 out of 104

17. Helsinki University of Technology Systems Analysis Laboratory 17 Application to road pavement projects (5/6) n Rank ordering, n 109 portfolios n 127 core projects n 32 exterior projects n Augmentation: some 30 out of 64

18. Helsinki University of Technology Systems Analysis Laboratory 18 Application to road pavement projects (6/6) n Rank order centroid  variation, n 4 portfolios n 152 core projects n 60 exterior projects n Augmentation: some 5 out of 11 n 4 projects from the optimal portfolio at w roc are sensitive to the variation

19. Helsinki University of Technology Systems Analysis Laboratory 19 Recent applications of RPM n Road pavement project selection n 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 n 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 n 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. Helsinki University of Technology Systems Analysis Laboratory 20 Conclusions (1/2) n 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) n Transparency –Simple and transparent model –Intuitive performance measures on different units of analysis n 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 n Robustness through integrated sensitivity analysis

21. Helsinki University of Technology Systems Analysis Laboratory 21 Conclusions (2/2) n Groups statements through the use of intervals –Negotiation over the borderline projects –Select a portfolio that best satisfies all views n 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. Helsinki University of Technology Systems Analysis Laboratory 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 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.

23. Helsinki University of Technology Systems Analysis Laboratory 23 Gradual selection in RPM Decision rules, e.g. minimax regret Not selected Narrower intervals 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 Selected Core Border Exterior