Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Measurable Value Functions for Project
Portfolio Selection and Resource
Allocation
Dr. Juuso Liesiö
Systems Analysis Laboratory
Aalto University
P.O. Box 11100, 00076 Aalto, Finland
http://www.sal.tkk.fi
juuso.liesio@aalto.fi
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Multi-criteria project portfolio selection
 Choose a subset (=a portfolio) of projects from a large set
of proposals
– Projects evaluated on multiple criteria
– Resource and other portfolio constraints
– Maximize multi-criteria portfolio value subject to resource constraints
→ Zero-one linear programming problem
 Applications
– Healthcare (Kleinmuntz, 2007), R&D (Golabi et al., 1981), infrastructure
asset management (Liesiö et al., 2007), military (Ewing et al., 2006),
strategy development (Lindstedt et al., 2008) etc.
 Additive-linear portfolio value function widely used
 We derive a more general class of portfolio value functions
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Standard additive-linear portfolio value
Project 1 X11
Project j
Xji
...
Xjn
...
– Does not often hold in practice (Kleinmuntz, 2007;
Mild and Salo, 2009)
Project m Xm1
...
...
 Constant marginal value on each criterion
...
– wi : ‘Importance’ weight of criterion i
Xj1
...
– vi(xji ) ϵ [0,1]: The value of project j w.r.t. crit. i (score)
...
...
– xji ϵ Xji : Performance of project j w.r.t. criterion i
... X1i ... X1n
... Xmi ... Xmn
→m x n attributes in total
– E.g., as financial performance increases other criteria
become relatively more important
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Example: choosing
forest sites for
conservation
Additive-linear portfolio value model
Scoring
Weighting
w1=0.6
w2=0.3
w3=0.1
Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Measurable value functions (Dyer and Sarin, 1979)
 V represents two preference relations
1. Portfolio xa is preferred to portfolio xb
2. A change from xb to xa is preferred to a change from xd to xc
→ Measurable value is “cardinal” or “captures strength of preference”
 If the subset of attributes X’ is…
… Preference Independent (PI), then preference order (1.) of levels of X’ does
not depend on the levels of the other attributes
… Weak Difference Independent (WDI), then preference order (2.) of changes
between levels of X’ does not depend on the levels of the other attributes
– WDI implies PI
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Additive-linear portfolio VF: assumptions
A1. Equitable treatment of projects
Portfolio value does not change if project
indexing is changed
Project 1 X11
...
Xjn
...
Project m Xm1
Xji
...
A4. Xj1 x…x Xjn is WDI for each j=1,…,m
...
...
Each criterion is a portfolio performance measure
Xj1
...
A3. X1i x…x Xmi is WDI for each i=1,…,n
Project j
...
...
A2. Each Xji is WDI
Makes project scoring possible
... X1i ... X1n
... Xmi ... Xmn
Value of including a project into the portfolio does
not depend on other projects in the portfolio
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Weaker preference assumptions
A1. Equitable treatment of projects
Project 1 X11
Xji
...
...
Xjn
...
Project m Xm1
...
...
Each criterion is a portfolio performance measure
Xj1
...
A3. X1i x…x Xmi is WDI for each i=1,…,n
Project j
...
A2. Each Xji is WDI
Makes project scoring possible
... X1i ... X1n
...
Portfolio value does not change if project
indexing is changed
... Xmi ... Xmn
 Theorem. A1-A3 hold iff V(x)=f(V1(xJ1),...,Vn(xJn)), where
– V1..,Vn are the symmetric multilinear criterion specific portfolio value functions
and xJi =(x1i,…,xmi)T
– f is a multilinear function (multiplicative and additive function are special cases)
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Symmetric multilinear criterion specific portfolio VF
 vi(xji )ϵ[0,1]: value of project j w.r.t. criterion i (score)
 wi (.): weighting function for criterion i
– wi(k)=Vi(xJi), where portfolio xJi has k projects at the most preferred performance
level in criterion i and m-k projects at the least preferred performance level
– E.g. k sites with “Endangered species = 100” and 50-k sites with “Endangered
species = 0”
 If wi(k+1)-wi(k)=constant for all k, then Vi reduces to the linear
criterion specific value (sum of scores)
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Multilinear vs. linear criterion specific value (1/2)

m=10 projects

X-axis: sum of scores
∑vi(xji)

Y-axis: sym. multilinear
Portfolio x with scores
(6/10,...,6/10)
crit. specific value Vi(xJi)

Black dots: weighting
function wi(k), k=1,…,10

Portfolio x with scores
(1,1,1,1,1,1,0,0,0,0)
Gray: range of possible
values for Vi when the
sum of scores is fixed
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Multilinear vs. linear criterion specific value (2/2)

m=50 projects

X-axis: sum of scores

Y-axis: sym. multilinear value

Black dots: wi(k), k=1,…,50

Gray: range of Vi
gi
→ When m is large and wi is
‘smooth’, a function gi can be
chosen such that
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Additive-linear portfolio value model
Example revisited
Scoring
Weighting
w1=0.6
w2=0.3
w3=0.1
Example revisited
Additive-multilinear portfolio value model
Scoring
Weighting
Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Conclusions
 If the project’s value depends on how it “fits” the portfolio, then
– The criterion specific portfolio value functions are symmetric multilinear
– These are aggregated using a multilinear, a multiplicative or an additive function
 Use of the additive-multilinear VF straightforward in decision
support processes based on the additive-linear VF
– No need to re-do/change criteria, feasibility constraints or project scoring
– Criterion weights depend on the portfolio’s performance w.r.t. to the criterion
 Optimization of portfolio value
– Implicit enumeration when the number of projects < 100
– MILP approximation for large additive-multilinear problems (piecewise linear gi )
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Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
J. Liesiö: Measurable Multiattribute Value Functions for Portfolio Decision Analysis
http://www.sal.tkk.fi/publications/pdf-files/mlie11.pdf
References:
Dyer, J., Sarin, R., (1979). Measurable Multiattribute Value Functions, Operations Research, Vol. 27, pp.
810-822.
Ewing Jr., P.L., Tarantino, W., Parnell, G.S., (2006). Use of Decision Analysis in the Army Base
Realignment and Closure (BRAC) 2005 Military Value Analysis, Decision Analysis, Vol. 3, pp. 33-49.
Golabi, K., Kirkwood, C.W., Sicherman, A., (1981). Selecting a Portfolio of Solar Energy Projects Using
Multiattribute Preference Theory, Management Science, Vol. 27, pp. 174-189.
Kleinmuntz, D.N., (2007). Resource Allocation Decisions, in Edwards, W., Miles, R.F. & von Winterfeldt,
D. (Eds.); Advances in Decision Analysis, Cambridge University Press.
Liesiö, J., Mild, p., Salo, A., (2007). Preference Programming for Robust Portfolio Modeling and Project
Selection, European Journal of Operational Research, Vol. 181, pp. 1488-1505.
Lindstedt, M., Liesiö, J., Salo, A., (2008). Participatory Development of a Strategic Product Portfolio in a
Telecommunication Company, International Journal of Technology Management, Vol. 42, pp. 250-266.
Mild P., Salo, A., (2009). Combining a Multiattribute Value Function with an Optimization Model: An
Application to Dynamic Resource Allocation for Infrastructure Maintenance, Decision Analysis, Vol. 6,
pp. 139-152.
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