a practical multicriteria methodology for assessing risky public investments

A practical multicriteria methodology for assessing riskypublic investments

Michel Beuthe*, Louis Eeckhoudt, Giuseppe Scannella

GTM - Groupe Transport et MobiliteÂ, FaculteÂs Universitaires Catholiques de Mons, 151 chausseÂe de Binche, B-7000

Mons, Belgium

Abstract

This paper proposes a simple, practical but also realistic, methodology of multicriteria decision aid forthe assessment of public investment projects which takes into account the uncertainty of projects'measures.

It follows the general framework of the expected utility developed by von Neumann and Morgensternand its operational modelling by an additive non-linear function proposed by Keeney and Rai�a.However, it builds upon the linear goal programming model of Jacquet-LagreÁ ze and Siskos which o�ersseveral convenient computational procedures for building non-linear (albeit, piecewise linear) partialutility functions re¯ecting, as well as possible, the decision-maker's preferences.

In order to simplify as much as possible the process of decision making, the paper proposes analternative method, named Quasi-UTA, whereby the decision maker chooses directly for each criterion a(piecewise linear) partial utility function with only two parameters. These are the respective relativeweights and a curvature parameter that models risk aversion or proneness. The method substantiallyreduces the number of questions to be answered by the decision-maker.

In a context of budget constraint, the paper seeks to calibrate the utility function in equivalent moneyvalue, and shows how to compute such a value in multicriteria analysis. Money valuation has theadvantage of permitting the use of standard ®nancial rules of assessment. The paper also shows how tocompute the cost of a project's uncertainty, i.e. its risk premium, as proposed in the ®nancial theory ofrisk.

The method also suggests a procedure to generate and compute each criterion's random distributionas a triangular, rectangular or discrete function. An example illustrates the use of themethodology. # 2000 Elsevier Science Ltd. All rights reserved.

Socio-Economic Planning Sciences 34 (2000) 121±139

0038-0121/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.PII: S0038-0121(99)00021-X

www.elsevier.com/locate/orms

* Corresponding author. Tel.: +32-65-32-32-96; fax: +32-65-31-56-91.E-mail address: gt&[email protected] (M. Beuthe).

1. Introduction

The valuation of public projects faces at least two di�cult problems. First, it has to integratecriteria that are not naturally measured in the same units. Second, most measures are notcertain but stochastic.As is well known, the ®rst problem can be solved by recourse to multicriteria analysis

techniques, while the second is usually dealt with by a sensitivity analysis of some parametersand measures. However, sensitivity analysis, does not, in principle, completely solve thedecision-maker's problem. It tells how the valuation can be a�ected by various stochasticelements, but abandons the decision maker (DM) with the question of how much weightshould be given to the risk involved in a project.The two problems are interlocked in many ways, as will be seen in this paper. Obviously,

many analyses have previously attempted to deal with them. As their list would be rather long,we will just mention here one of the more important: the operational modeling by Keeney andRai�a [2] of an additive non-linear utility function based on the expected utility theory of vonNeumann and Morgenstern [3]. Its full implementation is rather lengthy and burdensome and,for that reason, not very often applied.The present paper proposes a simpler multicriteria procedure. It follows the previous

authors' leads by using an expected utility framework and the modeling of an additive non-linear function. The latter is built by borrowing from the UTA (Utility Additive) linear goalprogramming model of Jacquet-LagreÂ ze and Siskos [4,5] which o�ers several convenientcomputational procedures. Moreover, focusing on the valuation of uncertainty, we show howto derive equivalent money values from multicriteria utility functions and how to compute thecost of uncertainty a�ecting a particular project.Section 2 begins by reviewing the reasons that favor a project's valuation in monetary units.

Section 3 sets the general framework of uncertainty analysis applied in the current paper. Itssubsection 3.3 focuses on computation of the risk premium (RP), while subsection 3.4 proposesa simple procedure for generating appropriate distribution functions for each criterion.Subsection 4.1 brie¯y reviews the UTA methodology while Subsection 4.2 proposes a simpli®ed``Quasi-UTA'' procedure with two-parameter partial utility functions. Subsection 4.3 illustratesthe methodology with an example. The conclusion will brie¯y draw some recommendations forpublic investment decision making, and outline directions for future research.

2. The money yardstick in multicriteria analysis

In matters of public investment, the political DM who wishes to promote the welfare of thepopulation should accept a project implementation if its general public utility is at least asgreat as either (1) the utility of spending public funds on other projects, or (2) the utility of themoney involved for taxpayers. Prices and money under conditions of perfect competitionprovide measures of both costs and consumers' (relative) marginal utilities. The budgetinvolved in a project is thus also a measure of the marginal utility of forgone projects andconsumption. As a result, monetary values provide a coherent and convenient yardstick formeasuring and comparing the utility of a project, i.e. the utility of all its impacts, to itsbudgetary cost.

M. Beuthe et al. / Socio-Economic Planning Sciences 34 (2000) 121±139122

Beyond the well known problems associated with the adjustment of market prices under non-competitive conditions, it remains to be seen whether all impacts can be valued in money terms.Actually, the choices to be made between di�erent types of consumption, investment andsaving, implicitly de®ne the importance and the monetary value attached to di�erent objectives:safety, gains of time, employment, industrial policy, etc. Given the presence of a budgetconstraint, there is an implicit price for everything, a price that translates the importance givenby the DM to the objectives, services or results he/she wishes to obtain. These prices are knownas opportunity costs or shadow prices, to distinguish them from the market prices. Since there isan opportunity cost while there may not be a market price, we recommend that, as far aspossible, prices and money values be the yardstick used for valuing a project's impacts.Since, in many public infrastructure projects, impacts are not marketed, there is the issue of

how to evaluate these impacts in money values. Actually, this is what social cost±bene®tanalysis (SCBA) attempts as it strives to value time and congestion, injuries, pollution, etc. inmoney values. The basis for such valuation is the willingness to pay or to accept compensationas derived from observed choices and behaviors, or deduced from statements of value declaredby those interviewed. Despite the potential strengths of this approach, SCBA is unable toprovide a full account of selected impacts in public decision making. Examples includeindustrial, land use and regional policies, which are generally beyond the private citizen's directresponsibility and control. To the extent that such objectives of a political nature are not takeninto account, there is a need for going beyond SCBA.Nevertheless, this does not imply that the monetary yardstick should be discarded.

Multicriteria analysis (MCA) provides the framework and tools to estimate the opportunitycosts of interest here, i.e. values imputed to selected qualitative or political objectives in termsof forgone investments. The multicriteria arena o�ers various methods for eliciting a DM'spreferences among projects based on the relative importance is given to the various criteria. Indoing so, it builds an index of preference or utility via a weighted aggregation of severalcriteria. The index can also be calibrated so that the valuation is made in money by referenceto those criteria that are naturally de®ned in such terms. Thus, as in cost±bene®t analysis(CBA), MCA can provide an estimation of a project's net present money value with allimpacts being taken into account.With such a money valuation of projects, it is then possible to keep unchanged the basic rules

for accepting a project in CBA: a project can be accepted if its net present value, all impacts'values included, is positive. If a choice must be made between several projects, the ratio of thenet present value to the budgetary cost of each project should be used as the rank orderingindex. This would guarantee that the total net present value of all projects implementable withthe available total budget will be maximized. An additional advantage of a money valuation ofall impacts is that it permits the computation of a project's risk in such terms.

3. Uncertainty

3.1. Importance in decision making

The outcome of a project, for instance, its net present value as estimated by a SCBA, or anequivalent value provided by a MCA, is often a�ected by some degree of uncertainty. Three

M. Beuthe et al. / Socio-Economic Planning Sciences 34 (2000) 121±139 123

main reasons explain the presence of risk or uncertainty in decision making: the inevitableimperfection of statistical estimations, the di�culty of forecasting within incomplete and/ordynamic systems, and the basic ignorance of the future. If the DM is risk averse, he/she shouldcertainly identify and seek to account for risk in decision making. In the context of a publicproject, the DM represents the state and its citizens. His/her attitude towards risk should thusaccount for the size of the project compared to the country's gross income.For small projects that are not correlated with the country's gross income, Arrow and Lind

[6] have shown that the risk could be neglected if it is actually shared by a large number ofcitizens. In that case, the state, which plays a role similar to a mutual insurance fund, canmake decisions without attention to risk, i.e. it can take a risk-neutral attitude in decision-making. However, this neglects the fact that, beside the usual ®nancial costs and bene®ts, theremay be substantial external e�ects that are not as well distributed over the total population, orare not distributed at all but a�ect everyone in the same way [7]. These are also components ofthe global risk of a project that should be taken into account.On the other hand, the risk of a large-scale project cannot be neglected: its risk cannot be

spread over a su�cient number of people in many cases, particularly when some impacts arenot evenly distributed over the population. Moreover, the project may a�ect the entireeconomy as well as other projects, so that it generates additional risk through interdependence.The distribution of impacts across segments of the population and over regions is a matter

of equity in distribution in the income sense of the word. It also determines the level of risk ofa project as well as who bears it. This should be a matter of serious concern for the DM.

3.2. Risk aversion, cost of uncertainty and non-linearity

The expected utility theory of von Neumann and Morgenstern [3] is still the reference modelfor risk analysis in decision making, and is used here as the framework for the proposedmethodology. It assumes that the utility function is strictly concave if there is any degree ofrisk aversion. It follows that the expected utility, taken as a measure of a risky project's utility,has a lower value than the utility of a certain project with outcome equal to the weightedaverage outcome of the project. Hence, there is a loss of utility resulting from the project'srisk.As will be explained in subsection 3.3, if one of the outcomes is valued in monetary terms, it

is possible to compute the money equivalent values of both the project and the loss of utility,i.e. cost of the risk in the project. It amounts to the di�erence between the money values of theuncertain outcome of the project and the certain project which has a utility equal to theexpected utility of the uncertain project. In the ®nancial literature, this di�erence is called theRP since it is the maximum that one would pay for insurance against the risk involved. ThisRP1 is a measure of the willingness to pay for averting the risk. It is what we need foradjusting the equivalent money value (EMV) of an uncertain project in ®nding its certainmoney equivalent.

1 There is an analogous concept, the option price, which represents the willingness to pay when the e�ects of anuncertain project are compared to an uncertain initial situation [8].


A linear utility function is the most convenient form for a multicriteria analysis and is thusoften assumed. But, in order to avoid the risk neutrality associated with such a function, anon-linear form must be used. We will here follow Keeney and Rai�a [2] in proposing anadditive non-linear speci®cation.It is necessary to note, though, that there are basically two types of utility functions. The

usual one, which is applied in most texts of micro-economic theory, is derived under conditionsof certainty. It expresses the preferences of the DM when confronted with certain outcomes. Inthe more advanced theoretical literature, it is called a ``value function''. When it is used forcomputing an expected utility, it implies that the utility of the uncertain outcome is theweighted average of the utilities that would be obtained from certain outcomes. This approachwas ®rst proposed by Bernoulli to handle the problem of risk [9]. From an operational point ofview it means, concretely, that to estimate such a value function we must focus on the DM'spreferences between projects with certain outcomes, i.e. without consideration of a range ofoutcomes for any speci®c project.In the context of uncertainty, a ``utility function'' incorporates whatever attitude towards

risk taking the DM may have in mind: risk aversion, risk neutrality or risk proneness.Actually, it can be thought of as a transformation of the value function that accounts forattitude towards risk in the assessment of the alternatives confronted by the DM. It impliesthat, when estimating the utility function, questions addressed by the analyst to the DM mustrefer to uncertain outcomes, or lotteries, as they are more generally referred to in the literature.This is obviously a much more di�cult task, particularly in an analysis with several criteria.A possible procedure could be as follows: ®rst, build a value function of some form; then,

transform it into a utility function by questioning the DM on the utility he/she attaches todi�erent lotteries of values. One may wonder, though, whether this might be too burdensomefor most DMs. Moreover, this procedure neglects the possibility that the DM's risk attitudevaries over criteria.If an attempt is made to incorporate some risk analysis by way of computing an expected

utility, we do not believe that it is necessary to go as far as generating a full-¯edged utilityfunction with the long and di�cult questionnaires it involves. Given the current practice ofpublic project valuation, we feel that an important step forward would already be achieved ifan attempt was made to simply obtain non-linear value functions.While the methodological distinction between a value and a utility function is assumed in

order to avoid theoretical confusion and controversy, it will not be necessary to maintain sucha distinction throughout our discussion. In what follows, then, only the more commonterminology of a utility function will be used.It is important to note that speci®cation of an additive utility function is based on the

assumption of mutual preferential independence. This means that, if two projects arecharacterized by the same values for some criteria, the preferences between them do notdepend on these given values, but only on the level of the remaining criteria. This is a ratherstrong condition which suggests, for example, that the willingness to pay for reducing pollutionought not depend on the level of regional development. We should think that generally it doesso. However, since we are concerned only with the additional utility obtained fromimplementing a particular project, it is probably easier to meet this condition: in the aboveexample, it can be thought that the willingness to pay for reducing pollution does not depend


as much on the project's impact on regional development as it does on the level itself. Also, itcan be argued that the utility provided by a project can be seen as a utility di�erential, whichis de®ned mathematically as a weighted addition of marginal utilities, i.e. as impacts' utilities2.Two approaches for generating additive multiattribute utility functions can be distinguished.

A number of methods separately build each partial utility function, and then, in a second step,estimate the weights that link the functions. This approach was initially proposed by Keeneyand Rai�a [2], but can be implemented in di�erent ways. The more recent MACBETH methodby Bana e Costa and Vansnick [13,14] is probably one of the most convenient as it applies aquestionnaire methodology with verbal propositions that seeks a good approximation ofinterval-scaled preferences in certainty. Such a value function measuring strength of preferencecannot, however, be taken as a von Neumann±Morgenstern utility function for valuing riskyactions. While a utility function measures the strength of preference, asking questions incertainty based on strength of preference cannot extract risk attitudes3.Other methods proceed from stated global preferences between projects with linear goal

programming models. Among these, the UTA method proposed by Jacquet-LagreÁ ze and Siskos[4,5], is probably the most developed and useful. For a number of reasons, to be explained inSection 4, this is the approach that will be followed in the proposed assessment methodology.

3.3. Risk premium computation in a multicriteria framework

Let U(A ) be an additive utility function of a project de®ned by A=(M, X ), a vector of twocriteria, where M is the present value of the project as computed by a classical CBA, and X isits environmental impact measured in physical units:

U�A� � u�M � � v�X �:The EMV of A, M(A ) is de®ned as:

M�A� � uÿ1�u�M � � v�X ��,where uÿ1 is the inverse of the partial utility function of M. In Fig. 1, M(A ) corresponds tothe intercept of indi�erence curve U8(A ) on the M axis4.In a situation of uncertainty, with random MÄ and XÄ , the certain EMV of project AÄ, MÃ is

de®ned by the condition that the expected utility of M(AÄ ) is equal to the utility of MÃ :

E�uuÿ1

ÿu� ~M� � v� ~X�

�� E

�u� ~M� � v� ~X�

�� u�M̂�:

Indeed, the DM would then be indi�erent between the amount MÃ and the random project AÄ.The RP, which corresponds to the cost of uncertainty, is then equal to the di�erence

2 For a more detailed discussion, see Chapter 6 of the APAS/Road/3 report [10]. It is interesting to note that, inthe similar ®eld of conjoint analysis, Green and Srinivasan [11,12] argue that the additive compensatory model islikely to predict well even if the decision process is more complex or non-compensatory.3 On this topic, see Bell and Rai�a [15] and Bouyssou and Vansnick [16] who discuss under which restricted con-

ditions equivalence of both types of functions can be found.4 Given the speci®cation, indi�erence curves have an intercept on each axis.


between M(A-), i.e. EMV computed at the average value of AÄ, and the certain MÃ of the

project, i.e. RP=M(A-)- MÃ . It is thus the di�erence between the money value of the project for

M-and X

-and the money value MÃ . RP can be interpreted as the maximum amount that the

DM would be ready to pay in order to escape the uncertain situation.It is worth noting that this multicriteria RP is a coherent concept since:

. MÃ and RP are unique;

. if both u(M ) and v(X ) are linear so that there is no risk aversion, then RP = 0 ; and

. if u(M ) or/and v(X ) are strictly concave, and neither strictly convex, then RP > 0, becauseu(M

-)>E[u(MÄ )] or/and v(X

-)> E[v(XÄ )], and the inverse inequalities are excluded.

An illustration of an EMV and RP computation can be found in Section 4.3, where anumerical example of the proposed Quasi-UTA method is given. As will be seen in that case, itis possible that the computed M(A )s fall outside the variation interval of the scale used for Mover which u(M ) is de®ned in a particular MCA. Rather than computing these values bysimple extrapolation of u(M ) beyond this range, we propose to modify the theoretical methodof M(A )'s computation as follows:

M̂ � �uÿ1ÿE�u� ~M� � v� ~X��

�and M� �A� � �uÿ1

ÿu� �M� � v� �X�

�,

where �uÿ1�M max ÿM minu�M max � , and Mmin and Mmax are, respectively, the minimum and maximum

values on the scale of MÄ . Hence, u-ÿ1 is the inverse of the average slope of u(MÄ ) over itsvariation interval.The use of the average slope u-, rather than slope of u(M ) at M

-, is made in order to obtain a

risk aversion measure independent of a particular project's M-. This RP retains the above

characteristics.

Fig. 1. Equivalent money value.


3.4. Distribution

Regardless of the method used to account for the risks inherent in a project, a properanalysis requires systematic estimation of the uncertainty a�ecting all relevant variables andparameters including their range of variation, and the probability of their realization.If statistical estimates are not available, these elements should be assessed by the analyst, in

some cases with the help of outside experts, on a subjective basis. In order to facilitate thiscollection of information, a questionnaire, to be addressed to the promoters and analysts of aproject, should be constructed. The outline of such a questionnaire is proposed in Appendix A.Since the multicriteria function is taken to be additive in the present case, each distribution

is assumed to be independent. However, account should be taken, as much as possible, of allthose factors that a�ect their distribution and, particularly, of the so-called systematic risk. Thelatter, which corresponds, for instance, to the possibility of an earthquake or of an economicrecession, is as much a part of the overall risk as any other factor speci®c to a project.Considering the di�culty of the task, it would be wise to limit the requested information to aminimum. On the basis of the overall assessment framework and the modeling correspondingto it, the requested information should focus solely on the criteria or their additivecomponents.The questions should be constructed on the basis of a triangular density function, a

rectangular one, or a discrete distribution. Under the two ®rst hypotheses, only a fewquestions, that appear not too di�cult, are necessary to generate the full distribution. Theyshould concern (1) a lower limit value under which the variable would not be likely to fall, (2)the value of the mode (or the mean) of the distribution, and (3) the upper limit value abovewhich the variable would not be likely to rise. The likelihood that the variable would gobeyond the two limits should also be indicated.A triangular density function may be taken as a convenient approximation of the normal

density, but it may also handle cases where the distribution is not symmetrical. Thus, it can bea useful approximation of many continuous distributions, with the rectangular one as a limitcase. The option of a discrete distribution should also be o�ered as there may be qualitativecriteria with but a few values.

4. UTA methods for building a non-linear additive utility function

4.1. UTA general method

The UTA method aims at building a preference index or utility function of the form

Ui � u1�Xi1� � u2�Xi2� � . . .� un�Xin�,where Xij is the value taken by the jth criterion in the ith project. Each ui(Xij ) is a partialutility function that can be non-linear in a piecewise linear approximation over a set ofintervals. Actually, the method ®nds the values taken by the uj(Xij )'s at a number of equallyspaced values Xij(k ) of the criterion. The utility values, which are taken within an interval


between two successive end-points, Xij(k ) and Xij(k + 1), then correspond to a linearcombination of the values taken by ui(Xij ) at these end-points.The estimation made by UTA is based on simple statements by the DM of global

preferences between projects in a sample. The problem is set as a linear programming model,where the constraints indicate the order of preference between projects. For instance, if projecti is preferred to project k, the following constraint would hold:

Sj

�uj�Xij � ÿ uj�Xkj �

� ei ÿ eked �a�

where: ei and ek, which cannot be negative, are error terms resulting from estimation of theDM's rank ordering; and d is a small positive number.This is a goal-programming type constraint such that the constraint is not absolute, but

embodies a goal that the additive utility function should try to achieve: in the case of (1), it isthe respect of the preference order between projects i and k. Another set of constraints requiresthat the uj(Xij ) functions be monotonically increasing. The objective of the program is then tominimize the Siei over all projects in the sample.Like in all goal-programming models, it is possible that the solution will not be unique. It is

thus recommended that one use the average of the optimal utility functions. Further, there isno problem extrapolating the function to additional projects of the same nature, if the sampleis representative. Finally, the sample may be partly composed of ®ctitious, but realistic,projects, in order to elicit from the DM more complete information about project preferences,as it is almost always done in conjoint analysis [11,12].Key advantages of the UTA method are that it assumes a utility function of a more general

form, and that it requires a limited amount of information from the DM. Moreover, since allpartial utilities are generated simultaneously on the basis of global preference statements, themethod accounts for the interdependence that may exist between various criteria. Furthermore,the method, sometimes described as an ordinal regression method, produces weight coe�cients.Like in a linear regression, these are a�ected by correlation between criteria and biased by theabsence of some variables, for instance a product of two criteria. Thus, if the preferences arein¯uenced by some interdependence between criteria, this phenomenon is taken into account bythe weights. This advantageous ``ad hoc'' feature of the UTA method is illustrated through aseries of simulations in [17]. On the other hand, it must be recognized that the method doesnot probe the DM about each partial function.Finally, calibration in terms of the money value of the SCBA result, say Xi1, should not

present any di�culty, as it is easy to impose that u1(Xi1)=Xi1 if (1) it can be assumed thatthere is no risk aversion with respect to Xi1, or (2) it is possible to derive EMVs on the basis ofthe utility function, as explained in Section 3.3.This is just an outline of the general approach, of which there are several versions. For

instance, it is possible to introduce additional constraints about the strength of preferencesbetween projects or the concavity of the functions. Detailed analyses of the method and itsmain variants are provided in Beuthe and Scannella [17,18]. Other useful references includeVincke [19] and Pomerol and Barba-Romero [20].At least two convenient PC versions of the UTA method exist: PREFCALC by Jacquet-

LagreÁ ze [21], and, more recently, UTA+ by Kostkowski and Slowinski [22]. The UTA+


software provides a number of features that help both the DM and the analyst explore possibleadjustments to the utility function and to run sensitivity analyses. In this regard, it speci®es aconcave utility function, if that is seen as useful. There also exists a UTA-based MulticriteriaDecision Aiding System, named MINORA (Multicriteria INteractive Ordinal RegressionAnalysis) by Siskos et al. [23]. This system o�ers a feedback procedure in case ofinconsistencies between the DM and the preference model provided by UTA. Actually, thesesoftware programs are decision support systems that permit an interactive approach to thebuilding of a utility function.

4.2. A more practical approach: Quasi-UTA

Use of the full-¯edged UTA method is recommended but, like other complete multicriteriatechniques, it implies a somewhat cumbersome process of eliciting preferences between criteriaand projects from the DM. An alternative, more practical approach, would be to let the DMdirectly choose the partial utility function relative to each criterion from a set of functions.Such a methodology is used in the decision system MIIDAS (Multicriteria InteractiveIntelligent Decision Aiding System) by Siskos et al. [24], where di�erent types of curves arevisually presented to the DM. However, their mathematical expression involves threeparameters, plus the weight found by running the UTA model.Here, a more convenient mathematical speci®cation is proposed, a sort of recursive

`exponential' function, that involves only two parameters to de®ne the relative weights withrespect to each other and their curvature. The formula is ¯exible enough to represent verydi�erent attitudes towards risk, from strong risk aversion (strongly concave function) to strongrisk love (strongly convex function). Naturally, the choice of parameters can be presentedgraphically with typical values associated with meaningful verbal statements.Using the same notation as above, let us consider a criterion Xi de®ned over a total interval

between Xi(1) and Xi(n ), which is divided in (nÿ 1) equal smaller intervals by n bounds, and apartial utility function v(Xi(k )) de®ned at the kth bound of the scale. Then, v is de®ned at eachbound by the following ®nite geometric series:

v�Xi�1�� 0,

v�Xi�2�� v�Xi�1�� g1 � g,

v�Xi�3�� v�Xi�2�� g2 � g� g2 � g�1� g�,

. . .

v�Xi�n�� v�Xi�nÿ 1�� gnÿ1 � g� g2 � g3 � . . .� gnÿ1 � g�1� g� g2 � . . .� gnÿ2�,which corresponds to the more general de®nition:


8>><>>:v�Xi�k�� 0 for k � 1,

v�Xi�k�� v�Xi�kÿ 1�� gkÿ1 �Xkÿ1j�1

g j � gXkÿ2j�0

g j for 1 < kEn:

For 1< kEn, this ®nite geometric sequence is given by:

v�Xi�k�� g � 1ÿ gkÿ1

1ÿ g� gÿ gk

1ÿ g, 8k and 8g:

But, if g=1 (the linear case), then v�Xi�k�� 00 : In order to obtain a valid result in this case,

we can take the limit of the function for g4 1 and apply L'HoÃ pital's rule [25], to obtain

lim g41

ÿgÿ gk

� 0�1ÿ g� 0 � lim g41

1ÿ kgkÿ1

ÿ1 � lim g41

ÿkgkÿ1 ÿ 1

�� kÿ 1

�)v�Xi�k�� kÿ 1 for g � 1:

This leads to the following de®nition of v:8>>>><>>>>:v�Xi�k�� gÿ gk

1ÿ g, 1EkEn and 8g 6� 1

v�Xi�k�� lim g41gÿ gk

1ÿ g� kÿ 1 1EkEn and g � 1:

With this speci®cation, the variation interval of v becomes �0, gÿgn1ÿg � but [0,n ÿ 1] in case of

g=1. Thus, we can de®ne a UTA-type function, ui(Xi(k )) with ui(Xi(1 ))=0 and aNi ÿ

1ui(Xi(n ))=1, where N is the number of criteria, as:8>>>><>>>>:ui�Xi�k��

gÿgk1ÿggÿgn1ÿg� u�i �

gÿ gk

gÿ gn� u�i , 1EkEn and 8g 6� 1

ui�Xi�k�� kÿ 1

nÿ 1, 1EkEn and g � 1,

where u �i is the UTA's scaling constant which is equal to the utility of the last bound on Xi

and therefore equal to the relative weight of Xi multiplied by [Xi(n )ÿXi(1)]. Graphicalexamples of this function are given in Fig. 2.However, if one needs to change the number of bounds dividing the variation interval of a

given criterion Xi, the formula must be adjusted in order to maintain the same curvature witha given g over the same interval as previously. This is because the number of bounds is used asa variable in the function in place of Xi itself.Thus, for two di�erent numbers of bounds, n and m, over a given interval of Xi, [Xi�,X

�i ], the

relation between the bounds' indices, k and p, must be such that k=p = 1 for Xi�, and k=n,p=m for X �i . This implies the following relation between k and p:


k � nÿ 1

mÿ 1p� mÿ n

mÿ 1or p � mÿ 1

nÿ 1k� nÿm

nÿ 1:

In fact, switching from the scale with n bounds to the scale with m bounds, ui(Xi( p )) must berede®ned as:

8>>>>>><>>>>>>:ui�Xi� p�� gÿ g

nÿ1mÿ1p�

mÿnmÿ1

gÿ gn� u�i , 1EpEm and 8g 6� 1

ui�Xi� p��

�nÿ 1

mÿ 1p� mÿ n

mÿ 1

�ÿ 1

nÿ 1, 1EpEm and g � 1:

This transformation is illustrated in Fig. 3.

Fig. 2. Examples of Quasi-UTA functions.


4.3. Example of Quasi-UTA

In this section, we present an application of the Quasi-UTA model to two simple projectswith only two impacts, the net present value and a environmental impact. They do notcorrespond to any real project, but they permit a su�cient illustration of the method.Table 1 gives the minimum and the maximum values of the physical scale on each impact, as

well as the two parameters' values which must be chosen by the DM. We note that, in thiscase, with g=0.25, the DM appears to exhibit more risk aversion with respect to the netpresent value result.Table 2 gives the impacts' data, which are needed to compute their triangular distribution.

They illustrate typical answers given to the questionnaire outlined in Appendix A.The corresponding results of the MCA are presented in Table 3. They are computed by a

program that uses the data of Tables 1 and 2 to set the partial utility functions and to estimatethe triangular distributions.The expected utilities, E[U(A )], give a valuation of the projects that takes into account their

Fig. 3. Utility function with di�erent numbers of breakpoints.

Table 1Physical scales and partial utility functions

Impact X Min Valuea X Max Valuea Weight Curvature parameter

Cost±Bene®t NPV 0 4 0.55 0.25

Environmental impact 0 4 0.45 0.50

a Note: Cost±Bene®t NPV in Billion ECU.


uncertainty. Thus, we see that, as a result of the greater risk a�ecting A1's NPV, A2 ispreferred to A1. In, contrast, if they were valued in terms of their utilities at the average, i.e.U(A

-)'s, without paying any attention to uncertainty, A1 would be preferred to A2.

Likewise, the multicriteria EMV of A2, MÃ 1=6.844 billion ECU is greater that MÃ 1=6.377billion ECU, even though the average M(A

-1)=7.090 billion is greater than M(A

-2)=6.982

billion. Again, the rank order of the two projects is reversed when their uncertainty isconsidered. Indeed, A1's RP (0.753) is much greater than A2's (0.137).The EMVs allow the use of standard ®nancial criteria. Obviously, both projects have

positive values and could be recommended. However, the budget of A1 being smaller, its®nancial ratio (0.253) is greater than that of A2 (0.201). Hence, it should be ranked ®rst.Nevertheless, if the total available budget permits the implementation of only one project, thesecond project could very well be chosen since its equivalent value is larger.

5. Conclusion

Decision-making on public investments is a convoluted process with many interacting actors.Even if we set aside problems related to the dynamics of the political process, as well as togroup decision making, the assessment of a project by one DM remains a di�cult matter bothat the theoretical and empirical levels.

Table 2Impact data in uncertainty

Project A1 Project A2

Cost±bene®t NPVa Environmental impact Cost±bene®t NPVa Environmental impact

Min 1 3 1 2.5

Mode 2.5 3.5 3 3.5Max 4.5 4.5 3.5 4Average 2.66 3.66 2.5 3.31

a 0 0 0 0.025b 0 0 0 0.025

a Note: Cost±Bene®t NPV in Billion ECU.

Table 3Results of the multicriteria risk analysis

Project Utilities Multicriteria EMVs (billion ECU)

U(A-) E[U(A )] Cost±Bene®t NPV (A

-) M(A

-) MÃ RP MÃ /Budget Ranking

A1 0.975 0.871 2.666 7.090 6.337 0.753 0.253 1

A2 0.960 0.941 2.500 6.982 6.844 0.137 0.201 2


Within a CBA, the direct valuation of some impacts remains problematic. Further, the

aggregation of all costs and bene®ts raises the sensitive question of the outcomes' distribution

across individuals. In many cases, CBA is unable to encompass all the impacts a project can

have on society, where some impacts may be so important to the DM that they ought to be

weighed separately. These considerations suggest then an assessment with several criteria,

which does not simplify the task. If most MCAs provide an ordering, or at least a classi®cation

of projects, they generally lack a clear-cut decision rule for project acceptance, in contrast with

CBA. Furthermore, eliciting a DM's preferences across criteria is a delicate task, particularly

when considering his/her attitude towards risk.

Uncertainty about some outcomes also creates a problem, particularly when the risk

involved in a project cannot be su�ciently distributed to be neglected. A sensitivity

analysis of the results generally does not completely solve the problem, since it leaves the

DM with the task of somehow assessing how much risk is worth taking.

In the present paper, we sought some answers to these questions by proposing a

methodological framework that is both practical and in accordance with basic theoretical

standards. Following the expected utility approach, we thus proposed a speci®cation of partial

utility functions, actually ``value'' functions, with only one curvature parameter. This

speci®cation permits a parsimonious approach in terms of the number of questions needed for

the DM to reveal his/her preferences. It also proposed a simple method for obtaining workable

approximations of the outcomes' distributions. Further, it showed how to properly compute

the EMV of multicriteria utility functions, so that, in a context of budget constraint, well

known ®nancial decision rules could be applied to better order and select projects. Importantly,

transforming utilities into EMV allows one to estimate a project's RP, i.e. the cost of the risk

involved as valued by the DM.

The proposed methodology implies some strong assumptions, but opens a practical and

reasonable approach to analyse the risks involved in projects. Providing a convenient

framework for estimating both distributions and partial utility functions, it should ease the

task of assessing risk and allow a deeper testing of what should be a crucial, but easily

forgotten, factor in public investment decision making. Actually, we submit that this

methodology is su�ciently accessible that analysts need not neglect a formal analysis of risks

taken by the DM in selecting projects. We are aware, nevertheless, that some ``side actors'' in a

decision process may be rather reluctant to provide necessary information about the uncertain

character of some impacts. This is perhaps the main obstacle to such an assessment procedure.

Further research on this methodology is certainly warranted. First, we are presently

developing software for its convenient and interactive application5. Second, even if the use of a

value function, rather than a strictly de®ned cardinal utility function, seems to be a reasonable

option in public investment decision making, we feel that the proposed speci®cation opens the

way to a more straightforward estimation of utilities under uncertainty. Currently, we are

preparing experimental and comparative validation of the proposed methodology.

5 To be called MUSTARD for Multicriteria Stochastic Aid for Ranking Decision. A User's Guidebook outliningthe main steps of the software will soon be available on our web site: http://www.fucam.gt&m.ac.be


Acknowledgements

This paper is one output of the research program EUNET of the European Commission onSocio-Economic and Spatial Impacts of Transport (see [1]). We wish to thank the FrenchCommunity of Belgium (FSRIU grant) and the Belgian `Services FeÂ deÂ raux des A�airesScienti®ques, Techniques et Culturelles' for granting ®nancial support to a part of thisresearch.

Appendix A. Questionnaire about risk identi®cation and measurement

A.1. Introductory remark

The purpose of this questionnaire is to determine the degree to which measures of eachcriterion introduced in the valuation of a project are a�ected by uncertainty. The informationcollected will be used to design a sensitivity analysis of the global results, and to assess the extentto which the risk characterizing a project changes its valuation in terms of its expected value.The questions seek to estimate a simple and convenient density function, either triangular or

rectangular, as an approximation of the real distribution of each criterion. The option of adiscrete distribution is also included. The triangular density can be taken as a practicalapproximation of the normal distribution, but it may not be symmetrical; its degenerated form isrectangular. The rectangular distribution can be useful when the analyst is unable to give aprecise characterization of the appropriate distribution for the concerned variable. The diagramsillustrate the distributions and de®ne the parameters on which the questionnaire will focus.


The questionnaire is sent to the leader of each research team working on the estimation of aparticular project's impact. If the team is working on several impacts or variables that will beadditively combined to form a single, or part of a single, criterion, responses are requested foreach component separately. An obvious example of this situation could be the usual componentsof a CBA criterion: fuel savings, time savings, accidents' material savings, etc. When theassessment framework and its various elements are well de®ned, additional questions relating toeach particular item may have to be added to the questionnaire.When responding to such a questionnaire, the analyst should account for all factors that

might in¯uence the value of a criterion. Those considered beyond the DM's control shouldalso be included, e.g. the possible in¯uences of an economic recession.

A.2. Identi®cation of the variables

1. For which criterion (or impact) will your estimation output be used?(Give simply the name of the criterion)

2. What is the output?(Give its name in concordance with the assessment framework)

3. Is it a sub-criterion that will be added to other sub-criteria, all being components of thesame main criterion?(Answer Yes or No)

4. Should it be taken as a stochastic variable because (1) it is the result of a statisticalestimation, or (2) because of the future's uncertainty ?(Answer No, 1, or 2 )

5. Should it be taken as a non-stochastic decision variable for which di�erent values could beconsidered, like a social discount rate or a legal prescription ?(Answer Yes or No)

6. Should it be taken as a constant, because it corresponds to a physical description devoid ofany signi®cant uncertainty ?(Answer Yes or No)

A.3. Distribution analysis

1. If it is a constant, give its value:2. If it is a continuous variable for which you have a statistical con®dence interval, give:

* its mode (m ), i.e. its most likely value:(note : if the distribution is symmetrical, the mode and the mean will be equal; if there

is no mode, the density function will be taken as rectangular)


* its mean (not necessary):* the lower limit (a) of the interval:* the upper limit (b) of the interval:* the probability level of the con®dence interval in % at each tail (a and b ):

(note: usually a and b are taken as equal)

3. If you do not have such statistically estimated information, or if the variable's uncertainvalue cannot be ascertained by a statistical procedure, give, on a subjective basis:* its mode (m ) , i.e. its most likely value (if any):

(note : if there is no mode, the density function will be taken as rectangular; if thedistribution is symmetrical, mode and mean are equal)

* its mean (not necessary):* the likely lower bound (a ) of the variable:* the subjective likelihood (1-a ) that the variable will have a value larger than a: (note: for

a rectangular density, a=b=0):* the likely upper bound (b ) of the variable:* the subjective likelihood (1-b ) that the variable will have a value lower than b:

4. If it is a discrete variable, indicate the numerical values it can take and their probabilities.5. If you are concerned with a non-stochastic decision variable, could you suggest values that

would deserve investigation in a sensitivity analysis?

A.4. Scenario analysis

1. If the variable (criterion or sub-criterion) is correlated with another variable of the valuationframework, give, to the best of your knowledge, the correlation (from ÿ1 to +1) betweenthem.

2. In words, suggest critical or likely scenarios involving speci®c values of the variables, andwhich you feel would be interesting to investigate.

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a practical multicriteria methodology for assessing risky public investments

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