MÁLEK, M. Stochastická optimalizace toků v sítích [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2017.

The author has completely fulfilled the goals defined in the specification of thesis. The diploma thesis is written in English, its length is over 80 pages and the text is divided into five numbered chapters. The theme of this diploma thesis is focused on stochastic optimization of network flows. In particular, the author dealt with the progressive hedging decomposition algorithm (PHA) and its effective implementation for selected problems. The chosen topic is challenging and required a profound self-study of recent research papers, especially these written by the principal stochastic programming authorities Wets and Rockafellar. However, at first, it was necessary to deepen the author's knowledge in the areas of graph theory, mathematical programming, network flow problems, probability theory and mathematical statistics. Next, he has focused on models of stochastic programming, and especially, on the issues of decomposition, and implementation of large-scale models. The student has presented the selection of his utilized knowledge in the initial chapters. The only critical point of the diploma thesis is the considerable heterogeneity and variability of the utilized notation. However, there is no doubt that as a supervisor, I could have recognized and pointed out this problem earlier. The author still has partially compensated this deficiency with a detailed list of used symbols with references to related chapters. The author has worked independently and, if necessary, he has consulted not only with the supervisor but also with other specialists that are listed in the acknowledgment of thesis. Chapter 4 begins with the original use of Farmer's Problem from stochastic programming monograph written by Birge and Louveaux for author's PHA testing, see Section 4.2. The detailed completion of the results presentation, including tables and figures, delights the reader's eye. The author has further modified the PHA for stochastic network flow problems including network design optimization, see Sections 4.1 and 4.3. Because these are integer programs, he needed to convert PHA for nonconvex cases into heuristics. I highly appreciate the software code realization in GAMS and C ++ although it is seldom mentioned in thesis. The achieved application results for real-world waste management data are successfully visualized. In conclusion, from the point of view of the Institute of Mathematics, I would like to regret the fact that the author prefers to work as an expert analyst in the energy industry before the choice of doctoral study. However, I wish him full success. At the same time, I appreciate that, despite this destiny choice, the author has contributed to the publication of main results of thesis in the WoS indexed series of Springer Lecture Notes. Based on the above, I recommend the thesis for defense.

Kritérium | Známka | Body | Slovní hodnocení |
---|---|---|---|

Splnění požadavků a cílů zadání | A | ||

Postup a rozsah řešení, adekvátnost použitých metod | A | ||

Vlastní přínos a originalita | A | ||

Schopnost interpretovat dosažené vysledky a vyvozovat z nich závěry | A | ||

Využitelnost výsledků v praxi nebo teorii | A | ||

Logické uspořádání práce a formální náležitosti | B | ||

Grafická, stylistická úprava a pravopis | A | ||

Práce s literaturou včetně citací | B | ||

Samostatnost studenta při zpracování tématu | A |

- Navrhovaná známka
- A

The thesis is concerned with stochastic optimization, and its application on artificial and real-world problems of network flows. Particular emphasis is put on large-scale issues. The first two chapters give a brief introduction to the Graph Theory and Mathematical Programming. In the latter one the author describes various reformulations of the problem. The issue of multi-stage stochastic programming is introduced as the basis for the following chapter dealing with Progressive Hedging algorithm (PHA). The PHA is afterwards applied, among others, as a solution method for large-scale mixed-integer problem designed at the Institute of Process Engineering. The theoretical part is appropriately followed by several examples and graphical schemes to make clear the described situation. Nevertheless, the thesis would benefit from improvements in formal accuracy. In some cases the notion is unclear (numerous symbols are not defined, e.g. what is “I” down on pg. 30?), and variables are confusing (duplicate marking, see “y_{j,s}” definition on pg. 37; “A” in Ch. 1-2 has about 4 different meanings). The reviewer wonders why the probability is denoted inconsistently through the text, i.e. “p_s” in eq. (3.7), “P_s” in (4.2), and “Pr_s” in (4.4). Extremely inconvenient is to use the variables “i”, “s”, and “is” in (4.4). Also, occasionally grammatical mistakes do not improve the readability. To the end, the bibliography style in not uniform; possibly reference to GAMS and AIMMS web pages should be added. On the other hand, besides of such minor inaccuracies, the thesis is written well, with high potential for the application in practical issues. The author extended the models discussed in his bachelor´s thesis first, and shows possibilities in solution of sizable mixed-integer problems. The topic is extremely relevant. Hereby, it should also emphasized the computational implementation of the task. The author combined modelling tools (GAMS, AIMMS) containing advanced solvers and C++ for his own implementation of PHA. The results are thoroughly discussed and followed with high quality visualization. The aims of the thesis were fulfilled and the reviewer recommends the thesis for defense.

Kritérium | Známka | Body | Slovní hodnocení |
---|---|---|---|

Splnění požadavků a cílů zadání | A | ||

Postup a rozsah řešení, adekvátnost použitých metod | A | ||

Vlastní přínos a originalita | A | ||

Schopnost interpretovat dosaž. vysledky a vyvozovat z nich závěry | A | ||

Využitelnost výsledků v praxi nebo teorii | A | ||

Logické uspořádání práce a formální náležitosti | C | ||

Grafická, stylistická úprava a pravopis | B | ||

Práce s literaturou včetně citací | B |

- Navrhovaná známka
- B

- According to the author’s “definition” on the page 14, a subgraph need not to be a graph. Please, explain the problem and suggest correction.
- What is the true difference between WS and IS reformulation? In both cases is the solution a random variable. How could be these approaches compared?
- In fig. 4.2.1 there is evident that the convergence of PHA is slow for large r. Nevertheless, r too small leads to some kind of oscillation which does not improve the convergence. Are there some general comments or suggestions to the choice of r (e.g. in the literature etc.)?
- It is unclear how were obtained the weights alpha_i in eq. (4.1), or beta_i on page 34, respectively. Have been the betas drawn from some probability distribution?
- Passing from eq. (2.9) to eq. (2.11), the feasible set C changes to the set R^n. Is it necessary to make such extension of the feasible set, i.e. is it possible to consider a “smaller” set (if C is bounded) according to the behavior of the random variable?

eVSKP id 101216