In applied mathematics, linear-fractional programming (LFP) formally is almost the same as linear programming (LP) but instead of linear objective function one has a ratio of two linear functions, subject to linear constraints. Informally, if linear programming determines the way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model and given some list of requirements represented as linear equations, in linear-fractional programming model we can achieve the best (highest) ratio of outcome to cost, i.e. highest efficiency.
For example, if in the frame of LP we maximize profit = income − cost and obtain maximal profit of 100 units (= $1100 of income − 1000$ of cost), then using LFP we can obtain only $10 of profit which requires only $50 of investment. Thus, in LP we have efficiency $100/$1000 = 0.1, at the same time LFP provides efficiency equal to $10/$50 = 0.2.
Linear-fractional programming can be used in the same real-world applications as LP, in various fields of study. Most extensively it is used in business and economic situations, especially in the situations of deficit of financial resources. Also LFP can be utilized for wide range of engineering problems. Some industries that use linear programming models including transportation, energy, telecommunications, and manufacturing, all of which may use LPF as well as LP.
References
- E.Bajalinov, Linear-Fractional Programming: Theory, Methods, Applications and Software. «Kluwer Academic Publishers», 2003.
Software
- WinGULF – educational interactive linear and linear-fractional programming solver with a lot of special options (pivoting, pricing, branching variables etc.).