Transcript Uniikki kuitu - Systeemianalyysin laboratorio, Aalto

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
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Helsinki University of Technology
Systems Analysis Laboratory
Multi-criteria capital budgeting (1/2)
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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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Helsinki University of Technology
Systems Analysis Laboratory
Multi-criteria capital budgeting (2/2)
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Find a feasible portfolio which maximizes the overall value
x j  X , j  1,...,m
– Criteria, i = 1,…,n  scores vij  [v]ij , weights w  w1 ,....,wn T
– Large number of projects
– Project value V ( x ) 
j
n
wv
i 1
j
i i
– Portfolio p  X , p  P  P( X ) , overall value V ( p) 
– Resources k = 1,…,q  resource consumption c
j
k
– Budget vector
C ( p) 
j
x j p
B  [B1,...,Bq ]T , the set of feasible portfolios PF  P
 C( x )   [c ,..., c ]
j
x j p
V ( x )
j
1
j T
q
 B p  PF
x j p
 With precise weights and scores the optimal portfolio is obtained as a solution
to the binary LP-problem max V ( p )
pPF
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Helsinki University of Technology
Systems Analysis Laboratory
Incomplete preference information (1/2)
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Set of feasible weights Sw  Sw0  w | wi  0,  wi  1 
– Linear constraints
– Several weight vectors are consistent with the given preference statements
– E.g. criterion 1 is the most important of three criteria
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S w  w  S w0 | w1  w2 , w1  w3
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– Interval sensitivity analysis (cf. Lindstedt et al., 2001)
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Interval scores Sv  R nm
– Lower and upper bounds for the criterion-specific scores of each project
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Sv  v  R n+m | vi  vij  vi j
j

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Helsinki University of Technology
Systems Analysis Laboratory
Incomplete preference information (2/2)
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Portfolio p dominates p’ ( p  p' ) iff
 minV ( p \ p' )  V ( p'\ p )  0
 V ( p)  V ( p' )w  S w , v  S v
 wS w

,
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maxV ( p \ p' )  V ( p'\ p)  0
w  S w , v  S s.t.V ( p)  V ( p' )

 wS w
where V ( p) 
n
 wi v , V ( p) 
x j  p i 1
j
i
n
j
w
v
 i i
x j  p i 1
– The value of projects included in both portfolios is canceled
 pairwise dominance check is an LP-problem
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The set of non-dominated portfolios
PN   p  PF | p '
p p '  PF 
– With precise scores and no a priori weight information (i.e. Sw  Sw0 ), the set of
non-dominated portfolios corresponds to the set of Pareto-optimal solutions
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Helsinki University of Technology
Systems Analysis Laboratory
Computation of non-dominated portfolios (1/2)
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Dominance checks require pairwise comparisons
Number of possible portfolios is high
– m projects lead to 2m possible portfolios, i.e. | P | 2m
– 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
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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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Helsinki University of Technology
Systems Analysis Laboratory
Computation of non-dominated portfolios (2/2)
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Outline of the algorithm
– Portfolios that use resources efficiently are stored in
– Projects x j are added one by one,
1) Let PN'  {{0
 },{x1}}
x1,...,xm
PN'
2) For j=2,…,m do
a) PN'  PN' { p  PF | p  x j  p' , p' PN' }
b) PN'  { p  PN' | p' PN' s.t. p'  p  C( p' )  C( p)}
3) Obtain
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PN  { p  PN' | p'  pp' PN' }
Effective implementation
– If P ' is sorted by portfolio cost, fewer pairwise comparisons are needed in 2b)
N
'
– The size of PN can be reduced by discarding portfolios that cannot end-up nondominated by adding projects
x j 1 ,...,xm
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Helsinki University of Technology
Systems Analysis Laboratory
Robust Portfolio Modeling (RPM)
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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
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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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Helsinki University of Technology
Systems Analysis Laboratory
RPM for project portfolio selection (1/4)
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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)
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Project-oriented selection
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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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Helsinki University of Technology
Systems Analysis Laboratory
RPM for project portfolio selection (2/4)
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Core index of a project
– Share of non-dominated portfolios in which a project is included
|{ p  PN | x j  p}|
CI ( x ) 
| PN |
j
– Project-specific performance measure derived in the portfolio context
» Accounts for competing projects, scarce resources and other portfolio constraints
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Core and exterior
– Core projects are included in all non-dominated portfolios, CI ( x j )  1
– Exterior projects are not included in any of the nd-portfolios, CI ( x j )  0
– Border line projects are included in some of the nd-portfolios, 0  CI ( x j )  1
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Helsinki University of Technology
Systems Analysis Laboratory
RPM for project portfolio selection (3/4)
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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
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Core and exterior expand with more complete information
– Additional information (s.t. Sw  Sw , Sv  Sv ) can reduce the set PN
– No new portfolio can become non-dominated
– Unique portfolio has no borderline projects
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Helsinki University of Technology
Systems Analysis Laboratory
RPM for project portfolio selection (4/4)
•Wide
intervals
•Loose weight
statements
Border line projects
“uncertain zone”
 Focus
Exterior projects
“Robust zone”
 Discard
Approach to promote robustness
through incomplete information
(integrated sensitivity analysis).
Account for group statements
Gradual selection:
Core
•Narrower intervals
•Stricter weights
Border
Exterior
Not selected
Large number
of projects.
Evaluated w.r.t.
multiple criteria.
Selected
Decision rules,
e.g. minimax
regret
Core projects
“Robust zone”
 Choose
Negotiation.
Manual iteration.
Heuristic rules.
Transparency w.r.t. individual projects
Tentative conclusions at any stage of the process
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Helsinki University of Technology
Systems Analysis Laboratory
Application to road pavement projects (1/6)
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Real-life data from Finnish Road Administration
– Selection of the annual pavement programme in one major road district
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Large set of m = 223 project proposals
– Generated by a specific road condition follow-up system
– Coherent road segments  proposals are considered independent
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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
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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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Helsinki University of Technology
Systems Analysis Laboratory
Application to road pavement projects (2/6)
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Illustrative data analysis with RPM tools
Three pre-set incomplete weight specifications
– No information S w0 
– Rank-ordering S wrank
 w | w  0,  w  1 
  w | w  w  w ,  w  1
i
i
1
2
3
i
– Rank order centroid wroc = (0.61, 0.28, 0.11) and 10% relative interval on each
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criterion S wroc  w | 0.9 wiroc  wi  1.1wiroc ,
– Set inclusion
Sw0  Swrank  Swroc
w
i
1

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Rank-ordering set by experts at Finnish Road Administration
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Complete score information
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Helsinki University of Technology
Systems Analysis Laboratory
Application to road pavement projects (3/6)
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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
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Prior decision as a reference
– Dominating solutions found
– Similar performance w.r.t. all criteria can be reached at 1.3M€ lower cost
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Positive feedback
– Transparent and simple model
– Use of incomplete preference information
– Downsizing the manual iteration task
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Helsinki University of Technology
Systems Analysis Laboratory
Application to road pavement projects (4/6)
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No information, Sw0
542 portfolios
103 core projects
16 exterior projects
Augmentation:
some 60 out of 104
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Helsinki University of Technology
Systems Analysis Laboratory
Application to road pavement projects (5/6)
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Rank ordering, Swrank
109 portfolios
127 core projects
32 exterior projects
Augmentation:
some 30 out of 64
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Helsinki University of Technology
Systems Analysis Laboratory
Application to road pavement projects (6/6)
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Rank order centroid
 variation, Swroc
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
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Helsinki University of Technology
Systems Analysis Laboratory
Recent applications of RPM
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Road pavement project selection
Strategic product portfolio selection
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–
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Ex post evaluation of an innovation programme
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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
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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Helsinki University of Technology
Systems Analysis Laboratory
Conclusions (1/2)
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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)
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Transparency
– Simple and transparent model
– Intuitive performance measures on different units of analysis
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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
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Robustness through integrated sensitivity analysis
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Helsinki University of Technology
Systems Analysis Laboratory
Conclusions (2/2)
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Groups statements through the use of intervals
– Negotiation over the borderline projects
– Select a portfolio that best satisfies all views
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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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Helsinki University of Technology
Systems Analysis Laboratory
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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Helsinki University of Technology
Systems Analysis Laboratory
Gradual selection in RPM
Core projects
“Robust zone”
 Choose
•Wide
intervals
•Loose weight
statements
Border line projects
“uncertain zone”
 Focus
Exterior projects
“Robust zone”
 Discard
Model robustness through
incomplete information
(cf. integrated sensitivity analysis).
Account for group statements
•Narrower
intervals
•Stricter
weights
Core
Border
Exterior
Not selected
Large number
of projects.
Evaluated w.r.t.
multiple criteria.
Selected
Decision rules,
e.g. minimax
regret
Negotiation.
Manual
iteration.
Gradual selection => transparency w.r.t. individual projects
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