PROBE–A multicriteria decision support system for portfolio robustness evaluation

PROBE–A multicriteria decision support system for portfolio robustness evaluation

Working Paper LSEOR 09.108

ISSN 2041-4668 (Online)

First published in Great Britain in 2009

by the Operational Research Group, Department of Management

London School of Economics and Political Science

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Working Paper No: LSEOR 09.108

Abstract: This paper deals with the problem of selecting a robust portfolio

of projects in the presence of limited resources, multiple criteria, different project interactions and several types of uncertainty We present a new decision support system, PROBE – Portfolio Robustness Evaluation, and the algorithms it implements PROBE identifies all efficient portfolios, either convex or non-convex efficient, depicting the respective Pareto frontier, within a given portfolio cost range, and allows performing in- depth interactive analysis of the robustness of selecting a proposed portfolio

Keywords: Portfolio decision analysis, resource allocation, portfolio robustness,

DSS

Working Paper LSEOR 09.108

1 Introduction

Suppose that a manager has several projects, each one expected to add value to his organization The manager has associated a forward cost for each project and realized that there is not enough money to fund them all Therefore, he wants to select a portfolio of projects providing the organization with the best value for money An exhaustive analysis of all possible portfolios, from the empty portfolio (in which no projects are funded and no benefits are realized) to the full portfolio (that would require all projects to be funded), would

be impractical even if the number of projects is not too big For example, 20 projects will give more than one million portfolios (precisely 220 = 1,048,576)

A practical selection strategy would be to prioritize the projects in decreasing order of their benefit-to-cost ratios and proceeding down the list until the available budget is exhausted (Lorie and Savage 1955, Edwards 1977, Kirkwood 1997, Kleinmuntz and Kleinmuntz 1999, Bana e Costa et al 2006, Buede and Bresnick 2007, Phillips and Bana e Costa 2007) The portfolio selected by this approach produces the highest benefit for the money spent, but may not be the one that produces the maximum benefit for the money available Alternatively, the optimization approach could be followed It consists in searching for the portfolio with the highest benefit for the budget available, by solving a (knapsack) mathematical programming problem (Martello and Toth 1990, Kellerer et al 2004) that maximizes cumulative benefit without exceeding the budget constraint (Weingartner 1963, Golabi et al 1981, Kirkwood 1997, Kleinmuntz 2007)

As shown in Section 2, the portfolio selected by each one of these two approaches, for the same fixed budget, is not necessarily the same, because the portfolio selected by the prioritization approach cannot include any project with a lower benefit-to-cost ratio (that is, a

less “productive” or “profitable” project - Cooper et al 1999, Brealey and Myers 2003) than

a non-selected project This is not necessarily the case of the portfolio selected by the optimization approach Arguments favoring each one of the two approaches are discussed in Section 2

Resource allocation decisions often require project portfolio selections involving multiple quantitative and qualitative benefit dimensions (or criteria) In a previous paper (Lourenço et al 2008), we studied commercial off-the-shelf software for multicriteria portfolio analysis based on additive aggregation of multiple benefit criteria: Equity (Catalyze 2008), HiPriority (Krysalis 2007), Logical Decisions Portfolio (Logical Decisions 2008) and Expert Choice Resource Aligner (Expert Choice 2007) Equity and HiPriority follow the

prioritization approach, Expert Choice Resource Aligner (Expert Choice 2007) follows the optimization approach, and Logical Decisions Portfolio implements the two approaches

Section 3 introduces PROBE – Portfolio Robustness Evaluation, a new decision

support system for multicriteria portfolio analysis that implements the optimization approach but also finds the solutions given by the prioritization approach When several benefit criteria are defined, PROBE calculates the benefit value of each project by an additive value model Therefore, the basic project inputs for a multicriteria portfolio analysis are the cost of each project and its value scores on each one of the benefit criteria, and the weights that capture trade-offs between criteria (see Section 3.1) Different types of project interactions or interdependencies can also be inputted (see Section 3.2)

For a resource allocation model defined with the data inputted by the user, PROBE is able to identify all efficient portfolios, distinguishing the convex from the non-convex efficient ones, and depicts the respective Pareto frontier, running the algorithms presented in Section 4.1 Several sources of uncertainty are often present in real-world resource allocation contexts, affecting the precision of some of the inputs Many times, a “best” portfolio is

selected on the basis of “best guess” input data only It would then be wise to analyze the

robustness of that “best” portfolio taking into account, simultaneously, uncertain data

affecting the costs and benefits of the portfolios, that is, imprecise project costs and benefit values and criteria weights Portfolio robustness evaluation (see Section 4.2) has been the core motivation for the conception of PROBE, because our previous study of commercial packages (Lourenço et al 2008) revealed that none of them addresses robustness analysis The example in Section 5 illustrates how a robustness analysis can be conducted with the DSS PROBE within a given uncertainty domain Some final remarks about our approach are presented in Section 6, namely a short comparison with the pioneer research of Liesiö et al (2007, 2008) on “robust portfolio modeling (RPM)”

2 Basic concepts and portfolio selection approaches

In this paper, portfolio selection is only concerned with a set X of m projects that worth funding, therefore assuming that project proposals that do not worth funding were screened out in a previous phase of the selection process Conceptually, the benefit value of a project that adds no value to a portfolio should be zero; consequently, the benefit value of a project that worth funding should be defined as the value that it adds to the portfolio Let cj > 0 and

vj > 0 be, respectively, the cost and the benefit value of project j of X (j = 1,…, m) and B the

budget available For simplicity, without loss of generality, we assume that the m projects of

X are presented in decreasing order by their benefit-to-cost ratio rj = vj/cj (j = 1,…, m) such

that rp ≥ rp+1, p = 1,…, m-1, as the four projects (1, 2, 3 and 4) in Table 1 are

Table 1 Benefit values, costs and benefit-to-cost ratios of four projects

Let vij be the value score of project j on the benefit criterion i, i = 1,…, n (n1) and

wi (wi  ) the weight of criterion i, i = 1,…, n (with 0 n1 i 1

Figure 1 Chart showing the portfolios that can be formed with the four projects

Note Efficient portfolios are represented by triangular dots and dominated portfolios by squared dots

0 2 4 6 8 10 12

The efficient portfolios are shown as triangular dots and form the efficient frontier (or Pareto-front), associated for simplicity with the dotted line in Figure 1, whereas the dashed line links only convex efficient portfolios (the convex efficient frontier, which is the convex hull of the Pareto-front) As can be seen in Figure 1, there are two efficient portfolios that are non-convex efficient (those that belong to the dotted line but not to the dashed line): portfolio {2} and portfolio {1, 2, 4} Given a fixed budget B, the prioritization approach selects the convex efficient portfolio formed by the projects j, j = 1,…, k with k ≤ m, such that

j c B

 In this approach, the notion of “value-for-money” of a project

(Bana e Costa et al 2006, Phillips and Bana e Costa 2007, Phillips 2007) or its

“bang-for-the-buck” (Cooper et al 2001, Brealey and Myers 2003, Buede and Bresnick 2007) is associated

with the slope of the benefit-to-cost triangle of each project, as shown in Figure 2 for the four projects of Table 1 The last column of Table 1 shows that the order of selection by prioritization would be: first project 1, then project 2, followed by project 3 and finally project 4 When the budget increases from 0 to 30 (see Figure 2), the sequence of selected portfolios, from the empty portfolio {} to the full portfolio {1, 2,3, 4}, starts by portfolio {1} for 4 ≤ B < 12, followed by portfolio {1,2} for 12 ≤ B < 22, and then portfolio {1,2,3}for 22

≤ B < 30

Figure 2 Cumulative cost versus cumulative benefit chart showing the portfolios formed by

the benefit-to-cost ratio approach

Notes The value-for-money of each project is given by the slope of its benefit-to-cost triangle The arrow in the value-for-money slopes box shows the direction of improvement of the benefit-to-cost value of the projects

The composition of each portfolio is shown in brackets next to the corresponding dot

Alternatively, the portfolio selected by the optimization approach is the optimal solution of the following mathematical programming problem (known as the knapsack problem):

0 2 4 6 8 10 12

j j j j

12 (see Figure 2) Figure 1 shows that both portfolios are efficient, but it seems that

optimization identifies a “better” portfolio than prioritization This is true for a manager only

concerned with getting more benefit (9 > 7), but not necessarily if the “bang-for-the-buck”

received also matters (9/20 < 7/12) Note that portfolio {1, 2, 4} includes project 4, which is

“less productive” than the non-selected project 3 (because r4 < r3 – see Table 1) Kirkwood

(1997, Chapter 8.1) and Kleinmuntz (2007) also briefly discuss pros and cons of these two approaches

Any efficient portfolio can be selected by the optimization approach, whereas the prioritization approach always selects a convex efficient portfolio When the optimal solution of the knapsack problem (4) is a convex efficient portfolio, the portfolio selected by prioritization is the same; but, when the optimal solution of problem (4) is a non-convex efficient portfolio, the portfolio selected by prioritization is the first convex efficient portfolio

at its left in the convex efficient frontier This portfolio could be found by constraining the knapsack problem in such a way that the optimal solution does not include any project with a lower benefit-to-cost ratio than a non-selected project.1

1 As proved by Dantzig (1957), a portfolio formed through the prioritization approach is included in the optimal solution of a (relaxed) knapsack problem in which the projects are assumed to be divisible (see also Martello and Toth 1979) The benefit-to- cost ratio algorithm is also designated by the “greedy algorithm for the knapsack problem” in (Kellerer et al 2004, Korte and Vygen 2006).

Finally, if there are at least two projects with the same benefit-to-cost ratio, some convex efficient portfolios are not identified by the prioritization approach However, the prioritization approach can easily deal with a large number of projects, contrary to knapsack optimization algorithms Indeed, the knapsack problem (4) is technically difficult to solve despite its straightforward structure, due to the integrality constraints xj{0,1}, j = 1,…, m.2

3 Introducing the DSS PROBE

3.1 The MCDA and PDA components and basic input data

PROBE is a multicriteria decision support system for portfolio robustness evaluation that integrates two main architectural components: a multicriteria decision analysis (MCDA) component and a portfolio decision analysis (PDA) component

The MCDA component allows the user to structure the benefit criteria in the form of a value tree and input data for the costs of the projects and their benefit scores on each bottom-level criterion of the value tree Let X be a specific set of projects j (j = 1,…, m) defined by

the user Even when uncertainty is present, PROBE always asks the user to input, for each project j, a (“best guess”) cost cj and (“best guess”) benefit value scores vij on each bottom-level criterion i, i = 1,…, n (n=1 if only one benefit dimension, such as NPV, is defined)

For a value tree with only one level of n>1 benefit criteria i (i = 1,…, n), (“best guess”)

weights wi (i = 1,…, n) should be introduced and PROBE computes the benefit value vj of each project j (j = 1,…, m) by applying the non-hierarchical additive model (1) If the value

tree has two or more levels below the root node, crisp weights are defined for the criteria at each level and PROBE uses a hierarchical value model to compute an aggregate benefit value

vj of each project j (j = 1,…, m) by applying model (1) bottom-up successively If a branch of

risk criteria is included in the value tree set of criteria, the vj of each project j is more

adequately designated by “risk-adjusted benefit” (Phillips and Bana e Costa 2007) For the

sake of simplicity, without loss of generality, all programs and algorithms presented in this paper assumes a non-hierarchical benefit model, which can be easily extended to the corresponding generic hierarchical formulation implemented in PROBE

2 The knapsack problem is considered to be a nondeterministic polynomial-time hard (NP-hard) problem (for a discussion on complexity and hardness see Garey and Johnson 1979) A significant number of exact and approximate resolution algorithms for this problem have been thoroughly studied (Martello and Toth 1990, Kellerer et al 2004)

For the given project costs cj and benefit scores vj (j = 1,…, m), the PDA component

solves the knapsack optimization problem (4) – with or without additional linear constraints

possibly added by the user to model project interactions (see Section 3.2) – for any fixed

budget B, finds all efficient portfolios, distinguishes convex from non-convex ones (see Section 4.1) – a functionality not included in the software packages analyzed in (Lourenço et

al 2008) – and displays the portion of the Pareto-front for an user-defined limited portfolio

cost range [ , ]B B (see section 4.1) When the number of projects of X is compatible with a reasonable computational time (see the Appendix), PROBE automatically displays the full efficient frontier, assuming by defaultB = 0 and

1

m j j

Concerning the modeling of uncertainty, PROBE allows the user to input: a set c of plausible cost ranges [ ,c cj j] such that cj cj cj (j = 1,…, m); a set v of plausible benefit scores ranges [ ,v vij ij] such that vij vij vij (i = 1,…, n; j = 1,…, m); linear relationships on

the weights (for example, weights rankings and/or weights ranges) defining a set w of feasible weights such that wi (i = 1,…, n) The MCDA component uses the additive w

model to calculate by optimization the feasible benefit value range [ ,v vj j]defined by v

w for each project j (j = 1,…, m) as follows: min 1

(available at http://sourceforge.net/projects/lpsolve/) to solve the optimization problems included in the PROBE algorithms presented in Sections 4.1 and 4.2 This solver is based on the revised simplex method and on the branch-and-bound algorithm to deal with integer variables

3.2 Inputting data for modeling project interactions

Synergies among projects

To address a synergy between the costs or/and the benefits of two projects s and t, an auxiliary project (s,t) must be added to X together with their conjoint cost c(s,t) and benefit values vi(s,t) on each benefit criterion i, i = 1,…,n PROBE then automatically defines an extra

binary variable x(s,t) such that x(s,t) = 1 if both projects are selected for the portfolio, or x(s,t) = 0 otherwise, and adds the following constraint to problem (4):

 

 

 

, , , , , ,

1,1,1,2

 Mutual inclusivity of two projects i and j

PROBE includes an interface that allows the user to add to problem (4) any other type

of linear constraint A feasible portfolio is a portfolio that respects all of the constraints introduced by the user

4 PROBE innovative functionalities

4.1 Finding all efficient portfolios within a given portfolio cost range

For project costs cj and benefit scores vj (j = 1,…, m) given by the MCDA component, and

supposing for now that no project interaction constraints were defined, the PDA component starts searching for the efficient portfolios, within a given portfolio cost range [ , ]B B , by solving problem (4) with B Next, problem (4) is again solved with B equal to the cost B

of the optimal portfolio previously found minus a small enough amount ; and so on while

B The algorithm designed to implement this process, FindEfficientPortfolios, presented B

in Figure 3, is also capable of identifying all possible multiple optimal solutions Finally, PROBE uses another algorithm, FindConvexEfficientPortfolios, presented in Figure 4, to differentiate convex efficient from non-convex efficient portfolios Additional linear constraints of the types described in Section 0 can easily be added to both algorithms to take project interactions into account when finding efficient portfolios

Figure 3 Algorithm FindEfficientPortfolios

Algorithm FindEfficientPortfolios(c, v, B , B )

Given the cost cj and benefit value vj of each project j (j= 1, ,m) of a set X of m projects, algorithm

FindEfficientPortfolios finds the whole set of efficient portfolios for a given portfolio cost range, defined by

its lower and upper bounds B and B , respectively The optimization problem inside each solve call is

presented including only the cost constraint, but other linear constraints can easily be added The efficient

portfolios found by the algorithm are stored in a matrix mEP with a number of rows equal to the number of

efficient portfolios and a number of columns equal to m+ 2 Each row stores the data of one efficient

portfolio, with its cost inserted in the first cell and its benefit value in the second one, and each of the other

m cells corresponding to each project j, j=1,…m, in such a way that 1 is inserted in cell j+2 if project j is

included in the portfolio, or 0 otherwise When searching for multiple optimal portfolios with the same

cost, each time an optimal portfolio is found a constraint is added to the optimization problem to prevent

that portfolio to be found again; the set of these constraints is designated by NRS At the end, the efficient

portfolios stored in matrix mEP are sorted by increasing order of cost

initialization

:

B  ; stop := false; duplicate := false; := 10 B -6 ; r := 0

search for efficient portfolios

while (stop = false) do

if (duplicate = false) then

j

B   c x ; duplicate := true end if

* :

else if (duplicate = true)

remove the NRS constraints from the optimization problem

* 1

j

B  c x   ; duplicate := false else