Directory UMM :Data Elmu:jurnal:S:Socio-Economic Planning Sciences:Vol34.Issue1.Mar2000:

Network DEA

Rolf FaÈre

a,b,

*, Shawna Grosskopf

b

a

Department of Agricultural and Resource Economics, Oregon State University, Corvallis, OR 97331-3612, USA b

Department of Economics, Oregon State University, Corvallis, OR 97331-3612, USA

Abstract

Most traditional DEA models treat their reference technologies as black boxes. Our network models, developed for the Swedish Institute for Health Economics (IHE), allow us to look into these boxes and to evaluate organizational performance and its component performance. The very general structure of the network model allows us to apply this model to a variety of situations: intermediate products, allocation of budgets or ®xed factors and certain (time separable) dynamic systems. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

The traditional models for Data Envelopment Analysis (hereafter DEA)-type performance measurement are based on thinking about production as a ``black box''. Inputs are transformed in this box into outputs. The actual transformation process is generally not modeled explicitly; rather, one simply speci®es what enters the box and what exits. This is, in fact, one of the advantages of DEA Ð it reveals rather than imposes the structure of the transformation process. Nevertheless, when researchers apply DEA to speci®c industries or situations, they have often added more structure to the model to better suit the application. Examples abound and include, among others: two (or more) stage models, hybrid models, cone-assurance regions, windows, etc. The reader is referred to Charnes et al. [1] (see especially pp. 425±36).

The variations on DEA mentioned above are generally intended to customize the model to suit the application. Here, we focus, in a somewhat more general way, on the transformation process in the black box. The general formulation we use is a network model, which has

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proved fruitful in engineering and operations research applications, among others. More speci®cally, in building on work by Shephard and FaÈre [2,3] on dynamic production correspondences, FaÈre and Grosskopf [4] developed a sequence of network models that can be used to address various re®nements of the standard DEA1models.

In this paper, we present three network models. The ®rst, used by FaÈre et al. [8] to study allocation of farmland to various crops, allows for allocation of a (®xed) factor or input among alternative uses. This general structure could also be used to introduce allocation of a budget or allocation of resources across units or branches. The second model, used by FaÈre and Whittaker [9], explicitly models intermediate products, i.e., products produced and used inside the technology. This model is also used by FaÈre et al. [10] to study alternative organizational structures of a multiplant ®rm. The third network formulation is a dynamic DEA model in which some outputs at period t are inputs in the next period, t+ 1. This provides an alternative to the dynamic DEA models proposed by Sengupta [11]. The dynamic DEA model introduced here is used by FaÈre and Grosskopf [12] to study the dynamic eciency of APEC (Asian-Paci®c Economic Community) countries.

Our goal is to show the usefulness of network DEA by bringing together a sequence of such models in one paper. We begin by discussing the network technology in a ``heuristic'' way, illustrating the various models with simple diagrams. Next, we formalize these technologies by specifying them as a series of linear inequality constraints, familiar from DEA. Then, we turn to the speci®cation of those models as performance measures based on distance functions which may be estimated using traditional DEA or Farrell [13] eciency measures. A summary concludes and discusses policy implications and directions for further research.

2. The heuristics of network models

In this section, we provide a heuristic road map to the di€erent network models discussed in the current paper. We denote inputs by x=(x1,. . .,xN) and outputs by y=(y1,. . .,yM). The

simple static non-network model, often referred to as the ``black box,'' is illustrated ®rst (see Fig. 1).

Here, inputs x are employed in the production process P to produce output y. P may be modeled, in the simplest case, by a production function or as a DEA model in more complex cases, as we illustrate in the next section. Independent of how P is modeled, there is no information about what is taking place within the production process P. Only the transformation of inputs into outputsx4yis modeled.

This static model can also be used to measure performance over time, as in Fig. 2. The comparative static model takes technology and inputs as ®xed and exogenous in each period, however (disembodied) technical change can occur over time. This idea has been used to model productivity change in a DEA framework, as in FaÈre et al. [14]. They show how to use DEA

1

to compute and decompose Malmquist productivity indexes into changes in eciency (``catching up'') and technical change (shifts in the frontier). See also FaÈre et al. [15].

Next, we turn to a related model that introduces the ``linkage'' among those processes characterizing network models. Let us assume that there are two production processes,P1 and P2, each producing an output vector y1 and y2, respectively. Moreover, assume that the two processes use the same source of inputs x. In this case, one can analyze the allocation of x to P1 and P2. In particular, ifx1 is employed by P1 and x2 by P2, then their sum cannot exceed x, i.e., x1+x2Ex. This type of model is known in agricultural economics as a model with ®xed (x) but allocatable inputs (Fig. 3).

As an example, assume that y1is the output of corn andy2 is the output of soybeans. Then, land and other resources x can be allocated to production of y1 and y2 under the constraint that total use of inputs does not exceed the given resources x. This model may be used to determine the optimal (output-maximizing, revenue maximizing or pro®t maximizing) allocation of land to crops. It could also be used to simulate the e€ect of set-aside programs, for example.

The static network model, which we will discuss next, attempts to analyze the ``inside'' of a production process and to explicitly model the black box. Subprocesses and intermediate goods are the two new concepts required to model a network. In Fig. 4, adopted from FaÈre and Grosskopf [4], we describe a network with three subprocesses, 1, 2 and 3. To these, a source 0 and an outlet or ``sink'' 4 are added. The source gives the network exogenous inputs x, which

Fig. 1. The static technology.

are allocated to the subprocesses ₀ix, i= 1,2,3 with the constraint 1₀x‡2₀x‡3₀xEx. The lower ``prescript'' refers to the source of the input whereas the upper ``prescript'' is the destination or point of use of the input.

The subprocess 1 produces the output vector y1, of which some are intermediate outputs 3₁y and some are ®nal outputs 4₁y, i.e., y1E3₁y‡4₁y. The notation means that process 1, the lower index, produces intermediate outputs that are inputs in process 3, i.e., 3₁y. In addition, some of its outputs are ®nal outputs, here represented with the upper index 4. In practice, intermediate outputs may also be ®nal outputs, as in the case of spare parts. Subprocess or node 3, uses network exogenous inputs 3₀x as well as intermediate inputs produced in subprocess 1 and 2, i.e.,3₁yand 3₂y, respectively, to produce ®nal outputs 4₃y.

The sink 4 collects all ®nal outputs from the network, i.e., yˆ4₁y‡4₂y‡4₃y. By appropriately adding zeros in the vectors, the problem of dimensionality is avoided. This type of network model has been used to study the organization of Swedish Pharmacies, where the subprocesses may represent di€erent types of pharmacies (at hospitals vs more standard commercial outlets or regional headquarters which distribute drugs to local pharmacies, for example). As another example, nodes 1 and 2 might represent production units that send their ``seconds'' to node 3, which is the factory outlet. Nodes 1 and 2 send their ®nal products directly to market (4).

We now proceed to show how the network notion can be used to model dynamic production processes or technologies. Suppose we have two periods and two production processes with period-speci®c inputs and outputs. In addition, we assume that some of the output in the ®rst period is used as input in the second, i.e., some outputs are time-intermediate products (Fig. 5). The two production processes are denoted byPt and Pt+ 1. Each uses time-speci®c inputs xt and xt+ 1 to produce time-speci®c ®nal outputs fyt and fyt+ 1. In addition, Pt produces intermediate outputs that are used as inputs in Pt+ 1. Pt also uses inputs from the earlier period tÿ1, namely iytÿ1. The total output from the dynamic model consists of ®nal outputs (fyt+fyt+1) and intermediate inputs (iyt, iyt+ 1). Clearly, there is a trade-o€ between producing ®nal outputs today (t) or tomorrow (t+ 1). Note that this model can be applied to study dynamic eciencies, as demonstrated by FaÈre and Grosskopf [12]. In their application, country level aggregate production is allocated between consumption (the ®nal outputs) and investment (the intermediate output). They solve the model for the optimal path of consumption and investment, which maximizes discounted aggregate consumption over the time horizon.

3. The formalistics of network models

The heuristic network models of the previous section are next formalized. In particular, we model the networks as the constraint sets or reference technologies of DEA models. These models have proven very useful for measuring eciency and productivity. As we shall see, the network DEA model is not a single model but a family of models, with the common feature of having linear constraints. In the next section, we will add an objective to these models in order to transform them into DEA performance measures.

n= 1,. . ._{,N,m= 1,}. . ._{,M, k= 1,}. . ._{,K are required to satisfy certain properties (see Appendix} A). To formulate the DEA model from the data (xk,yk), we need to introduce intensity variables zk, k= 1,. . .,K, one for each observation or activity k. These nonnegative variables tell us to what extent a particular DMU is involved in the production of outputs. The basic model, written in terms of an output set, is:

P…x† ˆ

The heuristics of this model are shown in Fig. 1. Although it satis®es certain properties, such as constant returns to scale and free disposability of inputs and outputs (again, see Appendix A), nothing particular can be said ex ante about its internal structure. For example, we cannot determine how inputs are allocated to the production of the various outputs, or if intermediate products are produced.

Our model ``Fixed but Allocative Inputs'' gives us some insights into how inputs may be shared by di€erent processes, here, modeled by two output sets P1 and P2. For simplicity, we assume that only the ®rst input can be allocated between the two processes or nodes, and that the others are preassigned to a speci®c process for each DMU. Here, superscripts refer to the process,P1 or P2. In this case, the heuristic model of Fig. 3 can be formalized as:

P

The main di€erence between the two models (1) and (2) is that in (2) the allocation of the ®rst inputxÃ1 between the two subprocesses is not given a priori like the other inputs xÃ12,. . .xÃ1N, xÃ22,. . .xÃ2N. Rather, we only require that the allocation across subprocesses be feasible, which is

the interpretation of the last inequality in (2). The model in (1) does not provide for such allocation, since it can be viewed as an aggregation of (2) that obscures the subprocesses.

intermediate goods are necessarily consumed or used up within the network; they may be ®nal output as well. Again, spare parts is a typical example of the latter.

We next illustrate the network in Fig. 4, which has three producing subprocesses, a source, and an outlet, giving us a total of ®ve nodes (0,. . ._{,4). Let us denote total available} (exogenous) inputs by x and let ₀ix, i= 1,2,3 denote the amount of the vector of (exogenous) inputs that is allocated to nodei. The source node models the constraints for the allocation of the exogenous inputs; in particular:

xeX

3

iˆ1

i

0x …3†

or

xne

1 0xn‡

2 0xn‡

3

0xn, nˆ1, . . ., N:

Denote the vector of outputs produced by subprocess or subtechnology i and delivered to node j by j_iy. Returning to Fig. 4, we see that the total production of node 1 is 3₁y‡4₁y, where

3

1y is its output of intermediate products and 4

1y is its ®nal output. Node 1 does not use any

intermediate products as inputs. Node or subprocess 3, however, uses inputs from both node 1 …3₁y† and node 2 …3₂y† as well as exogenous inputs 3₀x. This node produces only ®nal outputs 4₃y. The outlet or collection node 4, given that each subtechnology produces distinct output vectors,4_jy$RMj₊, j= 1,2,3, where M=M1+M2+M3, can be written as

yˆ

4 1y,

4 2y,

4 3y

: …4†

If we don't insist that each node produce distinct outputs, total production can be written as the sum P3

jˆ1 4

jy of the individual nodes' outputs. The appropriate number of zeros must be

added.

The piecewise linear or DEA technology associated with k= 1,. . ._{,K observations may be} written in terms of the output set as:

P…x† ˆ fyˆ

Distribution of exogenous inputs …l† 1

0xn‡

distribution of the exogenous inputs among subtechnologies, whereas the standard model does not. Furthermore, the network model allows us to explicitly model intermediate inputs, whereas the standard model does not.

FaÈre and Grosskopf [3] have shown that if the subtechnologies Pj, j= 1,2,3, satisfy properties such as free disposability of inputs and outputs and constant returns to scale then, so too, will the network technology (5).

Turning to the dynamic network model, illustrated by Fig. 5, we note that it consists of two distinct subtechnologies, Pt and Pt+ 1, one for each period. These subtechnologies are interactive in a way similar to the nodes in the network model above. The output from Pt consists of ®nal output fyt and intermediate output iyt. The latter is used as input at t+ 1. Thus, iyt is the ``investment'' from period t. The other inputs are the exogenous inputs at each period xt and xt+ 1. Although not considered here, some of these may be storable i.e., they can be used in a later period (see FaÈre and Grosskopf [4]). The technology illustrated in Fig. 5 can be expressed as a dynamic activity analysis or DEA model as follows:

P

xt, xt‡1, iytÿ1

ˆ f

f

yt,

f

yt‡1‡iyt‡1 :

f

yt_m‡iyt_m

EX

K

kˆ1

zt_k

f

yt_km‡iyt_m

, mˆ1, . . ._{, M,} Fig. 4. The network technology.

XK

kˆ1

zt_kxt_knExt_n, nˆ1, . . ._{, N,}

XK

kˆ1

zt_kiyt_kmÿ1Eiyt_mÿ1, mˆ1, . . ._{, M,}

zt_ke0,kˆ1, . . ._{, K,}

f

yt_m‡1‡iyt_m‡1

EX

K

kˆ1

zt_k‡1

f

yt_km‡1‡iyt_km‡1

, mˆ1, . . ._{, M,}

XK

kˆ1

z_kt‡1xt_kn‡1Ex_nt‡1, nˆ1, . . ._{, N,}

XK

kˆ1

zt_k‡1iyt_kmEiyt_m, mˆ1, . . ._{, M,}

zt_k‡1e0,kˆ1, . . ._{, Kg: …6†}

The dynamic DEA model (6) is related to the network model (5) in the sense that both have multiple subtechnologies or nodes, and both have intermediate products, making the subtechnologies interdependent. FaÈre and Grosskopf [5] have shown that if the subtechnologies Pt and Pt+1 satisfy properties such as free disposability of inputs and outputs and constant returns to scale, then so, too, does the dynamic model (6), i.e., the dynamic model inherits properties from the subtechnologies.

4. Performance measures

The performance of a particular ®rm, observation, or DMU k'_{, can, in principle, be} evaluated under two di€erent situations: ®rst, when no price information is available, and second, when prices are known. Of course, intermediate cases, when some prices are available, are also possible. Here, we address the no-price situation, and use distance functions as our yardsticks. Standard DEA or Farrell [13] technical eciency measures are closely related to distance functions, as we show below. The most general of these functions is the technology directional distance function, which is de®ned on the technologyP(x) by

~

D…x, y;gx,gy† ˆ max

where (ÿgx, gy) is the direction in which the distance is measured.2 Less general, but better known, are Shephard's [13] input and output distance functions, which are de®ned as

Di…y, x† ˆ max

l:y2P…x=l† …8†

and

D0…x, y† ˆ min

y:…y=y† 2P…x† , …9†

respectively. The input and output distance functions are special cases of the directional distance function (7). In particular, if we choose the direction (ÿgx,gy) to be (x,0), then (7) takes the form

~

D…x, y; x, 0† ˆ1ÿ1=Di…y, x† …10†

i.e., the directional distance function becomes the input distance function. Similarly, if we take (ÿgx,gy)=(0,y), then

~

D…x, y;0,y† ˆ …1=D0…x, y†† ÿ1 …11†

and the output distance function is obtained. The relation between the input and output distance function is given by

D0…x, y† ˆ1=Di…y, x† …12†

provided the technology exhibits constant returns to scale, i.e.,P(lx)=lP(x), l> 0.

We may illustrate the three distance functions (7), (8) and (9) as shown in Fig. 6. To do this, we introduce the graph of technologyT. It is de®ned in terms of output sets as

Tˆn…x, y†:y2P…x†, x2_RN_‡o: …13†

Clearly,P(x) can be retrieved fromTas

P…x† ˆ

y:…x, y† 2T : …14†

The reference technology is given by T, the area between the ray from the origin and the x-axis. When the observation (x,y) is evaluated by the directional distance function, the optimal value is attained at A whereb=D(x, y;~ gx,gy). The output distance function brings (x,y) to B whereD0(x,y)=y while the input distance function takes (x,y) to C whereDi(y,x)=l.

To make our example a lot more concrete, suppose that the observed (x,y) is (2,1), and we choose the direction vector to be (ÿ2,1). If the frontier of technology is described by y=x, then point B would have coordinates (2,2) and point C coordinates (1,1). The associated ecient point using the directional distance function (point A) would have a value of (4/3,4/3). The corresponding value of beta would be 1/3 for the directional distance function. The output

2

distance function in this case would have a value of 1/2 while the input distance function would have a value of 2.

For our ®rst illustration of performance measurement, we choose to evaluate ®rm or DMU k' relative to the static technology (1) by means of the Farrell output eciency measure [13]. This measure is the reciprocal of the output distance function,3namely:

ÿ

D0 ÿ

xk0, yk0 ÿ1ˆF0 ÿ

xk0, yk0 ˆ maxy

s:t: yyk0_mE

XK

kˆ1

zkykm, mˆ1, . . ., M,

XK

kˆ1

zkxknExk0_n, nˆ1, . . ., N,

zke0,kˆ1, . . ., K:

…15†

F0(xk

'

,yk') is the solution to the linear programming problem (15). A value equal to one signals eciency while a value larger than one signals ineciency. Output could be proportionally increased by (F0(xk

'

yk')-1) if the DMUk'_{is inecient.}

A more sophisticated measure than the Farrell measure is the Malmquist productivity index [14]. As a measure of productivity change, it tells us how much the ratio of aggregate output to aggregate input (an index of average product) has changed between any two time periods. This index can be input or output oriented, dependent on whether input or output distance functions are used. Here, we discuss the output oriented case and start with an illustration adopted from FaÈre and Grosskopf [4] (see Fig. 7).

The two input±output observations (xt,yt) and (xt+ 1,yt+ 1) belong to technologiesTt and Fig. 6. Distance functions.

3

Tt+1, respectively. The eciency change between periods is represented by:

Eq. (16) captures the change in distance/di€erence between observed production and the frontier Ð sometimes referred to as a measure of ``catching up''. Countries or ®rms that are engaged in successful learning either by doing or by imitation would exhibit improvements (values greater than one) in this component of productivity change.

Technical change is measured by:

TECHˆ

Note that Eq. (17) captures, for example, shifts in the frontier due to innovation.

The product of eciency and technical change is the Malmquist productivity measure, given by Eq. (18). relative to the two reference technologies, Tt and Tt+ 1, respectively. These are indicated by the superscripts on the distance functions,Dt0 and Dt0+ 1, respectively.

We next turn to the dynamic network model (6), which was used by FaÈre and Grosskopf [12] to compute dynamic eciency. Here, we need to resolve the question of whether a single overall scaling vs a year-by-year scaling should be used for the model. In the ®rst case, the

®nal output for each year, here fyt and fyt+ 1, would be scaled with the same scalar,y. In the second case, two scalars,yt and yt+ 1would be used and the objective function would be their sum. FaÈre and Grosskopf [12] used the latter formulation as they were interested in comparing the dynamic model with a sequence of static models. In either case, eciency can be estimated using the LP techniques familiar from DEA.

5. Summary

This paper has shown the ¯exibility of the DEA modeling framework by focusing on network DEA. These DEA models allow the researcher to study the ``inside'' of the usual black box technology both in static and dynamic settings. They accomplish this by providing a very general framework for specifying (endogenizing) the inner workings of the black box. The basic idea of the network model is thus to ``connect'' processes Ð providing a single model framework for stage production (with intermediate products, for example) or multi-period production. This situation has typically been handled in the DEA literature as a rather ad hoc series of DEA problems, or through the use of multiple stages.

The ``links'' between processes (or nodes) in the network model may also be used to analyze the allocation of resources across various units or processes. Examples include the allocation of a ®xed resource, such as land at farm level at a given point in time, over various crop uses; and the allocation of a school budget across various budgetary categories. The links can also be used to solve for the ``optimal'' linkages and for the structure of units Ð in cases where a ®rm has many plants or locations, etc.

All of these scenarios Ð intermediate goods, allocatable resources, and multi-period production Ð may be represented as network models. This then allows the researcher to endogenize allocations across space, time, etc. What we have also shown is that such models are readily estimatable using standard LP formulations familiar from the DEA literature. Our hope is that this rather simple framework will prove useful to those who use DEA, while moving researchers in other arenas to investigate the power of DEA.

Acknowledgements

The authors are grateful to Dr B. Parker for his ``red-pen adjustments.''

Appendix A

Assume there are k= 1,. . ._{,K DMU:s. These can be di€erent ®rms or a ®rm at di€erent} times. Each DMU is characterized by its input and output vector (xk,yk)=(xk1,. . .,xkN,yk1,. . .,ykM). The coecients (xkn,ykm), m= 1,. . .,M, n= 1,. . .,N,

…i† xkne0,ykme0,kˆ1, . . ., K, nˆ1, . . ., N, mˆ1, . . ., M,

…ii† X

K

kˆ1

xkn>0,nˆ1, . . .,N:

…iii† X

N

nˆ1

xkn>0, kˆ1, . . .,K:

…iv† X

K

kˆ1

ykm >0,mˆ1, . . ., M:

…v† X

M

mˆ1

ykm >0,kˆ1, . . ., K:

The conditions in (i) merely state that inputs and outputs are non-negative numbers, but need not all be positive. The requirement (ii) means that each input is used in at least one activity. The third condition says that each activity uses at least one input. The ®rst of the two output conditions (iv) requires that each output is produced by some activity while (v) states that each activity produces some output.

The model of (1) satis®es the properties:

1. P(0)={O} no ``free lunch''

2. yey1$P(x)= >y$P(x) free (strong) disposability of outputs 3. x1ex= >P(x1)uP(x) free (strong) disposability of inputs 4. P(lx)=lP(x),l> 0 constant returns to scale.

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