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Technically and cost-efficient centralized allocations in data envelopment analysis
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Technically and cost-efficient centralized allocations in data envelopment analysis Giovanni Cesaroni Department for public administration, Via del Sudario, 49 - 00186, Rome, Italy
ARTICLE INFO
ABSTRACT
Keywords: Dea Structural efficiency Industrial organization Ray average productivity Ray average cost
In data envelopment analysis and with a variable returns to scale production-technology, we apply Banker's [2] approach to determine the relationship between technically and cost-efficient industry structures, featuring reallocation of outputs and a variable number of firms. The interpretation based on the most productive and optimal scale-size notions allows us to both establish an inequality relationship between the corresponding industry-efficiency measures and provide adequate information on optimal solutions. At the applicative level, we introduce an exact algorithm to solve related non-linear programming problems, thus providing the decision maker with an accurate method for computing and comparing the input and output mixes and the optimal number of units obtained in the two allocations. Empirical illustration, given with reference to the Italian localpublic-transit sector and employing a multiple inputs and outputs technology, reveals striking differences with regard to the managerial and regulatory implications of the two centralized allocations.
1. Introduction Notwithstanding the early definition of the structural efficiency of a group of firms advanced by Farrell [[15], pp. 261–262] in his seminal contribution, the vast amount of literature on the non-parametric measurement of productive efficiency initiated by the model of Charnes et al. [9] has been almost exclusively concerned with the efficiency of the individual firm or decision-making unit (for a comprehensive review, see Emrouznejad et al. [11]). Only in recent years has the technical efficiency of a group of units been consistently investigated and developed. Førsund and Hjalmarsson [16] proposed to implement Farrell's notion - the output-weighted average of individual efficiencies - by evaluating the technical efficiency of the average firm. This approach was later applied to multiple-output production by Ylvinger [35], who employed DEA models to obtain technical efficiency measures based on the determination of input/output shadow prices under the assumptions of constant returns to scale (CRS) and no “reallocation of inputs across firms”. The studies cited so far have as a common characteristic that of not taking into account both the distinction between aggregate and individual production technologies and the possibility of reallocation of inputs among the existing firms. In this respect, an important progress is made by Li and Cheng [21], who analyze the output-oriented technical efficiency of the industry by making explicit the role of these two factors in a shadow price DEA model. They prove that in a
convex technology the structural efficiency of the industry in a multiple input and output technology can be taken to be equivalent to the technical efficiency of the average unit, and that measures in Ref. [35] are a particular case of their general solution. Moreover, the shadow price vector of the optimal output provides a general decomposition of structural efficiency, which clarifies that the divergence between the industry and the sum of individual technical-efficiency measures may be due to allocative inefficiencies of individual firms (see [21, p. 715]). Similar conclusions are reached by Nesterenko and Zelenyuk [26], in a setting which differs from the former only in the fact that output prices are exogenously given and efficiency is defined in terms of maximal revenue. In Refs. [21,26], convexity of the technology implies that the industry-benchmark is the same for each of the existing firms. Inspired by the model of industry efficiency proposed in Golany and Tamir [17], more recently, a new strand of literature has come to investigate the technically efficient “centralized resource allocation” of a group of units operating under the same central management by focusing mainly on the effects of specific and different rules of reallocation of inputs/outputs among the controlled units: each of these may be projected on a different point along the frontier of the individual technology (see, e.g., Lozano and Villa [22,23], Asmild et al. [1], Fang [12]). When compared to the previous literature, this strand does not introduce an aggregate technology, thus failing to properly determine the industry optimum while not addressing the issues linked to the divergence between group and individual efficiency measures (see the
E-mail address: [email protected]. https://doi.org/10.1016/j.seps.2019.100734 Received 1 October 2018; Received in revised form 7 May 2019; Accepted 12 August 2019 0038-0121/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Giovanni Cesaroni, Socio-Economic Planning Sciences, https://doi.org/10.1016/j.seps.2019.100734
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puzzles in [21, p. 714]). In the context of technical-efficiency analysis with reallocation, so far discussed, we may remark that the determination of the optimal number of firms in DEA as a part of the industry optimum - under variable returns to scale (VRS) - was first addressed by Ray and Hu [31] in a single-output production technology, and it was later solved by Peyrache [27] and Mar-Molinero et al. [25] in a general multipleoutput setting but without offering a clear economic interpretation of the centralized allocation. All of these contributions took a mixed-integer programming (MIP) approach to solving the optimization problem in question. The same approach is also taken in Peyrache [28], where cost-constrained technical efficiency of the industry is considered in an environment with input price segmentation.1 As far as the cost efficiency of an industry is concerned, fewer results regarding non-parametric production analysis are available in the literature. To be more precise, some important conclusions regarding the aggregation of individual cost efficiencies (Färe et al. [14]) and the determination of input shadow prices (Kuosmanen et al. [20], Fang and Li [13]) have been reached for a given number of units when reallocation of outputs across them is not possible. However, it is only in Ray [30] and Cesaroni [6] that a cost-efficient centralized allocation featuring a variable number of firms and reallocation of outputs comes to be analyzed. With respect to Ref. [30], which is more empirical in nature, [6] offers a theoretical characterization of the centralized allocation by considering: the distinction between individual and industry technologies, the general form of the allocation problem2 and the characteristics of its global optimum. In addition, [6] provides a decomposition of the industry efficiency measure similar to that of Li and Cheng [21] along with its rigorous economic interpretation based on the ray average cost. Even from the computational point of view, [6] follows a different approach by introducing a specific algorithm to solve the non-linear programming problem, where instead [30] transforms the same problem into a mixed-integer one. To the best of our knowledge, no contribution has so far either dealt with the relation linking technically and cost-efficient centralized allocations under a variable number of firms or proposed a robust alternative to the MIP method, which might be affected by computational inaccuracy due to NP-completeness and also not deliver information on multiple optimal solutions. We finally remark that another missing point in the debate concerns the actual relationship between CRS and VRS technical optima in an industry with a multiple-output technology (see, e.g., Refs. [21,25–27,30]), whereas the single-output case was discussed in Ref. [31]. This article contributes the literature on centralized allocations in several ways by adopting Banker's [2] approach, and its extension to cost analysis made by Cesaroni and Giovannola [7]. First, we provide a clear interpretation of technically efficient centralized allocation as a most productive scale size (MPSS) in a multiple inputs and outputs technology, which is currently absent from the literature and will prove to be essential for our subsequent conclusions. Second, we characterize the relation of CRS to VRS optimum. Third, we ascertain the relationship linking technical and cost efficiency industry measures. Fourth, we propose a rigorous approach to the computation of optimal centralized allocations, which is based on the properties of the ray average productivity and ray average cost functions, and compare it with the MIP approach. Finally, we provide the regulator (centralized decisionmaker) with some empirical evidence on the differences between
technically and cost efficient industry allocations, and on the relationship between CRS and VRS solutions - which is currently not available in the literature. The paper is structured as follows. Section 2 introduces the model and derives the programming problem associated with technically efficient centralized allocation. Section 3 provides interpretation of this optimal allocation and illustrates its relation to the cost-optimal analogue. Sections 4.1 deal with the key issue of the global optimum of the two centralized allocations and introduces a suitable computational procedure, along with a comparison of the two competing computational-approaches. Section 4.2 illustrates the empirical application. Section 5 briefly summarizes the relevance of our findings. 2. The model and the industry technical-efficiency problem 2.1. The model We suppose that the industry is made up of n firms featuring the same convex VRS production technology as given in the DEA model of Banker et al. [3], also known as the BCC model. Introducing notation, we have n observations, indexed by j (j = 1, ..., n) , using inputs x ij (i = 1, ..., m) to produce s outputs, yrj (r = 1, ..., s ) . The observed input 0 and yj = (y1j , ..., ysj ) 0 , reand output vectors are xj = (x1j, ..., xmj ) spectively, where the prime indicates the transposition operation.3 Given the postulates of convexity, free disposability of inputs and outputs and minimum extrapolation, the individual production possibility set can be expressed as n
T=
(x‚y)
n j xj
j=1
x‚
n j yj
j=1
y‚
j
= 1,
j
0; j
J
j=1
where J = {1, ..., n} . The observed industry's input and output vectors are X 0
and Y0
n y j=1 j ,
(1) n x j =1 j
respectively. According to Refs. [21,26], the tech-
nology at the industry level T IND is the sum of the individual possibility sets, thus we have
T IND = nT
(2)
as a result of the summation of identical convex sets. Note that this kind of aggregate technology, contrary to that based on the sum of individual input-requirement sets (employed in Ref. [14]), allows for reallocation of output across firms in the industry.4 Moreover, T is not necessarily included in T IND . As far as input prices are concerned, we will assume that each unit faces the same exogenously-given vector p = (p1 , ..., pm ) > 0 . This may correspond to either a situation in which different branches/offices are controlled by a central authority (e.g., a parent company or a publicsector department) which buys and provides inputs to them, or one in which the regulator adopts a set of input market prices - or rather a function of these indicating their shadow prices - to gauge the efficiency of the industry. Especially with regard to this latter case, it can be remarked that Koopmans [19] revealed that the uniformity of prices across producers is a necessary and sufficient condition for Pareto efficiency (see [20, pp. 736–737]). We also wish to point out that the assumption in question is necessary to establish an unambiguous relationship between technical and cost efficiency industry measures (see Appendix A.2). As far as the case of input prices varying across units is concerned, to the best of our knowledge, the literature currently does
1 However, we note that the aggregate technology implicit in the solution proposed in this contribution, which is based on indirect output sets, is nonconvex as a result of the summation of non-identical convex sets. Therefore, strictly speaking, this model goes beyond the scope of our paper. 2 We remark, in particular, that Ray [30] does not offer a formal argument for the optimality of the allocation in which different units share the same input and output vector (see, ibid., fn. 6, p. 73).
3 The vector-inequality sign means that at least one element of the vector is strictly greater than zero. 4 Contrary to what holds in (1), a convex input-requirement set does not allow for a convex combination of the output vectors of the observed units. This implies that each firm can at most produce its observed output vector (cf. [20, p. 742]).
2
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not offer a solution to the complex optimization problem which arises with reallocation of output over a variable number of firms (see Appendix D). Finally, we remark that, even if this solution were available, the assumption of unit-specific and known input-prices might not make sense: in fact, the known input-price vectors of the existing units do not necessarily apply to the new optimal units resulting from the efficient organization, because these latter may not correspond to observed units - as a consequence of the convexity of T.5
MIP problem. Although the literature does not discuss the consequences of this transformation, we will argue below that this choice may imply a cost. Second, if we denote by asterisks an optimal solution to (3a), we can note that (k y , k x ) TIND as long as the additivity assumption underlying (2) is referred to k firms, instead of n . 3. Interpretation of the technical-efficiency measure and relationship with its cost analogue
2.2. The industry technical-efficiency problem
3.1. Interpretation of the technical-efficient centralized allocation
To put the problem in its general form, we introduce the concept of technically efficient centralized allocation, or technically efficient organization of the industry:
A clear economic interpretation of technical structural-efficiency in DEA is at present not available. In fact, in a single-output technology with a variable number of firms, Ray and Hu [31, p. 11] suggest that the corresponding centralized allocation does not maximize the average productivity and therefore does not coincide with an MPSS. In a general multi-output setting and with a given number of firms, Li and Cheng [21] decompose the technical structural-efficiency measure without an average-productivity role being called for. The issue we raise is relevant not only from the interpretative/managerial point of view but, as we will reveal in Section 4.1.1, it is also crucial for the exact determination of optimal solutions. Banker [2, p. 37 and ff.] introduced the concept of MPSS in a VRS production technology as “the scale size at which the outputs produced ‘per unit’ of the inputs is maximized”, that is, the production possibility which maximizes the ray average productivity7 of a given input and output mix. By using expression (10) in Cesaroni and Giovannola [7, pp. 124] and Podinovski's [29, p.233] formulation of the optimization problem, we have the following definition.
Definition 1. The technically efficient organization of the industry is a set of production possibilities (xh , yh) T , h = 1, ..., k which minimizes in the same proportion the use of inputs for the industry output vector y0. Note that h is an index for the generic element of the set and k is an integer variable. In other words, the technical-efficient organization of an industry is that allocation of the observed industry output vector among a set of efficient points to be determined, {(xh ‚yh)} , which achieves the greatest proportional reduction in all inputs. The optimization problem posed by Definition 1 is
min {(x h, yh)} k
s.t.
X0
xh h=1
Definition 2. A given production possibility (xh, yh) only if
k
yh
Y0
(3)
h= 1
min
= 1. where x h yh Observe that this general formulation of the problem, allowing in principle for more than one efficient point, cannot be found in either Maindiratta6 [24] or the contributions of Ray and Hu [31] and MarMolinero et al. [25]. An output-oriented version of this formulation was first introduced by Ray [32]. As far as the form of the solution is concerned, we obtain important information by means of the following re-formulation of problem (3) n j=1
h j xj ,
n j=1
h j yj and
n j=1
h j
k yh
X0
The above definition obviously apply to any VRS production set,
such as T IND . Being
h, m
= max i
{ } and, xim xih
h, m
= max r
yrh 8 yrm
we point out
that expression h, m h, m is the ray average productivity of (xh, yh) re1 lative to (xm, ym) , and that its reciprocal, , indicates the ray h, m
k xh
h, m
m
CRS technical-efficiency score; b) the production possibility m yielding maximal productivity for an input-output mix h which is not an MPSS, i.e. the minimizer of program (4) when h, m h, m < 1, is an MPSS. Given these preliminaries, we can now investigate the characterization of a technical-efficient centralized allocation. An initial important property is established by the following Proposition.
(3a)
j x j yh n j=1 j
{ }
average productivity of (xm, ym) relative to (xh, yh) . Note that Definition 2 implies that for an MPSS h, m h, m 1. Moreover, the Proof of Proposition 1 and Corollaries 2 and 3 in Banker [2] can be used to show that: a) min h, m h, m = CRS , where CRS is the
Y0
n j=1
(4)
where subscript m denotes a generic point of T.
where (xh, yh) is the arithmetic average of k efficient-points, i.e.
xh
=1
(xm, ym)
min k, (xh, yh) s.t.
h, m h, m
T is an MPSS if and
n j=1
j yj
with
h
j
j k h=1 k
, which belongs to T
= 1. because of This means that convexity of the technology reduces problem (3) to the search for a single efficient point being replicated k times, a property which is not valid for non-convex production technologies such as that of the free disposal hull (FDH, Deprins et al. [10]). Therefore, it is natural to interpret k as the optimal number of firms in a technicalefficient centralized allocation. With regard to problem (3a), two preliminary remarks are in order. First, it clearly is a non-linear integer programming problem in k and in the j s, but - as shown in Ray and Hu [31] and especially in Peyrache [27] and Mar-Molinero et al. [25] - it can easily be reformulated as a
Proposition 1. An optimal solution to (3a), (k x , k y ) , maximizes the ray average productivity of (k x h, k yh) with respect to the aggregate input and output mix (X 0, Y0). Proof. See Appendix A.1. This important result, which is new to the literature, clarifies that in a VRS industry-technology the technically efficient centralized solution provides the aggregate input and output mix with the maximum ray 7 The term indicates average productivity in a multi-output technology (see Podinovski [29, p. 241]). 8 For the equivalence of our notation with that used by Banker [2], observe that δ = β and γ = 1/α.
5
Cf. Table B1 in Appendix B. 6 Maindiratta [24, p. 62], dealing with size efficiency of a single firm, sets a problem which may be taken as an imprecise formulation of our problem (3). 3
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average productivity compatible with an integer k, where this can be expressed as 1 (see Appendix A.1.). This means that, by following Banker and Thrall [4], a rigorous economic interpretation of the technical inefficiency of the allocation of (X 0, Y0) among the existing firms can be given in terms of its technical and scale radial inefficiency. The failure of this allocation in maximizing the ray average productivity of (X 0, Y0) is the economic factor which explains what has been so far generically termed as “reallocative efficiency” or “organizational efficiency” (cf., respectively, Li and Cheng [21] and ten Raa [34]). The next remark will be useful for understanding the relationship between VRS and CRS solutions.
of times, where kc can be interpreted as Baumol et al. [5]'s “efficient number of firms”. The associated industry cost-efficiency measure is kc px c R 0 = pX , where (x c , y c) can be interpreted as the production possi0 bility which minimizes the ray average cost of the aggregate input and output mix (X 0, Y0) subject to the integer constraint on the optimal number of firms (see Section 3.1 in Cesaroni [6]). Moreover, it can be immediately checked that remarks analogous to Remarks 1 and 2 also hold for problem (5). We can conclude that (x , y ) and (x c , y c) maximize and minimize, respectively, the ray average productivity and the ray average cost of (X 0, Y0) subject to the integer constraint on the optimal number of firms. This consideration introduces a central Proposition of this paper.
Remark 1. The CRS technical-efficiency problem9 is equivalent to problem (3a) without the integer constraint being imposed on k.
Proposition 3. In a VRS centralized allocation the cost-efficiency measure is not greater than the technical-efficiency measure, R 0 .
Accordingly, another remark can be immediately derived Remark 2. In a technically efficient centralized allocation
CRS
Proof. See Appendix A.2.
.
The intuition of this general result, which holds irrespectively of the values taken by the two distinct optimal-numbers of firms involved (k and kc ), relies on the fact that - with respect to - R 0 includes an allocative component. As a result of Proposition 3, the same relationship linking primal and dual individual efficiency-measures in an individual technology - where the integer constraint does not apply (see Corollary 2 in [7, p. 125]) - also holds for aggregate VRS technologies, where this constraint is present. We may remark, as shown in the Proof, that this conclusion depends strictly upon the interpretation of as the ray average productivity of (X 0, Y0) .
Obviously, the equality sign will hold when a CRS optimal solution implies an integer k. Moreover, it should be now evident that the VRS optimal solution is an integer approximation of the CRS one, this implies that the difference CRS can be very small in applications with a sufficiently-large number of firms (for an explanation see Section 4.1.2). On the basis of Remark 2, we can clarify the issue of the interpretation of the optimal centralized allocation (k x , k y ) as an MPSS by means of the following Proposition Proposition 2. The allocation (k x , k y ) is not necessarily the MPSS of (X 0, Y0) . Proof. It follows immediately from
1
1 CRS
4. Solution methods and empirical illustration
, i.e. (k x , k y ) does not
4.1. Solution methods
necessarily ensure maximal productivity for (X 0, Y0) as a consequence of the integer constraint on k. Q.E.D. For a graphical illustration, see Fig. 1.
In previous sections we presented a theoretical characterization of both technical-efficient centralized allocation and its relation to the cost-efficient analogue. However, from an operational point of view, the main difficulty concerns the computation of the solution of the integer non-linear programming problem involved by the two centralized allocations (see, respectively, problem (3a) above, and problem (4b) in Ref. [6]). In this regard, the approach commonly followed in the literature is that of transforming these non-linear problems into MIP problems, as clearly illustrated in Refs. [25,27,30,31]. Contrary to this approach, [6] proposes a computation procedure which preserves the non-linear nature of the programming problem and has its logical foundation in the theoretical characterization of the centralized allocations as ray-average-productivity and ray-averagecost optima, i.e. as MPSS and OSS respectively. The computation procedure in question can also be applied to problem (3a), which is now handled as a two-stage problem where the first stage is
Finally, note that (k x , k y ) has the same ray average productivity as (x , y ) : the properties established by Propositions 1 and 2 also hold for this latter allocation, which belongs to the individual technology T. Summing up, we can conclude this section by emphasizing that a VRS technical-efficient centralized allocation can be considered an integer-constrained MPSS of (X 0, Y0) (Proposition 1), but that the same allocation is not necessarily an MPSS of (X 0, Y0) (Proposition 2). These conclusions show that not only is a specific maximization of productivity implied in the VRS aggregate technology, but the possible divergence from the absolute maximum may also be limited, because it is exclusively due to presence of the integer constraint (see Section 4.2, and Appendix B). 3.2. The relationship between technically and cost-efficient centralized allocations
min (k ) j
The same line of reasoning employed in Section 2.2 can be followed to show that in a VRS technology the cost-efficient centralized allocation can be obtained as a solution of the following problem (see Cesaroni [6], p. 39)
s.t.
k:
min k px j, k
s.t. k y
n
min k
n
where x = j = 1 j xj and y = j = 1 j yj , with j = 1 j = 1. An optimal solution to (5), (kc x c , kc y c ) , has the same characterization ascertained above for its technical-efficient counterpart: a single optimal scale size (OSS), (x c , y c) T , being replicated an integer number n
x h and yh
X 0 Y0 , k k
Y0 , k
T
(6a)
and the second stage is
(5)
Y0
X0 k
(k )
(6b)
where in both problems k = 1, ..., n 0 ; n 0 is a pre-determined integer value, (k ) is the optimal value determined in (6a) and k is the solution to (6b). The first stage solves the technical efficiency problem for each notional average-unit corresponding to a given k, while the second stage selects the value k which yields the minimum technical-efficiency
9 In the CRS case, which is obtained by dropping the constraint in (1), the individual and the industry technologies coincide.
4
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Fig. 1. Individual and aggregate technologies, optimal allocations.
score among those computed at stage one. With regard to (6a), note that not all of the notional average-units will belong to T, because this is ensured only for k = n, so that a key element of the solution procedure will involve checking whether this condition holds10. To determine a solution to the two-stage problem at hand we propose the following algorithm:
(
productivity curve, outside the interval in which its minimal value is reached, implies that once that the proposed algorithm finds a solution, this is actually the sought global-minimum (the same outcome may not be given when the behavior of this curve is not monotonic, as in the non-convex case).11 Therefore, Proposition 4 establishes a property of accuracy of our solution method. Note, moreover, that this property necessarily depends on Proposition 1, that is the interpretation of the technically efficient centralized allocation as an integer-constrained MPSS. The only practical difficulty with our algorithm lies in choosing a suitable range for k in order to be reasonably sure of finding the desired minimum and avoid repeating the procedure. As regards this problem, observe that it can arise only in conditions which yield k > n so that as an initial choice one can try n0 = 2n . If this should turn out to not be sufficient, the algorithm can be repeated in the enlarged interval given by n0 = 3n , and so on. The same accuracy in the determination of an optimal solution is granted by the analogous algorithm for the computation of the costefficient centralized allocation: convexity ensures the same monotonic property to the ray average cost of a cost-efficient point which is not an OSS (see [6, p. 41], and Appendix C for a more detailed argument).
)
1) Construct a set of fictitious observations k0 , k0 with k = 1, ..., n 0 , and add this to the original set of observations which determine T. 2) Considering T as the reference technology, i.e. the frontier determined by the actual set of firms, perform technical-efficiency analysis on the enlarged set of observations. 3) Exclude fictitious observations featuring an efficiency score greater than one: i.e. they do not belong to T. 4) Perform technical-efficiency analysis on the new set of observations. This completes the solution to the first stage (6a) by determining the efficient point - and the associated efficiency score (k ) - corresponding to each fictitious observation. 5) The solution to the industry problem (second stage, 6b) is given by the efficient point - and the associated k - which ensures the minimal (k ) among those obtained at stage 4). X
Y
As for step 5), we would like to underline that the detection of the minimum in the range of k chosen in step 1) is made unambiguous by the form that the ray average productivity curve (i.e. the value function defined in Podinovski [29, p. 245]) assumes due to the assumption of convexity of T.
4.1.1. Computational simplification and comparison with alternative approaches In this section, we deal with the two issues represented, respectively, by the additional cost in terms of computational burden of the precision ensured by our method and the evaluation of its features relative to those of alternative solution methods. As for the first question, observe that within a DEA softwarepackage that allows for the specification of the reference technology as a subset of units, the algorithmic sequence from step (2) to step (5) boils down to a single step procedure (i.e., a single linear-programming problem). In fact, this software can be applied to estimate technical-
Proposition 4. The algorithm made up of steps 1) to 5) determines an optimal solution to problem (3a). Proof. See Appendix A.3. In other words, the monotonic behavior of the ray-average10 It can be easily shown that last condition in (6a) holds if and only if (x 0, y0) kT . Therefore, it ensures that all of the relevant points of the VRS aggregate technology are being considered.
11 An empirical comparison of the behavior of this curve in convex and nonconvex technologies is recently offered in Cesaroni et al. [8].
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efficiency scores of the enlarged and ordered set of observations, made up of n + n0 units :then the optimal solution is readily determined by the fictitious observation having a minimal score lower than 1. Thus, the only inconvenience may eventually be represented by the necessity of adjusting the initial guess of n0 (on the contrary, cf. Appendix B). This possible difficulty can be circumvented by taking Remark 1 into account. According to this remark, the solution of problem (3a), k , belongs to a neighborhood of the CRS solution, kCRS , where the size of this integer-interval will generally depend on the occurrence of multiple solutions: it will be determined by two consecutive integers when a single solution occurs, while it will generally be larger when additional solutions arise. Therefore, our algorithm can be reduced to the solution of two linear-programming (LP) programs: the first - relative to the original set of observations - determining the CRS solution for observation (X 0, Y0) , the second - relative to the set of the original ob-
(
represents the VRS industry technology with n = 2 . The industry's input and output mix is point E, with each of its integer contractions - i.e., the fictitious observations considered in the first stage of our algorithm lying in the interior of the line segment OE. More precisely, F and G are, respectively, the 1 and 1 contractions, i.e. corresponding to k = 2 and 2 3 k = 3. Note that higher-order contractions (e.g., with k 5) do not belong to T. The fictitious observation ensuring the minimal (k ) is F, corresponding to k = 2. This determines the maximal ray average productivity compatible with the integer constraint (see Proposition 1): it is given as the slope of the straight line OH, passing through the intersection of F’F and the frontier segment AB. However, as remarked by Proposition 2, the projection of F on the frontier AB is not necessarily the point of T ensuring the global maximum of productivity: in fact, the MPSS is given by point B, whose contraction factor is 1 < kCRS < 2 , where kCRS is the scale-factor implicit in the CRS centralized allocation. kCRS is 1, so that in Observe, however, that the upper bound of k applications with a large n the difference between the two points along the line segment OE which generate the VRS and the CRS optimal allocations can be small, because of the proximity of 1 and 1 (see,
)
servations enlarged by a few fictitious units k0 , k0 taken in a symmetrical neighborhood of kCRS - for the computation of the VRS solution. Observe, that the addition of these few additional units does not affect either the computational complexity or the solution time of the LP VRS problem (cf. Appendix B). As for the second question, we can now point out that our method has two main advantages over the MIP approach. First, the MIP solution does not allow to test for the presence of multiple solutions, while our method allows for this by supplying information on the behavior of ray average productivity in a neighborhood of kCRS (see Section 4.2 and Appendix B). Note that this specific information is of some value to the decision maker: because of the existence of adjustment costs, in some cases it will be optimal to choose either the minimal or the maximal k belonging to the set of different multiple solutions. Second, being based on two simple LP problems, our solution is not affected by computation problems regarding k , which conversely may arise in complex MIP problems within large data sets. In these cases, it is possible that, as a consequence of the NP-completeness of the MIP problem12 this approach has no effective criterion to check whether the computed solutions are optimal or not. Therefore, computation errors may in principle affect the MIP determination of both technically and cost efficient centralized allocations, while this is not true of our solution. For these reasons, we can conclude that our approach is more general and that it can be used to test the robustness of MIP results. Finally, it can be easily verified that our proposed method is optimal also when compared to a standard LP framework. Herein, in the case of a single VRS solution there would in fact arise the need to solve three LP programs (the CRS problem, the VRS problems associated to the two given integers closest to kCRS ), against the two programs required by our method. Moreover, in the case of multiple VRS solutions, the number of LP programs to be solved would become unknown and greater than three, because of its dependence on the multiplicity of solutions (see Section 4.2). X
Y
Section 4.2).
k
k CRS
4.2. Empirical application We will apply the proposed computational approach to the secondary data set employed in Ref. [6], which is made up by a representative sample of 43 companies operating in the Italian local public transit industry in the year 2012. Observations consider a three input-two output production technology: the outputs are vehicle-kilometres travelled and the size of the transport network (expressed as the area served by a company); the inputs consist of the number of company staff, the number of vehicles, and the quantity of a composite commodity representing the consumption of fuel, energy, materials and spare parts. Each company faces a specific input-price vector made up of: the unit-wage of its staff p1, the unit-depreciation of its stock of vehicles p2, and a price index for the composite intermediate commodity p3. In the computation of the cost efficient centralized allocation, we will assume that the regulator intends to estimate the industry configuration corresponding to two different input-price vectors: the first, given as the arithmetic average of input prices, is (p1 , p2 , p3 ) = (43188.16, 13718.38, 1.02) with prices denominated in Euro, the second - considering the average price weighted by the inputproportion over the industry's aggregate is (p1 , p2 , p3 ) = (45798.52, 15849.02, 0.99) . The summary statistics of companies’ observed input-output data are given in Table 1 (the units of measure of inputs and outputs are shown in brackets).where the mean describes the average unit. The industry's input and output mix can be derived by multiplying the mean by n = 43. To begin, we apply the algorithm of Section 4.1 to the estimation13 of the technical-efficient centralized allocation by considering n0 = 86, i.e. k [1,86]. However, a global minimum is not reached in this range because the ray average productivity of the fictitious observation, (k ) , is still decreasing at k = 86, and for this reason we try with n0 = 129 thus determining the range k [1,129] - but the outcome is the same. Only by enlarging the range to k [1, 516] do we find the global [482, 485] with (k ) = 0.9571. Fig. 2 depicts the minimum at k behavior of (k ) in a neighborhood of. k . We may remark that Fig. 2 is also noteworthy because it illustrates empirically, in an industry setting, the behavior of the ray-averageproductivity curve described by Theorem 5 of Ref. [29]. The slow pace
4.1.2. Graphical representation A simple representation in a one input-one output space will be of great help in illustrating the difference between individual and aggregate technologies, the relation of the VRS to the CRS technology and the functioning of the algorithm introduced in the preceding section. In Fig. 1, the technologies are represented by continuous lines: the frontier of the CRS technology, TCRS , is given by the straight line passing through points 0 and B; the frontier of the individual VRS technology, T, is given by the polyline ABC and by its horizontal extension from point C to the right; the analogous polyline passing through points 2A, 2B and 2C 12
Kannan and Monma [18] prove the NP-completeness of the MIP problem. NP-completeness implies that no known polynomial-time algorithm is available for the solution of a MIP problem, while such an algorithm is available for linear-programming problems. The heuristic methods employed by the MIP solution may thus imply an error.
13 Computations have been performed by means of two R-packages: Benchmarking 0.26 for technical and cost efficiency estimates, and lpSolveApi 5.5.2.0 for MIP.
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Table 1 Input-Output Data. Summary statistics. Inputs
Min Mean Max Stdev
Outputs
Staff (Number)
Vehicles (Number)
Commodity (Number)
Vehicle-Km (Millions)
Service Area (Km^2)
89 1311 11990 2307.91
43 526.28 4263 756.73
1191685.17 14049094.19 108985785.70 22281459.13
1.84 20.76 161.59 32.76
37.86 2091.91 11687.06 2672
Fig. 3. Centralized cost efficiency problem: ray average cost in a neighborhood of.kc .
yields a ray-average-productivity component ( allocative component
(
ps 0 pX 0
0, c
)
0, c 0, c )
= 0.9680 and an
= 0.4179, thus pointing out the abso-
lute relevance that allocative inefficiency may reach in the analysis of optimal industry configurations. In this context, the solution of the MIP problem yields results identical to those of our approach. With regard to the comparison of technical and cost-efficient centralized allocations, note that differences are widespread and pronounced. In fact, not only do efficiency measures substantially differ: (k ) = 0.9571 > R0 = 0.5501, but even the corresponding industry configurations - k , (x , y ) and kc , (x c , y c) - turn out to be quite diverse, as illustrated in Table 2. The large divergence in the optimal number of firms determines a considerably larger optimal scale-size in the case of the cost efficient allocation compared to technically efficient organization. The input and output mixes also differ: a substantially-higher density (ratio of vehiclekm to service area), higher vehicles-per-staff and commodity-per-vehicle ratios characterize the cost efficient allocation when compared to the technically efficient one. Note that in the former allocation, the greater consumption-per-vehicle matches the more intense use of vehicles suggested by the greater vehicle-per-staff ratio. With regard to the cost-efficient organization determined by the weighted-average input price vector, we obtain results that are practically coincident with those of Table 2, as shown in Table 3. In the local-public-transit industry, an optimal centralized allocation has a straightforward application as far as some regulatory-policy issues are concerned. In fact, in competitive-tendering procedures which the local authorities use to select firms, it can be used to set both the standard costs, which act as reserve-prices, and the two main dimensions of the service contract: the total vehicle-kilometres to be supplied and the service area to be covered. Finally, we remark that the huge difference in the optimal number of firms/service contracts featured in the two centralized allocations - may have non-trivial implications as far as the adjustment costs from the actual to the optimal configuration are concerned.
Fig. 2. Centralized technical efficiency problem: ray average productivity in a neighborhood of.k .
of ray average productivity in attaining a minimum, when k increases, is responsible for the high optimal number of firms. With regard to the relation of the VRS to the CRS solution, we find that in this data set the two solutions are practically coincident, because = CRS = 0.9571 and kCRS = 485.18. It is interesting to compare the above-described results with those stemming from the corresponding MIP problem. Here we obtain kMIP = 485 and MIP (kMIP ) = 0.9571, results that are similar to those of our method, exception made for the missing information regarding the occurrence of multiple solutions in the interval [482, 485]. The difference between multiple solutions, along with the practical coincidence between the VRS and CRS solutions, is further illustrated by means of an additional data set in Appendix B. As far as the cost-efficient centralized allocation is concerned, by applying the specific algorithm described in Ref. [6], the minimum industry cost is reached at kc = 97 , which determines a cost-efficiency measure R 0 = 0.5501. The CRS solution is practically coincident with the VRS, because of both an identical value of the cost-efficiency measure and kc , CRS = 97.0362 . Fig. 3 illustrates the behavior of the ray average cost in a neighborhood of kc . Decomposition (6) of the cost efficiency measure, in Appendix A.2, Table 2 Comparison of optimal industry structures. Inputs
Cost-efficient Organization Technical-efficient Organization
Outputs
Staff (Number)
Vehicles (Number)
Commodity (Number)
Vehicle-Km (Millions)
Service Area (Km^2)
243.54 90.10
176.33 43.24
6036765.90 1199623.93
9.22 1.85
2281.30 813.98
7
Firms (Number)
97 482
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Table 3 Weighted input-price vector - Cost efficient optimal structure. Inputs
Cost-efficient Organization
Outputs
Staff (Number)
Vehicles (Number)
Commodity (Number)
Vehicle-Km (Millions)
Service Area (Km^2)
243.13
176.08
6028702.01
9.20
2276.86
5. Concluding remarks
Firms (Number)
97
used to test the robustness of MIP solutions. The empirical application of our proposed computation procedure has revealed deep differences between the technically and the cost-efficient centralized allocations, as regards input and output mixes as well as the optimal number of operating units, which is an important finding from a managerial and regulatory point of view. Moreover, in both data sets employed, we have ascertained the occurrence of multiple solutions to the centralized technical efficiency problem, a circumstance that explains the coincidence between the estimates of ray average productivity in CRS and VRS solutions.
This paper has introduced a unified framework for the analysis and the determination of the relationship between technically and cost-efficient centralized allocations featuring reallocation of outputs and a variable number of firms. The use of Banker's [2] conceptual approach resulted not only in a clear economic interpretation of these optimal allocations and in an inequality relationship linking the corresponding efficiency measures, but also in a rigorous method of estimation - which delivers complete information on optimal solutions and can also be Appendix A A.1. Proof of Proposition 1
From the definition of ray average productivity given in (4) and from the inequality constraints of problem (3a), which hold in a solution with the
equality sign at least for one vector-component, we have
0, h
= max i
{ }= kxih Xi0
and
0, h
= max r
{ } = 1. Therefore, Yr 0 kyrh
ciprocal of the ray average productivity of (k x h, k yh) relative to (X 0, Y0) , is minimized in an optimal solution
0, h 0, h
= , which is the re-
. Q.E.D.
A.2. Proof of Proposition 3 For a generic solution of problem (5), (x c , yc ) ,
0, c
= max r
{ } = k , because of the inequality-constraint holding with the equality sign at least for Yr 0 yrh
k px
one vector-component. Then, by applying equation (12) in [7, p. 124], R 0 = pX c - the ray-average-cost ratio of (X 0, Y0) with respect to (x c , yc ) - can 0 be decomposed into the algebraic sum of a ray-average-productivity and an allocative component
R0 = where
0, c 0, c
0, h
ps0 pX 0
= max i
0, c
{ }, s xic Xi0
(6) 0
=
0, c X 0
xc
0 is a non-negative input-slack vector and R 0
min R 0 . In a solution to problem (3a), c
= min m
0, m 0, m
where its minimizer m does not necessarily coincide with c , the minimizer of R 0 in problem (5). c we have R 0 < . Q.E.D. . Second, when m First, when m = c , expression (6) makes evident that R 0 As a comment to this result, we point out that decomposition (6) obtains only under the assumption of firms facing an identical price vector, therefore Proposition 3 may not necessarily hold under the alternative assumption of a varying input-price vector. A.3. Proof of Proposition 4 Consider the lowest value of k such that corresponding efficient-point,
(
(
X (k min ) k 0 , min
)
X 0 Y0 , k T , which we denote k Y0 resulting from step 4), is kmin
)
as k min . The Proof of Proposition 1 can then be employed to show that the a radial efficient point as defined in Ref. [29]. Then, Theorem 5 in [29, p.
247] can be applied to prove that for each k > k min the ray average productivity of this point is strictly decreasing until a global minimum is reached at k , while it becomes strictly increasing for each k > k . Q.E.D. Appendix B
The occurrence of multiple solutions to the centralized technical efficiency problem is further investigated by means of a secondary data set employed in Ref. [7], made up of 246 observations (see, ibid., section 6.2 and appendix B for a detailed description). The solution to the CRS-LP problem gives CRS = 0.4129 and kCRS = 149.56, while - by adding ten fictitious observations in a neighborhood of kCRS [146, 150]. As far as the VRS-MIP solution is concerned, we obtain MIP = 0.4129 and kMIP = 149 , our VRS-LP solution yields = 0.4129 and k which coincides with our estimates - exception being made for the information regarding the existence of multiple solutions. Table B.1 shows the differences between the multiple optimal solutions to the VRS problem, as determined by our method.
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Table B.1
Comparison of multiple VRS solutions. Peers Dmu 11
Dmu 17
Dmu 160
Dmu 226
Dmu 234
0.008912 0.006366 0.003854 0.001376 0
0 0 0 0 0.005227
0.066824 0.066094 0.065374 0.064663 0.063236
0.709215 0.706083 0.702993 0.699944 0.696846
0.215049 0.221457 0.227779 0.234016 0.234691
k* 146 147 148 149 150
It is interesting to observe that the solutions corresponding to 149 and 150 differ in a non-negligible way from that corresponding to 146, this confirms the usefulness of our method in providing information which is conversely not available in the MIP approach. Finally, we remark that, in this data set, the extended version of our algorithm would have not required the adjustment of the initial estimate of n0 . Appendix C In [6, p. 41] it is affirmed that the convexity of T and the positivity of the cost at any observed non-null output vector - assumptions that hold in our model (see Section 2.1) - imply a u-shape form of the ray-average-cost curve of a given output mix. This ensures that the algorithm for the industry cost-optimal allocation determines an optimal solution to problem (5). Here we detail this argument. Shephard [33] shows that under convexity of the technology the cost function is non-decreasing and convex in outputs. Therefore, in our model Y0 at given prices p - the cost function associated to the contraction of the given output mix Y0 can be illustrated as follows.where y with > 0 and point A represents the optimal scale size. The algorithm presented in [6, p. 41] determines a sequence of points along the cost function belonging to the polyline which goes from C to B. The ray average cost of a generic point in the sequence is given as the slope of line segments such as OC′ and OB’. Therefore, Fig. 4 clearly shows that the ray average cost of the industry's output mix is monotonically decreasing either in an contraction or in an expansion towards the optimal scale size A. Note that this conclusion is not altered by the possibility of multiple optimal-scale sizes, i.e. the case in which a segment of the cost function is tangent to the straight line OA.
Fig. 4. Cost function of an industry's output mix.
Appendix D Assume that to each production unit can be associated a specific input-price vector, then in T the general problem of minimizing the industry cost - by means of a reallocation of the aggregate output over an unknown number of optimal scale sizes - can be expressed as 9
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G. Cesaroni k p x h=1 h h
min
{(x h, yh)} k
s.t.
yh
Y0
(5a)
h=1
= 1, and k is an integer to be determined. where x h and yh Problem (5a) is clearly indeterminate, because its objective function is a sum of linear functions whose number k is unknown. Contrary to what happens in the case of uniform input-price vectors ph = p (see Ref. [6], p. 39), the transformation used for the formulation of problem (3a) is not of any help in this context. In fact, considering the arithmetic average of the k optimal-scales we obtain n j=1
k
k
h j xj ,
n j=1
n j=1
h j yj
h j
k
ph x h h=1
ph xh h=1
k
h j
n
x with j where x h and j = 1 j = 1. h=1 k j=1 j j Therefore, problem (5a) remains indeterminate and requires the devise of a heuristic procedure to determine an approximated solution. In our opinion, this is the reason why this specific problem has not yet been dealt with in the literature: as a noteworthy example, the recent contribution of Ray [30, p.73] assumes ph = p for any set of units whose aggregate output must be reallocated among an unknown number of units. n
References [20]
[1] Asmild M, Paradi J, Pastor J. Centralized resource allocation BCC models. Omega 2009;37:40–9. [2] Banker RD. Estimating most productive scale size using data envelopment analysis. Eur J Oper Res 1984;17:35–44. [3] Banker RD, Charnes A, Cooper WW. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 1984;30:1078–92. [4] Banker RD, Thrall RM. Estimation of returns to scale using data envelopment analysis. Eur J Oper Res 1992;62:74–84. [5] Baumol WJ, Panzar JC, Willig RD. Contestable markets and the theory of industry structure. New York: Harcourt Brace Jovanovich; 1982. [6] Cesaroni G. Industry cost efficiency in data envelopment analysis. Soc Econ Plan Sci 2018;61:37–43. [7] Cesaroni G, Giovannola D. Average-cost efficiency and optimal scale sizes in nonparametric analysis. Eur J Oper Res 2015;242:121–33. [8] Cesaroni G, Kerstens K, Van de Woestyne I. Global and local scale characteristics in convex and nonconvex nonparametric technologies: a first empirical exploration. Eur J Oper Res 2017;259:576–86. [9] Charnes A, Cooper WW, Rhodes E. Measuring the efficiency of decision making units. Eur J Oper Res 1978;2:429–44. [10] Deprins D, Simar L, Tulkens H. Measuring labor efficiency in post offices. In: Marchand M, Pestieau P, Tulkens H, editors. The performance of public enterprises: concepts and measurements. Amsterdam: North Holland; 1984. p. 243–67. [11] Emrouznejad A, Parker B, Tavares G. Evaluation of research in efficiency and productivity: a survey and analysis of the first 30 years of scholarly literature in DEA. Soc Econ Plan Sci 2008;42:151–7. [12] Fang L. A generalized DEA model for centralized resource allocation. Eur J Oper Res 2013;228:405–12. [13] Fang L, Li H. Cost efficiency in data envelopment analysis under the law of one price. Eur J Oper Res 2015;240:488–92. [14] Färe R, Grosskopf S, Zelenyuk V. Aggregation of cost efficiency: indicators and indexes across firms. Acad Econ Pap 2004;32:395–411. [15] Farrell MJ. The measurement of productive efficiency. J R Stat Soc Ser A 1957;120(3):253–81. [16] Førsund F, Hjalmarsson L. Generalized Farrell measures of efficiency: an application to milk processing in Swedish dairy plants. Econ J 1979;89:294–315. [17] Golany B, Tamir E. Evaluating efficiency-effectiveness-equality trade-offs: a data envelopment analysis approach. Manag Sci 1995;41:1172–84. [18] Kannan R, Monma CL. On the computational complexity of integer programming problems. In: Henn R, Korte B, Oettli W, editors. Optimization and operations research. Lecture notes in economics and mathematical systems, vol 157. Berlin: Springer; 1978. p. 161–72. [19] Koopmans TC. An analysis of production as an efficient combination of activities.
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In: Koopmans TC, editor. Activity analysis of production and allocation, Cowles commission for research in economics, monograph n. 13. New York: Wiley; 1951. Kuosmanen T, Cherchye L, Sipiläinen T. The law of one price in data envelopment analysis: restricting weight flexibility across firms. Eur J Oper Res 2006;170:735–57. Li SK, Cheng YS. Solving the puzzles of structural efficiency. Eur J Oper Res 2007;180:713–22. Lozano S, Villa G. Centralized resource allocation using data envelopment analysis. J Prod Anal 2004;22:143–61. Lozano S, Villa G. Centralized DEA models with the possibility of downsizing. J Oper Res Soc 2005;56:357–64. Maindiratta A. Largest size-efficient scale and size efficiencies of decision making units in data envelopment analysis. J Econom 1990;46:39–56. Mar-Molinero C, Prior D, Segovia M, Portillo F. On centralized resource utilization and its reallocation by using DEA. Ann Oper Res 2014;221:273–83. Nesterenko V, Zelenyuk V. Measuring potential gains from reallocation of resources. J Prod Anal 2007;28:107–16. Peyrache A. Industry structural inefficiency and potential gains from mergers and break-ups: a comprehensive approach. Eur J Oper Res 2013;230:422–30. Peyrache A. Cost constrained industry inefficiency. Eur J Oper Res 2015;247:996–1002. Podinovski V. Efficiency and global scale characteristics on the “No free lunch” assumption only. J Prod Anal 2004;22:227–57. Ray SC. Cost efficiency in an Indian bank branch network: a centralized resource allocation model. Omega 2016;65:69–81. Ray S, Hu X. On the technically efficient organization of an industry. A study of US airlines. J Prod Anal 1997;8:5–18. Ray SC. Data envelopment analysis: theory and techniques for economics and operations research. Cambridge, United Kingdom: Cambridge University Press; 2004. Shephard RW. Theory of cost and production functions. Princeton, New Jersey: Princeton University Press; 1970. ten Raa T. Benchmarking and industry performance. J Prod Anal 2011;36:285–92. Ylvinger S. Industry performance and structural efficiency measures: solutions to problems in firm models. Eur J Oper Res 2000;121:164–74.
Giovanni Cesaroni is Senior Economist at the Prime Minister's Office since 2000. After receiving his PhD degree from the University of Rome “La Sapienza” in 1998, he was appointed Assistant Professor in economics at the University of Rome ‘Tor Vergata’ in 1999. In the period 2008–2014, he has been a member of various spending review units of the Italian Government. His current research focuses on public-administration organization and applied microeconomics, while former research dealt with macroeconomics and financial intermediation. His scientific articles have been published in national and international academic journals. He was awarded the certificate of excellence in reviewing by the European Journal of Operational Research.
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Socio-Economic Planning Sciences journal homepage: www.elsevier.com/locate/seps
Technically and cost-efficient centralized allocations in data envelopment analysis Giovanni Cesaroni Department for public administration, Via del Sudario, 49 - 00186, Rome, Italy
ARTICLE INFO
ABSTRACT
Keywords: Dea Structural efficiency Industrial organization Ray average productivity Ray average cost
In data envelopment analysis and with a variable returns to scale production-technology, we apply Banker's [2] approach to determine the relationship between technically and cost-efficient industry structures, featuring reallocation of outputs and a variable number of firms. The interpretation based on the most productive and optimal scale-size notions allows us to both establish an inequality relationship between the corresponding industry-efficiency measures and provide adequate information on optimal solutions. At the applicative level, we introduce an exact algorithm to solve related non-linear programming problems, thus providing the decision maker with an accurate method for computing and comparing the input and output mixes and the optimal number of units obtained in the two allocations. Empirical illustration, given with reference to the Italian localpublic-transit sector and employing a multiple inputs and outputs technology, reveals striking differences with regard to the managerial and regulatory implications of the two centralized allocations.
1. Introduction Notwithstanding the early definition of the structural efficiency of a group of firms advanced by Farrell [[15], pp. 261–262] in his seminal contribution, the vast amount of literature on the non-parametric measurement of productive efficiency initiated by the model of Charnes et al. [9] has been almost exclusively concerned with the efficiency of the individual firm or decision-making unit (for a comprehensive review, see Emrouznejad et al. [11]). Only in recent years has the technical efficiency of a group of units been consistently investigated and developed. Førsund and Hjalmarsson [16] proposed to implement Farrell's notion - the output-weighted average of individual efficiencies - by evaluating the technical efficiency of the average firm. This approach was later applied to multiple-output production by Ylvinger [35], who employed DEA models to obtain technical efficiency measures based on the determination of input/output shadow prices under the assumptions of constant returns to scale (CRS) and no “reallocation of inputs across firms”. The studies cited so far have as a common characteristic that of not taking into account both the distinction between aggregate and individual production technologies and the possibility of reallocation of inputs among the existing firms. In this respect, an important progress is made by Li and Cheng [21], who analyze the output-oriented technical efficiency of the industry by making explicit the role of these two factors in a shadow price DEA model. They prove that in a
convex technology the structural efficiency of the industry in a multiple input and output technology can be taken to be equivalent to the technical efficiency of the average unit, and that measures in Ref. [35] are a particular case of their general solution. Moreover, the shadow price vector of the optimal output provides a general decomposition of structural efficiency, which clarifies that the divergence between the industry and the sum of individual technical-efficiency measures may be due to allocative inefficiencies of individual firms (see [21, p. 715]). Similar conclusions are reached by Nesterenko and Zelenyuk [26], in a setting which differs from the former only in the fact that output prices are exogenously given and efficiency is defined in terms of maximal revenue. In Refs. [21,26], convexity of the technology implies that the industry-benchmark is the same for each of the existing firms. Inspired by the model of industry efficiency proposed in Golany and Tamir [17], more recently, a new strand of literature has come to investigate the technically efficient “centralized resource allocation” of a group of units operating under the same central management by focusing mainly on the effects of specific and different rules of reallocation of inputs/outputs among the controlled units: each of these may be projected on a different point along the frontier of the individual technology (see, e.g., Lozano and Villa [22,23], Asmild et al. [1], Fang [12]). When compared to the previous literature, this strand does not introduce an aggregate technology, thus failing to properly determine the industry optimum while not addressing the issues linked to the divergence between group and individual efficiency measures (see the
E-mail address: [email protected]. https://doi.org/10.1016/j.seps.2019.100734 Received 1 October 2018; Received in revised form 7 May 2019; Accepted 12 August 2019 0038-0121/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Giovanni Cesaroni, Socio-Economic Planning Sciences, https://doi.org/10.1016/j.seps.2019.100734
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puzzles in [21, p. 714]). In the context of technical-efficiency analysis with reallocation, so far discussed, we may remark that the determination of the optimal number of firms in DEA as a part of the industry optimum - under variable returns to scale (VRS) - was first addressed by Ray and Hu [31] in a single-output production technology, and it was later solved by Peyrache [27] and Mar-Molinero et al. [25] in a general multipleoutput setting but without offering a clear economic interpretation of the centralized allocation. All of these contributions took a mixed-integer programming (MIP) approach to solving the optimization problem in question. The same approach is also taken in Peyrache [28], where cost-constrained technical efficiency of the industry is considered in an environment with input price segmentation.1 As far as the cost efficiency of an industry is concerned, fewer results regarding non-parametric production analysis are available in the literature. To be more precise, some important conclusions regarding the aggregation of individual cost efficiencies (Färe et al. [14]) and the determination of input shadow prices (Kuosmanen et al. [20], Fang and Li [13]) have been reached for a given number of units when reallocation of outputs across them is not possible. However, it is only in Ray [30] and Cesaroni [6] that a cost-efficient centralized allocation featuring a variable number of firms and reallocation of outputs comes to be analyzed. With respect to Ref. [30], which is more empirical in nature, [6] offers a theoretical characterization of the centralized allocation by considering: the distinction between individual and industry technologies, the general form of the allocation problem2 and the characteristics of its global optimum. In addition, [6] provides a decomposition of the industry efficiency measure similar to that of Li and Cheng [21] along with its rigorous economic interpretation based on the ray average cost. Even from the computational point of view, [6] follows a different approach by introducing a specific algorithm to solve the non-linear programming problem, where instead [30] transforms the same problem into a mixed-integer one. To the best of our knowledge, no contribution has so far either dealt with the relation linking technically and cost-efficient centralized allocations under a variable number of firms or proposed a robust alternative to the MIP method, which might be affected by computational inaccuracy due to NP-completeness and also not deliver information on multiple optimal solutions. We finally remark that another missing point in the debate concerns the actual relationship between CRS and VRS technical optima in an industry with a multiple-output technology (see, e.g., Refs. [21,25–27,30]), whereas the single-output case was discussed in Ref. [31]. This article contributes the literature on centralized allocations in several ways by adopting Banker's [2] approach, and its extension to cost analysis made by Cesaroni and Giovannola [7]. First, we provide a clear interpretation of technically efficient centralized allocation as a most productive scale size (MPSS) in a multiple inputs and outputs technology, which is currently absent from the literature and will prove to be essential for our subsequent conclusions. Second, we characterize the relation of CRS to VRS optimum. Third, we ascertain the relationship linking technical and cost efficiency industry measures. Fourth, we propose a rigorous approach to the computation of optimal centralized allocations, which is based on the properties of the ray average productivity and ray average cost functions, and compare it with the MIP approach. Finally, we provide the regulator (centralized decisionmaker) with some empirical evidence on the differences between
technically and cost efficient industry allocations, and on the relationship between CRS and VRS solutions - which is currently not available in the literature. The paper is structured as follows. Section 2 introduces the model and derives the programming problem associated with technically efficient centralized allocation. Section 3 provides interpretation of this optimal allocation and illustrates its relation to the cost-optimal analogue. Sections 4.1 deal with the key issue of the global optimum of the two centralized allocations and introduces a suitable computational procedure, along with a comparison of the two competing computational-approaches. Section 4.2 illustrates the empirical application. Section 5 briefly summarizes the relevance of our findings. 2. The model and the industry technical-efficiency problem 2.1. The model We suppose that the industry is made up of n firms featuring the same convex VRS production technology as given in the DEA model of Banker et al. [3], also known as the BCC model. Introducing notation, we have n observations, indexed by j (j = 1, ..., n) , using inputs x ij (i = 1, ..., m) to produce s outputs, yrj (r = 1, ..., s ) . The observed input 0 and yj = (y1j , ..., ysj ) 0 , reand output vectors are xj = (x1j, ..., xmj ) spectively, where the prime indicates the transposition operation.3 Given the postulates of convexity, free disposability of inputs and outputs and minimum extrapolation, the individual production possibility set can be expressed as n
T=
(x‚y)
n j xj
j=1
x‚
n j yj
j=1
y‚
j
= 1,
j
0; j
J
j=1
where J = {1, ..., n} . The observed industry's input and output vectors are X 0
and Y0
n y j=1 j ,
(1) n x j =1 j
respectively. According to Refs. [21,26], the tech-
nology at the industry level T IND is the sum of the individual possibility sets, thus we have
T IND = nT
(2)
as a result of the summation of identical convex sets. Note that this kind of aggregate technology, contrary to that based on the sum of individual input-requirement sets (employed in Ref. [14]), allows for reallocation of output across firms in the industry.4 Moreover, T is not necessarily included in T IND . As far as input prices are concerned, we will assume that each unit faces the same exogenously-given vector p = (p1 , ..., pm ) > 0 . This may correspond to either a situation in which different branches/offices are controlled by a central authority (e.g., a parent company or a publicsector department) which buys and provides inputs to them, or one in which the regulator adopts a set of input market prices - or rather a function of these indicating their shadow prices - to gauge the efficiency of the industry. Especially with regard to this latter case, it can be remarked that Koopmans [19] revealed that the uniformity of prices across producers is a necessary and sufficient condition for Pareto efficiency (see [20, pp. 736–737]). We also wish to point out that the assumption in question is necessary to establish an unambiguous relationship between technical and cost efficiency industry measures (see Appendix A.2). As far as the case of input prices varying across units is concerned, to the best of our knowledge, the literature currently does
1 However, we note that the aggregate technology implicit in the solution proposed in this contribution, which is based on indirect output sets, is nonconvex as a result of the summation of non-identical convex sets. Therefore, strictly speaking, this model goes beyond the scope of our paper. 2 We remark, in particular, that Ray [30] does not offer a formal argument for the optimality of the allocation in which different units share the same input and output vector (see, ibid., fn. 6, p. 73).
3 The vector-inequality sign means that at least one element of the vector is strictly greater than zero. 4 Contrary to what holds in (1), a convex input-requirement set does not allow for a convex combination of the output vectors of the observed units. This implies that each firm can at most produce its observed output vector (cf. [20, p. 742]).
2
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not offer a solution to the complex optimization problem which arises with reallocation of output over a variable number of firms (see Appendix D). Finally, we remark that, even if this solution were available, the assumption of unit-specific and known input-prices might not make sense: in fact, the known input-price vectors of the existing units do not necessarily apply to the new optimal units resulting from the efficient organization, because these latter may not correspond to observed units - as a consequence of the convexity of T.5
MIP problem. Although the literature does not discuss the consequences of this transformation, we will argue below that this choice may imply a cost. Second, if we denote by asterisks an optimal solution to (3a), we can note that (k y , k x ) TIND as long as the additivity assumption underlying (2) is referred to k firms, instead of n . 3. Interpretation of the technical-efficiency measure and relationship with its cost analogue
2.2. The industry technical-efficiency problem
3.1. Interpretation of the technical-efficient centralized allocation
To put the problem in its general form, we introduce the concept of technically efficient centralized allocation, or technically efficient organization of the industry:
A clear economic interpretation of technical structural-efficiency in DEA is at present not available. In fact, in a single-output technology with a variable number of firms, Ray and Hu [31, p. 11] suggest that the corresponding centralized allocation does not maximize the average productivity and therefore does not coincide with an MPSS. In a general multi-output setting and with a given number of firms, Li and Cheng [21] decompose the technical structural-efficiency measure without an average-productivity role being called for. The issue we raise is relevant not only from the interpretative/managerial point of view but, as we will reveal in Section 4.1.1, it is also crucial for the exact determination of optimal solutions. Banker [2, p. 37 and ff.] introduced the concept of MPSS in a VRS production technology as “the scale size at which the outputs produced ‘per unit’ of the inputs is maximized”, that is, the production possibility which maximizes the ray average productivity7 of a given input and output mix. By using expression (10) in Cesaroni and Giovannola [7, pp. 124] and Podinovski's [29, p.233] formulation of the optimization problem, we have the following definition.
Definition 1. The technically efficient organization of the industry is a set of production possibilities (xh , yh) T , h = 1, ..., k which minimizes in the same proportion the use of inputs for the industry output vector y0. Note that h is an index for the generic element of the set and k is an integer variable. In other words, the technical-efficient organization of an industry is that allocation of the observed industry output vector among a set of efficient points to be determined, {(xh ‚yh)} , which achieves the greatest proportional reduction in all inputs. The optimization problem posed by Definition 1 is
min {(x h, yh)} k
s.t.
X0
xh h=1
Definition 2. A given production possibility (xh, yh) only if
k
yh
Y0
(3)
h= 1
min
= 1. where x h yh Observe that this general formulation of the problem, allowing in principle for more than one efficient point, cannot be found in either Maindiratta6 [24] or the contributions of Ray and Hu [31] and MarMolinero et al. [25]. An output-oriented version of this formulation was first introduced by Ray [32]. As far as the form of the solution is concerned, we obtain important information by means of the following re-formulation of problem (3) n j=1
h j xj ,
n j=1
h j yj and
n j=1
h j
k yh
X0
The above definition obviously apply to any VRS production set,
such as T IND . Being
h, m
= max i
{ } and, xim xih
h, m
= max r
yrh 8 yrm
we point out
that expression h, m h, m is the ray average productivity of (xh, yh) re1 lative to (xm, ym) , and that its reciprocal, , indicates the ray h, m
k xh
h, m
m
CRS technical-efficiency score; b) the production possibility m yielding maximal productivity for an input-output mix h which is not an MPSS, i.e. the minimizer of program (4) when h, m h, m < 1, is an MPSS. Given these preliminaries, we can now investigate the characterization of a technical-efficient centralized allocation. An initial important property is established by the following Proposition.
(3a)
j x j yh n j=1 j
{ }
average productivity of (xm, ym) relative to (xh, yh) . Note that Definition 2 implies that for an MPSS h, m h, m 1. Moreover, the Proof of Proposition 1 and Corollaries 2 and 3 in Banker [2] can be used to show that: a) min h, m h, m = CRS , where CRS is the
Y0
n j=1
(4)
where subscript m denotes a generic point of T.
where (xh, yh) is the arithmetic average of k efficient-points, i.e.
xh
=1
(xm, ym)
min k, (xh, yh) s.t.
h, m h, m
T is an MPSS if and
n j=1
j yj
with
h
j
j k h=1 k
, which belongs to T
= 1. because of This means that convexity of the technology reduces problem (3) to the search for a single efficient point being replicated k times, a property which is not valid for non-convex production technologies such as that of the free disposal hull (FDH, Deprins et al. [10]). Therefore, it is natural to interpret k as the optimal number of firms in a technicalefficient centralized allocation. With regard to problem (3a), two preliminary remarks are in order. First, it clearly is a non-linear integer programming problem in k and in the j s, but - as shown in Ray and Hu [31] and especially in Peyrache [27] and Mar-Molinero et al. [25] - it can easily be reformulated as a
Proposition 1. An optimal solution to (3a), (k x , k y ) , maximizes the ray average productivity of (k x h, k yh) with respect to the aggregate input and output mix (X 0, Y0). Proof. See Appendix A.1. This important result, which is new to the literature, clarifies that in a VRS industry-technology the technically efficient centralized solution provides the aggregate input and output mix with the maximum ray 7 The term indicates average productivity in a multi-output technology (see Podinovski [29, p. 241]). 8 For the equivalence of our notation with that used by Banker [2], observe that δ = β and γ = 1/α.
5
Cf. Table B1 in Appendix B. 6 Maindiratta [24, p. 62], dealing with size efficiency of a single firm, sets a problem which may be taken as an imprecise formulation of our problem (3). 3
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average productivity compatible with an integer k, where this can be expressed as 1 (see Appendix A.1.). This means that, by following Banker and Thrall [4], a rigorous economic interpretation of the technical inefficiency of the allocation of (X 0, Y0) among the existing firms can be given in terms of its technical and scale radial inefficiency. The failure of this allocation in maximizing the ray average productivity of (X 0, Y0) is the economic factor which explains what has been so far generically termed as “reallocative efficiency” or “organizational efficiency” (cf., respectively, Li and Cheng [21] and ten Raa [34]). The next remark will be useful for understanding the relationship between VRS and CRS solutions.
of times, where kc can be interpreted as Baumol et al. [5]'s “efficient number of firms”. The associated industry cost-efficiency measure is kc px c R 0 = pX , where (x c , y c) can be interpreted as the production possi0 bility which minimizes the ray average cost of the aggregate input and output mix (X 0, Y0) subject to the integer constraint on the optimal number of firms (see Section 3.1 in Cesaroni [6]). Moreover, it can be immediately checked that remarks analogous to Remarks 1 and 2 also hold for problem (5). We can conclude that (x , y ) and (x c , y c) maximize and minimize, respectively, the ray average productivity and the ray average cost of (X 0, Y0) subject to the integer constraint on the optimal number of firms. This consideration introduces a central Proposition of this paper.
Remark 1. The CRS technical-efficiency problem9 is equivalent to problem (3a) without the integer constraint being imposed on k.
Proposition 3. In a VRS centralized allocation the cost-efficiency measure is not greater than the technical-efficiency measure, R 0 .
Accordingly, another remark can be immediately derived Remark 2. In a technically efficient centralized allocation
CRS
Proof. See Appendix A.2.
.
The intuition of this general result, which holds irrespectively of the values taken by the two distinct optimal-numbers of firms involved (k and kc ), relies on the fact that - with respect to - R 0 includes an allocative component. As a result of Proposition 3, the same relationship linking primal and dual individual efficiency-measures in an individual technology - where the integer constraint does not apply (see Corollary 2 in [7, p. 125]) - also holds for aggregate VRS technologies, where this constraint is present. We may remark, as shown in the Proof, that this conclusion depends strictly upon the interpretation of as the ray average productivity of (X 0, Y0) .
Obviously, the equality sign will hold when a CRS optimal solution implies an integer k. Moreover, it should be now evident that the VRS optimal solution is an integer approximation of the CRS one, this implies that the difference CRS can be very small in applications with a sufficiently-large number of firms (for an explanation see Section 4.1.2). On the basis of Remark 2, we can clarify the issue of the interpretation of the optimal centralized allocation (k x , k y ) as an MPSS by means of the following Proposition Proposition 2. The allocation (k x , k y ) is not necessarily the MPSS of (X 0, Y0) . Proof. It follows immediately from
1
1 CRS
4. Solution methods and empirical illustration
, i.e. (k x , k y ) does not
4.1. Solution methods
necessarily ensure maximal productivity for (X 0, Y0) as a consequence of the integer constraint on k. Q.E.D. For a graphical illustration, see Fig. 1.
In previous sections we presented a theoretical characterization of both technical-efficient centralized allocation and its relation to the cost-efficient analogue. However, from an operational point of view, the main difficulty concerns the computation of the solution of the integer non-linear programming problem involved by the two centralized allocations (see, respectively, problem (3a) above, and problem (4b) in Ref. [6]). In this regard, the approach commonly followed in the literature is that of transforming these non-linear problems into MIP problems, as clearly illustrated in Refs. [25,27,30,31]. Contrary to this approach, [6] proposes a computation procedure which preserves the non-linear nature of the programming problem and has its logical foundation in the theoretical characterization of the centralized allocations as ray-average-productivity and ray-averagecost optima, i.e. as MPSS and OSS respectively. The computation procedure in question can also be applied to problem (3a), which is now handled as a two-stage problem where the first stage is
Finally, note that (k x , k y ) has the same ray average productivity as (x , y ) : the properties established by Propositions 1 and 2 also hold for this latter allocation, which belongs to the individual technology T. Summing up, we can conclude this section by emphasizing that a VRS technical-efficient centralized allocation can be considered an integer-constrained MPSS of (X 0, Y0) (Proposition 1), but that the same allocation is not necessarily an MPSS of (X 0, Y0) (Proposition 2). These conclusions show that not only is a specific maximization of productivity implied in the VRS aggregate technology, but the possible divergence from the absolute maximum may also be limited, because it is exclusively due to presence of the integer constraint (see Section 4.2, and Appendix B). 3.2. The relationship between technically and cost-efficient centralized allocations
min (k ) j
The same line of reasoning employed in Section 2.2 can be followed to show that in a VRS technology the cost-efficient centralized allocation can be obtained as a solution of the following problem (see Cesaroni [6], p. 39)
s.t.
k:
min k px j, k
s.t. k y
n
min k
n
where x = j = 1 j xj and y = j = 1 j yj , with j = 1 j = 1. An optimal solution to (5), (kc x c , kc y c ) , has the same characterization ascertained above for its technical-efficient counterpart: a single optimal scale size (OSS), (x c , y c) T , being replicated an integer number n
x h and yh
X 0 Y0 , k k
Y0 , k
T
(6a)
and the second stage is
(5)
Y0
X0 k
(k )
(6b)
where in both problems k = 1, ..., n 0 ; n 0 is a pre-determined integer value, (k ) is the optimal value determined in (6a) and k is the solution to (6b). The first stage solves the technical efficiency problem for each notional average-unit corresponding to a given k, while the second stage selects the value k which yields the minimum technical-efficiency
9 In the CRS case, which is obtained by dropping the constraint in (1), the individual and the industry technologies coincide.
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Fig. 1. Individual and aggregate technologies, optimal allocations.
score among those computed at stage one. With regard to (6a), note that not all of the notional average-units will belong to T, because this is ensured only for k = n, so that a key element of the solution procedure will involve checking whether this condition holds10. To determine a solution to the two-stage problem at hand we propose the following algorithm:
(
productivity curve, outside the interval in which its minimal value is reached, implies that once that the proposed algorithm finds a solution, this is actually the sought global-minimum (the same outcome may not be given when the behavior of this curve is not monotonic, as in the non-convex case).11 Therefore, Proposition 4 establishes a property of accuracy of our solution method. Note, moreover, that this property necessarily depends on Proposition 1, that is the interpretation of the technically efficient centralized allocation as an integer-constrained MPSS. The only practical difficulty with our algorithm lies in choosing a suitable range for k in order to be reasonably sure of finding the desired minimum and avoid repeating the procedure. As regards this problem, observe that it can arise only in conditions which yield k > n so that as an initial choice one can try n0 = 2n . If this should turn out to not be sufficient, the algorithm can be repeated in the enlarged interval given by n0 = 3n , and so on. The same accuracy in the determination of an optimal solution is granted by the analogous algorithm for the computation of the costefficient centralized allocation: convexity ensures the same monotonic property to the ray average cost of a cost-efficient point which is not an OSS (see [6, p. 41], and Appendix C for a more detailed argument).
)
1) Construct a set of fictitious observations k0 , k0 with k = 1, ..., n 0 , and add this to the original set of observations which determine T. 2) Considering T as the reference technology, i.e. the frontier determined by the actual set of firms, perform technical-efficiency analysis on the enlarged set of observations. 3) Exclude fictitious observations featuring an efficiency score greater than one: i.e. they do not belong to T. 4) Perform technical-efficiency analysis on the new set of observations. This completes the solution to the first stage (6a) by determining the efficient point - and the associated efficiency score (k ) - corresponding to each fictitious observation. 5) The solution to the industry problem (second stage, 6b) is given by the efficient point - and the associated k - which ensures the minimal (k ) among those obtained at stage 4). X
Y
As for step 5), we would like to underline that the detection of the minimum in the range of k chosen in step 1) is made unambiguous by the form that the ray average productivity curve (i.e. the value function defined in Podinovski [29, p. 245]) assumes due to the assumption of convexity of T.
4.1.1. Computational simplification and comparison with alternative approaches In this section, we deal with the two issues represented, respectively, by the additional cost in terms of computational burden of the precision ensured by our method and the evaluation of its features relative to those of alternative solution methods. As for the first question, observe that within a DEA softwarepackage that allows for the specification of the reference technology as a subset of units, the algorithmic sequence from step (2) to step (5) boils down to a single step procedure (i.e., a single linear-programming problem). In fact, this software can be applied to estimate technical-
Proposition 4. The algorithm made up of steps 1) to 5) determines an optimal solution to problem (3a). Proof. See Appendix A.3. In other words, the monotonic behavior of the ray-average10 It can be easily shown that last condition in (6a) holds if and only if (x 0, y0) kT . Therefore, it ensures that all of the relevant points of the VRS aggregate technology are being considered.
11 An empirical comparison of the behavior of this curve in convex and nonconvex technologies is recently offered in Cesaroni et al. [8].
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efficiency scores of the enlarged and ordered set of observations, made up of n + n0 units :then the optimal solution is readily determined by the fictitious observation having a minimal score lower than 1. Thus, the only inconvenience may eventually be represented by the necessity of adjusting the initial guess of n0 (on the contrary, cf. Appendix B). This possible difficulty can be circumvented by taking Remark 1 into account. According to this remark, the solution of problem (3a), k , belongs to a neighborhood of the CRS solution, kCRS , where the size of this integer-interval will generally depend on the occurrence of multiple solutions: it will be determined by two consecutive integers when a single solution occurs, while it will generally be larger when additional solutions arise. Therefore, our algorithm can be reduced to the solution of two linear-programming (LP) programs: the first - relative to the original set of observations - determining the CRS solution for observation (X 0, Y0) , the second - relative to the set of the original ob-
(
represents the VRS industry technology with n = 2 . The industry's input and output mix is point E, with each of its integer contractions - i.e., the fictitious observations considered in the first stage of our algorithm lying in the interior of the line segment OE. More precisely, F and G are, respectively, the 1 and 1 contractions, i.e. corresponding to k = 2 and 2 3 k = 3. Note that higher-order contractions (e.g., with k 5) do not belong to T. The fictitious observation ensuring the minimal (k ) is F, corresponding to k = 2. This determines the maximal ray average productivity compatible with the integer constraint (see Proposition 1): it is given as the slope of the straight line OH, passing through the intersection of F’F and the frontier segment AB. However, as remarked by Proposition 2, the projection of F on the frontier AB is not necessarily the point of T ensuring the global maximum of productivity: in fact, the MPSS is given by point B, whose contraction factor is 1 < kCRS < 2 , where kCRS is the scale-factor implicit in the CRS centralized allocation. kCRS is 1, so that in Observe, however, that the upper bound of k applications with a large n the difference between the two points along the line segment OE which generate the VRS and the CRS optimal allocations can be small, because of the proximity of 1 and 1 (see,
)
servations enlarged by a few fictitious units k0 , k0 taken in a symmetrical neighborhood of kCRS - for the computation of the VRS solution. Observe, that the addition of these few additional units does not affect either the computational complexity or the solution time of the LP VRS problem (cf. Appendix B). As for the second question, we can now point out that our method has two main advantages over the MIP approach. First, the MIP solution does not allow to test for the presence of multiple solutions, while our method allows for this by supplying information on the behavior of ray average productivity in a neighborhood of kCRS (see Section 4.2 and Appendix B). Note that this specific information is of some value to the decision maker: because of the existence of adjustment costs, in some cases it will be optimal to choose either the minimal or the maximal k belonging to the set of different multiple solutions. Second, being based on two simple LP problems, our solution is not affected by computation problems regarding k , which conversely may arise in complex MIP problems within large data sets. In these cases, it is possible that, as a consequence of the NP-completeness of the MIP problem12 this approach has no effective criterion to check whether the computed solutions are optimal or not. Therefore, computation errors may in principle affect the MIP determination of both technically and cost efficient centralized allocations, while this is not true of our solution. For these reasons, we can conclude that our approach is more general and that it can be used to test the robustness of MIP results. Finally, it can be easily verified that our proposed method is optimal also when compared to a standard LP framework. Herein, in the case of a single VRS solution there would in fact arise the need to solve three LP programs (the CRS problem, the VRS problems associated to the two given integers closest to kCRS ), against the two programs required by our method. Moreover, in the case of multiple VRS solutions, the number of LP programs to be solved would become unknown and greater than three, because of its dependence on the multiplicity of solutions (see Section 4.2). X
Y
Section 4.2).
k
k CRS
4.2. Empirical application We will apply the proposed computational approach to the secondary data set employed in Ref. [6], which is made up by a representative sample of 43 companies operating in the Italian local public transit industry in the year 2012. Observations consider a three input-two output production technology: the outputs are vehicle-kilometres travelled and the size of the transport network (expressed as the area served by a company); the inputs consist of the number of company staff, the number of vehicles, and the quantity of a composite commodity representing the consumption of fuel, energy, materials and spare parts. Each company faces a specific input-price vector made up of: the unit-wage of its staff p1, the unit-depreciation of its stock of vehicles p2, and a price index for the composite intermediate commodity p3. In the computation of the cost efficient centralized allocation, we will assume that the regulator intends to estimate the industry configuration corresponding to two different input-price vectors: the first, given as the arithmetic average of input prices, is (p1 , p2 , p3 ) = (43188.16, 13718.38, 1.02) with prices denominated in Euro, the second - considering the average price weighted by the inputproportion over the industry's aggregate is (p1 , p2 , p3 ) = (45798.52, 15849.02, 0.99) . The summary statistics of companies’ observed input-output data are given in Table 1 (the units of measure of inputs and outputs are shown in brackets).where the mean describes the average unit. The industry's input and output mix can be derived by multiplying the mean by n = 43. To begin, we apply the algorithm of Section 4.1 to the estimation13 of the technical-efficient centralized allocation by considering n0 = 86, i.e. k [1,86]. However, a global minimum is not reached in this range because the ray average productivity of the fictitious observation, (k ) , is still decreasing at k = 86, and for this reason we try with n0 = 129 thus determining the range k [1,129] - but the outcome is the same. Only by enlarging the range to k [1, 516] do we find the global [482, 485] with (k ) = 0.9571. Fig. 2 depicts the minimum at k behavior of (k ) in a neighborhood of. k . We may remark that Fig. 2 is also noteworthy because it illustrates empirically, in an industry setting, the behavior of the ray-averageproductivity curve described by Theorem 5 of Ref. [29]. The slow pace
4.1.2. Graphical representation A simple representation in a one input-one output space will be of great help in illustrating the difference between individual and aggregate technologies, the relation of the VRS to the CRS technology and the functioning of the algorithm introduced in the preceding section. In Fig. 1, the technologies are represented by continuous lines: the frontier of the CRS technology, TCRS , is given by the straight line passing through points 0 and B; the frontier of the individual VRS technology, T, is given by the polyline ABC and by its horizontal extension from point C to the right; the analogous polyline passing through points 2A, 2B and 2C 12
Kannan and Monma [18] prove the NP-completeness of the MIP problem. NP-completeness implies that no known polynomial-time algorithm is available for the solution of a MIP problem, while such an algorithm is available for linear-programming problems. The heuristic methods employed by the MIP solution may thus imply an error.
13 Computations have been performed by means of two R-packages: Benchmarking 0.26 for technical and cost efficiency estimates, and lpSolveApi 5.5.2.0 for MIP.
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Table 1 Input-Output Data. Summary statistics. Inputs
Min Mean Max Stdev
Outputs
Staff (Number)
Vehicles (Number)
Commodity (Number)
Vehicle-Km (Millions)
Service Area (Km^2)
89 1311 11990 2307.91
43 526.28 4263 756.73
1191685.17 14049094.19 108985785.70 22281459.13
1.84 20.76 161.59 32.76
37.86 2091.91 11687.06 2672
Fig. 3. Centralized cost efficiency problem: ray average cost in a neighborhood of.kc .
yields a ray-average-productivity component ( allocative component
(
ps 0 pX 0
0, c
)
0, c 0, c )
= 0.9680 and an
= 0.4179, thus pointing out the abso-
lute relevance that allocative inefficiency may reach in the analysis of optimal industry configurations. In this context, the solution of the MIP problem yields results identical to those of our approach. With regard to the comparison of technical and cost-efficient centralized allocations, note that differences are widespread and pronounced. In fact, not only do efficiency measures substantially differ: (k ) = 0.9571 > R0 = 0.5501, but even the corresponding industry configurations - k , (x , y ) and kc , (x c , y c) - turn out to be quite diverse, as illustrated in Table 2. The large divergence in the optimal number of firms determines a considerably larger optimal scale-size in the case of the cost efficient allocation compared to technically efficient organization. The input and output mixes also differ: a substantially-higher density (ratio of vehiclekm to service area), higher vehicles-per-staff and commodity-per-vehicle ratios characterize the cost efficient allocation when compared to the technically efficient one. Note that in the former allocation, the greater consumption-per-vehicle matches the more intense use of vehicles suggested by the greater vehicle-per-staff ratio. With regard to the cost-efficient organization determined by the weighted-average input price vector, we obtain results that are practically coincident with those of Table 2, as shown in Table 3. In the local-public-transit industry, an optimal centralized allocation has a straightforward application as far as some regulatory-policy issues are concerned. In fact, in competitive-tendering procedures which the local authorities use to select firms, it can be used to set both the standard costs, which act as reserve-prices, and the two main dimensions of the service contract: the total vehicle-kilometres to be supplied and the service area to be covered. Finally, we remark that the huge difference in the optimal number of firms/service contracts featured in the two centralized allocations - may have non-trivial implications as far as the adjustment costs from the actual to the optimal configuration are concerned.
Fig. 2. Centralized technical efficiency problem: ray average productivity in a neighborhood of.k .
of ray average productivity in attaining a minimum, when k increases, is responsible for the high optimal number of firms. With regard to the relation of the VRS to the CRS solution, we find that in this data set the two solutions are practically coincident, because = CRS = 0.9571 and kCRS = 485.18. It is interesting to compare the above-described results with those stemming from the corresponding MIP problem. Here we obtain kMIP = 485 and MIP (kMIP ) = 0.9571, results that are similar to those of our method, exception made for the missing information regarding the occurrence of multiple solutions in the interval [482, 485]. The difference between multiple solutions, along with the practical coincidence between the VRS and CRS solutions, is further illustrated by means of an additional data set in Appendix B. As far as the cost-efficient centralized allocation is concerned, by applying the specific algorithm described in Ref. [6], the minimum industry cost is reached at kc = 97 , which determines a cost-efficiency measure R 0 = 0.5501. The CRS solution is practically coincident with the VRS, because of both an identical value of the cost-efficiency measure and kc , CRS = 97.0362 . Fig. 3 illustrates the behavior of the ray average cost in a neighborhood of kc . Decomposition (6) of the cost efficiency measure, in Appendix A.2, Table 2 Comparison of optimal industry structures. Inputs
Cost-efficient Organization Technical-efficient Organization
Outputs
Staff (Number)
Vehicles (Number)
Commodity (Number)
Vehicle-Km (Millions)
Service Area (Km^2)
243.54 90.10
176.33 43.24
6036765.90 1199623.93
9.22 1.85
2281.30 813.98
7
Firms (Number)
97 482
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Table 3 Weighted input-price vector - Cost efficient optimal structure. Inputs
Cost-efficient Organization
Outputs
Staff (Number)
Vehicles (Number)
Commodity (Number)
Vehicle-Km (Millions)
Service Area (Km^2)
243.13
176.08
6028702.01
9.20
2276.86
5. Concluding remarks
Firms (Number)
97
used to test the robustness of MIP solutions. The empirical application of our proposed computation procedure has revealed deep differences between the technically and the cost-efficient centralized allocations, as regards input and output mixes as well as the optimal number of operating units, which is an important finding from a managerial and regulatory point of view. Moreover, in both data sets employed, we have ascertained the occurrence of multiple solutions to the centralized technical efficiency problem, a circumstance that explains the coincidence between the estimates of ray average productivity in CRS and VRS solutions.
This paper has introduced a unified framework for the analysis and the determination of the relationship between technically and cost-efficient centralized allocations featuring reallocation of outputs and a variable number of firms. The use of Banker's [2] conceptual approach resulted not only in a clear economic interpretation of these optimal allocations and in an inequality relationship linking the corresponding efficiency measures, but also in a rigorous method of estimation - which delivers complete information on optimal solutions and can also be Appendix A A.1. Proof of Proposition 1
From the definition of ray average productivity given in (4) and from the inequality constraints of problem (3a), which hold in a solution with the
equality sign at least for one vector-component, we have
0, h
= max i
{ }= kxih Xi0
and
0, h
= max r
{ } = 1. Therefore, Yr 0 kyrh
ciprocal of the ray average productivity of (k x h, k yh) relative to (X 0, Y0) , is minimized in an optimal solution
0, h 0, h
= , which is the re-
. Q.E.D.
A.2. Proof of Proposition 3 For a generic solution of problem (5), (x c , yc ) ,
0, c
= max r
{ } = k , because of the inequality-constraint holding with the equality sign at least for Yr 0 yrh
k px
one vector-component. Then, by applying equation (12) in [7, p. 124], R 0 = pX c - the ray-average-cost ratio of (X 0, Y0) with respect to (x c , yc ) - can 0 be decomposed into the algebraic sum of a ray-average-productivity and an allocative component
R0 = where
0, c 0, c
0, h
ps0 pX 0
= max i
0, c
{ }, s xic Xi0
(6) 0
=
0, c X 0
xc
0 is a non-negative input-slack vector and R 0
min R 0 . In a solution to problem (3a), c
= min m
0, m 0, m
where its minimizer m does not necessarily coincide with c , the minimizer of R 0 in problem (5). c we have R 0 < . Q.E.D. . Second, when m First, when m = c , expression (6) makes evident that R 0 As a comment to this result, we point out that decomposition (6) obtains only under the assumption of firms facing an identical price vector, therefore Proposition 3 may not necessarily hold under the alternative assumption of a varying input-price vector. A.3. Proof of Proposition 4 Consider the lowest value of k such that corresponding efficient-point,
(
(
X (k min ) k 0 , min
)
X 0 Y0 , k T , which we denote k Y0 resulting from step 4), is kmin
)
as k min . The Proof of Proposition 1 can then be employed to show that the a radial efficient point as defined in Ref. [29]. Then, Theorem 5 in [29, p.
247] can be applied to prove that for each k > k min the ray average productivity of this point is strictly decreasing until a global minimum is reached at k , while it becomes strictly increasing for each k > k . Q.E.D. Appendix B
The occurrence of multiple solutions to the centralized technical efficiency problem is further investigated by means of a secondary data set employed in Ref. [7], made up of 246 observations (see, ibid., section 6.2 and appendix B for a detailed description). The solution to the CRS-LP problem gives CRS = 0.4129 and kCRS = 149.56, while - by adding ten fictitious observations in a neighborhood of kCRS [146, 150]. As far as the VRS-MIP solution is concerned, we obtain MIP = 0.4129 and kMIP = 149 , our VRS-LP solution yields = 0.4129 and k which coincides with our estimates - exception being made for the information regarding the existence of multiple solutions. Table B.1 shows the differences between the multiple optimal solutions to the VRS problem, as determined by our method.
8
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G. Cesaroni
Table B.1
Comparison of multiple VRS solutions. Peers Dmu 11
Dmu 17
Dmu 160
Dmu 226
Dmu 234
0.008912 0.006366 0.003854 0.001376 0
0 0 0 0 0.005227
0.066824 0.066094 0.065374 0.064663 0.063236
0.709215 0.706083 0.702993 0.699944 0.696846
0.215049 0.221457 0.227779 0.234016 0.234691
k* 146 147 148 149 150
It is interesting to observe that the solutions corresponding to 149 and 150 differ in a non-negligible way from that corresponding to 146, this confirms the usefulness of our method in providing information which is conversely not available in the MIP approach. Finally, we remark that, in this data set, the extended version of our algorithm would have not required the adjustment of the initial estimate of n0 . Appendix C In [6, p. 41] it is affirmed that the convexity of T and the positivity of the cost at any observed non-null output vector - assumptions that hold in our model (see Section 2.1) - imply a u-shape form of the ray-average-cost curve of a given output mix. This ensures that the algorithm for the industry cost-optimal allocation determines an optimal solution to problem (5). Here we detail this argument. Shephard [33] shows that under convexity of the technology the cost function is non-decreasing and convex in outputs. Therefore, in our model Y0 at given prices p - the cost function associated to the contraction of the given output mix Y0 can be illustrated as follows.where y with > 0 and point A represents the optimal scale size. The algorithm presented in [6, p. 41] determines a sequence of points along the cost function belonging to the polyline which goes from C to B. The ray average cost of a generic point in the sequence is given as the slope of line segments such as OC′ and OB’. Therefore, Fig. 4 clearly shows that the ray average cost of the industry's output mix is monotonically decreasing either in an contraction or in an expansion towards the optimal scale size A. Note that this conclusion is not altered by the possibility of multiple optimal-scale sizes, i.e. the case in which a segment of the cost function is tangent to the straight line OA.
Fig. 4. Cost function of an industry's output mix.
Appendix D Assume that to each production unit can be associated a specific input-price vector, then in T the general problem of minimizing the industry cost - by means of a reallocation of the aggregate output over an unknown number of optimal scale sizes - can be expressed as 9
Socio-Economic Planning Sciences xxx (xxxx) xxxx
G. Cesaroni k p x h=1 h h
min
{(x h, yh)} k
s.t.
yh
Y0
(5a)
h=1
= 1, and k is an integer to be determined. where x h and yh Problem (5a) is clearly indeterminate, because its objective function is a sum of linear functions whose number k is unknown. Contrary to what happens in the case of uniform input-price vectors ph = p (see Ref. [6], p. 39), the transformation used for the formulation of problem (3a) is not of any help in this context. In fact, considering the arithmetic average of the k optimal-scales we obtain n j=1
k
k
h j xj ,
n j=1
n j=1
h j yj
h j
k
ph x h h=1
ph xh h=1
k
h j
n
x with j where x h and j = 1 j = 1. h=1 k j=1 j j Therefore, problem (5a) remains indeterminate and requires the devise of a heuristic procedure to determine an approximated solution. In our opinion, this is the reason why this specific problem has not yet been dealt with in the literature: as a noteworthy example, the recent contribution of Ray [30, p.73] assumes ph = p for any set of units whose aggregate output must be reallocated among an unknown number of units. n
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Giovanni Cesaroni is Senior Economist at the Prime Minister's Office since 2000. After receiving his PhD degree from the University of Rome “La Sapienza” in 1998, he was appointed Assistant Professor in economics at the University of Rome ‘Tor Vergata’ in 1999. In the period 2008–2014, he has been a member of various spending review units of the Italian Government. His current research focuses on public-administration organization and applied microeconomics, while former research dealt with macroeconomics and financial intermediation. His scientific articles have been published in national and international academic journals. He was awarded the certificate of excellence in reviewing by the European Journal of Operational Research.
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