UDC 330:519.8
Iu. V. Ignatova,
Ph.D. in Economics, Associate Professor, Associate Professor of the Economic and Mathematical Modelling Department,
Kyiv National Economic University named after Vadym Hetman, Kyiv
T. L. Kmytiuk,
Ph.D. in Economics, Senior lecturer of the Economic and Mathematical Modelling Department,
Kyiv National Economic University named after Vadym Hetman, Kyiv
PRICING MODELLING IN THE AIR TRANSPORT MARKET
Ю. В. Ігнатова,
к. е. н., доцент, доцент кафедри економікоматематичного моделювання,
ДВНЗ «Київський національний економічний університет імені Вадима Гетьмана»
Т. Л. Кмитюк,
к. е. н., старший викладач кафедри економікоматематичного моделювання,
ДВНЗ «Київський національний економічний університет імені Вадима Гетьмана»
МОДЕЛЮВАННЯ ЦІНОУТВОРЕННЯ НА РИНКУ АВІАПЕРЕВЕЗЕНЬ
One of the main conditions for a rational organization of transport activity and its profitability is the competent, qualitative and systematic formation of prices for transport services. In addition, where it is impossible to create a competitive environment, conduct a balanced tariff policy of the state. Transport fare is a fee for moving services. Its role in the activity of this or that type of transport company is difficult to overestimate, since the level of tariffs depends on the profitability, and hence the financial stability of competitiveness. The article proposes a series of mathematical models for forecasting average annual tariffs for passenger air transportation on the domestic USA market. Descriptive statistics and simulation modelling tools were used in constructing models. Since the basis of the simulation model is the discrete law of distribution of the average annual prices for airline tickets, the prospect of further research will be finding and justifying the continuous law of price distribution and the construction of mathematical models on this basis.
Одним з головних умов раціональної організації діяльності транспорту та його прибутковості є грамотне, якісне і планомірне формування цін на транспортні послуги, а там, де неможливе створення конкурентного середовища, проведення виваженої тарифної політики держави. Транспортний тариф  це плата за послуги з переміщення. Його роль у діяльності того чи іншого виду транспорту транспортного підприємства важко переоцінити, оскільки від рівня тарифів залежить прибутковість, а значить, і фінансова стабільність конкурентоспроможність. В статті запропоновано ряд математичних моделей прогнозування середньорічних тарифів на пасажирські авіаперевезення на внутрішньому ринку США. При побудові моделей використовувались інструментарії описової статистики та імітаційного моделювання. Так як основою імітаційної моделі є дискретний закон розподілу середньорічних цін на авіаквитки, то перспективою подальших досліджень буде знаходження та обґрунтування неперервного закону розподілу ціни та побудова математичних моделей на цій основі.
Keywords: pricing, passenger air transportation, descriptive statistics, simulation.
Ключові слова: ціноутворення, пасажирські авіаперевезення, описова статистика, імітаційне моделювання.
Formulation of the problem. Passenger transportation from one point to another is a kind of commodity. As with any other product, the price of the carriage is affected by demand, supply and costs. The peculiarity of air transport, which distinguishes it from terrestrial species, is the speed of passenger transportation, taking into account all the time that the passenger should spend on a passenger from the moment of departure from the place of departure to the destination. For short distances (up to 500 km), rail and road transport have an advantage over air transport, with much less time spent on ground handling. However, with increasing travel distances, passengers prefer air transport. Moreover, the price for tickets is also influenced by the use of passengers by air.
The purpose of the paper is to study the field of passenger air transportation, analyze the pricing of passenger transportation and analyze the general trends in airfare prices on an example of American airlines, starting from 1995 to 2016.
The analysis of recent research and publications. The subject of the pricing modeling in the air transport market was studied by such scientists as, G. Samoilenko, U. Kuidich, N.I. Kabushkin, and others. Analysis of research papers and gives reason to believe that this area is not explored to this day.
The object of research is the domestic market of American air travel.
The presentation of main material and results of the research. After analyzing a number of literary sources [16], we conclude that the demand for airfare depends on such factors:
 seasonality – there is a high season when many people want to fly: New Year, May holidays, summer holidays, but there is a low season when people fly less. In winter, on the contrary, in many areas prices are lower;
 days of the week: usually on weekdays the demand is lower than during the day off and the ticket price may be lower;
 time of day: there are more and less popular flights;
 reservation dates: airlines are seeking to sell their flight tickets as early as possible, so they encourage passengers to buy them in advance.
Airlines plan how many seats and at what price to sell. For example, for sale at a minimum cost is allocated 20 seats. Once they are booked, you will be offered a bigger price and so on. In turn, the offer for airline tickets depends on:
 quantity and quality of competitive offers from airlines in a specific direction: the more carriers operate from point A to point B, the higher the competition  and now more cheap offers  special tariffs, sales, promotions;
 the presence or absence of communication with alternative modes of transport, for example, by rail.
Airlines are, in essence, manufacturers of complex and integrated services, "air transportation", and in this case, to "production costs" can be credited: the type of aircraft, the number of seats, predictable loading by passengers; the cost of air navigation services; cost of airport services; flight range, fuel consumption per kilometer; personnel costs, crew training, staff placement in the host country; other expenses. In addition, the ticket price can be influenced by the marketing strategy of the company.
Airfare is the price at which the airline transports passengers from airport A to airport B under certain conditions.
There are several classifications of passenger air transport tariffs known by carriers. Distinguish tariffs published and confidential. The published airline tariffs are controlled by the International Air Transport Association. They are used mainly for the calculation of complex routes with the involvement of several carriers. Tickets for published airline tariffs can be purchased anywhere in the world, but the price is usually high.
Confidential tariffs are formed by airlines on their own, taking into account the demand, competition and strategy of the company in a separate direction. Such tariffs are a commercial secret for competitors. They are operated on the airline itself and its agents.
Such tariffs are a commercial secret for competitors, they operate on the airline itself and its agents. Confidential tariffs apply, for example, when a person buys a round trip at one airline.
Separately, it is necessary to consider this kind of air transportation as low cost and their ways of setting prices for tickets. Air lowcost carriers, as well as Lowcost airlines  airlines that provide air transportation services for passengers at prices lower than the traditional airlines. The native land of the concept of lowcost  the USA, namely the company Southwest Airlines, from which it spread to Europe in the early 1990's and further in many countries around the world.
The pricing in budget airlines is influenced by many factors. The first is the high loading of planes. One plane in Lowestoft performs a lot of flights a day, and the company is sometimes willing to sell a ticket for 1 euro, if only to run it half way.
The second factor  a plane ticket does not include any additional services. Baggage, food, drinks, even a comfortable place for the feet  all at an additional cost.
The third point is the use of airports. Cheap airlines are often based at small airports, and the main transport hubs are either bypassed or rented for half an hour. Smaller airports in smaller cities are much cheaper, and some are even willing to pay extra to the carrier, hoping that in this way a tourist will be brought to the city.
Let's turn to the analysis of the domestic market of American air travel. Based on available statistics for 19952016, pricing trends suggest a steady decrease in the cost of airfares, as can be seen from Fig. 1.
Fig. 1. Annual U.S. Domestic Average Itinerary Fare in 2016 Constant Dollars
Source: Built by authors on the basis of [1]
We will analyze the above prices using the descriptive statistics toolkit. To this end, we will construct a BoxWhisker graph in Fig. 2, and we will analyze it in detail.
Fig. 2. Boxplot of the Annual U.S. Domestic Average Itinerary Fare in Current Dollars
Source: Built by authors on the basis of [1]
Analyze the time series of average airfare prices. Thus, according to fig. 2, for the last 10 years, the minimum ticket price was $ 277, 25% of all tickets cost less than $ 309.2. 50% have a price lower than $ 324.5, and the other half is more than this number. On average, 1 domestic ticket costs 331.1 dollars, and the maximum cost is 391.0, while 75% of tickets are cheaper than 357.2 dollars.
Based on the conducted statistical analysis, it is advisable to analyze the trend line and determine the general tendency of prices for domestic flights.
After analyzing Fig. 1, assume that this trend is linear, so let's express the dependence of the price of the ticket on the period of the linear function.
Using the tool pair linear regression [710] we obtain the trend equation
(1)
This equation is a prediction equation that can be used to predict future prices. It's enough to just put the desired year into formula ().
Fig. 3. Annual U.S. Domestic Average Itinerary Fare and trend line in 2016 Constant Dollars
Source: built by authors on [1]
Then, we will try to construct a forecast for the average annual price for airfares for 20172025, using the equation (1):
Table 1.
The forecast for the average annual price for airline tickets for 20172025 on the basis of the trend line
Year 
Fare 
2017 
353.885 
2018 
349.778 
2019 
345.671 
2020 
341.564 
2021 
337.457 
2022 
333.350 
2023 
329.243 
2024 
325.136 
2025 
321.029 
Source: calculated by the authors independently by the formula (1) and data [1]
However, the disadvantage of this approach is that the price will always fall, while global fuel price trends, in turn, indicate a return. So, we will try to predict the average prices for air tickets on the basis of a discrete law of distribution of historical prices for airline tickets.
Using the toolkit of descriptive statistics, we define the discrete law of the distribution of the random value of the price of the ticket. To do this, we build a histogram of frequency distribution for airfares.
To determine the number of histogram intervals, use the Sturges’ rule [7]:
(2)
Where n is the number of intervals, N is the volume of choice.
To determine the size of the intervals (step) , set the maximum and minimum prices for the ticket. Then we get:
.
Determine the likelihood of the price of a particular interval, which is calculated in Table 2.
Table 2.
Interval number 
Lower limit prices 
The top price limit 
Average price 
The frequency of falling prices in the interval 
The probability of a price in the interval 
Cumulative 
1 
343 
368 
355 
3 
0,136364 
0 
2 
368 
392 
380 
10 
0,454545 
0,136 
3 
392 
417 
405 
2 
0,090909 
0,59 
4 
417 
442 
429 
3 
0,136364 
0,682 
5 
442 
467 
454 
4 
0,181818 
0,818 
The sum 



22 
1 

Source: calculated by the authors independently according to [1]
Then the density function of the distribution of average prices for airfares will look like in Fig. 4.
Fig. 4. The function of density distribution of average airfare prices
Source: calculated by the authors independently according to [1]
Consequently, taking for x the random value of the price of the ticket, we obtain the function of density distribution of prices:
(3)
Then the integral function of the price distribution density will look
(4)
Next we will determine the price forecast for the next month. For this purpose, we will apply the simulation modeling tool based on the discrete probability distribution (3)  (4) found.
Let the probability of occurrence of a price in one or another interval for 1 day can be randomly distributed in the range from 0 to 1. In particular, to develop the first simulation model, we will conduct 22 experiments on the appearance of the indicated probability, which are given in Table. 3.
Table 3.
The predicted price is found on the basis of the simulation modeling
Experiment No. 
Probability appearance of the price on the interval 
Average price 
1 
0,880 
454 
2 
0,252 
380 
3 
0,598 
405 
4 
0,593 
405 
… 
… 
… 
21 
0,968 
454 
22 
0,889 
454 
Source: calculated by the authors themselves
In table the 3 field of the average price corresponds to the integral probability distribution function F(x). For example, for the first experiment, the probability 0.88 falls in the interval from 0.818 to 1, then the price for the ticket corresponds to the lower limit of the interval and will equal 454. Similarly, the rest of the experiments were carried out.
According to the results of simulation of the prices for air tickets table 3, in the first forecast period, it can be either 454 days, or 380, or 405, etc. Then on average in one day the price will be
$. (5)
The root mean square deviation, in this case, will be (х)=37 $.
Similarly, we will simulate new probabilities of the emergence of a new price for the second forecast period, which are presented in Table 4.
Table 4.
Estimated prices based on simulation modeling
Experiment No. 
Probability the appearance of prices on the interval 
Price 
1 
0,280912 
380,0 
2 
0,22536 
380,0 
3 
0,732077 
454,2 
4 
0,400446 
429 
… 
… 

21 
0,9497 
380,0 
22 
0,995383 
380,0 
Source: calculated by the authors themselves
That is, the price in the second forecast day can be either 380, or 454, or 429, etc. Then, on average for the second day, the price will be 385 with a meansquare deviation of 3.38.
Similarly, using the simulation model with the "tightening" of the random variable x the price of the airline ticket to the discrete law of probability distribution, we obtain a forecast for the next 9 years.
Table 5.
Year of the forecast 
Price for the ticket 
2017 
429,5 
2018 
404,7 
2019 
355,2 
2020 
380,0 
2021 

2022 

2023 

2024 

2025 

Source: calculated by the authors independently on the basis of (5)
Compare the results:
Table 6.
Year of the forecast 
Forecast by the regression model 
Forecast by the simulation model 
2017 
353.885 
404,7 
2018 
349.778 
429,5 
2019 
345.671 
355,2 
2020 
341.564 
380,0 
2021 
337.457 
396,7 
2022 
333.350 
360,25 
2023 
329.243 
374,8 
2024 
325.136 
418,04 
2025 
321.029 
355,22 
Conclusions and perspectives of further research. One of the main conditions for rational organization of transport activity and its profitability is the competent, qualitative and systematic formation of prices for transport services, where it is impossible to create a competitive environment, conduct a balanced tariff policy of the state. Transport fare is a transfer fee. Its role in the activities of this or that type of transport company can not be overestimated, since the level of tariffs depends on the profitability, and hence, financial stability, survival in the competition. The article proposes a series of mathematical models for forecasting average annual tariffs for passenger air transportation on the domestic US market. When constructing models, tools used for descriptive statistics and simulation modeling were used. Since the basis of the simulation model is the discrete law of distribution of average annual prices for airline tickets, the prospect of further research will be finding and justifying the continuous law of price distribution and the construction of mathematical models on this basis.
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Стаття надійшла до редакції 11.09.2017 р.