EMSRtrc: relaxation of booking limits by total revenue control for expected marginal seat revenue

Airlines are the companies whose products are perishable, meaning they cannot be stocked and sold at a later date. It is challenging for companies to sell their products as efficiently as possible in a limited period. Therefore, revenue management (RM) methodologies are required to achieve a satisfactory income level. According to the International Air Transport Association (IATA) Annual Review 2018, the average profit per passenger in the airline industry was only 9.27 USD in 2017. Considering the prices of flight tickets, the profit rates are significantly low in comparison with other sectors, making RM much more critical for the airline industry. According to IATA's Airline Industry Financial Forecast report, the losses of airline companies have constantly been increasing since the beginning of 2020 due to the Covid-19 pandemic. The Annual Review 2021 of IATA mentioned that the total operating revenue of airline companies decreased more than 60% in the second quarter of 2021 compared with the same quarter of 2019.

Airline revenue management (ARM) is concerned with managing airplane seat inventory. Airline companies have different fare levels for the same transportation services as hotels in the tourism sector. Airlines prefer to fill their planes with higher-fare customers since demand uncertainty is still present in the market. Accordingly, companies try to fill their planes with lower-fare customers to avoid opportunity costs incurred by empty seats. The airlines must decide how many lower-fare seats to sell while ensuring that they have enough seats left to sell to higher-fare passengers (An et al., 2021). When determining different fare levels in an airplane, the limited seats should be distributed to predetermined fare levels as efficiently as possible, since any unsold seats or opportunity costs resulting from selling too many low-fare seats are undesirable. In the literature, there are many optimization models such as dynamic programming (Kunnumkal and Topaloğlu, 2010; Selçuk and Avşar, 2019), linear programming (Möller et al., 2004; Liu and van Ryzin, 2008), bid price control (Akan and Ata, 2009; Talluri and van Ryzin, 1998), and nested booking limits (van Ryzin and Vulcano, 2008; Luo et al., 2020). Since the problem space is enormous and human-related aspects are involved, algorithms are becoming more complicated to tackle complex problems arising in the ARM. In the literature, there is a severe need to develop simpler, efficient, and effective solution approaches.

EMSR, developed by Belobaba (1987), is one of the excellent examples of simple and effective heuristic methods. It has been widely studied and many applications have been reported in the literature (Razan et al. 2020; Buyruk and Güner 2021). EMSR is easy to understand and apply, and it is computationally less demanding. Additionally, the EMSR algorithm yields better revenue levels in comparison with its counterparts. Researchers generally use EMSR for benchmarking purposes and reported results to show its superior performance (Gosavii et al., 2002; Lawhead and Gosavi, 2019).

Airline companies operate hundreds of flights per day. The aim of these companies is to earn the highest income from these flights. For this, the distribution of the current seat capacity to different fare levels gains importance and this process needs to be done accurately and quickly. In this study, an extended EMSR model, namely Expected Marginal Seat Revenue Total Revenue Control (EMSRtrc), has been developed. The proposed model keeps the simplicity of the original EMSR while producing higher returns. The main rationale behind the proposed model is that after a prespecified revenue target is reached, the model allows available seats to be sold at the lowest price, in order to increase revenue because selling vacant seats at a higher price level might cause the seats to be unsold, which results in lower revenue. The proposed EMSRtrc improves the canonical EMSR and contributes to the ARM literature by providing a practical solution approach.

When the ARM literature is examined, it is seen that the studies focus on four areas: forecasting, overbooking, seat inventory control, and fare pricing. The proposed ARM method is within the scope of the domain of seat inventory control. The studies in this area deal with the determination of seat protection levels for each fare level. To the best of our knowledge, this is the first paper that handles revenue values as a target in the related literature.

The determination of the target revenue value was proposed in this study for the first time. With the proposed target revenue method, airline companies not only earn higher revenues but also allows them to make dynamic decisions regarding ticket prices and seat protection levels during the sales period.

The remainder of the present work is organized as follows: A literature review is given in Sect. 2. In Sect. 3, the methodology is discussed. Section 4 focuses on a numerical example, in which comparisons with different ARM methods have been provided. The results are compared in Sect. 5 and discussions are given in Sect. 6. Finally, concluding remarks are given in Sect. 7.

In the airline industry, the studies on ARM were first started in the 1970s with the Airline Deregulation Act in the United States. Since then, many studies have been conducted and optimization models have been developed. The first study was conducted by Littlewood (1972), and after that, Belobaba (1987) developed the EMSR method with the help of Littlewood’s rule. Belobaba (1992) developed a new version of the EMSR method with some new additions and after this work, the new version was named EMSRb, while the old version was renamed as EMSRa. The EMSR method is still the most easily applicable and preferred ARM method by the airline industry. Additionally, scientists often work on EMSR for improvement. Wollmer (1992), for example, made a comparison for the model that he developed with EMSR under different conditions. Boyd and Kallasen (2004) used EMSR to compare priceable and yieldable environment conditions. They used the EMSRb for simulating the yieldable environments. Another paper on this issue belongs to Weatherford (2004), who developed a model called EMSU, aiming to decrease the EMSR model's risk and compared it with EMSR. Fiig et al. (2010) presented the EMSRb-MR model, which was improved based on the EMSRb. This new model could be applied to single leg and network revenue management. Gönsch et al. (2013) proposed a heuristic method for the ARM model based on the single-leg EMSRa. According to the experiments in the paper, the results of the the new method are better than the existing ARM methods. Tavana and Weatherford (2017) proposed a new ARM algorithm called the EMSRc and compared their model with the EMSRa, the EMSRb, and the EMSRb-MR in unrestricted and restricted fare environments. According to the results, the EMSRc outperforms the EMSRa, the EMSRb, and the EMSRb-MR, especially under the condition of an unrestricted fare environment. Kyparisis and Koulamas (2018) addressed the revenue management problem of a single-leg two-cabin airline where business and economy cabins are flexibly divided. Using three random demand distributions, it was tried to determine the most suitable cabin section and the most reasonable fares. One of the other studies on EMSR belongs by Banciu et al. (2019). A dependency between fare levels was considered while the original EMSR calculated booking limits with the assumption of independence between fare levels. Additionally, different distributions for the demand were attempted in their study. Luo et al. (2020) proposed a generalized nesting policy (GNP) that can enrich the family of nesting policies. A mathematical model for the nesting control under GNP was suggested, in which the nesting policy and booking limits were both taken as the decision variables. Bondoux et al. (2020) presented a new airline Revenue Management System based on Reinforcement Learning, which does not require a demand forecaster. Yazdi et al. (2020) published a case study on the revenue management strategy of the airline industry in Iran. A mathematical model was proposed and solved with a binary differential evolution algorithm. Razan et al. (2020) published a bibliometric analysis article on revenue management in the airline industry. Shihab and Wei (2021) developed DRL (Deep Reinforcement Learning), learning customers' behaviour and other important dynamics of ARM, and compared DRL with the EMSRb. It was determined that DRL earns more revenue than the EMSRb. Also, DRL gets high load factors near a hundred percent. Apart from the airline industry, Seo et al. (2021) applied the EMSRb to compare their model developed for sea cargo management. There are other optimization models for ARM in the literature. Venkataraman et al. (2021) sought to determine an optimal reservation policy that considers the dynamics of group behavior regarding cancelations and refunds. They compared their model with the traditional model under various scenarios and demonstrated that they achieved better total revenue. Escovar-Álvarez and Belobaba (2022) explored revenue management strategies for accepting additional bookings in economy fare classes when some premium cabin seats were expected to be vacant. They proposed two heuristics for this situation. They have shown that the recommended methods increase total revenue by up to 1.1%. Duduke and Venkataraman (2022) proposed a new class of products called flexible products with preferences. They focused on the development of capacity control techniques when flexible products were offered along with specific products. They developed a heuristic to determine a booking and seat allocation policy for all forms of flexible products. Lee and Hersh (1993) presented the advantages of using a dynamic approach against nested methods for the demand uncertainty. Bid price control is an approach used for network revenue management, a popular strategy for the ARM. The first study carried out on bid price control was authored by R. W. Simpson in 1989 (Talluri and van Ryzin, 1998). After that, Williamson (1992) inquired in his Ph.D. thesis on bid price control.

3.1 Expected marginal seat revenue (EMSR)

As mentioned earlier, EMSR was proposed by Belobaba (1987). The model’s logic is based on Littlewood’s (1972) rule. The model is easy to understand and apply. Moreover, it gives satisfactory results on seat inventory management problems. Therefore, EMSR is used very often in the literature and sector. According to the model, the distribution of the seats among fare levels should be nested so that there will not be any rejected highest fare passengers.

EMSR tries to protect the seats against low fare passengers, which facilitates selling more seats to the high fare passengers. While defining the number of seats protected from the low fare level passengers, the following inequality is applied. On the left-hand side of the formula in the inequality (1), the expected marginal revenue of selling S^th seat in fare level i is written. On the right-hand side, the fare of level i is multiplied by the probability of having a demand of S for level i. The inequality explains that it is illogical to protect the S^th seat if the expected marginal revenue does not exceed the fare level of i + 1.

where, S_i: the number of seats protected from lower levels in level i;_i: fare of level i;(S_i): the probability of having a demand of S for level i.

Let us explain inequality (1) with a numerical example to make it more tangible. Assume that there are two fare levels: F₁ = $50 and F₂ = $30. If the probability of having the 10^th passenger at fare level 1 is 0.5, then the expected marginal revenue from selling the 10th seat is $25. At this point, the expected marginal revenue is less than F₂. Therefore, it does not make sense to protect the 10th seat from the fare level F₂. However, assume that the probability that the 9th passenger is at the F₁ fare level is 0.6. As a result, the expected marginal revenue of selling the 9th seat from the F₁ fare level is $30. Since it is equal to the F₂ fare level, it is logical to protect the 9th seat for the F₁ fare level from the F₂ fare level. In this way, EMSR defines the booking limits of each fare level. These booking limits are nested, meaning that the highest fare level has no limit, while lower fare levels have limitations according to the protected number of seats. Nine seats are covered from the F₂ fare level for the same example. If there are 20 seats in the aircraft, just 11 seats can be sold from the F₂ fare level. Therefore, the booking limit of the F₂ fare level is 11, while the booking limit of the F₁ fare level is 20 (which means no limit for the F₁ fare level).

EMSR has two popular types called EMSRa and EMSRb. The EMSRa calculates the protection fare levels by comparing each class with the fare levels lower than itself. However, the EMSRb develops an artificial fare level by taking approximate values of lower fare levels. Therefore, the total protection level for one fare level can be calculated. For the application of EMSR, the booking limits determined employing protection levels are used to accept or reject any incoming passenger. In the literature, it is assumed that the EMSRb yields better results than the EMSRa. However, some researchers argue that better results with EMSRb are controversial However, the EMSRb is more commonly used than the EMSRa compared with EMSR.

3.2 Expected marginal seat revenue—total revenue control (EMSRtrc)

In this study, a new model based on EMSR is developed. Under this subtitle, this new ARM method will be explained. The new model is called EMSRtrc. Typically, the EMSR method only considers the booking limits of fare levels. If the number of seats sold from any fare level is equal to its booking limit, the model does not accept any more passengers from this fare level. Additionally, the protected seats are unsold if the targeted level’s passenger does not arrive during the sales period. In other words, the protected seats are either sold to customers of higher levels or unsold to any passengers at all. If the same example given above is handled again, just 11 seats are sold from the F₂ fare level. After this limit, passengers who want to buy a flight ticket with the F₂ fare level will be rejected. Only the passengers willing to pay the F₁ fare level will be accepted. If no one buys a seat from the F₁ fare level, the remaining 9 seats will not be sold to anyone. As a result, it is risky to protect seats because it may cause empty seats in the aircraft. Belobaba (2015) states that any empty seat in an aircraft is a loss for airline companies. If empty seats can be sold for even $1, it is better than no sales at all.

EMSR calculates the expected revenue of each fare level for each seat sale. Thus, the total expected revenue can be estimated when all booking limits, fares, and probabilities are known. As a result, EMSR promises a total revenue to the airline company when all booking limits are applied. Airline companies earn the promised revenue by using EMSR. According to the probability of selling Sth seat from fare level i, EMSR gives a probabilistic revenue for each seat. Sometimes airlines can reach more revenue than promised, and sometimes less. This depends on the performance of the sales period. At this point, our proposed EMSRtrc comes in handy. When the sales period goes well, and the revenue exceeds the expected level, the EMSRtrc offers a closed lower fare level for sale. For the example given before, a flight with two fare levels has a booking limit of 11 seats for the F₂ fare level. Let us assume that the total expected revenue by selling 11 seats from the F₂ fare level and 9 seats from the F₁ fare level is $550. This means EMSR promises $550 revenue to the airline company if the booking limit of 11 seats is applied to the F₂ fare level and all the seats are sold. The formula that gives the $550 is given in formula 2. For each sale, the fare level of i is multiplied by the probability of selling the Sth seat from fare level i. All expected marginal revenue values are summed. As a result, total expected revenue is found.

where, Si: the number of seats protected from lower levels in level i; Fi: fare of fare level i; (Si): the probability of having a demand of S for level i.

However, this goal is reached when 11 seats are sold from the F₂ and 5 from the F₁ (30*11 + 50*5 = $580). After this point, it is logical to open the F₂ fare level again as the objective revenue by applying EMSR is achieved. The sales period goes better than expected when the objective revenue is gained before the booking limit. However, this does not mean that the period will go well in the next sales periods. The closed fare level can be opened, so as not to risk ending up with empty seats in the aircraft. The flowchart of the EMSRtrc, according to the example, is given in Fig. 1. The mathematical explanation of the algorithm is given as follows:

where, \(m\): number of fare levels; \({F}_{i}\): fare of the fare level i; b_i: number of seats sold to fare level i; \({EMSR}_{ji}\): Total expected revenue, n_i: total booking limit for fare level i; a_i: number of seats sold to fare level i.

The flowchart of the EMSRtrc algorithm

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In Formula 3, it is explained that if the total revenue is greater than or equal to EMSR, there can be a relaxation for n_i. Therefore, a_i can be greater than n_i. Note here that EMSRtrc does not calculate the n_i value with its algorithm. The model performs according to the booking limits taken from another EMSR model and its objective revenues. In other words, n_i is calculated by another EMSR model, such as EMSRb. The model calculates the total revenue limits according to the booking limits of EMSRb and the start sales period. It keeps the accepted passengers under the control of booking limits and objective revenue thresholds. However, these thresholds should not be confused with the bid price control, as the bid price control defines a threshold for each flight ticket. However, EMSRtrc’s revenue thresholds are defined by the summation of EMSR values. Before each sale, the booking limit is controlled, and the passengers are accepted if the booking limit is not exceeded. After the sale, the total revenue is checked. If the total objective revenue is reached, one lower level is opened for sale again until the next booking limit. All these steps are explained in Fig. 1 visually.

In the next section, three examples will be examined to better understand EMSRtrc and the performance of the algorithm will be demonstrated. Accordingly, EMRStrc will be compared with EMRSa and EMSRb. Moreover, the method developed by Tavana and Weatherford (2017), which is called EMSRc, will also be compared with EMSRtrc.

In this section, three examples will be examined along with the discussions. The examples are derived from Tavana and Weatherford (2017). According to their work, the EMSRc performs better in the unrestricted fare environment when the passengers arrive orderly. An unrestricted fare environment means that passengers purchase the cheapest available flight ticket all the time. Additionally, it is an ordered arriving system when the low fare passengers come first, and high fare passengers arrive later.

Compared with EMSRc, the fare environment is unlimited, and passengers come regularly. Additionally, there is a situation in which there are different wage levels in our examples, but there is no service difference between the wage levels. There is no business or economy class where service levels differ. All seats of the flight will receive the same transportation service under the same conditions. Another assumption of the numerical examples concerns the distribution of demand. In all examples, the demand is normally distributed, and they are all independent demands for each wage level. There is no overbooking, raising fare levels, cancelling a sale, or group booking. The total number of seats sold was limited by an authorized reservation limit.

All models used for comparison in this study were coded in C# programming language, and the problems were solved with Microsoft Visual Studio Express 2015. There are six patterns for each problem except the first one, generated, one for the EMSRa, one for the EMSRb, one for the EMSRc, and three for the EMSRtrc. Three different versions of EMSRtrc were used, which are EMSRtrc-(a), EMSRtrc-(b), and EMSRtrc-(c), respectively. This is because each version is built according to different booking limits and total objective revenues of a different EMSR model. These are EMSRtrc-(a) simulation for EMSRa, EMSRtrc-(b) simulation for EMSRb, and EMSRtrc-(c) simulation for EMSRc. For each example, 20 different demand scenarios were randomly generated according to the distribution type of fare level. In each problem, the same 20 scenarios were applied to the original EMSR models to see the actual performances of the models (every problem has different scenarios, but models within the problem use the same 20 scenarios). Each scenario was run, and the results were recorded. There are four models for the first problem with two fare levels because the EMSRa and EMSRb give the same booking limit result when there are two fare levels.

The data required for the first problem are given in Table 1. The total seat capacity of this problem is 50. The scenarios created for each model were run, and the results are given in Table 2. The models named the EMSRtrc-(a) and the EMSRtrc-(b) represent the EMSRtrc with the booking limits and total objective revenues of the EMSRa and EMSRb, respectively. The EMSRtrc-(c) is the EMSRtrc built upon the outputs of the EMSRc. The maximum, minimum, and approximate revenues are shown in Table 2. The values in the table were taken from the simulations built. According to the approximate revenue, the best models are EMSRtrc-(a) and the EMSRtrc-(b), as shown in Table 2. Similarly, this model provides the best value in the minimum return category.

If we consider scenario 17 to clarify the rationality of the EMSRtrc better, for this scenario, the EMSRtrc-(a), the EMSRtrc-(b), and the EMSRtrc-(c) outperformed the others. In this example, the booking limit for the H class is calculated as 32 from the EMSRa, the EMSRb, and 30 by the EMSRc. Therefore the maximum number of sold tickets from the H class is 32 for the EMSRa simulation, the EMSRb simulation, and 30 for the EMSRc simulations. For the 17th scenario, the EMSRa and EMSRb sold 14 Y class tickets and 32 H class tickets. In these two models, 4 seats remain empty. The EMSRc sold 15 tickets from Y class and 30 H class tickets. In the EMSRc model, five seats remained unoccupied. However, the EMSRtrc-(a) and the EMSRtrc-(b) sold 15 Y class tickets and 34 H class tickets. For the EMSRtrc-(c) the number of sold tickets from class Y is 16, and from class, H is 34. The idea behind the EMSRtrc can be seen through this scenario. When the total objective revenue is reached, the EMSRtrc opens class H again, while the other models protect the seats from class H. In this way, the algorithm does not wait for the Y class passengers and the probability of having an empty seat decreases. Additionally, the total objective revenue is reached even if the booking limits are crashed. As more seats are sold, more revenue is gained since the EMSRtrc gives more opportunities to H class passengers.

The second example has six fare levels. The data for the example are given in Table 3. There are six different ARM models to compare in this example. In this example, the capacity was 65 seats, and 20 scenarios were created according to the mean and standard deviation data. For the second example, Tavana and Weatherford (2017) divided the sale period into four sub-periods. The percentage of passengers arriving according to their fare level differs in each sub-period. The demand arrival distribution is given in Table 4. For example, the probability of customers coming from the fare levels 1 and 2 is zero in the first period. In Table 5, the results of the second example are given, where the best performances for approximate, maximum, and minimum revenue belong to the EMSRtrc models, respectively.

In the last example, models were run for eight fare classes. The capacity was determined as 65 seats, as in the second example. The demands were normally distributed, and data about the example are shown in Table 6. The example's demand arrival distribution was determined as given in Table 7. Twenty scenarios were run for comparison, and the results are provided in Table 8. One more time, the best revenue values were seen under the lines of the EMSRtrc models.

The numerical example part comprises three different examples solved by different EMSR models. Each EMSR model solves all instances. For all three problems, the proposed EMSRtrc model exhibits superior performance. A more detailed discussion related to the results is provided in the next section.

The EMSRtrc is an ARM model that relaxes the booking limits of the regular EMSR algorithms. While relaxing the booking limits, does not worsen the total objective revenue of the system. Simultaneously, it makes the system avoid the risk of having many empty seats in the aircraft. The EMSRtrc sells more seats from the lower fare levels, but it does not decrease the total revenue in the end. In the preceding section, three examples are handled for the EMSRa, b, c, and EMSRtrc. It is seen that the EMSRtrc has better results than the others in most cases in terms of total revenue. When the suggested versions of the EMSRtrc are evaluated, it is seen that EMSRtrc-(c) provides the highest return in the top revenue category in all three examples compared to with EMSRtrc-(a) and EMSRtrc-(b). This is because EMSRtrc-(c) protects more seats in higher fare classes than EMSRtrc-(a) and EMSRtrc-(b). As can be seen from the results, a lower maximum income was obtained for the first example problem, while a higher maximum income was achieved for the second and third example problems. However, considering the minimum revenue, the new models proposed produce outstanding results. It promises a better minimum revenue of 9900, 11,619, and 2537%, respectively, for the three sample problems solved.

Load factors of each model are given in Table 9. Loading factors are higher than in EMSRa, b, and c, as the EMSRtrc, allows the seats at the high fare level to be sold at lower prices. As can be understood from the load factors, some seats are empty at the end of the sales period. Considering the third numerical example, the approximate number of empty seats is 4.65 for the EMSRa, 7.25 for the EMSRb, and 9.25 for the EMSRc. However, the approximate number of empty seats is 1.3 for the EMSRtrc-(a), 2.65 for the EMSRtrc-(b), and 7.85 for the EMSRtrc-(c). Regarding the approximate number of empty seats, the highest number of emptied seats was observed in the EMSRtrc-(c) version. Although fewer seats were sold in this version, it achieved the highest revenue in all three examples as the maximum revenue compared with the other two versions. As mentioned earlier, EMSRtrc-(c) protects more seats in higher price classes. The results clearly show that the proposed EMSRtrc model outperforms other three EMSR models.

In Table 10, percentages of each model's total objective revenue are provided. For example, the EMSRa has a total objective revenue of $43046.04 for the second example. The EMSRa model managed to exceed target revenue in 5 of 20 scenarios. That means the EMSRa can reach its total objective revenue by 25%. However, the EMSRtrc-(a) exceeds the target revenue in seven of 20 scenarios. As a result, the EMSRtrc-(a)’s percentage is higher than the EMSRa by 35%. Except for three cases, the EMSRtrc has better percentage values in all comparisons.

In this study, each sample problem was run 20 times for each EMSR model, and the minimum, approximate and maximum revenues obtained were reported. Note that standard deviations are important measures to assess the robustness of the models. The standard deviations for the problems solved are given in Table 11. When the standard deviation values were examined, it was seen that the proposed EMSRtrc-(a) model had the smallest standard deviation values among the three sample problems. According to these values, the proposed EMSRtrc-(a) model is more stable in producing good results.

The critical issue for airline companies is that gain high revenue from each flight. The proposed ARM model achieved a higher revenue target than the traditional methods. Simultaneously, a high rate of up to 98% was achieved in the seat load factor rate. The airline companies earn higher revenues with the proposed model, while customers can receive lower prices. It can be challenging to determine the best model among EMSRtrc, EMSRtrc-(a), EMSRtrc-(b), and EMSRtrc-(c). For example, EMSRtrc-(a) is the best model according to approximate revenue outputs in the second example, while EMSRtrc-(b) is the best in the last example. We state that the model results are dependent on the instances and problem parameters. Because the proposed model is highly practical, all the variants of the proposed models can be easily applied and the best model can be chosen.

The proposed ARM model is quite simple, applicable and based on the idea that airline companies set a satisfactory revenue target they expect from any flight and maintain the set seat protection levels throughout the sales period until that target is reached. It proposes to sell the remaining seats, albeit at a low price, in order to increase its revenue when the set target is achieved. In other words, it can be said that while the seat prices are determined by the airline company at the beginning of the sales period, the seat prices are determined by the customers after the determined target is reached. Therefore, the airline companies can earn more revenue than the target revenue it has set. However, this can sometimes cause airline companies to earn less revenue. Namely, after reaching the predetermined revenue target, it could perhaps gain a higher revenue if the set seat protection levels regarding unsold seats are maintained. However, in this case, the seats may not be sold. The proposed ARM model ensures an acceptable revenue for airline companies by balancing the number of empty seats and lower revenue.

One of the decisions that need to be made regarding the proposed method is to determine the target revenue of the airline companies. What will be the airline's target of revenue from the flight? The value of the target revenue is difficult to determine. Different approaches can be employed to determine it. As the first approach, the airline company can calculate the expected revenue using the demand distribution, fare levels, and the number of seats at each fare level and use this value as the target of revenue. In the second approach, the approximate cost of the flight can be determined by the airline company and this value or adding a certain percentage profit to this value can be utilized as the target value.

On the other hand, the present work was compared with state-of-the-art algorithms. In order to perform fair comparisons, some assumptions and settings were adopted from the literature, which might limit the research scope. For further studies, the performance of the EMSRtrc under the conditions of a restricted fare environment, independent demand, and demand distributions different from the normal distribution can be investigated. furthermore, the ability to overbook or cancelation can be further analyzed. These conditions can complicate the problem, but they will bring the problem closer to ARM problems of the actual world. There are studies to make conditions more realistic in the literature to obtain better solutions for ARM problems. Weatherford and Ratliff (2010) mentioned the importance of dependent demand for the success of ARM methods. It was stated that taking the demand dependency of fare levels into account, the result of EMSR booking limits could change. This paper referred that dependent demand revenue management tools gave more than 5% revenue improvements consistently, according to the study of Gallego et al. (2009).

Another discussion is about demand forecasting for ARM. In that study, the demand distribution is accepted as the normal distribution. However, there are studies on demand forecasting in the literature, which may better represent real-life problems. Gautam et al. (2021) mentioned the importance and validity of demand forecasting in their article. They stated the significance of considering the historical demand data of airlines and other dimensions like market size and market share for demand forecasting. As a result, conditions of the ARM problem can be changed to simulate the real-life better. However, the present work aims to make the problem easier to understand, handle and solve. The same algorithm can be evaluated with more realistic problems in future studies.

In today's information age, the rapid developments in technology and the rapid increase in the use of the internet have led to the formation of a large number and variety of data in the digital field. The processing of this data, which is called big data, and its analysis and conversion into useful information has become an important source of information for businesses. Using this information, decision-makers have started to make more accurate decisions. Likewise, the behaviors and consumption preferences of customers who prefer the airline constitute a large data set. The key to accurate and fast airline revenue management depends heavily on the best analysis of this data set and the accurate prediction of future customer behavior and preferences. Among the prediction techniques, the use of artificial intelligence, which has been very successful in recent years, has increased considerably. More accurate and consistent results can be achieved with the use of artificial intelligence in airline revenue management.

In this study, a new EMSR method was developed by relaxing the booking limits by objective revenue values of the EMSRa, b, and c. In the unrestricted fare environment and orderly arriving passengers, the EMSRa, b, c, and the EMSRtrc were simulated through three different examples. According to the results, the EMSRtrc gives better revenue values in all examples. When the results are examined in detail, it is realized that the EMSRtrc performs better according to total revenue and load factor. Besides, it is seen that the EMSRtrc has better rates for the frequency of reaching the objective revenue of EMSR models.

Another important point is that the new ARM model provides convenience in terms of understanding and application. Like other EMSR models, the EMSRtrc has a simple logic. After a ticket is sold, it checks whether the total target revenue has been reached. Through this control, it decides whether or not to sell the unsold seats in the higher fare levels at a lower price. As a result, the proposed model is feasible and realistic for ARM.

Only single-leg flights were considered in this study. However, in reality there just are no single-leg flights. A flight can have both single-leg passengers and continuing passengers. In this case, more complex analyses are required. In this sense, the proposed method may be insufficient.

This study is limited to determining the number of seats for each fare level and increasing the seat sales revenue of the airline company. Other components of the ARM were excluded from this study. The proposed model was applied only to single-leg flights. However, the model can be generalized to multi-legged flights. In this study, we assumed that the demand is normally distributed. By extending this assumption, the behavior of the model can be examined for different distributions. In addition to the airline sector, the proposed models are applicable to other industries with perishable products, such as the hotel and cruise ship industries.