We present an integrated data envelopment analysis (DEA) and Malmquist productivity index (MPI) to evaluate the performance of decision making units (DMUs) by using a DIRECTIONAL DISTANCE FUNCTION with undesirable interval outputs. The MPI calculation is performed to compare the efficiency of the DMUs in distinct time periods. The uncertainty inherent in real-world problems is considered by using the best and worst-case scenarios, defining an interval for the MPI and reflecting the DMUs’ advancement or regress. The optimal solution of the robust model lies in the efficiency interval, i. e., it is always equal to or less than the optimal solution in the optimistic case and equal to or greater than the optimal solution in the pessimistic case. We also present a case study in the banking industry to demonstrate applicability and efficacy of the proposed integrated approach.

In conventional DEA methods, all inputs and outputs are assumed to be continuous. However, the implied presumption of continuous data may not maintain an acceptable level of precision in practical data. Many practical situations such as number of teachers in a school, number of students in a college, or research papers only take integer values. Presented work presumes subsets of input and output variables in traditional , ve-pronged axioms in DEA model to be integer values for the , rst time in the , eld. An additional e, ort in this work expands axioms and a new model is presented. This paper presents a novel two phase model that in the , rst phase produces more precise e, ciency values and the best benchmark in its second phase, i. e. the nearest integer point to the e, ciency boundary is selected based on constant returns to scale. A case study is presented that demonstrates a comparison of e, ciency values obtained with this model compared to prior models that is show in table 1. And in Table 2, the values of the slack are compared with each other.

Background: The ongoing advancements in modern society have negatively impacted air quality, and India is one of the worst affected countries. This study aimed to evaluate the efficiency of maintaining air quality in 10 major Indian cities. Methods: The present study employed a DIRECTIONAL DISTANCE FUNCTION (DDF) within the framework of data envelopment analysis (DEA) to evaluate the efficiency of 10 major cities including Chennai, Delhi, Bengaluru, Ahmedabad, Hyderabad, Jaipur, Lucknow, Patna, Gurugram, and Thiruvananthapuram from January 01, 2018 to December 31, 2019. Results: The results indicate that air pollution is a significant issue in most cities in India. Thiruvananthapuram, Bengaluru, and Chennai were identified as the most efficient cities in terms of air quality for both 2018 and 2019 whereas Ahmedabad was noted as a purely inefficient city during the same period. Moreover, it was revealed that cities in the northern (Delhi, Lucknow, Patna), western (Ahmedabad), and northwestern (Jaipur, Gurugram) parts of India had higher levels of air pollution compared to the southern (Chennai, Bengaluru, Hyderabad, Thiruvananthapuram) part of India. Conclusion: There are significant disparities in air quality efficiency among the cities, revealing that southern cities perform better than their northern, western, and northwestern counterparts. It emphasizes the need for targeted interventions to improve air quality, particularly in cities like Delhi, Ahmedabad, and Jaipur.

In the recent years, environmental pollution, specially, air pollution, has become one of the main concerns of the international community and those are affected health of living organisms and natural ecosystems. In the recent past, environmental pollution has imposed significant human and financial costs on society. One of the most pollutant sectors is the industrial sector and between the polluting industries, the high energy consumer industries have imposed a significant part of environmental pollution. Hence, assessment of environmental performance in these industries is important. This paper using a DIRECTIONAL DISTANCE FUNCTION approach and (SBM) approach has evaluated the environmental performance and technical efficiency in the other non-metallic mineral products and production of basic metals industry, respectively. The results indicate that the other non-metallic mineral products industry Although technically more efficient, but have a lower environmental performance and Production industrial facilities, industrial Production cement, lime and plaster and asphalt manufacturing industry had the lowest environmental performance.

This study evaluates decision-making units (DMUs) that produce  unavoidable undesirable outputs,  which negatively affect performance. Previous studies by researchers have explored   approaches for calculating the minimum undesirable outputs, presenting the abatement range for undesirable outputs and assessing DMU efficiency, yet challenges remain.  To address this, we  introduce  a new method based on the DIRECTIONAL DISTANCE FUNCTION  model with individual-proportion weak disposability. The proposed method calculates the minimum unavoidable undesirable outputs, evaluates DMU  efficiency  for both undesirable and minimum undesirable outputs, separates the effects  of desirable and undesirable outputs on inefficiency, and resolves the limitations of earlier methods. Finally, the proposed method is applied to practical examples to demonstrate its superiority over existing approaches.

In recent years, environmental problems have increased and manufacturing is one of the sectors that emits polluting gases, so energy efficiency and environmental efficiency of industries, has attracted a lot of attention. The purpose of this study is to assess the environmental efficiency in industrial sector of Iran's provinces for natural gas consumption. For this purpose, while using data envelopment analysis and DIRECTIONAL DISTANCE FUNCTION, the environmental effects of natural gas consumption were investigated. Labor, capital and natural gas consumption were considered as inputs and carbon dioxide from natural gas consumption and value added were considered as undesirable and desirable outputs, respectively. The results indicate that the environmental efficiency of industries in the provinces of Iran is low. In 1397, the industries of Tehran and Kerman provinces had the best environmental performance. Regional analysis also indicates that the provinces located in area one of the country's divisions have the highest level of environmental efficiency. Findings of this study also confirm the significant dispersion in the efficiency of different areas.

Objective: In recent years, high inflation has caused many problems for companies. The financial turmoil has caused companies to lose their value due to the imposition of capital costs and the inability to attract foreign capital and to get a poor rating compared to their competitors. Companies try to satisfy shareholders and stakeholders by reducing risk and increasing company value. Stakeholder concerns about financial turmoil are seen as a positive factor in ameliorating financial turmoil. Therefore, it can lead managers to find the right solution. There are several ways to reduce company risk and avoid financial turmoil. In the current economy, one of the most important ways to satisfy shareholders, stakeholders, and institutions is to provide activities related to social responsibility and pay attention to it. Today, corporate social responsibility is considered one of the most important financial issues. One of the ways out of these problems is to increase the social responsibility performance of the company. Considering the various results presented in this field, it seems necessary to conduct more research in this field to obtain reasonable results. Most of these problems are affected by life cycle stages. So; Compliance with social responsibility at different stages can help the company's sustainability. On the other hand, the impact of social responsibility on financial volatility depends on economic changes and developments, which vary throughout the life cycle. And the impact of CSR may not be the same in all stages of the life cycle. In this research, two goals have been pursued, one is to investigate the effect of social responsibility performance on financial turbulence without considering the life cycle. Another is to examine the impact of corporate social responsibility performance on financial turmoil over the life cycle. Therefore, the purpose of this study is to investigate the effect of social responsibility performance on financial turbulence during the company's life cycle for a period of ten years. Methods: The statistical sample of this research includes 112 active listed companies during the period of 2009 to 2019. To measure the performance variable of the company's social responsibility, using the website of Tehran Stock Exchange and the website of Kodal, about 36, 000 pages of reports of the activities of the board of directors during the considered years were studied, and the scores of each dimension of the company's social responsibility were inserted in an Excel file. and finally, using the mathematical method (directed DISTANCE FUNCTION), the social responsibility performance of companies was evaluated. Berger et al., Almeida, Campello and Altman models have been used to measure financial turbulence. Research hypotheses have been tested using panel data and fixed effects by multivariate regression statistical method. Results: The results show that the performance of social responsibility either in general or in detail, regardless of the company's life cycle, alone cannot reduce financial turmoil. The variable combination of the company's social responsibility performance with the life cycle in the stages of growth and maturity shows a significant and decreasing effect on financial turmoil. In fact, it can be said that corporate social responsibility performance and life cycle jointly reduce financial turmoil. The combination of social responsibility performance with the life cycle has a positive and significant effect on financial turbulence in the recession stage and has no effect in the decline stage. Conclusion: Companies can reduce business risk and financial turbulence through social responsibility management. Companies that have a high social responsibility performance have more credibility. This makes it easier for companies to access financial resources. Therefore, companies try to attract the opinion of their shareholders and stakeholders in different ways, in order to attract more capital and suffer less financial turmoil. As companies compete more in the stages of growth and maturity than in other stages of the life cycle, they pay more attention to social responsibility; Therefore; Based on the results, it can be concluded that the life cycle of the company and the performance of social responsibility jointly reduce financial turmoil. The results of the research showed that the life cycle moderates the relationship between social responsibility performance and financial turmoil and plays an important role. In such a way the companies that are in the maturity stage can reduce the financial turmoil by increasing the performance of social responsibility. Therefore, it seems that the results can be of great help to investors and stakeholders. Therefore, in order to reduce the risk of their investment, investors and users are suggested to examine the companies in terms of their life cycle and choose companies that are in the maturity period and have high financial resources, and also enjoy high social responsibility performance. Because investing in such companies will reduce the investment risk and the investor's concern will be reduced.