It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. Archimedean semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. We revisit this problem by using some well-known formulations concerning regular hexagons. Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.
#Circles in rectangle optimization code
We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation.
![circles in rectangle optimization circles in rectangle optimization](https://study.com/cimages/multimages/16/circle_of_radius_3_with_inscribed_rectangel880922437940175876.png)
Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes.
![circles in rectangle optimization circles in rectangle optimization](https://wallup.net/wp-content/uploads/2016/01/268218-digital_art-abstract-minimalism-geometry-painting-paint_splatter-rectangle-square-pattern.jpg)
The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications.