T his paper considers mathematical optimization for the multistage train formation problem, which at the core is the allocation of classification yard formation tracks to outbound freight trains, subject to realistic constraints on train scheduling, arrival and departure timeliness, and track capacity. The problem formulation allows the temporary storage of freight cars on a dedicated mixed-usage track. This real-world practice increases the capacity of the yard, measured in the number of simultaneous trains that can be successfully handled. Two optimization models are proposed and evaluated for the multistage train formation problem. The first one is a column-based integer programming model, which is solved using branch and price. The second model is a simplified reformulation of the first model as an arc-indexed integer linear program, which has the same linear programming relaxation as the first model. Both models are adapted for rolling horizon planning and evaluated on a five-month historical data set from the largest freight yard in Scandinavia. From this data set, 784 instances of different types and lengths, spanning from two to five days, were created. In contrast to earlier approaches, all instances could be solved to optimality using the two models. In the experiments, the arc-indexed model proved optimality on average twice as fast as the column-based model for the independent instances, and three times faster for the rolling horizon instances. For the arc-indexed model, the average solution time for a reasonably sized planning horizon of three days was 16 seconds. Regardless of size, no instance took longer than eight minutes to be solved. The results indicate that optimization approaches are suitable alternatives for scheduling and track allocation at classification yards.

Eye rheum is a physiological discharge, which accumulates at the medial angle of the healthy eye soon after opening in the morning. Microscopic evaluation of eye rheum revealed the presence of viable neutrophils, bacteria, epithelial cells, and particles, aggregated by neutrophil extracellular traps. We observed that in the evening, during eye closure, high C5a recruited neutrophils to the tear film and activated them. In this hypoxic area rich in CO2, neutrophils fight microbial aggressors by degranulation. Immediately after eye opening, the microenvironment of the ocular surface changes, the milieu gets normoxic, and loss of CO2 induces subtle alkalinization of tear film. These conditions favored the formation of neutrophil extracellular traps (NETs) that initially covers the ocular surface and tend to aggregate by eyelid blinking. These aggregated neutrophil extracellular traps (aggNETs) are known as eye rheum and contain several viable neutrophils, epithelial cells, dust particles, and crystals packed together by NETs. Similar to aggNETs induced by monosodium urate crystals, the eye rheum shows a robust proteolytic activity that degraded inflammatory mediators before clinically overt inflammation occur. Finally, the eye rheum passively floats with the tear flow to the medial angle of the eye for disposal. We conclude that the aggNETs‐based eye rheum promotes cleaning of the ocular surface and ameliorates the inflammation on the neutrophil‐rich ocular surfaces.

Integer Programming (IP) has been used to model educational timetabling problems since the very early days of Operations Research. It is well recognized that these IP models in general are hard to solve, and this area of research is dominated by heuristic solution approaches. In this paper a Two-Stage Decomposition of an IP model for a practical case of high school timetabling is shown. This particular timetabling problem consists of assigning lectures to both a timeslot and a classroom, which is modeled using a very large amount of binary variables. The decomposition splits this model into two separate problems (Stage I and Stage II) with far less variables. These two separate problems are solved in sequence, such that the solution for the Stage I model is given as input to the Stage II model, implying that irreversible decisions are made in Stage I. However, the objective of the Stage II model is partly incorporated in the Stage I model by exploiting that Stage II can be seen as a minimum weight maximum matching problem in a bipartite graph. This theoretically strengthens the decomposition in terms of global optimality. The approach relies on Hall's theorem for the existence of matchings in bipartite graphs, which in its basic form yields an exponential amount of constraints in the Stage I model. However, it is shown that only a small subset of these constraints is needed, making the decomposition tractable in practice for IP solvers. To evaluate the decomposition, 100 real-life problem instances from the database of the high school ERP system Lectio are used. Computational results show that the decomposition performs significantly better than solving the original IP, in terms of both found solutions and bounds.

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