In this work, we study linear programming problems on time scales. This approach unifies discrete and continuous linear programming models and extends them to other cases "in between." After a brief introduction to time scales, we formulate the primal as well as the dual time scales linear programming models. Next, we establish and prove the weak duality theorem and the optimality conditions theorem for arbitrary time scales, while the strong duality theorem is established for isolated time scales. Finally, examples are given in order to illustrate the effectiveness of the presented results.
In this paper, we present a new formulation for the Leontief production model using quantum calculus analogue. This formulation unifies discrete and continuous Leontief production models. Also, the classical Leontief production model is obtained by choosing q = 1. In addition, briefly give an introduction to quantum calculus. We present the formulation for continuous Leontief production models as well as quantum calculus models. Moreover, we establish the weak duality theorem and the strong duality theorem for quantum calculus analogue. Furthermore, using the objective functions for the primal and the dual quantum calculus models, we can easily obtain upper and lower bounds for the value of production at any production plan. Finally, examples are provided in order to illustrate the given results.
Separated linear programming problems can be used to model a wide range of real-world applications such as in communications, manufacturing, transportation, and so on. In this paper, we investigate novel formulations for two classes of these problems using the methodology of time scales. As a special case, we obtain the classical separated continuous-time model and the state-constrained separated continuous-time model. We establish some of the fundamental theorems such as the weak duality theorem and the optimality condition on arbitrary time scales, while the strong duality theorem is presented for isolated time scales. Examples are given to demonstrate our new results
In this paper, we derive a new formulation for an optimal investment allocation in N-regional economic model using quantum calculus analogue. This model is described as an optimal control model and formulated in both primal and dual models using quantum calculus formulation. This formulation is an extension of regional economic models. Also, the new formulation provides an exact optimal investment allocation. In addition, the classical regional economic model is obtained by choosing q=1. Furthermore, we formulate the primal and the dual regional economic models in quantum calculus. Moreover, we present a new version of the duality theorems for quantum calculus case. Finally, example is provided and solved using MATLAB. in order to show the given new results.
Network flow optimization has a wide range of real world applications such as in transportation, in electric, in civil engineering, in industrial engineering and in communication networks and so on. In the minimum cost network flow model, the goal is to find the values of the decisions variables that minimize the total cost of flows over the given network. In this work, a new formulation of flow optimization in dynamic networks using time scales approach has been presented. The continuous network model and the quantum case model are also obtained as special cases. The formulation has been given for both dynamic models and time scales models. Moreover, the new approach provides the exact optimal solution for this type of optimization problems. Furthermore, a new version of some duality theorems for time scales flow optimization in dynamic networks has been introduced.
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