Fractional Programming: Theory, Methods

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The objective function in a linear-fractional problem is both quasiconcave and quasiconvex (hence quasilinear) with a monotone property, pseudoconvexity, which is a stronger property than quasiconvexity. A linear-fractional objective function is both pseudoconvex and pseudoconcave, hence pseudolinear. Since an LFP can be transformed to an LP, it can be solved using any LP solution method, such as the simplex algorithm (of George B. Dantzig),[5][6][7][8] the criss-cross algorithm,[9] or interior-point methods.

Abbas and Moulaï [1] and Gupta and Malhotra [3] have presented each a technique to generate the efficient set of a MOILFP based on the same principle: solving a sequence of integer linear fractional programs (ILFPs) until the stopping criterion they propose is met. The first ILFP solved is defined by optimizing one of the objective functions subject to the entire feasible set; then a cutting plane is recursively added to eliminate the current optimal solution and eventually the solutions lying on an adjacent edge. The choice of this latter makes the main difference between these two methods. Unfortunately, they have both the same inconvenience that is scanning almost the whole search space which is very expensive in time and memory space.

A fractional programming problem arises in many types of optimization problems such as portfolio selection, production, information theory, and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system. Many economic, noneconomic, and indirect applications of fractional programming problem have also been given by Bector [1], Bector and Chandra [2], Craven [3], Mond and Weir [4], Stancu-Minasian [5], Schaible and Ibaraki [6], Ahmad et al. [7], Ahmad and Sharma [8], and Gulati et al. [9].

More recent works on linear fractional programming theory and methods can be found in 1, 2. The suggested method in this paper depends mainly on the updated method in iterative manner then the optimality condition for a given basic feasible solution of (LFP) is defined.

Shortcut methods were presented in this paper to produce sensitivity analysis of linear fractional programming models. Four different topics on sensitivity were taken into account: changes in the model parameters, i.e., changes on objective function coefficients, Changes in the Right-Hand-Side values of the constraints, Changes in the Left-Hand-Side values of the constraints, and Adding a new decision variable.

This research is a significant contribution in the sense that it will assist the management and business students at different universities in making correct decisions by using very short and easy-to-calculate methods concerning the sensitivity analysis of linear fractional programming problems.

From the literature that we observed, many authors tried to solve multiple-objective probabilistic programming problem by finding the deterministic equivalent. In addition some authors solved multiple-objective fractional programming problems by using classical method of fractional programming by transforming into equivalent multiple-objective programming problem. All these methods take time and even difficult to find the deterministic equivalence when the parameters involve some discrete random variables like hyper geometric and pascal distributions. 2b1af7f3a8