Solving a home-care districting problem in an urban setting

This article describes a districting study undertaken for the Côte-des-Neiges local community health clinic in Montreal. A territory must be partitioned into six districts by suitably grouping territorial basic units. Five districting criteria must be respected: indivisibility of basic units, respect for borough boundaries, connectivity, visiting personnel mobility, and workload equilibrium. The last two criteria are combined into a single objective function and the problem is solved by means of a tabu search technique that iteratively moves a basic unit to an adjacent district or swaps two basic units between adjacent districts. The problem was solved and the clinic management confirmed its satisfaction after a 2 year implementation period.     

Multistart method with estimation scheme for global satisfycing problems

We present a multistart method for solving global satisfycing problems. The method uses data generated by linearly converging local algorithms to estimate the cost value at the local minimum to which the local search is converging. When the estimate indicates that the local search is converging to a value higher than the satisfycing value, the local search is interrupted and a new local search is initiated from a randomly generated point. When the satisfycing problem is difficult and the estimation scheme is fairly accurate, the new method is superior over a straightforward adaptation of classical multistart methods.     

District design for arc-routing applications

In this paper we address the problem of district design for the organisation of arc-routing activities. In particular, the focus is on operations like winter gritting and road maintenance. The problem involves how to allocate the road network edges to a set of depots with given locations. The collection of edges assigned to a facility forms a district in which routes have to be designed that start and end at the facility. Apart from the ability to support good arc routing, well-designed districts for road-maintenance operations should have the road network to be serviced connected and should define clear geographical boundaries. We present three districting heuristics and evaluate the quality of the partitions by solving capacitated arc routing problems in the districts, and by comparing the solution values with a multi-depot CARP cutting plane lower bound. Our experiments reveal that based on global information about the distribution system (ie the number of facilities or districts, the average edge demand and the vehicle capacity) and by using simple guidelines, an adequate districting policy may be selected.     

A parallel stochastic method for solving linearly constrained concave global minimization problems

A parallel stochastic algorithm is presented for solving the linearly constrained concave global minimization problem. The algorithm is a multistart method and makes use of a Bayesian stopping rule to identify the global minimum with high probability. Computational results are presented for more than 200 problems on a Cray X-MP EA/464 supercomputer.     

MultiGLODS: global and local multiobjective optimization using direct search

The optimization of multimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global Pareto fronts for multiobjective multimodal derivative-free optimization problems. The proposed algorithm alternates between initializing new searches, using a multistart strategy, and exploring promising subregions, resorting to directional direct search. Components of the objective function are not aggregated and new points are accepted using the concept of Pareto dominance. The initialized searches are not all conducted until the end, merging when they start to be close to each other. The convergence of the method is analyzed under the common assumptions of directional direct search. Numerical experiments show its ability to generate approximations to the different Pareto fronts of a given problem.     

Combining a Local Search and Grover’s Algorithm in Black-Box Global Optimization

Grover’s quantum algorithm promises a quadratic acceleration for any problem formulable as a search. For unstructured search problems, its implementation and performance are well understood. The curse of dimensionality and the intractability of the general global optimization problem require any identifiable structure or regularity to be incorporated into a solution method. This paper addresses the application of Grover’s algorithm when a local search technique is available, thereby combining the quadratic acceleration with the acceleration seen in the multistart method. The author thanks Dr. Bill Baritompa for helpful discussions.     

Optimal partisan districting on planar geographies

We show that optimal partisan districting and majority securing districting in the plane with geographical constraints are NP-complete problems. We provide a polynomial time algorithm for determining an optimal partisan districting for a simplified version of the problem. In addition, we give possible explanations for why finding an optimal partisan districting for real-life problems cannot be guaranteed.     

Global optimization with a limited solution time

In this paper the box constrained global optimization problem in presence of a limited solution time is considered. A method is studied based on a combination of multistart and singlestart which implies a decision sequence on the number of random points to be generated. Search strategies are numerically illustrated. Criteria are introduced to measure the performance of solution methods for the problem class. Moreover, the performance of search strategies, specifically the efficiency of generating random points is analyzed.     

Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

This paper describes and experimentally compares five rather different multistart tabu search strategies for the unconstrained binary quadratic optimization problem: a random restart procedure, an application of a deterministic heuristic to specially constructed subproblems, an application of a randomized procedure to the full problem, a constructive procedure using tabu search adaptive memory, and an approach based on solving perturbed problems. In the solution improvement phase a modification of a standard tabu search implementation is used. A computational trick applied to this modification – mapping of the current solution to the zero vector – allowed to significantly reduce the time complexity of the search. Computational results are provided for the 25 largest problem instances from the OR-Library and, in addition, for the 18 randomly generated larger and more dense problems. For 9 instances from the OR-Library new best solutions were found.     

Attraction probabilities in variable neighborhood search

Empirical evidence demonstrates that when the same local search operator is used, variable neighborhood search consistently outperforms random multistart local search on all types of combinatorial and global optimization problems tested. In this paper we suggest that this superiority in performance may be explained by the distribution of the attraction basins around a current solution as a function of the distance from the solution. We illustrate with a well-known instance of the multisource Weber problem that the “attraction probabilities” for finding better solutions can be orders of magnitude larger in neighborhoods that are close to the current solution. The paper also discusses the global convergence properties of both general methods in the context of attraction probabilities.     

The rollon–rolloff waste collection vehicle routing problem with time windows

This study introduces a rollon–rolloff waste collection vehicle routing problem involving large containers that accumulate huge amounts of garbage at construction sites and shopping districts. In this problem, tractors move one container at a time between customer locations, a depot, disposal facilities, and container storage yards. The complicated constraints discussed in this study arise from having multiple disposal facilities, multiple container storage yards, seven service types of customer demands, different time windows for customer demands and facilities, various types and sizes of containers, and the lunch break of tractor drivers. In addition, real-world issues, such as changing service types, multiple demands at a customer’s location, and tractors with different work schedules, are dealt with. This study proposes a large neighborhood search based iterative heuristic approach consisting of several algorithms for the problem. The effectiveness of the proposed methods is demonstrated by computational experiments using benchmark data, some instances of which are derived from real-world problems.     

On the analysis of a simple evolutionary algorithm on quadratic pseudo-boolean functions

Evolutionary algorithms are randomized search heuristics, which are often used as function optimizers. In this paper the well-known (1+1) Evolutionary Algorithm ((1+1) EA) and its multistart variants are studied. Several results on the expected runtime of the (1+1) EA on linear or unimodal functions have already been presented by other authors. This paper is focused on quadratic pseudo-boolean functions, i.e., polynomials of degree 2, a class of functions containing NP-hard optimization problems. Subclasses of the class of all quadratic functions are identified where the (1+1) EA is efficient, for other subclasses the (1+1) EA has exponential expected runtime, but a large enough success probability within polynomial time such that a multistart variant of the (1+1) EA is efficient. Finally, a particular quadratic function is identified where the EA and its multistart variants fail in polynomial time with overwhelming probability.     

Performance of a genetic algorithm for the graph partitioning problem

The performance of the genetic algorithm (GA) for the graph partitioning problem (GPP) is investigated by comparison with standard heuristics on well-known benchmark graphs. In general, there is a case where a practical performance of a conventional genetic approach, which performs only simple operations without a local search strategy, is not sufficient. However, it is known that a combination of GA and local search can produce better solutions. From this practice, we incorporate a simple local search algorithm into the GA. In particular, the search ability of the GA is compared with standard heuristics such as multistart local search and simulated annealing, which use the same neighborhood structure of the simple local search, for solving the GPP. Experimental results show that the GA performs better than its competitors.     

A robust and efficient algorithm for planar competitive location problems

In this paper we empirically analyze several algorithms for solving a Huff-like competitive location and design model for profit maximization in the plane. In particular, an exact interval branch-and-bound method and a multistart heuristic already proposed in the literature are compared with uego (Universal Evolutionary Global Optimizer), a recent evolutionary algorithm. Both the multistart heuristic and uego use a Weiszfeld-like algorithm as local search procedure. The computational study shows that uego is superior to the multistart heuristic, and that by properly fine-tuning its parameters it usually (in the computational study, always) find the global optimal solution, and this in much less time than the interval branch-and-bound method. Furthermore, uego can solve much larger problems than the interval method.     

Hybrid Metaheuristics for the Vehicle Routing Problem with Stochastic Demands

This article analyzes the performance of metaheuristics on the vehicle routing problem with stochastic demands (VRPSD). The problem is known to have a computationally demanding objective function, which could turn to be infeasible when large instances are considered. Fast approximations of the objective function are therefore appealing because they would allow for an extended exploration of the search space. We explore the hybridization of the metaheuristic by means of two objective functions which are surrogate measures of the exact solution quality. Particularly helpful for some metaheuristics is the objective function derived from the traveling salesman problem (TSP), a closely related problem. In the light of this observation, we analyze possible extensions of the metaheuristics which take the hybridized solution approach VRPSD-TSP even further and report about experimental results on different types of instances. We show that, for the instances tested, two hybridized versions of iterated local search and evolutionary algorithm attain better solutions than state-of-the-art algorithms.     

A Fast,Network-based,Hybrid Heuristic for the Assignment of Students to Schools

In an era of declining and fluctuating enrolments, the determination of appropriate school sizes and student assignments poses difficult problems for administrators. For two decades, researchers have worked with variants of a linear programming districting model which minimizes total weighted distance as all students are assigned to schools constrained by both capacity and racial balance limitations. This model could aid in the generation of districting alternatives, but is difficult to solve in realistic applications with generalized solution procedures because of overall problem size. This has prompted the development of fast, network-based models, solvable with special algorithms and codes. We show, however, that these approaches all contain major shortcomings. We present a fast, hybrid heuristic based upon two network representations. Computational results demonstrate this to be a very fast solution approach for the general case of this districting problem.     

A nonmonotone GRASP

A greedy randomized adaptive search procedure (GRASP) is an iterative multistart metaheuristic for difficult combinatorial optimization problems. Each GRASP iteration consists of two phases: a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Repeated applications of the construction procedure yields different starting solutions for the local search and the best overall solution is kept as the result. The GRASP local search applies iterative improvement until a locally optimal solution is found. During this phase, starting from the current solution an improving neighbor solution is accepted and considered as the new current solution. In this paper, we propose a variant of the GRASP framework that uses a new “nonmonotone” strategy to explore the neighborhood of the current solution. We formally state the convergence of the nonmonotone local search to a locally optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP on three classical hard combinatorial optimization problems: the maximum cut problem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and the quadratic assignment problem (QAP).     

Adaptive memory in multistart heuristics for multicommodity network design

This paper focuses on the use of different memory strategies to improve multistart methods. A network design problem in which the costs are given by discrete stepwise increasing cost functions of the capacities installed in the edges is used to illustrate the contributions of adaptive memory and vocabulary building strategies. Heuristics based on shortest path and maximum flow algorithms are combined with adaptive memory in order to obtain an approximate solution to the problem in the framework of a multistart algorithm. Furthermore, a vocabulary building intensification mechanism supported by the resolution of a linear program is also explored. Numerical experiments have shown that the proposed algorithm obtained the best known solutions for some instances in the literature. These results show the contribution of each memory component and the effectiveness of their combination.     

A Simulated Annealing Genetic Algorithm for the Electrical Power Districting Problem

Due to a variety of political, economic, and technological factors, many national electricity industries around the globe are transforming from non-competitive monopolies with centralized systems to decentralized operations with competitive business units. A key challenge faced by energy restructuring specialists at the World Bank is trying to simultaneously optimize the various criteria one can use to judge the fairness and commercial viability of a particular power districting plan. This research introduces and tests a new algorithm for solving the electrical power districting problem in the context of the Republic of Ghana and using a random test problem generator. We show that our mimetic algorithm, the Simulated Annealing Genetic Algorithm, outperforms a well-known Parallel Simulated Annealing heuristic on this new and interesting problem manifested by the deregulation of electricity markets.     

The mixed capacitated general routing problem under uncertainty

We study the General Routing Problem defined on a mixed graph and with stochastic demands. The problem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clients whose demand is not deterministically known. Since each vehicle has a limited capacity, the demand uncertainty occurring at some clients affects the satisfaction of the capacity constraints, that, hence, become stochastic. The contribution of this paper is twofold: firstly we present a chance-constrained integer programming formulation of the problem for which a deterministic equivalent is derived. The introduction of uncertainty into the problem poses severe computational challenges addressed by the design of a branch-and-cut algorithm, for the exact solution of limited size instances, and of a heuristic solution approach exploring promising parts of the search space. The effectiveness of the solution approaches is shown on a probabilistically constrained version of the benchmark instances proposed in the literature for the mixed capacitated general routing problem.