# Operational Analysis

- Credits: 6
- Ending: Examination
- Range: 2P + 2C
- Semester: winter
- Year: 2
- Faculty of Business Economics with seat in Košice

## Teachers

## Included in study programs

**Teaching results**

The aim of the course is to present knowledge of graph theory as a modern mathematical discipline with extensive application in practice with emphasis on the application of this theory to optimize tasks in management / economics. Furthermore, students will acquire knowledge of solving linear programming problems and master the solution of problems using various variations of the simplex method.

Knowledge:

about network planning, solving time projects, identification of critical project paths,

about non-time network projects, time-cost projects, linear programming tasks. with emphasis on solving the transport task. Solving the problem of linear programming with emphasis on the simplex method.

Skills:

identify critical project paths and address the likelihood of project completion in the required time,

solve time-cost projects, linear programming tasks with emphasis on solving the transport task,

solve linear programming problems by simplex method,

evaluate basic microeconomic phenomena and processes using empirical and quantitative approaches,

search, process and analyze microeconomic information from various sources and apply it to practical case studies.

Competences:

ability to formulate and solve network planning tasks using CPM, PERT methods for planning, solving time projects and identifying critical project paths,

ability to identify the possibilities of using non-time network projects, time-cost projects, linear programming tasks,

ability to think abstractly and analytically economically.

**Indicative content**

Lectures:

1. Introduction to the subject. Contents. Sequence of studies. Forms of study of the subject. Literature. Basic concepts of graph theory Basic concepts of network planning. Time network projects - common procedures in CPM and PERT method

2. Network graph analysis. Calculation of project duration. Calculation of time reserve in network graph nodes. Calculation of time reserves for project activities. Identification of critical paths of the project. Monitoring the implementation of projects. Resource management

3. Specific procedures in the PERT method. Calculation of the average project duration

4. Calculation of the probability of project implementation within the set deadline. Calculation of the project duration for a given value of the probability of its completion.

5. Time-cost CPM method. Calculation of the project duration in the normal mode of all activities. Calculation of project duration in the limit mode of all activities. Calculation of minimum costs for the marginal duration of the project. Calculation of the cheapest project for the required duration

6. Non-time network projects.

7. Introduction to linear programming. Formulation and solution of the transport task.

8. Methods of determining the primary solution, optimality test, unbalanced traffic problems.

9. Assignment problem and methods of its solution.

10. General role of linear programming. Simplex method of ÚLP solution. Simplex algorithm, natural basis. Additional variables, artificial base method. Duality in linear programming, duality theorems.

11. Solution of primary and dual problems by primary simplex algorithm.

12. Dual simplex algorithm. Integer programming, methods of cutting surfaces. Gomory algorithm I. Gomory algorithm II. Combinatorial methods for solving integer programming problems.

13. Land's and Doig's method.

Exercises:

1. Introductory exercise. Requirements for meeting the conditions of the mid - term evaluation. Graph theory - basic concepts

2. Finding a critical path in the network - CPM

3. Search for a critical path in a network with a stochastic duration of activities (PERT).

4. Non-time network projects (minimum voltage tree) / (maximum network flow).

5. Non-time network projects - business traveler method.

6. TEST 45 minutes (only full-time study). Finding a critical path in the network and cost analysis.

7. Formulation of the task of linear programming. Examples of simple ÚLP.

8. Balanced transport tasks, unbalanced transport tasks, finding initial solutions.

9. Simplex method - natural base, artificial base method, additional variables - 1.

10. Simplex method - natural base, artificial base method, additional variables - 2.

10. TEST 45 minutes. Assignment problem.

12. Formulation of dual tasks. Determining the solution of primary and dual tasks. Dual simplex algorithm.

13. Gomory algorithm. Correction of ongoing evaluation.

**Support literature**

Elementary literature:

1. BREZINA, Ivan - IVANIČOVÁ, Zlatica - PEKÁR, Juraj. Operačná analýza. Bratislava : Iura Edition, 2007. Ekonómia. ISBN 978-80-8078-176-7.

2. BREZINA, Ivan - PEKÁR, Juraj. Operačná analýza v podnikovej praxi. Bratislava : Vydavateľstvo EKONÓM, 2014. ISBN 978-80-225-4012-4.

3. BREZINA, Ivan - PEKÁR, Juraj. Úvod do operačného výskumu II.. Bratislava : Letra Edu, 2019. ISBN 978-80-89962-28-0.

4. FRONC, M.: Operačná analýza I. – Bratislava : alfa, 1989.

5. FRONCOVÁ, H. – LINDA, B.: Operačná analýza – Návody na cvičenia. – Bratislava : Alfa, 1988.

6. IVANIČOVÁ, Z. – B BREZINA, I. – P PEKÁR, J.: Operačný výskum + CD ROM. Bratislava : Iura Edition, 2002. 292 s. ISBN: 80-89047-43-2

7. JENDROĽ S., MIHÓK P.: Diskrétna matematika I (Úvod do kombinatoriky a teórie grafov), UPJŠ, Košice,1993.

8. KOŘENÁŘ, V. – LAGOVÁ, M. – JABLONSKÝ, J. – DLOUHÝ, M.: Optimalizační metody. Praha : VŠE, 2003. 188 s. ISBN: 80-245-0609-2

9. LAŠČIAK, A. a kol.: Optimálne programovanie. – Bratislava : alfa, 1991. 600 s. MDT 05.012.12(075.8)

10. PLESNIK, J.: Grafove algoritmy. Veda, Bratislava 1983

11. PLESNÍK, J. – DUPAČOVA, J. – VLACH, M: Linearne programovanie. – Bratislava : alfa, 1990. 320 s., ISBN 80-05-00679-9

12. RAČKO, J.: Základy operačnej analýzy 1, Manažment projektov (sieťová analýza) – Liptovský Mikuláš : Vojenská akadémia, 1998, 100 s. ISBN 80-8040-081-4

13. SAKÁL, P. – JERZ, V.: Operačná analýza v praxi manažéra. Trnava : SP Synergia, 2003. 342 s. ISBN: 80-968734-3-1

14. SAKÁL, P. – JERZ, V.: Operačná analýza v praxi manažéra II. Trnava : SP Synergia, 2006. 360 s. ISBN: 80-969390-5-X

15. SEDLÁČEK: Úvod do teórie grafu, Académia, Praha, 1977.

16. MATOUŠEK J., NEŠETŘIL: Kapitoly z diskrétní matematiky, Matfyzpress, vydavateľství MFF UK, Praha, 1996.

17. WILLIAMS, H. P.: Model Solving in Mathematical Programming. – New York : John Wiley & Sons, 1992.

Supplementary literature:

18. FENDEK, M. – MLYNAROVIČ, V.: Optimálne programovanie I. Bratislava : ES VŠE, 1989.

19. FENDEK, M.: Nelineárne optimalizačné modely a metódy. Bratislava : EKONÓM, 1998.

20. GASS, S., I.: Lineárne programovanie, Bratislava : alfa, 1972

21. JENDROĽ, S. – MIHÓK, P.: Diskrétna matematika I (Úvod do kombinatoriky a teórie grafov). Košice : UPJŠ, 1993.

22. MATOUŠEK J. – NEŠETŘIL: Kapitoly z diskrétní matematik.y. Praha : Matfyzpress, vydavatelství MFF UK, 1996.

23. MURTAGH, B.A.: Advanced Linear Programming, Computation and Practice. NewYork : McGraw Hill, 1981.

24. PLESNÍK, J.: Grafové algoritmy. – Bratislava : Veda 1983.

25. PITEL, J.: Ekonomicko-matematické metódy, Bratislava : Príroda, 1988

26. SEDLÁČEK: Úvod do teórie grafu. – Praha : Académia, 1977.

**Syllabus**

Lectures: 1. Introduction to the subject. Contents. Sequence of studies. Forms of study of the subject. Literature. Basic concepts of graph theory Basic concepts of network planning. Time network projects - common procedures in CPM and PERT method 2. Network graph analysis. Calculation of project duration. Calculation of time reserve in network graph nodes. Calculation of time reserves for project activities. Identification of critical paths of the project. Monitoring the implementation of projects. Resource management 3. Specific procedures in the PERT method. Calculation of the average project duration 4. Calculation of the probability of project implementation within the set deadline. Calculation of the project duration for a given value of the probability of its completion. 5. Time-cost CPM method. Calculation of the project duration in the normal mode of all activities. Calculation of project duration in the limit mode of all activities. Calculation of minimum costs for the marginal duration of the project. Calculation of the cheapest project for the required duration 6. Non-time network projects. 7. Introduction to linear programming. Formulation and solution of the transport task. 8. Methods of determining the primary solution, optimality test, unbalanced traffic problems. 9. Assignment problem and methods of its solution. 10. General role of linear programming. Simplex method of ÚLP solution. Simplex algorithm, natural basis. Additional variables, artificial base method. Duality in linear programming, duality theorems. 11. Solution of primary and dual problems by primary simplex algorithm. 12. Dual simplex algorithm. Integer programming, methods of cutting surfaces. Gomory algorithm I. Gomory algorithm II. Combinatorial methods for solving integer programming problems. 13. Land's and Doig's method. Exercises: 1. Introductory exercise. Requirements for meeting the conditions of the mid - term evaluation. Graph theory - basic concepts 2. Finding a critical path in the network - CPM 3. Search for a critical path in a network with a stochastic duration of activities (PERT). 4. Non-time network projects (minimum voltage tree) / (maximum network flow). 5. Non-time network projects - business traveler method. 6. TEST 45 minutes (only full-time study). Finding a critical path in the network and cost analysis. 7. Formulation of the task of linear programming. Examples of simple ÚLP. 8. Balanced transport tasks, unbalanced transport tasks, finding initial solutions. 9. Simplex method - natural base, artificial base method, additional variables - 1. 10. Simplex method - natural base, artificial base method, additional variables - 2. 10. TEST 45 minutes. Assignment problem. 12. Formulation of dual tasks. Determining the solution of primary and dual tasks. Dual simplex algorithm. 13. Gomory algorithm. Correction of ongoing evaluation.

**Requirements to complete the course**

individual work, reports, written work, continuous tests, combined exam

Continuous assessment: 40%

activity in exercises / consultations (ES) and continuous verification of knowledge -10%

result of semester tests -10%

evaluation of reports from 3 assigned topics from lectures - 10%.

evaluation of written semester work – project -10%

Result of the final combined exam: 60% (written exam and oral part of the exam)

Note:

The condition for taking the exam is uploading 3 reports from individually determined topics to the student, examples from exercises and written semester work (project) into the platform).

**Student workload**

participation in lectures and seminars - 52 hours

preparation for active forms of teaching - 20 hours

elaboration of reports from lectures, examples for exercises and written semester work - project - 30 hours

preparation for the continuous semester test - 10 hours

preparation for the final exam test and oral exam - 44 hours

Total: 156 hours

**Language whose command is required to complete the course**

Slovak

Date of approval: 22.02.2022

Date of the latest change: 20.12.2022