Fixed Cost Allocation Based on DEA Cross Efficiency Considering Semi-Additive Production Technology: An Application to Bank Branches

محورهای موضوعی : Financial Mathematics

Javad Gerami ¹ ( Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran. )

تاریخ دریافت : 1401/01/17 تاریخ پذیرش : 1401/06/18 تاریخ انتشار : 1402/12/07

کلید واژه: Semi-additive production technology, Banking, Data envelopment analysis, Fixed cost allocation,

چکیده مقاله :

In the real world, situations frequently occur when we want to allocate a fixed cost between a set of decision-making units (DMUs) such as institutions, organizations. In this paper, we use the data envelopment analysis (DEA) technique to allocate fixed costs among DMUs. First, we introduce semi-additive production technology in DEA and present efficiency evaluation models in this technology. In estimating the frontier of this technology, in addition to the observed DMUs, the set of all aggregations of these DMUs are also used. In the following, we propose an interactive process for fixed cost allocation between DMUs in DEA based on the concept of cross-efficiency. We show that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency scores corresponding to all DMUs are improved at each stage of the interactive process. We also illustrate the proposed approach with a numerical example. The proposed approaches are demonstrated using an application of the fixed cost allocation problem for branches of commercial banks. Finally, we bring the results of the research.

چکیده انگلیسی:

In the real world, situations frequently occur when we want to allocate a fixed cost between a set of decision-making units (DMUs) such as institutions, organizations. In this paper, we use the data envelopment analysis (DEA) technique to allocate fixed costs among DMUs. First, we introduce semi-additive production technology in DEA and present efficiency evaluation models in this technology. In estimating the frontier of this technology, in addition to the observed DMUs, the set of all aggregations of these DMUs are also used. In the following, we propose an interactive process for fixed cost allocation between DMUs in DEA based on the concept of cross-efficiency. We show that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency scores corresponding to all DMUs are improved at each stage of the interactive process. We also illustrate the proposed approach with a numerical example. The proposed approaches are demonstrated using an application of the fixed cost allocation problem for branches of commercial banks. Finally, we bring the results of the research.

منابع و مأخذ:

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متن کامل:



Original Research

Fixed Cost Allocation Based on DEA Cross Efficiency Considering Semi-Additive Production Technology: An Application to Bank Branches

Javad Gerami*

Department of Mathematics, Shiraz branch, Islamic Azad University, Shiraz, Iran.

 

Article Info Article history: Received 2022-04-06 Accepted 2022-09-09   Keywords: Data Envelopment Analysis Semi-Additive Production Technology Fixed Cost Allocation Banking		Abstract
		In the real world, situations frequently occur when we want to allocate a fixed cost between a set of decision-making units (DMUs) such as institutions, organizations. In this paper, we use the data envelopment analysis (DEA) technique to allocate fixed costs among DMUs. First, we introduce semi-additive production technology in DEA and present efficiency evaluation models in this technology. In estimating the frontier of this technology, in addition to the observed DMUs, the set of all aggregations of these DMUs are also used. In the following, we propose an interactive process for fixed cost allocation between DMUs in DEA based on the concept of cross-efficiency. We show that our proposed iterative approach is always feasible, and ensures that all DMUs become efficient after the fixed cost is allocated as an additional input measure. The cross-efficiency scores ​​corresponding to all DMUs are improved at each stage of the interactive process. We also illustrate the proposed approach with a numerical example. The proposed approaches are demonstrated using an application of the fixed cost allocation problem for branches of commercial banks. Finally, we bring the results of the research.

 

 

1 Introduction

 

Efficiency analysis plays an important role in decision-making processes and is an important issue for improving the performance of organizations. Several different techniques have been proposed for efficiency analysis. One of these techniques is DEA and based on mathematical programming. At first, it was developed by Charnes et al. [5]. This method estimates the efficiency of a set of DMUs with multiple inputs and outputs. DEA obtain efficiency score from each DMU by making a set that called production possibility set (PPS). DEA accept the underlying assumptions for estimating the frontier of PPS, and consider the frontier of PPS as the efficiency frontier. DMUs located on this frontier are referred to as efficient DMUs and other DMUs are inefficient. By accepting different assumptions, different production technologies have been proposed to measure efficiency. Charnes et al. [5] accepted the property constant returns to scale (CRS) for reference technology, in order to estimate both technical and scale efficiencies. Banker et al. [3] introduced another production technology by accepting the property of variable returns-to-scale (VRS). This set based on the convex hull of individual units and strong disposability assumption. Fare et al. [11] used nonincreasing (decreasing) returns-to-scale (DRS) technology to estimate efficiency. In a different framework from DEA models, Koopmans [18] introduced another form of nonincreasing returns-to-scale technology. Grosskopf [16] showed Koopman’s technology in DEA framework and pointed out that it includes the sum of the individual units and comprises Fare et al.’s technology. Deprins et al. [9] introduced Free Disposal Hall (FDH) technology, which is a non-convex PPS. This model is equivalent to the model of Banker et al. [3] if only binary intensity variables are considered. Green and Cook [15] presented a non-convex PPS known as free coordination hall (FCH). In order to evaluate the performance of DMUs, in addition to the observed units, they also used the aggregations DMUs corresponding to these DMUs. As mentioned, some of the production technologies in the DEA literature include aggregated DMUs. In some of the DEA literature, especially axiomatic-based work, inclusion of aggregated DMUs is referred to as the additivity assumption. This assumption states that if the two observed DMUs A and B can be produced, then the newly created DMU can also be produced as A + B. Another assumption in estimating the frontier of PPS is semi-additive assumption. Ghiyasi [13] used semi-additive assumption and introduced a new PPS that consider aggregations of the set of DMUs. He measured the performance of DMUs based on individually observed units and the aggregations DMUs corresponding to these DMUs. Hence, this technology creates a larger competitive environment than other technologies for DMUs to reach the efficiency frontier. The two general assumptions of additivity and semi-additive can be assumed to be distinct. In accordance with additivity assumption, new aggregated DMUs can be exact multiples of the original observed units, but not in semi-additive technology, and we have to use different units to create new aggregated DMUs. According to the additivity assumption, if unit A belongs to the PPS, then the new aggregated DMUs as 2A or 3A also belong to the PPS, but this will not be in accordance with semi-additive assumption, and if A and B are two distinct DMUs, then A + B also belongs to the PPS in accordance with semi-additive assumption. [13] We must note that all production technologies accepting general additivity presume specific returns to scale properties. In the real world, however, unless we have knowledge regarding the returns to scale of a technology, we cannot impose any specific such properties concerning that technology. One of the main contributions of the semi-additive technology to the literature is that it addresses the returns to scale problem. Namely, the semi- additive technology does not presume any returns to scale requirements. [14]

The main problem with semi-additive technology is the resulting computational complexity due to the large number of aggregated DMUs created based on the observed DMUs. However, in many real-world applications, we may not need to consider all the DMUs and be able to solve the above problem. But Ghiyasi and Cook [14] solved the above problem, and they proposed a new PPS that is equal to the original semi- additive technology based on the power set of all DMUs. They showed that the efficiency evaluation model based on the new semi- additive technology no longer needs to consider all the aggregated DMUs in the power set corresponding to the on individually observed units and proposed the semi- additive technology in a new form based only on the observed units. They showed that the new model proposed for calculating efficiency in semi- additive technology significantly reduces the amount of computation and can be easily used. [14] One of the important applications of DEA is the issue of fixed cost allocation among DMUs. In many real management applications, we have to allocate a total fixed cost between a set of competitive DMUs. Cook and Kress [7] first presented the issue of fixed cost allocation in the context of DEA. They hypothesized that fixed costs could be considered as a new input measure for all DMUs. The basis of two principles efficiency invariance and pareto-minimality presented a fair cost allocation scheme by solving several linear programming models. But in general, their approach was to find a unique efficient unit by cone ratio approach, thus avoiding a multiple cost fixed allocation. Tohidi and Tohidnia [23] measure the interval industry cost efficiency score in DEA. They extend the concept of "cost minimizing industry structure" and develop two DEA models for dealing with imprecise data. Also, they propose an approach to compute the industry cost efficiency measure in the presence of interval data. They show that the value obtained by the proposed approach is an interval value. The lower bound and upper bound of the interval industry cost efficiency measure are computed and then decomposed into three components to examine the relationship between them and the lower and upper bounds of the individual interval cost efficiency measures. They define the cost-efficient organization of the industry as a set of DMUs, which minimizes the total cost of producing the interval industry output vector. Mozaffari et al. [22] presents two strategies for allocating fixed costs with undesirable data. In the first strategy, DMU first determines the minimum and maximum shares that it can receive from the fixed resources while the efficiency of that DMU and other DMUs re-mains the same after receiving the fixed resources. Also, the decision maker chooses the fixed cost for each DMU between the minimum and maxi-mum cost values proposed. In the second strategy, the allocation of fixed costs is done using the CCR multiplicative model with undesirable data. The effectiveness of both methods is examined by an applied study on the commercial banks.

Beasly [4] proposed a nonlinear programming model to achieve a unique cost fixed allocation by maximizing the average efficiency across all DMUs. Cook and Zhu [8] presented a new approach of feasible cost fixed allocation (but not optimal) in the output oriented based on the efficiency invariance principle and the proposed approach by Cook and Kress [7]. Li et al [19] presented a new approach to the fixed cost allocation problem based on DEA and degree of satisfaction. They provided a unique cost fixed allocation, and at the end of the proposed algorithm, all DMUs were efficient by considering the amount of dedicated cost fixed allocation as an additional input for all DMUs. They showed that the approach presented by them is equivalent to the proportional sharing method in one dimension, and in the multidimensional state, cost fixed allocation may not be unique. Du et al. [10] presented the issue of fixed cost and resource allocation based on the concept of cross-efficiency. They proposed an interactive algorithm for establishing the concept of cross-efficiency to provide a fixed cost allocation scheme between all cost DMUs. They hypothesized that the amount of cross-efficiency proportional to each of the DMUs at each stage of the proposed algorithm would be non-decreasing, and that at the end of the algorithm all DMUs would be cross-efficient. They considered the fixed cost proportional to each of the DMUs as an additional input. They then presented the issue of resource allocation based on the concept of cross-efficiency and presented another interactive algorithm in this regard.

Li et al. [20] proposed a new data envelopment analysis-based approach for fixed cost allocation. They suggest a non-egoistic principle which states that each DMU should propose its allocation proposal in such a way that the maximal cost would be allocated to itself. Also, a preferred allocation scheme should assign each DMU at most its non-egoistic allocation and lead to efficiency scores at least as high as the efficiency scores based on non-egoistic allocations. They integrate a goal programming method with DEA methodology to propose a new model under a set of common weights. The final allocation scheme is determined in such a way that the efficiency scores are maximized for all DMUs through minimizing the total deviation to goal efficiencies. Li et al. [21] developed allocating a fixed cost based on a DEA-game cross efficiency approach. They approach the fixed cost allocation problem by explicitly considering both competition and cooperation relationships among DMUs. They integrate cooperative game theory and the cross-efficiency method to propose a DEA-game cross efficiency approach to generate a unique and fair allocation plan. Based on the proposed approach by them, each DMU is considered as a player and a super-additive characteristic function is defined for coalitions of DMUs. In the following, the Shapley value is calculated for each DMU and accordingly associated common weights are optimized to determine the final allocation plan. Since the cross-efficiency method considers peer appraisal and the cooperative game theory allows for equitable negotiations, all DMUs are supposed to reach a consensus on the equitable allocation scheme through their novel approach. An et al. [1] proposed fixed cost allocation for two-stage systems with cooperative relationship using DEA. They developed an approach for fixed-cost allocation issues of two-stage systems by considering a cooperative relationship among DMUs. They integrate cooperative game theory and the DEA methodology to generate a unique and fair allocation plan. The results confirm that each DMU can maximize its relative efficiency to one by a series of optimal variables after the fixed cost allocation. A unique nucleolus solution can be generated through a feasible computation algorithm. An et al. [2] proposed fixed cost allocation based on the principle of efficiency invariance in two-stage systems. They proposed a fixed cost allocation approach for basic two-stage systems based on the principle of efficiency invariance and then extend it to general two-stage systems. They developed a fixed cost allocation under the overall condition of efficiency invariance when the two stages have a cooperative relationship. Then, the model of fixed cost allocation under the divisional condition of efficiency invariance wherein the two stages have a noncooperative relationship is studied. Chu et al. [6] developed DEA-based fixed cost allocation in two-stage systems based on the leader-follower and satisfaction degree bargaining game approaches. They proposed a new fixed cost allocation approach for allocating a fixed cost among DMUs with two-stage structures under the framework of DEA. They give the set of possible fixed cost allocations, and prove that all DMUs can be overall efficient when evaluated by a common set of weights after fixed cost allocation. Secondly, from a centralized point of view, they consider the competition between the DMUs’ two stages in fixed cost allocation and regard these two kinds of stages as two unions. They incorporate leader-follower models to propose a fixed cost allocation approach to handle the situation in which the two unions make decisions sequentially.  Izadikhah [17] proposes a new two stage BAM model and further evaluates the banks and financial institutes in Tehran stock exchange by considering the financial ratios. Conventional DEA models consider each firm as black box and don’t note into the inner activities. Two-stage data envelopment analysis has been researched by a number of authors that evaluate each firm by considering the inner operations. He proposes a new variant of two stage DEA models and further evaluates the banks and financial institutes in Tehran stock exchange by considering the financial ratios. In this paper, we present the issue of fixed cost allocation in semi-additive technology based on the concept of cross-efficiency. The proposed model for calculating cross-efficiency considers the fixed cost allocation as an additional input. According to the nonlinear form of the proposed model, we bring the necessary transformations for linearization of the model. To present an optimal fixed cost allocation plan, we propose an interactive algorithm. At each stage of algorithm, we improve the cross-efficiency scores corresponding to all DMUs, and when the algorithm terminates, we can obtain the optimal fixed cost allocation corresponding to each of the DMUs based on the concept of cross-efficiency. It can be said that the contribution of this paper is as follows. 1) In this paper, we present efficiency evaluation models in semi-additive technology based on the both envelopment and multiple models in DEA. 2) We present a cost fixed allocation scheme in semi-additive technology, based on the concept of cross-efficiency. 3) We apply the proposed approach in this paper for the data set of 18 branches of a bank in China.

The remainder of the paper is organized as follows. In the second section, we introduce semi-additive technology in theory and geometry. In the third second, we present the issue of fixed cost allocation in semi-additive technology. In the fourth section, we illustrate the proposed approach with a numerical example. In the fifth section, we show the application of the proposed approach for the set of bank branches in China, and in the sixth section, we bring the results of the research.

 

2 Semi-Additive Production Technology

 

Suppose that there are   producing the same set of  outputs by consuming the same set of  inputs. Let ,  denote observed DMUs and its  input and  output from this DMU are denoted by , , , and , , , respectively. We define the possibility of general production as follows.

Ghiyasi [13] introduced semi-additive production technology. To introduce the above technology, we first introduce the following assumptions in creating production technology.

S1. Feasibility of observations.

This assumption implies that all observed DMUs belong to the production technology, i.e.

, .

S2. Free (strong) disposability.

This assumption states that if  and if a point  is such that, , , then .

S3. Convexity.

This assumption states that if , , then  for all .

S4. Radial rescaling.

This assumption states that if , then , for all .

S5. Semi-additive.

This assumption implies that , , then , that .

S6. Minimum extrapolation.

This assumption states that the set  is the smallest set that holds in the above assumptions. In other words, the  set is the subscription of all sets of production technologies that have the above properties.

We now consider the power set corresponding to the set  as the set  . This set includes all subsets of the  set. Let set  . It should be noted that the set   is the power set corresponding to the set  that includes all subsets of the set  except the empty set namely this production technology does not include the origin. denote empty set. The set  has  members.

By accepting the assumptions S1-S6, we can present the PPS in semi-additive technology as follows.

 

.

The efficiency evaluation model of the unit under evaluation, i.e., in the input oriented based on semi-additive technology, will be as follows.

(1)

 

Now according to the definition of the set , model (2) will be as follows.

.

(2)

 

As can be seen, model (3) has  constraints and  variables. To solve model (3), we must form all the aggregated DMUs corresponding to the observed DMUs. As stated earlier. In this regard, we must create all aggregated DMUs from the set  or the power set corresponding to the set , which includes all subsets of the set  except the empty set. Therefore, it is very difficult to calculate the collective inputs and outputs for each member of the set . In this regard, Ghiyasi and Cook [14] showed that we can present the PPS under semi-additive condition, i.e. , as follows.

,               ,                           , , .

(3)

The efficiency evaluation model of the unit under evaluation, i.e., in the input oriented based on semi-additive technology based on the the new PPs namely  will be as follows.

 

(4)

Now according to the definition of the set , model (5) will be as follows.

.

(5)

 

Definition 1.  is called semi-additive efficiency in evaluation with model (6), if , otherwise it is called a semi-additive inefficient DMU.

As can be seen, model (6) has , constraints and only  variables. To solve model (3), we no longer need to create all the aggregated DMUs corresponding to the observed DMUs, and the efficiency is calculated only on the basis of the existing observed DMUs. Model (6) in evaluating the efficiency of the unit under evaluation i.e. , significantly reduces the amount of calculations compared to model (1). Ghiyasi and Cook [14] showed that . 

For illustrate PPS under semi-additive technology, we consider three DMUs as follows.

، ،  and .

In this case, we can form aggregated DMUs as follows.

, , , .

In the state one input (Input-axis) and one output (Output-axis), and all three original DMUs are efficient. The PPS under constant returns to scale (CCR) property is the region restricted by the Input-axis and the right-hand side of the line starting from the origin and passing the B (the dash–dot line). CRS technology is the biggest technology that includes all the other technologies. The bounded region by the Input-axis starting from  and the segment A–B–C and the horizontal extension from B is corresponding to the PPS of the BCC technology and has VRS property. The PPS under semi-additive assumption is bigger than the PPS of the BCC model and is bounded by the Input-axis starting from  passing the segment of A–B–G–E and horizontal extension from E.

Now, consider DMU . This DMU is efficient in variable returns to scale technology, while it is inefficient in semi-additive technology. In order to evaluate the efficiency of this DMU, we can solve model (6). This model depicts DMU  at point  on the efficiency frontier of the PPS corresponding to semi-additive technology. As shown in Fig. 1, the efficiency score is calculated as the ratio .  and , represent the image of the points  and  on the Input-axis, respectively.

 

3 Fixed Cost Allocation in Semi-Additive Technology

 

Suppose we have  decision units as . The input and output vectors corresponding to  are as  and 

respectively.

As stated in the second section, the efficiency score of the unit under evaluation, i.e.   in semi-additive technology can be obtained using model (6). 

Given that model (6) is a linear programming model, we can also get the efficiency score of the unit under evaluation based on the dual of this model as follows.

 

,                     ,                                                         , , .

(6)

 

In the above model, the multiples ,  are related to output, input constraints of model (6) respectively. Also, we consider the multiples ,  corresponding to constraints ,   in model (6), respectively.

Model (7) is presented in the fractional form as follows.

,  ,         ,                 , , , ,                 , , .

(7)

 

Definition 2.  is called semi-additive efficiency in evaluation with model (7), if the optimal objective function score of model (7) is equal to one, otherwise it is called a semi-additive inefficient DMU.

 

         Fig. 1: Semi-Additive Technology.

 

Definition 3.  is called semi-additive efficiency in evaluation with model (8), if , otherwise it is called a semi-additive inefficient DMU.

Suppose that a fixed cost R is to be distributed among all . Then, we can allocate amount a cost , , to each , , in a way that . If the shared cost R is treated as a new input, then the efficiency for , becomes

, ,                           , , , ,                                 , , .

(8)

For calculate the post-allocation efficiency score for , we consider the following fractional mathematical program.

 

In model (9), , is a parameter. At first, for solving model (9), Initially, we consider the value , as the semi-additive efficiency score of  which is obtained by solving model (8). In other words, we can put . is the optimal objective function of model (8), when we evaluate .

Note that model (9) is a non-linear program. In order to convert model (9) into a linear program, we use the Charnes–Cooper transformation and let  be a new variable as .

, ,                          , , ,                           , , , ,                            , , .

(9)

 

After solving model (10), suppose we obtain a set of optimal values of , ,, , , , corresponding to . Based on this set, the semi-additive k-cross efficiency for each , , is calculated as follows.

,  ,         ,  ,         ,         , , ,                 , , , ,                 ,  , .

(10)

 

Therefore, the semi-additive cross-efficiency score corresponding to , , is calculated as the average of all of its the semi-additive k-cross efficiencies as follows.

(11)

 

For the fixed cost allocation problem in the above strategy, we consider all DMUs. We determine the optimal solution of models (9) or (10) for each , and its maximum efficiency provided that the semi-additive cross-efficiency score of all DMUs do not be less of their current level. Now, we present an interactive algorithm to obtain an optimal fixed cost allocation scheme. In this regard, we first obtain the semi-additive cross-efficiency score corresponding to each of the DMUs based on model (10) and the value of parameter  in the first step of the proposed algorithm is equal to the optimal objective function score of model (8) in efficiency evaluation of , .

 

Table 1: An Algorithm for the Fixed Cost Allocation Process in Semi-Additive Technology.

(12)

Step One: Determine the efficiency scores of all DMUs. Solve model (8) and obtain a set of semi-additive efficiency scores . Let p = 1, , and go to the second step. Step Two: Determine the semi-additive cross-efficiency score of all DMUs. Solve model (10) for each . Let  , that , , , , , , represent optimal values for , , , ,, , when , respectively. Go to the third step. Step Three: Check the termination condition. If , for some j, where  is a specified small positive value, in this case, let , and go to step two. If , for all j, then go to the fourth step. Step Fourth: Determine the allocated fixed cost. For each , as the unit under evaluation, let the optimal allocation plan follows. , , we select the average of this allocations, that is  , , as the final amount of fixed cost R which is distributed to , .   In the next steps, we solve the model (10) for  and get the semi-additive cross-efficiency score of all DMUs based on equation (12). This process continues until the termination condition of the algorithm is met. We consider the termination condition of the algorithm as , this implies that the semi-additive cross-efficiency score for each of , , does not improve. At the end of the proposed algorithm, we obtain a fair allocation of a fixed cost among all DMUs. We now present the proposed algorithm as follows.   Theorem 1. Model (9) is always feasible for each . Proof. At first, we show that when , or equivalently, when , model (9) is feasible. Suppose , , , , , , be an optimal solution to the semi-additive fractional model (8) in evaluation . In this case, we have  . Now we put , , ,, Then we have , therefore , . If we put , , , , and  be any value between ,  , then   . This indicates that model (9) is feasible when . Now, we prove that if , and model (9) be feasible then when , model (9) is feasible also. Suppose  show an optimal solution to model (9) for  when , then we have . Put , ,, . Then we have , therefore , . If we put , , , , , and  be any value between  , , then  , , it show that , is a feasible solution to model (9) when . Therefore, model (9) is always feasible for any DMU and the proof is complete. Theorem 2. For each, , then  are non-decreasing for , and , that  is semi-additive efficiency score corresponding to , . Proof. Suppose  show an optimal solution to model (9) for  when , then, we have . Thus, for any , , , are non-decreasing, and satisfy  and the proof is complete. Theorem 3. If we consider the fixed cost as an additional input, then for all , the self-evaluated efficiency, or the optimal objective of model (9) for each , , is equals one, namely, . Proof. We consider the following program   Suppose , , , , be an optimal solution to model (13) in evaluation . Then we have  and , . Also, we have , and --, for any . If we put , , , , , , and  be any value between  , , then   , , , it show that , is a feasible solution to model (9) that makes  has efficiency score of one. When algorithm terminates, the cross-efficiency score corresponding to for any ,  , is equal to one. Also, all efficient DMUs become efficient after the fixed cost allocation, in the event that we consider the amounts of fixed cost allocated as an additional input for all DMUs.   4 Numerical Example In this section, we use the proposed approach in this paper for the data set in Table 2. This data includes 12 DMUs, each with three inputs and outputs. This data set was previously used in Cook and Kress [7], Beasley [4], and Cook and Zhu [8]. As in Beasley [4]. We allocate a fixed cost of 100 among the 12 DMUs. The last column of Table 2 shows the efficiency scores ​​of each DMUs based on constant returns to scale, variable returns to scale, and semi-additive technologies in the input oriented. In this paper, considering that the proposed approach is based on a semi-additive technology, we examine the results of this model. As shown in the last column of Table 2, units 4, 5, 8, 9, 11, and 12 are efficient and the other units are inefficient. We now use the algorithm presented in section 3 to find an optimal fixed cost allocation. For this purpose, we set the value . Table 3 shows the specific fixed cost values ​​based on equation (11) in iteration 13. The last row of Table 3 shows the amount of fixed costs allocated to 12 DMUs. If we increase the number of iterations of the algorithm, we can obtain a fairer fixed cost allocation. Table 4 shows the specific fixed cost values ​​based on equation (11) at the end of the algorithm after 55 iterations. As can be seen, the largest allocated cost is related to DMU 9 and the lowest fixed cost is related to DMUs 1 and 7, which is equal to zero. In Table 5, we compare the results of the proposed approach in this paper for the allocation of fixed costs with the results of other previous approaches that based on the constant returns to scale technology. But the proposed approach in this paper is presented in semi-additive technology. As previously stated, all 12 DMUs are efficient at the end of the algorithm, taking into account the specific fixed cost as a new input for all DMUs. As can be seen in semi-additive technology, we can obtain a different allocation scheme than the proposed previous approaches in this field. As can be seen in Table 5, units 9 and 11 have the same input, but based on the proposed approach in this paper, these units have different specific fixed cost amounts. This is a logical result because these two units have different output levels. But based on the approach provided by Cook and Kress [7], they have the same amount of specific fixed cost. We can have similar interpretations for DMUs 10 and 12. The proposed approach in this paper may not lead to a unique cost allocation scheme. However, if we want to obtain a unique fixed cost allocation scheme, we can impose additional restrictions on the fixed cost amount allocated ​​to the DMUs in model (9). For example, we can put , , , . In this case, by solving model (9), the values ​​of fixed costs specific to other DMUs are obtained as: , , ,, , , , , , , ,  , which are different from the results obtained from previous approaches as Beasley [4], and Cook and Zhu [8], Du et al. [10] based on models based on constant returns to scale technology.   Table 2: Numerical Example.                               , ,                            , , , ,                            , , . (13) DMU Input1 Input2 Input3 Output1 Output2 CCR efficiency BCC efficiency Semi efficiency 1 350 39 9 67 751 0.7567 0.8292 0.8292 2 298 26 8 73 611 0.923 0.9348 0.9348 3 422 31 7 75 584 0.747 0.7483 0.7483 4 281 16 9 70 665 1 1 1 5 301 16 6 75 445 1 1 1 6 360 29 17 83 1070 0.9612 1 0.9612 7 540 18 10 72 457 0.8604 0.8889 0.8889 8 276 33 5 78 590 1 1 1 9 323 25 5 75 1074 1 1 1 10 444 64 6 74 1072 0.8318 0.8333 0.8333 11 323 25 5 25 350 0.3333 1 1 12 444 64 6 104 1199 1 1 1   Table 3 shows the cross cost fixed allocation scores obtained on the step fourth of proposed algorithm in repetition 13. Also, last row of Table 3 shows the amount of fixed cost R=100 distributed to all DMUs obtained on the step fourth of proposed algorithm in repetition 13.       Table 3: The Cross Cost Fixed Allocation (In Repetition 13). DMU The cross cost fixed allocation 1 0.000 7.293 3.616 15.567 12.824 3.903 0.000 8.365 37.119 1.482 1.104 8.728 2 0.000 6.794 2.746 16.225 11.693 6.624 0.000 6.676 38.990 1.305 0.973 7.975 3 0.000 6.800 2.730 16.254 11.690 6.701 0.000 6.656 39.008 1.269 0.946 7.946 4 0.000 6.781 2.780 16.166 11.698 6.471 0.000 6.714 38.955 1.378 1.027 8.032 5 0.000 6.833 2.642 16.408 11.678 7.100 0.000 6.557 39.099 1.080 0.805 7.798 6 0.000 6.686 3.026 15.732 11.733 5.345 0.000 6.994 38.697 1.912 1.425 8.452 7 0.000 6.712 2.888 16.245 11.583 7.276 0.816 6.254 38.665 1.063 0.792 7.707 8 0.000 6.798 2.734 16.248 11.691 6.685 0.000 6.660 39.004 1.277 0.952 7.953 9 0.000 6.685 3.029 15.728 11.734 5.334 0.000 6.996 38.694 1.917 1.429 8.455 10 0.000 6.685 3.028 15.727 11.732 5.337 0.000 6.994 38.695 1.918 1.427 8.457 11 0.000 6.345 2.586 15.870 10.882 7.485 0.000 6.050 39.522 2.163 0.000 9.096 12 6.473 12.522 6.373 14.289 16.242 0.000 0.000 15.245 14.444 0.000 14.413 0.000 Mean 0.539 7.244 3.181 15.871 12.098 5.688 0.068 7.513 36.741 1.397 2.107 7.550   Table 4 shows the final amount of fixed cost R=100 distributed to all DMUs obtained on the step fourth of proposed algorithm in final repetition. Also, last row of Table 3 shows the optimal amount of fixed cost R=100 distributed to all DMUs obtained on the step fourth of proposed algorithm when the algorithm stops.   Table 4: The Optimal Cross Cost Fixed Allocation (At the End of the Algorithm). DMU The optimal cross cost fixed allocation 1 0.000 6.926 3.149 15.867 12.179 4.744 0.000 7.376 38.336 1.517 2.018 7.889 2 0.000 6.173 2.701 15.498 10.686 7.017 0.000 6.067 39.466 2.735 0.000 9.659 3 0.000 6.172 2.701 15.498 10.686 7.016 0.000 6.067 39.466 2.736 0.000 9.660 4 0.000 6.173 2.701 15.499 10.686 7.017 0.000 6.067 39.465 2.734 0.000 9.658 5 0.000 6.171 2.701 15.496 10.685 7.014 0.000 6.066 39.466 2.738 0.000 9.662 6 0.000 6.176 2.699 15.504 10.689 7.023 0.000 6.067 39.465 2.726 0.000 9.651 7 0.000 6.171 2.701 15.496 10.685 7.014 0.000 6.066 39.466 2.738 0.000 9.662 8 0.000 7.032 2.145 17.287 11.602 9.409 0.000 5.993 39.587 0.000 0.000 6.945 9 0.000 7.032 2.146 17.287 11.603 9.408 0.000 5.994 39.586 0.000 0.000 6.946 10 0.000 7.031 2.147 17.284 11.601 9.404 0.000 5.994 39.586 0.004 0.000 6.950 11 0.000 7.032 2.145 17.287 11.601 9.410 0.000 5.992 39.587 0.000 0.000 6.945 12 0.000 7.406 2.489 17.678 12.404 8.401 0.000 6.715 40.403 0.000 0.793 3.712 Mean 0.000 6.624 2.535 16.307 11.259 7.740 0.000 6.205 39.490 1.494 0.234 8.111   Table 5: Fixed Cost Allocation Results for the Different Approaches. DMU Our approach Feng et al. [12] cook and kress [7] beasly [4] cook and zhu [8] Du et al. [10] Li et al. [21] 1 0 8.09 14.52 6.78 11.22 5.79 5.54 2 6.624 7.36 6.74 7.21 0 7.95 7.53 3 2.535 8.91 9.32 6.83 16.95 6.54 7.35 4 16.307 6.94 5.6 8.47 0 11.1 7.87 5 11.259 7.13 5.79 7.08 0 8.69 6.38 6 7.74 9 8.15 10.06 15.43 13.49 11.5 7 0 9.77 8.86 5.09 0 7.1 5.9 8 6.205 7.41 6.26 7.74 0 6.83 7.77 9 39.49 8.2 7.31 15.11 17.62 16.68 11.9 10 1.494 10.21 10.08 10.08 21.15 5.42 11.38 11 0.234 5.46 7.31 1.58 17.62 0 2.74 12 8.111 11.52 10.08 13.97 0 10.41 14.14   5 Empirical Study In this section, we apply the proposed approach to data set was previously used in Li, et al. [21]. This data is involving a city commercial bank in Chengdu, Sichuan Province of China, which has 18 branches. As stated in Li, et al. [21], the city commercial bank was charged 29 million CNY for information and technology maintenance. The headquarters of that city commercial bank intends to distribute the total maintenance charge among its 18 branches. We consider each branch is considered as a homogeneous and independent DMU, and the total fixed cost is R = 2900 units (1 unit = 10 thousand CNY). Li et al. [21] consider three inputs and three outputs. The input and output variables have come in Table 6. Li et al. [21] consider the inputs as: staff (x1), that refers to human resource investment and manpower; Second input is fixed assets (x2), referring to the asset value of physical capital that can be used for business activities; Third input is operation costs (x3), that refers to the costs generated during the bank operations other than the labor costs. Outputs include deposit operating amount (y1), loan operating amount (y2) (i.e. total score of loans given by the bank), and revenue income (y3), that takes into account both interest income and non-interest income. Table 6: Input and Output Variables Regarding Branches (Li et al. [21]). Input/output Variable Unit Input Staffs (x1) Person Fixed assets (x2) 10 thousand CNY Operation costs except for the labor costs (x3) 10 thousand CNY Output Deposits (y1) 10 thousand CNY Loans (y2) 10 thousand CNY Income (y3) 10 thousand CNY   The values of inputs and outputs of city commercial bank branches are given in Table 7.   Table 7: The Dataset of 18 Branches of the City Commercial Bank (Li et al. [21]). DMU I1 I2 I3 O1 O2 O3 CCR efficiency BCC efficiency Semi efficiency 1 62 1822 1361 140117 130288 5260 1 1 1 2 80 1833 1565 213774 145761 10773 1 1 1 3 129 3595 1378 194084 130556 8006 0.6245 0.6249 0.6249 4 62 1978 333 87876 49454 4479 0.8173 1 1 5 89 2138 549 107091 60872 5897 0.6766 0.697 0.697 6 84 1910 704 97472 94310 3849 0.6672 0.8112 0.8112 7 36 1234 840 114001 80019 5292 1 1 1 8 172 4348 959 366423 306926 12479 1 1 1 9 62 879 1253 107393 86485 5132 1 1 1 10 53 2566 483 69691 43907 3869 0.5483 0.7478 0.7478 11 92 1348 419 148458 87193 7234 1 1 1 12 39 1229 513 83752 40046 3984 0.8343 1 1 13 144 4640 1323 223539 211466 10655 0.7922 0.8177 0.7922 14 47 2248 670 70555 65110 2205 0.6931 0.7967 0.7967 15 39 1571 362 99143 66736 5271 1 1 1 16 56 1635 669 112513 79366 5202 0.814 0.873 0.873 17 34 939 867 87660 56157 3000 0.8962 1 1 18 58 1807 419 88334 67160 4171 0.6994 0.8668 0.8668 Based on the data in Table 7, we have a series of CCR efficiencies through CCR model, as shown in the eighth column of Table 7. We can learn from Table 7 that seven bank branches are identified as efficient (1, 2, 7, 8, 9, 11, and 15), while the other eleven bank branches are inefficient. Based on the BCC model, as shown in the ninth column of Table 7. We can learn from Table 7 that ten bank branches are identified as efficient (1,2, 4, 7, 8, 9, 11, 12, 15, and 17), while the other eight bank branches are inefficient. In this paper, given that the proposed approach is based on a semi-additive technology, we consider the results in this technology. The results are similar to the results of BCC model, except that the efficiency score for the thirteen units of the two models are different. The efficiency scores for unit thirteen in technologies variable returns to scale and semi-additive are equal to 0.8177, 0.7922, respectively. Now, we apply the algorithm presented in section 3 to find an optimal fixed cost allocation. For this purpose, we set the value . For comparison purposes, the fixed cost allocation scheme from Li et al. [20] are also listed in in Table 8. We have given the corresponding rank for each unit based on the fixed cost allocated to each DMU based on both approaches. From Table 8, the fixed cost allocation scheme obtained from the proposed approach is feasible as each DMU is allocated a positive fixed cost. DMUs 8, 13 and 2 have the best rankings based on the approach presented in this paper, respectively. DMUs 8, 13 and 2 have the best rankings based on the approach presented in this paper and the proposed approach provided by Li et al. [21], respectively. The results of Li et al. [21] based on the constant returns to scale technology, but the proposed approach in this paper is presented in semi-additive technology.   Table 8: Fixed Cost Allocation Results for the Different Approaches for Bank Branches. DMU Our approach Rank Li et al. [21] Rank DMU Our approach Rank Li et al. [21] Rank 1 132.297 8 157.41 6 10 0.000 16 80.18 17 2 297.140 3 287.63 3 11 248.044 4 197.79 5 3 49.738 13 202.67 4 12 74.160 11 96.96 15 4 91.821 10 107.2 14 13 304.743 2 293.7 2 5 44.387 14 137.98 10 14 0.000 16 57.24 18 6 58.190 12 110.81 12 15 227.663 5 140.81 9 7 175.690 6 144.92 7 16 151.588 7 140.21 8 8 896.739 1 416.6 1 17 28.476 15 81.78 16 9 0.000 16 135.6 11 18 119.322 9 110.52 13   Table 9 shows the final amount of fixed cost R=2900 distributed to all DMUs obtained on the step fourth of proposed algorithm in final repetition. Also, last row of Table 9 show the optimal amount of fixed cost R=2900 distributed to all DMUs obtained on the step fourth of proposed algorithm when the algorithm stops.   6 Conclusion In this paper, we first introduce semi-additive technology by introducing the underlying assumptions for estimating the frontier of PPS. Also, we presented the performance evaluation model in semi-additive technology in the form of fractions and multiples. In the following, we present an interactive process for fixed cost allocation based on the concept of cross-efficiency in DEA. The proposed approach is always feasible. Cross-efficiency scores ​​in semi-additive technology corresponding to all DMUs are improved at each stage of the interactive process, and at the end of the algorithm, if the fixed cost amount assigned to each DMUs is added as a new input to all DMU, these units are efficient in semi-additive technology. Although the proposed approach may not lead to a unique cost allocation plan, however, by imposing constraints on the corresponding cost amounts of some DMUs, we can obtain a unique cost allocation plan corresponding to each of the DMUs. The proposed algorithm is convergent in the interactive process. Due to the linearization of models, it can be easily solved with conventional optimization software. As future work, we can use the proposed new models in semi-additive technology to determine the returns to scale class of DMUs. We can also obtain a fixed cost allocation scheme for new aggregate DMUs and extend the proposed approach to other data structures in DEA, such as a two-stage network structure.   Table 9: The Optimal Cross Cost Fixed Allocation to Bank Branches (At the End of the Algorithm).