Linear-fractional programming: Difference between revisions

Revision as of 09:45, 11 April 2011

Revision as of 10:09, 11 April 2011

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In [[mathematical optimization]], '''linear-fractional programming (LFP)''' is a generalization of [[linear programming]] (LP). Whereas the objective function in linear programs are [[linear functional|linear functions]], the objective function in a linear-fractional program is a ratio of two [[affinelinear function]]sfunctions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function one.

Informally,Both linear programming determinesand thelinear-fractional wayprogramming torepresent achieveoptimization theproblems bestusing outcomelinear (suchequations asand maximumlinear profitinequalities, orwhich lowestfor cost)each inproblem-instance define a givenfeasible mathematicalset. modelFractional andlinear givenprograms have a somericher listset of requirementsobjective representedfunctions. asInformally, linear equationsprogramming computes a policy delivering the best outcome, such as maximum profit or lowest cost. In contrast, a linear-fractional programming model is used to achieve the highest ''ratio'' of outcome to cost, i.e.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 (= $1100 of income − $1000 of cost). 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, which requires only $50 of investment however.