Hybridizing exact methods and metaheuristics: A taxonomy

The interest about hybrid optimization methods has grown for the last few years. Indeed, more and more papers about cooperation between heuristics and exact techniques are published. In this paper, we propose to extend an existing taxonomy for hybrid methods involving heuristic approaches in order to consider cooperative schemes between exact methods and metaheuristics. First, we propose some natural approaches for the different schemes of cooperation encountered, and we analyse, for each model, some examples taken from the literature. Then we recall and complement the proposed grammar and provide an annotated bibliography.     

Solving the capacitated location-routing problem

This is a summary of the main results presented in the author’s Ph.D thesis, available at http://prodhonc.free.fr/homepage. This thesis, written in French, was supervised by Christian Prins and Roberto Wolfler-Calvo, and defended on 16 October 2006 at the Université de Technologie de Troyes. Several new approaches are proposed to solve the capacitated location-routing problem (CLRP): heuristic, cooperative and exact methods. Their performances are tested on various kinds of instances with capacitated vehicles and capacitated or uncapacitated depots.      

Scheduling with earliness–tardiness penalties and parallel machines

This is a summary of the author’s PhD thesis supervised by Francis Sourd and Philippe Chrétienne and defended on 30 January 2007 at the Université Pierre et Marie Curie, Paris. The thesis is written in French and is available from the author upon request. This work is about scheduling on parallel machines in order to minimize the total sum of earliness and tardiness costs. To solve some variants of this problem we propose: an exact method based on continuous relaxations of convex reformulations derived from a 0–1 quadratic program; a heuristic algorithm that relies on a new exponential size neighborhood search; finally, a lower bound method based on a polynomial time solution of a preemptive scheduling problem for which the cost functions of the jobs have been changed into so called position costs functions. Partial funding provided by CONACyT (Mexican Council for Science&Technology).     

Exact and heuristic methodologies for scheduling in hospitals: problems, formulations and algorithms

This text summarizes the PhD thesis defended by the author in January 2006 under the supervision of Professor Erik Demeulemeester at the Katholieke Universiteit Leuven. The thesis is written in English and is available from the author’s website (http://www.econ.kuleuven.be/jeroen.belien). In this research we propose a number of exact and heuristic algorithms for various scheduling problems encountered in hospitals. The emphasis lies on the design of new methodologies as well as on the applicability of the algorithms in real-life environments. The main contributions include a new decomposition approach for a particular class of staff scheduling problems, an extensive study of master surgery scheduling algorithms that aim at leveling the resultant bed occupancy and an innovative method for integrating nurse and surgery scheduling.      

Z-eigenvalue methods for a global polynomial optimization problem

As a global polynomial optimization problem, the best rank-one approximation to higher order tensors has extensive engineering and statistical applications. Different from traditional optimization solution methods, in this paper, we propose some Z-eigenvalue methods for solving this problem. We first propose a direct Z-eigenvalue method for this problem when the dimension is two. In multidimensional case, by a conventional descent optimization method, we may find a local minimizer of this problem. Then, by using orthogonal transformations, we convert the underlying supersymmetric tensor to a pseudo-canonical form, which has the same E-eigenvalues and some zero entries. Based upon these, we propose a direct orthogonal transformation Z-eigenvalue method for this problem in the case of order three and dimension three. In the case of order three and higher dimension, we propose a heuristic orthogonal transformation Z-eigenvalue method by improving the local minimum with the lower-dimensional Z-eigenvalue methods, and a heuristic cross-hill Z-eigenvalue method by using the two-dimensional Z-eigenvalue method to find more local minimizers. Numerical experiments show that our methods are efficient and promising. This work is supported by the Research Grant Council of Hong Kong and the Natural Science Foundation of China (Grant No. 10771120).     

Solving the Vehicle Routing Problem with Stochastic Demands using the Cross-Entropy Method

An alternate formulation of the classical vehicle routing problem with stochastic demands (VRPSD) is considered. We propose a new heuristic method to solve the problem, based on the Cross-Entropy method. In order to better estimate the objective function at each point in the domain, we incorporate Monte Carlo sampling. This creates many practical issues, especially the decision as to when to draw new samples and how many samples to use. We also develop a framework for obtaining exact solutions and tight lower bounds for the problem under various conditions, which include specific families of demand distributions. This is used to assess the performance of the algorithm. Finally, numerical results are presented for various problem instances to illustrate the ideas.     

Dynamic management of a portfolio of orders

We survey the main results of the PhD Thesis presented by the author in December 2003 at The Institut National Polytechnique de Grenoble. This work was supervised by Prof. Lionel Dupont and assistant professor Christophe Rapine. The thesis is written in French and is available at http: //gilco.inpg.fr. In this work, we present results on the problem of selection and scheduling of orders in a make to order production environment. We proposed exact methods and heuristic approaches for the resolution of the problem in a static context and a static iterative algorithm for a dynamic context.Received: November 2004, AMS classification: 90C27, 90B35     

Exact and heuristic algorithms for routing problems

This is a summary of the authors PhD thesis supervised by Daniele Vigo and defended on 30 March 2010, at the Università di Bologna. The thesis is written in English and is available from the author upon request. Several rich routing problems attaining to the transportation area have been studied. “Simple” algorithms have been proposed to solve them, both exact and heuristic, producing high quality solutions and transferrable methods.     

COSEARCH: A Parallel Cooperative Metaheuristic

In order to design a well-balanced metaheuristic for robustness, we propose the COSEARCH approach which manages the cooperation of complementary heuristic methods via an adaptive memory which contains a history of the search already done. In this paper, we present the idiosyncrasies of the COSEARCH approach and its application for solving large scale instances of the quadratic assignment problem (QAP). We propose an original design of the adaptive memory in order to focus on high quality regions of the search and avoid attractive but deceptive areas. For the QAP, we have hybridized three heuristic agents of complementary behaviours: a Tabu Search is used as the main search algorithm, a Genetic Algorithm is in charge of the diversification and a Kick Operator is applied to intensify the search. The evaluations have been executed on large scale network of workstations via a parallel environment which supports fault tolerance and adaptive dynamic scheduling of tasks.     

Inverse integer optimization with an imperfect observation

This study develops and evaluates methods for inverse integer optimization problems with an imperfect observation where the unknown parameters are the cost coefficients. We propose a cutting plane algorithm for this problem and compare it to a heuristic which solves the inverse of the linear relaxation of the forward problem. We then propose a hybrid approach that initializes the cutting plane algorithm from the solution of the heuristic.     

Exact methods for large-scale multi-period financial planning problems

A relevant financial planning problem is the periodical rebalance of a portfolio of assets such that the portfolio’s total value exhibits certain characteristics. This problem can be modelled using a transition graph G to represent the future state space evolution of the corresponding economy and mathematically formulated as a linear programming problem. We present two different mathematical formulations of the problem. The first considers explicitly the set of the possible scenarios (scenario-based approach), while the second considers implicitly the whole set of scenarios provided by the graph G (graph-based approach). Unfortunately, for both the formulations the size of the corresponding linear programs can be huge even for simple financial problems. However, the graph-based approach seems to be a more powerful model, since it allows to consider a huge number of scenarios in a very compact formulation. The purpose of this paper is to present both heuristic and exact methods for the solution of large-scale multi-period financial planning problems using the graph-based model. In particular, in this paper we propose lower and upper bounds and three exact methods based on column, row and column/row generation, respectively. Since the methods based on column/row generation exploits simultaneously both the primal and the dual structure of the problem we call it Criss-Cross generation method. Computational results are given to prove the effectiveness of the proposed methods.      

Comparing branch-and-price algorithms for the Multi-Commodity k-splittable Maximum Flow Problem

The Multi-Commodity k-splittable Maximum Flow Problem consists in routing as much flow as possible through a capacitated network such that each commodity uses at most k paths and the capacities are satisfied. The problem appears in telecommunications, specifically when considering Multi-Protocol Label Switching. The problem has previously been solved to optimality through branch-and-price. In this paper we propose two exact solution methods both based on an alternative decomposition. The two methods differ in their branching strategy. The first method, which branches on forbidden edge sequences, shows some performance difficulty due to large search trees. The second method, which branches on forbidden and forced edge sequences, demonstrates much better performance. The latter also outperforms a leading exact solution method from the literature. Furthermore, a heuristic algorithm is presented. The heuristic is fast and yields good solution values.     

Enhanced mixed integer programming techniques and routing problems

This is a summary of the author’s PhD thesis supervised by Andrea Lodi and Paolo Toth and defended on 16 April 2009 at the Università di Bologna. The thesis is written in English and is available from the author upon request. This work is focused on Mixed Integer Programming (MIP). In particular, the first part of the thesis deals with general purpose cutting planes, which are probably the key ingredient behind the success of the current generation of MIP solvers. The second part is instead focused on the heuristic and exact exploitation of integer programming techniques for hard combinatorial optimization problems in the context of routing applications.     

A column generation approach to train timetabling on a corridor

We propose heuristic and exact algorithms for the (periodic and non-periodic) train timetabling problem on a corridor that are based on the solution of the LP relaxation of an ILP formulation in which each variable corresponds to a full timetable for a train. This is in contrast with previous approaches to the same problem, which were based on ILP formulations in which each variable is associated with a departure and/or arrival of a train at a specific station in a specific time instant, whose LP relaxation is too expensive to be solved exactly. Experimental results on real-world instances of the problem show that the proposed approach is capable of producing heuristic solutions of better quality than those obtained by these previous approaches, and of solving some small-size instances to proven optimality.      

A pseudo-polynomial heuristic for path-constrained discrete-time Markovian-target search

We propose a new heuristic for the single-searcher path-constrained discrete-time Markovian-target search. The algorithm minimizes an approximate, instead of exact, nondetection probability computed from the conditional probability that reflects the search history over the time windows of a fixed length, l. Having a pseudo-polynomial complexity, it can solve, in reasonable time, the instances an order of magnitude larger than those solved in the previous studies. By an asymptotic analysis relying on the fast-mixing Markov chain, we show that the relative error of the approximation exponentially diminishes as l increases and the experimental results confirm the analysis. The experiment also reveals a correlation very close to 1 between the approximate and exact nondetection probability of a search path. This means that the heuristic produces near-optimal search paths.     

A breadth-first search applied to the minimization of the open stacks

This paper presents a heuristic for the minimization of the open stacks problem (MOSP). The proposed heuristic is based on a simple breadth-first search in MOSP graphs and two new greedy rules to overcome errors. The performance of the proposed heuristic is compared with the best exact and heuristic methods available in the literature. The results show that in addition to the suggested heuristic having much shorter running times than the exact algorithm, the error gap between them is small for a substantial proportion of almost 4500 benchmark instances taken from the literature. The proposed heuristic also has a more robust behaviour than the best heuristic for the MOSP, although less accurate. The proposed heuristic therefore constitutes a viable and cost-effective alternative for solving or obtaining good upper bounds for the MOSP.     

Does migration cost influence cooperation among success-driven individuals?

Migration as an important social factor has been recently considered in evolutionary games on graphs. However, the migration-related cost is largely ignored in previous works, which may indeed influence individual migration decision in human society. Here we propose a model of the success-driven migration with migration costs where individuals decide whether to migrate or not according to the migration cost and expected payoff. We consider two different calculation schemes for the migration cost, i.e., distance-dependent and distance-independent costs, and study their effects on the evolution of cooperation in the spatial prisoner’s dilemma, respectively. It is found that although the migration cost inhibits the migration of individuals, it does not necessarily lead to the suppression of cooperation. We explain the phenomenon by investigating the spatial patterns of cooperators and defectors. Interestingly, the curves of cooperation exhibit step structures and the corresponding heuristic analysis is provided. Our work complements previous studies and deepens the understanding regarding the success-driven migration on the evolution of cooperation.     

A Discrete Lagrangian-Based Global-Search Method for Solving Satisfiability Problems

Satisfiability is a class of NP-complete problems that model a wide range of real-world applications. These problems are difficult to solve because they have many local minima in their search space, often trapping greedy search methods that utilize some form of descent. In this paper, we propose a new discrete Lagrange-multiplier-based global-search method (DLM) for solving satisfiability problems. We derive new approaches for applying Lagrangian methods in discrete space, we show that an equilibrium is reached when a feasible assignment to the original problem is found and present heuristic algorithms to look for equilibrium points. Our method and analysis provides a theoretical foundation and generalization of local search schemes that optimize the objective alone and penalty-based schemes that optimize the constraints alone. In contrast to local search methods that restart from a new starting point when a search reaches a local trap, the Lagrange multipliers in DLM provide a force to lead the search out of a local minimum and move it in the direction provided by the Lagrange multipliers. In contrast to penalty-based schemes that rely only on the weights of violated constraints to escape from local minima, DLM also uses the value of an objective function (in this case the number of violated constraints) to provide further guidance. The dynamic shift in emphasis between the objective and the constraints, depending on their relative values, is the key of Lagrangian methods. One of the major advantages of DLM is that it has very few algorithmic parameters to be tuned by users. Besides the search procedure can be made deterministic and the results reproducible. We demonstrate our method by applying it to solve an extensive set of benchmark problems archived in DIMACS of Rutgers University. DLM often performs better than the best existing methods and can achieve an order-of-magnitude speed-up for some problems.     

Location-routing: Issues,models and methods

This paper is a survey of location-routing: a relatively new branch of locational analysis that takes into account vehicle routing aspects. We propose a classification scheme and look at a number of problem variants. Both exact and heuristic algorithms are investigated. Finally, some suggestions for future research are presented.