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Alma Andabaka
MATHEMATICS IN TOURISM AND HOTEL MANAGEMENT STUDIES
Summary: This paper presents the role of mathematics in the study of tourism and hotel management and emphasises its importance in developing the capacity to resolve tasks. It gives individual examples of its application in practice, showing that mathematical methods are indispensable for making business decisions. Accordingly, the objective of mathematics is to train students to make the most of their knowledge in mathematics to resolve everyday business problems and tasks. Thus, human resources are created which will be competent to successfully address new challenges on the market.
Key words: mathematics, tourism management, hotel management
_______________________________
Alma Andabaka, prof, UTILUS Business School for Tourism and Hotel Management, Zagreb, Ul. Ivana Trnskoga 3
INTRODUCTION
The aim of the study of tourism and hotel management is to educate managers for middle, high and top-level management functions in tourism companies and hotels, and other business systems. In order for this aim to be achieved successfully, it is necessary to train students to analyse economic factors, indicators and the results of business activities, so that they might contribute to the development of the business environment. They will be successful only to the extent they succeed in acquiring the knowledge and skills needed in business economics and economic analyses. It is precisely the acquisition of knowledge and the development of skills that represent the focus of the Mathematics for Economists course in the study of tourism and hotel management.
However, the success of classes in mathematics depends on many factors. The classes must spark interest in the subject, which is not always easy to achieve (Kurnik, 2002). Mathematics is considered to be one of the difficult subjects, which requires uninterrupted and continuous work, and a lot of time, zest and effort. If students are not ready for such high-intensity focus, mathematics can pose some difficulty. It is not infrequent that students begin with only modest knowledge in mathematics, so they must go through many areas once again to reach the level where they are able to follow the syllabus. When Euclid, a great mathematician in Ancient Greece living from around 340 to 287 BC, was holding his first lecture to a group of students, at the end of the lecture one of them asked: "And what are we to do with maths in real life?" Euclid said nothing. Half an hour later, he sent one of his slaves to give the student a gold coin and he discharged him from school. Interesting enough, pupils and students are still known to ask the very same question raised in the 3rd century BC. However, we should not be offended and remain silent like Euclid, but we must clearly say that mathematics is all around us. Lev Nikolayevich Tolstoy, a great Russian writer (1828-1910) even said: "A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction!"
Mathematics is there to help in everyday life, in science and the arts (there is a close connection between maths and music, painting and sculpting), in trade, industry, etc. It is a powerful, concise and unambiguous means of communication, explanation and evaluation. No matter how complicated, we must all agree that it would be much harder to live without it. Plato, the Ancient Greek mathematician and philosopher (c. 427 c. 348 B.C.) said: "Not all people need all things, but as for calculus, it is needed not only by all of us together, but also by each single person. Anyone who does not know how to do sums or at least count must be deleted from the count of men, because otherwise there is no friendship amongst merchants, love amongst neighbours or servants in the municipality, just as fairness cannot dwell in justice perpetually!"
Therefore, it is necessary to immediately set clear goals that must be achieved through the syllabus of Mathematics for Economists in the study of tourism and hotel management. In this way, students are encouraged to repeat the things they left out in their previous education or did not sufficiently grasp. It is very important to provide other various forms of supplementary or extra activities (in the form of seminars, exercises, consultations, and the like), which have proven to be very effective in acquiring knowledge in mathematics.
MATHEMATICS A DISCIPLINE WHICH DEVELOPS THE ABILITY TO RESOLVE PROBLEMS
The difficulty of the subject lies in the fact that students, despite their great yearning to learn what is taught in the subject Mathematics for Economists, must already have a certain level of knowledge to enable them to do so. Mathematical contents are intertwined, so lack of understanding of one part of the syllabus leads to a lack of understanding of other contents that build upon this part. Consequently, mathematics can neither be taught from that moment onwards, nor can everything that should have been learnt be left to oblivion. In fact, such content should have been learnt and assimilated in secondary school. Fully aware of this, all teachers who work in colleges and in higher education in general must start off at a tempo which enables each student to repeat what was forgotten or poorly learnt in their education to date.
One of the ways for this gentle introduction is percentages, which are something surely well-known to students. They know or can easily grasp the formulae used, so mostly what needs work on here is how to apply them in everyday life and work. In this area, it is possible to instruct students on how to resolve tasks. The basic thing is for the students to understand the task, so that when they are instructed to start with the key elements, they will ultimately find the whole easier to resolve. The task should be viewed from all angles, and the students should try to make a connection between the problem being resolved and the knowledge they have previously acquired. They should be instructed to remember what helped them in similar situations, so that they can recognise what is familiar, and then actually put this to use. This is how students grasp how to resolve the task, and then they can proceed with the mathematical (algebraic) operations known to be practical.
In resolving a particular task, it is necessary to keep checking whether everything offered has been used and whether all the important elements used in the task have been taken into account. Finally, one should make sure that each individual step has been made correctly, through formal or intuitive reasoning, and preferably both. Once again, it is necessary to review detailed solutions and try to simplify them as far as possible. Results obtained in a particular task should be applied in other tasks, where it is necessary to analyse the intertwining points and the peculiarities of each individual step.
In any case, if a student develops the habit of examining and checking his results in the said way, he can acquire focused and usable knowledge which can be used to develop problem-solving skills. In this way, the student will become capable of applying mathematical procedures in the economy and observe the extent to which mathematics is deeply rooted in economic issues.
Resolving tasks and problems is a basic human activity. Our conscious reasoning mostly deals with tasks, our thoughts all strive to reach a goal, and we therefore seek out the means and ways to resolve a task. In order to accomplish the task set before us, we must know at least something of the subject to which it is related. We must pick out the necessary details or segments of our knowledge which, although it does exist, remains unresponsive at first. Hence, the details must be gathered. It should be mentioned that some people are more successful than others in achieving their goals and in resolving problems. However, finding a solution depends on what we manage to extract from the memory.
For example, students will find it much easier to tackle a task involving a consumer loan if they remember the arithmetic string and its characteristics, or when they connect a task in compound interest with the geometric string and its features. Such connections make the student realise that everything learnt is necessary to advance in resolving problems. In addition, it is easier to move forwards if we hope that we shall soon become more successful. To teach a student how to resolve tasks means giving him the will to find common features in how to process all types of problems.
We cannot expect to easily resolve any task that is worthy of effort. Helen Keller (1880-1968) once said: "Man would not be able to fully appreciate the beauty and richness of his knowledge if he did not need to overcome a certain degree of difficulty in gaining it. Reaching a peak would not be half as beautiful if dark valleys did not need to be crossed on that path."
Any student who learns this during his or her studies will know how to tackle problems at the place of work. He or she will be aware that any task faced will be better understood if more intensive time is spent working on it. By advancing in the work, new elements will be added to those observed in the beginning, thus ultimately creating a more successful solution.
APPLIED MATHEMATICS IN TOURISM AND THE HOTEL INDUSTRY
Just as for any other discipline, the Bologna Declaration recommends that mathematics should be taught as a one-semester subject. This is almost inconceivable if one wants students to reach an adequate level of knowledge. I would say this particularly relates to the study of tourism and the hotel industry in the educational institutions specialising in such a type of education. Students who opt for such studies as a general rule have relatively poorer knowledge of mathematics, which they usually explain by saying that mathematics is not important for their profession. Without going into the objectivity of such a standpoint, I would like to point out that in the studies concerned it is more logical to teach mathematics in two semesters (as one course or two separate courses, for example as Maths 1 and Maths 2).
Although, in this article at least, it is not possible to cover all the areas in which maths is applied in the field of tourism and the hotel industry, the following simple examples serve to demonstrate how the use of mathematical tools is today an essential element in making decisions concerning planning and business operations in the tourism sector.
One of the basic starting points in any analysis of tourism is to make an assessment of tourism demand, not only from the aspect of its size, but also the dynamics or movements in terms of changes in the economic categories on which it depends. By using differential calculus it is possible to calculate the extent of the response in tourism demand to changes in income or changes in the prices of specific services in the tourism market. In this way we can calculate the rate of change in tourism demand if income increases by 1%, and thus calculate the income elasticity of tourism demand. Alternatively, we can calculate the rate of change in tourism demand if the price of a specific service in the tourism market rises by 1%, and thus show the price elasticity of tourism demand.
Assessing the profit which a tourism facility can generate in the season ahead is based on the assessment of tourism income and the related costs. If the maximum income generated by a hotel which operates all year round is known, where capacity utilisation is 100% and the average price of an overnight stay and breakfast is given, it is possible to tabulate the ranges of income depending on the degree of capacity utilisation by using the calculus chain rule (supposing that income grows proportionately to the increase in the degree of capacity utilisation, that is, where the average price of overnight stay and breakfast is always the same).
For example, hotel "X", which is open all year round, generates at full capacity 1,500 overnight stays with breakfast at HRK 400, so the maximum income (1,500 x 400) is HRK600,000. Table 1 shows a comparison of the income flow depending on the degree of utilisation of the hotel's accommodation capacities.
Table 1. Income flow depending on the degree of utilisation of the hotel's accommodation capacity
Degree of capacity utilisationNumber of overnight stays with breakfastIncome in HRK0%0010%15060,00020%300120,00030%450180,00040%600240,00050%750300,00060%900360,00070%1050420,00080%1200480,00090%1350540,000100%1500600,000Source: Information is fictitious, exclusively for illustration
By knowing the amount of fixed and variable costs, it is easy to calculate the level of capacity utilisation which the hotel should reach to cover the aggregate operating costs, that is, the number of overnight stays needed to reach break-even point.
When a hotel chain is considering to expand its activity by opening new capacities, where it is necessary to take into account several factors, such as the price of building a new hotel, the number of employees, the available capital, the expected annual income of the hotel, legal limitations and the like, it is necessary to use more complex mathematical tools. In an effort to pinpoint the optimum solution to a problem based on a specific criterion, it is not appropriate to use the classic mathematical apparatus of differential and integral calculus (Nerali, 2003). With the method of linear programming used for the mathematical modelling of a problem and finding the optimum solution, while taking into account the goals of the organisation and limitations in terms of resources, it is possible to set a programme to expand a particular hotel chain to ensure the maximum expected annual income.
In these times of recession where each sector is facing the very difficult task of maintaining economic activity at the level of previous years, it is clear that everyone in the tourism sector will work on finding various ways of making their services even more attractive and available. It is precisely with the goal of ensuring easy financing of the use of tourism products that tourism agencies apply the model of approving consumer loans without any cash deposits which the beneficiaries then repay over a period of several months. The use of the said model enables tourism services to be paid for in instalments, but requires employees to know the basic elements of financial mathematics in order to determine the optimum duration of repayment and other terms of the approved loan. In addition, tourism agencies can use short-term loans for bridging short-term non-liquidity, that is, to be able to cover the material costs of the operation.
Advertising is one of the key marketing tools used by tourism agencies to present their products and services. It is very important to assess the impact of promotional messages on potential clients. The use of the exponential function provides an answer to such questions. For example, where it is expected that the percentage of viewers (g) who will respond to a commercial message about a new tourism offer after t days should work according to the formula: g(t)=0.7-e-0.2t, the tourism agency can expect that after 10 days 56.5% of viewers will respond to the message.
From the macroeconomic aspect, tourism has special significance for the development of the national economy precisely because of the multiplying effect of tourism demand on other economic sectors. By using the input-output analysis, it is possible to calculate the impact of changes in final consumption on the production of specific sectors. Matrix multipliers are tools to analyse the impact of total final consumption or its individual components on: (1) the value of production in individual sectors; (2) the demand for input from specific sectors; (3) depreciation; (4) added value (salaries, taxes, profit); (5) investments, etc. in individual sectors (Jur
i, 2000).
CONCLUSION
In view of the current situation on the market, there is evident need to have new human resources in the tourism and the hotel industry that can understand movements in the economy, resolve problems and find adequate solutions by using and applying knowledge and skills acquired during their studies. Students holding a degree in tourism and hotel management may assume responsible work posts in the economic activity concerned, which means they will be expected to know how to draw up and implement business strategies pertaining to their working environment. Bearing in mind that mathematics has become the language of modern analytical economics, it is precisely knowledge in mathematics that is a suitable instrument for achieving such goals in any economic sector, including tourism and the hotel industry.
LITERATURE
Gusi, I. (1995). Matemati
ki rje
nik. Zagreb: Element.
Jur
i, Lj. (2000). ''Razvitak input-output analize u Hrvatskoj''. Ekonomski pregled, 51 (11-12): 1313-1333.
Kurnik, Z. (2002). ''Historicizam''. Matematika i akola, no. 17: 52-58.
Nerali, L. (2003). Uvod u matemati
ko programiranje 1. Zagreb: Element.
Polya, G. (1956). Kako u rijeaiti matemati
ki zadatak. Zagreb: `kolska knjiga.
Simon, Carl P. & L. Blume. (1994). Mathematics for Economists. New York: W. W. Norton & Company, Inc.
`ego, B. (2005). Matematika za ekonomiste. Zagreb: Narodne Novine.
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@B*ph.nPPPPPQQQ2Q8QRQTQQQQQ.R^R`RbRdRfRhRhSjSvSTWghn".0pDFHLğƟȟ9ǺǸǱǱαααΦh82hICJaJ!jh82hI0JCJUaJhjhImHsHh'&xhIUjhNhI0JUhNhIhIjhI0JUhNhVuhNh:gZhNh;E6hNhhNh;E3isation and the required quality of professional human resources for the management of nautical tourism ports in the function of sustainable development. The testing of the model was conducted within the field of potential solutions, by which the linear model of the organisation was shown as the optimal model for organising the operation of Northern Adriatic nautical ports. For more information, see: Dundovi, (2005), Overview of the MSc thesis by Mirjana Kova
i: Model organizacije sjevernojadranskih luka nauti
kog turizma u funkciji odr~ivog razvoja. In: Pomorstvo - ISSN 1332-0718 19, pp. 317-320. UDK 656.61.
It is precisely by using the input-output model that Ljubo Jur
i made an analysis of the multiplying effects of Croatian tourism on other sectors of the Croatian economy, the dependence of Croatian tourism on imports, and the net foreign currency inflow in 1998. For more information, see: ''The multiplying effects of Croatian tourism''. Acta Turistica (10) no. 2, Ekonomski fakultet, Zagreb, 1998, and ''The import dependence of Croatian tourism''. Acta Turistica (12) no. 1, Ekonomski fakultet, Zagreb, 2000.
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