An Efficient Algorithm for Solving the Applied Problem Scheduling Optimization of a Parallel-Sequential System

The problem of optimal regulation of airline fleet schedules by reassigning aircraft to flights is considered. The optimal regulation of schedules is to create or change them in such a way that minimizes system losses due to current violations. As an estimate of losses, the total deviation of the adjusted schedule from the spetified departure schedules of aircraft is used. It is shown that the described technological system belongs to the category of parallel-sequential systems. Accordingly, the considered system control problem is NP-hard and does not have effective algorithms for exact solution. A brief overview of approaches to solving its modifications and related fleet management tasks is given. The original formal formulation is given, the decomposition of the problem is justified, and an algorithm for its approximate solution is presented. An illustrative example is given and comparative statistics of testing software implementations of the decomposition algorithm of the schedule control problem on real data are reflected, proving the actual effectiveness of the developed tools.

Signaling Pathways in Immune Responses with Incomplete Information: A Novel Model Founded on Coupled Uniform Sequential Reaction Systems

A novel mathematical model for determining the forms of signaling pathways, ranging from linear structures to networks, on the basis of single-mode and multimodal time courses of immune responses under the condition of incomplete information, was elaborated. The subcellular immune processes and systems lacking the kinetic parameters still remain, in general, undetermined by deterministic quantitative models, which has resulted in the resignation from these models in favor of the so-called kinetically agnostic models. The model proposed in this paper intends to give a deterministic description of signaling pathways implicated in immune responses with unknown kinetic parameters. To do that, uniform sequential reaction systems and coupled sequential reaction systems were constructed using systems of differential equations and the novel notions of uniform functions, transition, coupling and branch operations founded on the assumptions of a uniform distribution of unknown rate constants, mass action kinetics, and first-order kinetics. Reaction step-dependent analytical solutions to the uniform sequential systems were expressed by the means of the incomplete gamma function and unambiguously characterized using, amongst others, the Lambert W function, which were necessary to obtain strict analytical solutions to those systems. These systems were validated by comparing their solutions with those obtained for ordinary sequential systems, regarded as the systems with complete information. The numerical solutions to the uniform sequential systems obtained by the Lederberg-Marquardt method were compared with those obtained by the Lambert W function. The signaling pathways were determined by analyzing the uniform sequential and coupled sequential reaction systems that describe the time courses of multimodal immune responses. Bi- and tri-modal immune processes representing temporal expressions of the transcription factor T-bet, the IL-12Rβ2 receptor, the IL-4, Tbx21 and Twist1 genes—all in helper T cells, calcium ion in 2G12.1 T cell hybridoma, c-Fos sumoylation in serum-stimulated HeLa cells, and IFN-γ-secreting cytotoxic T cells were modeled. Application of the coupled uniform sequential systems to the above immune processes allowed accurate approximations of their time behavior and to determine the structures of the signaling pathways and to predict some. A notion of the fractional reaction step, and its meaning in revealing the coupled sequential systems behind the seeming sequential ones, was introduced and analyzed. A consistent approach to the different forms of signaling pathways, beginning from the linear sequential systems through coupled (nonlinear) sequential ones to networks, was presented together with an algebraic insight into those systems.