3.1. Concept Behind the Proposed Approach

This study proposes an approach based on output-to-input ratio indicators, which can also be extended to input-to-output ratio indicators. To ensure generalizability, all possible combinations of outputs and inputs are considered. DEA users can specify their desired output-to-input combinations to construct the necessary ratio-based indicators. The proposed approach identifies concrete improvement targets for outputs while maintaining the inputs constant. Therefore, improvement targets are presented as concrete output values to ensure their practical applicability. When these targets are converted into ratio indicators, they are guaranteed to be efficient. Because the proposed approach operates within the DEA-R framework, DEA-R efficiency is ensured, thereby validating the rationality and appropriateness of improvement targets. To achieve these objectives, the proposed approach for identifying improvement targets in DEA-R is developed based on the following three key ideas and procedures.

• Calculate the efficient DMUs within the given set \(\boldsymbol{R}\) by model 2.

• Construct \(E_\text{R}\) using the calculated efficient DMUs as follows:

  \begin{align} E_\text{R}=\left\{\boldsymbol{r} \left|\begin{array}{rl} &\left( \begin{array}{rl} &\boldsymbol{R}_{E}\symbf{\lambda} - \boldsymbol{r}=\boldsymbol{0}, \boldsymbol{1} ^\top \symbf{\lambda} =1,\\ &\boldsymbol{u}_1-\boldsymbol{u}_2+\boldsymbol{1}=\boldsymbol{0},\\ &-\boldsymbol{R}_E^{\top}\boldsymbol{u}_{2}-\boldsymbol{u}_{3} + \boldsymbol{1}u_4 - \boldsymbol{1}u_5= 0,\\ &\symbf{\lambda} ^\top\boldsymbol{u}_3=0,\\ &\symbf{\lambda} , \boldsymbol{u}_1, \boldsymbol{u}_2, \boldsymbol{u}_3 \geq \boldsymbol{0}, u_4, u_5 \geq 0\\ \end{array}\right)\\ &\text{~~~has a solution}~\boldsymbol{u}_1, \boldsymbol{u}_2, \boldsymbol{u}_3, u_4, u_5, \symbf{\lambda} . \end{array}\right. \right\} \end{align}

  The construction of \(E_\text{R}\) follows that of \(E\) in DEA, as described in ²⁴.

• Search for improvement targets for inefficient DMUs on the constructed \(E_\text{R}\).

Because the DEA-R model handles data in the form of ratios (either \({x}/{y}\) or \({y}/{x}\)), the slack generated by the DEA-R model is also proportional. Therefore, it is unclear how to specifically improve individual inputs or outputs. In this approach, we aim to provide concrete improvement targets for the output \(\boldsymbol{y}_o\) while maintaining input \(\boldsymbol{x}_o\) constant. However, even when the inputs are held constant, ambiguity remains when multiple ratio indicators involve the same output; specifically, it is unclear which ratio indicator should serve as the basis for determining the output improvement target. To resolve this issue, we incorporate the following constraint (Eq. 6) into our approach. For illustrative purposes, the formulation is presented using an arbitrary output \(y_{\check{s}o}\) and two arbitrary inputs, \(x_{\tilde{m}o}\) and \(x_{\bar{m}o}\), as an example.

When identifying improvement targets, the objective function can be flexibly designed to suit different scenarios. For instance, as in the case of the shortest-distance DEA model, which has recently received increasing attention ^19,20,21,25, the objective function may aim to minimize slacks. In this study, we adopt this scenario, and the formulation with \(y_{\check{s}o}\) as the reference output and \(x_{\tilde{m}o}\) and \(x_{\bar{m}o}\) as the selected inputs is presented as follows:

Alternatively, the objective function of model 8 can be formulated similar to that of the RAM model, so that the resulting objective value lies within \([0,1]\) and can be interpreted as an efficiency score. It is also possible to identify improvement targets without setting any objective function, simply by solving a system of equations. In the following section, we illustrate the proposed approach with two simple examples, while also addressing irregular situations that may arise when identifying improvement targets and providing corresponding solutions.

3.2. Illustration with Two Concrete Examples

3.2.1. Example with Two Inputs and One Output

Using a numerical example with two inputs and one output presented in Table 1, we identify improvement targets for inefficient DMUs using the proposed approach. First, we solve model 2 and confirm that DMUs C, D, and E are DEA-R efficient. These three efficient DMUs are then used to construct the efficient frontier \(E_\text{R}\). The numerical example is illustrated in Fig. 1 in which the line segments CE and DE represent the efficient frontier \(E_\text{R}\). In this example, because only one output \(y_{1o}\) exists, the constraint corresponding to Eq. 6 is given as follows:

\begin{align} r^{11}_o{x_{1o}} = r^{12}_o{x_{2o}}. \label{eq:addedU0020constraintsU00201} \end{align}

The computational results are presented in Table 2. For DMU\(_A\), \(\boldsymbol{r}^*=(r^{{11}^*}, r^{{12}^*})^\top=(1.22, 1.22)^\top\). Therefore, the improvement target for \(y_{1A}\) is \(r^{{11}^*}x_{1A}=r^{{12}^*}x_{2A}=1.22\), and that for DMU\(_A\) is DMU\(_{A^*} = (1, 1, 1.22)^\top\). However, for DMU\(_B\), no solution exists for model 8. This indicates that owing to the nature of the data, no point exists on \(E_\text{R}\) that satisfies the constraint 9 for DMU\(_B\). As shown in Fig. 1, DMU\(_B\) is not enveloped by \(E_\text{R}\); this is likely the cause. To address this irregular case, one practical option is to search for a suitable improvement target among the current efficient DMUs for DMU\(_B\). Because the current formulation focuses on minimizing slacks (the shortest distance), selecting the closest efficient DMU to DMU\(_B\), such as DMU\(_C\), as the improvement target for DMU\(_B\) appears to be a reasonable strategy.

3.2.2. Example with Two Inputs and Two Outputs

Using a numerical example with two inputs and two outputs presented in Table 3, we again apply the proposed approach to identify improvement targets for inefficient DMUs. We begin by solving model 2, from which DMUs A, C, and D are observed to be DEA-R efficient. These three efficient DMUs are then used to construct the efficient frontier \(E_\text{R}\). Because this example involves two outputs, to identify the improvement targets using \(y_{2o}\) as the reference, the corresponding additional constraint is as follows:

\begin{align} r^{21}x_{1o}=r^{22}x_{2o}. \end{align}

If \(y_{1o}\) is used as the reference, the corresponding additional constraint becomes

\begin{align} r^{11}x_{1o}=r^{12}x_{2o}. \end{align}

Using \(y_{2o}\) as the reference, the computational results are presented in Table 4. The computational results using \(y_{1o}\) as the reference are omitted because of the space constraints. Because \(y_{2o}\) is used as the reference, the improvement target for \(y_{2o}\) is uniquely determined from the optimal solution of model 8. However, the improvement target for \(y_{1o}\) is not necessarily unique. To ensure the efficiency of the identified improvement targets, the improvement target for \(y_{1o}\) is defined as \(\max\{r^{{11}^*}x_{1o}, r^{{12}^*}x_{2o}\}\). For example, for DMU\(_B\), \(\boldsymbol{r}^*=(r^{{11}^*}, r^{{12}^*}, r^{{21}^*}, r^{{22}^*})^\top=(8.26, 4.40, 1.92, 1.15)^\top\). Therefore, the improvement target for \(y_{1B}\) is \(\max\{ r^{{11}^*}x_{1B}, r^{{12}^*}x_{2B}\} = \max\{24.78, 22.00\} = 24.78\). The improvement target for \(y_{2B}\) is \(r^{{21}^*}x_{1B}=r^{{22}^*}x_{2B}=1.92 \times 3 = 1.15 \times 5 \fallingdotseq 5.76\).⁽²⁾ The computational details for DMU\(_E\) are omitted because of their similarity to those of DMU\(_B\).

Nevertheless, selecting the maximum value can alter \(E_\text{R}\). Therefore, it is necessary not only to verify whether the inefficient DMU becomes efficient after converting the identified improvement targets into ratio indicators, but also to check whether the originally efficient DMUs remain efficient. Accordingly, the efficiency of each DMU is examined. As presented in Table 5, DMU\(_C\), which was originally efficient, becomes inefficient. Following the aforementioned procedure, the improvement target for the DMU\(_C\) can to be identified, and the results are presented in Table 6. Subsequently, the efficiency verification is conducted again, and the results are presented in Table 7. Because both improvement targets and originally efficient DMUs are efficient when converted into ratio indicators, the improvement target identification process is complete.

3.3. Implementation of the Proposed Approach

The proposed approach can be formalized as the following algorithm.

1. Step 1.

   Solve model 2 to identify DEA-R efficient DMUs and construct the efficient frontier \(E_\text{R}\). Proceed to Step 2.

2. Step 2.

   Use the efficient frontier \(E_\text{R}\) constructed in Step 1, select a reference output (\(y_{1o}\)) and two reference inputs (\(x_{\tilde mo}\) and \(x_{\bar mo}\)), and solve model 8 for each inefficient DMU. Obtain

   \begin{align} \boldsymbol{r}^*&= \bigl(r^{{11}^*}, r^{{12}^*}, \dots , r^{{1m}^*},\nonumber\\ &\phantom{=~}~~~r^{{21}^*}, r^{{22}^*}, \dots ,r^{{2m}^*},\nonumber\\ &\phantom{=~}~~~\dots , \nonumber\\ &\phantom{=~}~~~r^{{s1}^*}, r^{{s2}^*}, \dots , r^{{sm}^*}\bigr)^\top, \end{align}

   to identify output-based improvement targets. Proceed to Step 3.

3. Step 3.

   Classify into two cases based on whether model 8 has a feasible solution.

   • If a solution exists for each inefficient DMU, proceed to Step 4 to identify the corresponding improvement targets.

   • If no solution exists for DMU\(_k\), then a direct improvement target is assigned among the current efficient DMUs depending on the scenario. The improvement target for DMU\(_k\) without solutions have been identified. Proceed to Step 4 to identify improvement targets for the other DMUs with solutions.

4. Step 4.

   For each inefficient DMU with a solution obtained in Step 3:

   • If there is one output (\(s = 1\)), the improvement target for \(y_{1o}\) is determined using

     \begin{align} r^{{1\tilde{m}}^*} x_{\tilde{m}o} = r^{{1\bar{m}}^*} x_{\bar{m}o}. \end{align}

     The algorithm ends, and improvement targets have been identified for all inefficient DMUs.

   • If there are multiple outputs (\(s \geq 2\)): The selection of the reference output depends on the user or the scenario. Here, we identify the output improvement targets using a general method based on the following additional constraint (\(y_{1o}\) as the reference output).

     \begin{align} r^{{1\tilde{m}}} x_{\tilde{m}o} = r^{{1\bar{m}}} x_{\bar{m}o}. \end{align}
     1. (1)

        Use \(r^{{1\tilde{m}}^*}\) and \(r^{{1\bar{m}}^*}\) obtained in Step 2 to identify the improvement target for \(y_{1o}\), denoted as \(y_{1o}^*\). Because of the selection of \(y_{1o}\) as the reference, \(y_{1o}^* = r^{{1\tilde{m}}^*} x_{\tilde{m}o} = r^{{1\bar{m}}^*} x_{\bar{m}o}\).

     2. (2)

        Use \(r^{{2\tilde{m}}^*}\) and \(r^{{2\bar{m}}^*}\) obtained in Step 2 to identify the improvement target for \(y_{2o}\), denoted as \(y_{2o}^*\). If \(r^{{2\tilde{m}}^*} x_{\tilde{m}o} = r^{{2\bar{m}}^*} x_{\bar{m}o}\), it is accepted as the improvement target for \(y_{2o}\); otherwise, the larger value is selected as the improvement target for \(y_{2o}\). In other word, \(y_{2o}^* = \max\{r^{{2\tilde{m}}^*} x_{\tilde{m}o}, r^{{2\bar{m}}^*} x_{\bar{m}o}\}\).

     3. (3)

        This procedure is applied to other outputs as well, and the efficient frontier \(E_\text{R}\) may be changed and extended accordingly. Proceed to Step 5.

5. Step 5.

   Verify whether the ratio indicators constructed using the identified improvement targets lie on \(E_\text{R}\).

   • If they do, the algorithm ends, and improvement targets have been identified for all inefficient DMUs.

   • If not, return to Step 1 and repeat the process for the corresponding DMUs.

4.1. Dataset and Experimental Setup

In this study, we employ the real-world dataset provided in ¹⁰ that was originally used for DEA-R analysis. The dataset consists of 21 hospitals characterized by two inputs and three outputs. The inputs are the numbers of beds and physicians, and the outputs are the numbers of outpatients, inpatients, and surgeries. The statistical summary of the dataset is presented in Table 8. Using the approach proposed in Section 3, we identify the improvement targets for the 21 hospitals by alternately selecting each of the three outputs as the reference output. Consequently, three types of improvement targets are identified for each hospital (hereafter referred to as a DMU). The statistical information on these improvement targets is also presented in Table 8. The differences among the three types of improvement targets, which arise from the selection of the reference output, are further examined later through density plots and sensitivity analysis.

4.2. Improvement Targets Identified Using the Proposed Approach

The proposed approach is then applied to the dataset, and the details of the three computational procedures, which alternately select three outputs to identify the improvement targets, are presented in Table 9. Although DMUs exist for which no feasible solution is obtained, no expansion of the feasible set \(E_\mathrm{R}\) is observed. Thus, irrespective of the output selected as the reference, the proposed approach is able to identify improvement targets for all inefficient DMUs within a single iteration.

In existing DEA-R models, improvement targets are typically provided only in the proportional form. Because a single input or output simultaneously contributes to multiple proportional indicators, converting these targets into non-proportional forms is inherently difficult. For instance, Gerami et al. ¹⁰ proposed ratio-based variants of the SBM model; however, their models did not provide concrete input and output improvement targets. Mozaffari et al. ⁸ proposed a central resource allocation model based on DEA-R to derive input and output improvement targets for inefficient DMUs. However, when these targets are converted into ratio indicators, they might become inefficient; this is inconsistent with the efficiency concept underlying DEA-R. Related studies by Mozaffari et al. ^22,23 focused on identifying efficient surfaces and reference DMUs for benchmarking purposes but did not explicitly provide concrete input and output improvement targets. In contrast, the approach proposed in this study is capable of generating concrete (non-proportional) improvement targets for either inputs or outputs. Furthermore, their efficiency is guaranteed when these improvement targets are converted into ratio indicators.

Figure 2 presents density plots comparing the observed values and the corresponding improvement targets identified from the three computational procedures. These plots provide a visual comparison of both the deviations between the observed values and targets, and the differences among the three types of improvement targets. The distributions of the improvement targets are visually very similar across the three procedures, suggesting that the selection of the reference output has only a limited impact on the resulting improvement targets. This visual observation is further quantitatively examined using the sensitivity analyses described in Section 4.3.

4.3. Sensitivity Analysis

To quantitatively assess the similarity among the improvement targets identified from the three computational procedures, we propose a new indicator called the target consistency index (TCI). Let \(t_j^{(k)}\) denote the improvement target for DMU\(_j\) identified from the \(k\)-th computational procedure \((k=1,2,3)\), and let \(d_j\) represent the observed input and output levels of DMU\(_j\). The TCI for DMU\(_j\) is defined as follows:

The numerator represents the average pairwise distance among the three improvement targets, and the denominator represents the average distance from the observed DMU to each improvement target. In this study, the absolute value is used as the distance measure. This selection is consistent with the objective function of model 8 used in our numerical experiment that minimizes the sum of slacks. Because the slack corresponds to absolute deviations from the efficient frontier, the absolute distance appropriately reflects the notion of distance embedded in the optimization model.

4.3.2. Numerical Results

Table 10 presents the TCI for all 21 DMUs. The minimum value is \(0\), the maximum value is \(1.883\), and the average value (\(\mu\)) is \(0.529\), with a standard deviation (\(\sigma\)) of \(0.437\). The first and third quartiles are \(0.272\) and \(0.677\), respectively, indicating that the majority of DMUs exhibit TCI values well below 1.403 (\(\mu + 2\sigma\)). The relatively large TCI values (\(1.883 > 1.403\)) observed for DMU\(_{18}\) can be explained by the discussion in Step 3 of the proposed approach in Section 3.3.⁽³⁾ Owing to the structure of the dataset, the DMU\(_{18}\) corresponds to an irregular case in which no feasible solution is obtained, similar to DMU\(_B\) illustrated in Fig. 1. Consequently, a direct improvement target is assigned to DMU\(_{18}\) from the set of currently efficient DMUs, depending on the selected shortest-distance scenario. Although the improvement target is still identified based on the shortest-distance principle, both input and output adjustments are required for the irregular DMU\(_{18}\). As a result, its TCI values are relatively larger than those of the DMUs that involve only output adjustments according to the proposed approach.

Figure 3 illustrates the distribution of TCI values across the 21 DMUs. Overall, the results suggest that the improvement targets identified from the three computational procedures, each corresponding to a different selection of reference output, are highly consistent for most DMUs. Although the DMU\(_{18}\) exhibits relatively large TCI values, the overall distribution confirms that the differences among the three improvement targets are generally minor. Therefore, the numerical evidence supports the claim that the proposed approach yield largely similar improvement targets, despite differences in the selection of the reference output.