Linear bilevel programming with upper level constraints depending on the lower level solution

A penalty function method based on Kuhn–Tucker condition for solving linear bilevel programming

Global convergent algorithm for the bilevel linear fractional-linear programming based on modified convex simplex method

This paper considers a particular case of linear bilevel programming problems with one leader and multiple followers. In this model, the followers are independent, meaning that the objective function and the set of constraints of each follower only include the leader's variables and his own variables. We prove that this problem can be reformulated into a linear bilevel problem with one leader and one follower by defining an adequate second level objective function and constraint region. In the second part of the paper we show that the results on the optimality of the linear bilevel problem with multiple independent followers presented in Shi et al. [The kth-best approach for linear bilevel multi-follower programming, J. Global Optim. 33, 563---578 (2005)] are based on a misconstruction of the inducible region.

In Operations Research, linear-fractional programming is considered as the generalization of linear programming problem. While in a linear programming the objective function is a linear function, and in a linear-fractional programming the objective function is the ratio of two linear or non-linear functions. The majority of the algorithm used for solving the linear fractional programming problem relies upon the classical simplex method. In this paper, we have proposed a new algorithm for solving a linear fractional programming problem in which the objective function is a combination of linear fractional function, while constraint functions are in the form of linear inequalities. Our proposed algorithm is based on the extension of the method, which is used to solve linear programming problems with linear constraints. The primary intent behind developing this method is that we did not need to transform the linear fractional programming problem into linear programming problem, and also it helps in finding out the feasible region via a sequence of points in the direction that improves the feasibility of the fractional objective function. Numerical examples are given to illustrate the use of these proposed methods. Lastly, to demonstrate the efficacy of the proposed algorithm, we have compared the findings obtained with other approaches to display our algorithm's efficacy.

This article considers a bi-level linear programming with single valued trapezoidal fuzzy neutrosophic cost coefficient matrix and Pythagorean fuzzy parameters in the set of constraints both in the right and left sides. Based on the score functions of the neutrosophic numbers and Pythagorean fuzzy numbers, the model is changed to the corresponding crisp bi-level linear programming (BLP) problem. This problem is designated as a Pythagorean fuzzy bi-level linear programming (PFBLP) problem under neutrosophic environment. Kuhn-Tucker's conditions for optimality are necessary and sufficient for the existence of the optimal solution to a BLP problem. Using the suggested methodology, the problem is formulated as a single-objective non-linear programming problem with several variables and constraints. Two typical numerical examples are examined to illustrate the proposed approach.

Multi-objective linear bi-level multi-follower programming problem (MOBLMFPP) is a special case of two level hierarchical programming problems in which the second level hierarchy includes multiple followers. This paper presents an alternate method based on fuzzy goal programming approach for the solution of multi-objective linear bi-level multi-follower programming (MOLBMFP) problem in which there is no sharing of information among followers. In the proposed FGP model formulation, each of objective functions of each level (leader and follower's level of MOLBMFPP) as well as decision variables at each level are characterised into fuzzy goals. Suitable linear membership functions are defined for each objective function and decision variables. Then minimising the sum of the negative deviational variables of both levels, the highest membership value of each of fuzzy goals is obtained. Existence of compromise optimal solution for MOLBMFPP is established in context of proposed approach. A numerical example is illustrated in support of proposed methodology. A comparative analysis is also carried out in order to show efficiency of proposed method over earlier method for the solution of MOLBMFPP.

Multi-objective linear bi-level multi-follower programming problem (MOBLMFPP) is a special case of two level hierarchical programming problems in which the second level hierarchy includes multiple followers. This paper presents an alternate method based on fuzzy goal programming approach for the solution of multi-objective linear bi-level multi-follower programming (MOLBMFP) problem in which there is no sharing of information among followers. In the proposed FGP model formulation, each of objective functions of each level (leader and follower's level of MOLBMFPP) as well as decision variables at each level are characterised into fuzzy goals. Suitable linear membership functions are defined for each objective function and decision variables. Then minimising the sum of the negative deviational variables of both levels, the highest membership value of each of fuzzy goals is obtained. Existence of compromise optimal solution for MOLBMFPP is established in context of proposed approach. A numerical example is illustrated in support of proposed methodology. A comparative analysis is also carried out in order to show efficiency of proposed method over earlier method for the solution of MOLBMFPP.

On the definition of linear bilevel programming solution

The revised harmonious fuzzy technique (RHFT) is a method used to solve fuzzy optimization problems. It was capitalized as an extension of the classical linear programming technique to handle constraints and objectives that are fuzzy. The harmonious fuzzy technique HFT aims to find a solution that satisfies the uncertain restraints and optimizes the uncertain objectives while taking into account the uncertainty or fuzziness of the problem parameters. This work demonstrates how the RHFT can be utilized to dexterously solve “fully fuzzy multi-goal linear fractional programming (FFMOLFP) problems”. Initially, the FFMOLFP problem can be converted to “single goal linear fractional programming (SOLFP) problems” consuming the modified brittle linear technique. Second, the RHFT is applied to converted brittle problems into linear programming problem, which follow, “the single-goal problem” is made on so on applied the revised harmonious fuzzy for apiece level. at the end, the obtained LPP will be solved by applied the simplex algorithm. To illustrate the application of this method, two examples will be provided. Also, the numerical results are simulated by comparing between proposed method and efficient ranking function methods for fully fuzzy linear fractional programming problems FFLFPP

In this paper, we investigate the relationship between a certain class of linear bilevel multifollower programming problems and multiple objective programming. We introduce two multiple objective linear programming problems with different objective functions and the same constraint region. We show that the extreme points of the set of efficient solutions for both problems are the same as those of the set of feasible solutions to the linear bilevel multi-follower programming problem. Based on this relationship, a new algorithm to find an optimal solution for the linear bilevel multifollower programming problem is developed. Some numerical examples are presented to show the feasibility of the proposed algorithm.

A global optimization algorithm for solving the bi-level linear fractional programming problem

A method of constructing test problems for linear bilevel programming problems is presented. The method selects a vertex of the feasible region, ‘far away’ from the solution of the relaxed linear programming problem, as the global solution of the bilevel problem. A predetermined number of constraints are systematically selected to be assigned to the lower problem. The proposed method requires only local vertex search and solutions to linear programs.

Integrating goal programming, Kuhn–Tucker conditions, and penalty function approaches to solve linear bi-level programming problems

This paper proposes a method to solve a bi-level linear fuzzy fractional programming problem comprising all its constants and coefficients expressed in form of trapezoidal fuzzy numbers. Fuzzy $$\alpha , \beta $$ -cuts are respectively used in the objective functions and constraints to equivalently transform the bi-level fuzzy optimization into bi-level interval valued form. Subsequently, a bi-level bi-objective linear fractional programming problem is generated. Change of variable method is implemented to construct linear fuzzy membership functions at upper and lower level. Fuzzy goal programming eliminating over deviations from aspiration level of fuzzy membership functions along with a proposed modified linearization process of fractional functions are together used to determine the compromise solution of the problem. To illustrate and justify the feasibility of the proposed method, an existing numerical problem is solved and the results obtained are comparatively analyzed.

This paper deals with a class of mathematical programming problems that includes linear and nonlinear programming problems in a particular form. First, a linear programming problem is considered, and the possibility of deriving its direct complete solution in terms of traditional mathematics without using known iterative computational procedures and algorithms of linear programming, such as the simplex method, is studied. Direct solutions to the problem in the case of minimal dimension with a reduced set of constraints are proposed. It is shown that the derivation of such solutions, as dimension increases, becomes a very complicated problem with increasing dimension and, therefore, is hardly feasible. Some examples of other linear and nonlinear programming problems, which can be obtained from the above-considered problem by means of isomorphic transformations, are presented. The main definitions and preliminary results of tropical mathematics, which are required for the subsequent description and application of tropical optimization methods, are then outlined. A tropical optimization problem is formulated, and direct complete solutions of this problem and of its special cases are given. The above-formulated linear and nonlinear programming problems are reduced to a tropical optimization problem to provide their direct complete solution in terms of tropical mathematics. The solution of the linear programming problem with a reduced set of constraints is written in terms of traditional mathematics.