Bilevel Linear Programming Research Articles

Global efficiency for multiobjective bilevel programming problems under generalized invexity

In this paper, we consider a nonlinear optimistic bilevel programming problem \(({\textit{NMBP}})\) where the upper-level is a vector optimization problem and the lower-level is a scalar optimization problem. By using the Karush–Kuhn–Tucker conditions associated to the lower-level problem, we reformulate the bilevel programming problem into a nonlinear multiobjective single-level programming problem with equality and inequality constraints \(({\textit{MP}})\). Similarly to Dempe and Dutta (2012), we establish relationships between the problems \(({\textit{NMBP}})\) and \(({\textit{MP}})\). We prove that under appropriate constraint qualification and convexity assumptions, global (weakly or properly) efficient solutions of \(({\textit{MP}})\) correspond to global (weakly or properly) efficient solutions of \(({\textit{NMBP}})\). We establish Fritz John type necessary efficiency conditions for \(({\textit{NMBP}})\) without using any constraint qualification. Furthermore, we obtain (Fritz John type) sufficient efficiency conditions for a feasible point of \(({\textit{MP}})\) corresponds to a (weakly or properly) efficient solution for the bilevel problem \(({\textit{NMBP}})\) under various forms of generalized invexity and infineness. Moreover, a linear multiobjective bilevel programming problem is studied and sufficient efficiency conditions are derived. To illustrate the obtained results some examples are given.

Bilevel linear programming with fuzzy random variables through absolute deviation minimisation

This paper considers bilevel linear programming with fuzzy random variables. Through the introduction of a novel optimisation model that is to minimise the absolute deviation of degree of possibility and necessity that each fuzzy goal is fulfilled, the original bilevel programming problems with fuzzy random variables can be transformed into deterministic bilevel fractional programming problems. It is shown that the extended Stackelberg solutions can be obtained by solving the transformed bilevel linear programming problem through the combined use of the variable transformation method and the Kth best algorithm. A numerical example of bilevel purchase problem for food retailing is provided to illustrate the usefulness of the proposed methods.

An Efficient Ranking Technique for Intuitionistic Fuzzy Numbers with Its Application in Chance Constrained Bilevel Programming

The aim of this paper is to develop a new ranking technique for intuitionistic fuzzy numbers using the method of defuzzification based on probability density function of the corresponding membership function, as well as the complement of nonmembership function. Using the proposed ranking technique a methodology for solving linear bilevel fuzzy stochastic programming problem involving normal intuitionistic fuzzy numbers is developed. In the solution process each objective is solved independently to set the individual goal value of the objectives of the decision makers and thereby constructing fuzzy membership goal of the objectives of each decision maker. Finally, a fuzzy goal programming approach is considered to achieve the highest membership degree to the extent possible of each of the membership goals of the decision makers in the decision making context. Illustrative numerical examples are provided to demonstrate the applicability of the proposed methodology and the achieved results are compared with existing techniques.

Sequential Coordination of Transmission Expansion Planning With Strategic Generation Investments

This paper proposes mathematical models for sequential coordination of transmission expansion planning with strategic generation investments. The proactive and reactive coordinations are modeled and studied. The interaction between transmission company (Transco) and strategic generation companies (Gencos) is modeled using the sequential-move game. This is while the interaction between the strategic Gencos is modeled as a simultaneous-move game. In the proactive coordination, the Transco expands its future transmission capacities taking into account the strategic investments by Gencos. In the reactive coordination, strategic Gencos move first and expand their future generation capacities and then Transco expands the transmission capacity. The proactive coordination is modeled as a mixed-integer bilevel linear program (MIBLP) and the reactive coordination is modeled as a mixed-integer linear program (MILP). The MIBLP has binary variables in both upper and lower levels. The Moore–Bard algorithm is parallelized and used to solve the MIBLP. The mathematical models and the parallelized Moore–Bard algorithm are tested on 3-bus and 6-bus example systems and the modified IEEE-RTS96. Also, the IEEE 118-bus test system is studied using a heuristic version of the Moore–Bard algorithm.

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.

Mixed-integer bilevel optimization for capacity planning with rational markets

We formulate the capacity expansion planning as a bilevel optimization to model the hierarchical decision structure involving industrial producers and consumers. The formulation is a mixed-integer bilevel linear program in which the upper level maximizes the profit of a producer and the lower level minimizes the cost paid by markets. The upper-level problem includes mixed-integer variables that establish the expansion plan; the lower level problem is an LP that decides demands assignments. We reformulate the bilevel optimization as a single-level problem using two different approaches: KKT reformulation and duality-based reformulation. We analyze the performance of these reformulations and compare their results with the expansion plans obtained from the traditional single-level formulation. For the solution of large-scale problems, we propose improvements on the duality-based reformulation that allows reducing the number of variables and constraints. The formulations and the solution methods are illustrated with examples from the air separation industry.

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.

Solving linear bilevel programs via a new neural network

A new neural network model containing fewer neurons is proposed for the linear bilevel programs. The Lyapunov stabilityand convergence of the neural network are proved. To illustrate the feasibility of the neural network approach, we give thecorresponding numerical experiments. The comparison results show that the new neural network approach is feasible and haslesser iterations to obtain the optimal solution.

The Improvement of Optimality Test over Possible Reaction Set in Bilevel Linear Optimization with Ambiguous Objective Function of the Follower

Verifying a rational response is the most crucial step in searching for an optimal solution in bilevel linear programming. Such verification is even difficult in a model with ambiguous objective function of the follower who reacts rationally to a leader’s decision. In our model, we assume that the ambiguous coefficient vector of follower lies in a convex polytope and we formulate bilevel linear programming with the ambiguous objective function of the follower as a special three-level programming problem. We use thek-th best method that sequentially enumerates a solution and examine whether it is the best of all possible reactions. The optimality test process over possible reactions in lower-level problems usually encounters degenerate bases that become obstacles to verifying the optimality of an enumerated solution efficiently. To accelerate optimality verification, we propose search strategies and the evaluation of basic possible reactions adjacent to a degenerate basic solution. We introduce these methods in both local and global optimality testing, confirming the effectiveness of our proposed methods in numerical experiments.

Solving bi-level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm

Bi-level linear programming, consisting of upper level and lower level objectives, is a technique for modeling decentralized decision. This study presents a hybrid of immune genetic algorithm and vector-controlled particle swarm optimization (IGVPSO) to solve the bi-level linear programming problem (BLPP). It is applied to a supply chain model that is a BLPP. Using four problems from the literature and the supply chain distribution models, the computational results indicate that the proposed method is superior to some algorithms.

Measuring the technical efficiency of local banks in UAE using rough bi-level linear programming technique

The aim of this paper is to use a bi-level linear programming technique with rough parameters in the constraints, for measuring the technical efficiency of local banks in UAE and Egypt, while the proposed linear objective functions will be maximized for different goals. Based on Dauer's and Krueger's goal programmingmethod, the described approach was developed to deal with the bi-level decision-making problem. The concept of tolerance membership function together was used to generate the optimal solution for the problem under investigation. Also an auxiliary problem is discussed to illustrate the functionality of the proposed approach.

A New Method To Solve Bi-Level Quadratic Linear Fractional Programming Problems

In this paper, we have developed a new method to solve bi-level quadratic linear fractional programming (BLQLFP) problems in which the upper-level objective function is quadratic and the lower-level objective function is linear fractional. In this method a BLQLFP problem is transformed into an equivalent single-level quadratic programming (QP) problem with linear constraints by forcing the duality gap of the lower-level problem to zero. Then by obtaining all vertices of the constraint region of the dual of the lower-level problem, which is a convex polyhedron, the single-level QP problem is converted into a series of finite number of QP problems with linear constraints which can be solved by any standard method for solving a QP. The best among the optimal solutions gives the desired optimal solution for the original bi-level programming (BLP) problem. Theoretical results have been illustrated with the help of a numerical example.

Integrating Goal Programming, Taylor Series, Kuhn-Tucker Conditions, and Penalty Function Approaches to Solve Linear Fractional Bi-level Programming Problems

In this paper, we integrate goal programming (GP), Tay- lor Series, Kuhn-Tucker conditions and Penalty Function approaches to solve linear fractional bi-level programming (LFBLP)problems. As we know, the Taylor Series is having the property of transforming fractional functions to a polynomial. In the present article by Taylor Series we obtain polynomial objective functions which are equivalent to fractional objective functions. Then on using the Kuhn-Tucker optimality condition of the lower level problem, we transform the linear bilevel programming problem into a corresponding single level programming. The complemen- tary and slackness condition of the lower level problem is appended to the upper level objective with a penalty, that can be reduce to a single objective function. In the other words, suitable transformations can be applied to formulate FBLP problems. Finally a numerical example is given to illustrate the complexity of the procedure to the solution.

Enhanced exact algorithms for discrete bilevel linear problems

We address a particular class of bilevel linear programming problems in which all the variables are discrete. The main computational complexities are analyzed and two enhanced exact algorithms are proposed. The rationale behind these two algorithms is described and a modified version is presented for both. A common test bed is used to assess their computational efficiency along with a comparison with an existing benchmark algorithm.

A bi-level multi-choice programming problem

A Bi-level linear programming problem is treated as a multi-objective optimisation problem where the decision is taken by two different decision makers who are at two different levels. In this paper we consider a bi-level linear programming problem where some of the cost coefficient of the objectives, and some of the right hand side parameters of the constraints are multi-choice parameters. The aim of this paper is to establish a suitable solution procedure to solve the stated bi-level programming problem. To tackle the multi-choice parameters of the bi-level programming problem, we use some interpolating polynomials. Multi-choice parameters are replaced with interpolating polynomials. Then we use fuzzy programming method to solve the transformed bi-level programming problem. We present a numerical example to illustrate the solution procedure of the bi-level linear programming problem involving some multi-choice parameters.

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.

Upper and lower bounds for the optimal values of the interval bilevel linear programming problem

In this paper, we will investigate the interval bilevel linear programming (IBLP) problem. Recently, Calvete et al. have proposed two algorithms to find the worst and the best (upper and lower bounds) optimal values of the leader objective function in the bilevel linear programming (BLP) problem when the coefficients of the leader and the follower objective functions are interval. Through some examples, we will first show that the algorithm to find the worst optimal value of the leader objective function does not always yield a correct solution. Then, after investigating its drawbacks, we will propose a revised algorithm with which the previous examples will yield correct solutions. Finally, it will be extended to the general BLP problem wherein all the coefficients are interval. It is, of course, possible to easily find the upper and the lower bounds for the lower level objective function too.

An algorithm based on particle swarm optimization for multiobjective bilevel linear problems

This paper presents an improved multiple objective particle swarm optimization (MOPSO) algorithm to solve bilevel linear programming problems with multiple objective functions at the upper level. The algorithm aims to produce a good approximation of the entire Pareto front of the problem. We have previously designed a MOPSO algorithm for the same class of problems, in which several techniques for the global best selection were tested, including a new one. The algorithm revealed a good convergence towards the Pareto front but the diversity of the solutions was a drawback. The algorithm we propose herein uses a hybrid strategy for the global best selection and an adaptive mutation operator. The incorporation of these mechanisms led to an improved algorithm, which also showed better overall performance than considering alternative options usually employed in MOPSO algorithms. The algorithm and computational results are presented.

A new method for strong-weak linear bilevel programming problem

We first propose an exact penalty method to solve strong-weak linearbilevel programming problem (for short, SWLBP) for every fixedcooperation degree from the follower. Then, we prove that the solution ofpenalized problem is also that of the original problem under someconditions. Furthermore, we give some properties of the optimalvalue function (as a function of the follower's cooperation degree)of SWLBP. Finally, we develop a method to acquire the criticalpoints of the optimal value function without enumerating all valuesof the cooperation degree from the follower, and thus this functionis also achieved. Numerical results show that the proposedmethods are feasible.

Cost efficiency in data envelopment analysis under the law of one price

To impose the law of one price (LoOP) restrictions, which state that all firms face the same input prices, Kuosmanen, Cherchye, and Sipiläinen (2006) developed the top-down and bottom-up approaches to maximizing the industry-level cost efficiency. However, the optimal input shadow prices generated by the above approaches need not be unique, which influences the distribution of the efficiency indices at the individual firm level. To solve this problem, in this paper, we developed a pair of two-level mathematical programming models to calculate the upper and lower bounds of cost efficiency for each firm in the case of non-unique LoOP prices while keeping the industry cost efficiency optimal. Furthermore, a base-enumerating algorithm is proposed to solve the lower bound models of the cost efficiency measure, which are bi-level linear programs and NP-hard problems. Lastly, a numerical example is used to demonstrate the proposed approach.