Inverse DEA in two-stage systems based on allocative efficiency

Abstract

Inputs and outputs of Decision Making Units (DMUs) are estimated by the Inverse Data Envelopment Analysis (InvDEA) models, while their relative efficiency scores remain unchanged. But, in some cases, cost/price information of the inputs and outputs are available. This paper employs the input and output cost/price information, including the generalized InvDEA concept in two-stage structures. To this end, it proposes a four-stage method to deal with the InvDEA concept, for estimating the inputs and outputs of the DMUs with a two-stage network structure method, while the allocative efficiency scores of all the units remain stable. Eventually, an empirical example is rendered to illustrate the competence of the method which is presented.

Keywords

Inverse DEA network DEA two-stage network cost efficiency input/output estimation

1 Introduction

The classical DEA methods use inputs and outputs to estimate the efficiency score of the DMU under evaluation. From another point of view, the InvDEA technique which was introduced by Wie et al. [34] aims at responding to this question, that, if among a group of DMUs, we increase certain inputs for a particular unit and assume that the DMU maintains its current efficiency level with respect to other units, how many more outputs could the unit produce? Or else, if the outputs need to be increased to a certain level and the efficiency of the unit remains unchanged, how many more inputs should be provided for the unit? Numerous studies have been conducted on the InvDEA concept. Yan et al. in [35] discussed the InvDEA problem with preference to cone constraints. Then in [19], Jahanshahloo et al. developed the presented method by Yan et al. [35] by utilizing InvDEA to estimate the output levels of a DMU, when some or all of its input entities are increased and its current efficiency level is improved. In addition, Hadi-Vencheh and Foroughi [16] suggested a method in which an increase in some inputs (outputs) and a decrease, due to some of the other inputs (outputs) are taken into account at the same time. Furthermore, in [24] Lertworasirikul et al. developed an inverse BCC (InvBCC) model that can preserve the relative efficiency values of all DMUs in a new production possibility set (PPS), composed of all DMUs and the perturbed DMUs with new input and output values. In [9], Eyni et al. discussed the InvDEA with preference to cone constraints in a manner that, the undesirable inputs and outputs are simultaneously present in the DMUs. In addition, Ghiyasi [14] developed a theoretical background of the InvDEA, with pollution generating technology that is capable of dealing with undesirable outputs. Subsequently, An et al. [4] proposed two-stage inverse data envelopment analysis models, with undesirable outputs, to formulate resource plans for 16 listed Chinese commercial banks, with increased outputs and an overall efficiency which remains unchanged on a short term basis.

Since the existing radial based DEA models neglect slacks, while evaluating the overall efficiency level of DMUs, in [18], Hu et al. proposed a model for a situation where the investigated DMU does not have a slack. The revised model can preserve a radial efficiency index, as well as eliminate all slacks. Moreover, Ghobadi [15] deals with the inverse DEA using the non-radial Enhanced Russell (ER)-measure in the presence of fuzzy data. In addition, Zhang and Gui [36] proposed the concept of InvDEA known as inverse non-radial DEA. They constructed the mathematical formula of an inverse slack based model (SBM) which can overcome the error caused, by ignoring the slacks.

Since the symptoms of climate change become more prevalent, in [8], Emrouznejad et al. introduced an InvDEA method for the allocation of CO₂ emissions, targeting to reduce these emissions into varied two-digit manufacturing industries and different regions. They addressed the CO₂ emission reduction in phases consisting of three-stages. Moreover, a new InvDEA model for optimizing greenhouse gas emission (GHG) was introduced by Wegner and Amin [33]. The proposed model minimizes the overall GHG emissions by a set of DMUs in order to produce a certain level of outputs, so that the DMUs at least maintain the status of its accessible performance.

InvDEA can be applied in managerial environment aspects. In [11], Gattoufi et al. for example, took advantage of the InvDEA issue for merging banks. They suggested a novel application of InvDEA in strategic decision making regarding mergers and acquisition in banking; as in some cases, of the suggested method, the merger may drop out of the PPS. In [3], Amin et al. proposed a method to anticipate as to whether a merger is generating a minor or major consolidation in a market. Moreover, Amin and Al-Muharrami [1] introduced a new InvDEA method for mergers with negative data. Then in [17], Hassanzadeh et al. presented two input oriented and output oriented inverse semi-oriented radial models, which are applied to determine resource allocations and investment strategies for assessing the sustainability of countries. Their proposed models can deal handle with both, positive and negative data. Recently, the problem of target setting in a merger has been addressed by Amin et al. in [2]. They considered the InvDEA method as a multi-objective problem and then utilized the goal programming (GP) approach for the M&A problem, when there is a preference for saving the specific problem.

As was mentioned earlier, classical DEA models use inputs and outputs data in order to assess the efficiency scores of DMUs, which are known as black boxes. But, in some cases DMUs may have intermediate products. Two-stage network systems which consist of two divisions are connected together with intermediate measures. A two-stage network system consumes exogenous inputs to produce outputs of the first stage, called intermediate products. Then the intermediate products are used as the inputs of the second stage, to produce the outputs of the second stage, which are also the final outputs of the whole system. There are many studies in relevance with the two-stage network concept. Seiford and Zhu [28] utilized the independent model to assess the efficiency scores of the first stage, and the second stage, including the whole system of a number of 55 US commercial banks. The connections and relationships between stages were not considered in their study. Moreover, Kao and Hwang [22] investigated the efficiency decomposition in a two-stage network system, by taking a series of relationships of the two sub-processes into account, for measuring the efficiencies. Furthermore in [31], Wang et al. provided an investigation of the monotonicity for the decomposition of weights, in a two-stage DEA model, with shared resource flows and found that, the weight in such a model, was unbiased in relative to the second stage. The utilization of constant weights, in such a model, is able to improve the discrimination of the efficient DMUs. Recently in [32], Wang et al. developed a high-tech industrial evaluation framework of technological innovation efficiency, based on a two-stage network DEA, constructed with shared inputs, additional intermediate inputs, and free intermediate outputs.

In some cases, information on costs for the inputs and outputs is available. The main issue is to minimize the overall costs of outputs. For example, Lozano [25] presented some models for computing technical, scale, cost and allocative efficiency scores in homogenous networks of processes. In addition, Shi et al. in [29] proposed a two-stage cost efficiency DEA model that minimizes the cost of the hypothetical DMU, while maintaining the overall merger efficiency by comparing its minimal total cost with its actual cost. Furthermore, Ghiyasi [13] incorporated the concepts of cost efficiency and InvDEA. He proposed a model that deals with the InvDEA problem when price information is available. The proposed model is based on the cost efficiency problem and preserves technical and cost efficiency scores of DMUs unchanged, simultaneously. As a matter of fact, the allocative efficiency of all DMUs would stay unchanged. In addition Soleimani-Chamkhorami et al. in [26] introduced the new models which are based on InvDEA for preserving cost and revenue efficiency, when data is changed. Moreover in [27], Soleimani-Chamkhorami et al. proposed a ranking system on the basis of InvDEA, which enables the researcher to rank the efficient DMUs in an appropriate manner.

There are some other approaches that incorporate the concepts of InvDEA and network DEA. For instance, a network-dynamic input-oriented RAM model and its inverse for assessing sustainability of supply chains were developed in [20] by Kalantary et al. The proposed model changes both inputs and outputs of DMUs so that the efficiency scores of DMUs would remain unchanged. In addition, Kalantary et al. [21] proposed a network-dynamic DEA model to assess the sustainability of supply chains in multiple periods. Then, they introduced an inverse network DEA model in a dynamic context.

It is for the first time that, this paper incorporates the inverse two-stage network DEA and allocative efficiency concepts. Then, the InvDEA concept is generalized to the two-stage structures in the presence of inputs and outputs, as well as cost price information. To this end, this paper employs the method proposed by Ghiyasi [13] and suggests inverse cost and revenue efficiency DEA models for inputs and outputs estimation in two-stage network systems. The proposed method would like to answer the following inverse DEA questions:

If amongst a group of comparable DMUs, with a two-stage network structure, the output levels of a unit increase to a certain level, how much more inputs are required with respect to the unchanged technical and cost efficiency scores of all the DMUs?

If amidst a group of comparable DMUs consisting of a two-stage network structure, the input levels of a unit increase to a certain level, how much more outputs are produced so that in order revenue efficiency of all units precede to remain unchanged?

Eventually, an empirical example is given to illustrate the capability or capacity of the presented method.

The rest of this paper is outlined as follows: In Section 2, we review some basic concepts of DEA, InvDEA, the basic two-stage networks and cost and revenue efficiency models. The inverse cost and inverse revenue efficiency DEA models in two-stage network systems is presented in Section 3. Finally, to examine the proposed model, a case study is presented in Section 4. Section 5 renders our conclusions and suggests future research guidelines.

2 Preliminaries

In this section the concepts of DEA, InvDEA, the basic two-stage networks and cost and revenue efficiency are reviewed.

2.1 DEA

DEA, is a mathematical approach to evaluate the performance of DMUs with multiple inputs and outputs, which was proposed by Charnes et al. [5]. Let us assume that there are n DMUs to evaluate (DMU_j, j = 1, . . . . , n), which consume m inputs (x_ij, i = 1, . . . , m) to produce s outputs (y_rj, r = 1, . . . , s). The unit under evaluation (x_o, y_o) is called DMU_o. The input oriented DEA model was proposed by Charnes, Cooper and Rhodes, known as CCR and is presented as follows: $\begin{array}{l} θ_{o}^{*} = \min θ_{o} \\ s . t . \sum_{j = 1}^{n} λ_{j} x_{i j} \leq θ_{o} x_{i o}, i = 1, ..., m \\ \sum_{j = 1}^{n} λ_{j} y_{r j} \geq y_{r o}, r = 1, ..., s \\ λ_{j} \geq 0, j = 1, ..., n . \end{array}$ (1)

In the optimal solution of model (1), if $θ_{o}^{*} = 1$ , then ${DMU}_{o}$ is called CCR efficient, otherwise ${DMU}_{o}$ is inefficient. Similar to model (1), the following model measures the output-oriented efficiency score of ${DMU}_{o}$ : $\begin{matrix} φ_{o}^{*} = max φ_{o} \\ s . t . \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq φ_{o} y_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, j = 1, . . ., n . \end{matrix}$ (2)

2.2 Inverse DEA

InvDEA is a useful method for the estimation of the inputs and outputs of a DMU. It was firstly proposed by Wei et al. in [34] so as to solve this problem; that, if among a group of comparable DMUs, the output (input) levels of DMU_o increase, how much more inputs (outputs) should the unit consume (produce) in order that the efficiency score of the unit, $θ_{o}^{*}$ remains unchanged? To solve this problem, let us suppose that the outputs of DMU_o are changed from y_ro (r = 1, . . . , s) to $\begin{matrix} β_{ro} = y_{ro} + Δ y_{ro} (r = 1, . . ., s) \end{matrix}$ . We need to estimate the input vector $\begin{matrix} α_{i} = x_{i} + Δ x_{i} (i = 1, . . ., m) \end{matrix}$ such that, the efficiency scores of DMU_o would still be $θ_{o}^{*}$ as obtained from model (1). The InvDEA model is: $\begin{matrix} min (α_{1}, α_{2}, . . . ., α_{m}) \\ s . t . \sum_{i = 1}^{m} λ_{j} x_{ij} \leq θ_{o}^{*} α_{i}, i = 1, . . ., m \\ \sum_{r = 1}^{s} λ_{j} y_{ij} \geq β_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, j = 1, . . ., n \\ α_{i} \geq 0, i = 1, . . ., m, \end{matrix}$ (3)

Where, all x_ij (i = 1, . . . , m), y_rj (r = 1, . . . , s) and β_ro (r = 1, . . . , s) are given and we need to obtain α_i (i = 1, . . . , m) s.

Now, assume that (λ, α) is a feasible solution of model (3). If there is no feasible solution $(\bar{λ}, \bar{α})$ such that, for all i = 1, . . . , m, $\bar{α_{i}} \leq α_{i}$ , then we say that (λ, α) is a weak efficient solution for model (3). It has been proven by Ghiyasi [12] that if a revision for DMU_o from (x_o, y_o) to (α_o, β_o) is considered and it is assumed that (λ_o, α_o) is a weak efficient solution of model (3), then the efficiency scores of all DMUs would stay unchanged after the revision.

There are several methods to solve the MOLP model (3) [30]. Assume that all the inputs are weighed (priced) and the weights (values) are known. Let w_i (i = 1, . . . , m) be the value weight for i^th input. To solve model (3), the weighed sum method is taken under consideration: $\begin{matrix} min \sum_{i = 1}^{m} w_{i} α_{i} \\ s . t . \sum_{i = 1}^{m} λ_{j} x_{ij} \leq θ_{o}^{*} α_{i}, i = 1, . . ., m \\ \sum_{r = 1}^{s} λ_{j} y_{ij} \geq β_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, j = 1, . . ., n \\ α_{i} \geq 0, i = 1, . . ., m . \end{matrix}$ (4)

By solving the above single objective programming model, we can attain new input levels.

2.3 The basic two-stage network

In the basic two-stage network, where all the inputs x_ij (i = 1, . . . , m), supplied from outside are consumed by the first stage, to produce the intermediate products z_gj (g = 1, . . . , h) and for the second stage to produce the final outputs y_rj (r = 1, . . . , s) [23]. The first stage does not produce the final outputs and the second stage does not consume the exogenous inputs. The structure of the basic two-stage system is depicted in Fig. 1.

Fig. 1

Structure of the basic two-stage network.

There are some perspectives to evaluate the efficiency score of network systems [22]. Based on the structure of the basic two-stage system shown in Fig. 1, the input-oriented model proposed by Kao and Hwang [22], under constant returns to scale (CRS) in the multiplier form is: $\begin{matrix} E_{o}^{input} = max \sum_{r = 1}^{s} u_{r} y_{ro} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{io} = 1 \\ system constraints : \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} \leq 0, j = 1, . . ., n \\ division constraints : \\ \sum_{g = 1}^{h} w_{g} z_{gj} - \sum_{i = 1}^{m} v_{i} x_{ij} \leq 0, j = 1, . . ., n \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{g = 1}^{h} w_{g} z_{gj} \leq 0, j = 1, . . ., n \\ u_{r} \geq 0, r = 1, . . ., s \\ v_{i} \geq 0, i = 1, . . ., m \\ w_{g} \geq 0, g = 1, . . ., h . \end{matrix}$ (5)

Optimally, the system efficiency in the input oriented form $(E_{o}^{input})$ and the stage efficiencies $(E_{o}^{(1)}$ and $E_{o}^{(2)})$ , based on the constrains of model (5), can be expressed as:

$\begin{matrix} E_{o}^{input} = \frac{\sum_{r = 1}^{s} u_{r}^{*} y_{ro}}{\sum_{i = 1}^{m} v_{i}^{*} x_{io}}, E_{o}^{(1) input} = \frac{\sum_{g = 1}^{h} w_{g}^{*} z_{go}}{\sum_{i = 1}^{m} v_{i}^{*} x_{io}} \\ and E_{o}^{(2) input} = \frac{\sum_{r = 1}^{s} u_{r}^{*} y_{ro}}{\sum_{g = 1}^{h} w_{g}^{*} z_{go}} . \end{matrix}$ (6)

The system efficiency is the product of the two stage efficiencies. The dual form of model (5) is as follows: $\begin{matrix} E_{o}^{input} = min θ_{o} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq θ_{o} x_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \end{matrix}$ $\begin{matrix} \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n . \end{matrix}$ (7)

The output oriented version of model (5) is: $\begin{matrix} E_{o}^{output} = min \sum_{r = 1}^{s} v_{i} x_{io} \\ s . t . \sum_{r = 1}^{s} u_{r} y_{ro} = 1 \\ system constraints : \\ \sum_{i = 1}^{m} v_{i} x_{ij} - \sum_{r = 1}^{s} u_{r} y_{rj} \geq 0, j = 1, . . ., n \\ division constraints : \\ \sum_{i = 1}^{m} v_{i} x_{ij} - \sum_{g = 1}^{h} w_{g} z_{gj} \geq 0, j = 1, . . ., n \\ \sum_{g = 1}^{h} w_{g} z_{gj} - \sum_{r = 1}^{s} u_{r} y_{rj} \geq 0, j = 1, . . ., n \\ u_{r} \geq 0, r = 1, . . ., s \\ v_{i} \geq 0, i = 1, . . ., m \\ w_{g} \geq 0, g = 1, . . ., h . \end{matrix}$ (8)

The dual form of model (8) is as follows: $\begin{matrix} E_{o}^{output} = max φ_{o} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq φ_{o} y_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n, \end{matrix}$ (9) which evaluates the output-oriented technical efficiency score of the two-stage network system.

2.4 Cost and revenue efficiency DEA models

Traditional DEA models use inputs and outputs of DMUs to evaluate the efficiency score of units. But, in some cases, the prices or weights of inputs are known and we need to estimate the minimum cost of inputs. Assume $c_{i} \in ℝ^{+} (i = 1, . . ., m)$ is the input price (weight). Then, the observed cost of DMU_o with input-output vector (x_o, y_o) can be computed by $c^{t} x_{o} = \sum_{i = 1}^{m} c_{i} x_{io}$ . In order to obtain the cost efficiency score of (x_o, y_o), initially the following model is solved [10]: $\begin{matrix} c^{t} x^{*} = min \sum_{i = 1}^{m} c_{i} x_{i} \\ s . t . \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{i}, i = 1, . . ., m \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, j = 1, . . ., n \\ x_{i} \geq 0, i = 1, . . ., m, \end{matrix}$ (10)

Then, the cost efficiency score of (x_o, y_o) which is the ratio of the optimal cost to the actual cost can be obtained by: ${CE}_{o} = \frac{c^{t} x^{*}}{c^{t} x_{o}} = \frac{\sum_{i = 1}^{m} c_{i} x_{i}^{*}}{\sum_{i = 1}^{m} c_{i} x_{io}} .$ (11)

Definition 1. If the cost efficiency score (CE_o) of DMU_o is equal to 1, then DMU_o is called cost efficient, otherwise DMU_o is cost inefficient.

Now, assume $p_{r} \in ℝ^{+} (r = 1, . . ., s)$ is output price (weight). So, the production revenue of DMU_o can be computed by $p^{t} y_{o} = \sum_{r = 1}^{s} p_{r} y_{ro}$ . In order to obtain the revenue efficiency score of DMU_o, first, the following model is solved [10]: $\begin{matrix} p^{t} y^{*} = max \sum_{i = 1}^{m} p_{r} y_{r} \\ s . t . \sum_{j = 1}^{n} λ_{j} x_{ij} \leq x_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{r}, r = 1, . . ., s \\ λ_{j} \geq 0, j = 1, . . ., n \\ y_{r} \geq 0, r = 1, . . ., s . \end{matrix}$ (12)

Next, the revenue efficiency score of (x_o, y_o) which is the ratio of the optimal revenue to the actual revenue can be obtained by: ${RE}_{o} = \frac{p^{t} y_{o}}{p^{t} y^{*}} = \frac{\sum_{r = 1}^{s} p_{r} y_{o}}{\sum_{r = 1}^{s} p_{r} y_{r}^{*}} .$ (13)

Definition 2. If the revenue efficiency score (RE_o) of DMU_o is equal to 1, then DMU_o is called revenue efficient, otherwise, DMU_o is revenue inefficient.

Definition 3. The ratio of cost efficiency in relative to technical efficiency is called allocative efficiency, that is ${AE}_{o} = \frac{{CE}_{o}}{θ_{o}}$ . If AE_o is equal to 1, DMU_o is called allocative efficient, otherwise, DMU_o is allocative inefficient.

3 The Inverse cost and revenue efficiency models in two stage network systems

As was previously mentioned, Ghiyasi [13] proposed the inverse cost and revenue efficiency models, which estimated the inputs and outputs levels of DMUs, in order to keep the cost and revenue efficiency scores of units unchanged, respectively. The preceding study, did not take into account the inputs and outputs cost information, in estimating the outputs and inputs of the DMUs with two-stage network structures. Therefore, the main goal of this paper is to estimate the inputs and outputs level of units, so that the technical and cost efficiency values would remain unchanged in the two-stage network systems. In this section, we propose two methods which conduct inverse cost and revenue efficiency concepts in two-stage network structures.

3.1 Inverse two-stage cost efficiency model

In this section, we are attempting to answer this question, that, if among a group of comparable DMUs with two-stage network structure, we increase the output levels of DMUs, how many inputs are required in order that the technical and cost efficiency scores of DMUs stay unchanged? To respond to this question, we follow the undergoing steps:

Step 1: Solve model (7) and suppose that $(E_{o}^{input}, λ^{*}, μ^{*})$ is the optimal solution.

Step 2: Assume $c_{i} \in ℝ^{+} (i = 1, . . ., m)$ are input prices of the whole system (and also the input prices of stage1). Then, find the cost efficiency score of DMU_o before perturbation, by using the following two-stage network cost efficiency model: $\begin{matrix} c^{t} x^{*} = min \sum_{i = 1}^{m} c_{i} x_{i} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{i}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n \\ x_{i} \geq 0, i = 1, . . ., m . \end{matrix}$ (14)

Using $(x_{i}^{*}, λ^{*}, μ^{*})$ as an optimal solution of model (14), the cost efficiency score of DMU_o is calculated by (11).

Step 3: Assume the output level of DMU_o perturbs from y_o to β_o = y_o + Δy_o. Solve the following cost efficiency model with new output levels: $\begin{matrix} c^{t} \tilde{x} = min \sum_{i = 1}^{m} c_{i} x_{i} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{i}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro} + Δ y_{ro} = β_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n \\ x_{i} \geq 0, i = 1, . . ., m . \end{matrix}$ (15)

Assume that $({\tilde{x}}_{i}, {\tilde{λ}}^{*}, {\tilde{μ}}^{*})$ is an optimal solution of model (15).

Step 4: Now, we are going to find the minimum levels of inputs, in which the cost efficiency scores of DMUs remain unchanged by solving the following model: $\begin{matrix} min (α_{1}, α_{2}, . . ., α_{m}) \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq E_{o}^{input} α_{i}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro} + Δ y_{ro} = β_{ro}, r = 1, . . ., s \\ c^{t} α = \frac{c^{t} \tilde{x}}{{CE}_{o}} \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n \\ α_{i} \geq 0, i = 1, . . ., m . \end{matrix}$ (16)

In model (16), the first set of constraints guarantees that the efficiency score of the DMU under evaluation stays unchanged. The first three constraint sets also ensure production possibility. Finally, the fourth constraint set, ensures that the cost efficiency of DMU_o remains unchanged. Note that model (16) is an MOLP and can be inverted to a single objective form so as to be solved.

Theorem 1. Let us assume that, DMU_o with two-stage network structure, perturbs its output level from y_o to β_o = y_o + Δy_o. If $E_{o}^{input}$ is the optimal value of model (7) and (λ, μ, α) is a weak efficient solution of the MOLP model (16); then, the technical and cost efficiency of all the DMUs remain unchanged after changing the inputs and outputs of DMU_o.

Proof. Consider the following problem: $\begin{matrix} E_{o}^{*} = min θ_{o} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq θ_{o} α_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq β_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n . \end{matrix}$ (17)

We want to show that $E_{o}^{*} = E_{o}^{input}$ . Since (λ, μ, α) is a weak efficient solution of the MOLP model (16), it satisfies the following conditions: $\begin{matrix} \sum_{j = 1}^{n} μ_{j} x_{ij} \leq E_{o}^{input} α_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro} + Δ y_{ro} = β_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n . \end{matrix}$ (18)

From where it seems that $E_{o}^{*}$ is a feasible solution of model (19), and hence, we have $E_{o}^{*} \leq E_{o}^{input}$ . So, it has been illustrated that $E_{o}^{*} E_{o}^{input}$ . In assuming $E_{o}^{*} < E_{o}^{input}$ ; then, the following condition is held: $E_{o}^{*} = {tE}_{o}^{input}, 0 \leq t \leq 1 .$ (19)

So, we have the subsequent constraints: $\begin{matrix} \sum_{j = 1}^{n} μ_{j} x_{ij} \leq E_{o}^{*} α_{io} = {tE}_{o}^{input} α_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro} + Δ y_{ro} = β_{ro}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n . \end{matrix}$ (20)

It means that (λ, μ, tα_o) is a feasible solution for the MOLP model (16) and this contradicts the assumption that (λ, μ, α_o) is a weak efficient solution for the MOLP model (18). Then, the technical efficiency score of perturbed DMU_o is equal to the technical efficiency score of the original DMU_o and we have $E_{o}^{*} = E_{o}^{input}$ .

Following the proof of theorem by Ghiyasi [13], we see that the cost efficiency of DMU_o stays unchanged.□

Remark 1. Since the technical and cost efficiency scores of DMU_o stay unchanged, the allocative efficiency of DMU_o which is the ratio of cost efficiency in respect to the technical efficiency, stays unchanged after DMU_o perturbs to new input-output levels.

3.2 Inverse two-stage revenue efficiency model

In this section, an inverse revenue efficiency model is proposed in the two-stage network systems, targeting to respond to this question; that, if, among a group of comparable DMUs, the inputs of DMU_o perturb from x_o to α_o = x_o + Δx_o, how much more outputs are produced in order that, the technical and revenue efficiency scores of all the DMUs would remain unchanged? To answer this question, follow these steps:

Step 1: Find the output-oriented efficiency score of the DMUs by solving model (9) and suppose that $(φ_{o}^{output}, λ^{*}, μ^{*})$ is the optimal solution.

Step 2: Find the revenue efficiency of DMU_o before perturbation, by using the following two-stage revenue efficiency model: $\begin{matrix} p^{t} y^{*} = max \sum_{i = 1}^{m} p_{r} y_{r} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{r}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n \\ y_{r} \geq 0, r = 1, . . ., s . \end{matrix}$ (21)

Theorem 2. The model (21) is feasible and its optimal value is finite.

Proof. We argue these two objects as follows.

As can be seen, (λ, μ, y) = (λ_o = 1, λ_j = 0, j = 1, . . . , n, j ≠ o, μ_o = 1, μ_j = 0, j = 1, . . . , n, j ≠ o, y_r = y_ro, r = 1, . . . , s) is a feasible solution of model (21). Therefore, model (21) is feasible.

According to the constraints $\sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{r} (r = 1, . . ., s)$ , the $\sum_{j = 1}^{n} λ_{j} y_{rj}$ , gives an upper bound on the non-negative variable y_r (r = 1, . . . , s). On the other hand, by the constraints $\sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{io} (i = 1, . . ., m)$ , the variables μ_j (j = 1, . . . , n) are bounded. Hence, for all j (j = 1, . . . , n) , λ_j cannot be infinite. Otherwise, the constraints j (j = 1, . . . , n) , λ_j $\sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, (g = 1, . . ., h)$ are not satisfied. Therefore, model (21) is bounded.

Hence, model (21) is always feasible and has finite optimal solution. □

Assume that $(x_{i}^{*}, λ^{*}, μ^{*})$ is an optimal solution of model (21).

Step 3: Assume the input level of DMU_o perturbs from x_o to x_o + Δx_o = α_o. Now the following two-stage revenue efficiency model with new input level of DMU_o is solved: $\begin{matrix} p^{t} \tilde{y} = max \sum_{i = 1}^{m} p_{r} y_{r} \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{io} + Δ x_{io} = α_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{r}, r = 1, . . ., s \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n \\ y_{r} \geq 0, r = 1, . . ., s . \end{matrix}$ (22)

Step 4: Finally, to find the maximum level of outputs so that the revenue efficiency scores of the DMUs would stay unchanged, the following model is solved: $\begin{matrix} max (β_{1}, β_{2}, . . ., β_{s}) \\ s . t . \sum_{j = 1}^{n} μ_{j} x_{ij} \leq x_{io} + Δ x_{io} = α_{io}, i = 1, . . ., m \\ \sum_{j = 1}^{n} (μ_{j} - λ_{j}) z_{gj} \geq 0, g = 1, . . ., h \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq φ_{o}^{output} β_{r}, r = 1, . . ., s \end{matrix}$ $\begin{matrix} β_{r} \geq y_{ro}, r = 1, . . ., s \\ p^{t} β = {RE}_{o} \times p^{t} \tilde{y} \\ λ_{j} \geq 0, μ_{j} \geq 0, j = 1, . . ., n . \end{matrix}$ (23)

Theorem 3. Assume that DMU_o with two-stage network structure, perturbs its input level from x_o to α_o = x_o + Δx_o. If $E_{o}^{output}$ is the optimal value of model (9) and (λ, μ, β) is a weak efficient solution of the MOLP model (23), then the technical and revenue efficiency scores of all the DMUs stay unchanged after DMU_o change to new input-output level.

Proof. It is similar to the proof of Theorem 1 with some revision. □

4 Case study

In this section, we examine our proposed model for a data set of 27 banks from Chen and Zhu [6]. The data is given in Table 1. Fixed assets, IT budget and the number of employees are three indicators which are considered as inputs (second, third and fourth column) to produce deposits as an intermediate product (fifth column). Profit and fraction of loans recovered are the final outputs (columns six and seven).

Table 1
Data set of Chen and Zhu [6]

Banks Fixed assets ($ billion) (x₁) IT budget ($ billion) (x₂) # of employees (thousand) (x₃) Deposits ($ billion) (z₁) Profit ($ billion) (y₁) Fraction of loans recovered (y₂)

1 0.713 0.15 13.3 14.478 0.232 0.986

2 1.071 0.17 16.9 19.502 0.34 0.986

3 1.224 0.235 24 20.952 0.363 0.986

4 0.363 0.211 15.6 13.902 0.211 0.982

5 0.409 0.133 18.485 15.206 0.237 0.984

6 5.846 0.497 56.42 81.186 1.103 0.955

7 0.918 0.06 56.42 81.186 1.103 0.986

8 1.235 0.071 12 11.441 0.199 0.985

9 18.12 1.5 89.51 124.072 1.858 0.972

10 1.821 0.12 19.8 17.425 0.274 0.983

11 1.915 0.12 19.8 17.425 0.274 0.983

12 0.874 0.05 13.1 14.342 0.177 0.985

13 6.918 0.37 12.5 32.491 0.648 0.945

14 4.432 0.44 41.9 47.653 0.639 0.979

15 4.504 0.431 41.1 52.63 0.741 0.981

16 1.241 0.11 14.4 17.493 0.243 0.988

17 0.45 0.053 7.6 9.512 0.067 0.98

18 5.892 0.345 15.5 42.469 1.002 0.948

19 0.973 0.128 12.6 18.987 0.243 0.985

20 0.444 0.055 5.9 7.546 0.153 0.987

21 0.508 0.057 5.7 7.595 0.123 0.987

22 0.37 0.098 14.1 16.906 0.233 0.981

23 0.395 0.104 14.6 17.264 0.263 0.983

24 2.68 0.206 19.6 36.43 0.601 0.982

25 0.781 0.067 10.5 11.581 0.12 0.987

26 0.872 0.1 12.1 22.207 0.248 0.972

27 1.757 0.0106 12.7 20.67 0.253 0.988

Banks	Fixed assets ($ billion) (x₁)	IT budget ($ billion) (x₂)	# of employees (thousand) (x₃)	Deposits ($ billion) (z₁)	Profit ($ billion) (y₁)	Fraction of loans recovered (y₂)
1	0.713	0.15	13.3	14.478	0.232	0.986
2	1.071	0.17	16.9	19.502	0.34	0.986
3	1.224	0.235	24	20.952	0.363	0.986
4	0.363	0.211	15.6	13.902	0.211	0.982
5	0.409	0.133	18.485	15.206	0.237	0.984
6	5.846	0.497	56.42	81.186	1.103	0.955
7	0.918	0.06	56.42	81.186	1.103	0.986
8	1.235	0.071	12	11.441	0.199	0.985
9	18.12	1.5	89.51	124.072	1.858	0.972
10	1.821	0.12	19.8	17.425	0.274	0.983
11	1.915	0.12	19.8	17.425	0.274	0.983
12	0.874	0.05	13.1	14.342	0.177	0.985
13	6.918	0.37	12.5	32.491	0.648	0.945
14	4.432	0.44	41.9	47.653	0.639	0.979
15	4.504	0.431	41.1	52.63	0.741	0.981
16	1.241	0.11	14.4	17.493	0.243	0.988
17	0.45	0.053	7.6	9.512	0.067	0.98
18	5.892	0.345	15.5	42.469	1.002	0.948
19	0.973	0.128	12.6	18.987	0.243	0.985
20	0.444	0.055	5.9	7.546	0.153	0.987
21	0.508	0.057	5.7	7.595	0.123	0.987
22	0.37	0.098	14.1	16.906	0.233	0.981
23	0.395	0.104	14.6	17.264	0.263	0.983
24	2.68	0.206	19.6	36.43	0.601	0.982
25	0.781	0.067	10.5	11.581	0.12	0.987
26	0.872	0.1	12.1	22.207	0.248	0.972
27	1.757	0.0106	12.7	20.67	0.253	0.988

Assume that (c₁, c₂, c₃) = (2, 3, 4) is the unit input-price vector. Let us increase the output levels of all the DMUs by 10 percent. Now we are interested in finding the required input levels of all the DMUs in order to keep the cost efficiency score unchanged (the results are shown in columns eight to ten in Table 2). First we locate the technical and cost efficiencies of all units by solving models (7) and (11), respectively. The technical and cost efficiencies of all 27 units are depicted in the second and third columns of Table 2. As is seen in Table 2, DMU7 is allocative efficient.

Table 2

Technical, cost and revenue efficiencies and inputs changes of 27 banks based on inverse two-stage cost and revenue efficiency models

Banks				Two - StageNetwork									Black - Box
	Technical Efficiency $(E_{o}^{input})$	Cost Efficiency (CE_o)	Allocative Efficiency (AE_o)	Fixed asset changes (Δx₁)	IT budget changes (Δx₂)	Number of employees changes (Δx₃)	α ₁	α ₂	α ₃	$E_{o}^{input}$	CE _o	AE _o	Δx ₁	Δx ₂	Δx ₃	α ₁	α ₂	α ₃
1	0.659	0.463	0.702	–0.025	–0.089	0.024	0.014	0.011	0.173	1.000	0.914	0.914	0.036	–0.048	0.019	0.075	0.052	0.168
2	0.749	0.543	0.725	–0.041	–0.099	0.034	0.018	0.014	0.222	1.000	0.919	0.919	0.013	–0.062	0.044	0.072	0.051	0.233
3	0.573	0.419	0.732	–0.043	–0.137	0.042	0.025	0.020	0.310	0.778	0.694	0.892	0.024	–0.092	0.049	0.092	0.065	0.317
4	0.535	0.353	0.661	–0.004	–0.128	0.019	0.016	0.012	0.194	1.000	0.724	0.724	0.054	–0.089	–0.020	0.074	0.052	0.155
5	0.507	0.404	0.797	–0.004	–0.074	0.023	0.018	0.015	0.229	1.000	0.787	0.787	0.052	–0.037	–0.036	0.074	0.052	0.170
6	0.689	0.514	0.746	–0.260	–0.282	0.154	0.063	0.050	0.784	0.853	0.659	0.772	–0.257	–0.037	–0.036	0.074	0.052	0.170
7	0.779	0.779	1.000	0.005	0.004	0.063	0.056	0.044	0.693	1.000	1.000	1.000	0.005	0.004	0.063	0.056	0.044	0.693
8	0.586	0.474	0.809	–0.055	–0.037	0.033	0.013	0.011	0.167	1.000	1.000	1.000	0.007	0.005	0.013	0.075	0.052	0.147
9	0.647	0.400	0.618	–0.887	–0.911	0.407	0.113	0.089	1.407	0.777	0.513	0.660	–0.879	–0.905	0.502	0.121	0.095	1.502
10	0.418	0.355	0.848	–0.215	–0.082	0.122	0.048	0.038	0.059	0.581	0.530	0.912	–0.146	–0.035	0.102	0.117	0.085	0.575
11	0.415	0.351	0.846	–0.229	–0.082	0.127	0.048	0.038	0.600	0.581	0.525	0.904	–0.159	–0.035	0.102	0.117	0.085	0.575
12	0.432	0.385	0.890	–0.096	–0.026	0.059	0.030	0.024	0.372	0.881	0.659	0.749	–0.044	0.008	–0.057	0.082	0.58	0.256
13	0.900	0.541	0.601	–0.947	–0.328	0.356	0.053	0.042	0.654	1.000	0.694	0.694	–0.939	–0.322	0.457	0.061	0.048	0.755
14	0.438	0.336	0.768	–0.646	–0.356	0.324	0.106	0.084	1.324	0.534	0.432	0.808	–0.640	–0.352	0.394	0.112	0.088	1.394
15	0.507	0.390	0.770	–0.658	–0.347	0.328	0.107	0.084	1.328	0.617	0.501	0.811	–0.652	–0.342	0.399	0.112	0.089	1.399
16	0.257	0.237	0.921	–0.142	–0.056	0.124	0.069	0.054	0.859	0.404	0.366	0.906	–0.038	0.014	0.006	0.173	0.124	0.740
17	0.220	0.206	0.938	–0.041	–0.025	0.054	0.035	0.028	0.442	0.834	0.435	0.522	0.013	0.009	–0.212	0.089	0.062	0.176
18	0.795	0.581	0.730	–0.908	–0.272	0.354	0.092	0.073	1.145	0.951	0.745	0.784	–0.901	–0.267	0.438	0.099	0.078	1.229
19	0.280	0.238	0.847	–0.300	–0.078	0.146	0.063	0.050	0.789	0.411	0.368	0.894	–0.194	–0.007	0.087	0.169	0.121	0.730
20	0.382	0.324	0.849	–0.136	–0.032	0.064	0.029	0.023	0.365	0.802	0586	0.731	–0.074	0.009	–0.053	0.091	0.064	0.248
21	0.315	0.259	0.822	–0.161	–0.034	0.065	0.029	0.023	0.356	0.773	0.513	0.663	–0.093	0.010	–0.077	0.096	0.067	0.214
22	0.259	0.243	0.939	–0.072	–0.046	0.099	0.066	0.052	0.818	0.452	0.380	0.842	0.016	0.012	–0.080	0.154	0.110	0.639
23	0.282	0.264	0.936	–0.079	–0.050	0.104	0.068	0.054	0.849	0.427	0.399	0.934	0.013	0.011	0.009	0.161	0.115	0.754
24	0.422	0.327	0.739	–0.901	–0.128	0.237	0.099	0.078	1.237	0.552	0.420	0.760	–0.898	–0.125	0.271	0.102	0.081	1.271
25	0.118	0.098	0.833	–0.370	–0.008	0.099	0.074	0.059	0.926	0.562	0.197	0.350	–0.312	0.025	–0.539	0.133	0.092	0.288
26	0.208	0.175	0.843	–0.409	–0.031	0.135	0.087	0.069	1.087	0.397	0.268	0.675	–0.324	0.023	–0.184	0.172	0.123	0.769
27	0.217	0.151	0.695	–0.915	0.057	0.060	0.085	0.067	1.060	1.000	0.231	0.231	–0.931	0.039	–0.689	0.069	0.050	0.311

Let us consider DMU25 with the lowest technical and cost efficiency score of 0.118 and 0.098, respectively. By a 10 percent perturbation in its output levels, its first and second input levels will decrease by 0.370 and 0.008 units, respectively. Then, its third input level would increase by 0.099 units.

DMU13 has the highest technical efficiency score of 0.900. After perturbation in its output levels by 10 percent, its first and second input levels would decrease by 0.947 and 0.328 units, respectively. But, its third input level would increase by 0.356 units.

Now, consider DMU7 which is allocative efficient. It would have a new input vector (0.056, 0.044 and 0.693) after a 10 percent perturbation in its output levels.

Now, in assuming that, p = (2, 3) is the price of outputs and let us increase the input levels of units by 2 percent; we are going to find the new output levels of all DMUs, in order to keep the output technical and revenue efficiency scores unchanged (the results are given in the sixth and seventh columns of Table 5). Primarily, we acquire the output oriented technical efficiency and revenue efficiency scores of all the DMUs, by utilizing models (9) and (13), respectively. As can be seen, in the second and third columns of Table 5, none of the units are technical and revenue efficient.

Consider DMU1 with output and revenue efficiency scores of 1.516 and 0.578, respectively. Through a 2 percent increase in its input levels, it is observed that, its first and second output levels increase by 0.002 and 0.020 units, respectively. So, the new output levels is (0.127 and 1.018) after perturbation.

Now, contemplate on DMU26 with a lowest allocative efficiency score of 0.097. We perturb its inputs level by 2 percent and see that the first and second output increases by 0.005 and 0.019 units, respectively. DMU17 has no changes in the second output level.

Now we compare our inverse two-stage cost efficiency model with the black-box point of view. For this purpose, we omit the intermediate product column of DMUs in Table 3 (z₁) and treat each DMU as a black-box. The results of the black-box perspective are shown in the columns 11–19 in Table 2.

Table 3

Output oriented technical and revenue efficiencies, Output changes and new outputs level

Banks	Two - StageNetwork						Black - Box
	Output technical efficiency $(φ_{o}^{output})$	Revenue efficiency (RE_o)	Profit changes (Δy₁)	Fraction of loans recovered changes (Δy₂)	β ₁	β ₂	$φ_{o}^{output}$	RE _o	Δy ₁	Δy ₂	β ₁	β ₂
1	1.516	0.578	0.002	0.020	0.127	1.018	1.000	1.000	0.002	0.020	0.127	1.018
2	1.335	0.461	0.004	0.020	0.187	1.008	1.000	0.798	0.004	0.020	0.187	1.008
3	1.745	0.333	0.004	0.020	1.199	1.008	1.285	0.609	0.004	0.020	0.199	1.008
4	1.871	0.505	0.002	0.020	0.116	1.004	1.000	1.000	0.002	0.020	0.116	1.004
5	1.971	0.431	0.003	0.020	0.130	1.006	1.000	1.000	0.003	0.020	0.130	1.006
6	1.451	0.159	0.012	0.019	0.606	0.976	1.172	0.272	0.012	0.019	0.606	0.976
7	1.283	0.184	0.012	0.020	0.606	1.008	1.000	1.000	0.012	0.020	0.606	1.008
8	1.705	0.586	0.002	0.020	0.109	1.007	1.000	1.000	0.002	0.020	0.109	1.007
9	1.546	0.108	0.020	0.019	1.020	0.993	1.286	0.208	0.020	0.019	1.020	0.993
10	2.391	0.181	0.005	0.020	0.279	1.005	1.720	0.416	0.005	0.020	0.279	1.005
11	2.409	0.180	0.005	0.020	0.279	1.005	1.720	0.416	0.005	0.020	0.279	1.005
12	2.314	0.274	0.004	0.020	0.180	1.007	1.136	0.863	0.033	0.000	0.210	0.987
13	1.111	0.194	0.013	0.019	0.660	0.966	1.000	0.585	0.013	0.019	0.660	0.966
14	2.282	0.098	0.013	0.020	0.650	1.001	1.872	0.178	0.013	0.020	0.650	1.001
15	1.974	0.103	0.015	0.020	0.754	1.003	1.620	0.187	0.015	0.020	0.754	1.003
16	3.891	0.124	0.005	0.020	0.247	1.010	2.472	0.403	0.031	0.003	0.273	0.993
17	4.543	0.215	0.031	0.000	0.098	0.982	1.199	0.728	0.031	0.000	0.098	0.982
18	1.258	0.130	0.020	0.019	1.020	0.969	1.052	0.259	0.020	0.019	1.020	0.969
19	3.566	0.134	0.005	0.020	0.248	1.007	2.433	0.367	0.005	0.020	0.248	1.007
20	2.618	0.276	0.003	0.020	0.156	1.009	1.246	0.798	0.033	0.000	0.186	0.989
21	3.176	0.278	0.002	0.020	0.125	1.009	1.294	0.766	0.032	0.000	0.155	0.989
22	3.858	0.128	0.005	0.020	0.238	1.003	2.214	0.436	0.034	0.000	0.267	0.983
23	3.545	0.126	0.020	0.005	0.268	1.005	2.340	0.422	0.032	0.002	0.295	0.987
24	2.261	0.103	0.012	0.020	0.613	1.004	1.811	0.277	0.012	0.020	0.613	1.004
25	8.461	0.107	0.002	0.020	0.122	1.009	1.778	0.508	0.032	0.000	0.152	0.989
26	4.817	0.097	0.005	0.019	0.253	0.993	2.520	0.397	0.005	0.019	0.253	0.993
27	4.602	0.101	0.005	0.020	0.258	1.010	1.000	1.000	0.005	0.020	0.258	1.010

Consider DMU1 with the technical and cost efficiency scores of 0.659 and 0.463, respectively. When we consider DMU1 as a black-box without considering the intermediate product (z₁), the technical and cost efficiency scores become 1.000 and 0.914, respectively. As is seen, both technical and cost efficiency scores increase. In addition to what has been said, the results are too far from the real world. In the two-stage system, when the output levels are perturbed by 10 percent, the first and second inputs decrease to 0.014 and 0.011, respectively and the third input increases to 0.173. In the black-box perspective, however, the first and third input rise to 0.075 and 0.168, respectively and the second input falls to 0.052.

Now our inverse two-stage revenue efficiency model and the black-box model are compared. The results of the black-box perspective are shown in columns 8–13 in Table 3. Consider DMU23 with the output-oriented technical efficiency and revenue efficiency scores of 3.545 and 0.126, respectively. When we consider it as a black-box without considering the intermediate product (z₁) the output-oriented technical and revenue efficiency scores become 2.340 and 0.422, respectively. Let the input levels of units increase by 2 percent. In the two-stage structure, the new output levels become (0.268, 1.005), and in the black-box structure it becomes (0.295, 987).

5 Conclusion

In this paper the inverse DEA methods was generalized for estimating the inputs and outputs for the two-stage network systems, in the presence of cost and price information. The proposed methods deal with inputs and outputs cost information, estimating these levels, for the unit under evaluation, in order to keep the technical, cost and revenue efficiency scores unchanged. In the proposed MOLP model, the decision maker’s preferences can be considered in input (outputs) weights in the output (inputs) estimation procedure. The proposed method was applied to an empirical example in the presence of data for cost and price information.

We used model (7) to determine the efficiency of two-stage of DMUs under constant returns to scale (CRS) assumption. But, this model is a non-linear model under variable returns to scale (VRS) case assumption [7]. Therefore, the major limitation of our method is its inability to handle the numerous situations wherein the VRS model structure is required.

A stream of future research can extend our framework to other DEA network structures, two-stage structures with negative or imprecise data. The proposed method can also be extended as a future research, in cases of uncertain and stochastic cost/price information for the two-stage network systems also, supply chain structures.

Footnotes

Acknowledgments

The authors would like to thank the anonymous reviewers and the editor-in-chief for their constructive comments and suggestions.

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