Linear-fractional programming

In mathematical optimization, linear-fractional programming is a generalization of linear programming. Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function one. Both linear programming and linear-fractional programming represent optimization problems using linear equations and linear inequalities, which for each problem-instance define a feasible set. Fractional linear programs have a richer set of objective functions. Informally, linear programming computes a policy delivering the best outcome, such as maximum profit or lowest cost. In contrast, a linear-fractional programming is used to achieve the highest ratio of outcome to cost, the ratio representing the highest efficiency. For example, in the context of LP we maximize the objective function profit = income − cost and might obtain maximal profit of $100.

Thus, in LP we have an efficiency of $100/$1000 = 0.1. Using LFP we might obtain an efficiency of $10/$50 = 0.2 with a profit of only $10, but only requiring $50 of investment. Formally, a linear-fractional program is defined as the problem of maximizing a ratio of affine functions over a polyhedron, maximize c T x + α d T x + β subject to A x ≤ b, where x ∈ R n represents the vector of variables to be determined, c, d ∈ R n and b ∈ R m are vectors of coefficients, A ∈ R m × n is a matrix of coefficients and α, β ∈ R are constants; the constraints have to restrict the feasible region to, i.e. the region on which the denominator is positive. Alternatively, the denominator of the objective function has to be negative in the entire feasible region. Under the assumption that the feasible region is non-empty and bounded, the Charnes-Cooper transformation y = 1 d T x + β ⋅ x; the solution for y and t yields the solution of the original problem as x = 1 t y. Let the dual variables associated with the constraints A y − b t ≤ 0 and d T y + β t − 1 = 0 be denoted by u and λ, respectively.

The dual of the LFP above is minimize λ subject to A T u + λ d = c − b T u + λ β ≥ α u ∈ R + m, λ ∈ R, {\

North Greenville University

North Greenville University is private Baptist university in Tigerville, South Carolina. It is associated with the South Carolina Baptist Convention and the Southern Baptist Convention and accredited by the Southern Association of Colleges and Schools; the institution awards bachelor's, master's, doctoral degrees. NGU was founded in 1892 as a non-government school by private individuals and named North Greenville High School, the first high school in the northern portion of Greenville County. Land for the school was donated by Benjamin F. Neves, it was operated by the North Greenville Baptist Association, was set up to expand educational offerings in the mountainous northern portion of Greenville County. The school received a state charter in 1904, it was taken over by the Southern Baptist Convention's Home Mission Board a year and renamed North Greenville Baptist Academy in 1915. The North Greenville Baptist Association reassumed control of the school in 1929. In 1934, the academy was expanded to include a junior college.

In 1949, it was transferred to the South Carolina Baptist Convention, which renamed the school North Greenville Junior College a year later. In 1957, it was accredited as a two-year college, high school courses were dropped altogether, it was renamed North Greenville College in 1972. NGC began offering its first junior- and senior-level classes in 1992, in Christian studies and church music and added a teacher education program in 1997. NGU began granting master's degrees as well. Dr. Mac Brunson, 1978, Senior Pastor of Valleydale Baptist Church, Alabama.