Optimal value bounds in interval fractional linear programming and revenue efficiency measuring
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Optimal value bounds in interval fractional linear programming and revenue efficiency measuring Amin Mostafaee1 · Milan Hladík2 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper deals with the fractional linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. A method is provided for the situation in which the feasible set is described by a linear interval system. Moreover, certain dependencies between the coefficients in the nominators and denominators can be involved. Also, we extend this approach for situations in which the same vector appears in different terms in nominators and denominators. The applicability of the approaches developed is illustrated in the context of the analysis of hospital performance. Keywords Linear interval systems · Fractional linear programming · Optimal value range · Interval matrix · Dependent data Mathematics Subject Classification 90C31 · 90B50 · 65G40
1 Introduction Throughout the paper, we consider a fractional linear programming problem in the form min
B
p T x + q1T y + c p T x + q2T y + d
, s.t. (x, y) ∈ M(A, B, b),
(1)
Milan Hladík [email protected] Amin Mostafaee [email protected]
1
Department of Mathematics, College of Science, Islamic Azad University, North Tehran Branch, Tehran, Iran
2
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 11800 Prague, Czech Republic
123
A. Mostafaee, M. Hladík
where M(A, B, b) is a convex polyhedral set described by linear constraints with constraint matrices A and B, and the right-hand side vector b. This is a general model since M(A, B, b) can be characterized by linear equations, inequalities or both. In the subsequent sections, we will consider particular cases. We investigate the effects of independent and simultaneous variations of (possible all) coefficients in prescribed interval domains on the optimal value. In particular, we are interested in determining the best and the worst optimal value. ˇ State-of-the-art This problem is well studied in linear programming (Cerný and Hladík 2016; Chinneck and Ramadan 2000; Fiedler et al. 2006; Hladík 2012, 2014; Mostafaee et al. 2016), but there are considerable less results in the nonlinear case. A general approach to solving interval valued nonlinear program was proposed in Hladík (2011), and particular quadratic problems, e.g., in Li et al. (2015, 2016). A specific case of generalized fractional linear programming with variable coefficients was investigated in Hladík (2010), and the case of intervals in the objective function in Borza et al. (2012) and Effati and Pakdaman (2012). Some duality theorems in fractional programming with (not only) interval uncertainty were stated in Jeyakumar et al. (2013). The related problem of inverse fractional linear programming was studied, e.g., in Jain and Arya (2013). In a fuzzy environment, fractional linear programming was disc
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