Multicomponent Efficiency Measurement and Shared Inputs in Data Envelopment Analysis: An Application to Sales and Service Performance in Bank Branches

Transcription

1 Multicomponent Efficiency Measurement and Shared Inputs in Data Envelopment Analysis: An Application to Sales and Service Performance in Ban Branches Wade D. Coo Schulich School of Business, Yor University, Toronto, Ontario, Canada M3J 1P3 Moez Hababou Schulich School of Business, Yor University, Toronto, Ontario, Canada M3J 1P3 Hans J. H. Tuenter Schulich School of Business, Yor University, Toronto, Ontario, Canada M3J 1P3 Abstract. In most applications of DEA presented in the literature, the models presented are designed to obtain a single measure of efficiency. In many instances however, the decision maing units involved may perform several different and clearly identifiable functions, or can be separated into different components. In such situations, inputs, in particular resources, are often shared among those functions. This sharing phenomenon will commonly present the technical difficulty of how to disaggregate an overall measure into component parts. In the present paper, we extend the usual DEA structure to one that determines a best resource split to optimize the aggregate efficiency score. The particular application area investigated is that involving the sales and service functions within the branches of a ban. An illustrative application of the methodology to a sample of branches from a major Canadian ban is given. Keywords: DEA, Bans, Sales and Service, Composite performance measures, shared resources Supported under CIBC project #[192251], funded jointly by the Canadian Imperial Ban of Commerce and The Natural Sciences and Engineering Research Council of Canada. c 2000 Kluwer Academic Publishers. Printed in the Netherlands. INPUTS-2.tex; 4/05/2000; 0:43; p.1

2 2 Coo, Hababou and Tuenter 1. Introduction Bans are presently evolving fromtheir traditional role as reactive monetary intermediaries, and service providers, toward a more general and proactive function as universal financial agents with a distinct sales culture. This new status has resulted in the introduction of a broad range of financial products to the maret place. Under the Canadian Ban Act of 1991, it became legal for an institution to engage in a broad range of financial activities. Technology has contributed as well to the changes that bans are undergoing; there is now a range of convenient customer access points such as ATMs (Automatic Teller Machines), debit cards, telephone- and PC baning, to name a few. Bans generate profits from two main sources (1) interest income, which captures the spread realized on loans and traditional activities, and (2) non-interest income from fees and financial services activities. While historically interest income was the principal source of profits for the ban, the importance of non-interest income has grown significantly over time. It is interesting to note that the profitability ratio, that is the profit as a percentage of assets, has increased dramatically since Specifically, for the period , the ratio ranged from0.24% to 0.79%, with an average of 0.43%; the corresponding figures for the period are 0.59% to 1.90% with an average of 1.20%. This dramatic change has been due in part to the revised regulations in the Ban Act, and partially to improved access to financial services, coupled with a more active sales orientation. Performance measurement, using tools such as Data Envelopment Analysis (DEA), as proposed by Charnes et al. (1978), has tended to concentrate on achieving a single measure for each m em ber of a set of decision maing units (DMUs). In most applications a single measure of production or profit efficiency, provided by the DEA methodology has been an adequate and useful means of comparing units and identifying best performance. This has been particularly true in the case of bans, where the primary candidates for DMUs are branches, and in their traditional setting, product and prices have tended to be undifferentiated. Numerous studies of ban-branch efficiency using DEA have been conducted over the past 15 years see, for instance, Charnes et al. (1990), Oral and Yolalan (1990), Schaffnit et al. (1997), Sherman and Gold (1985), and Sherman and Ladino (1995). There is now a desire to create value-added customer segments by identifying their specific needs. The new challenge is to optimize resource allocation, with most of the industry now allocating 60 80% of its human capital to customers and marets that represent less than 20% of its customer base. There is a growing need to view per- INPUTS-2.tex; 4/05/2000; 0:43; p.2

3 Multicomponent Efficiency Measurement and Shared Inputs in DEA 3 formance in a more disaggregated sense, paying specific attention to different components of the operation. These components include different classes of products or sales activities, such as mutual funds and mortgages, and different elements of service. By measuring a branch s performance on each of a set of such components, particular areas of strength and weaness can be identified and addressed, where necessary. In the sections to follow we present a model for deriving an aggregate measure of ban-branch performance, with accompanying component measures that mae up that aggregate value. The technical difficulty surrounding the development of an appropriate model has to do with the presence of shared resources on the input side, and mechanisms for allocating such resources to the individual components. The idea of measuring efficiency relative to certain subprocesses or components of a DMU is not new. Färe and Grossopf (1996), for example, loo at a multistage process wherein intermediate products or outputs at one stage, can be both final products and inputs to later stages of production. Those authors are not explicitly interested in obtaining measures of efficiency at each stage, but rather are concerned with overall efficiency measurement, whereby the networ structure of the intermediate activity explicitly enters into the model description. Hence, they are able to provide a better representation of the technology than would a blac box input and final output model, see Färe and Grossopf (1996). Another example is due to Färe and Primart (1984) and involves the evaluation of efficiency of a set of multiplant firms as DMUs, while at the same time measuring the efficiency of plants within firms. Along the same line, Coo et al. (1998) develop a DEA model for hierarchically arranged DMUs. These applications of multicomponent efficiency measurement do not involve shared resources as does the situation examined herein. The wor of Beasley (1995) on separating teaching and research, most closely compares to the present application, although we show herein that our treatment of shared resources leads to a linear rather than a nonlinear model. Section 2 begins with the basic idea of a production model as a ratio of aggregate outputs to aggregate inputs, with fixed input and output prices. This concept is then extended to a flexible (multiplier or price) optimization model, as in the original DEA methodology, pioneered by Charnes et al. (1978). Section 3 modifies the conventional DEA model for ban-branch performance by providing a methodology for splitting shared inputs among the identified components. For development purposes, we concentrate on two specific components, namely service specific and product specific sales activities. Section 4 discusses INPUTS-2.tex; 4/05/2000; 0:43; p.3

4 4 Coo, Hababou and Tuenter approximation procedures for solving the (nonlinear) model, and applies the technique to an example involving 20 branches. Discussion and conclusions appear in Section A Production Model for Ban-Branch Performance Measuring Performance with Fixed, Known Multipliers In evaluating the performance of ban branches in terms of production efficiency, it is recognized that each branch produces multiple outputs and consumes multiple inputs. This is depicted in Figure 1. X.. Branch.. Y Figure 1. Production process for an individual DMU Seen as a production process, one measure of performance for any branch is given by the aggregate output divided by the aggregate input. In technical terms, if Y =(y j )=(y 1,y 2,...,y s ) is the vector of s outputs, and X =(x i )=(x 1,x 2,...,x r ) the vector of r inputs for branch, one can represent the efficiency (output per unit of input) by the expression s r e = µy /νx = µ j y j / ν i x i, where µ =(µ 1,...,µ s )andν =(ν 1,...,ν r ) are vectors of multipliers for outputs and inputs, respectively. In a production setting, µ j would represent the processing time per unit of output j, andν i the price per unit of resource i. As an example, suppose all branch transactions are classified into two groups, sales and service, with annual sales Y 1 = 400(thousands) units and service Y 2 = 450(thousands) units. Furthermore, unit processing times are given by µ 1 = 0.73(hours) per sales transaction, and µ 2 =0.51(hours) per service transaction. Assume as well, that there are two inputs, say two types of staff, in amounts X 1 =10andX 2 = 5. Let annual salaries for these two types be given by $30(thousand) and $35(thousand) per FTE. Then, the performance rating for the branch would be computed as e =( )/( ) = 521.5/475 = 1.09 It should be pointed out that in place of the ratings e,asmeasuresof performance of branches, one could as well use the relative ratings j=1 ē = e / max {e }. i=1 INPUTS-2.tex; 4/05/2000; 0:43; p.4

5 Multicomponent Efficiency Measurement and Shared Inputs in DEA 5 Specifically, if we scale each rating down by dividing it by the rating for the best performing branch, the resulting scaled values ē provide an equivalent relative picture of performance. In this manner, all ratings are at or below 1, with the best performing branches having a score of 1. Measuring Performance when Multipliers are Flexible In case multipliers µ and ν are unnown, or can be estimated only within some limits, the above definition of performance must be modified. Charnes et al. (1978) proposed measuring the relative efficiencies of a set of K decision maing units by solving, for each DMU the mathematical programming problem e o = m ax µy o /νx o (1a) st µy /νx 1, (1b) µ j,ν i ε, i, j (1c) The essential idea behind this model is to afford each branch the opportunity to present its efficiency picture in the most favorable light possible. Hence, each DMU is allowed to choose multipliers that maximize its efficiency score. This (input oriented) model assumes constant returns-to-scale, and provides a radial measure of efficiency. As indicated earlier, it is reasonable to use measures of efficiency which have been scaled not to exceed unity, hence constraints (1b). In their original formulation, Charnes et al. (1978) used the infinitesimal ε to bound all multipliers µ and ν away fromzero, as indicated by (1c). Other limitations on permissible multipliers may be imposed. Other radial models have been developed, including output-oriented models, wherein one minimizes the ratio of aggregate inputs to aggregate outputs; the choice of models depends upon what is being measured at the time. Baner et al. (1984) have extended this methodology to permit variable returns-to-scale. 3. A Multicomponent Performance Measurement Model With the increased emphasis on sales and the differentiation of products and customer segments, there is a need to provide a performance measurement tool with component-based information as part of the aggregate efficiency score. INPUTS-2.tex; 4/05/2000; 0:43; p.5

6 6 Coo, Hababou and Tuenter Multiple Functions and Shared Resources While one may wish to measure the performance of several components of the DMU, we will, for purposes of development in this paper, assume that transactions can be separated into exactly two distinct classes: service and sales. It should be emphasized that this split is not always transparent; the opening of a mortgage loan would generally be classified as a sales transaction, although there are service activities that must be performed from time to time pertaining to that loan, such as loan renewal. Thus, a particular transaction may contain both sales and service components. Care should, therefore, be exercised in clearly delineating those activities that belong to each function. Furthermore, one would generally need to separate those sales activities that are volume related (and pertain to specific products), from those that involve the selling part of the sales activities. The latter would include reviewing customer portfolios, answering customer requests on various products, and so on. The former would involve the transaction tass performed after the customer has chosen a particular product. In summary, the selling aspect of sales does not relate to specific sales products while the transaction part of sales is product specific. In this paper we consider only those sales activities that are product or volume specific. We tae up the non-volume related activities in a later paper. For notational purposes, let (Y 1,Y2 ) denote the sets of service and sales transactions, respectively, i.e. the two sets of outputs are Y 1 =(y 1 1,...,y 1 J 1 ) and Y 2 =(y ,y 2 J 2 ). On the input side, this split is more complex. Some resources can be designated as dedicated service inputs, some as dedicated to sales, and still others are shared by the two functions. If, for example, branch staff are classified as Sales, Service, and Support, we can, for illustrative purposes, assume that Support staff are shared by the two functions while the other two classes are dedicated. In some branches this distinction may be less clear than in others. Technology resources may as well be classified as shared. A revised schematic of the production process for a particular DMU is given in Figure 2. Here, X 1,X2 and X s denote I 1,I 2 and I s -dimensional vectors of service dedicated-, sales dedicated-, and shared inputs, respectively. Some portion α i (0 α i 1) of the shared resource x s i is allocated to the service function of DMU, with the remainder (1 α i ) being allocated to sales. In the model to be developed in the following section, α i is a decision variable to be set by the DMU. At least two difficulties arise in attempting to capture a measure of performance of the DMU on both service and sales functions within INPUTS-2.tex; 4/05/2000; 0:43; p.6

7 Multicomponent Efficiency Measurement and Shared Inputs in DEA 7 Service X 1 X s X Branch X 1 X s X 2 X s.... Sales Figure 2. Production Process for a DMU with Shared Resources.... Y 1 Y 2 some overall efficiency measure. First, if one attempts to derive an overall measure of performance that somehow incorporates sales and service components, the importance of the components of X s relative to one another and relative to the dedicated resources X 1 and X 2 (as reflected in the ν-vectors ν 1,ν 2 and ν s ), may be different when considering the impact of X s on Y 1 as compared to its impact on Y 2.For example, consider the simple case of one staff type for each dedicated class (X 1 =no.servicestaff,x 2 = no. sales staff), and two resources, support staff and available technology, as shared inputs. One may argue that in evaluating service efficiency, technology is more important than support staff. As an example, a constraint such as ν2 s 2νs 1 might be imposed. On the other hand, if technology such as ATMs play a minor role in sales, then a constraint such as ν2 s 0.3νs 1 may be an accurate reflection of the importance of the two shared resources relative to one another. Clearly, these constraints are infeasible if imposed simultaneously. Moreover, even if this issue could be resolved, there would be no clear way of separating the resulting aggregate measure into separate sales and service indicators. A second difficulty arises if instead of developing an aggregate measure, one attempts to derive separate measures of performance relative to sales and service, with the intention of combining these separate measures into an aggregate score after the fact. The problem here is that the shared resources X s would need to be apportioned to these two functions in some manner consistent with their usage in creating the outputs of the functions. With any shared resources, however, branches do not generally maintain a record of the usage split at the function level. Consequently, a mechanism is needed to split shared resources across functions in some equitable manner. To motivate the development, reconsider Figure 2, but with the shared resources X s allocated to the two functions according to pro- INPUTS-2.tex; 4/05/2000; 0:43; p.7

8 8 Coo, Hababou and Tuenter Service X 1.. X 1 αx s.... Y 1 X s.. Branch X 2 X 2.. (1 α)x s Sales.. Y 2.. Figure 3. Splitting Shared Resources portionality variables α 1 i, as depicted in Figure 3. The issue of how α i should be derived is discussed below. Let α =(α 1,α 2,...,α Is ) T denote the column vector of proportionality variables, and let αx s denote the column vector (α 1 x s 1,α 2x s 2,...,α I s x s I s ) T. Further, we let (1 α)x s denote the column vector ( (1 α 1 )x s 1, (1 α 2)x s 2,...,(1 α I s )x s ) T I s. The Aggregate Performance Measure FromFigure 3 one can argue that since the total bundles of outputs Y 1 and Y 2 are produced fromthe inputs X 1,X2 and X s,ameasureof aggregate performance e a can be represented by: e a = µ 1 Y 1 + µ2 Y 2 ν 1 X 1 + νs 1 (αx s )+ν s 2 ((1 α)x s )+ν 2 X 2. (2) For this representation the vectors of multipliers µ l and ν t would be determined in a DEA manner to be discussed below. The rationale for allowing for the possibility of different vectors ν s 1 and ν s 2 for the shared service and sales resources, respectively, is that the relative importance of the components of X s in generating Y 1 may be different than their importance in generating Y 2. This was discussed earlier. In this manner, we avoid the possibility of infeasibilities created by possibly conflicting restrictions on the multipliers ν s. There is yet another rationale for permitting ν s 1 and ν s 2 to be different multiplier vectors. It can be argued that normally in a DEA analysis there is no clear connection between subsets of outputs and subsets of inputs. In this event, it is certainly the case that ν s 1 and ν s 2 should be the same vectors since they pertain to the same inputs (for example, support staff). When a direct lin can be made between such subsets of INPUTS-2.tex; 4/05/2000; 0:43; p.8

9 Multicomponent Efficiency Measurement and Shared Inputs in DEA 9 input and output bundles, however, one might then attempt to impose some form of lining constraints as discussed in earlier literature. We do this in the model discussed below. Such constraints may only be feasible if ν s 1 and ν s 2 are, in fact, permitted to be different vectors. Function-Specific Performance Measures From e a, performance measures for DMU that capture service and sales efficiency would appear to be appropriately represented by e 1 and e 2, respectively, as defined by: e 1 = µ 1 Y 1 ν 1 X 1 + νs 1 (αx s ) (3) and e 2 = µ 2 Y 2 ν s 2 ((1 α)x s )+ν 2 X 2. (4) PROPERTY 1. The aggregate performance measure e a combination of the service and sales measures. is a convex Specifically e a = β e 1 +(1 β )e 2,whereβ is the portion of all inputs utilized in e 1 (applied to the service component), i.e. β = [ν 1 X 1 + νs 1(αX s )] [ν 1 X 1 + νs 1 (αx s )+ν s 2 ((1 α)x s )+ν 2 X 2 ]. The aggregate measure is, therefore, a weighted average of the performance across the various functions of the organization, as one would intuitively expect. Fromthis property it is seen that a DMU will be deemed efficient, if and only if it is efficient in both service and sales components. Again we point to the importance of separate vectors ν s 1,ν s 2 being permitted in the aggregate measure (2). If ν s 1 and ν s 2 are forced to be the same in (2), yet are permitted to be different in (3) and (4), then no connection between the aggregate and function-specific measures, as per Property 1, can be made. Derivation of e a,e1,e2 The defined measures are based upon proportionality variables α which will be treated as DMU-specific variables. Thus, it will be at the discretion of each DMU to allocate X s across the two functions. Furthermore, the model will mae the necessary provisions to ensure that all three measures are appropriately scaled, specifically they will not exceed unity. INPUTS-2.tex; 4/05/2000; 0:43; p.9

10 10 Coo, Hababou and Tuenter Consider the following mathematical programming model: max e a o Subject to: e a 1, e 1 1, e 2 1, 0 α i 1, i (5) (µ 1,µ 2 ) Ω 1 (ν 1,ν s 1,ν s 2,ν 2 ) Ω 2 µ 1 j,µ2 j,ν1 i,ν2 i,νs 1 i,ν s 2 i δ, i, j In this formulation, the objective is to maximize the aggregate efficiency rating for each DMU o, while ensuring that the function level ratings (for sales and service) do not exceed 1. We replace ε by δ here to denote the fact that an absolute lower bound δ may be in effect. The sets Ω 1 and Ω 2 are assurance regions (see Thompson et al. (1990)) defined by any restrictions imposed on the multipliers. Similar wor was done by Beasley and Wong (1990). The set Ω 1 may, for example, contain ratio constraints on the components µ 1 j and µ2 j (the output multipliers), dictated by ranges on transaction processing times. The region Ω 2 would be defined by any restrictions expressing the relative importance of the various inputs pertaining to their impacts on outputs. More will be said regarding such assurance regions later. In general, (5) is a constrained version of the original model of Charnes et al. (1978) wherein lining constraints that connect output and input bundles are present. An Alternative Formulation Model (5) can be reduced to a non-ratio format in the usual manner of Charnes and Cooper (1962). Specifically, mae the transformation t = ( ν 1 X 1 o + νs1 (αx s1 o )+νs2 ((1 α)x s2 o )+ν2 X 2 o ) 1, and let v = tν and u = tµ. It is noted that due to scaling we may restrict the virtual input (i.e., 1/t) to be bounded above by some constant C. Without loss of generality we may choose C=1. Thus, t 1, and because of the multiplier restrictions u, v δt, it is advantageous to choose t as small as possible. Thus, for all practical purposes we may set t = 1, and retain the same variables ν and µ in the non-ratio model as were used above in the ratio format. Proceeding in the manner of INPUTS-2.tex; 4/05/2000; 0:43; p.10

11 Multicomponent Efficiency Measurement and Shared Inputs in DEA 11 Charnes and Cooper (1962), our model (5) can be expressed in the form e a o = m ax µ 1 Yo 1 + µ 2 Yo 2 s.t. ν 1 Xo 1 + νs 1(αXo s1)+νs 2((1 α)x s 2 o )+ν2 Xo 2 =1 µ 1 Y 1 + µ2 Y 2 ν1 X 1 νs 1 (αx s) νs 2 (1 α)x s ν2 X 2 0, µ 1 Y 1 ν1 X 1 νs 1(αX 2) 0, µ 2 Y 2 νs2 ((1 α)x 2) ν2 X 2 0, 0 α i 1, i (µ 1,µ 2 ) Ω 1, (ν 1,ν s 1,ν s 2,ν 2 ) Ω 2 µ j,ν j δ, i, j (6) Since α i is a decision variable, this problemis clearly nonlinear. If we mae the change of variables ν s1 = αν s1 and ν s2 =(1 α)ν s2,then problem(6) reduces to the following form: e a o = m ax µ1 Yo 1 + µ2 Yo 2 s.t. ν 1 Xo 1 + ν s 1 Xo s + ν s 2 Xo s + ν 2 Xo 2 =1 µ 1 Y 1 + µ2 Y 2 ν1 X 1 νs1 X s νs 2X s ν2 X 2 0, µ 1 Y 1 ν1 X 1 νs 1 X s 0, µ 2 Y 2 νs 2 X s ν2 X 2 0, 0 α i 1, i (µ 1,µ 2 ) Ω 1, (ν 1, ν s1, ν s2,ν 2 ) Ω 2 µ 1 j,µ2 j,ν1 i,ν2 i δ ν s 1 i α i δ, ν s 2 i (1 α i )δ (7) The formof Ω 2 depends upon how Ω 2 is structured. Clearly, if Ω 2 is the full real space, as is the case when no additional restrictions are imposed on the input multipliers, then (7) is a linear programming problem whose solution will immediately yield a solution to the nonlinear model (6). In the case that Ω 2 is a proper subset of the real space, defined by restrictions on the input multipliers, then (7) may or may not be linear. We consider various types of restrictions on the vectors ν, and their impact on the linearity of Ω 2, hence model formulation (7). Again, we point out that this model is similar to that developed by Beasley (1995) for analyzing the efficiency of universities in terms of teaching and research. In that case the same vector ν s was used INPUTS-2.tex; 4/05/2000; 0:43; p.11

12 12 Coo, Hababou and Tuenter for both functions (teaching and research), rather than allowing for different multipliers for vectors on the two components. As a result, Beasley s model does not have an LP equivalent. Types of Constraints in Ω 2 1. Absolute bounds on the components of (ν 1,ν 2,ν s 1,ν s 2 ) In the case of upper and lower bounds of the form δ 1 νi e δ 2, where e =1, 2,s 1,s 2,then Ω 2 will consist of linear restrictions since, for example, δ 1 ν s 1 i δ 2 becomes αδ 1 ν s 1 i αδ Share of total virtual input occupied by a particular subset of inputs Here, we might have constraints of the form ν s 1 (αx s ) ν s 1 (αx s )+ν s 2 ((1 α)x s ) c. Again, such constraints are linear and do not result in nonlinear restrictions in Ω Ratio constraints Restrictions of the cone-ratio variety, see Charnes et al. (1990), may result in nonlinearities in Ω 2, depending upon which components of the ν-vectors are compared. Specifically, cone-ratio restrictions that do not involve ν s 1 or ν s 2 will result in linear constraints in Ω 2, for instance the cone-ratio restriction νi 1 1 /νi 2 2 c can be rewritten as the linear constraint νi 1 1 cνi 2 2. Ratio constraints on the multipliers of the shared resources will render Ω 2 nonlinear. For example, restrictions of the form are transformed to α i1 ν s 1 i 1 α i2 ν s c α i 1 1 i 2 α i2 ν s 1 i 1 ν s 1 i 2 c, or ν s 1 i 1 ν s c α i 1, 1 i 2 α i2 in order to tae account of the sharing of resources between sales and service activities. Special Cases The extent to which both shared and dedicated resources exist can vary fromone situation to another. There can be special circumstances where, for example, there are no dedicated resources and all resources INPUTS-2.tex; 4/05/2000; 0:43; p.12

13 Multicomponent Efficiency Measurement and Shared Inputs in DEA 13 are shared. This does not change the general structure of the constrained DEA model (5), nor the requirement that component measures must fall out of the results. One special case is worth noting, namely, when no shared resources are present, and only resources dedicated to the separate components are involved. In this situation, (5) is completely separable in the sense that one can derive the individual component measures e 1 o and e 2 o by two separate DEA analyses; one for sales and one for service. The overall aggregate measure e a o is then a convex combination of these two measures. In the following section an application of this multi-component model to a set of ban branches is provided. Due to the presence of ratio constraints of this latter type in the example, the resulting model is nonlinear. In a practical setting with a large number of ban branches to evaluate, solving a quadratic programming problem for each would probably prove to be problematic. A linear relaxation of this nonlinear model is discussed, and outputs from the example are presented. Such a relaxation would prove to be more tractable in the situation where many DMUs are present. 4. An Application The model presented herein evolved from an earlier conventional DEA study of branch efficiency in a major Canadian ban. A total of approximately 1300 branches was involved, with the aim of the study being to identify benchmar branches for purposes of establishing cost targets. While data on several hundred different transactions is available from ban records, thirteen of the major ones (some grouped) account for approximately 80% of branch worload, and were used as outputs in the analysis. The only inputs considered in that study were personnel counts. Time studies were conducted previously on a small sample of typical branches, and provide ranges on unit processing times for all transactions. These ranges were the basis for the cone-ratio constraints on output multipliers for the DEA runs performed. One result of the aforementioned study was that members of the branches identified as being efficient were those that were primarily service oriented units specifically those with low levels of activity on the sales side while being very efficient in terms of routine counter transactions. The clear desire of the organization was a methodology that could provide a measure of performance on both components as well as an overall efficiency score. In this way one can identify not only those branches that are underperforming, but also the component that is weaest. INPUTS-2.tex; 4/05/2000; 0:43; p.13

14 14 Coo, Hababou and Tuenter Table I. Input- and Output measures used in an application of the model Inputs Outputs FSE # service staff MDP # counter level deposits FSA # sales staff MTR # transfers between accounts FSU # support staff RSP # retirement savings plan openings FOT# other staff MOR # mortgage accounts opened The model discussed in Section 3 was applied to a dataset of 20 branches out of the full set of ban branches. These were all chosen from one district. For purposes of illustration only, a subset of transaction types was chosen as outputs, and only personnel counts were used as inputs. The chosen input- and output measures that are used in the application are summarized in Table I. The relevant data for a one year period is displayed in Table II. Restrictions imposed To provide for a realistic picture of branch performance, a number of restrictions were imposed: Type 1: Ratio constraints on multipliers Ratio constraints of the form a µ i1 /µ i2 b on output multipliers were imposed to reflect processing times. Ratio constraints on the shared input multipliers were applied to reflect the relative importance of the two inputs (support and other staff) that are split between sales and service. Type 2: Limitations on α i It is generally the case that some bounds need to be imposed on the fraction α i of shared resource i being allocated to service activities. For illustrative purposes the range 1/3 α i 2/3 was chosen. Type 3: Constraints on the ratios of total service inputs to total inputs Here constraints are imposed to restrict the portion of virtual inputs being allocated to the service component. Recalling the definition of β in Property 1, restrictions were imposed on the range over which β could vary. For present purposes the limits 1/3 β 2/3 were applied. While the same limits were used for all branches in the example herein, it may be the case that different ranges would apply to different classes of branches. Large urban branches may allocate different mixes of resources to sales than small or mid-size branches. INPUTS-2.tex; 4/05/2000; 0:43; p.14

15 Multicomponent Efficiency Measurement and Shared Inputs in DEA 15 Model Relaxation The model presented in the previous section is nonlinear in the presence of ratio constraints (Type 1) on shared input multipliers. Specifically, when we impose constraints a ν s 1 1 /νs 1 2 b, these tae the form a α 1 νs1 α 2 ν s 1 1 α 1 2 α 2 b α 1 α 2 in the presence of the transformation discussed in Section 3. To render the model more tractable, various linear relaxations are possible. One approach attempted was iterative. Specifically, in the first stage all α i are assumed to be equal for any given branch (i.e., α i = d,asingle variable), and the resulting linear problemwas solved to determine a starting solution. This yields an optimal solution (µ (1),ν (1),α (1) ). Fixing µ = µ (1) and ν = ν (1), the second stage derives a best set of α i (2) relative to the constants µ (1) and ν (1). In subsequent stages one alternately fixes either α (n) or the pair (µ (n),ν (n)), and optimizes (5) on the other. One of the difficulties encountered with this method was that many iterations were required in order to converge to a solution that was reasonably close to the optimum. An alternative and somewhat more practical method was investigated. This amounted to choosing a grid of points in each α i range. In the present case, each of the two α i ranged from0.25 to 0.75 and the grid of 5 values 0.25, 0.35, 0.45, 0.65, 0.75 was used. Recall that α 1 is the percentage of support staff allocated to service transactions and that α 2 represents the split of other staff. This resulted in 5 5=25 different combinations for the tuple (α 1,α 2 ). Given the relatively small sample size in this particular example (20 DMUs), the problemcan easily be treated directly in its nonlinear form, and was solved using a standard spreadsheet solver. Results A proper evaluation of data such as that in Table II is complicated by the fact that the sales component is a two-level process as discussed earlier. The ranges for average processing times, as reflected in the cone-ratio constraints imposed upon the output multipliers, pertain only to the second of these two levels, namely the transaction part of sales. These average times do not account for the level of effort required to transact the sale. This effort would involve activities such as interaction with customers, review of portfolios, etc. To compensate for the understated values of the µ j components, one must either scale INPUTS-2.tex; 4/05/2000; 0:43; p.15

16 16 Coo, Hababou and Tuenter up these values, or adjust (downward) the resources (inputs) alloted to the sales component. The latter option becomes problematic in that the portion of sales resources not allocated to the transaction part of sales is left as unassigned inputs (i.e., they appear to not contribute to any of the outputs). In the present situation, the former option of scaling up the sales output multipliers was chosen. The scaling factor γ, defined as the ratio of the Total Sales effort to the Transaction effort was based on an estimate provided by the organization. The ranges provided for µ j,namelya µ j b, were replaced by scaled ranges γa µ j γb. The resulting aggregate, service and sales efficiency scores are displayed in Table III. It is noted that only one of the branches (#11), is efficient in the aggregate sense, that is in both sales and service. Clearly, branches may be efficient in one component only, such as is the case for branches #12 and #18. The respective α 1 and α 2 values are also shown. 5. Conclusions The DEA model presented here can be used for the analysis of any real situation where a significant number of inputs and outputs are included, but where management views the production process as having only a few maret contact activities. This solution allows for proper allocation of shared resources to output components chosen by management. The provision of component efficiency scores facilitates managerial action only on those components where the DMU is underperforming. While a DMU may only be efficient in an aggregate sense if all components operate efficiently, the application of this DEA model should save considerable managerial effort by concentrating on the real problemareas. Hence, decisions based on these results should be easier to accept by the DMUs in the field. With the new emphasis on sales within the baning industry, there is a need to view performance along component lines. While an aggregate measure of performance is useful for identifying best overall achievement, separate measures for different services within the organization may be essential, if particular areas of strength and weaness are to be highlighted. This is particularly true, not only of the sales armof the ban branch, but also of the different elements of the sales component, such as different classes of investment products. A technical complexity associated with multi-component efficiency measurement is the presence of shared resources. When it is unclear as to the precise allocation of such resources among a set of component services, it is necessary to provide a mechanism for splitting such re- INPUTS-2.tex; 4/05/2000; 0:43; p.16

17 Multicomponent Efficiency Measurement and Shared Inputs in DEA 17 sources in some rational way. A nonlinear optimization model is given for accomplishing this split. While this paper has concentrated on pure transaction or volume related components, future research will examine non-volume related sales activities. References Baner, R. D., A. Charnes, and W. W. Cooper: 1984, Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science 30(9), Beasley, J. E.: 1995, Determining Teaching and Research Efficiencies. Journal of the Operational Research Society 46(4), Beasley, J. E. and Y.-H. B. Wong: 1990, Restricting weight flexibility in Data Envelopment Analysis. Journal of the Operational Research Society 41(9), Charnes, A. and W. W. Cooper: 1962, Programming with linear fractional functionals. Naval Research Logistics Quarterly 9, Charnes, A., W. W. Cooper, Z. M. Huang, and D. B. Sun: 1990, Polyhedral Cone- Ratio DEA Models with an Illustrative Application to Large Commercial Bans. Journal of Econometrics 46(1-2), Charnes, A., W. W. Cooper, and E. L. Rhodes: 1978, Measuring the Efficiency of Decision Maing Units. European Journal of Operational Research 2(6), Coo, W. D., D. Chai, J. Doyle, and R. Green: 1998, Hierarchies and groups in DEA. Journal of Productivity Analysis 10(2), Färe, R. and S. Grossopf: 1996, Productivity and intermediate products: a frontier approach. Economic Letters 50(1), Färe, R. and D. Primart: 1984, Efficiency measures for multiplant firms. Operations Research Letters 3(5), Oral, M. and R. Yolalan: 1990, An Empirical Study on Measuring Operational Efficiency and Profitability of Ban Branches. European Journal of Operational Research 46(3), Schaffnit, C., D. Rosen, and J. C. Paradi: 1997, Best Practice Analysis of Ban Branches: an Application of Data Envelopment Analysis in a Large Canadian Ban. European Journal of Operational Research 98(2). Sherman, H. D. and F. Gold: 1985, Ban Branch Operating Efficiency: Evaluation with Data Envelopment Analysis. Journal of Baning and Finance 9(2), Sherman, H. D. and G. Ladino: 1995, Managing Ban Productivity Using Data Envelopment Analysis (DEA). Interfaces 25(2), Thompson, R. G., L. N. Langemeier, C.-T. Lee, E. Lee, and R. M. Thrall: 1990, The Role of Multiplier Bounds in Efficiency Analysis with Application to Kansas Farming. Journal of Econometrics 46(1-2), INPUTS-2.tex; 4/05/2000; 0:43; p.17

18 18 Coo, Hababou and Tuenter Table II. Branch Data for a selection of 20 ban branches service outputs sales outputs inputs shared inputs DMU MDP MTR RSP MOR FSE FSA FSU FOT INPUTS-2.tex; 4/05/2000; 0:43; p.18

19 Multicomponent Efficiency Measurement and Shared Inputs in DEA 19 Table III. Efficiency Scores and optimal split of shared resources Efficiencies Shared Staff Aggregate Service Sales Support Other DMU e a e 1 e 2 α 1 α l INPUTS-2.tex; 4/05/2000; 0:43; p.19

20 INPUTS-2.tex; 4/05/2000; 0:43; p.20