scholarly journals Linear programming problems on time scales

2018 ◽  
Vol 12 (1) ◽  
pp. 192-204 ◽  
Author(s):  
Rasheed Al-Salih ◽  
Martin Bohner
Keyword(s):  
Linear Programming ◽  
Optimality Conditions ◽  
Time Scales ◽  
Duality Theorem ◽  
Strong Duality ◽  
Programming Models ◽  
Weak Duality ◽  
Continuous Linear ◽  
Arbitrary Time ◽  
Linear Programming Problems

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.

scholarly journals Separated and state-constrained separated linear programming problems on time scales

2019 ◽  
Vol 38 (4) ◽  
pp. 181-195 ◽  
Author(s):  
Rasheed Al-Salih ◽  
Martin J. Bohner
Keyword(s):  
Linear Programming ◽  
Time Scales ◽  
Continuous Time ◽  
Duality Theorem ◽  
Strong Duality ◽  
Continuous Time Model ◽  
Time Model ◽  
Wide Range ◽  
Linear Programming Problems ◽  
Fundamental Theorems

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

Sensitivity Analysis on Linear Programming Problems with Trapezoidal Fuzzy Variables

2013 ◽  
pp. 64-82
Author(s):  
Seyed Hadi Nasseri ◽  
Ali Ebrahimnejad
Keyword(s):  
Linear Programming ◽  
Simplex Algorithm ◽  
Programming Models ◽  
Fuzzy Data ◽  
Fuzzy Variables ◽  
Fuzzy Environment ◽  
Dual Simplex ◽  
Linear Programming Problems ◽  
Simplex Tableau ◽  
Dual Simplex Algorithm

In the real word, there are many problems which have linear programming models and sometimes it is necessary to formulate these models with parameters of uncertainty. Many numbers from these problems are linear programming problems with fuzzy variables. Some authors considered these problems and have developed various methods for solving these problems. Recently, Mahdavi-Amiri and Nasseri (2007) considered linear programming problems with trapezoidal fuzzy data and/or variables and stated a fuzzy simplex algorithm to solve these problems. Moreover, they developed the duality results in fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables. Here, the authors show that this presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or post optimality) analysis using primal simplex tableaus.

scholarly journals An improved three-step method for solving the interval linear programming problems

2018 ◽  
Vol 28 (4) ◽  
pp. 435-451 ◽  
Author(s):  
Mehdi Allahdadi ◽  
Chongyang Deng
Keyword(s):  
Linear Programming ◽  
Optimality Conditions ◽  
Optimality Condition ◽  
Solution Space ◽  
Interval Linear Programming ◽  
Step Method ◽  
Feasibility Condition ◽  
Extra Step ◽  
Interval Linear ◽  
Linear Programming Problems

Feasibility condition, which ensures that the solution space does not violate any constraints, and optimality condition, which guarantees that all points of the solution space are optimal, are very significant conditions for the solution space of interval linear programming (ILP) problems. Among the existing methods for ILP problems, the best-worst cases (BWC) method and two-step method (TSM) do not ensure feasibility condition, while the modified ILP (MILP), robust TSM (RTSM), improved TSM (ITSM), and three-step method (ThSM) guarantee feasibility condition, whose solution spaces may not be completely optimal. We propose an improved ThSM (IThSM) for ILP problems, which ensures both feasibility and optimality conditions, i.e., we introduce an extra step to optimality.

scholarly journals Approximate dual and approximate vector variational inequality for multiobjective optimization

1989 ◽  
Vol 47 (3) ◽  
pp. 418-423 ◽  
Author(s):  
G.–Y. Chen ◽  
B. D. Craven
Keyword(s):  
Variational Inequality ◽  
Multiobjective Optimization ◽  
Duality Theorem ◽  
Optimization Problem ◽  
Strong Duality ◽  
Vector Variational Inequality ◽  
Weak Duality ◽  
Multiobjective Optimization Problem ◽  
Feasible Set

AbstractAn approximate dual is proposed for a multiobjective optimization problem. The approximate dual has a finite feasible set, and is constructed without using a perturbation. An approximate weak duality theorem and an approximate strong duality theorem are obtained, and also an approximate variational inequality condition for efficient multiobjective solutions.

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