A data envelopment analytic model for assessing the relative efficiency of the selling function

European Journal of Operational Research 53 (1991) 189 205 North-Holland

189

Theory and Methodology

A data envelopment analytic model for assessing the relative efficiency of the selling function Jayashree

Mahajan

Department of Marketing, University of Arizona, Tucson, A Z 85721, USA Abstract: This paper presents a data envelopment analytic model for assessing the relative efficiency of sales units that simultaneously incorporates multiple sales outcomes, controllable and uncontrollable resources, and environmental factors. The model enables comparisons among a reference set of sales units engaged in selling the same product/service by deriving a single summary measure of relative sales efficiency. In addition, it provides insights into modifications that are necessary in order to enhance relative efficiency of an individual sales unit. Conditions when the sales unit has additional control over resources are explored and the effects on relative efficiency are examined. An illustration of the model in the context of sales units from a sample of insurance companies demonstrates the critical features of the model. Keywords: Marketing, sales performance, DEA, fractional programming

1. Introduction In recent years there has been a substantial body of research that has addressed a variety of critical decisions necessary for managing the selling function. These include such fundamental questions as: How many salespeople should we employ? What is the most efficient approach to territory design? How should sales effort be allocated? What kind of compensation plans should be i m p l e m e n t e d ? Should external or internal salespeople be used? The thrust of this research effort has been to ensure that the selling function is efficiently managed as some aspect of efficiency is inherent in most theoretical frameworks and models in the sales management literature (e.g., Anderson, 1985; Basu et al., 1985; Beswick and Received September 1988; revised November 1989

Cravens, 1977; Lal and Staelin, 1986; Lodish, 1975; Ryans and Weinberg, 1979; Zoltners and Sinha, 1983). Despite this underlying goal, the crucial issue of how sales efficiency should be measured has not been examined. This issue is important as, in addition to having an indirect impact on the above decisions, the assessment of sales efficiency is central to developing control systems for evaluating the salesforce (Anderson and Oliver, 1987). Further, the selling function consumes a sizeable portion of the marketing budget, so that the assessment of sales efficiency is relevant in determining whether these resources are being used productively. Hence, it would be useful to assess whether the selling function is being performed efficiently and, if necessary, to determine what specific changes are required to improve inefficient performance. There have been a number of studies that have

0377-2217/91/$03.50 '~; 1991 - Elsevier Science Publishers B.V. (North-Holland)

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J. Mahajan / A data envelopment analytic modelfor the sellingfunction

examined the issue of measuring marketing productivity (e.g., Bucklin, 1978; Bonoma and Clarke, 1987); however, there has been little research that has explicitly focused on measuring sales productivity. The only study that examines this issue is Behrman and Perrauit (1982); however the emphasis is on generating a multi-item scale for measuring sales performance at the individual salesperson level. In practice individual or multiple sales outcomes such as actual sales, gross margin, contribution to profit, or market share have been widely used. However, the use of individual or multiple outcomes creates difficulties when sales performance is better than average according to some outcomes and poorer than average according to others. An alternative approach is to develop a composite indicator of multiple performance criteria although this necessitates weighing a large number of conflicting criteria. Further, ratio analysis, such as a sales cost ratio, has limited use when factors that are not under the control of management but impact performance (e.g., low territory potential) need to be considered. Least square estimation methods such as regression analysis are not appropriate when analyzing outliers such as maximally efficient or inefficient sales organizations; and, they do not provide an indication of the changes that are necessary for an inefficient sales organization to become efficient. In sum, dispite the existence of the variety of approaches, there is little consensus as to the appropriate method to use or what factors need to be considered, " . . . sales managers are not universally agreed upon any corresponding measurement (of efficiency), or even upon which activities should be considered as input and output" (Henry, 1975, p. 88). Given the importance of assessing sales efficiency and the limitations associated with prior approaches it would be useful to develop an approach which would provide a comprehensive measure of efficiency (i.e., one which incorporates multiple a n d / o r conflicting criteria). Additionally, such an approach should also help to diagnose causes of inefficiencies and suggest ways by which these inefficiencies can be reduced. A contemporary management science approach for the assessment of relative efficiency is Data Envelopment Analysis (DEA) (Charnes, Cooper and Rhodes, 1978). DEA converts multiple inputs and outputs into

a single summary measure of performance for a decision making unit (DMU) by utilizing a fractional linear program. In order to assess the relative efficiency for a specific DMU, the values of inputs and outputs for similar DMUs are evaluated without requiring any a priori judgement as to their relative importance. Further, DEA does not impose any restriction on the functional form of the relationships between the inputs and outputs (i.e., it is assumed that all inputs are jointly used to generate a set of outputs). DEA is particularly appropriate for identifying inefficient DMUs as well as characterizing the nature of the production function for each DMU. For example, it has been demonstrated that ratio analysis performed poorly in detecting inefficiencies in comparison to DEA despite the use of numerous ratios (Sherman, 1984). A comparison of DEA and the translog cost function method indicated that DEA provides a richer and more diverse characterization of the production functions of hospitals as both increasing and decreasing returns to scale could be inferred. The translog method, on the other hand, suggested that only constant returns to scale were present (Banker, Conrad and Strauss, 1986). DEA also performs well in identifying the sources and amounts of inefficiencies that would be undetected using alternative methods such as regression analysis. A comparison of regression analysis and DEA demonstrated that, although both procedures perform well in their overall evaluation of efficiency for a set of hospitals, in the case of individual hospitals regression analysis did not perform as well as DEA in isolating the underlying sources and amounts of inefficiencies (Bowlin et al., 1985). An important difference between regression analysis and DEA is that while the former optimizes across all DMUs, DEA optimizes on each D M U using a series of optimizations. This difference suggests that while regression might be used when general characteristics of all DMUs are of interest, DEA is useful when considering an individual D M U in comparison to other similar units. Recent developments suggest that there have been a number of studies that focus on conceptual issues such as the treatment of fixed inputs and outputs (Banker and Morey, 1986a) and the use of categorical variables (Banker and Morey, 1986b). In addition, several successful applications of DEA

J. Mahajan / A data envelopment analytic model for the selling function

have been reported such as estimating efficient production surfaces (Banker and Maindiratta, 1986), evaluating economic planning models (Macmillan, 1986), examining the economic impact of information technology (Chismar and Kriebel, 1985), planning and control of school districts (Bessent et al., 1982) and evaluating the administrative efficiency of courts (Lewin, Morey and Cook, 1982). Note that while this approach has been advocated in the marketing literature (see Charnes et al. 1985b), as such this is one of the first studies to incorporate these developments in controlling and evaluating specific marketing activities. The remainder of the paper is organized as follows. The underlying assumptions and definition of terms necessary to formulate the proposed model are discussed in Section 2. In Section 3 the data envelopment analytic model for assessing sales efficiency along with analytical interpretations stemming from the model are presented. Section 4 examines the model and its interpretations under conditions of additional control over resources and Section 5 illustrates the model using data collected from a sample of insurance firms. Finally, the managerial implications of the study and potential research directions for the future are discussed in Section 6.

2. Assumptions and definitions Critical to developing a model for assessing the relative efficiency of the selling function are the theoretical underpinnings and assumptions that relate to the unit of analysis, concepts of relative efficiency, sales outcomes and resources and the issue of control over resources. These are discussed below. Definition of a sales unit. The selling function is conceptualized in terms of a sales unit. A specific sales unit b is defined as a unit that is organized to deliver a specific service, apply a functional expertise or serve a geographical area. For example, a branch operation, an agency or sales territory could be the sales unit under investigation provided the above criteria are met. Since efficiency is likely to vary depending on the specific context facing each sales unit, sales unit b is assumed to belong to a reference group of sales units (k = 1. . . . . K). The reference group of sales

191

units is defined as a set of sales units engaged in selling the same product or service. Concept of relative efficiency. Efficiency for a specific sales unit b is defined as a sales productivity measure that reflects the relationship between sales outcomes and resource needs for the sales unit. As an absolute measure of efficiency is difficult to evaluate without the existence of norms and standards, the emphasis is on relative efficiency (Zh), i.e., the efficiency of a sales unit in comparison to the reference group of sales units. Further, Pareto optimality conditions are assumed to underlie the notion of relative efficiency (i.e., a sales unit is relatively efficient in comparison to other similar units, if and only if, no other unit produces either a higher amount of sales outcomes with the same amount of resources, or uses a lesser amount of resources to generate the same amount of sales outcomes). Such optimality conditions seem reasonable as relative efficiency is to be derived for a comparable set of sales units. Given that each sales unit is likely to be characterized by differing returns to scale depending on such factors as external market conditions, the functional form of the relationship between the sales outcomes and resources could be difficult to specify a priori. Further, sales units characterized by increasing returns to scale could be relatively more efficient than sales units characterized by decreasing returns to scale so that efficiency may be distorted due to this aspect. Hence, adjusting for the 'scale effects' of sales unit b (/~) is essential in deriving a comparable efficiency measure across sales units. Specific sales resources and outcomes. In identifying the specific sales resources and outcomes an open systems view is adopted. According to classical organization theorists the view was that the organization (or their subunits) were relatively closed systems. As a result, the focus was on internal resources and inputs with little attention given to factors beyond the control of the unit. More recently, current views argue for an open systems view so that organizations need to be efficient in a 'larger' context (Miles, 1984). The implications of an open system perspective are that the external and internal (organization) environment are as much a source of constraints and contingencies for the efficiency of the unit as resources over which an individual unit has control. In this paper, an open systems view is as-

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J. Mahajan / A data envelopment analytic model for the selling function

s u m e d so that the relative efficiency of a sales unit is a function of resources which are c o n t r o l l e d b y the sales unit (i.e., c o n t r o l l a b l e resources); resources that are allocated from other subunits or from higher levels in the o r g a n i z a t i o n a l h i e r a r c h y (i.e., allocative resources) and, external c o n d i t i o n s ( e n v i r o n m e n t a l factors). This d i c h o t o m y between c o n t r o l l a b l e a n d u n c o n t r o l l a b l e resources has also been used in p r i o r research (e.g., in d e v e l o p i n g territory sales response; see R y a n s a n d W e i n b e r g , 1979). C o n t r o l l a b l e resources ( c = 1 . . . . . C ) include any resource inputs over which the sales unit has direct control such as the n u m b e r of p r o d u c t s handled, a m o u n t of advertising or types of incentives. Further, allocative resources ( a = 1 . . . . . A) include factors that are not directly c o n t r o l l e d by the sales unit but facilitate the selling effort of a p a r t i c u l a r sales unit (e.g., the extent of s u p p o r t p r o v i d e d by the m a r k e t i n g research d e p a r t m e n t ) . E n v i r o n m e n t a l factors (e = 1 . . . . . E ) reflect characteristics of the m a r k e t that i m p a c t p e r f o r m a n c e (e.g., the n u m b e r of c o m p e t i t o r s or m a r k e t p o t e n tial) a n d can only be influenced b y the strategic p o s i t i o n of the sales unit. Specific sales o u t c o m e s (i = 1 . . . . . N ) reflect the o u t p u t s g e n e r a t e d by the units' selling effort such as total sales or n u m b e r of new accounts. It is a s s u m e d that the set of sales units have i n f o r m a t i o n a b o u t specific sales outcomes ( ~ k ) , c o n t r o l l a b l e resources (Xck), allocative resources ( X l a k ) a n d e n v i r o n m e n t a l factors ( X 2 . k ) . In other words, the sales o u t c o m e s a n d resources are o b s e r v e d variables. The ' w e i g h t s ' associated with the sales outc o m e s (u,), c o n t r o l l a b l e resources (v,), allocative resources (vlu), a n d e n v i r o n m e n t a l factors ( v 2 e ) are the decision variables (i.e., d e t e r m i n e d from the model p r e s e n t e d in the next section) a n d are derived so as to present a specific sales unit b in the best possible manner. Issue of control over resources. It is a s s u m e d that there are c o n d i t i o n s when each sales unit can acquire control over resources that were previously allocated to it from other s u b u n i t s or higher levels in the organization. Hence, each sales unit w o u l d have c o m p l e t e discretion over these resources once they are allocated so that they can be c h a n g e d as necessary. Similarly, it is a s s u m e d that there are c o n d i t i o n s when the sales units can influence specific e n v i r o n m e n t a l c o n d i t i o n s due to its strategic p o s i t i o n in the market. N o t e that this

resource a l l o c a t i o n issue is distinct from how sales effort should be a l l o c a t e d to specific sales territories or p r o d u c t lines (Lodish, 1975; Z o l t n e r s a n d Sinha, 1983) a n d i n s t e a d focuses on if the sales unit should have c o n t r o l over resources that were previously m a n a g e d at a higher level in the organization hierarchy. These a s s u m p t i o n s a n d definitions form the basis of the m o d e l which is presented in the next section.

3. Development of the sales unit efficiency model The discussion in the earlier section suggested the following n o t a t i o n which is essential to developing the model. Let: N = N u m b e r of sales o u t c o m e s (i =

1 . . . . . N), A

= N u m b e r of allocative resource i n p u t s ( a = l . . . . . A), E = N u m b e r of e n v i r o n m e n t a l factors (e =1 ..... E), C = N u m b e r of c o n t r o l l a b l e resource inputs (c = 1 . . . . . C), K = N u m b e r of sales units ( k = 1 . . . . . K ) , ~k = O b s e r v e d value of sales o u t c o m e i for sales unit k, X l , k = O b s e r v e d value of allocative resource i n p u t a for sales unit k, X2,, k = O b s e r v e d value of e n v i r o n m e n t a l factor e for sales unit k, X, k = O b s e r v e d value of c o n t r o l l a b l e resource i n p u t c for sales unit k, b = Sales unit for which efficiency is b e i n g assessed. T h e decision variables are: Zh = Relative efficiency of sales unit b, u~ = ' W e i g h t ' for the sales o u t c o m e i, vl, = ' W e i g h t ' for allocative resource i n p u t a,

v2 e

= ' W e i g h t ' for e n v i r o n m e n t a l factor input e, vc = ' W e i g h t ' for c o n t r o l l a b l e resource inp u t c, Uh = T h e ' s c a l e effect' a t t r i b u t a b l e to sales unit b. In d e v e l o p i n g the m o d e l o n l y factors within the control of the sales unit (i.e., c o n t r o l l a b l e resource inputs) are directly considered, while u n c o n t r o l l a ble resources (i.e. allocative resource i n p u t s a n d e n v i r o n m e n t a l factors) are j u d g e d as having an

J. Mahajan / A data en~eloprnent analytic model for the selling function

indirect effect on relative efficiency. The objective of the model is to maximize the sales productivity for sales unit b which is the ratio of sales outcomes to the controllable resource inputs. This productivity ratio is adjusted by (a) ratio of uncontrollable sales resources to controllable resources, and (b) ratio of 'scale effects' to controllable resources. This is equivalent to adjusting the sales outcomes for the uncontrollable resources and 'scale effects' directly (see, for example Banker and Morey, 1986a). As noted earlier, the adjustment for 'scale effects' is necessary so as to derive a comparable efficiency measure. An adjustment for uncontrollable resources is essential as an open systems perspective is adopted so that both controllable and uncontrollable resources need to be considered in determining the relative efficiency of a sales unit. Hence, the relative efficiency for a specific sales unit needs to be adjusted for any inefficiencies attributable to the lack of control over allocative resources and environmental factors. As a resulL in the formulation presented below, the numerator of the objective function (equation (la)) for a sales unit b is the adjusted sales outcomes (for the allocative resource inputs, environmental factors and 'scale effects'). On dividing this numerator by the controllable resource inputs~ the objective function can be interpreted as maximizing the ratio of adjusted sales outcomes to controllable resource inputs utilizing by sales unit b. An alternative formulation of the same objective function would be as follows. In the numerator, the sales outcomes adjusted for scale effects could be included; while in the denominator, the controllable resource inputs adjusted for uncontrollable resource inputs and environmental factors could be included. Although this type of objective would be similar to the traditional D E A model objective (i.e., outputs divided by inputs), such a formulation is not in line with the definition of sales productivity used in this study (i.e., the ratio of sales outcomes to controllable resource inputs, as mentioned before). This definition appears to be logical since it assesses the rate of net outputs to each controllable resource input utilized for each sales unit and hence, measures the productivity over the controllable sales inputs. Further, note that this argument is also consistent with that of Banker and Morey (1986) in formulating the D E A model with exogenously fixed inputs.

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Constraint set (lb) restricts this ratio for all sales units (including unit b) to be less than or equal to 1 as it represents a sales productivity measure. This ensures that a feasible solution to the problem always exists such that relative efficiency for sales unit b will lie between a value of zero and one (i.e., 0 < Z h < 1). The strict positivity of the 'weights' to be derived for the sales outcomes (u i) and controllable resource inputs (u,) is enforced in constraint set (lc) by introducing an infinitesimal constant e (e.g., e = 10 ~'). This ensures that the sales outcomes and controllable resources have a direct impact on the relative efficiency of the sales unit. The non-negativity of the 'weights' to be determined for the allocative resources ( u l , ) and environmental factors (u2e) is incorporated in constraint set (ld). These weights are constrained to be non-negative so that values of v l = t,2,. = 0 are possible, a condition that corresponds to the assumption of 'free disposability' in economics literature ( K o o p m a n s , 1951). Finally, in constraint (le), G, is unrestricted in sign so as to ensure that increasing, constant and decreasing returns to scale can be incorporated in this formulation. The basic non-linear p r o g r a m m i n g ( N L P ) formulation of the model is, maximize

Zh=[ ~

,=l~ vl"Xl"h C e--I

J

c

1

subject to

E u,Y,k- E vl,,Xl,,k- E u2.X2~,k- Uh i=1

a=l

e=l

C

/~uX,

k_
for a l l k ,

(lb)

C=I

u i, c',.>e for all i, c, ul,,, u2,, >_0 for all a, U~, unrestricted.

(lc) e,

(ld) (le)

When deriving a solution to p r o g r a m N L P , the value of Z h is independent of the measurement units in which the observed sales outcomes, resources (controllable or allocative) and environmental factors are stated as long as the measure-

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d. Mahajan / A data envelopment analytic model for the selling function

ment units are identical for each sales unit (Charnes and Cooper, 1985). This model derives the relative efficiency for an individual sales unit b by determining the optimal 'weights' for the sales outcomes and sales resources (i.e., ui, v l , , 0 2 e and G). In other words, these 'weights' are determined so as to maximize the individual sales unit's efficiency subject to the condition imposed by constraint set (lb) (i.e., the relative efficiency of any other sales unit with these same 'weights' does not exceed 1). Hence, these 'weights' are specific to sales unit b (i.e., they are likely to vary across the set of sales units) and can be interpreted as the changes in relative efficiency that can be achieved due to a unit increase in a sales o u t c o m e or a unit decrease in a controllable sales resource (Lewin and Morey, 1981). At first glance the model seems to imply that the optimal values of the 'weights' associated with the uncontrollable resources (i.e., vl a and v2e) should be zero in order to maximize the objective function (equation (la)) for a specific sales unit b. However, since these 'weights' represent inefficiencies attributable to the lack of control over allocative resources and environmental factors, for some of the K sales unit under consideration they could be non-zero. This can be shown by comparing the formulation N L P with another formulation which does not include the uncontrollable resources. Such a comparison would show that the relative efficiency of the b using program N L P is greater than or equal to the relative efficiency of the same unit c o m p u t e d using a formulation which does not include the uncontrollable resources. This suggests that the 'weights' v l , and v2~ could be positive in a n u m b e r of solutions. Since program N L P is a non-linear, non-convex mathematical p r o g r a m it is computationally complex and hence, difficult to solve. However, by carrying out a transformation of variables as follows, an equivalent linear program can be derived as follows, Define: c g2= 1 /[g,.=,(v,x,.)l,

F, = ~2ui for i = 1 . . . . . N, /G = I2vl~ for a = 1 . . . . . A, Ce = I2v2e for e = 1 . . . . . E, O,=I2v, for c = l . . . . . C, q~b = I2Ub. Then multiplying the numerator and denominators in program N L P by I2 and adding the condi-

tion: I2[F.c=lvxCch c ] = 1, program L P can be formulated as follows: N

maximize

A

Z b = Y" F, Y,b -- ~ i=1

~ Xl,h

a=l E

-

E ~eX2eh-- ~h

(2a)

e=l

subject to N

A

E F,Y,k i=1

E

£ IGXlak - E ~egZek -- ~b a=l

e=l

C

< Y'~ VeX, k

for all k,

(2b)

c=l C

Y~ O,.X,+ = 1,

(2c)

c=l

Fi,O,.>_e

for a l l i , c,

(2d)

/ ~ , '~e > 0

for all a, e,

(2e)

~h

unrestricted.

(2f)

Program L P needs to be solved a total of K times so as to derive the relative efficiency of each sales unit (k = 1 . . . . . K). Since this p r o g r a m contains K + C + N + I constraints and A + E + C + N variables if A + E < K + I (i.e., the total n u m b e r of allocative resources and environmental factors is less then the total n u m b e r of sales units in the reference set), it is computationally attractive to solve the dual of p r o g r a m L P (i.e., D L P ) which is as follows: minimize

DZb = Qb - e

Si+ + ~ S~i=1

c=l

(3a) subject to K

Y'~ Y,kck -- Si+ > Y~h for all i,

(3b)

k=l K

Xl,k~ k < Xl~b

for all a,

(3c)

e,

(3d)

k=l K

Y'~ X2ek'rk <_ X2eb

for

all

k=l K

~_, Xck'rk + S~7 - QbX~+ <_0 k=l

for all c,

(3e)

J. Mahajan / A data envelopment analytic model for the selling function K

• ~ = 1,

(3f)

k=l

"ck, S, +, S,_ >_0 for a l l i , c, k,

(3g)

Qh unrestricted.

(3h)

Analytical interpretations The model developed in this section provides several analytical interpretations that enables: (a) the identification of the relatively efficient and inefficient sales units; (b) the isolation of the source of the inefficiencies (if they exist); and (c) the determination of the potential changes in the controllable resource inputs and sales outcomes that are necessary to make a relatively inefficient sales unit more efficient. First, sales unit b is relatively inefficient if and only if Zh* < 1 (or DZ~" < 1). This implies that all the efficient units (i.e., all k such that Zk* = 1) can obtain the same sales outcomes as unit b by using fewer controllable resources. Alternatively, this interpretation also suggests that the efficient sales units can secure higher sales outcomes by using the same controllable resources as the inefficient sales units. For example, either the efficient sales units generate the same dollar sales and net margin as the inefficient ones by using fewer salespeople and lower amounts of advertising; or the efficient sales units generate higher sales and net marginus using the same n u m b e r of salespeople and a m o u n t of advertising as the inefficient ones. Second, since the sales unit efficiency model makes no assumption about the functional form of the efficient frontier for any of the sales units, whether an individual sales unit is exhibiting increasing, decreasing or constant returns to scale can be examined (Banker, Charnes and Cooper, 1984). A sales unit exhibits increasing (decreasing) returns to scale if an additional unit of a resource input results in a greater (lesser) than proportional increase the sales outcomes; while a sales unit exhibits constant returns to scale when an additional unit of a resource results in a proportional increase in the sales outcomes. The returns to scale for sales unit b can be characterized by examining the optimal value of the dual variable associated with constraint (3f) (i.e., ~h*) as follows: (a) If ~h* < 0 ¢~ increasing returns to scale;

195

(b) If ~h* > 0 ,=, decreasing returns to scale; (c) If ~ ' = 0 ¢* constant returns to scale. Third, the optimal solution of p r o g r a m D L P also indicates the necessary increases in the sales outcomes and required decreases in controllable resources for an inefficient sales unit to become relatively efficient (Charnes and Cooper, 1985). This interpretation states that simultaneous changes are required in the sales outcomes and controllable resources to make a sales unit relatively efficient. This would be a viable in circumstances where the sales unit can perform well with little additional effort such as when a competitor is not active in an existing market or if the sales unit has a d o m i n a n t position in a market with high sales potential. The specific changes are as follows. Let Q~, S, +*, S [ * represent the optimal solution of program D L P for sales unit b such that DZ~" < 1 (i.e., unit b is relatively inefficient). Then for sales unit b to become efficient in comparison to the other sales units (i.e., so that DZt* = 1), the following simultaneous changes in the sales outcomes and controllable resources (estimated from constraint sets (3b) and (3e), respectively) are necessary: (a) Increase all sales outcomes i to new levels ~ such that: ~=

Y,h + S , +*

for a l l i ;

(b) decrease all controllable resource inputs c to new levels )(,!h such that:

X,!h = Q~ X, h - S , 7 *

for all c.

In a n u m b e r of instances the sales unit may not be in a position to decrease the use of these resources and at the same time increase the sales outcomes. Hence, it would be useful to examine the impact of a unit change in either an individual sales outcomes or a controllable resource on relative efficiency. This is examined in the next two interpretations. Fourth, in situations where the sales unit has several conflicting outcomes (such as the n u m b e r of new customers and the n u m b e r of repeat orders) that need to be achieved, it would be useful to identify specific outcomes that need to be increased and by how m u c h relative efficiency would change. An optimal solution to p r o g r a m D L P provides estimates of the changes in relative efficiency due to a unit increase in a single sales

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J. Mahajan / A dataenvelopment analytic model for thesellingfunction

outcome. This can be accomplished by examining the dual variables associated with constraint set (3b) (i.e., F,) which also represent the optimal 'weights' for the sales outcomes. Assume that for an inefficient sales unit that Si+* = 0 for any one sales outcome i; and F , * > 0 (i.e., the slack is equal to 0 for any one outcome in constraints set (3b) and that the dual variable associated with this constraint is positive). Then the efficiency of this inefficient unit can be increased at the rate F,* for a unit change in sales outcome i (Lewin and Morey, 1981). Note that this holds only for the range of R H S values for this constraint within which the basis remains unchanged. This is mathematically shown as follows. If DZ~" < 1 then: ~'h* = 0 ( s i n c e % is non-basic) ~ ,-,k:l'~x"~-* Y,k is constant for any i ~ an increase in Y,h reduces Si+* from constraint (3b) = an increase in DZ~'. Finally, of even greater significance is to identify the impact of a unit change in a single controllable resource input on the relative efficiency of the sales unit as this would provide guidelines on which resources should be controlled. This requires a reformulation of the above model so that the rate of controllable resources used per unit of net surplus generated for sales unit b is minimized. Hence, program N L P 2 formulated in Appendix 1 is the reciprocal version of NLP. Programs LP2 and D L P 2 (also presented in Appendix 1) are derived in a similar manner as programs LP and DLP, respectively. An optimal solution to program D L P 2 provides estimates of the changes in relative efficiency due to a unit increase in a single sales resource. This can be accomplished by examining the dual variables associated with constraint set (3b) (i.e., (9) which represent the optimal 'weights' for the controllable resources. Then, based on program DLP2, note the following. Assume that for an inefficient unit, S,7" = 0 for any controllable resource input ' c ' and that 69* > 0 (i.e., the slack is equal to 0 for any one controllable resource input in constraint set (7e) and the dual variable associated with this constraint is positive). Then the efficiency of this inefficient unit can be increased at the rate @,~' for a unit decrease in controllable resource input c. Once again, as in the case of the sales outcomes, this only holds for the range of right hand side values for this resource in constraint (Be) within which the basis remains unchanged. The model presented in this section has pro-

vided several insights as to the sources of relative inefficiency as well as the potential changes required to make a sales unit efficient. In the next section, the model is examined under conditions when the sales unit has control over resources that were previously considered uncontrollable.

4. The e f f e c t o f control over aliocative r e s o u r c e s In this section the impact of control over allocative resources on the relative efficiency of a sales unit is examined. It is assumed that control over allocative resources implies that the sales unit can alter the a m o u n t of such resources consumed rather than having the amounts fixed at a higher level in the organization. Hence, the situation when, due to the increasing emphasis on regional markets, each individual sales unit has control over allocative resources (i.e., decentralized decision making) is examined. For example, the market area covered or the a m o u n t of advertising performed could be determined at the individual unit level rather than at a higher level in the organization. In order to assess relative efficiency when the sales unit has control over resources that were previously allocated from higher levels or other subunits, program N L P 1 needs to be formulated as presented below. The formulation is similar to program N L P discussed earlier except that the allocative resources are treated like the controllable resources (see equation (4a) and constraint sets (4b) and (4e)). Hence, the objective function in this case (equation (4a)) focuses on maximizing the net surplus per unit of controllable and allocative resources for sales unit b; while constraint set (4b) restricts this value to be a m a x i m u m of 1 for all sales units. Constraint set (4c) enforces the strict positivity of the determined 'weights' for the sales outcomes as well as the controllable and allocative resources; while constraint set (4d) ensures the non-negativity of the d e t e r m i n e d 'weights' for the environmental factors. Finally, as with program NLP, Uh is unrestricted in sign. Maximize ZI,,=

~ u,Y,,,- Y[ v2eX2, h i=1

e=l

/ [ c.~-v'X''+Lvl'Xl~h], 1 .=~

Uh

] (4a)

J. Mahajan / A data envelopment analytic model for the selling function

subject to

u , < k - F., v2,X2,,~ - G i=1

e=l

/

vcXck + kc=l

vaXak

<_ 1

for all k ,

a~l

(4b) u,, v i, vl, >_ e

v2,.>_0

for all i, c, a,

for all e,

U/, unrestricted.

(4c)

197

and ZI~ = D Z I ~ . Note that D Z ~ < 1 in program DLP1 since sales unit b is itself one of the K reference units. However, an optimal solution to program DLP1 is a feasible solution to program DLP since the feasible region of the latter program encompasses the feasible region of the former program. Hence, by solving program DLP, we can achieve a lower efficiency rating for sales unit b (i.e., D Z ~ <_ DZI'~ ~ Z ~ < ZI~, ). E~

(4d) (4e)

Using a similar transformation of variables as before, program NLP1 can be reformulated as a linear program so that it is computationally tractable. Note that all the earlier analytical interpretations for programs NLP, LP and DLP also hold for this case. In addition, since the allocative resources are now controllable, a reformulation of this program will also indicate the increase in efficiency that can be achieved by a unit decrease in any one allocative resource input (to carry this out it would be necessary to derive the reciprocal formulation of program NLP1, as was done earlier for program NLP in Appendix 1). It is important to note that control over allocative resources, in addition to control over other resources, has the potential to enhance the relative efficiency of each of the sales units. The fundamental argument is that each sales unit, due to their proximity to the customers, is likely to be more informed about the market conditions which determine their resource needs. As a result, they are better at determining their resource requirements as opposed to another subunit or a higher level authority. This is shown below. Lemma. The control over allocative resources by all the sales units resuhs in a relative efficiency of each sales unit that is greater than or equal to the relative efficiency o f the same unit when there was no control over these resources. This implies that Z I ~ >_ Z~' or, in other words, the efficienc)' rating derived for sales unit b using program NLP1 is greater than or equal to the efficiency rating derived f o r the same unit using program NLP. Proof. To prove this lemma, we refer to program DLP1 in Appendix 2 and compare this to program DLP in Section 3. Now at optimality, Z~ = DZI*

5. An empirical illustration The sales unit efficiency model is illustrated by applying it to data collected from branch operations of 33 insurance companies. In this illustration individual units of competing firms are used, however, the reference set of sales units could consists of units within a single organization or cooperating units external to the organization. Note that the emphasis in this section is on illustrating the potential use of the model rather than on describing a specific implementation. Further, the relative efficiency of a sales unit is assessed in terms of a sales productivity ratio and not based on the specific values of a single output or input. In this illustration, the branch operations were considered as sales units provided that they employed at least five salespeople. In order to control for extraneous sources of variation, the sample was selected judgmentally from a single state. The data consisted of key informant reports (i.e., branch managers) on the relevant variables. Table 1 presents an operationalization of each of the variable used in the illustration. Specifically, important sales outcomes were identified as the average level of premiums generated by a typical salesperson (LEVPRO) and the expected increase in premiums for the branch (INCPRE). The size of the branch as reflected by the number of salespeople employed (SIZE), the number of product offerings (PRODUCT), the extent of advertising performed by the branch (ADVERT) and the number of incentives offered to the salespeople (INCENT) were considered as controllable resources for the sales units. Note that the sales outcomes are assessed in average/ percentage terms while the controllable resources are measured in absolute terms. Although this may appear to focus on scale rather than effi-

J. Mahajan / A data envelopment analytic model for the selling function

198 Table 1 Operationalization of variables

Sales outcomes:

Controllable resources:

Allocative resource: Environmental factor:

Variable

Operationalization

Average level of premiums (LEVPRO) Growth in premiums (INCPRE) Size of the salesforce (SIZE) Number of product Offerings (PRODUCTS) Extent of advertising (ADVERT) Number of incentives (INCENT) Number of counties (COUNT) Level of competition (COMPT)

The amount of premiums generated by a typical agent in a given year. The percentage growth in total premiums for the branch for a given year. The number of agents that were employed on a full-time basis in a given year. The number of products/services available to the customers. A multi-item scale assessing the extent of local advertising. The number of incentive schemes offered by the branch to the sales people. The number of counties from which 85% of premiums are generated. The approximate number of insurance branches in the primary trading area i.e., in COUNT).

ciency measures, we are of the opinion that such measures incorporate scale in efficiency computations. Hence, we argue that for larger branches (in terms of the controllable resources) to be as efficient as smaller branches, the former should not only generate higher absolute sales outcomes but should also generate higher average/percentage sales outcomes. The number of counties in which the branch operated ( C O U N T ) was considered as the alloc-

ative resource as the size of the territory operated in was determined at a higher level in the organization. Finally, the number of competitors within the branch's operating area (COMPT) was treated as the environmental factor as it reflects an external market condition that could only be influenced by the strategic positioning of the branch in a territory. In operationalizing the sales unit efficiency model, the allocative resource and environmental

Table 2 Slack analysis of inefficient sales units Sales unit I.D.

Rel. eff. score

Returns to scale a

2 3 7 9 10 11 12 17 18 23 25 28 30 31

0.7323 0.6765 0.7091 0.9366 0.6904 0.8051 0.5740 0.7647 0.8662 0.8316 0.7507 0.5831 0.9594 0.8311

D D D D D I D D D D D D I D

Resource input slacks

Sales outcome slacks

SIZE

PRODUCTS

INCENT

ADVERT

LEVPRO

INCPRE

0.00 21.38 0.00 0.00 19.78 5.93 0.00 20.78 55.94 0.00 0.00 0.00 0.00 10.93

0.00 0.00 0.44 0.00 0.00 3.40 0.00 2.00 0.00 1.76 0.23 0.00 3.78 0.79

0.00 0.00 0.00 0.26 0.00 0.00 0.00 3.61 0.30 0.00 0.00 0.00 5.17 0.00

14.10 0.00 0.07 3.20 0.00 0.74 0.00 0.00 0.00 0.00 0.00 1.13 7.73 0.00

3.54 2.92 9.55 5.18 0.36 27.57 0.00 22.84 0.00 26.68 0.00 0.00 32.84 24.44

0.00 0.00 2.33 0.00 0.00 0.00 7.54 0.00 0.00 0.00 78.61 6.36 0.00 0.00

D =- decreasing returns to scale; I -= increasing returns to scale.

J. Mahajan / A data envelopment analytic model for the selling function

factor are initially treated as uncontrollable variables. The solution for the models was derived by formulating the model as program DLP (see Section 3) and using the mathematical programming library XMP (Marsten, 1981). Table 2 displays the relative efficiency scores, the returns to scale and the slack values for each inefficient sales unit. The major findings discussed below correspond to each of the analytical interpretations presented in Section 3. First, the branch I.D. and relative efficiency scores shown in columns 1 and 2, respectively, of Table 2 indicate that there were 14 inefficient branches out of 33 (i.e., 42%). This suggests that the efficient branches obtained higher sales outcomes using the same controllable resources as any of the other branches listed in Table 2; or that the efficient branches obtained the same sales outcomes using fewer controllable resources than the inefficient ones. Specifically, branches 7 and 9 are cases in point. For example, branch used the same quantity of controllable resources as branch 19 but generated lesser outputs (i.e., LEVPRO and INCPRE) than the latter branch. On the other hand, branch 9 generated identical outcomes as branch 21 but utilized greater controllable resources (i.e., I N C E N T and ADVERT) than the latter branch in generating the outcomes. Second, in order to determine the returns to scale for these units the sign associated with the dual variable of constraint (3f) (i.e., q)h*) was

199

examined. Based on the interpretation presented earlier in Section 3, the returns to scale for each sales units was determined and these are presented in the third column of Table 2. The results suggest that overall there were decreasing returns to scale for most of the inefficient sales units. A potential explanation is that since the branches operate in highly competitive (as reflected by the number of competitors (i.e, COMPT) in a branch's primary trading area) and mature markets augmenting the controllable resources is likely to lead to a less than proportionate increase in the sales outcomes. In other words, deploying more salespeople, increasing the amount of advertising, the number of product offerings or the incentives offered to the salespeople does not enhance the growth or average premiums generated proportionately. Third, the slack values in columns 4 through 9 indicate the simultaneous changes required in the sales outcomes a n d / o r controllable resources to make a unit efficient (the new sales outcome value is equal to the slack value plus the original sales outcome value, i.e., Y,~ = Y,h + S, , for all i; and the new controllable resource value is equal to the relative efficiency score multiplied by the current value of the resource minus the slack value for the resource, i.e., X,}h = Q~ X, h - S,7", for all c. For example, for branch 12 to become relatively efficient, I N C P R E should be increased to 119.54 (i.e., 112 + 7.54) from" the current level of 112. Further, for branch 18 SIZE should be decreased from the

Table 3 Dual prices for the controllable resources and sales outcomes for inefficient sales units Resource inputs

Sales

Sales outcomes

unit I.D.

LEVPRO

INCPRE

SIZE

P R O D U C TS

INCENT

ADVFRT

2 3 7 9 10 11 12 17 18 23 25 28 30 31

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0006 0.0000 0.0037 0.0000 0.0011 0.0052 0.0000 0.0000

0.0038 0.0032 0.0000 0.0078 0.0032 0.0195 0.0000 0.0088 0.0011 0.0032 0.0000 0.0000 0.0379 0.0037

0.0086 0.0000 0.0709 0.0284 0.0000 0.0000 0.0006 0.0000 0.0000 0.0207 0.0055 0.0083 0.0406 0.0000

0.0361 0.0137 0.0000 0.0634 0.0153 0.0000 0.0062 0.0000 0.0187 0.0000 0.0000 0.0184 0.0000 0.0000

0.0421 0.0156 0.0267 0.0000 0.0174 0.1949 0.0402 0.0000 0.0000 0.0248 0.0228 0.0230 0.0000 0.0126

0.0000 0.0277 0.0000 0.0000 0.0296 0.0000 0.0155 0.0402 0.0583 0.0381 0.0267 0.0000 0.0000 0.0471

200

J. Mahajan / A data envelopment analytic model for the selling function

current level of 90 to 22 (i.e., (0.8662 * 9 0 ) 55.94); and, I N C E N T should be decreased from the current level of 5 to 4.03 (i.e., (0.8662 * 5) 0.30). For all the other branches both the sales outcomes and controllable resources need to be changed. As an illustration, for branch 28 to become relatively efficient, A D V E R T should be decreased to 11.12 (i.e., (0.5831 * 21) - 1.13) from a current level of 21; and I N C P R E should be increased to 126.36 (i.e., 120 + 6.36) from a current level of 120. Although the decreases/increases in sales outcomes/resources may appear to be unrealistic due to non-integer values, this does not limit the usefulness of the findings as the new levels of the outcomes/resources could be respecified in more meaningful terms by rounding to integer values. Fourth, the dual prices for the sales outcomes, shown in the second and third columns of Table 3, indicate the change in relative efficiency corresponding to a unit increase in a sales outcome (these were computed using program DLP presented in Section 3). For instance, for branch 23 a unit increase in I N C P R E (keeping all other other resources and sales outcomes constant) will increase the relative efficiency of the same unit to 0.7537 (i.e., 0.7507 + 0.0032). Although increasing sales outcomes enhances relative efficiency of the branch, it is likely to be difficult to achieve in practice. A more viable approach would be examine the change in relative efficiency due to a unit decrease in the resources used. The dual prices for the controllable resources, shown in columns 4 through 7 in Table 3, indicate the change in relative efficiency corresponding to a unit decrease in a controllable resource (these were computed based on program DLP2 presented in Appendix 1). For instance, for branch 23 a unit decrease in SIZE (keeping all other sales outcomes and resources constant) will increase the relative efficiency of this unit to 0.7714 (i.e., 0.7507 + 0.0207). Note that the dual prices for both the controllable resources and sales outcomes are zero if there is a positive slack associated with the same variable (see Table 2 for slack values). In other words, this suggests that a marginal gain in relative efficiency can only result from changing those sales outcomes and resources that are currently used at an optimal level. For example, if branch 28 is unable to simultaneously decrease A D V E R T to 11.12 and increase I N C P R E to

Table 4 Relative efficiency for i n c r e a s e d c o n t r o l o v e r a l l o c a t i v e f a c t o r s Sales u n i t

R e l a t i v e efficiency a

I.D.

Model 1

Model 2

2 3h 7 9 h 10 b 11 b

0.7323 0.6765 0.7091 0.9366 0.6904 0.8051

0.7323 0.7532 0.7091 0.9396 0.7251 0.8357

12 17 18 23 25 28 30

0.5740 0.7647 0.8662 0.8316 0.7507 0.5831 0.9594

0.5740 1.0000 0.8675 0.8316 0.7590 0.5868 1.0000

0.8311

0.8311

31

b b b b b

a M o d e l 1 = n o c o n t r o l over a l l o c a t i v e r e s o u r c e s a n d e n v i r o n mental factors, Model 2 = control over allocative resources b u t n o t e n v i r o n m e n t a l factors. b F o r these sales units, relative e f f i c i e n c y i n c r e a s e d d u e to c o n t r o l o v e r allocative r e s o u r c e s .

126.56, then the alternative approach would be to incrementally enhance relative efficiency by a single unit decrease in the number of salespeople, products or incentives or a unit increase in the average level of premiums generated.

Relatioe efficiency and additional control In order to examine the changes in relative efficiency due to the branches' control over the allocative resources, the model was reformulated (as suggested in Section 4) so that each branch could control the number of counties it operated in. In operationalizing the model, program DLP1 (see Appendix 1) was used. The relative efficiency scores for the inefficient branches with no control and control over this allocative resource are shown in columns 2 and 3 of Table 4 (these were computed using program DLP from Section 1 and program DLP1 in Appendix 2, respectively). The findings show that the efficiency scores for 9 of the 13 (69%) inefficient branches increased due to additional control over the allocative resource COUNT. This result is consistent with the proposition over allocative resources can increase the relative efficiency of a sales unit. Further, two sales units (17 and 30) become relatively efficient

J. Mahajan / A data envelopment analytic model for the selling function

as compared to the other sales units (i.e., ZI~' = 1). Finally, note that the returns to scale, the simultaneous changes (i.e., slack values) and individual unit changes (i.e., dual prices) in sales outcomes and resources can also examined under these additional control conditions. In sum, this illustration identified relatively inefficient sales branches and suggested the simultaneous changes necessary in the controllable resources and sales outcomes for an inefficient branch to become efficient. As an alternative strategy, the illustration also specified the marginal gain in relative efficiency that can be derived through a unit change in a controllable resource or sales outcome. Finally, the illustration demonstrated that additional control over resources can result in an increase in relative efficiency sales branch.

6. Implications and conclusions This paper presented a data envelopment analytic model for assessing the critical issue of assessing the relative efficiency of the selling function. While there has been a substantial body of literature on identifying factors that impact the performance of the selling function (Churchill et al., 1985), the question of how to measure sales efficiency has not been addressed. This study was a first attempt at throwing light on this important issue. There are several unique features of the model that enhances its managerial relevance in assessing the relative efficiency of the sales unit. First, it analytically identifies when the selling function is performed efficiently or inefficiently as compared to performance in other organizations. While the empirical illustration compared selling functions across organizations, it could be used to compare the efficiency of the selling function within one organization (i.e., different branches or territories). Second, the model derives a single summary measure of relative efficiency which could be used for performance evaluations and making resource allocation decisions. Third~ the model incorporates multiple a n d / o r conflicting performance criteria (i.e. sales volume, turnover or absenteeism) and multiple resource

201

factors (i.e, extent of training or incentive plans used). In addition, these criteria and factors can have different measurement units; for example, sales outcomes could be expressed in units such as number of new accounts while resources might be measured in dollar amounts. This is an important feature of the model as in such situations converting the factors to a common scale of measurement could be difficult. Fourth, the model can incorporate multiple environmental factors not under the control of management (i.e. market potential or degree of competition). This is a key issue as in order to ensure equity in the evaluation process aspects such as territory characteristics need to be considered. In addition, the model can incorporate qualitative factors such as satisfaction of the salespeople as sales outcomes or skill levels as resource inputs. Fifth, it does not requires specification of the functional forms regarding the relationship between sales outcomes and resources but, on the other hand, it indicates to the manager whether the sales unit is faced with increasing, decreasing or constant returns to scale. More importantly, the relative efficiency scores are adjusted for scale effects as well as the effects of uncontrollable resources and environmental factors so that it provides a more accurate assessment of the performance of the uni't. Sixth, through the use of sensitivity analysis the model provides insight into factors that contribute to relative efficiency. This is the significance when the manager has conflicting outcomes that need to be achieved or there are limited resources. Finally, the model can also be used to identify changes in efficiency for a sales unit over time. Hence, it can be used to assess the impact of changes in sales outcomes and resources made at a point in time on the efficiency of the unit at some future point in time. In addition to the potential managerial uses discussed above, the model developed in this paper can also be regarded as a diagnostic tool for providing an explanation of the potential causes of inefficiencies. In this context, it provides a single summary measure of efficiency which facilitates the comparison of several sales units. Such a measure can also be used to develop a meaningful classification scheme. For example, by examining all inefficient sales units, common fac-

202

J.

Mahajan / A data envelopment analytic model for the selling function

tors which characterize such units can be identified. There are several directions for the future that stem from this paper, two of which are as follows. First, the model and approach presented in this paper could be used as a basis for developing a hierarchical set of models that would provide information about sales outcomes and resources at each level of the hierarchy. The goal would be to provide an optimal theory of resource control. An implicit assumption in resource allocation studies is that the decision process for resource allocation takes place in stages that reflect the organization hierarchy (Rao and Sabavala, 1986). However, existing control procedures that exist may not be the optimal one. While this paper examined the effects of delegating control over resources to the sales units it did not explore control of different resources at distinct levels. Second, the model presented in this paper could be extended to incorporate changes in the environment that reflect discrete events (Banker and Morey, 1986b). One such development is the pervasiveness of new information technology (Little, 1987). Hence, the model could form the basis for conceptualizing efficiency gains from the introduction of information technology (e.g., sales productivity systems) in the selling function. In conclusion, the paper presented and illustrated a model for assessing the relative efficiency of sales units that incorporates multiple a n d / o r conflicting resources and outcomes. The analytical interpretations of the model provided a relative efficiency rating, the individual and simultaneous changes that are necessary in the sales outcomes and resource inputs and the returns to scale for the sales unit. While the sales unit efficiency model focused on efficiency by considering the direct impact of controllable resources, the model was reformulated to account for increased control of resources. The reformulation showed that as greater control is exercised by the sales unit, there is a potential for improving relative efficiency of the unit. The model was illustrated using data collected from a sample of sales units of insurance firms. Specifically, the illustration indicated that relative efficiency can be increased by simultaneous or individual changes in the sales outcomes and resources, and by assuming additional control over resources and environmental conditions.

Appendix I In this appendix, equivalent formulations to programs NLP, LP and D L P are presented. These can be used to estimate the changes in relative efficiency due to changes in a single resource input. In deriving these formulations the following condition is assumed to hold: N

A

E

Y[ u,Y,a.- ~. v l u X l . k i=1

a=l

E V 2 e X 2 e k - Uh > 0 e=l

for all k. Based on this assumption, program NLP2 is formulated as follows: minimize

Z2 b = y" v,.X,,h/ c=l

uiY, h i

vl. Xlub a=l

- E v2eXZeb- Ub e=l

]

(Sa)

subject to

E v,,XcJ

u~Y,k- E v l . X l ~ k

c=l

i

a=l

]

- ~ V2eX2ek--Ub >1 e=l

for a l l k ,

for all i, c,

u,, v~.> e v l , , v2 e > 0

(5b) (5c)

for all a, e.

(5d)

Note that program NLP2 is actually the reciprocal version of program NLP, and hence, it follows that at optimality Z3~' = 1/(Zb* ). Now as with program NLP, the following transformation of variables is carried out so as to derive an equivalent linearized version of program NLP3. Define: A

uiY,-h- ~ v l s X l . b a=l

-

V2eXZebe=l

Fi ~ ff~ui

for i = 1 . . . . . N, for a = 1 . . . . . A,

¢, = ~2v2 for e = 1 . . . . . E, O, = gay2 for c = 1 . . . . , C , O h = I2Uh.

Uh

,

J. Mahajan / A data envelopment analytic model,for the selling function Then multiplying the numerator and denominators in program NLP3 by 12 and adding the consistency condition, uiY.,-

vl. XI,, h -

t

v2~X2,, h - Uh

a=l

e~l

=1,

program LP2 can be formulated as follows: C

minimize subject

Z2 h = E

(0,. X.h)

(6a)

to

c

203

To estimate the change in relative efficiency attributable to a unit decrease in any resource input, the following procedure can be used. Note that for an inefficient sales unit b (i.e., D Z 2 ' ~ > 1 in formulation DLP2) %* = 0. Hence, in constraint set (7e), Y~ff_lXlcj/r~ is constant. As a result, any change in the observed value of controllable resource c (i.e., X,h) is only a change in the R H S of this constraint. This implies that the efficiency of sales unit b can be increased at the rate O,* (i.e., the dual variable associated with controllable resource c), over the range of X,q, for which the current basis remains unchanged.

N

E o,

- E 1;,

+ E .° xl.

t--1

c--1

a=l

Appendix 2

E

+ Y~ ~e X2,.k + q~h > 1

(6b)

for all k ,

e=l N

,4

E

E l~iYth- E ]~taXluh- E ~')eXaeb- eb = 1, i=1

a

1

e=l

(6c) F,, O,>__e for a l l i , c,

(6d)

~,,~,,>__0

(6e)

for all a, e,

~h unrestricted.

(6f)

In this appendix, a dual formulation is presented which is equivalent to program NLP1. This formulation (program DLP1) is derived by carrying out a transformation of variables on problem NLP1 (similar to that used to obtain programs LP and D L P from program N L P in Section 3) and is as follows: minimize

The dual of program LP2 (i.e., DLP2) is as fol-

s?+ Zs,:+

D Z 1 h = Q1 h - e [

lOWS:

i=l

c=l

(8a) maximize

DZ2 h = Q2 h + e

J

Si + +

C'=

S,7

subject to

(7a)

K

Y~ Y , : ' k - S, + > ~h

subject to

k

K

Y'. Y , : k -- S, + - Q 2 h Y , h > 0

for all i,

(7b)

k=l

for all i,

(8b)

1 K

Y~ X l . # r k + Sly- - Q l h X l . h

< 0

for all a,

k-1

K

Y'~ X l , , : k

- Q2hXI,, b < 0

for all a,

(8c)

(7c)

k=l

K

K

Y'~ X2ekr ~. < X2eh

Y'~ X 2 , . k ~ k -- Q 2 h X 2 e h

< 0

for all e,

(7d)

k=l

K

K

Y'~ X,.krk + S,F <

X,..

for all c,

(7e)

k=l

E rk - Q2h = 0,

(7f)

k=l

Q2 b unrestricted.

X c : k + S,7 - Q l h X , . b < 0

for all c,

(8e)

k=l K

K

~'k, Si+, S,5 > 0

(8d)

for all e,

k=l

for a l l i , c, k,

~

(8f)

~'k=l,

k=l

(7g)

~'k, S,+, S1£-, S f

(7h)

Q1 h unrestricted.

>0

for a l l i , a , c, k ,

(8g) (8h)

204

J. Mahajan / A data envelopment analytic model for the selling function

Acknowledgements The author appreciates the many constructive comments and suggestions provided by Dipankar Chakravarti, Gilbert A. Churchill Jr., Erich A. Joachimsthaler, Vijay Mahajan, Chakravarthi Narasimhan, Asoo J. Vakharia and an anonymous reviewer.

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