Fuzzy Bilevel Linear Programming Research Articles

Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems

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.

A method for solving interval type-2 triangular fuzzy bilevel linear programming problem

In this paper, we consider the bilevel linear programming problem (BLPP) where all the coefficients in the problem are interval type-2 triangular fuzzy numbers (IT2TFNs). First, we convert a BLPP with IT2TFN parameters to an interval BLPP. In the next step, we solve BLPPs and obtain optimal solution as an IT2TFN.

An approach based on reliability-based possibility degree of interval for solving general interval bilevel linear programming problem

This paper presents a new method based on reliability-based possibility degree of interval to handle the general interval bilevel linear programming problem involving interval coefficients in both objective functions and constraints. Considering reliability of the uncertain constraints, the interval inequality constraints are first converted into their deterministic equivalent forms by virtue of the reliability-based possibility degree of interval, and then the original problem is transformed into a bilevel linear programming with interval coefficients in the upper and lower level objective functions only. Then, the notion of the optimal solution of the problem is given by means of a type of the interval order relation. Based on this concept, the transformed problem is reduced into a deterministic bilevel programming with the aid of linear combination method. Furthermore, the proposed method is extended to deal with the fuzzy bilevel linear programming problem through the nearest interval approximation. Finally, three numerical examples are given to illustrate the effectiveness of the proposed approach.

Interactive programming approach for solving the fully fuzzy bilevel linear programming problem

In this paper, a fully fuzzy bilevel linear programming problem in which all the coefficients and decision variables of both objective functions and the constraints are expressed as fuzzy numbers is addressed. The purpose of this paper is to develop an interactive programming approach to obtain a balanced solution of the fully fuzzy bilevel programming problem. To this end, we first define the feasible region of the problem, and give a concept of the fuzzy optimal solution of the problem. Based on a fuzzy relationship to rank fuzzy numbers, the fully fuzzy bilevel programming problem can be converted into a deterministic one under different feasibility degrees of the constraints. Taking into account a tradeoff between better objective function values and higher feasibility degrees of the constraints, an interactive programming approach is presented to solve the fully fuzzy bilevel programming problem. Finally, several numerical examples are provided to demonstrate the feasibility of the proposed method.

Solving the Fully Fuzzy Bilevel Linear Programming Problem through Deviation Degree Measures and a Ranking Function Method

This paper is concerned with a class of fully fuzzy bilevel linear programming problems where all the coefficients and decision variables of both objective functions and the constraints are fuzzy numbers. A new approach based on deviation degree measures and a ranking function method is proposed to solve these problems. We first introduce concepts of the feasible region and the fuzzy optimal solution of a fully fuzzy bilevel linear programming problem. In order to obtain a fuzzy optimal solution of the problem, we apply deviation degree measures to deal with the fuzzy constraints and use a ranking function method of fuzzy numbers to rank the upper and lower level fuzzy objective functions. Then the fully fuzzy bilevel linear programming problem can be transformed into a deterministic bilevel programming problem. Considering the overall balance between improving objective function values and decreasing allowed deviation degrees, the computational procedure for finding a fuzzy optimal solution is proposed. Finally, a numerical example is provided to illustrate the proposed approach. The results indicate that the proposed approach gives a better optimal solution in comparison with the existing method.

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.

A Novel Method for Solving the Fully Fuzzy Bilevel Linear Programming Problem

We address a fully fuzzy bilevel linear programming problem in which all the coefficients and variables of both objective functions and constraints are expressed as fuzzy numbers. This paper is to develop a new method to deal with the fully fuzzy bilevel linear programming problem by applying interval programming method. To this end, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of the problem into a finite number ofα-level sets. By usingα-level sets of fuzzy numbers, the fully fuzzy bilevel linear programming problem is transformed into an interval bilevel linear programming problem for eachα-level set. The main idea to solve the obtained interval bilevel linear programming problem is to convert the problem into two deterministic subproblems which correspond to the lower and upper bounds of the upper level objective function. Based on theKth-best algorithm, the two subproblems can be solved sequentially. Based on a series ofα-level sets, we introduce a linear piecewise trapezoidal fuzzy number to approximate the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem. Finally, a numerical example is provided to demonstrate the feasibility of the proposed approach.

Solving the Fuzzy BiLevel Linear Programming with Multiple Followers through Structured Element Method

The optimal solution of fuzzy bilevel linear programming with multiple followers (MFFBLP) model is shown to be equivalent to the optimal solution of the bilevel linear programming with multiple followers by using fuzzy structured element theory. The optimal solution to this model is found out by adopting the Kuhn-Tucker approach. Finally, an illustrative numerical example for this model is also provided to demonstrate the feasibility and efficiency of the proposed method.

BILEVEL LINEAR PROGRAMMING WITH FUZZY PARAMETERS

Bilevel linear programming is a decision making problem with a two-level decentralized organization. The is in the upper level and the follower, in the lower. Making a decision at one level affects that at the other one. In this paper, bilevel linear programming with inexact parameters has been studied and a method is proposed to solve a fuzzy bilevel linear programming using interval bilevel linear programming. Bilevel linear programming (BLP), first introduced by Stackelberg (22), is a nested optimization problem including a leader and a follower problem. Making a decision at one level, affects the objective function and the decision space of the other and the order is from top to bottom. Kth-best (6), Branch and bound (4) and Complementary pivot (13) are among the most important methods presented for the solution of this problem. In real cases, the parameters of an optimization problem may not be exact, i.e. they may be interval or fuzzy; same is the case with BLP problems. It is worth mentioning that when the parameters of a Fuzzy Bilevel Linear Programming (FBLP) problem are fuzzy quantities, the objective functions are fuzzy quantities too. Recently, Calvete et al. (9) have presented two algorithms, similar to the Kth-best, for the calculation of the best and the worst optimal values of the Interval Bilevel Linear Programming (IBLP) objective function. They are based on the ordering of the extreme points. In the IBLP problem, investigated by the above researchers, only the coefficients of the leader and the follower objective functions are interval; the other coefficients are ordinary. These algorithms can be extended to the solution of the IBLP problems wherein all parameters are interval. In this paper, using -cut, an IBLP is obtained from a fuzzy one. The best and the worst optimal values of the objective function will be found first, and then a linear piecewise trapezoidal approximate fuzzy number will be presented for the optimal value of the leader objective function related to the FBLP problem.

Random fuzzy bilevel linear programming through possibility-based fractile model

This paper focuses on random fuzzy noncooperative bilevel linear programming problems. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy bilevel programming problems are reduced to deterministic ones. Extended concepts of Stackelberg solutions are introduced and computational methods are also presented. A numerical example is provided to illustrate the proposed method.

Random fuzzy bilevel linear programming through possibility-based value at risk model

This article considers bilevel linear programming problems where random fuzzy variables are contained in objective functions and constraints. In order to construct a new optimization criterion under fuzziness and randomness, the concept of value at risk and possibility theory are incorporated. The purpose of the proposed decision making model is to optimize possibility-based values at risk. It is shown that the original bilevel programming problems involving random fuzzy variables are transformed into deterministic problems. The characteristic of the proposed model is that the corresponding Stackelberg problem is exactly solved by using nonlinear bilevel programming techniques under some convexity properties. A simple numerical example is provided to show the applicability of the proposed methodology to real-world hierarchical problems.

Fuzzy random bilevel linear programming through expectation optimization using possibility and necessity

In this paper, assuming noncooperative behavior of the decision makers, solution methods for decision-making problems in hierarchical organizations under fuzzy random environments are presented. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random noncooperative bilevel linear programming problems. Considering the possibility and necessity measure that each objective function fulfills, the corresponding fuzzy goal, the fuzzy random bilevel linear programming problems to minimize each objective function with fuzzy random variables, are transformed into stochastic bilevel programming problems to maximize the degree of possibility and necessity that each fuzzy goal is fulfilled. Through expectation optimization in stochastic programming, which is suitable for risk-neutral decision makers, the transformed stochastic bilevel programming problems can be reduced to deterministic bilevel programming problems. For the transformed problems, extended concepts of Stackelberg solutions are introduced and computational methods are also presented. It is shown that the extended Stackelberg solutions can be obtained through the combined use of the variable transformation method and the Kth best algorithm for bilevel linear programming problems. A numerical example is provided to illustrate the proposed methods.