Supermatrix-Analysis As a Method of Measuring Interdependent Relative Importance Weights in Customer Satisfaction Research

ABSTRACT - The importance of quality and customer satisfaction has dramatically acceleratd over the past twenty years. Given the need to maintain high quality, practitioners and academics have devoted increasing effort on how to measure quality or customer satisfaction. Three popular methods that are currently used for determining which attributes are the most important to satisfy the consumer are #gap analysis’, linear regression of the overall satisfaction rating on the ratings for the attributes, and conjoint analysis. Unfortunately, these methods have shortcomings. While regression analysis is able to accommodate linear relationships among the predictor variables in principle conjoint analysis assumes the absence of such interdependencies. However, multicollinearity often is severe in satisfaction regression models and causes unstable parameter estimates. This paper presents a new method which is to address the aforementioned problems in satisfaction measurement. We outline the application of the supermatrix-approach to determine the relative importance weights of interdependent service attributes. The empirical example is drawn from the tourism industry.

Citation:

Frank Huber, Marc Fischer, and Andreas Herrmann (2000) ,"Supermatrix-Analysis As a Method of Measuring Interdependent Relative Importance Weights in Customer Satisfaction Research", in NA - Advances in Consumer Research Volume 27, eds. Stephen J. Hoch and Robert J. Meyer, Provo, UT : Association for Consumer Research, Pages: 92-99.

Advances in Consumer Research Volume 27, 2000      Pages 92-99

SUPERMATRIX-ANALYSIS AS A METHOD OF MEASURING INTERDEPENDENT RELATIVE IMPORTANCE WEIGHTS IN CUSTOMER SATISFACTION RESEARCH

Frank Huber, University of Mannheim

Marc Fischer, University of Mannheim

Andreas Herrmann, University of Mainz

ABSTRACT -

The importance of quality and customer satisfaction has dramatically acceleratd over the past twenty years. Given the need to maintain high quality, practitioners and academics have devoted increasing effort on how to measure quality or customer satisfaction. Three popular methods that are currently used for determining which attributes are the most important to satisfy the consumer are #gap analysis’, linear regression of the overall satisfaction rating on the ratings for the attributes, and conjoint analysis. Unfortunately, these methods have shortcomings. While regression analysis is able to accommodate linear relationships among the predictor variables in principle conjoint analysis assumes the absence of such interdependencies. However, multicollinearity often is severe in satisfaction regression models and causes unstable parameter estimates. This paper presents a new method which is to address the aforementioned problems in satisfaction measurement. We outline the application of the supermatrix-approach to determine the relative importance weights of interdependent service attributes. The empirical example is drawn from the tourism industry.

1. INTRODUCTION

The importance of quality and customer satisfaction has dramatically accelerated over the past twenty years (Johnson 1997, p. 3). Given the need to maintain high quality, practitioners and academics have devoted increasing effort on how to measure quality or customer satisfaction (Cronin/Taylor 1992, 1994; Danaher/Mattsson 1994; Parasuraman et al. 1985, 1988, 1991; Rust/Zahorik 1993; Rust/Oliver 1994; Rust/Zahorik/Keiningham 1995; Teas 1993, 1994). Indeed Peterson and Wilson (1992) report that in the past 20 years over 15.000 articles have been written on measuring customer satisfaction.

Three popular methods that are currently used for determining which attributes are the most important to satisfy the consumer are #gap analysis’ and linear regression of the overall satisfaction rating on the ratings for the attributes and conjoint analysis. Unfortunately, these methods have shortcomings. One purpose of this contribution is to contrast existing and develop a new method for determining which attributes are the most important in the tourism sector for increasing customer satisfaction.

2. CURRENT USED METHODS TO MEASURE CUSTOMER SATISFACTION AND THEIR LIMITATIONS

One method for determining the relative importance of service or product attributes is to measure customer expectations or ideals (Oliver 1997) and calculate the gap between the expected and actual service or product (Parasuraman 1988, 1991; Teas 1993). Gaps can be calculated for each attribute separately and the attribute with the biggest gap is deemed the one upon which to focus attention. In addition to the fact that attribute interaction is ignored, the clear problem with this approach is that some product or service attributes may have large gaps but these attributes may not be very important to customers (Hemmasi et al. 1994). A second issue (Johnson 1997) pertains to the nature of the expectations used as a basis for the disconfirmation. There is confusion as to whether expectations constitute a prediction as to what a consumption experience will be (#will’ expectation), or a preference as to what a consumption experience should be (#should’ expectation).

Another way to determine the relative importance of product or service attributes is to measure only performance levels rather than expectations and performance (Cronin/Taylor 1992). The linear compensatory model is then assumed and operationalized by regressing overall satisfaction on the performance scores of the attributes (Danaher/Mattsson 1994; Rust 1994). Either linear or logistic regression are suitable and often give the same results (Ostrowski et al. 1993). The intuition here is that the attribute with the greatest slope parameter will result in the largest increase in overall satisfaction per unit increase in attribute performance. Furthermore, as long as the range of the scales to measure the attributes are equal, the estimated slope coefficient for an attribute is proportional to the relative importance of that attribute. Unfortunately, multicollinearity problems are rather common in regression analysis of customer satisfaction survey data (Rust et al. 1994). To handle this problem either principal components regression (Rust/Zahorik 1993) or equity estimator have been used with mixed success (Rust et al. 1994). A more successful way to solve the multicollinearity problem seems to be to apply the Partial Least Squares regression with latent variables (Helland 1988; 1990).

A common and powerful way to operationalize multiple measurements on respondents for varying attribute levels is via conjoint analysis (Green/Rao 1971). The advantages of conjoint analysis are, that it enables us to model a response surface for satisfaction across a full range of attributes levels, at eiter the individual or aggregate level. Lastly, conjoint analysis with an orthogonal design has zero correlation among the attributes and relative importance weights which reflect true importance (and add to 100%) can be calculated. This contrasts with using linear regression analysis, where one of the difficulties is that many of the attributes are correlated, causing multicollinearity problems and possibly masking the effects of some attributes.

However, there are a number of problems with the conjoint approach that have now become apparent. One of these is the distortion in computed utility values that arises in circumstances where abstract attributes are evaluated against more concrete variables. This can lead to dramatic underestimation of the overall contribution or importance of abstract #issues’ (Pinnell 1994). This paper argues that this limits the scope of classical conjoint studies.

Moreover, conjoint analysis should not be applied if there are interdependencies between individual features. AHP and supermatrix analysis shall now be presented as two methods by means of which the importance of attributes to consumers can be determined and which, by applying the supermatrix, allow interdependencies between attributes and explicitly take them into account when calculating the importance of the product or service attributes. The special feature of this improved AHP is that supermatrix analysis allows dependencies between attributes and takes them explicitly into account when calculating importances. To demonstrate the effectiveness of this approach, we tested the new measurement concept in a real-life situation in collaboration with 8 travel agencies.

FIGURE 1

GROSSLY SIMPLIFIED STRUCTURE OF AN EXEMPLARY AHP HIERARCHY

3. AHP AND THE SUPERMATRIX METHOD AS TECHNIQUES FOR DETERMINING THE RELATIVE IMPORTANCE OF QUALITY ATTRIBUTES

The AHP was developed in the United States in the 1970s by mathematician Thomas L. Saaty as a method for solving poorly structured complex decision problems (Saaty 1980). The method was as yet used, for example, to solve multicriterial decision problems, for planning and resource allocation (Saaty/Kearns 1985), for cost-benefit analyses (Saaty/Kearns 1985) and for forecasting purposes (Saaty 1991). The AHP includes five steps (Saaty 1980, 1996).

In a first step, the decision problem is split up in its decision elements and structured hierarchically. Figure 1 shows a grossly simplified form of a 3-step hierarchy (Zahedi 1986), p. 97 and Haedrich/Tomczak 1988, p.636).

In a second step, the priorities of the individual elements are determined by the decision-maker using paired comparisons. The decision-maker has to compare a pair of elements in a level of a hierarchy with respect to a parent element in the level above referring to their importance. The relevant question is: "Which of two elements (e.g. goals, criteria, attributes, alternatives) is more important with respect to an element at the next higher level and by what measure?" This is evaluated using a scale of nine developed and tested by Saaty.

The results of the comparative judgements on the basis of a scale of nine are summarized in matrices. Such a paired comparison matrix thus contains the paired judgements of all elements in one level with respect to an element at the superior level. Figure 2 is a diagrammatic view of a paired comparison matrix A with the values a_ij for all i, j=1, 2, ..., n for the paired comparisons of the criteria on the second level of the exemplary hierarchy from Figure 2 with regard to the goal (Haedrich/Kuss/Kreilkamp 1986, p. 123).

In a third step, the relative weights (or priorities) of the elements are calculated from these paired judgements using the eigenvector derivation procedure. The eigenvectors resulting from this calculation contain the priorities of the attributes sought after (Cf. Saaty 1996 for a detailed mathematical derivation). In a fourt step, consistency of the decision-maker’s judgements is checked using a measure of inconsistency (Saaty 1980). If the result is a no longer tolerable value of inconsistency, the decision-maker can reconsider his judgements and revise them, if required. In a fifth and last step the relative local priorities of each element are aggregated and condensed into global priorities for the whole hierarchy. These indicate the importance of each element at all levels with regard to the objective of the hierarchy. They therefore represent the desired importances of the attributes.

Application of the AHP, however, is subject to specific premises. For example, there should neither be any dependencies between attributes at one hierarchy level nor between attributes of higher order or subordinate levels. In addition, it should be possible to structure the decision problem as a hierarchy (Saaty 1996, Xu/Wang 1994). As in reality many application problems are based on structures that do not fulfil these premises, Saaty developed supermatrix analysis. This method enables its user to consider all dependencies between attributes perceived and, moreover, to solve problems that have a complex and non-hierarchical structure (Saaty 1980, 1996).

When applying the supermatrix method, the individual elements of the system (attributes, properties, etc.) are first pooled into clusters. Now two types of dependencies can be distinguished (Saaty/Takizawa 1986, Saaty 1994). Those between elements of a cluster are called inner dependencies. They are graphically represented by a loop on the respective cluster. Interdependencies between one or more elements of one cluster and elements of other clusters are called outer dependencies and shown as arrows between groups (cf. Fig. 3).

In this way, decision problems that are based on complex structures with diverse interdependencies can be represented as networks. The hierarchy known from the AHP constitutes the special case of such network, a linear network (cf. Fig. 3; Saaty 1986, p. 232; Saaty 1987).

The general structure of a supermatrix can be derived from the graphic representation of a decision problem. All clusters and their elements are on the left-hand side of the matrix and in its header. To create this initial supermatrix, eigenvectors are calculated as in the AHP from paired comparison matrices by solving the eigenvector derivation problem. These are pooled for each cluster in an n by x by n matrix that is called a block matrix (Saaty 1994). The block matrices are entered in the supermatrix at the respective positions depending on what dependencies there are between clusters or their elements. The values in the columns of the supermatrix reflect the influence that the elements of the clusters on the left-hand side of the matrix exert on the elements of the clusters in the header of the matrix.

FIGURE 2

DIAGRAMMATIC VIEW OF A PAIRED COMPARISON MATRIX

FIGURE 3

SYSTEMS WITH INNER AND OUTER DEPENDENCIES

The values on the principal diagonal of the supermatrix represent the inner dependencies. If clusters have neither inner nor outer dependencies the respective block matrices in the supermatrix are filled with zero vectors (Saaty 1994). If we assume a system with N clusters c₁, c₂,,c_N and call the elements of a cluster c_k e_k1, e_k2,,e_kNk, where _Nk indicates their number, the supermatrix in its general form looks as shown in Figure 4 (Saaty 1980):

The general form of a block matrix is given as:

EQUATION

Each column of w_ij shows the relative influence that the elements of the i-th cluster have on each element of the j-th cluster.

Figure 5 shows an example of a graph and the accordingly structured supermatrix (Schoner/Wedley/Choo 1993, p. 390). We assume that the elements of cluster 1 are criteria and the elements of cluster 2 are alternatives of a decision problem.

If there were no inner dependencies between alternatives and the criteria in Fig. 5, their respective block matrices on the principal diagonal of the supermatrix wold be filled with zero vectors. If there were no feedback between alternatives and criteria, the top right block matrix in the supermatrix would have to be filled with zero vectors (Schoner/Wedley/Choo 1993).

It is the goal of the supermatrix method to calculate the priorities of each element or attribute in such a way that all interdependencies within the system related to the respective element are reflected in this calculation. Similar to the AHP, the priorities are calculated by solving the eigenvector derivation problem. But this is on condition that the initial supermatrix is a stochastic matrix. Therefore, the blocks of the initial supermatrix have to be weighted prior to calculating the final supermatrix, so that the values in each column all total one. The priority of each block matrix should reflect its importance within the system (Saaty 1996).

FIGURE 4

FIGURE 5

GRAPH AND ASSOCIATED SUPERMATRIX WITH BLOCK MATRICES

4. USE OF THE AHP AND THE SUPERMATRIX METHOD FOR DETERMINING THE IMPORTANCE OF QUALITY ATTRIBUTES WHEN BOOKING A FLIGHT

4.1. On the relevance of the research object

The dramatic changes that characterize the tourist industry were recently described as follows by Peter Landsberger, chairman of the DER management board, on a DER meeting held in Dublin: "We do not have to look for structural change lying ahead, we have been in the centre of it since three years (quoted from Niedecken/Chierek 1997, p. 1). Commission earnings for travel agents have dropped while their expenses have increased. For example, percentage return on sales dropped to 0.3 percent from 1992 to 1995, and the profit percentage dropped from 8.4 to 2.7 percent in 1995." (quoted from Niedecken 1997b, p. 10). In this context, it remains to be seen what the consequences of the new commission system installed by Lufthansa will be; further reductions in earnings can be expected. [A processing cost analysis in travel agencies revealed that the average total cost of booking a flight is DM 53,83 (at full cost). As the commision earnings per flight are DM 49,33 there is a difference of approx. DM 4,50 per ticket unless this is compensated under Lufthansa's Partner Plus programme. Cf. Haas/Jegminat (1997), p.1 and pp. 8ff.]

Another reason for declining earnings is the growing number of travel agencies while the industry’s turnover is stagnant (Jegminat 1996, p. 1 and p. 8). The number of travel agencies increased by 10% to 17,500 from 1994 to 1996 (Niedecken 1997b, pp. 10-12). Statistically, this would mean an average turnover per travel agency of about DM 2 million per year while the assumed break-even point is approx. DM 3 million (Niedecken 1997b, p. 10).

These changes resulting from competition are accompanied by structural changes that may challenge the role of travel agents in the future or deteriorate their market situation: the new media, first of all the online services and the Internet. They enable service providers and travel organizers to contact their customers directly while excluding travel agents (Mayer 1996, p. 229f). As technical know-how is required in connection with the Internet, it is also quite likely that poachers from outside the industry enter the market and further aggravate the market situation using distribution-specific competence (Schertler 1994, p. 555f). This is confirmed by the appearance on the market of Microsoft software corporation with its Expedia travel agency (http://www.expedia.com).

Possible responses aimed at securing the economic situation are differentiation and a market niche strategy (Kreilkamp 1995, S. 26). This is to be achieved either by additional and improved service or focusing on a few target groups. Endeavours of a travel agency to appear different from a competitor result in reviewing and possibly restructuring the range of services offered. The major questions in this context are what the product should look like and which individual services should be included. To improve the prospects of success when restructuring one’s own range of services requires that appropriate studies are carried out prior to making any decisions in this respect.

For the first time, the supermatrix method will be used below to answer the question how to optmize products. It is the purpose of this study to point to alternative courses of action for travel agents, in particular, travel agencies that specialize in selling flights. The empirical study aims at identifying those services that customers expect to get from a travel agencyBbeginning from the first contact.

4.2. Operating procedure

Before the object of study can be structured graphically in a first step of the AHP and the supermatrix method, the attributes and properties that are relevant to the customer’s perception of quality have to be determined. A list of service attributes which can be expected to influence the travelers’ perception of quality was compiled for this study. The relevant literature on tourism marketing was studied for this purpose. Then the list was presented to 18 experts from 8 travel offices and to 81 customers’, which were choosen at random in 8 different travel agencies with the request to make supplements. Considerations to limit the number of alternative travel opportunities led us to restrict the study to just one area of destination: North America. Furthermore, the prices the inquirers had in mind could be taken into account by defining an area of destination.

The quality criteria obtained in this way were taken as the basis of a factor analysis. The factor analysis was on the one hand to reduce the 35 quality criteria to only a few factors that subsequently were to form the second level of the AHP hierarchy. On the other hand, it was to determine which criteria represent the mass transit service quality as perceived by the respondents particularly well. The importance of all criteria had to be assessed using an ordinal scale of seven. The questionnaire was handed to 90 travellers for answering.

A Kaiser-Meyer-Olkin criterion with a value of 0.699 was obtained after calculating the correlation matrix, indicating that the data quality justified another factor analysis. The principal component method was used to extract the factors. Initially, 9 factors were determined that together account for 82.7% of the overall scattering of the variables. Since the number of attributes to be compared when using the AHP or the supermatrix method should be limited to a maximum of six due to the limited information processing capacity of the respondents (Saaty 1977), the number of factors to be extracted in this study was reduced to six. This seemed possible because these six factors together account for 70.3% of the overall variable scattering and the communality for each variable is relatively high.

Attributes were reduced to the decisive attributes for forming preferences by menas of direct dual questioning (Tscheulin 1992, p. 96). Respondents were to identify the most important travel agency services and relate them to competitors because a service offered by all travel agencies has only limited potential for discriminating among travel agents. This questionnaire was handed to 41 respondents selected at random from the customers of two travel agencies specializing in air passages.

Not all variables (properties) were included in the main study to prevent the participants in the study from excessive strain. Instead, three properties were selected for each factor (attribute). The criteria for this selection were the factor loadings of the properties as determined in the factor analysis. The attributes (factors) and properties (variables) now form the elements of the network system the object of study is based on. The first step of using the AHP and the supermatrix is the visual structuring of the application problem. The system we find here is a hierarchical system without feedback and with inner dependencies at the second level (cf. Fig. 6).

As the loop at the cluster of the second level shows, there are dependencies between some attributes of this level. For example, there is a connection between the drivers’ behaviour and the reliability of the tram, the safety the passengers feel and their sense of well-being. As such dependencis are not allowed when using AHP, the supermatrix is used to correctly calculate the priorities of the attributes taking into account the given dependencies.

25 respondents were asked to mark with arrows in the diagram in Fig. 7 (Griffin/Hauser, 1993) between which attributes they think there are dependencies and how an attribute influences another attribute to identify dependencies between attributes. All 25 respondents made identical statements. The following diagram can be drawn to show lines of influence (the arrows indicate the from-to direction).

A questionnaire was developed based on these findings. Passengers had to compare pairs of attributes with regard to their importance for the desired quality of service and pairs of properties with regard to their importance for their higher order attributes. The respondents were to answer the following question: "Which of the two attributes is more important to you when using public mass transit services, and how much more important is one attribute as compared to the other?" The latter statement was to be made by checking an AHP scale of nine. Priorities that later became part of the supermatrix as weighting vectors were to be calculated during data analysis from these statements made by the respondents.

The focus of the second part of the questionnaire was on determining dependencies between attributes (cf. Fig. 7). The respondents were asked again to compare pairs of attributes. This time they were asked, however, which of two attributes has a greater influence on a third attribute and how much stronger this influence is. For example, passengers were asked whether the consulting performance or the communication has a greater influence on the travellers’ evaluation of the tourism agency. The weighting vectors determined were placed in the respective columns of the supermatrix in the course of the data analysis.

The data was collected from 155 at randomly choosen customers. So the customers were just asked as measuring customer satisfaction requires product experience and as it was not the purpose of this study to develop an offensive but rather a defensive marketing strategy. This is to say the study was made to determine and to increase customer satisfaction and the customers’ assessment of quality and to build up barriers against change.

FIGURE 6

STRUCTURE OF THE NETWORK FOR DETERMINING THE PRIORITY COMPONENT

FIGURE 7

INNER DEPENDENCIES BETWEEN ATTRIBUTES OF THE SECOND HIERARCHY LEVEL

4.3. Data analysis

The local AHP priorities of the attributes and properties as well as the values of attribute dependencies were calculated based on the data thus obtained using the EXPERT CHOICE software package, release 9.0. Then the initial supermatrix was created based on the aggregated values from all respondents. The local AHP weighting vectors and the weighting vectors of the inner dependencies were entered in the supermatrix at their respective positions. To guarantee stochasticity, the blocks of the supermatrix have then to be weighted in such a way that the columns of the matrix all total one. Subsequently the final supermatrix can be calculated. In the present case, it should be calculated using the equation (I B W)^-1 (where I represents the identity matrix). The results after using the supermatrix method are shown in figure 8.

The respondents were contacted once again two week after the study was carried out. This time the focus was on determining a target hierarchy using the classical AHP. As a comparison of the results shows, the two methods yield different priorities. As dependencies between second-level attributes were taken into account, the importance of consulting (0,15) and availability (0,04) was increased, whereas the importance of the price dropped (0,51). There was no change regarding the range of services.

FIGURE 8

FINAL SUPERMATRIX RESULTS

5. CRITICAL ASSESSMENT OF AHP AND THE SUPERMATRIX METHD AS REGARDS THEIR SUITABILITY FOR DETERMINING RELATIVE PRIORITIES

Use of AHP to determine the importance of service attributes holds a number of benefits for the user. The respondents are forced due to the paired comparisons to make compromises when stating the importance of individual attributes and properties. This prevents distortion of the actual importance of attributes by the phenomenon of aspiration level inflation when determining priorities. Moreover, consistency of the comparative judgements can be verified. According to a study by Tscheulin, the forecasting validity of AHP is greater than that of conjoint analysis if the respondents have understood the task (Tscheulin 1992).

The advantages of AHP listed here also apply to the supermatrix method. This method has one more benefit in that it can take into account dependencies between attributes of the object of study when determining priorities. Thus the importance of the drivers’ behaviour was much increased in this study after applying the supermatrix method.

A clear disadvantage is that including the supermatrix entails an increase in the number of the paired comparisons required. This also increases the amount of time required for an interview.

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