Stochastic Weights in Multiattribute Decision Making

We assume that cost differences involved in producing brands with different levels of type A attributes may be considerable, while for type B attributes it is negligible (see Appendix A for a discussion of this assumption). We concentrate on type B attributes in this paper primarily because we can separate the problem of finding the optimal combination of attributes from the problem of setting the optimal price (again, see Appendix A for a more detailed discussion of this point).

The preference model described above can be used to formulate the firm's optimization problem which enables it to determine the best combination of attributes for its brands. Since the primary concern of this paper is the validation of the stochastic parameter model, we relegate the formulation of the firm's optimization problem to Appendix B.

THE STOCHASTIC ATTRIBUTE WEIGHT MODEL

The solution of the optimization problem described in Appendix B depends upon the values of the preference model parameters. These parameters have typically been assumed to remain constant. However, as was argued earlier, it seems more reasonable to allow these parameters to vary across purchase situations. Consequently, we present below a model with stochastic parameters. To keep the model simple, we assume that the consumer's ideal point does not change. However, we allow his weight vector to shift from situation to situation. As the weights vary, the rank orderings, in general, will not remain the same since the D_ik's (defined in equation (2)) will change for each situation. Changes in the rank orderings can result in changes in the most-preferred brand. The proportion of situations in which a particular brand attains the most-preferred position will constitute our prediction of how often the consumer will choose that brand. To validate the model, we need only to compare these predictions with the consumer's actual choice patterns.

TABLE

Let w₁, w₂, and w₃ represent the individual's weights on the three attributes and let the sum of the weights equal 1.0 for simplicity. The first step in the analysis is the specification (for every pair of brands) of the constraints which express the fact that one brand of the pair is preferred to the other. Starting with brands A and B, the constraint specifying that A is preferred to B is as follows:

(D_"-D_B) < 0 <=> (-2w₁ + w₂ + w₃) < 0 <=> w₁ > 1/3

Similarly,

(D_"-D_C) < 0 <=> (-w₁+ w₂) < 0 <=> w₂ < w₁

(D_B-D_C) < 0 <=> (w₁ - w₃) < 0 <=> w₂ < (1 - 2w₁).

These constraints define ranges of weights within which a certain rank order would hold as illustrated in Figure 3. For instance, the area PQRS in Figure 3 represents: (1) w₁ > 1/3, implying A is preferred to B, (2) w₂ > w₁, implying A is preferred to C, and (3) w₂ < (1 - 2w₁), implying C is preferred to B. This gives rise to the rank order [ACB](which indicates that brand A is preferred to brand C and brand C is preferred to brand B). The basic logic of the analysis is unaffected if, instead of the City Block metric used in the above constraints, we use the Euclidean metric described in equation (2) (see Pekelman and Sen, 1974, for a discussion of the appropriate metric). The next step of the analysis involves the prediction of the frequency with which each brand is chosen by the consumer. This requires the determination of a distribution for the attribute weights. Initially; we will assume a uniform distribution of weights. Given this assumption, the choice frequency for each brand for a three-attribute problem is given by the ratio of the sum of the areas where the brand is most preferred to the total area. For example, the computation of Pr{A}, the probability of choosing A, involves first the summation of the areas where A is the most-preferred brand: areas PQRS and QTR in Figure 3. Pr{A} is then computed as the ratio of this area sum, PQTS, to the total area, UVS. Using this model, the predicted probabilities are:

Pr{A} = 0.39, Pr{B} = 0.44, and Pr{C} = 0.17.

The final step is to compare these predictions with the consumer's actual choice behavior as computed from the record of his purchases in a diary panel. If, for example, the consumer's purchase history indicated that he bought brand A 40 per cent of the time, brand B 45 per cent of the time, and brand C 15 per cent of the time, our predictions (Pr{A} = 0.39, Pr{B} = 0.44, and Pr{C} = 0.17) would compare very favorably with actual behavior.

FIGURE 3

RANGES OF ATTRIBUTE WEIGHTS DEFINING PREFERENCE RANK ORDERINGS

We also present a second (and weaker) version of the model in which we attempt to incorporate information about unrealizable rankings since, for a given consumer, not all possible rankings are realizable. For example, if we consider three objects, the consumer may state that two of the six possible rankings have not occurred in the past and are highly unlikely to occur in the future. Such a statement would make sense if, for example, the consumer could not visualize any circumstances in which one of the three objects would be his most-preferred alternative. As shown below, unrealizable rankings modify our brand choice prediction in a simple manner. Clearly, this version of the model amounts to what we may call "partial prediction." Here we predict just part of the choice distribution, the realizable rankings, while the other part of the distribution, the unrealizable rankings, is given as input.

Returning to our numerical example, assume that the consumer states that the rank order [BAC] is unrealizable. This amounts to redefining the "total area" in Figure 3 as (UVS - QVT). The modified predictions are:

Pr{A} = 0.44, Pr{B} = 0.37, and Pr{C} = 0.19.

Formal Representation of the Stochastic Weight Model

The formal representation of the model utilizes the following notation:

s_u = a discrete situation in the situation space S, u = 1, 2, ..., h where h denotes the number of situations.

w_u = a vector of attribute weights corresponding to s_u (one to one). w_u is an n-dimensional vector in w-space where n equals the number of attributes.

R_r = a particular rank order of all objects, r = 1, 2, . . ., q! where q denotes the number of objects.

W_r = the set of vectors, w_u, leading to the rank order, R_r. The set W_r contains a_r members. In other words, a_r weight vectors lead to the rank order R_r. Clearly, a_r can equal zero.

c_k = event of having object k (k = 1, 2, . . ., q) as the most preferred object. c_k represents the decision to choose object k.

R_k= the set of all R_r's such that the most preferred brand in R_r is k. Let b_k be the number of members in this set. b_k will range between 0 and (q-1)! (In this model, ties are not allowed).

W_k = the set of  R_k defined in terms of attribute weights, i.e.,

EQUATION   (15)

Pr(s_u) = probability that situation s will occur. This is also equal to the probability that the weight vector, w_u, will occur. Thus, Pr(s_u) = Pr (w_u).

Pr(R_r) = probability that rank R_r would be realized.

EQUATION

Pr(c_k) = probability that decision c_k would be realized.

EQUATION   (16)

Our ultimate objective is to compare the computed values of Pr(Ck) with the consumer's actual frequency of choosing brand k.

MODEL TESTING

The two data sets required for both versions of the model are:

1. Relative frequency of purchase of the various brands.

2. The distances between the ideal points and the locations of the q brands on each of the n attributes (the d_ijk of equation (1)).

The first data set can be obtained from existing consumer panel data. The second set of data should be collected from the same consumer group participating in the panel by direct questioning. For the weaker version of our model, there is an additional data require-merit, i.e., the specification of the set of unrealizable rankings. This can be obtained by direct questioning in the same questionnaire used to obtain the brand distances.

The M.R.C.A. Data

The Market Research Corporation of America (M.R.C.A.) has made available to us the coffee purchasing records of a panel of 900 households. The panel was operated by M.R.C.A. between October, 1972 and June, 1973 in Phoenix, Arizona. The panel members were chosen after ensuring that they had purchased both instant and regular coffee. From this panel data, the relative frequency of purchase of the leading coffee brands in Phoenix can be easily computed.

In addition to the panel data, we have available the responses of 542 of the 900 panel members to a series of questions dealing with their perceptions of the 11 leading coffee brands in Phoenix in terms of six product attributes. These 11 brands accounted for 70 per cent of all coffee purchases in Phoenix.

The purchase panel data and the brand perception responses provide us with the information needed to test the stochastic weight model presented in this paper.

We cannot test the weaker version of the model because it was not possible to gather data on a complex concept such as "unrealizable rankings" in a mail questionnaire. Such information can probably be gathered only through personal interviews. The questionnaire also provides information on brand preferences. Some limited lifestyle information is also available. In addition, M. R. C.A. has provided detailed demographic data on each panel family. The data were made available to us very recently. Thus, it has not been possible to present any empirical results in this paper.

Evaluation of the M.R.C.A. Data

Coffee data are particularly appropriate for testing a model of situational behavior since the choice of instant versus regular coffee is likely to be situation oriented. Further, coffee is a frequently-purchased product which will result in a relatively large number of purchases per family. This will provide us with the sample size necessary to compute market shares for the various brands with a reasonable degree of confidence.

However, it is important to point out some of the potential problems in using panel data to test the stochastic weight model. Our model cannot distinguish between brand switching due to marketing factors (such as price and promotion) and brand switching due to changing attribute weights. Hence, we expect that the model will perform better for those consumers who are not very sensitive to marketing variables and yet switch between brands frequently (see Blattberg and Sen, 1974, for descriptions of such consumers). Even for these consumers, the brand switching could result not only from changing attribute weights but also from changing ideal points, changing brand perceptions due to company advertising, and purchases made by different members of the family (Davis, 1970). For the purpose of testing the stochastic weight model, we will have to assume that these variables do not have a systematic effect on the brand choice frequencies.

APPENDIX A

ATTRIBUTE TYPES AND THEIR ECONOMIC IMPLICATIONS

EQUATION   (3)

for the entire range of z.. Type B attributes (Figure 1b), on the other hand, are characterized by:

EQUATION   (4)

where z_j^* equals the ideal level of attribute j.

There exists another important difference between these two types of attributes. The cost differences involved in producing brands with different levels of type A attributes may be considerable, while for type B attributes it is negligible. To produce brands of cereal with different amounts of sugar within a limited range would usually involve negligible differences in cost. (Naturally, once we move out of this range, we may have technological problems which could increase the cost sharply.) Similarly, the manufacture of soft drinks with different levels of carbonation would not generally involve different costs. However, the addition of safety mechanisms in a car or the improvement of its miles per gallon performance would generally involve a considerable increase in cost.

FIGURE 1a

UTILITY FUNCTION FOR TYPE A ATTRIBUTES

FIGURE 1b

UTILITY FUNCTION FOR TYPE B ATTRIBUTES

These ideas are illustrated in Figures 2a and 2b which show demand and marginal cost (MC) curves for four brands: 1, 2, 3, and 4. The "level of attributes" increases from brand 1 to brand 4. In accordance with our discussion above, we make the following two assumptions: (1) for type A attributes (Figure 2a), each brand has a different MC curve (MC₄ > MC₃ > MC₂ > MC₁), and (2) for type B attributes (Figure 2b), each brand has an identical MC curve (MC₄ = MC₃ = MC₂ = MC₁ = MC). For both types of attributes each brand has a different demand curve. For type A attributes, demand increases with the "level of attributes" in the brand, i.e., D₄ > D₃ > D₂ > D₁. However, for type B attributes, D₂ > D₃ > D₄ > D₁. Note that brand 2 has the highest demand while brand 4, with the highest "level of attributes," is superseded in demand by both 2 and 3.

It is obvious, therefore, that if the firm produces a single brand, it would select the one with the highest demand in Figure 2b while it may select any of the levels (depending upon the corresponding MC) in Figure 2a. Essentially, the proposed research examines certain elements of a model for finding the highest demand function when type B attributes are involved.

Given the assumption that there exists an identical MC for all attribute combinations, decisions regarding price and the optimal combination of attributes are separable; i.e., the decisions can be made sequentially. In other words, we wish to identify first the attribute combination which yields the highest demand curve and then determine the price.

FIGURE 2a

DEMAND-COST RELATIONSHIPS FOR TYPE A ATTRIBUTES

FIGURE 2b

DEMAND-COST RELATIONSHIPS FOR TYPE B ATTRIBUTES

APPENDIX B

FORMULATION OF THE FIRM'S OPTIMIZATION PROBLEM

Let P_ik be the probability that individual i would purchase brand k during a specified period of time.

P_ik = f_ik(D_i1, D_i2, ..., D_ik, ..., D_iq;   p₁, p₂, ..., p_k, ..., p_q)    (5)

where p₁, p₂, ..., p_q are the prices of the q brands and where

EQUATIONS   (6)  -  (8)

EQUATIONS   (10)  -  (13)

max E_i r_if _ik(D_i1, ..., D_iq, D_ik; p₁, ..., p_q, p_k)   (14)

where

D_ik = E_j(x_ij - y_ij)²w_ij for all k=1, 2, ..., q, k  .

Note that the parameters r_i, y_kj, x_ij, w_ij and, as a result, D_ik, are known. Pekelman and Sen (1974) show how the weights, w_ij, for all i, j can be numerically derived. The parameters y_kj for all k, j are generally obtained directly from the various individuals in the sample. The parameters x_ij for all i, j can be numerically computed using a technique described by Srinivasan and Shocker (1973). The decision variables for the firm are the attribute locations of the new product expressed by (y_k1,y_k2, . . ., y_kn).

Optimal solutions of the problem posed by expression (14) are difficult to obtain numerically. Consequently, a practical approach is to search for an optimal or near-optimal solution by forming a grid on the attribute space (Shocker and Srinivasan, 1974). Of course, as the number of attributes increases the grid increases exponentially, thereby increasing the search costs considerably. But since the number of attributes is usually no larger than four (Pekelman and Sen, 1974) the search would be feasible. Also note that in general we would not search over the entire attribute space. Usually, technological constraints would restrict the search to a limited region. The problem can be easily extended to deal with the case when the firm wishes to introduce, on the described space, multiple brands. The gain achieved by introducing an additional brand can then be compared to the cost incurred by this action.

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