Malmquist productivity index for two-stage network systems under data uncertainty: A real-world case study

Seyed Ehsan Shojaie; Seyed Jafar Sadjadi; Reza Tavakkoli-Moghaddam

doi:10.1371/journal.pone.0307277

Abstract

The measurement of productivity change in decision-making units (DMUs) is crucial for assessing their performance and supporting efficient decision-making processes. In this paper, we propose a new approach for measuring productivity change using the Malmquist productivity index (MPI) within the context of two-stage network data envelopment analysis (TSNDEA) under data uncertainty. The two-stage network structure represents a realistic model for DMUs in various fields, such as insurance companies, bank branches, and mutual funds. However, traditional DEA models do not adequately address the issue of data uncertainty, which can significantly impact the accuracy of productivity measurements. To address this limitation, we integrate the MPI methodology with an uncertain programming framework to tackle data uncertainty in the productivity change measurement process. Our proposed approach enables the evaluation of productivity change by capturing both technical efficiency and technological progress over time. By incorporating fuzzy mathematical programming into the DEA framework, we model the inherent uncertainty in input and output data more effectively, enhancing the robustness and reliability of productivity measurements. The utilization of the proposed approach provides decision-makers with a comprehensive analysis of productivity change in DMUs, allowing for better identification of efficiency improvements or potential areas for enhancement. The findings from our study can enhance the decision-making process and facilitate more informed resource allocation strategies in real-world applications.

Citation: Shojaie SE, Sadjadi SJ, Tavakkoli-Moghaddam R (2024) Malmquist productivity index for two-stage network systems under data uncertainty: A real-world case study. PLoS ONE 19(7): e0307277. https://doi.org/10.1371/journal.pone.0307277

Editor: Nhat-Luong Nhieu, National Kaohsiung University of Science and Technology, TAIWAN

Received: February 13, 2024; Accepted: July 2, 2024; Published: July 18, 2024

Copyright: © 2024 Shojaie et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript.

Funding: The author(s) received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

Efficiency and productivity assessments are crucial for organizations seeking continuous improvement and growth in today’s highly competitive and dynamic business environment [1–3]. Various methodologies and techniques have been developed to evaluate the performance of decision-making units (DMUs) in different areas and applications such as agriculture, banking, communication, education, energy, finance, fishery, forestry, healthcare, insurance, manufacturing, power, supply chain, transportation, and tourism [4–6]. One widely used approach is the data envelopment analysis (DEA), which measures the relative efficiency of DMUs based on their input-output characteristics [7–9]. However, traditional and conventional DEA models may not be suitable for evaluating the performance of complex systems characterized by two-stage networks and data uncertainty [10–14].

In recent years, researchers have recognized the importance of extending DEA models to analyze the productivity change in two-stage network systems [15–17]. Two-stage networks refer to interconnected systems with intermediate processes or stages between the input and output stages. These systems are prevalent in real-world case studies and applications such as airlines, bank branches, electricity distribution companies, food supply chain, healthcare supply chain, insurance companies, manufacturing companies, mutual funds, research and development projects, and universities, where multiple stages are involved in the production or service delivery process [18–20]. The presence of intermediate stages in these networks allows for more efficient resource allocation, coordination of activities, and optimization of overall system performance. By understanding and analyzing the dynamics of two-stage networks, researchers can identify bottlenecks, improve process efficiency, and enhance overall system productivity.

The Malmquist productivity index (MPI) is a well-known DEA-based tool used to measure productivity change over time [21–23]. It calculates the changes in efficiency and technical progress for individual DMUs and aggregates them at the industry or country level. However, the traditional MPI does not account for data uncertainty, which is inherent in real-world situations and problems [24–26]. Data uncertainty and ambiguity arise due to various sources, including measurement errors, missing data, and imprecise estimates. Ignoring data uncertainty can lead to biased and unreliable productivity assessments. Hence, there is a need to develop effective productivity measurement techniques that consider data uncertainty in the context of two-stage network systems.

To address this research gap, this paper proposes the application of credibility-based fuzzy network data envelopment analysis (CFNDEA) approach integrated with Malmquist productivity index to evaluate the productivity change of two-stage network systems under data uncertainty. Fuzzy network DEA models extend the traditional DEA framework by incorporating fuzzy set theory to handle imprecision and vagueness in input-output data [27–29]. Credibility theory, on the other hand, deals with uncertainty by assigning credibility weights to observations based on their accuracy and reliability [30–32]. The objectives of this research paper are threefold: I) to develop a fuzzy network DEA approach for evaluating the efficiency and productivity change of two-stage network DMUs; II) to integrate credibility theory into the fuzzy network DEA model to account for data uncertainty; and III) to apply the proposed methodology to a real case study involving Iranian mutual funds.

Mutual funds are investment vehicles that pool money from multiple investors to invest in a diversified portfolio of securities such as stocks, bonds, or a combination of both [33–35]. They are managed by professional fund managers who make investment decisions on behalf of the investors. Mutual funds offer individual investors access to a wide range of investment opportunities with the potential for diversification and professional management. Investors can buy shares or units of mutual funds, which represent their ownership in the fund’s assets. The mutual fund’s performance is typically measured by its net asset value (NAV), which is the value of the fund’s assets minus its liabilities, divided by the number of outstanding shares or units. Investors can buy and sell mutual fund shares at the fund’s NAV, which is calculated at the end of each trading day.

Performance evaluation of mutual funds is important as it helps investors assess the fund’s historical returns, risk levels, and consistency. It aids in comparing different funds and making informed investment decisions. Evaluating performance also helps investors track fund managers’ skills and determine if the fund aligns with their financial goals and risk tolerance [36–38]. The proposed fuzzy network DEA model, integrated with credibility theory, will enable us to evaluate the productivity change of the mutual funds network under data uncertainty. By considering imprecise and unreliable data, the model will provide more robust and accurate performance assessments. It should be explained that the main advantages of the current research can be summarized as follows:

I) Proposing a new Malmquist productivity index that is capable of being used for network systems in the presence of uncertain data. II) Measuring efficiency score, efficiency change, technological change, and productivity change of two-stage DMUs under fuzzy panel data. III) The linearity of the proposed credibility-based fuzzy network DEA approach. IV) The unique efficiency decomposition of two-stage DMUs under data ambiguity. V) The discriminatory power of the credibility-based fuzzy network model exceeds that of a classical network DEA model. VI) The proposed uncertain network Malmquist productivity index is applied for the dynamic performance evaluation of mutual funds. VII) Identifying the state of productivity of mutual funds both in general and in terms of sub-units during two consecutive time periods. VIII) The sensitivity analysis of results from the proposed CFNDEA approach and uncertain network MPI is examined under different confidence levels.

The remainder of this research paper is organized as follows: Section 2 provides a two-stage network DEA approach based on additive efficiency decomposition. Section 3 delineates the classification of the literature review and underscores significant gaps in the existing literature. Section 4 presents the proposed methodology, including the formulation of the fuzzy network DEA model and the integration of credibility theory. Section 5 proposes a new Malmquist productivity index for two-stage network systems under data uncertainty. Section 6 outlines the case study design and data collection process as well as the results and discussion. Finally, Section 7 introduces the conclusion and future research directions.

2. Literature review

In this section, an extensive review of the literature is conducted from two distinct perspectives: theoretical (focusing on network DEA and fuzzy chance-constrained programming) and practical (applying network DEA in the context of mutual funds). Furthermore, the study outlines the identified gaps in the existing literature that serve as the focal point of this research. Table 1 summarizes the main characteristics of the fuzzy chance-constrained network DEA researches considering different aspects such as uncertain measure types, including possibility (POS), necessity (NEC), credibility (CR), and general fuzzy (GF). Moreover, a nuanced categorization of mutual fund performance assessment through network DEA approach is depicted comprehensively in Table 2.

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Table 1. Network data envelopment analysis and fuzzy chance-constrained programming: A literature review.

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Table 2. Network data envelopment analysis and mutual funds: A literature review.

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As evident in Tables 1 and 2, the goal of this research is to fill the current research gap in the literature by introducing a novel and efficient Malmquist productivity index for mutual funds operating within two-stage network systems under data uncertainty. To achieve this goal, the research utilizes the concepts of the Malmquist productivity index, network data envelopment analysis approach, additive efficiency decomposition technique, credibility theory, and fuzzy chance-constrained programming. Finally, the applicability and efficacy of the proposed uncertain network Malmquist productivity index are demonstrated through a real case study involving Iranian mutual funds.

3. The network DEA approach: Additive efficiency decomposition

Fig 1 visually depicts a generic two-stage framework, comprising a collection of S homogeneous decision-making units DMU_s (s = 1,…,S). In the initial stage, each DMU possesses P inputs I_ps (p = 1,…,P), while Y outputs L_ys (y = 1,…,Y) (referred to as leakage variables) exit the system. Additionally, G intermediate variables C_gs (g = 1,…G) establish a connection between the first and second stages. Moving on to the second stage, there are X additional inputs A_xs(x = 1,…,X) and ultimately Q outputs Q_qs (q = 1,…,Q). It should be explained that the non-negative weights α_p (p = 1,…,P), λ_y (y = 1,…,Y), ω_g (g = 1,…,G), μ_x (x = 1,…,X), and β_q (q = 1,…,Q) are assigned to the I_ps (p = 1,…,P), L_ys (y = 1,…,Y), C_gs (g = 1,…,G), A_xs (x = 1,…,X), and Q_qs (q = 1,…,Q), respectively.

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Fig 1. The graphical presentation of general two-stage structure.

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The present study utilizes the additive efficiency decomposition approach as a fundamental nonparametric network DEA method for evaluating the performance of both the overall system and its sub-components (stages 1 and 2). This approach proves to be suitable for analyzing a two-stage structure with a leakage variable in the first stage and additional inputs in the second stage [61]. It is worth mentioning that the additive efficiency decomposition approach holds a prominent position among the various methods employed in the field of network data envelopment analysis [62]. The subsequent section provides a detailed explanation of the modeling process utilizing an additive efficiency decomposition approach for a general two-stage structure. Referring to Fig 1, the efficiency score of the initial stage and subsequent stage for DMU_k (the DMU under investigation) can be computed using Models (1) and (2) correspondingly: (1) (2)

In line with the concept proposed by Chen et al. [61], the overall efficiency of the typical two-stage network procedure can be represented mathematically by Eq (3): (3)

Please be aware that, in Eq (3), and represent user-defined weights, which contribute to the magnitude of . To elaborate, and exemplify the significance of first stage and second stage in relation to the system’s overall performance. Consequently, the efficiency score of DMU_k is determined by solving Model (4) in the ensuing manner: (4)

As evident from Model (4), this particular model cannot be transformed into a linear program using the standard Charnes & Cooper [63] conversion technique. In order to resolve this challenge, a proposed solution by Chen et al. [61] involves defining and as Eqs (5) and (6) correspondingly: (5) (6)

Therefore, employing the aforementioned equations will lead to the transformation of Model (4) into Model (7), which can be expressed in the following manner: (7)

Now, through the utilization of the Charnes & Cooper [63] conversion technique, the existing Model (7) can be transformed into a linear programming representation known as Model (8): (8)

Please note that the optimal multipliers derived from Model (8) may not have a unique solution. As a result, the decomposition of the overall efficiency described in Eq (3) will not be unique either. Accordingly, Kao & Hwang [64] proposed a technique to identify a group of multipliers that yield the highest efficiency score for either stage 1 or stage 2, while still preserving the overall efficiency score. If we assume that the efficiency of the first stage carries more significance for the decision maker, we can estimate by solving Model (9), while optimizing through Model (8): (9)

As Model (9) is classified as a linear fractional program, through the application of conversion technique of Charnes & Cooper [63], this specific model can be considered equivalent to Model (10): (10)

Once is determined through the utilization of the Model (10), the efficiency score for the second stage can be acquired by employing Eq (11): (11)

It should be emphasized that the optimal weights and are derived from Model (8) by implementing Eqs (5) and (6). In contrast, if we consider the stage 2 to be more significant, the efficiency of stages 2 and 1 will be evaluated using a comparable approach.

4. The credibility-based fuzzy network DEA approach

In this particular section, we will introduce a novel FNDEA approach designed to handle imprecise and vague data. Notably, our proposed fuzzy network DEA model, adopts an additive efficiency decomposition approach and assumes that first stage holds greater significance. Consequently, we make the assumption that all the data (, , , and ) involved can be considered to be approximately known, and can be represented by triangular membership functions denoted as fuzzy numbers , , , , and in which , , , , and . To account for the ambiguity in the available data, we can reformulate Models (8) and (10) for fuzzy observations into Models (12) and (13) correspondingly: (12) (13)

As evident from the Models (12) and (13), notable modifications have occurred in both the objective function and equality constraints. However, it is important to note that these alterations do not affect the optimal solutions proposed by the models [65, 66]. Consequently, to effectively handle imprecise and ambiguous data within Models (12) and (13), we shall employ the principles of credibility theory [67] and chance-constrained programming [68]. It should be explained that the credibility (Cr) measure of {†} is defined on the possibility space (Ξ,P(Ξ),Pos) as the average of its possibility (Pos) and necessity (Nec) measures that is presented in Eq (14): (14)

Let be a triangular fuzzy variable on the possibility space (Ξ,P(Ξ),Pos) and η be a crisp number. The credibility of fuzzy events and can be specified using Eqs (15) and (16), respectively: (15) (16)

In accordance with the credibility measure, the process of transforming fuzzy chance constraints into their equivalent crisp constraints at a specific confidence level ξ can be precisely defined utilizing Eqs (17) and (18) correspondingly [32, 65]: (17) (18)

In light of the credibility measure and chance-constrained programming, the Models (12) and (13) have been redefined as Models (19) and (20) correspondingly: (19) (20)

Based on Eqs (17) and (18), if the confidence levels are above or below 0.5, the crisp counterpart of fuzzy chance constraints would differ. It is important to mention that by utilizing a binary variable Ω and a significantly large value Γ, one can streamline the process of linearization of incompatible constraints and seamlessly incorporate fuzzy NDEA models for ξ≤0.5 and ξ>0.5. As a result, we introduce the credibility-based fuzzy network DEA approach for determining the overall efficiency score of a DMU_k considering fuzzy data. This model is denoted as Model (21): (21)

Next, we proceed to estimate the efficiency score of the first stage considering the introduction of fuzzy data. This estimation is achieved by solving Model (22), with the values of being obtained from Model (21): (22)

1

Ultimately, upon deriving the value of from Model (22), we can proceed to assess the efficiency score of second stage by employing Eq (23): (23)

It is important to acknowledge that the efficiency scores derived from the proposed credibility-based fuzzy network DEA approach can surpass a value of one.

5. The proposed Malmquist productivity index

The Malmquist productivity index, is a productivity measurement tool that evaluates the change in productivity over time. It takes into account the efficiency change and technological change components to analyze productivity growth or decline [21]. The integration of MPI and DEA involves using the efficiency scores obtained from DEA as inputs to calculate the Malmquist productivity index. This allows for a more comprehensive assessment of productivity change by considering both efficiency improvements and technological advancements. By combining the two approaches, the integration of MPI and DEA provides a more robust analysis of productivity change over time. It helps identify the sources of productivity growth or decline, determine the contribution of efficiency improvements versus technological progress, and compare the performance of different DMUs over multiple periods.

In this section, we introduce an improved framework for calculating the Malmquist productivity index specifically designed for two-stage network systems. Our approach takes into account the presence of fuzzy data, which enables a more comprehensive and accurate analysis. By addressing the limitations of traditional methodologies, we enhance the evaluation and comparison of productivity change in complex network systems. Moreover, by considering fuzzy data, our methodology enables a more nuanced and complete understanding of performance variations within network systems. This enhanced understanding allows organizations to make more informed decisions and formulate effective strategies to improve productivity. The calculation of the Malmquist productivity index for two-stage network systems under fuzzy data involves a multi-step process. Here are the key steps:

Step 1: Determine the specific time period for which you want to calculate the Malmquist productivity index. It could be a year, a quarter, or any other defined time frame.

Step 2: Collect the essential data regarding inputs and outputs for the decision-making units that are intended for analysis.

Step 3: Apply the proposed credibility-based fuzzy network data envelopment analysis approach to calculate the efficiency scores for the two-stage DMUs in the given time period.

Step 4: Calculate the components of the MPI for each DMU. The MPI consists of two components: efficiency change and technological change.

Efficiency Change: Measure the change in efficiency for each DMU between time periods t and t+1. This is done by comparing the efficiency scores obtained from DEA for the two time periods. Notably, in order to derive the overall efficiency score of a DMU_k for two time periods, the proposed credibility-based fuzzy network DEA approach is executed on a dataset that pertains to time periods t and t+1. Our method utilizes Models (24) and (25) to evaluate the overall efficiency score of the DMU_k for time periods t and t+1, respectively: (24) (25)

Technological Change: Assess the technological progress between time periods t and t+1. This is determined by comparing the production frontiers of the two time periods. Accordingly, in order to derive the technological change of a DMU_k for two time periods t and t+1, Models (26) and (27) are applied as follows: (26) (27)

In the same manner, the mentioned results are calculated for the first and second stages, and the corresponding models and relationships are available in the S1 Appendix.

Step 5: Combine the efficiency change and technological change components to compute the credibility-based fuzzy network Malmquist productivity index (CFNMPI) for each DMU. The CFNMPI is calculated as the geometric mean of the efficiency change and technological change.

(28)

(29)

(30)

Step 6: Analyze the CFNMPI values to understand the productivity change over time. It is crucial to note that the assessment of productivity change for the DMU under analysis depends on the value of the CFNMPI. This value can fall into three categories: greater than one, equal to one, or less than one:

CFNMPI_k(ξ)>1: If the CFNMPI exceeds one, it signifies a boost in productivity during the examined timeframe. This implies that the DMU has witnessed favorable enhancements in both efficiency and technology, ultimately leading to an upsurge in productivity when compared to the base period.
CFNMPI_k(ξ) = 1: When the CFNMPI value is one, it indicates that there has been no alteration in productivity during the examined timeframe. This implies that the efficiency and technology of the DMU have consistently remained unchanged, leading to the maintenance of the same level of productivity as in the reference period.
CFNMPI_k(ξ)<1: If the CFNMPI falls below one, it indicates a drop in productivity during the evaluated period. This suggests that the DMU has encountered unfavorable developments in efficiency and technology, leading to reduced productivity when compared to the reference period.

6. A real-life application: Mutual funds

In this section, the proposed credibility-based fuzzy network Malmquist productivity index will be implemented for 10 mutual funds from Iranian capital market. Mutual funds hold a crucial position in the capital market as significant financial institutions. They effectively manage the funds entrusted by investors, directing them towards specified investment plans. As a result, each individual who invests in these funds is entitled to a portion of the investment gains and is also exposed to potential risks. These distributions and vulnerabilities are commensurate with the respective interests that investors hold in the mutual funds. It should be noted that the functioning of mutual funds can be perceived as a dual-phase procedure. Initially, the management of mutual funds endeavors to allure investments from individuals, constituting the first stage (Operational Management). Subsequently, their attention shifts to the creation of the most advantageous portfolio, which forms the second stage (Portfolio Management). A visual representation of the empirical framework illustrating the activities of mutual funds is presented in Fig 2.

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Fig 2. The graphical presentation of mutual funds structure.

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Tables 3 and 4 display the data sets related to the 10 Iranian mutual funds (IMFs) for two consecutive time periods. Accordingly, the results of the proposed credibility-based fuzzy NDEA approach for , , , and are introduced in Tables 5 to 8, respectively. Finally, the results for efficiency change, technological change, and CFNMPI of IMFs at different confidence levels, including 0%, 20%, 40%, 60%, 80%, and 100%, are calculated in Tables 9 to 11, respectively:

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Table 3. Data set of 10 Iranian mutual funds: Inputs and intermediate measure.

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Table 4. Data set of 10 Iranian mutual funds: Additional inputs and output.

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Table 5. The results of CFNDEA approach:

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Table 6. The results of CFNDEA approach:

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Table 7. The results of CFNDEA approach:

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Table 8. The results of CFNDEA approach:

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Table 9. The results of efficiency change of IMFs based on the CFNDEA approach.

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Table 10. The results of technological change of IMFs based on the CFNDEA approach.

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Table 11. The results of credibility-based fuzzy network Malmquist productivity index.

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According to the results in in Table 11, it is clear that only two mutual funds, IMF-05 and IMF-10, have significantly improved their productivity in all stages, including overall and stages 1 and 2, during the specified period. As a result, by analyzing the strategies, practices, and processes implemented by IMF-05 and IMF-10, other mutual funds can identify areas where they can make improvements. This could include factors such as investment selection, risk management, portfolio diversification, and operational efficiency. Benchmarking allows mutual funds to identify best practices and set performance goals based on the success of top-performing funds. By learning from the achievements of IMF-05 and IMF-10, other funds can aim to enhance their productivity and ultimately deliver better results for their investors. The findings of this research will contribute to the existing literature on productivity assessment in two-stage network systems. Moreover, they will provide valuable insights for decision-makers in the mutual fund industry, enabling them to identify areas of improvement and implement strategies to enhance efficiency and productivity. Notably, the practical application of the uncertain network Malmquist productivity index results for policymakers in the context of mutual funds could involve:

Performance Assessment: Policymakers can utilize the Malmquist Productivity Index to evaluate the efficiency and productivity changes over time in mutual funds. This can help identify underperforming funds that may require intervention or support.
Resource Allocation: By analyzing the productivity changes using the index, policymakers can make informed decisions on resource allocation within the mutual fund industry. Funds showing productivity improvements could be encouraged or incentivized, while those lagging behind may require attention.
Regulatory Oversight: The Malmquist Productivity Index results can assist policymakers in setting regulatory standards and benchmarks for mutual funds. By comparing fund performance using this index, policymakers can implement regulations that promote efficiency and competitiveness in the industry.
Policy Formulation: Policymakers can use the insights from the Malmquist Productivity Index to develop policies aimed at enhancing overall productivity and innovation within the mutual fund sector. This could involve introducing measures to support technological advancements, streamline operations, or foster collaboration among funds for mutual benefit.
Investor Protection: Policymakers can leverage the index results to ensure investor protection by monitoring the productivity and efficiency of mutual funds. This can help in safeguarding investor interests and maintaining the stability and transparency of the market.
By incorporating the findings from the uncertain network Malmquist productivity index analysis into policy decisions, policymakers can work towards fostering a more efficient, competitive, and sustainable mutual fund industry.

7. Conclusions and future research directions

This research paper investigated the application of the Malmquist productivity index in analyzing the productivity change of two-stage network systems, specifically focusing on mutual funds. The study utilized the fuzzy network data envelopment analysis approach and incorporated the principles of credibility theory to account for data uncertainty. The findings of this research highlight the significance of considering the specific characteristics of mutual funds as two-stage decision-making units when assessing productivity change. The Malmquist productivity index proved to be a valuable tool in capturing and quantifying these changes over time. Moreover, by incorporating fuzzy set theory and credibility theory, the analysis accounted for data uncertainty and enhanced the accuracy of the results. The research contributes to the existing literature on productivity measurement by offering a novel approach for evaluating the performance of mutual funds within the context of two-stage network systems. The findings provide valuable insights for fund managers, investors, and policymakers in understanding the dynamics of mutual fund productivity and making informed decisions. Overall, this research contributes to the advancement of productivity measurement methods and offers a comprehensive and effective framework for analyzing the performance of two-stage network decision-making units under data ambiguity. Future research can build upon this study by expanding the application of the Malmquist productivity index to other sectors or industries with similar characteristics of two-stage network systems [69–71]. Furthermore, the Malmquist productivity index can be extended to address special data scenarios, including those involving negative data [72–74]. It is also suggested to use other commonly used approaches in the field of uncertain programming to deal with other types of uncertainty [75–77].

Supporting information

S1 Appendix.

https://doi.org/10.1371/journal.pone.0307277.s001

(DOCX)

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Figures

Abstract

1. Introduction

2. Literature review

3. The network DEA approach: Additive efficiency decomposition

4. The credibility-based fuzzy network DEA approach

5. The proposed Malmquist productivity index

6. A real-life application: Mutual funds

7. Conclusions and future research directions

Supporting information

S1 Appendix.

References