Network Flows in Logistics (FSI-SNF-A)

Academic year 2026/2027
Supervisor: RNDr. Pavel Popela, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
Course type: departmental course
Aims of the course unit:

 Emphasis is placed on gaining deep knowledge of models and methods for solving network optimization problems with a focus on logistics applications. The objectives are aimed at problem analysis, creation of mathematical models including their notation, finding suitable transformations, selection, modification, and implementation of algorithms. The mentioned models and methods are supported by the explanation of selected theoretical knowledge. 

The course is intended for logistics students and may also be useful for students of applied sciences and engineering. Students will gain deeper knowledge of network flows and integer optimization in relation to logistics problems. They will also gain an understanding of the application of network optimization models in typical applications. 

 
Learning outcomes and competences:
 
Prerequisites:

Knowledge of basic concepts of linear optimization and graph theory (following the course SGA-A Graphs and Algorithms) is assumed.  

 
Course contents:

The course focuses on basic network models and methods for solving logistics problems. The lectures build on linear optimization models and deepen and concretize the understanding of the following general principles of mathematical optimization: understanding the network problem, constructing a network model, considering the possible integrality of variables, finding, analyzing, and interpreting the optimal solution. The course mainly covers the issue of network flows (typical problems, LP model formulations, graph formulations, special algorithms for solving problems). The lectures also include an introduction to the related issue of integer programming (problem formulations, integer flows, network designs, indicator variables, selected algorithms, and their software implementations). A specific case of the distribution of green micromobility resources to their rental locations, their relocation, and service will be discussed, leading to a multi-criteria integer optimization problem on a Time-Space network. A specific computational algorithm will be proposed, and its complexity will be discussed.  
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:

The exam is written and includes formulation, computational, and theoretical questions. An oral discussion follows the written work. Attendance is monitored through active student participation in solving problems, and missed classes are compensated by independently solving assigned tasks.  
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

1. Motivational problems and basics of modeling network problems.


2. Transportation problems, their LP modeling, and their (software) solutions.


3. LP models of minimum flow problems in networks and their (software) solutions.


4. Special minimum flow problems in networks (LP models for finding the shortest path and for the assignment problem). Maximum flow problem in networks.


5. Pitfalls of solving network problems using the simplex method and their solutions.


6.-7. Efficient methods for solving selected network problems.


8. Integrality of solutions in network problems.


9.-10. Mathematical formulation of the multi-criteria integer optimization problem on a Time-Space network.


11.-13. Design of a computational algorithm, implementation, and complexity.
    Exercise

1. Examples of logistics problems and their modeling as network problems.


2. Examples of transportation problems, their LP modeling, and their (software) solutions in a modeling language.


3. Examples of LP models of minimum flow problems in networks and their (software) solutions in a modeling language.


4. Examples of special minimum flow problems in networks (LP models for finding the shortest path and for the assignment problem). Example of the maximum flow problem in networks.


5. Examples of problems in solving network problems using the simplex method and their solutions.


6.-7. Efficient methods for solving selected network problems and examples of their use.


8. Examples of the integrality of solutions in network problems.


9.-10. Formulation of the multi-criteria integer optimization problem in a modeling environment.


11.-13. Design and implementation of solution algorithms (transformation to single-objective, Pareto frontier).
Literature - fundamental:
1. Ghiani, G. et al. Introduction to Logistics Systems Management, 3rd edition, Wiley, 2022.
2. Bazaraa, M.S. et al. Linear Programming and Network Flows, 4th edition, Wiley, 2010.
3. Wolsey, L. Integer Programming, 2nd Edition, Wiley, 2020.
4. Williamson, D.P. Network Flow Algorithms, Cambridge University Press, 2019.
5. Garcia-Diaz A. and Phillips, D.T. Fundamentals of Network Analysis and Flow Optimization, Virtualbook Publishing, 2022.
Literature - recommended:
1. Ghiani, G. et al. Introduction to Logistics Systems Management, 3rd edition, Wiley, 2022.
2.

Bazaraa, M.S. et al. Linear Programming and Network Flows, 4th edition, Wiley, 2010.
3. Sofer, A. et al. Linear and Nonlinear Optimization (Chapter 8 Network Problems), 2nd Edition, Society for Industrial and Applied Mathematics, 2009.
4. Williams, H.P. Model Building in Mathematical Programming, 5th edition, Wiley, 2013.
5. Nemhauser, G.L. and Wolsey, L.A. Integer and Combinatorial Optimization, 1st Edition, Wiley, 1988. 
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-LAN-A full-time study --- no specialisation -- Cr,Ex 5 Compulsory 2 1 S
C-AKR-P full-time study CLS -- Cr,Ex 5 Elective 1 1 S