Ph.D., senior lecturer E.A. Shirobakina
Ph.D., Associate Professor N.V. Stetsenko
Ph.D., Senior Lecturer T.V. Khovanskaya
Ph.D., Associate Professor I.V. Abdrakhmanova
Volgograd State Academy of Physical Culture, Volgograd

Keywords: sports manager, linear programming, managerial decision.

Qualifications of sports managers is the most important factor in the complex process of reformation of the physical culture and sports movement in our country. The labor market analysis has revealed that, firstly, there is a great shortage of specialists in the sphere of sports management, even at the time when graduates of such specializations are being trained in higher educational institutions (L.A. Gromova, V.V. Timchenko, 2013). Secondly, the enhancement of targeted training of sports managers within the system of physical education largely depends on the introduction in the educational process of the new and radically reformed subjects, current information, technologies and training techniques, aimed at the formation of professional competences (S.I. Smirnov, 2014 , N.V. Kandaurova, 2014, O.V. Lomovtseva, A.Yu. Goloshumov, 2014 et al). Thirdly, professional incompetency of managerial and administrative staff in the field of PC&S is still a real-life and burning problem. Obviously, it is only individuals, who have received special vocational (including additional) training and retraining, who are able to fulfill their professional duties more successfully (S.G. Seyranov, 2002, A.A. Bazhenov, 2003).

Physical culture and sport belong to the sphere of non-material production, its product promotes good health in people, contributes to the formation of their socio-cultural activity. PC&S economics is the science that studies special techniques of practical problem solving in the field of sports activities associated with the use of various resources, both in individual sports organizations and society in general. The economics of PC&S involves different mathematical tools, methods of data representation and information computer technologies [2, 3].

The development of sports management in modern times is attended with the intensive use of computer technologies and computer equipment. It is informatization that provides the major transformation of work performed by a sports manager, as well as the integration of two types of activity - professional and information. The dialogue mode during problem formulation, the ability to evaluate the results and change conditions for problem solving while interacting with the computer make it possible to reduce the time spent on problem solution.

Technology-based approach has long dominated the issue of using technical aids as an auxiliary tool that expedites and simplifies work. In recent years, attention has been increasingly focused on the social component of the information process within the frame of which the models of information support of various activities of a sports manager are being developed (S.S. Filippov, V.V. Yermilova, 2009). The data component becomes an essential part of decision making within the management system of any organization, since incorrect decisions are often due to incorrect information. Sports and management information is a totality of the necessary for making of managerial decisions data on the status and processes occurring in physical education and sports organizations and the environment, in which conditions it operates (I.I. Pereverzin, 2004). Herewith, if used for solution of practical problems, information can transform into knowledge, but in this case it is to have an adequate structural, semantic and logical organization.

When implementing managerial functions, a manager of a sports organization has to make a large number of decisions. In this regard, the problem of development of managerial decision making is the most important and relevant in the management theory. Decision making is selection of a particular action from a variety of options. In this context, sports activities are not an exception, for example, when there is a need to gather a team, choose the playing tactics, organize athlete's diet, etc. In such cases, before making a decision, it is required to conduct a preliminary analysis. At the same time, the more complex the upcoming activity, the more important it is to anticipate the consequences and drop the unacceptable options to find the most satisfactory ones and so minimize the degree of subjectivity of a decision. There is a variety of methods of managerial decision making. One of the simplest ones in the conceptual aspect is the method of linear programming. This feature enables to widely use it in sports practice.

In order to solve the linear programming problems it is generally required to develop a mathematical model: select variables, set up the constraint system, formulate the objective function and, if necessary, some additional conditions (define the boundaries, within which the values of the decision variables may vary in the optimal solution). Problem solution is deemed feasible if all the constraints and boundary conditions are met.

One can find the optimal solutions to the linear programming problems in the software package MS Excel, using the Solver Add-in, this not only simplifies the process, but also makes the results more reliable. For instance, while solving a balanced problem the algorithm is realized by performing the following operations:

1) to determine the unknowns (variables) and their number, and tabularize these tasks in an Excel sheet;
2) to perform all intermediate calculations, in which case, as a rule, a row / column for assignments and the stock of products are calculated by the formula "SUM";
3) to select a separate cell and enter an optimization criterion (objective function) - the total prime cost of all the work done is calculated by the formula "SUMPRODUCT";
4) to select the cell containing the calculated value of the objective function, open the dialogue box Solver and find a solution, having indicated as a preliminary: the cell with the objective function, the type of optimization, the cells changing in the process of solution and the constraints imposed on the variables.

Linear programming problems relate to activity optimization. First and foremost, among these are the solutions to the following problems:

• Transportation - shipping cost reduction;
• Assignments - distribution of jobs among employees to achieve the best result at minimum cost;
• Product mix - maximum quantity of goods produced using limited stocks of raw materials;
• Diet (blending) - formulation of a diet of specified quality (diet or blending) at minimum cost;
• Resources - a variety of problems associated with the optimal resource allocation.

Each problem is matched by a situation in practice of a sports manager. The purpose of our study was to find and/or write the linear programming problems corresponding to the given problems that could contribute to managerial decision making, as well as minimization of cost and maximization of profit of a sports organization. Consider the examples of such problems.

A special type of mathematical linear programming problem, in which one must find the optimum allocation of similar objects at minimum shipping cost is called the "transportation problem". Since hotel and transport services for athletes and teams are included into the package of services offered to consumers, a sports manager is to know and be able to formulate the transportation problem and minimize customer service costs [4].

Transportation problem. Guests and athletes arrive from different regions by different means of transport: at the airport "Domodedovo" - 11 people, at the airport "Vnukovo" - 15, at the Paveletskiy station - 40, and at the bus station - 19. Three hotels in different areas were booked - in the first one - 30 (0, 5, 10, 12), in the second one - 32 (14, 15, 0, 7), in the third one - 23 (16, 15, 9, 20) rooms. To minimize the cost of transfer of the competitors. The cost of transferring of one person from the place of arrival to the hotel is enclosed in a parenthesis.
Among the problems to be solved in management, there is a sort when to perform some similar work it is necessary to designate executive agents. It is required to split the work among the executives in such a way as to maximize (minimize) the total efficiency (inefficiency) criterion of execution of all jobs.

The problems like that are also called the "assignment problems" and can be considered as a special case of the transportation problem. If the number of executives equals the number of jobs done, then this is a balanced problem, otherwise - unbalanced. In case of the balanced assignment problem two conditions are met: each executive agent performs one job only, and each job is performed by one executive only. There are two options for the unbalanced problem: if the number of executives exceeds the number of jobs done and if the number of executives is less than the number of jobs done. In both cases, in order to solve the unbalanced problem it is reduced to the balanced one by means of the introduction of false jobs or false executive agents with zero values. Situations like that often arise in organizations, for example, in cases when the number of sports groups exceeds the number of instructors. Such problems can be aimed both at cost minimization and profit maximization. Consider the example of the balanced problem of appointment of instructors.

Cost-minimization assignment problem. There are 5 instructors and 7 groups of trainees. Each instructor is able to conduct some particular types of training, their hourly rate is presented in the table. Plan the workout in such a way that each instructor could conduct two types of workouts only, and the total cost of hourly rate was minimal.

Profit-maximization assignment problem. There are 7 instructors and 5 groups of trainees. Each instructor is able to conduct some particular types of training, the profit made by each instructor is represented in the table. It is required to distribute the applications among the instructors so that the total value was maximum, and if, in compliance with loading, each instructor can conduct up to 3 workouts per day, all types of training classes should be daily conducted in 2 groups of trainees.

The main strategic purpose of any commercial enterprise, including those related to physical culture and sport, is to maximize profits earned on the services, provided that customers are fully satisfied. The cost of the services offered by the company depends on various factors: company image, service availability and appeal, comprehensive range and ability to efficiently use the available equipment.

Scheduling problem. Suppose in a fitness club there are 24 fitness instructor per shift, each of them can carry out both group and personal trainings. Moreover, these instructors conduct 25 group (consisting of 10 people each) trainings or 100 personal ones per shift, or both, defined by the acceptable labor efforts. The number of lockers in the locker room, simulators and halls allow no more than 70 clients at a time. Personal training is known to cost 5 times as much as the group one. It is necessary to include in the schedule the maximum number of group and personal trainings in order to realize maximum profit.

Besides, the basic functions fulfilled by sports managers (planning, organization, management and analysis), their activity includes the specific ones - the technical skill to establish a relationship between physical loads, diet and so on. [1].

Athletes' diet quality and dietary regime, as well as the training process, are the essential components of their good shape. The main feature of athlete's diet is an increased caloric intake, which corresponds to an increased energy consumption. Also, in order to preserve health and working capacity, it is required to consume a certain amount of nutrients per day (vitamins, fats, trace elements, proteins, water, etc.). For different sports there are different nutritional requirements for an athlete. For instance, in strength sports, daily protein allowance should be such as to provide not less than 2 grams of protein per one kilogram of athlete's body mass. The task of a manager in the field of physical culture and sport is to determine not only the amount of nutrients needed for an athlete to meet the daily requirements, but also the cheapest diet. An enlightened manager in the field of physical education and sport should be able to develop a mathematical model that would correspond to the actual problem like this [5, 6].

All sports that require strength, speed, endurance, flexibility and motor coordination are nominally divided into several groups: team sports (basketball, volleyball, football), complex coordination sports (artistic gymnastics, rhythmic gymnastics, aerobics, acrobatics), cyclic sports (swimming, rowing, track and field athletics) and martial arts (boxing, weightlifting). Depending on the group, differentiated and individual adjustment of diet is required to increase performance and faster recovery after exercises. In view of the mentioned sports we selected the necessary data to write the daily diet problems with an indication of the demand rate of nutrients optimal for this kind of sport.

The diet problem. How many products of each kind does an athlete need to meet his nutritional and vitamin requirements at minimum cost?
Input data: 1) Eating occasion - breakfast; 2) Athlete's specialization - artistic gymnastics; 3) Athlete's weight - 60 kg; 4) Foodstuffs: a) barley (6 rubles. per 100 g); b) cheese "Rossiyskiy" (25 rubles. per 100 g); 5) Nutrients: a) proteins; b) fats; b) carbohydrates; 6) Vitamins: a) potassium; b) calcium.
Fans form the main group of consumers of sports events. The most spectacular sports club performances are the most visited. If athletes' skills are insufficient, audience appeal fades and thus profits decrease. To improve this situation, it is necessary to invest in new athletes [1].

Athlete recruitment problem. A trainer needs more athletes to gather a football team. What is the optimal number and type of players (i.e. cheap and reliable) to be gathered, if a player from the "Spartak" team is worth $10,000, and from "Uralmash" - $8000. In addition, each player from the "Spartak" team possesses speed, endurance and technique rated 5, 3, and 4 points, respectively, and a player from the "Uralmash" team possesses the same qualities rated 2, 4 and 3 points. The overall index of teamed-up athletes should be as follows: at least 90 in technique; at least 75 in endurance; at least 80 in speed.

In the stadiums and arenas, during sports events, fans are often offered souvenirs for sale - badges, flags, T-shirts, caps, pennants with symbols, reference books and other business-mascots. Souvenirs sales enable a sports club to produce extra income, as well as help attract fans, individuate a team, create its image [2].

Profit-maximization souvenir problem. To produce two kinds of souvenirs - A and B, three types of raw materials are used. The production of one souvenir A requires 12, 4 and 3 units of raw materials 1, 2 and 3 respectively. The production of one souvenir B requires 4, 4, and 12 units of the same types of raw materials respectively. There are 300, 150 and 252 units of raw materials 1, 2 and 3 in stock. The profit on sales of souvenirs A and B equals $30 and $40 respectively. It is necessary to work out a business plan, in which sales proceeds will be maximum, provided that there should be more souvenirs B rather than souvenirs A.

Cost-minimization souvenir problem. To present the souvenirs after the competition, it is necessary to form two kinds of gifts (for athletes who outstood with their performances). The 1st gift includes 3 baseball caps, 4 badges/pennants and a T-shirt, and the 2nd one - 2 baseball caps, 6 badges/pennants and 3 T-shirts. It is known that the sponsors of the competition provided 10 baseball caps, 20 badges/pennants and 7 T-shirts. The 1st gift costs $3, and the 2nd one - $4. What kind and how many gifts need to be formed in order to use all the resources and minimize cost?

Conclusions. The presented problems cannot fully correspond to the actual problems that a sports manager solves during his professional activity, but the knowledge and skills resulting from the study of this material will enable to obtain the information necessary for making managerial decisions in order to minimize costs and maximize profits.

We assume that solution of linear programming problems and writing of own ones will contribute to the formation of the administrative and economic knowledge of students of the specialization "Sports management". All interested are welcome to discuss this topic and offer own position on the matter.

References

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Corresponding author: shirobackina_prepod@mail.ru