UDK: APPLICATION OF MATHEMATICAL METHODS IN LOGISTICS Maryna Naumova, Daria Baz

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1 UDK: APPLICATION OF MATHEMATICAL METHODS IN LOGISTICS Maryna Naumova, Daria Baz ЗАСТОСУВАННЯ МАТЕМАТИЧНИХ МЕТОДІВ В ЛОГИСТИЦІ Марина Наумова, Дар я Базь Summary. Modern logistics is involved in traffic management issues of material, financial, documentary, information flows. It is necessary to integrate these flows throughout the supply chain and optimize its operations. Finding the optimal solution requires the analysis and accounting of a number of factors that can not be done without the use of advanced mathematical methods. The article provides an overview of the main economic and mathematical models and methods used in logistics. A cost minimization model of trade company activity is made. The network of branches with a minimum cost of functioning is created. The optimal number of sales representatives for the each branch and the annual cost of maintenance of work of branches and staff travel are calculated. Keywords: logistics, supply chain, mathematical methods in logistics, cost minimization model, optimal solution. Анотація. Сучасна логістика займається питаннями управління рухом матеріального, фінансового, документального, інформаційного потоків. Необхідно інтегрувати всі ці потоки в єдиний ланцюжок і оптимізувати його діяльність. Пошук оптимального рішення вимагає аналізу і врахування безлічі чинників, що неможливо здійснити без застосування сучасних математичних методів. У статті проведено огляд основних економікоматематичних моделей і методів, які використовуються в логістиці. Побудована модель мінімізації витрат діяльності фірми. Визначено оптимальний розподіл поїздок, оптимальну кількість торгових

2 представників для кожної встановленої філії і щорічні витрати на забезпечення роботи філій і поїздки співробітників. Ключові слова: логістика, ланцюжки поставок, математичні методи в логістиці, модель мінімізації затрат, оптимальний розв язок. Problem statement. In today's market, enterprises should first of all follow the interests of consumers but not by their own interests. This purpose should be reflected in the maximum customer satisfaction. The combination of a decent standard of quality goods and services and the other consumer properties means for the consumer a satisfaction of his needs. However, the important factor is the cost of goods and services, which depends on the level of costs arising from the production and sale. Reducing the costs is possible only with the use of different methods of logistics directly to the cycle of movement of goods and services. Modern logistics is involved in traffic management issues of material, financial, documentary, information (including the service) flows. Logistics is the science of market relations. It occurs when between the supplier and consumer companies appear very different forms of ownership, and everyone is trying to achieve their personal results. There is a need for integration of all streams in a single chain. The most typical problems of logistics in Ukraine is the lack of efficient management of supply chain, immature transport network, lack of network terminal and logistics systems, inadequate technical support, lack of knowledge in the field of increasing the efficiency of warehouse management and warehouse complexes. Supply chain optimization and automation of warehouse complexes using warehouse management systems are able to bring the most tangible benefits to businesses. Logistics is the management tool of industrial and commercial activity, which uses special logistics concept and economic-mathematical methods. Application of mathematics in economics is one of the most important areas in the development of economics and business, including logistics. Logistics has reached

3 a level where the application of mathematical methods become not only possible, but also necessary in the theory and practice. Finding the optimal solution requires the analysis and taking into account of a number of factors that cannot be done without using of advanced mathematical methods. Therefore, today, problem in the application of existing and development of new mathematical methods to improve the economic efficiency of logistics systems has become topical. Researches of many economists are devoted to the questions of logistics, for example Harrison A., van Hoek R. [1, Harrison, van Hoek, 2008], Chatham R. and Wilding R.D. [2, van Hoek, Chatham, Wilding, 2002], White A. [3, Harrison, White, 2006], Chiani G., Laporte G., Musmanno R. [4, Chiani, Laporte, Musmanno, 2004], Fisher M.L. and Rosenwein M.B. [5, Fisher, Rosenwein, 1989], Arntzen B.C., Brown G.G., Harrison T.P., Trafton L.L. [8, Arntzen, Brown, Harrison, Trafton, 1995], Huang G.H., Baetz B.W., Patry G.C., Terluk V. [9, Huang, Baetz, Patry, Terluk, 1996], Barnhart C., Schneur R.R. [10, Barnhart, Schneur, 1996], Anikin B.A. [11, Anikin, 2005], Mastjaeva I.N. [12, Mastjaeva, 2004] and others. These economists have proposed various approaches to the study of problems of logistics. At the same time, the ever-growing needs of the modern world require improving logistics methods to solve problems Ukrainian logistics. The purpose of this article is reviewing the basic mathematical methods in logistics, as well as the construction of a cost minimization model. The main material. Demand for logistics services in the region, as well as in Ukraine in general, only grows with each passing year. This is caused by the growth of the economy and increased competition in the consumer and industrial markets. Logistical activities extend from the moment of need for your product or service and the time to meet this need. All functions and operations need to be planned, managed and coordinated as a whole. All processes in the framework of the individual functions are coordinated with each other and create reserves of the

4 total cost reducing. The various activities are the basis of the logistics work: purchase of raw materials, materials procurement, production, product sales, transportation, information technology, as well as various control systems stock, product and service quality, planning and so on. Nowadays, the implementation of logistics tasks is possible only with the use of different economic and mathematical methods that allow you to optimize the activities of individual elements of the logistics system and the integration of these elements into a single structure. High precision of calculations in solving logistical problems is based on mathematical modeling of the learning process. Problems of logistics can be solved quickly and efficiently only with the help of mathematical methods and computers. The main models used in logistics are: 1) The optimum size of the parties supply (Wilson formulas); 2) Location of the supply bases (optimization model); 3) Attaching the consumer businesses to the supply bases (the gravity model); 4) Cross-industry flows (model input-output balance); 5) Product acceptance models; 6) Correlation-regression models; 7) Model of logistic systems (warehouses, shops, etc.); 8) Transport problem; 9) The problem of the assortment production load; 10) Network models (network graphics); 11) Applications of the game theory; 12) Models of periodic oscillations of logistic variables (demand, sales, material consumption). A set of mathematical methods in logistics involves a wide range of branches of mathematics, namely: classical mathematical analysis, probability theory and mathematical statistics, queuing theory, linear programming, reliability

5 theory, game theory, harmonic analysis. The method of linear programming, for example, can be used for: finding the optimal number of trips on the routes during a residence time (the problem to minimize the loss of working time); determination the best options for promoting uniform freight traffic from the sources of their generation to the destination (problems of minimize transportation costs); developing the optimal strategies for carriers orientation for specific group of customers (on a selected segment of logistics services); making rational routes (problems of minimize idle runs); efficient distribution of transport, loading and unloading facilities along the routes of supply chains (problems of maximum utilization of the working time of vehicles and working hours of loading and unloading mechanisms, and others.). Consider the cost minimization model of firm activity. Leaders of a trading company have decided to establish branches. It is expected that the final selection of the locations of the branches must be carried out on the basis of the following alternative set of potential locations: Lviv, Vinnitsa, Zaporizhia and Poltava. The main customers of the company are located in 16 regions. The expected number of trips to each region, and the average cost per visit of each alternative location (city) are presented in Table 1. The annual fixed costs for the operation of the branch in Lviv up UAH, Vinnitsa UAH, Zaporizhia UAH and Poltava UAH. Table 1: Annual number of trading company trips and travel costs for potential branches Travel costs (UAH) Region Lviv Vinnytsia Zaporizhia Poltava Number of trips Chernihiv Dnipropetrovsk

6 Ivano- Frankivsk Kharkiv Kherson Khmelnytskyi Kiev Kirovohrad Luhansk Mykolaiv Odessa Rivne Sumy Ternopil Zakarpattia Zhytomyr Annual fixed cost made by the authors It is expected that each sales representative makes up to 25 trips during the year. There are no restrictions on the number of sales representatives in each branch. The aim of the problem is the formation of a network of branches with a minimum cost of functioning. It is also necessary to determine the optimal number of representatives for the each branch and the annual cost of branches work maintenance and staff trips. We introduce the denotation: n the number of potential branches; m the number of regions where the company's customers are located;

7 T the annual number of trips in j th region ( j 1, m); j f i fixed costs of providing work of i th branch in one year; c ij the cost of a trip from i th branch to the j th region; t ij the number of trips from i th branch to the j th region; y i 1, if the branch is created, 0, otherwise. Then, the objective function will have the form: and restrictions: n n m, (1) Z f y c t min n i 1 i i ij ij i 1 i 1 j 1 tij Tj j 1,..., m, (2) y 0,1 i 1,..., n. (3) i As a result of applying "Solver" in Excel we get the following distribution of trips (Table 2). Table 2: The network of branches with a minimum cost of functioning Region Lviv Vinnytsia Zaporizhia Poltava Restrictions Chernihiv = 40 Dnipropetrovsk = 35 Ivano- Frankivsk = 100 Kharkiv = 25 Kherson = 40 Khmelnytskyi = 25 Kiev = 50 Kirovohrad = 30 Luhansk = 50 Mykolaiv = 65

8 Odessa = 40 Rivne = 30 Sumy = 20 Ternopil = 30 Zakarpattia = 40 Zhytomyr = 55 Z made by the authors Further, we define the optimal number of sales representatives for the each branch and the annual cost of maintenance of work of branches and staff trips. Expenses for sales representative s trips make up m j 1 ct ij ij, and the total costs f y m c t. i i ij ij j 1 The calculation results are presented in Table 3. Table 3: The number of sales representatives and the costs Lviv Vinnytsia Zaporizhia Poltava

9 Total Expenses for trips Total costs made by the authors Results of the solution show in what area of the branch of the company it is advisable to send representatives. For example, region sales representatives need to go in equal number from two branches: Zaporizhia and Poltava to Luhansk. Table 3 shows the calculation of the optimal number of representatives in each branch and the annual cost of maintenance of work of branches and staff trips. Conclusions. Success in business depends not only on the individual company's performance, but also on its partners suppliers, dealers, distributors, carriers, freight forwarders, etc. The need to ensure linkages of different tasks, functions and processes requires an inclusive, comprehensive and integrated approach based on the principles of logistics. Ensuring coherence of the scale as the business organizations and government agencies, as well as science and education, provide a decline in the level of costs on a national scale. This serves the interests of consumers and is a real step forward in improving the competitiveness of Ukrainian goods and services. References: [1] Harrison A., van Hoek R. (2008). Logistics management and strategy: competing through the supply chain. Pearson Education Limited. 343 p.

10 [2] Van Hoek R.I., Chatham R., Wilding R.D. (2002). People in supply chains: the critical dimension // Supply Chain Management. An International Journal. Vol. 7, No. 3. P [3] Harrison A., White A. (2006). Intelligent distribution and logistics // IEE Proceedings of Intelligent Transportation Systems. Vol. 153, No. 2. P [4] Chiani G., Laporte G., Musmanno R. (2004). Introduction to logistics systems planning and control (Wiley Interscience Series in Systems and Optimization). 360 p. [5] Fisher M.L., Rosenwein M.B. (1989). An interactive optimization system for bulk-cargo ship scheduling. Naval Research Logistics. P [6] Logistika (Logistics): ucheb. / pod red. B.A. Anikina. M.: INFRA-M, p. [7] Mastjaeva I.N. Matematicheskie metody i modeli v logistike (Mathematical methods and models in logistics). M., p. [8] Arntzen B.C., Brown G.G., Harrison T.P., Trafton L.L. (1995). Global supply chain management at Digital Equipment Corporation. Interfaces 25 (1). P [9] Huang G.H., Baetz B.W., Patry G.C., Terluk V. (1996). Capacity panning for an integrated waste management system under uncertainty: a North American case study // Waste Management. 15. P [10] Barnhart C., Schneur R.R. (1996). Air network design for express shipment service // Operations Research, 44. P Biography of the authors: 1. Name: Maryna Naumova. Academic titles: PhD in Physics and Mathematics, Associate Professor. Organization: Donetsk National University, Vinnytsia, Ukraine. A personal naumova.maryna@gmail.com

11 2. Name: Baz Daria. Academic titles: student. Organization: Donetsk National University, Vinnytsia, Ukraine