$\mathbb{No \ hay \ cosa \ mas \ feliz \ en \ el \ mundo \ que \ ver \ tu \ sonrisa \ mi \ Miffy}$

\(\mathbb{No \ hay \ cosa \ mas \ feliz \ en \ el \ mundo \ que \ ver \ tu \ sonrisa \ mi \ Miffy}\)

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What is that？

Let us pay attention to a common problem that we often meet in daily life：

There are \(n\) different commodities. Each commodity has two attributes, one for value \(v,v>0\), the other for cost \(w, w>0\)

. Now, you should to choose some of them to let the cost performance highest.

Mathematically, if we let \(V = {\large \sum \limits _{i = 1}^{n}} v(i)x(i)\), \(W = {\large \sum \limits _{i = 1}^{n}} w(i)x(i)\), the answer will change to \(\dfrac {V} {W}\). Noticed every element \(x(i)\) of function \(x\)

, we stipulate that \(x(i)\) only can equal to \(0\) or \(1\). They respectively indicate whether the commodity is taken or not.

The Fractional Programming is such a solution to these kind of problem.

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How to do it？

Let the integer \(D\) equals to \(\dfrac {V} {W}\). Because the situation \(W\) equal to zero is meaningless, so it is easy to find that \(DW = V\)

. Deform this equation, \(DW - V = 0\).