Journal of Numerical Mathematics and Stochastics, 11 (1) : 1-18, 2018 © JNM@S http://www.jnmas.org/jnmas11-1.pdf Euclidean Press, LLC Online: ISSN 2151-2302 Linear Fractional Programming Problems on Time Scales R. AL-SALIH ∗ , and M. BOHNER Missouri University of Science and Technology, Rolla, MO, USA; E-mail: bohner@mst.edu Abstract. In this paper, we develop a time scales approach to formulate and solve linear fractional programming problems. This time scales approach unifies the discrete and continuous linear fractional programming models and extends them to other cases “in between”. Our approach enables us to derive a pair of primal and dual linear fractional programming models on arbitrary time scales. We also establish and prove the weak duality theorem and the optimality condition for arbitrary time scales, while the strong duality theorem is established for isolated time scales. Examples are provided to illustrate the presented theory. Key words: Time Scales, Linear Fractional Programming Problem, Primal Problem, Dual Problem, Weak Duality Theorem, Optimality Condition, Strong Duality Theorem. AMS Subject Classifications: 90C05, 90C11, 90C32 1. Introduction It is well known that discrete-time linear programming problems have numerous applications in areas such as portfolio optimization, crew scheduling, manufacturing, transportation, telecommunication, agriculture, and so on. Continuous-time linear programming problems were first studied by Bellman [5] as a bottleneck process. He established the weak duality theorem and optimality conditions. A computational approach has been presented by Bellman and Dreyfus [6]. The strong duality theorem was studied by Tyndall [29, 30] and Levinson [25]. Grinold [23] has established strong duality without discretizing the continuous problem. A numerical solution to continuous-time linear programming was considered by Buie and Abrham [21]. Wen, Lur, and Lai [35] have presented an approximation approach to solve continuous-time problems. ____________________ ∗ Current address: University of Sumer, Statistics Department, Al-Rifa’i, Thi-Qar, Iraq. 1