Introduction

In recent years, the logistics industry has been booming, which is of great significance for the social and economic development and improvement of people's livelihoods. Evaluating the efficiency of the logistics industry has become the focus of recent research. How to accurately assess logistics efficiency to identify production issues and provide support for optimizing logistics efficiency has become a current research challenge. Data envelopment analysis (DEA), a nonparametric methodology, is a common method for evaluating efficiency scores of a set of comparable decision-making units (DMUs). Charnes et al.1 proposed the first DEA model named Charnes, Cooper, Rhodes (CCR) model. The CCR model is defined as the ratio of weighted outputs to weighted inputs under the constant returns to scale (CRS) assumption. Banker et al.2 developed a DEA model under the variable returns to scale (VRS) condition, which was known as Banker, Charnes, Cooper (BCC) model. Both the CCR model and the BCC model ignore input excesses and output shortfalls that are represented by non-zero slacks. To solve the problem, Charnes et al.3 proposed an additive model by introducing slacks of inputs and outputs, which were involved in the objective function. The slacks-based measure (SBM)4, the range adjusted measure (RAM)5, and the bounded adjusted measure (BAM)6 are all additive models with different objective functions.

The RAM model was introduced by Cooper et al.5. It addresses the boundary issues that may arise in traditional DEA models. In traditional DEA models, a problem exists where insufficient or inadequate DMUs in the sample may lead to inaccurate efficiency assessment results. The RAM model resolves this issue by adjusting the range of efficiency assessment. It no longer relies on the best DMU in the sample, but instead expands the range of evaluation to a wider range to ensure the accuracy and reliability of the assessment results. Compared to traditional CCR1 and BCC2 radial models, the RAM model can also analyze and capture potential non-radial changes in inputs and outputs.

RAM can be coded as VRS, is unit-invariant, and can handle large variations in scale7. Sueyoshi and Sekitani8 emphasized that the RAM model has an advantage over other DEA models when negative values exist in the dataset. However, the RAM model has two drawbacks. One is that it only applies under the VRS condition and not under other conditions, such as the CRS. The other is that the RAM's range bounds may be too tight under the definition of the existing lower bound of inputs and upper bound of outputs. In this condition, the resulting production possibility set (PPS) may exclude the original projections of inefficient units, which leads to lower efficiency assessment. The aim of this paper is to propose a modified RAM (MRAM) model for accurately evaluating the efficiency of the logistics industry.

The organization of this paper is as follows: “Literature review” section elaborates on the current research status from two aspects: logistics industry efficiency and RAM. “Methods” section introduces the research methodology of this paper. Firstly, the RAM-CRS model is proposed, which is then expanded to apply to any returns to scale (ARS) situations. Next, MRAM is established, and its properties are also analyzed. “A case study” section analyzes a case study using the MRAM model. “Discussions” section compares MRAM with RAM-CRS and the additive model respectively. Additionally, the applications of the MRAM in other fields are also discussed. “Conclusion” section provides a summary of the entire paper.

Literature review

Logistics industry efficiency

Researchers have proposed or applied numerous methods for evaluating logistics industry efficiency and have achieved significant progress. Wu et al.9 utilized CCR and BCC models to evaluate some logistics firms in China. Markovits-Somogyi and Bokor10 proposed a DEA pairwise comparison method for measuring the logistics efficiency of European countries. Cao11 combined the DEA method with the Bayes method for addressing the limitation of DEA on resource allocation in advance. Martí et al.12 proposed a method based on the CCR model to assess the logistics performance index. Li et al.13 combined a four-stage DEA model with a Tobit model for measuring port logistics efficiency in China. Zhang and Xiang14 applied a three-stage DEA model to measure the technical efficiency, pure technical efficiency, and scale efficiency of railway logistics. Yu et al.15 combined DEA with Grey forecasting model to predict the sustainability performance of logistics companies. Liu et al.16 applied a non-radial method to assess the unified efficiency of logistics companies in China.

In addition to single-stage production processes, scholars have also studied the multi-stage efficiency of the logistics industry. For example, Hong17 developed a two-stage DEA model for measuring the efficiency of supply chains. Wang et al.18 proposed an enhanced dynamic network DEA, which was developed by the CCR and BCC models, to assess the efficiency of the Internet Plus Logistics sector. Momeni et al.19 introduced a multi objective additive network DEA model to evaluate the efficiency of reverse logistics providers in supply chains. Andrejić20 proposed a new method, consisting of the CCR model, efficiency decomposition models, and game-theory-based models, for assessing the performance of retail supply chains.

Although many researchers have employed various methods to measure logistics efficiency, most have utilized radial models such as CCR and BCC models, while few have utilized non-radial models to assess efficiency. This represents a current deficiency in research on logistics efficiency.

RAM

Cooper et al.5 first developed the RAM model. Its properties, such as “translation” and “units invariant”, were presented and discussed. To estimate the consumption efficiency empirically, Lee et al.21 combined the RAM model and the free disposal hull (FDH) model to develop a discrete RAM model, which can report the discreteness of the consumer choice. Kleine and Sebastian22 proposed a general RAM approach using a general DEA framework. Sueyoshi et al.23 incorporated undesirable outputs extended into the basic RAM model. Then Sueyoshi and Goto24 applied the RAM model to combine desirable outputs and undesirable outputs in a unified treatment. Qi et al.25 employed the RAM model with natural and managerial disposability to assess port unified efficiency. Tavassoli et al.26 applied the RAM model with strong complementary slackness condition and discriminant analysis to assess airlines. Their method overcome the issue of traditional DEA methods failing to fully utilize fully use all inputs and outputs. Chen et al.27 discussed the truck restriction policy in China by utilizing the RAM method. Chiu et al.28 developed a context-dependent RAM model, where the performance of DMUs is evaluated in different contexts. Under the weak-G disposability, Cui and Yu29 proposed a dynamic RAM for assessing the dynamic efficiency. Tsang et al.30 introduced a RAM-based Malmquist productivity index to estimate dynamic production efficiency.

With the advancement of RAM technology, the network DEA framework is gradually being introduced. Avkiran and McCrystal7 extended the RAM model to a network RAM model through a network DEA framework. Heydari et al.31 presented a fully fuzzy network RAM model in the fully fuzzy framework to evaluate the performance of airlines. Kalantary et al.32 developed a network dynamic RAM model and then extended it to an inverse version. Tsolas33 presented a two-stage DEA modeling framework for assessing the credit risk, where the RAM and a common set of weights of RAM are applied. Tavassoli et al.34 elucidated the interconnection between various PPS in the network RAM model, focusing on both internal and external viewpoints. Mousavi et al.35 proved that the RAM-returns-to-scale method could classify DMUs and determine the increasing/decreasing trends of returns to scale.

Despite significant advancements in RAM methodology and its widespread application, there is currently no literature studying the RAM model under the CRS assumption, nor is there any research addressing the modified range issues of the RAM model.

Methods

RAM under any returns to scale

Assume that there are n DMUs involved in the logistics industry. Every \(DMU_{j} (j = 1,2, \ldots ,n)\) uses I inputs to produce S outputs, which is reported in Fig. 1. \(x_{ij}^{{}} (i = 1,2, \ldots ,I_{{}} )\) represents the ith input \(y_{rj}^{{}} (r = 1,2, \ldots ,S_{{}} )\) indicates the rth output. The RAM model is developed based on the additive model. Under the assumption of VRS, Cooper et al.5 initially introduced the RAM model, which has the following form.

$$\begin{aligned} & E^{RAM - VRS} = \min \;\;1 - \frac{1}{I + S}\left( {\sum\limits_{i = 1}^{I} {\frac{{s_{io}^{ - } }}{{R_{i}^{ - } }} + \sum\limits_{r = 1}^{S} {\frac{{s_{ro}^{ + } }}{{R_{r}^{ + } }}} } } \right) \\ & s.t. \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } + s_{io}^{ - } = x_{io}^{{}} ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } - s_{ro}^{ + } = y_{ro}^{{}} ,\;\;r = 1,2, \ldots ,S \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} = 1} ,\;j = 1,2, \ldots ,n \\ & s_{io}^{ - } \ge 0,\;s_{ro}^{ + } \ge 0,\;\;\lambda_{j} \ge 0,\;j = 1,2, \ldots ,n \\ \end{aligned}$$
(1)

where \(s_{io}^{ - } ,\;s_{ro}^{ + }\) are the slacks of the ith input and the rth output, respectively. \(R_{i}^{ - }\) denotes the range of the ith input. \(R_{i}^{ - }\) = \(\overline{x}_{i} - \underline {x}_{i}^{{}}\), where \(\overline{x}_{i}\) = max (\(x_{ij}^{{}}\)) and \(\underline {x}_{i}^{{}}\) = min (\(x_{ij}^{{}}\)). \(R_{r}^{ + }\) denotes the range of the rth output. \(R_{r}^{ + }\) = \(\overline{y}_{r} - \underline {y}_{r}\), where \(\overline{y}_{r}\) = max (\(y_{rj}^{{}}\)) and \(\underline {y}_{r}\) = min (\(y_{rj}^{{}}\)). \(\lambda_{j}^{{}}\) denotes the intensity. Constraints \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} + s_{io}^{ - } = x_{io}^{{}} }\) and \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } - s_{ro}^{ + } = y_{ro}^{{}}\) correspond to inputs and outputs, respectively. The constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } = 1\) represents the assumption of VRS.

Figure 1
figure 1

Production process of a DMU.

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It should be emphasized that Model (1) is only applicable under the VRS condition and not under CRS. Deleting the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } = 1\) does not yield the RAM-CRS model, and in such cases, the efficiency of DMUs may be less than 0. Now, we give the formulas of the RAM-CRS.

$$\begin{aligned} & E^{RAM - CRS} = \min \;\;1 - \frac{1}{I + S}\left( {\sum\limits_{i = 1}^{I} {\frac{{s_{io}^{ - } }}{{R_{i}^{ - } }} + \sum\limits_{r = 1}^{S} {\frac{{s_{ro}^{ + } }}{{R_{r}^{ + } }}} } } \right) \\ & s.t. \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } = x_{io}^{{}} - s_{io}^{ - } ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } = y_{ro}^{{}} + s_{ro}^{ + } ,\;\;r = 1,2, \ldots ,S \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } \ge x_{io}^{{}} - R_{i}^{ - } ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \le R_{r}^{ + } + y_{ro}^{{}} ,\;\;r = 1,2, \ldots ,S \\ & s_{io}^{ - } ,\;s_{ro}^{ + } ,\lambda_{j} \ge 0 \\ \end{aligned}$$
(2)

In this way, we obtain the RAM-CRS model. Constraints \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } \ge x_{io}^{{}} - R_{i}^{ - }\) and \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \le R_{r}^{ + } + y_{ro}^{{}}\) are applied to ensure the efficiency projection of point \((x_{o}^{{}} ,y_{o}^{{}} )\) , given by \(\left( {\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} x_{o}^{{}} } + R_{{}}^{ - } ,\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} y_{o}^{{}} } - R_{{}}^{ + } } \right)\), belongs to the range-bounded PPS. When \(s_{io}^{ - } ,\;s_{ro}^{ + } ,\lambda_{j}\) reach their optimal values, there exist:

$$0 \le E^{RAM - CRS} = \;1 - \frac{1}{I + S}\left( {\sum\limits_{i = 1}^{I} {\frac{{x_{io}^{{}} - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} x_{ij}^{{}} } }}{{R_{i}^{ - } }} + \sum\limits_{r = 1}^{S} {\frac{{\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} y_{rj}^{{}} } - y_{ro}^{{}} }}{{R_{r}^{ + } }}} } } \right) \le 1$$
(3)
$$s_{io}^{* - } = x_{io}^{{}} - \sum\limits_{j = 1}^{n} {\lambda_{j}^{*} x_{ij}^{{}} } \le R_{i}^{ - }$$
(4)
$$s_{ro}^{* + } = \sum\limits_{j = 1}^{n} {\lambda_{j}^{*} y_{rj}^{{}} } - y_{ro}^{{}} \le R_{r}^{ + }$$
(5)

We present the RAM model under the assumption of CRS. There are also other two conditions: non-increasing returns to scale (NIRS) and non-decreasing returns to scale (NDRS). As Cooper et al.6 pointed out, the constraint on \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} }\) must be modified if we want to restrict the RAM to satisfy NIRS or NDRS. If we want to introduce the RAM under these two assumptions, the following constraints should be considered.

$${\text{NIRS}}:\sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} } \le 1$$
(6)
$${\text{NDRS}}:\sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} } \ge 1$$
(7)

Here, we can propose the RAM model under the assumption of ARS. It can be represented as the following form.

$$\begin{aligned} & E^{RAM - ARS} = \min \;\;1 - \frac{1}{I + S}\left( {\sum\limits_{i = 1}^{I} {\frac{{s_{io}^{ - } }}{{R_{i}^{ - } }} + \sum\limits_{r = 1}^{S} {\frac{{s_{ro}^{ + } }}{{R_{r}^{ + } }}} } } \right) \\ & s.t. \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } = x_{io}^{{}} - s_{io}^{ - } ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } = y_{ro}^{{}} + s_{ro}^{ + } ,\;\;r = 1,2, \ldots ,S \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } \ge x_{io}^{{}} - R_{i}^{ - } ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \le R_{r}^{ + } + y_{ro}^{{}} ,\;\;r = 1,2, \ldots ,S \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} \ge 0} \;\;\;or\;\;\;\sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} \le 1\;\;or\;\;\sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} \ge 1} } \; \\ & s_{io}^{ - } ,\;s_{ro}^{ + } ,\lambda_{j} \ge 0 \\ \end{aligned}$$
(8)

If the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } \ge 0\) holds true, then Model (8) represents the RAM-CRS model. If the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } \le 1\) holds true, then Model (8) indicates the RAM-NIRS model. If the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } \ge 1\) holds true, then Model (8) indicates the RAM-NDRS model. It should be specified that for the RAM-NIRS model, with the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } \le 1\), there always exists \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \le \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} \overline{y}_{r}^{{}} } \le \overline{y}_{r}^{{}} \le \overline{y}_{r}^{{}} + y_{ro}^{{}} - \underline {y}_{r}^{{}}\). This means that the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \le R_{r}^{ + } + y_{ro}^{{}}\) can be deleted in RAM-NIRS model. Similarly, for the RAM-NDRS model, the inequality \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } \ge x_{io}^{{}} - R_{i}^{ - }\) always exists when \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } \ge 1\) holds on. It illustrates that the constraint for inputs can be deleted. RAM-NIRS is under the CRS assumption. It applies to situation where increasing the scale of inputs and outputs leads to decreased cost efficiency. RAM-NDRS is under the NDRS assumption, which indicates that it applies to situation where increasing the scale of inputs and outputs does not lead to decreased cost efficiency. RAM-CRS is applicable when the scale of inputs and outputs increases proportionally, and cost efficiency remains unchanged.

The modified RAM

Pastor et al.36 proved that the lower bound of inputs \(\underline {x}_{i}^{{}}\) = min (\(x_{ij}^{{}}\)) and upper bound of outputs \(\overline{y}_{r}\) = max (\(y_{rj}^{{}}\)) are tight for DMUs. Now we consider making adjustments to the ranges of inputs and outputs. Our goal is to define new ranges of inputs and outputs, thereby easing the restrictions on the existing ranges. As mentioned above, the ranges of inputs and outputs can be represented as \(R_{i}^{ - }\) = \(\overline{x}_{i} - \underline {x}_{i}^{{}}\) and \(R_{r}^{ + }\) = \(\overline{y}_{r} - \underline {y}_{r}\), respectively. We attempt to find a way to relax the ranges. When dealing with the bounds of the BAM model, Pastor et al.36 utilized the CRS additive models to define the lower bounds of inputs and upper bounds of outputs. Similar to their approach, we introduce radial models to modify the ranges.

Now, the following input-oriented model is used to redefine the lower bound of inputs.

$$\begin{aligned} & \min \;\theta_{o} \\ & s.t.\;\sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} = \theta_{o} x_{{i_{o} o}}^{{}} - } s_{{i_{o} o}}^{ - } ,\;i = i_{o} \\ & \sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} \le x_{io}^{{}} } ,\;i \ne i_{o} ,i = 1, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} \ge y_{ro}^{{}} } ,r = 1, \ldots ,S \\ & 0 \le \theta_{o} \le 1,\lambda_{j} \ge 0,s_{{i_{o} o}}^{ - } \ge 0 \\ \end{aligned}$$
(9)

\(\theta_{o}\) represents the ioth input efficiency evaluated by the input-oriented model for \(DMU_{o}\). Define \(\underline {x}_{i}{\prime} = \min \left\{ {\theta_{o} x_{{i_{o} j}}^{{}} - s_{{i_{o} j}}^{ - } } \right\},j = 1, \ldots ,n\) and \(R_{i}^{M - } = \overline{x}_{i} - \underline {x}_{i}{\prime}\). \(R_{i}^{M - }\) represents the modified range of inputs, which is used to replace \(R_{i}^{ - }\).

The above model is built upon different inputs. Each input \(i_{o}\) has different efficiency \(\theta_{o}\). In input-oriented CCR model1, there exists \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} x_{ij} \le \theta_{o} x_{io}^{{}} }\). For the observed input \(i_{o}\), \(s_{{i_{o} o}}^{ - }\) is introduced to set the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} x_{ij} = \theta_{o} x_{{i_{o} o}}^{{}} } - s_{{i_{o} o}}^{ - }\) for avoiding the weak efficiency. Constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} x_{ij} \le x_{io}^{{}} } ,\;i \ne i_{o}\) is related to inputs other than \(i_{o}\). Constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} y_{rj} \ge y_{ro}^{{}} }\) is to restrict outputs. The mathematical significance of the above model is to find appropriate values of \(\theta_{o}\) such that, when \(s_{io}^{ - }\) attains optimal value \(s_{io}^{* - } = x_{io}^{{}} - \sum\nolimits_{j = 1}^{n} {\lambda_{j}^{*} x_{ij}^{{}} }\), the condition \(\frac{{s_{io}^{* - } }}{{R_{i}^{M - } }} \le 1\) holds true. Overall, this measure can keep efficiency values between 0 and 1 under the assumption of CRS.

Similarly, the following output-oriented model is used to define the upper bound of outputs.

$$\begin{aligned} & \max \;\varphi_{o} \\ & s.t.\;\sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} = \varphi_{o} y_{{r_{o} o}}^{{}} + s_{{r_{o} o}}^{ + } } ,r = r_{o} \\ & \sum\limits_{j = 1}^{n} {\lambda_{j} y_{rj} \ge y_{ro}^{{}} } ,r \ne r_{o} ,r = 1, \ldots ,S \\ & \sum\limits_{j = 1}^{n} {\lambda_{j} x_{ij} \le x_{io}^{{}} } ,i = 1, \ldots ,I \\ & \varphi_{o} \ge 1,\lambda_{j} \ge 0,s_{{r_{o} o}}^{ + } \ge 0 \\ \end{aligned}$$
(10)

\(\varphi_{o}\) is the roth output efficiency under the output-oriented model for \(DMU_{o}\). Define \(\overline{y}_{r}^{\prime} = \max \left\{ {\varphi_{o} y_{{r_{o} j}}^{{}} + s_{{r_{o} j}}^{* + } } \right\},j = 1, \ldots ,n\) and \(R_{r}^{M + } = \overline{y}_{r}^{\prime} - \underline {y}_{r}\). For the observed input \(r_{o}\), we use the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} y_{rj} = \varphi_{o} y_{{r_{o} o}}^{{}} } + s_{{r_{o} o}}^{ + }\) to replace the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} y_{rj} \ge \varphi_{o} y_{{r_{o} o}}^{{}} }\) in the output-oriented CCR model. Constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} y_{rj} \ge y_{ro}^{{}} } ,r \ne r_{o}\) corresponds to outputs other than \(r_{o}\). Constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j} x_{ij} \le x_{io}^{{}} }\) is to restrict inputs. If \(s_{ro}^{ + }\) reaches optimal value \(s_{ro}^{* + }\), the inequity \(s_{ro}^{* + } \le R_{r}^{M + }\) holds true.

Here, we obtain the new ranges of inputs and outputs: \(R_{i}^{M - }\) and \(R_{r}^{M + }\). Based on the new definition of ranges, we can propose the following MRAM model.

$$\begin{aligned} & E^{M} = \min \;\;1 - \frac{1}{I + S}\left( {\sum\limits_{i = 1}^{I} {\frac{{s_{io}^{ - } }}{{R_{i}^{M - } }} + \sum\limits_{r = 1}^{S} {\frac{{s_{ro}^{ + } }}{{R_{r}^{M + } }}} } } \right) \\ & s.t. \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } + s_{io}^{ - } = x_{io}^{{}} ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } - s_{ro}^{ + } = y_{ro}^{{}} ,\;\;r = 1,2, \ldots ,S \\ & s_{io}^{ - } \ge 0,\;s_{ro}^{ + } \ge 0,\;\;\lambda_{j} \ge 0,\;j = 1,2, \ldots ,n \\ \end{aligned}$$
(11)

The MRAM is under the assumption of CRS, which illustrates that it applies to the condition where increasing the scale of inputs and outputs maintains constant cost efficiency. Compared to the RAM-CRS model, the modified ranges enable the MRAM model to delete constraints \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } \ge x_{io}^{{}} - R_{i}^{ - }\) and \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \le R_{r}^{ + } + y_{ro}^{{}}\). Compared to the RAM-VRS model, MRAM can break free from the constraint \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} } = 1\), which the RAM-VRS cannot achieve. In fact, when constraints in Model (11) holds, that is, \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } + s_{io}^{ - } = x_{io}^{{}}\) and \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } - s_{ro}^{ + } = y_{ro}^{{}}\), it necessarily implies \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } \le x_{io}^{{}}\) and \(\sum\nolimits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } \ge y_{ro}^{{}}\). This means that all constraints in Models (9) and (10) also necessarily hold. This is also why we construct Models (9) and (10) in this manner. It ensures that \(i_{o}\) and \(r_{o}\) apply not only to Models (9) and (10) but also to Model (11).

Now, we discuss properties of MRAM. For the MRAM model, there exist \(s_{io}^{* - } \le \overline{x}_{i} - \underline {x}_{i}^{\prime} = R_{i}^{M - }\) and \(s_{ro}^{* + } \le \overline{y}_{r}^{\prime} - \underline {y}_{r} = R_{i}^{M + }\) . So \(0 \le 1 - \frac{1}{I + S}\left( {\sum\nolimits_{i = 1}^{I} {\frac{{s_{io}^{* - } }}{{R_{i}^{M - } }} + \sum\nolimits_{r = 1}^{S} {\frac{{s_{ro}^{* + } }}{{R_{r}^{M + } }}} } } \right) \le 1\) holds true. If and only if \(s_{io}^{* - } = s_{ro}^{* + } = 0\), \(E^{M}\) can achieve the value of 1, at which point the \(DMU_{o}\) is fully efficient. If and only if \(s_{io}^{* - } = R_{i}^{M - }\) and \(s_{ro}^{* + } = R_{r}^{M + }\), \(E^{M}\) can achieve the value of 0. In this condition, the \(DMU_{o}\) is fully inefficient.

The MRAM model is also units invariant, which is consistent with the RAM model and BAM model. The RAM-VRS model is translation invariant. The translation invariant property cannot be achieved by any CRS model, so CRS excludes the translation invariant property36. At this point, the MRAM cannot achieve translation invariance.

The MRAM exhibits strong monotonicity, which was discussed by Cooper et al.6.

The summary of the properties of the MRAM is as follows.

  1. (1)

    \(0 \le E^{M} \le 1\)

  2. (2)

    \(E^{M} = \left\{ \begin{gathered} 0 \Leftrightarrow DMU_{O} \;{\text{is}}\;{\text{fully}}\;{\text{inefficient}}{.}\; \hfill \\ 1 \Leftrightarrow DMU_{O} \;{\text{ is}}\;{\text{fully}}\;{\text{efficient}}{.} \hfill \\ \end{gathered} \right.\)

  3. (3)

    \(E^{M}\) is invariant to \(\left\{ \begin{gathered} {\text{alternative }}\;{\text{optima}}. \hfill \\ {\text{units}}\;{\text{of}}\;{\text{measurement}}\;{\text{of}}\;{\text{inputs}}\;{\text{and}}\;{\text{outputs}}{.} \hfill \\ \end{gathered} \right.\)

  4. (4)

    \(E^{M}\) is strongly monotonic.

A case study

Data

In the previous section, we established the MRAM model by modifying the bounds and discussed some of its properties. To demonstrate the practicality of MRAM, we will verify it through a case study. We collect production data from the logistics industry in 18 provinces of China in 2022 for assessing the logistics efficiency of different provinces. Figure 2 illustrates the logistics production process of each province. Table 1 reports the indicators of inputs and outputs involved in the logistics industry production process. We select 3 inputs and 3 outputs as evaluation indicators. Inputs include “Mileage of transport route”, “Number of logistics employees”, and “Number of logistics companies”, while outputs include “Freight volume”, “Freight turnover”, and “Value-added of logistics industry”. The specific data for these indicators are summarized in Table 2.

Figure 2
figure 2

Logistics industry production process.

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Table 1 Indicators.
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Table 2 Data of the logistics industry.
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Evaluation results

Applying Model (11) to assess the performance of these DMUs, the results are summarized in Table 3. It can be observed that the efficiency of 10 DMUs is 1, namely Hebei, Inner Mongolia, Jiangsu, Zhejiang, Anhui, Fujian, Shandong, Henan, Guangdong, and Hainan. This indicates that they are all efficient. It should be noted that their optimal slack variables are all 0, which is also a condition for achieving fully efficient. This also satisfies Property (2) of MRAM. The efficiency of Shanxi, Jiangxi, Hubei, and Guangxi is above 0.9. For the remaining DMUs, the performance of Heilongjiang is the worst, with an efficiency score of only 0.8239. On average, there are 6 provinces below the mean, namely Liaoning, Jilin, Heilongjiang, Jiangxi, Hubei, and Hunan. Therefore, Heilongjiang, which exhibits the poorest performance, should promptly adjust the quantities of inputs and outputs to optimize production strategies and enhance logistics efficiency. Other provinces below the average efficiency level, including Liaoning, Jilin, Jiangxi, Hubei, and Hunan, should also alter their production status to achieve the production efficiency levels of those fully efficient provinces, such as Shandong and Guangdong.

Table 3 Evaluation results of Model (11).
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We also present evaluation results of Model (9) and Model (10), which are shown in Table 4. It can be observed that there are 10 efficient provinces selected by both the input model and the output model, namely Hebei, Inner Mongolia, Jiangsu, Zhejiang, Anhui, Fujian, Shandong, Henan, Guangdong, and Hainan. This aligns with the selection results of MRAM. The other 8 DMUs are all inefficient. We can draw a conclusion that the efficient DMUs under both the input-oriented model and the output-oriented model remain efficient in MRAM.

Table 4 Evaluation results of Model (9) and Model (10).
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Table 5 presents the modified bounds of inputs and outputs. The first three rows represent the lower bounds of three inputs, and the last 3 rows represent the upper bounds of three outputs. The modified bound of “Mileage of transport route” is 37,067.45, and no projection of units reaches it. There is also no projection of units reaching the modified bounds of “Number of logistics employees” and “Number of logistics companies”. The modified upper bound of “Freight volume” is 394,061, which is reached by an efficient unit Anhui. The modified upper bound of “Freight turnover” is only reached by an efficient DMU Guangdong. The modified upper bound of “Value-added of logistics industry” is only reached by Shandong, which is also an efficient DMU.

Table 5 Modified bounds.
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Discussions

Comparison with RAM-CRS

To make comparisons, we utilize Model (2) to compute the efficiency under the RAM-CRS. Table 6 summarizes the calculation results. The conclusion can be drawn that the efficiency scores of Hebei, Inner Mongolia, Jiangsu, Zhejiang, Anhui, Fujian, Shandong, Henan, Guangdong, and Hainan are all 1, which indicates that they are fully efficient. 3 provinces have logistics efficiencies between 0.9 and 1, namely Shanxi, Jiangxi, and Guangxi. The efficiency scores of the other provinces are all below 0.9, with Heilongjiang being the worst performer, with an efficiency of only 0.8083. Compared with the mean value of 0.9540, the efficiency scores of Liaoning, Jilin, Heilongjiang, Jiangxi, Hubei, and Hunan are below the mean, while the efficiency scores of other provinces are above the mean.

Table 6 Evaluation results of Model (2).
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Table 7 displays the range bounds of inputs and outputs. All the range bounds are reached. The lower bounds of “Mileage of transport route”, “Number of logistics employees”, and “Number of logistics companies” are all reached by Hainan, an efficient unit. The upper bound of “Freight volume” is reached by Anhui, while the upper bounds of “Freight turnover” and “Value-added of logistics industry” are reached by Guangdong and Shandong respectively.

Table 7 Range bounds.
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Figure 3 shows the comparison of range bounds and modified bounds. It is evident that the range bounds of the three inputs are significantly larger than the modified bounds. As for the three outputs, their upper bounds are equal in both cases. The range bounds are noticeably lower than modified bounds, which indicates that range bounds are tighter for DMUs. Such a tight range reduces the difficulty for the projection of DMUs to reach it. This implies that the range bounds impose tighter constraints on DMUs, which also aligns with the conclusion reached by Poster et al.36: tighter bounds increase the number of projections able to reach them.

Figure 3
figure 3

Comparison of range bounds and modified bounds.

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Figure 4 compares efficiency scores of the MRAM with efficiency scores of the RAM-CRS. It can be observed that the efficiency scores of Shanxi, Liaoning, Jilin, Heilongjiang, Jiangxi, Hubei, Hunan, and Guangxi under MRAM are slightly higher than under RAM-CRS. The efficiency scores of Hebei, Inner Mongolia, Jiangsu, Zhejiang, Anhui, Fujian, Shandong, Henan, Guangdong, and Hainan are all 1 in both cases, which indicates that there is no difference. Therefore, overall, the calculation results of RAM-CRS are lower than those of MRAM. The reason is that the range bounds under RAM-CRS are too tight, resulting in efficiency values at a relatively low level. Specifically, for outputs, there exist \(R_{1}^{M + } = R_{1}^{ + }\), \(R_{2}^{M + } = R_{2}^{ + }\), \(R_{3}^{M + } = R_{3}^{ + }\). For inputs, there exist \(R_{1}^{M - } > R_{1}^{ - }\), \(R_{2}^{M - } > R_{2}^{ - }\),\(R_{3}^{M - } > R_{3}^{ - }\). Under this condition, \(\frac{1}{3 + 3}\left( {\sum\nolimits_{i = 1}^{3} {\frac{{s_{io}^{ - } }}{{R_{i}^{M - } }} + \sum\nolimits_{r = 1}^{3} {\frac{{s_{ro}^{ + } }}{{R_{r}^{M + } }}} } } \right) \le \frac{1}{3 + 3}\left( {\sum\nolimits_{i = 1}^{3} {\frac{{s_{io}^{ - } }}{{R_{i}^{ - } }} + \sum\nolimits_{r = 1}^{3} {\frac{{s_{ro}^{ + } }}{{R_{r}^{ + } }}} } } \right)\) holds true. Then \(1 - \frac{1}{3 + 3}\left( {\sum\nolimits_{i = 1}^{3} {\frac{{s_{io}^{ - } }}{{R_{i}^{M - } }} + \sum\nolimits_{r = 1}^{3} {\frac{{s_{ro}^{ + } }}{{R_{r}^{M + } }}} } } \right) \ge 1 - \frac{1}{3 + 3}\left( {\sum\nolimits_{i = 1}^{3} {\frac{{s_{io}^{ - } }}{{R_{i}^{ - } }} + \sum\nolimits_{r = 1}^{3} {\frac{{s_{ro}^{ + } }}{{R_{r}^{ + } }}} } } \right)\) can be acquired. In this case, the efficiency under RAM-CRS is lower than that under MRAM. The modified bounds appropriately alleviate this restriction.

Figure 4
figure 4

Comparison of efficiency scores.

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Comparison with additive model

To further highlight the advantages of MRAM, we also introduce the additive model for comparison. The additive model-CRS has the following formula:

$$\begin{aligned} & \max \;\sum\limits_{i = 1}^{I} {s_{io}^{ - } + \sum\limits_{r = 1}^{S} {s_{ro}^{ + } } } \\ & s.t. \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} x_{ij}^{{}} } + s_{io}^{ - } = x_{io}^{{}} ,\;\;i = 1,2, \ldots ,I \\ & \sum\limits_{j = 1}^{n} {\lambda_{j}^{{}} y_{rj}^{{}} } - s_{ro}^{ + } = y_{ro}^{{}} ,\;\;r = 1,2, \ldots ,S \\ & s_{io}^{ - } \ge 0,\;s_{ro}^{ + } \ge 0,\;\;\lambda_{j} \ge 0,\;j = 1,2, \ldots ,n \\ \end{aligned}$$
(12)

Since the additive model cannot directly obtain the efficiency of DMUs, we can calculate the efficiency of Model (12) using the following formula:

$$E^{ADD} = \;\frac{{1 - \frac{1}{I}\left( {\sum\nolimits_{i = 1}^{I} {\frac{{s_{io}^{* - } }}{{x_{io} }}} } \right)}}{{1 + \frac{1}{S}\left( {\sum\nolimits_{r = 1}^{S} {\frac{{s_{ro}^{* + } }}{{y_{ro} }}} } \right)}}$$
(13)

where \(s_{io}^{* - }\) , \(s_{ro}^{* + }\) denotes the optimal values in Model (12).

Using the above method, we can obtain efficiency under the additive model. Figure 5 summarizes the comparison results.

Figure 5
figure 5

Comparison results.

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It can be observed that Shanxi, Liaoning, Jilin, Heilongjiang, Jiangxi, Hubei, Hunan and Guangxi have lower efficiency under the additive model compared to MRAM. In particular, the differences between Jilin, Heilongjiang, and Hunan under both conditions are quite significant. The efficiency values for other DMUs under both models are 1, which indicates that there are no differences under the two methods. From the data, the production efficiency of these DMUs should be higher, but their efficiency under the additive model is significantly lower, which is unreasonable.

In summary, the efficiency under the additive model is significantly lower than under the MRAM model. This suggests that using the additive model to assess the performance of DMUs may underestimate efficiency values.

Applications in other fields

In fact, DEA is a nonparametric mathematical tool, which is not limited to specific industries. Based on the characteristics of the industry, suitable input and output variables can be selected to apply our model for efficiency evaluation. This paper applies the MRAM model to the logistics industry. Besides logistics, MRAM is also applicable to other fields, such as evaluating agricultural system efficiency, assessing ecological system efficiency, and studying the production processes of banks, among others. For instance, Fukuyama and Tan37 applied DEA method to study the loan loss reserves in banking production process. Li et al.38 evaluated the performance of ecological systems in China. Future research could utilize the MRAM model to study production processes in various domains.

Assume the following inputs and outputs exist in a certain field. Table 8 shows specific data on inputs and outputs. The data is randomly generated.

Table 8 Data.
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Applying MRAM model, we can obtain the efficiency of these DMUs. The results are shown in Table 9.

Table 9 Results.
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It can be observed that the efficiency values of DMUs 2, 3, 4, 5, 6, and 9 are 1, which indicates that they are all highly efficient. DMUs 1, 7, 8, and 10 have efficiency scores less than 1, which reports that they are inefficient. Among them, DMU 8 performs the worst, with an efficiency of only 0.6. This suggests that DMU 8 needs to adjust its production strategy promptly, identify weak production areas, and optimize production efficiency.

In general, MRAM can be applied not only to the logistics industry but also to other industries under the premise of selecting appropriate inputs and outputs. Therefore, the MRAM model is an effective method for efficiency assessment.

Conclusion

Assessing the efficiency of the logistics industry is currently a focal point. How to accurately assess logistics efficiency is a challenge facing researchers today. The main deficiency in existing research on logistics efficiency is that most studies utilize radial models, with few using non-radial models. There are two main deficiencies in existing research on RAM. Firstly, the current RAM model only applies to the VRS condition. Secondly, there is a lack of literature studying the bounds issues of the RAM model.

To address these deficiencies, this paper first proposed a RAM model under the CRS condition, and it was then extended to a RAM model under any returns to scale. Next, we introduced a modified RAM model, redefining the upper bound of inputs and the lower bound of outputs through input-oriented and output-oriented models. New ranges of inputs and outputs were presented.

To test the practicality of the new model, we evaluated the logistics efficiency of 18 provinces and analyzed calculation results, which demonstrated that the MRAM model could be used to assess logistics efficiency in practice. For comparison, we also used the RAM-CRS model to calculate the logistics efficiency of 18 provinces. It showed that the calculation results of RAM-CRS were lower than those of MRAM. This indicated that the bounds before modification were too tight for DMUs, which resulted in lower efficiency. The modified bounds overcome this deficiency. To further compare, we also used the additive model to calculate logistics efficiency. The results indicated that efficiency under the additive model is significantly lower than under MRAM. Applying the additive model may lead to an underestimation of efficiency values. Furthermore, applications of MRAM in other fields were discussed.

The production process in the logistics industry involves two special elements, namely dual-role factors and undesirable outputs. Dual-role factors can play the input and output roles simultaneously39. For undesirable outputs, managers often prefer them to be as minimal as possible. In future research, dual-role factors and undesirable outputs can be incorporated into the scope of MRAM studies, thus forming a new model to evaluate the performance of more complex production processes.