Aggregate Utility Factor Model: A Concept for Modeling Pair-wise Dependent Attributes in Multiattribute Utility Theory

Transcription

1 Universität Bayreuth Rechts- und Wirtschaftswissenschaftliche Fakultät Wirtschaftswissenschaftliche Diskussionspapiere Aggregate Utility Factor Model: A Concept for Modeling Pair-wise Dependent Attributes in Multiattribute Utility Theory Johannes Siebert Discussion Paper June 2010 ISSN Correspondence address: University of Bayreuth Department of Law and Economics Institute for Production and Industrial Management D Bayreuth Phone: +49 (0) Fax: +49 (0) Johannes.Siebert@uni-bayreuth.de Johannes.Siebert@gmx.de

2 Abstract A brief review of the art of modeling in MAUT shows that virtually all used model variants are based on an additive decision rule. Limitations of MAUT with respect to modeling attribute interactions are identified and analyzed. Since in practice virtually any target system contains interactive attributes, a real multiplicative model is presented that extends MAUT such that attribute interactions can be accounted for. The utility of an attribute level is expressed by a utility factor that is formed as a ratio of the utility of the considered attribute level to the utility of an average evaluated attribute. The product of the utility factors is decomposed by means of a Taylor series and modified such that interactions can be modeled conform to corresponding preferences. A straightforward step-wise procedure for separately eliciting utility factors for attribute levels and interaction coefficients for quantifying complementary interactions is provided. Keywords: Decision Theory, Multiattribute Utility Theory, Modeling Interactions, Multiplicative Model, Preference Independence, Aggregated Utility Factor Model.

3 1 Introduction The multiattribute utility theory (MAUT) is widely used for achieving rational decisions that reflect as closely as possible the preferences of a decision maker (DM). The target system of objectives is formed by a set of attributes. Elicitation of the DM s preferences determines utility levels for each attribute level. The utility levels for any one attribute form the utility function for that attribute. The utility functions are also called attribute utilities and are multiplied with weights that serve to express the relative importance of the attributes. Under the constraint of attribute utility independence, the aggregated utility is expressible as a linear combination of weighted attribute utilities (Keeney and Raiffa 1976). The most general model is the so called multilinear model. It contains in addition to the attribute weights interaction coefficients for modeling interaction between two or more attributes. The so called multiplicative model is derived from the multilinear model by setting all interaction coefficients equal to one constant, while the purely additive model can be derived by setting this constant to zero (Keeney 1974). In practice there is virtually no target system without at least some interactive attributes (Bell 1979, Keeney 1981, von Winterfeldt and Edwards 1986). For example, when buying a house, the attributes attractive architecture and a pleasant garden are complementarily interacting since `pleasant garden accentuates an `attractive architecture and vice versa (Goodwin and Wright 2004). Similarly, when selecting a manager, there exists a strengthening complementary interaction between the attributes qualifications and leadership characteristics. In both examples, a simple addition of the attribute utilities would be insufficient. Candidates possessing useful qualifications and promising leadership characteristics have an appreciable added value. Therefore, clearly the attributes attractive architecture and pleasant garden as well as the attributes qualifications and leadership characteristics are not independent since the contribution to the overall utility by one attribute depends on the utility level of the other attribute. In other words, the preference for one attribute does depend on the preference for another attribute. This dependence of attributes is the basic assumption and contemporaneously the Achilles heel of MAUT since it requires preference independence. Abbas (2009) concludes: If these utility independence conditions do not hold, then we are faced with the need to incorporate utility dependence among the attributes into our analysis and summarizes methods proposed in particular by Farquhar (1975), Keeney and Raiffa (1976), Kirkwood (1976), Bell (1979, Keeney (1981), as well as Abbas and Howard (2005)

4 Farquhar (1975) establishes a fundamental decomposition theorem for multiattribute utility functions. The independence conditions in this theorem are based on conditional preference orders derived from fractional hypercubes. These hypercubes are used to generate a variety of attribute independence conditions that are necessary and sufficient for various decompositions, among them also non-additive one that contain some non-separable interaction terms and are therefore applicable to decision problems not covered by earlier models. Keeney and Raiffa (1976) construct a multiattribute utility function that includes utility dependence. To that purpose, they construct curves along which U(x 1,, x n ) is constant. Kirkwood (1976) introduces and defines the concept of parametric dependence of preferences for multiattribute consequences. In essence, marginal utility functions are expressed as functions of dependent parameters and then used to represent the utility function. Bell (1979) constructs multidimensional utility distributions introducing interpolation schemes between utilities that are determined on grid spanned by the attribute utilities. The procedure starts with the corner points and allows refinements if the extra work is justified by the importance of the decision. Keeney (1981) suggests a problem-oriented approach. Based on an analysis of the causes of dependencies in a given set of attributes, the attributes are divided into components or transformed into an alternative set of attributes that is structurally related to the original set such that the new set has a lesser degree of dependence. Abbas and Howard (2005) introduce the so called attribute dominance utility functions. Dominance means here that the utility U(x 1,, x i* =x i*,min,,x n ) assumes the value zero for all possible levels of X ii*. They show that the construction of multiattribute utility functions that incorporate utility dependence is greatly simplified when attribute dominance utility conditions exist. These conditions allow introducing marginal utility functions adopting and adapting decomposition techniques known in distribution theory. Based on this paper, Abbas (2009) proposes the use of Archimedean utility copulas that link marginal utility functions under certain transformations that make partial utility independence among the attributes possible. These proposals provide more flexibility in the preference structure and lessen the degree of dependence. They require in addition a considerable amount of effort in preference elicitation and strictly speaking, most of them only produce fairly accurate solutions. All variants of MAUT models are in essence based on an additive decision rule for utility aggregation. The independence conditions are needed for preserving the transitivity of the utility functions on an interval scale level (Keeney and Raiffa, 1976). Therefore, consistent modeling of interactions in MAUT is hardly possible

5 This paper extends MAUT by suggesting a real multiplicative model and a general mathematical decomposition that includes interactions. The model is termed Aggregated Utility Factor Model (AUFM). It is based on a modified Taylor series of the product of utility factors for all attributes and a simple aggregation algorithm. The modified series contains a primary part that accounts for the contribution of single attributes to an aggregated utility factor (AUF) and a secondary part that includes pair-wise interaction among attributes. Furthermore, a straightforward step-wise procedure for eliciting utility factors and interaction coefficients separately for quantifying complementary interactions is provided. This paper is organized as follows. Section 2 summarizes the state of the research in MAUT, which shows that all commonly used model variants are based on an additive decision rule and analyzes the limitations of MAUT with respect to modeling attribute interactions. Section 3.1 introduces the concept of an utility factor and the multiplicative modeling in MAUT (MM in MAUT) that yields a product of utility factors. A Taylor series is used in Section 3.2 for decomposing this product, and Section 3.3 introduces the aggregated utility factor as a conclusive modification of this decomposition. The aggregation model is described in Section 3.4 and the elicitation and modeling is explicated in Section 3.5. Section 4 summarizes and draws conclusions. 2 Multiattribute Utility Theory: Scope and Limitations 2.1 Basic concepts and modeling The nomenclature used for MAUT is non-uniform in literature. In essence the author follows the nomenclature introduced by Keeney and Raiffa (1976). A decision in economics serves to select one or more out of given alternatives A j, j[1,m], where m is the number of given A j. The decision-relevant properties of A j are called attributes X i, i[1,n]. Each X i must have at least two attribute levels x i,1 and x i,2. All scale levels are permitted. For example, when buying a used car, the attributes color and fuel consumption might be considered among others. The attribute levels of color are two or more colors on a nominal scale. The attribute levels of fuel consumption cover a continuous finite domain on an interval scale. An alternative A j is completely described by definite attribute levels for all attributes, e.g. A j (x 1,j = red, x 2,j = 0.06 l/km, ), uncertainties associated with attribute levels, for instance expressed by probability distributions, are not considered in this paper. Attribute levels for A j are written as x i,j, irrespective of their scale level. In MAUT it is tacitly assumed that the set of - 4 -

6 selected attributes is complete, i.e. all decision-relevant properties of the alternatives can be expressed. In that sense, the set of attributes is a target system for the objectives of the decision. A utility level u i (x i ) is associated with each attribute level x i. The utility level is a real function of the attribute levels. Thus, the domain of u i (x i ) coincides with the domain of X i. The range of u i (x i ) is normalized to the interval [0.0,1.0]. The relative importance of the attributes X i is expressed by weights w i, i[1,n]. The weights are finite positive real numbers. The utility of an alternative A j is a real valued function of w i and u i,j, termed U j. The range of U j is normalized to the interval [0.0, 1.0]. The term aggregated utility is also used for U j. It is noted that U is a measurable function if and only if the utility functions u for all attribute are measurable functions, cf. Section 2.2. In this case, measurable is used synonymously with on interval scale level. The models for MAUT contain the weighted utilities of attributes and their products. The products contain real scaling constants that scale their relative importance. They express the contribution to utility that depends jointly on all attributes in such a product; therefore they are called interaction coefficients,,. The set of the real valued weights, utility levels, and the interaction coefficients reflect the preferences of the DM. The functional dependence of U j on its arguments w i and u i,j as well as interaction coefficients results from modeling the preference aggregation. The basic concepts of MAUT are the decomposition of the measurable aggregated utility independent of each other determined weights and utility levels for all attributes and their compensatory composition using aggregation models (von Winterfeldt and Edwards 1986). This concept allows to consider a large number of attributes with reasonable effort. However, modeling eliciting preferences and choosing as well as interpreting interaction coefficients are subject to some constraints, see Section 2.2. Following Keeney and Raiffa (1976), the most general model of MAUT is multilinear. The aggregated utility U j is given by (1),,,,, where U multilinear, w, u and the,,, are the aggregated utility, attribute weights, utility levels and interaction coefficients, respectively, as defined above. The last term can also be expressed as by the following:,,,,,. (2) - 5 -

7 Since the range of the aggregated utility extends by definition from zero to one, it obtains a constraint for normalization by setting all u i,j in Formula (2) equal to one: 1. (3) If all scaling constants in Formula (1) are replaced such that, up to a, this yields the multiplicative aggregation model which, in a compact form, is given by, 1, 1. (4) The normalization constraint, using the same argument as for Formula (3), is given by 1 1. (5) Finally, setting all interaction coefficients in Formula (1) to zero leads to the additive aggregation model that is given by,, with the normalization constraint 1. (6) Both models, the multiplicative and the additive, are derived from the multilinear model by restricting the number of the interaction coefficients in Formula (1) (Keeney 1974). It is important to note that the multilinear model and its variants are in essence based on an additive approach, since they all are expressed by a sum over various terms (Farquhar 1975). Therefore the author uses the term additive modeling (AM) in MAUT for these models. 2.2 Necessary and sufficient conditions for additive modeling in MAUT The decomposition of the aggregated utility in weights and utility levels as well as the compensatory composition presupposes that the attributes and the utility levels fulfill some conditions (Abbas 2009). Several degrees and different formulations of independence are found in the literature (Keeney and Raiffa 1976, Dyer and Sarin 1979, von Winterfeldt and Edwards 1986, Miyamoto and Wakker 1996). Therefore, these conditions are briefly recapitulated here. The so called preference independence is given if the preference, i.e. the assigned utility for any given level of a considerable attribute, does not depend on the levels of the other attributes. Accordingly, (1) An attribute X i* is preferentially independent of an attribute X ki* if u i* (x i* ) for any possible attribute level x i* is the same for all possible levels x ki*

8 (2) Preference independence is given if all attributes X i*, i*[1,,n], are preferentially independent from all attributes X k, k[1,, i*1,i*+1,, n]. Considering the pair of alternatives I I,I,,,I,,,I and II II,II,I,,,II,,,II,I and denoting the corresponding aggregated utilities by U I and U II allows a compact formal definition of the so called difference independence: (1) An attribute X i* is difference independent if for all possible attribute levels x k,i ki* and all transitive relations, i.e. for <, =, >,, and, generically represented by, holds that U I U II if u i*,i (x i*,i ) u i*,ii (x i*,ii ). (2) Difference independence is given if all attributes X i*, i*[1,, n], are difference independent. These conditions assure that the utility levels of attribute levels and thereby the aggregated utility levels are measurable functions since they are transitive which therefore permit the relations <, =, >,, and. If preference independence and difference independence are given, then a compensatory utility tradeoff between the utility levels of any pair of attribute levels (, ) is possible. This allows that alternatives compensate a low utility level for one attribute level by a high utility level for a different attribute level ; where i 1 i 2 (Labreuche and Grabisch 2007). This compensation is one reason for considering an additive composition of the weighted utility levels as the utility level for an alternative. Difference independence warrants that the utility levels for any attribute are measurable on an interval scale level. Since the weights of the attributes are also measurable on an interval scale level, it follows that an additive composition of the weighted utility levels also yields a measureable aggregated utility level. Keeney and Raiffa (1976), derive an additive decision rule from preference independence and some additional presuppositions that are not relevant here. French (1986) conjectures that even given strong preference independence an additive decision rule is required for an appropriate mapping of the preferences. 2.3 Elicitation of preferences The DM s first task is to select an appropriate set of attributes that form a target system for the objectives, unless the given objectives of a well established set of attributes is available - 7 -

9 (Tsetlin and Winkler 2007). This selection implicitly reflects preferences. Theory does not provide specific guidance on selecting attributes, but it does provide some constraints on a selection. Mathematically interpreted, the attributes are the dimensions of a decision space and should be independent of each other since preference independence would otherwise be violated. The DM expresses his preferences by assigning utility levels to all levels of the attributes and values to the weights of the attributes. The assignment is in essence a difference value measurement since higher values enhance and lower values decrease the expected utility. When using multilinear or multiplicative models, the complete expression of preferences requires the assignment of values to the interaction coefficients; see Keeney (1974) and Dyer and Sarin (1979). The supposition of preference independence allows the assignment of utility levels to levels of any attribute independently of the utility levels for the levels of all other attributes as well as the assignment of weights independently of each other. In such a separate assignment, the required efforts of the DM are greatly decreased (Farquhar 1975). Assignments of a utility level to a specific attribute level x i or of weights to attributes express subjective judgments. A survey on methods for constructing utility functions ranging from simple or bisectional rating methods to fairly complex indifference methods is found in Keeney and Raiffa (1976) as well as in von Winterfeldt and Edwards (1986). Therefore, to introduce the problem it is sufficient to describe the so called direct rating method which in itself is quite straightforward. When an attribute has only a small number m of levels it is feasible to assign utility levels u i (x i,m ) for the few possible levels x i,1,,x i,m. However, if m is large or if x i is continuous in the domain of the attribute it is preferable to use a utility function for this attribute. The simplest procedure is to use the direct rating method in order to obtain an appropriate subset of attribute levels and to construct a utility function using interpolation schemes. The use of spline functions yields a continuous utility function. The methods for assessing weights (which is being used synonymously with tradeoffs) range from approximation procedures like ratio or swing weighting (von Winterfeldt and Edwards 1986) to indifference procedures and pricing-out methods (Keeney and Raiffa 1976). The assessment of weights for the attributes is necessary since utility levels for any attribute have by definition the range [0.0, 1.0] and therefore any attribute would be of equal importance. This in turn would rarely reflect the DM s preferences (von Winterfeldt and Edwards 1986). The weights can mathematically be interpreted as conversion factors that - 8 -

10 serve to use the same global unit for utility levels with all attributes that span the decision space. However, the usage of such a global unit in eliciting preferences and consequently the usage of different ranges for the utility levels appears far more difficult and error prone than a separated elicitation. 2.4 Complementary Interactions Various and quite different aspects are subsumed under the term of interactions between attributes. Attempts for modeling interactions within MAUT using either multiplicative or multilinear aggregation focus almost exclusively on complementary interactions for expressing synergetic effects. Since this section describes additive modeling MAUT, only complementary interactions are discussed. For instance, when selecting a manager the DM considers among others the attributes on qualifications and leadership characteristics. The DM prefers a candidate whose utility levels for both of these attributes are above average more than a simple addition of these utilities would indicate. In other words, the DM sees a strengthening complementary interaction between these attributes. In the additive modeling in MAUT the average utility level is simply the midpoint of the range, u[0.0,1.0], i.e., 0.5. Hence u > 0.5 is above average and u < 0.5 is below average (Labreuche and Grabisch 2006). For a more detailed analysis, three different cases are to be discussed: Case I (both utility levels are above average), Case II (both utility levels are below average), and Case III (one utility level is above average and the other is below average). Interaction terms are provided by the multilinear and the multiplicative model but not by the most widely used additive model (Grabisch 1996, Goodwin and WrigFht 2004, Tsetlin and Winkler 2009). In Case I, multilinear and the multiplicative model quantify a positive intensifying effect by a term that increases approximately quadratic with increasing utility levels. In Case II, the complementary interaction is modeled with the relative absence of a positive intensifying effect (Siebert 2009). In Case III, the models produce a slight increase of utility due to the complementary interaction. This modus operandi appears to be ad hoc, not based on an economical interpretation, and the handling of Case II and Case III does not appear to be conclusive (Siebert 2009). In analogy to the increase of utility in Case I, a marked decrease of utility is expected in Case II. It is plausible and consistent to request that the decrease in Case II is somehow reciprocal to increase in Case I. In Case III, it is surprising that the models slightly enhance the utility. Why should the binary interaction - 9 -

11 overcompensate a below average utility for one attribute level because the utility for the other attribute level is above average? This is not conclusive. Rather, it appears reasonable to request that an appropriate model overrides the interaction in Case III. In summary, a consistent model accounts for complementary interactions by enhancing utility in Case I, decreasing it in Case II, and leaving it unchanged in Case III. When modeling influences of interactions on the aggregated utility it is helpful to distinguish between primary and secondary parts. The primary part is the contribution of terms to the aggregated utility that solely depend on one attribute level, i.e. it is part of all models that result from AM in MAUT and given by,. The secondary part is the contribution of terms to the aggregated utility that depend on the levels of two attributes. Higher order terms (tertiary, quaternary, etc.) are not considered here. The number of interaction coefficients in the multilinear model depends on the number of attributes; see Formula (1). For instance, the target system formed by n = 10 attributes would entail having 1022 interaction coefficients. An elicitation of all these interaction coefficients under the normalization constraint given in Formula (3) is not manageable (Bell 1979, von Winterfeldt and Edwards 1986). At a first glance, the multiplicative model, see Formula (4), seems to be preferable since it uses one interaction coefficient only. The terms obtained from product in Formula (4) can be sorted into n groups. The first group is the primary part, i.e.,, that does not contain the interaction coefficient. The second group is the secondary part, i.e. a sum over products of two weighted utility levels and the interaction coefficient (von Winterfeldt and Edwards 1986). The third group the a sum over products of three weighted utility levels and the squared interaction coefficient and so forth up to the last group which is the product over all n weighted utility levels and I n1. The value of the interaction coefficient is not freely elicited since the weights and the interaction coefficient have to comply with the normalization constraint given in Formula (5). As discussed in Section 2.2, the attributes that form the target system should be independent of each other, otherwise preference independence would be violated and neither the single attribute nor the aggregated utility would be a measurable function. However, in practice, as exemplified with the manager attributes qualifications and leadership

12 characteristics, the DM s preferences can only be met by modeling complementary interactions. According to several authors, a target system without such interactions seems to be an exemption (Bell 1979, Keeney 1981, von Winterfeldt and Edwards 1986). For instance, Keeney and Raiffa (1976) consider the requirement of preference independence as unrealistic. 2.5 Modeling interactions in MAUT The discussion of complementary interactions in Section 2.4 has shown that using additive model variants of MAUT, i.e. U multilinear and U multiplicative, produce puzzling results that do not allow a conclusive interpretation. Furthermore, the large number of possible interaction coefficients in U multilinear would have to be reduced in practice (von Winterfeldt and Edwards 1986). It is plausible to consider only pair-wise complementary interactions, since preference elicitation for the interactions among three or more attributes appears to be very difficult. It can also be argued that in practice interactions only concern a few attributes. The simplest case for demonstrating this is a target system with three attributes and an interaction between two pairs attributes, i.e. between X 1 and X 2 as well as X 2 and X 3. Equation (1) can be reduced to,,,,,,,, (7) and is subject to the constraint 1,,. The utility functions and the weights can be elicited as described in Section 2.3. However, the interaction coefficients are constrained. This means the preference cannot be expressed independently. Given the weights, it is only possible to express a relative importance of I 1,2 / I 2,3. This limitation is even worse when using the so called multiplicative model: and is subject to the constraint 1,,. The utility functions and the weights can be elicited as described in Section 2.3. However, if is equal to one, then,,. This means the preference for interactions cannot be expressed independently, unless the w i are rescaled. This situation is even worse when using the so called multiplicative model:,,,,,,, (8) since here the cannot have the value one, unless I = 0. If 1, then the interaction coefficient I is defined by the given weights, irrespective of the DM s judgment on interaction (von Winterfeldt and Edwards 1986). Furthermore, accepting this procedure as an ad hoc approximation is not conclusive since this procedure unavoidably violates preference

13 independence. However, preference independence is a necessary condition for obtaining measurable utility functions. In summary, using the additive decision rule as required for achieving a measurable utility function forbids the inclusion of complementary attribute interactions. The discussion of complementary interactions in Section 2.4 has shown that a consistent model should account for them by enhancing utility in Case I, decreasing it in Case II, and leaving it unchanged in Case III. Furthermore, it was identified that the treatment of Case I and Case II should be somehow reciprocal. These properties can be obtained by introducing a multiplicative decision rule. This means that the utility levels for attribute levels are to be defined as utility factors. Consequently, in eliciting preferences the DM would not use an interval scale but a ratio scale instead and therefore, the tradeoff would be expressed as a ratio. As will be derived in Section 3, the simple multiplication of utility factors for the attribute levels forms an aggregated utility factor. Both, the utility factors for the attribute levels and the aggregated utility factor are then measurable functions on a ratio scale. As also derived in Section 3, interactions can be accounted for without constraints. Models based on the multiplicative decision rule and utility factors are termed multiplicative models (MM) in MAUT, while models based on the additive decision rule and utility functions are termed additive models (AM) in MAUT. 3 Aggregated Utility Factor Model 3.1 Utility factor Multiplicative Modeling in MAUT When using AM in MAUT, both the range of the single attribute utilities as well as the range of the aggregated utility extends from zero to one. MM in MAUT is based on quite a different measurement of scale and interpretation. Here the utility level of an attribute X i is interpreted as the ratio to an average utility level of the attribute. The qualifier average is not used to denote the mean of given or expected alternatives, but rather like a grade as it is used at school or at a university, i.e. grade A (excellent), B (above average), C (average), D (below average), or F (fail). However, the mean grade in a class could deviate significantly from C. The level x i of the attribute X i to which an average utility level is ascribed is called reference point x i,ref. The set of attribute values (x 1,ref,, x n,ref ) is termed global reference point. The utility level in the context of MM in MAUT is termed utility factor and the symbol p(x i ) is used, where p(x i ) is a real valued function. The range of p(x i ) is limited to p(x i )(0,),

14 i.e. only finite real values greater than zero are admissible. The domain of p(x i ) is the range of the level x i of the attribute X i. For the sake of simplicity, p(x i ) is abbreviated as p i. Different attributes may be expressed on different scale levels. However, levels of a specific attribute must be on the same scale level. Irrespective of the scale level of an attribute, the scale for the associated utility factor is the ratio scale. Therefore, the DM would assign a utility factor 2 p(x i,ref ) to an alternative j with an attribute level x i,j if the DM expects that this alternative is twice as useful as an alternative with an attribute level x i,ref. Vice versa, the DM would assign a utility factor of 1 / 2 p(x i,ref ) to an alternative j with an attribute level x i,j if the DM expects that this alternative is half as useful as an alternative with an attribute level x i,ref. In MM in MAUT, the aggregation of the preferences is achieved by multiplication of the utility factors for all attributes. For this aggregated utility factor the symbol is used. Furthermore, is a function of the utility factors for all attributes i[1,n] for any given alternative j:,,,,,. (9) An alternative that is averagely evaluated with respect to all attributes is called the alternative at the global reference point, i.e.,,,,. Assigning the number one to the average utility level of an attribute, i.e. p(x i,ref ) =: 1, then yields consistently the value one for the aggregated utility level at the global reference point, and j > 1 then refers to an above average alternative j, whereas j < 1 refers to a below average alternative j. The real multiplicative model in MAUT as formulated in Equation (9) implies that the change of j due to a change in any one utility factor, to, does depend on the value of all other utility factors: Δ,,,,,,,,,,,,,,,. (10) In general, this given interdependence needs to be modified for adequately expressing preferences of the DM. Thus, as modeled in Equation (9) is termed unmodified aggregated utility factor (unmodified AUF). A Taylor series of at the global reference point is used as a basis for these modifications. 3.2 Representation of the Utility Factor by a Taylor Series The Taylor series of a function is used in mathematics to describe the local behavior of a function at a given set of arguments. In applied sciences it is also used to approximate a function that is difficult to evaluate by a function that can easily be evaluated. The Taylor

15 series for a function f(x) at the point x 0 is given as a power series of x = xx 0, f(x)=, where the coefficients c k are the k th order derivatives of the function, evaluated at x 0 and divided by k!; note that k is a positive integer and 0! = 1! = 1 and for k > 1 it follows that k! = 2 3 (k1) k. For example, consider the function f(x) = exp(x): ; where!. (11) In practice, only terms up to a low order are used. The possible deviation due to neglecting terms is called remainder R. Thus, for small values of x it is sufficient to use two terms only: e e 1. (12) The range of the values of the remainder can be evaluated in this simple case if the domain of exp(x) is given. A Taylor series can also be used to represent a function of more than one variable, such as the unmodified AUF obtained from MM in MAUT. The general form of the series is preserved, but the derivate with respect to one variable is replaced by a sum over partial derivates. The Taylor series for the unmodified AUF at the global reference point is given by:!,,, ; (13) where p i = p i p i,ref = p i 1. Furthermore, using that, and, 0 (14) and introducing p i = (p i 1), the Taylor series for the unmodified AUF can be rewritten as:,,,,,. (15) Equations (15) and (1) show an almost isomeric structure if the partial derivates evaluated at p i = p i,ref = 1 are formally considered as interaction coefficients:,

16 , (16).,. Using Equation (16) and remembering that g(p 1,ref,,p n,ref ) = 1, the unmodified AUF for a given alternative in MM of MAUT can be expressed as:,,, 1,,,,,,. (17) Equation (17) shows that the Taylor series of the unmodified AUF for a given alternative in MM of MAUT is a sum of terms that depend on an increasing number of attributes. When demonstrating this important result, it is sufficient to consider the simplest case with only two attributes. Equation (17) reduces then to,,, 1,,,,. (18) Without modification, all interaction coefficients as can be inferred from Equation (17) have the value of one and Equation (18) reduces to:,,, 1,,,, 1, 1, 1, 1, 1 1,, 2,,,, 1,,. (19) This illustrates how the full Taylor series reproduces the value of the unmodified AUF correctly. 3.3 Basic idea of the Aggregated Utility Factor Model The basic idea of the Aggregated Utility Factor Model (AUFM) is to use the Taylor series as a starting point and interpret and modify its terms for an appropriate modeling of the DM s preferences. In obtaining a theoretical interpretation, it is helpful to rearrange Equation (18) to:

17 k0 k1 k2 gˆ j 11 p1, j p2, j p1, ji1,2p2, j. Reference value Primary part1 Primary part2 Secondary part1,2 The first term results from the k = 0 term of the Taylor series. It reflects the global reference point and the corresponding value of its contribution to the aggregated utility factor which is one. The contributions of the k = 1 terms of the Taylor series are defined as the primary parts, the contribution of the k = 2 term is defined as the secondary part. (20) If the number of attributes were three or even more, Equation (16) would yield tertiary or even quaternary terms and so forth. This in turn would require techniques for considering joint interactions between three, four, or even more attributes. However, there is almost no well established method to deal with interacting criteria, and usually people tend to avoid the problem by constructing independent criteria (or criteria that are supposed to be so) (Grabisch 1996). Thus, it is already problematic to suggest pair-wise interactions. A model that uses higher order interactions would therefore barely be accepted by DMs (von Winterfeld and Edwards 1986, Figueira, Greco et al. 2009). For this reason, the Taylor series of the aggregated utility factor is only extended to second order, i.e. k max = 2. The primary part reflects the difference between the utility factor for an alternative and an alternative at the global reference point due to the linear deviation of its utility factors for the attribute levels from the corresponding reference values. The contribution by any one attribute is independent from those of all others. This contribution is plausible in the context of modeling, therefore modification is not necessary. Since the interaction coefficients I i formally account for the interaction of an attribute with itself, their values are set to one without a loss in generality. The value of the contribution of the primary parts to the unmodified AUF is then simply the sum p 1,j +p 2,j, or for n attributes p 1,j +p 2,j + +p n,j. The secondary part reflects contributions to the unmodified AUF that are approximately quadratic deviations. While keeping I 1,2 constant, they depend on the product of the deviations of the utility factor for an alternative from an average for two attributes. The value of the secondary part to the aggregated utility factor is then given by,s P,,,. (21) If the reduced Taylor series is still to reflect the preferences of the DM, these contributions need to be modified. As shown below, these modifications require not only a new interpretation of the interaction coefficients, but also an algorithm for modeling the cases I through III, as requested in Section

18 To underline these differences, the aggregated utility factor, as obtained from this reduced modified Taylor series, is simply termed aggregated utility factor (AUF). To clearly distinguish it from the unmodified AUF, the symbol g is used instead of the symbol. Names and the meaning of the p i,j, p i,j, and, are not affected. The aim of the modification is to provide the basis for an easy to use procedure for the DM for expressing his judgment on possible interactions such that AUF allows a discrimination of all given alternatives in accordance with the DM s preferences. 3.4 Aggregation of the Utility Factors The values of all interaction coefficients in the unmodified AUF are equal to one, see Equation (17). By modifying the values of the when using the (modified) AUF it is possible to change the relative contribution of the secondary part to the AUF. Therefore, the interaction coefficients are now considered as adjustable parameters. The value of can be set to zero if the attributes and do not interact and a value greater than zero can be used to model a strengthening complementary interaction between these attributes. This weighting is demonstrated for the simple case with only two attributes. Consider an alternative j with utility factors p 1,j = 1.75 and p 2,j = 1.50, the values of the primary parts are p 1 = 0.75 and p 2 = Inserting these values in Formula (19) yields to: 1.75, 1.50;, , (22) In Figure 1, the values of the various contributions are represented by the ratio of the corresponding areas to the area ascribed to the value at the reference point. Figure 1 shows that the area ascribed to the secondary part is proportional to the product of interaction coefficient I 1,2 and the primary parts. The relative contribution of the secondary part to the AUF is greater than zero and proportional to the product of the primary parts

19 Figure 1: Influence of the interaction coefficient on AUF in Case I. Figure 1 demonstrates Case I, i.e. the utility factors used are both greater than one. If both utility factors are less than one, this is called Case II. Consider for instance p 1,j = 0.25 and p 2,j = 0.50 or equivalently p 1,j = 0.75 and p 2,j = Inserting these values in Formula (19) yields: 0.25, 0.50;, , ,. (23) The DM s preferences can require low values for the interaction coefficients. However, inspection of Equation (23) shows that g j (0.25, 0,50; I 1,2 ) < 0 unless I 1,2 > This means that such an alternative could not be handled. In Section 2.4 it was argued that the decrease in Case II should be somehow reciprocal to increase in Case I. This is achieved by a simple algorithm. For an alternative j for which all, =,, i.e. 4, 2;, follows from the defining Equation (9) that,. Indeed, when using p 1,j = 4 1 = 3 and p 2,j = 2 1 = 1, Equation (23) yields: 0.25, 0.50;, , (24) As required, any value greater or equal than zero for I 1,2, now yields a positive AUF. For instance using the value 0.00, 0.75, and 1.00 for I 1,2, the values 0.200, 0.138, and result

20 for g j, respectively. This demonstrates that the AUF decreases in this case with an increasing value of the interaction coefficient, as is logically expected. Finally, the term Case III is used for an alternative for which p 1,j 1 and p 2,j < 1, or vice versa. When examining this case it is helpful to rewrite Equation (19) using the p i,j instead of the p i,j :,,, ;, 1, 1, 1, 1,, 1. (25) The requirement that the value of g j must be greater than zero puts a constraint on allowed values of the interaction coefficient that can be inferred be rearranging Equation (25) and setting g j = 0:,,,. (26),, Since p 1,j 1 and 0 < p 2,j < 1 it follows that I 1,2 > 1 for all allowed values of p 1,j and p 2,j, except for p 1,j = 1 where Equation (25) would be reduced to,,, ;,,. These considerations show that only I 1,2 = 1 is admitted for all possible values of p 1,j 1 and p 2,j < 1. For this value, the restricted Taylor series for two attributes reproduces the unmodified AUF that is given by the product of p 1,j and p 2,j, cf. Equation (19). This result shows that the model leaves the AUF unchanged in Case III as requested above, cf. Section 2.4. The extension to more than two attributes is straightforward for the cases I and II. Case I now pertains to alternatives where p i,j 1 for all i[1,n] and the AUF is obtained by using the restricted Taylor series directly. Case II now pertains to alternatives where p i,j < 1 for all i[1,n] and the AUF is obtained by generalizing Equation (24) to:,,,, ;,,,,,,,, ;,,,,. (27) For treating Case III, some p i,j 1 and some p i,j < 1, it is helpful to use the Heaviside step function H(x), that assumes the value zero for x < 0 and the value one for x 0. This allows defining, that contains utility factors p i,j 1 and, that contains utility factors p i,j < 1:, , (28)

21 and, is obtained by replacing p i, and by their inverse, respectively. The AUF is then obtained as the product of, and,. For example, consider an alternative j for five attributes and p i,j 1 for i = 1, 3, and 5 and p i,j < 1 for i = 2 and 4. The AUF is then given by:,,,, ;,,,,,,,,,,,,,,,,. (29),,,,, Inspection of the equation shows that for this alternative j the interaction coefficients I 1,2, I 1,4, I 2,3, I 2,5, I 3,4, and I 4,5 do not influence the AUF. To see that the AUF is a continuous function of the utility factors and the interaction coefficients, assume the following: p 1,j = 1+, p 1,j* = 1, and p i1,j = p 1,j*. From Equation (29) follows:,,,,,,,,,, (30),,,,, where, and the squared bracket in the numerator contains all cofactors for,. For computing g j*,, is written as a Taylor series,, 1 and:,,,,,,,,,,,,,,. (31) Inspection of Equations (30) and (31) shows that lim lim. An extension to tertiary and higher order interactions is very straightforward by simply extending Equation (28). However, the additional effort in the elicitation procedures may be unjustified. 3.5 Elicitation and modeling of preferences Elicitation procedures should be as simple as possible and adequately map the DM s preferences (Little 1970, Dyer, Fishburn et al. 1992). Simplicity is requested by the DM, who is generally not interested in theory, but rather purely interested in the handling of the tools. Adequate mapping is vital because without it, decisions based on the model could not reliably reflect preferences. The AUFM allows eliciting utility factors and interaction coefficients separately. The utility factors are elicited independently for each single attribute and the interaction coefficients independently for each possible pair of attributes. Furthermore, the AUFM uses step-wise elicitation procedures that implement well-known and simple techniques that considerably reduce the required effort of the DM. The procedure to be used can be divided into three tasks:

22 1. The first task is to consider two propositions, for example, A and B, and to select one of three statements: (1) A is more useful than B, (2) A and B are equally useful, or (3) A is less useful than B. 2. In case of selecting statement 1 or 3, the linguistic variables more or less are to be quantified by a factor F AB such that g(b) = g(a) F AB or alternatively p(b) = p(a) F AB. Procedures used in context with the Analytical Hierarchy Process (see Saaty 1980) might be adopted and adapted. 3. The third task is to use interpolation schemes. Two important properties of the AUF allow this strategy and warrant adequate mapping of the preference. Firstly the AUF is defined for any combination of possible levels of all attributes. For example, B is by definition given by the above introduced simple formula g(b) = g(a) F AB. When using A at the global reference point, the value of g(a) is equal to one, and the formula reduces to g(b) = F AB. Secondly the AUF is separable in a primary and a secondary part. The primary part does not contain an interaction coefficient. Hence the elicitation procedure can consider the attributes independently from each other. The secondary part describes the interdependence of each two attributes by means of interaction coefficients that do not depend on each other; e.g. I 1,2 is independent of I 1,3. The first part of the elicitation procedure is eliciting utility factors for each single attribute X i, i[1,n]. This is subdivided into three steps. The first step is to identify the reference point x i,ref. The second step is determining the attribute levels x i,+ and x i, for which p(x i,+ ) and p(x i, ) assume the highest and the lowest value, respectively. The third step is to generate an interpolation scheme. The first step has already been described in Section 3.1 and 3.2. The utility level at the reference point is set to the value of one, i.e. p(x i,ref ) =: 1. Such a reference point can be found, irrespectively of the scale level of the attribute levels. For instance, considering the quasinominal attribute color when buying a used car, the DM could select black as a reference point and express his higher or lower preference by utility factors that are greater or lower than one, respectively. It is noted that as an attribute level an ordinal, an interval or a ratio level scale x i,ref is not necessarily given by the median, the mean, or the geometrical mean, respectively. In the second step, the DM compares the utility of attribute levels with the utility level at the reference point. Note that the step does indeed only require solving the described two tasks listed above. Here, proposition A is the reference point and proposition B is any other

23 level of that attribute. Therefore, the DM is merely asked to state: I know (believe, think, judge or conjecture) that proposition B is F AB times more (less) useful than proposition A. Clearly, this procedure allows identifying the attribute levels values x i,+ and x i, where the maximum and the minimum of the utility factor for this attribute occur. The abbreviations p i,max and p i,min are used for p(x i,+ ) and p(x i, ), respectively. It is noted that for attribute level at an ordinal, an interval, or a ratio level scale x i, is not necessarily given by the first/last, the highest/lowest values of attribute levels. With respect to the used car, the DM could for instance assign p i,max to red and p i,min to gray. In an empirical study most participants selected a range between p i,min = 1/4 and p i,max = 4. This is equivalent to saying that the most preferable alternative is about sixteen times more useful than the least preferable one. In steps one and two the DM has already inspected the given levels of an attribute for identifying values x i,+, x i,ref, and x i,. The effort to assign a utility level to the remaining levels of the attribute under study in the third step depends on their scale level. For instance, given a nominal scale level for the attribute color the DM selects the utility factor for color c such that p(gray) < p(c) < p(red). Given an ordinal scale level for m ordered levels of attributes, the DM could use the same procedure as described for the nominal scale if m is small. Otherwise, if m is large, the simplest approach would be to interpret the ordered set of attribute levels as discrete points on an interval scale, i.e. to interpret them as integers. Given an interval or quasi-interval, such as large ordered set, the simplest procedure is the use of a log-linear interpolation. This corresponds to a linear interpolation on an interval scale as proposed for MAUT by Bell (1979). Table 1 summarizes the three steps for an alternative with m = 60 discrete values on an interval scale. The first column contains labels for the corresponding rows. The first row, labeled AL (attribute levels), displays the attribute levels. The second row, labeled 0 th, shows the results of the first step, i.e. the determination of the reference point, in this case 21, and the second step, i.e. the determination of x i,+ and x i,, in this case 5 and 52, and the corresponding values of p min = p(x=5) and p max = p(x=52). It is noted that there is only one reference point, whereas the value for p min and in rare cases also the value for p max could be assigned to more than one attribute level. In this demonstration p min = 1/4 and p max = 4 have been chosen. The rows labeled 1 st and 2 nd show how the third step is carried out by two iterations. Since in this example p min and p max are not assigned to x min = 1 and x max = m, the utility factors p(x 1 ) and p(x m ) are to be determined. In this demonstration p(x=1) = 1/3 and p(x=60) = 2 have been chosen. It is up to the DM to spend more time and to assign utility