A New Method of Sensitivity Analysis of Returns to Scale in Two-Stage Network; A Case Study of the Insurance Industry in Iran
Subject Areas : Financial MathematicsMaryam Sarparast 1 , Farhad Hosseinzadeh Lotfi 2 * , Alireza Amirteimoori 3 , Mohsen Rostamy-Malkhalifeh 4
1 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
3 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.
4 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
Keywords: Two-Stage Networks , Sensitivity Analysis, Returns to Scale , Data Envelopment Analysis , Insurance Industry,
Abstract :
One important issue in data envelopment analysis (DEA) which has been studied by many researchers is returns to scale (RTS). The authors developed DEA models to evaluate the efficiency of two-stage networks in returns to scale variable and introduced a new definition to determine return to scale classification in two-stage networks. The current article proposed an approach for determining the stability region of returns to scale classification in two-stage network DEA. The data were collected from insurance companies in Iran in 2019. We consider the insurance industry process as a two-stage network; the stage of marketing and that of investment. The effectiveness of insurance companies was evaluated, and, after determining the classification of returns to scale, we found a sustainability interval to classify returns to their scale.
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Original Research
A New Method of Sensitivity Analysis of Returns to Scale in Two-Stage Network; A Case Study of the Insurance Industry in Iran
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Maryam Sarparasta, Farhad Hosseinzadeh Lotfi a,*, Alireza Amirteimoorib, Mohsen Rostami Malkhalifeha
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aDepartment of Mathematics, Science and Research Branch, Islamic Azad University,Tehran, Iran. bDepartment of Mathematics, Rasht Branch, Islamic Azad University,Rasht, Iran
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Article Info Article history: Received 2021-11-30 Accepted 2022-08-14
Keywords: Two-Stage Networks Sensitivity Analysis Returns to Scale Data Envelopment Analysis Insurance Industry
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| Abstract | |
One important issue in data envelopment analysis (DEA) which has been studied by many researchers is returns to scale (RTS). The authors developed DEA models to evaluate the efficiency of two-stage networks in returns to scale variable and introduced a new definition to determine return to scale classification in two-stage networks. The current article proposed an approach for determining the stability region of returns to scale classification in two-stage network DEA. The data were collected from insurance companies in Iran in 2019. We consider the insurance industry process as a two-stage network; the stage of marketing and that of investment. The effectiveness of insurance companies was evaluated, and, after determining the classification of returns to scale, we found a sustainability interval to classify returns to their scale. | |||
1 Introduction
Data envelopment analysis (DEA) is one of the methods used to measure relative efficiency of peer decision-making units (DMUs) that have multiple homogeneous inputs and outputs introduced by Charnes et al. [3]. They discussed a non-parametric approach to identify the best performance in a set of DMUs and presented CCR model. BCC model offered by Banker et al [1]. This model with production frontiers measured the existing decision-making units by the convex hull. Later on the DEA was used as a measurement tool in different fields such as management, economics etc. During this period of model development, the economic concept of returns to scale (RTS) has also been widely studied within different frameworks provided by these methods. In the literature of classical economics, returns to scale describes the behavior of the rate of increase in single output relative to the associated increase in the inputs. If output increases by the same proportional change as all inputs change, then there are constant returns to scale (CRS). If output increases by less than the proportional change in inputs, there are decreasing returns to scale (DRS). If output increases by more than the proportional change in inputs, there are increasing returns to scale (IRS). Banker et al [2] described RTS concept for multiple-output cases using DEA. They focused on the sign of u0 from BCC model for returns to scale classification. Färe and Grosskopf [6] provided a two-stage method to recognize returns to scale classification based on CCR and BCC models. Banker and Lovell [4] prepared a new method to determine returns to scale in DEA based on the envelopment form of the BCC model. Consequently, many scholars studied returns to scale in DEA. For example, Khodabakhshi et al [13] provided an Additive model method to estimate returns to scale in both stochastic and fuzzy data envelopment analyses. One of the interesting research topics is network DEA and many studies have been conducted in this regard. In traditional DEA models, DMUs were considered as a black box and the efficiency evaluation was limited by the final outputs and initial inputs. Färe and Grosskopf [5] presented a network model that measured the efficiency of the entire system due to its sub-units. Subsequently, the researchers recruited many networks in areas such as intermediate products, allocation of budgets, fixed factors, dynamic systems and etc. Kao and Hwang [8] examined the structure of the two-stage networks and provided a new model to evaluate the efficiencies of the whole process. This model described series of relationships between the whole process and the two sub-processes. Khaleghi et al. [12] studied the structure of the two-stage systems. The aim of their study was to determine the returns to scale (RTS) classification and scale elasticity (SE) in two-stage DEA. Sarparast et al. [28] presented diverse approaches to deal with two-stage networks which evaluated the efficiency of two-stage networks in variable returns to scale and introduced a new definition of the types of returns to scale in two-stage networks and also methods to determining the type of returns to scale of efficient units. Researchers provided DEA models for sensitivity analysis of returns to scale in the two-stage network DMUs. Peykani et al [25] introduced novel robust data envelopment analysis models capable of being investigated in the presence of discrete and continuous uncertainties. A year laer, in 2019, they [27] presented a new approach, FDEA, for scale efficiency and stock ranking. Put differently, the very model was offered to measure the efficiency of stocks when negative data and uncertainties within input/output parameters exist (Peykani et al. [23, 24, 26]. Khodakaram et al. [14], in their article “concurrent estimation of efficiency, effectiveness, and returns to scale” studied the efficiency, effectiveness and return to scale of DMUs simultaneously. Neralic and Wendell [22], also, provided an algorithm approach to sensitivity in DEA for the CCR and additive models that provide sufficient conditions that preserve the efficiency of the input and/or outputs of DMUs. Nastion and et al [21] prepared an article entitled “sensitivity analysis in data envelopment analysis for interval data remains insure and improve the efficiency of DEA modeling and presented a model to calculate the lower and upper limits for each DMU. Kang et al [7] presented an article in this study proposes a hybrid two-stage network model and a mixed network DEA with the shared-inputs model to jointly measure the efficiency and effectiveness of a metro transport system. Performance is determined by the hybrid two-stage network DEA with the shared-inputs model to account for the non-storable service feature. To solve the problems of non-linearity, obtaining a total return greater than one, the need to assign variable weights to combine the divisive returns, adopt a fixed weight to combine the divisive returns, and Inability to find efficient two-stage DMUs in the network contribute network DEA, Khoveynia et al [15] have proposed an input-output-oriented linear model to measure the overall efficiency of two-stage DMUs with shared resources.
Tavassoli et al [29] formulated a Fuzzy Network DEA (FNDEA) model to assess the efficiency of Iran's EDNs components with sustainability, considerations and uncertain data. In order to utilize all input and output criteria, this study also proposes a fuzzy linear programming model to determine the optimal lower bound to all input and output weights. Furthermore, some appropriate policies are suggested based on the strengths and weaknesses of each EDN to improve its efficiency. Michalia et al. [19] examined the applicability of the subsampling bootstrap procedure in the approximation of the asymptotic distribution of the DEA estimator when the production process has a network structure, and in the presence of undesirable factors. Evidence on the performance of subsampling bootstrap is obtained through Monte Carlo experiments for the case of two-stage series structures, where overall and stage efficiency scores are calculated using the additive decomposition approach. Results indicate great sensitivity both to the sample and subsample size, as well as to the data generating process. Lianga et al [16] provided, for the first time, the production process of manufacturing industry is modeled as a network system integrated by AI technology development stage, AI application stage and AI upgrade stage because by optimizing production and industrial structure, artificial intelligence (AI) is considered to play a key role in low carbon manufacturing. Understanding the performance of AI driven low carbon manufacturing is of great significance to achieve carbon emission reduction targets and sustainable development of resources. Then, an interactive three stage network DEA model with ratio data is developed to evaluate the manufacturing industry in China from 2016 to 2019. Show that many regions perform well in the AI application stage while most of them have low AI technology development and AI upgrade performance.
In this study we propose a new method for sensitivity analysis of returns to scale and present a new model based on Kao and Hwang’s model [8] and concepts presented by Sarparast et al. [28]. The rest of the paper is organized as follows: Section 2 presents the basic DEA model and the generic two-stage process and a review of studies conducted by Kao and Hwang [8] and Sarparast et al. [28], then we proceed to introduce a new method for returns to scale classification and provide a new model for sensitivity analysis of returns to scale classification in two-stage network systems in Section 3. In section 4, we use data from Sarparast et al.’s [28] article and compare two methods. After that, two methods are used to analyze the data of insurance companies in Iran in 2019, and the results of the two methods are compared together.
2 Background
2.1. The Basic Concepts of DEA
Suppose that there is a set of DMUs consisting of DMU1, DMU2, …, DMUn, where each DMUj produces s outputs 
using m inputs 
. The CCR input-oriented radial efficiency of DMUo (xo, yo) is obtained by solving the following model:
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The dual form of the model (1) is obtained from the same data which then is used in the model (2).
Where u and v are non-negative variables corresponding to constraints. Model (1) and (2) are respectively called envelopment form and multiplier form of CCR model (Charnes et al. [3]). DMUo is CCR-efficient if and only if the optimal solutions obtained from model (1) and (2) are equal to 1. Then, on the basis of all optimal lambda solutions to (1), the CCR RTS method can be expressed as (Banker and Thrall. [2]):
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