Scaling production and improving efficiency in DEA: an interactive approach
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Abstract
DEA models help a DMU to detect its (in)efficiency and to improve activities, if necessary. Efficiency is only one economic aim for a decisionmaker; however, up or downsizing might be a second one. Improving efficiency is the main topic in DEA; the longterm strategy towards the right production size should attract our attention as well. Not always the management of a DMU primarily focuses on technical efficiency but rather is interested in gaining scale effects. In this paper, a formula for returns to scale (RTS) is developed, and this formula is even applicable for interior points of technology. Particularly, technical and scale inefficient DMUs need sophisticated instruments to improve their situation. Considering RTS as well as efficiency, in this paper, we give an advice for each DMU to find an economically reliable path from its actual situation to better activities and finally to most productive scale size (mpss), perhaps. For realizing this path, we propose an interactive algorithm, thus harmonizing the scientific findings and the interests of the management. Small numerical examples illustrate such paths for selected DMUs; an empirical application in theatre management completes the contribution.
Keywords
Data envelopment analysis Returns to scale Efficiency Upsizing/downsizing mpssIntroduction

constant RTS,

decreasing RTS, and

increasing RTS.
Data envelopment analysis (DEA) as a nonparametric approach permits the approximation of the efficient boundary of technologies. This approximation takes place via gathered data of inputs and outputs for the socalled decisionmaking units (DMUs), that is, classical DEA; numerous theoretical papers and applications prove its value, see, for instance, Bashiri et al. (2013), Shokrollahpour et al. (2016) and Ziari (2016) or for a overwhelming survey Emrouznejad and Yang (2017).
Already in their pioneering work, the authors in Banker et al. (1984) also tackled the problem of the DMUs’ RTS. They proved the sign of a variable u in the socalled multiplier form of inputoriented DEA to indicate the RTS situation—rather than the RTS measure—of a DMU, and they restricted their analyses to efficient rather than inefficient units, only. Roughly speaking, we have constant/nondecreasing/non increasing RTS iff optimal \(u =\)/\(\geqq\)/\(\leqq 0\). These results were generalized in Banker and Thrall (1992) or Sahoo et al. (2016) for efficient production points with variable u’s, i.e., nonunique RTS situations. The authors in Førsund (1996), Førsund and Hjalmarsson (2004), Førsund et al. (2007), and Fukuyama (2000) advanced the RTS context by models of neoclassical production theory and developed equations of scale elasticity for efficient and nonefficient DMUs; an overview and a discussion regarding the concept of RTS are provided in Tone and Sahoo (2003) or Jahanshahloo and SoleimaniDamaneh (2004). In the present paper, we give an elegant and comprehensible proof and an easy interpretation of such scale elasticity, for efficient and nonefficient DMUs. Generally speaking, there are several ways for classifying a DMU’s RTS situation in DEA—either envelopment or multiplier form driven ways. However, the RTS measure is based on an optimal solution of the multiplier form; in this sense, the abovementioned authors provide equivalent representations of the respective formula for calculating such scale elasticities.
A DMU which is informed about its relative inefficiency wants to react by input reduction, output increase, or both: Classical DEA theory recommends to proceed against the boundary of technology; for some activity planning procedures cf. Du et al. (2010), Du and Liang (2012), Homayounfar et al. (2014), Zhang et al. (2015) and Tohidi and Khodadadi (2013). DEA software supports such instructions and helps the user to take respective actions. In all these calculations, scale elasticities are not taken into account. However, the additional knowledge of scale elasticities should encourage the DMUs also to make use of scaling effects. Consequently, if a DMU wants to maintain its BCC efficiency, then for increasing RTS it should upsize its production as any increasing inputs result in disproportionally higher increase of outputs. Decreasing RTS rather recommends downsizing due to lower disproportionality. Therefore, each DMU pursues two objectives: improvement of efficiency and upsizing or downsizing production. Mostly if not always the way to technology boundary is hard to realize. Labour law or social restrictions might forbid such rigid alterations. Is there a more convenient way towards the right production size and efficiency? It is, and the hitherto necessary information is available even for interior points of technology.
For the combination of efficiency and scaling improvement, we develop an (inputoriented) interactive algorithm, thus harmonizing scientific findings and managerial interests of the DMU under consideration. Stepwise it improves its efficiency and its scale size and hence runs through a path from its original activity towards mpss, if possible. The DMU’s awareness of inefficiency sometimes demands unrealistic input reduction. Rather it—the DMU—might communicate its disposition for a more reasonable reduction and the algorithm should respect this information.
Not always realizing the path from the actual activity of the DMU to most productive scale size (mpss) is an easy job. An impressive example for a vulnerable theatre scenery illustrates this issue.
The paper is organized as follows. In the second section, we present preliminaries of DEA. In the next section, two forms of activity changes are given, one maintaining BCC efficiency and one maintaining CCR efficiency. The following section is dedicated to situations of nonunique RTS. The next section provides the central topic of this paper: an interactive and iterative algorithm towards mpss. In the following section, an empirical example illustrates the new method. For 30 German theatres, efficiencies and RTS are calculated, and for a selected theatre, the proposed method is outlined. The last section contains a short resume and delineates prospects of further research.
Preliminaries
Koopmans’s activity analyses are the roots of DEA, cf. Koopmans (1951). Activities are processes which transform objects in other objects. If such objects are material or immaterial goods, such a process by definition is a production process, cf. Frisch (1965), p. 3. The set of all such processes is the production possibility set or technology \(\text{T}\), for short. The activity over a certain time period transforms the input \(\mathbf{x} \in \mathbb {R}^M_+\) into the output \(\mathbf{y} \in \mathbb {R}^S_+\) and thus characterizes the performance of a DMU, such as a project, a corporation, or even a nonprofit utility. Once a technology is determined, DEA theory allows for the efficiency measurement of any activity. We restrict our attention to inputoriented efficiency measures and omit output orientation, cf. Banker et al. (1984), however. Partial inefficiencies which might occur with radial input reduction will not be considered here, either; the reader is referred to Charnes and Cooper (1984) or again Banker et al. (1984).
The authors in Charnes et al. (1978), following Debreu (1951), and Farell (1957) developed linear optimization problems (LOPs) measuring efficiency.
In the DEA literature, models (1.1), (2.1) and (1.2), (2.2) often are named by the acronyms CCR and BCC, due to their creators Charnes, Cooper, Rhodes and Banker, Charnes, Cooper, respectively. Whenever convenient, we follow such practice.
Activity changes
Activity change under constant BCC efficiency
Theorem 1
Proof
Conclusion 1

\(\varepsilon _k= \delta \ \, {\text{for}} \ {u}_k^{*} = 0 ; \ \mathbf{y} \, \mathrm{changes \ radially \ to \ the \ same \ amount \ as} \ \mathbf{x}\) does \(\Longleftrightarrow\) constant RTS
\(\varepsilon _k> \delta \ \mathrm{{{for}}} \ {u}_k^{*} > 0 ; \ \mathbf{y} \ \mathrm{changes \ radially \ to \ a \ greater \ amount \ than} \ \mathbf{x}\) does \(\Longleftrightarrow\) increasing RTS
\(\varepsilon _k< \delta \ \mathrm{{{for}}} \ {u}_k^{*} < 0 ; \ \mathbf{y} \ \mathrm{changes \ radially \ to \ a \ smaller \ amount \ than }\ \mathbf{x}\) does \(\Longleftrightarrow\) decreasing RTS.

All statements are valid for an efficient (\(g_k^{*} = 1\)) as well as for an inefficient (\(g_k^{*} < 1\)) DMU k.
The measure of RTS for an activity \((\mathbf{x}_k, \mathbf{y}_k)\) is \(\frac{ \mathbf{u}_k^{*\mathsf {T}}{} \mathbf{y}_k+{{u}_k^{*}}}{\mathbf{u}_k^{*\mathsf {T}}{} \mathbf{y}_k}\), rather than \({u}_k^{*}\). This measure is not only a function of \({u}_k^{*}\), but also of prices \(\mathbf{u}_k^{*}\) and outputs \(\mathbf{y}_k\). For equivalent equations, confer again Førsund et al. (2007). The authors in Podinovski et al. (2009) grabbed this question again and simplified mathematical derivations for either case, envelopment, and multiplier form, the socalled direct and indirect approach.
Corollary 1
Moving on such a hyperplane means moving under constant BCC efficiency. As soon as input projection of the virtual activity \((\mathbf{x}, \mathbf{y})\) falls on another facet of technology than that of \((\mathbf{x}_k, \mathbf{y}_k)\), efficiency will change, of course. More on that in Sect. “Improving scale size and efficiency: an interactive approach” or for calculating efficiency stability regions, see, for example, Zamani and Borzouei (2016).
Activity change under constant CCR efficiency

maintaining BCC efficiency in Sect. “Activity change under constant BCC efficiency” and

maintaining CCR efficiency in Sect. “Activity change under constant CCR efficiency”.
Before doing so, however, we need some statements on nonunique RTS. Sometimes a DMU must decide under aggravated conditions whether it should expand or reduce activity. In those cases, nonunique RTS play a decisive role.
Nonunique RTS
Conclusion 2
If \(0 < {u}_{k}^{\phantom{.}} \leqq {u}_{k} \leqq {u}_{k}^{+}\), then for all \({u}_k\) increasing RTS prevail at (\(\mathbf{x}_k, \mathbf{y}_k\) ); if \({u}_{k}^{\phantom{.}} \leqq {u}_{k} \leqq {u}_k^{+}<\) 0, then decreasing RTS. The case \({u}_{k}^{\phantom{.}} \leqq 0 \leqq {u}_{k}^{+}\) yields decreasing, increasing and specially constant RTS. This property remains valid for all activities \((\mathbf{x}, \mathbf{y}) \in \text{T}\) located on the hyperplane at efficiency level \(g_k^{*}\) like in (5).
Example
For the activities of 7 DMUs, we have the data in Table 1. Table 2 contains the optimal weights, efficiencies, and RTS of the DMUs; the shaded area in Fig. 1 shows the respective technology. In addition, Fig. 1 illustrates nonunique RTS.
Inputs/outputs of seven DMUs
DMU 1  DMU 2  DMU 3  DMU 4  DMU 5  DMU 6  DMU 7  

Input  2  \(\frac{5}{2}\)  5  8  8  3  3 
Output  1  3  7  9  8  2  \(\frac{1}{2}\) 
Efficiency, optimal weights, and RTS of seven DMUs
\({{g}}_k^{*}\)  \(\mathbf{v}^+_k\), \(\mathbf{u}^+_k\), \({u}^+_k\)  \(\mathbf{v}^{\phantom{.}}_k\), \(\mathbf{u}^{\phantom{.}}_k\), \({u}^{\phantom{.}}_{k}\)  \(\varepsilon _k^{+} = \delta \ \frac{\mathbf{u}_{k}^{+\mathsf{T}}{} \mathbf{y}_k+{{u}_{k}^{+}}}{\mathbf{u}_{k}^{+\mathsf{T}}{} \mathbf{y}_k}\)  \(\varepsilon _k^{\phantom{.}} = \delta \ \frac{\mathbf{u}_{k}^{\mathsf{T}}{} \mathbf{y}_k+{{u}_{k}^{{\phantom{.}}}}}{\mathbf{u}_{k}^{\mathsf{T}}{} \mathbf{y}_k}\)  

DMU 1  1  \(\frac{1}{2}\), 0, 1  \(\frac{1}{2}\), \(\frac{1}{8}\), \(\frac{7}{8}\)  \(\varepsilon _1^{+} = \delta \cdot \infty\)  \(\varepsilon _1^{{\phantom{.}}} = \delta \cdot 8\) 
DMU 2  1  \(\frac{2}{5}\), \(\frac{1}{10}\), \(\frac{7}{10}\)  \(\frac{2}{5}\), \(\frac{1}{4}\), \(\frac{1}{4}\)  \(\varepsilon _2^{+} = \delta \cdot \frac{10}{3}\)  \(\varepsilon _2^{{\phantom{.}}} = \delta \cdot \frac{4}{3}\) 
DMU 3  1  \(\frac{1}{5}\), \(\frac{1}{8}\), \(\frac{1}{8}\)  \(\frac{1}{5}\), \(\frac{2}{7}\), \(1\)  \(\varepsilon _3^{+} = \delta \cdot \frac{8}{7}\)  \(\varepsilon _3^{{\phantom{.}}} = \delta \cdot \frac{10}{21}\) 
DMU 4  1  \(\frac{1}{8}\), \(\frac{1}{5}\), \(\frac{2}{3}\)  \(\frac{1}{8}\), \(\infty\), \(\infty\)  \(\varepsilon _4^{+} = \delta \cdot \frac{16}{27}\)  \(\varepsilon _4^{{\phantom{.}}} = \delta \frac{\infty  \infty }{\infty } =: \delta \cdot 0\) 
DMU 5  \(\frac{13}{16}\)  \(\frac{1}{8}\), \(\frac{3}{16}\), \(\frac{11}{16}\)  \(\frac{1}{8}\), \(\frac{3}{16}\), \(\frac{11}{16}\)  \(\varepsilon _5^{+} = \delta \cdot \frac{13}{24}\)  \(\varepsilon _5^{{\phantom{.}}} = \delta \cdot \frac{13}{24}\) 
DMU 6  \(\frac{3}{4}\)  \(\frac{1}{3}\), \(\frac{1}{12}\), \(\frac{7}{12}\)  \(\frac{1}{3}\), \(\frac{1}{12}\), \(\frac{7}{12}\)  \(\varepsilon _6^{+} = \delta \cdot \frac{9}{2}\)  \(\varepsilon _6^{{\phantom{.}}} = \delta \cdot \frac{9}{2}\) 
DMU 7  \(\frac{2}{3}\)  \(\frac{1}{3}\), 0, \(\frac{2}{3}\)  \(\frac{1}{3}\), 0, \(\frac{2}{3}\)  \(\varepsilon _7^{+} = \delta \cdot \infty\)  \(\varepsilon _7^{{\phantom{.}}} = \delta \cdot \infty\) 
DMU 6 under model (2.2) performs with efficiency \(g_6^{*} = \frac{3}{4}\). Problem (9) yields a unique \({u}^_{6} = {u}^+_{6} = \frac{7}{12}\). The dashed line captioned with \({u}_{6} = \frac{7}{12}\) in Fig. 1 shows the hyperplane of DMU 6 at efficiency level \(g_6^{*}\). From equation \({g}^*_{6} = \frac{\mathbf{u}_{6} \cdot {2}+{{u}_{6}}}{\mathbf{v}_{6} \cdot {3}}\) or \({\mathbf{u}_{6} \cdot {2}+{{u}_{6}}}  \frac{3}{4} \cdot {\mathbf{v}_{6} \cdot {3}} = 0\), respectively, we get: \(\mathbf{u}_{6} = \frac{1}{12}\), \(\mathbf{v}_{6} = \frac{1}{3}\). The hyperplane of DMU 6 which is illustrated in Fig. 1 is the equation \({\frac{1}{12}\mathbf{y} + \frac{7}{12}  \frac{3}{4} \cdot \frac{1}{3}{} \mathbf{x}} = 0\). Note that the slope of this equation is 3—whereas with \(0.1 \cdot \frac{1/12 \cdot {2}+7/12}{1/12 \cdot {2}} = 0.45\), the RTS is 45% for a 10% increase. DMU 6 operates under strictly increasing RTS. \(\diamond\)
Improving scale size and efficiency: an interactive approach
From Eqs. (2.2) and (4), DMU k calculates its relative efficiency and its RTS. Even if the respective activity is an interior point of technology, the DMU knows the exact rate of radial output/input change to keep BCC efficiency constant, cf. Eq. (3\(^{\prime }\)) and the subsequent text. But what must be done to improve productivity and simultaneously make the right scaling decision? After all, Eq. (4) is a recommendation of upsizing or downsizing production, see Conclusion 1. However, how far should this production sizing go. Banker (1984) formulated the concept of mpss activity. An activity has most productive scale size, when CCR and BCC efficiency coincide and are equal to 1. The author showed a way for each DMU how to reach this goal in one step without taking interactive communication with a DMU into account.
Here, the situation is different: How can even an inefficient DMU find its way stepwise to mpss avoiding this lack? Before giving a general answer to this question, we study the example again.
Example
(continued).

downsize activities due to decreasing RTS; as \({u}^+_5 = {u}^{\phantom{.}}_5 = \frac{11}{16} < 0\),

improve efficiency from actual \({{g}}_5^{*} = \frac{13}{16}\).
If DMU 5 runs on the dotted line, it realizes downsizing on its (in)efficiency hyperplane, i.e., with constant BCC efficiency. It should stop when its input projection falls on another facet of technology and check its situation with respect to new weights.
So far downsizing; and next the goal efficiency improvement. We propose an interactive step. DMU 5 must find out its input reduction potential: What is the minimal input portion just to meet the benchmark \(\tilde{\mathbf{y}}_5\)? Choose a number \(g_5^* \leqq \text{f}_5 \leqq 1\), such that \(\hat{\mathbf{x}}_5 = \tilde{\mathbf{x}}_5 \cdot \text{f}_5\) suffices to produce \(\tilde{\mathbf{y}}_5\). Make (\(\hat{\mathbf{x}}_5, \hat{\mathbf{y}}_5 = \tilde{\mathbf{y}}_5\)) the new activity. For \(\text{f}_5 = g_5^*\), DMU 5 becomes efficient, and for \(\text{f}_5 = 1\), it has no input reduction potential at all. For \(g_5^*< \text{f}_5 < 1\), it remains inefficient, but improves efficiency from \(g_5^*\) to \(\frac{g_5^*}{\text{f}_5}\). In Fig. 2, \(\text{f}_5 = g_5^*\) even makes it a mpss.
Now, consider DMU 6. A first iteration step concerning upsizing yields input \(\tilde{\mathbf{x}}_6 = \frac{10}{3}\) and output \(\tilde{\mathbf{y}}_6 = 3\). For an exemplary demonstration, we assume a radial input reduction factor \(\text{f}_6 = \frac{17}{20}\) resulting in \(\hat{\mathbf{x}}_6 = \tilde{\mathbf{x}}_6 \cdot \frac{17}{20} = \frac{17}{6}\) and \(\hat{\mathbf{y}}_6 = \tilde{\mathbf{y}}_6 = 3\). This activity is a start point for a second iteration. A second upsizing results in \(\tilde{\mathbf{x}}_6 = \frac{17}{3}\) and \(\tilde{\mathbf{y}}_6 = 7\), and a second reduction step with \(\text{f}_6 = \frac{15}{17}\) yields (\(\hat{\mathbf{x}}_6 = 5, \hat{\mathbf{y}}_6\) = 7). In addition, this makes DMU 6 mpss.
Whether or not DMU 6 finally reaches mpss obviously depends on its readiness for respective input reductions. In this case, the iterative procedure results in scale efficiency 1 and even mpss.\(\diamond\)
 1.
\(\mathbf{u}_k^{*\mathsf {T}}{} \mathbf{y}_k\) might be 0 and this makes \(\frac{\mathbf{u}_k^{*\mathsf {T}}{} \mathbf{y}_k + {u}^{*}_k}{\mathbf{u}_k^{*\mathsf {T}}{} \mathbf{y}_k}\) indefinite, cf. Theorem 1. Consider DMU 7: Its input projection falls on the vertical facet with \(\mathbf{u}^{*}_7 = 0\). Consequently, RTS becomes indefinite, and hence, Eq. (6) is undefined. In such cases, the ideal path on BCC(in)efficiency hyperplanes, as was demonstrated for DMUs 5 and 6, is blocked.
 2.
Assume that DMU 7—besides the abovementioned problem—succeeded in realizing activity (3, 1), see Fig. 1. Its efficiency remains \(\frac{2}{3}\), but now, Eq. (6) permits different weight systems for this efficiency. Which of these weight systems DMU 7 should select for further activity change remains an open question. In addition, this problem worsens, the more BCC efficiency hyperplanes pass through (3, 1). For highdimensional DEA, this causes a severe problem.
 1.
Solve (1.2) with optimal value of objective function \(\theta _k^{*}\).
 2.
Parameterize the 1 in the convexity restriction of (1.2). Let \([r^{},r^{+}]\) be the range of this sensitivity analysis. Make \((\hat{\mathbf{x}}_k, \hat{\mathbf{y}}_k)^{\pm } = (\mathbf{x}_k, \mathbf{y}_k)/r^{\pm }\) like in (8) and respective comments. Mind the fact that the scaling direction, \(r^{}\) or \(r^{+}\), depends on the RTS, including \(\infty\) and 0, cf. Table 2.
 3.
Solve (1.2) for \((\hat{\mathbf{x}}_k, \hat{\mathbf{y}}_k)\) with optimal efficiency \(\hat{\theta^*_k }\). Make \((\tilde{\mathbf{x}}_k, \tilde{\mathbf{y}}_k) = (\hat{\mathbf{x}}_k \cdot {\hat{\theta^*_k }}/{{\theta _k^{*}}}, \hat{\mathbf{y}}_k)\). This step traces \((\hat{\mathbf{x}}_k, \hat{\mathbf{y}}_k)\) back to BCC(in)efficiency hyperplane of DMU k at level \({\theta }_k^{*}\)!

up or downsizing with constant efficiency,

interactive efficiency improvement,
So far the algorithm. How a DMU in the medium or in the long term can realize such alterations of its activities is due to its economical environment and its change management. These questions are beyond of the scope of this paper.
Interactive improvement of activities in theatres
A rich theatre scenery is considered the basis of a broad cultural supply for the society, worldwide. The German theatre landscape follows this tradition. Therefore, in the season 2013/14, 142 public theatres attracted 21 million visitors offering more than 67,000 events.

number of seats

personnel expenses (million euro)

events

productions

visitors
Inputs/outputs, efficiencies, and RTS of 30 theatres
DMU  Seats  Pers. exp.  Events  Prod.  Visitors  \({g_k^{**}}\)  \({g_k^{*}}\)  RTS 

BI  3622  18.351  631  35  221,051  0.8940  1  DRS 
C  2792  24.683  785  75  193,889  0.8370  1  DRS 
CB  1236  17.791  431  43  121,126  0.7760  0.7810  DRS 
CO  1661  11.495  413  37  88,178  0.8350  0.9040  IRS 
DA  1674  27.752  644  51  236,710  0.8640  0.8860  DRS 
DE  1645  15.476  922  51  150,798  1  1  CRS 
DO  2856  31.868  672  54  231,292  0.6200  0.6400  DRS 
G  2049  15.277  759  59  135,465  1  1  CRS 
GI  1098  11.693  432  40  117,475  1  1  CRS 
H  2361  49.844  1261  81  409,431  1  1  CRS 
HBV  803  12.006  532  37  128,535  1  1  CRS 
HD  4831  19.143  1144  48  212,407  1  1  CRS 
HZ  940  6.843  243  21  94,447  1  1  CRS 
KA  1923  39.899  971  70  315,162  0.9660  0.9910  DRS 
KI  3744  25.201  807  46  225,099  0.7160  0.7700  DRS 
KL  933  16.106  485  30  119,520  0.7960  0.8610  IRS 
KO  3208  12.432  270  25  85,860  0.5790  0.6220  IRS 
KS  3812  26.958  700  47  222,660  0.6350  0.6960  DRS 
MD  1455  22.422  895  67  172,226  0.9990  1  DRS 
MS  1230  13.104  569  43  138,680  0.9920  0.9930  IRS 
MZ  1286  18.251  525  44  195,519  0.9860  1  DRS 
OL  2258  20.646  734  54  179,742  0.7890  0.8160  DRS 
OS  2170  15.335  649  47  171,159  0.9400  0.9870  DRS 
PF  2375  10.613  422  27  128,899  0.9330  0.9600  DRS 
R  1688  15.315  605  33  164,773  0.9010  0.9380  DRS 
S  5236  81.157  809  64  491,316  0.5820  1  DRS 
SB  2874  25.029  573  26  188,917  0.5910  0.6320  DRS 
SN  2821  18.772  810  72  193,132  1  1  CRS 
WI  2926  29.215  918  72  340,817  0.9590  1  DRS 
WU  2120  13.481  429  30  129,125  0.7450  0.7680  DRS 
Interactive iterations for DMU OL
Iteration  \((\mathbf{x}_{\text{OL}}, \mathbf{y}_{\text{OL}})\)  \({g_{\text{OL}}^{**}}\)  \({g_{\text{OL}}^{*}}\)  SE  \(r^+\)  \(f_{\text{OL}}\) 

OL  (2258, 20.646, 734, 54, 179742)  0.7890  0.8160  0.9669  –  – 
1  (2055, 18.789, 686, 50, 167930)  0.8104  0.8325  0.9735  1.0703  0.98 
2  (1856, 16.970, 637, 47, 155880)  0.8329  0.8494  0.9806  1.0773  0.98 
3  (1661, 15.187, 586, 43, 143600)  0.8573  0.8668  0.9890  1.0856  0.98 
4  (1470, 13.442, 535, 39, 131070)  0.8845  0.8845  1  1.0956  0.98 
From this table, we learn that the reduction process of activities for Oldenburg’s public would be very painful. First, reducing activities stepwise by \(r^+ = 1.0703\), 1.0773, 1.0856, and 1.0956 plus input reduction of \((10.98) \times 100 = 2\)% in each step means an irresponsible sellout of cultural quality in town. And such a sellout very likely will cause an angry protest in Oldenburg’s population. Even worse, this reduction does not make OL theatre efficient at all. All together, this strenuous effort over 4 planning periods results in 88.45% CCR and BCC efficiency and a 100% scale efficiency. However, mpss is still far away....
Data envelopment analysis is a suitable instrument for a DMU to detect its weaknesses and its improvement potentials. Whether or not a DMU can realize such findings depends on the surrounding conditions and the DMU’s change management, however. Whether or not a theatre like the one in Oldenburg will follow recommendations to reduce activities is by no means a DEA question but rather a political issue. We hope that the audience in this town will have many years of vivid sensations with its theatre.
Conclusion and the road ahead
In this contribution, radial returns to scale are measured for all DMUs along their respective (in)efficiency hyperplanes. This measure involves not only the variable u but also outputs and their corresponding virtual prices. The measure is valid for efficient and inefficient activities and can be applied even to cases with nonunique u’s. In other words: even for an interior point of the technology, unique and nonunique returns are measurable without any projection upon the technology boundary. This measure for each DMU is a handy instrument to evaluate consequences of radial upsizing and downsizing. Each time, such upsizing or downsizing is realized, the DMU might again check its efficiency and its returns to scale and take action to improve its situation. In this paper, we propose an algorithm to support DMUs in finding an economically reliable path towards mpss. This algorithm interactively communicates with the decisionmaker to avoid unrealistic steps of activity improvements. In an application, such steps might be modified due to environmental conditions. These modifications are a worthwhile focus for future research.
Crossefficiencies are considered an interesting approach to evaluate DMUs’ efficiencies from the point of view of other DMUs. Whenever a supervising institution dismisses selfappraisal as a valid concept, crosswise evaluations might help to find a peer, a weight system acceptable for all DMUs. Earlier and recent DEA literature reports on such peerappraisal concepts confer, e.g., Doyle and Green (1994), Rödder and Reucher (2012). Once such a concept is accepted and a peer is selected, his weight system not only appraises all DMUs efficiencies but becomes the transfer price system of the whole group. Do there exist crossRTS similar to crossefficiencies and which consequences do such crossRTS have upon a DMUs scalesizing. Such questions could be the issue of further research.
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