EFFICIENCY DECOMPOSITION IN A THREE-STAGE NETWORK STRUCTURE: COOPERATIVE DEA, NASH BARGAINING GAME MODELS AND CONIC RELAXATIONS

. In this paper, for evaluating the efficiency in a three-stage DEA structure we use the additive and the multiplicative cooperative models that comply with the cooperation paradigm in the organizations, where for improving efficiency of system, stages cooperate together. Since the overall efficiency from the cooperative models may not be unique and consequently the stages’ efficiencies, then we combine them with the Nash bargaining game approach that besides maximizing efficiency scores for stages and the whole system, provides a unique and fair efficiency decomposition. Second order programming relaxation of the proposed nonlinear models are given in contrast to the parametric linear models in the literature. Finally, the effectiveness of the proposed models are illustrated with two numerical examples.

two latter studies considered the exogenous inputs and outputs in the network structure. Also, models of both latter studies can only deal with the assumption of CRS. To make an equilibrium between desirable outputs and undesirable outputs in a two-stage sustainable manufacturing process, Wu et al. [41] proposed a Nash bargaining game model. Recently, using the cooperative approach in a three-stage network with external inputs and outputs, Amirkhan et al. [1] decomposed the overall efficiency into the stages' efficiencies. They developed a three-stage DEA model based on the Nash bargaining game that provided a unique and fair decomposition of the overall efficiency among the stages. To solve the proposed nonlinear bargaining model, they utilized a heuristic search approach. However, to calculate the overall efficiency in the three-stage structure with external inputs and outputs, their proposed method has neglected the existence of intermediate products. Using the concept borrowed from Nash bargaining game, Mahmoudi et al. [31] also handled insufficient number of DMUs and inputs issue in the network structure.
Utilizing the exogenous inputs and outputs in the network structure, such as the outputs that are sent directly to the outside from the first stage and the additional inputs to intermediate stages, make the multi-stage DEA models highly nonlinear. Applying Charnes-Cooper transformations to such models usually lead to parametric linear programs for which heuristic methods often are used [27]. To surmount this issue, Chen et al. [9] developed a second-order cone program (SOCP) approach to solve nonlinear two-stage DEA models. Their approach only ensures a feasible solution instead of the global optimal solution. Recently in another study, Chen et al. [12] pointed out weakness of Chen et al. [9]'s approach and introduced a conic relaxation model that generates feasible approximations and tighter upper bounds on the overall efficiency. In this paper, we consider a more general form of the three-stage structure studied by Amirkhan et al. [1] and Despotis et al. [14] based on the AED and MED in the cooperative game context. Our first aim is to maximize the stages' efficiencies and the overall efficiency of such structures. First, we maximize the overall efficiency in both AED and MED models, then the stages' efficiencies are derived from the weights that are obtained by AED and MED models. The second aim of this paper is to achieve a fair and unique efficiency decomposition of the overall efficiency into the stages' efficiencies. To this end, we utilize the Nash bargaining game approach. Since the proposed models cannot be linearized using Charnes-Cooper transformation [7], we present their SOCP relaxations. To sum up, compared with the previous studies in the multi-stage structure, the proposed approaches in this paper has four merits. First, we use network structure under variable returns to scale (VRS). Second, to maximize all stages' efficiencies, we study both AED and MED under cooperative relationship between the three stages of system. Each stage in the cooperative approach tries to achieve weights to maximize its efficiency, but cannot effect the weights of the other stages. In other words, each stage's weights are independent of the other stages' weights. Third, there are structures where the stages' weights may depend on the weights of the other stages. Thus, to make an equilibrium between stages' efficiencies, we utilize Nash bargaining game approach. Fourth, to solve the three nonlinear models, we use SOCP relaxations.
We begin Section 2 with the conventional BCC model on the three-stage structure and continue with the introduction of models based on cooperative game approach and their SOCP formulations. In Section 3, we propose the Nash bargaining game model to improve the stages' efficiencies, and present its SOCP formulation. In Section 4, we generalize the proposed bargaining model to the multi-stage structure. In Section 5, two numerical examples are used to compare the proposed models. Finally, the conclusions are given in the last section. The notations throughout the paper are summarized in Table 1.

Cooperative efficiency evaluation in a three-stage system
Consider the three-stage serial structure as depicted in Figure 1. The conventional BCC model for measuring the whole system performance under VRS can be expressed as follows [3]: The outputs matrix of all DMUs Figure 1. Three-stage system.
If we set 0 = 0, then model (2.1) converts to the CCR model [8]. Indeed, this variable characterizes the intercept and relaxes the condition that the production function must pass from the orgin [26]. Obviously, intermediate products produced and consumed within the system are not considered in model (2.1), and system's performance is measured from the viewpoint of outside. Since system's performance depends on the intermediate products that are not observable from outside and their weak performance may lead to overall inefficiency of system, thus we measure efficiency from the viewpoint of inside. To evaluate the overall efficiency of DMU under evaluation (DMU ) according to AED, we consider the weighted sum of stages' efficiencies as follows: where 0 ≤ , , ≤ 1 are the stages' weights. The overall efficiency score for DMU using the AED network DEA model (2.2) relies on the predetermined weights which is either provided by the decision-makers or determined by the model. Since cooperative approach considers same priorities for the stages, thus we consider = = = 1 3 in model (2.2). Further, since model (2.2) is nonlinear, we present its conic relaxation formulation. The following lemma play key roles in the conic relaxation formulations.
In the sequel, inspired by Chen [12], a conic relaxation of model (2.2) is given. To do so, suppose that is still an optimal solution and the overall efficiency remains unchanged. On the other hand, are now between zero and one. Thus, we add the following constraints to model (2.2) without losing its optimal solution. Thus in the sequel, we consider the following model: Proof. For simplicity, we denote the feasible set of model (2.4) by Ω. At first, we solve the following three optimization problems: and 3 = max Now, we convert model (2.4) from a maximization problem to a minimization problem: Then, let In the sequel, to determine intervals containing , = 1, 2, 3 and , = 1, 2, 3, the following linear programming problems are solved: Let the minimum and maximum of models (2.10), (2.12) and (2.14) be , = 1, 2, 3 and , = 1, 2, 3, respectively. Likewise, the reciprocal of the maximum and minimum of models (2.11), (2.13) and (2.15) be , = 1, 2, 3 and , = 1, 2, 3, respectively. Then, we have ≤ ≤ , = 1, 2, 3 and ≤ ≤ , = 1, 2, 3. Thus, model (2.9) is equivalent to (2.16) Further, let , = 1, 2, 3 be such that Thus, model (2.16) is equivalent to Then Using these inequalities, the first set of constraints are relaxed. Further, using Lemma 2.1, the third, fifth and seventh constraints can be presented as conic forms. Thus the relaxed form of (2.17) becomes: Thus, we get model (2.5).
The MED is the other approach for decomposing the overall efficiency in the network structures that has been used in [1,42] for different multi-stage structures. The MED model for the overall efficiency of the structure as given in Figure 1 is as follows: (2.20) Proof. For simplicity, we denote the feasible set of model (2.19) by Ψ. Let In the sequel, to determine intervals containing , = 1, 2, 3 and , = 1, 2, 3, the following linear programming problems are solved: (2.28) Now, let , = 1, 2, 3 and be such that ≤ , = 1, 2, 3, Thus, model (2.28) is equivalent to min s.t. (2.29) Further, we know that Similar to the AED case Using these inequalities, the first and second set of constraints are relaxed. Furthur, using Lemma 2.1, the fourth, sixth and eighth constraints can be presented as conic forms. Thus, the relaxed form of (2.29) becomes: min s.t.

Efficiency decomposition based on bargaining game
Although cooperative models determine stages' efficiencies and the overall efficiency with common weights and their aim is to maximize the efficiency of system, the overall efficiency derived from these models may not be unique, and thus the stages' efficiencies may also be non-unique [29]. On the other hand, each stage in the cooperative approach tries to achieve weights to maximize its efficiency but cannot effect the weights of the other stages. In other words, each stage's weights are independent of the weights of the other stages, while there exist structures where stages' weights may depend on the weights of other stages [20]. To overcome this issue, we utilize Nash bargaining game approach that has been used in several network DEA models [31,39,45]. First, let us briefly discuss this approach. The set of three persons in the bargaining are shown by = {1, 2, 3}, and a payoff vector is an element of the payoff space 3 , which is the three-dimensional Euclidean space. A feasible set and a breakdown point − → are a subset of the payoff space and an element of the payoff space, respectively. Then a bargaining problem is the triple (︁ , , − → )︁ consisting of persons, feasible set, and breakdown point. Nash approach requires the convexity and compactness of the feasible set, furher it should contain some payoff vector such that each person's payoff is at least as large as the person's breakdown payoff. Also, Nash [33,34] argued that a solution should satisfy the following four properties: Pareto efficiency, invariance with respect to affine transformation, independence of irrelevant alternatives, and symmetry. After all, he showed that there always exists a unique solution, called the Nash solution, which satisfies the abovementioned four properties and can be obtaind by solving the following problem: where − → is the payment vector for the persons, and and are the th elements of − → and − → , respectively. In the proposed structure, we consider the stages, the efficiency of each stage and the minimum achievable efficiencies for the stages respectively, as players, the payoff function and the breakdown points. To begin bargaining between stages, we obtain a breakdown point for each stage. In the process, one would not bargain with the other stage if its efficiency is less than the minimum efficiency, else the efficiency of the whole system would decrease. In fact, the benefit is distributed between the players based on their competition and bargaining power. To estimate the breakdown points, we apply the approach of Du et al. [16] that the minimum achievable efficiency for each stage can be obtained by adding a virtual DMU such that max = max { } and min = min { } for each stage. Since each DMU uses maximum amount of inputs to produce the minimum amount of outputs, efficiency scores calculated by DEA for (︀ max , min )︀ represents the least achievable efficiency for each stage. Thus, we can formulate the following Nash bargaining game model for measuring efficincy decomposition: Proof. The proof is similar to the proof of Theorem 2.3.
Now the stages' efficiency scores that are unique can be computed as the follows: Figure 2. General multi-stage series structure.

General multi-stage structure
In this section, we consider a general multi-stage series peocess, as depicted in Figure 2 [14]. The overall efficiency based on Nash bargaining game approach is obtained by the following model: Proof. The proof is similar to the proof of Theorem 3.2.

=1
. Notice that, this relationship also holds for models based on AED and MED in the cooperative context.

Numerical examples
In this section, we consider two numerical examples to illustrate the advantages of the proposed approaches. -Example 1. The data for this example is taken from Despotis et al. [14] consists of 30 DMUs with a three-stage structure, as depicted in Figure 1. The breakdown points of each stage are obtained as (︀ 1 min , 2 min , 3 min )︀ = (0.2698, 0.3853, 0.4149) by Du et al. [16] method. Table 2 shows the overall efficiency of four different models. The second column indicates the efficiency scores of BCC model (2.1), where the internal structure is neglected. In the third and forth columns, we can see the overall efficiency derived from AED model (2.5) and MED model (2.20), respectively. The last column of Table 2 also shows the overall efficiency scores derived from model (3.2). The BCC model shows the efficiency scores of 28 DMUs equal to one, while by considering the internal performance of system in cooperative model (2.5), only three DMUs (11, 13 and 22) and in model (3.2), only DMU22 are efficient. Also, DMU8 is the most efficient DMU in cooperative model (2.20). Further, we can see in Table 2 that DMU1 has the lowest efficiency score in model (2.1), in contrast to the cooperative models (2.5) and (2.20) and bargaining model (3.2) indicate DMU4 as the most inefficient DMU. Figure 3 also shows the overall efficiency distribution of four models. The results of the stages' efficiencies are reported in Table 3. Columns 2-4 and columns 5-7 show the stages' efficiencies derived from the weights of AED and MED models in the coopeartive context, respectively. The results show that AED and MED scores are almost consistent. Also, the stages' efficiency scores arising from the weights of bargaining model (3.2) are given in the last three columns of Table 3. Although the results of model (3.2) show a significant decrease in the number of efficient units, it has led to equilibrium between stages' efficiency. For instance, the number of efficient DMUs of stage 2 has been decreased from 14 DMUs in the cooperative model (2.5) to 7 units in the bargaining model (3.2). While we can see the efficiency scores of stage 3 are improved. Unlike both cooperative models, the bargaining model (3.2) makes an equilibrium between the stages. In general, depending on the managers' targets, whether to maximize efficiency or to achieve an equilibrium, each of these two approaches may be useful.
The data set for this example is taken from Tone and Tsutsui [38] consists of 10 vertically integrated power companies in the US with three stages: Generation, Transmission and Distribution. The number of employees ( 1 ) is stage 1's input to genetate electric power ( 1 ), which is output from Generation stage and input to Transmission stage. Also, the number of employees ( 2 ) as another input of stage 2 produces electric power sold to large customers ( 2 ). In addition, electric power sent ( 2 ) as output from the Transmission stage is inputted to stage 3 and alongside the number of employees ( 3 ) generate electric power sold to small customers ( 3 ). Note that this three-stage process is a special case of the one given in Figure 1 Table 4, respectively. We see 8 DMUs (No. 1,3,4,5,7,8,9 and 10) in model (2.1) that are efficient, while the results of models (2.5), (2.20) and (3.2) show that none of the DMUs are efficient. Table 5 also reports the stages' efficiency scores of three different models in the presence of intermediate products. Columns 2-4 show the stages' efficiencies of the AED model (2.5). Also, we can see the stages' efficiency results derived from model (2.20) in columns 5-7. The last three columns of this table also shows the efficiency scores of model (3.2). For example, for DMUs 2, 6, 7 and 9, we can see that 2 of (3.2) is higher than 2 of both cooperative models. Moreover, we see that compared with two cooperative models (2.5) and (2.20), in spite of the efficiency decrease of some DMUs, the second stages' efficiencies mean are increased in the bargaining model (3.2). Indeed, using the bargaining game approach leads to increase transmission in the power companies. As the

Conclusions
Conventional DEA models ignore the internal structure of DMUs and treat them as black box. In this paper, to identify the internal performance of system, we studied a three-stage network structure containing exogenous inputs and outputs. First, for the overall efficiency decomposition into stages' efficiency, we proposed cooperative AED and MED based models. Since the proposed models are nonlinear and the use of Charnes-Cooper transformation lead to parametric linear models, we used conic relaxation approach to solve them. On the other hand, in order to make an equilibrium based on the negotiation between the stages in the production process, we applied Nash bargaining game approach on the cooperative model that besides maintaining advantages of the cooperative model, provides a unique and fair decomposition of the overall efficiency. For the proposed nonlinear bargaining model a conic relaxation is presented instead of a parametric approach used in [1] for a special structure. The results of both multiplicative cooperative and bargaining models on two numerical examples show that the overall efficiency is equal to the product of the stages' efficiencies. Also, the overall efficiency of additive cooperative model is equal to arithmetic mean of stages' efficiencies. In general, depending on the managers' targets in an organization, whether to maximize the stages' efficiencies or to made equilibrium and fair allocation, each of these two approaches can be applied.