Carmen Galé

A new approach for solving linear bilevel problems using genetic algorithms

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. This paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron. Taking into account the existence of an extreme point of the polyhedron which solves the problem, the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron. The numerical results show the efficiency of the proposed algorithm. In addition, this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear.

Bilevel model for production-distribution planning solved by using ant colony optimization

This paper addresses a hierarchical production-distribution planning problem. There are two different decision makers controlling the production and the distribution processes, respectively, that do not cooperate because of different optimization strategies. The distribution company, which is the leader of the hierarchical process, controls the allocation of retailers to each depot and the routes which serve them. In order to supply items to retailers, the distribution company orders from the manufacturing company the items which have to be available at the depots. The manufacturing company, which is the follower of the hierarchical process, reacts to these orders deciding which manufacturing plants will produce them. A bilevel program is proposed to model the problem and an ant colony optimization based approach is developed to solve the bilevel model. In order to construct a feasible solution, the procedure uses ants to compute the routes of a feasible solution of the associated multi-depot vehicle route problem. Then, under the given data on depot needs, the corresponding production problem of the manufacturing company is solved. Global pheromone trail updating is based on the leader objective function, which involves costs of sending items from depots to retailers and costs of acquiring items from manufacturing plants and unloading them into depots. A computational experiment is carried out to analyze the performance of the algorithm.

A goal programming approach to vehicle routing problems with soft time windows

The classical vehicle routing problem involves designing a set of routes for a fleet of vehicles based at one central depot that is required to serve a number of geographically dispersed customers, while minimizing the total travel distance or the total distribution cost. Each route originates and terminates at the central depot and customers demands are known. In many practical distribution problems, besides a hard time window associated with each customer, defining a time interval in which the customer should be served, managers establish multiple objectives to be considered, like avoiding underutilization of labor and vehicle capacity, while meeting the preferences of customers regarding the time of the day in which they would like to be served (soft time windows). This work investigates the use of goal programming to model these problems. To solve the model, an enumeration-followed-by-optimization approach is proposed which first computes feasible routes and then selects the set of best ones. Computational results show that this approach is adequate for medium-sized delivery problems.

Linear bilevel programs with multiple objectives at the upper level

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. Focus of the paper is on general bilevel optimization problems with multiple objectives at the upper level of decision making. When all objective functions are linear and constraints at both levels define polyhedra, it is proved that the set of efficient solutions is non-empty. Taking into account the properties of the feasible region of the bilevel problem, some methods of computing efficient solutions are given based on both weighted sum scalarization and scalarization techniques. All the methods result in solving linear bilevel problems with a single objective function at each level.

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.

The bilevel linear/linear fractional programming problem

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. In this paper we consider the bilevel linear/linear fractional programming problem in which the objective function of the first level is linear, the objective function of the second level is linear fractional and the feasible region is a polyhedron. For this problem we prove that an optimal solution can be found which is an extreme point of the polyhedron. Moreover, taking into account the relationship between feasible solutions to the problem and bases of the technological coefficient submatrix associated to variables of the second level, an enumerative algorithm is proposed that finds a global optimum to the problem.

Solving linear fractional bilevel programs

In this paper, we prove that an optimal solution to the linear fractional bilevel programming problem occurs at a boundary feasible extreme point. Hence, the Kth-best algorithm can be proposed to solve the problem. This property also applies to quasiconcave bilevel problems provided that the first level objective function is explicitly quasimonotonic.

A genetic algorithm for solving linear fractional bilevel problems

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. In this paper a genetic algorithm is proposed for the class of bilevel problems in which both level objective functions are linear fractional and the common constraint region is a bounded polyhedron. The algorithm associates chromosomes with extreme points of the polyhedron and searches for a feasible solution close to the optimal solution by proposing efficient crossover and mutation procedures. The computational study shows a good performance of the algorithm, both in terms of solution quality and computational time.

A Multiobjective Bilevel Program for Production-Distribution Planning in a Supply Chain

Production-distribution planning problems in a supply chain are complex and generally involve several decision makers. If there is a principal firm which controls the integrated production-distribution process or multiple firms assumed to collaborate in order to achieve common goals, then standard mathematical programs with one or multiple objectives can be used to address them. However, very often production-distribution planning problems involve decision makers at two distinct levels with a hierarchical relationship between them. This might be systems in which a principal firm, at the upper-level of the decision process, controls the distribution centers and seeks to minimize transportation costs from the suppliers to the warehouses and from these to the retailers. At the lower-level, each manufacturing plant, on receiving the order of the company, seeks to minimize its operating costs. This paper addresses these systems and proposes bilevel optimization to model them. If a single firm controls the manufacturing plants, the resulting model is a linear bilevel optimization program. When manufacturing plants have more objectives to consider than just operating costs, a linear/linear multiobjective bilevel program is proposed to model the system. In both cases, the optimal solution is achieved at an extreme point of the constraint region.

A note on ‘bilevel linear fractional programming problem’☆

Abstract In this note, the bilevel linear fractional programming problem is considered. We examine a previous paper published by Thirwani and Arora [Cah. CERO 35 (1993) 135]. We show that several proofs in that paper are incorrect and provide alternative proofs.