In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Where the objective function of linear programs are linear functions, the objective function of a linear-fractional program is a ratio of two linear functions. In other words, a linear program is a fractional-linear program in which the denominator is the constant function having the value one everywhere.
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.
References
- Bajalinov, E. B. (2003). Linear-Fractional Programming: Theory, Methods, Applications and Software. Boston: Kluwer Academic Publishers.
- Barros, Ana Isabel (1998). Discrete and fractional programming techniques for ___location models. Combinatorial Optimization. Vol. 3. Dordrecht: Kluwer Academic Publishers. pp. xviii+178. ISBN 0-7923-5002-2. MR 1626973.
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- Craven, B. D. (1988). Fractional programming. Sigma Series in Applied Mathematics. Vol. 4. Berlin: Heldermann Verlag. p. 145. ISBN 3-88538-404-3. MR 0949209.
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- Martos, Béla (1975). Nonlinear programming: Theory and methods. Amsterdam-Oxford: North-Holland Publishing Co. p. 279. ISBN 0-7204-2817-3. MR 0496692.
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- Stancu-Minasian, I. M. (1997). Fractional programming: Theory, methods and applications. Mathematics and its applications. Vol. 409. Dordrecht: Kluwer Academic Publishers Group. pp. viii+418. ISBN 0-7923-4580-0. MR 1472981.
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Software
- WinGULF – educational interactive linear and linear-fractional programming solver with a lot of special options (pivoting, pricing, branching variables etc.).