Bilevel Linear Programming 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.

Farm planning in nitrate sensitive agricultural areas

One of the main aims of the Rural Development Plan under the EU Common Agricultural Policy is the protection of nitrate sensitive areas through agri-environmental schemes. This paper presents a mathematical programming model for farm planning in agricultural areas that are sensitive to nitrates. A bilevel linear programming (BLP) model is developed, that can achieve the optimal farm production plan assuming two conflicting goals: the maximisation of farm gross margin and the minimisation of fertilisers’ use. The first goal is pursued by farmers, and comprises the first level of BLP. The second goal is pursued by society, through the government, and comprises the second level of BLP. The model is applied to an agricultural area in Northern Greece, which belongs to the nitrate sensitive areas scheme of the Greek Rural Development Plan 2007–2013. The model is further used to simulate the impacts of the measure under two scenarios proposed for farms located in nitrate sensitive areas. The result shows that the model can achieve the two goals set by increasing gross margin and reducing fertilisers use.

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.

Linear bilevel programming with interval coefficients

In this paper, we address linear bilevel programs when the coefficients of both objective functions are interval numbers. The focus is on the optimal value range problem which consists of computing the best and worst optimal objective function values and determining the settings of the interval coefficients which provide these values. We prove by examples that, in general, there is no precise way of systematizing the specific values of the interval coefficients that can be used to compute the best and worst possible optimal solutions. Taking into account the properties of linear bilevel problems, we prove that these two optimal solutions occur at extreme points of the polyhedron defined by the common constraints. Moreover, we develop two algorithms based on ranking extreme points that allow us to compute them as well as determining settings of the interval coefficients which provide the optimal value range.

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.

Exploring the Steel Bar Logistics Value Chain in China

The aim of the paper is to present a conceptual exploration on how a newly emerged distribution service business in China is developed, interpreted and modelled. The study draws on a recent case from a fast expanding 3PL company in China and frames the case with a conceptual flow-model and interprets it mathematically by using a linear bi-level programming algorism. It finds that recent advances in the steel bar logistics practices in China have been phenomenal. The emerging business model is set to define the standard for the industrial sector. The logistics process and structure changes were primarily driven by not only the higher returns on investment to the participating members individually, but also by that of the total supply chain. The emerged steel bar logistics service model studied here represents a new business evolution in the 3PL service industry ever seen in China. The application of bi-level programming algorithm in assessing returns on investment and the total supply chain value has been original.

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.

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

A global convergent algorithm is proposed to solve bilevel linear fractional-linear programming, which is a special class of bilevel programming. In our algorithm, replacing the lower level problem by its dual gap equaling to zero, the bilevel linear fractional-linear programming is transformed into a traditional single level programming problem, which can be transformed into a series of linear fractional programming problem. Thus, the modified convex simplex method is used to solve the infinite linear fractional programming to obtain the global convergent solution of the original bilevel linear fractional-linear programming. Finally, an example demonstrates the feasibility of the proposed algorithm.

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.

A genetic algorithm for bi-level linear programming dynamic network design problem

Solving the Bi-level Linear Programming Network Design Problem (BLPNDP), which is typically regarded as a strongly NP-complete problem, with traditional mathematical programming algorithms is computationally intractable. A majority of the research efforts have been focused on tackling this problem using meta-heuristics. However, the evaluation functions in most meta-heuristic approaches involve dynamic traffic simulation when evaluating feasible capacity expansion policies. This commonly used evaluation function becomes a major bottleneck constraining the applicability of meta-heuristics as it is computationally expensive and analytically complicated. To address this issue, we present a modified Genetic Algorithm (GA) to solve the BLPNDP with a novel evaluation function which reduces the number of dynamic traffic simulations needed. The heuristic exploits the underlying linear programming structure of the BLPNDP and employs dual variable approximation techniques to estimate the evaluation function instead of dynamic traffic simulation. Empirical analyses are conducted on three networks of varying sizes to examine the performance of the GA based heuristic. From the experiments conducted, the proposed GA based heuristic outperforms the traditional GA. Also the proposed method of estimating the evaluation function is flexible and can be used within the framework of many meta-heuristic approaches.

DC programming and DCA for globally solving the value-at-risk

The value-at-risk is an important risk measure that has been used extensively in recent years in portfolio selection and in risk analysis. This problem, with its known bilevel linear program, is reformulated as a polyhedral DC program with the help of exact penalty techniques in DC programming and solved by DCA. To check globality of computed solutions, a global method combining the local algorithm DCA with a well adapted branch-and-bound algorithm is investigated. An illustrative example and numerical simulations are reported, which show the robustness, the globality and the efficiency of DCA.

An exact penalty on bilevel programs with linear vector optimization lower level

We are interested in a class of linear bilevel programs where the upper level is a linear scalar optimization problem and the lower level is a linear multi-objective optimization problem. We approach this problem via an exact penalty method. Then, we propose an algorithm illustrated by numerical examples.