An effective VNS for the capacitated p-median problem

In the capacitated p-median problem (CPMP), a set of n customers is to be partitioned into p disjoint clusters, such that the total dissimilarity within each cluster is minimized subject to constraints on maximum cluster capacity. Dissimilarity of a cluster is the sum of the dissimilarities between each customer who belongs to the cluster and the median associated with the cluster. An effective variable neighbourhood search heuristic for this problem is proposed. The heuristic is characterized by the use of easily computed lower bounds to assess whether undertaking computationally expensive calculation of the worth of moves, within the neighbourhood search, is necessary. The small proportion of moves that need to be assessed fully are then evaluated by an exact solution of a relatively small subproblem. Computational results on five standard sets of benchmark problem instances show that the heuristic finds all the best-known solutions. For one instance, the previously best-known solution is improved, if only marginally.     

Density Based Problem Space Search for the Capacitated Clustering p-Median Problem

In the Capacitated Clustering Problem (CCP), a given set of n weighted points is to be partitioned into p clusters such that, the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centers that minimises total scatter of points allocated to them. In this paper a new constructive method, a general framework to improve the performance of greedy constructive heuristics, and a problem space search procedure for the CCP are proposed. The constructive heuristic finds patterns of natural subgrouping in the input data using concept of density of points. Elements of adaptive computation and periodic construction–deconstruction concepts are implemented within the constructive heuristic to develop a general framework for building efficient heuristics. The problem-space search procedure is based on perturbations of input data for which a controlled perturbation strategy, intensification and diversification strategies are developed. The implemented algorithms are compared with existing methods on a standard set of bench-marks and on new sets of large-sized instances. The results illustrate the strengths of our algorithms in terms of solution quality and computational efficiency.     

Worst-case analysis of demand point aggregation for the Euclidean p-median problem

Solving large-scale p-median problems is usually time consuming. People often aggregate the demand points in a large-scale p-median problem to reduce its problem size and make it easier to solve. Most traditional research on demand point aggregation is either experimental or assuming uniformly distributed demand points in analytical studies. In this paper, we study demand point aggregation for planar p-median problem when demand points are arbitrarily distributed. Efficient demand aggregation approaches are proposed with the corresponding attainable worst-case aggregation error bounds measured. We demonstrate that these demand aggregation approaches introduce smaller worst-case aggregation error bounds than that of the honeycomb heuristic [Papadimitriou, C.H., 1981. Worst-case and probabilistic analysis of a geometric location problem. SIAM Journal on Computing 10, 542–557] when demand points are arbitrarily distributed. We also conduct numerical experiments to show their effectiveness.     

A simple heuristic for the p-centre problem

We describe a simple heuristic for determining the p-centre of a finite set of weighted points in an arbitrary metric space. The computational effort is O(np) for an n-point set. We show that the ratio of the objective function value of the heuristic solution to that of the optimum is bounded by min(3, 1 + α), where α is the maximum weight divided by the minimum weight of points in the set.     

Searching for a counterfeit coin with b-balance

We consider the classical problem of searching for a heavier coin in a set of n coins, n-1 of which have the same weight. The weighing device is b-balance which is the generalization of two-arms balance. The minimum numbers of weighings are determined exactly for worst-case sequential algorithm, average-case sequential algorithm, worst-case predetermined algorithm, average-case predetermined algorithm.We also investigate the above search model with additional constraint: each weighing is only allowed to use the coins that are still in doubt. We present a worst-case optimal sequential algorithm and an average-case optimal sequential algorithm requiring the minimum numbers of weighings.     

Some heuristic methods for solving p-median problems with a coverage constraint

The aim of this paper is to solve p-median problems with an additional coverage constraint. These problems arise in location applications, when the trade-off between distance and coverage is being calculated. Three kinds of heuristic algorithms are developed. First, local search procedures are designed both for constructing and improving feasible solutions. Second, a multistart GRASP heuristic is developed, based on the previous local search methods. Third, by employing Lagrangean relaxation methods, a very efficient Lagrangean heuristic algorithm is designed, which extends the well known algorithm of Handler and Zang, for constrained shortest path problems, to constrained p-median problems. Finally, a comparison of the computational efficiency of the developed methods is made between a variety of problems of different sizes.     

A new distributionally robust p-hub median problem with uncertain carbon emissions and its tractable approximation method

The p-hub median problem is to determine the optimal location for p hubs and assign the remaining nodes to hubs so as to minimize the total transportation costs. Under the carbon cap-and-trade policy, we study this problem by addressing the uncertain carbon emissions from the transportation, where the probability distributions of the uncertain carbon emissions are only partially available. A novel distributionally robust optimization model with the ambiguous chance constraint is developed for the uncapacitated single allocation p-hub median problem. The proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, which is computationally intractable for general ambiguity sets. To solve this hard optimization model, we discuss the safe approximation to the ambiguous chance constraint in the following two types of ambiguity sets. The first ambiguity set includes the probability distributions with the bounded perturbations with zero means. In this case, we can turn the ambiguous chance constraint into its computable form based on tractable approximation method. The second ambiguity set is the family of Gaussian perturbations with partial knowledge of expectations and variances. Under this situation, we obtain the deterministic equivalent form of the ambiguous chance constraint. Finally, we validate the proposed optimization model via a case study from Southeast Asia and CAB data set. The numerical experiments indicate that the optimal solutions depend heavily on the distribution information of carbon emissions. In addition, the comparison with the classical robust optimization method shows that the proposed distributionally robust optimization method can avoid over-conservative solutions by incorporating partial probability distribution information. Compared with the stochastic optimization method, the proposed method pays a small price to depict the uncertainty of probability distribution. Compared with the deterministic model, the proposed method generates the new robust optimal solution under uncertain carbon emissions.     

Robust Optimization for the Connected Facility Location Problem

In this paper we present a robust optimization (RO) model for the Connected Facility Location (ConFL) problem within the framework introduced by Bertsimas and Sim [Bertsimas, D. and M. Sim, Robust discrete optimization and network flows, Mathematical Programming 98 (2003), pp. 49–71], and show how to use a heuristic in conjunction with a lower bounding mechanism to rapidly find high-quality solutions. The use of a heuristic and a lower bound mechanism within this RO approach decreases significantly its computational time and broadens its applicability to other NP-hard problems. Here we present some of our computational results that attest to the efficiency of the approach, particularly on the Robust ConFL problem.     

Mathematical programming approaches for generating p-efficient points

Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.     

A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems

In this paper we present a genetic algorithm-based heuristic especially for the weighted maximum independent set problem (IS). The proposed approach treats also some equivalent combinatorial optimization problems. We introduce several modifications to the basic genetic algorithm, by (i) using a crossover called two-fusion operator which creates two new different children and (ii) replacing the mutation operator by the heuristic-feasibility operator tailored specifically for the weighted independent set. The performance of our algorithm was evaluated on several randomly generated problem instances for the weighted independent set and on some instances of the DIMACS Workshop for the particular case: the unweighted maximum clique problem. Computational results show that the proposed approach is able to produce high-quality solutions within reasonable computational times. This algorithm is easily parallelizable and this is one of its important features.     

The p-median problem under uncertainty

Consider the need to currently locate p facilities but it is possible that up to q additional facilities will have to be located in the future. There are known probabilities that 0 ? r ? q facilities will need to be located. The p-median problem under uncertainty is to find the location of p facilities such that the expected value of the objective function in the future is minimized. The problem is formulated on a graph, properties of it are proven, an integer programming formulation is constructed, and heuristic algorithms are suggested for its solution. The heuristic algorithms are modified to reduce the run time by about two orders of magnitude with minimal effect on the quality of the solution. Optimal solutions for many problems are found effectively by CPLEX. Computational results using the heuristic algorithms are presented.     

Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness

Consider a single machine and a set of n jobs that are available for processing at time 0. Job j has a processing time p_j, a due date d_j and a weight w_j. We consider bi-criteria scheduling problems involving the maximum weighted tardiness and the number of tardy jobs. We give NP-hardness proofs for the scheduling problems when either one of the two criteria is the primary criterion and the other one is the secondary criterion. These results answer two open questions posed by Lee and Vairaktarakis in 1993. We consider complexity relationships between the various problems, give polynomial-time algorithms for some special cases, and propose fast heuristics for the general case. The effectiveness of the heuristics is measured by empirical study. Our results show that one heuristic performs extremely well compared to optimal solutions.     

Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values

We prove boundary asymptotics to solutions of weighted p-Laplacian equations that take infinite value on the boundary of a bounded domain. Uniqueness of such solutions would then follow as a consequence. Our results extend previously known results by allowing weights that are unbounded in the domain.     

The directional p-median problem: Definition,complexity, and algorithms

An instance of a p-median problem gives n demand points. The objective is to locate p supply points in order to minimize the total distance of the demand points to their nearest supply point. p-Median is polynomially solvable in one dimension but NP-hard in two or more dimensions, when either the Euclidean or the rectilinear distance measure is used. In this paper, we treat the p-median problem under a new distance measure, the directional rectilinear distance, which requires the assigned supply point for a given demand point to lie above and to the right of it. In a previous work, we showed that the directional p-median problem is polynomially solvable in one dimension; we give here an improved solution through reformulating the problem as a special case of the constrained shortest path problem. We have previously proven that the problem is NP-complete in two or more dimensions; we present here an efficient heuristic to solve it. Compared to the robust Teitz and Bart heuristic, our heuristic enjoys substantial speedup while sacrificing little in terms of solution quality, making it an ideal choice for real-world applications with thousands of demand points.     

A tutorial proof that extreme points are basic feasible solutions

Given a set of m linear equations in n unknowns with the requirement that the solution space be nonnegative, a simple, heuristic proof is offered which shows that the extreme points of the set of feasible solutions are also basic feasible solutions. This proof can be used in many text treatments of Linear Programming which omit the proof on the grounds that it is too difficult to prove.     

Scheduling a single batch-processing machine with arbitrary job sizes and incompatible job families: An ant colony framework

This paper investigates the first attempt on the batch-processing machine scheduling problem, where the machine can process multiple jobs simultaneously, using an ant colony optimization metaheuristic. We consider the scheduling problem of a single batch-processing machine with incompatible job families and the performance measure of minimizing total weighted completion time. Jobs of a given family have an identical processing time and are characterized by arbitrary sizes and weights. Based on a number of developed heuristic approaches, we propose an ant colony framework (ACF) in two versions, which are distinguished by the type of embedded heuristic information. Each version is also investigated in two formats, that is the pure ACF and the hybridized ACF. To verify the performance of our framework, comparisons are made based on using a set of well-known existing heuristic and meta-heuristic algorithms taken from the literature, on a diverse set of artificially generated test problem instances. Computational results show the high performance of the proposed framework and signify its ability to outperform the comparator algorithms in most cases as the problem size increases.     

A heuristic for direct consumer participation in primary health care planning

Recent attempts at consumer participation in the health care planning process have proved weak in their ability to responsively account for consumer health welfare. This can be attributed, in large part, to the mechanisms employed for identifying and utilizing the consumer's health care views and preferences. A heuristic planning procedure designed to overcome these problems by directly incorporating consumer preferences is developed. It identifies that (primary) health care delivery system which maximizes total incremental health benefit to a community subject to a prespecified budget constraint. The model assumes a methodology (previously developed by the author) for measuring, in aggregable units, the benefit, B_ip, from some health care facility p as perceived by some consumer i. Application of the procedure and subsequent sensitivity analyses demonstrate its ability to generate valid solutions that are robust to disturbances in the planning system.     

Likelihood robust optimization for data-driven problems

We consider optimal decision-making problems in an uncertain environment. In particular, we consider the case in which the distribution of the input is unknown, yet there is some historical data drawn from the distribution. In this paper, we propose a new type of distributionally robust optimization model called the likelihood robust optimization (LRO) model for this class of problems. In contrast to previous work on distributionally robust optimization that focuses on certain parameters (e.g., mean, variance, etc.) of the input distribution, we exploit the historical data and define the accessible distribution set to contain only those distributions that make the observed data achieve a certain level of likelihood. Then we formulate the targeting problem as one of optimizing the expected value of the objective function under the worst-case distribution in that set. Our model avoids the over-conservativeness of some prior robust approaches by ruling out unrealistic distributions while maintaining robustness of the solution for any statistically likely outcomes. We present statistical analyses of our model using Bayesian statistics and empirical likelihood theory. Specifically, we prove the asymptotic behavior of our distribution set and establish the relationship between our model and other distributionally robust models. To test the performance of our model, we apply it to the newsvendor problem and the portfolio selection problem. The test results show that the solutions of our model indeed have desirable performance.     

A fast approximation algorithm for the multicovering problem

The multicovering problem is: MIN cx subject to Ax≥b, xϵ{0, 1}ⁿ, where A is a matrix whose elements are all zero or one and b is a vector of positive integers. We present a fast heuristic for this important class of problems and analyze its worst-case performance: the ratio of the heuristic value to the optimum does not exceed the maximum row sum of the matrix A. The heuristic algorithm also provides a feasible dual solution vector that is useful in generating lower bounds on the value of the optimum.     

The p-median problem: A survey of metaheuristic approaches

The p-median problem is one of the basic models in discrete location theory. As with most location problems, it is classified as NP-hard, and so, heuristic methods are usually used to solve it. Metaheuristics are frameworks for building heuristics. In this survey, we examine the p-median, with the aim of providing an overview on advances in solving it using recent procedures based on metaheuristic rules.