What is it about?
In the last few decades, uncertainty linear programming (ULP) has received important attention among researchers and industrials. In this paper the ULP problem in which all parameters and/or decision variables are specified in terms of interval-valued intuitionistic fuzzy (IVIF) numbers is considered. First, we introduce a new method for solving the LP problem, denoted by IVIFLP, in which all parameters are IVIF numbers. Through using this method, the IVIFLP problem is broken down into nine smaller crisp linear problems (CLPs). Second, we modified the proposed method for solving an LP problem, denoted by FIVIFLP, in which all parameters and decision variables are IVIF numbers. An additional bounded variables constraint is added for the CLPs in which the optimization variables of all lower problems are considered as parameters at the upper problems. According to the reduction technique based on a linear combination between variables, the CLPs in the modified method are reduced to two CLPs, giving the most and the least favorable value of the objective function. The proposed methods are illustrated numerically. Finally, we explored the shortcomings of Bharati and Singh’s method (2018) for solving transportation problems in an IVIF environment and applied the proposed methods to solve such problems.
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Why is it important?
The main contributions of our presented methods are as follows: A new approach is presented to address the interval-valued intuitionistic fuzzy linear programming (IVIFLP) problem, in which some or all coefficients of the objective function and constraints are specified in terms of IVIF numbers. Utilizing the interval method [44], the IVIFLP problem is transformed into nine crisp linear problems that are solved using classical linear problem algorithms. The proposed approach is illustrated by means of transportation problems under the IVIF environment, and the obtained results are compared with the method of Bharati and Singh [45]. The proposed approach is extended to address the fully interval-valued intuitionistic fuzzy linear programming (FIVIFLP) problem, in which all decision variables and parameters are characterized by IVIF numbers. Utilizing the interval method [44], the FIVIFLP problem is transformed into nine crisp linear problems such that an additional bounded variables constraint is added for the crisp problems in which the optimization variables of all lower problems are considered as parameters at the upper problems. The extended approach is illustrated by means of transportation problems under a fully IVIF environment, and the obtained results are compared with the method of Bharati and Singh [45]. According to the reduction technique based on a linear combination of variables, the crisp problems in the extended method are reduced to two CLPs that give the most and the least favorable values of the objective function. The proposed approaches overcome the shortcomings of Bharati and Singh’s method [45].
Perspectives
1. The proposed approaches deal with the linear programming problem in which some or all parameters and/or decision variables are IVIF numbers. 2. Each IVIFLP and FIVIFLP problem is broken down into nine smaller crisp linear problems, which are treated one at a time using classical linear problem algorithms. 3. In contrast with the existing method, the proposed approaches provide an IVIF optimal solution. 4. In contrast with the existing method, the proposed approach can be used to find the IVIF optimal solution of IVIFLP and FIVIFLP problems with equality and inequality constraints. 5. The extended approach is novel and unique for solving the fully IVIFLP problem and fully IVIF transportation problem due to the nonexistence of any other method in the literature for solving such problems. 6. Application examples are provided to prove the strong discrimination capability of the proposed method in TPs. 7. The TP approach given in Bharati and Singh [45] transforms a balanced IVIFT problem into an unbalanced IVIFT problem; hence, Bharati and Singh’s approach [45] fails to find the IVIF optimal solution of the considered problem. 8. The method of dealing with TPs (Bharati and Singh [45]) fails whenever the variables are not real numbers, such as IVIFNs. However, in practical situations, the variables involved may not be crisp. In the extended approach, the variables are also IVIFNs. 9. The proposed approaches mention nonnegative variables and cost. 10. The proposed approaches are simple to understand and easy to apply. 11. The proposed approaches provided an appropriate best IVIF solution to various uncertainty linear programming models having parameters and/or IVIF numbers.
Dr. Eman Fathy
Faculty of Science, Helwan University
Read the Original
This page is a summary of: A new method for solving the linear programming problem in an interval-valued intuitionistic fuzzy environment, Alexandria Engineering Journal, December 2022, Elsevier,
DOI: 10.1016/j.aej.2022.03.077.
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