A Two-Stage Rough Data Envelopment Analysis and Its Application in Three-Level Supply Chain Performance Evaluation

In the era of globalization, supply chain management (SCM) plays an important role in the business administration field due to the increase in the number of competitors in market globalization. At the beginning of the 1990s, SCM appeared and was concerned with the planning of supply and demand, managing the production and manufacturing, controlling the inventory level, and distributing the products. Performance evaluation is the ability of a firm to produce maximum outputs by using a set of available inputs. The firm is efficient if it achieves maximum output with a set of inputs [1]. The measure of efficiency is depended on the objective of the government, e.g., minimizing the use of resources (input-oriented), minimizing cost, maximizing productivity (output-oriented), or maximizing profit. from this perspective, supply chain efficiency is how the firm utilizes the resources to achieve its goal in the whole supply chain and it’s all levels.

Data Envelopment Analysis is one of the most popular methods in performance efficiency evaluation. DEA is a data-oriented approach. It is a nonparametric mathematical-based tool that transforms multiple inputs into multiple outputs to evaluate the efficiency and performance of a set of entities DMUs. The result from the DEA study is classified all DMUs as either “efficient” or “inefficient”. Efficiency is defined simply as a ratio when the case has a single input and output. However, in real-life, any organization have multiple inputs and output. The efficiency of a DMU is measured relative to all other DMUs. Each DMU is calculated and analyzed to get the maximum performance for each unit separately.

Chames, Cooper, and Rhodes, (CCR model) [2] consider the fundamental DEA model, built to measure the efficiency of DMUs. The obtained efficiency is not absolute efficiency but relative efficiency to other DMUs which are measured. CCR model is considered the birth of DEA and is still the most widely used type of DEA model. After that, Banker, Charnes, & Cooper 1984 [3] developed the BCC model which is VRS. It uses only when required to check for increasing or decreasing returns because its interpretations for results are complicated. On the contrary, the CCR model is most commonly used because it is easy and better in interpreting results. Recently, the DEA models are becoming popular for evaluating the relative efficiency in different sectors (e.g., education, healthcare, economy, etc.). It determines the efficiency based on its inputs and outputs and compares it with the other DMUs involved in the analysis.

In the traditional DEA, the supply chain is treated as a black box where the initial inputs from the first stage and the final outputs from the last stage only are considered. That is, all the intermediate stages are neglected which makes the efficiency evaluation not accurate and the results may not make sense. Therefore, the evaluation in the form of a network (i.e. evaluate the DMUs in multi-stages not evaluate all the processes as one stage, by considering only the initial inputs and final outputs) is considered a good solution for solving this problem.

A two-stage DEA model was developed to measure the internal efficiency of any sub-systems [4]. The traditional DEA model was modified to can measure the efficiency of the series relationship of the two sub-processes within the whole process. In Two-stage DEA models, the first stage consumes some inputs and produces some outputs. The outputs of the first stage are used as the inputs of the second stage. These are called the intermediate variables.

The anther limitation in applying traditional DEA to measure the efficiency of a supply chain is uncertainty. In the traditional DEA, all variables are required to be deterministic, and this is not the case in most real-life problems. Unfortunately, the most of observed data are vague, imprecise, and incomplete. Usually, data is vague like the production amount. Most of the time there is no exact amount of products produced, but it depends on estimations. In addition, data may be imprecise like time, cost of production, and cost of transportation. That makes the efficiency evaluation process results inaccurate and does not represent reality. After stochastic DEA and fuzzy DEA. Rough DEA is a new tool used to deal with uncertainty. So in our study, To overcome this problem. we use Rough DEA to can treat the rough uncertain environment in the supply chain.

In this paper, we integrate two-stage DEA and rough set theory to produce a two-stage RDEA model that overcomes the most two popular problems faced by decision-makers in the efficiency evaluation of any system. That is the Uncertainty and vague data, and the multi-stage systems with dependent relations like supply chains. In the previous studies, the two-stage DEA and rough DEA are used separately. This paper aims to develop a model that can open the black box and evaluate the complex systems (e.g. supply chain) with internal relations between its stages. As well as, can be treated with vague, uncertain, imprecise data for inputs and outputs values. In the two-stage RDEA, we use the α-optimistic and α-pessimistic method which transforms rough variables into a deterministic form which is necessary for the DEA method. After that, we use Wang and chin [5] method to transform two-stage into one stage to evaluate the efficiency of the whole system and all its subsystem comprehensively. Our proposed model will apply to a three-level supply chain of a cement company to show its ability to treat with multi-stage supply chain with rough inputs and outputs variables.

The rest of the paper is represented as follows: section 2. shows a background about two-stage DEA and rough DEA. In section 3, we introduce the proposed model (i.e., Two-stage RDEA) in detail and show how merging Two-stage DEA and rough set theory to produce our proposed model. In section 4, we apply our proposed model on a three-level supply chain example to evaluate the efficiency of the whole supply chain and its two stages. Finally, section 5 concludes the work and discusses future works.

3.1 The combination of two-stage DEA and rough set theory

In the section, we show how can merge two-stage DEA and rough set theory. The combined model can use to solve the problem of black-box efficiency evaluation and uncertain data in inputs and outputs. By evaluating the whole system and its subsystems with considering vague and incomplete data.

A two-stage RDEA consists of merging Eq. (2) and Eq. (5) to produce: 

$\theta_{0}^{*}=\operatorname{Max}_{1} \sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d 0}+\lambda_{2} \sum_{r=1}^{s} u_{r} \hat{\mathrm{y}}_{r 0}$

s.t.

$\lambda_{1} \sum_{i=1}^{m} v_{i} \hat{x}_{i 0}+\lambda_{2} \sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d 0}=1$

$\sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d j}-\sum_{i=1}^{m} v_{i} \hat{x}_{i j} \leq 0(j=1,2,3, \ldots, n)$

$\sum_{r=1}^{s} u_{r} \hat{\mathrm{y}}_{r j}-\sum_{d=1}^{D} \eta_{d} \hat{z}_{d j} \leq 0(j=1,2,3, \ldots, n)$

$v_{i}, \eta_{d}, u_{r} \geq 0$                    (6)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s.

We use the α-optimistic and α-pessimistic method to convert the rough variables $\hat{X} j, \hat{Y} j, \hat{Z} j$ to deterministic variables, this is done through two steps.

In the first step, where 0.5˂α≤1, then $\theta_{0}^{\inf (\alpha)} \geq \theta_{0}^{\sup (\alpha)}$. Therefore, Eq. (6) will convert to an interval programming model:

$\theta_{0}^{*}=M a x \lambda_{1} \sum_{d=1}^{D} \eta_{d}\left[Z_{d 0}^{\sup (\alpha)}, Z_{d 0}^{\inf (\alpha)}\right]$

$+\lambda_{2} \sum_{r=1}^{s} u_{r}\left[Y_{r 0}^{\sup (\alpha)}, Y_{r 0}^{\inf (\alpha)}\right]$

s.t. $\lambda_{1} \sum_{i=1}^{m} v_{i}\left[X_{i 0}^{\sup (\alpha)}, X_{i 0}^{\inf (\alpha)}\right]+\lambda_{2} \sum_{d=1}^{m} \eta_{d}\left[Z_{d 0}^{\sup (\alpha)}, Z_{d 0}^{\inf (\alpha)}\right]=1$

$\sum_{d=1}^{D} \eta_{d}\left[Z_{d j}^{\sup (\alpha)}, Z_{d j}^{\inf (\alpha)}\right]-\sum_{i=1}^{m} v_{i}\left[X_{i j}^{\sup (\alpha)}, X_{i j}^{\inf (\alpha)}\right] \leq 0$

$\sum_{r=1}^{s} u_{r}\left[Y_{r j}^{\sup (\alpha)}, Y_{r j}^{\inf (\alpha)}\right]-\sum_{d=1}^{D} \eta_{d}\left[Z_{d j}^{\sup (\alpha)}, Z_{d j}^{\inf (\alpha)}\right] \leq 0$

$v_{i}, \eta_{d}, u_{r} \geq 0$          (7)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s; j=1, 2, 3, …, n.

The second step, convert the interval programming Eq. (7) to a deterministic linear programming model. Consequently, Eq. (7) divide into two deterministic linear programming to get the lower bound ($\theta_{0}^{\sup (\alpha)}$) and upper bound $\left(\theta_{0}^{\inf (\alpha)}\right)$.

To get the minimum linear programming of Eq. (7) $\left(\theta_{0}^{\sup (\alpha)}\right)$, under trust level $\alpha$. Let DMU₀ be the evaluated DMU and will compare it with the rest of the DMUs. In the worst case, the inputs for DMU₀ have maximum values and the outputs have the minimum values compared with the other DMUs. Consequently, the lower bound $\theta_{0}^{\sup (\alpha)}$ donated as:

$\theta_{0}^{\sup (\alpha)}=\operatorname{Max}_{1} \sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}+\lambda_{2} \sum_{r=1}^{s} u_{r} Y_{r 0}^{\sup (\alpha)}$

s.t. $\lambda_{1} \sum_{i=1}^{m} v_{i} X_{i 0}^{\inf (\alpha)}+\lambda_{2} \sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}=1$

$\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}-\sum_{i=1}^{m} v_{i} X_{i 0}^{\inf (\alpha)} \leq 0(j=0)$

$\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\inf (\alpha)}-\sum_{i=1}^{m} v_{i} X_{i j}^{\sup (\alpha)} \leq 0(j=1,2,3, \ldots, n)$

$\sum_{r=r f 1}^{s} u_{r} Y_{r 0}^{\sup (\alpha)}-\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)} \leq 0(j=0)$

$\sum_{r=1}^{s} u_{r} Y_{r j}^{\inf (\alpha)}-\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\sup (\alpha)} \leq 0(j=1,2,3, \ldots, n)$

$v_{i}, \eta_{d}, u_{r} \geq 0$                    (8)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s.

Furthermore, to get the maximum linear programming of Eq. (7) $\left(\theta_{0}^{\inf (\alpha)}\right)$, under trust level $\alpha$. Let DMU₀ be the evaluated DMU and will compare it with the rest of the DMUs. In the best case, the inputs for DMU₀ have minimum values and the outputs have the maximum values compared with the other DMUs. Consequently, the upper bound $\theta_{0}^{\inf (\alpha)}$ donated as:

$\theta_{0}^{\inf (\alpha)}=\operatorname{Max} \lambda_{1} \sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}+\lambda_{2} \sum_{r=1}^{s} u_{r} Y_{r 0}^{\inf (\alpha)}$

s.t. $\lambda_{1} \sum_{i=1}^{m} v_{i} X_{i 0}^{\sup (\alpha)}+\lambda_{2} \sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}=1$

$\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}-\sum_{i=1}^{m} v_{i} X_{i 0}^{\sup (\alpha)} \leq 0(j=0)$

$\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\sup (\alpha)}-\sum_{i=1}^{m} v_{i} X_{i j}^{\inf (\alpha)} \leq 0(j=1,2,3, \ldots, n)$

$\sum_{r=1}^{s} u_{r} Y_{r 0}^{\inf (\alpha)}-\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)} \leq 0(j=0)$

$\sum_{r=1}^{s} u_{r} Y_{r j}^{\sup (\alpha)}-\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\inf (\alpha)} \leq 0(j=1,2,3, \ldots, n)$

$v_{i}, \eta_{d}, u_{r} \geq 0$                   (9)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s.

From solving linear programming (8) and (9), we obtain the relative efficiency interval [θ^sup(α),θ^inf(α)] for DMU₀ compared with the n-1 DMUs.

3.2 A modified two-stage RDEA in measuring the efficiency for three-level supply chains

This study aims to apply our proposed model that can use to get a comprehensive efficiency for the entire supply chain which contains multi-level subchains. We apply our model in three-level supply chains that contain suppliers, manufacturers, and distributors as shown in Figure 3.

3.png

Figure 3. Three-level supply chain

In the previous section, we discuss how to measure the efficiency of a supply chain that contains two stages and has some rough variables. Here, we introduce how we generalize model (6) to measure the efficiency of multi-level supply chains comprehensively with rough variables. The modified model will be as:

$\theta_{0}^{*}=\operatorname{Max} \lambda_{1}\left(\sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d 0}+\sum_{r=1}^{s} u_{r} \hat{\mathrm{y}}_{r 0}\right)$

$+\lambda_{2}\left(\sum_{r=1}^{s} u_{r} \hat{\mathrm{y}}_{r 0}+\sum_{t=1}^{T} \alpha_{t} \hat{w}_{t 0}\right)$

s. t. $\lambda_{1}\left(\sum_{i=1}^{m} v_{i} \hat{x}_{i 0}+\sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d 0}\right)+\lambda_{2}\left(\sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d 0}+\sum_{r=1}^{m} u_{r} \hat{\mathrm{y}}_{r 0}\right)$

$\left(\sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d j}+\sum_{r=1}^{m} u_{r} \hat{\mathrm{y}}_{r j}\right)-\left(\sum_{d=1}^{D} \eta_{d} \hat{\mathrm{z}}_{d j}+\sum_{i=1}^{m} v_{i} \hat{x}_{i j}\right) \leq 0$

$\left(\sum_{r=1}^{s} u_{r} \hat{\mathrm{y}}_{r j}+\sum_{t=1}^{T} \alpha_{t} \hat{w}_{t j}\right)-\left(\sum_{r=1}^{s} u_{r} \hat{\mathrm{y}}_{r j}+\sum_{d=1}^{D} \eta_{d} \hat{z}_{d j}\right) \leq 0$

$v_{i}, \eta_{d}, u_{r}, \alpha_{t} \geq 0$   (10)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s; t=1, 2, 3, …, T.

Model (10) will transform into an interval programming model:

$\begin{aligned} \theta_{0}^{*}=\operatorname{Max}_{1}\left(\sum_{d=1}^{D} \eta_{d}\left[Z_{d 0}^{\sup (\alpha)}, Z_{d 0}^{\inf (\alpha)}\right]\right.\\ &\left.+\sum_{r=1}^{s} u_{r}\left[Y_{r 0}^{\sup (\alpha)}, Y_{r 0}^{\inf (\alpha)}\right]\right) \\ &+\lambda_{2}\left(\sum_{r=1}^{s} u_{r}\left[Y_{d 0}^{\sup (\alpha)}, Y_{d 0}^{\inf (\alpha)}\right]\right.\\ &\left.+\sum_{t=1}^{T} \alpha_{t}\left[W_{t 0}^{\sup (\alpha)}, W_{t 0}^{\inf (\alpha)}\right]\right) \end{aligned}$   

s. t. $\lambda_{1}\left(\sum_{i=1}^{m} v_{i}\left[X_{i 0}^{\sup (\alpha)}, X_{i 0}^{\inf (\alpha)}\right]+\sum_{d=1}^{D} \eta_{d}\left[Z_{d 0}^{\sup (\alpha)}, Z_{d 0}^{\inf (\alpha)}\right]\right)+\lambda_{2}\left(\sum_{d=1}^{D} \eta_{d}\left[Z_{d 0}^{\sup (\alpha)}, Z_{d 0}^{\inf (\alpha)}\right]+\sum_{r=1}^{m} u_{r}\left[Y_{r 0}^{\sup (\alpha)}, Y_{r 0}^{\inf (\alpha)}\right]\right)=1$   

$\left(\sum_{d=1}^{D} \eta_{d}\left[Z_{d j}^{\sup (\alpha)}, Z_{d j}^{\inf (\alpha)}\right]+\sum_{r=1}^{m} u_{r}\left[Y_{r j}^{\sup (\alpha)}, Y_{r j}^{\inf (\alpha)}\right]\right)-\left(\sum_{d=1}^{D} \eta_{d}\left[Z_{d j}^{\sup (\alpha)}, Z_{d j}^{\inf (\alpha)}\right]+\sum_{i=1}^{m} v_{i}\left[X_{i j}^{\sup (\alpha)}, X_{i j}^{\inf (\alpha)}\right]\right) \leq 0$ 

$\left(\sum_{r=1}^{s} u_{r}\left[Y_{r j}^{\sup (\alpha)}, Y_{r j}^{\inf (\alpha)}\right]+\sum_{t=1}^{T} \alpha_{t}\left[W_{t j}^{\sup (\alpha)}, W_{t j}^{\inf (\alpha)}\right]\right)-\left(\sum_{r=1}^{s} u_{r}\left[Y_{r j}^{\sup (\alpha)}, Y_{r j}^{\inf (\alpha)}\right]+\sum_{d=1}^{D} \eta_{d}\left[Z_{d j}^{\sup (\alpha)}, Z_{d j}^{\inf (\alpha)}\right]\right) \leq 0$

$v_{i}, \eta_{d}, u_{r}, \alpha_{t} \geq 0$    (11)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s; t=1, 2, 3, …, T; j=1, 2, 3, …, n.

Model (11) will convert into minimum linear programing and maximum linear programming.

The $\theta_{0}^{\sup (\alpha) *}$ donated as:

 $\theta_{0}^{\sup (\alpha)^{*}}=\operatorname{Max} \lambda_{1}\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}+\sum_{r=1}^{s} u_{r} Y_{r 0}^{\sup (\alpha)}\right)+\lambda_{2}\left(\sum_{r=1}^{s} u_{r} Y_{r 0}^{\sup (\alpha)},+\sum_{t=1}^{T} \alpha_{t} W_{t 0}^{\sup (\alpha)}\right)$

s.t. $\lambda_{1}\left(\sum_{i=1}^{m} v_{i} X_{i 0}^{\inf (\alpha)}+\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}\right)+\lambda_{2}\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}+\sum_{r=1}^{m} u_{r} Y_{r 0}^{\inf (\alpha)}\right)=1$

$\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}+\sum_{r=1}^{m} u_{r} Y_{r 0}^{\sup (\alpha)}\right)-\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}+\sum_{i=1}^{m} v_{i} X_{i 0}^{\inf (\alpha)}\right) \leq 0(j=0)$

$\left(\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\inf (\alpha)}+\sum_{r=1}^{m} u_{r} Y_{r j}^{\inf (\alpha)}\right)-\left(\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\sup (\alpha)}+\sum_{i=1}^{m} v_{i} X_{i j}^{\sup (\alpha)}\right) \leq 0$

$\left(\sum_{r=1}^{s} u_{r} Y_{r 0}^{\sup (\alpha)}+\sum_{t=1}^{T} \alpha_{t} W_{t 0}^{\sup (\alpha)}\right)-\left(\sum_{r=1}^{s} u_{r} Y_{r 0}^{\inf (\alpha)}+\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}\right) \leq 0(j=0)$

$\left(\sum_{r=1}^{s} u_{r} Y_{r j}^{\inf (\alpha)}+\sum_{t=1}^{T} \alpha_{t} W_{t j}^{\inf (\alpha)}\right)-\left(\sum_{r=1}^{s} u_{r} Y_{r j}^{\sup (\alpha)}+\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\sup (\alpha)}\right) \leq 0$

$v_{i}, \eta_{d}, u_{r}, \alpha_{t} \geq 0$         (12)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s; t=1, 2, 3, …, T; j=1, 2, 3, …, n.

The $\theta_{0}^{\inf (\alpha) *}$ donated as:

$\theta_{0}^{\inf (\alpha)^{*}}=\operatorname{Max} \lambda_{1}\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}+\sum_{r=1}^{s} u_{r} Y_{r 0}^{\inf (\alpha)}\right)+\lambda_{2}\left(\sum_{r=1}^{s} u_{r} Y_{r 0}^{\inf (\alpha)},+\sum_{t=1}^{T} \alpha_{t} W_{t 0}^{\inf (\alpha)}\right)$   

s.t. $\lambda_{1}\left(\sum_{i=1}^{m} v_{i} X_{i 0}^{\sup (\alpha)}+\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}\right)+\lambda_{2}\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}+\sum_{r=1}^{m} u_{r} Y_{r 0}^{\sup (\alpha)}\right)=1$

$\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\inf (\alpha)}+\sum_{r=1}^{m} u_{r} Y_{r 0}^{\inf (\alpha)}\right)-\left(\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}+\sum_{i=1}^{m} v_{i} X_{i 0}^{\sup (\alpha)}\right) \leq 0(j=0$

$\left(\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\sup (\alpha)}+\sum_{r=1}^{m} u_{r} Y_{r j}^{\sup (\alpha)}\right)-\left(\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\inf (\alpha)}+\sum_{i=1}^{m} v_{i} X_{i j}^{\inf (\alpha)}\right) \leq 0$

$\left(\sum_{r=1}^{s} u_{r} Y_{t 0}^{\inf (\alpha)}+\sum_{t=1}^{T} \alpha_{t} W_{t 0}^{\inf (\alpha)}\right)-\left(\sum_{r=1}^{s} u_{r} Y_{r 0}^{\sup (\alpha)}+\sum_{d=1}^{D} \eta_{d} Z_{d 0}^{\sup (\alpha)}\right) \leq 0(j=0)$

$\left(\sum_{r=1}^{S} u_{r} Y_{r j}^{\sup (\alpha)}+\sum_{t=1}^{T} \alpha_{t} W_{t j}^{\sup (\alpha)}\right)-\left(\sum_{r=1}^{S} u_{r} Y_{r j}^{\inf (\alpha)}+\sum_{d=1}^{D} \eta_{d} Z_{d j}^{\inf (\alpha)}\right) \leq 0$ 

$v_{i}, \eta_{d}, u_{r}, \alpha_{t} \geq 0$          (13)

where, i=1, 2, 3, …, m; d=1, 2, 3, …, D; r=1, 2, 3, …, s; t=1, 2, 3, …, T; j=1, 2, 3, …, n.

The comprehensive evaluation process for the three-level supply chain in Figure 3 will be summarized in three steps:

First step: split the three-level supply chain into two subchains (suppliers-manufacturers) and (manufacturers- distributors). In this step, we evaluate the efficiency of each subchain separately. Figure 4 and Figure 5 are the transformations for the suppliers-manufacturers subchain and manufacturers-distributors subchain, respectively. Then evaluate each subchain independently by using Eq. (8) and (9), respectively.

Second step: transform the whole three-level supply chain into one stage Figure 6. Then evaluate the whole supply chain by using Eq. (12) and (13).

Based on the results from step one (i.e., efficiency evaluation for each subchain) and step two (i.e., efficiency evaluation for the whole supply chain), we can get a comprehensive efficiency evaluation for each DMUs and determine which DMU is the best.

Third step: is to rank all DMUs (supply chains) according to the MRA method will discuss in the next section.

$\max \left(r_{i}\right)=\max _{j \neq 1}\left\{\theta_{j}^{\inf (\alpha)}\right\}-\theta_{i}^{\sup (\alpha)}$

4.png

Figure 4. Two-stage supplier and manufacturer supply chain converted into the one-stage supply chain

5.png

Figure 5. Two-stage manufacturer and distributors supply chain converted into the one-stage supply chain

6.png

Figure 6. Three-level suppliers, manufacturers, and distributors supply chain converted into the one-stage supply chain

3.3 The ranking method for efficiency DMUs

After we get the efficiency interval for all DMUs, we should rank the efficiency intervals for DMUs. Because in case having more than one DMU is efficient, we can determine which DMU is the most efficient, and order the rest efficient DMUs based on the difference between its upper and lower bound intervals. In this study, we use the MRA approach introduced by Wang et al. [13] to rank the efficient DMUs intervals. In the beginning, we should notice that the three-level supply chain will be efficient only if:

The upper limit $\left(\theta_{0}^{\inf (\alpha) *}\right)$ for all sub-chains=1.

The upper limit $\left(\theta_{0}^{\inf (\alpha) *}\right)$ for the whole supply chain =1.

The MRA approach is introduced as follows:

Compute the maximum loss of efficiency for each efficient DMUs and choose the maximum loss of efficiency (regret).

Compare the regret function for all efficiency intervals and order the DMUs from the minimum regret (no loss of efficiency) to the maximum regret. 

$\min _{i}\left\{\max \left(r_{i}\right)\right\}=\min _{i}\left\{\max \left[\max _{j \neq 1}\left(\alpha_{j}^{\inf (\alpha)}\right)-\theta_{i}^{\sup (\alpha)}, 0\right]\right\}$