Ranking discovered rules from data mining with multiple criteria by data envelopment analysis

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Ranking discovered rules from data mining with multiple criteria

by data envelopment analysis

Mu-Chen Chen

*

Institute of Traﬃc and Transportation, National Chiao Tung University 4F, No. 118, Section 1, Chung Hsiao W. Road, Taipei 10012, Taiwan, ROC

Abstract

In data mining applications, it is important to develop evaluation methods for selecting quality and profitable rules. This paper utilizes a non-parametric approach, Data Envelopment Analysis (DEA), to estimate and rank the efficiency of association rules with multiple criteria. The interestingness of association rules is conventionally measured based on support and confidence. For specific applications, domain knowledge can be further designed as measures to evaluate the discovered rules. For example, in market basket analysis, the product value and cross-selling profit associated with the association rule can serve as essential measures to rule interestingness. In this paper, these domain measures are also included in the rule ranking procedure for selecting valuable rules for implementation. An exam-ple of market basket analysis is applied to illustrate the DEA based methodology for measuring the efficiency of association rules with multiple criteria.

Keywords: Data mining; Association rules; Interestingness; Data envelopment analysis; Multiple criteria

1. Introduction

Data mining techniques have become widespread in business. Moreover, various rules may be obtained using data mining techniques, and only a small number of these rules may be selected for implementation due, at least in part, to limitations of budget and resources. Association rule mining diﬀers from traditional machine learning tech-niques by permitting decision makers to pick from the many potential models that can be supported by the data (Webb & Zhang, 2005). Generally, association rule mining discovers all rules that meet certain sets of criteria or con-straints, such as minimum support and minimum conﬁ-dence, rather than generating a single model that best matches the data.

Evaluating the interestingness or usefulness of associa-tion rules is important in data mining. In many business

applications, it is necessary to rank rules from data mining due to the number of quality rules (Tan & Kumar, 2000) and business resource constraint (Choi, Ahn, & Kim, 2005). Selecting the more valuable rules for implementation increases the possibility of success in data mining. For example, in market basket analysis, understanding which products are usually bought together by customers and how the cross-selling promotions are beneficial to sellers both attract marketing analysts. The former makes sellers to provide appropriate products by considering the customers’ preferences, and the later allows sellers to gain increased profits by considering the sellers’ profits. Customers’ preferences can be measured based on support and confidence in association rules. On the other hand, seller profits can be assessed using domain related measures such as sale profit and cross-selling profit associated with the association rules.

Since high value products are relatively uncommonly bought by customers, a rule that is proﬁtable to sellers may not be discovered by setting constraints of minimum support and minimum conﬁdence in the mining process.

*

Tel.: +886 2 2349 4967; fax: +886 2 2349 4953. E-mail address:ittchen@mail.nctu.edu.tw

www.elsevier.com/locate/eswa Expert Systems with Applications 33 (2007) 1110–1116

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Cohen et al. (2000) described a good example of this, namely the Ketel vodka and Beluga caviar problem. Although, most customers infrequently buy either of these two products, and they rarely appear in frequent itemsets, their profits may be potentially higher than many lower value products that are more frequently bought. Another example regarding the interesting infrequent itemsets is described inTao, Murtagh, and Farid (2003). The associa-tion rule of [wine) salmon, 1%, 80%] may be more inter-esting to analysts than [bread) milk, 3%, 80%] despite the first rule having lower support. The items in the first rule typically are associated with more profit per unit sale. From the examples of Ketel vodka and Beluga caviar and wine and salmon, infrequent itemsets may be interesting for certain applications provided that domain information is considered (Tao et al., 2003; Webb & Zhang, 2005). How-ever, the traditional association rule mining algorithms (Agrawal, Imielinski, & Swami, 1993; Srikant & Agrawal, 1997) cannot classify such infrequent products to interest-ing itemsets since the subjective domain knowledge is ignored. A lower threshold can be set to identify the infre-quent itemsets with a high value. However, numerous asso-ciation rules are consequently generated, and it is extremely difficult for analysts to select the useful rules between them. In previous studies dealing with the discovery of subjec-tively interesting association rules, most approaches require manual input or interaction by asking users to explicitly distinguish between interesting and uninteresting rules (Liu, Hsu, Chen, & Ma, 2000). Liu et al. briefly reviewed these existing approaches. The measures of inter-estingness are specified as constraints in the mining process, and only the rules that satisfied these constraints are retrieved. Klemetinen, Mannila, Ronkainen, Toivonen, and Verkamo (1994) proposed an item constraint, which describes the occurrence of certain items in the conditional (right hand side) and consequent (left hand side) parts.

Srikant, Vu, and Agrawal (1997) also proposed a mining algorithm that considered the item and item hierarchy con-straints speciﬁed by analysts. Moreover, Lakshmanan, Han, and Pang (1998) extended the approach developed by Srikant et al. to consider much more complicated con-straints, including domain, class, and SQL-style aggregate constraints. The approach developed by Ng et al. can sup-port constraint based, human-centered exploratory mining of association rules.Goethals and Van den Bussche (2000)

also proposed an interactive approach based on querying conditions within the association rule mining process.

Liu et al. (2000)proposed an approach to assist analysts in ﬁnding interesting rules from a set of mined association rules by analyzing the rules using the domain information. The mined rules are then ranked according to two subjec-tive interestingness measures, unexpectedness and action-ability. The degree of unexpectedness of rules can be measured by the extent to which they surprise the analyst (Liu & Hsu, 1996; Silberschatz & Tuzhilin, 1996). Mean-while, the degree of actionability can be measured by the extent to which analysts can use the discovered rules to

their advantage. The system developed by Liu et al. (2000) is an interactive and iterative post-processing tech-nique. This system ﬁrst asks analysts to specify their exist-ing domain knowledge, and then analyzes the discovered rules to identify the potentially interesting ones. However, Liu et al. focused on unexpected rules, which are measured by unexpectedness.

Choi et al. (2005) proposed a group decision making approach based on Analytic Hierarchy Process (AHP) to rank the association rules generated from data mining. This approach would construct a consensus provided that a group of managers work together to select discovered. The rule quality can be improved by considering both objective criteria and subjective preferences of managers. However, this approach encounters a problem of requiring considerable human interaction to ﬁnd out the weights of criteria by aggregating the opinions of various managers.

Most existing association rule mining algorithms take the measure of large support to ﬁnd frequent itemsets, and all items are considered to have equal weight (Tao et al., 2003). Therefore, these approaches are unsuitable for discovering the interesting infrequent itemsets as described in the above two examples. Tao et al. developed an approach that used an improved model of weighted sup-port. In the approach of weighted association rule mining, itemsets are no longer simply counted as they appear in a transaction, and the subjective measures (e.g., proﬁt) are also adopted for rule evaluation.

Most of the abovementioned approaches focus on com-putation eﬃciency by embedding the subjective constraints in the mining procedure to prune the search space. How-ever, a huge amount of subjective domain knowledge may exist, which can be considered as potential subjective constraints and interestingness measures. It is sophisticated to determine the subjective constraints and interestingness measures before discovering some rules. Provided that the constraints are not adequately stated, the interesting rules may not be discovered after the mining procedure. Additionally, rule interestingness may be a relative mea-sure, but not an absolute one. Generally, decision makers can suitably select interesting rules for implementation after making comparisons between some potential rules.

In data mining, it is substantial to bring together the sta-tistic based rule extraction and profit based action to meet the enterprises’ objectives (Wang, Zhou, & Han, 2002). This paper aims at using a non-parametric approach, Data Envelopment Analysis (DEA), to estimate and rank the efficiency (interestingness or usefulness) of association rules with multiple criteria. The interestingness of association rules is measured by multiple criteria involving support, confidence and domain related measures. This paper uses DEA as a post-processing approach. After the rules have been discovered from the association rule mining algo-rithms, DEA is used to rank those discovered rules based on the specified criteria. The remainder of this paper is organized as follows. Section 2 introduces the concept of association rules. Section3then presents the DEA method.

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Furthermore, the proposed approach is described in Section 4. Next, an example of market basket analysis is illustrated in Section5. Finally, Section6makes a conclu-sion to this paper.

2. Association rules

Association rule mining discovers the relationships between items from the set of transactions. These relation-ships can be expressed by association rules such as [i1) i2,

i3support = 3.5%, conﬁdence = 45%]. This association rule

implies that 3.5% of all the transactions under analysis show that items i1, i2and i3 appear jointly. A conﬁdence

of 45% indicates that 45% of the transactions containing i1 also contain i2 and i3. Associations may include any

number of items on either side of the rule.

The problem of mining association rules is formally sta-ted as follows (Agrawal et al., 1993; Srikant & Agrawal, 1997). Let I = {i1, i2, . . . , im} denote a set of literals, namely

items. Moreover, let D represent a set of transactions, where each transaction T is a set of items such that T I. A unique identifier, namely TID, is associated with each transaction. A transaction T is said to contain X, a set of some items in I, if X T. An association rule implies the form X) Y, where X I, Y I and X \ Y = ;. The rule X) Y holds in the transaction set D with confidence, c, where c% of transactions in D that contain X also con-tain Y. The rule has support, s, in the transaction set D if s% of transactions in D contain X[ Y. An efficient algo-rithm is required that restricts the search space and checks only a subset of all association rules, yet does not miss important rules. The Apriori algorithm developed by Agra-wal et al. (1993)andSrikant and Agrawal (1997)is such an algorithm. However, the interestingness of rule is only based on support and confidence. The Apriori algorithm is described as follows:

(1) L1= ﬁnd_large_1-itemsets;

(2) for (k = 2; Lk15;; k++) do begin

(3) Ck= apriori_gen (Lk1); // new candidates

(4) forall TID T2 D do begin

(5) CT= subset (Ck, T); // candidates contained in T

(6) forall candidates C2 CT do

(7) C.count++; (8) end

(9) Lk= {C2 Ckj C.count/no_of_dataP minimum

sup-port threshold} (10) end

(11) Return L =[kLk.

In the above Apriori algorithm, the apriori_gen procedure generates candidates of itemset and then uses the minimum support criterion to eliminate infrequent itemsets. The apriori_gen procedure performs two actions, namely, join and prune, which are discussed in Han and Kamber (2001). In join step, Lk1 is joined with Lk1 to

generate potential candidates of itemset. The prune step

uses the minimum support criterion to remove candidates of itemset that are not frequent. In fact, expanding an item-set reduces its support. A k-itemitem-set can only be frequent if all of its (k 1)-subsets are also frequent, consequently apriori_gen only generates candidates with this property, a situation easily achievable given the set Lk1.

Association rule mining is a popular technique for mar-ket basmar-ket analysis, which typically aims at ﬁnding buying patterns for supermarket, mail-order and other customers. By mining association rules, marketing analysts try to ﬁnd sets of products that are frequently bought together, so that certain other items can be inferred from a shopping cart containing particular items. Association rules can often be used to design marketing promotions, for exam-ple, by appropriately arranging products on a supermarket shelf and by directly suggesting to customers items that may be of interest.

With the constant collection and storage of considerable quantities of business data, association rules are discovered from the domain databases and applied in many areas, such as marketing, logistics and manufacturing (Chen, 2003; Chen, Chiu, & Chang, 2005a; Chen, Huang, Chen, & Wu, 2005b; Chen & Wu, 2005; Chen & Lin, in press; Wang et al., 2004). In the areas of marketing, advertising and sales, corporations have found they can beneﬁt enor-mously if implicit and previously unknown customer buy-ing and callbuy-ing patterns can be discovered from large volumes of business data (Chen, Han, & Yu, 1996).

Generally, support and confidence are taken as two measures to evaluate the interestingness of association rules (Agrawal et al., 1993; Srikant & Agrawal, 1997). Association rules are regarded as interesting if their sup-port and confidence are greater than the user-specified min-imum support and minmin-imum confidence, respectively. In data mining, it is important but difficult to appropriately determine these two thresholds of interestingness. Data miners usually specify these thresholds in an arbitrary man-ner. Numerous algorithms for finding association rules have been developed in previous studies (Hipp, Gu¨nter, & Nakhaeizadeh, 2000). However, relatively little literature has attempted to employ the application-specific criteria for setting the threshold of association rules.

3. Data envelopment analysis

In 1978, Data Envelopment Analysis (DEA) was initi-ated by Charnes, Cooper and Rhodes (CCR), and they demonstrated how to change a fractional linear measure of efficiency into a linear programming model (Charnes, Cooper, & Rhodes, 1978). DEA was defined by Charnes et al. as: a mathematical programming model applied to observational data, which provides a new method of obtaining empirical estimates of extremal relations-such as the production functions and/or efficient production possibility surfaces that are fundamental to modern eco-nomics. Researchers have developed several DEA models by theoretically broadening the CCR model (e.g.,Charnes,

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Cooper, Seiford, & Stutz, 1982; Charnes, Cooper, Golany, Seiford, & Stutz, 1985; Cook & Kress, 1990; Hashimoto, 1997; Obata & Ishii, 2003).

DEA was originally designed to mathematically mea-sure decision making units (DMUs) with multiple inputs and outputs in terms of relative eﬃciency (i.e., the ratio of total weighted output to total weighted input). However, no obvious production function exists for aggregating the data in its entirety (Adler, Friedman, & Sinuany-Stern, 2002). Cook and Kress (1990) introduced a theoretical extension of DEA to analyze ranked voting data. In the ranked voting system, each candidate (DMU) is regarded as having multiple outputs (ranked votes) and only input with amount unity, i.e., the pure output DEA model (Hashimoto, 1997). In the approach developed by Cook and Kress, preference scores are estimated without initially imposing any ﬁxed weights. The score of each candidate (DMU score) is calculated based on its most favorable weights (Obata & Ishii, 2003).Adler et al. (2002)reviewed some ranking methods in DEA.

The preference score, Zi, of candidate (DMU) i is the

weighted sum of votes with certain weights. The mathemat-ical model of the ranked voting system in DEA is formu-lated as follows (Cook & Kress, 1990):

Maximize X k j¼1 wjvoj ð1Þ Subject to: Xk j¼1 wjvij61; i¼ 1; 2; . . . ; m; ð2Þ wj wjþ1P dðj; eÞ; j¼ 1; 2; . . . ; k 1; ð3Þ wjP dðk; eÞ; ð4Þ

where wjdenotes the weight of the jth place; vijrepresents

the number of jth place votes of candidate i (i = 1, 2, . . . , m, j = 1, 2, . . . , k); and d(•,e), known as the discrim-ination intensity function, is nonnegative and nondecreas-ing in e and satisﬁes d(•,0) = 0. Parameter e is nonnegative. The above mathematical model is resolved for each can-didate o, o = 1, 2, . . . , m. The resulting objective value is the preference score of candidate o. Constraints(3)ensure that the vote of the higher place may have a greater importance than that of the lower place. Several candidates may achieve a maximum preference score of 1 once the linear programs (1)–(4) are resolved for all candidates. Candi-dates with preference score 1 are called eﬃcient candiCandi-dates. Without setting the priorities of criteria, Constraints(3)are relaxed.

DEA frequently generates several efficient candidates (Obata & Ishii, 2003). The set of efficient candidates is the top group of DMUs, but no efficient DMU can be dis-tinguished as the winner among this group. It is necessary to further discriminate these efficient candidates. To dis-criminate efficient candidates, the discriminant method proposed by Obata and Ishii (2003) is adopted in this

paper. The discriminant model for eﬃcient candidates is formulated as follows (Obata & Ishii, 2003):

Minimize X k j¼1 wj ð5Þ Subject to: Xk j¼1 wjvoj¼ 1; ð6Þ Xk j¼1

wjvij61; for all efficient i6¼ o; ð7Þ

wj wjþ1P dðj; eÞ; j¼ 1; 2; . . . ; k 1; ð8Þ

wjP dðk; eÞ: ð9Þ

Obata and Ishii’s discriminant model, (5)–(9), is also a linear program. The preference score of the second stage (Z0

i) is obtained as a reciprocal of the optimal value, that

is Z0_i¼ 1=Pk_j¼1wj. This discriminant model does not

employ any information about ineﬃcient candidates, and thus the problem of varying the rank of eﬃcient candidates does not occur (Obata & Ishii, 2003). Simi-lar to Cook and Kress’s DEA model, (1)–(4) Constraints

(8) are relaxed if the priorities of criteria are not speciﬁed.

4. Proposed post-processing approach

The interestingness of a rule can be use to ﬁlter a large number of rules and report only those which may be useful to decision makers (Mitra, Pal, & Mitra, 2002). The thresh-olds of support and conﬁdence are selected by only considering the database perspective. However, the inter-estingness of an association rule is commonly applica-tion-dependent (Srikant et al., 1997). The domain information in application areas can potentially provide useful criteria for picking important rules, and can be adopted to improve the rule selection procedure.

Association rule mining contains no theoretically optimal thresholds to filter rules and patterns. The idea of efficiency in DEA is a comparative concept ( Serrano-Cinca, Fuertes-Calle’n, & Mar-Molinero, 2005). As dis-cussed above, the idea of interestingness resembles the comparative concept of efficiency, and no absolute measure exists. In DEA, the set of efficient candidates is taken as the leading set of candidates. All candidates (association rules) can be ranked in decreasing order of preference score, and interesting rules then can be selected by setting a threshold of preference score or selecting rules with first N highest preference scores. This paper further adopts Obata and Ishii’s discriminant model, (5)–(9), is further adopted to discriminate the efficient candidates (i.e., the interesting association rules), which are generated in Cook and Kress’s DEA model,(1)–(4). The discriminant model can be used to identify one winner (the most favorable association rule) if such a winner exists.

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With the above discussion, the proposed post-processing approach for ﬁnding the interesting association rule is sche-matically illustrated in the ﬂow chart in Fig. 1. The pro-posed approach is described as follows:

Step 1. Input data for association rule mining.

Step 2. Mine association rules by using the Apriori algo-rithm with minimum support and minimum conﬁdence.

Step 3. Determine subjective interestingness measures by further considering the domain related knowledge. Step 4. Calculate the preference scores of association rules discovered in Step 2 by using Cook and Kress’s DEA model, (1)–(4).

Step 5. Discriminate the eﬃcient association rules found in Step 3 by using Obata and Ishii’s discriminant model, (5)–(9).

Step 6. Select rules for implementation by considering the reference scores generated in Step 5 and domain related knowledge.

In Step 2 of the proposed post-processing approach, the Apriori algorithm (Agrawal et al., 1993; Srikant & Agra-wal, 1997) is initially used to discover the association rules only with the thresholds of support and conﬁdence. These two thresholds can be set relatively lower since this paper

intends to ﬁnd some infrequent itemsets, which may be of interest to analysts by considering subjective domain related knowledge. The interestingness of association rules generated in Step 2 can be further judged by the following DEA approach in Steps 4 and 5.

The traditional Apriori algorithm cannot classify the infrequent items to interesting itemsets since the subjective domain knowledge is ignored. A huge amount of subjective domain knowledge may exist, which can be considered as potential subjective constraints and measures for evaluat-ing association rules. Followevaluat-ing the discovery and report-ing of some rules, a data miner can select the subjective interestingness measures in Step 3. In market basket anal-ysis, understanding which products are usually bought together by customers and which products are beneficial to sellers are both interesting subjects for marketing ana-lysts. The former can be measured in terms of support and confidence in association rules. In this paper, the sub-jective measures of sellers’ profits are evaluated in terms of itemset value and cross-selling profit corresponding to the association rules. For association rules like X) Y, four criteria are jointly used for rule evaluation as follows:

Support: The support, s, is the percentage of transac-tions that contain X[ Y (Agrawal et al., 1993). It takes the form

s¼ P ðX [ YÞ ð10Þ

Conﬁdence: The conﬁdence, c, is the ratio of the percent-age of transactions that contain X[ Y to the percentage of transactions that contain X (Agrawal et al., 1993). It takes the form

c¼PðX [ YÞ

PðXÞ ¼ P ðY XÞj ð11Þ Itemset value: The itemset value, v, is the sum of values of all items (vg’s) in the itemset, X[ Y, and can be

calcu-lated by v¼ X

g2X[Y

vg ð12Þ

Cross-selling profit: For the rule X) Y, the cross-selling is described as recommending that customers purchase Y, if they have bought X. Therefore, the cross-selling profit is the sum of the profits of all items (ph’s) in Y, and can be

calculated by p¼X

h2Y

p_h ð13Þ

In the above four measures, support and conﬁdence are generated by the Apriori algorithm. Meanwhile, the other two measures of itemset value and cross-selling proﬁt are subjectively selected by analysts. In Step 4, these four mea-sures are used as criteria in Cook and Kress’s DEA model,

(1)–(4), to calculate the preference scores for all association rules generated in Step 2. Provided that there exist more than one eﬃcient association rules with preference score 1, the algorithm proceeds to Step 5 to further discriminate Input data

Mine association rules

Find rule preference scores by DEA

Discriminate efficient rules

Select rules for implementation Determine subjective

measures

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these rules by using Obata and Ishii’s discriminant model,

(5)–(9). Finally, the rankings of all rules can be obtained, and analysts accordingly select useful rules for im-plementation.

5. Illustrative example

An example of market basket data is used to illustrate the proposed approach presented in Section 4. Association rules first are discovered by the Apriori algorithm, in which minimum support and minimum confidence are set to 1.0% and 10.0%, respectively. Forty-six rules then are identified

in Step 2. The itemset values and cross-selling proﬁts for these 46 rules are then calculated, as summarized inTable 1. Preference scores (Zi) for each rule listed inTable 1are

calculated by Cook and Kress’s DEA model. In this paper, d(•,e) = 0 is adopted. Additionally, the priorities of criteria are not specified. Table 1 reveals 11 efficient association rules with preference score 1. These 11 efficient rules are further analyzed using Obata and Ishii’s discriminant model. According to the preference scores (Z0

i) obtained

by solving Obata and Ishii’s discriminant model, these 11 rules are ranked in decreasing order inTable 2.

The interestingness of a rule is measured by two types of preference scores (Ziand Z0i), which are essential measures

for ﬁltering a number of rules and report only those which are most interesting to decision makers. The criteria of sup-port and conﬁdence only consider the database perspective. According to these two criteria, marketing analysts are most likely to pick rules 1 and/or 12 to design their promo-tion campaigns. However, with the above discussion, the interestingness of an association rule is generally applica-tion-dependent, and the domain information in application areas can potentially provide useful criteria for picking important rules.

For example, this paper also considers two subjective measures of itemset value and cross-selling profit. Observ-ing the results of Cook and Kress’s DEA model and Obata and Ishii’s discriminant model based on four criteria, analysts likely select rules 26, 22 and/or 18 to design the marketing activities. From Tables 1 and 2, rules 1 and 12 are also identified as efficient candidates (interesting rules), while these two rules have relatively lower preference scores in Obata and Ishii’s discriminant model.

6. Conclusion

Association rule discovery is one of the popular techniques recently developed in the area of data mining. Evaluating the interestingness or usefulness of association rules is an essential task in data mining applications. In market basket analysis, marketing analysts are no longer satisﬁed by a set of rules or patterns discovered by a data

Table 1

Summary of results of Cook and Kress’s model Rule no. Support (%) Conﬁdence (%) Itemset value Cross-selling proﬁt Preference score (Zi) 1 3.87 40.09 337.00 25.66 1.00 2 1.42 18.17 501.00 11.63 0.78 3 2.83 17.64 345.00 11.29 0.84 4 2.34 30.83 163.00 19.73 0.71 5 2.63 23.90 325.00 15.30 0.78 6 1.19 55.65 436.00 35.61 1.00 7 1.19 47.42 598.00 30.35 1.00 8 1.19 15.70 436.00 52.91 0.69 9 1.19 10.82 598.00 36.45 0.85 10 1.19 12.32 436.00 20.08 0.67 11 1.19 12.32 598.00 40.04 0.85 12 3.87 38.08 337.00 103.97 1.00 13 1.18 15.09 710.00 41.19 0.99 14 2.44 15.22 554.00 41.56 1.00 15 2.14 28.21 372.00 77.02 0.78 16 2.51 22.81 534.00 62.26 0.99 17 1.19 50.92 436.00 139.02 1.00 18 1.19 45.25 598.00 123.52 1.00 19 1.19 11.70 436.00 43.54 0.67 20 1.19 11.70 598.00 62.50 0.88 21 1.42 13.99 501.00 61.16 0.79 22 1.18 12.23 710.00 53.45 1.00 23 1.50 13.64 698.00 59.59 1.00 24 2.83 27.82 345.00 78.17 0.84 25 2.44 25.27 554.00 71.00 1.00 26 1.25 15.97 718.00 44.87 1.00 27 1.22 34.89 339.00 98.04 0.75 28 1.30 35.12 435.00 98.68 0.81 29 1.42 33.81 534.00 95.01 0.90 30 1.91 25.26 380.00 70.97 0.75 31 1.43 37.14 618.00 104.35 1.00 32 2.38 21.63 542.00 60.78 0.98 33 1.18 30.24 366.00 84.98 0.70 34 1.23 29.36 626.00 82.51 0.96 35 1.58 22.65 354.00 63.64 0.67 36 2.34 22.99 163.00 22.76 0.60 37 2.14 22.14 372.00 21.92 0.75 38 1.91 11.94 380.00 11.82 0.72 39 2.03 18.42 360.00 18.23 0.72 40 1.19 30.73 436.00 30.43 0.75 41 2.63 25.87 325.00 67.52 0.78 42 2.51 25.98 534.00 67.81 0.99 43 1.50 19.16 698.00 50.02 1.00 44 2.38 14.85 542.00 38.75 0.98 45 2.03 26.73 360.00 69.78 0.75 46 1.19 30.73 598.00 80.22 0.93 Table 2

Summary of results of Obata and Ishii’s model Rule no. Support (%) Conﬁdence (%) Itemset value Cross-selling proﬁt Preference score (Z0 i) 26 1.25 15.97 718.00 44.87 718.00 22 1.18 12.23 710.00 53.45 393.23 18 1.19 45.25 598.00 123.52 306.12 17 1.19 50.92 436.00 139.02 164.95 7 1.19 47.42 598.00 30.35 2.04 23 1.50 13.64 698.00 59.59 1.17 6 1.19 55.65 436.00 35.61 0.79 43 1.50 19.16 698.00 50.02 0.26 31 1.43 37.14 618.00 104.35 0.16 12 3.87 38.08 337.00 103.97 0.12 1 3.87 40.09 337.00 25.66 0.10

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mining algorithm. Instead, marketing analysts wish to develop rules or patterns that are ranked with respect to certain criteria. The complexity of rule evaluation and selec-tion is diﬃcult for analysts. The tradiselec-tional approaches usu-ally ignore the subjective domain knowledge in selecting useful rules. To meet the requirements of marketing ana-lysts, Data Envelopment Analysis (DEA) is used in this paper to evaluate the eﬃciency (interestingness or useful-ness) of association rules with multiple criteria, including subjective domain related measures. The proposed approach provides more insights into the rules discovered and can assist rule evaluation and selection.

Acknowledgements

The authors would like to thank the National Science Council of the Republic of China, Taiwan for ﬁnancially supporting this research under Contract No. NSC 95-2416-H-009-034-MY3.

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