Operational performance of Indian passenger airlines using hierarchical categorical DEA approach

Abstract

The challenges faced by the Indian Aviation market are complex and perplexing for the decision-makers in this sector. This industry has faced debilitating losses during COVID-19 pandemic times. The travel restrictions have been gradually lifted worldwide, and therefore the airlines can expect to witness a growth in domestic and international air passenger traffic during 2021–22. The losses incurred during the pandemic times need to be recovered. Thus, there is an urgent need to strategically plan and improve the operations so that the airlines can perform efficiently and earn revenues. Airlines in India work under heterogeneous environments depending upon their segment of operations. Some of the airlines operate for only domestic passengers and others for both domestic and international passengers. Therefore these airlines must be assessed as per their homogeneous peer group to obtain substantial results. The proposed study has discussed an approach to study airlines' operational performance using hierarchical categorical data envelopment analysis (DEA). The airlines have been categorized based upon their segment of operations. The efficiency of the airlines has been evaluated for the period 2014–19 as per their categorical input value. Window analysis has also been performed for a meaningful analysis of the results. The outcomes of the study will be helpful for policymakers to improvise the present working model of the industry and alleviate their performance.

Introduction

Data Envelopment Analysis (DEA), a non-parametric technique, is based on the framework of (Farrell 1957), later extended by Charnes et al. (1978) and Banker et al. (1984). Techniques of DEA have been employed to measure the relative efficiency of homologous entities called decision-making units (DMUs) with multiple inputs and outputs. The complexity of the relationships between multiple inputs and outputs, where the units are frequently not comparable, is where DEA has an edge over other mathematical programming and statistical techniques. Thus, DEA offers an alternative to the traditional approaches employed for this purpose. Further, DEA allows us to estimate efficient performance without making restricted assumptions about the underlying knowledge of production function. This attribute of DEA for efficiency evaluations makes it to be more advantageous for the policy makers and managers in taking decisions for improvements.

DEA with a range of existential settings has been extensively used in identifying the technical inefficiencies of various service sectors and industries such as transport, banking, medical, tourism, or any production sector. In the transport sector, DEA has been used to assess the efficiency of land, air, and water transport. It helps to determine their performance, analyze the factors affecting their working efficiency, and provide them the benchmarks for their growth. Hence, policymakers can plan their strategies as per the results obtained from various DEA models. DEA techniques have been applied to assess airlines' operational performance by estimating their relative distance from the non-parametric production frontier. Efficiencies calculated are comparable to the best performing airline. Hence makes DEA a powerful tool for comparative efficiency analysis.

The basic assumption of DEA is to evaluate the homologous DMUs, which implies that they are working under the same circumstances and are involved in the same process. Haas and Murphy in (2003) have stated that an analyst may want to assess the DMUs, which are heterogeneous. They also observed many articles in the DEA literature to assess the non-homogeneous evaluation set (e.g., airlines serving domestic or international sectors, banks working in rural, urban, or tourist regions, production plants in the same industry producing different products). Banker and Morey in (1986) had introduced the use of categorical variables in the basic Banker, Charnes, and Cooper 1984 model (Banker et al. 1984). They have illustrated the use of categorical variables in DEA using actual data. They also mentioned that categorical variables could be controllable or non-controllable by decision-makers. Controllable variables include all those over which decision-makers have direct control and can be modified. Non-controllable variables include those which are not in control of the decision-makers of any organization. Kamakura has overcome the limitations of the original model of Banker and Morey (1986). However, revision in the mixed-integer model noted in Kamakura (1988) does not allow for virtual DMUs that combine the different categorical variable levels. Kamakura's work was revised by Rousseau and Semple in Rousseau and Semple (1993). They formulated a new linear programming model to handle DEA categorical variables, which annihilated the earlier models' shortcomings. Later on, it was further investigated and extended by many researchers.

In the literature, Soteriou and Zenios in (1999) have categorized 144 banks into homogeneous groups as per their working environment and size. They have evaluated their efficiency within the groups and compared the scores of the groups to benchmark the performance drivers simultaneously. In 2016, Min and Joo (2016) assessed airlines' comparative performance after the strategic alliances among global airlines using categorical DEA. They have also mentioned that DEA can be employed to DMUs with categorical input values such as hierarchical levels, customer feedback scores, level of competition, etc. In 2018 Karadayi and Ekinci (2019) have assessed the R&D performance of European Union countries using categorical DEA. The aviation sector is a transport sector that includes different categories of operating airlines like cargo or passenger, domestic or international, public or private. These airlines work under a heterogeneous environment and can be categorized based on their working environment. Evaluating airlines' operational performance by classifying them in homologous groups will prove to be a better approach for efficiency evaluation and benchmarking.

The Indian airline industry had started its operations in the year 1911. Since then, it has been functional with its complex structures and expensive operational costs. Airline companies function under extreme stress as they have to satisfy their customers by providing them with world-class services and at the same time have to earn revenues to sustain themselves in the global market. The ongoing pandemic situation has made this even more difficult for players in this sector. Thus, an airline company must strategize and plan its policies carefully. Since the aviation sector is affected by many factors and the decision-making process is very complicated, they must arrange their actions according to the inferences obtained from the research-based models. However, the research related to the performance evaluation in Indian aviation sector though being in an infancy stage has been studied by a few researchers. The efficiency assessments of all Indian airlines for the period 2006–2010 were studied by Jain and Natarajan (2015). Selvan et al. (2016) conducted an analysis of the performance of Indian airlines from 2010 to 2014. They employed the super-efficiency DEA model to rank the airlines after using the two-stage DEA to assess overall efficiency. Seth et al. (2019, 2021) have assessed the operational performance of passenger airlines operating in India using traditional and two-stage approaches of DEA respectively.

While formulating a model for performance evaluation of decision-making units, one has first to identify whether the working environment of decision-making units is the same or not. It may happen that some of the DMUs are working in the highly competitive market, and others are in their typical business situation. Assessing the performance of all the DMUs as "scratch" competitors would be unfair to those in a highly competitive position. This complication can be handled using a hierarchical category. DMUs working in a highly competitive environment are classified as category 1. Then those working in a normal situation are classified as category 2 and then according to the working environment. Others are also classified similarly, and DMUs in the most advantageous situation fall in the highest category group, say (h). In this manner, a set of the categorical input variable is created with input values 1, 2,…, h. DMUs with categorical input value 1 are evaluated within the group, DMUs with category input value 2 are evaluated in reference to categories 1 and 2. Similarly, DMUs with higher categories are evaluated in reference to the DMUs falling in preceding categories.

For the Indian airline industry, passenger airlines provide only domestic services or domestic and international services. It would be unfair and unjustified to assess the operational efficiency of all passenger airlines on the same grounds irrespective of whether they are working for the domestic sector or international sector only or both the sectors. The present study is a novel approach to evaluate the operational performance of passenger airlines using DEA with non-controllable hierarchical categorical variables. The study comprises the performance evaluation of all private and government passenger airlines regarding whether they are working in the domestic sector or in both domestic and international.

The rest of the paper is trailed as follows. Section 2 provides a detailed explanation of the proposed work with the data descriptive statistics and the variables used in the study. Section 3 describes the mathematical formulation of the proposed model to assess the DMUs with hierarchical categories. Section 4 presents the results obtained from the five-year data of Indian passenger airlines. Section 5 concludes the present study with future research scope.

Research methodology

Present work includes the formulation of the DEA conventional models with hierarchical category levels and input orientation. They are further used to assess the evaluation of operational efficiency of Indian passenger airlines, namely Air India, Air India Express, Alliance Air, Air Asia, Air Deccan, Air Odisha, Go Air, Indigo, Jet Airways, Jet Lite, Spice Jet, True Jet, Vistara, Zoom Air, Air Heritage, Star Air for 2014–19. Efficiency evaluation of passenger airlines as per their categorical input value together with DEA conventional models in input orientation can prove to be the best approach as they take care that the airline is dealing in which sector. Data has been extracted from the reports available on the Directorate General of Civil Aviation (DGCA), an Indian Ministry of Civil Aviation office.

The procedure adopted for data collection and analysis can be summarized as:

1. Data Collection Airlines’ operating statistics has been extracted from the source: https://dgca.gov.in/digigov-portal/

2. Variable Selection The process of choosing relevant input and output variables for analysis is based on the elements responsible for the performance of airline firms. Prior investigations into the appraisal of the effectiveness of airlines has seen a range of input and output drivers. The economic structure, airline administrative practices and market performance all have a significant impact on an airline's operational performance. Thus, the variables chosen for the study can be divided into three categories based on their structural, marketing, and administrative drivers. Due to the limitation in the availability of the data, the following four widely used variables are selected for the study.

   • i.
     Financial Parameter: Operating cost
   • ii.
     Administration Parameter: Fleet size
   • iii.
     Financial Parameter: Passenger revenue
   • iv.
     Marketing Parameter: Passenger load factor
     Joo and Fowler (2014), Kottas et al. (2018), Saranga and Nagpal (2016) have also used these variables in their study. Table 1 below contains a brief description of the variables used in the study.

3. Data Analysis Airlines have been classified as per their domain of service. Those working in the domestic sector are only classified as category 1, and airlines providing service in domestic and international sectors are classified as category 2. Airlines are divided into two sub-groups as per the categorical input value (1, 2). In this efficiency evaluation process, category 1 airlines are evaluated within the group only, and category 2 airlines are evaluated in reference to airlines of category 1 and 2. Airlines in category 1 are considered at a lower level, and category 2 airlines are at a higher level. Descriptive statistics of the data for the period 2014–19 have been provided in Table 2. Proposed models have been executed on the data for performance analysis. Window analysis with three windows of length three has been done for the period 2014–19. The two categories of airlines have been compared bilaterally, and the airlines under study have been ranked as per their performance.

Table 1.

Variables that are used in the study

Variables	Definition
Input Variables
Operating Cost (OC)	It includes all the expenses which an airline company is outlaying for the smooth functioning of its operations
Fleet Size (FS)	It represents the total number of aircraft, including owned and leased, available at the end of the year
Output Variables
Passenger Revenue (PR)	It includes the revenue generated from revenue-paying passengers and extra baggage charges
Passenger Load Factor (PLF)	It determines the capacity utilization of any airline company. It is calculated by dividing the revenue passenger kilometers by the available seat kilometers

Table 2.

Descriptive statistics of the data used in the study

		Maximum	Minimum	Average	Standard Deviation
2014–15	FS	107	3	40	40.64234
	OC	226,854.4	2681.9	70,003.76	83,179.95
	PLF	82.4	53.7	75.64	8.360885
	PR	165,895.9	556.8	57,450.5	63,902.4
2015–16	FS	108	2	39.81818	42.59923
	OC	199,085.4	1151.5	60,557.51	74,627.98
	PLF	90.6	66.5	79.48182	6.515829
	PR	181,661.2	645.1	56,013.17	67,378.14
2016–17	FS	131	1	40.5	45.1931
	OC	215,615.9	24.7	63,722.77	80,185.05
	PLF	91.6	68.8	79.81667	6.559831
	PR	182,434	14	56,388.21	68,998.15
2017–18	FS	159	2	41.28571	49.64794
	OC	246,617.4	55.33	64,706.2	88,946.62
	PLF	93.6	27.2	76.82143	16.48559
	PR	213,423.4	0.48	57,086.57	76,528.45
2018–19	FS	217	1	51.45455	63.77142
	OC	291,689.2	20.9	78,193.46	103,898.8
	PLF	91.9	48.5	76.57273	13.25924
	PR	269,473.6	3	64,975.59	86,639.44

Mathematical model

Let $\mathrm{M}=\left\{1,2,,,\cdots ,,,\mathrm{n}\right\}$ be the set of the total number of airlines under study. Let ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ be the group of all those airlines with categorical input values 1 and 2, respectively. Then $\mathrm{M}={\mathrm{M}}_1\cup {\mathrm{M}}_2$ and ${\mathrm{M}}_1\cap {\mathrm{M}}_2=\varnothing$. The input-oriented constant and variable returns to scale DEA conventional models with categorical data are proposed in the present study. Let $\mathrm{X}=\left\{{\mathrm{X}}_1,,,{\mathrm{X}}_2,,,\cdots ,,,{\mathrm{X}}_{\mathrm{n}}\right\}$ and $\mathrm{Y}=\left\{{\mathrm{Y}}_1,,,{\mathrm{Y}}_2,,,\cdots ,,,{\mathrm{Y}}_{\mathrm{n}}\right\}$ be $\mathrm{m}\times \mathrm{n}$ and $\mathrm{s}\times \mathrm{n}$ input and output matrix, respectively. Let $\mathrm{V}$ and $\mathrm{U}$ be input and output weight row vectors. The proposed mathematical models for evaluating the efficiency performance of each $k^{\mathit{th}}$ airline are defined below.

Categorical input-oriented Charnes, Cooper, and Rhodes (CCR) model (CCR_I_CAT)

The multiplier form of the input-oriented categorical CCR (CCR_I_CAT) model, which evaluates the efficiency of the kth airline, is formulated below:

$${\left\{\begin{array}{c}\underset{\text{V},\text{U}}{\text{Maximize}}\mathrm{\theta }=U{\text{Y}}_{\text{k}}\\ \text{subject to}\\ \frac{{\text{UY}}_{\text{r}}}{{\text{VX}}_{\text{r}}}\le 1\left(\forall \text{r}\in {\text{M}}_1,\text{if k}\in {\text{M}}_1\text{otherwise}\forall \text{r}\in \text{M}\right)\\ V{\text{X}}_{\text{k}}=1\\ U,V\ge 0\end{array}\right\}}_{\text{Model}1}$$

Categorical input-oriented Banker, Charnes and Cooper (BCC) model (BCC_I_CAT)

The multiplier form of the input-oriented categorical BCC (BCC_I_CAT) model, which evaluates the efficiency of the kth airline, is formulated below:

$${\left\{\begin{array}{c}\underset{\text{V},\text{U}}{\text{Maximize}}\mathrm{\theta }=U{\text{Y}}_{\text{k}}-{\text{U}}_0\\ \text{subject to}\\ \frac{{\text{UY}}_{\text{r}}-{\text{U}}_0}{{\text{VX}}_{\text{r}}}\le 1\left(\forall \text{r}\in {\text{M}}_1,\text{if k}\in {\text{M}}_1\text{otherwise}\forall \text{r}\in \text{M}\right)\\ V{\text{X}}_{\text{k}}=1\\ U,V\ge 0\\ {\text{U}}_0\text{unrestricted}\end{array}\right\}}_{\text{Model}2}$$

The implementation of the proposed models can be strategically summarized in the following algorithm.

Results and discussion

Efficiency scores

The efficiency scores of the proposed input oriented categorical DEA models (CCR_I_CAT) and (BCC_I_CAT) are presented in Table 3 below for the passenger airlines operating in 2014–19. The objective of the input-oriented model is to minimize the inputs with the production of at least given output levels.

Table 3.

DEA Categorical Efficiency Analysis of passenger airlines for the period 2014–19

		2014–15		2015–16		2016–17		2017–18		2018–19
S No	DMU	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT
1	Air India	1	1	0.88	0.88	0.845	0.85	0.83	0.84	1	1
2	Air India Express	1	1	1	1	1	1	1	1	1	1
3	Alliance Air	1	1	0.785	0.81	0.631	0.631	0.696	0.73	0.75	0.85
4	Air Asia	1	1	0.9	0.93	0.86	1	0.93	0.99	1	1
5	Air Deccan	*	*	*	*	*	*	1	1	*	*
6	Air Odisha	*	*	*	*	*	*	1	1	*	*
7	Go Air	1	1	1	1	1	1	1	1	0.99	1
8	Indigo	0.91	1	0.88	1	0.91	1	0.95	1	0.86	1
9	Jet Airways	0.99	1	1	1	1	1	1	1	*	*
10	Jet Lite	1	1	1	1	1	1	1	1	*	*
11	Spice Jet	0.91	0.91	0.73	1	0.84	1	0.92	1	0.84	1
12	True Jet	*	*	1	1	0.62	1	1	1	1	1
13	Vistara	0.76	1	0.6	0.61	0.68	0.68	0.77	0.77	1	1
14	Zoom Air	*	*	*	*	1	1	1	1	*	*
15	Air Heritage	*	*	*	*	*	*	*	*	1	1
16	Star Air	*	*	*	*	*	*	*	*	0.56	1

*Airline was not operating for that year

Following are the critical conclusions from Table 3:

1. Efficiency scores obtained from the (BCC_I_CAT) model are more than the (CCR_I_CAT) model values for all the passenger airlines from 2014 to 19 since the (BCC_I_CAT) model has low discriminating power in efficiency evaluation.

2. Airlines operating since 2014 are Air India, Air India Express, Alliance Air, Air Asia, Go Air, Indigo, Jet Airways, Jet Lite, Spice Jet, True Jet, and Vistara.

3. Air India Express is the only airline that is relatively globally and locally efficient during the study period.

4. It has been observed that for the years 2014 and 2018, airlines have obtained the maximum efficiency score. However, for 2015–18, Air India and Alliance Air airlines have shown a declining trend in performance while Indigo, Spice Jet, and Vistara have exhibited a positive performance trend.

Airlines, namely Air Deccan, Air Odisha, Star Air, Air Heritage, Zoom Air have joined the market for very little time, and some of them are not even operating today. Therefore, the performance evaluation of other experienced market players with these airlines does not provide conclusive results.

Window analysis

Window analysis is a unique approach to use DEA in a time series mode. This method is a moving average analysis pattern. A DMU in each period is treated as a different DMU. The performance of DMU is compared with its performance in other periods and compared with the performance of other DMUs in the same period. The variation of efficiency of DMUs over time can help in making essential conclusions. Since the number of airlines operating during the study period is few, for the period 2014–19, window analysis with categorical data has been done to obtain the relative efficiency scores. Operational efficiencies of each airline have been obtained by using the combination of proposed categorical models with window analysis for each window. Window analysis has been done for the eight airlines, namely Air India, Air India Express, Alliance Air, Air Asia, Go Air, Indigo, Spice Jet, and Vistara, to overcome the limitation of fewer operating airlines during the period 2014–19. These have been working in the aviation market since 2014 and are still operational. Table 4 presents the results of window analysis for the period of study. (CCR_I_CAT) and (BCC_I_CAT) models have been used for calculating the global and local efficiency for each window. Three windows of length three have been formed, and average values for each year have been mentioned in Table 4. Table 4 also contains the average global and local efficiency of each airline.

Table 4.

Window Analysis Results using (CCR_I_CAT) and (BCC_I_CAT) models

S No	DMU	Category	2014–15		2015–16		2016–17		2017–18		2018–19		Average		No. of times efficient
			CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT	CCR_I_CAT	BCC_I_CAT
1	Air India	2	0.983	1	0.919	0.99	0.901	0.961	0.914	0.969	1	1	0.9434	0.984	1	1
2	Air India Express	2	1	1	1	1	0.939	0.956	0.951	0.96	1	1	0.978	0.9832	3	3
3	Alliance Air	1	0.992	1	1	1	0.904	0.909	0.913	1	0.861	0.884	0.934	0.9586	1	3
4	Air Asia	1	1	1	0.997	1	0.988	1	0.977	0.985	0.921	0.924	0.9766	0.9818	1	3
5	Go Air	1	1	1	0.997	1	0.994	1	1	1	0.789	1	0.956	1	2	5
6	Indigo	2	0.866	0.974	0.881	1	0.852	0.982	0.874	1	0.824	1	0.8594	0.9912	0	3
7	Spice Jet	2	0.879	0.889	0.725	1	0.791	0.9904	0.826	1	0.769	0.884	0.798	0.95268	0	2
8	Vistara	1	0.759	1	0.684	0.707	0.767	0.773	0.792	0.796	0.932	0.936	0.7868	0.8424	0	1

Following key points has been noticed.

1. None of the airlines is globally efficient for the period 2014–19.

2. Both Air India Express and Air Asia airlines have achieved the highest global efficiency (0.98 approx).

3. Low performers for the period 2014–19 are Spice jet and Vistara, but Vistara's individual years' performance shows that it has been progressing since 2016.

4. The last two columns of Table 4 show the number of times airlines can take a position on efficiency frontiers of (CCR_I_CAT) and (BCC_I_CAT) models.

5. Air India Express for three years (2014, 2015, and 2018) was both globally and locally efficient, which shows that it is efficiently utilizing the input resources to obtain the given outputs.

6. Go Air airline operating for the domestic sector is locally efficient throughout 2014–19 and globally efficient for 2014 and 2017.

The performance chart reveals the trend followed by the eight airlines for the period 2014–19. Overall, Air Asia is the only airline that is showing a declining performance trend in the case of both (CCR_I_CAT) and (BCC_I_CAT) model scores, while others have mixed performance scores over the period.

Bilateral DEA

Further DEA-bilateral comparison model (Bilateral-CCR-I) has been applied to test the significant difference in efficiency between category 1 and 2 airlines. Bilateral DEA assesses the efficiency of each airline of category 1(2) with the airlines of category 2(1). This approach is beneficial when two different groups are based on different frontiers.

Bilateral comparison model, which provides sharp discrimination between two groups performance, has revealed that category 1 airlines have outperformed the category 2 airlines for the years 2014–15, 2015–16, 2016–17 and 2017–18 but in 2018–19, there was no significant difference in the performance of both the categories. It can be concluded that airlines providing only domestic passenger services were performing better for 2014–18. Table 5 presents the ranking order of the airlines for the year 2014 to 2019 using the Bilateral-CCR-I model followed by a rank chart representing the airlines for each year as per their performance.

Table 5.

Ranking order from Bilateral DEA Comparison using Bilateral-CCR-I model

S. No	DMU	2014–15	2015–16	2016–17	2017–18	2018–19
1	Air India	7	9	12	14	6
2	Air India Express	5	6	6	9	4
3	Alliance Air	2	2	2	5	3
4	Air Asia	1	3	3	7	5
5	Air Deccan	*	*	*	3	*
6	Air Odisha	*	*	*	1	*
7	Go Air	9	11	8	13	11
8	Indigo	6	8	9	10	8
9	Jet Airways	8	7	7	11	*
10	Jet Lite	4	4	4	6	*
11	Spice Jet	10	10	10	12	9
12	True Jet	*	1	1	4	2
13	Vistara	3	5	5	8	7
14	Zoom Air	*	*	11	2	*
15	Air Heritage	*	*	*	*	10
16	Star Air	*	*	*	*	1

*Airline was not operating for that year

Conclusion and future scope

Indian passenger airlines belong to two different systems based on their customer market. One system provides service to the domestic sector, and the other provides service to domestic and international sectors. Since their customer dealing markets are different, relative assessment of all the Indian passenger airlines with a single production frontier would be unfair, and results obtained will not be substantial. Thus DEA with categorical DMUs has been used to assess the operational performance of Indian passenger airlines for the period 2014–19. The present study is a novel approach comprising the mathematical formulation of the (CCR_I_CAT) and (BCC_I_CAT) models that have been validated through an algorithm. Airlines have been given categorical input value 1 if they work for the domestic sector only and categorical input value 2 if they work for domestic and international sectors. The proposed models have been corroborated on the real data set of Indian passenger airlines operating in their respective categories.

Operating statistics for the period 2014–19 have been obtained from the website of DGCA. It has been discovered that some of the airlines, namely Air Odisha, Air Deccan, and Zoom Air, just started operating during the period of study stopped its operation in the year 2018–19. Big airline companies, namely Jet Airways and Jet Lite, have only witnessed their failure during this period. The Indian aviation industry has seen many changes across the period 2014–19. Therefore the data obtained was not easy to handle as the number of operating airlines was less with airlines operating in different sectors. But the limitation of fewer operating airlines has been addressed using the technique of Window analysis with the (CCR_I_CAT) and (BCC_I_CAT) models. Thus, the results obtained are conclusive and clearly show each airline's performance trend over the period 2014–19. A bilateral comparison has been made to compare the performance of airlines working in the domestic sector with the airlines operating in both domestic and international sectors. It has been found that airlines working for the domestic industry only performed better during 2014–19.

Without tasting the flavors of failure, no one can enjoy the flavors of success. The success of any company or person depends on how the company or person has dealt with past failures. Results obtained in Sect. 4 indicate that many airline companies have witnessed a downfall or loss. Thus, there is a dire need to learn from failures and redesign the operational model for the Indian airline industry. The present study has dealt with input orientation models, and in the future, output orientation models can also be formulated with constant and variable returns to scale. Sensitivity analysis of the input and output variables can also be done in the future with the proposed models.

Funding

The authors have not received any funding for this study.

Declarations

Conflict of interest

The authors declare that there is no conflict of interest.