The Robust Malmquist Productivity Index: A Framework for Measuring Productivity Changes over Time Under Uncertainty

Abstract

The purpose of this study is to propose a novel approach for measuring productivity changes in decision-making units (DMUs) over time and evaluating the performance of each DMU under uncertainty in terms of progress, regression, and stagnation. To achieve this, the Malmquist productivity index (MPI) and the data envelopment analysis (DEA) models are extended, and a new productivity index capable of handling uncertain data are introduced through a robust optimization approach. Robust optimization is recognized as one of the most applicable and effective methods in uncertain programming. The implementation and calculation of the proposed index are demonstrated using data from 15 actively traded stocks in the petroleum products industry on the Tehran stock exchange over two consecutive years. The results reveal that a significant number of stocks exhibit an unfavorable trend, marked by a decline in productivity. The findings highlight the efficacy and effectiveness of the proposed robust Malmquist productivity index (RMPI) in measuring and identifying productivity trends for each stock under data uncertainty.

1. Introduction

Productivity measurement serves as a cornerstone for assessing the performance and efficiency of decision-making units (DMUs) across a wide range of industries [1,2,3]. It provides valuable insights into how effectively resources are being utilized and how operational processes can be optimized to achieve better outcomes. Among the various methodologies developed for this purpose, the Malmquist productivity index (MPI) has emerged as a widely adopted tool for evaluating productivity changes over time [4,5,6,7]. The combination of the MPI and data envelopment analysis (DEA) allows analysts to decompose productivity growth into two key components: efficiency change, which measures the ability of a DMU to enhance its performance relative to the production frontier, and technological change, which captures shifts in the frontier itself resulting from innovation or technological advancements [8,9,10].

Despite its widespread application, the traditional Malmquist productivity index framework is not without limitations. One of its most significant shortcomings is its inability to adequately handle environments characterized by data uncertainty. In real-world scenarios, uncertainty is an inherent challenge that can stem from volatile market conditions, incomplete or imprecise data, and measurement errors [11,12,13,14]. These factors can distort productivity assessments, leading to unreliable conclusions and suboptimal decision-making. Consequently, developing resilient analytical frameworks has become essential to address these limitations, ensuring reliable measurements and actionable insights despite data fluctuations, model instability, and other uncertainty-induced challenges in productivity evaluation.

It should be noted that robust optimization (RO) has emerged as a powerful paradigm for addressing uncertainty in optimization problems, where uncertainty manifests in both continuous (e.g., parameter fluctuations) and discrete (e.g., scenario variations) forms [15,16,17,18]. This approach provides computationally tractable solutions while guaranteeing feasibility under uncertainty. The field of robust optimization has evolved several prominent methodologies: scenario-based robust optimization (SBRO) [19] for discrete scenario sets; p-robust optimization (PRO) [20] for probabilistic constraints; min-max robust optimization (MMRO) [21] for worst-case immunization; and regret robust optimization (RRO) [22] for relative performance. Advanced variants include convex robust optimization (CRO) [23,24,25] for convex uncertainty sets, light robust optimization (LRO) [26] for balanced conservatism, and distributionally robust optimization (DRO) [27,28,29] for ambiguous distributions. Adaptive approaches like adaptive robust optimization (ARO) [30,31,32] and adaptive distributionally robust optimization (ADRO) [33] enable dynamic decision-making, while data-driven robust optimization (DDRO) [34,35,36] leverages empirical data to inform uncertainty sets.

To bridge this gap, this study proposes the robust Malmquist productivity index (RMPI), an innovative framework designed to measure productivity changes over time while explicitly accounting for uncertainty. The RMPI builds upon the foundational principles of the traditional MPI but integrates robust optimization techniques to enhance its applicability in uncertain environments. By embedding robust optimization into the MPI framework, the RMPI offers a more resilient and adaptive methodology for analyzing productivity dynamics. This approach not only mitigates the risks associated with data uncertainty but also provides decision-makers with a more accurate representation of productivity trends, enabling them to identify areas of progress, regression, or stagnation with greater confidence. Notably, robust optimization is a powerful mathematical approach that incorporates uncertainty directly into the modeling process, ensuring that the resulting productivity assessments remain stable and reliable even when input data are subject to variability or imprecision.

The RMPI is particularly well-suited for industries where uncertainty is a dominant factor, such as financial markets, where stock performance is influenced by unpredictable economic conditions, or manufacturing sectors, where supply chain disruptions and fluctuating demand can significantly impact productivity. By addressing the limitations of traditional productivity measurement tools, the RMPI represents a significant advancement in the field, offering a robust and practical solution for evaluating productivity in complex and uncertain environments. The practical applicability of the RMPI is demonstrated through an empirical analysis of 15 actively traded stocks in the chemical industry on the Tehran stock exchange (TSE) over two consecutive years. The findings reveal significant insights into the productivity trends of these stocks, highlighting the effectiveness of the RMPI in identifying progress, regression, and stagnation under data uncertainty. The results underscore the importance of adopting robust methodologies in productivity analysis, particularly in industries where uncertainty is a prevailing factor.

The key contributions and advantages of this research include the following: (1) a systematic review of the Malmquist productivity index under uncertainty, comparing key literature to highlight trends, methodological approaches, gaps, and theoretical-practical implications; (2) advancement of uncertain MPI literature through development of a comprehensive framework incorporating both envelopment (production possibility set) and multiplier (relative efficiency) forms; (3) integration of robust optimization into MPI under deep uncertainty, accommodating both pessimistic and optimistic perspectives; (4) formulation of a linear robust Malmquist productivity index variant that enhances computational efficiency and facilitates practical implementation; (5) effective tracking of decision-making units’ productivity changes under deep uncertainty conditions, significantly broadening practical applicability; and (6) validation of the framework’s real-world relevance through successful application to stock market analysis.

The remainder of this paper is organized as follows. Section 2 presents a comprehensive systematic review of existing research on the MPI under uncertainty. Through a comparative analysis of seminal studies, this section identifies critical methodological trends, theoretical developments, and current research gaps while examining both theoretical frameworks and practical implementations. Section 3 provides a comprehensive background of the study, delving into the foundational concepts and methodologies that underpin the research. Specifically, it explores data envelopment analysis, a widely used non-parametric method for evaluating the efficiency of decision-making units, the Malmquist productivity index, a tool for measuring productivity changes over time, and robust optimization, a mathematical approach designed to address uncertainty in optimization problems. These concepts collectively form the theoretical basis for the proposed methodology. In Section 4, a novel approach termed the robust Malmquist productivity index is introduced. This innovative methodology extends the traditional Malmquist productivity index by incorporating robust optimization techniques to account for uncertainty in the measurement of productivity changes over time. The proposed method is designed to provide more reliable and resilient productivity assessments, particularly in environments where data variability and uncertainty are prevalent. Detailed mathematical formulations, algorithmic steps, and theoretical insights are provided to elucidate the methodology. Section 5 demonstrates the practical application of the proposed robust Malmquist productivity index through a real case study conducted on the Tehran stock exchange. This empirical analysis evaluates the productivity changes of 15 listed companies from a financial perspective over a specified period, providing valuable insights into the effectiveness and applicability of the proposed methodology in real-world scenarios. Finally, Section 6 concludes the study by summarizing the key findings, contributions, and limitations of the research. It also outlines potential directions for future work, including further refinements to the methodology and applications to other domains where productivity measurement under uncertainty is critical.

2. Literature Review and Research Gaps

This section presents a comprehensive literature review on the Malmquist productivity index in uncertain environments, highlighting methodological approaches and key research gaps.

Table 1. Summary of the literature on the Malmquist productivity index under uncertain conditions.

Year	Research	DEA Model	Form		Uncertain Programming Approach						Application
			EF	MF	FO	IP	SO	GS	RO	UT
2006	Jahanshahloo et al. [37]	Charnes-Cooper-Rhodes	✓		✓	✓					Banking
2007	Hosseinzadeh Lotfi and Ghasemi [38]	Charnes-Cooper-Rhodes	✓			✓					Telecommunication
2007	Jahanshahloo et al. [39]	Charnes-Cooper-Rhodes	✓		✓						Insurance
2011	Emrouznejad et al. [40]	Overall Profit	✓		✓	✓					Numerical Example
2012	Hatami-Marbini et al. [41]	Charnes-Cooper-Rhodes Banker-Charnes-Cooper		✓	✓						Healthcare
2012	Khalili-Damghani and Hosseinzadeh Lotfi [42]	Charnes-Cooper-Rhodes	✓		✓						Traffic Monitoring and Control
2013	Payan and Shariff [43]	Charnes-Cooper-Rhodes	✓		✓						Social Security Organizations
2015	Oruc [44]	Charnes-Cooper-Rhodes		✓				✓			Numerical Example
2017	Aghayi and Maleki [45]	Directional Distance Function		✓		✓					Numerical Example
2018	Atici et al. [46]	Charnes-Cooper-Rhodes		✓	✓						Agriculture
2018	Khalili-Damghani and Haji-Sami [47]	Charnes-Cooper-Rhodes	✓			✓					Energy
2018	Peykani et al. [48]	Charnes-Cooper-Rhodes	✓		✓						Healthcare
2019	Aghayi et al. [49]	Directional Distance Function		✓		✓			✓		Banking
2019	Lee and Prabhu [50]	Directional Distance Function	✓						✓		Innovation
2020	Akbarian [51]	Overall Profit	✓			✓					Numerical Example
2021	Peykani and Seyed Esmaeili [52]	Charnes-Cooper-Rhodes	✓		✓						Numerical Example
2022	Hatami-Marbini et al. [53]	Charnes-Cooper-Rhodes		✓					✓		Oil Refinery
2022	Seyed Esmaeili et al. [54]	Network DEA	✓			✓					Insurance
2023	Ait Sidhoum [55]	Directional Distance Function	✓				✓				Agriculture
2023	Pourmahmoud and Bagheri [56]	Charnes-Cooper-Rhodes Banker-Charnes-Cooper	✓							✓	Healthcare
2023	Wu and Sheng [57]	Charnes-Cooper-Rhodes	✓							✓	Environment
2024	Amirteimoori et al. [58]	Banker-Charnes-Cooper	✓				✓				Banking
2024	Shakouri and Salahi [59]	Network DEA		✓			✓		✓		Petroleum
2024	Shojaie et al. [60]	Network DEA		✓	✓						Mutual Fund
2025	Qu et al. [61]	Network DEA		✓					✓		Energy
The Current Research		Charnes-Cooper-Rhodes	✓	✓					✓		Stock Market

systematically summarizes the critical features of selected studies, such as the applied DEA model, model form (envelopment form (EF) or multiplier form (MF)), uncertain programming (UP) techniques, and relevant application domains. Notably, the examined uncertain programming methodologies encompass fuzzy optimization (FO), interval programming (IP), stochastic optimization (SO), grey systems (GS), robust optimization (RO), and uncertainty theory (UT). By analyzing these studies, this review identifies limitations in existing frameworks and underscores the need for further research to address uncertainties in productivity measurement.

As evident in Table 1, the majority of studies have employed fuzzy optimization and interval programming approaches to address uncertainty, while only a limited number have adopted robust optimization techniques. However, recent years have witnessed a growing trend in utilizing alternative uncertain programming methodologies. It is also noteworthy that none of the reviewed studies have developed the Malmquist productivity index in both envelopment and multiplier forms, nor have they applied it in the stock market context. This study aims to bridge these gaps by incorporating the aforementioned aspects into the proposed framework.

3. Theoretical Background

In this section, the background of the study is presented, encompassing key concepts such as data envelopment analysis, the Malmquist productivity index, and robust optimization. These foundational methodologies provide the theoretical basis for the research, enabling a comprehensive understanding of productivity measurement and its application under conditions of uncertainty.

3.1. Data Envelopment Analysis

Data envelopment analysis was initially introduced by Charnes et al. [62], building upon the foundational work of Farrell [63]. DEA is a non-parametric methodology designed to assess the relative efficiency and performance of homogeneous DMUs [64,65,66]. The technique is widely used for benchmarking and ranking DMUs by evaluating their efficiency in converting inputs into outputs [67,68,69]. The original DEA model, developed by Charnes et al. [62], operates under the assumption of constant returns to scale (CRS) and is commonly referred to as the CCR model, named after its creators. This model serves as a cornerstone in the field of efficiency analysis, providing a framework for measuring productivity and identifying best practices among comparable entities. Later, Banker et al. [70] extended the CCR model by introducing the concept of variable returns to scale (VRS), which accounts for situations where DMUs operate at different scales of production. This development, known as the BCC model, enhanced the flexibility and applicability of data envelopment analysis, allowing for more nuanced efficiency evaluations in diverse real-world scenarios [71].

To grasp the framework of data envelopment analysis modeling, imagine a situation involving a set of comparable decision-making units ${\mathrm{DMU}}_j(j=1,\dots ,n)$, where each DMU employs a set of inputs $x_{ij}(i=1,\dots ,m)$ to produce a set of outputs $y_{rj}(r=1,\dots ,s)$. The DMU under evaluation is typically identified by a specific subscript ${\mathrm{DMU}}_0$. In this context, non-negative weights, represented as $v_i(i=1,\dots ,m)$ for inputs and $u_r(r=1,\dots ,s)$ for outputs, are assigned to quantify the relative significance of each input and output in the efficiency assessment process. Accordingly, the input-oriented CCR model is presented in both envelopment and multiplier forms, denoted as Models (1) and (2), respectively:

$$\mathrm{Min} \theta$$

(1)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}}\le \theta x_{i0},\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}}\ge y_{r0},\hspace{1em}\forall r$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

$$\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}}$$

(2)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}}=1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}}\le 0,\hspace{1em}\forall j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

The envelopment form in DEA constructs an efficient frontier by enveloping the data points, which represents the production possibility set (PPS). It identifies efficient DMUs located on the frontier and measures the distance of other units from this boundary. In contrast, the multiplicative form evaluates relative efficiency by maximizing the ratio of weighted outputs to inputs, offering a performance measure based on the optimal weights assigned to inputs and outputs.

3.2. Malmquist Productivity Index

Measuring the productivity changes in DMUs and identifying their performance trends over time is a highly significant research and practical topic. By calculating productivity changes, it is possible to determine whether a specific unit has shown progress, regression, or stagnation in the current period compared to the previous period, relative to other units [72,73,74]. It is important to note that one of the most widely used approaches for measuring productivity changes in DMUs over two consecutive time periods is the integration of the MPI and DEA models. This combined approach enables the calculation of efficiency changes and technological changes in various units over time. Färe et al. [75] utilized the MPI and DEA to measure productivity changes. They proposed this index by considering two time periods, $t$ and $t+1$, and calculating efficiency changes and technological changes between these periods.

It should be noted that in order to measure the Malmquist productivity index, it is necessary to solve four models based on the DEA approach. Therefore, by considering the envelopment form of the input-oriented CCR model of data envelopment analysis as the base model for calculating the Malmquist productivity index, to obtain efficiency changes, it is sufficient to solve the aforementioned model for two time periods, $t$ and $t+1$, in the form of Models (3) and (4), and then calculate the results:

$${\mathrm{\Upsilon }}_0^t\left(x_0^t,y_0^t\right)=\mathrm{Min} \theta$$

(3)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^t}\le \theta x_{i0}^t,\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^t}\ge y_{r0}^t,\hspace{1em}\forall r$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

$${\mathrm{\Upsilon }}_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Min} \theta$$

(4)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^{t+1}}\le \theta x_{i0}^{t+1},\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^{t+1}}\ge y_{r0}^{t+1},\hspace{1em}\forall r$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

Additionally, to calculate technological changes, Models (5) and (6) are solved. It should be noted that Model (5) represents the distance of the DMU under evaluation at time $t+1$ relative to the efficiency frontier at time $t$, while Model (6) represents the distance of the DMU under evaluation at time $t$ relative to the efficiency frontier at time $t+1$.

$${\mathrm{\Upsilon }}_0^t\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Min} \theta$$

(5)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^t}\le \theta x_{i0}^{t+1},\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^t}\ge y_{r0}^{t+1},\hspace{1em}\forall r$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

$${\mathrm{\Upsilon }}_0^{t+1}\left(x_0^t,y_0^t\right)=\mathrm{Min} \theta$$

(6)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^{t+1}}\le \theta x_{i0}^t,\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^{t+1}}\ge y_{r0}^t,\hspace{1em}\forall r$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

Upon solving the aforementioned envelopment models, the results derived from Models (3) through (6) are incorporated into Equation (7) to measure the Malmquist productivity index:

$${\mathrm{MPI}}_0^E=\sqrt{{\displaystyle \frac{{\mathrm{\Upsilon }}_0^t\left(x_0^{t+1},y_0^{t+1}\right){\mathrm{\Upsilon }}_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)}{{\mathrm{\Upsilon }}_0^t\left(x_0^t,y_0^t\right){\mathrm{\Upsilon }}_0^{t+1}\left(x_0^t,y_0^t\right)}}}$$

(7)

In a similar manner, by adopting the multiplier form of the input-oriented CCR model in data envelopment analysis as the base model for calculating the Malmquist productivity index, efficiency changes can be derived by solving Models (8) and (9). These models allow for a comprehensive evaluation of productivity shifts, offering insights into how technological advancements and efficiency gains contribute to overall performance. By comparing the efficiency of DMUs across different periods, the MPI can identify whether changes are due to improvements in efficiency or shifts in technology.

$${\Lambda }_0^t\left(x_0^t,y_0^t\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^t}$$

(8)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^t}=1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^t}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^t}\le 0,\hspace{1em}\forall j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

$${\Lambda }_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^{t+1}}$$

(9)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^{t+1}}=1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^{t+1}}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^{t+1}}\le 0,\hspace{1em}\forall j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

Furthermore, to assess technological changes, Models (10) and (11) are employed and solved. These models are specifically designed to isolate and quantify shifts in technology over time, providing a clearer understanding of how advancements in technology impact productivity.

$${\Lambda }_0^t\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^{t+1}}$$

(10)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^{t+1}}=1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^t}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^t}\le 0,\hspace{1em}\forall j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

$${\Lambda }_0^{t+1}\left(x_0^t,y_0^t\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^t}$$

(11)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^t}=1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^{t+1}}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^{t+1}}\le 0,\hspace{1em}\forall j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

After solving the previously discussed multiplier models, the outcomes obtained from Models (8) to (11) are integrated into Equation (12) to compute the Malmquist productivity index. This process combines the efficiency and technological change data from the earlier models, allowing for a comprehensive measurement of productivity shifts over time.

$${\mathrm{MPI}}_0^M=\sqrt{{\displaystyle \frac{{\Lambda }_0^t\left(x_0^{t+1},y_0^{t+1}\right){\Lambda }_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)}{{\Lambda }_0^t\left(x_0^t,y_0^t\right){\Lambda }_0^{t+1}\left(x_0^t,y_0^t\right)}}}$$

(12)

It is important to note that based on the value of the Malmquist productivity index, which can be greater than, equal to, or less than one, the productivity trend of the unit under evaluation is interpreted as follows: If the MPI is greater than one ($\mathrm{MPI}>1$), it indicates an increase and improvement in the unit’s productivity. If the index is equal to one ($\mathrm{MPI}=1$), it signifies stagnation and no change in productivity. Conversely, if the index is less than one ($\mathrm{MPI}<1$), it reflects a decline and regression in the unit’s productivity.

3.3. Robust Optimization

Robust optimization is a widely recognized and practical methodology for addressing uncertainty in diverse real-world applications and optimization problems [76,77,78,79]. Owing to its effectiveness, this approach has been successfully implemented across various domains, including data envelopment analysis, where it has demonstrated particular efficacy in managing data uncertainty while ensuring solution reliability [80,81,82,83,84].

The foundational work in this field was introduced by Soyster [27], who developed a robust optimization approach based on a “Box” uncertainty set to handle continuous uncertainty. While Soyster’s method is formulated as a linear programming problem, it tends to be overly conservative by adopting the worst-case values for uncertain parameters. To address this limitation, Ben-Tal and Nemirovski [28] proposed an advanced robust formulation using a “Box and Ellipsoidal” uncertainty set, allowing decision-makers to adjust the conservatism level through a parameter. However, their approach results in a nonlinear programming problem, which can pose challenges in real-world applications. Building on these developments, Bertsimas and Sim [29] introduced an innovative robust optimization approach based on a “Box and Polyhedral” uncertainty set. This method retains the linear programming structure while enabling decision-makers to control the conservatism level via a parameter, offering a more flexible and computationally efficient solution [85,86,87,88,89]. To illustrate the robust approach proposed by Bertsimas and Sim [29], consider the following linear programming problem:

$$\mathrm{Max} {\displaystyle \sum _b{\mathcal{l}}_b{†}_b}$$

(13)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _b{\tilde{\Delta }}_{ab}}{†}_b\le {\tau }_a,\hspace{1em}\forall a$$

$${†}_b\ge 0$$

Let ${\tilde{\Delta }}_{ab}$ represent the uncertain parameter in constraint $a$, and let ${\mho }_a$ denote the set of coefficients in constraint $a$ that are subject to uncertainty. Each uncertain coefficient ${\tilde{\Delta }}_{ab},b\in {\mho }_a$ is modeled as a symmetric and bounded random variable, taking values within the interval $[{\Delta }_{ab}-{\widehat{\Delta }}_{ab},{\Delta }_{ab}+{\widehat{\Delta }}_{ab}]$, where ${\Delta }_{ab}$ represents the nominal value at the center of the interval, and ${\widehat{\Delta }}_{ab}$ denotes the maximum perturbation of the uncertain parameter ${\tilde{\Delta }}_{ab},b\in {\mho }_a$. Additionally, ${\widehat{\Delta }}_{ab}$ can be interpreted as ${\widehat{\Delta }}_{ab}=\varpi {\Delta }_{ab}$ that controls the percentage deviation $\varpi$ from the nominal value. Using the robust optimization framework proposed by Bertsimas and Sim [29], the robust counterpart of Model (13) is formulated as Equation (14):

$$\left\{\begin{array}{l}{\displaystyle \sum _b{\Delta }_{ab}{†}_b}+{\xi }_a{\Gamma }_a+{\displaystyle \sum _{b\in {\mho }_a}{\eta }_{ab}}\le {\tau }_a,\hspace{1em}\forall a\\ {\xi }_a+{\eta }_{ab}\ge \varpi {\Delta }_{ab}{†}_b,\hspace{1em}\forall a,b\in {\mho }_a\\ \xi ,\eta \ge 0\end{array}\right.$$

(14)

It is important to highlight that in the robust optimization framework of Bertsimas and Sim [29], the parameter $\Gamma$ serves as a tunable budget of robustness that governs the model’s conservatism. Setting $\Gamma =0$ reduces the model to a deterministic optimization problem, where all uncertain parameters are fixed at their nominal values, effectively eliminating robustness considerations. Conversely, when $\Gamma$ attains its maximum value, equal to the number of uncertain parameters in a given constraint, the solution becomes fully robust, yielding the most conservative worst-case scenario. In this extreme case, the model’s solution aligns precisely with Soyster’s [27] classical robust optimization approach, which provides complete immunization against uncertainty but often at the cost of excessive conservatism. The key advantage of the Bertsimas and Sim’s [29] method lies in its flexibility, allowing decision-makers to strategically select $\Gamma$ between these two bounds (0 and $\left|{\mho }_a\right|$) to achieve an optimal trade-off between robustness and solution performance based on their specific risk tolerance and operational requirements. This adjustable robustness framework enables more practical and computationally tractable solutions compared to traditional overly conservative approaches.

To ensure that the probability of violating constraint $a$ does not exceed $Q$, it is sufficient to select ${\Gamma }_a$ such that it is greater than or equal to the value specified in Equation (15). Here, $\Phi$ represents the cumulative distribution function (CDF) of the standard normal distribution, while $d$ denotes the number of uncertain parameters associated with constraint $a$ [85]. This condition ensures robustness by accounting for the uncertainty in the parameters, thereby maintaining the constraint’s feasibility with a confidence level of at least $1-Q$.

$$1-Q_a=1-\Phi ({\displaystyle \frac{{\Gamma }_a-1}{\sqrt{d}}})\iff {\Gamma }_a=1+{\Phi }_{(1-Q_a)}^{-1}\sqrt{d}$$

(15)

A comprehensive sensitivity analysis examining different uncertainty conditions proves invaluable for evaluating model performance across scenarios. By testing varying budget of robustness (0%, 25%, 75%, and 100%) alongside perturbation levels (1% and 10%), we can systematically assess the model’s behavior. The 0% budget of robustness represents a deterministic approach where all parameters assume nominal values, while 100% corresponds to maximum conservatism that accounts for all potential parameter variations. Intermediate values (25%, 75%) enable balanced solutions between optimality and protection against uncertainty. The perturbation levels (1% for minor fluctuations, 10% for significant variations) further test the model’s resilience to different magnitudes of uncertainty. This dual-axis analysis provides critical insights into the following: (1) the model’s stability across uncertainty regimes; (2) optimal trade-offs between solution quality and robustness; and (3) practical parameter selection for real-world applications where both computational efficiency and reliable performance are essential.

4. The Proposed Robust Malmquist Productivity Index

As discussed in the previous section, the traditional Malmquist productivity index is incapable of being applied in the presence of data uncertainty. Consequently, the conventional MPI is unsuitable for real-world problems where data are affected by uncertainty. To address this limitation, the objective of this section is to introduce a novel Malmquist productivity index designed for implementation in uncertain environments.

It should be emphasized that in classical DEA, the envelopment and multiplier forms are dual to each other. Under optimality and deterministic conditions, both forms yield identical efficiency scores. However, in the presence of uncertainty, particularly in their robust optimization counterparts, this equivalence no longer holds. Specifically, as data uncertainty increases, the robust optimization approach dictates that the solution of the linear programming model deteriorates; that is, the objective value decreases in maximization problems and increases in minimization problems.

Consequently, under uncertainty, the efficiency scores computed by the robust input-oriented CCR model diverge between the envelopment and multiplier forms. More precisely, the envelopment form produces optimistic results (with a minimization objective where increased uncertainty leads to higher efficiency scores), whereas the multiplier form yields pessimistic results (with a maximization objective where increased uncertainty reduces efficiency).

Given this dichotomy, the present study is the first to systematically examine uncertainty in both envelopment and multiplier forms through optimistic and pessimistic frameworks, measuring productivity changes under varying uncertainty levels. The proposed approach ensures that the evaluation status of decision-making units is more reliable, comprehensive, and statistically robust.

Finally, owing to the linear nature of the robust counterpart and the flexibility to fine-tune the conservatism level in the robust optimization approach developed by Bertsimas and Sim [29], this methodology will be employed in this study to address uncertainty within the proposed Malmquist productivity index framework. In the following subsections, the robust Malmquist productivity index is developed under both the envelopment and multiplier forms, respectively.

4.1. The Proposed Envelopment RMPI

This subsection presents an envelopment-based version of the robust Malmquist productivity index, developed using the envelopment form of the data envelopment analysis approach. Accordingly, the robust frameworks for Models (3) through (6) are presented in detail in the following section, respectively:

$${\Theta }_0^t\left(x_0^t,y_0^t\right)=\mathrm{Min} \theta$$

(16)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^t}+{\xi }_i^x{\Gamma }_i^x+{\displaystyle \sum _{j=1}^n{\beta }_{ij}}+{\beta }_{i0}\le \theta x_{i0}^t,\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^t}-{\xi }_r^y{\Gamma }_r^y-{\displaystyle \sum _{j=1}^n{\alpha }_{rj}}-{\alpha }_{r0}\ge y_{r0}^t,\hspace{1em}\forall r$$

$${\xi }_i^x+{\beta }_{i0}\ge \varpi x_{i0}^t{v}_i,\hspace{1em}\forall i\in {\Omega }_{\beta }$$

$${\xi }_i^x+{\beta }_{ij}\ge \varpi x_{ij}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_r^y+{\alpha }_{r0}\ge \varpi y_{r0}^t{u}_r,\hspace{1em}\forall r\in {\Omega }_{\alpha }$$

$${\xi }_r^y+{\alpha }_{rj}\ge \varpi y_{rj}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_i^x,{\xi }_r^y\ge 0,\hspace{1em}\forall i,r$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

$${\Theta }_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Min} \theta$$

(17)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^{t+1}}+{\xi }_i^x{\Gamma }_i^x+{\displaystyle \sum _{j=1}^n{\beta }_{ij}}+{\beta }_{i0}\le \theta x_{i0}^{t+1},\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^{t+1}}-{\xi }_r^y{\Gamma }_r^y-{\displaystyle \sum _{j=1}^n{\alpha }_{rj}}-{\alpha }_{r0}\ge y_{r0}^{t+1},\hspace{1em}\forall r$$

$${\xi }_i^x+{\beta }_{i0}\ge \varpi x_{i0}^{t+1}{v}_i,\hspace{1em}\forall i\in {\Omega }_{\beta }$$

$${\xi }_i^x+{\beta }_{ij}\ge \varpi x_{ij}^{t+1}{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_r^y+{\alpha }_{r0}\ge \varpi y_{r0}^{t+1}{u}_r,\hspace{1em}\forall r\in {\Omega }_{\alpha }$$

$${\xi }_r^y+{\alpha }_{rj}\ge \varpi y_{rj}^{t+1}{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_i^x,{\xi }_r^y\ge 0,\hspace{1em}\forall i,r$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

$${\Theta }_0^t\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Min} \theta$$

(18)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^t}+{\xi }_i^x{\Gamma }_i^x+{\displaystyle \sum _{j=1}^n{\beta }_{ij}}+{\beta }_{i0}\le \theta x_{i0}^{t+1},\hspace{1em}\forall i.$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^t}-{\xi }_r^y{\Gamma }_r^y-{\displaystyle \sum _{j=1}^n{\alpha }_{rj}}-{\alpha }_{r0}\ge y_{r0}^{t+1},\hspace{1em}\forall r$$

$${\xi }_i^x+{\beta }_{i0}\ge \varpi x_{i0}^{t+1}{v}_i,\hspace{1em}\forall i\in {\Omega }_{\beta }$$

$${\xi }_i^x+{\beta }_{ij}\ge \varpi x_{ij}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_r^y+{\alpha }_{r0}\ge \varpi y_{r0}^{t+1}{u}_r,\hspace{1em}\forall r\in {\Omega }_{\alpha }$$

$${\xi }_r^y+{\alpha }_{rj}\ge \varpi y_{rj}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_i^x,{\xi }_r^y\ge 0,\hspace{1em}\forall i,r$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

$${\Theta }_0^{t+1}\left(x_0^t,y_0^t\right)=\mathrm{Min} \theta$$

(19)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{j=1}^n{\lambda }_j{x}_{ij}^t}+{\xi }_i^x{\Gamma }_i^x+{\displaystyle \sum _{j=1}^n{\beta }_{ij}}+{\beta }_{i0}\le \theta x_{i0}^{t+1},\hspace{1em}\forall i$$

$${\displaystyle \sum _{j=1}^n{\lambda }_j{y}_{rj}^t}-{\xi }_r^y{\Gamma }_r^y-{\displaystyle \sum _{j=1}^n{\alpha }_{rj}}-{\alpha }_{r0}\ge y_{r0}^{t+1},\hspace{1em}\forall r$$

$${\xi }_i^x+{\beta }_{i0}\ge \varpi x_{i0}^{t+1}{v}_i,\hspace{1em}\forall i\in {\Omega }_{\beta }$$

$${\xi }_i^x+{\beta }_{ij}\ge \varpi x_{ij}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_r^y+{\alpha }_{r0}\ge \varpi y_{r0}^{t+1}{u}_r,\hspace{1em}\forall r\in {\Omega }_{\alpha }$$

$${\xi }_r^y+{\alpha }_{rj}\ge \varpi y_{rj}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_i^x,{\xi }_r^y\ge 0,\hspace{1em}\forall i,r$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$${\lambda }_j\ge 0,\hspace{1em}\forall j$$

Upon solving the robust envelopment DEA models discussed earlier, the results obtained from Models (16) to (19) under a specific robustness budget and perturbation level are integrated into Equation (20) to compute the robust Malmquist productivity index in its envelopment form:

$${\mathrm{RMPI}}_0^E\left(\Gamma ,\varpi \right)=\sqrt{{\displaystyle \frac{{\Theta }_0^t\left(x_0^{t+1},y_0^{t+1}\right){\Theta }_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)}{{\Theta }_0^t\left(x_0^t,y_0^t\right){\Theta }_0^{t+1}\left(x_0^t,y_0^t\right)}}}$$

(20)

It should be noted that in robust envelopment models, as uncertainty increases, efficiency values are expected to rise. This suggests that the structure of the proposed approach is, in essence, optimistic.

4.2. The Proposed Multiplier RMPI

This subsection introduces a multiplier-based version of the robust Malmquist productivity index, developed using the multiplier form of the data envelopment analysis approach. Accordingly, the robust frameworks for Models (8) through (11) are presented in detail in the following section, respectively:

$${\Psi }_0^t\left(x_0^t,y_0^t\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^t}-{\xi }_0^y{\Gamma }_0^y-{\displaystyle \sum _{r=1}^s{\alpha }_{r0}}$$

(21)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^t}+{\xi }_0^x{\Gamma }_0^x+{\displaystyle \sum _{i=1}^m{\beta }_{i0}}\le 1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^t}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^t}+{\xi }_j{\Gamma }_j+{\displaystyle \sum _{r=1}^s{\alpha }_{rj}}+{\displaystyle \sum _{i=1}^m{\beta }_{ij}}\le 0,\hspace{1em}\forall j$$

$${\xi }_0^y+{\alpha }_{r0}\ge \varpi y_{r0}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_0^x+{\beta }_{i0}\ge \varpi x_{i0}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_j+{\alpha }_{rj}\ge \varpi y_{rj}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_j+{\beta }_{ij}\ge \varpi x_{ij}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_0^y,{\xi }_0^x,{\xi }_j\ge 0,\hspace{1em}\forall j$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

$${\Psi }_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^{t+1}}-{\xi }_0^y{\Gamma }_0^y-{\displaystyle \sum _{r=1}^s{\alpha }_{r0}}$$

(22)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^{t+1}}+{\xi }_0^x{\Gamma }_0^x+{\displaystyle \sum _{i=1}^m{\beta }_{i0}}\le 1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^{t+1}}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^{t+1}}+{\xi }_j{\Gamma }_j+{\displaystyle \sum _{r=1}^s{\alpha }_{rj}}+{\displaystyle \sum _{i=1}^m{\beta }_{ij}}\le 0,\hspace{1em}\forall j$$

$${\xi }_0^y+{\alpha }_{r0}\ge \varpi {\widehat{y}}_{r0}^{t+1}{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_0^x+{\beta }_{i0}\ge \varpi {\widehat{x}}_{i0}^{t+1}{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_j+{\alpha }_{rj}\ge \varpi {\widehat{y}}_{rj}^{t+1}{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_j+{\beta }_{ij}\ge \varpi {\widehat{x}}_{ij}^{t+1}{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_0^y,{\xi }_0^x,{\xi }_j\ge 0,\hspace{1em}\forall j$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

$${\Psi }_0^t\left(x_0^{t+1},y_0^{t+1}\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^{t+1}}-{\xi }_0^y{\Gamma }_0^y-{\displaystyle \sum _{r=1}^s{\alpha }_{r0}}$$

(23)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^{t+1}}+{\xi }_0^x{\Gamma }_0^x+{\displaystyle \sum _{i=1}^m{\beta }_{i0}}\le 1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^t}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^t}+{\xi }_j{\Gamma }_j+{\displaystyle \sum _{r=1}^s{\alpha }_{rj}}+{\displaystyle \sum _{i=1}^m{\beta }_{ij}}\le 0,\hspace{1em}\forall j$$

$${\xi }_0^y+{\alpha }_{r0}\ge \varpi {\widehat{y}}_{r0}^{t+1}{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_0^x+{\beta }_{i0}\ge \varpi {\widehat{x}}_{i0}^{t+1}{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_j+{\alpha }_{rj}\ge \varpi {\widehat{y}}_{rj}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_j+{\beta }_{ij}\ge \varpi {\widehat{x}}_{ij}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_0^y,{\xi }_0^x,{\xi }_j\ge 0,\hspace{1em}\forall j$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

$${\Psi }_0^{t+1}\left(x_0^t,y_0^t\right)=\mathrm{Max} {\displaystyle \sum _{r=1}^s{u}_r{y}_{r0}^t}-{\xi }_0^y{\Gamma }_0^y-{\displaystyle \sum _{r=1}^s{\alpha }_{r0}}$$

(24)

$$\mathrm{S}.\mathrm{t}. {\displaystyle \sum _{i=1}^m{v}_i{x}_{i0}^t}+{\xi }_0^x{\Gamma }_0^x+{\displaystyle \sum _{i=1}^m{\beta }_{i0}}\le 1$$

$${\displaystyle \sum _{r=1}^s{u}_r{y}_{rj}^{t+1}}-{\displaystyle \sum _{i=1}^m{v}_i{x}_{ij}^{t+1}}+{\xi }_j{\Gamma }_j+{\displaystyle \sum _{r=1}^s{\alpha }_{rj}}+{\displaystyle \sum _{i=1}^m{\beta }_{ij}}\le 0,\hspace{1em}\forall j$$

$${\xi }_0^y+{\alpha }_{r0}\ge \varpi {\widehat{y}}_{r0}^t{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_0^x+{\beta }_{i0}\ge \varpi {\widehat{x}}_{i0}^t{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_j+{\alpha }_{rj}\ge \varpi {\widehat{y}}_{rj}^{t+1}{u}_r,\hspace{1em}\forall j,r\in {\Omega }_{\alpha }$$

$${\xi }_j+{\beta }_{ij}\ge \varpi {\widehat{x}}_{ij}^{t+1}{v}_i,\hspace{1em}\forall j,i\in {\Omega }_{\beta }$$

$${\xi }_0^y,{\xi }_0^x,{\xi }_j\ge 0,\hspace{1em}\forall j$$

$${\alpha }_{r0},{\alpha }_{rj}\ge 0,\hspace{1em}\forall r,j$$

$${\beta }_{i0},{\beta }_{ij}\ge 0,\hspace{1em}\forall i,j$$

$$u_r,v_i\ge 0,\hspace{1em}\forall r,i$$

After solving the previously discussed robust multiplier DEA models, the results obtained from Models (21) to (24) under a specific robustness budget and perturbation level are integrated into Equation (25) to compute the robust Malmquist productivity index in its multiplier form:

$${\mathrm{RMPI}}_0^M\left(\Gamma ,\varpi \right)=\sqrt{{\displaystyle \frac{{\Psi }_0^t\left(x_0^{t+1},y_0^{t+1}\right){\Psi }_0^{t+1}\left(x_0^{t+1},y_0^{t+1}\right)}{{\Psi }_0^t\left(x_0^t,y_0^t\right){\Psi }_0^{t+1}\left(x_0^t,y_0^t\right)}}}$$

(25)

It is important to note that in robust multiplier models, as uncertainty increases, efficiency values are expected to decrease. This indicates that the structure of the proposed approach is, in essence, pessimistic.

5. Case Study: Tehran Stock Exchange

In this section, the proposed two versions of the envelopment and multiplier forms of the robust Malmquist productivity index are implemented in a real case study involving 15 companies from the petroleum products industry of the Tehran stock exchange. Accordingly, the study incorporates three inputs, such as the following: I (1): Current Ratio; I (2): Asset Turnover; and I (3): Solvency Ratio-II, as well as two outputs, including: O (1): Return on Assets and O (2): Return on Equity. These metrics will be used to evaluate the performance and productivity of the companies within the framework of the proposed RMPI. The real dataset extracted for two consecutive years, 2015 and 2016, is presented in

Table 2. The Dataset for the First Year.

Stocks	Inputs			Outputs
	I (1)	I (2)	I (3)	O (1)	O (2)
Stock 01	1.793	6.053	0.664	13.838	23.027
Stock 02	0.536	3.332	1.680	8.679	23.263
Stock 03	0.411	3.752	6.501	1.797	13.479
Stock 04	1.260	6.209	1.196	10.212	22.428
Stock 05	1.311	2.402	1.478	38.539	95.513
Stock 06	1.174	6.305	4.012	5.127	25.694
Stock 07	1.210	1.757	4.328	5.923	31.555
Stock 08	0.748	3.950	2.088	7.456	23.023
Stock 09	1.691	0.248	0.726	11.446	54.885
Stock 10	3.435	0.104	0.203	6.664	8.015
Stock 11	1.775	1.511	0.977	29.772	58.854
Stock 12	1.085	0.742	2.544	19.071	67.587
Stock 13	1.922	1.337	1.680	21.337	57.188
Stock 14	1.204	2.011	1.664	26.676	71.055
Stock 15	1.253	1.423	2.049	21.847	66.609

and

Table 3. The Dataset for the Second Year.

Stocks	Inputs			Outputs
	I (1)	I (2)	I (3)	O (1)	O (2)
Stock 01	1.269	3.775	0.944	12.904	25.084
Stock 02	1.038	1.910	2.234	1.132	3.662
Stock 03	0.460	2.230	4.623	0.467	2.624
Stock 04	1.384	3.634	0.852	7.900	14.635
Stock 05	0.709	1.649	3.802	4.898	23.522
Stock 06	1.317	4.897	2.986	4.694	18.706
Stock 07	1.134	2.237	2.683	8.641	28.172
Stock 08	0.342	2.253	15.061	0.199	3.193
Stock 09	2.424	0.277	0.477	16.122	15.373
Stock 10	3.807	0.285	0.152	25.611	29.516
Stock 11	1.322	1.238	1.852	16.712	47.660
Stock 12	1.120	0.613	2.867	13.575	52.498
Stock 13	1.994	1.195	1.619	21.173	55.455
Stock 14	0.915	1.394	4.015	8.102	40.634
Stock 15	1.130	1.073	2.883	11.561	44.896

, respectively. These tables provide the necessary data for the analysis and implementation of the proposed RMPI, enabling a comparative evaluation of the performance and productivity of the 15 companies from the TSE over the specified time period.

5.1. The Envelopment Approach

In this subsection, the results of Models (16) to (19) under different budgets of robustness and perturbation levels are presented in

Table 4. The results of Model (16) ${\Theta }_0^t(x_0^t,y_0^t)$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.68082	0.70861	1.01703	0.70861	1.01703	0.70861	1.01703
Stock 02	0.59572	0.62003	0.88990	0.62003	0.88990	0.62003	0.88990
Stock 03	0.45015	0.46852	0.67244	0.46852	0.67244	0.46852	0.67244
Stock 04	0.31786	0.33083	0.47482	0.33083	0.47482	0.33083	0.47482
Stock 05	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 06	0.30040	0.31266	0.44875	0.31266	0.44875	0.31266	0.44875
Stock 07	0.37600	0.39134	0.56167	0.39134	0.56167	0.39134	0.56167
Stock 08	0.42247	0.43972	0.63110	0.43972	0.63110	0.43972	0.63110
Stock 09	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 10	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 11	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 12	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 13	0.73689	0.76573	1.08159	0.76697	1.10079	0.76697	1.10079
Stock 14	0.82695	0.86070	1.23532	0.86070	1.23532	0.86070	1.23532
Stock 15	0.82882	0.86264	1.23794	0.86264	1.23811	0.86264	1.23811

,

Table 5. The results of Model (17) ${\Theta }_0^{t+1}(x_0^{t+1},y_0^{t+1})$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.98961	1.03000	1.46420	1.03000	1.47830	1.03000	1.47830
Stock 02	0.08767	0.09125	0.13096	0.09125	0.13096	0.09125	0.13096
Stock 03	0.12170	0.12666	0.18179	0.12666	0.18179	0.12666	0.18179
Stock 04	0.59281	0.61701	0.86583	0.61701	0.88556	0.61701	0.88556
Stock 05	0.70779	0.73667	1.05731	0.73667	1.05731	0.73667	1.05731
Stock 06	0.32177	0.33490	0.48067	0.33490	0.48067	0.33490	0.48067
Stock 07	0.61319	0.63822	0.91600	0.63822	0.91600	0.63822	0.91600
Stock 08	0.19918	0.20731	0.29754	0.20731	0.29754	0.20731	0.29754
Stock 09	0.93386	0.97197	1.38868	0.97197	1.39503	0.97197	1.39503
Stock 10	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 11	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 12	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 13	1.00000	1.04081	1.49383	1.04081	1.49383	1.04081	1.49383
Stock 14	0.94742	0.98609	1.41529	0.98609	1.41529	0.98609	1.41529
Stock 15	0.84906	0.88372	1.26835	0.88372	1.26835	0.88372	1.26835

,

Table 6. The results of Model (18) ${\Theta }_0^t(x_0^{t+1},y_0^{t+1})$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.48408	0.50384	0.72314.	0.50384	0.72314	0.50384	0.72314
Stock 02	0.04842	0.05040	0.07234	0.05040	0.07234	0.05040	0.07234
Stock 03	0.07830	0.08149	0.11696	0.08149	0.11696	0.08149	0.11696
Stock 04	0.31365	0.32645	0.46853	0.32645	0.46853	0.32645	0.46853
Stock 05	0.45537	0.47396	0.68025	0.47396	0.68025	0.47396	0.68025
Stock 06	0.19496	0.20291	0.29123	0.20291	0.29123	0.20291	0.29123
Stock 07	0.34099	0.35491	0.50938	0.35491	0.50938	0.35491	0.50938
Stock 08	0.12815	0.13338	0.19143	0.13338	0.19143	0.13338	0.19143
Stock 09	1.56479	1.62865	2.32924	1.62865	2.33752	1.62865	2.33752
Stock 10	5.13268	5.34215	7.66733	5.34215	7.66733	5.34215	7.66733
Stock 11	0.66623	0.69285	0.98676	0.69342	0.99523	0.69342	0.99523
Stock 12	0.85678	0.89175	1.27988	0.89175	1.27988	0.89175	1.27988
Stock 13	0.77352	0.80509	1.13538	0.80509	1.15551	0.80509	1.15551
Stock 14	0.63427	0.66016	0.94749	0.66016	0.94749	0.66016	0.94749
Stock 15	0.61374	0.63879	0.91682	0.63879	0.91682	0.63879	0.91682

and

Table 7. The results of Model (19) ${\Theta }_0^{t+1}(x_0^t,y_0^t)$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.91959	0.95712	1.34665	0.95712	1.37371	0.95712	1.37371
Stock 02	1.28088	1.33315	1.91341	1.33315	1.91341	1.33315	1.91341
Stock 03	0.69967	0.72822	1.04518	0.72822	1.04518	0.72822	1.04518
Stock 04	0.73083	0.76065	1.09143	0.76065	1.09173	0.76065	1.09173
Stock 05	2.51212	2.61465	3.75210	2.61465	3.75268	2.61465	3.75268
Stock 06	0.46692	0.48597	0.69749	0.48597	0.69749	0.48597	0.69749
Stock 07	0.55636	0.57907	0.83111	0.57907	0.83111	0.57907	0.83111
Stock 08	0.79639	0.82889	1.18966	0.82889	1.18966	0.82889	1.18966
Stock 09	2.36949	2.46620	3.53961	2.46620	3.53961	2.46620	3.53961
Stock 10	0.74415	0.77452	1.11162	0.77452	1.11162	0.77452	1.11162
Stock 11	1.80349	1.87709	2.63266	1.87709	2.69410	1.87709	2.69410
Stock 12	1.43115	1.48956	2.13790	1.48956	2.13790	1.48956	2.13790
Stock 13	1.03744	1.07978	1.54976	1.07978	1.54976	1.07978	1.54976
Stock 14	1.76168	1.83358	2.63162	1.83358	2.63165	1.83358	2.63165
Stock 15	1.39076	1.44752	2.07755	1.44752	2.07755	1.44752	2.07755

, respectively:

Then, by using Equation (20), the robust Malmquist productivity index under the envelopment form is calculated for specific budgets of robustness, including 0%, 25%, 75%, and 100%, as well as two perturbation levels, including 1% and 10%. The results are presented in

Table 8. The results of Proposed Envelopment RMPI.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.87474	0.87474	0.87926	0.87474	0.87474	0.87474	0.87474
Stock 02	0.07459	0.07459	0.07459	0.07459	0.07459	0.07459	0.07459
Stock 03	0.17394	0.17394	0.17394	0.17394	0.17394	0.17394	0.17394
Stock 04	0.89466	0.89466	0.88475	0.89466	0.89466	0.89466	0.89466
Stock 05	0.35819	0.35819	0.35822	0.35819	0.35819	0.35819	0.35819
Stock 06	0.66876	0.66876	0.66876	0.66876	0.66876	0.66876	0.66876
Stock 07	0.99977	0.99977	0.99977	0.99977	0.99977	0.99977	0.99977
Stock 08	0.27543	0.27543	0.27543	0.27543	0.27543	0.27543	0.27543
Stock 09	0.78531	0.78531	0.78213	0.78531	0.78531	0.78531	0.78531
Stock 10	2.62629	2.62629	2.62629	2.62629	2.62629	2.62629	2.62629
Stock 11	0.60779	0.60754	0.61222	0.60779	0.60779	0.60779	0.60779
Stock 12	0.77373	0.77373	0.77373	0.77373	0.77373	0.77373	0.77373
Stock 13	1.00589	1.00671	1.00590	1.00590	1.00590	1.00590	1.00590
Stock 14	0.64225	0.64225	0.64226	0.64225	0.64225	0.64225	0.64225
Stock 15	0.67237	0.67237	0.67241	0.67237	0.67237	0.67237	0.67237

. This table provides a comprehensive overview of the productivity changes and performance trends of the 15 companies from the Tehran stock exchange under varying levels of robustness and uncertainty, offering valuable insights into the sensitivity and reliability of the productivity index across different scenarios.

As illustrated in Table 8, the majority of the 15 companies experienced a decline in productivity, with only two companies, namely Company 10 and Company 13, demonstrating progress. Additionally, the proposed Malmquist productivity index for Company 7 is very close to 1, indicating negligible change in its productivity level. This suggests that Company 7 maintained its productivity without significant improvement or deterioration during the evaluation period. The results highlight the varying performance trends among the companies, with most facing challenges in enhancing their productivity, while only a few managed to achieve positive growth. These findings underscore the importance of identifying and addressing the factors contributing to productivity declines, as well as understanding the strategies that enabled Companies 10 and 13 to achieve progress. Further analysis could provide deeper insights into the drivers of productivity changes and support the development of targeted interventions to improve overall performance across the companies.

5.2. The Multipler Approach

In a similar manner, in this subsection, the results of Models (21) to (24) under different budgets of robustness and perturbation levels are presented in

Table 9. The results of Model (21) ${\Psi }_0^t(x_0^t,y_0^t)$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.68082	0.66413	0.52947	0.65421	0.45655	0.65413	0.45576
Stock 02	0.59572	0.58099	0.46225	0.57236	0.41364	0.57236	0.39879
Stock 03	0.45015	0.43902	0.34929	0.43250	0.30134	0.43250	0.30134
Stock 04	0.31786	0.31060	0.25381	0.30596	0.22235	0.30539	0.21278
Stock 05	1.00000	0.98883	0.89527	0.97861	0.80880	0.96079	0.66942
Stock 06	0.30040	0.29298	0.23310	0.28862	0.20110	0.28862	0.20110
Stock 07	0.37600	0.36744	0.29975	0.36195	0.25643	0.36125	0.25170
Stock 08	0.42247	0.41203	0.32782	0.40591	0.28281	0.40591	0.28281
Stock 09	1.00000	0.98883	0.89527	0.97861	0.80880	0.96079	0.66942
Stock 10	1.00000	0.98386	0.85235	0.97208	0.75243	0.96079	0.66942
Stock 11	1.00000	0.98565	0.86818	0.97428	0.76968	0.96079	0.66942
Stock 12	1.00000	0.98873	0.89367	0.97837	0.80443	0.96079	0.66942
Stock 13	0.73689	0.72347	0.61488	0.71401	0.53585	0.70800	0.49329
Stock 14	0.82695	0.80831	0.71436	0.79623	0.63252	0.79452	0.55358
Stock 15	0.82882	0.81315	0.71106	0.80259	0.63674	0.79632	0.55483

,

Table 10. The results of Model (22) ${\Psi }_0^{t+1}(x_0^{t+1},y_0^{t+1})$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.98961	0.96874	0.80172	0.95427	0.68643	0.95080	0.66247
Stock 02	0.08767	0.08559	0.07095	0.08440	0.06328	0.08423	0.05869
Stock 03	0.12170	0.11869	0.09443	0.11693	0.08147	0.11693	0.08147
Stock 04	0.59281	0.58007	0.47791	0.57140	0.40947	0.56957	0.39684
Stock 05	0.70779	0.69029	0.54921	0.68003	0.47381	0.68003	0.47381
Stock 06	0.32177	0.31534	0.26380	0.31062	0.22849	0.30916	0.21540
Stock 07	0.61319	0.59865	0.48905	0.59036	0.43606	0.58915	0.41048
Stock 08	0.19918	0.19426	0.15455	0.19137	0.13334	0.19137	0.13334
Stock 09	0.93386	0.91198	0.73874	0.89860	0.63418	0.89724	0.62515
Stock 10	1.00000	0.98883	0.89527	0.97861	0.80880	0.96079	0.66942
Stock 11	1.00000	0.98743	0.87645	0.97630	0.78435	0.96079	0.66942
Stock 12	1.00000	0.98883	0.89527	0.97861	0.80880	0.96079	0.66942
Stock 13	1.00000	0.98835	0.88942	0.97771	0.79889	0.96079	0.66942
Stock 14	0.94742	0.92400	0.73515	0.91027	0.63423	0.91027	0.63423
Stock 15	0.84906	0.83305	0.71199	0.82196	0.62968	0.81577	0.56838

,

Table 11. The results of Model (23) ${\Psi }_0^t(x_0^{t+1},y_0^{t+1})$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.48408	0.47322	0.38729	0.46614	0.33291	0.46510	0.32406
Stock 02	0.04842	0.04729	0.03872	0.04659	0.03312	0.04653	0.03242
Stock 03	0.07830	0.07636	0.06075	0.07523	0.05241	0.07523	0.05241
Stock 04	0.31365	0.30671	0.25177	0.30213	0.21731	0.30135	0.20996
Stock 05	0.45537	0.44412	0.35335	0.43752	0.30484	0.43752	0.30484
Stock 06	0.19496	0.19014	0.15128	0.18731	0.13051	0.18731	0.13051
Stock 07	0.34099	0.33256	0.26855	0.32762	0.22967	0.32762	0.22827
Stock 08	0.12815	0.12498	0.09944	0.12312	0.08579	0.12312	0.08579
Stock 09	1.56479	1.53449	1.29130	1.51388	1.12226	1.50343	1.04750
Stock 10	5.13268	5.00579	3.99889	4.93142	3.48352	4.93142	3.43593
Stock 11	0.66623	0.65401	0.56559	0.64576	0.50628	0.64010	0.44599
Stock 12	0.85678	0.83923	0.71507	0.82654	0.63466	0.82318	0.57355
Stock 13	0.77352	0.75735	0.64246	0.74671	0.55928	0.74319	0.51781
Stock 14	0.63427	0.61988	0.50605	0.61062	0.43289	0.60940	0.42459
Stock 15	0.61374	0.59942	0.48654	0.59046	0.42365	0.58967	0.41085

and

Table 12. The results of Model (24) ${\Psi }_0^{t+1}(x_0^t,y_0^t)$.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.91959	0.89893	0.73430	0.88549	0.64449	0.88353	0.61560
Stock 02	1.28088	1.24921	0.99559	1.23065	0.86718	1.23065	0.85745
Stock 03	0.69967	0.68237	0.54291	0.46837	0.67223	0.67223	0.46837
Stock 04	0.73083	0.71491	0.58855	0.70423	0.50612	0.70217	0.48923
Stock 05	2.51212	2.45833	2.05903	2.42159	1.82558	2.41362	1.68167
Stock 06	0.46692	0.45537	0.36230	0.44861	0.31256	0.44861	0.31256
Stock 07	0.55636	0.54261	0.43171	0.53455	0.37244	0.53455	0.37244
Stock 08	0.79639	0.77746	0.62427	0.76666	0.54443	0.76516	0.53312
Stock 09	2.36949	2.31515	1.95646	2.28364	1.69591	2.27658	1.58619
Stock 10	0.74415	0.72575	0.58708	0.71497	0.52631	0.71497	0.49815
Stock 11	1.80349	1.76429	1.47117	1.73793	1.29288	1.73277	1.20729
Stock 12	1.43115	1.39731	1.22894	1.37807	1.09543	1.37504	0.95805
Stock 13	1.03744	1.01665	0.89728	1.00344	0.79619	0.99676	0.69449
Stock 14	1.76168	1.72475	1.48450	1.69898	1.31578	1.69260	1.17931
Stock 15	1.39076	1.36528	1.16156	1.34931	1.03085	1.33622	0.93100

, respectively:

Using Equation (25), the robust Malmquist productivity index under the multiplier form is computed for specific robustness budgets, including 0%, 25%, 75%, and 100%, as well as two perturbation levels, 1% and 10%. These calculations are systematically summarized in

Table 13. The results of Proposed Multiplier RMPI.

Stocks	Γ = 0%	Γ = 25%		Γ = 50%		Γ = 100%
	ϖ = 0%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%	ϖ = 1%	ϖ = 10%
Stock 01	0.87474	0.87629	0.89366	0.87628	0.88127	0.87474	0.87474
Stock 02	0.07459	0.07468	0.07726	0.07471	0.07644	0.07459	0.07459
Stock 03	0.17394	0.17394	0.17394	0.20838	0.14519	0.17394	0.17394
Stock 04	0.89466	0.89511	0.89747	0.89511	0.88921	0.89466	0.89466
Stock 05	0.35819	0.35513	0.32446	0.35433	0.31276	0.35819	0.35819
Stock 06	0.66876	0.67038	0.68741	0.67034	0.68879	0.66876	0.66876
Stock 07	0.99977	0.99928	1.00741	0.99984	1.02403	0.99977	0.99977
Stock 08	0.27543	0.27530	0.27404	0.27517	0.27256	0.27543	0.27543
Stock 09	0.78531	0.78185	0.73798	0.78020	0.72033	0.78531	0.78531
Stock 10	2.62629	2.63292	2.67479	2.63510	2.66734	2.62629	2.62629
Stock 11	0.60779	0.60940	0.62298	0.61019	0.63171	0.60779	0.60779
Stock 12	0.77373	0.77503	0.76348	0.77455	0.76323	0.77373	0.77373
Stock 13	1.00589	1.00880	1.01770	1.00945	1.02336	1.00590	1.00590
Stock 14	0.64225	0.64097	0.59230	0.64100	0.57435	0.64225	0.64225
Stock 15	0.67237	0.67067	0.64762	0.66945	0.63750	0.67237	0.67237

, which provides a comprehensive overview of the productivity trends and performance variations across the evaluated companies. The table highlights the sensitivity of the productivity index to different levels of robustness and perturbation, offering critical insights into how uncertainties and varying degrees of conservatism influence the results. This analysis enables a deeper understanding of the stability and reliability of the productivity measurements under diverse scenarios.

As observed in Table 13, the results obtained from the multiplier approach align closely with those derived from the envelopment approach, further validating the credibility and reliability of the findings. Among the 15 companies analyzed, only Companies 10 and 13 exhibited productivity improvements over the two evaluated time periods, suggesting that these firms implemented effective strategies to enhance their operational efficiency. In contrast, Company 7 showed no significant change in its productivity levels, indicating a state of stability without notable progress or decline.

The consistency between the two methodologies not only reinforces the robustness of the analysis but also highlights the methodological soundness of the proposed frameworks. These findings provide valuable insights into the productivity dynamics within the petroleum products industry of the Tehran stock exchange, emphasizing the challenges most companies face in achieving productivity growth. The results also underscore the importance of identifying and replicating the successful strategies employed by Companies 10 and 13 to drive industry-wide improvements.

Finally, to conduct a deeper analysis of the reasons behind declining productivity in most joint-stock companies, this study examines data from two consecutive time periods. The findings reveal that only Company 10 exhibited a significant increase in both output metrics (ROA and ROE), whereas the outputs of all other companies, except for two cases, showed a downward trend compared to the previous year, including some cases of sharp decline. Consequently, Company 10 achieved the highest productivity index, while most other companies regressed due to unfavorable output trends.

6. Conclusions and Future Research Directions

This paper contributes to the existing literature by offering a comprehensive framework for productivity measurement that is both theoretically sound and practically applicable. The proposed robust Malmquist productivity index not only enhances the accuracy of productivity assessments but also provides decision-makers with a valuable tool for strategic planning and performance evaluation in uncertain environments. By integrating robust optimization techniques with traditional productivity analysis, this study addresses the limitations of conventional methods and offers a more resilient approach to measuring productivity under varying levels of uncertainty. The application of this framework to the petroleum products industry of the Tehran stock exchange demonstrates its effectiveness in identifying productivity trends, highlighting areas of improvement, and supporting data-driven decision-making. The findings reveal that only a few companies managed to achieve productivity growth, emphasizing the need for targeted strategies to enhance operational efficiency across the industry. Looking ahead, future research could explore the application of network data envelopment analysis (NDEA) to incorporate the network structure of decision-making units [90,91,92,93,94,95]. This approach would allow for a more nuanced analysis of productivity by considering the interdependencies and interactions within complex organizational structures [96,97,98,99,100]. By extending the current framework to include network DEA, researchers can gain deeper insights into the internal processes and sub-unit efficiencies that drive overall productivity, further advancing the field of productivity measurement under uncertainty. Such advancements would not only refine the analytical tools available to decision-makers but also provide a more holistic understanding of productivity dynamics in interconnected systems. Furthermore, other powerful and effective robust optimization approaches, such as scenario-based robust optimization [101,102] and data-driven robust optimization [103,104], can be employed to address various types of data uncertainties.