An application of data envelopment analysis in telephone officesevaluation with partial data

Computers & Operations Research 26 (1999) 59—72

An application of data envelopment analysis in telephone offices evaluation with partial data Soung-Hie Kim!,*,1, Choong-Gyoo Park",2, Kyung-Sam Park#,3 !Graduate School of Management, Korea Advanced Institute of Science and Technology, 207-43 Cheongryangri, Dongdaemoon, Seoul 130-012, South Korea "Graduate School of Management, Korea Advanced Institute of Science and Technology, 207-43 Cheongryangri, Dongdaemoon, Seoul 130-012, South Korea #Graduate School of Business, University of Texas at Austin, Austin, TX 78712, USA Received November 1997; received in revised form April 1998

Scope and purpose A procedure for handling both linear partial data on factor values and its weight preferences in data envelopment analysis (DEA) structure is presented. DEA is a methodology for driving the relative efficiencies of organizations or decision making units (DMUs) where there are multiple incommensurate inputs and outputs. The usual setting for many DEA applications involves a set of similar DMUs, for each of which there is an observable and measurable set of inputs and outputs. In some applications, however, a number of factors may be measurable only on partial data such as ordinal rankings and ratio bounds, owing mainly to

*Corresponding author. Tel.: 82-2-958-3611; fax.: 82-2-958-3604; e-mail: [email protected] 1 Soung-Hie Kim is a professor of Graduate School of Management at the Korea Advanced Institute of Science and Technology (KAIST). He holds a B.S. from Seoul National University, a M.S. from the University of Missouri-Columbia, and a Ph.D. in Engineering-Economic Systems from Stanford University. His teaching and research specialties are in the fields of decision theory, multi-criteria decision, decision support systems, group decision, technological forecasting, and business reengineering. He has published numerous papers, which have appeared in Computers and Operations Research, European Journal of Operational Research, Journal of the Operational Research Society, Naval Research ¸ogistics, Information and Decision ¹echnologies, ¹echnological Forecasting and Social Change, Expert Systems with Applications, Applied Artificial Intelligence, etc. 2 Choong-Gyoo Park is a senior researcher at Korea Telecom and doctoral candidate of Graduate School of Management at KAIST. He received his M.S. in Industrial Engineering from KAIST and his B.S. in Industrial Engineering from Seoul National University. His research interests include data envelopment analysis, multi-criteria decision, CALS/EC, and information system analysis/evaluation. 3 Kyung-Sam Park is a visiting scholar in the MSIS Department, Graduate School of Business, University of Texas at Austin, Austin. He holds a B.S. from Busan University, and a M.S. in Industrial Engineering from KAIST, and a Ph.D. in Industrial Engineering from KAIST. His research interests are multi-criteria decision, decision theory, data envelopment analysis, information system analysis/evaluation, and mathematical programming. 0305-0548/98/$19.00#0.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 8 ) 0 0 0 4 1 - 0

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intangible attributes to reflect social and environmental impacts. A model dealing with these partial data is presented and applied to the efficiency evaluation of telephone offices. Abstract The presence of partial data motivates the need to investigate how such factors can be incorporated into the existing measurement models. In this paper, a procedure is proposed for incorporating a set of factors with partial data into the DEA structure and restricting factor weights. The first DEA formulation is a complicated non-linear model issued from the set of partial data. In order to transform the first formulation into an ordinary linear programming model, both a linear scale transformation and variable change technique are used. The resulting linear programming model is then applied to the efficiency evaluation of telephone offices. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Data envelopment analysis; Partial data; Telephone office

1. Introduction Data envelopment analysis (DEA) is a methodology for driving the relative efficiencies of organizations or decision making units (DMUs) where there are multiple incommensurate inputs and outputs. It provides a means for assessing the relative efficiencies of DMUs with minimal prior assumptions on input—output relations in these units. After initial studies by Charnes et al. [4], DEA methods have subsequently been developed and extended (for a review, see Cooper et al. [10] and Seiford [15]). There have been numerous applications of DEA to various fields (see [15, 16]). Through these applications, DEA has become an accepted approach for assessing efficiency in a wide range of cases. The usual setting in most DEA applications involves a set of similar DMUs, for each of which there is an observable and measurable set of inputs and outputs. In some applications, however, certain inputs and/or outputs may be measurable only on linear partial data such as rankings and bound descriptions, rather than specific numerable values. In an application that we will assess the efficiencies of telephone offices (see Section 3), two of the five output factors (i.e. the operation/maintenance level of telecommunication facilities and the degree of customer satisfaction) are measurable on ordinal scales. The ordinal data on the output factors in this application motivates the need to investigate how such factors can be incorporated into existing measurement models. As far as we know, there is no method and application to deal with ordinal data in DEA structure, except for two papers of Cook et al. [8, 9]. While their initial model [8] permits the inclusion of a single ordinal factor, their later model [9] allows us to include multiple ordinal factors. These approaches focus mainly on dealing with ranking-styled-data on factors. However, the other styles of factor values, such as ratio bounds, may be elicited from the DEA domain experts (although those are not the case in our application). For example, DMU 1 produces an output between 60 and 70% in comparison with DMU 2. In contrast to these approaches, proposed in this paper is a new method being able to include any style of linear partial data (see Section 2). In the meantime, there existed a number of approaches to the inclusion of additional information on factor weights: constant bounds [11, 13], rankings [1, 12], assurance region [17, 18], cone-ratio

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[6, 7], and proportion on virtual input/output [2, 19]. Restricting factor weights is aimed at controlling some kinds of flexibility that are intrinsic to DEA. Because optimization in DEA aims to accord each DMU the best efficiency rating (under the given constraints), DEA allows much flexibility in the determination of the weights of inputs and outputs when assessing the relative efficiency of a DMU. As the efficiency of each DMU is solved separately, the set of weights derived will be widely different for the various DMUs. This can lead some DMUs to be assessed by only a small subset of their inputs and outputs, while their remaining inputs and outputs are all but ignored practically. Roll and Golany [14] criticized such results on several grounds. The model proposed in this paper permits partial factor values and thus possesses their own flexibility to make matters worse. It is, therefore, much more required that this model includes some additional constraints on the factor weights so as to increase the efficiency discrimination power of the DMUs. In our application, 11 of the 33 DMUs were efficient in the absence of additional weight constraints. Restricting the weights with linear partial constraints reduced the set of efficient DMUs from 11 to 6. In order to impose importance relations among factor weights, it is noted that the factor values must generally be commensurable across the various factors . This can be carried out through a normalization of the observed values on each factor, based on the units-invariance property of DEA (for the proof, see [3]), a property which plays an important role in translating a non-linear programming model into as ordinary linear one.

2. DEA model with partial data As shown in CCR [4, 5], DEA efficiency can be computed by using the model below: max z " + k y 0 r ro r|R s.t.

+ k y !+ u x )0 ∀j, r rj i ij r|R i|I

(1)

+ u x "1, i io i|I k , u *e ∀r, i, r i where x and y represent the observed values for input i3I and output r3R for each of j3J ij rj DMUs, u and k are factor weights (or virtual multipliers) for input i and output r respectively, and i r e is a non-Archimedean infinitesimal. The subscript j"0 denotes the DMU to be evaluated by placing it in the objective while leaving it in the constraints. When * is used to denote an optimal value throughout the paper, the condition for full DEA efficiency becomes z*"1 in (1). 0 Let us consider another DEA model with weight restrictions. Denote (`-RDRD and (~-RDID, say assurance region (AR), to be the admissible sets of worth vectors k"(k ) and u"(u ) specified r i by prior judgments (i.e. constraints of the weights), respectively. An example of the permissible sets is given by (`"Mk"(k )Da )k /k )b for some r, a (b are positive constantsN. The r r r r`1 r r r

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AR-DEA model can be expressed as max n " + k y r ro 0 r|R s.t.

+ k y !+ u x )0 ∀j, r rj i ij r|R i|I

(2)

+ u x "1, i io i|I k"(k )3(`, r u"(u )3(~, i k, u*0, where n* indicates AR-efficiency for DMU 0. If n*"1, then j"0 is called an AR-efficient DMU. 0 0 Assume, without loss of generality, that the factor values (x , y ) are known only on linear ij rj partial data. Let #`-RDJD be the permissible set of worth vectors y such that #`"My "(y )D the r r r r rj linear partial judgment on y for a given rN. #~-RDJD of x "(x ) follows similarly. Then a DEA rj i i ij model annexing the constraints x 3#~ and y 3#` to model (2) becomes r i r i max n " + k y 0 r ro r|R s.t.

+ k y !+ u x )0 ∀j, r rj i ij r|R i|I

(3)

+ u x "1, i io i|I k"(k )3(`, r u"(u )3(~, i x "(x )3#~ ∀i, i ij i y "(y )3#` ∀r. r r rj k, u*0, Observe that model (3) is a non-linear programming (NLP) formulation in which there exist the sum-product forms of both the factor values and its weights partially known. We will examine a resolution technique, translation of NLP (3) into ordinary LP. Let u denote a real-valued function for the factor k3I or R such that f P[0, 1] where f ,x k kj kj kj or y . An example of such u is a linear scale transformation function: kj k u ( f )"f /max M f N for each k, k kj kj j kj

(4)

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where u ( ) )3[0, 1]. As in (4) all factor values are transformed in a linear or proportional way so k that the relative order in the magnitude of the values remains equal. Let f 0"max M f N for a given kj j kj k, then u ( f 0 )"1. There may exist multiple u ( f 0) for each k. k kj k kj Now, consider the following model: max n " + k u (y ) 0 r r ro r|R s.t.

+ k u (y )!+ u u (x ))0 ∀j, r r rj i i ij r|R i|I

(5)

+ u u x "1, i i io i|I k"(k )3(`, r u"(u )3(~, i u (x )"Mu (x )3H~ ∀i, i i i i ij u (y )"Mu (y )3H` ∀r, r r r r rj k, u*0. Model (5) is a modified model of (3), i.e., this includes the variables of u (x )3H~ and u (y )3 H`, r i r rj i ij instead of x 3H~ and y 3H` as in (3). By virtue of the units-invariance property of DEA, the r i r i optimal solution of (3) equals the optimal solution of (5). Let X "u ) u (x ), ½ "k ) u (y ). ij i i ij rj r r rj

(6)

Then all of X and ½ become non-negative variables. Denote ij rj X 0"max MX N, ½ 0"max M½ 0N ij ij rj j j rj

(7)

where respective j0 corresponds to j of x0 and y0 . Then, clearly ij rj u "X 0 , k "½ 0 r rj i ij

(8)

since u (x0 )"u (y0 )"1. Thus, the constraints k , u *e in model (5) can be translated into ½ 0 , i ij r rj r i rj X 0*e. We further have u (x )"X /X 0 and u (y )"½ /½ 0 from (6) and (8). The sets H~ and i i ij ij ij r rj rj rj ij H` in (5) can be transformed into H~ and H` , respectively, in which the components are also r i r linear partial forms, where H~ and H` are the permissible sets of worth vectors X "(X ) and r i ij i ½ "(½ ), respectively. Consequently, NLP (3) can be converted into the following ordinary LP r rj

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through NLP (5): max n " + ½ 0 ro r|R s.t.

+ ½ !+ X )0 ∀j, rj ij r|R i|I

(9)

+ X "1, io i|I (X 0)3W~ ij (½ 0)3W`, rj X "(X )3H~ ∀i, i i ij ½ "(½ )3H` ∀r, r r rj X , ½ *0 ∀i, r. i r Let us illustrate the translation of NLP into LP with some numerical examples, which is a procedure of transformations: (#~ , #` )P(H~ , H` )P(H~ , H` ), namely, the first judgement i r i r i r (#~ , #` ) in (3) is translated into (H~ , H` ) in (5), and then (H~ , H` ) in (5) is also translated into i r i r i r (H~ , H` ) in (9). Note that occasionally (H~ , H` ) for some i and r would be the first judgement of i r i r the DEA evaluator. Another one is ((~, (`) P (W~, W`), i.e., the first judgement ((~, (`) about factor weights in (3) can be expressed by (W~, W`). Note that indeed both H~ and W~ (both i H` and W`) are homogeneous sets to have the same variables X (½ ), but described separately r ij ij here for the sake of convenience. Let a set of DMUs J"M1, 2, 3N, inputs I"M1, 2N, and output R"MaN. Suppose that the first assessments are Mx "60, x "60, x "30N (precise data), Mx )x )x N (ranking data), 11 12 13 21 22 23 and M0.5)y /y )0.6, 1.25)y /y )1.60N (ratio bound data). !2 !1 !3 !1 A linear scale transformation technique being adapted as in (4), the precise data are translated into Mu (x )"u (x )"1, u (x )"0.5N, and the ranking data into Mu (x ))u (x )) 1 11 1 12 1 13 2 21 2 22 u (x )"1N. In the ratio bound data, taking the reciprocal of the second placed inequalities yields 2 23 0.625)y /y )0.80, and multiplying these inequalities by the first placed inequalities results in !1 !3 0.3125)y /y )0.480. Now set u (y )"1 (i.e., adapt linear scale transformation), then we !2 !3 ! !3 obtain 0.625)u (y ))0.80 and 0.3125)u (y ))0.480. ! !1 ! !2 Finally, the constraints in (H~ , H` ) are derived from those in (H~ , H` ). In the precise data, r i r i X 0 can be both X and X . Set X 0"X arbitrary, then we get MX "X , X "0.5 X N 1j 11 12 1j 11 11 12 13 11 (since u (x )"X /X 0). In the ranking data, set X 0"X , then we have MX )X )X N. 1 13 13 1j 2j 23 21 22 23 Analogously, we have M0.625½ )½ )0.80½ , 0.3125½ )½ )0.480½ N in the ratio !3 !1 !3 !3 !2 !3 bound data. Next, suppose that the DEA evaluator provides a ranking information on u such that (~"Mu"(u , u ) D u *u N. Observing u "X 0"X and u "X 0"X , we can ex1 2 1 2 1 1j 11 2 2+ 23 press the set (~ as W~"M(X , X ) D X *X N. As another example, if (~"Mu D 0.5)u /u 11 23 11 23 2 1 )0.6N, we have W~"M(X , X )D0.5X )X )0.6X N. 11 23 11 23 11

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Model (9) permits to include partial information on all factor values and their weights. This can be solved by using any LP package for obtaining the relative efficiency measures of DMUs.

3. An application to the efficiency evaluation of telephone offices The telecommunications industry is largely composed of two parts: one is a telecommunications equipment supplier and the other is a service provider. Korea Telecom, which is a telecommunications service provider, has played an overwhelming role in leading the information society in Korea. This application attempts to evaluate 33 telephone offices under Seoul regional headquarters which is the biggest of ten regional headquarters’ of Korea Telecom. 3.1. Selection of inputs and outputs Inputs to evaluate the efficiencies of the telephone offices largely consist of three factors: manpower, operating costs, and telecommunications facilities (e.g. switching systems, transmission systems, and cables). In many applications of DEA, manpower and operating costs have been typical inputs. A representative measure regarding the magnitude of the telecommunications facilities is the number of telephone lines installed in telephone offices. All of the three inputs are characterized as tangible factors, thereby having numerical values as shown in Table 1. These selected inputs are summarized as follows: X1: Manpower (MP). The number of regular employees who perform the common tasks of telephone offices, excluding the personnel to carry out special tasks such as data communication processing, CATV-related services, and the investigation of various demands for the telecommunications services. X2: Operating costs (OC). Various costs except for interest cost, equipment investment, depreciation cost, and labor cost (because the manpower factor reflects the labor cost). X3: Number of telephone lines (NTL). The maximum number of telephone lines installed in each telephone office, that is, the maximum capacity of switching systems to connect subscribers. Telephone offices provide various telecommunications services for customers with telecommunications facilities. The common services in all of the 33 telephone offices include local, long-distance, and international telephone services. To measure each of the three common services, we use local, long-distance, and international revenues as outputs. Meanwhile, Korea Telecom provides some other services such as leased-line services, CATV-related services, and satellite services, however, for which only some particular offices are responsible. Hence the three factors below are used in terms of monetary outputs and their numerical data are given in Table 1. Y1: Local revenues (LR). The total revenues of local telephone services in each office in 1996. It includes the revenues of local public pay-phone services. Y2: Long-distance revenues (LDR). The total revenues of long-distance telephone services in each office in 1996. The revenues of long-distance public pay-phone services is covered. Y3: International revenues (IR). The total revenues of international telephone services in each office in 1996. It includes the revenues of international public pay-phone services. The outputs related to the operation/maintenance of telecommunications facilities may be represented with the rate of call completion and the number of failures per 100 subscribers which

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Table 1 Data set of thirty three telephone offices DMU

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D20 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30 D31 D32 D33 MAX

Man Power

Operating Costs (millions $)

Local Revenues (million $)

(X2)

Number of Telephone Lines in thousand (X3)

(X1) 239 261 170 290 200 283 286 375 301 333 346 175 217 441 204 216 347 288 185 242 234 204 356 292 141 220 298 261 216 171 123 89 109 441

International Revenues (million $)

(Y1)

LongDistance Revenues (million $) (Y2)

(Y3)

Oper. Maint. Customer Level of Satisfaction Telecom. Facilities (Y4)! (Y5)"

7.03 3.94 2.10 4.54 3.99 4.65 6.54 6.22 4.82 6.87 6.46 2.06 4.11 7.71 3.64 3.24 5.65 4.66 3.37 5.12 2.52 4.24 7.95 4.52 5.21 6.09 3.44 4.30 3.86 2.45 1.72 0.88 1.35 7.95

158 163 90 201 140 214 197 314 257 235 244 112 131 214 163 154 301 212 178 270 126 174 299 236 63 179 225 213 156 150 61 42 57 314

47.99 37.47 20.70 41.82 33.44 42.43 47.03 55.48 49.20 47.12 49.43 20.43 29.41 61.20 32.27 32.81 59.01 42.27 32.95 65.06 31.55 32.47 66.04 49.97 21.48 47.94 42.35 41.70 31.57 24.09 11.97 6.40 10.57 66.04

16.67 14.11 6.80 11.07 9.81 11.34 14.62 16.39 16.15 13.86 15.88 4.95 11.39 25.59 9.57 11.46 17.82 14.52 9.46 24.57 8.55 11.15 22.25 14.77 9.76 17.25 11.14 11.13 11.89 9.08 4.78 3.18 3.43 25.59

34.04 19.97 12.64 6.27 6.49 5.16 13.04 7.31 6.33 6.51 8.87 1.67 4.38 33.01 3.65 9.02 8.19 7.33 2.91 20.72 7.27 2.95 14.91 6.35 16.26 22.09 4.25 4.68 10.48 2.60 2.95 1.48 2.00 34.04

6 8 15 11 4 13 25 20 26 28 16 2 1 18 19 9 5 10 27 17 7 12 21 22 23 3 30 14 31 24 29 32 33 —

4 3 3 2 4 2 4 5 3 4 4 1 4 3 2 4 5 2 4 5 4 3 4 3 4 4 4 1 3 1 5 1 2 —

!Ordinal ranks (1"the best; 33"the worst). "Five ordinal scales (1"the best group; 5"the worst group).

are important measures for telecommunications service providers. In Korea Telecom, however, the rate of call completion depends on types or models of the switching systems and regional characteristics (e.g. residential area or business area). The number of failures depends highly on regional characteristics, too: Telephone offices located in new towns, business areas, or apartment areas, where telephone lines are buried under ground, have less failures than those located in old

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residential areas, where telephone lines on the telephone poles are easily damaged by natural disaster. For these reasons, the direct use of the existing data both on the call completion rate and the number of failures per 100 subscribers are unfair over the telephone offices facing various situations. Y4: The operation/maintenance level of telecommunications facilities (OMLTF). The regional headquarters evaluates OMLTF for all subsidiary telephone offices once a year. The evaluation criteria include call completion rate, the number of failures per 100 subscribers, repair activities of failures, and countermeasures for serious failures. The results of the evaluation are usually reported in ordinal rankings of all telephone offices (see Table 1). Finally, an output factor is considered. Y5: The degree of customer satisfaction (CS). It is one of the most important managerial goals in telecommunications service industries. Inherently this factor may be an intangible attribute. The regional headquarters evaluates the degree of CS for all subsidiary telephone offices twice a month by using monitors. The aggregated score on each telephone office is classified as five ordinal scales (see Table 1), which reflects the front line staff’s kindness and knowledge necessary to provide customers with various telecommunications services, the availability of convenient facilities, the maintenance level of public pay-phones and booths, and so on. 3.2. Initial DEA results We first compute the DEA efficiencies of all DMUs without an additional set of weight restriction. The factor values specified in ordinal number (Y4, Y5) are imposed by the forms of strict rankings: In Y4, these are ½ !½ *e; ½ !½ *e;2 , ½ !½ *e; ½ *e. 4,13 4,12 4,12 4,26 4,32 4,33 4,33 In Y5, ½ "½ "½ "½ ; ½ !½ *e; ½ "½ "½ "½ "½ ; 5,12 5,28 5,30 5,32 5,32 5,4 5,4 5,6 5,15 5,18 5,33 ½ !½ *e; 2, ½ !½ *e; ½ "½ "½ "½ ;½ *e. 5,33 5,2 5,27 5,8 5,8 5,17 5,20 5,31 5,31 The second column of Table 2 shows the efficiency scores for 33 DMUs (where we use e"10~5) in which 19 of 33 DMUs are efficient. That there are too many efficient DMUs (about 60% for all DMUs) might be due to the inclusion of two ordinal factors whose values can be highly flexible: Consider an ordinal relation ½ !½ *e where ½ (½ ) is an output of DMU A (B). The e may be A B A B indicative of value flexibility. If e+0, then ½ +½ in the efficiency evaluations of DMUs A and A B B (because each DMU seeks its maximal efficiency score separately), thereby the relation ½ !½ *e does not give any impact on efficiency discrimination between A and B. So we can A B infer that for larger values of e, the ordinal relation contributes more to the efficiency discrimination. (In this sense, Cook et al. [9] suggested to use maximal e of its possible values within the DEA formulation.) Too many efficient DMUs may directly result from using a small e"10~5 in this study. As a matter of fact, it is quite difficult to set up an appropriate value of e (but being able to use a sensitivity analysis). In the next subsection, another approach is suggested to impose restrictions on ordinal relations in order to prevent them from being useless. 3.3. Use of ratio scale bounds The k y "½ is the credited worth of output r to DMU k. In the case of cardinal factors, the r rk rk ratio o "k y /k y "½ /½ is a fixed quantity regardless of the size of k . However, this rk r rk r rk`1 rk rk`1 3

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Table 2 Efficiency scores DMU

Strict ranking (No weight restriction)

Ratio scale bounds (No weight restriction)

Weight restriction & Ratio scale bounds

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D20 D21 D22 D23 D24 D25 D26 D27 D28 D29 D30 D31 D32 D33

1 1 1 0.9409 1 0.9165 0.8689 0.7211 0.8877 0.7585 0.7816 1 1 1 1 1 0.9124 0.9466 1 1 1 1 0.8647 0.9604 1 1 0.9799 0.9734 0.9517 1 1 1 1

1 1 1 0.8644 0.9851 0.8340 0.8611 0.7211 0.8254 0.7485 0.7734 1 1 1 0.8467 0.9456 0.8610 0.8418 0.7878 1 1 0.7786 0.8482 0.8941 1 1 0.9789 0.8836 0.8151 0.9283 0.7939 1 0.8812

1 0.9875 1 0.6721 0.8534 0.6415 0.6431 0.5340 0.6410 0.5063 0.5928 1 0.9251 0.9011 0.6724 0.8679 0.7234 0.7511 0.5869 1 0.8761 0.6817 0.6818 0.6627 0.8612 1 0.5554 0.7193 0.7278 0.8360 0.6307 1 0.7069

obvious fixed ratio property inherent to cardinal factors is absent in ordinal factors. The ratios of the ordinal factors are clearly variable quantities. It, therefore, seems reasonable to restrict the ratio variables on the ordinal factors. From this point of view, the restrictions will be in the form of bounds on ½ /½ and ½ /½ in this application, where DMU j has the next rank position of 4j 4k 5j 5k DMU k. Although there may be various methods to decide the upper and lower bounds, we determine them on the basis of the ratios of max My N/min My N in the cardinal outputs. That is, the j rj j rj ratio of the value of the highest ranked DMU to that of the lowest ranked DMU in any ordinal factor is set to be between the maximum and minimum of the ratios of the highest value to the lowest one in the cardinal factors.

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In Table 1, the minimum and maximum ratios are 8.04 (in Y2) and 23.04 (in Y3) respectively. Thus, the restrictions on the ordinal factors Y4 and Y5 are given by 8.04) max M½ N/min M½ N)23.04 and 8.04)max M½ N/min M½ N)23.04, where max is the first j 4j j 4j j 5j j 5j j (highest) ranked DMU and min the last (lowest) ranked DMU. We further consider the restricj tions of all ½ /½ , and denote their lower and upper bounds to be ¸ and º . We use ¸ "1.68 5j 5k 5 5 5 and º "2.19 from the fact that ¸(5~1)"8.04 and º(5~1)"23.04. In Y4, ¸ and º are set as 1.07 5 5 5 4 4 and 1.10 respectively for all ½ /½ . 4j 4k Column 3 in Table 2 shows the result of efficiency scores with the constraints in the form of the ratio scale bounds above. By applying ratio scale bounds instead of strict ranking to the ordinal factors, the number of efficient DMUs decreased by eight, and the eight DMUs are less expected to be efficient from the point of view of the evaluator (i.e., the manager of the regional headquarters). For example, DMU 33 was efficient in the absence of the ratio scale bounds, even though it consumed more of all inputs than DMU 32. However DMU33 produced less of two ordinal outputs, same amount of the output of Y2, and more of two outputs of Y1, Y3. As described in the previous subsection, the strict ranking restrictions with ultimately small e are not enough to contribute and make DMU 33 inefficient. Using ratio bound restrictions eliminates DMUs, less expected to be efficient, from the set of efficient offices in this application domain. 3.4. Restriction of weights There were still many efficient DMUs (i.e., 11 of 33) in the eyes of the manager who is going to reduce the set of efficient offices, to the possible extent, and perform stringent management so that Korea Telecom becomes a competitive telecommunications service provider. As an alternative, we try to impose additional weight restrictions. For deriving intuitive judgments, we discussed with the regional headquarters and 33 telephone offices with a number of questions regarding the relative importance of inputs and outputs. For instance, when we asked the relation between the weights of MP (X1), u , and OC (X2), u , we MP OC obtained this ratio which ranged from 80 to 300%, through the various responses of each office. In summary, the following relations are elicited: 0.8)u /u )3.0 (0.8X )X )3.0X ), MP OC 2,23 1,14 2,23 u /u )1 (X )X ), NTL OC 3,8 2,23 1)k

/k )2 (½ )½ )2½ ), LDR LR 1,23 2,14 1,23

1)k /k )2 (½ )½ )2½ ), IR LR 1,23 3,1 1,23 3.5)(k #k #k )/k )8.0 (3.5½ )½ #½ #½ )8.0½ ), LR LDR IR OMLTF 4,13 1,23 2,14 3,1 4,13 7)(k #k #k )/k )16 (7½ )½ #½ #½ )16½ ), LR LDR IR CS 5,12 1,23 2,14 3,1 5,12 where the first judgment on weights is translated into the relation in the parentheses, based on Equation (8), for example, u is given by X 0"X , which is directly used for AR-efficiency MP 1j 1,14 computation.

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These weight constraints being incorporated into model (9), the AR-efficiency scores of the DMUs can be obtained as shown in the last column of Table 2, where the restrictions of the ratio bounds are used for the ordinal factors denoted in the previous subsection. The number of efficient DMUs is reduced to six from eleven. In terms of finding, without weight restrictions in this application, there existed some weights whose values were exceedingly small (i.e., near zero), thereby the corresponding factors were practically neglected. If a DMU has a poorer value on an input (output) factor than the other input (output) factors in comparison with the other DMUs in each factor, then this DMU may have a small weight on the corresponding factor (nevertheless this DMU might be efficient by ignoring the poor factor). Imposing the weight restrictions blocked such a phenomenon to ignore some factors to a great extent (but not completely), and still remained DMUs, which have weighting structures coinciding with the weight restrictions, as efficient (i.e., DMUs 1, 3, 12, 20, 26, 32). As an example, Table 3 shows the weighting structures of sharply reduced DMUs (10, 24, 27) in efficiency score (more than 0.23). DMU 27 has relatively inferior values on all the inputs except for OC and outputs except for LR and CS (refer to Table 1) compared to those of the other DMUs, but this DMU can achieve a high-efficiency score (0.9789) by ignoring the poor factors. However, its efficiency score is sharply reduced by 0.9789—0.5554"0.4235 (see Table 2) by imposing the weight restrictions and hence not neglecting its inferior factor values. Finally, the telephone offices are classified into two groups (large size: 16, small size: 16) in terms of the magnitude of MP and NTL respectively, to check whether the size of telephone offices has any impact on efficiency. The T-test results are summarized in Table 4. Under no weight restrictions, there is no significant difference of efficiency between two groups in the case of MP but significant difference in the case of NTL, however, under the weight restrictions, small telephone offices are more efficient than large ones, which implies that large telephone offices are less productive on outputs than small ones. Consequently, it might be necessary to establish a policy to reduce MP and/or reinforce the marketing activities of large size telephone offices. Table 3 Weighting Structures of sharply reduced DMUs in efficiency score Variable

MP OC NTL LR LDR IR OMLTF CS

DMU 10

DMU 24

DMU 27

Without weight restrictions

With weight restrictions

Without weight restrictions

With weight restrictions

Without weight restrictions

With weight restrictions

0.0942 0.2855 0.9114 10.274 0! 0 0 0.0731

0.3609 0.4511 0.4511 0.2446 0.4715 0.2446 0.1201 0.0818

0 1.7470 0 1.1418 0 0 0 0.0840

0.4325 0.5406 0.5406 0.2811 0.5622 0.2811 0.1646 0.0949

0 2.3118 0 1.5054 0 0 0 0.0640

0.5583 0.6979 0.4477 0.4284 0.4284 0.4284 0.1606 0.1247

!The 0 denotes e("10~5).

S.-H. Kim et al. / Computers & Operations Research 26 (1999) 59—72

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Table 4 T-test results for comparing the means of efficiency scores Criteria of Categorization

Without weight restrictions

With weight restrictions

MP NTL

1.65 2.14!

2.46! 3.02!

!The superscript represents that the values are greater than T-value (1.70) at a (significance level)"5% and l (degree of freedom)"30.

4. Conclusions This paper has presented a procedure for handling both linear partial data on factor values and its weight preferences within the DEA structure. The first formulation of the CCR model becomes a complicated NLP formulation because of the involvement of the sum-product forms of the unknown factor values and the unknown factor weights. In order to translate the NLP formulation into the corresponding LP, both a linear scale transformation and variable change technique were used. The variable change technique enabled to convert the product of two different variables (i.e. factor value and its weight) into the single attributed variable. It was necessary to impose additional restrictions on ordinal factor values and its weights in order to make a result more acceptable and practicable. For this purpose, we used ratio scale bounds for restricting ordinal factor values, and used weight restrictions that were the domain expert’s intuitive judgments on the relative importance of inputs and outputs. The procedure was applied to the efficiency evaluation of the telephone offices with three inputs and five outputs, where two of five outputs are ordinal factors. When the values of the ordinal factors are given in the forms of strict ranking, nineteen of thirty-three DMUs are efficient. By applying ratio scale bounds instead of strict ranking to the ordinal factors, efficient DMUs decreased to eleven, a result more sensible in the eyes of the evaluator. Imposing the additional weight restrictions for more stringent evaluation, the number of efficient DMUs reduced from eleven to six. Investigating the T-test results to check whether the size of telephone offices has any impact on efficiency, we found that small telephone offices are more efficient than large ones under the weight restrictions. This implies that large telephone offices are less productive on outputs than small ones. References [1] Ali AI, Cook WD, Seiford LM. Strict vs. weak ordinal relations for multipliers in data envelopment analysis. Management Science 1991;37/6:733—8. [2] Beasley JE. Comparing university departments. OMEGA International Journal of Management Science 1990;18/2:171—83. [3] Charnes A, Cooper WW. Preface to topics in data envelopment analysis. Annals of Operations Research 1985;2:59—94. [4] Charnes A, Cooper WW, Rhodes E. Measuring the efficiency of decision making units. European Journal of Operational Research 1978;2:429—44.

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