The Multi-attribute Utility Method
5.2 Multi-Attribute Utility Methodology and Techniques
Important concepts of the theory are described in Table 5.1. With Multi-Attribute Utility theory, the overall evaluation of an alternative is defined as a weighted addition of its values with respect to its relevant attributes. This technique requires the decision-maker to evaluate the alternatives on each value dimension (called attribute) separately. For example, apartment is an attribute level of the attribute type of dwelling. Next, the decision-maker assigns relative weights to the various attributes that express the trade-off between attributes. Values and weights are then combined and aggregated by means of a formal model that generates an overall evaluation of each alternative (Von Winterfeldt and Edwards 1986, p. 6). The lin-ear additive preference function is mostly used but other functions are also possi-ble (see, for example, Keeney and Raiffa 1976). Important contributors to this field of research are Keeney and Raiffa (1976) and Von Winterfeldt and Edwards (1986).
Although the practical application of a multi-attribute utility method may vary, all such procedures include the following steps (Von Winterfeldt and Edwards 1986, p. 273):
1. Defining alternatives and value-relevant attributes;
2. Evaluating each alternative separately on each attribute;
3. Assigning relative weights to the attributes;
4. Aggregating the weights of attributes and the single-attribute evaluations of alternatives to obtain an overall evaluation of alternatives;
5. Perform sensitivity analyses and make recommendations.
According to Von Winterfeldt and Edwards (1986), all approaches are identical in steps 1 and 5, but may differ in the procedures for eliciting single-attribute evalu-ations and weights and in the models for aggregation. In the next section, these steps will be described in more detail.
5.2.1 Step 1: Define Alternatives and Value-Relevant Attributes
The first step in the analysis is to determine the available alternatives and their most salient attributes. For example, in the case of housing these alternatives could be available dwellings in a specific region. Note that alternatives need not necessarily be objects, but could almost be everything as long as it concerns a decision problem at hand. Next, salient attributes have to be chosen on which the alternatives will be judged. The set of attributes has to be complete, operational, decomposable, non-redundant and minimal (Keeney and Raiffa 1976, p. 50). Complete means that all important aspects of the decision problem have to be covered. The term operational is used to indicate that the attributes must have the ability to be meaningfully used in the analysis. Decomposable refers to simplifying aspects of the evaluation pro-cess by breaking it down into parts. Non-redundant means that no double counting of aspects should take place. Finally, the number of attributes should be kept as
Table 5.1 Important concepts in Multi-Attribute Utility theory
Concept Description
Alternatives Options where the decision-maker has to choose from, for example, various available dwellings.
Attributes Important (‘salient’) characteristics of the alternatives, for example, “dwelling type” and “number of rooms”.
Attribute levels Levels of the attributes. For example, “2 rooms” is a level of the attribute “number of rooms”.
Attribute value The numerical value that is attached to a particular attribute level.
A higher value is generally related to more attractiveness.
Importance score A numerical value that indicates the importance of each attribute.
A higher score is generally related to more importance.
Weight The importance score after transformation such that, for each respondent, all attribute weights add up to one.
Single-attribute utility The numerical strength of preference of an attribute level. It results from the multiplication of the attribute value with the attribute weight.
Combination rule The rule that is used to aggregate over the single-attribute utilities.
Usually, the simple additive rule is applied: the single-attribute utilities are simply added to obtain the multi-attribute utility.
Multi-attribute utility The numerical strength of preference of an alternative. It results from the aggregation of single-attribute utilities.
small as possible in order to prevent factors such as boredom, fatigue and confusion.
The identification of the right attributes is a very important step in the procedure.
Unfortunately, as Keeney and Raiffa (1976, p. 64) point out, these attributes are not simply handed to the researchers in an envelope at the beginning of the study. It requires a thorough search into the aspects that are most important to the particular decision problem. Tools to find salient attributes can be, for example, face-to-face interviews with experts, focus group interviews with experts, a literature search and methods such as Decision Plan Nets and the Repertory Grid method.
In choosing the salient attributes, Von Winterfeldt and Edwards (p. 220) advise constructing a scale that represents some natural quantitative attribute of the evalua-tion object, a so-called natural scale. Examples of such natural scales in the domain of housing are the size of the backyard in meters and the number of rooms. When a natural scale is not possible or available, a qualitative scale can be constructed that defines the attribute, its endpoints and possibly some intermediate marker points using verbal descriptions (for example: a quiet/lively/busy neighborhood). It may be useful to associate numbers with qualitative descriptions that are at least ordinally related to the decision maker’s preferences for the levels of the qualitative scale.
However, even a natural scale may not be a satisfying value scale. The relation-ship between the natural scale (for example, 2, 3, 4 rooms) and the value scale (for example, 2 rooms = good, 3 rooms = better, 4 rooms = best) may not be linear or may even be non-monotone. A monotone natural scale has a value that increases or decreases monotonically with it, such that more is always better or always worse (note that the relationship need not be perfectly linear). Monotonicity can be violated for some attributes. For example, there may be a maximum to the preferred number of rooms. Above some maximum number the added value may become negative as more space may not compensate for more maintenance.
Value functions may be linear in the sense that the distance in value between con-secutive attribute levels is about equal. However, value functions can also be concave or convex. For example, the value function of the attribute number of rooms may be concave as the added value of each additional room may be worth less than the previ-ous one (a decreasing marginal evaluation, see, for example, Keeney and Raiffa 1976, p. 88). This means that the step from four to five rooms may be appreciated more than the step from five to six rooms, and so forth. Von Winterfeldt and Edwards (1986, p. 237) advise, in cases where concave or convex functions are observed, reformu-lating the evaluation problem in order to produce linear or near-linear value functions by carefully selecting or creating a scale. This can be done, for example, by setting limits to the minimum and maximum attribute levels to be evaluated. The argument for this is that a value function is linear in the natural scale that most closely reflects the value concerns to which it is related. For example, to prevent decreasing added value, a maximum can be set on the number of rooms in order to obtain an interval with a more or less linear relationship between value and size.
Thus, one must carefully select and construct the attribute scales that are used to explore the attribute values. However, even after careful selection and the construc-tion of natural scales, value funcconstruc-tions may not be perfectly monotone linear. In such cases, curve fitting procedures can be used, such as exponential or polynomial
functions, to estimate the attribute level values. After fitting such a function, values at scale points for which no assessments were made, can be interpolated.
5.2.2 Step 2: Evaluate Each Alternative Separately on Each Attribute
After determining the salient attributes, the alternatives have to be evaluated for each of these attributes. For example, what is the value of an alternative consisting of a dwelling with three rooms? And with four rooms? Von Winterfeldt and Edwards (1986, chapters 7 and 8) present a variety of methods that can be used to elicit single-attribute value functions and utility functions. Here, some often-used methods will be described.
With direct rating, the respondent considers at least three stimuli: two stimuli that are used as endpoints or anchors and one that is used to elicit the numerical judgment. A “bad” or least preferred stimulus is arbitrarily assigned the value of 0 and a “good” or most preferred stimulus has been given the value of 100. The inter-mediate stimulus (or stimuli) is judged in relation to these anchor points. The rela-tive spacing between points reflects the strength of one stimulus over another. For example, if a dwelling with one room is used as the lower anchor (0) and a dwelling with six rooms is used as the higher anchor (100), the value of a dwelling with two, three, four or five rooms can be determined between these extremes.
With the difference standard sequences technique the respondent identifies a sequence of stimuli that are equally spaced in value. For example, backyard length of 5, 10, 13 and 15 m may be equivalent to a value of 0.25, 0.50, 0.75 en 1.0, respectively. With the bisection method, the most and least preferred option are identified and a midpoint stimulus is found that is equidistant in value from both extremes. For example, the respondent is asked which number of rooms is least preferred (value = 0) and most preferred (value = 100). Next, the number of rooms that lies in value exactly between these anchors is determined.
5.2.3 Step 3: Assign Relative Weights to the Attributes
After evaluating all alternatives on all salient attributes, the importance of each of the attributes is determined and the weights are calculated. A number of techniques is available to determine the importance of the attributes. See, for example, Von Winterfeldt and Edwards for an overview (1986, chapter 8, p. 274). Well-known methods are ranking, direct rating, ratio estimation and the method of swing weights. With ranking, the respondent is asked to rank all attributes in order of importance. An example of direct rating is to divide 100 points over the attri-butes so that the number of points reflects the relative importance of the attriattri-butes.
For the ratio estimation method, first the least important attribute is determined.
Next, the respondent is asked how much more important each attribute is when related to the least important one. Finally, for swing weighting, the respondent is asked how much an attribute contributes to the overall value of the alternatives rela-tive to other attributes. Usually with this method the respondent is provided with a choice between profiles reflecting the worst and best levels of each attribute. The respondent is asked to indicate which of the differences between the worst and the best level (called swings) contributes most in overall value. Then, the extent to which the value swings differ between attributes is assessed by letting the respon-dent assign a score to the relative significance of the range when compared to the most important range. For example, a profile consisting of a dwelling with two rooms and a backyard of 5 m (worst levels) is compared to a profile consisting of a dwelling with five rooms and a backyard of 15 m (best levels). The respondent first indicates that the swing from two to five rooms is more important than the swing from a backyard of 5–15 m in length. Next, the respondent indicates that this swing is four times less important than the swing from two to five rooms.
After scores have been collected for the importance of the attributes, these are transformed into weights by dividing, for every respondent, the rating of each attri-bute by the sum of all ratings (Von Winterfeldt and Edwards 1986, p. 281):
'
where w'i is the not-normalized ratio weight and wi the normalized weight. Hereby individual weights for each attribute are obtained that add to 1, as is conventional in Multi-Attribute Utility theory (Von Winterfeldt and Edwards 1986). Assume, for example, that a respondent has the following importance scores for eight attributes:
20, 30, 40, 50, 30, 60, 70, 20. The sum of these ratings is 320. The weight for the first attribute is therefore: 20/320 = 0.06. The other weights are calculated in the same way.
5.2.4 Step 4: Aggregate the Weights of Attributes and the Single-Attribute Evaluations of Alternatives to Obtain an Overall Evaluation of Alternatives
Weights and single-attribute values or functions can be aggregated using a variety of models. The weighted linear additive preference function is the most commonly applied aggregation method (Von Winterfeldt and Edwards 1986, p. 275). Other els are, for example, the multiplicative model and the multi-linear model. These mod-els are not frequently used. With the weighted linear additive function, the overall evaluation of an alternative is calculated by multiplying the weight by the attribute value for each attribute and summing these weighted attribute values over all attributes
(Payne et al. 1993). It is assumed that the alternative with the highest overall evaluation will be chosen. The weighted linear additive preference function processes all of the relevant information. Furthermore, the conflict among values is assumed to be addressed and resolved by explicitly considering the extent to which one is willing to trade off attribute values, as reflected by the relative importance or weights. This simple preference function is a compensatory combination rule as a low value of one attribute can, at least partially, be compensated by higher values on one or more of the remaining attributes. The multi-attribute utility for alternative x is:
1
( ) n i i( ),i
i
v x w v x
=
=
å
(5.2)where v xi( )i is the value of alternative x on the ith attribute, wi is the importance weight of the ith attribute, and n is the number of different attributes (Von Winterfeldt and Edwards 1986, p. 263, p. 275).
In theory, there are various techniques to elicit values, they can be combined with different techniques for calculating weights and they can be aggregated with a number of models. Thus, the application of Multi-Attribute Utility theory can differ considerably between studies. However, in practice, a very limited combination of techniques is used. The most important procedure is termed SMART (Simple Multi-Attribute Rating Technique) (Edwards and Newman 1982). This procedure is simple and easily applicable; it consists of the direct rating technique for eliciting values combined with the ratio estimation technique for calculating weights and the weighted linear additive preference function.
5.2.5 Step 5: Perform Sensitivity Analyses and Make Recommendations
In the fourth step, multi-attribute utilities for all alternatives have been calculated.
The alternative with the highest multi-attribute utility should be the preferred choice. In the last step of the procedure, sensitivity analyses are carried out to evaluate the stability of the results. The impact of different values and weights on the multi-attribute utilities of the available alternatives can be determined. One way to obtain different values and weights is by using different elicitation methods. For example, both the direct rating technique and bisection method could be used to obtain values. Multi-attribute utilities could then be calculated twice: once using the values obtained with the direct rating technique and once with the values obtained with the bisection method. The resulting multi-attribute utilities for the available alternatives can be compared and the robustness of the results can be determined.
Another way to obtain insight into the stability of the results is to use a different weighting technique. For example, the equal weight function can be applied. This technique simplifies the choice process by ignoring information about the relative importance of each attribute (Jia et al. 1998; Bettman et al. 2006, p. 329). In doing
so the method assumes that all attributes have equal weight. An overall value for each alternative is obtained by simply summing the values for each attribute for that alternative. For example, for eight attributes, the weight of each attribute is 1/8 = 0.125. Other weighting techniques include the rank reciprocal rule and the rank sum weighing procedure (Von Winterfeldt and Edwards 1986).
In this section Multi-Attribute Utility theory has been explained in more detail.
Note that this was just a short description as Multi-Attribute Utility theory is a method that has been developed and used in many research domains and can be quite complicated. Interested readers are referred to, for example, Keeney and Raiffa (1976) and Von Winterfeldt and Edwards (1986). Also, computer programs have been developed to help decision-makers in structuring the problem. In the next section, some studies that have used the Multi-Attribute Utility method in the field of housing will be discussed.