DIRECTIONAL SCALE ELASTICITY CONSIDERING THE MANAGEMENT PREFERENCE OF DECISION-MAKERS

Most data envelopment analysis (DEA) studies on scale elasticity (SE) and returns to scale (RTS) of efficient units arise from the traditional definitions of them in economics, which is based on measuring radial changes in outputs caused by the simultaneous change in all inputs. In actual multiple inputs/outputs activities, the goals of expanding inputs are not only to obtain increases in outputs, but also to expect the proportions of such increases consistent with the management preference of decision-makers. However, the management preference is usually not radial changes in outputs. With the latter goal into consideration, this paper proposes the directional SE and RTS in a general formula for multi-output activities, and offers a DEA-based model for the formula of directional SE at any point on the DEA frontier, which is straightforward and requires no simplifying assumptions. Finally, the empirical part employs the data of 16 basic research institutions in Chinese Academy of Sciences (CAS) to illustrate the superiority of the proposed theories and methods. Mathematics Subject Classification. 90B30, 90B50, 90C08, 90C90. Received October 20, 2021. Accepted August 14, 2021.

The structures of these contributions are organized as follows. In Section 2, we propose the generalized definitions of directional RTS and directional SE considering the management preference of decision-makers. In Section 3, we offer a DEA-based approach to derive the formulae for directional SE of the DMU located at the DEA frontier. In Section 4, we provide an empirical example consisting of data from 16 basic institutes of CAS in 2016 to illustrate the proposed theories and models. The last section contains the conclusion and delineates prospects of further research.

Definitions of directional RTS and directional SE
In this section, we propose the generalized definitions RTS and SE with consideration of the management preference of the decision-makers. Suppose there are observed DMUs. X = ( 1 , . . . , ) and Y = ( 1 , . . . , ) represent the input vector and the output vector that of the -th DMU, respectively. Under the assumption of variable returns to scale (VRS), the production possibility set (PPS) can be formulated as: Before discussing the scale issues, it should be kept in mind that RTS and SE is a local property and applies to boundary (frontier) points. This argument is emphasized in most studies: examples include Banker and Thrall [4], Førsund and Hjalmarsson [11], Yang and Liu [25] and Zelenyuk [27] among others. In the studies, the definition of SE is only associated to the efficient units on the frontier, and therefore it does not depend on the efficiency measure. For an inefficient DMU that is a point in the interior of the PPS is first represented by a benchmark (projection) of the frontier, in which case some explicit efficiency measures are needed, and thereafter the discussion of the scale issue is targeted at this benchmark. Podinovski et al. [19] extended the SE to the inefficient DMUs by integrating the input and output radial efficiency measures into the formulae of SE, which is essentially in line with the idea of the typical literature mentioned above. Such benchmarks for inefficient DMUs in Podinovski et al. [19] are set as the input and output radial projections.
Analogous to this idea, we employ the Farrell measures of efficiency from input and output orientation [10] to realize the radial projections for a target DMU(X 0 , Y 0 ), which have been commonly used as the benchmarks for inefficient DMUs in the discussion of SE using DEA methods. Such projections can be determined by solving the following input-and output-oriented BCC models, respectively. Here is a non-Archimedean small positive number, and the vectors s − = ( − 1 , . . . , − ) and s + = ( + 1 , . . . , + ) represent the input slacks and output slacks, respectively. The DMU(X 0 , Y 0 ) is called "strongly efficient" if and only if the optimal solution ( * , * , s +* , s −* ) (( * , * , s +* , s −* )) obtained from model (2.2) ((2.3)) satisfies * = 1 ( * = 1) and has no slacks (s +* = 0, s −* = 0). Otherwise, the DMU(X 0 , Y 0 ) can be brought into the strongly efficient status by the BCC-I and BCC-O projections as defined by [BCC-I-P] For convenience, we define the projection point of DMU(X 0 , Y 0 ) derived from different efficiency measures uniformly as DMU(X * 0 , Y * 0 ), where if the observation point DMU(X 0 , Y 0 ) is strongly efficient, then it is consistent with DMU(X * 0 , Y * 0 ). It is noted that this projection can be also determined by some alternative efficiency measures (e.g., directional radial measures). In fact, the above efficiency measures only provide a measure of the distance from the inefficient DMU to the efficient frontier, but do not serve to explain why a DMU is inefficient [11]. It is not difficult to find that projecting inefficient DMUs to the efficient frontier and exploring the SE and RTS of their projection points are two different processes. The general definitions of RTS and SE can be treated as the idea of measuring the change in outputs caused by the change in inputs on the efficient frontier. This change can be radial or non-radial, depending on the management preferences of decision-makers. This paper is concerned with how to incorporate the management preferences into the SE and RTS of a strongly efficient DMU or the strongly efficient projection of an inefficient DMU, and the resulting SE and RTS are called as the direction SE and directional RTS. To address this problem, in what follows we dwell mainly on both radial projections under input and output orientations of each DMU in the definition and derivation of directional SE and directional RTS.
We now propose the definitions of directional SE and directional RTS accounting for the management preference of decision-makers towards this strongly efficient DMU. As mentioned earlier, when considering the management preference of decision-makers in the analyses of RTS and SE, we need to generalize the concept of SE and RTS to accommodate any proportional change of outputs and not only in the proportional change as in the traditional SE measure. Notionally as Yang and Liu [25], we first introduce the direction of input changes and the direction of output changes to describe the situation that inputs and outputs change non-proportionally. For a strongly efficient DMU(X * 0 , Y * 0 ), let the vector = ( 1 , 2 , . . . , ) ( ≥ 0, = 1, . . . , ) represents the direction of input changes, and the vector = ( 1 , 2 , . . . , ) ( ≥ 0, = 1, . . . , ) represents the direction of output changes, which essentially describes the management preference of decision-makers. Now, along these given directions, increasing inputs X * 0 by proportion in the direction , the maximal proportion ( ) of increase in outputs Y * 0 possible in the direction is allowed by the function ( )/ , where = ( 1 , 2 , . . . , ) and = ( 1 ( ), 2 ( ), . . . , ( )). Similar to the classic definition of SE, at any boundary point let At this point, its directional SE can be defined as the ratio of the marginal productivity d‖ ‖ d‖ ‖ to the average productivity ‖ ‖+1 ‖ ‖+1 when → 0 (see, e.g., [25]). Then , where = 0 and ( ) = 0, the directional SE derived from formula (2.8) can be represented as follows: Further, if we let ∑︀ =1 = and ∑︀ =1 = (where and are arbitrary positive numbers) be the completely generalized direction setting (as in [2]), then the proposed directional SE can be rewritten as By comparing the proposed directional SE and the classic SE (i.e., d ( )/d ), we find that the proposed directional SE is / time of the classic SE. To maintain consistency with the classic SE in the form, we consider the normalization condition of = and = for standardizing our directional vector. In this setting, the proposed and classic SEs can be uniformly expressed as Now, the rationale of the formulae (2.11) is that a marginal increase in the inputs by a scale factor (where → 0) in the direction , corresponds to the maximum increase of the outputs by a scale factor × (X * 0 , Y * 0 ) in the direction for the DMU(X * 0 , Y * 0 ). Therefore, the directional SE measure in formula (2.11) has "reciprocal" interpretation with the classic case, i.e., the value greater than 1 indicates about increasing directional RTS, equal to 1 indicates about constant directional RTS, and less than 1 indicates decreasing directional RTS.

Measurement of directional SE
In Section 3, we propose a DEA-based approach to derive the formulae for the directional SE for the target DMU(X * 0 , Y * 0 ). In view of both the definitions of traditional SE and directional SE, we assume henceforth that DMU(X * 0 , Y * 0 ) is strongly efficient with respect to . If not, we project it onto the strongly frontier of using model (2.2) (or model (2.3)), and the directional SE can be measured through its strongly efficient projection ((2.4) or (2.5)).
Based on the definition (2.6) for the strongly efficient DMU(X * 0 , Y * 0 ), the function ( ) is determined by solving the following linear program: To prove Theorem 3.1, we use the concept of the directional derivatives of the optimal value function in linear programming problems. Podinovski et al. [19] formulated this in the form of Proposition 3.2.
be defined as the optimal value in the following linear program formulated in terms of the vector X ∈ (assuming it has an optimal solution), where is some subset of {1, . . . , }: Assume a non-zero vector d is a feasible direction at b. (This means (3.2) as an optimal solution with vector b ′ = b + d on its right-hand side for all sufficiently small > 0.) Then, the directional derivative ′ (b; d) of at b in the direction d exists and is equal to Proof of Theorem 3.1. Program (3.1) is a special case of (3.2) and its optimal value can be represented as particularly. According to Proposition 3.2, the right-hand side of (0), that is, ′ Then, consider the dual model of program (3.1) with = 0 is as follows: where e is a row vector with all its elements being equal to one. Let Ψ represent the set of all the optimal solutions of program (3.6). We follow with interest the directional RTS measurement of strongly efficient DMUs and the strongly efficient projections of inefficient DMUs, and it's easy to get (0) = 0, for each optimal solution from Ψ, and we have vX * 0 −uY * 0 + 0 = 0. From the program (3.6), the formulae (3.4) and (3.5) can be expressed by: Intuitively, Theorem 3.1 provides the formulae for directional SE under VRS assumption. Based on Theorem 3.1, we further construct the two auxiliary linear programs (3.7) and (3.8) to calculate the minimum and maximum values of v( ∘ X * 0 ): From the objective function of programs (3.7) and (3.8) we can obtain the numerical values of the right-and left-directional SE in the direction of = ( 1 , . . . , ) (inputs) and = ( 1 , . . . , ) (outputs). Accordingly, the directional RTS to the "right" and the "left" of DMU(X * 0 , Y * 0 ) can be determined as follows: ) and ( 1 , . . . , ) ; (iii) if min{v( ∘ X * 0 )} < 1 holds, then directional RTS to the "right" of DMU(X * 0 , Y * 0 ) is decreasing in the direction of ( 1 , . . . , ) and ( 1 , . . . , ) .
It is worth emphasizing that our formulae of the directional SE should be applied to the strongly efficient unit with respect to the DEA frontier. For the inefficient DMU, one might not be able to distinguish whether the output increases resulting from the input increases is due to the improvement of technology inefficiency or the economies of scale, so it is necessary to analyze the directional SE and directional RTS for the target DMU under the premise of technology efficiency.

Directional SE and directional RTS analysis
In this section, to illustrate our proposed method, we analyze an empirical dataset consisting of 16 basic research institutes in the CAS in 2016. The data set consists of two inputs (research staff and research expenditures) and three outputs (external research funding, high SCI publications and granted patents). Table 1 lists the input and output datasets of these 16 basic research institutes in 2016.
As discussed previously, the analyses of directional SE and directional RTS are aimed at a strongly efficient DMU of . We start by identifying the scale states with respect to strongly efficient projections under the input  Table 2, and there are six institutes are efficient. As shown in Table 2, the corresponding projections are reported in columns 3-7. Then, we apply the auxiliary models (3.7) and (3.8) to calculate the directional SE to the "left" and "right" for the input-oriented projections of these DMUs. Here, for comparison purposes, we set three different directions of output changes, which respectively represent different management preferences of decision-makers: Case 1: 1 = 0.75, 2 = 0.75, 2 = 1.5; Case 2: 1 = 1, 2 = 1, 2 = 1; Case 3: 1 = 1.25, 2 = 1.25, 2 = 0.5.
The above three directions can be expressed as different management preferences of decision-makers with respect to the three outputs: external research funding, high SCI publications, and granted patents. In Case 1, the given direction of output changes represents the management preference that the decision-maker prefers to prioritize increasing patent quantity compared to the other two outputs; the given direction in Case 2 can be interpreted that the decision-maker has an equal priority on the increase of all three outputs. Contrary to the management preference assumed in Case 1, the direction given by Case 3 indicates that the decision-maker prioritizes the increases in external research funding and high SCI publications over granted patents. Based on the three sets of output directions, using Theorem 3.1 proposed in this paper, we can calculate the values of right-hand and left-hand directional SE for 16 institutes. Here, we take the results for the input-oriented projections of DMU 1 and DMU 2 as an example to conduct an in-depth explanation.
Tables 3 and 4 display the directional RTS and directional SE to the "right" and "left" for the input-oriented projections of DMU 1 and DMU 2 in the three cases, respectively. To present these results intuitively, we offer Figures 1-6 to show the RTS status for the input-oriented projections of DMU 1 and DMU 2 in different input change directions under these cases. Figures 1-3 show the results regarding the input-oriented projection of DMU 1, and Figures 4-6 show the results with respect to the input-oriented projection of DMU 2.
From the results in Figures 1-6 and Tables 3, 4, for the input-oriented projections of DMU 1 and DMU 2, we can find that the directional SE and directional RTS have obvious differences due to the different management preferences of decision-makers. Several observations can be made regarding the above results on these inputoriented projections. Table 3 and Figures 1-3 report the directional SE and directional RTS for the input-oriented projection of DMU 1. The following conclusions emerge. From the results of the right directional RTS in Figure 1a, it can be observed intuitively for the input-oriented projection of DMU 1, if the proportion of the increases in Staff and Res. Expen. locates in Region 2 (including radial proportion), increasing directional RTS prevails. On the contrary, if the proportion of the inputs' increases locates in Region 3, decreasing directional RTS prevails. As mentioned earlier, the management preference indicated in Case 1 is that the decision-maker prefers to increasing patent quantity versus external research funding and high SCI publications. Considering such management preference, the DMU 1 which has been realized strongly efficient in PPS under the input orientation is suggested to increase the two inputs in these proportions that are located in Region 2. From the results of left directional RTS in Figure 1b, It indicates that increasing directional RTS prevails at the input-oriented projection of DMU 1 for any proportion of the decreases in Staff and Res. Expen. Similar to Case 1, the results with respect to Cases 2 and 3 can be observed intuitively in Figures 2 and 3, which are not discussed in detail here.
Comparing the results of Figures 1, 2 and 3, we can easily find that the SE and RTS for the input-oriented projection of DMU 1 are different if the decision-maker have different management preference. Cases 1, 2 and 3 represent three different management preferences of decision-makers with respect to the three outputs: external research funding, high SCI publications, and granted patents, respectively. Here, as an example of the results of increasing inputs in equal proportions (i.e., 1 = 2 = 1), the calculation results of the right directional SE for the input-oriented projection of DMU 1 in the three cases are 1.23, 1.03 and 0.82, and the corresponding directional RTS status are increasing, increasing, and decreasing, respectively (see column 4 and column 6 of Tab. 3). Therefore, the SE and RTS need to consider different management preferences of decision-makers in practice.
In the meanwhile, looking at the results of Case 1 in Table 3, we can find that the best SE for the inputoriented projection of DMU 1 is achieved as the proportion of increased in Staff and Res. Expen. is 0.9:1.1 ( 1 = 0.9 and 2 = 1.1), not as the equal-proportion ( 1 = 1.0 and 2 = 1.0) of increases in the two inputs.    This result confirms our supposition above whereby an equal-proportional change in all inputs may not be the best strategy for practical management. It is possible that an unequal-proportional increase in inputs may contribute to a lager SE for a strongly efficient DMU, especially concerning the management preferences of decision-makers. For DMU 2, since it is a strongly efficient DMU of PPS, that is, we can directly analyze the RTS status in terms of its current inputs and outputs. According to the results of the right-hand and left-hand directional RTS (see Figs. 4-6), we can observe that the directional RTS to the "right" and "left" of DMU 2 in the give directions of inputs and outputs are approximately equal. Thus, it can be summarized briefly that on the basis of existing inputs, if the proportion of Staff and Res. Expen. increases (or decreases) of DMU 2 locates in Region 1 of Figures 4, 5 and 6 (corresponding to Cases 1, 2, and 3), increasing directional RTS prevails. If the proportion of increases (or decreases) in the two inputs locates in Region 2, constant directional RTS prevails. If this proportion locates in Region 3, constant directional RTS prevails. In addition, under the three cases, we note that the feasible ranges for the proportion of increases (or decreases) in Staff and Res. Expen. corresponding to   Table 4, for example, we can find that in Case 1, if the proportion of increases in the two inputs is 1.2:0.8 ( 1 = 1.2 and 2 = 0.8), increasing directional RTS prevails on DMU 2. However, in Case 3, if the two inputs increase in this proportion, decreasing directional RTS prevails. The results further verify our supposition above that the SE and RTS are different under different management preference of the decision-makers.
In the following, let us analyze the scale issue of the strongly efficient projections of 16 DMUs under the output orientation. Table 5 reports the output-oriented inefficiency scores and projections computed by model (2.3).
Looking at Table 5, it is clear that the six strongly efficient DMUs under the input orientation remain strongly efficient under the output orientation. While for the inefficient DMUs, a significant difference can be found in their strongly efficient projections under these two orientations. As we discussed previously, the directional SE and directional RTS are measured at the strongly efficient DMU or the strongly efficient projections (i.e., benchmarks) of inefficient DMUs in this contribution.
By employing models (3.7) and (3.8), the "left" and "right" directional SE and directional RTS are calculated similarly based on the output-oriented projections of the 16 DMUs. Given that DMU 2 is strongly efficient under both input and output orientations, the same results on the directional SE and directional RTS (as in Tab. 4) are not repeatedly presented. Also taking the output-oriented projection of DMU 1 as example for a detailed explanation, the results are shown in Table 6.
According to results in Table 6, we can find that in these given directions of input change and output change, the values of the directional SE (see columns 4 and 5 of Tab. 6) at the output-oriented projection of DMU 1 are little different from the results at its input-oriented projection (see columns 4 and 5 of Tab. 3), and naturally the resulting directional RTS at these two projections are similar. Therefore, we make no repeat explanation.
Consequently, according to the analyses of directional SE and RTS proposed in this paper, research institutions can identify the status of increasing, constant and decreasing directional RTS with respect to their own management preference, and then adjust the proportion of inputs based on this. From a management point of view, the directional SE and directional RTS parameter is of particular importance to develop a feasible strategy for input adjustment and effectively avoid blind investment and resource waste.

Comparison analysis with Yang's method
Furthermore, we have also provided a comparison analysis on the directional SE and RTS computed by the proposed method and Yang's method (see [25]). As introduced previously, the underlying idea of our method is to derive the formulate of the directional SE based on the directional derivatives of the optimal value function in linear programming programs, and present two auxiliary linear programs (models (3.7) and (3.8)) to obtain the exact values of the right-and left-hand directional SE. Yang's method mainly adopts the basic idea of the finite difference method (FDM) to provide an approximate estimate for the directional SE by testing the ratio of the amount of change of outputs on the efficient frontier in a specified direction caused by a change in a small enough amount 0 of inputs in a specified direction.
For comparison, we use Yang's method to estimate the directional SE and directional RTS for the inputoriented and output-oriented projections of DMU 1, respectively. Following Yang and Liu [25], this small enough amount for the input change is set to 0 = 1 −6 . Table 7 lists these estimated directional SE and directional RTS for the input-oriented projections of DMU 1, and Table 8 reports the results regarding the output-oriented projections of DMU 1.
As we can see from Tables 7 and 8, we notice that Yang's method used for estimating the directional SE suffers from the infeasibility problems in some directions, and clearly the corresponding RTS status in these directions cannot be identified using their method. Specifically, for the input-oriented projection of DMU 1, the incidences of the infeasibility problem are 21.05% and 29.82% when using Yang's method to estimate the leftand right-hand directional SE, respectively. For the output-oriented projection of DMU 1, the incidences of the infeasibility problem for estimating the left-and right-hand SE are 15.79% and 5.26%, respectively. Compared to the results computed by our method (see the results listed Tabs. 3 and 6), the exact values of the directional SE are obtained explicitly in all the above directions without any infeasibility problem.
Furthermore, the proposed method used for calculating the directional SE is not subject to any simplifying assumptions, such as the setting on the small enough quantity of the input change in FDM methods, and actually the setting on this quantity may perturb the estimations of directional SE and directional RTS. Finally, the proposed method provides a straightforward derivation for the directional scale elasticity, where all calculations   are based on the simple linear programming. Hence, this method can be easily extended to a more generalized case and has a low computational cost.

Concluding remarks
The most existing approaches for quantitative analysis of RTS follow the concepts of the traditional RTS and SE in economics, i.e., the radial change in outputs resulting from the radial changes of all inputs. In actual multiple-input and multiple-output activities, due to the constraints of actual management needs and objective conditions, the goals of expanding inputs are not only to obtain increases in outputs, but also to expect the proportions of such increases consistent with the management preference of the decision-makers, which is usually not radial changes in outputs.
This paper introduces the direction of output changes to describe the management preference of decisionmakers, and extends the concepts of directional SE and directional RTS in multi-output activities. Then, we propose a general DEA-based approach to numerically calculate the directional SE at any point on the DEA frontier. In this approach, the formula derivation of the directional SE requires no simplifying assumptions and can be easily extended to more general cases. Through the proposed definitions and measures of directional SE, one can identify the status of RTS in any given directions of input changes and output changes. Finally, we employ the data from 16 basic research institutions in CAS as the empirical example to illustrate the advantages of the analyses of directional SE and directional RTS. The results show that the proposed directional SE and directional RTS provide more precise information on how to adjust inputs to maximize output increases consistent with the decision maker's management preference.