Omega 37 (2009) 395–410www.elsevier.com/locate/omega

Missing consequences in multiattribute utility theory�

Antonio Jiménez∗, Alfonso Mateos, Sixto Ríos-InsuaDepartment of Artificial Intelligence, Technical University of Madrid (UPM), Campus de Montegancedo s/n, 28660 Madrid, Spain

Received 2 June 2006; accepted 27 April 2007Available online 18 May 2007

Abstract

This paper addresses how to deal with decision alternatives with missing consequences in multicriteria decision-makingproblems. We propose disregarding the attributes for which a decision alternative provides no consequence by redistributing theirrespective weights throughout the objective hierarchy in favor of a straightforward idea: the assignation of the respective attributerange as a default value for missing consequences due to possible uncertainty about the decision alternative consequences. Inboth cases, decision alternatives are evaluated by means of an additive multi-attribute utility model. An illustrative example ofthe restoration of radionuclide contaminated aquatic ecosystems is shown.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Multicriteria decision-making; Decision support systems; Environmental studies

1. Introduction

Multiattribute expected utility theory (MAUT) can beconsidered as a leading paradigm for normative deci-sion theory. However, MAUT calls for decision-makers(DMs) to provide all the information describing thedecision situation in order to build an objective hier-archy that includes all the relevant aspects of the re-spective problem [1,2]. Also DMs have to identify thedecision alternatives to be analyzed, as well as theirimpact or consequences on the attributes under consid-eration, in advance [3,4]. Finally, they have to assessboth the component scalar utility functions ui , whichrepresent the DMs’ preferences over the possible at-tribute values [5,6], and the weights that represent the

� This manuscript was processed by Area Editor B. Lev.∗ Corresponding author. Tel.: +34 91 3367438;

fax: +34 91 3524819.E-mail addresses: ajimenez@fi.upm.es (A. Jiménez),

amateos@fi.upm.es (A. Mateos), srios@fi.upm.es (S. Ríos-Insua).

0305-0483/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.omega.2007.04.003

relative importance of attributes. Moreover, indepen-dence conditions must be met to get an appropriate func-tional form for the global utility function u includingthese components [2].

These information requirements can obviously be fartoo strict in many practical situations [7]. This leads tothe search for approximations that capture the notion ofcomplexity not as imperfect knowledge but as indecid-edness or uncertainty about the world around us.

Consequently, incomplete information about inputparameters has been incorporated into the decision-making process. Given an objective’s hierarchy withattributes Xj , j = 1, . . . , n, the consequences of adecision alternative Si , i = 1, . . . , m, can be de-scribed under uncertainty by a vector of ranges[xiL, xiU]= ([xiL

1 , xiU1 ], . . . , [xiL

n , xiUn ]), where xiL

j and

xiUj are the lower (L) and upper (U) levels of attribute

Xj for Si and assuming a probability distribution overeach range [8].

On the other hand, DMs can be allowed to provideimprecise responses to the probability questions they

396 A. Jiménez et al. / Omega 37 (2009) 395–410

have to answer when assessing preferences. This leadsto classes of utility functions [9–11] and weight in-tervals [9,12–14]. This is less stressful on experts andmakes the elicitation procedures suitable for groupdecision-making, because individual conflicting judg-ments in a group of DMs can be easily captured throughranges [15].

The introduction of uncertainty about the decisionalternative consequences and imprecision concerningcomponent utility and weight elicitation will lead toimprecise results, i.e. overall utility intervals. In manycases, this will produce very overlapped utility inter-vals and, consequently, the output information will notbe meaningful enough to definitively recommend a de-cision alternative. So-called decision-making with par-tial information [16,17] which makes use of impreciseinputs, plays a very important role under such circum-stances. The assessment of nondominated and poten-tially optimal decision alternatives could be useful forrejecting bad alternatives [18–20], and the use of simu-lation techniques for sensitivity analysis could providemore meaningful information, see [21,22].

The problem we address in this paper is how to dealwith missing consequences in different attributes forthe decision alternatives under consideration. Generally,there are three possible levels of missing knowledgeabout the alternative consequences or performances.Firstly, DMs may know the alternative’s exact perfor-mance for all the criteria and a precise value is pro-vided for each alternative. An intermediate situation iswhere a probability distribution is provided (uniform,normal. . .) for the criteria and a Monte Carlo simula-tion is performed to evaluate the considered alternatives(boxplot diagrams), as in Logical Decisions [23,24], oruncertainty is represented by fuzzy numbers [25,26].Finally, there is an extreme situation where there is noknowledge whatsoever of how several alternatives willperform for some attributes, i.e. neither precise perfor-mance nor a probability distribution can be specified.This is the situation considered in the paper. Note that al-though most decision support systems based on MAUT,like Logical Decisions [23], VISA [27] or WINPRE[12], allow for imprecise information in both alternativeconsequences and/or preferences, they do not accountfor missing consequences.

Our experience in complex decision-making prob-lems through our participation in several European re-search projects, such as MOIRA [28], COMETES [29],and EVANET-HYDRA, shows that although the expertsare in most cases capable of providing the precise or im-precise decision alternative performances for all the at-tributes under consideration (based on previous studies,

the development of ad hoc simulation or predictionmodels, or just the DM’s knowledge and experience),it is not always possible to identify decision alternativeconsequences, especially when an attribute has an as-sociated subjective scale. Additionally, if the analysiswere to include a new decision alternative for which noprevious studies had been performed, its consequenceson some attributes would not necessarily be known.

Several questions arise under these circumstanceswithin MAUT. For example, how should a decisionalternative with missing consequences be evaluated?Could we just ignore weights for attributes where noconsequence is provided? Should these weights beredistributed through the objective’s hierarchy? Or, isit possible to assign default values to missing con-sequences? These and other questions that will ariseduring the analysis are the central topic of this paper.

Missing data concerning alternatives has been exam-ined in other multicriteria decision-making methodolo-gies. Some examples follow.

The other most applied method, apart from MAUT,when the information on the criteria is mainly cardi-nal is the analytical hierarchy process (AHP), where acomplete compensation between attributes is possible.AHP [30] requires the determination of pairwise com-parison matrixes to derive objective weights and alter-native scores. The entry in row i and column j indicateshow much more important objective (alternative) i isthan objective (alternative) j. Thus, n(n − 1)/2 ques-tions must be answered for each matrix on the basisof an integer-valued 1–9 scale, each entry being an ap-proximation to the weights ratio of the n items beingcompared. Then, the score at each level of the hierar-chy can be calculated as a weighted sum of the lowerlevel scores. Within this context, alternative modes ofquestioning were proposed in [31] as an extension ofthe eigenvector approach, allowing the DM to say “Idon’t know” to some of the questions being asked, andto approximate nonlinear functions of the weights ratio.Ref. [32] deals with missing data in AHP too.

Rough sets proved to be very useful for analyzing de-cision problems concerning objects described in a datatable by a set of condition attributes and by a set ofdecision attributes, [33,34]. Greco et al. [35] proposesan extension of the rough set methodology when thedata table is not complete because some data are miss-ing. The adaptation concerns both the classical roughset approach based on indiscernibility relations and anew rough set approach based on dominance relations.While the first approach deals with multiattribute clas-sification problems, the second deals with multicrite-ria sorting problems. Both approaches work like the

A. Jiménez et al. / Omega 37 (2009) 395–410 397

original approach when the data table is complete, in-ducing a generalized description of the information con-tained in the decision table in terms of “if. . ., then. . .”decision rules. Also, [36] deals with missing data in therough set analysis of multicriteria decision problems.

Outranking methods [37–39] overcome the assump-tion of “complete compensation” and the existenceof a true ranking of alternatives, which only needs tobe discovered. They assume that preferences are notconstant in time, are not unambiguous and are not inde-pendent of the process of analysis. The outranking re-lation is the result of pairwise comparisons between thealternatives with respect to each criterion. The outrank-ing relation does not only rely on strict preference andindifference, but also explicitly allows for incompara-bility and weak preference. Within this context, [40]suggests the utilization of the PROMETHEE methodwith trapezoidal fuzzy intervals. The missing valuescould be identified in processing and compared with agiven value through a specifically enhanced preferencefunction. For the consideration of missing values in thepairwise comparisons, the whole interval [0, 1] insteadof a crisp preference value is the suggested degree ofpreference. This indicates that no statement about thepreference is possible on the basis of the given data.

In stochastic multicriteria acceptability analysis(SMAA) the DMs need not express their preferencesexplicitly or implicitly. Instead, the method is basedon exploring the weight space in order to describe thescores that would make each alternative the preferredone. The SMAA-2 method [41] extends the originalSMAA by considering all ranks in the analysis, andSMAA-O [42] is designed for problems where criteriainformation for some or all criteria are ordinal. Inac-curate or uncertain criteria values are represented byprobability distributions from which the method com-putes confidence factors describing the reliability ofthe analysis. Situations where partial preference infor-mation is available are also considered, and the resultsare presented graphically. Although different ways tohandle dependent uncertainties within SMAA-2 havebeen analyzed [43] we have no knowledge of missingdata concerning criteria values having been dealt withinthis methodology.

In the next section we consider the possibility of dis-regarding the attributes for which a decision alternativeprovides no consequence, leading to the redistributionof their respective weights throughout the objective hier-archy, in favor of a straightforward idea: the assignationof the respective attribute range as a default value formissing consequences due to possible uncertainty aboutthe decision alternative consequences. In Section 3,

we describe an example of a complex decision-makingproblem. This example is used to illustrate the abovepossibilities in Section 4. Finally, some conclusions aregiven in Section 5.

2. Dealing with missing consequences

Let us suppose that we have a decision-making prob-lem whose objective hierarchy has three levels and eightlowest-level objectives, see Fig. 1. A set of eight at-tributes associated with the lowest-level objectives hasbeen established, denoted by X1, . . . , X8, respectively.The lower (L) and upper (U) bounds for the attributeranges are precisely known and denoted by rL

j and rUj

(j = 1, . . . , 8). This is the most common situation: therange of possible values for all attributes are known,but there are alternatives whose performances are un-known for some attributes. Another less probable situ-ation is when the range of possible values is unknownfor an attribute but a sample of values is available.Then, on the basis of the central limit theorem, we canassume that the distribution of the attribute values isnormal with mean, �, that one in the available sampleand variance, �2, that one in the available sample di-vided by the sample size (high enough). Thus, the inter-val [� − 1.96�, � + 1.96�] would be considered as theattribute range, because it is well known that the prob-ability of a known value within this interval is 0.95.A third even less probable, but not impossible, situationis when attribute values are not available either. Then,standard normal distribution could be considered, andthe attribute range would be [−2, 2], where −2 is theworst attribute value and 2 the best.

Three decision alternatives have been identified inadvance, S1, S2, and S3, whose consequences for theattributes under consideration can be described underuncertainty by

S1 : ([x1L1 , x1U

1 ], [x1L2 , x1U

2 ], [x1L3 , x1U

3 ], [x1L4 , x1U

4 ],[x1L

5 , x1U5 ], [x1L

6 , x1U6 ], [x1L

7 , x1U7 ], [x1L

8 , x1U8 ]),

S2 : ([x2L1 , x2U

1 ], m.c., m.c., [x2L4 , x2U

4 ],[x2L

5 , x2U5 ], m.c., [x2L

7 , x2U7 ], [x2L

8 , x2U8 ]),

S3 : (m.c., [x3L2 , x3U

2 ], [x3L3 , x3U

3 ], [x3L4 , x3U

4 ],[x3L

5 , x3U5 ], [x3L

6 , x3U6 ], [x3L

7 , x3U7 ], m.c.),

where xiLj and xiU

j are the lower (L) and upper (U) levels

of attribute Xj for Si , and m.c. (missing consequence)indicates that the consequence of the respective decisionalternative on the respective attribute is unknown.

398 A. Jiménez et al. / Omega 37 (2009) 395–410

Fig. 1. Generic objective hierarchy with three levels and nine attributes.

We assume that a weight elicitation procedure, suchas trade-offs weighting [2], has been applied to calculatethe weights representing the relative importance of ob-jectives throughout the hierarchy and the DM has beenallowed to provide imprecise responses to the proba-bility questions he or she was asked [9,10]. This leadsto normalized weight intervals and average normalizedweights (shown in Fig. 1).

The normalization process [2] is as follows: let usassume that we are assessing the relative importancefor objectives 2.1 and 2.2, and the weight intervals[pL

21, pU21] and [pL

22, pU22] have been elicited from the

DMs’ responses, respectively. Then, the respective av-erage normalized weights are

w21 = pL21 + pU

21∑2i=1(p

L2i + pU

2i )and

w22 = pL22 + pU

22∑2i=1(p

L2i + pU

2i ),

and the normalized weight intervals are

[wL21, w

U21] =

[2 × w21 × pL

21

pL21 + pU

21

,2 × w21 × pU

21

pL21 + pU

21

]and

[wL22, w

U22] =

[2 × w22 × pL

22

pL22 + pU

22

,2 × w22 × pU

22

pL22 + pU

22

].

Different normalization techniques for multicriteriaproblems are described in [16].

Attribute weights over the decision, used in the de-cision alternatives evaluation process, are then assessed

rLj

rUj

0

1

Xj

x1Lj

x1Uj

x1Mj

uU(x1U)j j

uL(x1L)j j

uM(x1M)j j

Fig. 2. Component utility function class.

by multiplying the respective average weights and nor-malized weight interval bounds of the objectives in thepath from the root (overall objective) to each leaf (at-tribute). For example, the average normalized weightskj are

k1 = w1, k2 = w2 × w21 × w211,

k3 = w2 × w21 × w212, k4 = w2 × w22,

k5 = w3 × w31 × w311, k6 = w3 × w31 × w312,

k7 = w3 × w31 × w313, k8 = w3 × w32,

with∑8

j=1 kj = 1. To get the normalized weight inter-

vals [kLj , kU

j ] we would proceed similarly.Finally, we assume that component utilities have been

assessed for the eight attributes under consideration. Asin the weight elicitation, incomplete information hasbeen allowed, leading in this case to the class of utilityfunctions shown in Fig. 2. Note that uL

j and uUj are the

A. Jiménez et al. / Omega 37 (2009) 395–410 399

lower and upper bounds of the utility class for attributeXj , respectively, while uM

j is the utility function thatpasses through the center of the class.

Now, the different decision alternatives under consid-eration can be evaluated by means of an additive mul-tiattribute utility function, whose appearance is

u(Si) =n∑

j=1

kjuj (xij ),

where kj is the j th attribute weight, xij is the conse-

quence for decision alternative Si in the j th attribute,and uj (x

ij ) is the utility associated with the above con-

sequence. The additive form is sometimes a good ap-proximation for the reasons described in [44,45].

As we allow uncertainty about the decision alter-native consequences and incomplete information aboutthe component utility and weight elicitation, the addi-tive model is then used to assess average overall utili-ties, on which the ranking of decision alternatives willbe based and minimum and maximum overall utilities,which give further insight into the robustness of thisranking. For example, the overall utilities for S1 withno missing consequences are

umin(S1) =

8∑j=1

kLj uL

j (x1Lj ),

uavg(S1) =

8∑j=1

kjuMj (x1M

j ),

umax(S1) =

8∑j=1

kUj uU

j (x1Uj ),

where x1Mj = (x1L

j + x1Uj )/2, see Fig. 2.

However, decision alternatives S2 and S3 have miss-ing consequences for different attributes. To deal withmissing consequences we propose assigning the respec-tive attribute range as a default value for missing conse-quences due to possible uncertainty about the decisionalternative consequences.

As, thanks to consequence intervals for the attributesunder consideration, uncertainty about the decisionalternative consequences is allowed, missing conse-quences could be represented by the respective attributerange, [rL

j , rUj ], assuming that all attribute values are

possible and equiprobable, because if we do not knowthe ways in which an event can occur (and thereforehave no reason to believe that it is more likely to occurin one way than another), the event is equally likely tooccur in any way.

Therefore, taking into account the component utilitiesshown in Fig. 2, we can state uL

j (rLj )=0 and uU

j (rUj )=1,

and the overall utilities for decision alternatives S2 andS3 are

umin(S2) =

∑j∈{1,4,5,7,8}

kLj uL

j (x2Lj ),

uavg(S2) =

∑j∈{1,4,5,7,8}

kjuMj (x2M

j )

+ k2/2 + k3/2 + k6/2,

umax(S2) =

∑j∈{1,4,5,7,8}

kUj uU

j (x2Uj ) + kU

2 + kU3 + kU

6 ,

umin(S3) =

7∑j=2

kLj uL

j (x3Lj ),

uavg(S3) =

7∑j=2

kjuMj (x3M

j ) + k1/2 + k8/2,

umax(S3) =

7∑j=2

kUj uU

j (x3Uj ) + kU

1 + kU8 .

Note that uMj (rM

j ) = uMj ((rL

j + rUj )/2) = 1

2 . In the caseof convex or concave utility functions, the above utilitywould not be 1

2 .From a computational point of view, the introduc-

tion of uncertainty about some decision alternativeconsequences will lead to wider overall utility inter-vals, [umin(S

i), umax(Si)]. Additionally, the higher the

weight for these attributes, the wider the overall utilityintervals output will be. Note that incomplete informa-tion has also been allowed for component utility andweight elicitation.

Wide overall utility intervals will probably lead tooverlapped intervals. Consequently, the output infor-mation will not be meaningful enough to definitivelyrecommend a decision alternative. The so-called deci-sion making with partial information, which makes useof imprecise inputs, plays a very important role underthese circumstances. The assessment of nondominatedand potentially optimal decision alternatives could beuseful for rejecting bad alternatives, and the use of sim-ulation techniques could provide more meaningful in-formation, as shown in the example in the next section.

Note that one alternative is dominated by anotherone if its overall utility is lower for any combination

400 A. Jiménez et al. / Omega 37 (2009) 395–410

of the imprecise parameters (weights, component utili-ties, and alternative consequences). Hence, an alterna-tive is nondominated if it is not dominated by any otherone, and it is potentially optimal if it is best ranked forat least one combination of the imprecise parameters[18–20].

If there were many decision alternatives with miss-ing consequences for several attributes, the impreci-sion concerning the results would be so high that noconclusions could be reached from the alternatives anddecision-making with partial information. However, thiswould mean that so little is known about the decisionalternatives that it would make no sense to analyzethem.

We analyze the possibility of not taking into accountthe respective attributes with missing consequences inthe evaluation of each decision alternative against theoption of considering the respective attribute range as adefault value for missing consequences. This, however,is not a straightforward thing to do because each at-tribute has a bearing (weight) on the associated decision.If they are not taken into account, it is as if their con-sequences were to be the worst attribute values. In thiscase, their utility is equal to 0, and 0 is also the productof the respective attribute weights. This does not matchup with reality. Therefore, the weights for attributeswith missing consequences should be redistributedthroughout the objective hierarchy. Consequently, dif-ferent objective hierarchies and decision weights forthe same attribute could be used to evaluate decisionalternatives with missing consequences for differentattributes.

Decision alternative S2 has missing consequences forattributes X2, X3, and X6. Table 1 lists the objectiveweights updated as a result of the redistribution.

After local objective weights have been redistributedthroughout the hierarchy, attribute weights for the deci-sion are reassessed. For example, the new average nor-malized weights k′

j are

k′1 = k1 = w1,

k′4 = w2 × w′

22 = w2,

k′5 = w3 × w31 × w′

311,

k′7 = w3 × w31 × w′

313,

k′8 = k8 = w3 × w32,

with k′1 + k′

4 + k′5 + k′

7 + k′8 = 1. To get the nor-

malized weight intervals [k′Lj , k′U

j ], we would proceedsimilarly.

Finally, the overall utilities for S2 are

umin(S2) =

∑j∈{1,4,5,7,8}

k′Lj uL

j (x2Lj ),

uavg(S2) =

∑j∈{1,4,5,7,8}

k′j u

Mj (x2M

j ),

umax(S2) =

∑j∈{1,4,5,7,8}

k′Uj uU

j (x2Uj ).

On the other hand, decision alternative S3 has miss-ing consequences for attributes X1 and X8. Table 2 il-lustrates the objective weights updated as a result of theredistribution.

Attribute weights for the decision are reassessed, andthe final overall utilities for S3 are

umin(S3) =

7∑j=2

k′′Lj uL

j (x3Lj ),

uavg(S3) =

7∑j=2

k′′j uM

j (x3Mj ),

umax(S3) =

7∑j=2

k′′Uj uU

j (x3Uj ).

This possibility for dealing with missing conse-quences has drawbacks from a logical point of view.For instance, if the consequence for an attribute witha high weight is unknown, the evaluation of that deci-sion alternative would output more imprecise results.However, when redistributing weights throughout theobjective’s hierarchy, the level of results imprecisionremains. On the other hand, decision alternatives withmissing consequences for key factors (highly weightedattributes) could tend to have high (low) overall utili-ties if they provide good (bad) consequences for lessimportant attributes whose weights have substantiallyincreased due to the redistribution. However, if the con-sequences of attributes of little importance are missingthe above negative effect would be negligible.

Additionally, if weights are redistributed and producedifferent objective hierarchies for evaluating the deci-sion alternatives, it would not be possible to solve therespective linear optimization problems to assess non-dominated and potentially optimal decision alternatives[20], which is especially useful for very overlappedoverall utility intervals, [umin(S

i), umax(Si)].

3. Restoration of a radionuclide contaminatedaquatic ecosystem

This complex decision-making problem is concernedwith evaluating intervention strategies for restoring

A. Jiménez et al. / Omega 37 (2009) 395–410 401

Table 1Weight redistribution for decision alternative S2

Average normalized weight Normalized weight interval

w′22 = 1 [w′L

22 = 1, w′U22 = 1]

w′311 = w311 + w312 × w311

w311 + w313

[w′L

311 = 2 × w′311 × wL

311

wL311 + wU

311

, w′U311 = 2 × w′

311 × wU311

wL311 + wU

311

]

w′313 = w313 + w312 × w313

w311 + w313

[w′L

313 = 2 × w′313 × wL

313

wL313 + wU

313

, w′U313 = 2 × w′

313 × wS313

wL313 + wU

313

]

“′” denotes the updated value after redistribution.

Table 2Weight redistribution for decision alternative S3

Average normalized weight Normalized weight interval

w′′2 = w2 + w1 × w2

w2 + w3

[w′′L

2 = 2 × w′′2 × wL

2

wL2 + wU

2

, w′′U2 = 2 × w′′

2 × wU2

wL2 + wU

2

]

w′′3 = w3 + w1 × w3

w2 + w3

[w′′L

3 = 2 × w′′3 ∗ wL

3

wL3 + wU

3

, w′′L3 = 2 ∗ w′′

3 × wU3

wL3 + wU

3

]

w′′31 = 1 [w′′L

31 = 1, w′′U31 = 1]

“′′” denotes the updated value after redistribution.

a radionuclide contaminated aquatic ecosystem. Suchproblems have been studied in depth in the Europeanprojects in which we have participated: MOIRA [28],COMETES [29], and EVANET-HYDRA. As a result ofthe above projects a synthetic, flexible, and user-friendlycomputerized decision support system, MOIRA, wasdeveloped and implemented.

The MOIRA system integrates several modules,including a GIS (geographical information system)database together with a complete set of reliable, val-idated models [29] to predict the temporal behaviorof radionuclides in the fresh water environment andthe ecological, social, and economic impacts of theintervention strategies as the basic pillars on which thedecision should be based.

The multiattribute analysis module of the MOIRAsystem is an adaptation of the generic multiattributeanalysis (GMAA) system, [9] and [10]. The GMAAsystem (http://www.dia.fi.upm.es/∼ajimenez/GMAA)accounts for uncertainty about the strategy conse-quences. It also admits incomplete information aboutDM preferences and has been adapted to test the redis-tribution option for dealing with missing consequences.

Throughout the example, the data provided by pre-diction models will be used in the GMAA system toevaluate the intervention strategies for restoring a ra-dionuclide contaminated aquatic ecosystem, and then,

missing consequences will be introduced to analyze thepossibilities pointed out in the above section.

Specifically, the scenario to be analyzed is Lake Svy-atoye, an undrained lake within a region that was heav-ily contaminated after the Chernobyl accident. It is lo-cated 237 km from the Chernobyl Nuclear Power Plantand 30 km southeast of Kostyukovichy (Belarus).

Fig. 3 shows the constructed hierarchy associatedwith the Lake Svyatoye scenario. Environmental im-pact (Environ Imp) is one of the main objectives of thedecision analysis. It is divided into Lake Ecosystem In-dex (L.E.I.), a simple, rational approach for measuringthe ecological status of a lake [46], and dose to fish(Dose to Fish).

Social impact (Social Imp) is handled by two sub-objectives: to minimize impact on health, dose to man,and living restrictions (Living Restr). When minimizingimpact on health, we focus on the dose critical indi-vidual (Dose Crit In), who should never receive levelsabove thresholds for early health effects, and collectivedose (Coll Dose), which will be linearly related to theincrease in the risk of developing serious latent effects.As regards living restrictions, other impacts are takeninto consideration. These include countermeasuresaffecting the direct consumption of fish for food orits processing in the food industry, drinking water andwater used by the food industry, the use of water for

402 A. Jiménez et al. / Omega 37 (2009) 395–410

Fig. 3. An objective hierarchy for Lake Svyatoye.

crops irrigation, and the recreational uses of waterbodies. For all these objectives, the attributes will bethe amount of fish affected by restrictions (Amountof Fish), as well as the duration of such restrictions(Ban Duration).

Finally, economic impact (Economic Im) has been di-vided into direct effects (Direct Eff), which include thecosts generated by the different bans or restrictions onnormal living conditions, and intangible effects (IntangEff), like loss-of-image and adverse market reactionsfor the affected area. Direct effects can be sub-dividedinto costs to economy (Cost to Econ) and cost of appli-cation (Applic Cost), i.e. costs of chemical and physicalremedial countermeasures.

Note that attribute cost of image (X9) is subjective,which means that there is no objective quantity, and asubjective index has been constructed associated with ascale from 0 to 1.

The candidate intervention strategies for applicationto restore the aquatic ecosystem are shown in Table 3,alongside a short description. The consequences un-der uncertainty for the seven intervention strategies de-scribed above are shown in Table 4.

Next, the DM’s preferences were quantified by meansof the GMAA system, which includes four proceduresfor assessing component utilities that are selected de-pending on the level of knowledge and features of theattribute under consideration, and two procedures foreliciting weights, weight elicitation based on trade-offsand direct assignment, see [9,47]. Note that imprecision

concerning DM responses is allowed in all proceduresthrough ranges of responses to the probability questionthe DM is asked. Fig. 4 shows the average normalizedattribute weights and the normalized attribute weightintervals, indicators of the influence of the individualcriteria on the decision.

Finally, intervention strategies were evaluated bymeans of an additive multiattribute utility model, lead-ing to the overall utilities shown in Fig. 5. The rankingof the intervention strategies for Lake Svyatoye isbased on average utilities, i.e. the vertical white lineson each bar. The best ranked intervention strategy isS2 (Potash) with an average overall utility of 0.799,followed by S7 (Wetland Liming) (0.753), and S5 (LakeLiming) (0.752), while the worst ranked is S6 (SedimentRemoval) with a utility of 0.608.

Looking at the overlapped utility intervals (robust-ness of the strategies ranking), however, we can con-clude that the information gathered by this evaluationis not meaningful enough to definitively recommend anintervention strategy.

Nevertheless, only intervention strategies S2, S3, S4,S5, and S7 are nondominated and potentially optimal.Therefore, we can definitely discard S1 and S6 from fur-ther consideration. Note that an alternative is not dom-inated by another one when its overall utility is higherfor any combination of the imprecise input parame-ters (weights, component utilities, and strategy con-sequences) and is potentially optimal when it is bestranked for at least one combination of the imprecise

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Table 3Candidate intervention strategies for the Lake Svyatoye

Intervention strategies Description

S1: No action Natural evolution of the situation without interventionS2: Potash 15 tonnes of potassium chloride added to the lake in April 1987S3: Fertilizer 800 kg of fertilizer added to the lake between April and July 1987, 1988, 1989, 1990, i.e. 200 kg/monthS4: Food ban Automatic fish consumption ban when 137Cs content in fish is higher than 1000 Bq/kgS5: Lake liming 15 tonnes of lime added to the lake in April 1987S6: Sediment removal 250, 000 m2 of sediments removed from the lake down to a depth of 10 cm from May to June 1990S7: Wetland liming 30 tonnes of lime added to the catchment in May 1987

Table 4Impact of the seven intervention strategies for lake Svyatoye

Strategy Attributes

X1 X2 × 102 X3 X4 × 103 X5 X6 X7 × 103 X8 X9

S1 [1.25, 1.41] [15, 33.4] [200, 246] [1.9, 2.46] [0, 0] [0, 0] [0, 0] [0, 0] [0.0, 0.1]S2 [1.20, 1.46] [8, 11] [80, 132] [0.8, 1.36] [0, 0] [0, 0] [0, 0] [600, 800] [0.7, 0.9]S3 [1.60, 2.14] [18, 27.4] [180, 244] [1.6, 2.54] [0, 0] [0, 0] [0, 0] [103, 1660] [0.7, 0.9]S4 [1.25, 1.41] [15, 33.4] [10, 12.8] [0.1, 0.158] [8, 14] [130, 140] [2.5, 4] [0, 0] [0.3, 0.5]S5 [1.40, 1.68] [16, 28.6] [180, 234] [1.8, 2.24] [0, 0] [0, 0] [0, 0] [103, 1500] [0.7, 0.9]S6 [1.20, 1.46] [13, 19.8] [100, 172] [1, 1.68] [0, 0] [0, 0] [0, 0] [75 × 105, 107] [0.6, 0.8]S7 [1.20, 1.70] [20, 27.2] [180, 256] [1.7, 2.56] [0, 0] [0, 0] [0, 0] [2000, 3020] [0.6, 0.8]

Fig. 4. Imprecise attribute weights for Lake Svyatoye.

input parameters. Linear optimization problems must besolved to assess nondominated and potentially optimalalternatives, as described in [20].

Monte Carlo simulation techniques for sensitivityanalysis allow simultaneous changes to the weightsand generate results that can be easily analyzed

statistically to provide more insight into the multiat-tribute model recommendations. We propose selectingweights at random using a computer simulation pro-gram so that the results of many combinations ofweights, including a complete ranking, can be exploredefficiently.

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Fig. 5. Imprecise overall utilities for intervention strategies without missing consequences.

Fig. 6. Multiple boxplot without missing consequences.

The system displays a multiple boxplot for the alter-natives and computes several statistics (mode, min, 25thpercentile, . . . , mean, std. deviation) about the rankingsof each alternative. All this information can be usefulfor discarding some strategies. Fig. 6 shows the mul-tiple boxplot for a simulation where attribute weightsare randomly assigned values taking into account theweight intervals provided by the DMs in weight elicita-tion, i.e. the attribute weight intervals shown in Fig. 4.

In this multiple boxplot, only two intervention strate-gies are best ranked for at least one combination ofweights, S2 (Potash) and S4 (Food ban), but the worstclassification for both are second and seventh, respec-tively. Moreover, the mean classifications are 1.065and 3.785, respectively. The intervention strategy thatshould definitely be recommended appears to be S2

(Potash).

4. Missing consequences analysis

After having detailed the complex decision-makingproblem with consequences under uncertainty for theseven intervention strategies, we now consider the sameproblem, this time including missing consequences onthe attribute with a subjective Cost of Image scale forsome intervention strategies. Specifically, experts thattook part in the EVANET project plenary session as-sumed that instead of having the consequences listedin Table 4, the consequences of intervention strategiesS2 (Potash), S3 (Fertilizer), S5 (Lake liming), and S7

(Wetland liming) were unknown.The average normalized weight for Cost of Im-

age is 0.108, while its normalized weight interval is[0.048, 0.192], see Fig. 4. If all attributes were equallyimportant, their average normalized weight would be

A. Jiménez et al. / Omega 37 (2009) 395–410 405

Fig. 7. Imprecise overall utilities using the attribute range as missing consequences.

0.111. Therefore, we can consider this situation asbeing where the impact of the attribute with missingconsequences on the decision is neither high, nor low,it is medium. Situations when impact is high and lowwill be analyzed afterwards.

If we consider the respective attribute range as adefault value for missing consequences, as the attributerange for Cost of Image is [0, 1], strategy consequencesfor this attribute used to evaluate intervention strategiesare:

S1 S2 S3 S4 S5 S6 S7

Cost ofImage

[0, 0.1] [0, 1] [0, 1] [0.3, 0.5] [0, 1] [0.6, 0.8] [0, 1]

Note that the attribute range is only used for inter-vention strategies with missing data, while the remain-ing strategies retain the original consequences shown inTable 4.

The resulting overall utilities, also including utilitieswhen there are no missing consequences, are shownin Fig. 7. As was expected, overall utility intervals forintervention strategies with missing consequences be-come wider. For example, the overall utility interval for

S2 (Potash) was [0.601, 1.075] with no missing conse-quences, and has become [0.546, 1.088], 0.068 wider.We also find that there are variations in average over-all utilities in these intervention strategies, leading tochanges in the strategy ranking. This is because the av-erage consequences for these strategies on the Cost ofImage attribute are 0.5 when the attribute range is usedas a consequence. This does not match up with the con-sequences shown in Table 4. On the other hand, overallutilities for intervention strategies with no missing con-sequences are the same.

Now, all strategies are nondominated and potentiallyoptimal. Therefore, no intervention strategy can be dis-carded. There is no doubt that this is because of thehigher uncertainty about the intervention strategy con-sequences caused by the utilization of the attribute rangeas a missing consequence for the Cost of Image attribute.

Finally, Monte Carlo simulation techniques outputthe multiple boxplot shown in Fig. 8.

If we compare the results in Figs. 8 and 6 (withoutmissing consequences), we again find that the results aremore imprecise results. Now, three intervention strate-gies are best ranked for at least one combination ofweights, S2 (Potash), S4 (Food ban), and S7 (Wetlandliming). However, while the worst classification for S2

406 A. Jiménez et al. / Omega 37 (2009) 395–410

Fig. 8. Multiple boxplot using attribute ranges as missing consequences.

Fig. 9. Weight redistribution throughout the objective hierarchy.

and S7 is third, it is sixth for S4. Taking into account themean classifications for S2 and S7 (1.434 and 2.759),we conclude that the intervention strategy that shoulddefinitely be recommended is again S2 (Potash).

Now, let us demonstrate that the redistribution ofweights throughout the objective hierarchy is not a goodoption. We do not know the Cost of Image attributewhen evaluating intervention strategies S2, S3, S5, andS7, and redistributing its weight throughout the objec-tive hierarchy. Note that for the remaining interventionstrategies their impact is as shown in Table 4.

Fig. 9 shows average normalized weights after re-distribution throughout the objective hierarchy, and theresulting imprecise attribute weights for the decision.

Looking at the new attribute weights, we realize thatthe weights for cost to economy and cost of applica-tion have increased, with updated average normalizedweights 0.072 and 0.288, respectively. The values of theremaining attribute weights are the same because they

are not affected by redistribution. The results of the in-tervention strategies evaluation are shown in Fig. 10.

Now, the average overall utilities for interventionstrategies with missing consequences are higher, andthey are the four best ranked strategies. This is becausethey provide good consequences for attributes whoseweight has increased after the redistribution, i.e. cost toeconomy and cost of application. Therefore, less knowl-edge about intervention strategies has not led to moreimprecision about overall utilities, as was the case ofthe results in Fig. 7, but has made strategies with miss-ing consequences more desirable. The multiple boxplotfor Monte Carlo simulation performance confirms this,see Fig. 11.

Moreover, the higher the weight for Cost of Image,the more desirable these strategies would become. Forexample, if we consider the attribute weights shownin Fig. 12, where the average normalized weight forcost of image is 0.252, the results of Monte Carlo

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Fig. 10. Imprecise overall utilities redistributing weights throughout the objective hierarchy.

Fig. 11. Multiple boxplot after weight redistribution.

Fig. 12. New imprecise attribute weights.

simulation techniques without missing consequencesand redistributing weights are as shown in Fig. 13.

Looking at Fig. 13, we find that strategies with miss-ing consequences have become more desirable than the

others. For example, S7 has best, mean and worst clas-sifications second, third and sixth, respectively, with-out missing consequences, while these classificationsare second, second and third, respectively, after weight

408 A. Jiménez et al. / Omega 37 (2009) 395–410

Fig. 13. Multiple boxplots for a higher Cost of Image weight.

redistribution. At the opposite end of the scale, thebest classification for S1, with no missing consequencesis second, whereas it is always classified worst afterredistribution.

The overrated strategies with missing consequencesmight have been underrated if they had provided poorconsequences for attributes whose weight had increasedafter redistribution.

Another negative effect of weight redistribution is asfollows. Let us suppose we have two lowest-level objec-tives stemming from the same upper-level objective withlocal average normalized weights of 0.9 and 0.1, repre-senting their relative importance. The decision weightson the attribute associated with these lowest-levelobjectives, assessed using the above relative weights,are 0.45 and 0.05, respectively. In this case, when redis-tributing the weight for the first attribute, the decisionweight for the second becomes 0.5, because this is theonly objective stemming from the upper-level objec-tive. This means that an attribute that has practically noinfluence on the decision for the DMs has become akey attribute. This does not match up with the DMs’preferences. Moreover, the strategy over- or underratingeffect would be high.

This problem appears to be solved by a nonhierar-chical redistribution, i.e. directly proportional to the re-maining attribute weights on the decision. For exam-ple, if we redistribute the weight of the Cost of Imageattribute in this way, the resulting average normalizedweight over the decision would be

X1 X2 X3 X4 X5 X6 X7 X8

Avg. norm.

weight

0.1726 0.1895 0.1525 0.231 0.0661 0.0605 0.0605 0.066

However, the resulting weights do again not matchup with the DMs’ preferences. For example, DMs

stated that the relative importance of Social Impactwith respect to the environmental and economic im-pacts is 0.44. This should be equal to the summatoriumof decision weights on attributes stemming from socialimpact, but 0.1525(X3) + 0.231(X4) + 0.0661(X5) +0.0605(X6) = 0.5101.

5. Conclusions

Multiattribute expected utility theory calls for DMsto provide all the information describing the decisionsituation. This means identifying the decision alterna-tives to be analyzed in advance, as well as their impacton the attributes under consideration and uncertainty, ifnecessary. However, it can be too strict in some practi-cal situations, where the impact of decision alternativeson some attribute may be unknown for several reasons.

In this paper we consider the situation where there isthe least knowledge of the alternative consequences orperformances, i.e. when there is no knowledge whatso-ever of the performance of several alternatives for someattributes, i.e. neither a precise performance nor a prob-ability distribution can be specified.

In this situation we have compared a well-knownidea, the use of the respective attribute range as a defaultvalue for missing consequences, i.e. all attribute valuesare possible and equiprobable, against a weight redistri-bution throughout the objective hierarchy, i.e. not tak-ing into account the respective attributes with missingconsequences to evaluate each decision alternative.

We have demonstrated that weight redistributionthroughout the objective hierarchy contradicts reality.First, uncertainty about the results of the evaluationof decision alternatives with missing consequences isnot higher, as it should be. On the other hand, it couldlead to decision alternatives with missing consequences

A. Jiménez et al. / Omega 37 (2009) 395–410 409

being over- or underrated depending on the hierarchystructure, especially when not many attributes are in-volved in the redistribution. Moreover, attributes withalmost no influence on the decision could play a veryimportant role as a consequence of redistribution, con-tradicting DMs’ preferences. Such an over- or under-rating effect may also appear with a nonhierarchicalredistribution.

On the other hand, we have shown that although theincreased uncertainty about results caused by the useof attribute ranges as consequences for attributes withmissing values might appear at first glance to be a bigdrawback, decision-making with partial information canbe useful for definitively discarding poor decision alter-natives by assessing nondominated and potentially op-timal alternatives. Also, it can give further insight intothe robustness of the results, as shown throughout thepaper using Monte Carlo simulation.

Acknowledgments

Spanish Ministry of Education and Science projectsTSI2004-06801-C04-04 and MTM2005-25611-E, andMadrid regional government project S-0505/TIC/0230.

References

[1] Brownlow SA, Watson SR. Structuring multi-attribute valuehierarchies. Journal of the Operational Research Society1987;38:309–17.

[2] Keeney RL, Raiffa H. Decisions with multiple objectives:preferences and value tradeoffs. New York: Wiley; 1976.

[3] Clemen RT. Making hard decisions. An introduction to decisionanalysis. Boston: PWS-KENT Publishing Company; 1986.

[4] Kirkwood CW. Strategic decision making. Multiobjectivedecision analysis with spreadsheets. Belmont: Duxbury Press;1997.

[5] Fishburn PC. Utility theory for decision making. New York:Wiley; 1970.

[6] Savage LJ. The foundations of statistics. New York: Wiley;1954.

[7] Stewart TJ. A critical survey on the status of multiple criteriadecision making: theory and practice. Omega 1992;20:569–86.

[8] Fishburn P. Analysis of decisions with incomplete knowledgeof probabilities. Operations Research 1965;13:217–37.

[9] Jiménez A, Ríos-Insua S, Mateos S. A decision supportsystem for multiattribute utility evaluation based on impreciseassignments. Decision Support Systems 2003;36(1):65–79.

[10] Jiménez A, Ríos-Insua S, Mateos S. A generic multi-attribute analysis system. Computers & Operations Research2006;33(4):1081–101.

[11] Weber M. Decision making with incomplete information.European Journal of Operational Research 1987;28:44–57.

[12] Mustajoki J, Salo AA, Hämäläinen RP. Decision support byinterval smart/swing—a method to incorporate uncertainty intomultiattribute analysis. Technical report. Helsinki: HelsinkiUniversity of Technology, System Analysis Laboratory; 2003.

[13] Saaty TL, Vargas L. Uncertainty and rank order in the analytichierarchy process. European Journal of Operational Research1987;32:107–17.

[14] Salo AA, Hämäläinen RP. Preference programming throughapproximate ratio comparisons. European Journal ofOperational Research 1995;82:458–75.

[15] Bose U, Davey AM, Olson DL. Multi-attribute utility methodsin group decision making: past applications and potential forinclusion in GDSS. Omega 1997;25(6):691–706.

[16] Rietveld P. Multiple objective decision methods and regionalplanning. Studies in regional science and urban economics, vol.7. Amsterdam: North-Holland; 1980.

[17] Ríos Insua D, French S. A framework for sensitivity analysisin discrete multi-objective decision-making. European Journalof Operational Research 1991;54:176–90.

[18] Eum YS, Park KS, Kim SH. Establishing dominanceand potential optimality in multi-criteria analysis withimprecise weight and value. Computers & Operations Research2001;28:397–409.

[19] Lee KS, Park KS, Kim SH. Dominance, potential optimality,imprecise information and hierarchical structure in multi-criteriaanalysis. Computers & Operational Research 2002;29:1267–81.

[20] Mateos A, Ríos-Insua S, Jiménez A. Dominance, potentialoptimality and alternative ranking in an imprecise multi-attribute model. Journal of the Operations Research Society2007;58:326–36.

[21] Butler J, Jia J, Dyer J. Simulation techniques for the sensitivityanalysis of multi-criteria decision models. European Journal ofOperational Research 1997;103:531–46.

[22] Jiménez A, Ríos-Insua S, Mateos S. Monte-Carlo simulationtechniques in a multi-attribute decision support system. In:Proceedings of the 12th IASTED International Conference onApplied Simulation and Modelling; 2003. p. 85–90.

[23] Logical Decisions, V. 5.0, 1014 Wood Lily Dr, Golden, USA,1998.

[24] Canbolat Y, Chelst K, Garg N. Combining decision tree andMAUT for selecting a country for a global manufacturingfacility. Omega 2007;35(3):312–25.

[25] Carlsson C, Fullér R. Fuzzy reasoning in decision making andoptimization. Studies in fuzziness and soft computing series.Berlin, Heidelberg: Springer; 2001.

[26] Chiou H-K, Tzeng G-H, Cheng D-C. Evaluating sustainablefishing development strategies using fuzzy MCDM approach.Omega 2005;35(3):312–25.

[27] VISA for Windows, V.4. SPV Software Products, 6 BonalySteading, Edinburgh, UK, 1995.

[28] Appelgren A, Bergström U, Brittain J, Gallego E, Håkanson L,Heling R, et al., editors. An outline of a model-based expertsystem to identify optimal remedial strategies for restoringcontaminated aquatic ecosystem: the project MOIRA. ReportRT/AMB/96/17. Rome: ENEA; 1996.

[29] Monte L, Brittain J, Håkanson L, Gallego G, Zheleznyak M,Voitsekhovitch O, et al., editors. Implementing computerisedmethodologies to evaluate the effectiveness of countermeasuresfor restoring radionuclide contaminated fresh water ecosystems.Report RT/AMB/2001/28. Rome: ENEA; 2001.

[30] Saaty TL. Axiomatic foundation of the analytic hierarchyprocess. Management Science 1986;32:841–55.

[31] Harker PT. Alternative modes of questioning in the analytichierarchy process. Mathematical Modeling 1987;9:353–60.

[32] Kwiesielewicz M, van Uden E. Ranking decision variants bysubjective paired comparisons in cases with incomplete data.

410 A. Jiménez et al. / Omega 37 (2009) 395–410

Lecture notes in computer science, vol. 2669, Computationalscience and its applications (ICCSA 2003). Berlin; Springer:2003. p. 208–15.

[33] Pawlak Z. Rough sets. International Journal of Information &Computer Sciences 1982;11:341–56.

[34] Pawlak Z, Rough sets. Theoretical aspects of reasoning aboutdata. Dordrecht: Kluwer Academic Publishers; 1991.

[35] Greco S, Matarazzo B, Slowinski R. Handling missing valuesin rough set analysis of multi-attribute and multi-criteriadecision problems. In: Zhong N, Skowron A, Ohsuga S, editors.New directions in rough sets, data mining, and granular-softcomputing. Lecture notes in artificial intelligence, vol. 1711,Berlin: Springer; 1999. p. 146–57.

[36] Greco S, Matarazzo B, Slowinski R. Dealing with missingdata in rough set analysis of multi-attribute and multi-criteriadecision problems. In: Zanakis SH, Doukidis G, Zopounidis C,editors. Decision making: recent developments and worldwideapplications. Dordrecht: Kluwer Academic Publishers; 2000.p. 295–316.

[37] Brans JP, Mareschal B. Promethee methods. In: Multiple criteriadecision analysis. State of the art survey. New York: Springer;2005.

[38] Figueira J, Mousseau V, Roy B. Electre methods. In: Multiplecriteria decision analysis. State of the art survey. New York:Springer; 2005.

[39] Papadopoulos A, Karagiannidis A. Application of the multi-riteria analysis method Electre III for the optimizationof decentralised energy systems. Omega 2006, 10.1016/j.omega.2006.01.004.

[40] Geldermann J, Rentz O. Integrated technique assessment withimprecise information as a support for the identification of bestavailable techniques (BAT). OR-Spektrum 2001;23:137–57.

[41] Lahdelma R, Salminen P. SMAA-2: stochastic multicriteriaacceptability analysis for group decision making. OperationsResearch 2001;49:444–54.

[42] Lahdelma R, Miettinen K, Salminen P. Ordinal criteriain stochastic multicriteria acceptability analysis (SMAA).European Journal of Operational Research 2003;147(1):117–27.

[43] Lahdelma R, Makkonen S, Salminen P. Two ways to handledependent uncertainties in multi-criteria decision problems.Omega 2006; 10.1016/j.omega.2006.08.005.

[44] Raiffa H. The art and science of negotiation. Cambridge:Harvard University Press; 1982.

[45] Stewart TJ. Robustness of additive value function methodin MCDM. Journal of Multi-Criteria Decision Analysis1996;5:301–9.

[46] Håkanson L, Gallego E, Ríos-Insua S. The application ofthe lake ecosystem index in multi-attribute decision analysisin radioecology. Journal of Environmental Radioactivity2000;49(3):319–44.

[47] Jiménez A, Ríos-Insua S, Mateos A, Gallego E. Multiattributeevaluation module of the MOIRA system. Reference guide& user’s manual. Report CTN17-99. Madrid: Cátedra deTecnología Nuclear, ETSII, Universidad Politécnica de Madrid;2001.