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Abstrac t— Our interest here is in multi-criteria decision-making when we use a fuzzy measure to capture informationabout the importances and relationships between the criteria.We describe the use of an integral, such as the Choquet or Sugenointegral, to evaluate the overall satisfaction of each of theavailable alternatives. We discuss three measures particularlyuseful for these multi–criteria decision problems, the additive,cardinality–based and possibility measures. We note that theusefulness of these measures is a result of the fact that for each ofthese, the measure's values for any subset just depends on a smallnumber of parameters. We then consider the situation in whichwe have some imprecision in these underlying parameters. Weshow how to represent this imprecision in the underlyingparameters using a Dempster-Shafer belief structure. We thenconsider the evaluation of alternatives under this kind ofimprecision using the Choquet, Sugeno and Median typeaggregations. As a result of the imprecision in the parametersour overall evaluation for the alternatives, rather then beingsimple scalar values are imprecise, they are intervals. Wediscuss some methods for associating a scalar value with aninterval. One notable method here is what we refer to as GoldenRule aggregation.

Index Terms— Multi-Criteria, Fuzzy Measure,Imprecision Modeling, Aggregation

1. INTRODUCTION

Problems in which the selection of a best alternative from aset of possibilities is based upon its satisfaction to acollection of multiple criteria are pervasive [1-7]. Oneapproach to this task is to use a fuzzy measure to captureinformation of the importances and relationships between thecriteria [8-10]. We recall the measure associates with eachsubset of criteria a degree of importance. Once having themeasure, use is then made of an integral, such as the Choquetor Sugeno integral, to evaluate the overall satisfaction of eachof the available alternatives and then select the alternative withthe best overall satisfaction [11]. Three measures, particularlyuseful for these multi–criteria decision problems are theadditive, cardinality-based and possibility measures. The

Manuscript received Feb 6, 2013. This work was supported by theOffice of Naval Research, the Army Research Office MURI Program andthe Distinguished Scientist Fellowship Program at King Saud University..Ronald R. Yager is with the Machine Intelligence Institute, Iona College,New Rochelle, NY 10805 USA and is a Visiting Distinguished Scientist atKing Saud University, Riyadh, Saudi Arabia (e-mail: yager@panix.com ).Naif Alajlan is with the ALISR Laboratory, College of Computer andInformation Sciences, King Saud University Riyadh Saudi Arabia (e-mail:najlan@ksu.edu.sa)

usefulness of these three measures is based on the fact that foreach of these measures its value for any subset just depends ona small number of parameters. In this work we consider thesituation in which we have some imprecision in theseunderlying parameters. We shown how to represent thisimprecision in the underlying parameters using a Dempster-Shafer belief structure [12-15]. We then consider theevaluation of alternatives, under parameter imprecision, usingthe Choquet, Sugeno and Median type aggregations. As aresult of the imprecision in the parameters our overallevaluation for the alternatives rather then being simple scalarvalues are imprecise, they are intervals. In order to selectbetween these criteria we must associate with each interval ascalar. We discuss some methods for associating a scalarvalue with an interval. One notable method introduced here iswhat we refer to as Golden Rule aggregation.

2. AGGREGATING MULTI-CRITERIA USING A FUZZYMEASURE AND CHOQUET INTEGRAL

In multiple criteria decision-making we have a collection ofcriteria of interest to the decision maker C = {C1, …, Cn}.We also have a set of alternative actions {x1, …, xn} fromamong which we must select the best based on theirsatisfactions to the criteria. For any alternative x the valueCi(x) ∈ [0, 1] indicates the degree to which x satisfies thecriteria Ci. In the classic case of multi-criteria decision-making [5-7] we assume that the decision maker provides a set

αi ∈[0, 1] having αii=1

n∑ = 1 where αi is the importance

associated the criteria Ci. Using this information we cancalculate the degree to which an alternative x satisfies the

collection of criteria, S(x) = αii=1

n∑ Ci(x). Using the function

S we then select the alternative that has the largest value forS(x).

A more general approach to the representation of thecriteria importance information in multi-criteria decision-making can be obtained with the aid of a monotonic setmeasure, also called a fuzzy measure [8, 16-18]. Assume C =

{C1, …, Cn} is a collection of criteria. A monotonic set(fuzzy) measure µ on C is a mapping µ: 2C → [0, 1] such

Multi-Criteria Decision Making WithImprecise Importance Weights

Ronald R. Yager, Fellow IEEE, and Naif Alajlan

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that: 1) µ(∅) = 0, 2) µ(C) = 1 and 3) For any two subsets A ⊆

B we have µ(A) ≤ µ(B)Here then for any subset A, µ(A) ∈ [0, 1] indicates a

degree of importance associated with the subset A of criteria.A prototypical example of this is an additive set measures[17]. Here if µ({Cj}) = αj then µ(A) = α j

C j∈A∑ . Since µ(C)

= 1 then Σjαj = 1.Another example of this type of measure is a possibility

measure [19]. For this measure we have the special propertythat µ(A ∪ B) = Max[µ(A), µ(B)]. It can be shown [20] thatthis is equivalent to a measure µ in which for each Cj we havea value πj such that µ({Cj}) = πj and for any A we have µ(A)= MaxCj∈A

[πj]. In this case we require that each πj ∈ [0, 1] and

Maxj [πj] = 1.Another example of this type of measure is a cardinality-

based measure. In this case the measure of a subset A justdepends on the cardinality of A. Here we require a set of n

values rj ∈ [0, 1] with rjj=1

n∑ = 1 and define µ(A) = rj

j=1

Card(A)∑ .

We observe that if A ⊂ B, then µ(A) ≤ µ(B). Essentially rjis the gain in importance weight is going from satisfying j - 1criteria to satisfying j criteria.

While in general the specification of a fuzzy measurerequires a large number of parameters to define µ(A), 2n, thesepreceding three measures are very special in that all they needare n parameters to uniquely specify the measure for any subsetof C.

Once having a measure associated with the importances ofthe criteria a most common way of evaluating the overallsatisfaction of an alternative is to use the Choquet integral [21,11]. Assume for alternate x we let Ci(x) ∈ [0, 1] be thecriteria satisfaction. Let ind be an index function so that ind(j)is the index of the jth most satisfied criteria. Thus hereCind(1)(x) ≥  Cind(2)(x) ≥ …. ≥ Cind(n)(x). Furthermore weshall let Hj = {Cind(k)/ k = 1 to j}, it is the subset of j mostsatisfied criteria for alternative x. Using this we calculate

Choqµ(x) = (µ(H j)j=1

n∑ - µ(Hj - 1))Cind(j)(x)

We note that if we let wj = µ(Hj) - µ(Hj - 1) then Choqµ(x) =

wjj=1

n∑ Cind(j)(x). It be easily shown that all wj ∈ [0, 1] and

wjj=1

n∑ = 1. It is a weighted average of the criteria

satisfactions.We note that the Choquet integral can be equivalently

expressed as

Choqµ(x) = j=1

n∑ (Cind(j)(x) - Cind(j+1)(x)) µ(Hj)

with the understanding that Cind(n+1)(x) = 0.In the special case where µ is the additive measure then we

easily see that wj = µ( Hj) – µ( Hj - 1) = αind(j) and hence

Choqµ(x) = wind( j)j=1

n∑ Cind(j)x = αi

j=1

n∑ Ci(x).

In this case we get the classic approach to the determination ofmulti-criteria satisfaction by alternative x.

In the case of the cardinality-based measure we see that Hj

= {Cind(k)/k = 1 to j} has j elements thus µ(Hj) = rjK=1

j∑ .

With Choqµ(x) = µj=1

n∑ (Hj) - µ(Hj-1) Cind(j)(x)=

wjj=1

n∑ Cind(j)(x) and since for the cardinality based measure wj

= µ(Hj) - µ(Hj-1) = rkk=1

j∑ - rk

k=1

j−1∑ = rj then

Choqµ(x) = wjj=1

n∑ Cind( j)(x) = rj

j=1

n∑ Cind( j)(x)

Here wj are again a collection of weights such that wj ∈ [0, 1]

and wjj=1

n∑ = 1. We observe that wj = rj is independent of x.

We note that is formulation is equivalent to the OWA operatorintroduced by Yager [22, 23].

Two special cases of the preceding are worth pointing out.The first case which we denote as µ* is one in which, r1 = 1and rj = 0 for all j = 2 to n. In this case w1 = 1 and wj = 0 for

j ≠ 1. Using this we see that Choqµ* (x) = wjj=1

n∑ Cind(j)(x)

= Cind(1)(x) = Maxi[Ci(x)]. Here we get the satisfaction ofthe most satisfied criteria by alternative x.

The second special case is denoted µ* is one in which rn =1 and rj = 0 for all j = 1 to n-1. In this case wn = 1 and wj = 0for j ≠ n. Using this we see that

Choqµ* (x) = Cind(n) = Mini(Ci(x))

Intermediate to these two is a measure µK such that rK = 1 andrj = 0 for j ≠ K. In this case we can show that ChoqµK (x) =

Cind(K)(x). It is equal to the value of the K most satisfiedcriteria.

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We now consider the case of a possibility type measure.Here µ({Ci) = πi and µ(A) = Max

Ci∈A[πi ] . Here we observe that

µ(Hj) = Maxk=1 to k

[πind(k)] . In this case

Choqµ(x) =  wjj=1

n∑ Cind(j)(x) with wj = µ(Hj) - µ(Hj - 1) is

slightly more complex then in the procedure because the wj aremore complex. We observe that wj = 0 if πind(j) ≤ Maxk = 1 to j-1[πind(k)] wj = πind(j) - Maxk = 1 to j-1[πind(k)] otherwiseWe can more succinctly express this as

wj = (πind(j) - Maxk = 1 to j-1[πind(k)]) ∨ 0

An alternative expression is wj = (πind(j) - wkk=1

j−1∑ ) ∨ 0

We note that the only restriction on the πi other thanlying in the unit interval is that at least one of the πi = 1. Weobserve that if all πi = 1 then w1 = 1 and Choqµ(x) =Maxi[Ci(x)]. In the case where all of the πi = 0 except πK = 1then we see Choqµ(x) = CK(x).

Assume π1 = 1 and π2 = 1 and all other πi = 0, thenChoqµ(x) = Max[C1(x), C2(x)]

Assume π1 = 1 and π2 = β < 1. We each see that ifC1(x) > C2(x) then Choqµ(x) = C1(x) and if C2(x) > C1(x)then Choqµ(x) = β C2(x) + (1 - β)C1(x),

3. MODELING IMPRECISE WEIGHTS USING BELIEFSTRUCTURES

In the preceding we looked at a number of approaches for themodeling of multi-criteria decision functions based on ameasure whose complete specification was determined using asmall number of parameters. Central to these was theavailability of precise values for the associated parameters, inthe case of the additive measures the αj’s, in the case of thecardinality base measure the rj's and in the case of thepossibility type measure the πj's. Now we want to considerthe situation in which there is a possibility for someimprecision, uncertainty, in our knowledge of the associatedparameters. We shall first start with the case of additive typeof measure and extend this to include the possibility ofuncertainties in the parameters.

Let us first recall that measures in addition to being usedfor the representation of information in multi-criteria decision-making are also used to represent information about uncertainvariables [24, 25]. In the case of providing information aboutan uncertain variable if V is a variable taking its value in theset X = {x1, …, xn}, then the interpretation of the measureµ :  2X → [0, 1] is such that µ(A) indicates the anticipationthat V lies in A. A prototypical example of uncertainty is

probability theory. In this case the measure used is an additivemeasure, where the probabilities play the same role as theimportance weights, the αj, in multi-criteria decision-making.A well-known approach for the modeling of impression inprobabilistic information is the Dempster-Shafer model [12,26]. Here we shall investigate the role of a Dempster–Shafertype model in representing imprecise information in the multi-criteria model.

Assume we have a set C = {C1, …, Cn} of criteria ofinterest in a problem. We now let F1, …, Fq be a collectionof non-empty subsets of C called focal elements. We notethat no requirements are made on the focal elements other thanthey are non-empty. We now associate with each subset Fi a

value αi ∈ [0, 1] and require that αii=1

q∑ = 1. In the following

we will use m to indicate this Dempster-Shafer type structure.Our interpretation of this model is that αi is an amount ofimportance weight that is allocated in some unknown mannerto the criteria in Fi. This is the basis of our imprecision inthe important weights.

Our concern here is with the determination of the value ofthe Choquet integral in the face of the Dempster-Shaferrepresentation m of our allocation of importance weights.Here we assume that Ci(x) ∈ [0, 1] is the satisfaction ofcriteria Ci by alternate x. Our object here is to calculateChoqm(C1(x), …, Cn(x)). We recall in the case of using theDempster-Shafer approach for modeling impression inprobabilistic uncertainty, where we can't exactly calculate theprobabilities of events, we obtain only intervals in which theylie. The same is true of our use of Dempster–Shafer model formulti-criteria decision-making, we can only obtain intervalswithin which the payoffs associated with alternatives will lie.Here we shall obtain upper and lower bounds forChoqm(C1(x), …, Cn(x)). In particularChoqm(C1(x), …, Cn(x)) ∈ [Low–Choqm(C1(x), …, Cn(x)),

Upp–Choqm(C1(x), …, Cn(x))]In the following, when appropriate, we shall use the termsUppm(x) and Lowm(x) as short hands for Upp–Choqm(C1(x),…, Cm(x)) and Low–Choqm(C1(x), …, C1m(x)) respectively.

We now let ind be an index function such that ind(j) is theindex of the criteria with the jth largest satisfaction underalternative x. Here Cind(1)(x) = Maxk[Ck(x)].

We now express

Uppm(x) = wind( j)j=1

n∑ Cind(j)(x)

where

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wind(j) = αii=1

q∑ (Fi(Cind(j)) ∧ (1 - Maxk = 1to j-1Fi(Cind(k))))

We note that Fi(Cind(j)) = 1 if Cind(j) ∈Fi and Fi(Cind(j)) = 0if Cind(j) ∉ Fi.

Here we see using this we are trying to assign as much ofthe weights as possible to those criteria with the mostsatisfaction, this is kind of optimistic.

To obtain Lowm(x) we let pind be an index function sothat pind(j) is the index of the criteria with the jth smallest satisfaction under x, here Cind(1)(x) = Mink[Ck(x)]. Usingthis we express

Lowm(x) = wpind( j)j=1

n∑ Cpind(j)(x)

Where wpind(j)

= αii=1

q∑ (Fi(Cpind(j)) ∧ (1-Maxk =1to j-1Fi(Cpind(k))))

Here we are assigning as much of the importance associatedwith Fi to those criteria with the least satisfaction.

In passing we note ind and pind are related bypind(j) = n - ind(j) + 1

The following example illustrates the above approachExample: Assume four criteria C1, C2, C3, C4 where foralternative x we have

C1(x) = 0.2, C2(x) = 1, C3(x) = 0.6, C 4 = 0.5In this case we have that ind is such that

ind(1) = 2, ind(2) = 3, ind(3) = 4 and ind(4) = 1From this we get pind as

pind(1) = 1, pind(2) = 4, pind(3) = 3 and pind(4) = 2We assume information about importance is carried by a beliefstructure with three focal elements

F1 = {C1, C2}, F3 = {C2, C3, C4} and F3 = {C1, C4}wherem(F1) = α1 = 0.3, m(F2) = α2 = 0.5 and m(F 3) = α2 = 0.2

We see that

Uppm(x) = wind( j)j=1

4∑ Cind(j) = wind(1)C2(x) + wind(2)C3(x)

+ wind(3)C4(x) + wind(4)C1(x)We calculate the wind(j) as follows

wind(1) = α1 + α2 = 0.8wind(2) = 0wind(3) = α3 = 0.2wind(4) = 0

ThusUppm(x) = 0.8c2(x) + 0.2C4(x) = 0.8(1) + (0.2) (0.5) = 0.9

To calculate Low(x) we have

Lowm(x) = wpind( j)i=1

4∑ Cpind(j)(x) = wpind(1)C1(x) +

wpind(2)C4 + wpind(3)C3(x) + wpind(4)c2(x)wpind(1) = α1 + α3 = 0.5wpind(2) = α2 = 0.5wpind(3) = 0wpind(4) = 0

From this we have

Lowm(x) = 0.5C1(x) + 0.5C4(x) = 12

(0.3 + 0.5) = 0.4

Then in this case Choqm(x) ∈ [0.4, 0.9].Given a Dempster-Shafer type allocation of the

importance weights we can express a degree of precision as

Pm = 1

αi Card(Fi )i=1

q∑

We see that if all Fi are singletons, all the Cind(Fi) = 1, andsince Σiαi = 1 then Pm = 1. At the other extreme is whereα1 = 1 and F1 = {C1, …, Cn} in this case Pm = 1/n.

The following provides semantics for a situation in whichthe preceding Dempster-Shafer representation would beappropriate. Consider a multi-criteria decision problem inwhich we have a decision maker indicating that they areinterested in the satisfying the criteria Fi for i = 1 to q whereαi is the importance they attribute to Fi. However, for theobjects being considered the criteria Fi are not directlymeasurable, they are too abstract. However, there exists anadditional set of criteria/features, Cj for j = 1 to n associatedwith the alternatives, that can be measured, for any alternativex, Cj(x) is the degree of satisfaction. Additionally the decisionmaker can associate with each of the Fi a subset Fi of the Cjwhose satisfactions are directly related to the satisfaction ofFi. However, in this environment we know no more thenthat the satisfactions to the Cj in Fi are good for thesatisfaction to the Fi.

A simple example would be when a person is consideringtaking a new job that requires them moving to a new location.Here they may consider three meta-criteria, location good forthe children, job good for career and good place to live. Thesewould constitute the Fi. These in turn would need to be morespecifically defined in terms of subsets of measurable criteriathat could contribute to their satisfaction, the Fi.

4. IMPRECISION IN CARDINALITY-BASED MEASUREWe now consider the modeling of imprecision in the weightsin the case of a cardinality–based measure using the frameworkof a Dempster-Shafer type structure. Here we let R = {r1, …,rn} indicate the n weights associated with a cardinality-basedmeasure where rj denotes the weight associated with the jth

most satisfied criteria. We now describe a Dempster–Shafertype structure, mR, defined on R consisting of a collection

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G1, …, Gq of subsets of R called focal elements wheremR(Gi) = βi so that βi ∈ [0, 1] and Σi βi = 1. Here this beliefstructure is indicating that the amount of weight mR(Gi) isshared among the rk’s that are contained in Gi.

Assume Ck(x) is the degree of satisfaction to criteria Ckby alternative x. Here again using the Choquet integral leadsus to an interval type solution.

ChoqmR(C1(x), …, Cn(x)) ∈ [Low–ChoqmR(C1(x), …,Cn(x)), Upp–ChoqmR(C1(x), …, Cn(x))]

Let ind be an index function so that ind(j) is the index of thejth most satisfied criteria. In this case we get that

Upp–ChoqmR(C1(x), …, Cn(x)) = wind( j)j=1

n∑ Cind(j)(x)

where

wind(j) = βii=1

q∑ [Gi(rj) ∧ (1 - Maxk = 1 to j-1Gi(rk))]

Here we see in this formulation that we are trying to push asmuch of the weights to the wind(j)'s associated with the moresatisfied criteria, the smaller j.

We obtain Low–ChoqmR(C1(x), …, Cn(x))] as follows

Low–ChoqmR(C1(x), …, Cn(x))] =

wind( j)j=1

n∑ Cind(j)(x)

where

wind( j) = βi

i=1

q∑ [Gi(rj) ∧ (1 - Max

k= j+1 to nGi(rk))]

Here we see in this that we are trying to push as much ofthe weights to the wind(j)'s associated with the least satisfiedcriteria, the larger j.

The following example illustrates this calculationExample: Here we have four criteria C1, C2, C3, C4. Foralternative x we have

C1(x) = 0.3, C2(x) = 1, C3(x) = 0.6 , C4(x) = 0.5We shall assume

G1 = {r1, r3}, G2 = {r2, r4} and G3 = {r1, r2, r3, r4}with

β1 = 0.5, β 2 = 0.3 and β 3 = 0.2Here we have

Cind(1)(x) = C2(x) = 1Cind(2)(x) = C3(x) = 0.6Cind(3)(x) = C4(x) = 0.5Cind(4)(x) = C1(x) = 0.2

From this we calculatewind(1) = 0.5 + 0.2 = 0.7wind(2) = 0.3wind(3) = Wind(4) = 0

From this we get

Upp–ChoqmR(C1(x), …, C4(x))= (0.7)(1) + (0.3)0.6 = 0.88For the lower case we obtain

wind(4) = 0.3 + 0.2 = 0.5

wind(3) = 0.5 = 0.5

wind(2) = 0

wind(1) = 0

In this case we haveLow–ChoqmR(C1(x), …, Cn(x)) = (0)(1) + (0)(0.6) + (5)(0.3)

+(3)(0.3) = 12

(0.8) = 0.4

Thus hereChoqmR(C1(x), …, Cn(x)) ∈ [0.4, 0.8]

5. IMPRECISION WITH POSSIBILISTIC MEASUREIn the case in which the underlying measure carrying theinformation about the importance of criteria is a possibilitiesmeasure the representation of imprecise information requires adifferent structure.. Here we shall associate with each criteriaCi a value bi ∈ [0, 1] with the understanding that πi ≤ bi.Thus bi is an upper bound on µ(Ci). Furthermore we requirethat at least one of bi = 1.

Again in this case we letChoqπ(C1(x), …, Cn(x)) ∈ [Low–Choqπ(C1(x), …, Cn(x)),

Upp–Choqπ(C1(x), …, Cn(x))]Here

Upp–Choqπ(C1(x), …, Cn(x)) = wind( j)j=1

n∑ Cind(j)(x)

where ind(j) is the index of the jth largest Ci(x). Furthermorewind(j) = (bind(j) - Max

k= 1 to j−1[bind(k)]) ∨ 0

Similarly

Low–Choqπ(C1(x), …, Cn(x))]= wpind( j)j=1

n∑ Cpind(j)(x)

here pind(j) is the index of the jth least satisfaction criteria. Inthis case

wpind(j) = (bpind(j) - Maxk= 1 to j−1

[bpind(k)]) ∨ 0

The following example illustrates this approachExample: Assume four criteria C1, C2, C3, C4 where foralternative x we have

C1(x) = 0.3, C2(x) = 1, C3(x) = 0.6, C4(x) = 0.5We assume

b1 = 0.5, b2 = 0.7, b3 = 1 and b4 = 1In this case

ind(1) = 2, ind(2) = 3, ind(3) = 4, ind(4) = 1Using this we

wind(1) = bind(1) = 0.7wind(2) = (bind(2) - 0.7) ∨ 0 = 1 - 0.7) = 0.3wind(3) = 0

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wind(4) = 0We now obtain

Upp–Choqπ(C1(x), C2(x), C3(x), C4(x))) = (0.7)(1) +(0.3)(0.6) = 0.7 + 0.18 = 0.88

For calculatingLow–Choqπ(C1(x), C2(x), C3(x), C4(x)) =

wpind( j)j=1

4∑ Cpind(j)(x)

wherewpind(j) = (bpind(j) - Max

k= 1 to j−1[bpind(k)]) ∨ 0

we first note thatpind(1) = 1, pind(2) = 4, pind(3) = 3, pind(4) = 2

Using this we calculatewpind(1) = bpind(1) = b1 = 0.5wpind(2) - (bpind(2) - 0.5) ∨ 0 = b4 - 0.5) = 0.5wpind(3) = wpind(4) = 0

Here then

Low–Choqπ(C1(x), C2(x), C3(x), C4(x)) = 12

(0.3) = 12

(0.5)

= 0.4This here

Choqπ(C1(x), C2(x), C3(x), C4(x)) ∈[0.4, 0.88]

6 USING SUGENO AND MEDIANIn the preceding we used a measure µ on a space of criteria C ={C1, …, Cn} to capture our information above theimportances associated with the criteria. Using this with Ci(x)∈ [0, 1] as the satisfaction of criteria Ci by x we calculated theoverall satisfaction using the Choquet integral as Choqµ(x) =

wjj=1

n∑ Cind(j)(x) where Cind(j) is the satisfaction of jth most

satisfied criteria and wj = µ(Hj) - µ(Hj) where Hj is the set ofthe j most satisfied criteria. Other integrals can be used toobtain this overall satisfaction [27, 10, 11]. One of these isthe Sugeno integral [8]. In this case

Sugµ(x) = Maxj=1 to n

[µ(Hj) ∧ Cind(j)(x)]

here ∧ indicates the Min operator.Let us look at this for the special cases of the measures

that we discussed earlier. Assume µ is a simple additivemeasure where the parameter αi corresponds to µ(Ci). Herethen if ind(j) is the index of the jth most satisfied criteria thenαind(j) is its associated parameter, µ(Cind(j)) . Since

Hj = {Cind(k)/k = 1 to j} then µ(Hj) = αind(k)k=1

j∑ and here

Sugµ(x) = Maxj=1 to n

[ αind(k)k=1

j∑ ∧ Cind(j)(x)]

In the case where µ is a cardinality based measure we have

µ(Hj) = rkk=1

r∑ . From this we get

Sugµ(x) = Maxj=1 to n

[ rkk=1

j∑ ∧ Cind(j)(x)]

In the case of the possibility measure we have µ(Hj) =Maxk=1 to j

[πind(k)]. From this we see

Sugµ(x) = Maxj=1 to n

[ Maxk=1 to j

[πind(k)] ∧ Cind(j)(x)]

However a little algebra shows that this possibilistic casetakes a very simple form. Let us denote T(j) =Maxk=1 to j

[πind(k)] then Sugµ(x) = Maxj=1 to n

[T(j) ∧ Cind(j)(x)].

However let us now observe that T(j) = T(j - 1) ∨ πind(j)where ∨ indicates the max operator. Consider any term T(j*)such that T(j*) ≠ πind(j*) then T(j* - 1) > πind(j*). In thiscase since Cind(j* - 1)(x) ≥ Cind(j*)(x) then

T(j*) ∧ Cind(j*)(x) ≤ T(j* - 1) ∧ Cind(j* - 1)(x)and hence we can remove the term T(j*) ∧ cind(j*)(x)

Thus the only terms T(j) ∧ Cind(j)(x) they contribute toMaxj=1ton

[T(j) ∧ Cind(j)(x)] are these for where T(j) = πind(j).

From these we see a sample forSugµ(x) = Max

j=1ton[πind(j) ∧ Cind(j)x] = Max

i=1ton[π i ∧ Ci(x)]

Another possible approach is the median type operator.Here again letting µ(Hj) be the measure of the subset of the jmost satisfied criteria we observe that µ(Hj) ≥ µ(k) if j ≥ k.Now let j* be such that µ(Hj* - 1) < 0.5 ≤ µ(Hj*). Usingthis we obtain

Medµ(x) = Cind(j*)(x)It is the j* most satisfied criteria.

The calculation of Medµ(x) essentially requires thedetermination of j*, a task that is particularly simple for thethree special measures.

1) The first step in all cases is to order the Ci(x) indecreasing order

Cind(1)(x) ≥ Cind(2)(x) ≥ …,≥ Cind(n)(x)2) The second step is to determine j*. W describe this for

each of the measures in the followinga) Additive measure: Find j* so that

αind(k)k=1

j*−1∑ < 0.5 ≤ αind(k)

k=1

j*

∑

b) Cardinal-based measure. Find j* so that

rkk=1

j*−1∑ < 0.5 ≤ rk

k=1

j*

∑

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c) Possibility measure: Find j* so thatMax

k= 1 to j*−1[πind(k)] < 0.5 ≤ Max

k= 1 to j*[πind(k)]

3) Once having j* we easily obtainMedµ(x) = Cind(j*)(x)

Here we make some observations1) For the cardinality-based measure j* is independent of

the ordering of values of Ci(x). It will be the same for allalternatives. It is just depends on the rk.

2) In the case of possibility measure j* is the minimalk so that πind(k) ≥ 0.5. In this possibilistic case weessentially order the πI via index function

πind(1), πind(2), ………, πind(n)then starting from the left j* is equal ind(k) where αind(k) isfirst value equal or greater then 0.5

We note that a more general form of median can beobtained by replacing 0.5 by any value λ  ∈ [0, 1]. Weobserve that if λ → 1 then we get MiniCi(x) and if λ → 0 weget Maxi[Ci(x)].

It is interesting to observe that each of these approacheshas particular benefits in case of one of the measure. We notethat in the case of the additive measure the use of the Choquet

integral results in the simple form Choqµ(x) = αii=1

n∑ Ci(x).

Here no ordering is necessary and also no aggregation of theαi.

We note that in the case of the possibilistic measure, theuse of the Sugeno integral results in the simple form Sugµ(x)

= Maxi=1

n[πi ∧ Ci(x)]. Here again there is no ordering necessary

and also no aggregation is of πi is required.Consider now the case of the cardinality-based measure

with the median type integral. Here for any alternative x wehave

Med(x)i = Cind(j*)(x)it is always the j* the most satisfied criteria. Here for eachalternative we just need to order the criteria satisfaction andthen select the j* largest.

. 7. INCLUDING IMPRECISION WITH SUGENO ANDMEDIAN AGGREGATION

In preceding we discussed ways of representing imprecisionwith respect to the parameters associated with the additive,possibilistic and cardinality-based measures. Let us investigatewhat happens when we use the Sugeno and Median typeaggregations. Here we shall assume a collection C = {C1, …,Cn} of criteria.

In the case of additive measure the parameters are the αi.Here we represented our imprecision using a Dempster-Shafertype structure with focal elements F1, …, Fq and associated

with each is m(Fi) ≥ 0 where Σim(Fi) = 1 with theunderstanding that the amount of importance m(Fi) can bedistributed among the criteria in Fi. Let us consider theSugeno integral in this case. Here we shall obtain an interval

Sugm(C1(x), …, Cn(x)) = [Low–Sugm(C1(x), …, Cn(x)),Upp–Sugm(C1(x), …, Cn(x))]

In calculating Upp–Sugm(C1(x), …, Cn(x)), we let ind(j) bethe index of the jth most satisfied criteria. Using this we let

αind( j) = m(Fi )j=1

q∑ (Fi(Cind(j))) ∧ (1 -

Maxk= 1 to j−1

(Fi(Cind(k)))))

From this we obtain

Upp–Sugm(C1(x), …, Cn(x))) = Maxj=1 to n

[(

α ind(k)k=1

j∑ ) ∧

Cind(j)(x)]To calculate Low–Sugm(C1(x), …, Cn(x)) we shall let

pind(j) indicate the index of the jth least satisfied criteria.Using this we calculate

tpind( j) = m(Fi )

j=1

n∑ (Fi(Cpind(j))) ∧ (1 - Maxk = 1 to j -

1(Fi(Cpind(k))))From this we let

αind( j) =

tpind(n − j + 1) ). Using this we get

Low–Sugm(C1(x), …, Cn(x)) = Maxj=1 to n

[

αind(k)k=1

j∑ ∧

Cind(j)(x)]In the case of the median type aggregation if we have

imprecision in the additive measure we proceed in ananalogous manner

Medm(C1(x), …, Cn(x)) = [Low–Medm(C1(x), …, Cn(x)),Upp–Medm(C1(x), …, Cn(x))]

Here we calculate αind( j) and α ind( j) as in the preceding.

Using these in the case of Upp–Medm(C1(x), …, Cn(x)) we

calculate

j* so that

αind( j)j=1

j*−1∑ < 0.5 ≤ αind( j)

j=1

j*

∑ . Using

this we obtain Upp–Medm(C1(x), …, Cn(x)) = Cind( j*)(x) .

In the case of Low–Medm(C1(x), …, Cn(x)) we calculate j *

so that

αind( j)j=1

j*−1∑ ≤ 0.5 ≤

α ind( j)j=1

j*

∑ . Using this we obtain

Low–Medm(C1(x), …, Cn(x)) =

Cind( j* )

(x)

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We consider the case of cardinality-based measure. In thiscase the parameters are rj, for j = 1 to n. Here we let R = {r1,…, rn}. We shall assume the same modeling for theuncertainty in these parameters as in the earlier case, aDempster-Shafer belief structure mR . Here then with we letG1, …, Gq be of subsets of R called focal elements. Here wehave mR(Gi) = βi where βi ∈ [0, 1] and Σβi = 1. Ourunderstanding is that the value βi can be allocated in any wayto the elements in Gi. Again we let Cind(j)(x) be the degree ofsatisfaction of the jth most satisfied criteria.

Here we shall obtain in the face of this imprecision in ourknowledge of the values of parameters R that for alternative xits degree of satisfaction using the Sugeno integral is anintervalSugmR(C1(x), …, Cn(x)) = [Low–SugmR(C1(x), …, Cn(x)),

Upp–SugmR(C1(x), …, Cn(x))]Here we let Hj be collection of the j most satisfied criteria.We now let Poss(Hj/Gi) be such that

Poss(Hj/Gi) = 0 if Hj ∩ Ei = ∅Poss(Hj/Gi) = 1 if Hj ∩ Ei ≠ ∅

Using this we define

µ(H j) = βii=1

q∑ (Poss(Hj/Gj) ∧ (1 - Max

k= 1 to j−1[Poss[Hj/Gk)]))

and

µ(H j) = βi

i=1

q∑ (Poss(Hj/Gj) ∧ (1 - Max

k= j+1 to q[Poss[Hj/Gk)]))

Using these value for criteriaLow–SugmR(C1(x), …, Cn(x)) = Maxj[

µ(H j) ∧ Cind(j)(x)]

Upp–SugmR(C1(x), …, Cn(x)) = Maxj[ µ(H j) ∧ Cind(j)(x)]

In the case of median type integral we obtain when thereis an imprecision in our knowledge of the cardinality-basedmeasure that the degree of satisfaction by alternative x isMedmR(C1(x), …, Cn(x)) = [Low–MedmR(C1(x), …, Cn(x)),

Upp–MedmR(C1(x), …, Cn(x))]To obtain these we shall again use

µ and

µ . Using

µ we

find j such that µ(H j−1)  <  0 .5  ≤  

µ(H j ) and using

µ we

find j such that

µ(H j−1) < 0.5 ≤

µ(H j) . Having j and j

we obtainLow–MedmR(C1(x), …, Cn(x))=

Cind(j) (x)

Upp–MedmR(C1(x), …, Cn(x))= Cind( j) (x)

We now turn to the case of an imprecise possibility typemeasure. Here we shall assume two pairs of π’s, πi and

π i

so that πi ≥ π i for i = 1 to n and at least one pair ( πi π

i ) has

both equal to 1. We call πi the upper possibility and π i the

lower possibility. We say that the true πI ∈ [ π i , πi ]. Again

if Cind(j)(x) is the value of the jth most satisfied criteria wedefine Hj = {Cind(k)/k = 1 to j}. We also let

π ind( j) and

πind( j) indicate the values of π and for the jth most satisfied

criteria, Cind(j). Using this we define µ(H j) =

Maxk= 1 to j

[ πind(k) ] and µ(H j) = Max

k= 1 to j [

π ind(k) ]. Having

these values we obtainSugπ(C1(x), …, Cn(x)) = [Low–Sugπ(C1(x), …, Cn(x)),

Upp–Sugπ(C1(x), …, Cn(x))]whereLow–Sugπ(C1(x), …, Cn(x)) = Max

j=1 to n[ µ(H j) ∧ Cind(j)(x)]

Upp–Sugπ(C1(x), …, Cn(x)) = Maxj=1 to n

[ µ(H j) ∧ Cind(j)x]

In the case of the median type integral we proceed as wedid earlier. We calculate

j so

µ(H j−1) < 0.5 ≤

µ(H j) and

j  so that µ(H j−1) < 0.5 ≤

µ(H j ) . Using these we get

Medπ(C1(x), …, Cn(x))= [ Cind(j) (x),

Cind( j) (x)]

8. COMPARING IMPRECISE ALTERNATIVESATISFACTIONS USING REPRESENTATIVE VALUES

In all the situations in which we had some imprecision in ourknowledge of the parameters associated with the measurecarrying information about the multi-criteria aggregationimperative we obtained an interval for the degree of satisfactionof an alternative xi to the multi-criteria, Sat(xi) = [ai, bi].Here of course ai ≤ bi and both lie in the unit interval. Thushere Sat(xi) is a sub–interval of unit interval I. In thefollowing we shall use the notation [II] to indicate the set of all

intervals of the unit interval [II] = {[a, b]} such that a ≤ b and

a, b ∈[0, 1]. Thus Sat(xi) ∈[II].

In decision making environments we are faced with theproblem of selecting one alternative as the best. Here then webecome faced with the problem of comparing intervals.Clearly if [ai, bi] and [aj, bj] are two intervals so that aj ≥ bithen we can say xj is preferred to xi we shall denote this [aj,bj] [ai, bi]. In order to be able to select alternative xk as thebest from a collection {x1, , xn} we must have [ak, bk] [ai, bi] for all i ≠ k. Generally this is a very difficultrequirement to meet. In order to circumvent this difficulty acommon approach is to associate with each interval arepresentative scalar value, Rep(xi), and compare thealternatives with respect to this representative value. Thus ifRep(xj) > Rep(xi) we shall say alternative xj is preferred to xi.

In the following using the notation [II] to indicate the setof all intervals of the unit interval we can formally expressthat Rep is a mapping Rep: [II] → [0, 1], it maps intervals ofthe unit interval into numbers in the unit interval.

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One required feature of this representative value Rep isthat it is consistent with the comparison of the intervals, if[aj, bj] [ai, bi] then Rep(xj) ≥ Rep(xi). A more generaldesired property, which includes the preceding as a special case,is what we shall refer to as monotonicity. In order to definethis concept of monotonicity we introduce one notationalconvenience. Assume [aj, bj] and [ai, bi] are two intervals weshall see that [aj, bj] ≥ [ai, bi] if aj ≥ ai and bj ≥ bi. Pleasenote the difference between [aj, bj] ≥ [ai, bi] and [aj, bj] [ai,bi], in particular [aj, bj] [ai, bi] requires that aj ≥ bi. Wefurther note that if [aj, bj] [ai, bi] then aj ≥ bi and hence aj≥ ai and bj  ≥   bi thus if [aj, bj] [ai, bi] it surely satisfies[aj, bj] ≥ [ai, bi].We now provide a definition for monotonicityDefinition: A function Rep: [II] → [0, 1] is said to be

monotonic if Rep([aj, bj]) ≥ Rep([ai, bi]) when [aj, bj] ≥ [ai,bi]. Thus we require that any function Rep used to comparealternatives be monotonic. Furthermore since we have shownthat if [aj, bj] [ai, bi] then [aj, bj] ≥ [ai, bi] because of themonotonicity of Rep if [aj, bj] [ai, bi] then Rep([aj, bj]) ≥Rep([ai, bi]) and thus we always get the correct preferencebetween xj and xi.

In choosing a function Rep to adjudicate betweenimprecise payoffs in addition to it being monotonic it shouldreflect, as much as possible, the decision maker's preferencesand attitude with respect to the comparison of imprecisepayoffs. Thus there is some freedom in selecting the functionRep, allowing some room for subjectivity. In thisenvironment it is useful to have multiple possible choices forthe representative function, particularly if we can have someintuitive understanding of the underlying decision makingattitude that is reflected in the choice the of a representativefunction.

One common choice for the representative value of theseintervals is the mid-point

Rep(xi) = mi =ai + bi2

.

This is clearly monotonic. In some cases, the use of thissimple representative for the interval may lead to results thatmay not fully reflect the preferences for some decisionmakers.

a

mi

m j

0 1

0 1

ai bi

bja j

Figure 1. Example of two intervals associatedwith xi and xj

Consider the situation shown in figure 1. We see that herewhile both alternative xi and xj have the same mid-pointvalue, mi = mj, the ranges for their possible degrees ofsatisfactions are clearly different. We see the range(variability) for alternative xj, rj = bj - aj, is much larger thenthat for xi, ri =  bi  –   ai. Some decision makers may beuncomfortable with the large variability in xj and want theirrepresentative value to reflect their preference for xi in thissituation. On the other hand some other decision makersmay like the larger availability of possibilities in xj as itoffers more opportunity of getting a larger degree ofsatisfaction

In the following we shall suggest a more generalformulation for the representative value of alternative whichallows for a more sophisticated representative value whichincludes the mean value of the interval, mj, the range ofinterval, rj, and a subjective parameter, λ ∈ [-1, 1], reflectingthe decision makers attitude toward the resolution ofuncertainty. Using these we suggest a formulation

Rep(xj) = mj + λ rj2

We see this is equivalent to

Rep(xj) = 12

((bj + aj) + λ (bj - aj)).

In this formulation the parameter λ can be viewed as beingdrawn from scale such that positive values, λ > 0, indicate anoptimistic attitude toward a resolution of the uncertaintyassociated with the satisfaction by an alternative. On the otherhand the negative values, λ < 0, indicates a pessimisticattitude toward a resolution of the uncertainty associated withthe satisfaction by an alternative. A value of λ = 0 reflects adecision makers neutrality on this issue. Anotherinterpretation of the scale from which λ is drawn is onerepresenting certainty versus opportunity. Negative valuesindicating a decision maker's preference for certainty whilepositive values indicate a preference for opportunity.

Let us look at this form of representative value for somenotable choices of λ . If λ = 0 we get that Rep(xj) = mj =12

(bj + aj), our original suggestion. If λ = 1 then

Rep(xj) = 12

((bj + aj) + (bj - aj)) = bj

and if λ = -1 then Rep(xj) = 12

((bj + aj) - (bj - aj)) = aj.

In using Rep(xj) = mj + λ rj2

we that for positive values

of λ the larger rj the bigger Rep(xj) while for negative valuesof λ the smaller rj the bigger Rep(xj). Here we can concludethat optimistic decision makers prefer wider ranges, biggervalues for rj, while pessimistic decision makers prefer narrowervalues for rj. Thus given two alternatives with the same valuefor m the optimist, λ > 0, would prefer the alternative with

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the larger rj and the pessimist, λ < 0, would prefer thealternative with the smaller rj

We must now assure ourselves of the monotonicity ofthis formulation for the Rep function. Consider two intervals[aj, bj] and [ai, bi] such that aj = ai + d and bj = bi + e whered and e ≥ 0. In this case

Rep(xi) = 12

((bi + ai) + λ (bi - ai))

Rep(xj) = 12

((bj + aj) + λ (bj - aj))

= 12

((bi + e + ai + d) + λ (bi + e - (ai + d)))

Since [aj, bj] ≥ [ai, bi] then monotonicity requires thatRep(xj) ≥ Rep(xi). Let us look at this.

Rep(xj) - Rep(xi) = 12

(e + d + λ(e - d))

Rep(xj) - Rep(xi) = 12

(e(1 + λ) + d(1 - λ))

Since λ ∈ [-1, 1] then 1 + λ ∈ [0, 2] and (1 - λ) ∈ [0, 2] andhence both e(1 + λ) and d(1 - λ) ) are non-negative and thusRep(xj) - Rep(xi) ≥ 0. Thus monotonicity is satisfied.

9. GOLDEN RULE BASED REPRESENTATIVE VALUESAnother approach to the construction of the function Rep isto use a rule-base specification of Rep and implement itusing the Takagi-Sugeno approach to fuzzy systemsmodeling [28, 29]. Here we illustrate this with using thefollowing four rules capturing the decision maker's preferenceregarding the formulation of Rep.If the mean is large and the range is small then Rep(x) = 1If the mean is large and the range is large then Rep(x) = 0.5If the mean is small and the range is large then Rep(x) = 0.5If the mean is small and the range is small then Rep(x) = 0

Here the decision maker is expressing their preferenceregarding uncertainty interval as: if the mean of the intervalis large I am satisfied and I prefer no opportunities fordeviation. On the other hand if the mean is small I want alarge range. This can be seen as a kind of golden rule, "ifyou have don't be greedy and if you don't have try to dobetter"

Here for simplicity we represent large and small assimple linear fuzzy sets L and S defined on the unit intervalwere L(y) = y and S(z) = 1 - z. If x is an alternative with

payoff [a, b], m = a + b2

and r = b - a then using the Takagi-

Sugeno implementation [28, 29] of this rule base we get

Rep(x) = m(1− r) + (0.5)mr + (0.5)(1−m)r + 0(1−m)(1− r)m(1− r) +mr + (1−m)r + (1−m)(1− r)

Since m(1− r) +mr + (1−m)r + (1−m)(1− r) = 1 then we get

Rep(x) = m(1− r) + (0.5)mr + (0.5)(1−m)r = m + 12

r - mr

Replacing m and r respectively with a + b2

and b - a we get

Rep(x) = 12

((a + b) + (b - a) - (a + b)(b-a)) = 12

(2b - b2 + a2)

Monotonicity requires that dRe p(x)da

≥ 0 and

dRe p(x)db

≥ 0 we see here that

dRe p(x)da

= a ≥ 0 and dRe p(x)db

= 1 - b ≥ 0

hence the formulation is monotonic.

Consider the formulation Rep(x) = m + 12

r - mr we see

that with a little algebra

Rep(x) = m + ( 12

- m)r

This last formulation invites comparison with preceding

formulation Rep(x) = m + 12λr. We see in the formulation

Rep(x) = m + 12λr the range r contributes to Rep in a manner

depending on the parameter λ, for positive λ the range addsvalue and for negative λ the range reduces value. In the case

where Rep(x) = m + ( 12

- m)r the range r contributes to Rep

in a manner depending on the mean m, for mean greater then12

the range reduces value and for mean less then 12

the range

adds value. We shall refer to the formulation Rep(x) = m +

( 12

- m)r as Golden-Rule Aggregation.

It is interesting to observe that

m + ( 12

- m)r = (1 - r)m +r 12

thus here Rep is weighted average of mean m and 12

were the

weights depend on r, the range.

10. CONCLUSIONWe focused on the problem of multi-criteria decision-making.We discussed the use of a fuzzy measure to captureinformation bout the importances and relationships betweenthe criteria. We described the role of an integral, such as theChoquet or Sugeno integral, to evaluate the overallsatisfaction of each of the available alternatives. Weintroduced three measures particularly useful for thesemulti–criteria decision problems, the additive,cardinality–based and possibility measures. We noted that theusefulness of these measures is a result of the fact that foreach of these the measures values for any subset just dependson a small number of parameters. We then considered thesituation in which we have some imprecision in theseunderlying parameters. We showed how to represent theseimprecision's in the underlying parameters using a Dempster-Shafer belief structure. We then considered the evaluation ofalternatives under imprecision using the Choquet, Sugeno andMedian type aggregations. As a result of the imprecision in

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the parameters our overall evaluation for the alternativesrather then being simple scalar values are imprecise, they areintervals. We discussed some methods for associating ascalar value with an interval. One notable method introducedhere is the Golden Rule aggregation.

. ACKNOWLEDGEMENTThis work has been supported by an ONR grant award. This work has alsobeen supported by a Multidisciplinary University Research Initiative (MURI)grant(Number W911NF-09-1-0392) issued by the US ArmyResearch Office (ARO). The authors would like to acknowledge the supportfrom the Distinguished Scientist Fellowship Program at King Saud University..

REFERENCES[1]. Yager, R. R., "Multiple objective decision making using fuzzy sets,"International J. of Man-Machine Studies 9, 375-382, 1977.[2]. Kahraman, C., Fuzzy Multi-Criteria Decision Making: Theory andApplications with Recent Developments, Springer: Heidelberg, 2008.[3]. Greco, S., Ehrgott, M. and Figueira, Trends in Multiple Criteria DecisionAnalysis, Springer: Heidelberg, 2010.[4]. Pedrycz, W., Ekel, P. and Parreiras, R., Fuzzy Multicriteria Decision-Making: Models, Methods and Applications, John Wiley: New York, 2010.[5]. Triantaphyllou, E., Multi-Criteria Decision Making Methods: AComparative Study, Kluwer: Dordrecht, 2010.[6]. Koksalan, M., Wallenius, J. and Zionts, S., Multiple Criteria DecisionMaking: From Early History to the 21st Century, World Scientific Publishing:Singapore, 2011.[7]. Tzeng, G. H. and Huang, J. J., Multiple Attribute Decision Making, CRCPress: Boca Raton, 2011.[8]. Sugeno, M., "Fuzzy measures and fuzzy integrals: a survey," in FuzzyAutomata and Decision Process, Gupta, M.M., Saridis, G.N. & Gaines, B.R.(eds.), Amsterdam: North-Holland Pub, 89-102, 1977.[9]. Murofushi, T. and Sugeno, M., "Theory of fuzzy measures:representations, the Choquet integral and null sets," J. Math Anal. Appl 159,532-549, 1991.[10]. Grabisch, M., Murofushi, T. and Sugeno, M., Fuzzy Measures andIntegrals, Physica-Verlag: Heidelberg, 2000.[11]. Wang, Z., Yang, R. and Leung, K.-S., Nonlinear Integrals and TheirApplications in Data Mining, World Scientific: Singapore, 2010.[12]. Yager, R. R. and Liu, L., (A. P. Dempster and G. Shafer, AdvisoryEditors) Classic Works of the Dempster-Shafer Theory of Belief Functions,Springer: Heidelberg, 2008.[13]. Liu, L. and Yager, R. R., "Classic works of the Dempster-Shafer theoryof belief functions: an introduction," in Classic Works of the Dempster-Shafer Theory of Belief Functions, edited by Yager, R. R. and Liu, L.,Springer: Heidelberg, 1-34, 2008.[14]. Dempster, A. P., "The Dempster-Shafer calculus for statisticians,"International Journal of Approximate Reasoning 48, 365-377, 2008.[15]. Yager, R. R., Kacprzyk, J. and Fedrizzi, M., Advances in theDempster-Shafer Theory of Evidence, John Wiley & Sons: New York, 1994.[16]. Murofushi, T. and Sugeno, M., "Fuzzy measures and fuzzy integrals,"in Fuzzy Measures and Integrals, edited by Grabisch, M., Murofushi, T. andSugeno, M., Physica-Verlag: Heidelberg, 3-41, 2000.[17]. Klir, G. J., Uncertainty and Information, John Wiley & Sons: NewYork, 2006.[18]. Wang, Z. and Klir, G. J., Generalized Measure Theory, Springer: NewYork, 2009.[19]. Zadeh, L. A., "Possibility theory and its application to informationanalysis," in Proceedings of International Colloquium on Information Theory,CNRS, Paris, 173-182, 1978.[20]. Dubois, D. and Prade, H., Possibility Theory : An Approach toComputerized Processing of Uncertainty, Plenum Press: New York, 1988.[21]. Choquet, G., "Theory of Capacities," Annales de l'Institut Fourier 5,131-295, 1953.[22]. Yager, R. R., "On ordered weighted averaging aggregation operatorsin multi-criteria decision making," IEEE Transactions on Systems, Man andCybernetics 18, 183-190, 1988.[23]. Yager, R. R., Kacprzyk, J. and Beliakov, G., Recent Developments inthe Ordered Weighted Averaging Operators: Theory and Practice, Springer:Berlin, 2011.[24]. Yager, R. R., "A measure based approach to the fusion of possibilisticand probabilistic uncertainty," Fuzzy Optimization and Decision Making 10,91-113, 2011.[25]. Yager, R. R. and Alajlan, N., "Measure based representation ofuncertain information," Fuzzy Optimization and Decision Making 11, 363-385, 2012.[26]. Yager, R. R. and Filev, D. P., "Using Dempster-Shafer structures toprovide probabilistic outputs in fuzzy systems modeling," In CombiningExperimentation and Theory: A Homage to Abe Mamdani, edited by Trillas,

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Ronald R. Yager has worked in the area ofmachine intelligence for over twenty-five years.He has published over 500 papers and fifteenbooks in areas related to fuzzy sets, decisionmaking under uncertainty and the fusion ofinformation. He is among the world’s top 1%most highly cited researchers with over 7000citations. He was a recipient of the IEEEComputational Intelligence Society Pioneeraward in Fuzzy Systems. Dr. Yager is a fellowof the IEEE, the New York Academy ofSciences and the Fuzzy Systems Association.He was given a lifetime achievement award by

the Polish Academy of Sciences for his contributions. He served at theNational Science Foundation as program director in the InformationSciences program. He was a NASA/Stanford visiting fellow and a researchassociate at the University of California, Berkeley. He has been a lecturerat NATO Advanced Study Institutes. He is a distinguished honoraryprofessor at the Aalborg University Denmark. He is distinguished visitingscientist at King Saud University, Riyadh, Saudi Arabia. He received hisundergraduate degree from the City College of New York and his Ph. D.from the Polytechnic University of New York. Currently, he is Director ofthe Machine Intelligence Institute and Professor of Information Systems atIona College. He is editor and chief of the International Journal ofIntelligent Systems. He serves on the editorial board of numeroustechnology journals.

Naif Alajlan received his B.Sc. (with first class honors)and M.Sc. degrees in Electrical Engineering from

King Saud University, Riyadh, Saudi Arabia in 1998and 2003, respectively. He received his PhD degreefrom the University of Waterloo, Waterloo, Ontario,Canada in 2006. He worked as a Systems andControl engineer in the Saudi Basic Industries Co.(SABIC), Riyadh from 1998-2000, then joined the

EElectrical Engineering Dept., King Saud University aslecturer from 2000-2003 and assistant professor from

2006-2011. Since 2011, he works as an associateprofessor at the Computer Engineering Department,King Saud University, Riyadh, Saudi Arabia. His

current research interests include shape retrieval, pattern recognition,remote sensing and image processing. He authored and co-authored morethan 23 journal papers (some with high impact factors) and 12 conferencepapers. He is the founder and director of the Innovation Center and theAdvanced Lab for Intelligent Systems Research at King Saud University..