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It is inevitable for a manager to consider the performance effects of each component of a multi-stage financial equity capital. These components serve as inputs in the first stage to raise investments. The investments, as outputs of the first stage, become inputs for the second stage and are used in bank services, such as bank facilities, which are outputs of the second stage. Therefore, when evaluating bank performance, the connectivity between the stages must be considered; otherwise, efficiency may not be calculated correctly. Traditional methods often assess multi-stage systems as black boxes, neglecting the potential connectivity that may exist among the stages. We delve into the system and propose models to improve overall efficiency and the efficiency of each stage. Additionally, the continuity and relationships among stages introduce numerous variables and constraints to linear programming for evaluating the entire system. A centralized approach calculates the efficiency score of units simultaneously by solving only one linear programming problem, significantly reducing computational complexity. This approach, especially in large organizations, is commonly employed by central managers. In this paper, we introduce a centralized method for evaluating units with a multi-stage structure. We apply the proposed models to evaluate the efficiencies of bank branches and insurance companies, demonstrating the superiority of the improved network approach and centralized method in enhancing overall system efficiency. Bank branches typically have a two-stage structure, involving labor, physical capital, and other factors.IntroductionBank branches operate under the supervision of a central management team. The central manager, acting as the decision-maker, allocates resources such as labor and financial equity capital as inputs for these branches. The goal is to optimize the overall efficiency of the branches by minimizing the total consumption of resources while maximizing the desired outputs, such as security investments. A common approach to enhancing the performance of banks involves evaluating each branch separately. However, this method does not guarantee the minimization of total resource consumption and can be time-consuming. Since all bank branches are under the control of central management, the decision-maker can optimize the efficiency scores of branches by allocating resources to them simultaneously. This approach, known as centralized Data Envelopment Analysis (DEA), is particularly relevant when certain variables are controlled by a central authority, such as a Head Office, rather than individual unit managers. DEA is a mathematical programming technique used to assess the performance of homogeneous Decision Making Units (DMUs). However, in cases where DMUs have a network structure, such as banks, where the outputs of one division or sub-process serve as inputs for the next sub-process, traditional DEA models treat two-stage DMUs as black boxes and overlook potential connectivity among the stages. In our approach, we consider the internal activities within the system and propose a non-radial model to optimize multi-stage DMUs by taking into account the connectivity among the stages. Furthermore, in previous network DEA models, constraints related to intermediate activities were treated as inequalities, which, as we will demonstrate in this paper, can lead to contradictions in optimality. We address this issue by carefully considering the connectivity among stages. The presence of connectivity among stages introduces numerous variables and constraints to the corresponding model. This model, when used to measure the overall efficiency scores of all DMUs, would traditionally require solving as many problems as there are DMUs, which can be highly time-consuming. In our paper, we introduce a centralized approach that measures the efficiency scores of multi-stage structure DMUs by solving only one linear programming problem. We have applied these proposed models to evaluate bank branches and insurance companies. This approach provides a more comprehensive and efficient way to assess and improve the performance of multi-stage organizations like banks, taking into account the interconnected nature of their operations.MethodologyWe employ the Data Envelopment Analysis approach to evaluate systems with a multi-stage structure, often referred to as a network structure. Traditional DEA models treat two-stage DMUs as black boxes and overlook the potential for connectivity among these stages. In contrast, we delve into the internal activities of the system and propose a model that optimizes multi-stage DMUs by considering the interconnections among the stages. Moreover, in previous models designed to assess network systems, constraints related to intermediate activities were typically treated as inequalities, which could lead to inconsistencies in optimization. In our approach, we enhance these constraints associated with intermediate activities to ensure more robust optimization. Additionally, we apply a centralized approach to allocate resources to DMUs, allowing for the simultaneous optimization of the efficiency scores of all DMUs through the solution of a single linear programming problem. This centralized method streamlines resource allocation and improves the overall efficiency of the DMUs.ResultsWe evaluated 20 bank branches, treating them as 20 DMUs with a two-stage structure. In the first stage, inputs included paid interest, personnel costs, paid interest related to foreign currency transactions, and personnel costs related to foreign currency transactions. The first stage produced intermediate outputs in the form of raised funds and raised funds related to foreign currency transactions. In the second stage, the outputs consisted of loans and common incomes. Notably, some loans in the second stage might become non-performing, where borrowers are unable to make full or even partial repayments. To address this, we considered non-performing loans as undesirable or bad outputs and transformed them into inverse values to treat them as good outputs. To calculate the efficiency scores of the bank branches, we employed both our improved network model and the traditional DEA approach. Our network-based method revealed that many of the bank branches under evaluation were inefficient, in contrast to the traditional method, which inaccurately identified many of the bank branches as efficient. Subsequently, we extended our network method to a centralized case, significantly reducing computation time. The network-based assessment of bank branches took nearly 5 seconds, whereas solving the centralized model required only 0.1 second. In addition to evaluating bank branches, we applied our methods to assess insurance companies. The results demonstrated that our model provided more accurate efficiency scores compared to previous network-based approaches.ConclusionIn multi-stage production systems, the production process comprises several stages. Banks, for example, operate with a network structure in which labor, physical capital, and financial equity capital serve as inputs in the first stage to generate deposits as intermediate outputs. In the second stage, these banks utilize the deposits obtained from the first stage to create loans and security investments. We have introduced models to assess the efficiency of each stage, whether it's the first, intermediate, or final stage, individually. Additionally, we have developed a non-radial SBM model designed for evaluating DMUs with multi-stage structures. The Centralized DEA approach is a valuable method for central managers, particularly in large organizations like bank branches, to allocate resources effectively. We have extended our network-based method to a centralized approach, allowing us to calculate efficiency scores by solving just one linear programming problem. The results obtained from applying our proposed models to evaluate bank branches and insurance companies, both exhibiting network structures as DMUs, demonstrate the superiority of the network centralized approach over previous models.

Managers tend to improve the use of resources (inputs) of organizations to obtain the most productivity. Additionally, many industrial units have multi-stage structure in which the output of one stage is the input to the next stage. This paper, for the first time, presents data envelopment analysis (DEA) approaches to obtain the most productivity in two-stage decision making units (DMUs). Radial and non-radial models, by considering internal activities in system, are proposed to evaluate network DMUs and radial model is developed to identify most productive scale size (MPSS) pattern. Proposed models are applied to optimize the performance of bank branches as units with two-stage structure. Results show efficiency scores and also improvements are needed in costs and paid interests (inputs) to get values of incomes and loans (outputs) which results in the most productivity. This study provides managers with information to propose better strategies to improve not only the overall performance but also the efficiency of each stage.