Linear Bilevel Programming Problem Research Articles

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

In this paper, we obtain the solution to bi-level linear fractional programming problem (BLFP) by means of an optimization algorithm based on the duality gap of the lower level problem. In our algorithm, the bi-level linear fractional programming problem is transformed into an equivalent single level programming problem by forcing the dual gap of the lower level problem to zero. Then, by obtaining all vertices of a polyhedron, the single level programming problem can be converted into a series of linear fractional programming problems. Finally, the performance of the proposed algorithm is tested on a set of examples taken from the literature.

A LINEAR BILEVEL PROGRAMMING PROBLEM FOR OBTAINING THE CLOSEST TARGETS AND MINIMUM DISTANCE OF A UNIT FROM THE STRONG EFFICIENT FRONTIER

Data envelopment analysis (DEA) can be used for assessing the relative efficiency of a number of operating units, finding, for each unit, a target operating point lying on the strong efficient frontier. Most DEA models project an inefficient unit onto a most distant target, which makes its attainment more difficult. In this paper, a linear bilevel programming problem for obtaining the closest targets and minimum distance of a unit from the strong efficient frontier by different norms is provided. The idea behind this approach is that closer targets determine less demanding levels of operation for the inputs and outputs of the units to perform efficiently. Finally, it will be shown that the proposed method is an extension of the existing methods.

Computing the Pareto frontier of a bi-objective bi-level linear problem using a multiobjective mixed-integer programming algorithm

In this article, we study the bi-level linear programming problem with multiple objective functions on the upper level (with particular focus on the bi-objective case) and a single objective function on the lower level. We have restricted our attention to this type of problem because the consideration of several objectives at the lower level raises additional issues for the bi-level decision process resulting from the difficulty of anticipating a decision from the lower level decision maker. We examine some properties of the problem and propose a methodological approach based on the reformulation of the problem as a multiobjective mixed 0–1 linear programming problem. The basic idea consists in applying a reference point algorithm that has been originally developed as an interactive procedure for multiobjective mixed-integer programming. This approach further enables characterization of the whole Pareto frontier in the bi-objective case. Two illustrative numerical examples are included to show the viability of the proposed methodology.

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.

Comments on: Algorithms for linear programming with linear complementarity constraints

This paper addresses the solution of linear programming problems that further incorporate complementarity restrictions involving designated pairs of complementary variables, as well as its generalization where the objective function is a nonlinear continuously differentiable function. Such problems subsume mixed linear complementary problems, absolute value programs, the linear bilevel programming problem, and the bilinear programming problem as well as more general nonconvex quadratic programs, among many others. The existing research on these problems focuses on finding strongly stationary points, or more general B-stationary points, both of which are necessary characterizations for local minima. However, computing B-stationary points is more combinatorial since it involves characterizing solutions that turn out to be Karush–Kuhn–Tucker (KKT) points for an entire family of nonlinear programs. In special nondegenerate cases (where at least one of each complementary pair is nonzero in any complementary solution) or under a special linear independence constraint qualification, B-stationary points also turn out to be strongly stationary. Among several algorithms for computing such strongly stationary or B-stationary points, the most competitive method to-date is a complementary active-set algorithm (CASET) and its advanced variants. In several studies, it has been verified that this procedure works well, even for degenerate problems, outperforming other nonlinear programming approaches such as penalty methods, non-smooth optimization techniques, interior point methods, and sequential quadratic programming algorithms. The CASET algorithm requires an initial complementary feasible solution to start with, which can always be computed by using an enumerative method.

Bilevel Linear Fractional Programming Problem based on Fuzzy Goal Programming Approach

This paper presents fuzzy goal programming approach for bilevel linear fractional programming problem with a single decision maker at the upper level and a single decision maker at the lower level. Here, each level has single objective function, which are fractional in nature and the system constraints are linear functions. In the proposed approach, we first construct fractional membership functions by determining individual best solution of the objective functions subject to the system constraints. The fractional membership functions are then transformed into equivalent linear membership functions by first order Taylor polynomial series. Since the objectives of both level decision makers are potentially conflicting in nature, a possible relaxation of both level decisions is considered for avoiding decision deadlock. Then, the fuzzy goal programming approach is used for achieving highest degree of each of the membership goals to the maximum possible by minimizing the negative deviational variables. To demonstrate the efficiency of the proposed approach, an illustrative numerical example is solved and Euclidean distance function is used to obtain compromise optimal solution. General Terms Bi-level programming.

A hybrid of genetic algorithm and particle swarm optimization for solving bi-level linear programming problem – A case study on supply chain model

The main goal of supply chain management is to coordinate and collaborate the supply chain partners seamlessly. On the other hand, bi-level linear programming is a technique for modeling decentralized decision. It consists of the upper level and lower level objectives. Thus, this paper intends to apply bi-level linear programming to supply chain distribution problem and develop an efficient method based on hybrid of genetic algorithm (GA) and particle swarm optimization (PSO). The performance of the proposed method is ascertained by comparing the results with GA and PSO using four problems in the literature and a supply chain distribution model.

Solving Bilevel Linear Multiobjective Programming Problems

This study addresses bilevel linear multi-objective problem issues i.e the special case of bilevel linear programming problems where each decision maker has several objective functions conflicting with each other. We introduce an artificial multi-objective linear programming problem of which resolution can permit to generate the whole feasible set of the upper level decisions. Based on this result and depending if the leader can evaluate or not his preferences for his different objective functions, two approaches for obtaining Pareto- optimal solutions are presented.

A New Approach To Solve Multi-objective Linear Bilevel Programming Problems

Many problems in sciences and industry such as signal optimization, traffic assignment, economic market,… have been modeled, or their mathematical models can be approximated, by linear bilevel programming (LBLP) problems, where in each level one must optimize some objective functions. In this paper, we use fuzzy set theory and fuzzy programming to convert the multi-objective linear bilevel programming (MOLBLP) problem to a linear bilevel programming problem, then we extend the Kth-best method to solve the final LBLP problem. The existence of optimal solution, and the convergence of this approach, are important issues that are considered in this article. A numerical example is illustrated to show the efficiency of the new approach.

Stackelberg solutions for fuzzy random bilevel linear programming through level sets and probability maximization

This paper considers Stackelberg solutions for bilevel linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random bilevel linear programming problem, α-level sets of fuzzy random variables are introduced and an α-stochastic bilevel linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic bilevel linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through probability maximization in stochastic programming, the transformed stochastic bilevel programming problem can be reduced to a deterministic bilevel programming problem. An extended concept of Stackelberg solution is introduced and a computational method is also presented. A numerical example is provided to illustrate the proposed method.

Numerical solution of a linear bilevel problem

The linear bilevel programming problem in the optimistic formulation is studied. It is reduced to an optimization problem with a nonconvex constraint in the form of a d.c. function (that is, the difference of two convex functions). For this problem, local and global search methods are developed. Numerical experiments performed for numerous specially generated problems, including large-scale ones, demonstrate the efficiency of the proposed approach.

A neural network approach for solving linear bilevel programming problem

A novel neural network approach is proposed for solving linear bilevel programming problem. The proposed neural network is proved to be Lyapunov stable and capable of generating optimal solution to the linear bilevel programming problem. The numerical result shows that the neural network approach is feasible and efficient.

An interval number programming approach for bilevel linear programming problem

In this paper, we propose a mathematical formulation of bilevel linear programming problem to deal with an interval number programming approach. Bilevel linear programming problem is usually viewed as a problem with two decision makers at two different hierarchical levels. The upper-level decision maker, the leader, selects his or her decision vector first and the lower decision maker, the follower, selects his or her afterward based on the decisions of the upper level. In mathematical programming problem, the coefficients in the objective function and the constraint functions are always determined as crisp values. In practice, however, there are many decision situations where the objective functions and/or the constraints are uncertain to some degree. Over the last two decades, interval programming based on the interval analysis has been developed as a useful and simple method to deal with this type of uncertainty. The interval numbers will be in both of the objective function and the constraints. An Illustrative numerical example is provided to clarify the proposed approach.

Application of particle swarm optimization algorithm for solving bi-level linear programming problem

Bi-level linear programming is a technique for modeling decentralized decision. It consists of the upper-level and lower-level objectives. This paper attempts to develop an efficient method based on particle swarm optimization (PSO) algorithm with swarm intelligence. The performance of the proposed method is ascertained by comparing the results with genetic algorithm (GA) using four problems in the literature and an example of supply chain model. The results illustrate that the PSO algorithm outperforms GA in accuracy.

A Solution Methodology of Bi-Level Linear Programming Based on Genetic Algorithm

Problem statement: We deal with the bi-level linear programming problem. A bi-level programming problem is formulated for a problem in which two Decision-Makers (DMs) make decisions successively. Approach: In this research we studied and designs a Genetic Algorithm (GA) of Bi-Level Linear Programming Problems (BLPP) by constructing the fitness function of the upper-level programming problems based on the definition of the feasible degree. This GA avoids the use of penalty function to deal with the constraints, by changing the randomly generated initial population into an initial population satisfying the constraints in order to improve the ability of the GA to deal with the constraints. Also we designed software to solve this problem. A comparative study between proposed method and previous methods through numerical results of some examples. Finally, parametric information of the GA was introduced. Results: Results of the study showed that the proposed method is feasible and more efficient to solve (BLPP), also there exist package to solve (BLPP) problem. Conclusion: This GA avoids the use of penalty function to deal with the constraints, by changing the randomly generated initial population into an initial population satisfying the constraints in order to improve the ability of the GA to deal with the constraints.

A method for solving bilevel linear programming problems

This paper presents an approach for solving bilevel linear programming problems (BLPP). It is based on the result that an optimal solution to the BLPP is reachable at an extreme point of the underlying region. Consequently, we develop a pivot technique to find the global optimal solution on an expanded tableau that represents the data of the BLPP. The pivot technique allows to rank in increasing order the outer level objective function value until

An adaptive genetic algorithm for solving bilevel linear programming problem

Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this problem. Of all the algorithms, the genetic algorithm is an alternative to conventional approaches to find the solution of the bilevel linear programming. In this paper, we describe an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes may be infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references.

Weighting method for bi-level linear fractional programming problems

In this paper we consider the solution of a bi-level linear fractional programming problem (BLLFPP) by weighting method. A non-dominated solution set is obtained by this method. In this article decision makers (DMs) provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control. We convert the hierarchical system into scalar optimization problem (SOP) by finding proper weights using the analytic hierarchy process (AHP) so that objective functions of both levels can be combined into one objective function. Here the relative weights represent the relative importance of the objective functions.

New Branch-and-Cut Algorithm for Bilevel Linear Programming

Linear mixed 0---1 integer programming problems may be reformulated as equivalent continuous bilevel linear programming (BLP) problems. We exploit these equivalences to transpose the concept of mixed 0---1 Gomory cuts to BLP. The first phase of our new algorithm generates Gomory-like cuts. The second phase consists of a branch-and-bound procedure to ensure finite termination with a global optimal solution. Different features of the algorithm, in particular, the cut selection and branching criteria are studied in details. We propose also a set of algorithmic tests and procedures to improve the method. Finally, we illustrate the performance through numerical experiments. Our algorithm outperforms pure branch-and-bound when tested on a series of randomly generated problems.

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

Bi-level programming, a tool for modeling decentralized decisions, consists of the objective(s) of the leader at its first level and the follower(s) at the second level. Three level programming results when the second level itself is a bi-level programming which helps us develop the idea of bi-level programming to multi-level programs with any number of levels. There are many approaches for encountering with bi(multi)-level problems. The method introduced by Chenggen Shi et al. [Chenggen Shi, Jie Lu, Guangquan Zhang, An extended Kuhn–Tucker approach for linear bi-level programming, Applied Mathematics and Computation 162 (2005) 51–63] uses extended Kuhn–Tucker conditions to solve bi-level programming problems. In this paper, we integrate goal programming (GP), Kuhn–Tucker conditions (KKT) and Penalty Function approaches to solve linear bi-level programming problems. By using a numerical example we illustrate efficiency of our methodology.