If the rectangle is $257 \times 157$ and the radius of a circle is $\sqrt \approx 36592.5$. (Also, if the rectangle is only $2m \cdot r$ units tall, we can alternate columns with $m$ and $m-1$ circles.) So if you want the triangular packing to have $m$ circles in each column, and $n$ columns, then the rectangle must be at least $(2m 1) \cdot r$ units tall and $(2 (n-1)\sqrt3) \cdot r$ units long. Each pair of vertical blue lines is a distance $r \sqrt 3$ apart, and they're still a distance $r$ from the edges. If the circles have radius $r$, then each pair of horizontal red lines is a distance $r$ apart, and they're a distance $r$ from the edges. Giving the profit of each circle is: P(a) = 200 - 200/a (a is the area of the circle)Ĭonsider the following diagram of a triangular packing: So my question is: Did I calculate it in a correct way? Are there any other more effective calculation methods?īecause in later question, it asks me to find the area of the circle to so that we get the maximum profit. However, I find my math calculation kinda inefficient, long, and not correct in any other cases. > That means in this case, i can fit in 43*72= 3096 circlesĢ) Then I try triangular pattern, which can fit more circles, 3575 circles. ![]() I had the height 157/d (diameter) -> I got about 43.999 -> So along the height, i can place 43 circle.I had the width 257/d (diameter) -> I got about 72.024 -> So along the width, i can place 72 circle.So, i try to pack as many as possible (taking this website as reference):ġ) First, I tried to place them in rectangular pattern: After a lot of research, I found out that there are no optimal solution. Resultados numéricos son presentados para demostrar la eficiencia del enfoque propuesto y realizar una comparación con los resultados conocidos.I'm asked to pack the maximum number of 10m^2 circle into a 257 x 157m rectangle. El problema de empaquetamiento se escribe entonces, como un problema de optimización 0–1 a gran escala y es resuelto con software comercial. Se propone una nueva formulación basada en el uso de una malla regular que cubre el contenedor y donde se considera a los nodos de la malla como posiciones potenciales para la asignación de centros de los círculos. Frecuentemente, el problema es formulado como un problema de optimización continua no convexo que es resuelto con técnicas heurísticas combinadas con procedimientos de búsqueda local. Este problema tiene numerosas aplicaciones dentro de la logística, incluyendo la producción y empaquetado para la industria textil, naval, automotriz, aeroespacial y la industria de alimentos. El objetivo es maximizar el número (ponderado) de círculos dentro del contenedor o minimizar el desperdicio de espacio dentro del mismo. Se considera el problema de empaquetar un número limitado de círculos de radios diferentes en un contenedor rectangular de dimensiones fijas. Numerical results are presented to demonstrate the efficiency of the proposed approach. Nesting circles inside one another is also considered. Two families of valid inequalities are proposed to strengthening the formulation. The resulting binary problem is then solved by the commercial software. The binary variables represent the assignment of centers to the nodes of the grid. The packing problem is then stated as a large scale linear 0–1 optimization problem. A new formulation is proposed based on using a regular grid approximated the container and considering the nodes of the grid as potential positions for assigning centers of the circles. ![]() Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with the local search procedures. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace and food industries. ![]() The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. ![]() A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered.
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