A generalization of the weighted set covering problem

We study a generalization of the weighted set covering problem where every element needs to be covered multiple times. When no set contains more than two elements, we can solve the problem in polynomial time by solving a corresponding weighted perfect b‐matching problem. In general, we may use a polynomial‐time greedy heuristic similar to the one for the classical weighted set covering problem studied by D.S. Johnson [Approximation algorithms for combinatorial problems, J Comput Syst Sci 9 (1974), 256–278], L. Lovasz [On the ratio of optimal integral and fractional covers, Discrete Math 13 (1975), 383–390], and V. Chvatal [A greedy heuristic for the set‐covering problem, Math Oper Res 4(3) (1979), 233–235] to get an approximate solution for the problem. We find a worst‐case bound for the heuristic similar to that for the classical problem. In addition, we introduce a general type of probability distribution for the population of the problem instances and prove that the greedy heuristic is asymptotically optimal for instances drawn from such a distribution. We also conduct computational studies to compare solutions resulting from running the heuristic and from running the commercial integer programming solver CPLEX on problem instances drawn from a more specific type of distribution. The results clearly exemplify benefits of using the greedy heuristic when problem instances are large. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

An efficient heuristic for large set covering problems

A heuristic solution procedure for set covering is presented that works well for large, relatively dense problems. In addition, a confidence interval is established about the unknown global optimum. Results are presented for 30 large randomly generated problems.     

The graph association problem: Mathematical models and a lagrangian heuristic

Graph association is the problem of merging many graphs that collectively describe a set of possibly repetitive entities and relationships into a single graph that contains unique entities and relationships. As a form of data association, graph association can be used to identify when two sensors are observing the same object so information from both sensors can be combined and analyzed in a meaningful and consistent way. Graph association between two graphs is related to the problem of graph matching, and between multiple graphs it is related to the common labeling of a graph set (also known as multiple graph matching) problem. This article contribution is to formulate graph association as a binary linear program and introduce a heuristic for solving multiple graph association using a Lagrangian relaxation approach to address issues with between‐graph transitivity requirements. The algorithms are tested on a representative dataset. The developed model formulation was found to accurately solve the graph association problem. Furthermore, the Lagrangian heuristic was found to solve the developed model within 3% of optimal on many problem instances, and found better solutions to large problems than is possible by directly using CPLEX. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Models and algorithms for shuffling problems in steel plants

In this article, we study item shuffling (IS) problems arising in the logistics system of steel production. An IS problem here is to optimize shuffling operations needed in retrieving a sequence of steel items from a warehouse served by a crane. There are two types of such problems, plate shuffling problems (PSP) and coil shuffling problems (CSP), considering the item shapes. The PSP is modeled as a container storage location assignment problem. For CSP, a novel linear integer programming model is formulated considering the practical stacking and shuffling features. Several valid inequalities are constructed to accelerate the solving of the models. Some properties of optimal solutions of PSP and CSP are also derived. Because of the strong NP‐hardness of the problems, we consider some special cases of them and propose polynomial time algorithms to obtain optimal solutions for these cases. A greedy heuristic is proposed to solve the general problems and its worst‐case performances on both PSP and CSP are analyzed. A tabu search (TS) method with a tabu list of variable length is proposed to further improve the heuristic solutions. Without considering the crane traveling distance, we then construct a rolling variable horizon heuristic for the problems. Numerical experiments show that the proposed heuristic algorithms and the TS method are effective. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

An asymptotically optimal greedy heuristic for the multiperiod single‐sourcing problem: The cyclic case

The dynamics of the environment in which supply chains evolve requires that companies frequently redesign their logistics distribution networks. In this paper we address a multiperiod single‐sourcing problem that can be used as a strategic tool for evaluating the costs of logistics network designs in a dynamic environment. The distribution networks that we consider consist of a set of production and storage facilities, and a set of customers who do not hold inventories. The facilities face production capacities, and each customer's demand needs to be delivered by a single facility in each period. We deal with the assignment of customers to facilities, as well as the location, timing, and size of inventories. In addition, to mitigate start and end‐of‐study effects, we view the planning period as a typical future one, which will repeat itself. This leads to a cyclic model, in which starting and ending inventories are equal. Based on an assignment formulation of the problem, we propose a greedy heuristic, and prove that this greedy heuristic is asymptotically feasible and optimal in a probabilistic sense. We illustrate the behavior of the greedy heuristic, as well as some improvements where the greedy heuristic is used as the starting point of a local interchange procedure, on a set of randomly generated test problems. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 412–437, 2003     

An exact algorithm for the maximal covering problem

This article develops a robust, exact algorithm for the maximal covering problem (MCP) using dual-based solution methods and greedy heuristics in branch and bound. Based on tests using randomly generated problems with problem parameters similar to those in the existing literature, the hybrid approach developed in this work appears to be effective over a wide range of MCP model parameters. The method is further validated on problems constructed from three real-world data sets. The extensive computational study compares the new method with other existing exact methods using problems that are as big, or larger than, those used in previous work on MCP. The results show that the proposed method is effective in most instances of MCP. In particular, it is shown that bounding schemes using Lagrangian relaxation are effective on MCP as a method of obtaining both exact and heuristic solutions. © 1996 John Wiley & Sons, Inc.     

Analysis of the greedy approach in problems of maximum k-coverage

In this paper, we consider a general covering problem in which k subsets are to be selected such that their union covers as large a weight of objects from a universal set of elements as possible. Each subset selected must satisfy some structural constraints. We analyze the quality of a k-stage covering algorithm that relies, at each stage, on greedily selecting a subset that gives maximum improvement in terms of overall coverage. We show that such greedily constructed solutions are guaranteed to be within a factor of 1 − 1/e of the optimal solution. In some cases, selecting a best solution at each stage may itself be difficult; we show that if a β-approximate best solution is chosen at each stage, then the overall solution constructed is guaranteed to be within a factor of 1 − 1/e^β of the optimal. Our results also yield a simple proof that the number of subsets used by the greedy approach to achieve entire coverage of the universal set is within a logarithmic factor of the optimal number of subsets. Examples of problems that fall into the family of general covering problems considered, and for which the algorithmic results apply, are discussed. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 615–627, 1998     

Benders' cuts guided large neighborhood search for the traveling umpire problem

This article introduces the use of Benders' cuts to guide a large neighborhood search to solve the traveling umpire problem, a sports scheduling problem inspired by the real‐life needs of the officials of a sports league. At each time slot, a greedy matching heuristic is used to construct a schedule. When an infeasibility is recognized first a single step backtracking is tried to resolve the infeasibility. If unsuccessful, Benders' cuts are generated to guide a large neighborhood search to ensure feasibility and to improve the solution. Realizing the inherent symmetry present in the problem, a large family of cuts are generated and their effectiveness is tested. The resulting approach is able to find better solutions to many instances of this problem. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Using grasp to solve the component grouping problem

Component grouping problems, a type of set-partitioning problem, arise in a number of different manufacturing and material logistics application areas. For example, in circuit board assembly, robotic work cells can be used to insert components onto a number of different types of circuit boards. Each type of circuit board requires particular components, with some components appearing on more than one type. The problem is to decide which components should be assigned to each work cell in order to minimize the number of visits by circuit boards to work cells. We describe two new heuristics for this problem, based on so-called greedy random adaptive search procedures (GRASP). With GRASP, a local search technique is replicated many times with different starting points. The starting points are determined by a greedy procedure with a probabilistic aspect. The best result is then kept as the solution. Computational experiments on problems based on data from actual manufacturing processes indicate that these GRASP methods outperform, both in speed and in solution quality, an earlier, network-flow-based heuristic. We also describe techniques for generating lower bounds for the component grouping problem, based on the combinatorial structure of a problem instance. The lower bounds for our real-world test problems averaged within 7%-8% of the heuristic solutions. Similar results are obtained for larger, randomly generated problems. © 1994 John Wiley & Sons. Inc.     

The fixed charge problem

The fixed charge problem is a nonlinear programming problem of practical interest in business and industry. Yet, until now no computationally feasible exact method of solution for large problems had been developed. In this paper an exact algorithm is presented which is computationally feasible for large problems. The algorithm is based upon a branch and bound approach, with the additional feature that the amount of computer storage required remains constant throughout (for a problem of any given size). Also presented are three suboptimal heuristic algorithms which are of interest because, although they do not guarantee that the true optimal solution will be found, they usually yield very good solutions and are extremely rapid techniques. Computational results are described for several of the heuristic methods and for the branch and bound algorithm.     

The maximum collection problem with time-dependent rewards

We consider a routing problem where the objective is to maximize the sum of the rewards collected at the nodes visited. Node rewards are decreasing linear functions of time. Time is spent when traveling between pairs of nodes, and while visiting the nodes. We propose a penalty-based greedy (heuristic) algorithm and a branch-and-bound (optimal) algorithm for this problem. The heuristic is very effective in obtaining good solutions. We can solve problems with up to 20 nodes optimally on a microcomputer using the branch-and-bound algorithm. We report our computational experience with this problem. © 1996 John Wiley & Sons, Inc.     

A heuristic with tie breaking for certain 0–1 integer programming models

A heuristic for 0–1 integer programming is proposed that features a specific rule for breaking ties that occur when attempting to determine a variable to set to 1 during a given iteration. It is tested on a large number of small- to moderate-sized randomly generated generalized set-packing models. Solutions are compared to those obtained using an existing well-regarded heuristic and to solutions to the linear programming relaxations. Results indicate that the proposed heuristic outperforms the existing heuristic except for models in which the number of constraints is large relative to the number of variables. In this case, it performs on par with the existing heuristic. Results also indicate that use of a specific rule for tie breaking can be very effective, especially for low-density models in which the number of variables is large relative to the number of constraints.     

The continuous‐time single‐sourcing problem with capacity expansion opportunities

This paper develops a new model for allocating demand from retailers (or customers) to a set of production/storage facilities. A producer manufactures a product in multiple production facilities, and faces demand from a set of retailers. The objective is to decide which of the production facilities should satisfy each retailer's demand, in order minimize total production, inventory holding, and assignment costs (where the latter may include, for instance, variable production costs and transportation costs). Demand occurs continuously in time at a deterministic rate at each retailer, while each production facility faces fixed‐charge production costs and linear holding costs. We first consider an uncapacitated model, which we generalize to allow for production or storage capacities. We then explore situations with capacity expansion opportunities. Our solution approach employs a column generation procedure, as well as greedy and local improvement heuristic approaches. A broad class of randomly generated test problems demonstrates that these heuristics find high quality solutions for this large‐scale cross‐facility planning problem using a modest amount of computation time. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Heuristics for the location of inspection stations on a network

This article considers the preventive flow interception problem (FIP) on a network. Given a directed network with known origin‐destination path flows, each generating a certain amount of risk, the preventive FIP consists of optimally locating m facilities on the network in order to maximize the total risk reduction. A greedy search heuristic as well as several variants of an ascent search heuristic and of a tabu search heuristic are presented for the FIP. Computational results indicate that the best versions of the latter heuristics consistently produce optimal or near optimal solutions on test problems. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 287–303, 2000     

Using hybrid metaheuristics for the one‐way and two‐way network design problem

A network with traffic between nodes is known. The links of the network can be designed either as two‐way links or as one‐way links in either direction. The problem is to find the best configuration of the network which minimizes total travel time for all users. Branch and bound optimal algorithms are practical only for small networks (up to 15 nodes). Effective simulated annealing and genetic algorithms are proposed for the solution of larger problems. Both the simulated annealing and the genetic algorithms propose innovative approaches. These innovative ideas can be used in the implementation of these heuristic algorithms for other problems as well. Additional tabu search iterations are applied on the best results obtained by these two procedures. The special genetic algorithm was found to be the best for solving a set of test problems. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 449–463, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10026     

Scheduling of depalletizing and truck loading operations in a food distribution system

This paper studies a scheduling problem arising in a beef distribution system where pallets of various types of beef products in the warehouse are first depalletized and then individual cases are loaded via conveyors to the trucks which deliver beef products to various customers. Given each customer's demand for each type of beef, the problem is to find a depalletizing and truck loading schedule that fills all the demands at a minimum total cost. We first show that the general problem where there are multiple trucks and each truck covers multiple customers is strongly NP‐hard. Then we propose polynomial‐time algorithms for the case where there are multiple trucks, each covering only one customer, and the case where there is only one truck covering multiple customers. We also develop an optimal dynamic programming algorithm and a heuristic for solving the general problem. By comparing to the optimal solutions generated by the dynamic programming algorithm, the heuristic is shown to be capable of generating near optimal solutions quickly. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Class scheduling algorithms for Navy training schools

The problem of developing good schedules for Navy C-Schools has been modeled as a combinatorial optimization problem. The only complicating feature of the problem is that classes must be grouped together into sequences known as pipelines. An ideal schedule will have all classes in a pipeline scheduled in consecutive weeks. The objective is to eliminate the nonproductive time spent by sailors at C-Schools who are waiting for the next class in a pipeline. In this investigation an implicit enumeration procedure for this problem was developed. The key component of our algorithm is a specialized greedy algorithm which is used to obtain a good initial incumbent. Often this initial incumbent is either an optimal schedule or a near optimal schedule. In an empirical analysis with the only other competing software system, our greedy heuristic found equivalent or better solutions in substantially less computer time. This greedy heuristic was extended and modified for the A-School scheduling problem and was found to be superior to its only competitor. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 533–551, 1998     

Non‐greedy heuristics and augmented neural networks for the open‐shop scheduling problem

In this paper we propose some non‐greedy heuristics and develop an Augmented‐Neural‐Network (AugNN) formulation for solving the classical open‐shop scheduling problem (OSSP). AugNN is a neural network based meta‐heuristic approach that allows integration of domain‐specific knowledge. The OSSP is framed as a neural network with multiple layers of jobs and machines. Input, output and activation functions are designed to enforce the problem constraints and embed known heuristics to generate a good feasible solution fast. Suitable learning strategies are applied to obtain better neighborhood solutions iteratively. The new heuristics and the AugNN formulation are tested on several benchmark problem instances in the literature and on some new problem instances generated in this study. The results are very competitive with other meta‐heuristic approaches, both in terms of solution quality and computational times. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Worst‐case performance of approximation algorithms for tool management problems

Since the introduction of flexible manufacturing systems, researchers have investigated various planning and scheduling problems faced by the users of such systems. Several of these problems are not encountered in more classical production settings, and so‐called tool management problems appear to be among the more fundamental ones of these problems. Most tool management problems are hard to solve, so that numerous approximate solution techniques have been proposed to tackle them. In this paper, we investigate the quality of such algorithms by means of worst‐case analysis. We consider several polynomial‐time approximation algorithms described in the literature, and we show that all these algorithms exhibit rather poor worst‐case behavior. We also study the complexity of solving tool management problems approximately. In this respect, we investigate the interrelationships among tool management problems, as well as their relationships with other well‐known combinatorial problems such as the maximum clique problem or the set covering problem, and we prove several negative results on the approximability of various tool management problems. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 445–462, 1999     

图论中独立支配集的最佳求解算法研究

通过对图论中独立集和支配集的深入研究,提出了独立支配集的概念,论证了独立支配集同极大独立集及极小支配集之间的内在联系,并在此基础上给出了独立支配集的最佳求解算法,从而圆满地解决了图论中独立集及支配集的求解问题,对图的着色及匹配等问题的研究均有相当重要的借鉴意义。