A variable neighborhood search with an effective local search for uncapacitated multilevel lot-sizing problems

In this study, we improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. The improvement is twofold. First, we developed an effective local search method known as the Ancestors Depth-first Traversal Search (ADTS), which can be embedded in the VNS to significantly improve the solution quality. Second, we proposed a common and efficient approach for the rapid calculation of the cost change for the VNS and other generate-and-test algorithms. The new VNS algorithm was tested against 176 benchmark problems of different scales (small, medium, and large). The experimental results show that the new VNS algorithm outperforms all of the existing algorithms in the literature for solving uncapacitated MLLS problems because it was able to find all optimal solutions (100%) for 96 small-sized problems and new best-known solutions for 5 of 40 medium-sized problems and for 30 of 40 large-sized problems.     

General variable neighborhood search for the continuous optimization

We suggest a new heuristic for solving unconstrained continuous optimization problems. It is based on a generalized version of the variable neighborhood search metaheuristic. Different neighborhoods and distributions, induced from different metrics are ranked and used to get random points in the shaking step. We also propose VNS for solving constrained optimization problems. The constraints are handled using exterior point penalty functions within an algorithm that combines sequential and exact penalty transformations. The extensive computer analysis that includes the comparison with genetic algorithm and some other approaches on standard test functions are given. With our approach we obtain encouraging results.     

A variable neighborhood search heuristic for periodic routing problems

The aim of this paper is to propose a new heuristic for the Periodic Vehicle Routing Problem (PVRP) without time windows. The PVRP extends the classical Vehicle Routing Problem (VRP) to a planning horizon of several days. Each customer requires a certain number of visits within this time horizon while there is some flexibility on the exact days of the visits. Hence, one has to choose the visit days for each customer and to solve a VRP for each day. Our method is based on Variable Neighborhood Search (VNS). Computational results are presented, that show that our approach is competitive and even outperforms existing solution procedures proposed in the literature. Also considered is the special case of a single vehicle, i.e. the Periodic Traveling Salesman Problem (PTSP). It is shown that slight changes of the proposed VNS procedure is also competitive for the PTSP.     

A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem

We present a variable neighborhood search approach for solving the one-commodity pickup-and-delivery travelling salesman problem. It is characterized by a set of customers such that each of the customers either supplies (pickup customers) or demands (delivery customers) a given amount of a single product, and by a vehicle, whose given capacity must not be exceeded, that starts at the depot and must visit each customer only once. The objective is to minimize the total length of the tour. Thus, the considered problem includes checking the existence of a feasible travelling salesman’s tour and designing the optimal travelling salesman’s tour, which are both NP-hard problems. We adapt a collection of neighborhood structures, k-opt, double-bridge and insertion operators mainly used for solving the classical travelling salesman problem. A binary indexed tree data structure is used, which enables efficient feasibility checking and updating of solutions in these neighborhoods. Our extensive computational analysis shows that the proposed variable neighborhood search based heuristics outperforms the best-known algorithms in terms of both the solution quality and computational efforts. Moreover, we improve the best-known solutions of all benchmark instances from the literature (with 200 to 500 customers). We are also able to solve instances with up to 1000 customers.     

A variable neighbourhood search algorithm for the open vehicle routing problem

In the open vehicle routing problem (OVRP), the objective is to minimise the number of vehicles and then minimise the total distance (or time) travelled. Each route starts at the depot and ends at a customer, visiting a number of customers, each once, en route, without returning to the depot. The demand of each customer must be completely fulfilled by a single vehicle. The total demand serviced by each vehicle must not exceed vehicle capacity. Additionally, in one variant of the problem, the travel time of each vehicle should not exceed an upper limit.     

Asymptotic properties of constrained Markov Decision Processes

We present in this paper several asymptotic properties of constrained Markov Decision Processes (MDPs) with a countable state space. We treat both the discounted and the expected average cost, with unbounded cost. We are interested in (1) the convergence of finite horizon MDPs to the infinite horizon MDP, (2) convergence of MDPs with a truncated state space to the problem with infinite state space, (3) convergence of MDPs as the discount factor goes to a limit. In all these cases we establish the convergence of optimal values and policies. Moreover, based on the optimal policy for the limiting problem, we construct policies which are almost optimal for the other (approximating) problems. Based on the convergence of MDPs with a truncated state space to the problem with infinite state space, we show that an optimal stationary policy exists such that the number of randomisations it uses is less or equal to the number of constraints plus one. We finally apply the results to a dynamic scheduling problem.This work was partially supported by the Chateaubriand fellowship from the French embassy in Israel and by the European Grant BRA-QMIPS of CEC DG XIII     

Using mathematical programming to solve Factored Markov Decision Processes with Imprecise Probabilities

This paper investigates Factored Markov Decision Processes with Imprecise Probabilities (MDPIPs); that is, Factored Markov Decision Processes (MDPs) where transition probabilities are imprecisely specified. We derive efficient approximate solutions for Factored MDPIPs based on mathematical programming. To do this, we extend previous linear programming approaches for linear approximations in Factored MDPs, resulting in a multilinear formulation for robust “maximin” linear approximations in Factored MDPIPs. By exploiting the factored structure in MDPIPs we are able to demonstrate orders of magnitude reduction in solution time over standard exact non-factored approaches, in exchange for relatively low approximation errors, on a difficult class of benchmark problems with millions of states.     

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.     

A reactive variable neighborhood tabu search for the heterogeneous fleet vehicle routing problem with time windows

This paper presents a solution methodology for the heterogeneous fleet vehicle routing problem with time windows. The objective is to minimize the total distribution costs, or similarly to determine the optimal fleet size and mix that minimizes both the total distance travelled by vehicles and the fixed vehicle costs, such that all problem’s constraints are satisfied. The problem is solved using a two-phase solution framework based upon a hybridized Tabu Search, within a new Reactive Variable Neighborhood Search metaheuristic algorithm. Computational experiments on benchmark data sets yield high quality solutions, illustrating the effectiveness of the approach and its applicability to realistic routing problems. This work is supported by the General Secretariat for Research and Technology of the Hellenic Ministry of Development under contract GSRT NM-67.     

A reduced variable neighborhood search algorithm for uncapacitated multilevel lot-sizing problems

Multilevel lot-sizing (MLLS) problems, which involve complicated product structures with interdependence among the items, play an important role in the material requirement planning (MRP) system of modern manufacturing/assembling lines. In this paper, we present a reduced variable neighborhood search (RVNS) algorithm and several implemental techniques for solving uncapacitated MLLS problems. Computational experiments are carried out on three classes of benchmark instances under different scales (small, medium, and large). Compared with the existing literature, RVNS shows good performance and robustness on a total of 176 tested instances. For the 96 small-sized instances, the RVNS algorithm can find 100% of the optimal solutions in less computational time; for the 40 medium-sized and the 40 large-sized instances, the RVNS algorithm is competitive against other methods, enjoying good effectiveness as well as high computational efficiency. In the calculations, RVNS updated 7 (17.5%) best known solutions for the medium-sized instances and 16 (40%) best known solutions for the large-sized instances.     

Variable neighborhood search for harmonic means clustering

Harmonic means clustering is a variant of minimum sum of squares clustering (which is sometimes called K-means clustering), designed to alleviate the dependance of the results on the choice of the initial solution. In the harmonic means clustering problem, the sum of harmonic averages of the distances from the data points to all cluster centroids is minimized. In this paper, we propose a variable neighborhood search heuristic for solving it. This heuristic has been tested on numerous datasets from the literature. It appears that our results compare favorably with recent ones from tabu search and simulated annealing heuristics.     

A variable selection method based on Tabu search for logistic regression models

A Tabu search method is proposed and analysed for selecting variables that are subsequently used in Logistic Regression Models. The aim is to find from among a set of m variables a smaller subset which enables the efficient classification of cases. Reducing dimensionality has some very well-known advantages that are summarized in literature. The specific problem consists in finding, for a small integer value of p, a subset of size p of the original set of variables that yields the greatest percentage of hits in Logistic Regression. The proposed Tabu search method performs a deep search in the solution space that alternates between a basic phase (that uses simple moves) and a diversification phase (to explore regions not previously visited). Testing shows that it obtains significantly better results than the Stepwise, Backward or Forward methods used by classic statistical packages. Some results of applying these methods are presented.     

A variable neighborhood search for the capacitated arc routing problem with intermediate facilities

The capacitated arc routing problem (CARP) focuses on servicing edges of an undirected network graph. A wide spectrum of applications like mail delivery, waste collection or street maintenance outlines the relevance of this problem. A realistic variant of the CARP arises from the need of intermediate facilities (IFs) to load up or unload the service vehicle and from tour length restrictions. The proposed Variable Neighborhood Search (VNS) is a simple and robust solution technique which tackles the basic problem as well as its extensions. The VNS shows excellent results on four different benchmark sets. Particularly, for all 120 instances the best known solution could be found and in 71 cases a new best solution was achieved.     

A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem

We present a new general variable neighborhood search approach for the uncapacitated single allocation p-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.     

An integer linear programming approach and a hybrid variable neighborhood search for the car sequencing problem

In this paper we present two major approaches to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line with respect to constraints of the form “no more than l_ccars are allowed to require a component c in any subsequence of m_cconsecutive cars”. The first method is an exact one based on integer linear programming (ILP). The second approach is hybrid: it uses ILP techniques within a general variable neighborhood search (VNS) framework for examining large neighborhoods. We tested the two methods on benchmark instances provided by CSP_LIB and the automobile manufacturer RENAULT for the ROADEF Challenge 2005. These tests reveal that our approaches are competitive to previous reported algorithms. For the CSP_LIB instances we were able to shorten the required computation time for reaching and proving optimality. Furthermore, we were able to obtain tight bounds on some of the ROADEF instances. For two of these instances the proposed ILP-method could provide new optimality proofs for already known solutions. For the VNS, the individual contributions of the used neighborhoods are also experimentally analyzed. Results highlight the significant impact of each structure. In particular the large ones examined using ILP techniques enhance the overall performance significantly, so that the hybrid approach clearly outperforms variants including only commonly defined neighborhoods.     

Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches. This work is supported by the RTN ADONET under grant 504438.     

A compromised large-scale neighborhood search heuristic for capacitated air cargo loading planning

Cost effectiveness is central to the air freight forwarders. In this work, we study how an air freight forwarder should plan its cargo loading in order to minimize the total freight cost given a limited number of rented containers. To solve the problem efficiently for practical implementation, we propose a new large-scale neighborhood search heuristic. The proposed large-scale neighborhood relaxes the subset-disjoint restriction made in the existing literature; the relaxation risks a possibility of infeasible exchanges while at the same time it avoids the potentially large amount of checking effort required to enforce the subset-disjoint restriction. An efficient procedure is then used to search for improvement in the neighborhood. We have also proposed a subproblem to address the difficulties caused by the fixed charges. The compromised large-scale neighborhood (CLSN) search heuristic has shown stably superior performance when compared with the traditional large-scale neighborhood search and the mixed integer programming model.     

Model and algorithm for inventory/routing decision in a three-echelon logistics system

This paper addresses an integrated inventory and routing problem in a three-echelon logistics system, which consists of a supplier, a central warehouse and a group of retailers. The inventory decision of each member and the routing decision among members of the system are made simultaneously, with the objective of minimizing the overall average cost of the system. A strategy named fixed partition and power-of-two (FP–POT) is proposed for the considered problem and a variable large neighborhood search (VLNS) algorithm, which is a special case of variable neighborhood search (VNS) algorithm, is developed. The efficiency of the strategy as well as the algorithm is illustrated by comparing computational results with a lower bound. The advantage of the proposed VLNS algorithm is further shown by getting better results for the problems in a two-echelon logistics system, which have been solved by a Tabu Search algorithm recently.     

A continuous variable neighborhood search heuristic for finding the three-dimensional structure of a molecule

We develop a continuous variable neighborhood search heuristic for minimizing the potential energy function of a molecule. Computing the global minimum of this function is very difficult because it has a large number of local minimizers which grows exponentially with molecule size. Experimental evidence shows that in the great majority of cases the global minimum potential energy of a given molecule corresponds to its three-dimensional structure and this structure is important because it dictates most of the properties of the molecule. Computational results for problems with up to 200 degrees of freedom are presented and favourable compared with other two existing methods from the literature.     

A hybrid heuristic algorithm for the open-pit-mining operational planning problem

This paper deals with the Open-Pit-Mining Operational Planning problem with dynamic truck allocation. The objective is to optimize mineral extraction in the mines by minimizing the number of mining trucks used to meet production goals and quality requirements. According to the literature, this problem is NP-hard, so a heuristic strategy is justified. We present a hybrid algorithm that combines characteristics of two metaheuristics: Greedy Randomized Adaptive Search Procedures and General Variable Neighborhood Search. The proposed algorithm was tested using a set of real-data problems and the results were validated by running the CPLEX optimizer with the same data. This solver used a mixed integer programming model also developed in this work. The computational experiments show that the proposed algorithm is very competitive, finding near optimal solutions (with a gap of less than 1%) in most instances, demanding short computing times.