Polytope

4 The Simplex Method As we have seen, a linear programming problem forms a convex polygon in the best possible scenario. It is imperative to obtain a process that would assist in determining the optimal solution without the need to examine the graphical representation. The need for an algorithm that would perform such process was essential in the early days of the formulations of linear programming problems. Although challenging, the task was accomplished by a mathematician of the twentieth century. In 1947, George Dantzig developed a process that assisted in computing optimal solutions for minimization and maximization linear programming problems, this method is known as the simplex method [6]. Regardless of his great discovery, the linear programming problem needed to be set up in canonical form, so that the process could be utilized. Dantzig’s discovery could be applied to optimize any given objective function, given that the structure was in canonical form. In addition to the proper set up, the number of variables needs to be larger than the number of constraints. Assume that there are m equations and n unknowns. If the number of equations is equal to the number of constraints, then the solution could be derived without considering the objective function. Thus n>m is a prerequisite for a linear programming problem to be optimized. As we begin the process for solving this type of problems, we must consider basic variables and nonbasic variables. The…

Many have questions like what are tessellations? Tessellations are the arrangement of small squares in a checkered or mosaic form. Tessellations are seen all around, but yet you don't notice them. Tessellations can be found in your house from the tiles in bathrooms, or even to the honeycomb of bees. There was a famous man with the name of M.C. Esher, and he was the first to use these tessellations in artwork. He became well known for them and being the first one to use them M.C. Esher became…

Tessellations , Tessellations are a arrangement of shapes closely fitted together , especially of polygons in a repeated patterns without gaps or overlapping . They make really pretty designs if you do it right . Another word for Tessellations is “tiling”.Tessellations are catching to the eye and they are found in the regular everyday patterns such as , bathroom floors , backsplashes , turtle shells , honeycombs, raspberries, quilts and fish . “Tessellations are also widely used in the design of…

composer, music theorist and architecture-engineer. So he has the authority of the relationship between the music and architecture. He described the process of composing music as analogues to navigating a “cosmic vessel sailing in the space of sound, across sonic constellations and galaxy” and explained human intelligence in the language of astrophysics. As an architect, he thought that the evolution of humanity had reached a cosmic stage requiring new forms of dwellings, intend to bring the…